Diplomarbeit

Cellular resolutions of monomial ideals

Stefan Tappe Matrikel-Nr.: 3660635 Universität Paderborn

Fachbereich 17 · Mathematik/Informatik

stefanta@uni-paderborn.de

Betreut von: apl. Prof. Dr. Uwe Nagel Abgabe: 11.06.2002

Contents

1. Introduction 4 2. Preparatory Results 6 2.1. Convex Geometry 6 2.2. Cell complexes 20 2.3. Topological cell complexes 34 2.4. Fine grading 41 3. Cellular resolutions 46 3.1. Cellular resolutions 47 3.2. The Taylor resolution 72 3.3. The hull resolution 78 3.4. The Scarf complex 91 3.5. Generic ideals 127 3.6. Reducing cellular resolutions 138 3.7. Cellular resolutions with MAPLE 150 References 166

3

1. Introduction

The topic of this diploma thesis are graded free resolutions of monomial ideals. The construction of free resolutions is an important objective in the Commutative Algebra. However, even for a polynomial ring there are only few classes of ideals for which free resolutions are known. For a finitely generated graded module M over a polynomial ring it is easy to compute the Hilbert function h_{M} from a finite free resolution. Since we can then derive the Hilbert polynomial p_{M} of the module, we obtain other important informations like the dimension and the degree of M. Finding a minimal free resolution even allows to decide whether the module is Cohen-Macaulay, or not. Free resolutions also play an important role in Algebraic Geometry. A projective scheme X ⊂ ℙ^{n} is given by a factor ring R/I with R = K[x_{0}, x_{1}, …, x_{n}], where K is a field and I its homogeneous ideal. The Hilbert Syzygy theorem guarantees that each factor ring R/I of this kind possesses a graded free resolution of R-moduls of finite length. The crucial point is to find such a resolution, or even the minimal free resolution. In general, this is a very difficult task. The goal of this thesis is to construct free resolutions of monomial ideals

I ⊂ R = K[x_{1}, …, x_{n}],

which are a special class of the homogeneous ideals. The reason for specializing to monomial ideals is that they have certain structural properties, like their uniquely determined, finite set of minimal generators, so one can hope to construct resolutions in a systematic way. Moreover, the monomial ideals are a class of ideals which does often appear, there are, for example, many interesting projective schemes described by a monomial ideal. This thesis is based on [8]. It contains more detailed proofs than [8] and adjusts some inaccuracies. Finally, this thesis has an amount of appropriate examples to illustrate the theory of cellular resolution, most of them not contained in [8]. A central idea is that a free resolution, which is in fact a complicated algebraic object, is encoded by a geometric object (and its decomposition), in which the minimal generators of the monomial ideal are represented by the vertices. Therefore, we obtain structural information about the resolution. The second chapter of the thesis deals with the prerequisites needed for the theory in Chapter 3. In Chapter 3, we define the cellular complex 𝔽_{X} of a monomial ideal I derived from a cell complex X. Cell complexes are introduced in Section 2.2. A cellular complex is a finite sequence of free R-modules which need not be a resolution. This complex possesses a ℤ^{n}-grading, the so-called fine grading. Consequently, we have to deal with general gradings in Section 4 of the second chapter. There are several reasons for considering this fine grading. ℤ^{n}-graded free modules are obtained from a cell complex in a natural way, the fine grading contains more information than the normal ℤ-grading, in particular, every ℤ^{n}-graded resolution can easily be transformed into a resolution with ℤ-grading. Finally, the ℤ^{n}-graded ideals are exactly the monomial ideals which form the topic of this thesis. As mentioned above, a cellular complex 𝔽_{X} need not be a resolution. Hence, we have to find criteria for the exactness of 𝔽_{X} (Theorem 3.1.11) and, provided 𝔽_{X} is a resolution, for the minimality (Proposition 3.1.14). It is an interesting observation that the exactness of a cellular complex 𝔽_{X} does not only depend on the cell complex X. For a given cell complex X it may also depend on the choice of the field K. A pertinent example will be given in Example 3.1.23. In the case that 𝔽_{X} is a resolution, we call it a cellular resolution. In order to test the exactness of a cellular complex 𝔽_{X}, it is important to check the acyclicity of cell complexes. Some preparations are recalled in the Sections 2 and 3 of the second chapter. While the results of Section 2.2 are proven with the help of Linear Algebra, we apply methods of Algebraic Topology and Convex Geometry in order to prove the results of Section 2.3. Convex Geometry is important throughout this thesis, therefore, some important concepts and results are explained in the first section of Chapter 2, as far as they will be needed for our purposes. After having developed the basic theory about cellular resolutions in Section 3.1, we discuss special kinds of cell complexes X which even yield an exact cellular complex 𝔽_{X}. The simplest example is the Taylor complex Δ whose cellular complex 𝔽_{Δ} is always a resolution for every choice of the field K (see Theorem 3.2.2). It is typically far from being minimal, but its canonical construction allows to derive a formula for the Hilbert function of I and R/I which can directly be used for any monomial ideal (see Proposition 3.2.6). If we manage to find a minimal resolution, we can decide, in particular, if R/I is Cohen-Macaulay. The Taylor resolution 𝔽_{Δ} is very analogous to the classical Koszul complex, but, unlike the Koszul complex, it is always exact. Provided the monomial ideal possesses s minimal generators, the geometric object encoding the Taylor resolution is just a simplex in ℝ^{s}. Next, we introduce the hull complex hull(I), a special complex which has better properties than the Taylor complex. As we pointed out, we want to assign a geometric object to a free resolution. The hull complex is obtained by finding the bounded faces of a certain unbounded polyhedron. Its construction makes use of the methods of Convex Geometry explained in the first section of Chapter 2. In Theorem 3.3.15 we prove that 𝔽_{hull(I)} is, like the Taylor resolution, a resolution for each monomial ideal, independent of the choice of the field K. This so-called hull resolution need not be minimal, but it is most often shorter than the Taylor resolution and it respects symmetry. We discuss several examples to get an idea how big the difference between the hull resolution and the minimal free resolution of a monomial ideal can be. A disadvantage of the hull complex is that it requires the computations of convex hulls. In order to reduce the effort of computation, we introduce the Scarf complex Δ_{I} whose cellular complex 𝔽_{Δ_{I}} need not be resolution. Nevertheless, the Scarf complex is interesting for our purpose because it can be computed easily and, as we prove in Theorem 3.4.11, it is always a part of the hull complex. This implies that some computations can be avoided if we want to determine the hull complex. After having presented some important examples of cellular complexes, we introduce a special class of monomial ideals, namely the generic. The generic ideals have the nice property that the Scarf complex Δ_{I}, which can easily be determined, coincides with the hull complex and yields a minimal free resolution (see Theorem 3.5.7). As a consequence, we prove that one minimal free resolution of every monomial I ⊂ K[x, y] is given by the Scarf complex. Almost all monomial ideals are generic, in the sense that those which fail to be generic lie on finitely many hyperplanes in the matrix space of exponents. However, many interesting monomial ideals are not generic, but, as indicated above, we illustrate and test the theory of cellular resolutions by choosing several appropriate non-generic examples. A cellular resolution, like the hull resolution or the Taylor resolution, need not be minimal. Hence, we reflect in Section 3.6 upon possibilities to reduce such a resolution in order to make it minimal. We propose two reduction strategies, successfully applied in three examples. After having dealed with some examples, it becomes desirable to have some kind of support. The author of this thesis has written a program, based on the Computer Algebra System MAPLE, to create cellular resolutions. It is explained in detail in the Section 3.7. The algorithms of the program apply the results described in this thesis, and the examples of this thesis have been checked by the help of this program. Some longer examples have been shortened, the interested reader finds the whole examples on the CD attached to this thesis.

2. Preparatory Results

In this chapter we develop the foundation on which the theory of cellular resolutions is built. We discuss some Convex Geometry (Section 2.1), cell complexes (Section 2.2), which are fundamental for constructing cellular complexes, and useful links between cell complexes, Geometry, and Topology (Section 2.3). Furthermore, we have to deal in Section 2.4 with general gradings of modules using an arbitrary abelian group G.

2.1. Convex Geometry.

In this section we collect some facts about Convex Geometry are important throughout this thesis. We define convex sets in ℝ^{n} and consider linear spaces, affine spaces and cones as special kinds of convex sets. Before this we take a look at some generalities:

Definition 2.1.1. For two points x = (x_{1}, …, x_{n}), y = (y_{1}, …, y_{n}) of the n-dimensional space ℝ^{n} we define the following relations

x ≤ y if x_{i} ≤ y_{i}, i = 1, …, n,
x < y if x ≤ y, x ≠ y,
x ◁ y if x_{i} < y_{i}, i = 1, …, n.

The sets

(x, y) = {(1 − λ) x + λ y : 0 < λ < 1},
(x, y] = {(1 − λ) x + λ y : 0 < λ ≤ 1},
[x, y) = {(1 − λ) x + λ y : 0 ≤ λ < 1},
[x, y] = {(1 − λ) x + λ y : 0 ≤ λ ≤ 1}

are said to be the open, half-open, and closed line segments with end points x and y. The sets

((x, y)) = {z ∈ ℝ^{n} : x ◁ z ◁ y},
((x, y]] = {z ∈ ℝ^{n} : x ◁ z ≤ y},
[[x, y)) = {z ∈ ℝ^{n} : x ≤ z ◁ y},
[[x, y]] = {z ∈ ℝ^{n} : x ≤ z ≤ y}

are called open, half-open, and closed parallel epipeds in ℝ^{n}. Both line segments and parallel epipeds generalize the real intervals of finite length. For a finite sequence of points x_{1}, …, x_{r} ∈ ℝ^{n} (r ≥ 2) we define

[x_{1}, …, x_{r}] = [x_{1}, x_{2}] ∪ … ∪ [x_{r−1}, x_{r}].

CELLULAR RESOLUTIONS OF MONOMIAL IDEALS 7

We denote the line through x and y by

L(x, y) = {(1 − λ) x + λ y : λ ∈ ℝ}.

The inner product for two points x, y ∈ ℝ^{n} is given by

⟨x, y⟩ = ∑_{i=1}^{n} x_{i} y_{i},

and the Euclidian norm for any x ∈ ℝ^{n} is

∥x∥ = \sqrt{⟨x, x⟩} = \sqrt{∑_{i=1}^{n} x_{i}^{2}}.

For x ∈ ℝ^{n} and r > 0 we define the open ball U(x, r) and the closed ball B(x, r) by

U(x, r) = {z ∈ ℝ^{n} : ∥z − x∥ < r},
B(x, r) = {z ∈ ℝ^{n} : ∥z − x∥ ≤ r}.

This figure shows the half-open parallel epiped (( (1, 2), (4, 4) ]]. Definition 2.1.2. A subset K of ℝ^{n} is convex if for any two points x, y ∈ K the line segment [x, y] belongs to K. Example 2.1.3. The following

is a non-convex set, while the triangle is convex.

We define affine spaces and cones in a similar way:

Definition 2.1.4. A subset A of ℝ^{n} is an affine space if for any two points x, y ∈ A the line L(x, y) belongs to A.

Definition 2.1.5. A subset C of ℝ^{n} is a cone if for any point x ∈ C the set {λ x : λ ≥ 0} belongs to C.

The following lemma shows that linear spaces in ℝ^{n} can be characterized similarly:

Lemma 2.1.6. A subset V of ℝ^{n} is a linear space if and only if 0 ∈ V, and for any two points x, y ∈ V the line L(x, y) belongs to V.

Proof. Let V be a linear space of ℝ^{n}. Then by definition 0 ∈ V, and for any x, y ∈ V, λ ∈ ℝ we have

λ x ∈ V,
x + y ∈ V.

This implies

(1 − λ) x + λ y ∈ V.

Now, assume that V is a set of ℝ^{n} satisfying 0 ∈ V and

(1 − λ) x + λ y ∈ V

for any x, y ∈ V, λ ∈ ℝ. By setting x = 0 we obtain for arbitrary λ ∈ ℝ, y ∈ V

λ y ∈ V.

By setting λ = \frac{1}{2} we get for any x, y ∈ V

\frac{1}{2} (x + y) = \frac{1}{2} x + \frac{1}{2} y ∈ V,

which implies

x + y ∈ V.

Hence, V is a linear space of ℝ^{n}. □

Remark 2.1.7. A linear space is an affine space as well as a cone. Affine spaces and cones are convex sets.

Lemma 2.1.8. The intersection of any non-empty family of

convex sets
affine spaces

cones

linear spaces

in ℝ^{n} is again

\{ \begin{matrix}convex \\ an & affine & space \\ a cone \\ a linear space \end{matrix} \} .

Proof. The proof is simple, thus omitted. □

According to the preceding lemma the following definition makes sense.

Definition 2.1.9. The

\{ \begin{matrix}convex hull \\ affine hull \\ positive hull \\ linear hull \end{matrix} \} ,

of a subset X ⊂ ℝ^{n}, denoted

\{ \begin{matrix}conv X \\ aff X \\ pos X \\ lin X \end{matrix} \} ,

10 STEFAN TAPPE

is the intersection of all

\{ \begin{matrix}convex & sets \\ affine & spaces \\ cones \\ linear & spaces \end{matrix} \} ,

which contain X. Definition 2.1.10. We say x ∈ ℝ^{n} is

\{ \begin{matrix}a convex combination \\ an affine combination \\ a positive combination \\ a linear combination \end{matrix} \} ,

of x_{1}, …, x_{r} ∈ ℝ^{n} if there are λ_{1}, …, λ_{r} ∈ ℝ such that

x = ∑_{j=1}^{r} λ_{j} x_{j}

and the condition(s)

\{ \begin{matrix}(1), (2) \\ (1) \\ (2) \\ (no & further & condition) \end{matrix} \} ,

hold(s), with (1),(2) being

(1) ∑_{j=1}^{r} λ_{j} = 1,
(2)  λ_{j} ≥ 0, j = 1, …, r.

Theorem 2.1.11. The

\{ \begin{matrix}convex & hull \\ affine & hull \\ positive & hull \\ linear & hull \end{matrix} \} ,

of a subset X ⊂ ℝ^{n} is the set of all

convex combinations
affine combinations
positive combinations
linear combinations

of finite subsets of X. Proof. For a proof see, for example, [1] or [2]. The statement for convex sets is known as Carathéodory’s Theorem. □

The next result is straightforward.

Lemma 2.1.12. Let X, Y ⊂ ℝ^{n} and ∗ ∈ {conv, aff, pos, lin}.

(i) X ⊂ Y implies ∗ X ⊂ ∗ Y.

(ii) X ⊂ ∗ X = ∗ (∗ X).

(iii) X is

a convex set

an affine space

a cone

a linear space

if and only if

\{ \begin{matrix}X = conv X \\ X = aff X \\ X = pos X \\ X = lin X \end{matrix} \} .

(iv) If X and Y are

\{ \begin{matrix}convex & sets \\ affine & spaces \\ cones \\ linear & spaces \end{matrix} \} ,

their sum X + Y is again

\{ \begin{matrix}convex \\ an & affine & space \\ a cone \\ a linear space \end{matrix} \} .

To every convex set in ℝ^{n} we want to attribute a dimension:

Let A ⊂ ℝ^{n} be an affine space. We pick x ∈ A; then there is a (unique) linear space V of ℝ^{n} (not depending on x) such that

A = x + V.

The vector space V is called the associated linear space of A.

Definition 2.1.13. Let A ⊂ ℝ^{n} be an affine space and V its associated linear space. We define the dimension of A as the vector space dimension of V:

dim A = dim_{ℝ} V.

By convention, dim ∅ = −1. For any convex set K we define its dimension by

dim K = dim (aff K).

It is clear that the dimensions of hulls can be computed by the rank of a suitable matrix.

12 STEFAN TAPPE

Lemma 2.1.14. Let x_{0}, x_{1}, …, x_{r} ∈ ℝ^{n}. Then we have

dim (lin {x_{1}, …, x_{r}}) = rk ( x_{1} ⋯ x_{r} )
dim (pos {x_{1}, …, x_{r}}) = rk ( x_{1} ⋯ x_{r} )
dim (conv {x_{0}, x_{1}, …, x_{r}}) = rk ( x_{1} − x_{0} ⋯ x_{r} − x_{0} )
dim (aff {x_{0}, x_{1}, …, x_{r}}) = rk ( x_{1} − x_{0} ⋯ x_{r} − x_{0} )

where rk( x_{1} ⋯ x_{r} ) denotes the rank of the matrix with x_{1}, …, x_{r} being its columns.

In order to illustrate the previous definitions we discuss an illustrating example: Example 2.1.15. Consider the four sets

V = lin {(5, 2)},
A = aff {(0, 2), (5, 4)},
K = conv {(−5, 1), (−2, 2), (−3, 4)},
C = pos {(2, −5), (1, −1)}.

Since V is the associated linear space of A, we can write A as

A = (0, 2) + V.

The dimensions can be computed by applying the preceding Lemma 2.1.14

dim V = rk ( \begin{matrix}5 \\ 2 \end{matrix} ) = 1,
dim A = rk ( \begin{matrix}5 \\ 2 \end{matrix} ) = 1,
dim K = rk ( \begin{matrix}3 & 2 \\ 1 & 3 \end{matrix} ) = 2,
dim C = rk ( \begin{matrix}2 & 1 \\ −5 & −1 \end{matrix} ) = 2.

Now we want to consider two kinds of convex sets, polyhedrons and polytopes, which are very important for our purpose. Beforehand, we recall the concepts hyperplane and half-space.

Definition 2.1.16. For any v ∈ ℝ^{n}, v ≠ 0, and β ∈ ℝ the set

H(v, β) = {x ∈ ℝ^{n} : ⟨v, x⟩ = β}

is called a hyperplane with normal vector v. The two sets

H(v, β)^{+} = {x ∈ ℝ^{n} : ⟨v, x⟩ ≥ β}

and

H(v, β)^{−} = {x ∈ ℝ^{n} : ⟨v, x⟩ ≤ β}

are called the (closed) half-spaces determined by H(v, β).

Hyperplanes of ℝ^{n} are affine spaces of dimension n − 1. Half-spaces are closed convex sets of dimension n.

Remark 2.1.17. For v ∈ ℝ^{n}, v ≠ 0 and c, β ∈ ℝ we have the following identities:

H(v, β) = H(c v, c β),  c ≠ 0,
H^{+}(v, β) = H^{+}(c v, c β), c > 0,
H^{−}(v, β) = H^{−}(c v, c β), c > 0,
H^{+}(v, β) = H^{−}(c v, c β), c < 0,
H^{−}(v, β) = H^{+}(c v, c β), c < 0.

Example 2.1.18. This figure shows

a hyperplane and one of the resulting half-spaces:

H_{1} = H((2, 1), −2),
H_{2} = H((2, 1), −2)^{+}.

Definition 2.1.19. A polytope is the convex hull of a finite set of points in ℝ^{n}.

14 STEFAN TAPPE

Definition 2.1.20. A polyhedral set or polyhedron is the intersection of a finite number of closed half-spaces. Obviously, polyhedra are closed convex sets, but of course they need not be bounded. Theorem 2.1.21. A subset of ℝ^{n} is a polytope if and only if it is a bounded polyhedron. Proof. This follows from the Theorems II.1.4 and II.1.5 of [1]. □ Example 2.1.22. We start with a polytope in ℝ^{2} It can be written as the convex hull of a finite set of points

P = conv {(0, 0), (3, 1), (5, 5), (2, 4)}.

Alternatively, it can be written as the intersection of a finite number of closed half-spaces

P = H((−1, 3), 0)^{+} ∩ H((−1, 3), 10)^{−} ∩ H((2, −1), 0)^{+} ∩ H((2, −1), 5)^{−}. Now we consider an unbounded polyhedron, which can be described as

P =H((1, 0), 1)^{+} ∩ H((3, 1), 10)^{+} ∩ H((1, 1), 6)^{+}
∩ H((1, 3), 10)^{+} ∩ H((0, 1), 1)^{+}.

Finally, we consider an octahedron, which is a polytope in ℝ^{3}:

It is the convex hull of its 6 vertices and also the intersection of 8 half-spaces.

Definition 2.1.23. Let P be a polyhedron and H(v, β) a hyperplane. Then H(v, β) is called a supporting hyperplane if

H(v, β) ∩ P ≠ ∅

and P is contained in one of the closed half-spaces determined by H(v, β). If H is a supporting hyperplane of P, then H(v, β) ∩ P is called a face of P.

It is convenient to consider the empty set and P as faces, the improper faces. All the other faces are called proper faces.

Theorem 2.1.24. The faces of a polyhedron (polytope) are again polyhedra (polytopes).

Proof. [1] Theorem II.1.1 states that the faces of a polytope are again polytopes. The analogous statement is trivial for polyhedra. □

A d-polyhedron P is a polyhedron of dimension d. Faces of dimension 0, 1, d − 2, d − 1 are called vertices, edges, subfacets and facets, respectively. We write

vert P

for the set of vertices of the polyhedron P.

Theorem 2.1.25. Let P be a polytope. Then we have:

(i) P is the convex hull of its vertices, that is

P = conv(vert P).

(ii) Each subfacet F of P is contained in precisely two facets F_{1} and F_{2} of P, and

F = F_{1} ∩ F_{2}.

Proof. We refer to Theorem II.1.2 for (i) and Theorem 3.1.6 for (ii). □

Theorem 2.1.26. Let P be a polyhedron. Then we have:

(i) P has only a finite number of faces.

(ii) Let F be a face of P and F′ a face of F. Then F′ is a face of P.

(iii) Any proper face of P is a face of some facet of P.

(iv) Let F_{i} ⊂ F_{k} be faces of the poyhedron P, i < k, such that F_{i} is an i-face and F_{k} a k-face. There is a chain

F_{i} ⊂ F_{i+1} ⊂ … ⊂ F_{k−1} ⊂ F_{k},

such that F_{j}, j = i, …, k, is a j-face of P.

Proof. We quote the following references (i) [2], Theorem 2.6.6. (ii) [1], Theorem II.1.7. (iii) [1], Theorem II.1.8. (iv) [1], Theorem II.1.9.

□

Example 2.1.27. In the following figure

H_{1} and H_{2} denote supporting hyperplanes of the polytope P. The corresponding faces are a vertex and an edge. The next figure shows a prism, again with two supporting hyperplanes:

The resulting faces are the base of the prism and a vertex of the base.

Definition 2.1.28. Let P be a polyhedron. Any subset of the finite set of faces of P is a partially ordered set (abbreviated: poset) under inclusion. F(P) denotes the poset of faces of P and F^{b}(P) denotes the subposet of bounded faces of the polyhedron P.

Definition 2.1.29. Let F, G be two posets. A bijective, order-preserving map

φ : F → G

is called an isomorphism of posets. Two posets F, G are isomorphic provided that there is an isomorphism

φ : F → G.

We say that a nonempty family (F_{i})_{i∈I} of posets is independent of i ∈ I, if for any two i, j ∈ I the two posets F_{i} and F_{j} are isomorphic.

The following proposition states that a finitely generated cone C is a polyhedron and that any face of C is again a cone.

Proposition 2.1.30. Let

C = pos {x_{1}, …, x_{r}}, x_{1}, …, x_{r} ∈ ℝ^{n},

be a finitely generated cone. Then C is a polyhedron and for any subset F ⊂ ℝ^{n}, F ≠ ∅, F ≠ {0}, F ≠ C the following conditions are equivalent:

(a) F ∈ F(C).

(b) There are J ⫋ {1, …, r}, J ≠ ∅ and v ∈ ℝ^{n}, v ≠ 0 such that

F = pos {x_{j} : j ∈ J}

and

⟨v, x_{j}⟩ = 0, j ∈ J,
⟨v, x_{j}⟩ > 0, j ∉ J.

In this case, H(v, 0) is a supporting hyperplane for C.

Proof. For the proof that C is a polyhedron, see, e.g., [1] or [2]. Now we prove that the conditions (a) and (b) are equivalent. Let F be a face of C as in the statement. There are v ∈ ℝ^{n} and β ∈ ℝ such that

H(v, β) ∩ C = F

and (without loss of generality)

C ⊂ H(v, β)^{+}.

We claim that

β = 0.

Indeed, assume β ≠ 0. Without loss of generality we consider the case β > 0. Fix an element x ∈ F and λ < 1. Then λ x belongs to the cone C, and we get

⟨v, λ x⟩ = λ \underbrace{⟨v, x⟩}_{=β} < β,

which contradicts

C ⊂ H(v, β)^{+}.

Without loss of generality, we have

⟨v, x_{j}⟩ ≥ 0, j = 1, …, r.

Define

J = {j ∈ {1, …, r} : ⟨v, x_{j}⟩ = 0} ⊂ {1, …, r}.

Suppose J = ∅. For an arbitrary element

x = ∑_{j=1}^{r} λ_{j} x_{j} ∈ C \ {0}

we get

⟨v, x⟩ = ∑_{j=1}^{r} λ_{j}⟨\underbrace{v, x_{j}⟩}_{>0} > 0,

which shows that

H(v, 0) ∩ C = {0}.

This contradicts the assumption F ≠ {0}. Therefore, we get J ≠ ∅. Note that H(v, 0) is also a cone because for x ∈ H(v, 0) we obtain

⟨v, λ x⟩ = λ \underbrace{⟨v, x⟩}_{=0} = 0, λ ≥ 0.

It is evident that

{x_{j} : j ∈ J} ⊂ H(v, 0) ∩ C.

Since H(v, 0) ∩ C is a cone, we have the inclusion

pos {x_{j} : j ∈ J} ⊂ H(v, 0) ∩ C.

by Lemma 2.1.12 (i),(iii). Now let

x = ∑_{j=1}^{r} λ_{j} x_{j} ∈ C ∩ H(v, 0).

We get

∑_{j∈J} λ_{j} \underbrace{⟨v, x_{j}⟩}_{=0} + ∑_{j∉J} λ_{j} \underbrace{⟨v, x_{j}⟩}_{>0} = 0

This implies

λ_{j} = 0, j ∉ J,

it follows that

x ∈ pos {x_{j} : j ∈ J}.

Therefore, the other inclusion

H(v, 0) ∩ C ⊂ pos {x_{j} : j ∈ J}

is proven, and we have shown that

F = pos {x_{j} : j ∈ J}.

We still have to verify the converse. The proof that

H(v, 0) ∩ C = F

can be taken from above.

For any

x = ∑_{j=1}^{r} λ_{j} x_{j} ∈ C

we have the inequalities

⟨v, x⟩ = ∑_{j=1}^{r} λ_{j} \underbrace{⟨v, x_{j}⟩}_{≥0} ≥ ∑_{j∈J} λ_{j} ≥ 0,

hence

C ⊂ H(v, 0)^{+}.

Consequently, F is a face of C. □

Our last subject in this section are projective transformations and how they act on polytopes.

Definition 2.1.31. Let A ∈ ℝ^{n,n}, b ∈ ℝ^{n}, v ∈ ℝ^{n}, β ∈ ℝ, such that A is regular and v ≠ 0. The map T : ℝ^{n} \ H(v, β) → ℝ^{n}

T x = \frac{Ax + b}{⟨v, x⟩ − β}

is called a projective transformation.

Lemma 2.1.32. Let T : ℝ^{n} \ H(v, β) → ℝ^{n} be a projective transformation and P a polytope in ℝ^{n} such that

P ∩ H(v, β) = ∅.

Then T P is again a polytope and

F(T P) = {T F : F ∈ F(P)}.

Proof. This follows from [2], Exercise 2.4.10(ii). □

Lemma 2.1.33. Let T : ℝ^{n} \ H(v, β) → ℝ^{n} be a projective transformation and let P a polytope in ℝ^{n} such that

P ∩ H(v, β)

is a nonempty face of P. Then P′ = T (P \ H(v, β)) is an unbounded polyhedron and its poset of bounded faces is

F^{b}(T P′) = {T F : F ∈ F(P), F ∩ H(v, β) = ∅}.

Proof. Without loss of generality we may assume that

P ⊂ H^{+}(v, β).

For ε > 0 let T_{ε} : ℝ^{n} \ H(v, β − ε) → ℝ^{n} be the projective transformation

T_{ε}x = \frac{Ax + b}{⟨v, x⟩ − (β − ε)}.

By Lemma 2.1.32 T_{ε}P is again a polytope for any ε > 0, and

F(T_{ε}P) = {T_{ε}F : F ∈ F(P)}.

The limit process ε → 0 yields the assertion. □

2.2. Cell complexes.

In Chapter 3 we use cell complexes in order to construct cellular resolutions. Thus, in this section, first we introduce simplicial complexes, then we define cell complexes, which generalize the simplicial complexes. Subsequently, we define the so-called augmented oriented chain complex C̃(X; K) for a cell complex X and a field K. For the theory of cellular resolutions it is important that we are able to check whether such a complex is acyclic. In the next two sections, we recall some results concerning the acyclicity of cell complexes.

Definition 2.2.1. Let V = {v_{1}, …, v_{s}} be a nonempty, finite set. A simplicial complex Δ on V is a collection of subsets of V such that F ∈ Δ whenever F ⊂ G for some G ∈ Δ, and such that {v_{i}} ∈ Δ for i = 1, …, n.

The elements of Δ are called cells. The dimension, dim F, of a cell F ∈ Δ is the number |F | − 1. The dimension of Δ is dim Δ = max {dim F : F ∈ Δ}. Note that ∅ is a cell (of dimension −1) of any simplicial complex Δ. By Δ^{i}, i = −1, …, dim Δ we denote the cells of dimension i in Δ. The sets Δ^{i} are called the components of Δ.

Example 2.2.2. Consider the two tables

Δ_{1}
Component  Cell
Δ_{1}^{−1}  ∅
{v_{1}}
{v_{2}}
Δ_{1}^{0}  {v_{3}}
{v_{4}}
{v_{1}, v_{2}}
{v_{1}, v_{4}}
Δ_{1}^{1}  {v_{2}, v_{3}}
{v_{2}, v_{4}}
{v_{3}, v_{4}}
{v_{1}, v_{2}, v_{4}}
Δ_{1}^{2}  {v_{2}, v_{3}, v_{4}}

and

Δ_{2}
Component  Cell
Δ_{2}^{−1}  ∅
{v_{1}}
Δ_{2}^{0}  {v_{2}}
{v_{4}}
{v_{1}, v_{2}}
Δ_{2}^{1}  {v_{1}, v_{4}}
{v_{2}, v_{4}}
Δ_{2}^{2}  {v_{1}, v_{2}, v_{4}}

One easily checks that Δ_{1} is a simplicial complex on the vertex set V = {v_{1}, v_{2}, v_{3}, v_{4}} while Δ_{2} is a simplicial complex on the vertex set V = {v_{1}, v_{2}, v_{4}}.

Given an arbitrary collection {F_{1}, …, F_{m}} of subsets of V, there is a (unique) smallest simplicial complex, denoted by ⟨F_{1}, …, F_{m}⟩, which contains all F_{i}. This simplicial complex is said to be generated by F_{1}, …, F_{m}. It consists of all subsets G ∈ V which are contained in some F_{i}. A simplicial complex Δ generated by one subset W of V, Δ = P(W) = ⟨W ⟩, is called a simplex. Δ is said to be the full simplex on the set of vertices W.

For instance, the simplicial complex Δ_{1} of Example 2.2.2 can be written as

Δ_{1} = ⟨{v_{1}, v_{2}, v_{4}}, {v_{2}, v_{3}, v_{4}}⟩,

and

Δ_{2} = ⟨{v_{1}, v_{2}, v_{4}}⟩ = P({v_{1}, v_{2}, v_{4}})

is the full simplex on V = {v_{1}, v_{2}, v_{4}}.

Before we generalize the simplicial complexes, we have to prove the following

Lemma 2.2.3. Let Δ be a simplicial complex. For any pair of cells F, G ∈ Δ such that

F ⊂ G and dim G = dim F + 2,

there are exactly two cells H_{1}, H_{2} ∈ Δ satisfying

F ⊂ H_{i} ⊂ G and dim H_{i} = dim F + 1 (i = 1, 2).

Proof. By assumption we have

F = {v_{j} : j ∈ J},
G = F ∪ {v_{j_{1}}, v_{j_{2}}}

for a subset J ⊂ {1, …, s} and suitable vertices v_{j_{1}}, v_{j_{2}}. Then, the two cells H_{1}, H_{2} ∈ Δ fulfilling the conditions of the lemma are

H_{1} = F ∪ {v_{j_{1}}} and H_{2} = F ∪ {v_{j_{2}}}.
□

Given an arbitrary collection X of subsets of a finite set V, the dimension of an element F ∈ X is

dim F = d − 1,

if

F_{0} = ∅ ⫋ F_{1} ⫋ … ⫋ F_{d−1} ⫋ F_{d} = F

is an ascending chain of maximal length of elements in X.

Note that this procedure coincides with the definition of the dimensions for a simplicial complex.

Definition 2.2.4. Let V = {v_{1}, …, v_{s}} be a nonempty, finite set and X a collection of subsets of V such that

∅ ∈ X,
{v_{i}} ∈ X, i = 1, …, n.

X is called a cell complex on V if for any pair F, G ∈ X with

F ⊂ G and dim G = dim F + 2,

there are exactly two H ∈ X satisfying

F ⊂ H ⊂ G and dim H = dim F + 1.

The elements of X are called cells. Cells of dimension i are said to be i-cells. The dimension of X is dim X = max {dim F : F ∈ X}. By X^{i}, i = −1, …, dim X we denote the cells of dimension i in X. The sets X^{i} are called the components of X. Subsequently, we will assume that every component X^{i} is given by X^{i} = {F_{1}^{i}, …, F_{|X^{i}|}^{i}}, in other words, they can be considered as being ordered. The cells of dimension 0 and 1 are called vertices and edges, respectively. The maximal cells under inclusion are called the facets of the cell complex. A cell F is a face of the cell F′ ≠ F if F ⊂ F′. So the last condition of the preceding definition can be reformulated as follows: If F ∈ X^{i} and F′ ∈ X^{i−2} is a face of F, then there are exactly two (i − 1)-cells such that F_{j} is a face of F and F′ is a face of F_{j} (j = 1, 2).

By Lemma 2.2.3 it is clear that simplicial complexes are a special class of cell complexes.

Remark 2.2.5. The definition of cell complexes given above does not coincide with the definition in [3], page 253. In the next section we will introduce topological cell complexes, whose definition is the same as the definition of cell complexes in [3]. We will show that each cell complex possessing a geometric realization (see the next section) can be identified with a topological cell complex. The reason for using this Definition 2.2.4 is that it provides a suitable data structure for MAPLE computations of cell complexes (see Section 3.7).

Now, let us consider an example for a cell complex which is not simplicial:

Example 2.2.6. The cell complex

X
Component  Cell
X^{−1} F_{1}^{−1} ∅
F_{1}^{0}  {v_{1}}
F_{2}^{0}  {v_{2}}
X^{0} F_{3}^{0} {v_{3}}
F_{4}^{0}  {v_{4}}
F_{1}^{1}  {v_{1}, v_{2}}
F_{2}^{1}  {v_{1}, v_{3}}
X^{1} F_{3}^{1} {v_{2}, v_{4}}
F_{4}^{1}  {v_{3}, v_{4}}
X^{2}  F_{1}^{2} {v_{1}, v_{2}, v_{3}, v_{4}}

is a non-simplicial cell complex.

After having defined a cell complex, we now introduce subcomplexes of a cell complex.

Definition 2.2.7. Let X be a cell complex on the vertex set V. A subset Y of X such that Y ∩ X^{0} ≠ ∅ is called a subcomplex, if for each F ∈ Y all the faces of F are contained in Y.

Lemma 2.2.8. Let X be a cell complex on the vertex set V.

(i) A subcomplex Y ⊂ X is itself a cell complex on a nonempty vertex set W ⊂ V with the property

Y^{i} ⊂ X^{i}, i = −1, …, dim Y.

(ii) For any i = 0, …, dim X

Y = X^{−1} ∪ … ∪ X^{i}

is a subcomplex of X. (iii) If X is simplicial, any subcomplex of X is again simplicial. (iv) If Y ⊂ X is a simplicial complex, then Y is a subcomplex of X.

(v) If Y is a subcomplex of X and Z a subcomplex of Y, then Z is also a subcomplex of X. (transitivity)

Proof. The easy proof is omitted. □

Example 2.2.9. We present two examples of subcomplexes

(i) In Example 2.2.2 Δ_{2} is a subcomplex of Δ_{1}.

(ii) The simplicial complex

Y
Component  Cell
Y^{−1} F_{1}^{−1} ∅
F_{1}^{0} {v_{1}}
F_{2}^{0} {v_{2}}
Y^{0}  F_{3}^{0} {v_{3}}
F_{4}^{0} {v_{4}}
F_{1}^{1} {v_{1}, v_{2}}
F_{2}^{1} {v_{1}, v_{3}}
Y^{1}  F_{3}^{1} {v_{2}, v_{4}}
F_{4}^{1} {v_{3}, v_{4}}

is a subcomplex of the cell complex X in Example 2.2.6, and we have the identity

Y = X^{−1} ∪ X^{0} ∪ X^{1}.

Before defining the augmented oriented cell complex, we have to examine incidence functions.

Definition 2.2.10. An incidence function on X is a function satisfying the following conditions:

(a) to each pair (F, F′) such that F ∈ X^{i} and F′ ∈ X^{i−1} for some i ≥ 0, ε assigns a number ε(F, F′) ∈ {−1, 0, 1};

(b) ε(F, F′) ≠ 0 ⟺ F′ is a face of F;

(c) ε(F, ∅) = 1 for all vertices F;

(d) if F ∈ X^{i} and F′ ∈ X^{i−2} is a face of F, then

ε(F, F_{1}) ε(F_{1}, F′) + ε(F, F_{2}) ε(F_{2}, F′) = 0

where F_{1} and F_{2} are those (i − 1)-cells such that F_{j} is a face of F and F′ is a face of F_{j} (j = 1, 2).

Lemma 2.2.11. Let X be a cell complex. Then there exists an incidence function on X.

Proof. This is Lemma IV.7.1 in [5] (in a different formulation). □

The proof of Theorem IV.7.2 of [5] shows that incidence functions can be constructed in a completely naive manner.

(i) One starts with X^{−1} ∪ X^{0} on which there is no choice for an incidence function according to property (c) of incidence functions.

(ii) If one has constructed an incidence function ε^{i} on

X^{−1} ∪ X^{0} ∪ … ∪ X^{i},

there is an incidence function ε^{i+1} on

X^{−1} ∪ X^{0} ∪ … ∪ X^{i+1}

whose restriction to

X^{−1} ∪ X^{0} ∪ … ∪ X^{i}

is just ε^{i}.

Under special circumstances, an incidence function is particularly easy to compute:

Lemma 2.2.12. Let X be a cell complex.

(i) If X is simplicial, an incidence function ε is given by

ε(F, F′) = (−1)^{k+1}

if

F = {v_{i_{1}}, …, v_{i_{m}}},
F′ = {v_{i_{1}}, …, v̂_{i_{k}}, …, v_{i_{m}}}, i_{1} < … < i_{m},

and

ε(F, F′) = 0

if F′ is not a face of F.

(ii) Let ε be an incidence function on X and let Y be a subcomplex of X. Then the restriction of ε to Y is an incidence function on Y.

Proof. We omit the easy proof. □

Example 2.2.13. Consider the complexes Δ_{1} and Δ_{2} from Example 2.2.2. Since Δ_{1} is simplicial, an incidence function is given by Lemma 2.2.12(i):

\begin{matrix}ε & \{v_{1}\} & \{v_{2}\} & \{v_{3}\} & \{v_{4}\} \\ ∅ & 1 & 1 & 1 & 1 \end{matrix}

\begin{matrix}ε & \{v_{1}, v_{2}\} & \{v_{1}, v_{4}\} & \{v_{2}, v_{3}\} & \{v_{2}, v_{4}\} & \{v_{3}, v_{4}\} \\ \{v_{1}\} & -1 & -1 & 0 & 0 & 0 \\ \{v_{2}\} & 1 & 0 & -1 & -1 & 0 \\ \{v_{3}\} & 0 & 0 & 1 & 0 & -1 \\ \{v_{4}\} & 0 & 1 & 0 & 1 & 1 \end{matrix}

\begin{matrix}ε & \{v_{1}, v_{2}, v_{4}\} & \{v_{2}, v_{3}, v_{4}\} \\ \{v_{1}, v_{2}\} & 1 & 0 \\ \{v_{1}, v_{4}\} & -1 & 0 \\ \{v_{2}, v_{3}\} & 0 & 1 \\ \{v_{2}, v_{4}\} & 1 & -1 \\ \{v_{3}, v_{4}\} & 0 & 1 \end{matrix}

An incidence function on Δ_{2} can be obtained by the restriction to ε on Y (see Lemma 2.2.12(ii)) or, since Δ_{2} also is simplicial, by using Lemma 2.2.12(i) again.

Now we construct an incidence function for the non-simplicial cell complex X in Example 2.2.6. We start by determining an incidence function on the simplicial subcomplex

Y = X^{−1} ∪ X^{0} ∪ X^{1}

of Example 2.2.9(ii).

\begin{matrix}ε & \{v_{1}\} & \{v_{2}\} & \{v_{3}\} & \{v_{4}\} \\ ∅ & 1 & 1 & 1 & 1 \end{matrix}

\begin{matrix}ε & \{v_{1}, v_{2}\} & \{v_{1}, v_{3}\} & \{v_{2}, v_{4}\} & \{v_{3}, v_{4}\} \\ \{v_{1}\} & -1 & -1 & 0 & 0 \\ \{v_{2}\} & 1 & 0 & -1 & 0 \\ \{v_{3}\} & 0 & 1 & 0 & -1 \\ \{v_{4}\} & 0 & 0 & 1 & 1 \end{matrix}

This can by completed to an incidence function on X:

\begin{matrix}ε & \{v_{1}, v_{2}, v_{3}, v_{4}\} \\ \{v_{1}, v_{2}\} & 1 \\ \{v_{1}, v_{3}\} & -1 \\ \{v_{2}, v_{4}\} & 1 \\ \{v_{3}, v_{4}\} & -1 \end{matrix}

Let δ : X → {−1, 1} be a function with δ(∅) = 1 and δ(F) = 1 for all vertices F. Then the function

ε′(F, F′) = δ(F′) ε(F, F′) δ(F)

is also an incidence function. On the other hand, all pairs ε, ε′ of incidence functions differ only by a ’sign’ δ:

Theorem 2.2.14. Let X be a cell complex with incidence functions ε and ε′. Then there is a map δ : X → {−1, 1} such that δ(∅) = 1 and

ε′(F, F′) = δ(F′) ε(F, F′) δ(F)

for all F ∈ X^{i}, F′ ∈ X^{i−1}, i = 0, …, dim X.

Proof. This is Theorem IV.7.2 of [5] (in a different formulation). □

Now we are ready to define the augmented oriented chain complex:

Definition 2.2.15. Let X be a cell complex of dimension d, ε an incidence function on X, and K an arbitrary field. For i = −1, …, d put

C̃_{i}(X; K) = K^{|X^{i}|}.

For i = 0, …, d denote by {f_{1}, …, f_{p}}, {g_{1}, …, g_{q}} the canonical bases of C̃_{i}(X; K) and C̃_{i−1}(X; K). We set

˜∂_{i}(f_{j}) = ∑_{k=1}^{q} ε(F_{j}^{i}, F_{k}^{i−1}) g_{k}, j = 1, …, p.

The augmented oriented chain complex C̃(X; K) is the following sequence of finite dimensional K-vector spaces

C̃(X; K) : 0 ⟶ C̃_{d}(X; K) \underrightarrow{˜∂_{d}} … \underrightarrow{˜∂_{0}} C̃_{−1}(X; K) ⟶ 0.

Consequently, the matrix of ˜∂_{i} with respect to the canonical bases is the q × p matrix

( ε(F_{j}^{i}, F_{k}^{i−1}) )_{{k=1...q \atop j=1...p}} .

In particular, the matrix of ˜∂_{0} with respect to the canonical bases is the 1 × s matrix

( 1 ⋯ 1 ) .

Example 2.2.16. We compute the augmented oriented chain complex for the cell complex X in Example 2.2.6, using the incidence function determined in Example 2.2.13.

C̃(X; K) : 0 ⟶ K ⟶^{( \begin{matrix}1 \\ −1 \\ 1 \\ −1 \end{matrix} )} K^{4} ⟶^{( \begin{matrix}−1 & −1 & 0 & 0 \\ 1 & 0 & −1 & 0 \\ 0 & 1 & 0 & −1 \\ 0 & 0 & 1 & 1 \end{matrix} )} K^{4} \underrightarrow{( 1 1 1 1 )} K ⟶ 0

We use the term augmented because one also considers the complex

C(X; K) : 0 ⟶ C_{d}(X; K) ⟶^{∂_{d}} … ⟶^{∂_{1}} C_{0}(X; K) ⟶ 0

derived from C̃(X; K) by deleting the vector space obtained from X^{−1}.

The next three Lemmas show, that the augmented oriented chain complex is well-defined. First of all, to justify the name augmented oriented chain complex, we have to prove that C̃(X; K) is a complex.

Lemma 2.2.17. C̃(X; K) is a complex of finitely generated K-vector spaces.

Proof. We have to show that

˜∂_{i} ∘ ˜∂_{i+1} = 0, i = 0, …, d − 1.

Let {f_{1}, …, f_{p}},{g_{1}, …, g_{q}} and {h_{1}, …, h_{r}} denote the canonical bases of C̃_{i+1}(X; K), C̃_{i}(X; K) and C̃_{i−1}(X; K). For j ∈ {1, …, p} we have

˜∂_{i} ∘ ˜∂_{i+1}(f_{j}) = ˜∂_{i} (∑_{k=1}^{q} ε(F_{j}^{i+1}, F_{k}^{i}) g_{k})

= ∑_{k=1}^{q} ε(F_{j}^{i+1}, F_{k}^{i}) ˜∂_{i}(g_{k})

= ∑_{k=1}^{q} ε(F_{j}^{i+1}, F_{k}^{i}) (∑_{l=1}^{r} ε(F_{k}^{i}, F_{l}^{i−1}) h_{l})

= ∑_{l=1}^{r} (∑_{k=1}^{q} ε(F_{j}^{i+1}, F_{k}^{i}) ε(F_{k}^{i}, F_{l}^{i−1})) h_{l}

Therefore, it suffices to show that

∑_{k=1}^{q} ε(F_{j}^{i+1}, F_{k}^{i}) ε(F_{k}^{i}, F_{l}^{i−1}) = 0, l = 1, …, r.

In the first case suppose F_{l}^{i−1} is not a face of F_{j}^{i+1}. This implies for each k ∈ {1, …, q} : F_{l}^{i−1} is not a face of F_{k}^{i} or F_{k}^{i} is not a face of F_{j}^{i+1}. Therefore, we get:

ε(F_{j}^{i+1}, F_{k}^{i}) ε(F_{k}^{i}, F_{l}^{i−1}) = 0, k = 1, …, q.

Now suppose that F_{l}^{i−1} is a face of F_{j}^{i+1}. There are exactly two cells F_{k_{1}}^{i}, F_{k_{2}}^{i} ∈ X^{i} such that F_{k_{m}}^{i} is a face of F_{j}^{i+1} and F_{l}^{i−1} is a face of F_{k_{m}}^{i} (m = 1, 2). Together with the properties (b) and (d) of incidence functions (see Definition 2.2.10), it follows

∑_{k=1}^{q} ε(F_{j}^{i+1}, F_{k}^{i}) ε(F_{k}^{i}, F_{l}^{i−1})

= ε(F_{j}^{i+1}, F_{k_{1}}^{i}) ε(F_{k_{1}}^{i}, F_{l}^{i−1}) + ε(F_{j}^{i+1}, F_{k_{2}}^{i}) ε(F_{k_{2}}^{i}, F_{l}^{i−1})
= 0.
□

The next Lemma shows that the dependence of C̃(X; K) on ε is inessential.

Lemma 2.2.18. Let X be a cell complex with incidence function ε. We obtain an isomorphic complex upon replacing ε by another incidence function ε′.

Proof. Let d = dim X. Denote by

C̃(X; K) : 0 ⟶ C̃_{d}(X; K) \underrightarrow{˜∂_{d}} … \underrightarrow{˜∂_{0}} C̃_{−1}(X; K) ⟶ 0

the augmented oriented chain complex upon ε and by

C̃(X; K) : 0 ⟶ C̃_{d}(X; K) \underrightarrow{φ̃_{d}} … \underrightarrow{φ̃_{0}} C̃_{−1}(X; K) ⟶ 0

the augmented oriented chain complex upon ε′. According to Theorem 2.2.14 there is a map δ : X → {−1, 1} such that δ(∅) = 1 and

ε′(F, F′) = δ(F′) ε(F, F′) δ(F)

for all F ∈ X^{i}, F′ ∈ X^{i−1}, i = 0, …, dim X. Now let us define an isomorphism between these two complexes. For i ∈ {−1, …, d} let p = |X^{i}|. We define ψ_{i} : C̃_{i}(X; K) → C̃_{i}(X; K) by

ψ_{i}(f_{j}) = δ(F_{j}^{i})f_{j} j = 1, …, p

where {f_{1}, …, f_{p}} denotes the canonical basis of K^{p}. Clearly, every ψ_{i} is an isomorphism. We have to check that each diagram is commutative. Fix i ∈ {0, …, d} and let p = |X^{i}|, q = |X^{i−1}|. We obtain the diagram

K^{p} ⟶^{A} K^{q}
↓^{C}  ↓^{D}
K^{p} ⟶^{B} K^{q}

where the maps are represented by the matrices

A = ( ε(F_{j}^{i}, F_{k}^{i−1}) )_{{k=1...q \atop j=1...p}} ,
B = ( ε′(F_{j}^{i}, F_{k}^{i−1}) )_{{k=1...q \atop j=1...p}} ,
C = diag ( δ(F_{j}^{i}) )_{j=1...p} ,
D = diag ( δ(F_{k}^{i−1}) )_{k=1...q} .

A straightforward computation provides

C^{−1}AD
= CAD
= diag ( δ(F_{j}^{i}) )_{j=1...p} · ( ε(F_{j}^{i}, F_{k}^{i−1}) )_{{k=1...q \atop j=1...p}} · diag ( δ(F_{k}^{i−1}) )_{k=1...q}
= ( δ(F_{j}^{i}) ε(F_{j}^{i}, F_{k}^{i−1}) δ(F_{k}^{i−1}) )_{{k=1...q \atop j=1...p}}
= ( ε′(F_{j}^{i}, F_{k}^{i−1}) )_{{k=1...q \atop j=1...p}}
= B.

Therefore, the diagram is commutative. □

For computing C̃(X; K) we assume that every component X^{i} of X is given by X^{i} = {F_{1}^{i}, …, F_{|X^{i}|}^{i}}. The next Lemma guarantees that we obtain an isomorphic complex upon another order of the elements of the X^{i}

Lemma 2.2.19. Let X and Y be cell complexes such that

dim X = dim Y

and

|X^{i}| = |Y^{i}|, i = −1, …, dim X.

Suppose that every component is given by

X^{i} = {F_{1}^{i}, …, F_{|X^{i}|}^{i}},
Y^{i} = {G_{1}^{i}, …, G_{|X^{i}|}^{i}}, i = −1, …, dim X.

If for any i ∈ {−1, …, dim X} there is a permutation

σ_{i} : {1, …, |X^{i}|} → {1, …, |X^{i}|}

such that

G_{j}^{i} = F_{σ_{i}(j)}^{i}, j = 1, …, |X^{i}|,

then the two complexes C̃(X; K) and C̃(Y; K) are isomorphic.

Proof. Let ε be an incidence function on X. First, we define an incidence function ε′ on Y: Let i ∈ {0, …, dim X}, p = |X^{i}|, q = |X^{i−1}|. For j ∈ {1, …, p} and k ∈ {1, …, q} we set

ε′(G_{j}^{i}, G_{k}^{i−1}) = ε(F_{σ_{i}(j)}^{i}, F_{σ_{i−1}(k)}^{i−1}).

ε′ is an incidence function on Y. According to Lemma 2.2.18, the choice of the incidence function is inessential. The two augmented oriented chain complexes are

C̃(X; K) : 0 ⟶ C̃_{d}(X; K) \underrightarrow{˜∂_{d}} … \underrightarrow{˜∂_{0}} C̃_{−1}(X; K) ⟶ 0

and

C̃(Y; K) : 0 ⟶ C̃_{d}(Y; K) \underrightarrow{φ̃_{d}} … \underrightarrow{φ̃_{0}} C̃_{−1}(Y; K) ⟶ 0.

Let i ∈ {−1, …, d} and p = |X^{i}|. Define ψ_{i} : C̃(X; K) → C̃(Y; K) by

ψ_{i}(f_{j}) = f_{σ_{i}(j)} j = 1, …, p

where {f_{1}, …, f_{p}} denotes the canonical basis of K^{p}. Evidently, each ψ_{i} is an isomorphism. Now fix an i ∈ {0, …, d}, let p = |X^{i}|, q = |X^{i−1}| and consider the following diagram

K^{p} ⟶^{A} K^{q}
↓^{C}  ↓^{D}
K^{p} ⟶^{B} K^{q}

The matrices with respect to the canonical bases are

A = ( ε(F_{j}^{i}, F_{k}^{i−1}) )_{{k=1...q \atop j=1...p}} ,
B = ( ε′(G_{j}^{i}, G_{k}^{i−1}) )_{{k=1...q \atop j=1...p}} ,
C = perm(σ_{i}),
D = perm(σ_{i−1}).

Recall that such a permutation matrix is defined by

perm(σ)[j, k] = \{ \begin{matrix}1 & k = σ(j) \\ 0 & k ≠ σ(j) \end{matrix}

The following computation

C^{−1}AD
= perm(σ_{i})^{−1} · ( ε(F_{j}^{i}, F_{k}^{i−1}) )_{{k=1...q \atop j=1...p}} · perm(σ_{i−1})
= perm(σ_{i}^{−1}) · ( ε(F_{j}^{i}, F_{k}^{i−1}) )_{{k=1...q \atop j=1...p}} · perm(σ_{i−1})
= ( ε(F_{σ_{i}(j)}^{i}, F_{σ_{i−1}(k)}^{i−1}) )_{{k=1...q \atop j=1...p}}
= ( ε′(G_{j}^{i}, G_{k}^{i−1}) )_{{k=1...q \atop j=1...p}}
= B

shows that the diagram is commutative. □

The preceding lemmas show that the concept of an augmented oriented chain complex is well-defined. It follows from the next lemma that we get an isomorphic complex by renumbering the vertices:

Lemma 2.2.20. Let X be a cell complex on the vertex set V = {v_{1}, …, v_{s}}. For a bijective map σ : {1, …, s} → {1, …, s} let Y be the cell complex obtained by applying σ on all the cells of X:

{v_{i_{1}}, …, v_{i_{m}}} ↦ {v_{σ(i_{1})}, …, v_{σ(i_{m})}}

Then the two complexes C̃(X; K) and C̃(Y; K) are isomorphic.

Proof. The cell complexes X and Y both have the same structure. The elements of the components X^{i} and Y^{i} need not be in the same order. Now we apply Lemma 2.2.19. □

It is convenient to consider X = {} as a cell complex. This so-called complex on no vertices yields the exact sequence

C̃(X; K) : … ⟶ 0 ⟶ 0 ⟶ …

Note that

˜∂_{0} : C̃_{0}(X; K) → C̃_{−1}(X; K)

is always an epimorphism. Therefore, C̃(X; K) is exact if and only if it is acyclic, and the following definition is reasonable:

Definition 2.2.21. A cell complex X is called acyclic over a field K if C̃(X; K) is acyclic.

Note that the acyclicity of a cell complex X may depend on the choice of the field K. A pertinent example will be given in Example 3.1.23.

Now we are looking for methods to decide whether a cell complex is acyclic. The next Lemma 2.2.22 provides a criterion using just Linear Algebra.

Lemma 2.2.22. Let

V_{•} : 0 ⟶ V_{d} ⟶^{φ_{d}} … ⟶^{φ_{2}} V_{1} ⟶^{φ_{1}} V_{0} ⟶ 0

be a complex of finite dimensional K-vector spaces with some choice of bases. Let A_{1}, A_{2}, …, A_{d} denote the matrices of φ_{0}, …, φ_{d} with respect to these bases. Then we have:

(i) V_{•} is exact if and only if

dim_{K} V_{0} = rk A_{1},
dim_{K} V_{j} = rk A_{j} + rk A_{j+1}, j = 1, …, d − 1,
dim_{K} V_{d} = rk A_{d}.

(ii) V_{•} is acyclic if and only if

dim_{K} V_{j} = rk A_{j} + rk A_{j+1}, j = 1, …, d − 1,
dim_{K} V_{d} = rk A_{d}.

(iii) If V_{•} is exact, then

∑_{i=0}^{d}(−1)^{i} dim_{K} V_{i} = 0.

Proof. (i) First of all, we have the equations

dim_{K} (im φ_{j}) = rk A_{j},  j = 1, …, d ,
dim_{K} (ker φ_{j}) = dim_{K} V_{j} − rk A_{j}, j = 1, …, d .

Assume that the conditions of (i) hold.

dim_{K} V_{0} = rk A_{1}

implies that φ_{1} is surjective, and

dim_{K} V_{d} = rk A_{d}

yields that φ_{d} is injective. For j = 1, …, d − 1 we obtain

dim_{K} (im φ_{j+1}) = rk A_{j+1} = dim_{K} V_{j} − rk A_{j} = dim_{K} (ker φ_{j}).

Moreover, we have

im φ_{j+1} ⊂ ker φ_{j},

since V_{•} is a complex, therefore

im φ_{j+1} = ker φ_{j}.

Hence, V_{•} is an exact sequence of K-vector spaces.

For the converse, assume that the complex V_{•} is exact. The surjectivity of φ_{1} implies

dim_{K} V_{0} = rk A_{1},

and the injectivity of φ_{d} shows that

dim_{K} V_{d} = rk A_{d}.

For j = 1, …, d − 1 we have

im φ_{j+1} = ker φ_{j},

which implies

dim (im φ_{j+1}) = dim (ker φ_{j}).

This equation can be written as

rk A_{j+1} = dim_{K} V_{j} − rk A_{j},

and we obtain

dim_{K} V_{j} = rk A_{j} + rk A_{j+1}.

(ii) is obvious.

(iii) Using the identities of (i) we obtain

∑_{i=0}^{d}(−1)^{i} dim_{K} V_{i}

= rk A_{1} + ∑_{i=1}^{d−1}(−1)^{i} (rk A_{i} + rk A_{i+1}) + (−1)^{d} rk A_{d}

= ∑_{i=1}^{d}(−1)^{i+1} rk A_{i} + ∑_{i=1}^{d}(−1)^{i} rk A_{i}
= 0.
□

Evidently, the conditions (i) and (ii) are equivalent for the augmented oriented chain complex C̃(X; K), they are satisfied if and only if the cell complex X is acyclic. Let us illustrate the usage of Lemma 2.2.22 by an example:

Example 2.2.23. Once again, we consider the cell complex X of Example 2.2.6. The augmented oriented chain complex has been computed in Example 2.2.16:

C̃(X; K) : 0 ⟶ K ⟶_{1}^{( \begin{matrix}1 \\ −1 \\ 1 \\ −1 \end{matrix} )} K^{4} ⟶_{3}^{( \begin{matrix}−1 & −1 & 0 & 0 \\ 1 & 0 & −1 & 0 \\ 0 & 1 & 0 & −1 \\ 0 & 0 & 1 & 1 \end{matrix} )} K^{4} \begin{matrix}( & 1 1 1 1 & ) \\ ⟶ \\ 1 \end{matrix} K ⟶ 0

We choose K = ℝ, the field of real numbers. The numbers below the arrows denote the ranks of the corresponding matrices. X is acyclic over ℝ by Lemma 2.2.22.

Corollary 2.2.24. Let X be a cell complex and K an arbitrary field. If

∑_{i=−1}^{dim X}(−1)^{i} |X^{i}| ≠ 0,

then X is not acyclic over K.

Proof. This is a direct consequence of Lemma 2.2.22(iii). □

This result has the advantage that we do not have to compute the augmented oriented chain complex. Moreover, the result does not depend on the choice of the field.

Example 2.2.25. Let Y be the cell complex of Example 2.2.9(ii). According to Corollary 2.2.24 Y is not acyclic over K for every field K.

Later on, we will consider the following special class of cell complexes:

Definition 2.2.26. Let P be a polyhedron in ℝ^{n} with x_{1}, …, x_{s} ∈ ℝ^{n} being its vertices:

vert P = {x_{1}, …, x_{s}}.

We set

X = {∅} ∪ {{v_{j} : j ∈ J} : J ⊂ {1, …, s}, J ≠ ∅,
∃F ∈ F^{b}(P) : vert F = {x_{j} : j ∈ J}}

X is a cell complex on the vertex set V = {v_{1}, …, v_{s}} by Theorem 2.1.25(ii). By Theorem 2.1.26(iv) the dimensions of its cells coincide with the dimensions of the corresponding faces of the polyhedron P. X is said to be the complex of bounded faces of the polyhedron P. If P is a polytope, we also call X the complex of faces of the polytope P.

We notice that the complex of bounded faces of a polyhedron is uniquely determined up to a renumbering of the vertices. For our purpose, this is sufficient (see Lemma 2.2.20). Therefore, the complex of bounded faces is well-defined.

Example 2.2.27. The complex of faces of the first polytope in Example 2.1.22 is the cell complex X of Example 2.2.6. The complex of bounded faces of the polyhedron in Example 2.1.22 is given by the simplicial complex

Δ

Component  Cell
Δ^{−1} F_{1}^{−1} ∅
F_{1}^{0} {v_{1}}
F_{2}^{0} {v_{2}}
Δ^{0}  F_{3}^{0} {v_{3}}
F_{4}^{0} {v_{4}}
F_{1}^{1} {v_{1}, v_{2}}
Δ^{1}  F_{2}^{1} {v_{2}, v_{3}}
F_{3}^{1} {v_{3}, v_{4}}

2.3. Topological cell complexes.

In this section, we obtain more criteria for the acyclicity of cell complexes by applying topological methods.

Definition 2.3.1. A topological cell complex is a non-empty topological space T together with a finite set X of subsets of T such that the following conditions are satisfied:

(i) T = ⋃_{F ∈X}F;

(ii) the subsets F ∈ X are pairwise disjoint; (iii) for each F ∈ X, F ≠ ∅ there exists a homeomorphism from a closed i-dimensional

ball B^{i} = \{x ∈ ℝ^{n} : ∥x∥ ≤ 1\} onto the closure \overline{F} of F which maps the open ball U^{i} = {x ∈ ℝ^{n} : ∥x∥ < 1} onto F;

(iv) ∅ ∈ X.

By the invariance of dimension the number i in (iii) is uniquely determined by F, we set dim F = i. One sets |X| = T. A cell F′ is a face of the cell F ≠ F′ if F′ ⊂ \overline{F}.

Note that this is the definition of cell complexes given in [3], page 253.

For topological cell complexes the following property, which follows from (i)-(iv), is an elementary theorem:

Theorem 2.3.2. Let X be a cell complex. If F ∈ X^{i} and F′ ∈ X^{i−2} is a face of F, then there are exactly two cells F_{1}, F_{2} ∈ X^{i−1} such that F_{j} is a face of F and F′ is a face of F_{j} (j = 1, 2).

Proof. See [3], page 253. □

We continue with some examples of topological cell complexes:

Example 2.3.3. First, we consider a topological cell complex consisting of two points:

|Y | = {(1, 4), (4, 1)} ⊂ ℝ^{2}
Component  Cell
Y^{−1} F_{1}^{−1} ∅
F_{1}^{0} {(1, 4)}
Y^{0}  F_{2}^{0} {(4, 1)}

Example 2.3.4. The following topological cell complex describes a rectangle:

|Y | = [[ (1, 1), (2, 2) ]] ⊂ ℝ^{2}
Component  Cell
Y^{−1} F_{1}^{−1} ∅
F_{1}^{0}  {(1, 1)}
F_{2}^{0}  {(1, 2)}
Y^{0} F_{3}^{0} {(2, 1)}
F_{4}^{0}  {(2, 2)}
F_{1}^{1} ( (1, 1), (1, 2) )
F_{2}^{1} ( (1, 1), (2, 1) )
Y^{1}  F_{3}^{1} ( (1, 2), (2, 2) )
F_{4}^{1} ( (2, 1), (2, 2) )
Y^{2}  F_{1}^{2} (( (1, 1), (2, 2) ))

The notations used in the list are explained in Definition 2.1.1.

Before we continue with examples of topological cell complexes, let us recall

Definition 2.3.5. Let M be a subset of ℝ^{n}. The relative interior of M is

relint M = {x ∈ ℝ^{n} : ∃r_{x} > 0 : B(x, r_{x}) ∩ aff M ⊂ M}.

For example, it is not difficult to see that the relative interior of the convex hull of a finite set {x_{1}, …, x_{r}} is the set of all convex combinations of {x_{1}, …, x_{r}} with no coefficient being equal to zero. The relative interior of a closed half-space derived from a hyperplane H(v, β) is an open half-space:

relint H^{+}(v, β) = {x ∈ ℝ^{n} : ⟨v, x⟩ > β},
relint H^{−}(v, β) = {x ∈ ℝ^{n} : ⟨v, x⟩ < β}.

Example 2.3.6. A further example for a topological cell complex is

|Y | = [[ (1, 1), (4, 4) ]] ⊂ ℝ^{2}

Component  Cell
Y^{−1} F_{1}^{−1} ∅
F_{1}^{0}  {(1, 1)}
F_{2}^{0}  {(1, 4)}
Y^{0} F_{3}^{0} {(4, 4)}
F_{4}^{0}  {(4, 1)}
F_{1}^{1}  ( (1, 1), (1, 4) )
F_{2}^{1}  ( (1, 1), (4, 1) )
Y^{1} F_{3}^{1} ( (1, 4), (4, 4) )
F_{4}^{1}  ( (1, 4), (4, 1) )
F_{5}^{1}  ( (4, 4), (4, 1) )
F_{1}^{2} (( (1, 1), (2, 2) )) ∩ relint H^{−}((1, 1), 5)
Y^{2}  F_{2}^{2} (( (1, 1), (2, 2) )) ∩ relint H^{+}((1, 1), 5)

Example 2.3.7. Finally, we consider the topological cell complex

|Y | = [(7, 1), (4, 2), (2, 4), (1, 7)] ⊂ ℝ^{2}

Component  Cell
Y^{−1} F_{1}^{−1} ∅
F_{1}^{0}  {(7, 1)}
F_{2}^{0}  {(4, 2)}
Y^{0} F_{3}^{0} {(2, 4)}
F_{4}^{0}  {(1, 7)}
F_{1}^{1}  ( (7, 1), (4, 2) )
Y^{1} F_{2}^{1} ( (4, 2), (2, 4) )
F_{3}^{1}  ( (2, 4), (1, 7) )

Now we point out how cell complexes are related to topological cell complexes. The first step is to define a geometric realization of a cell complex as a topological cell complex:

Definition 2.3.8. Let X be a cell complex. A topological cell complex Y is called a geometric realization of X if there is a dimension and inclusion-preserving bijective map

ρ : X → Y

i.e. dim F = dim ρ(F) for any cell F ∈ X, and F is a face of G if and only if ρ(F) is a face of ρ(G) for all pairs F, G ∈ X.

The next result often allows to find a geometric realization of a cell complex:

Lemma 2.3.9. Let X be a cell complex on the vertex set V. Given a map ρ : V → ℝ^{n} such that

(a) ρ is injective,

(b) dim F = dim (conv ρ(F)) for all F ∈ X,

(c) relint (conv ρ(F)) ∩ relint (conv ρ(G)) = ∅ for all F, G ∈ X, F ≠ G. Then

Y = {relint (conv ρ(F)) : F ∈ X}

is a geometric realization of X with the underlying topological space

T = ⋃_{F ∈X} relint (conv ρ(F)).

Proof. It is easy to check that the conditions of Definition 2.3.1 are fulfilled. □

Every simplicial complex Δ admits a geometric realization. Indeed, if V = {v_{1}, …, v_{s}} is the vertex set of Δ, and x_{1}, …, x_{s} are affinely independent elements in ℝ^{n} (which is the case if and only if x_{2} − x_{1}, …, x_{s} − x_{1} are linearly independent), then ρ : V → ℝ^{n} with ρ(v_{i}) = x_{i} for i = 1, …, n defines a geometric realization of Δ. The underlying topological space is the union of all

conv ρ(F)

with F being the facets of Δ.

It is fairly evident that also every complex of bounded faces of a polyhedron admits a geometric realization: Let P be a polyhedron with x_{1}, …, x_{s} ∈ ℝ^{n} being its vertices and X the complex of bounded faces of P on the vertex set V = {v_{1}, …, v_{s}}. A geometric realization is given by ρ : V → ℝ^{n} with ρ(v_{i}) = x_{i} for i = 1, …, n. The underlying topological space is

T = ⋃_{F ∈F^{b}(P)} relint F.

If P is a polytope, the underlying topological space is the polytope itself.

Example 2.3.10. The simplicial complex

Δ

Component  Cell
Δ^{−1}  F_{1}^{−1} ∅
F_{1}^{0} {v_{1}}
Δ^{0}  F_{2}^{0} {v_{2}}

has the topological cell complex of Example 2.3.3 as a geometric realization. For the cell complex X of Example 2.2.6 we have a geometric realization given by the topological cell complex of Example 2.3.4. A geometric realization of the simplicial complex Δ_{1} of Example 2.2.2 is the topological cell complex of Example 2.3.6. Finally, a geometric realization of the simplicial complex Δ of Example 2.2.27 is given by the topological cell complex in Example 2.3.7.

The next result shows that each cell complex X admitting a geometric realization may be identified with a topological cell complex:

Lemma 2.3.11. Let X be a cell complex and let Y, Z be geometric realizations of X. Then the underlying topological spaces |Y | and |Z| are homeomorphic.

Proof. See, e.g., [6] or [7]. □

It is not hard to find a geometric realization for the cell complex X of Example 2.2.6 with the first polytope of Example 2.1.22 as underlying topological space, which is homeomorphic to |Y | of Example 2.3.4.

Since contractibility and connection are invariant under homeomorphisms the last result allows us to define:

Definition 2.3.12. A cell complex X having a geometric realization Y is called contractible if the underlying topological space |Y | is contractible. X is said to be connected, if the underlying topological space |Y | is connected.

Now we get the desired result:

Theorem 2.3.13. A contractible cell complex is acyclic over K for an arbitrary field K.

Proof. For the proof we quote results from Homology Theory. Each singular homology group of a contractible topological space vanishes (see, e.g., [6] or [7]). Together with Theorem 6.2.3 in [3] the assertion follows. □

Recall that any convex set in ℝ^{n} is contractible.

Example 2.3.14. Once again, we consider the cell complex X, first discussed in Example 2.2.6. In Example 2.2.23 we have only shown that X is acyclic over ℝ. The underlying topological space of the geometric realization in Example 2.3.4 is a convex set in ℝ^{2}. Consequently, X is contractible and Theorem 2.3.13 yields that X is acyclic over K for every field K. We can also apply Theorem 2.3.13 to the simplicial complex Δ_{1} of Example 2.2.2, which is contractible, too (see Example 2.3.6). Without having computed C̃(Δ_{1}; K), we obtain that Δ_{1} is acyclic over K for an arbitrary choice of the field K.

Proposition 2.3.15. A simplex Δ is acyclic over every field K.

Proof. According to our preceding results Δ has a geometric realization. Since Δ has exactly one facet, the underlying topological space is a polytope, a convex set in ℝ^{n}. Therefore, it is contractible. We apply Theorem 2.3.13 to complete the proof. □

Example 2.3.16. The simplex Δ_{2} of Example 2.2.2 is acyclic over each field K.

The next result will be used in Chapter 3.

Proposition 2.3.17. The complex of bounded faces of a polyhedron is acyclic over K for every field K.

Proof. The complex of bounded faces of a polyhedron is known to be contractible, see, e.g., [11], Exercise 4.27(a). Now apply Theorem 2.3.13. □

Example 2.3.18. The simplicial complex Δ, computed in Example 2.2.27 is acyclic over every field K.

Proposition 2.3.19. Let X be a non-connected cell complex and let K be an arbitrary field. Then X is not acyclic over K.

Proof. Again we apply methods from the Singular Homology Theory of Algebraic Topology. The first singular homology group of a non-connected topological space does not vanish (see, e.g., [6] or [7]). Therefore, X is not acyclic over K by Theorem 6.2.3 in [3]. □

Example 2.3.20. The simplicial complex Δ of Example 2.3.10 is not acyclic since it is non-connected. Applying Corollary 2.2.24 leads to the same result.

2.4. Fine grading.

In this section, we are interested in ℤ^{n}-graded modules and maps. More generally, we consider gradings for rings and modules coming from an abelian group G. The reader will observe that the well-known ℤ-grading is a special case of this. We will not prove the following lemmas since their proof can be done the same way as for ℤ-grading. A central result is that the ℤ^{n}-graded ideals of the polynomial ring R = K[x_{1}, …, x_{n}] are just the monomial ideals, on which we focus in this thesis. It is no problem to transform a ℤ^{n}-graded resolution of a monomial ideal I into a ℤ-graded and a none-graded resolution. Hence, it makes sense to search for ℤ^{n}-graded resolutions. We also define the ℤ^{n}-graded Betti numbers and examine how they are related to the normal Betti numbers.

Definition 2.4.1. Let (G, +) be an abelian group. A G-graded ring is a ring R together with a decomposition

R = ⨁_{a∈G}[R]_{a}

(as a ℤ-module) such that

[R]_{a}[R]_{b} ⊂ [R]_{a+b}, a, b ∈ G.

A G-graded R-module M is a module over a G-graded ring R together with a decomposition

M = ⨁_{a∈G}[M]_{a}

(as a ℤ-module) such that

[R]_{a}[M]_{b} ⊂ [M]_{a+b}, a, b ∈ G.

One calls [M]_{a} the a-th homogeneous (or G-graded) component of M. The elements x ∈ [M]_{a} are called homogeneous (of degree a); those of [R]_{a} are called a-forms. According to this definition the zero element is homogeneous of arbitrary degree. The degree of x is denoted by deg x. An arbitrary element x ∈ M has a unique presentation x = ∑_{a} x_{a} as a sum of homogeneous elements x_{a} ∈ [M]_{a}. The elements x_{a} are called the homogeneous components of x.

Remark 2.4.2. [R]_{0} is a subring of R, all summands [M]_{a} are [R]_{0}-modules, and

M = ⨁_{a∈G}[M]_{a}

is a direct sum decomposition of M as an [R]_{0}-module.

Lemma 2.4.3. Let N be a submodule of the G-graded R-module M. Then the following conditions are equivalent:

(i) N has a direct sum decomposition

N = ⨁_{a∈G}[N]_{a}

such that [N]_{a} ⊂ [M]_{a} for all a ∈ G; (ii) N has the decompositon

N = ⨁_{a∈G}(N ∩ [M]_{a});

(iii) If x ∈ M is in N then all its homogeneous components x_{a} are in N; (iv) N is generated by homogeneous elements.

(v) M/N is a G-graded R-module possessing the grading

[M/N]_{a} = ([M]_{a} + N)/N.

Definition 2.4.4. The submodules of an G-graded R-module M satisfying the conditions of Lemma 2.4.3 are the G-graded submodules of M. The G-graded submodules of R are called G-graded ideals.

Definition 2.4.5. Let M, N be G-graded R-modules. A homomorphism of G-graded R-modules is an R-module homomorphism

φ : M → N

satisfying

φ([M]_{a}) ⊂ [N]_{a}, a ∈ G.

A sequence

M_{•} : … ⟶ M_{i+1} ⟶^{φ_{i+1}} M_{i} ⟶^{φ_{i}} M_{i−1} ⟶ …

is called a sequence of G-graded R-modules if for any i ∈ ℤ the homomorphism

φ_{i} : M_{i} → M_{i−1}

is a homomorphism of G-graded R-modules. Let M be a G-graded R-module. A G-graded free resolution of M is a free resolution of M, which is a sequence of G-graded R-modules.

Definition 2.4.6. Let φ : M → N be a homomorphism of G-graded R-modules. For any a ∈ G we define the degree a part of φ

[φ]_{a} : [M]_{a} → [N]_{a}

as the restriction of φ on [M]_{a}. [φ]_{a} is homomorphism of [R]_{0}-modules. For a ∈ G and a sequence

M_{•} : … ⟶ M_{i+1} ⟶^{φ_{i+1}} M_{i} ⟶^{φ_{i}} M_{i−1} ⟶ …

of G-graded R-modules we define the sequence of [R]_{0}-modules

[M_{•}]_{a} : … ⟶ [M_{i+1}]_{a} ⟶^{[φ_{i+1}]_{a}} [M_{i}]_{a} ⟶^{[φ_{i}]_{a}} [M_{i−1}]_{a} ⟶ …

and call it the degree a part of M_{•}.

Lemma 2.4.7. Let φ : M → N be a homomorphism of G-graded R-modules. Then ker φ is a G-graded submodule of M and im φ is a G-graded submodule of N. For any a ∈ G we have the identities

[ker φ]_{a} = ker([φ]_{a}),
[im φ]_{a} = im([φ]_{a}).

Definition 2.4.8. Let M be a G-graded R-module. For any a ∈ G we define M(a) as the G-graded R-module with the same module structure as M, but with the grading

[M(a)]_{b} = [M]_{a+b}, b ∈ G.

M(a) is called the module M with degree shift a.

Example 2.4.9. An arbitrary ring R has the trivial grading [R]_{0} = R obtained by G = {0}. For R = K[x_{1}, …, x_{n}] the standard ℤ-grading is given by the K-vector spaces

[R]_{b} = \{ \begin{matrix}⟨x^{a} : |a| = b⟩ & b ≥ 0 \\ 0 & b < 0 \end{matrix}

Note that

dim_{K}([R]_{b}) = \{ \begin{matrix}\binom{n+b−1}{b} & b ≥ 0 \\ 0 & b < 0 \end{matrix}

The polynomial ring R = K[x_{1}, …, x_{n}] does also have a ℤ^{n}-grading which we call the fine grading (or multi grading). We define the components of R as the K-vector spaces

[R]_{b} = \{ \begin{matrix}⟨x^{b}⟩ & b ≥ 0 \\ 0 & b ≱ 0 \end{matrix}

whose dimension is

dim_{K}([R]_{b}) = \{ \begin{matrix}1 & b ≥ 0 \\ 0 & b ≱ 0 \end{matrix}

Note that the ℤ^{n}-graded ideals in R are just the monomial ideals and the ℤ^{n}-graded prime ideals are just the finitely many ideals which are generated by subsets of {x_{1}, …, x_{n}}. For any G-graded ring R

F = ⨁_{j=1}^{p} R(−a_{j}), a_{1}, …, a_{p} ∈ G,

is a free G-graded R-module with the components

[F]_{b} = ⨁_{j=1}^{p}[R]_{b−a_{j}}, b ∈ G.

In particular,

F = R^{p}

is a free G-graded R-module, which contains the components

[F]_{b} = [R]_{b}^{p}, b ∈ G.

Now consider R = K[x_{1}, …, x_{n}] with the ℤ^{n}-grading above and let F be the free module

F = ⨁_{j=1}^{p} R(−a_{j}), a_{1}, …, a_{p} ∈ ℤ^{n},

with basis {f_{1}, …, f_{p}}. If there is an index j such that a_{j} ≤ b (b ∈ ℤ^{n}), then

{x^{b−a_{j}} f_{j} : a_{j} ≤ b}

is a basis of the K-vector space [F]_{b}. Otherwise [F]_{b} = 0. For a monomial ideal

I = (x^{a^{1}}, …, x^{a^{s}})

we obtain

{x^{b}}

as a K-basis for the b-th component [I]_{b}, provided that there exists an index j such that a_{j} ≤ b, and otherwise [I]_{b} = 0.

Remark 2.4.10. An objective of this thesis is the construction of free resolutions of monomial ideals I in the polynomial ring R = K[x_{1}, …, x_{n}]. Since monomial ideals admit a ℤ^{n}-grading it is reasonable to search for ℤ^{n}-graded resolutions of I. Let

𝔽_{1} : 0 ⟶ F_{d} ⟶^{φ_{d}} … ⟶^{φ_{1}} F_{0} ⟶^{φ_{0}} I ⟶ 0,

F_{i} = ⨁_{j=1}^{p_{i}} R(−a_{i}_{j}), i = 0, …, d

be a ℤ^{n}-graded resolution of I. Then

𝔽_{2} : 0 ⟶ G_{d} ⟶^{φ_{d}} … ⟶^{φ_{1}} G_{0} ⟶^{φ_{0}} I ⟶ 0,

G_{i} = ⨁_{j=1}^{p_{i}} R(−|a_{i}_{j}|), i = 0, …, d

is a ℤ-graded resolution of I, and

𝔽_{3} : 0 ⟶ H_{d} ⟶^{φ_{d}} … ⟶^{φ_{1}} H_{0} ⟶^{φ_{0}} I ⟶ 0,
H_{i} = R^{p_{i}}, i = 0, …, d

is a none-graded resolution of I.

Definition 2.4.11. It is known that every ℤ^{n}-graded, finitely generated module M over the polynomial ring

R = K[x_{1}, …, x_{n}]

possesses a minimal free ℤ^{n}-graded resolution

𝔽 : 0 ⟶ F_{d} ⟶ … ⟶ F_{0} ⟶ M ⟶ 0,

F_{i} = ⨁_{b∈ℤ^{n}} R^{β_{i,b}}(−b), i = 0, …, d,

and that the numbers β_{i,b} ∈ ℕ_{0} (by convention R^{0} = 0) do not depend on the choice of the resolution. Therefore, we can define the β_{i,b} = β_{i,b}^{M} as the ℤ^{n}-graded Betti numbers of I. Analogously, we define the ℤ-graded Betti numbers β_{i,j}, j ∈ ℤ and the none-graded Betti numbers β_{i} = rk F_{i} of the monomial ideal I.

Remark 2.4.12. We have the following identities:

β_{i,j} = ∑_{{b∈ℤ^{n} \atop |b|=j}} β_{i,b}, j ∈ ℤ,

β_{i} = ∑_{b∈ℤ^{n}} β_{i,b} = ∑_{j∈ℤ}β_{i,j}.

Example 2.4.13. The ideal

I = (x^{4}, xy, y^{3}) ⊂ R = K[x, y]

has the ℤ^{2}-graded resolution

𝔽_{1} : 0 ⟶ \begin{matrix}R(−4, −1) \\ ⊕ \\ R(−1, −3) \end{matrix} ⟶^{( \begin{matrix}−y & 0 \\ x^{3} & −y^{2} \\ 0 & x \end{matrix} )} \begin{matrix}R(−4, 0) \\ ⊕ \\ R(−1, −1) \\ ⊕ \\ R( 0, −3) \end{matrix} \underrightarrow{( x^{4} xy y^{3} )} I ⟶ 0,

which induces the ℤ-graded resolution

𝔽_{2} : 0 ⟶ \begin{matrix}R(−5) \\ ⊕ \\ R(−4) \end{matrix} ⟶^{( \begin{matrix}−y & 0 \\ x^{3} & −y^{2} \\ 0 & x \end{matrix} )} \begin{matrix}R(−4) \\ ⊕ \\ R(−2) \\ ⊕ \\ R(−3) \end{matrix} \underrightarrow{( x^{4} xy y^{3} )} I ⟶ 0

and the none-graded resolution

𝔽_{3} : 0 ⟶ R^{2} ⟶^{( \begin{matrix}−y & 0 \\ x^{3} & −y^{2} \\ 0 & x \end{matrix} )} R^{3} \underrightarrow{( x^{4} xy y^{3} )} I ⟶ 0.

We obtain the following table of Betti numbers:

Betti numbers of I

ℤ^{2}-Degree 0 1 ℤ-Degree 0 1 0 1
(1,1)  1 0  2  1 0
(0,3)  1 0  3  1 0
(1,3)  0 1  3 2
(4,0)  1 0  4  1 1
(4,1)  0 1  5  0 1

The remaining Betti numbers, not appearing in the table, are equal to zero.

Finally, we mention two more useful observations for graded sequences.

Lemma 2.4.14. Let M_{•} be a sequences of G-graded R-modules. M_{•} is

\{ \begin{matrix}a complex \\ acyclic \\ exact \end{matrix} \}

if and only if for all a ∈ G the degree a part [M_{•}]_{a} is

a complex

an acyclic sequence
an exact sequence

of [R]_{0}-modules.

Lemma 2.4.15. Let R be an arbitrary ring and M_{•}, N_{•} isomorphic sequences of Rmodules. Then M_{•} is

\{ \begin{matrix}a complex \\ acyclic \\ exact \end{matrix} \}

if and only the same holds for N_{•}.

3. Cellular resolutions

Our goal is to determine free resolutions of monomial ideals

I ⊂ R = K[x_{1}, …, x_{n}].

For any monomial ideal I = (x^{a^{1}}, …, x^{a^{s}}) our general assumption is that {x^{a^{1}}, …, x^{a^{s}}} is the uniquely determined set of minimal generators. In Section 3.1 we obtain a cellular complex 𝔽_{X} from a cell complex X. 𝔽_{X} need not be a resolution. Thus, we give a criterion for its exactness. If 𝔽_{X} is a resolution, we call it a cellular resolution. We will also find a criterion for the minimality of cellular resolutions. After having developed our basic theory, we discuss special kinds of such cellular resolutions as well as special kinds of monomial ideals. In Section 3.2 we introduce the Taylor resolution, and in Section 3.3 we present with the hull resolution of a monomial ideal I. As their names indicate, they are examples of resolutions. Both of them need not be minimal, but the hull resolution is most often shorter than the Taylor resolution. In Section 3.4 we investigate the Scarf complex. It turns out that the Scarf complex is a subcomplex of the hull complex. An interesting class of monomial ideals, namely the generic, is the topic of Section 3.5. For generic ideals the Scarf complex coincides with the hull complex and yields a minimal free resolution. Since cellular resolutions need not be minimal, we propose strategies in Section 3.6 how to reduce cellular resolutions in order to make them minimal. In Section 3.7 we describe how to construct cellular resolutions using the Computer Algebra System MAPLE.

3.1. Cellular resolutions.

In this section we develop our basic theory. We shall derive so-called cellular complexes 𝔽_{X} of free R-modules from cell complexes X.

Definition 3.1.1. Let a_{1}, …, a_{p} ∈ ℤ^{n} and let ρ, σ ∈ ℤ^{n} be the elements defined by

ρ_{i} = min {a_{1}_{i}, …, a_{p}_{i}}, i = 1, …, n,
σ_{i} = max {a_{1}_{i}, …, a_{p}_{i}}, i = 1, …, n.

The greatest common divisor and the least common multiple are defined as follows:

gcd {a_{1}, …, a_{p}} = ρ,
lcm {a_{1}, …, a_{p}} = σ.

Accordingly, we set

gcd {x^{a^{1}}, …, x^{a^{p}}} = x^{ρ},
lcm {x^{a^{1}}, …, x^{a^{p}}} = x^{σ}.

Definition 3.1.2. Let

I = (x^{a^{1}}, …, x^{a^{s}}) ⊂ R = K[x_{1}, …, x_{n}]

be a monomial ideal. Recall that we always assume that {x^{a^{1}}, …, x^{a^{s}}} is the uniquely determined set of minimal generators. We say that a cell complex X supports I if X is a cell complex on a vertex set V = {v_{1}, …, v_{s}}. In this case we define the degree of a cell F ∈ X \ {∅} by

deg F = lcm {a_{i} : v_{i} ∈ F }.

This choice of the grading provides that the vertices of X can be identified by the minimal generators of I. Hence, it makes sense to choose the vertices as follows:

v_{i} = x^{a^{i}}, i = 1, …, s.

Now we define the cellular complex 𝔽_{X}:

Definition 3.1.3. Let

I = (x^{a^{1}}, …, x^{a^{s}}) ⊂ R = K[x_{1}, …, x_{n}]

be a monomial ideal, X a cell complex of dimension d which supports I, and ε an incidence function on X. We define the cellular complex

𝔽_{X} : 0 ⟶ F_{d} ⟶^{∂_{d}} … ⟶^{∂_{1}} F_{0} ⟶^{∂_{0}} I ⟶ 0

as follows:

For i = 0, …, d let p = |X^{i}| and F_{i} be the free R-module

F_{i} = ⨁_{j=1}^{p} R(− deg(F_{j}^{i})).

For i = 1, …, d let p = |X^{i}|, q = |X^{i−1}| and denote the canonical bases of F_{i} and F_{i−1} by {f_{1}, …, f_{p}}, {g_{1}, …, g_{q}}. We set

∂_{i}(f_{j}) = ∑_{k=1}^{q} ε(F_{j}^{i}, F_{k}^{i−1}) x^{deg(F_{j}^{i})−deg(F_{k}^{i−1})} g_{k}, j = 1, …, p.

Therefore, the matrix of ∂_{i} with respect to the canonical bases is the q × p matrix

( ε(F_{j}^{i}, F_{k}^{i−1}) x^{deg(F_{j}^{i})−deg(F_{k}^{i−1})} )_{{k=1...q \atop j=1...p}} .

It is |X^{0}| = s, the number of minimal generators of I. Let {f_{1}, …, f_{s}} denote the canonical basis of F_{0}. The epimorphism ∂_{0} : F_{0} ⟶ I is given by

∂_{0}(f_{j}) = x^{a^{j}}, j = 1, …, s.

Hence, the matrix of ∂_{0} is the 1 × s matrix

( x^{a^{1}} … x^{a^{s}} ) .

We have to show that this construction is well-defined. The first thing we have to do, is to prove the following lemma:

Lemma 3.1.4. 𝔽_{X} is a sequence of ℤ^{n}-graded R-modules for every choice of an incidence function ε.

Proof. First of all, we have to check that the matrices for each ∂_{i} only contain elements of R. This is obvious for ∂_{0}. For i ∈ {1, …, d} fix

F_{j}^{i} ∈ X^{i} and F_{k}^{i−1} ∈ X^{i−1}, j ∈ {1, …, p}, k ∈ {1, …, q}.

If F_{k}^{i−1} is not a face of F_{j}^{i}, we obtain ε(F_{j}^{i}, F_{k}^{i−1}) = 0, therefore

ε(F_{j}^{i}, F_{k}^{i−1}) x^{deg(F_{j}^{i})−deg(F_{k}^{i−1})} = 0 ∈ R.

Otherwise, if F_{k}^{i−1} is a face of F_{j}^{i}, we have

deg(F_{k}^{i−1}) ≤ deg(F_{j}^{i}),

which implies

ε(F_{j}^{i}, F_{k}^{i−1}) x^{deg(F_{j}^{i})−deg(F_{k}^{i−1})} ∈ R.

CELLULAR RESOLUTIONS OF MONOMIAL IDEALS 49

Now let i ∈ {1, …, d}, p = |X^{i}|, q = |X^{i−1}| and b ∈ ℤ^{n}. We have to show that

∂_{i}([F_{i}]_{b}) ⊂ [F_{i−1}]_{b}.

According to Example 2.4.9 [F_{i}]_{b} and [F_{i−1}]_{b} can be written as

[F_{i}]_{b} = ⨁_{j=1}^{p}[R]_{b−deg(F_{j}^{i})},
[F_{i−1}]_{b} = ⨁_{k=1}^{q}[R]_{b−deg(F_{k}^{i−1})}.

Let {f_{1}, …, f_{p}} denote the canonical basis of [F_{i}]_{b} and let {g_{1}, …, g_{q}} denote the canonical basis of [F_{i−1}]_{b}. Consider any

h = ∑_{j=1}^{p} h_{j}f_{j} ∈ [F_{i}]_{b}.

Fix j ∈ {1, …, p}. If deg(F_{j}^{i}) ≰ b, then h_{j} = 0. Otherwise, if deg(F_{j}^{i}) ≤ b, we have

h_{j} = c x^{b−deg(F_{j}^{i})}, c ∈ K,

and

∂_{i}(h_{j}f_{j}) = h_{j} ∂_{i}(f_{j})

= c x^{b−deg(F_{j}^{i})} ∑_{k=1}^{q} ε(F_{j}^{i}, F_{k}^{i−1}) x^{deg(F_{j}^{i})−deg(F_{k}^{i−1})} g_{k}

= c ∑_{k=1}^{q} ε(F_{j}^{i}, F_{k}^{i−1}) x^{b−deg(F_{k}^{i−1})} g_{k} ∈ [F_{i−1}]_{b}.

We obtain

∂_{i}(h) ∈ [F_{i−1}]_{b}.

[F_{0}]_{b} can be described as

[F_{0}]_{b} = ⨁_{j=1}^{s}[R]_{b−a_{j}}.

Let {f_{1}, …, f_{s}} denote the canonical basis of [F_{0}]_{b}. Consider any

h = ∑_{j=1}^{s} h_{j}f_{j} ∈ [F_{0}]_{b}.

Let j ∈ {1, …, s}. If a_{j} ≰ b, then h_{j} = 0. Otherwise, if a_{j} ≤ b, we have

h_{j} = c x^{b−a_{j}}, c ∈ K,

and

∂_{0}(h_{j}f_{j}) = h_{j} ∂_{0}(f_{j}) = c x^{b−a_{j}} x^{a^{j}} = c x^{b} ∈ [I]_{b}.

Therefore,

∂_{0}(h) ∈ [I]_{b}.
□

Definition 3.1.5. Let b ∈ ℤ^{n}. If there is an index j ∈ {1, …, s} satisfying a_{j} ≤ b, we set

X_{≤b} = {∅} ∪ {F ∈ X \ {∅} : deg F ≤ b},
I_{≤b} = (x^{a^{j}} : j = 1, …, s, a_{j} ≤ b),
V_{≤b} = {v_{j} : j = 1, …, s, a_{j} ≤ b}.

Otherwise, we put

X_{≤b} = {}, I_{≤b} = {0}, V_{≤b} = ∅.

It is obvious that I_{≤b} is again a monomial ideal, which is contained in I, and that X_{≤b} is a subcomplex of X, which supports I_{≤b}, on the vertex set V_{≤b}.

Lemma 3.1.6. Let X be a cell complex which supports I, and b ∈ ℤ^{n}.

(i) If X is the full simplex on the vertex set V and V_{≤b} ≠ ∅, then X_{≤b} is the full simplex on the vertex set V_{≤b}:

X_{≤b} = P(V_{≤b}) = ⟨V_{≤b}⟩.

(ii) Suppose there is no j ∈ {1, …, s} such that a_{j} ≤ b, then

X_{≤b} = {}.

This is the case for b < gcd {a_{1}, …, a_{s}}. (iii) Assume that a_{j} ≤ b for all j ∈ {1, …, s}. This implies

X_{≤b} = X.

The condition holds for b ≥ lcm {a_{1}, …, a_{s}}. (iv) If there is exactly one index j ∈ {1, …, s} such that a_{j} ≤ b, then X_{≤b} is the

simplex

Δ

Component  Cell
Δ^{−1}  F_{1}^{−1} ∅
Δ^{0}  F_{1}^{0} {v_{j}}

This condition is satisfied for b = a_{j}.

(v) For every c ∈ ℤ^{n} we have

(X_{≤b})_{≤c} = X_{≤gcd {b,c}}.

(vi) For c = lcm {a_{j} : a_{j} ≤ b} we have

X_{≤b} = X_{≤c}.

Proof. The proof is simple and omitted. □

Now we need two technical Lemmas:

Lemma 3.1.7. For each b ∈ ℤ^{n} satisfying X = X_{≤b} the two sequences [𝔽_{X}]_{b} and C̃(X; K) are isomorphic .

Proof. We have to consider the sequences

[𝔽_{X}]_{b} : 0 ⟶ [F_{d}]_{b} ⟶^{[∂_{d}]_{b}} … ⟶^{[∂_{1}]_{b}} [F_{0}]_{b} ⟶^{[∂_{0}]_{b}} [I]_{b} ⟶ 0,
C̃(X; K) : 0 ⟶ C̃_{d}(X; K) ⟶^{φ_{d}} … ⟶^{φ_{1}} C̃_{0}(X; K) ⟶^{φ_{0}} C̃_{−1}(X; K) ⟶ 0.

Because of Lemma 2.2.18 we may assume that both [𝔽_{X}]_{b} and C̃(X; K) are obtained from the same incidence function ε. Let i ∈ {0, …, d}, p = |X^{i}|. We have

[F_{i}]_{b} = ⨁_{j=1}^{p}[R]_{b−deg(F_{j}^{i})}.

Let {f_{1}, …, f_{p}} denote the canonical basis of [F_{i}]_{b}. Then

{x^{b−deg(F_{j}^{i})}f_{j} : j = 1, …, p}

is a K-basis of [F_{i}]_{b}, and

{f_{j} : j = 1, …, p}

is a K-basis of C̃_{i}(X; K).

Define ψ_{i} : [F_{i}]_{b} → C̃_{i}(X; K) by

ψ_{i}(f_{j}) = f_{j}, j = 1, …, p.

Clearly, ψ_{i} is an isomorphism, whose matrix with respect to the bases above is the p × p unit matrix E_{p}. {x^{b}} is a K-basis of [I]_{b}, and {1} is a K-basis of C̃_{−1}(X; K). We define the isomorphism ψ_{−1} : [I]_{b} → C̃_{−1}(X; K) by

ψ_{−1}(x^{b}) = 1.

The matrix of this isomorphism is the unit matrix E_{1}. In order to see that every diagram is commutative, it suffices to show that for any i ∈ {0, …, d} the matrices of [∂_{i}]_{b} and φ_{i} with respect to the bases given above coincide. Fix i ∈ {1, …, d}. Let p = |X^{i}|, q = |X^{i−1}| and {f_{1}, …, f_{p}} denote the canonical basis with p coordinates. Then

{x^{b−deg(F_{j}^{i})}f_{j} : j = 1, …, p}

is a K-basis of [F_{i}]_{b}, and

{f_{j} : j = 1, …, p}

is a K-basis of C̃_{i}(X; K).

Analogously, let {g_{1}, …, g_{q}} denote the canonical basis with q elements. Then

{x^{b−deg(F_{k}^{i−1})}g_{k} : k = 1, …, q}

is a K-basis of [F_{i}]_{b}, and

{g_{k} : k = 1, …, q}

is a K-basis of C̃_{i−1}(X; K).

Denote by A and B the matrices of [∂_{i}]_{b} and φ_{i}, respectively. The matrices of ψ_{i} and ψ_{i−1} are the identity matrices E_{p} and E_{q}. Therefore, we get the diagram

[F_{i}]_{b} ⟶^{A} [F_{i−1}]_{b}
↓^{E_{p}}  ↓^{E_{q}}
K^{p} ⟶^{B} K^{q}

Obviously

B = ( ε(F_{j}^{i}, F_{k}^{i−1}) )_{{k=1...q \atop j=1...p}} .

Now we have

[∂_{i}]_{b}(x^{b−deg(F_{j}^{i})}f_{j})
= x^{b−deg(F_{j}^{i})} ∂_{i}(f_{j})
= x^{b−deg(F_{j}^{i})} ∑_{k=1}^{q} ε(F_{j}^{i}, F_{k}^{i−1}) x^{deg(F_{j}^{i})−deg(F_{k}^{i−1})} g_{k}

= ∑_{k=1}^{q} ε(F_{j}^{i}, F_{k}^{i−1}) x^{b−deg(F_{k}^{i−1})} g_{k}, j = 1, …, p.

Consequently, A = B, and we conclude that the diagram is commutative.

Now consider i = 0. Let {f_{1}, …, f_{s}} denote the canonical basis with s coordinates. Recall that s is the number of minimal generators of the monomial ideal I.

{x^{b−a_{j}}f_{j} : j = 1, …, s}

is a K-basis of [F_{0}]_{b}, and

{f_{j} : j = 1, …, s}

is a K-basis of C̃_{0}(X; K).

{x^{b}} and {1} are K-bases of [I]_{b} and C̃_{−1}(X; K). A and B denote the matrices of [∂_{0}]_{b} and φ_{0} with respect to the bases given above. The matrices of ψ_{0} and ψ_{−1} are the unit matrices E_{p} and E_{1}. Therefore, we have to consider the diagram

[F_{0}]_{b} ⟶^{A} [I]_{b}
↓^{E_{p}}  ↓^{E_{1}}
K^{s} ⟶^{B} K

It is clear that B = ( 1 ⋯ 1 ). Thus,

[∂_{0}]_{b}(x^{b−a_{j}}f_{j}) = x^{b−a_{j}} ∂_{0}(f_{j}) = x^{b−a_{j}}x^{a^{j}} = x^{b}

shows that

A = B = ( 1 ⋯ 1 ) ,

which completes the proof. □

Lemma 3.1.8. For any b ∈ ℤ^{n} the two sequences [𝔽_{X}]_{b} and [𝔽_{X_{≤b}}]_{b} are isomorphic.

Proof. We have to consider the two sequences

[𝔽_{X}]_{b} : 0 ⟶ [F_{d}]_{b} ⟶^{[∂_{d}]_{b}} … ⟶^{[∂_{1}]_{b}} [F_{0}]_{b} ⟶^{[∂_{0}]_{b}} [I]_{b} ⟶ 0,
[𝔽_{X_{≤b}}]_{b} : 0 ⟶ [G_{d}]_{b} ⟶^{[φ_{d}]_{b}} … ⟶^{[φ_{1}]_{b}} [G_{0}]_{b} ⟶^{[φ_{0}]_{b}} [I_{≤b}]_{b} ⟶ 0.

Without loss of generality we may assume that there is an index i ∈ {1, …, s} such that a_{i} ≤ b because otherwise both [𝔽_{X}]_{b} and [𝔽_{X_{≤b}}]_{b} coincide with the trivial complex

… ⟶ 0 ⟶ 0 ⟶ …

Choose an incidence function ε on X. The restriction of ε on X_{≤b} is an incidence function on X_{≤b} according to Lemma 2.2.12(ii). Because of the Lemmas 2.2.18 and 3.1.7 we may assume that the incidence function on X_{≤b} is given by this restriction. Let i ∈ {0, …, d}, p = |X^{i}|, p̃ = |X_{≤b}^{i}|. We have

[F_{i}]_{b} = ⨁_{j=1}^{p}[R]_{b−deg(F_{j}^{i})}.

If [F_{i}]_{b} = 0, then deg(F_{j}^{i}) > b for j = 1, …, p, therefore [G_{i}]_{b} = 0. Now, let us assume that [F_{i}]_{b} ≠ 0. Then, note that p̃ ≥ 1. Let X^{i} be given by X^{i} = {F_{1}^{i}, …, F_{p}^{i}}. There is a unique injective map σ_{i} : {1, …, p̃} → {1, …, p}, such that σ_{i}(1) < … < σ_{i}(p̃), X_{≤b}^{i} = {F_{σ_{i}(1)}^{i}, …, F_{σ_{i}(p̃)}^{i}} and

[G_{i}]_{b} = p̃ ; ⨁ ; j=1[R]_{b−deg(F_{σ_{i}(j)}^{i})}.

Let {f_{1}, …, f_{p}} denote the canonical basis with p coordinates and {f̃_{1}, …, f̃p̃} the canonical basis with p̃ coordinates. Then

{x^{b−deg(F_{σ_{i}(j)}^{i})}f_{σ_{i}(j)} : j = 1, …, p̃}

is a K-basis of [F_{i}]_{b}, and

{x^{b−deg(F_{σ_{i}(j)}^{i})}f̃_{j} : j = 1, …, p̃}

is a K-basis of [G_{i}]_{b}.

ψ_{i} : [F_{i}]_{b} → [G_{i}]_{b} defined by

ψ_{i}(x^{b−deg(F_{σ_{i}(j)}^{i})} f_{σ_{i}(j)}) = x^{b−deg(F_{σ_{i}(j)}^{i})} f̃_{j}, j = 1, …, p̃

is an isomorphism. Its matrix with respect to these bases is the p̃× p̃ unit matrix Ep̃.

[I]_{b} and [I_{≤b}]_{b} both have the K-basis {x^{b}}. We simply define ψ_{−1} : [I]_{b} → [I_{≤b}]_{b} by

ψ_{−1}(x^{b}) = x^{b},

whose matrix is again the unit matrix E_{1}. To see that every diagram is commutative, it suffices to show that for any i ∈ {0, …, d} the matrices of [∂_{i}]_{b} and [φ_{i}]_{b} with respect to the bases given above coincide. Fix i ∈ {1, …, d}. We may suppose that [F]_{i} ≠ 0 and [F]_{i−1} ≠ 0. Let p = |X^{i}|, p̃ = |X_{≤b}^{i}|, q = |X^{i−1}|, q̃ = |X_{≤b}^{i−1}| and {f_{1}, …, f_{p}} denote the canonical basis with p elements and {f̃_{1}, …, f̃p̃} the canonical basis with p̃ elements. Then

{x^{b−deg(F_{σ_{i}(j)}^{i})}f_{σ_{i}(j)} : j = 1, …, p̃}

is a K-basis of [F_{i}]_{b}, and

{x^{b−deg(F_{σ_{i}(j)}^{i})}f̃_{j} : j = 1, …, p̃}

is a K-basis of [G_{i}]_{b}.

Analogously, let {g_{1}, …, g_{q}} denote the canonical basis with q coordinates and {g̃_{1}, …, g̃q̃} the canonical basis with q̃ coordinates. Then

{x^{b−deg(F_{σ_{i−1}(k)}^{i−1})} g_{σ_{i−1}(k)} : k = 1, …, q̃}

is a K-basis of [F_{i−1}]_{b}, and

{x^{b−deg(F_{σ_{i−1}(k)}^{i−1})} g̃_{k} : k = 1, …, q̃}

is a K-basis of [G_{i−1}]_{b}. Denote by A and B the matrices of [∂_{i}]_{b} and [φ_{i}]_{b} with respect to these bases. The matrices of ψ_{i} and ψ_{i−1} are the identity matrices Ep̃ and Eq̃, respectively. Therefore, we have to consider the following diagram:

[F_{i}]_{b} ⟶^{A} [F_{i−1}]_{b}
↓^{Ep̃}  ↓^{Eq̃}
[G_{i}]_{b} ⟶^{B} [G_{i−1}]_{b}

Next, we determine the matrices A and B. The two chains of equalities

[∂_{i}]_{b}(x^{b−deg(F_{σ_{i}(j)}^{i})}f_{σ_{i}(j)})
= x^{b−deg(F_{σ_{i}(j)}^{i})} ∂_{i}(f_{σ_{i}(j)})
= x^{b−deg(F_{σ_{i}(j)}^{i})} ∑_{k=1}^{q} ε(F_{σ_{i}(j)}^{i}, F_{k}^{i−1}) x^{deg(F_{σ_{i}(j)}^{i})−deg(F_{k}^{i−1})} g_{k}

= ∑_{k=1}^{q} ε(F_{σ_{i}(j)}^{i}, F_{k}^{i−1}) x^{b−deg(F_{k}^{i−1})} g_{k}

= q̃ ; ∑ ; k=1 ε(F_{σ_{i}(j)}^{i}, F_{σ_{i−1}(k)}^{i−1}) x^{b−deg(F_{σ_{i−1}(k)}^{i−1})} g_{σ_{i−1}(k)}, j = 1, …, p̃,

and

[φ_{i}]_{b}(x^{b−deg(F_{σ_{i}(j)}^{i})}f̃_{j})
= x^{b−deg(F_{σ_{i}(j)}^{i})} φ_{i}(f̃_{j})
= x^{b−deg(F_{σ_{i}(j)}^{i})} q̃ ; ∑ ; k=1 ε(F_{σ_{i}(j)}^{i}, F_{σ_{i−1}(k)}^{i−1}) x^{deg(F_{σ_{i}(j)}^{i})−deg(F_{σ_{i−1}(k)}^{i−1})} g̃_{k}

= q̃ ; ∑ ; k=1 ε(F_{σ_{i}(j)}^{i}, F_{σ_{i−1}(k)}^{i−1}) x^{b−deg(F_{σ_{i−1}(k)}^{i−1})} g̃_{k}, j = 1, …, p̃,

show that

A = B = ( ε(F_{σ_{i}(j)}^{i}, F_{σ_{i−1}(k)}^{i−1}) )_{{k=1...q̃ \atop j=1...p̃}} ,

which yields that the diagram is commutative.

We still have to check the diagram for i = 0. We have s = |X^{0}|. Remember that s denotes the number of minimal generators of the ideal I. Let {f_{1}, …, f_{s}} denote the canonical basis with s elements and for s̃ = |X_{≤b}^{0}| let {f̃_{1}, …, f̃s̃} be the canonical basis with s̃ elements.

{x^{b−a_{σ_{0}(j)}}f_{σ_{0}(j)} : j = 1, …, s̃}

is a K-basis of [F_{0}]_{b}, and

{x^{b−a_{σ_{0}(j)}} f̃_{j} : j = 1, …, s̃}

is a K-basis of [G_{0}]_{b}.

[I]_{b} and [I_{≤b}]_{b} both have the K-basis {x^{b}}. A and B denote the matrices of [∂_{0}]_{b} and [φ_{0}]_{b} with respect to these bases. The matrices of ψ_{0} and ψ_{−1} are the identity matrices Ep̃ and E_{1} respectively. Therefore, we have to consider the following diagram:

[F_{0}]_{b} ⟶^{A} [I]_{b}
↓^{Ep̃}  ↓^{E_{1}}
[G_{0}]_{b} ⟶^{B} [I_{≤b}]_{b}

Again, we determine the matrices A and B. The two chains of equalities

[∂_{0}]_{b}(x^{b−a_{σ_{0}(j)}}f_{σ_{0}(j)}) = x^{b−a_{σ_{0}(j)}} ∂_{0}(f_{σ_{0}(j)}) = x^{b−a_{σ_{0}(j)}} x^{a^{σ_{0}(j)}} = x^{b}

and

[φ_{0}]_{b}(x^{b−a_{σ_{0}(j)}} f̃_{j}) = x^{b−a_{σ_{0}(j)}} φ_{0}(f̃_{j}) = x^{b−a_{σ_{0}(j)}} x^{a^{σ_{0}(j)}} = x^{b}

prove that

A = B = ( 1 ⋯ 1 ) .

Hence, the diagram is commutative. □

Proposition 3.1.9. For every b ∈ ℤ^{n} the degree b part [𝔽_{X}]_{b} and the augmented oriented chain complex C̃(X_{≤b}; K) are isomorphic.

Proof. According to Lemma 3.1.6(v) we have for every b ∈ ℤ^{n}

(X_{≤b})_{≤b} = X_{≤b}.

The Lemmas 3.1.7 and 3.1.8 yield

[𝔽_{X}]_{b} ≅ [𝔽_{X_{≤b}}]_{b} ≅ C̃(X_{≤b}; K).
□

Now we have completed all preparations which guarantee that the cellular complex 𝔽_{X} is well-defined: First of all, Lemma 3.1.4 proves that 𝔽_{X} is a sequence of ℤ^{n}-graded R-modules. Using the isomorphism [𝔽_{X}]_{b} ≅ C̃(X_{≤b}; K) (see Proposition 3.1.9) and the Lemmas 2.4.14 and 2.4.15 we get: 𝔽_{X} is a complex of ℤ^{n}-graded R-modules by Lemma 2.2.17. Therefore, the name cellular complex is justified. The dependence of 𝔽_{X} on the choice of the incidence function ε is inessential by Lemma 2.2.18. Lemma 2.2.19 ensures that we obtain an isomorphic complex by changing the order of the elements of the components X^{i} and according to Lemma 2.2.20 the order of the minimal generators of the monomial ideal is inessential.

Remark 3.1.10. Using the embedding I ⊂ R a cellular complex

𝔽_{X} : 0 ⟶ F_{d} ⟶ … ⟶ F_{0} ⟶ I ⟶ 0

gives the complex

𝔽 : 0 ⟶ F_{d} ⟶ … ⟶ F_{0} ⟶ R ⟶ R/I ⟶ 0

of ℤ^{n}-graded modules. 𝔽 is a free resolution of R/I if and only if 𝔽_{X} is a resolution of the monomial ideal I. It is minimal if and only if 𝔽_{X} is minimal.

Now we are looking for criteria to decide whether such a cellular complex 𝔽_{X} is exact, and, provided it is a resolution of I, if it is minimal. We will also obtain a formula for the Betti numbers and the Hilbert function of I.

Theorem 3.1.11. Let K be a field. The following statements are equivalent: (i) The cellular complex 𝔽_{X} is a free resolution of I; (ii) X_{≤b} is acyclic over K for the finitely many degrees b ∈ ℤ^{n} satisfying

gcd {a_{1}, …, a_{s}} ≤ b ≤ lcm {a_{1}, …, a_{s}}.

Proof. According to Lemma 2.4.14 the complex 𝔽_{X} is a free resolution of I if and only if for all b ∈ ℤ^{n} the degree b part [𝔽_{X}]_{b} is an exact sequence of K-vector spaces. It follows from Lemma 3.1.9 that this is the case if and only if X_{≤b} is acyclic for all degrees b ∈ ℤ^{n}. It remains to show that it suffices to check that this condition is satisfied for all degrees b ∈ ℤ^{n} such that

gcd {a_{1}, …, a_{s}} ≤ b ≤ lcm {a_{1}, …, a_{s}}.

Let b ∈ ℤ^{n}. Without loss of generality we may assume that X_{≤b} ≠ {} because in this case it is trivially acyclic. Consider

c = lcm {a_{j} : a_{j} ≤ b}.

We obtain

gcd {a_{1}, …, a_{s}} ≤ c ≤ lcm {a_{1}, …, a_{s}}

and by Lemma 3.1.6(vi)

X_{≤b} = X_{≤c}.
□

Definition 3.1.12. If the conditions of Theorem 3.1.11 are satisfied, we call 𝔽_{X} a cellular resolution of I.

Remark 3.1.13. The last criterion provides an algorithm for checking the exactness of 𝔽_{X}. Using the Computer Algebra System MAPLE, the author has written a program, which computes the cellular complex 𝔽_{X} for a given monomial ideal I and a supporting cell complex X. The program can also test properties of the cellular complex by using the last and the following criteria. A detailed description of this program will be given in the Section 3.7. As we have already mentioned, this is the reason for using a modified definition of cell complexes, a definition which can be realized as a data structure for the computer.

Proposition 3.1.14. A cellular resolution 𝔽_{X} is a minimal resolution of I if and only if any pair of cells F, F′ ∈ X, such that F′ is a face of F, has distinct degrees

deg F′ < deg F.

Proof. According to the construction of 𝔽_{X}, the elements of the matrices of the ∂_{i}, i = 1, …, dim X, which are not equal to zero, are of the form

ε(F, F′) x^{deg F −deg F′},

where F′ is a face of F. Therefore, the only field elements not being equal to zero that can appear in such a matrix are 1 and −1. The matrices of 𝔽_{X} do not contain the elements 1 and −1 if and only if any two cells F, F′ ∈ X, such that F′ is a face if F, have distinct degrees deg F′ < deg F. □

Corollary 3.1.15. The cellular complex 𝔽_{X} is a resolution of I if and only if the cellular complex 𝔽_{X_{≤b}} is a resolution of the monomial ideal I_{≤b} for all b ∈ ℤ^{n}. If 𝔽_{X} is a minimal resolution, 𝔽_{X_{≤b}} is minimal for all b ∈ ℤ^{n}.

Proof. This follows from Theorem 3.1.11 and the identity

(X_{≤b})_{≤c} = X_{gcd {b,c}}

of Lemma 3.1.6(v). If 𝔽_{X} is a minimal resolution, 𝔽_{X_{≤b}} remains minimal for all b ∈ ℤ^{n} by Proposition 3.1.14. □

Corollary 3.1.16. Let X be a cell complex which supports I. If

∑_{i=−1}^{dim X}(−1)^{i} |X^{i}| ≠ 0,

then 𝔽_{X} is not a resolution of the ideal I.

Proof. Apply Theorem 3.1.11 and Corollary 2.2.24. □

Corollary 3.1.17. Let 𝔽_{X} be a minimal free resolution of the monomial ideal I and i ∈ {0, …, dim X}. Then there are the following equations for the various Betti numbers of I:

β_{i,b} = |{F ∈ X^{i} : deg F = b}|, b ∈ ℤ^{n},
β_{i,j} = |{F ∈ X^{i} : | deg F | = j}|, j ∈ ℤ,
β_{i} = |X^{i}|.

Proposition 3.1.18. Let 𝔽_{X} be a free resolution of the monomial ideal

I ⊂ R = K[x_{1}, …, x_{n}].

Then the Hilbert functions of I and R/I can be computed as follows:

h_{I}(k) = ∑_{i=0}^{dim X}(−1)^{i} ∑_{j=1}^{|X^{i}|} \binom{n − 1 + k − | deg F_{j}^{i}|}{n − 1}, k ∈ ℤ,

h_{R/I}(k) = \binom{n − 1 + k}{n − 1} − h_{I}(k),  k ∈ ℤ.

Proof. Let

𝔽_{X} : 0 ⟶ F_{d} ⟶ … ⟶ F_{0} ⟶ I ⟶ 0

be the cellular resolution. Then we get

h_{I}(k) = ∑_{i=0}^{d}(−1)^{i} h_{F_{i}}(k), k ∈ ℤ.

The Hilbert function of the free module

F = ⨁_{j=1}^{p} R(−a_{j})

is

h_{I}(k) = ∑_{j=1}^{p} \binom{n − 1 + k − a_{j}}{n − 1},

which yields the asserted identity.

The second equation is an easy consequence of

h_{R/I}(k) = h_{R}(k) − h_{I}(k), k ∈ ℤ.
□

Remark 3.1.19. Applying the last Proposition 3.1.18, we obtain the Hilbert polynomials of I and R/I and, hence, their dimensions and degrees. The height of the monomial ideal I can be computed as

h(I) = n − dim (R/I).

Proposition 3.1.20. Let 𝔽_{X} be a minimal free resolution of I. Then we have: R/I is Cohen-Macaulay if and only if

dim (R/I) + dim X = n − 1.

Proof. It is known that R/I is Cohen-Macaulay if and only if the height of I is equal to the projective dimension of R/I:

h(I) = pd_{R}(R/I).

Because of the equalities

h(I) = n − dim (R/I),
pd_{R}(R/I) = dim X + 1,

we obtain as equivalent condition

n − dim (R/I) = dim X + 1,

hence

dim (R/I) + dim X = n − 1.
□

Now, the theory above will be illustrated by an example:

Example 3.1.21. Consider the monomial ideal

I = (x^{2}y, xz, yz^{2}, y^{2}) ⊂ R = K[x, y, z].

We discuss two simplicial complexes X, Y which support I:

X = ⟨{v_{1}, v_{2}, v_{4}}, {v_{2}, v_{3}, v_{4}}⟩
Component  Cell  Degree
X^{−1}  F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (2,1,0)
F_{2}^{0}  {v_{2}}  (1,0,1)
X^{0}  F_{3}^{0}  {v_{3}}  (0,1,2)
F_{4}^{0}  {v_{4}}  (0,2,0)
F_{1}^{1} {v_{1}, v_{2}} (2,1,1)
F_{2}^{1} {v_{1}, v_{4}} (2,2,0)
X^{1}  F_{3}^{1} {v_{2}, v_{3}} (1,1,2)
F_{4}^{1} {v_{2}, v_{4}} (1,2,1)
F_{5}^{1} {v_{3}, v_{4}} (0,2,2)
F_{1}^{2} {v_{1}, v_{2}, v_{4}} (2,2,1)
X^{2}  F_{2}^{2} {v_{2}, v_{3}, v_{4}} (1,2,2)

and

Y = ⟨{v_{1}, v_{2}, v_{3}}, {v_{1}, v_{3}, v_{4}}⟩

Component  Cell  Degree
Y^{−1}  F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (2,1,0)
F_{2}^{0}  {v_{2}}  (1,0,1)
Y^{0}  F_{3}^{0}  {v_{3}}  (0,1,2)
F_{4}^{0}  {v_{4}}  (0,2,0)
F_{1}^{1} {v_{1}, v_{2}} (2,1,1)
F_{2}^{1} {v_{1}, v_{3}} (2,1,2)
Y^{1}  F_{3}^{1} {v_{1}, v_{4}} (2,2,0)
F_{4}^{1} {v_{2}, v_{3}} (1,1,2)
F_{5}^{1} {v_{3}, v_{4}} (0,2,2)
F_{1}^{2} {v_{1}, v_{2}, v_{3}} (2,1,2)
Y^{2}  F_{2}^{2} {v_{1}, v_{3}, v_{4}} (2,2,2)

X already appeared as Δ_{1} in Example 2.2.2. We have computed an incidence function ε in Example 2.2.13. So we can determine the homomorphisms of the cellular complex (as described in Definition 3.1.3):

\begin{matrix}∂_{0} & \{v_{1}\} & \{v_{2}\} & \{v_{3}\} & \{v_{4}\} \\ ∅ & x^{2}y & xz & yz^{2} & y^{2} \end{matrix}

\begin{matrix}∂_{1} & \{v_{1}, v_{2}\} & \{v_{1}, v_{4}\} & \{v_{2}, v_{3}\} & \{v_{2}, v_{4}\} & \{v_{3}, v_{4}\} \\ \{v_{1}\} & −z & −y & 0 & 0 & 0 \\ \{v_{2}\} & xy & 0 & −yz & −y^{2} & 0 \\ \{v_{3}\} & 0 & 0 & x & 0 & −y \\ \{v_{4}\} & 0 & x^{2} & 0 & xz & z^{2} \end{matrix}

\begin{matrix}∂_{2} & \{v_{1}, v_{2}, v_{4}\} & \{v_{2}, v_{3}, v_{4}\} \\ \{v_{1}, v_{2}\} & y & 0 \\ \{v_{1}, v_{4}\} & −z & 0 \\ \{v_{2}, v_{3}\} & 0 & y \\ \{v_{2}, v_{4}\} & x & −z \\ \{v_{3}, v_{4}\} & 0 & x \end{matrix}

We obtain the cellular complex

𝔽_{X} : 0 ⟶ R^{2}(−5) ⟶^{( \begin{matrix}y & 0 \\ −z & 0 \\ 0 & y \\ x & −z \\ 0 & x \end{matrix} )} R^{5}(−4) ⟶^{( \begin{matrix}−z & −y & 0 & 0 & 0 \\ xy & 0 & −yz & −y^{2} & 0 \\ 0 & 0 & x & 0 & −y \\ 0 & x^{2} & 0 & xz & z^{2} \end{matrix} )} \begin{matrix}R(−3) \\ ⊕ \\ R(−2) \\ ⊕ \\ R(−3) \\ ⊕ \\ R(−2) \end{matrix} \underrightarrow{( x^{2}y xz yz^{2} y^{2} )} I ⟶ 0.

We claim that the cellular complex 𝔽_{X} is a minimal free resolution of I. By Theorem 3.1.11 we have to check that X_{≤b} is acyclic for each degree

(0, 0, 0) ≤ b ≤ (2, 2, 2).

We compute the augmented oriented chain complex C̃(X_{≤b}; K) for any such b by using the restriction of ε as an incidence function on X_{≤b} (see Lemma 2.2.12(ii)). The numbers below the arrows denote the rank of the matrices above the arrows for K = ℝ.

b = (2, 2, 2)

C̃(X_{≤b}; K) : 0 ⟶ K^{2} ⟶_{2}^{( \begin{matrix}1 & 0 \\ −1 & 0 \\ 0 & 1 \\ 1 & −1 \\ 0 & 1 \end{matrix} )} K^{5} ⟶_{3}^{( \begin{matrix}−1 & −1 & 0 & 0 & 0 \\ 1 & 0 & −1 & −1 & 0 \\ 0 & 0 & 1 & 0 & −1 \\ 0 & 1 & 0 & 1 & 1 \end{matrix} )} K^{4} \begin{matrix}( & 1 1 1 1 & ) \\ ⟶ \\ 1 \end{matrix} K ⟶ 0

b = (2, 2, 1)

C̃(X_{≤b}; K) : 0 ⟶ K ⟶_{1}^{( \begin{matrix}1 \\ −1 \\ 1 \end{matrix} )} K^{3} ⟶_{2}^{( \begin{matrix}−1 & −1 & 0 \\ 1 & 0 & −1 \\ 0 & 1 & 1 \end{matrix} )} K^{3} \begin{matrix}( & 1 1 1 & ) \\ ⟶ \\ 1 \end{matrix} K ⟶ 0

b = (1, 2, 2)

C̃(X_{≤b}; K) : 0 ⟶ K ⟶_{1}^{( \begin{matrix}1 \\ −1 \\ 1 \end{matrix} )} K^{3} ⟶_{2}^{( \begin{matrix}−1 & −1 & 0 \\ 1 & 0 & −1 \\ 0 & 1 & 1 \end{matrix} )} K^{3} \begin{matrix}( & 1 1 1 & ) \\ ⟶ \\ 1 \end{matrix} K ⟶ 0

b = (2, 1, 2)

C̃(X_{≤b}; K) : 0 ⟶ K^{2} ⟶_{2}^{( \begin{matrix}−1 & 0 \\ 1 & −1 \\ 0 & 1 \end{matrix} )} K^{3} \begin{matrix}( & 1 1 1 & ) \\ ⟶ \\ 1 \end{matrix} K ⟶ 0

b = (2, 1, 1)

C̃(X_{≤b}; K) : 0 ⟶ K ⟶_{1}^{( \begin{matrix}−1 \\ 1 \end{matrix} )} K^{2} \begin{matrix}( & 1 1 & ) \\ ⟶ \\ 1 \end{matrix} K ⟶ 0

b = (2, 2, 0)

C̃(X_{≤b}; K) : 0 ⟶ K ⟶_{1}^{( \begin{matrix}−1 \\ 1 \end{matrix} )} K^{2} \begin{matrix}( & 1 1 & ) \\ ⟶ \\ 1 \end{matrix} K ⟶ 0

b = (1, 1, 2)

C̃(X_{≤b}; K) : 0 ⟶ K ⟶_{1}^{( \begin{matrix}−1 \\ 1 \end{matrix} )} K^{2} \begin{matrix}( & 1 1 & ) \\ ⟶ \\ 1 \end{matrix} K ⟶ 0

b = (1, 2, 1)

C̃(X_{≤b}; K) : 0 ⟶ K ⟶_{1}^{( \begin{matrix}−1 \\ 1 \end{matrix} )} K^{2} \begin{matrix}( & 1 1 & ) \\ ⟶ \\ 1 \end{matrix} K ⟶ 0

b = (0, 2, 2)

C̃(X_{≤b}; K) : 0 ⟶ K ⟶_{1}^{( \begin{matrix}−1 \\ 1 \end{matrix} )} K^{2} \begin{matrix}( & 1 1 & ) \\ ⟶ \\ 1 \end{matrix} K ⟶ 0

b = (2, 1, 0)

C̃(X_{≤b}; K) : 0 ⟶ K \begin{matrix}( & 1 & ) \\ ⟶ \\ 1 \end{matrix} K ⟶ 0

b ∈ {(1, 0, 1), (1, 0, 2), (1, 1, 1), (2, 0, 1), (2, 0, 2)}

C̃(X_{≤b}; K) : 0 ⟶ K \begin{matrix}( & 1 & ) \\ ⟶ \\ 1 \end{matrix} K ⟶ 0

b = (0, 1, 2)

C̃(X_{≤b}; K) : 0 ⟶ K \begin{matrix}( & 1 & ) \\ ⟶ \\ 1 \end{matrix} K ⟶ 0

b ∈ {(0, 2, 0), (0, 2, 1), (1, 2, 0)}

C̃(X_{≤b}; K) : 0 ⟶ K \begin{matrix}( & 1 & ) \\ ⟶ \\ 1 \end{matrix} K ⟶ 0

For the remaining degrees b, which are

b ∈ {(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 1, 0),
(0, 1, 1), (1, 0, 0), (1, 1, 0), (2, 0, 0)},

we get the acyclic cell complex X_{≤b} = {}.

By Lemma, 2.2.22 each X_{≤b} is acyclic over ℝ. We get a better result by applying Theorem 2.3.13: As the geometric realizations show, the subcomplexes X_{≤b} are contractible, therefore X_{≤b} is acyclic over K for each degree b and each field K. 𝔽_{X} is a resolution of I for an arbitrary field K by Theorem 3.1.11. It follows from a check of the table of X that the condition of Proposition 3.1.14 is satisfied. Hence, the resolution 𝔽_{X} is minimal. This also follows by an inspection of the matrices describing the maps in 𝔽_{X}. By applying Corollary 3.1.17, we obtain the following table of Betti numbers:

Betti numbers of I

ℤ^{n}-Degree 0 1 2 ℤ-Degree 0 1 2 0 1 2
(0,2,0) 1 0 0
(1,0,1) 1 0 0 2 2 0 0
(0,1,2) 1 0 0
(2,1,0) 1 0 0 3 2 0 0
(0,2,2) 0 1 0
(1,1,2) 0 1 0  4 5 2
(1,2,1) 0 1 0 4 0 5 0
(2,1,1) 0 1 0
(2,2,0) 0 1 0
(1,2,2) 0 0 1
(2,2,1) 0 0 1  5  0 0 2

Now, consider the cellular complex 𝔽_{Y}. Our assertion is that 𝔽_{Y} is not a resolution of I. For b = (1, 2, 1) we get

the augmented oriented chain complex

C̃(Y_{≤b}; K) : 0 ⟶ K^{2} \begin{matrix}( & 1 1 & ) \\ ⟶ \\ 1 \end{matrix} K ⟶ 0.

Both Corollary 2.2.24 and Proposition 2.3.19 state that Y_{≤b} is not acyclic over K for every field K. Therefore,

𝔽_{Y} : 0 ⟶ \begin{matrix}R(−6) \\ ⊕ \\ R(−5) \end{matrix} ⟶ \begin{matrix}R(−5) \\ ⊕ \\ R^{4}(−4) \end{matrix} ⟶ \begin{matrix}R^{2}(−3) \\ ⊕ \\ R^{2}(−2) \end{matrix} ⟶ I ⟶ 0

is not a resolution.

We can determine the values of the Hilbert function h_{R/I} from the cell complex X according to Proposition 3.1.18 and obtain the Hilbert polynomial

p_{R/I} = 2.

CELLULAR RESOLUTIONS OF MONOMIAL IDEALS 69

Therefore, we get

dim (R/I) = 1,
deg (R/I) = 2,

and the height of I is

h(I) = 2.

R/I is not Cohen-Macaulay according to Proposition 3.1.20. Example 3.1.22. Consider

J = (xz, yz^{2}, y^{2}) ⊂ R = K[x, y, z].

Note that

J = I_{≤b}

with I being the monomial ideal of Example 3.1.21 and b = (1, 2, 2). Corollary 3.1.15 implies that

𝔽_{X_{≤b}} : 0 ⟶ R(−5) ⟶^{( \begin{matrix}y \\ −z \\ x \end{matrix} )} R^{3}(−4) ⟶^{( \begin{matrix}−yz & −y^{2} & 0 \\ x & 0 & −y \\ 0 & xz & z^{2} \end{matrix} )} \begin{matrix}R(−2) \\ ⊕ \\ R(−3) \\ ⊕ \\ R(−2) \end{matrix} \underrightarrow{( xz yz^{2} y^{2} )} J ⟶ 0

is a minimal free resolution of J. The next example illustrates that the exactness of a cellular complex 𝔽_{X} may depend on the choice of the field K. Example 3.1.23. We consider the ideal

I = (xyz, xyw, xzv, xuv, xuw, yzu, yuv, yvw, zuw, zvw)
⊂ R = K[x, y, z, u, v, w]

and the cell complex X given by the following table:

X
Component  Cell  Degree
X^{−1}  F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (1,1,1,0,0,0)
F_{2}^{0}  {v_{2}}  (1,1,0,0,0,1)
F_{3}^{0}  {v_{3}}  (1,0,1,0,1,0)
F_{4}^{0}  {v_{4}}  (1,0,0,1,1,0)
F_{5}^{0}  {v_{5}}  (1,0,0,1,0,1)
X^{0}  F_{6}^{0}  {v_{6}}  (0,1,1,1,0,0)
F_{7}^{0}  {v_{7}}  (0,1,0,1,1,0)
F_{8}^{0}  {v_{8}}  (0,1,0,0,1,1)
F_{9}^{0}  {v_{9}}  (0,0,1,1,0,1)
F_{10}^{0}  {v_{10}}  (0,0,1,0,1,1)
F_{1}^{1}  {v_{1}, v_{2}}  (1,1,1,0,0,1)
F_{2}^{1}  {v_{1}, v_{3}}  (1,1,1,0,1,0)
F_{3}^{1}  {v_{1}, v_{6}}  (1,1,1,1,0,0)
F_{4}^{1}  {v_{2}, v_{5}}  (1,1,0,1,0,1)
F_{5}^{1}  {v_{2}, v_{8}}  (1,1,0,0,1,1)
F_{6}^{1}  {v_{3}, v_{4}}  (1,0,1,1,1,0)
F_{7}^{1}  {v_{3}, v_{10}}  (1,0,1,0,1,1)
X^{1}  F_{8}^{1}  {v_{4}, v_{5}}  (1,0,0,1,1,1)
F_{9}^{1}  {v_{4}, v_{7}}  (1,1,0,1,1,0)
F_{10}^{1}  {v_{5}, v_{9}}  (1,0,1,1,0,1)
F_{11}^{1}  {v_{6}, v_{7}}  (0,1,1,1,1,0)
F_{12}^{1}  {v_{6}, v_{9}}  (0,1,1,1,0,1)
F_{13}^{1}  {v_{7}, v_{8}}  (0,1,0,1,1,1)
F_{14}^{1}  {v_{8}, v_{10}}  (0,1,1,0,1,1)
F_{15}^{1}  {v_{9}, v_{10}}  (0,0,1,1,1,1)
F_{1}^{2} {v_{1}, v_{2}, v_{3}, v_{8}, v_{10}} (1,1,1,0,1,1)
F_{2}^{2} {v_{1}, v_{2}, v_{5}, v_{6}, v_{9}} (1,1,1,1,0,1)
F_{3}^{2} {v_{1}, v_{3}, v_{4}, v_{6}, v_{7}} (1,1,1,1,1,0)
X^{2}  F_{4}^{2} {v_{2}, v_{4}, v_{5}, v_{7}, v_{8}} (1,1,0,1,1,1)
F_{5}^{2} {v_{3}, v_{4}, v_{5}, v_{9}, v_{10}} (1,0,1,1,1,1)
F_{6}^{2} {v_{6}, v_{7}, v_{8}, v_{9}, v_{10}} (0,1,1,1,1,1)

This cell complex has an interesting geometric realization in the form of the real projective plane ℝℙ^{2}:

We get a cellular complex

𝔽_{X} : 0 ⟶ R^{6}(−5) ⟶ R^{15}(−4) ⟶ R^{10}(−3) ⟶ I ⟶ 0.

Now, to decide whether 𝔽_{X} is a resolution, we study the augmented oriented chain complex

C̃(X; K) : 0 ⟶ K^{6} ⟶ K^{15} ⟶ K^{10} ⟶ K ⟶ 0.

We make a rank computation of the corresponding matrices over the fields

K = ℝ and K = 𝔽_{2}

and obtain

C̃(X; ℝ) : 0 ⟶ K^{6} ⟶_{6} K^{15} ⟶_{9} K^{10} ⟶_{1} K ⟶ 0,
C̃(X; 𝔽_{2}) : 0 ⟶ K^{6} ⟶_{5} K^{15} ⟶_{9} K^{10} ⟶_{1} K ⟶ 0.

By Lemma 2.2.22, C̃(X; ℝ) is acyclic, whereas C̃(X; 𝔽_{2}) is not acyclic. Theorem 3.1.11 yields that 𝔽_{X} is not a resolution of I if K = 𝔽_{2}. It can be checked that X_{≤b} is acyclic over ℝ for all other degrees b. Hence, 𝔽_{X} is a resolution by Theorem 3.1.11 if K = ℝ, it is minimal according to Proposition 3.1.14.

We would like to mention for the interested reader that the minimal free resolution for K = 𝔽_{2} is of the form

𝔽 : 0 ⟶ R(−6) ⟶ \begin{matrix}R^{6}(−5) \\ ⊕ \\ R(−6) \end{matrix} ⟶ R^{15}(−4) ⟶ R^{10}(−3) ⟶ I ⟶ 0.

The Hilbert polynomial if R/I is

p_{R/I} = 5X^{2} + 1.

We obtain

dim (R/I) = 3,
deg (R/I) = 10,
h(I) = 3.

According to Proposition 3.1.20, R/I is Cohen-Macaulay for K = ℝ, yet it is not for K = 𝔽_{2}.

So far we have developed the basic theory. Now, we focus on three kinds of supporting cell complexes: The Taylor complex Δ, the hull complex hull(I), and the Scarf complex Δ_{I}.

3.2. The Taylor resolution.

The simplest example of a cellular resolution is the Taylor resolution 𝔽_{Δ} for monomial ideals. It is always a resolution of I but typically far from being minimal. We continue to use the notation of Section 3.1. In particular, V represents the set of minimal generators of I.

Definition 3.2.1. Let Δ = P(V) = ⟨V ⟩ be the full simplex on the vertex set V. Δ is called the Taylor complex of I.

Theorem 3.2.2. The cellular complex F_{Δ} is a resolution of I for every field K.

Proof. Let K be an arbitrary field. By Theorem 3.1.11 we need to show that each subcomplex Δ_{≤b} of Δ is acyclic over K. This follows immediately if Δ_{≤b} = {}. Otherwise, by Lemma 3.1.6(i), Δ_{≤b} is the full simplex of some nonempty vertex set W ⊂ V, which is acyclic over K by Proposition 2.3.15. □

Definition 3.2.3. We call F_{Δ} the Taylor resolution of I.

The Taylor resolution F_{Δ} is typically very far from minimal if s ≫ n. By increasing the number of generators, the resolution F_{Δ} can be of arbitrary length, and its free modules can have arbitrary large ranks. But for a special class of ideals, the Taylor resolution F_{Δ} is always minimal:

Corollary 3.2.4. For a complete intersection I the Taylor resolution F_{Δ} is a minimal free resolution.

Proof. By Theorem 3.2.2 F_{Δ} is a resolution of I. It remains to show that F_{Δ} is minimal. The complete intersection

I = (f_{1}, …, f_{s}) = (x^{a^{1}}, …, x^{a^{s}})

is characterized by

gcd {f_{i}, f_{j}} = 1, i ≠ j.

Consequently, the degree of a cell F = {v_{j_{1}}, …, v_{j_{m}}}, j_{1} < … < j_{m}, can be computed as

deg F = ∑_{k=1}^{m} a_{j_{k}}.

This shows that any two pairs of cells F, F′ ∈ X, such that F′ is a face of F, have distinct degrees deg F′ < deg F, and the condition of Proposition 3.1.14 is satisfied. □

Example 3.1.22 shows that the minimality of the Taylor resolution F_{Δ} does not imply that I is a complete intersection.

For a complete intersection I = (f_{1}, …, f_{s}), the Taylor resolution F_{Δ} is particulary easy to compute: Since Δ is a simplex, we can use the incidence function ε, determined as in Lemma 2.2.12(i). The matrices of the homomorphism ∂_{i}, i = 1, …, s − 1 have elements of the form

ε(F, F′) x^{deg F −deg F′}, F, F′ ∈ Δ.

If F′ is a face of F, say,

F = {v_{j_{1}}, …, v_{j_{m}}}, j_{1} < … < j_{m},
F′ = {v_{j_{1}}, …, v̂_{j_{k}}, …, v_{j_{m}}},

we obtain

(−1)^{k+1}f_{j_{k}}.

The free modules in the complex 𝔽_{Δ} are given by

F_{i−1} = ⨁_{{J⊂\{1,...,s\} \atop |J|=i}} R (− ∑_{j∈J} a_{j}) , i = 1, …, s.

Their rank is given by

rk F_{i−1} = \binom{s}{i}, i = 1, …, s.

Since we have seen that for a complete intersection the Taylor resolution coincides with the Koszul complex, this is no surprise.

Now, we determine the Taylor resolutions for complete intersections of height 2, 3 and 4. The appearing degrees correspond to the prototype of a complete intersection of height n, the monomial ideal

I = (x_{1}, …, x_{n}) ⊂ R = K[x_{1}, …, x_{n}].

Example 3.2.5. For a complete intersection I = (f_{1}, f_{2}), the full the simplex Δ is

Δ = ⟨{v_{1}, v_{2}}⟩
Component  Cell  Degree
Δ^{−1}  F_{1}^{−1}  ∅
F_{1}^{0} {v_{1}} (1,0)
Δ^{0}  F_{2}^{0} {v_{2}} (0,1)
Δ^{1}  F_{1}^{1} {v_{1}, v_{2}} (1,1)

74 STEFAN TAPPE

Now, the homomorphisms can be computed as follows:

\begin{matrix}∂_{0} & \{v_{1}\} & \{v_{2}\} \\ ∅ & f_{1} & f_{2} \end{matrix}

\begin{matrix}∂_{1} & \{v_{1}, v_{2}\} \\ \{v_{1}\} & −f_{2} \\ \{v_{2}\} & f_{1} \end{matrix}

We obtain the Taylor resolution

𝔽_{Δ} : 0 ⟶ R ⟶^{( \begin{matrix}−f_{2} \\ f_{1} \end{matrix} )} R^{2} \underrightarrow{( f_{1} f_{2} )} I ⟶ 0.

Now, we consider a complete intersection I = (f_{1}, f_{2}, f_{3}) of height 3. The full simplex Δ is given by

Δ = ⟨{v_{1}, v_{2}, v_{3}}⟩
Component  Cell  Degree
Δ^{−1}  F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (1,0,0)
Δ^{0}  F_{2}^{0}  {v_{1}}  (0,1,0)
F_{3}^{0}  {v_{1}}  (0,0,1)
F_{1}^{1} {v_{1}, v_{2}} (1,1,0)
Δ^{1}  F_{2}^{1} {v_{1}, v_{3}} (1,0,1)
F_{3}^{1} {v_{2}, v_{3}} (0,1,1)
Δ^{2}  F_{1}^{2} {v_{1}, v_{2}, v_{3}} (1,1,1)

CELLULAR RESOLUTIONS OF MONOMIAL IDEALS 75

The homomorphisms are

\begin{matrix}∂_{0} & \{v_{1}\} & \{v_{2}\} & \{v_{3}\} \\ ∅ & f_{1} & f_{2} & f_{3} \end{matrix}

\begin{matrix}∂_{1} & \{v_{1}, v_{2}\} & \{v_{1}, v_{3}\} & \{v_{2}, v_{3}\} \\ \{v_{1}\} & −f_{2} & −f_{3} & 0 \\ \{v_{2}\} & f_{1} & 0 & −f_{3} \\ \{v_{3}\} & 0 & f_{1} & f_{2} \end{matrix}

\begin{matrix}∂_{2} & \{v_{1}, v_{2}, v_{3}\} \\ \{v_{1}, v_{2}\} & f_{3} \\ \{v_{1}, v_{3}\} & −f_{2} \\ \{v_{2}, v_{3}\} & f_{1} \end{matrix}

We get the Taylor resolution

𝔽_{Δ} : 0 ⟶ R ⟶^{( \begin{matrix}f_{3} \\ −f_{2} \\ f_{1} \end{matrix} )} R^{3} ⟶^{( \begin{matrix}−f_{2} & −f_{3} & 0 \\ f_{1} & 0 & −f_{3} \\ 0 & f_{1} & f_{2} \end{matrix} )} R^{3} \underrightarrow{( f_{1} f_{2} f_{3} )} I ⟶ 0.

At last, let I = (f_{1}, f_{2}, f_{3}, f_{4}) be a complete intersection. The full simplex Δ is determined by

Δ = ⟨{v_{1}, v_{2}, v_{3}, v_{4}}⟩

Component  Cell  Degree
Δ^{−1}  F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (1,0,0,0)
F_{2}^{0}  {v_{2}}  (0,1,0,0)
Δ^{0}  F_{3}^{0}  {v_{3}}  (0,0,1,0)
F_{4}^{0}  {v_{4}}  (0,0,0,1)
F_{1}^{1}  {v_{1}, v_{2}}  (1,1,0,0)
F_{2}^{1}  {v_{1}, v_{3}}  (1,0,1,0)
F_{3}^{1}  {v_{1}, v_{4}}  (1,0,0,1)
Δ^{1}  F_{4}^{1}  {v_{2}, v_{3}}  (0,1,1,0)
F_{5}^{1}  {v_{2}, v_{4}}  (0,1,0,1)
F_{6}^{1}  {v_{3}, v_{4}}  (0,0,1,1)
F_{1}^{2} {v_{1}, v_{2}, v_{3}} (1,1,1,0)
F_{2}^{2} {v_{1}, v_{2}, v_{4}} (1,1,0,1)
Δ^{2}  F_{3}^{2} {v_{1}, v_{3}, v_{4}} (1,0,1,1)
F_{4}^{2} {v_{2}, v_{3}, v_{4}} (0,1,1,1)
Δ^{3}  F_{1}^{3} {v_{1}, v_{2}, v_{3}, v_{4}} (1,1,1,1)

We obtain the homomorphisms

\begin{matrix}∂_{0} & \{v_{1}\} & \{v_{2}\} & \{v_{3}\} & \{v_{4}\} \\ ∅ & f_{1} & f_{2} & f_{3} & f_{4} \end{matrix}

\begin{matrix}∂_{1} & \{v_{1}, v_{2}\} & \{v_{1}, v_{3}\} & \{v_{1}, v_{4}\} & \{v_{2}, v_{3}\} & \{v_{2}, v_{4}\} & \{v_{3}, v_{4}\} \\ \{v_{1}\} & −f_{2} & −f_{3} & −f_{4} & 0 & 0 & 0 \\ \{v_{2}\} & f_{1} & 0 & 0 & −f_{3} & −f_{4} & 0 \\ \{v_{3}\} & 0 & f_{1} & 0 & f_{2} & 0 & −f_{4} \\ \{v_{4}\} & 0 & 0 & f_{1} & 0 & f_{2} & f_{3} \end{matrix}

\begin{matrix}∂_{2} & \{v_{1}, v_{2}, v_{3}\} & \{v_{1}, v_{2}, v_{4}\} & \{v_{1}, v_{3}, v_{4}\} & \{v_{2}, v_{3}, v_{4}\} \\ \{v_{1}, v_{2}\} & f_{3} & f_{4} & 0 & 0 \\ \{v_{1}, v_{3}\} & −f_{2} & 0 & f_{4} & 0 \\ \{v_{1}, v_{4}\} & 0 & −f_{2} & −f_{3} & 0 \\ \{v_{2}, v_{3}\} & f_{1} & 0 & 0 & f_{4} \\ \{v_{2}, v_{4}\} & 0 & f_{1} & 0 & −f_{3} \\ \{v_{3}, v_{4}\} & 0 & 0 & f_{1} & f_{2} \end{matrix}

\begin{matrix}∂_{3} & \{v_{1}, v_{2}, v_{3}, v_{4}\} \\ \{v_{1}, v_{2}, v_{3}\} & −f_{4} \\ \{v_{1}, v_{2}, v_{4}\} & f_{3} \\ \{v_{1}, v_{3}, v_{4}\} & −f_{2} \\ \{v_{2}, v_{3}, v_{4}\} & f_{1} \end{matrix}

and the resulting Taylor resolution

𝔽_{Δ} : 0 ⟶ R ⟶^{D} R^{4} ⟶^{C} R^{6} ⟶^{B} R^{4} ⟶^{A} I ⟶ 0,

A = (^{f_{1}} ^{f_{2}} ^{f_{3}} ^{f_{4}} ) ,  B = ( \begin{matrix}−f_{2} & −f_{3} & −f_{4} & 0 & 0 & 0 \\ f_{1} & 0 & 0 & −f_{3} & −f_{4} & 0 \\ 0 & f_{1} & 0 & f_{2} & 0 & −f_{4} \\ 0 & 0 & f_{1} & 0 & f_{2} & f_{3} \end{matrix} ) ,

C = ( \begin{matrix}f_{3} & f_{4} & 0 & 0 \\ −f_{2} & 0 & f_{4} & 0 \\ 0 & −f_{2} & −f_{3} & 0 \\ f_{1} & 0 & 0 & f_{4} \\ 0 & f_{1} & 0 & −f_{3} \\ 0 & 0 & f_{1} & f_{2} \end{matrix} ) , D = ( \begin{matrix}−f_{4} \\ f_{3} \\ −f_{2} \\ f_{1} \end{matrix} ) .

The following formula improves the result of Proposition 3.1.18 since it does not refer to a certain cell complex. Therefore, it can directly be used for an arbitrary monomial ideal. The formula is applied in the algorithm on the CD attached to this thesis, which computes the Hilbert function of a monomial ideal.

Proposition 3.2.6. Let

I = (x^{a^{1}}, …, x^{a^{s}}) ⊂ R = K[x_{1}, …, x_{n}]

be a monomial ideal. The following identities for the Hilbert functions of I and R/I hold:

h_{I}(k) = ∑_{i=1}^{s}(−1)^{i+1} ∑_{{J⊂\{1,...,s\} \atop |J|=i}} \binom{n − 1 + k − |lcm \{a_{j} : j ∈ J\}|}{n − 1}, k ∈ ℤ,

h_{R/I}(k) = \binom{n − 1 + k}{n − 1} − h_{I}(k),  k ∈ ℤ.

Proof. This is an immediate consequence of Proposition 3.1.18 and Theorem 3.2.2. □

3.3. The hull resolution.

For any monomial ideal I we consider a polyhedron P_{t}, whose bounded faces form the hull complex of I. We prove that the cellular complex derived from the hull complex is always a resolution of I for every field K. Therefore, we call it the hull resolution of I. It need not be minimal, but it is most often shorter than the Taylor resolution, and, unlike minimal resolutions, it respects symmetry.

Definition 3.3.1. For a ∈ ℤ^{n} and t ∈ ℝ we abbreviate

t^{a} = (t^{a_{1}}, …, t^{a_{n}}).

ℝ_{+}^{n} denotes the closed orthant consisting of all non-negative real vectors:

ℝ_{+}^{n} = {z ∈ ℝ^{n} : z ≥ 0}.

Definition 3.3.2. Let I be a monomial ideal in R = K[x_{1}, …, x_{n}] and t > (n + 1)!. We define P_{t} ⊂ ℝ^{n} to be the convex hull of the set consisting of all points t^{a} ∈ ℝ^{n}, for which the monomial x^{a} lies in I:

P_{t} = conv {t^{a} : x^{a} ∈ I}.

Our goal is to define the hull complex as the complex of bounded faces of the polyhedron P_{t} for t > (n + 1)!. Therefore, we have to show that P_{t} is a polyhedron and that the poset of bounded faces of P_{t} does not depend on t for t > (n + 1)!. This requires some preparations.

Proposition 3.3.3. We have the following identity:

P_{t} = ℝ_{+}^{n} + conv {t^{a^{1}}, …, t^{a^{s}}}.

Proof. For any t^{b} ∈ ℝ^{n} satisfying x^{b} ∈ I there is an index i ∈ {1, …, s} such that x^{a^{i}} divides x^{b}. This implies t^{a^{i}} ≤ t^{b}, and thus we can write

t^{b} = (t^{b} − t^{a^{i}}) + t^{a^{i}} ∈ ℝ_{+}^{n} + conv {t^{a^{1}}, …, t^{a^{s}}}.

Therefore, we have the inclusion

{t^{b} : x^{b} ∈ I} ⊂ ℝ_{+}^{n} + conv {t^{a^{1}}, …, t^{a^{s}}}.

Note that the sum on the right side is convex by Lemma 2.1.12(iv). We form the convex hull of the two sets and obtain

P_{t} ⊂ ℝ_{+}^{n} + conv {t^{a^{1}}, …, t^{a^{s}}}

by applying Lemma 2.1.12(i),(ii). For the opposite inclusion we first prove that

t^{a^{i}} + ℝ_{+}^{n} ⊂ P_{t}, i = 1, …, s.

Fix b = a_{i} and a non-negative real vector u. Choose a positive integer r such that

0 ≤ n u_{j} ≤ t^{b_{j}+r} − t^{b_{j}}, j = 1, …, n.

Denote by {e_{1}, …, e_{n}} the canonical basis of ℝ^{n}. Let C be the convex hull of the 2^{n} points

v_{J} = t^{b} + ∑_{j∈J}(t^{b_{j}+r} − t^{b_{j}}) e_{j},

where J runs over all subsets of {1, …, n}.

We set

λ_{j} = \frac{u_{j}}{t^{b_{j}+r} − t^{b_{j}}} , j = 1, …, n

and

μ = 1 − ∑_{j=1}^{n} λ_{j}.

Note that μ ≥ 0 because we have λ_{j} ≤ \frac{1}{n} for each j = 1, …, n. Now we get

t^{b} + u = (μ + ∑_{j=1}^{n} λ_{j}) t^{b} + ∑_{j=1}^{n} u_{j} e_{j}

= (μ + ∑_{j=1}^{n} λ_{j}) t^{b} + ∑_{j=1}^{n} λ_{j}(t^{b_{j}+r} − t^{b_{j}}) e_{j}

= μ t^{b} + ∑_{j=1}^{n} λ_{j}(t^{b} + (t^{b_{j}+r} − t^{b_{j}}) e_{j})

= μ v_{∅} + ∑_{j=1}^{n} λ_{j} v_{{j}}.

This describes t^{b} + u as a convex combination of the v_{J}. We conclude that

t^{b} + u ∈ C.

Now, we show that each v_{J} is an element of {t^{a} : x^{a} ∈ I}. Let c ∈ ℝ^{n} be the vector defined by

c_{j} = \{ \begin{matrix}b_{j} + r & j ∈ J \\ b_{j} & j ∉ J \end{matrix}

We obtain x^{c} ∈ I and

v_{J} = t^{b} + ∑_{j∈J}(t^{b_{j}+r} − t^{b_{j}}) e_{j}
= (t^{b_{1}}, …, t^{b_{n}}) + ∑_{j∈J}(0, …, t^{b_{j}+r} − t^{b_{j}}, …, 0)
= t^{c}.

It follows that

v_{J} ∈ {t^{a} : x^{a} ∈ I},

consequently, by applying Lemma 2.1.12(i)

t^{b} + u ∈ C ⊂ P_{t}.

80 STEFAN TAPPE

Now let λ_{1}, …, λ_{s} ≥ 0 such that

∑_{j=1}^{s} λ_{j} = 1.

We obtain

∑_{j=1}^{s} λ_{j} t^{a^{j}} + u = ∑_{j=1}^{s} λ_{j} t^{a^{j}} + \underbrace{(∑_{j=1}^{s} λ_{j})}_{=1} u
= ∑_{j=1}^{s} λ_{j} \underbrace{(t^{a^{j}} + u)}_{∈P_{t}} ∈ P_{t},

which shows the inclusion

conv {t^{a^{1}}, …, t^{a^{s}}} + ℝ_{+}^{n} ⊂ P_{t}.

□

With the help of this identity it is easy to visualize the set P_{t} for certain ideals. Example 3.3.4. For the maximal ideal

I = (x, y) ⊂ R = K[x, y]

the set P_{7} looks like

The ideal

I = (x, y)^{2} = (x^{2}, xy, y^{2}) ⊂ R = K[x, y]

yields the following set P_{7} Lemma 3.3.5. P_{t} is an unbounded polyhedron.

Proof. It is obvious that P_{t} is unbounded by the formula in Lemma 3.3.3. The convex hull

conv {t^{a^{1}}, …, t^{a^{s}}}

has a representation as the intersection of a finite number of closed half-spaces:

conv {t^{a^{1}}, …, t^{a^{s}}} = ⋂_{j=1}^{m} H_{j}.

Let J ⊂ {1, …, m} be the set consisting of all j for which there is a vector v_{j} ∈ ℝ^{n}, v_{j} ▷ 0 (for the notation see Definition 2.1.1), such that

H_{j} = H^{+}(v_{j}, 1).

Furthermore, we set

ρ_{i} = min {t^{a_{1}^{i}}, …, t^{a_{s}^{i}}}, i = 1, …, n.

Denote by {e_{1}, …, e_{n}} the canonical basis of ℝ^{n}. Our assertion is that

P_{t} = ⋂_{i=1}^{n} H^{+}(e_{i}, ρ_{i}) ∩ ⋂_{j∈J} H^{+}(v_{j}, 1),

which implies that P_{t} is a polyhedron.

Let

z = u + ∑_{k=1}^{s} λ_{k} t^{a^{k}} ∈ ℝ_{+}^{n} + conv {t^{a^{1}}, …, t^{a^{s}}} = P_{t}.

We get for i = 1, …, n

⟨e_{i}, z⟩ = \underbrace{⟨e_{i}, u⟩}_{≥0} + ∑_{k=1}^{s} λ_{k} \underbrace{⟨e_{i}, t^{a^{k}}⟩}_{≥ρ_{i}} ≥ ρ_{i},

and for j ∈ J we obtain

⟨v_{j}, z⟩ = \underbrace{⟨v_{j}, u⟩}_{≥0} + ∑_{k=1}^{s} λ_{k} \underbrace{⟨v_{j}, t^{a^{k}}⟩}_{≥1} ≥ 1.

Hence,

P_{t} ⊂ ⋂_{i=1}^{n} H^{+}(e_{i}, ρ_{i}) ∩ ⋂_{j∈J} H^{+}(v_{j}, 1).

Now consider z ∈ ℝ^{n} satisfying

⟨e_{i}, z⟩ ≥ ρ_{i}, i = 1, …, n,
⟨v_{j}, z⟩ ≥ 1, j ∈ J.

The following sequence (w_{k})_{k=1,...,n} is defined inductively. For each k = 1, …, n there is a uniquely determined w_{k} ≥ 0 such that

z − ∑_{l=1}^{k} w_{l} e_{l} ∈ ⋂_{i=1}^{n} H^{+}(e_{i}, ρ_{i}) ∩ ⋂_{j∈J} H^{+}(v_{j}, 1),

z − ∑_{l=1}^{k} w_{l} e_{l} − ε e_{k} ∉ ⋂_{i=1}^{n} H^{+}(e_{i}, ρ_{i}) ∩ ⋂_{j∈J} H^{+}(v_{j}, 1), ε > 0.

The following figure visualizes this procedure for two points in ℝ^{2}:

Define

w = (w_{1}, …, w_{n}) ∈ ℝ_{+}^{n}.

Then we obtain that

z − w ∈ conv {t^{a^{1}}, …, t^{a^{s}}},

and thus we can write z = (z − w) + w as an element of P_{t}. Therefore,

P_{t} ⊃ ⋂_{i=1}^{n} H^{+}(e_{i}, ρ_{i}) ∩ ⋂_{j∈J} H^{+}(v_{j}, 1).

□

Right now, with the knowledge of P_{t} being an unbounded polyhedron, we investigate the bounded faces of P_{t}.

Proposition 3.3.6. Let F ⊂ ℝ^{n}, F ≠ ∅. Then the following conditions are equivalent: (a) F ∈ F^{b}(P_{t}).

(b) There are J ⊂ {1, …, s}, J ≠ ∅ and v ∈ ℝ^{n}, v ▷ 0, such that

F = conv {t^{a^{j}} : j ∈ J}

and

⟨v, t^{a^{j}}⟩ = 1, j ∈ J,
⟨v, t^{a^{j}}⟩ > 1, j ∉ J.

In this case H(v, 1) is a supporting hyperplane for P_{t}. Proof. Let F be a bounded face of P_{t}. There exist v ∈ ℝ^{n} and β ∈ ℝ such that

H(v, β) ∩ P_{t} = F

and (without loss of generality)

P_{t} ⊂ H(v, β)^{+}.

We claim

v ▷ 0.

Suppose v ⋫ 0. Then we have v_{j} ≤ 0 for an index j ∈ {1, …, n}. Consider the two cases

(i) v_{j} < 0. Fix z ∈ F and c > 0. Then, according to Proposition 3.3.3

z + c e_{j} ∈ P_{t},

but

⟨v, z + c e_{j}⟩ = \underbrace{⟨v, z⟩}_{=β} + \underbrace{c v_{j}}_{<0} < β

shows that

z + c e_{j} ∉ H(v, β)^{+},

this is a contradiction. (ii) v_{j} = 0. Fix an element z ∈ F. Then, according to Proposition 3.3.3

z + c e_{j} ∈ P_{t}, c ≥ 0,

and

⟨v, z + c e_{j}⟩ = \underbrace{⟨v, z⟩}_{=β} + \underbrace{c v_{j}}_{=0} = β.

This implies

z + c e_{j} ∈ F, c ≥ 0.

Therefore, F is an unbounded face of P_{t}, which contradicts our hypothesis. Now we assert

β > 0.

Assume β ≤ 0. This yields

H(v, β) ∩ ℝ_{n}^{+} ⊂ {0},

84 STEFAN TAPPE

hence

H(v, β) ∩ P_{t} = ∅,

which is a contradiction. Without loss of generality we may assume that β = 1. We have

⟨v, t^{a^{j}}⟩ ≥ 1, j = 1, …, s.

Define

J = {j ∈ {1, …, s} : ⟨v, t^{a^{j}}⟩ = 1} ⊂ {1, …, s}.

Suppose J = ∅. For an arbitrary element

z = u + ∑_{j=1}^{s} λ_{j} t^{a^{j}} ∈ P_{t}

we get

⟨v, z⟩ = \underbrace{⟨v, u⟩}_{≥0} + ∑_{j=1}^{s} λ_{j} \underbrace{⟨v, t^{a^{j}}⟩}_{>1} > 1,

which shows that

H(v, 1) ∩ P_{t} = ∅.

This contradiction shows J ≠ ∅. It is evident that

{t^{a_{j}} : j ∈ J} ⊂ H(v, 1) ∩ P_{t}.

Since H(v, 1) ∩ P_{t} is convex, we have the inclusion

conv {t^{a^{j}} : j ∈ J} ⊂ H(v, 1) ∩ P_{t}

by Lemma 2.1.12 (i),(iii). Now let

z = u + ∑_{j=1}^{s} λ_{j} t^{a^{j}} ∈ H(v, 1) ∩ P_{t}.

We get

⟨v, z⟩ = \underbrace{⟨v, u⟩}_{≥0} + ∑_{j∈J} λ_{j} \underbrace{⟨v, t^{a^{j}}⟩}_{=1} + ∑_{j∉J} λ_{j} \underbrace{⟨v, t^{a^{j}}⟩}_{>1} = 1.

This implies

u = 0,
λ_{j} = 0, j ∉ J,

it follows that

z ∈ conv {t^{a^{j}} : j ∈ J}.

Consequently, the other inclusion

H(v, 1) ∩ P_{t} ⊂ conv {t^{a^{j}} : j ∈ J}

is done and we have shown that F = conv {t^{a^{j}} : j ∈ J}.

We still have to verify the converse. It is clear that F is bounded since it is the convex hull of a finite set of points. The proof that

H(v, 1) ∩ P_{t} = F

is similarly as above. For any

z = u + ∑_{j=1}^{s} λ_{j} t^{a^{j}} ∈ P_{t}

the inequality

⟨v, z⟩ = \underbrace{⟨v, u⟩}_{≥0} + ∑_{j=1}^{s} λ_{j} \underbrace{⟨v, t^{a^{j}}⟩}_{≥1} ≥ ∑_{j∈J} λ_{j} = 1

holds, consequently

P_{t} ⊂ H(v, 1)^{+}.

Therefore, F is a bounded face of P_{t}. □

Proposition 3.3.7. The vertices of the polyhedron P_{t} are precisely the points t^{a^{i}}, i = 1, …, s, that is

vert P_{t} = {t^{a^{i}} : i = 1, …, s}.

Proof. Fix i ∈ {1, …, s}, set b = a_{i} and

v = \frac{1}{n} t^{−b} ∈ ℝ^{n}.

We obtain

v ▷ 0

and

⟨v, t^{b}⟩ = \frac{1}{n} ∑_{k=1}^{n} t^{0} = 1.

For any other j ≠ i in {1, …, s} let c = a_{j}. Since the x^{a^{i}}, i = 1, …, s are minimal generators of I, we have b_{k} < c_{k} for some k ∈ {1, …, n}, thus

⟨v, t^{c}⟩ = \frac{1}{n} ∑_{k=1}^{n} t^{c_{k}−b_{k}} ≥ \frac{t}{n} > \frac{(n + 1)!}{n} > 1.

Recall that t is always assumed to be larger than (n + 1)!. The conditions of Proposition 3.3.6(b) are fulfilled. Therefore, t^{a^{i}} is a vertex of P_{t}.

By Proposition 3.3.6, P_{t} does not have vertices besides t^{a^{1}}, …, t^{a^{s}}. □

Definition 3.3.8. For t > (n + 1)! let C_{t} ⊂ ℝ^{n+1} be the cone

C_{t} = pos {(e_{1}, 0), …, (e_{n}, 0), (t^{a^{1}}, 1), …, (t^{a^{s}}, 1)}.

Denote by F ′(C_{t}) the set consisting of the empty face and the faces of C_{t}, which are generated as a cone by some nonempty subset of

{(t^{a^{1}}, 1), …, (t^{a^{s}}, 1)}.

Lemma 3.3.9. For any t > (n + 1)! the two posets F^{b}(P_{t}) and F ′(C_{t}) are isomorphic.

Proof. We start with defining a map

φ : F^{b}(P_{t}) → F ′(C_{t}).

Let F ≠ ∅ be an arbitrary bounded face of P_{t}. By Proposition 3.3.6 there are J ⊂ {1, …, s}, J ≠ ∅ and v ∈ ℝ^{n}, v ▷ 0, such that

F = conv {t^{a^{j}} : j ∈ J}

and

⟨v, t^{a^{j}}⟩ = 1,  j ∈ J,
⟨v, t^{a^{j}}⟩ > 1,  j ∉ J.

We obtain

⟨ (v, −1), (t^{a^{j}}, 1) ⟩ = 0,  j ∈ J,
⟨ (v, −1), (t^{a^{j}}, 1) ⟩ > 0,  j ∉ J,
⟨ (v, −1), (e_{i}, 0) ⟩ > 0,  i = 1, …, n.

Define

φ(F) = pos {(t^{a^{j}}, 1) : j ∈ J}.

φ(F) lies in F ′(C_{t}) by Proposition 2.1.30. Furthermore, we set

φ(∅) = ∅.

First, we show that φ is injective. For subsets J, K ⊂ {1, …, s} let

F = conv {t^{a^{j}} : j ∈ J},
G = conv {t^{a^{k}} : k ∈ K}

be nonempty bounded faces of P_{t} such that

φ(F) = φ(G).

Let

z = ∑_{j∈J} λ_{j} t^{a^{j}} ∈ F.

This implies that the convex combination

(z, 1) = ∑_{j∈J} λ_{j} (t^{a^{i}}, 1) ∈ φ(F)

can be written as a positive combination

(z, 1) = ∑_{k∈K} μ_{k} (t^{a^{k}}, 1) ∈ φ(G).

Obviously, this is even a convex combination because of the last component, and we obtain

z = ∑_{k∈K} μ_{k} t^{a^{k}} ∈ G.

The inclusion G ⊂ F can be shown analogously. Therefore, φ is injective. Next, let F ∈ F ′(C_{t}) be nonempty. Then, according to Proposition 2.1.30, there are J ⊂ {1, …, s}, J ≠ ∅, v ∈ ℝ^{n+1}, v ≠ 0 satisfying

⟨ v, (t^{a^{j}}, 1) ⟩ = 0,  j ∈ J,
⟨ v, (t^{a^{j}}, 1) ⟩ > 0,  j ∉ J,
⟨ v, (e_{i}, 0) ⟩ > 0,  i = 1, …, n.

The last inequalities imply

v_{i} > 0, i = 1, …, n,

hence, because of the first equation, we get

v_{n+1} < 0.

Without loss of generality we may assume that v_{n+1} = −1. For w = (v_{1}, …, v_{n}) ▷ 0 we get

⟨w, t^{a^{j}}⟩ = 1, j ∈ J,
⟨w, t^{a^{j}}⟩ > 1, j ∉ J.

So φ is surjective by Proposition 3.3.6. Finally, it is evident that φ is order-preserving. □

Lemma 3.3.10. Let a_{ij} be integers for 1 ≤ i, j ≤ r. Then the Laurent polynomial

f(t) = det (( t^{a_{ij}} )_{{i=1...r \atop j=1...r}} )

either vanishes identically or has no real roots for t > r!. Proof. Suppose that f is not zero and write

f(t) = c_{α} t^{α} + ∑_{β} c_{β} t^{β},

where the first term has the highest degree in t. Per definition, we have

f(t) = ∑_{σ∈S_{r}} \underbrace{sign(σ)}_{∈\{−1,1\}} ∏_{j=1}^{r} t^{a_{σ(j),j}}.

Since |S_{r}| = r!, we get

∑_{β} |c_{β}| < r!.

Now we have for t > r! the inequalities

|∑_{β} c_{β} t^{β}| ≤ ∑_{β} |c_{β}| t^{β} ≤ (∑_{β} |c_{β}|) t^{α−1} < r! t^{α−1} < t^{α} ≤ |c_{α} t^{α}|.

Therefore, f(t) ≠ 0 and sign(f(t)) = sign(c_{α}). □

Theorem 3.3.11. The subposet F^{b}(P_{t}) of bounded faces of the polyhedron P_{t} is independent (up to isomorphisms of posets) of t for t > (n + 1)!.

Proof. For t > (n + 1)! consider the matrix

A_{t} = ( \begin{matrix}e_{1} & ⋯ & e_{n} & t^{a^{1}} & ⋯ & t^{a^{s}} \\ 0 & ⋯ & 0 & 1 & ⋯ & 1 \end{matrix} )

By applying Laplace expansion, it follows from Lemma 3.3.10 that the sign of any minor of this matrix is independent of t for t > (n + 1)!. Therefore, the theory of oriented matroids (see, e.g., [11]) provides that the face poset F(C_{t}), and therefore the subposet F ′(C_{t}) of the cone C_{t} is independent of t for t > (n + 1)!. Now Lemma 3.3.9 completes the proof. □

Our preparations are completed and we are now ready to define the hull complex.

Definition 3.3.12. The hull complex of a monomial ideal I, denoted by hull(I), is the complex of bounded faces of the polyhedron P_{t} (see Definition 2.2.26) for t > (n + 1)!. Theorem 3.3.11 ensures that hull(I) is well-defined and depends only on I. hull(I) supports I by Proposition 3.3.7.

Remark 3.3.13. The hull complex of a monomial ideal I can be characterized as follows: Let t > (n + 1)! be fixed. Proposition 3.3.6 yields that for an arbitrary set J ⊂ {1, …, s}, J ≠ ∅, the cell

F = {v_{j} : j ∈ J} ∈ Δ

belongs to hull(I) if and only if there is a vector v ∈ ℝ^{n}, v ▷ 0, such that

⟨v, t^{a^{j}}⟩ = 1, j ∈ J,
⟨v, t^{a^{j}}⟩ > 1, j ∉ J.

Example 3.3.14. We consider the two ideals of Example 3.3.4. The hull complex of the maximal ideal

I = (x, y) ⊂ R = K[x, y]

is given by

hull(I)
Component  Cell  Degree
hull(I)^{−1} F_{1}^{−1}  ∅
F_{1}^{0} {v_{1}} (1,0)
hull(I)^{0} F_{2}^{0} {v_{2}} (0,1)
hull(I)^{1} F_{1}^{1} {v_{1}, v_{2}} (1,1)

CELLULAR RESOLUTIONS OF MONOMIAL IDEALS 89

We obtain the cellular complex

𝔽_{hull(I)} : 0 ⟶ R(−2) ⟶^{( \begin{matrix}−y \\ x \end{matrix} )} R^{2}(−1) \underrightarrow{( x y )} I ⟶ 0.

For the ideal

I = (x, y)^{2} = (x^{2}, xy, y^{2}) ⊂ R = K[x, y]

we get the hull complex

hull(I)
Component Cell Degree
hull(I)^{−1} F_{1}^{−1}  ∅
F_{1}^{0} {v_{1}} (2,0)
hull(I)^{0} F_{2}^{0} {v_{2}} (1,1)
F_{3}^{0} {v_{3}} (0,2)
F_{1}^{1} {v_{1}, v_{2}} (2,1)
hull(I)^{1} F_{2}^{1} {v_{2}, v_{3}} (1,2)

and the cellular complex

𝔽_{hull(I)} : 0 ⟶ R^{2}(−3) ⟶^{( \begin{matrix}−y & 0 \\ x & −y \\ 0 & x \end{matrix} )} R^{3}(−2) \underrightarrow{( x^{2} xy y^{2} )} I ⟶ 0.

It follows from the next Theorem 3.3.15 that the two cellular complexes are free resolutions. They are minimal by Proposition 3.1.14. In Example 3.5.14 we will determine minimal free resolutions for each ideal of the type

I = (x, y)^{n} ⊂ R = K[x, y], n ∈ ℕ.

Theorem 3.3.15. The cellular complex 𝔽_{hull(I)} is a free resolution of I for every field K.

Proof. Let K be a field and X = hull(I)_{≤b} for some degree b; by Theorem 3.1.11 we need to show that X is acyclic. This is immediate if X is empty or a single vertex. Otherwise choose t > (n + 1)! and let

v = t^{−b} ∈ ℝ^{n}.

We recall from Proposition 3.3.7 that the vertices of the polyhedron P_{t} are precisely the points t^{a^{i}}, i = 1, …, s. For an i ∈ {1, …, s} set c = a_{i}. Consider the two cases

(i) c ≤ b. We even obtain c < b because otherwise X would just be a single vertex. So

⟨v, t^{c}⟩ = ⟨t^{−b}, t^{c}⟩ < ⟨t^{−b}, t^{b}⟩ = n.

(ii) c ≰ b. We get c_{j} > b_{j} for some j ∈ {1, …, n}, therefore

⟨v, t^{c}⟩ = ⟨t^{−b}, t^{c}⟩ > t^{c_{j}−b_{j}} ≥ t > (n + 1)! > n.

Thus, the hyperplane H(v, n) separates the vertices of X from the remaining vertices of P_{t}. Hence, the set

P = P_{t} ∩ H^{−}(v, n)

is a polytope. Let T : ℝ^{n} \ H(v, n) → ℝ^{n} be a projective transformation (see Definition 2.1.31). By Lemma 2.1.33,

P′ = T (P \ H(v, n))

is an unbounded polyhedron and its poset of bounded faces is

F^{b}(P′) = {T F : F ∈ F(P), P ∩ H(v, n) = ∅}
= {T F : F ∈ F^{b}(P_{t}), deg F ≤ b}.

This expresses X as the complex of bounded faces of a polyhedron, and we are done by applying Proposition 2.3.17. □

Definition 3.3.16. We call 𝔽_{hull(I)} the hull resolution of I.

It is obvious that computing the hull complex is, unlike determining the Taylor complex of a monomial ideal I, a difficult task because it it requires the computations of convex hulls. The results of the next section provide some help in the computations. We mentioned that the hull resolution need not be minimal. An example will be given in the next section (Example 3.4.19).

3.4. The Scarf complex.

We define the Scarf complex Δ_{I}, which need not necessarily provide a resolution, but it is easily computed. The central result of this section is that the Scarf complex is a subcomplex of the hull complex. Consequently, a part of the hull complex of a monomial ideal I is given by determining its Scarf complex. Hence, the effort for computing the hull complex can be reduced.

Definition 3.4.1. For a monomial ideal I we define the Scarf complex Δ_{I} as

Δ_{I} = {∅} ∪ {F ∈ Δ, F ≠ ∅ : ∄G ∈ Δ, G ≠ ∅,
G ≠ F : deg F = deg G}.

As before, Δ denotes the Taylor complex of the ideal I.

Observe that the Scarf complex is easy to compute, but it can take a long time for a large number of generators.

Lemma 3.4.2. The Scarf complex Δ_{I} is a simplicial complex on V.

Proof. It is clear that {v_{i}} ∈ Δ_{I}, i = 1, …, s since each vertex inherits the degree of a minimal generator. Let F ∈ Δ_{I} and G ⊂ F. We have to show that G ∈ Δ_{I}. Without loss of generality we may assume that |F | ≥ 2 and |G| = |F | − 1. Suppose that G ∉ Δ_{I}. Then there is a subset H ⊂ V, H ≠ G, H ≠ ∅ such that

deg G = deg H.

We have

F ≠ H ∪ (F \ G),

because the corresponding equality implies that G ⊂ H ⊂ F, therefore F = H or G = H. It is evident that

deg(H ∪ (F \ G)) ≤ deg F.

Suppose

deg(H ∪ (F \ G)) < deg F.

Then there is an index j ∈ {1, …, n} such that

deg H_{j} < deg F_{j}

and

deg(F \ G)_{j} < deg F_{j}.

The second inequality yields deg F_{j} = deg G_{j}. Since deg G_{j} = deg H_{j}, this contradicts the first one. Thus,

deg F = deg(H ∪ (F \ G)),

but this implies, that F is not a cell of the Scarf complex Δ_{I}, which contradicts our hypothesis. □

Remark 3.4.3. It follows from Lemma 2.2.8(iv) that the Scarf complex Δ_{I} is a subcomplex of the Taylor complex Δ. Evidently, Δ_{I} supports I, therefore one can construct the cellular complex 𝔽_{Δ_{I}}. As the following examples show, it does not need to be exact, but if so, 𝔽_{Δ_{I}} is minimal by Proposition 3.1.14 because no two distinct cells of Δ_{I} have the same degree.

Example 3.4.4. Consider the monomial ideal

I = (x^{2}y, xz, yz^{2}, y^{2}) ⊂ R = K[x, y, z],

discussed in Example 3.1.21. The Taylor complex Δ is given by

Δ = ⟨{v_{1}, v_{2}, v_{3}, v_{4}}⟩
Component  Cell  Degree
Δ^{−1}  F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (2,1,0)
F_{2}^{0}  {v_{2}}  (1,0,1)
Δ^{0}  F_{3}^{0}  {v_{3}}  (0,1,2)
F_{4}^{0}  {v_{4}}  (0,2,0)
F_{1}^{1}  {v_{1}, v_{2}}  (2,1,1)
F_{2}^{1}  {v_{1}, v_{3}}  (2,1,2)
F_{3}^{1}  {v_{1}, v_{4}}  (2,2,0)
Δ^{1}  F_{4}^{1}  {v_{2}, v_{3}}  (1,1,2)
F_{5}^{1}  {v_{2}, v_{4}}  (1,2,1)
F_{6}^{1}  {v_{3}, v_{4}}  (0,2,2)
F_{1}^{2} {v_{1}, v_{2}, v_{3}} (2,1,2)
F_{2}^{2} {v_{1}, v_{2}, v_{4}} (2,2,1)
Δ^{2}  F_{3}^{2} {v_{1}, v_{3}, v_{4}} (2,2,2)
F_{4}^{2} {v_{2}, v_{3}, v_{4}} (1,2,2)
Δ^{3}  F_{1}^{3} {v_{1}, v_{2}, v_{3}, v_{4}} (2,2,2)

It is not difficult to see that the Scarf complex of I is the simplicial complex

Δ_{I} = ⟨{v_{1}, v_{2}, v_{4}}, {v_{2}, v_{3}, v_{4}}⟩,

also discussed in Example 3.1.21. According to this example, 𝔽_{Δ_{I}} is a minimal free resolution of I.

Example 3.4.5. Consider the monomial ideal

I = (xyz, xyw, xzv, xuv, xuw, yzu, yuv, yvw, zuw, zvw)
⊂ R = K[x, y, z, u, v, w]

and the cell complex X of Example 3.1.23. The Scarf complex of I is

Δ_{I} = X^{−1} ∪ X^{0} ∪ X^{1}.

The cellular complex

𝔽_{Δ_{I}} : 0 ⟶ R^{15}(−4) ⟶ R^{10}(−3) ⟶ I ⟶ 0

is not exact by Corollary 3.1.16.

Before we are able to prove the central result of this section, we have to deal with some technical lemmas.

Lemma 3.4.6. Let F ∈ Δ_{I} be a cell of the Scarf complex. Then

|F | ≤ n.

Proof. Let F = {v_{1}, …, v_{p}}. Suppose p > n. For each index j ∈ {1, …, n} fix an i_{j} ∈ {1, …, p} satisfying

deg {v_{i_{j}}}_{j} = deg F_{j}.

Define

M = {1, …, p} \ {i_{1}, …, i_{n}}.

M is not empty since p > n. For any i ∈ M we obtain

deg(F \ {v_{i}}) = deg F,

which is a contradiction. □

Lemma 3.4.7. Let F = {v_{1}, …, v_{p}} be a cell of the Scarf complex Δ_{I}. There is an injective map

σ_{0} : {1, …, p} → {1, …, n}

such that

a_{k}_{σ_{0}(i)} < a_{i}_{σ_{0}(i)}, 1 ≤ i, k ≤ p, i ≠ k.

For each l ∈ {1, …, n}, l ∉ im(σ_{0}) there is an injective map

σ_{l} : {1, …, p} → {1, …, n}

such that

l ∈ im(σ_{l}),
σ_{l}|_{{1,...,p}\{σ_{l}^{−1}(l)}} = σ_{0}|_{{1,...,p}\{σ_{l}^{−1}(l)}}

and

a_{k}_{σ_{l}(i)} < a_{i}_{σ_{l}(i)}, 1 ≤ i, k ≤ p, i ≠ k, σ_{l}(i) ≠ l,
a_{k}_{l} ≤ a_{σ_{l}^{−1}(l)}_{l}, 1 ≤ k ≤ p.

Proof. Lemma 3.4.6 yields that p ≤ n. Let i ∈ {1, …, p}. Suppose there is no j ∈ {1, …, n} such that

a_{i}_{j} = deg F_{j}

and

a_{k}_{j} < a_{i}_{j}, k ≠ i.

For every j ∈ {1, …, n} we get

a_{i}_{j} < deg F_{j}

or

a_{i}_{j} = a_{k}_{j} for an index k ≠ i.

We conclude that

deg F = deg(F \ {v_{i}}),

which is a contradiction. Hence, there is an index j ∈ {1, …, n} with the properties mentioned above, and we set

σ_{0}(i) = j.

Evidently, σ_{0} is injective and satisfies our conditions. For any l ∈ {1, …, n}, l ∉ im(σ_{0}) we have an index k ∈ {1, …, p} such that

a_{k}_{l} = deg F_{l}.

We define σ_{l} by

σ_{l}(i) = σ_{0}(i), i ∈ {1, …, p}, i ≠ k,
σ_{l}(k) = l.

It is obvious that σ_{l} is injective and fulfills the conditions of the Lemma. □

Lemma 3.4.8. Let a_{i,j} ∈ ℕ_{0} for 1 ≤ i, j ≤ p, p ≥ 2 and j_{0} ∈ {1, …, p} such that

a_{i,j} < a_{j,j}, 1 ≤ i, j ≤ p, j ≠ i, j ≠ j_{0},
a_{i,j_{0}} ≤ a_{j_{0},j_{0}}, 1 ≤ i ≤ p.

Then

A = ( X^{a_{i,j}} )_{{i=1...p \atop j=1...p}}

is a p × p matrix over the field of rational functions ℚ(X), and we have:

CELLULAR RESOLUTIONS OF MONOMIAL IDEALS 95

(a) det(A) is a monic polynomial in X of degree

∑_{k=1}^{p} a_{k,k}.

(b) For 1 ≤ i, j ≤ p we have

A^{−1}[i, j] = \frac{P}{det(A)},

with P being a polynomial in X whose degree is at most

∑_{{k=1 \atop k≠i}}^{p} a_{k,k}.

(c) For 1 ≤ i ≤ p we have

A^{−1}[i, i] = \frac{P}{det(A)},

with P being a monic polynomial in X of degree

∑_{{k=1 \atop k≠i}}^{p} a_{k,k}.

(d) For 1 ≤ j ≤ p, j ≠ j_{0} we obtain

A^{−1}[j_{0}, j] = \frac{P}{det(A)},

with P being a polynomial of degree less than

∑_{{k=1 \atop k≠j_{0}}}^{p} a_{k,k}.

(e) Let 1 ≤ i ≤ p, i ≠ j, i ≠ j_{0} satisfying

A^{−1}[i, j] = \frac{P}{det(A)},

with P being a polynomial of degree

∑_{{k=1 \atop k≠i}}^{p} a_{k,k}.

This implies j = j_{0}, and the leading coefficient of P has the absolute value 1. Proof. For any bijection σ ∈ S_{p} let m_{σ} be the monomial

m_{σ} = X^{d}, d = ∑_{k=1}^{p} a_{σ(k),k}.

96 STEFAN TAPPE

We get

det(A) = ∑_{σ∈S_{p}} sign(σ) ∏_{k=1}^{p} A[σ(k), k]
= ∑_{σ∈S_{p}} sign(σ) ∏_{k=1}^{p} X^{a_{σ(k),k}}
= ∑_{σ∈S_{p}} sign(σ) m_{σ}.

For the identity σ = id we obtain

sign(id) = 1

and

deg(m_{id}) = ∑_{k=1}^{p} a_{k,k}.

If σ ≠ id, there are two indices 1 ≤ k_{1}, k_{2} ≤ p satisfying

k_{1} ≠ k_{2}, k_{1} ≠ j_{0}, σ(k_{1}) ≠ k_{1}, σ(k_{2}) ≠ k_{2}.

Therefore,

a_{σ(k_{1}),k_{1}} < a_{k_{1},k_{1}},

and we conclude

deg(m_{σ}) = ∑_{k=1}^{p} a_{σ(k),k} = \underbrace{a_{σ(k_{1}),k_{1}}}_{<a_{k_{1},k_{1}}} + ∑_{{k=1 \atop k≠k_{1}}}^{p} \underbrace{a_{σ(k),k}}_{≤a_{k,k}}
< ∑_{k=1}^{p} a_{k,k} = deg(m_{id}),

which completes the proof of (a). It is known that for 1 ≤ i, j ≤ p

A^{−1}[i, j] = \frac{1}{det(A)} (−1)^{i+j} det(A(j | i))

where A(j | i) is the matrix obtained from A by deleting the j-th row and the i-th column. The determinant of A(j | i) can be computed as follows: For any bijective map

σ : {1, …, p} \ {i} → {1, …, p} \ {j}

let m_{σ} be the monomial

m_{σ} = X^{d}, d = ∑_{{k=1 \atop k≠i}}^{p} a_{σ(k),k}.

Then

det(A(j | i)) = ∑_{σ} sign(σ) ∏_{{k=1 \atop k≠i}}^{p} A[σ(k), k]

= ∑_{σ} sign(σ) ∏_{{k=1 \atop k≠i}}^{p} X^{a_{σ(k),k}}
= ∑_{σ} sign(σ) m_{σ},

where σ runs over all bijective maps as above. For any such σ we have

deg(m_{σ}) = ∑_{{k=1 \atop k≠i}}^{p} \underbrace{a_{σ(k),k}}_{≤a_{k,k}} ≤ ∑_{{k=1 \atop k≠i}}^{p} a_{k,k},

hence (b) is shown. For (c) we apply (a) and the formula

A^{−1}[i, i] = \frac{det(A(i | i))}{det(A)}.

(d) Fix any bijection

σ : {1, …, p} \ {j_{0}} → {1, …, p} \ {j}.

We have j ≠ j_{0}. This implies that i_{0} = σ^{−1}(j_{0}) exists, and we get

i_{0} ≠ j_{0}.

Consequently,

deg(m_{σ}) = ∑_{{k=1 \atop k≠j_{0}}}^{p} a_{σ(k),k} = \underbrace{a_{j_{0},i_{0}}}_{<a_{i_{0},i_{0}}} + ∑_{{k=1 \atop k≠j_{0},k≠i_{0}}}^{p} \underbrace{a_{σ(k),k}}_{≤a_{k,k}}

< ∑_{{k=1 \atop k≠j_{0}}}^{p} a_{k,k},

which completes the proof of (d).

(e) There is a bijective map σ : {1, …, p} \ {i} → {1, …, p} \ {j}

satisfying

deg(m_{σ}) = ∑_{{k=1 \atop k≠i}}^{p} a_{k,k} = a_{j_{0},j_{0}} + ∑_{{k=1 \atop k≠i,k≠j_{0}}}^{p} a_{k,k}.

From the definition of m_{σ} it follows that

deg(m_{σ}) = ∑_{{k=1 \atop k≠i}}^{p} a_{σ(k),k} = a_{σ(j_{0}),j_{0}} + ∑_{{k=1 \atop k≠i,k≠j_{0}}}^{p} a_{σ(k),k}.

98 STEFAN TAPPE

This yields

σ(k) = k, k ∈ {1, …, p} \ {i, j_{0}},

therefore σ is uniquely determined, which implies that the leading coefficient of P has the absolute value 1. We have σ(j_{0}) = i, which implies

im(σ) = {1, …, p} \ {j_{0}},

thus j = j_{0}. □

Lemma 3.4.9. Let a_{i,j} ∈ ℕ_{0} for 1 ≤ i, j ≤ p, p ≥ 2 such that

a_{i,j} < a_{j,j}, 1 ≤ i, j ≤ p, i ≠ j.

Once again, we consider the p × p matrix

A = ( X^{a_{i,j}} )_{{i=1...p \atop j=1...p}} .

The following statements hold:

(a) det(A) is a monic polynomial in X of degree

∑_{k=1}^{p} a_{k,k}.

(b) For 1 ≤ i ≤ p we have

A^{−1}[i, i] = \frac{P}{det(A)},

with P being a monic polynomial in X of degree

∑_{{k=1 \atop k≠i}}^{p} a_{k,k}.

(c) For 1 ≤ i, j ≤ p, i ≠ j we obtain

A^{−1}[i, j] = \frac{P}{det(A)},

with P being a polynomial of degree less than

∑_{{k=1 \atop k≠i}}^{p} a_{k,k}.

Proof. (a) is a special case of Lemma 3.4.8(a). For (b) we apply (a) and the formula

A^{−1}[i, i] = \frac{det(A(i | i))}{det(A)}.

(c) Let

σ : {1, …, p} \ {i} → {1, …, p} \ {j}

be any bijective map. We have i ≠ j, hence i_{0} = σ^{−1}(i) exists and we obtain i ≠ i_{0} and

deg(m_{σ}) = ∑_{{k=1 \atop k≠i}}^{p} a_{σ(k),k} = \underbrace{a_{i,i_{0}}}_{<a_{i_{0},i_{0}}} + ∑_{{k=1 \atop k≠i,k≠i_{0}}}^{p} \underbrace{a_{σ(k),k}}_{≤a_{k,k}} .

< ∑_{{k=1 \atop k≠i}}^{p} a_{k,k},

which completes the proof. □

Lemma 3.4.10. Let F ≠ ∅ be a cell of the Scarf complex Δ_{I} and let i ∈ {1, …, s} be such that v_{i} ∉ F. Then there is an index j ∈ {1, …, n} satisfying

deg F_{j} < a_{i}_{j}.

Proof. Assume

a_{i} ◁ deg F.

This implies

deg F = deg(F ∪ {v_{i}}),

which is a contradiction to F being a cell of Δ_{I}. □

Theorem 3.4.11. For any monomial ideal I, the Scarf complex Δ_{I} is a subcomplex of the hull complex hull(I).

Proof. Let F be a face of the Scarf complex Δ_{I}. If F is empty or a single vertex, it follows immediately that F ∈ hull(I). Therefore, we may assume that F = {v_{1}, …, v_{p}}, p ≥ 2. As in Lemma 3.4.7 we choose an injective map

σ_{0} : {1, …, p} → {1, …, n}

and further injective maps

σ_{l} : {1, …, p} → {1, …, n}, l ∉ im(σ_{0}).

The following p × p matrices are matrices over the field of rational functions ℚ(X):

A_{l} = ( X^{a_{i}^{σ_{l}(j)}} )_{{i=1...p \atop j=1...p}} , l ∈ \{0, …, n\}, l ∉ im(σ_{0}).

Note that A_{0} satisfies the conditions of Lemma 3.4.9 and the other A_{l} satisfy the conditions of Lemma 3.4.8. So these two lemmas ensure that the A_{l} are invertible over ℚ(X). For each l ∈ {0, …, n}, l ∉ im (σ_{0}) let ṽ^{σ_{l}} be the sum of the column vectors of the inverse matrix:

ṽ^{σ_{l}} = ∑_{j=1}^{p} A_{l}^{−1}_{•j}.

We augment the p-vectors ṽ^{σ_{l}} with additional zero coordinates and obtain an n-vector v^{σ_{l}} by setting

v_{i}^{σ_{l}} = \{ \begin{matrix}ṽ_{j}^{σ_{l}} & i = σ_{l}(j) \\ 0 & i ∉ im(σ_{l}) \end{matrix}

By taking a convex combination

v = λ_{0} v^{σ_{0}} + ∑_{l∉im(σ_{0})} λ_{l} v^{σ_{l}}

with all coefficients being non-zero, we obtain a vector v ∈ ℝ^{n}. If we set t ≫ 0 for X, this vector satisfies the conditions of Proposition 3.3.6(b) as we will show next: For each i = 1, …, p we have

⟨v, X^{a^{i}}⟩ = λ_{0} \underbrace{⟨v^{σ_{0}}, X^{a^{i}}⟩}_{=1} + ∑_{l∉im(σ_{0})} λ_{l} \underbrace{⟨v^{σ_{l}}, X^{a^{i}}⟩}_{=1} = 1.

The inner products are equal to 1 because the vectors v^{σ_{l}} are obtained from the inverse of the matrix A_{l}.

Both the Lemmas 3.4.8 and 3.4.9 yield that for all j ∈ {1, …, n}

v_{j} = c_{j} \frac{P_{j}}{Q_{j}}

with c_{j} > 0 and P_{j}, Q_{j} being a monic polynomials in X, such that

deg(Q_{j}) − deg(P_{j}) = deg F_{j}.

Therefore, we obtain

v ▷ 0

by setting t ≫ 0 for X. Now let i > p. According to Lemma 3.4.10 there is an index j ∈ {1, …, n} satisfying

deg F_{j} < a_{i}_{j},

and by setting t ≫ 0 for X we obtain:

⟨v, X^{a^{i}}⟩ ≥ v_{j} X^{a_{i_{j}}} = \frac{c_{j} X^{a_{i_{j}}} P_{j}}{Q_{j}} > 1.

We conclude that F is a cell of hull(I).

So we have seen that

Δ_{I} ⊂ hull(I).

It follows from Lemma 2.2.8(iv) that Δ_{I} is a subcomplex of hull(I) since Δ_{I} is simplicial by Lemma 3.4.2. □

The following example illustrates the constructions of the proof above:

Example 3.4.12. The Scarf complex of the monomial ideal

I = (x^{2}y, xz, yz^{2}, y^{2}) ⊂ K[x, y, z]

is (see Example 3.4.4) the simplicial complex

Δ_{I} = ⟨{v_{1}, v_{2}, v_{4}}, {v_{2}, v_{3}, v_{4}}⟩.

We would like to see that

F = {v_{1}, v_{2}, v_{4}}

and

G = {v_{1}, v_{2}}

belong to hull(I). For F = {v_{1}, v_{2}, v_{4}} we renumber v_{3} and v_{4} by setting

deg {v_{3}} = (0, 2, 0),
deg {v_{4}} = (0, 1, 2).

Hence, we may now consider F̃ = {v_{1}, v_{2}, v_{3}}. σ_{0} : {1, 2, 3} → {1, 2, 3} given by

σ_{0}(1) = 1, σ_{0}(2) = 3, σ_{0}(3) = 2.

is an injective map as in Lemma 3.4.7. The corresponding matrix is

A_{0} = ( \begin{matrix}X^{2} & 1 & X \\ X & X & 1 \\ 1 & 1 & X^{2} \end{matrix} ) ,

and its inverse

A_{0}^{−1} = ( \begin{matrix}\frac{1}{X^{2}−1} & −\frac{X}{X^{4}+X^{3}−X−1} & −\frac{1}{X^{3}−1} \\ −\frac{1}{X^{2}−1} & \frac{X}{X^{2}−1} & 0 \\ 0 & −\frac{1}{X^{3}−1} & \frac{X}{X^{3}−1} \end{matrix} ) .

We can directly compute the vector v as

v = v^{σ_{0}} = ( \frac{X}{(X^{2}+X+1)(X+1)} ; \frac{1}{X^{2}+X+1} ; \frac{1}{X+1} ) .

For X we set t = 25 and obtain

H ((\frac{25}{16926}, \frac{1}{651}, \frac{1}{26}) , 1)

as a supporting hyperplane.

Now consider G = {v_{1}, v_{2}}. It is not necessary to renumber the vertices. Injective maps σ_{0}, σ_{2} : {1, 2} → {1, 2, 3} as in Lemma 3.4.7 are given by

σ_{0}(1) = 1, σ_{0}(2) = 3,
σ_{2}(1) = 2, σ_{2}(2) = 3.

The corresponding matrices are

A_{0} = ( \begin{matrix}X^{2} & 1 \\ X & X \end{matrix} )

and

A_{2} = ( \begin{matrix}X & 1 \\ 1 & X \end{matrix} ) .

The inverse matrices are given by

A_{0}^{−1} = ( \begin{matrix}\frac{1}{X^{2}−1} & −\frac{1}{X(X^{2}−1)} \\ −\frac{1}{X^{2}−1} & \frac{X}{X^{2}−1} \end{matrix} )

and

A_{2}^{−1} = ( \begin{matrix}\frac{X}{X^{2}−1} & −\frac{1}{X^{2}−1} \\ −\frac{1}{X^{2}−1} & \frac{X}{X^{2}−1} \end{matrix} ) .

We obtain the vectors

v^{σ_{0}} = ( 0_{\frac{1}{X+1}}^{\frac{1}{(X+1)X}} )

and

v^{σ_{2}} = ( 0 ; \frac{1}{X+1} ; \frac{1}{X+1} ) .

We take as convex combination

v = \frac{1}{2}v^{σ_{0}} + \frac{1}{2}v^{σ_{2}} = ( \frac{1}{2} \frac{1}{(X+1)X} ; \frac{1}{2} \frac{1}{X+1} ; \frac{1}{X+1} ) ,

which leads, by setting t = 25 for X, to the supporting hyperplane

H ((\frac{1}{1300}, \frac{1}{52}, \frac{1}{26}) , 1) .

We have proven a result that helps us to determine the hull complex of a monomial ideal I. Now, we show a statement which is also of use for computing the hull complex.

Lemma 3.4.13. Let F ⊂ ℝ^{n} be a nonempty bounded face of the polyhedron P_{t}. Then there is a nonzero linear functional

f : ℝ^{n} → ℝ

which is nonnegative on ℝ_{+}^{n}, whose minimum over P_{t} is attained at the face F, and which fulfills the property

f(z_{1}) < f(z_{2}), z_{1} < z_{2}.

Proof. Choose an n-vector v ▷ 0 as in Proposition 3.3.6(b) for the face F and set

f : ℝ^{n} → ℝ,
f(z) = ⟨v, z⟩.

It is easy to verify that f satisfies the conditions of the lemma. □

CELLULAR RESOLUTIONS OF MONOMIAL IDEALS 103

Example 3.4.14. For the maximal ideal

I = (x, y) ⊂ R = K[x, y]

the polyhedron P_{7} is

It is obvious that

conv {(1, 7), (7, 1)} = ( (1, 7), (7, 1) )

is a bounded face of P_{7}. One can check that

v = (\frac{1}{8}, \frac{1}{8})

satisfies the conditions of Proposition 3.3.6(b). The supporting hyperplane

H(v, 1)

is shown in the figure. A linear functional

f : ℝ^{2} → ℝ

as in Lemma 3.4.13 is given by

f(z) = ⟨v, z⟩ = \frac{1}{8} (z_{1} + z_{2}).

Proposition 3.4.15. Let F ≠ ∅ be a cell of hull(I) and x^{b} ∈ I. Then there is an index j ∈ {1, …, n} such that

deg F_{j} ≤ b_{j}.

Proof. Let F = {v_{1}, …, v_{p}} and G_{t} ⊂ ℝ^{n} be the corresponding bounded face

G_{t} = conv {t^{a^{1}}, …, t^{a^{p}}}, t > (n + 1)!,

of the polyhedron P_{t}. Suppose that

b ◁ deg F.

Consider

w = \frac{1}{p} ∑_{i=1}^{p} X^{a^{i}}.

The jth coordinate of w is a polynomial in X of degree deg F_{j}. The jth coordinate of X^{b} is a monomial of strictly smaller degree. Hence,

t^{b} < w(t) for t ≫ 0.

Fix t > (n + 1)! such that

t^{b} < w(t).

We have w(t) ∈ G_{t} since it is a convex combination of the t^{a^{i}}. We choose a linear functional

f : ℝ^{n} → ℝ

as in Lemma 3.4.13. The minimum over P_{t} is attained at G_{t}. Therefore, we get

f(w(t)) ≤ f(t^{b}).

On the other hand, Lemma 3.4.13 yields

f(w(t)) > f(t^{b}),

which is a contradiction. □

Example 3.4.16. Again, we consider the monomial ideal

I = (x^{2}y, xz, yz^{2}, y^{2}) ⊂ R = K[x, y, z].

By Theorem 3.4.11 the Scarf complex Δ_{I}, determined in Example 3.4.4, is a subcomplex of hull(I). By Proposition 3.4.15 the remaining cells of the Taylor complex Δ

F_{2}^{1} = {v_{1}, v_{3}},
F_{1}^{2} = {v_{1}, v_{2}, v_{3}},
F_{3}^{2} = {v_{1}, v_{3}, v_{4}},
F_{1}^{3} = {v_{1}, v_{2}, v_{3}, v_{4}},

are not in hull(I) since

a_{2} = (1, 0, 1) ◁ (2, 1, 2) = deg {v_{1}, v_{3}} = deg {v_{1}, v_{2}, v_{3}},
a_{2} = (1, 0, 1) ◁ (2, 2, 2) = deg {v_{1}, v_{3}, v_{4}} = deg {v_{1}, v_{2}, v_{3}, v_{4}}.

Thus,

hull(I) = ⟨{v_{1}, v_{2}, v_{4}}, {v_{2}, v_{3}, v_{4}}⟩

is also the simplicial complex X of Example 3.1.21. We conclude that the hull complex and the Scarf complex of I coincide. The cellular complex 𝔽_{hull(I)} = 𝔽_{Δ_{I}} is a minimal free resolution of I. This property is characteristic for generic ideals introduced in the next Section 3.5. Indeed, I is such a generic ideal.

Remark 3.4.17. In general there are cells F = {v_{j} : j ∈ J} of the Taylor complex Δ_{I} to which neither Theorem 3.4.11 nor Proposition 3.4.15 apply. In this case, we proceed by determining the set of solutions for the system of linear equations

⟨v, t^{a^{j}}⟩ = 1, j ∈ J,

which is in fact done with the help of elementary Linear Algebra. Afterwards, we decide whether

conv {t^{a^{j}} : j ∈ J}

is a bounded face by applying the criterion of Proposition 3.3.6(b). This is made precisely in the next statement.

Lemma 3.4.18. For J ⊂ {1, …, s}, J ≠ ∅ consider the system of linear equations

⟨v, t^{a^{j}}⟩ = 1, j ∈ J

(i) The existence of a solution v ∈ ℝ^{n} satisfying the conditions

v ▷ 0,
⟨v, t^{a^{j}}⟩ > 1, j ∉ J,

implies that

{v_{j} : j ∈ J} ∈ hull(I).

(ii) If each solution v ∈ ℝ^{n} of the system does not satisfy the conditions

v ▷ 0,
⟨v, t^{a^{j}}⟩ > 1, j ∉ J,

we can conclude that

{v_{j} : j ∈ J} ∉ hull(I).

(iii) If the system possesses a uniquely determined solution v ∈ ℝ^{n} fulfilling

v ▷ 0,
⟨v, t^{a^{j}}⟩ > 1, j ∉ J,

we obtain

{v_{j} : j ∈ J} ∈ hull(I)

and F ∉ hull(I) for each F satisfying the inclusion F ⫌ {v_{j} : j ∈ J}. (iv) If there is a uniquely determined solution v ∈ ℝ^{n} which does not satisfy

v ▷ 0,
⟨v, t^{a^{j}}⟩ > 1, j ∉ J,

then

{v_{j} : j ∈ J} ∉ hull(I).

Furthermore, if

v ⋫ 0

or

⟨v, t^{a^{j}}⟩ < 1

for some j ∉ J, we even have

F ∉ hull(I) if F satisfies F ⊃ {v_{j} : j ∈ J}.

(v) If the system has no solution, we get

106 STEFAN TAPPE

F ∉ hull(I) if F ⊃ {v_{j} : j ∈ J}.

Proof. This follows from the characterization of the hull complex in Remark 3.3.13. □ Now that we have collected important results which simplify the computation of the hull complex, we can finally treat more advanced examples than the ones in Example 3.3.14. Example 3.4.19. Consider the monomial ideal

I = (x, y, z)^{2} = (x^{2}, y^{2}, z^{2}, xy, xz, yz)

⊂ R = K[x, y, z]

The Scarf complex is given by

Δ_{I}

Component  Cell  Degree
Δ_{I}^{−1}  F_{1}^{−1}  ∅
F_{1}^{0} {v_{1}} (2,0,0)
F_{2}^{0} {v_{2}} (0,2,0)
F_{3}^{0} {v_{3}} (0,0,2)
Δ_{I}^{0}  F_{4}^{0} {v_{4}} (1,1,0)
F_{5}^{0} {v_{5}} (1,0,1)
F_{6}^{0} {v_{6}} (0,1,1)
F_{1}^{1} {v_{1}, v_{4}} (2,1,0)
F_{2}^{1} {v_{1}, v_{5}} (2,0,1)
F_{3}^{1} {v_{2}, v_{4}} (1,2,0)
Δ_{I}^{1}  F_{4}^{1} {v_{2}, v_{6}} (0,2,1)
F_{5}^{1} {v_{3}, v_{5}} (1,0,2)
F_{6}^{1} {v_{3}, v_{6}} (0,1,2)

The cellular complex

𝔽_{Δ_{I}} : 0 ⟶ R^{6}(−3) ⟶ R^{6}(−2) ⟶ I ⟶ 0

is not a resolution by Corollary 3.1.16. We assert that hull(I) consists of four triangles combined into one large triangle:

hull(I)

Component Cell Degree
hull(I)^{−1} F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (2,0,0)
F_{2}^{0}  {v_{2}}  (0,2,0)
F_{3}^{0}  {v_{3}}  (0,0,2)
hull(I)^{0} F_{4}^{0}  {v_{4}}  (1,1,0)
F_{5}^{0}  {v_{5}}  (1,0,1)
F_{6}^{0}  {v_{6}}  (0,1,1)
F_{1}^{1} {v_{1}, v_{4}} (2,1,0)
F_{2}^{1} {v_{1}, v_{5}} (2,0,1)
F_{3}^{1} {v_{2}, v_{4}} (1,2,0)
F_{4}^{1} {v_{2}, v_{6}} (0,2,1)
hull(I)^{1} F_{5}^{1} {v_{3}, v_{5}} (1,0,2)
F_{6}^{1} {v_{3}, v_{6}} (0,1,2)
F_{7}^{1} {v_{4}, v_{5}} (1,1,1)
F_{8}^{1} {v_{4}, v_{6}} (1,1,1)
F_{9}^{1} {v_{5}, v_{6}} (1,1,1)
F_{1}^{2} {v_{1}, v_{4}, v_{5}} (2,1,1)
F_{2}^{2} {v_{2}, v_{4}, v_{6}} (1,2,1)
hull(I)^{2} F_{3}^{2} {v_{3}, v_{5}, v_{6}} (1,1,2)
F_{4}^{2} {v_{4}, v_{5}, v_{6}} (1,1,1)

By Theorem 3.4.11 each cell of Δ_{I} also belongs to hull(I). To find the other cells of hull(I), we choose t = 25 and obtain the following vertices:

t^{a^{1}} = (625, 1, 1),
t^{a^{2}} = (1, 625, 1),
t^{a^{3}} = (1, 1, 625),
t^{a^{4}} = (25, 25, 1),
t^{a^{5}} = (25, 1, 25),
t^{a^{6}} = (1, 25, 25).

For J = {4, 5} we obtain the system of linear equations

( \begin{matrix}25 & 25 & 1 \\ 25 & 1 & 25 \end{matrix} ) · v = 1.

The set of all solutions of this system is

\{v = ( \frac{1}{25} 0 0 ) + λ ( \begin{matrix}−\frac{26}{25} \\ 1 \\ 1 \end{matrix} ) : λ ∈ ℝ\} .

By setting λ = \frac{1}{30} we obtain the vector

v = (\frac{2}{375}, \frac{1}{30}, \frac{1}{30}) ,

which satisfies the conditions of Lemma 3.4.18(i). For reasons of symmetry we have

{v_{4}, v_{5}}, {v_{4}, v_{6}}, {v_{5}, v_{6}} ∈ hull(I).

Now let J = {1, 4, 5}. The corresponding system of linear equations is

( \begin{matrix}625 & 1 & 1 \\ 25 & 25 & 1 \\ 25 & 1 & 25 \end{matrix} ) · v = 1.

It has the uniquely determined solution

v = (\frac{1}{675}, \frac{1}{27}, \frac{1}{27}) .

The conditions of Lemma 3.4.18(iii) are satisfied. Symmetry yields

{v_{1}, v_{4}, v_{5}}, {v_{2}, v_{4}, v_{6}}, {v_{3}, v_{5}, v_{6}} ∈ hull(I),

and F ∉ hull(I) if F ⫌ {v_{1}, v_{4}, v_{5}}, F ⫌ {v_{2}, v_{4}, v_{6}}, or F ⫌ {v_{3}, v_{5}, v_{6}}.

For J = {4, 5, 6} we have the system

( \begin{matrix}25 & 25 & 1 \\ 25 & 1 & 25 \\ 1 & 25 & 25 \end{matrix} ) · v = 1,

which has the uniquely determined solution

v = (\frac{1}{51}, \frac{1}{51}, \frac{1}{51}) ,

satisfying the conditions of Lemma 3.4.18(iii). Therefore,

{v_{4}, v_{5}, v_{6}} ∈ hull(I),

and F ∉ hull(I) for F ⫌ {v_{4}, v_{5}, v_{6}}.

The system of linear equations for J = {1, 2} is given by

( \begin{matrix}625 & 1 & 1 \\ 1 & 625 & 1 \end{matrix} ) · v = 1.

In this case, the solutions of the system can be written as

\{v = ( \begin{matrix}0 \\ 0 \\ 1 \end{matrix} ) + λ ( \begin{matrix}1 \\ 1 \\ −626 \end{matrix} ) : λ ∈ ℝ\} .

CELLULAR RESOLUTIONS OF MONOMIAL IDEALS 109

Suppose there is such a vector v satisfying the conditions

v ▷ 0,
⟨v, t^{a^{j}}⟩ > 1, j ∈ {3, 4, 5, 6},

Then v ▷ 0 implies

λ > 0.

On the other hand, ⟨v, t^{a^{4}}⟩ > 1 implies

λ < 0.

Since the sets J = {1, 3} and J = {2, 3} can be treated analogously, we obtain by Lemma 3.4.18(ii)

{v_{1}, v_{2}}, {v_{1}, v_{3}}, {v_{2}, v_{3}} ∉ hull(I).

For J = {1, 6} we have to consider the system

( \begin{matrix}625 & 1 & 1 \\ 1 & 25 & 25 \end{matrix} ) · v = 1.

The set of all solutions is

\{v = ( \frac{1}{651} \frac{26}{651} 0 ) + λ ( \begin{matrix}0 \\ −1 \\ 1 \end{matrix} ) : λ ∈ ℝ\} .

Assume that there is a vector v from this set fulfilling

v ▷ 0,
⟨v, t^{a^{j}}⟩ > 1, j ∈ {2, 3, 4, 5},

⟨v, t^{a^{4}}⟩ > 1 implies

\frac{225}{217} − 24λ > 1,

and ⟨v, t^{a^{5}}⟩ > 1 implies

\frac{17}{217} + 24λ > 1.

Adding these two inequalities yields

\frac{242}{217} > 2.

This is a contradiction. Because of the symmetry, we get

{v_{1}, v_{6}}, {v_{2}, v_{5}}, {v_{3}, v_{4}} ∉ hull(I)

by applying Lemma 3.4.18(ii). For J = {1, 2, 3} we obtain the system of linear equations

( \begin{matrix}625 & 1 & 1 \\ 1 & 625 & 1 \\ 1 & 1 & 625 \end{matrix} ) · v = 1.

This system has the uniquely determined solution v = (\frac{1}{627}, \frac{1}{627}, \frac{1}{627}), for which the inequality

⟨v, t^{a^{4}}⟩ < 1.

holds. By Lemma 3.4.18(iv)

F ∉ hull(I) for F ⊃ {v_{1}, v_{2}, v_{3}}.

If J = {1, 2, 4}, we obtain the system of linear equations

( \begin{matrix}625 & 1 & 1 \\ 1 & 625 & 1 \\ 25 & 25 & 1 \end{matrix} ) · v = 1.

It has the uniquely determined solution

v = (0, 0, 1),

Since

v ⋫ 0,

we get, by using Lemma 3.4.18(iv) and the symmetry again F ∉ hull(I) if F ⊃ {v_{1}, v_{2}, v_{4}}, F ⊃ {v_{1}, v_{3}, v_{5}} or F ⊃ {v_{2}, v_{3}, v_{6}}. By considering J = {1, 2, 5} we get the system of linear equations

( \begin{matrix}625 & 1 & 1 \\ 1 & 625 & 1 \\ 25 & 1 & 25 \end{matrix} ) · v = 1.

It has exactly one solution, namely

v = (\frac{1}{651}, \frac{1}{651}, \frac{25}{651}).

Note that

⟨v, t^{a^{4}}⟩ < 1.

By applying Lemma 3.4.18(iv) and the symmetry, we obtain that F ∉ hull(I) for each F satisfying one of the following inclusions:

F ⊃ {v_{1}, v_{2}, v_{5}}, F ⊃ {v_{1}, v_{2}, v_{6}}, F ⊃ {v_{1}, v_{3}, v_{4}},

F ⊃ {v_{1}, v_{3}, v_{6}}, F ⊃ {v_{2}, v_{3}, v_{4}}, F ⊃ {v_{2}, v_{3}, v_{5}}.

J = {1, 4, 6} yields the system

( \begin{matrix}625 & 1 & 1 \\ 25 & 25 & 1 \\ 1 & 25 & 25 \end{matrix} ) · v = 1,

which has the uniquely determined solution

v = (\frac{1}{651}, \frac{25}{651}, \frac{1}{651}) .

Observe that

⟨v, t^{a^{5}}⟩ < 1.

The following sets can be treated analogously, we obtain, by applying Lemma 3.4.18(iv), that F ∉ hull(I) if F fulfills one of the following inclusions:

F ⊃ {v_{1}, v_{4}, v_{6}}, F ⊃ {v_{1}, v_{5}, v_{6}}, F ⊃ {v_{2}, v_{4}, v_{5}},

F ⊃ {v_{2}, v_{5}, v_{6}}, F ⊃ {v_{3}, v_{4}, v_{5}}, F ⊃ {v_{3}, v_{4}, v_{6}}.

The hull resolution

𝔽_{hull(I)} : 0 ⟶ \begin{matrix}R^{3}(−4) \\ ⊕ \\ R(−3) \end{matrix} ⟶ R^{9}(−3) ⟶ R^{6}(−2) ⟶ I ⟶ 0

is not minimal by Proposition 3.1.14 because

deg {v_{5}, v_{6}} = deg {v_{4}, v_{5}, v_{6}} = (1, 1, 1).

However, we observe that it is very close to being minimal. In Example 3.6.2 we reduce 𝔽_{hull(I)} such that it becomes a minimal free resolution.

Example 3.4.20. The ideal

I = (x, y) ∩ (z, w) = (xz, xw, yz, yw) ⊂ R = K[x, y, z, w]

defines a pair of skew lines in ℙ^{3}. The Scarf complex of I consists of the edges of a rectangle:

Δ_{I}
Component  Cell  Degree
Δ_{I}^{−1}  F_{1}^{−1}  ∅
F_{1}^{0} {v_{1}} (1,0,1,0)
F_{2}^{0} {v_{2}} (1,0,0,1)
Δ_{I}^{0}  F_{3}^{0} {v_{3}} (0,1,1,0)
F_{4}^{0} {v_{4}} (0,1,0,1)
F_{1}^{1} {v_{1}, v_{2}} (1,0,1,1)
F_{2}^{1} {v_{1}, v_{3}} (1,1,1,0)
Δ_{I}^{1}  F_{3}^{1} {v_{2}, v_{4}} (1,1,0,1)
F_{4}^{1} {v_{3}, v_{4}} (0,1,1,1)

The cellular complex

𝔽_{Δ_{I}} : 0 ⟶ R^{4}(−3) ⟶ R^{4}(−2) ⟶ I ⟶ 0

is not a resolution of I because of Corollary 3.1.16. The hull complex of I is a filled rectangle.

hull(I)
Dimension  Cell  Degree
hull(I)^{−1} F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (1,0,1,0)
F_{2}^{0}  {v_{2}}  (1,0,0,1)
hull(I)^{0} F_{3}^{0}  {v_{3}}  (0,1,1,0)
F_{4}^{0}  {v_{4}}  (0,1,0,1)
F_{1}^{1}  {v_{1}, v_{2}}  (1,0,1,1)
F_{2}^{1}  {v_{1}, v_{3}}  (1,1,1,0)
hull(I)^{1} F_{3}^{1}  {v_{2}, v_{4}}  (1,1,0,1)
F_{4}^{1}  {v_{3}, v_{4}}  (0,1,1,1)
hull(I)^{2} F_{1}^{2} {v_{1}, v_{2}, v_{3}, v_{4}} (1,1,1,1)

To prove this, we apply Theorem 3.4.11 and choose t = 121, which yields the following vertices:

t^{a^{1}} = (121, 1, 121, 1), t^{a^{2}} = (121, 1, 1, 121), t^{a^{3}} = (1, 121, 121, 1), t^{a^{4}} = (1, 121, 1, 121).

For J = {1, 2, 3, 4} we obtain the system of linear equations

( \begin{matrix}121 & 1 & 121 & 1 \\ 121 & 1 & 1 & 121 \\ 1 & 121 & 121 & 1 \\ 1 & 121 & 1 & 121 \end{matrix} ) · v = 1.

The set of all solutions is

\{v = ( \frac{1}{122} \frac{1}{122} 0 0 ) + λ ( \begin{matrix}−1 \\ −1 \\ 1 \\ 1 \end{matrix} ) : λ ∈ ℝ\} .

By setting λ = \frac{1}{1000} we obtain the vector

v = (\frac{439}{61000}, \frac{439}{61000}, \frac{1}{1000}, \frac{1}{1000}) , which satisfies the conditions of Lemma 3.4.18(i), and we get

{v_{1}, v_{2}, v_{3}, v_{4}} ∈ hull(I).

J = {1, 4} yields the system of linear equations

( \begin{matrix}121 & 1 & 121 & 1 \\ 1 & 121 & 1 & 121 \end{matrix} ) · v = 1.

It has the set of all solutions

\{v = ( \frac{1}{122} \frac{1}{122} 0 0 ) + λ_{1} ( \begin{matrix}−1 \\ 0 \\ 1 \\ 0 \end{matrix} ) + λ_{2} ( \begin{matrix}0 \\ −1 \\ 0 \\ 1 \end{matrix} ) : λ_{1}, λ_{2} ∈ ℝ\} .

Suppose there is such a vector v satisfying the conditions

v ▷ 0,
⟨v, t^{a^{j}}⟩ > 1, j ∈ {2, 3}.

⟨v, t^{a^{2}}⟩ > 1 implies

1 − 120λ_{1} + 120λ_{2} > 1,

and ⟨v, t^{a^{3}}⟩ > 1 implies

1 + 120λ_{1} − 120λ_{2} > 1.

Adding these two inequalities yields

2 > 2,

a contradiction. The case J = {2, 3} can be treated analogously. Hence, we get by applying Lemma 3.4.18(iii)

{v_{1}, v_{4}}, {v_{2}, v_{3}} ∉ hull(I).

For J = {1, 2, 3} the resulting system of linear equations is

( \begin{matrix}121 & 1 & 121 & 1 \\ 121 & 1 & 1 & 121 \\ 1 & 121 & 121 & 1 \end{matrix} ) · v = 1.

The set of all solutions is given by

\{v = ( \frac{1}{122} \frac{1}{122} 0 0 ) + λ ( \begin{matrix}−1 \\ −1 \\ 1 \\ 1 \end{matrix} ) : λ ∈ ℝ\} .

Suppose there is a vector v of this set fulfilling

v ▷ 0,
⟨v, t^{a^{j}}⟩ > 1, j = 4.

⟨v, t^{a_{4}}⟩ > 1 implies

1 > 1,

which is a contradiction. Analogously, one treats the other cases and obtains by using Lemma 3.4.18(ii)

{v_{1}, v_{2}, v_{3}}, {v_{1}, v_{2}, v_{4}}, {v_{1}, v_{3}, v_{4}}, {v_{2}, v_{3}, v_{4}} ∉ hull(I).

We have computed an incidence function ε on hull(I) in Example 2.2.13, which we use to determine the homomorphisms ∂_{0}, ∂_{1}, ∂_{2}:

\begin{matrix}∂_{0} & \{v_{1}\} & \{v_{2}\} & \{v_{3}\} & \{v_{4}\} \\ ∅ & xz & xw & yz & yw \end{matrix}

\begin{matrix}∂_{1} & \{v_{1}, v_{2}\} & \{v_{1}, v_{3}\} & \{v_{2}, v_{3}\} & \{v_{3}, v_{4}\} \\ \{v_{1}\} & −w & −y & 0 & 0 \\ \{v_{2}\} & z & 0 & −y & 0 \\ \{v_{3}\} & 0 & x & 0 & −w \\ \{v_{4}\} & 0 & 0 & x & z \end{matrix}

\begin{matrix}∂_{2} & \{v_{1}, v_{2}, v_{3}, v_{4}\} \\ \{v_{1}, v_{2}\} & y \\ \{v_{1}, v_{3}\} & −w \\ \{v_{2}, v_{3}\} & z \\ \{v_{3}, v_{4}\} & −x \end{matrix}

The hull resolution

𝔽_{hull(I)} : 0 → R(−4) ⟶^{( \begin{matrix}y \\ −w \\ z \\ −x \end{matrix} )} R^{4}(−3) ⟶^{( \begin{matrix}−w & −y & 0 & 0 \\ z & 0 & −y & 0 \\ 0 & x & 0 & −w \\ 0 & 0 & x & z \end{matrix} )} R^{4}(−2) \underrightarrow{( xz xw yz yw )} I → 0

is minimal according to Proposition 3.1.14. I is an example of an ideal for which the hull resolution is minimal, whereas the Scarf complex does not provide a resolution. This example also shows that the hull complex, unlike the Scarf complex, need not be simplicial. We obtain the following Betti numbers:

Betti numbers of I

ℤ^{4}-Degree 0 1 2 ℤ-Degree 0 1 2 0 1 2
(0,1,0,1) 1 0 0
(0,1,1,0) 1 0 0
(1,0,0,1) 1 0 0 2 4 0 0
(1,0,1,0) 1 0 0
(0,1,1,1) 0 1 0  4 4 1
(1,0,1,1) 0 1 0
(1,1,0,1) 0 1 0 3 0 4 0
(1,1,1,0) 0 1 0
(1,1,1,1) 0 0 1  4  0 0 1

The Hilbert polynomial of R/I is given by

p_{R/I} = 2X + 2.

CELLULAR RESOLUTIONS OF MONOMIAL IDEALS 115

Thus, we get:

dim (R/I) = 2, deg (R/I) = 2,

h(I) = 2.

Therefore, R/I is not Cohen-Macaulay by Proposition 3.1.20. Example 3.4.21. Consider the ideal

I = (x, y)^{2} ∩ (z, w) = (x^{2}z, x^{2}w, xyz, xyw, y^{2}z, y^{2}w)

⊂ R = K[x, y, z, w].

The hull complex of I consists of two rectangles sharing an edge.

hull(I)

Dimension Cell Degree
hull(I)^{−1} F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (2,0,1,0)
F_{2}^{0}  {v_{2}}  (2,0,0,1)
F_{3}^{0}  {v_{3}}  (1,1,1,0)
hull(I)^{0} F_{4}^{0}  {v_{4}}  (1,1,0,1)
F_{5}^{0}  {v_{5}}  (0,2,1,0)
F_{6}^{0}  {v_{6}}  (0,2,0,1)
F_{1}^{1}  {v_{1}, v_{2}}  (2,0,1,1)
F_{2}^{1}  {v_{1}, v_{3}}  (2,1,1,0)
F_{3}^{1}  {v_{2}, v_{4}}  (2,1,0,1)
hull(I)^{1} F_{4}^{1}  {v_{3}, v_{4}}  (1,1,1,1)
F_{5}^{1}  {v_{3}, v_{5}}  (1,2,1,0)
F_{6}^{1}  {v_{4}, v_{6}}  (1,2,0,1)
F_{7}^{1}  {v_{5}, v_{6}}  (0,2,1,1)
F_{1}^{2} {v_{1}, v_{2}, v_{3}, v_{4}} (2,1,1,1)
hull(I)^{2} F_{2}^{2} {v_{3}, v_{4}, v_{5}, v_{6}} (1,2,1,1)

We obtain a minimal free resolution 𝔽_{hull(I)} : 0 ⟶ R^{2}(−5) ⟶ R^{7}(−4) ⟶ R^{6}(−3) ⟶ I ⟶ 0.

116 STEFAN TAPPE

Example 3.4.22. Now consider

I = (x, y)^{3} ∩ (z, w)

= (x^{3}z, x^{3}w, x^{2}yz, x^{2}yw, xy^{2}z, xy^{2}w, y^{3}z, y^{3}w) ⊂ R = K[x, y, z, w].

The hull complex

hull(I)
Dimension Cell Degree
hull(I)^{−1} F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (3,0,1,0)
F_{2}^{0}  {v_{2}}  (3,0,0,1)
F_{3}^{0}  {v_{3}}  (2,1,1,0)
F_{4}^{0}  {v_{4}}  (2,1,0,1)
hull(I)^{0} F_{5}^{0}  {v_{5}}  (1,2,1,0)
F_{6}^{0}  {v_{6}}  (1,2,0,1)
F_{7}^{0}  {v_{7}}  (0,3,1,0)
F_{8}^{0}  {v_{8}}  (0,3,0,1)
F_{1}^{1}  {v_{1}, v_{2}}  (3,0,1,1)
F_{2}^{1}  {v_{1}, v_{3}}  (3,1,1,0)
F_{3}^{1}  {v_{2}, v_{4}}  (3,1,0,1)
F_{4}^{1}  {v_{3}, v_{4}}  (2,1,1,1)
F_{5}^{1}  {v_{3}, v_{5}}  (2,2,1,0)
hull(I)^{1} F_{6}^{1}  {v_{4}, v_{6}}  (2,2,0,1)
F_{7}^{1}  {v_{5}, v_{6}}  (1,2,1,1)
F_{8}^{1}  {v_{5}, v_{7}}  (1,3,1,0)
F_{9}^{1}  {v_{6}, v_{8}}  (1,3,0,1)
F_{10}^{1}  {v_{7}, v_{8}}  (0,3,1,1)
F_{1}^{2} {v_{1}, v_{2}, v_{3}, v_{4}} (3,1,1,1)
hull(I)^{2} F_{2}^{2} {v_{3}, v_{4}, v_{5}, v_{6}} (2,2,1,1)
F_{3}^{2} {v_{5}, v_{6}, v_{7}, v_{8}} (1,3,1,1)

yields a minimal free resolution 𝔽_{hull(I)} : 0 ⟶ R^{3}(−6) ⟶ R^{10}(−5) ⟶ R^{8}(−4) ⟶ I ⟶ 0.

CELLULAR RESOLUTIONS OF MONOMIAL IDEALS 117

Example 3.4.23. For the monomial ideal

I = (x, y)^{2} ∩ (z, w)^{2}

= (x^{2}z^{2}, x^{2}w^{2}, y^{2}z^{2}, y^{2}w^{2}, x^{2}zw, y^{2}zw, xyz^{2}, xyw^{2}, xyzw) ⊂ R = K[x, y, z, w].

the hull complex is given by

hull(I)
Dimension Cell Degree
hull(I)^{−1} F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (2,0,2,0)
F_{2}^{0}  {v_{2}}  (2,0,1,1)
F_{3}^{0}  {v_{3}}  (2,0,0,2)
F_{4}^{0}  {v_{4}}  (1,1,2,0)
hull(I)^{0} F_{5}^{0}  {v_{5}}  (1,1,1,1)
F_{6}^{0}  {v_{6}}  (1,1,0,2)
F_{7}^{0}  {v_{7}}  (0,2,2,0)
F_{8}^{0}  {v_{8}}  (0,2,1,1)
F_{9}^{0}  {v_{9}}  (0,2,0,2)
F_{1}^{1}  {v_{1}, v_{2}}  (2,0,2,1)
F_{2}^{1}  {v_{1}, v_{4}}  (2,1,2,0)
F_{3}^{1}  {v_{2}, v_{3}}  (2,0,1,2)
F_{4}^{1}  {v_{2}, v_{5}}  (2,1,1,1)
F_{5}^{1}  {v_{3}, v_{6}}  (2,1,0,2)
F_{6}^{1}  {v_{4}, v_{5}}  (1,1,2,1)
hull(I)^{1} F_{7}^{1}  {v_{4}, v_{7}}  (1,2,2,0)
F_{8}^{1}  {v_{5}, v_{6}}  (1,1,1,2)
F_{9}^{1}  {v_{5}, v_{8}}  (1,2,1,1)
F_{10}^{1}  {v_{6}, v_{9}}  (1,2,0,2)
F_{11}^{1}  {v_{7}, v_{8}}  (0,2,2,1)
F_{12}^{1}  {v_{8}, v_{9}}  (0,2,1,2)
F_{1}^{2} {v_{1}, v_{2}, v_{4}, v_{5}} (2,1,2,1)
F_{2}^{2} {v_{2}, v_{3}, v_{5}, v_{6}} (2,1,1,2)
hull(I)^{2} F_{3}^{2} {v_{4}, v_{5}, v_{7}, v_{8}} (1,2,2,1)
F_{4}^{2} {v_{5}, v_{6}, v_{8}, v_{9}} (1,2,1,2)

The hull resolution

𝔽_{hull(I)} : 0 ⟶ R^{4}(−6) ⟶ R^{12}(−5) ⟶ R^{9}(−4) ⟶ I ⟶ 0

is a minimal free resolution of I.

Example 3.4.24. Our last example of this type is the monomial ideal

I = (x, y)^{3} ∩ (z, w)^{2}

= (x^{3}z^{2}, x^{3}zw, x^{3}w^{2}, x^{2}yz^{2}, x^{2}yzw, x^{2}yw^{2},

xy^{2}z^{2}, xy^{2}zw, xy^{2}w^{2}, y^{3}z^{2}, y^{3}zw, y^{3}w^{2})
⊂ R = K[x, y, z, w].

The hull complex of I is

hull(I)
Dimension  Cell  Degree
hull(I)^{−1} F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (3,0,2,0)
F_{2}^{0}  {v_{2}}  (3,0,1,1)
F_{3}^{0}  {v_{3}}  (3,0,0,2)
F_{4}^{0}  {v_{4}}  (2,1,2,0)
F_{5}^{0}  {v_{5}}  (2,1,1,1)
F_{6}^{0}  {v_{6}}  (2,1,0,2)
hull(I)^{0} F_{7}^{0}  {v_{7}}  (1,2,2,0)
F_{8}^{0}  {v_{8}}  (1,2,1,1)
F_{9}^{0}  {v_{9}}  (1,2,0,2)
F_{10}^{0}  {v_{10}}  (0,3,2,0)
F_{11}^{0}  {v_{11}}  (0,3,1,1)
F_{12}^{0}  {v_{12}}  (0,3,0,2)
F_{1}^{1}  {v_{1}, v_{2}}  (3,0,2,1)
F_{2}^{1}  {v_{1}, v_{4}}  (3,1,2,0)
F_{3}^{1}  {v_{2}, v_{3}}  (3,0,1,2)
F_{4}^{1}  {v_{2}, v_{5}}  (3,1,1,1)
F_{5}^{1}  {v_{3}, v_{6}}  (3,1,0,2)
F_{6}^{1}  {v_{4}, v_{5}}  (2,1,2,1)
F_{7}^{1}  {v_{4}, v_{7}}  (2,2,2,0)
F_{8}^{1}  {v_{5}, v_{6}}  (2,1,1,2)
hull(I)^{1} F_{9}^{1}  {v_{5}, v_{8}}  (2,2,1,1)
F_{10}^{1}  {v_{6}, v_{9}}  (2,2,0,2)
F_{11}^{1}  {v_{7}, v_{8}}  (1,2,2,1)
F_{12}^{1}  {v_{7}, v_{10}}  (1,3,2,0)
F_{13}^{1}  {v_{8}, v_{9}}  (1,2,1,2)
F_{14}^{1}  {v_{8}, v_{11}}  (1,3,1,1)
F_{15}^{1}  {v_{9}, v_{12}}  (1,3,0,2)
F_{16}^{1}  {v_{10}, v_{11}}  (0,3,2,1)
F_{17}^{1}  {v_{11}, v_{12}}  (0,3,1,2)
F_{1}^{2} {v_{1}, v_{2}, v_{4}, v_{5}} (3,1,2,1)
F_{2}^{2} {v_{2}, v_{3}, v_{5}, v_{6}} (3,1,1,2)
F_{3}^{2} {v_{4}, v_{5}, v_{7}, v_{8}} (2,2,2,1)
hull(I)^{2} F_{4}^{2} {v_{5}, v_{6}, v_{8}, v_{9}} (2,2,1,2)
F_{5}^{2} {v_{7}, v_{8}, v_{10}, v_{11}} (1,3,2,1)
F_{6}^{2} {v_{8}, v_{9}, v_{11}, v_{12}} (1,3,1,2)

120 STEFAN TAPPE

We get a minimal free resolution

𝔽_{hull(I)} : 0 ⟶ R^{6}(−7) ⟶ R^{17}(−6) ⟶ R^{12}(−5) ⟶ I ⟶ 0.

The preceding examples support the following Conjecture 3.4.25. For every monomial ideal I of the type

I = (x, y)^{a} ∩ (z, w)^{b} ⊂ R = K[x, y, z, w], a, b ∈ ℕ,

the hull complex consists of a × b rectangles, and the hull resolution

𝔽_{hull(I)} : 0 ⟶ R^{ab}(−(a + b + 2)) ⟶ R^{a(b+1)+(a+1)b}(−(a + b + 1))
⟶ R^{(a+1)(b+1)}(−(a + b)) ⟶ I ⟶ 0

is minimal. Example 3.4.26. Consider the ideal

I = (x, y) ∩ (z, u) ∩ (v, w)

= (xzv, xzw, xuv, xuw, yzv, yzw, yuv, yuw)

⊂ R = K[x, y, z, u, v, w]

The hull complex of I is a cube:

hull(I)
Dimension  Cell  Degree
hull(I)^{−1} F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (1,0,1,0,1,0)
F_{2}^{0}  {v_{2}}  (1,0,1,0,0,1)
F_{3}^{0}  {v_{3}}  (1,0,0,1,1,0)
F_{4}^{0}  {v_{4}}  (1,0,0,1,0,1)
hull(I)^{0} F_{5}^{0}  {v_{5}}  (0,1,1,0,1,0)
F_{6}^{0}  {v_{6}}  (0,1,1,0,0,1)
F_{7}^{0}  {v_{7}}  (0,1,0,1,1,0)
F_{8}^{0}  {v_{8}}  (0,1,0,1,0,1)
F_{1}^{1}  {v_{1}, v_{2}}  (1,0,1,0,1,1)
F_{2}^{1}  {v_{1}, v_{3}}  (1,0,1,1,1,0)
F_{3}^{1}  {v_{1}, v_{5}}  (1,1,1,0,1,0)
F_{4}^{1}  {v_{2}, v_{4}}  (1,0,1,1,0,1)
F_{5}^{1}  {v_{2}, v_{6}}  (1,1,1,0,0,1)
F_{6}^{1}  {v_{3}, v_{4}}  (1,0,0,1,1,1)
hull(I)^{1} F_{7}^{1}  {v_{3}, v_{7}}  (1,1,0,1,1,0)
F_{8}^{1}  {v_{4}, v_{8}}  (1,1,0,1,0,1)
F_{9}^{1}  {v_{5}, v_{6}}  (0,1,1,0,1,1)
F_{10}^{1}  {v_{5}, v_{7}}  (0,1,1,1,1,0)
F_{11}^{1}  {v_{6}, v_{8}}  (0,1,1,1,0,1)
F_{12}^{1}  {v_{7}, v_{8}}  (0,1,0,1,1,1)
F_{1}^{2}  {v_{1}, v_{2}, v_{3}, v_{4}}  (1,0,1,1,1,1)
F_{2}^{2}  {v_{1}, v_{2}, v_{5}, v_{6}}  (1,1,1,0,1,1)
F_{3}^{2}  {v_{1}, v_{3}, v_{5}, v_{7}}  (1,1,1,1,1,0)
hull(I)^{2} F_{4}^{2}  {v_{2}, v_{4}, v_{6}, v_{8}}  (1,1,1,1,0,1)
F_{5}^{2}  {v_{3}, v_{4}, v_{7}, v_{8}}  (1,1,0,1,1,1)
F_{6}^{2}  {v_{5}, v_{6}, v_{7}, v_{8}}  (0,1,1,1,1,1)
hull(I)^{3} F_{1}^{3} {v_{1}, v_{2}, v_{3}, v_{4}, v_{5}, v_{6}, v_{7}, v_{8}} (1,1,1,1,1,1)

We obtain a minimal free resolution

𝔽_{hull(I)} : 0 ⟶ R(−6) ⟶ R^{6}(−5) ⟶ R^{12}(−4) ⟶ R^{8}(−3) ⟶ I ⟶ 0.

Example 3.4.27. For the monomial ideal

I = (x, y)^{2} ∩ (z, u) ∩ (v, w)

= (x^{2}zv, x^{2}zw, x^{2}uv, x^{2}uw, xyzv, xyzw,

xyuv, xyuw, y^{2}zv, y^{2}zw, y^{2}uv, y^{2}uw) ⊂ R = K[x, y, z, u, v, w]

the hull complex consists of two cubes.

hull(I)

Dimension Cell Degree
hull(I)^{−1} F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}} (2,0,1,0,1,0)
F_{2}^{0}  {v_{2}} (2,0,1,0,0,1)
F_{3}^{0}  {v_{3}} (2,0,0,1,1,0)
F_{4}^{0}  {v_{4}} (2,0,0,1,0,1)
F_{5}^{0}  {v_{5}} (1,1,1,0,1,0)
F_{6}^{0}  {v_{6}} (1,1,1,0,0,1)
hull(I)^{0} F_{7}^{0}  {v_{7}} (1,1,0,1,1,0)
F_{8}^{0}  {v_{8}} (1,1,0,1,0,1)
F_{9}^{0}  {v_{9}} (0,2,1,0,1,0)
F_{10}^{0} {v_{10}} (0,2,1,0,0,1)
F_{11}^{0} {v_{11}} (0,2,0,1,1,0)
F_{12}^{0} {v_{12}} (0,2,0,1,0,1)
F_{1}^{1} {v_{1}, v_{2}} (2,0,1,0,1,1)
F_{2}^{1} {v_{1}, v_{3}} (2,0,1,1,1,0)
F_{3}^{1} {v_{1}, v_{5}} (2,1,1,0,1,0)
F_{4}^{1} {v_{2}, v_{4}} (2,0,1,1,0,1)
F_{5}^{1} {v_{2}, v_{6}} (2,1,1,0,0,1)
F_{6}^{1} {v_{3}, v_{4}} (2,0,0,1,1,1)
F_{7}^{1} {v_{3}, v_{7}} (2,1,0,1,1,0)
F_{8}^{1} {v_{4}, v_{8}} (2,1,0,1,0,1)
F_{9}^{1} {v_{5}, v_{6}} (1,1,1,0,1,1)
F_{10}^{1} {v_{5}, v_{7}} (1,1,1,1,1,0)
hull(I)^{1} F_{11}^{1} {v_{5}, v_{9}} (1,2,1,0,1,0)
F_{12}^{1} {v_{6}, v_{8}} (1,1,1,1,0,1)
F_{13}^{1} {v_{6}, v_{10}} (1,2,1,0,0,1)
F_{14}^{1} {v_{7}, v_{8}} (1,1,0,1,1,1)
F_{15}^{1} {v_{7}, v_{11}} (1,2,0,1,1,0)
F_{16}^{1} {v_{8}, v_{12}} (1,2,0,1,0,1)
F_{17}^{1} {v_{9}, v_{10}} (0,2,1,0,1,1)
F_{18}^{1} {v_{9}, v_{11}} (0,2,1,1,1,0)
F_{19}^{1} {v_{10}, v_{12}} (0,2,1,1,0,1)
F_{20}^{1} {v_{11}, v_{12}} (0,2,0,1,1,1)

CELLULAR RESOLUTIONS OF MONOMIAL IDEALS 123

hull(I)

Dimension  Cell  Degree
F_{1}^{2}  {v_{1}, v_{2}, v_{3}, v_{4}}  (2,0,1,1,1,1)
F_{2}^{2}  {v_{1}, v_{2}, v_{5}, v_{6}}  (2,1,1,0,1,1)
F_{3}^{2}  {v_{1}, v_{3}, v_{5}, v_{7}}  (2,1,1,1,1,0)
F_{4}^{2}  {v_{2}, v_{4}, v_{6}, v_{8}}  (2,1,1,1,0,1)
F_{5}^{2}  {v_{3}, v_{4}, v_{7}, v_{8}}  (2,1,0,1,1,1)
hull(I)^{2} F_{6}^{2}  {v_{5}, v_{6}, v_{7}, v_{8}}  (1,1,1,1,1,1)
F_{7}^{2}  {v_{5}, v_{6}, v_{9}, v_{10}}  (1,2,1,0,1,1)
F_{8}^{2}  {v_{5}, v_{7}, v_{9}, v_{11}}  (1,2,1,1,1,0)
F_{9}^{2}  {v_{6}, v_{8}, v_{10}, v_{12}}  (1,2,1,1,0,1)
F_{10}^{2}  {v_{7}, v_{8}, v_{11}, v_{12}}  (1,2,0,1,1,1)
F_{11}^{2}  {v_{9}, v_{10}, v_{11}, v_{12}}  (0,2,1,1,1,1)
F_{1}^{3} {v_{1}, v_{2}, v_{3}, v_{4}, v_{5}, v_{6}, v_{7}, v_{8}} (2,1,1,1,1,1)
hull(I)^{3} F_{2}^{3} {v_{5}, v_{6}, v_{7}, v_{8}, v_{9}, v_{10}, v_{11}, v_{12}} (1,2,1,1,1,1)

The hull resolution

𝔽_{hull(I)} 0 ⟶ R^{2}(−7) ⟶ R^{11}(−6) ⟶ R^{20}(−5) ⟶ R^{12}(−4) ⟶ I ⟶ 0 is a minimal free resolution of I. Conjecture 3.4.28. For every monomial ideal I of the type

I = (x, y)^{a} ∩ (z, u)^{b} ∩ (v, w)^{c} ⊂ R = K[x, y, z, u, v, w], a, b, c ∈ ℕ,

the hull complex consists of a × b × c cubes, and the hull resolution

𝔽_{hull(I)} : 0 ⟶ R^{abc}(−(a + b + c + 3)) ⟶ R^{ab(c+1)+a(b+1)c+(a+1)bc}(−(a + b + c + 2))

⟶ R^{a(b+1)(c+1)+(a+1)b(c+1)+(a+1)(b+1)c}(−(a + b + c + 1)) ⟶ R^{(a+1)(b+1)(c+1)}(−(a + b + c)) ⟶ I ⟶ 0

is minimal. As a generalization of Conjecture 3.4.25 and Conjecture 3.4.28, we formulate Conjecture 3.4.29. For every monomial ideal I of the form

I = ⋂_{k=1}^{n}(x_{2k−1}, x_{2k})^{a_{k}} ⊂ K[x_{1}, …, x_{2n}], n, a_{1}, …, a_{n} ∈ ℕ,

124 STEFAN TAPPE

the hull resolution

𝔽_{hull(I)} : 0 ⟶ R^{p_{n}}(−(d + n)) ⟶ … ⟶ R^{p_{0}}(−d) ⟶ I ⟶ 0,

d = ∑_{k=1}^{n} a_{k},

p_{i} = ∑_{{J⊂\{1,...,n\} \atop |J|=i}} ∏_{k∈J} a_{k} ∏_{k∉J}(a_{k} + 1), i = 0, …, n,

is a minimal free resolution of I. The correctness of Conjecture 3.4.29 is also supported by Example 3.5.14, where we prove it for the case n = 1. Example 3.4.30. Consider

I = (x^{π(1)}y^{π(2)}z^{π(3)} : π ∈ S_{3})

= (xy^{2}z^{3}, x^{2}yz^{3}, x^{3}yz^{2}, x^{3}y^{2}z, x^{2}y^{3}z, xy^{3}z^{2}) ⊂ R = K[x, y, z],

derived from all permutations of {1, 2, 3}. The hull complex of I is a convex 6-gon.

X

Dimension  Cell  Degree
X^{−1}  F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (1,2,3)
F_{2}^{0}  {v_{2}}  (2,1,3)
F_{3}^{0}  {v_{3}}  (3,1,2)
X^{0}  F_{4}^{0}  {v_{4}}  (3,2,1)
F_{5}^{0}  {v_{5}}  (2,3,1)
F_{6}^{0}  {v_{6}}  (1,3,2)
F_{1}^{1}  {v_{1}, v_{2}}  (2,2,3)
F_{2}^{1}  {v_{1}, v_{6}}  (1,3,3)
F_{3}^{1}  {v_{2}, v_{3}}  (3,1,3)
X^{1}  F_{4}^{1}  {v_{3}, v_{4}}  (3,2,2)
F_{5}^{1}  {v_{4}, v_{5}}  (3,3,1)
F_{6}^{1}  {v_{5}, v_{6}}  (2,3,2)
X^{2}  F_{1}^{2} {v_{1}, v_{2}, v_{3}, v_{4}, v_{5}, v_{6}} (3,3,3)

The hull resolution

𝔽_{hull(I)} : 0 ⟶ R(−9) ⟶ R^{6}(−7) ⟶ R^{6}(−6) ⟶ I ⟶ 0

is a minimal free resolution of I.

Remark 3.4.31. Note that for each of the preceding Examples 3.4.20 - 3.4.30 the hull complex supports a minimal free resolution, whereas the Scarf complex consists of the vertices and the edges contained in the hull complex:

Δ_{I} = hull(I)^{−1} ∪ hull(I)^{0} ∪ hull(I)^{1}.

In the following two examples, the hull resolution is not minimal.

Example 3.4.32. The monomial ideal

I = (xz, xv, yv, yw, zw) ⊂ R = K[x, y, z, v, w]

is a Gorenstein ideal of height 3 generated by 5 monomials. The following cell complex X, which is a convex 5-gon,

X
Dimension  Cell  Degree
X^{−1}  F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (1,0,1,0,0)
F_{2}^{0}  {v_{2}}  (1,0,0,1,0)
X^{0}  F_{3}^{0}  {v_{3}}  (0,1,0,1,0)
F_{4}^{0}  {v_{4}}  (0,1,0,0,1)
F_{5}^{0}  {v_{5}}  (0,0,1,0,1)
F_{1}^{1}  {v_{1}, v_{2}}  (1,0,1,1,0)
F_{2}^{1}  {v_{1}, v_{5}}  (1,0,1,0,1)
X^{1}  F_{3}^{1}  {v_{2}, v_{3}}  (1,1,0,1,0)
F_{4}^{1}  {v_{3}, v_{4}}  (0,1,0,1,1)
F_{5}^{1}  {v_{4}, v_{5}}  (0,1,1,0,1)
X^{2}  F_{1}^{2} {v_{1}, v_{2}, v_{3}, v_{4}, v_{5}} (1,1,1,1,1)

yields a minimal free resolution

𝔽_{X} : 0 ⟶ R(−5) ⟶ R^{5}(−3) ⟶ R^{5}(−2) ⟶ I ⟶ 0.

126 STEFAN TAPPE

The reader may make sure that

Δ_{I} = X^{−1} ∪ X^{0} ∪ X^{1},

and that X is contained in the hull complex, that is X ⫋ hull(I).

Example 3.4.33. Now we consider a Gorenstein ideal of height 3 generated by 7 monomials:

I = (xyz, xyw, xvw, uvw, tuv, ztu, yzt) ⊂ R = K[x, y, z, t, u, v, w]

From the following cell complex X, which is a convex 7-gon,

X

Dimension  Cell  Degree
X^{−1}  F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (1,1,1,0,0,0,0)
F_{2}^{0}  {v_{2}}  (1,1,0,0,0,0,1)
F_{3}^{0}  {v_{3}}  (1,0,0,0,0,1,1)
X^{0}  F_{4}^{0}  {v_{4}}  (0,0,0,0,1,1,1)
F_{5}^{0}  {v_{5}}  (0,0,0,1,1,1,0)
F_{6}^{0}  {v_{6}}  (0,0,1,1,1,0,0)
F_{7}^{0}  {v_{7}}  (0,1,1,1,0,0,0)
F_{1}^{1}  {v_{1}, v_{2}}  (1,1,1,0,0,0,1)
F_{2}^{1}  {v_{1}, v_{7}}  (1,1,1,1,0,0,0)
F_{3}^{1}  {v_{2}, v_{3}}  (1,1,0,0,0,1,1)
X^{1}  F_{4}^{1}  {v_{3}, v_{4}}  (1,0,0,0,1,1,1)
F_{5}^{1}  {v_{4}, v_{5}}  (0,0,0,1,1,1,1)
F_{6}^{1}  {v_{5}, v_{6}}  (0,0,1,1,1,1,0)
F_{7}^{1}  {v_{6}, v_{7}}  (0,1,1,1,1,0,0)
X^{2}  F_{1}^{2} {v_{1}, v_{2}, v_{3}, v_{4}, v_{5}, v_{6}, v_{7}} (1,1,1,1,1,1,1)

we obtain a minimal free resolution

𝔽_{X} : 0 ⟶ R(−7) ⟶ R^{7}(−4) ⟶ R^{7}(−3) ⟶ I ⟶ 0.

Again, the reader may make sure that Δ_{I} = X^{−1} ∪ X^{0} ∪ X^{1}, and that X is contained in the hull complex:

X ⫋ hull(I).

Remark 3.4.34. Let I be any Gorenstein ideal of height 3 generated by s monomials. It is shown in [4], §6 that the minimal free resolution of I is the cellular resolution

𝔽_{X} : 0 ⟶ R ⟶ R^{s} ⟶ R^{s} ⟶ R ⟶ 0

supported on a convex s-gon.

3.5. Generic ideals.

In the last section we have observed that the cellular complex derived from the Scarf complex does not need to be a resolution. Now we consider a class of monomial ideals, the generic ideals, for which the Scarf complex Δ_{I} coincides with the hull complex and yields a minimal free resolution 𝔽_{Δ_{I}} = 𝔽_{hull(I)}.

Definition 3.5.1. Let R = K[x_{1}, …, x_{n}]. For a monomial x^{a} ∈ R we call

supp(x^{a}) = {i ∈ {1, …, n} : a_{i} ≠ 0}

the support of x^{a}. We say a moniomial x^{b} ∈ R strictly divides x^{a} ∈ R if x^{b} divides x^{a} and

supp(x^{a}/x^{b}) = supp(x^{a}).

Definition 3.5.2. A monomial ideal

I = (x^{a^{1}}, …, x^{a^{s}}) ⊂ R = K[x_{1}, …, x_{n}]

is called generic if the following condition holds: if two distinct minimal generators x^{a^{i}} and x^{a^{j}} have the same positive degree in some variable, there is a third generator x^{a^{l}} which strictly divides the least common multiple of x^{a^{i}} and x^{a^{j}}.

Remark 3.5.3. This definition of genericity is Definition 1.1 in [10]. It is more inclusive than the one given in [8]. The new definition of generic is justified by the completeness of the result in [8], Theorem 2.9.

Example 3.5.4. Consider the following monomial ideals in the polynomial ring R = K[x, y, z]:

I_{1} = (x^{2}z^{3}, x^{3}z^{2}, xyz, y^{2}),
I_{2} = (x^{2}y, xz, yz^{2}, y^{2}),
I_{3} = (x^{2}, y^{2}, z^{2}, xy, xz, yz).

The ideal I_{1} is generic because no two distinct minimal generators have the same positive degree in some variable. The minimal generators x^{2}y and yz^{2} of I_{2} both have degree 1 in y. The minimal generator xz strictly divides the least common multiple x^{2}yz^{2}. Hence, I_{2} is also generic. I_{3} is a non-generic ideal.

The following lemma is a little technical, but necessary to prove the next Theorem 3.5.7.

Definition 3.5.5. Let

I = (x^{a^{1}}, …, x^{a^{s}}) ⊂ R = K[x_{1}, …, x_{n}]

be a monomial ideal. Choose q ≤ s and set

G = {v_{1}, …, v_{q}}.

Without loss of generality, we may assume that

deg G_{i} > 0, i = 1, …, m,
deg G_{i} = 0, i = m + 1, …, n,

for an 1 ≤ m ≤ n. For b ∈ ℤ^{n} let b^{∗} ∈ ℤ^{m} be defined by

b_{i}^{∗} = b_{i}, i = 1, …, m.

I^{∗} denotes the monomial ideal

I^{∗} = (x^{a^{∗}^{1}}, …, x^{a^{∗}^{q}}) ⊂ K[x_{1}, …, x_{m}].

For any F ⊂ G we have the degree

deg^{∗} F = (deg F_{1}, …, deg F_{m}) ∈ ℤ^{m}.

Lemma 3.5.6. With the notation of the last Definition 3.5.5, let F ⊂ G be a cell of the hull complex hull(I). Then F is also contained in the hull complex hull(I^{∗}).

Proof. It is obvious that {x^{a^{∗}^{1}}, …, x^{a^{∗}^{q}}} is the set of minimal generators of I^{∗}. Without loss of generality, we may assume that F = {v_{1}, …, v_{p}}, p < q. Choose t > (n + 1)!. According to Proposition 3.3.6 there is a vector v ∈ ℝ^{n}, v ▷ 0, such that

⟨v, t^{a^{j}}⟩ = 1, j = 1, …, p,
⟨v, t^{a^{j}}⟩ > 1, j = p + 1, …, s.

For each j = 1, …, s the inner product can be written as

⟨v, t^{a^{j}}⟩ = ∑_{k=1}^{n} v_{k} t^{a_{j}^{k}}

= ∑_{k=1}^{m} v_{k} t^{a_{j}^{k}} + ∑_{k=m+1}^{n} v_{k} t^{0}

= ∑_{k=1}^{m} v_{k} t^{a_{j}^{k}} + ∑_{k=m+1}^{n} v_{k},

and we obtain

1 − ∑_{k=m+1}^{n} v_{k} > 0.

Now define w ∈ ℝ^{m} by

w_{i} = (1 − ∑_{k=m+1}^{n} v_{k})^{−1} v_{i}, i = 1, …, m.

Then we have w ▷ 0 and

⟨w, t^{a^{∗}^{j}}⟩ = 1, j = 1, …, p,
⟨w, t^{a^{∗}^{j}}⟩ > 1, j = p + 1, …, q.

The conditions of Proposition 3.3.6 are satisfied, therefore F ∈ hull(I^{∗}). □

Theorem 3.5.7. If I is a generic monomial ideal, then hull(I) coincides with the Scarf complex Δ_{I} of I, and the hull resolution

𝔽_{hull(I)} = 𝔽_{Δ_{I}}

is minimal.

Proof. Let F = {v_{1}, …, v_{p}}, p ≥ 2 be any cell of hull(I). Suppose that F is not a cell of Δ_{I}. Then there is a set G ⊂ V, G ≠ ∅, G ≠ F with deg G = deg F. We discuss the two cases

(i) G ⊂ F (ii) G ⊄ F

Consider first case (i). We have

deg F = deg {v_{1}, …, v̂_{i}, …, v_{p}}

for some i ∈ {1, …, p}. Following the terminology of Definition 3.5.5 we have, without loss of generality

deg F_{j} > 0, j = 1, …, m,
deg F_{j} = 0, j = m + 1, …, n,

for an 1 ≤ m ≤ n. Lemma 3.5.6 yields that F ∈ hull(I^{∗}). By Proposition 3.4.15 there is an index j ∈ {1, …, m} such that

deg^{∗} F_{j} ≤ a_{i}_{j}^{∗},

hence

a_{i}_{j} = a_{k}_{j}

for some k ∈ {1, …, p}, k ≠ i. deg^{∗} F ▷ 0 implies that a_{i}_{j} > 0. Hence, there is a third generator x^{a^{l}} which strictly divides the least common multiple of x^{a^{i}} and x^{a^{k}}, which implies

deg^{∗} {v_{l}} ◁ deg^{∗} F,

but Lemma 3.5.6 and Proposition 3.4.15 state that there is an index j ∈ {1, …, m} such that

deg^{∗} F_{j} ≤ a_{l}_{j}^{∗},

and we get a contradiction.

Consider now case (ii). There is an index q > p such that

deg F = deg {v_{1}, …, v_{p}, v_{q}}.

We may assume that q = p + 1. Again, we use the terminology of Definition 3.5.5, and get, without loss of generality

deg {v_{1}, …, v_{p}, v_{q}}_{j} > 0, j = 1, …, m,
deg {v_{1}, …, v_{p}, v_{q}}_{j} = 0, j = m + 1, …, n,

for an 1 ≤ m ≤ n. We claim

a_{q}^{∗} ◁ deg^{∗} F.

Assume

a_{q}_{j}^{∗} = deg^{∗} F_{j}

for some j ∈ {1, …, m}. Then

a_{q}_{j} = a_{i}_{j}

for some i ∈ {1, …, p}. deg^{∗} F ▷ 0 implies that a_{i}_{j} > 0. Consequently, we have a third generator x^{a^{l}} which strictly divides the least common multiple of x^{a^{q}} and x^{a^{i}}, which implies

deg^{∗} {v_{l}} ◁ deg^{∗} F.

But according to Lemma 3.5.6 and Proposition 3.4.15 there is an index j ∈ {1, …, m} such that

deg^{∗} F_{j} ≤ a_{l}_{j}^{∗},

which is a contradiction. So we have shown that

a_{q}^{∗} ◁ deg^{∗} F,

but Lemma 3.5.6 and Proposition 3.4.15 yield that there is an index j ∈ {1, …, m} satisfying

deg^{∗} F_{j} ≤ a_{q}_{j}^{∗}.

Again, we have a contradiction.

Hence, both (i) and (ii) lead to a contradiction, and we conclude that every cell of hull(I) is a cell of Δ_{I}. This implies

hull(I) = Δ_{I}

by Theorem 3.4.11. The resolution 𝔽_{Δ_{I}} is minimal because no two distinct cells in Δ_{I} have the same degree, therefore the condition of Proposition 3.1.14 is fulfilled. □

Example 3.5.8. Consider the generic ideal

I = (x^{2}z^{3}, x^{3}z^{2}, xyz, y^{2}) ⊂ R = K[x, y, z].

The Scarf complex Δ_{I} is a triangle connected to an edge:

CELLULAR RESOLUTIONS OF MONOMIAL IDEALS 131

Δ_{I}

Component  Cell  Degree
Δ_{I}^{−1}  F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (2,0,3)
F_{2}^{0}  {v_{2}}  (3,0,2)
Δ_{I}^{0}  F_{3}^{0}  {v_{3}}  (1,1,1)
F_{4}^{0}  {v_{4}}  (0,2,0)
F_{1}^{1} {v_{1}, v_{2}} (3,0,3)
F_{2}^{1} {v_{1}, v_{3}} (2,1,3)
Δ_{I}^{1}  F_{3}^{1} {v_{2}, v_{3}} (3,1,2)
F_{4}^{1} {v_{3}, v_{4}} (1,2,1)
Δ_{I}^{2}  F_{1}^{2} {v_{1}, v_{2}, v_{3}} (3,1,3)

The cellular complex

𝔽_{Δ_{I}} : 0 ⟶ R(−7) ⟶^{( \begin{matrix}y \\ −x \\ z \\ 0 \end{matrix} )} \begin{matrix}R^{3}(−6) \\ ⊕ \\ R(−4) \end{matrix} ⟶^{( \begin{matrix}−x & −y & 0 & 0 \\ z & 0 & −y & 0 \\ 0 & xz^{2} & x^{2}z & −y \\ 0 & 0 & 0 & xz \end{matrix} )} \begin{matrix}R^{2}(−5) \\ ⊕ \\ R(−3) \\ ⊕ \\ R(−2) \end{matrix} \underrightarrow{( x^{2}z^{3} x^{3}z^{2} xyz y^{2} )} I ⟶ 0

is a minimal free resolution of I by Theorem 3.5.7. Example 3.5.9. Let

I = (xyz, x^{4}y^{3}, x^{3}y^{5}, y^{4}z^{3}, y^{2}z^{4}, x^{2}z^{2}) ⊂ R = K[x, y, z].

The Scarf complex of this generic ideal consists of two triangles and an edge meeting at a vertex.

132 STEFAN TAPPE

Δ_{I}

Component  Cell  Degree
Δ_{I}^{−1}  F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (1,1,1)
F_{2}^{0}  {v_{2}}  (4,3,0)
F_{3}^{0}  {v_{3}}  (3,5,0)
Δ_{I}^{0}  F_{4}^{0}  {v_{4}}  (0,4,3)
F_{5}^{0}  {v_{5}}  (0,2,4)
F_{6}^{0}  {v_{6}}  (2,0,2)
F_{1}^{1} {v_{1}, v_{2}} (4,3,1)
F_{2}^{1} {v_{1}, v_{3}} (3,5,1)
F_{3}^{1} {v_{1}, v_{4}} (1,4,3)
Δ_{I}^{1}  F_{4}^{1} {v_{1}, v_{5}} (1,2,4)
F_{5}^{1} {v_{1}, v_{6}} (2,1,2)
F_{6}^{1} {v_{2}, v_{3}} (4,5,0)
F_{7}^{1} {v_{4}, v_{5}} (0,4,4)
F_{1}^{2} {v_{1}, v_{2}, v_{3}} (4,5,1)
Δ_{I}^{2}  F_{2}^{2} {v_{1}, v_{4}, v_{5}} (1,4,4)

By Theorem 3.5.7

𝔽_{Δ_{I}} : 0 ⟶ R^{2} ⟶ R^{7} ⟶ R^{6} ⟶ I ⟶ 0

is a minimal free resolution of I. Example 3.5.10. Consider the ideal

I = (x^{4}, y^{4}, z^{4}, xy^{2}z^{3}, x^{3}yz^{2}, x^{2}y^{3}z) ⊂ R = K[x, y, z].

This ideal is generic. The Scarf complex Δ_{I} of I is

Δ_{I}
Component  Cell  Degree
Δ_{I}^{−1}  F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (4,0,0)
F_{2}^{0}  {v_{2}}  (0,4,0)
F_{3}^{0}  {v_{3}}  (0,0,4)
Δ_{I}^{0}  F_{4}^{0}  {v_{4}}  (1,2,3)
F_{5}^{0}  {v_{5}}  (3,1,2)
F_{6}^{0}  {v_{6}}  (2,3,1)
F_{1}^{1} {v_{1}, v_{2}} (4,4,0)
F_{2}^{1} {v_{1}, v_{3}} (4,0,4)
F_{3}^{1} {v_{1}, v_{5}} (4,1,2)
F_{4}^{1} {v_{1}, v_{6}} (4,3,1)
F_{5}^{1} {v_{2}, v_{3}} (0,4,4)
F_{6}^{1} {v_{2}, v_{4}} (1,4,3)
Δ_{I}^{1}  F_{7}^{1} {v_{2}, v_{6}} (2,4,1)
F_{8}^{1} {v_{3}, v_{4}} (1,2,4)
F_{9}^{1} {v_{3}, v_{5}} (3,1,4)
F_{10}^{1} {v_{4}, v_{5}} (3,2,3)
F_{11}^{1} {v_{4}, v_{6}} (2,3,3)
F_{12}^{1} {v_{5}, v_{6}} (3,3,2)
F_{1}^{2} {v_{1}, v_{2}, v_{6}} (4,4,1)
F_{2}^{2} {v_{1}, v_{3}, v_{5}} (4,1,4)
F_{3}^{2} {v_{1}, v_{5}, v_{6}} (4,3,2)
Δ_{I}^{2}  F_{4}^{2} {v_{2}, v_{3}, v_{4}} (1,4,4)
F_{5}^{2} {v_{2}, v_{4}, v_{6}} (2,4,3)
F_{6}^{2} {v_{3}, v_{4}, v_{5}} (3,2,4)
F_{7}^{2} {v_{4}, v_{5}, v_{6}} (3,3,3)

We obtain a cellular complex

𝔽_{Δ_{I}} : 0 ⟶ R^{7}(−9) ⟶ \begin{matrix}R^{3}(−7) \\ ⊕ \\ R^{9}(−8) \end{matrix} ⟶ \begin{matrix}R^{3}(−4) \\ ⊕ \\ R^{3}(−6) \end{matrix} ⟶ I ⟶ 0,

which is a minimal free resolution according to Theorem 3.5.7.

Example 3.5.11. The monomial ideal

I = (x^{2}y, xz, yz^{2}, y^{2}) ⊂ R = K[x, y, z]

is generic. We have determined the Scarf complex Δ_{I} in Example 3.4.4. The resulting minimal free resolution

𝔽_{Δ_{I}} : 0 ⟶ R^{2}(−5) ⟶ R^{5}(−4) ⟶ \begin{matrix}R^{2}(−2) \\ ⊕ \\ R^{2}(−3) \end{matrix} ⟶ I ⟶ 0

coincides with the resolution 𝔽_{X} of Example 3.1.21.

Corollary 3.5.12. For any monomial ideal

I ⊂ R = K[x, y]

the cellular complex 𝔽_{Δ_{I}} derived from the Scarf complex is a minimal free resolution of the ideal I.

Proof. Let I = (x^{a_{1}}y^{b_{1}}, …, x^{a_{s}}y^{b_{s}}) with {x^{a_{1}}y^{b_{1}}, …, x^{a_{s}}y^{b_{s}}} being the set of minimal generators. We claim that I is generic. This is obvious for s = 1. Suppose that for s ≥ 2 there are i ≠ j such that a_{i} = a_{j}. Then it follows immediately that one of the generators x^{a_{i}}y^{b_{i}}, x^{a_{j}}y^{b_{j}} is not minimal, and we have a contradiction. Therefore, I is generic because no two distinct minimal generators have the same positive degree in some variable. The proof is completed by Theorem 3.5.7. □

Example 3.5.13. Consider

I = (x^{4}, xy, y^{3}) ⊂ R = K[x, y].

The Scarf complex Δ_{I} consists of two lines:

Δ_{I}
Component  Cell  Degree
Δ_{I}^{−1}  F_{1}^{−1}  ∅
F_{1}^{0} {v_{1}} (4,0)
Δ_{I}^{0}  F_{2}^{0} {v_{2}} (1,1)
F_{3}^{0} {v_{3}} (0,3)
F_{1}^{1} {v_{1}, v_{2}} (4,1)
Δ_{I}^{1}  F_{2}^{1} {v_{2}, v_{3}} (1,3)

By Corollary 3.5.12 𝔽_{Δ_{I}} is a minimal resolution of the ideal I. The reader is invited to verify that 𝔽_{Δ_{I}} coincides with the minimal free resolution presented in Example 2.4.13.

Example 3.5.14. For an arbitrary n ∈ ℕ consider

I = (x, y)^{n} = (x^{n}, x^{n−1}y, …, xy^{n−1}, y^{n})
= (x^{n−i+1}y^{i−1} : i = 1, …, n + 1) ⊂ R = K[x, y]

and the simplicial complex

X = ⟨{v_{i}, v_{i+1}} : i = 1, …, n⟩
= {∅} ∪ {{v_{i}} : i = 1, …, n + 1} ∪ {{v_{i}, v_{i+1}} : i = 1, …, n}.

We claim that X is the Scarf complex of I. First of all, note that for F = {v_{i_{1}}, …, v_{i_{m}}}, i_{1} < … < i_{m} the following is true:

deg F = (n − i_{1} + 1, i_{m} − 1),
| deg F | = n + i_{m} − i_{1}.

In particular, we have

deg {v_{i}} = (n − i + 1, i − 1),  | deg {v_{i}}| = n,
deg {v_{i}, v_{i+1}} = (n − i + 1, i),  | deg {v_{i}, v_{i+1}}| = n + 1.

Let F = {v_{i}} for an index i ∈ {1, …, n + 1} and G ⊂ V such that G ≠ ∅, G ≠ F. We consider the two cases

(i) G = {v_{j}} for some j ≠ i. Then, according to our degree formula above, we have

deg F ≠ deg G.

(ii) |G| ≥ 2. Then there are j < k such that {v_{j}, v_{k}} ⊂ G. We obtain

| deg F | = n,
| deg G| ≥ | deg {v_{j}, v_{k}}| = n + k − j > n,

therefore

deg F ≠ deg G.

So we have F ∈ Δ_{I}. Now, let F = {v_{i}, v_{i+1}} for an i ∈ {1, …, n} and G ⊂ V such that G ≠ ∅, G ≠ F. We may assume that |G| ≥ 2 since we have already treated the case |G| = 1 above. Again, we consider two cases (i) G = {v_{j}, v_{j+1}} for some j ≠ i. Then, it is evident that

deg F ≠ deg G.

(ii) Otherwise, there are j < k such that k − j > 1 and {v_{j}, v_{k}} ⊂ G, and we get

| deg F | = n + 1,
| deg G| ≥ | deg {v_{j}, v_{k}}| = n + k − j > n + 1.

Hence, we get

deg F ≠ deg G.

We conclude that F ∈ Δ_{I}. For the converse, choose F ⊂ V, F ∉ X. This implies |F | ≥ 2.

(i) |F | = 2. In this case there are i < k such that k − i > 1 and F = {v_{i}, v_{k}}. Choose an index j satisfying i < j < k and set G = {v_{i}, v_{j}, v_{k}}. These two cells have the same degree:

deg F = deg G = (n − i + 1, k − 1).

(ii) |F | > 2. Then F = {v_{i_{1}}, …, v_{i_{m}}}, i_{1} < … < i_{m}, m ≥ 3. For G = {v_{i_{1}}, v_{i_{m}}} we have G ≠ F, but

deg F = deg G = (n − i_{1} + 1, i_{m} − 1).

This implies that F ∉ Δ_{I}. Now, we have proven that

Δ_{I} = {∅} ∪ {{v_{i}} : i = 1, …, n + 1} ∪ {{v_{i}, v_{i+1}} : i = 1, …, n}.

The following figure shows a geometric realization of Δ_{I} for n = 9:

We determine an incidence function ε according to Lemma 2.2.12(i):

ε({v_{j}}, ∅) = 1, j = 1, …, n + 1 ,
ε({v_{j}, v_{j+1}}, {v_{j}}) = −1, j = 1, …, n ,
ε({v_{j}, v_{j+1}}, {v_{j+1}}) = 1, j = 1, …, n .

Let {f_{1}, …, f_{n}} and {g_{1}, …, g_{n+1}} denote the canonical bases of R^{n} and R^{n+1}, respectively. For j ∈ {1, …, n} we obtain

∂_{1}(f_{j}) = ∑_{k=1}^{n+1} ε({v_{j}, v_{j+1}}, {v_{k}}) x^{deg {v_{j},v_{j+1}}−deg {v_{k}}} g_{k}
= −x^{deg {v_{j},v_{j+1}}−deg {v_{j}}} g_{j} + x^{deg {v_{j},v_{j+1}}−deg {v_{j+1}}} g_{j+1}
= −x^{(n−j+1,j)−(n−j+1,j−1)} g_{j} + x^{(n−j+1,j)−(n−j,j)} g_{j+1}
= −y g_{j} + x g_{j+1}.

The cellular complex is

𝔽_{Δ_{I}} : 0 ⟶ R^{n} \underrightarrow{( \begin{matrix}−y & 0 & 0 & ⋯ & 0 \\ x & −y & 0 & ⋯ & 0 \\ 0 & x & −y & ⋯ & 0 \\ 0 & 0 & x & ... & ⋮ \\ ⋮ & ⋮ & ⋮ & ... & −y \\ 0 & 0 & 0 & ⋯ & x \end{matrix} )} R^{n+1} \underrightarrow{( _{x^{n}} _{x^{n−1}y} ⋯ _{xy^{n−1}} _{y^{n}} )} I ⟶ 0,

as complex of ℤ-graded modules

𝔽_{Δ_{I}} : 0 ⟶ R^{n}(−(n + 1)) ⟶ R^{n+1}(−n) ⟶ I ⟶ 0,

and as complex of ℤ^{n}-graded modules

𝔽_{Δ_{I}} : 0 ⟶ ⨁_{i=1}^{n} R(i − n − 1, −i) ⟶ ⨁_{i=1}^{n+1} R(i − n − 1, 1 − i) ⟶ I ⟶ 0.

𝔽_{Δ_{I}} is a minimal free resolution by Corollary 3.5.12.

This result generalizes Example 3.3.14, where we have treated the cases n = 1, 2. It gives also evidence for the correctness of Conjecture 3.4.29 since it proves the case n = 1 of this conjecture.

For a special class of ideals, the three considered types of cell complexes coincide and yield a minimal free resolution:

Corollary 3.5.15. If I is a complete intersection, then hull(I), the Scarf complex Δ_{I}, and the Taylor complex Δ coincide, and the cellular complex

𝔽_{hull(I)} = 𝔽_{Δ_{I}} = 𝔽_{Δ}

is a minimal free resolution of I.

Proof. The complete intersection

I = (f_{1}, …, f_{s}) = (x^{a^{1}}, …, x^{a^{s}})

is characterized by

gcd {f_{i}, f_{j}} = 1, i ≠ j.

Therefore, I is generic because no two distinct minimal generators have the same positive degree in some variable. Moreover, the degree of a cell F = {v_{j_{1}}, …, v_{j_{m}}}, j_{1} < … < j_{m}, can be computed as

deg F = ∑_{k=1}^{m} a_{j_{k}}.

We show that Δ_{I} = Δ. Let F = {v_{1}, …, v_{p}} be any cell of Δ. Suppose that F is not a cell of Δ_{I}. Then there is a set G ⊂ V, G ≠ ∅, G ≠ F with deg G = deg F. We discuss the two cases

(i) G ⊂ F (ii) G ⊄ F

Consider first case (i). We have

deg F = deg {v_{1}, …, v̂_{i}, …, v_{p}}

for some i ∈ {1, …, p}, which is a contradiction according to our degree formula above.

Consider now case (ii). There is an index q > p such that

deg F = deg {v_{1}, …, v_{p}, v_{q}}.

Again, we have a contradiction.

We conclude that every cell of Δ is a cell of Δ_{I}. Now apply Theorem 3.5.7. □

3.6. Reducing cellular resolutions.

In this section we discuss some strategies how to transform a cellular resolution 𝔽_{X} of a monomial ideal, which is not minimal, for example the Taylor resolution 𝔽_{Δ} or the hull resolution 𝔽_{hull(I)}, into a minimal cellular resolution. We are guided by Proposition 3.1.14 which states that a cellular resolution is not minimal as long as there is a pair of cells F, F′ ∈ X such that F′ is a face of F and deg F = deg F′, thus suggesting to delete such a pair. This is reasonable if F ∈ X^{i} and F′ ∈ X^{i−1} since the alternating sum of the cardinality of the components remains equal to zero (see Corollary 3.1.16). It is desirable to maintain the structure of a cell complex after such a step, and that the cellular complex derived from this new cell complex remains a resolution. In fact, it is not too bad if the cell complex structure gets lost in the meantime. The crucial is point that we obtain a cell complex Y after having deleted all pairs of the same degree, and the resulting cellular complex 𝔽_{Y} should be a resolution of I. If this is the case, 𝔽_{Y} is a minimal free resolution of the monomial ideal I. In the following, the procedure starts with cells of maximal dimension. Let us demonstrate this strategy with an example:

Example 3.6.1. We discuss the monomial ideal

I = (x^{2}z^{3}, x^{3}z^{2}, xyz, y^{2}) ⊂ R = K[x, y, z],

considered in Example 3.5.8. The Taylor complex Δ of I is

Δ = ⟨{v_{1}, v_{2}, v_{3}, v_{4}}⟩

Component  Cell  Degree Step
Δ^{−1}  F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (2,0,3)
F_{2}^{0}  {v_{2}}  (3,0,2)
Δ^{0}  F_{3}^{0}  {v_{3}}  (1,1,1)
F_{4}^{0}  {v_{4}}  (0,2,0)
F_{1}^{1}  {v_{1}, v_{2}}  (3,0,3)
F_{2}^{1}  {v_{1}, v_{3}}  (2,1,3)
F_{3}^{1}  {v_{1}, v_{4}}  (2,2,3) 3
Δ^{1}  F_{4}^{1}  {v_{2}, v_{3}}  (3,1,2)
F_{5}^{1}  {v_{2}, v_{4}}  (3,2,2) 2
F_{6}^{1}  {v_{3}, v_{4}}  (1,2,1)
F_{1}^{2} {v_{1}, v_{2}, v_{3}} (3,1,3)
F_{2}^{2} {v_{1}, v_{2}, v_{4}} (3,2,3) 1
Δ^{2}  F_{3}^{2} {v_{1}, v_{3}, v_{4}} (2,2,3) 3
F_{4}^{2} {v_{2}, v_{3}, v_{4}} (3,2,2) 2
Δ^{3}  F_{1}^{3} {v_{1}, v_{2}, v_{3}, v_{4}} (3,2,3) 1

The Taylor resolution

𝔽_{Δ} : 0 ⟶ R(−8) ⟶ \begin{matrix}R(−8) \\ ⊕ \\ R^{3}(−7) \end{matrix} ⟶ \begin{matrix}R^{2}(−7) \\ ⊕ \\ R^{3}(−6) \\ ⊕ \\ R(−4) \end{matrix} ⟶ \begin{matrix}R^{2}(−5) \\ ⊕ \\ R(−3) \\ ⊕ \\ R(−2) \end{matrix} ⟶ I ⟶ 0

is not minimal since

deg {v_{1}, v_{2}, v_{3}, v_{4}} = deg {v_{1}, v_{2}, v_{4}} = (3, 2, 3), deg {v_{2}, v_{3}, v_{4}} = deg {v_{2}, v_{4}} = (3, 2, 2), deg {v_{1}, v_{3}, v_{4}} = deg {v_{1}, v_{4}} = (2, 2, 3).

The steps of our reduction are already contained in that list. After the first step, we obtain the cell complex X The cellular complex

𝔽_{X} : 0 ⟶ R^{3}(−7) ⟶ \begin{matrix}R^{2}(−7) \\ ⊕ \\ R^{3}(−6) \\ ⊕ \\ R(−4) \end{matrix} ⟶ \begin{matrix}R^{2}(−5) \\ ⊕ \\ R(−3) \\ ⊕ \\ R(−2) \end{matrix} ⟶ I ⟶ 0

is also a resolution, but it is not minimal. We proceed with Step 2 and get the cell complex Y

The resulting cellular complex

𝔽_{Y} : 0 ⟶ R^{2}(−7) ⟶ \begin{matrix}R(−7) \\ ⊕ \\ R^{3}(−6) \\ ⊕ \\ R(−4) \end{matrix} ⟶ \begin{matrix}R^{2}(−5) \\ ⊕ \\ R(−3) \\ ⊕ \\ R(−2) \end{matrix} ⟶ I ⟶ 0

is a resolution of I, but it is still not minimal. Finally, after Step 3 we get the Scarf complex Δ_{I} of I.

According to Example 3.5.8

𝔽_{Δ_{I}} : 0 ⟶ R(−7) ⟶ \begin{matrix}R^{3}(−6) \\ ⊕ \\ R(−4) \end{matrix} ⟶ \begin{matrix}R^{2}(−5) \\ ⊕ \\ R(−3) \\ ⊕ \\ R(−2) \end{matrix} ⟶ I ⟶ 0

is a minimal free resolution of I.

Now we present another strategy how to get rid of a pair of cells of the same degree. Assume that F′ is a face of F, F ∈ X^{i}, F′ ∈ X^{i−1}, deg F = deg F′, and that there is exactly one other G ∈ X^{i} such that F′ is a face of G. Then, instead of just deleting F and F′, we may proceed as follows: Delete F, G, F′ and add F ∪ G ∈ X^{i} of degree

deg(F ∪ G) = lcm {deg F, deg G}.

Again, the alternating sum of the cardinality of the components remains equal to zero (see Corollary 3.1.16). The following example gives an illustration:

Example 3.6.2. As we have proven in Example 3.4.19, the hull complex of the ideal

I = (x, y, z)^{2} = (x^{2}, y^{2}, z^{2}, xy, xz, yz)

⊂ R = K[x, y, z]

is given by

hull(I)
Component  Cell  Degree Step
hull(I)^{−1} F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (2,0,0)
F_{2}^{0}  {v_{2}}  (0,2,0)
F_{3}^{0}  {v_{3}}  (0,0,2)
hull(I)^{0} F_{4}^{0}  {v_{4}}  (1,1,0)
F_{5}^{0}  {v_{5}}  (1,0,1)
F_{6}^{0}  {v_{6}}  (0,1,1)
F_{1}^{1} {v_{1}, v_{4}} (2,1,0)
F_{2}^{1} {v_{1}, v_{5}} (2,0,1)
F_{3}^{1} {v_{2}, v_{4}} (1,2,0)
F_{4}^{1} {v_{2}, v_{6}} (0,2,1)
hull(I)^{1} F_{5}^{1} {v_{3}, v_{5}} (1,0,2)
F_{6}^{1} {v_{3}, v_{6}} (0,1,2)
F_{7}^{1} {v_{4}, v_{5}} (1,1,1)
F_{8}^{1} {v_{4}, v_{6}} (1,1,1)
F_{9}^{1} {v_{5}, v_{6}} (1,1,1) 1
F_{1}^{2} {v_{1}, v_{4}, v_{5}} (2,1,1)
F_{2}^{2} {v_{2}, v_{4}, v_{6}} (1,2,1)
hull(I)^{2} F_{3}^{2} {v_{3}, v_{5}, v_{6}} (1,1,2)
F_{4}^{2} {v_{4}, v_{5}, v_{6}} (1,1,1) 1

142 STEFAN TAPPE

The hull resolution

𝔽_{hull(I)} : 0 ⟶ \begin{matrix}R^{3}(−4) \\ ⊕ \\ R(−3) \end{matrix} ⟶ R^{9}(−3) ⟶ R^{6}(−2) ⟶ I ⟶ 0

is not minimal because

deg({v_{4}, v_{5}, v_{6}}) = deg({v_{5}, v_{6}}) = (1, 1, 1).

The only other cell G ∈ hull(I)^{2} such that {v_{5}, v_{6}} is a face of G, is G = {v_{3}, v_{5}, v_{6}}. We proceed as described above and obtain the cell complex X

X

Component  Cell  Degree
X^{−1}  F_{1}^{−1}  ∅
F_{1}^{0}  {v_{1}}  (2,0,0)
F_{2}^{0}  {v_{2}}  (0,2,0)
F_{3}^{0}  {v_{3}}  (0,0,2)
X^{0}  F_{4}^{0}  {v_{4}}  (1,1,0)
F_{5}^{0}  {v_{5}}  (1,0,1)
F_{6}^{0}  {v_{6}}  (0,1,1)
F_{1}^{1}  {v_{1}, v_{4}}  (2,1,0)
F_{2}^{1}  {v_{1}, v_{5}}  (2,0,1)
F_{3}^{1}  {v_{2}, v_{4}}  (1,2,0)
F_{4}^{1}  {v_{2}, v_{6}}  (0,2,1)
X^{1}  F_{5}^{1}  {v_{3}, v_{5}}  (1,0,2)
F_{6}^{1}  {v_{3}, v_{6}}  (0,1,2)
F_{7}^{1}  {v_{4}, v_{5}}  (1,1,1)
F_{8}^{1}  {v_{4}, v_{6}}  (1,1,1)
F_{1}^{2} {v_{1}, v_{4}, v_{5}} (2,1,1)
X^{2}  F_{2}^{2} {v_{2}, v_{4}, v_{6}} (1,2,1)
F_{3}^{2} {v_{3}, v_{4}, v_{5}, v_{6}} (1,1,2)

The resulting cellular complex

𝔽_{X} : 0 ⟶ R^{3}(−4) ⟶ R^{8}(−3) ⟶ R^{6}(−2) ⟶ I ⟶ 0

is a minimal resolution of I by Theorem 3.1.11 and Proposition 3.1.14. The reader may make sure that just deleting the two cells does not work because the structure of a cell complex gets lost.

We can determine the values of the Hilbert function h_{R/I} by Proposition 3.1.18 or by Proposition 3.2.6. We obtain

h_{R/I}(0) = 1,
h_{R/I}(1) = 3,
h_{R/I}(k) = 0, k ≥ 2.

Thus, the Hilbert polynomial of R/I is

p_{R/I} = 0.

Therefore, we get

dim (R/I) = 0,
deg (R/I) = 4,

and the height of I is

h(I) = 3.

R/I is Cohen-Macaulay according to Proposition 3.1.20.

Next, we consider a difficult example:

Example 3.6.3. We consider the ideal

I = (xyz, xyw, xzv, xuv, xuw, yzu, yuv, yvw, zuw, zvw)
⊂ R = K[x, y, z, u, v, w]

of Example 3.1.23. The crucial point is that the cellular complex 𝔽_{X} from Example 3.1.23 is a minimal free resolution if K = ℝ, but it is no resolution for K = 𝔽_{2}.

The hull complex of I is

hull(I)
Cell  Dimension Degree Step
∅  -1
{v_{1}}  0  (1,1,1,0,0,0)
{v_{2}}  0  (1,1,0,0,0,1)
{v_{3}}  0  (1,0,1,0,1,0)
{v_{4}}  0  (1,0,0,1,1,0)
{v_{5}}  0  (1,0,0,1,0,1)
{v_{6}}  0  (0,1,1,1,0,0)
{v_{7}}  0  (0,1,0,1,1,0)
{v_{8}}  0  (0,1,0,0,1,1)
{v_{9}}  0  (0,0,1,1,0,1)
{v_{10}}  0  (0,0,1,0,1,1)
{v_{1}, v_{2}}  1  (1,1,1,0,0,1)
{v_{1}, v_{3}}  1  (1,1,1,0,1,0)
{v_{1}, v_{4}}  1  (1,1,1,1,1,0) D1
{v_{1}, v_{5}}  1  (1,1,1,1,0,1) D6
{v_{1}, v_{6}}  1  (1,1,1,1,0,0)
{v_{1}, v_{7}}  1  (1,1,1,1,1,0) D2
{v_{1}, v_{8}}  1  (1,1,1,0,1,1) D11
{v_{1}, v_{9}}  1  (1,1,1,1,0,1) D7
{v_{1}, v_{10}}  1  (1,1,1,0,1,1) D12
{v_{2}, v_{3}}  1  (1,1,1,0,1,1) D13
{v_{2}, v_{4}}  1  (1,1,0,1,1,1) D16
{v_{2}, v_{5}}  1  (1,1,0,1,0,1)
{v_{2}, v_{6}}  1  (1,1,1,1,0,1) D8
{v_{2}, v_{7}}  1  (1,1,0,1,1,1) D17
{v_{2}, v_{8}}  1  (1,1,0,0,1,1)
{v_{2}, v_{9}}  1  (1,1,1,1,0,1) D9
{v_{2}, v_{10}}  1  (1,1,1,0,1,1) D14
{v_{3}, v_{4}}  1  (1,0,1,1,1,0)
{v_{3}, v_{5}}  1  (1,0,1,1,1,1) D21
{v_{3}, v_{6}}  1  (1,1,1,1,1,0) D3
{v_{3}, v_{7}}  1  (1,1,1,1,1,0) D4
{v_{3}, v_{8}}  1  (1,1,1,0,1,1) D15
{v_{3}, v_{9}}  1  (1,0,1,1,1,1) D22
{v_{3}, v_{10}}  1  (1,0,1,0,1,1)
{v_{4}, v_{5}}  1  (1,0,0,1,1,1)
{v_{4}, v_{6}}  1  (1,1,1,1,1,0) D5
{v_{4}, v_{7}}  1  (1,1,0,1,1,0)
{v_{4}, v_{8}}  1  (1,1,0,1,1,1) D18
{v_{4}, v_{9}}  1  (1,0,1,1,1,1) D23
{v_{4}, v_{10}}  1  (1,0,1,1,1,1) D24
{v_{5}, v_{6}}  1  (1,1,1,1,0,1) D10
{v_{5}, v_{7}}  1  (1,1,0,1,1,1) D19
{v_{5}, v_{8}}  1  (1,1,0,1,1,1) D20

to be continued

CELLULAR RESOLUTIONS OF MONOMIAL IDEALS 145

continued from the last page

Cell  Dimension Degree Step
{v_{5}, v_{9}}  1  (1,0,1,1,0,1)
{v_{5}, v_{10}}  1  (1,0,1,1,1,1) D25
{v_{6}, v_{7}}  1  (0,1,1,1,1,0)
{v_{6}, v_{8}}  1  (0,1,1,1,1,1) D26
{v_{6}, v_{9}}  1  (0,1,1,1,0,1)
{v_{6}, v_{10}}  1  (0,1,1,1,1,1) D27
{v_{7}, v_{8}}  1  (0,1,0,1,1,1)
{v_{7}, v_{9}}  1  (0,1,1,1,1,1) D28
{v_{7}, v_{10}}  1  (0,1,1,1,1,1) D29
{v_{8}, v_{9}}  1  (0,1,1,1,1,1) D30
{v_{8}, v_{10}}  1  (0,1,1,0,1,1)
{v_{9}, v_{10}}  1  (0,0,1,1,1,1)
{v_{1}, v_{2}, v_{3}}  2  (1,1,1,0,1,1) D13
{v_{1}, v_{2}, v_{4}}  2  (1,1,1,1,1,1) C1
{v_{1}, v_{2}, v_{5}}  2  (1,1,1,1,0,1) D6
{v_{1}, v_{2}, v_{6}}  2  (1,1,1,1,0,1) D8
{v_{1}, v_{2}, v_{7}}  2  (1,1,1,1,1,1) C9
{v_{1}, v_{2}, v_{8}}  2  (1,1,1,0,1,1) D11
{v_{1}, v_{2}, v_{9}}  2  (1,1,1,1,0,1) D9
{v_{1}, v_{2}, v_{10}}  2  (1,1,1,0,1,1) D14
{v_{1}, v_{3}, v_{4}}  2  (1,1,1,1,1,0) D1
{v_{1}, v_{3}, v_{5}}  2  (1,1,1,1,1,1) C2
{v_{1}, v_{3}, v_{6}}  2  (1,1,1,1,1,0) D3
{v_{1}, v_{3}, v_{7}}  2  (1,1,1,1,1,0) D4
{v_{1}, v_{3}, v_{8}}  2  (1,1,1,0,1,1) C41
{v_{1}, v_{3}, v_{9}}  2  (1,1,1,1,1,1) C10
{v_{1}, v_{3}, v_{10}}  2  (1,1,1,0,1,1) D12
{v_{1}, v_{4}, v_{5}}  2  (1,1,1,1,1,1) C3
{v_{1}, v_{4}, v_{6}}  2  (1,1,1,1,1,0) C31
{v_{1}, v_{4}, v_{7}}  2  (1,1,1,1,1,0) D2
{v_{1}, v_{5}, v_{6}}  2  (1,1,1,1,0,1) C36
{v_{1}, v_{5}, v_{9}}  2  (1,1,1,1,0,1) D7
{v_{1}, v_{6}, v_{7}}  2  (1,1,1,1,1,0) C32
{v_{1}, v_{6}, v_{8}}  2  (1,1,1,1,1,1) C20
{v_{1}, v_{6}, v_{9}}  2  (1,1,1,1,0,1) C37
{v_{1}, v_{6}, v_{10}}  2  (1,1,1,1,1,1) C21
{v_{1}, v_{7}, v_{8}}  2  (1,1,1,1,1,1) C22
{v_{1}, v_{8}, v_{10}}  2  (1,1,1,0,1,1) C42
{v_{1}, v_{9}, v_{10}}  2  (1,1,1,1,1,1) C23
{v_{2}, v_{3}, v_{4}}  2  (1,1,1,1,1,1) C4
{v_{2}, v_{3}, v_{5}}  2  (1,1,1,1,1,1) C5
{v_{2}, v_{3}, v_{8}}  2  (1,1,1,0,1,1) D15
{v_{2}, v_{3}, v_{10}}  2  (1,1,1,0,1,1) C43
{v_{2}, v_{4}, v_{5}}  2  (1,1,0,1,1,1) D16

to be continued

146 STEFAN TAPPE

continued from the last page

Cell  Dimension Degree Step
{v_{2}, v_{4}, v_{7}}  2  (1,1,0,1,1,1) C46
{v_{2}, v_{4}, v_{8}}  2  (1,1,0,1,1,1) D18
{v_{2}, v_{5}, v_{6}}  2  (1,1,1,1,0,1) D10
{v_{2}, v_{5}, v_{7}}  2  (1,1,0,1,1,1) D17
{v_{2}, v_{5}, v_{8}}  2  (1,1,0,1,1,1) D20
{v_{2}, v_{5}, v_{9}}  2  (1,1,1,1,0,1) C39
{v_{2}, v_{5}, v_{10}}  2  (1,1,1,1,1,1) C24
{v_{2}, v_{6}, v_{7}}  2  (1,1,1,1,1,1) C6
{v_{2}, v_{6}, v_{8}}  2  (1,1,1,1,1,1) C11
{v_{2}, v_{6}, v_{9}}  2  (1,1,1,1,0,1) C38
{v_{2}, v_{7}, v_{8}}  2  (1,1,0,1,1,1) C47
{v_{2}, v_{8}, v_{9}}  2  (1,1,1,1,1,1) C16
{v_{2}, v_{8}, v_{10}}  2  (1,1,1,0,1,1) C44
{v_{2}, v_{9}, v_{10}}  2  (1,1,1,1,1,1) C25
{v_{3}, v_{4}, v_{5}}  2  (1,0,1,1,1,1) D21
{v_{3}, v_{4}, v_{6}}  2  (1,1,1,1,1,0) D5
{v_{3}, v_{4}, v_{7}}  2  (1,1,1,1,1,0) C34
{v_{3}, v_{4}, v_{8}}  2  (1,1,1,1,1,1) C12
{v_{3}, v_{4}, v_{9}}  2  (1,0,1,1,1,1) D22
{v_{3}, v_{4}, v_{10}}  2  (1,0,1,1,1,1) D24
{v_{3}, v_{5}, v_{9}}  2  (1,0,1,1,1,1) C51
{v_{3}, v_{5}, v_{10}}  2  (1,0,1,1,1,1) D25
{v_{3}, v_{6}, v_{7}}  2  (1,1,1,1,1,0) C33
{v_{3}, v_{6}, v_{9}}  2  (1,1,1,1,1,1) C7
{v_{3}, v_{6}, v_{10}}  2  (1,1,1,1,1,1) C13
{v_{3}, v_{7}, v_{8}}  2  (1,1,1,1,1,1) C26
{v_{3}, v_{7}, v_{10}}  2  (1,1,1,1,1,1) C27
{v_{3}, v_{8}, v_{10}}  2  (1,1,1,0,1,1) C45
{v_{3}, v_{9}, v_{10}}  2  (1,0,1,1,1,1) C52
{v_{4}, v_{5}, v_{6}}  2  (1,1,1,1,1,1) C14
{v_{4}, v_{5}, v_{7}}  2  (1,1,0,1,1,1) D19
{v_{4}, v_{5}, v_{8}}  2  (1,1,0,1,1,1) C50
{v_{4}, v_{5}, v_{9}}  2  (1,0,1,1,1,1) D23
{v_{4}, v_{5}, v_{10}}  2  (1,0,1,1,1,1) C54
{v_{4}, v_{6}, v_{7}}  2  (1,1,1,1,1,0) C35
{v_{4}, v_{6}, v_{9}}  2  (1,1,1,1,1,1) C17
{v_{4}, v_{7}, v_{8}}  2  (1,1,0,1,1,1) C48
{v_{4}, v_{7}, v_{9}}  2  (1,1,1,1,1,1) C19
{v_{4}, v_{7}, v_{10}}  2  (1,1,1,1,1,1) C28
{v_{4}, v_{8}, v_{10}}  2  (1,1,1,1,1,1) C29
{v_{4}, v_{9}, v_{10}}  2  (1,0,1,1,1,1) C53
{v_{5}, v_{6}, v_{7}}  2  (1,1,1,1,1,1) C8
{v_{5}, v_{6}, v_{9}}  2  (1,1,1,1,0,1) C40
{v_{5}, v_{7}, v_{8}}  2  (1,1,0,1,1,1) C49

to be continued

CELLULAR RESOLUTIONS OF MONOMIAL IDEALS 147

continued from the last page

Cell  Dimension Degree Step
{v_{5}, v_{7}, v_{9}}  2  (1,1,1,1,1,1) C15
{v_{5}, v_{8}, v_{9}}  2  (1,1,1,1,1,1) C18
{v_{5}, v_{8}, v_{10}}  2  (1,1,1,1,1,1) C30
{v_{5}, v_{9}, v_{10}}  2  (1,0,1,1,1,1) C55
{v_{6}, v_{7}, v_{8}}  2  (0,1,1,1,1,1) D26
{v_{6}, v_{7}, v_{9}}  2  (0,1,1,1,1,1) D28
{v_{6}, v_{7}, v_{10}}  2  (0,1,1,1,1,1) D29
{v_{6}, v_{8}, v_{9}}  2  (0,1,1,1,1,1) C56
{v_{6}, v_{8}, v_{10}}  2  (0,1,1,1,1,1) D27
{v_{6}, v_{9}, v_{10}}  2  (0,1,1,1,1,1) C57
{v_{7}, v_{8}, v_{9}}  2  (0,1,1,1,1,1) D30
{v_{7}, v_{8}, v_{10}}  2  (0,1,1,1,1,1) C59
{v_{7}, v_{9}, v_{10}}  2  (0,1,1,1,1,1) C58
{v_{8}, v_{9}, v_{10}}  2  (0,1,1,1,1,1) C60
{v_{1}, v_{2}, v_{3}, v_{4}}  3  (1,1,1,1,1,1) B1
{v_{1}, v_{2}, v_{3}, v_{5}}  3  (1,1,1,1,1,1) B2
{v_{1}, v_{2}, v_{3}, v_{8}}  3  (1,1,1,0,1,1) C41
{v_{1}, v_{2}, v_{3}, v_{10}}  3  (1,1,1,0,1,1) C43
{v_{1}, v_{2}, v_{4}, v_{5}}  3  (1,1,1,1,1,1) B3
{v_{1}, v_{2}, v_{4}, v_{7}}  3  (1,1,1,1,1,1) C1
{v_{1}, v_{2}, v_{5}, v_{6}}  3  (1,1,1,1,0,1) C36
{v_{1}, v_{2}, v_{5}, v_{9}}  3  (1,1,1,1,0,1) C39
{v_{1}, v_{2}, v_{6}, v_{7}}  3  (1,1,1,1,1,1) B4
{v_{1}, v_{2}, v_{6}, v_{8}}  3  (1,1,1,1,1,1) B5
{v_{1}, v_{2}, v_{6}, v_{9}}  3  (1,1,1,1,0,1) C38
{v_{1}, v_{2}, v_{7}, v_{8}}  3  (1,1,1,1,1,1) C9
{v_{1}, v_{2}, v_{8}, v_{10}}  3  (1,1,1,0,1,1) C44
{v_{1}, v_{2}, v_{9}, v_{10}}  3  (1,1,1,1,1,1) C25
{v_{1}, v_{3}, v_{4}, v_{5}}  3  (1,1,1,1,1,1) B6
{v_{1}, v_{3}, v_{4}, v_{6}}  3  (1,1,1,1,1,0) C31
{v_{1}, v_{3}, v_{4}, v_{7}}  3  (1,1,1,1,1,0) C34
{v_{1}, v_{3}, v_{5}, v_{9}}  3  (1,1,1,1,1,1) C2
{v_{1}, v_{3}, v_{6}, v_{7}}  3  (1,1,1,1,1,0) C33
{v_{1}, v_{3}, v_{6}, v_{9}}  3  (1,1,1,1,1,1) B8
{v_{1}, v_{3}, v_{6}, v_{10}}  3  (1,1,1,1,1,1) B7
{v_{1}, v_{3}, v_{7}, v_{8}}  3  (1,1,1,1,1,1) C22
{v_{1}, v_{3}, v_{8}, v_{10}}  3  (1,1,1,0,1,1) C42
{v_{1}, v_{3}, v_{9}, v_{10}}  3  (1,1,1,1,1,1) C10
{v_{1}, v_{4}, v_{5}, v_{6}}  3  (1,1,1,1,1,1) C3
{v_{1}, v_{4}, v_{6}, v_{7}}  3  (1,1,1,1,1,0) C32
{v_{1}, v_{5}, v_{6}, v_{9}}  3  (1,1,1,1,0,1) C37
{v_{1}, v_{6}, v_{7}, v_{8}}  3  (1,1,1,1,1,1) C20
{v_{1}, v_{6}, v_{8}, v_{10}}  3  (1,1,1,1,1,1) C21
{v_{1}, v_{6}, v_{9}, v_{10}}  3  (1,1,1,1,1,1) C23

to be continued

148 STEFAN TAPPE

continued from the last page

Cell  Dimension Degree Step
{v_{2}, v_{3}, v_{4}, v_{5}}  3  (1,1,1,1,1,1) B9
{v_{2}, v_{3}, v_{4}, v_{8}}  3  (1,1,1,1,1,1) C4
{v_{2}, v_{3}, v_{5}, v_{10}}  3  (1,1,1,1,1,1) C5
{v_{2}, v_{3}, v_{8}, v_{10}}  3  (1,1,1,0,1,1) C45
{v_{2}, v_{4}, v_{5}, v_{7}}  3  (1,1,0,1,1,1) C46
{v_{2}, v_{4}, v_{5}, v_{8}}  3  (1,1,0,1,1,1) C50
{v_{2}, v_{4}, v_{7}, v_{8}}  3  (1,1,0,1,1,1) C48
{v_{2}, v_{5}, v_{6}, v_{7}}  3  (1,1,1,1,1,1) B10
{v_{2}, v_{5}, v_{6}, v_{9}}  3  (1,1,1,1,0,1) C40
{v_{2}, v_{5}, v_{7}, v_{8}}  3  (1,1,0,1,1,1) C47
{v_{2}, v_{5}, v_{8}, v_{9}}  3  (1,1,1,1,1,1) B11
{v_{2}, v_{5}, v_{8}, v_{10}}  3  (1,1,1,1,1,1) C30
{v_{2}, v_{5}, v_{9}, v_{10}}  3  (1,1,1,1,1,1) C24
{v_{2}, v_{6}, v_{7}, v_{8}}  3  (1,1,1,1,1,1) C6
{v_{2}, v_{6}, v_{8}, v_{9}}  3  (1,1,1,1,1,1) C11
{v_{2}, v_{8}, v_{9}, v_{10}}  3  (1,1,1,1,1,1) C16
{v_{3}, v_{4}, v_{5}, v_{9}}  3  (1,0,1,1,1,1) C51
{v_{3}, v_{4}, v_{5}, v_{10}}  3  (1,0,1,1,1,1) C54
{v_{3}, v_{4}, v_{6}, v_{7}}  3  (1,1,1,1,1,0) C35
{v_{3}, v_{4}, v_{6}, v_{9}}  3  (1,1,1,1,1,1) B12
{v_{3}, v_{4}, v_{7}, v_{8}}  3  (1,1,1,1,1,1) B13
{v_{3}, v_{4}, v_{7}, v_{10}}  3  (1,1,1,1,1,1) C27
{v_{3}, v_{4}, v_{8}, v_{10}}  3  (1,1,1,1,1,1) C12
{v_{3}, v_{4}, v_{9}, v_{10}}  3  (1,0,1,1,1,1) C52
{v_{3}, v_{5}, v_{9}, v_{10}}  3  (1,0,1,1,1,1) C55
{v_{3}, v_{6}, v_{7}, v_{10}}  3  (1,1,1,1,1,1) C13
{v_{3}, v_{6}, v_{9}, v_{10}}  3  (1,1,1,1,1,1) C7
{v_{3}, v_{7}, v_{8}, v_{10}}  3  (1,1,1,1,1,1) C26
{v_{4}, v_{5}, v_{6}, v_{7}}  3  (1,1,1,1,1,1) B14
{v_{4}, v_{5}, v_{6}, v_{9}}  3  (1,1,1,1,1,1) C14
{v_{4}, v_{5}, v_{7}, v_{8}}  3  (1,1,0,1,1,1) C49
{v_{4}, v_{5}, v_{7}, v_{9}}  3  (1,1,1,1,1,1) B15
{v_{4}, v_{5}, v_{8}, v_{10}}  3  (1,1,1,1,1,1) C29
{v_{4}, v_{5}, v_{9}, v_{10}}  3  (1,0,1,1,1,1) C53
{v_{4}, v_{6}, v_{7}, v_{9}}  3  (1,1,1,1,1,1) C17
{v_{4}, v_{7}, v_{8}, v_{10}}  3  (1,1,1,1,1,1) C28
{v_{4}, v_{7}, v_{9}, v_{10}}  3  (1,1,1,1,1,1) C19
{v_{5}, v_{6}, v_{7}, v_{9}}  3  (1,1,1,1,1,1) C8
{v_{5}, v_{7}, v_{8}, v_{9}}  3  (1,1,1,1,1,1) C15
{v_{5}, v_{8}, v_{9}, v_{10}}  3  (1,1,1,1,1,1) C18
{v_{6}, v_{7}, v_{8}, v_{9}}  3  (0,1,1,1,1,1) C56
{v_{6}, v_{7}, v_{8}, v_{10}}  3  (0,1,1,1,1,1) C59
{v_{6}, v_{7}, v_{9}, v_{10}}  3  (0,1,1,1,1,1) C58
{v_{6}, v_{8}, v_{9}, v_{10}}  3  (0,1,1,1,1,1) C57

to be continued

continued from the last page

Cell  Dimension Degree Step
{v_{7}, v_{8}, v_{9}, v_{10}}  3  (0,1,1,1,1,1) C60
{v_{1}, v_{2}, v_{3}, v_{4}, v_{5}}  4  (1,1,1,1,1,1) A1
{v_{1}, v_{2}, v_{3}, v_{4}, v_{7}, v_{8}}  4  (1,1,1,1,1,1) B1
{v_{1}, v_{2}, v_{3}, v_{5}, v_{9}, v_{10}}  4  (1,1,1,1,1,1) B2
{v_{1}, v_{2}, v_{3}, v_{8}, v_{10}}  4  (1,1,1,0,1,1)
{v_{1}, v_{2}, v_{4}, v_{5}, v_{6}, v_{7}}  4  (1,1,1,1,1,1) B3
{v_{1}, v_{2}, v_{5}, v_{6}, v_{9}}  4  (1,1,1,1,0,1)
{v_{1}, v_{2}, v_{6}, v_{7}, v_{8}}  4  (1,1,1,1,1,1) B4
{v_{1}, v_{2}, v_{6}, v_{8}, v_{9}, v_{10}}  4  (1,1,1,1,1,1) B5
{v_{1}, v_{3}, v_{4}, v_{5}, v_{6}, v_{9}}  4  (1,1,1,1,1,1) B6
{v_{1}, v_{3}, v_{4}, v_{6}, v_{7}}  4  (1,1,1,1,1,0)
{v_{1}, v_{3}, v_{6}, v_{7}, v_{8}, v_{10}}  4  (1,1,1,1,1,1) B7
{v_{1}, v_{3}, v_{6}, v_{9}, v_{10}}  4  (1,1,1,1,1,1) B8
{v_{2}, v_{3}, v_{4}, v_{5}, v_{8}, v_{10}}  4  (1,1,1,1,1,1) B9
{v_{2}, v_{4}, v_{5}, v_{7}, v_{8}}  4  (1,1,0,1,1,1)
{v_{2}, v_{5}, v_{6}, v_{7}, v_{8}, v_{9}}  4  (1,1,1,1,1,1) B10
{v_{2}, v_{5}, v_{8}, v_{9}, v_{10}}  4  (1,1,1,1,1,1) B11
{v_{3}, v_{4}, v_{5}, v_{9}, v_{10}}  4  (1,0,1,1,1,1)
{v_{3}, v_{4}, v_{6}, v_{7}, v_{9}, v_{10}}  4  (1,1,1,1,1,1) B12
{v_{3}, v_{4}, v_{7}, v_{8}, v_{10}}  4  (1,1,1,1,1,1) B13
{v_{4}, v_{5}, v_{6}, v_{7}, v_{9}}  4  (1,1,1,1,1,1) B14
{v_{4}, v_{5}, v_{7}, v_{8}, v_{9}, v_{10}}  4  (1,1,1,1,1,1) B15
{v_{6}, v_{7}, v_{8}, v_{9}, v_{10}}  4  (0,1,1,1,1,1)
{v_{1}, v_{2}, v_{3}, v_{4}, v_{5}, v_{6}, v_{7}, v_{8}, v_{9}, v_{10}}  5  (1,1,1,1,1,1) A1

The proof that this table represents the hull complex of I is done with MAPLE. It is the file "examples.mws" (see also Section 3.7).

It follows from Theorem 3.3.15 that

𝔽_{hull(I)} : 0 → R(−6) → \begin{matrix}R^{6}(−5) \\ ⊕ \\ R^{16}(−6) \end{matrix} → \begin{matrix}R^{30}(−5) \\ ⊕ \\ R^{45}(−6) \end{matrix} → \begin{matrix}R^{60}(−5) \\ ⊕ \\ R^{30}(−6) \end{matrix} → \begin{matrix}R^{15}(−4) \\ ⊕ \\ R^{30}(−5) \end{matrix} → R^{10}(−3) → I → 0

is a resolution of I for every field K, but this resolution is obviously not minimal. All the steps A1-D30 contained in the list are done with our first strategy. We just delete pairs of cells having the same degree. After the steps A1-C19 we obtain a cellular complex

𝔽_{Y} : 0 → R^{6}(−5) → \begin{matrix}R^{30}(−5) \\ ⊕ \\ R^{11}(−6) \end{matrix} → \begin{matrix}R^{60}(−5) \\ ⊕ \\ R^{11}(−6) \end{matrix} → \begin{matrix}R^{15}(−4) \\ ⊕ \\ R^{30}(−5) \end{matrix} → R^{10}(−3) → I → 0.

It can be checked by Theorem 3.1.11 that 𝔽_{Y} is exact for K = ℝ and K = 𝔽_{2}.

After the steps A1-C30 we get the simplicial complex Z generated by all cells of the component X^{2}, where X is the cell complex of Example 3.1.23:

Z = ⟨{v_{1}, v_{2}, v_{3}, v_{8}, v_{10}}, {v_{1}, v_{2}, v_{5}, v_{6}, v_{9}}, {v_{1}, v_{3}, v_{4}, v_{6}, v_{7}},
{v_{2}, v_{4}, v_{5}, v_{7}, v_{8}}, {v_{3}, v_{4}, v_{5}, v_{9}, v_{10}}, {v_{6}, v_{7}, v_{8}, v_{9}, v_{10}}⟩

The cellular complex

𝔽_{Z} : 0 ⟶ R^{6}(−5) ⟶ R^{30}(−5) ⟶ R^{60}(−5) ⟶ \begin{matrix}R^{15}(−4) \\ ⊕ \\ R^{30}(−5) \end{matrix} ⟶ R^{10}(−3) ⟶ I ⟶ 0

is a resolution for K = ℝ, but not if K = 𝔽_{2}: For the augmented oriented chain complex we obtain the ranks

C̃(Z; ℝ) : 0 → K^{6} ⟶_{6} K^{30} ⟶_{24} K^{60} ⟶_{36} K^{45} ⟶_{9} K^{10} ⟶_{1} K → 0

and

C̃(Z; 𝔽_{2}) : 0 → K^{6} ⟶_{6} K^{30} ⟶_{24} K^{60} ⟶_{35} K^{45} ⟶_{9} K^{10} ⟶_{1} K → 0.

After having done all steps of the table, we obtain the cell complex X of Example 3.1.23, whose cellular complex

𝔽_{X} : 0 ⟶ R^{6}(−5) ⟶ R^{15}(−4) ⟶ R^{10}(−3) ⟶ I ⟶ 0.

is a resolution for K = ℝ, but not for K = 𝔽_{2}.

It is amazing that we were actually successful by just using our first strategy for such an extensive cell complex. This example indicates that it might be possible to reduce each cellular resolution of an arbitrary monomial ideal by using the two introduced strategies, provided the field has characteristic 0. We close this section about ’Reducing cellular resolutions’ by posing this as an unanswered question.

3.7. Cellular resolutions with MAPLE.

The author has written a program to create cellular resolutions using MAPLE, which proved to be very supportive. The reader will admit that it is hard, and sometimes even impossible, to compute examples just with paper and pencil. In the following, we give an overview of the functionality of this program. The MAPLE version, for which the procedures have been developed, is Maple 7. There are two MAPLE files on the CD attached to this thesis, "examples.mws", which contains all the examples collected in this theses, and "Example 3 6 3.mws", which is a proof that the hull complex of the ideal in Example 3.6.3 is the cell complex we have used there. After having started MAPLE, activate the new commands by

   > read(`cellular_resolutions.txt`):

Make sure that this file is contained in the search path of MAPLE.

First of all, we have to create a cell complex X supporting a monomial ideal I. This is done by the command

   > create_cellcomplex():

The required arguments are a list of the degrees of the minimal generators of I and a list containing the elements of the cell complex. The empty set and the vertices should not appear in this list because they will be completed automatically. The cell complex structure can also be written as Scarf in order to create the Scarf complex belonging to the monomial ideal, as Taylor (or Simplex) for the Taylor complex, or as simplicial together with a list of generators, to create a simplicial complex. To display the cell complex X, use

   > show_cellcomplex():

If the given set of sets does not have the structure of a cell complex, MAPLE will refuse to execute the following commands. MAPLE automatically creates an incidence function ε which can be shown by

   > show_incidence_function():

To display the cellular complex 𝔽_{X}, apply

   > show_cellular_complex():

There are two optional arguments: The grading can be chosen as fine (ℤ^{n}-grading), normal (ℤ-grading) and none. The default is without grading. One can also fix the variables of the polynomial ring by Polyring. The default are the variables x_{1}, …, x_{n}. Use

   > show_cc_matrices():

in order to display the matrices of the maps in 𝔽_{X} with respect to the canonical bases. It is possible to test the properties of 𝔽_{X} by the command

   > test_properties():

The possible statements are

“The cellular complex is a minimal resolution of I.” “The cellular complex is a resolution of I, but not minimal.” “The cellular complex is not a resolution of I.”

The algorithm applies the criteria of Theorem 3.1.11 and Proposition 3.1.14 together with the Linear Algebra methods discussed in Lemma 2.2.22. If 𝔽_{X} is not a resolution of I, MAPLE prints all the degrees b for which X_{≤b} is not acyclic. For the case that 𝔽_{X} is a minimal free resolution, the command

   > Betti_numbers():

creates a table of the Betti numbers of I.

   > Hilbert_function():

computes the Hilbert function of I using the formula of Proposition 3.2.6. The commands presented above shall now be demonstrated by two examples, the monomial ideal of Example 2.4.13

   > create_cellcomplex(Degrees=[[4,0],[1,1],[0,3]],Type="Scarf"):
   > show_cellcomplex():

Cell Dimension Degree
{}  −1
{v_{1}}  0  [4, 0]
{v_{2}}  0  [1, 1]
{v_{3}}  0  [0, 3]
{v_{2}, v_{1}}  1  [4, 1]
{v_{3}, v_{2}}  1  [1, 3]

> show_incidence_function():

[ \begin{matrix}ε & \{v_{1}\} & \{v_{2}\} & \{v_{3}\} \\ \{\} & 1 & 1 & 1 \end{matrix} ]

[ \begin{matrix}ε & \{v_{2}, v_{1}\} & \{v_{3}, v_{2}\} \\ \{v_{1}\} & −1 & 0 \\ \{v_{2}\} & 1 & −1 \\ \{v_{3}\} & 0 & 1 \end{matrix} ]

> show_cellular_complex(grading="fine",Polyring=[x,y]):

[ \begin{matrix}& & R(−4, −1) & & R(−4, 0) \\ 0 & − > & R(−1, −3) & − > & R(−1, −1) & − > & I & − > & 0 \\ & & & & R(0, −3) \end{matrix} ]
I = (x^{4}, x y, y^{3})

> show_cellular_complex(grading="normal",Polyring=[x,y]):

[ \begin{matrix}& & R(−5) & & R(−4) \\ 0 & − > & R(−4) & − > & R(−2) & − > & I & − > & 0 \\ & & & & R(−3) \end{matrix} ]
I = (x^{4}, x y, y^{3})

> show_cellular_complex(grading="none",Polyring=[x,y]):

[ 0 − > R^{2} − > R^{3} − > I − > 0 ]
I = (x^{4}, x y, y^{3})

> show_cc_matrices(Polyring=[x,y]):

[ x^{4} x y y^{3} ]
[ \begin{matrix}−y & 0 \\ x^{3} & −y^{2} \\ 0 & x \end{matrix} ]

> test_properties():

“The cellular complex is a minimal resolution of I.”

> Betti_numbers():

Betti numbers 0 1
[1, 1]  1 0
[0, 3]  1 0
[1, 3]  0 1
[4, 0]  1 0
[4, 1]  0 1
2  1 0
3  1 0
4  1 1
5  0 1
Total  3 2

and the skew lines discussed Example 3.4.20

> create_cellcomplex(Degrees=[[1,0,1,0],[1,0,0,1],[0,1,1,0],[0,1,0,1
> ]],Type=[[1,2],[1,3],[2,4],[3,4],[1,2,3,4]]):
> show_cellcomplex():

Cell  Dimension Degree
{}  −1
{v_{1}}  0  [1, 0, 1, 0]
{v_{2}}  0  [1, 0, 0, 1]
{v_{3}}  0  [0, 1, 1, 0]
{v_{4}}  0  [0, 1, 0, 1]
{v_{1}, v_{2}}  1  [1, 0, 1, 1]
{v_{1}, v_{3}}  1  [1, 1, 1, 0]
{v_{2}, v_{4}}  1  [1, 1, 0, 1]
{v_{3}, v_{4}}  1  [0, 1, 1, 1]
{v_{1}, v_{2}, v_{3}, v_{4}}  2  [1, 1, 1, 1]

> show_incidence_function():

[ \begin{matrix}ε & \{v_{1}\} & \{v_{2}\} & \{v_{3}\} & \{v_{4}\} \\ \{\} & 1 & 1 & 1 & 1 \end{matrix} ]

[ \begin{matrix}ε & \{v_{1}, v_{2}\} & \{v_{1}, v_{3}\} & \{v_{2}, v_{4}\} & \{v_{3}, v_{4}\} \\ \{v_{1}\} & 1 & 1 & 0 & 0 \\ \{v_{2}\} & −1 & 0 & 1 & 0 \\ \{v_{3}\} & 0 & −1 & 0 & 1 \\ \{v_{4}\} & 0 & 0 & −1 & −1 \end{matrix} ]

[ \begin{matrix}ε & \{v_{1}, v_{2}, v_{3}, v_{4}\} \\ \{v_{1}, v_{2}\} & 1 \\ \{v_{1}, v_{3}\} & −1 \\ \{v_{2}, v_{4}\} & 1 \\ \{v_{3}, v_{4}\} & −1 \end{matrix} ]

> show_cellular_complex(grading="none",Polyring=[x,y,z,w]):

154 STEFAN TAPPE

[ 0 − > R − > R^{4} − > R^{4} − > I − > 0 ]

I = (x z, x w, y z, y w)

> show_cc_matrices(Polyring=[x,y,z,w]):

[ x z x w y z y w ]
[ \begin{matrix}w & y & 0 & 0 \\ −z & 0 & y & 0 \\ 0 & −x & 0 & w \\ 0 & 0 & −x & −z \end{matrix} ]
[ \begin{matrix}y \\ −w \\ z \\ −x \end{matrix} ]

> test_properties():

“The cellular complex is a minimal resolution of I.”

> Betti_numbers():

Betti numbers 0 1 2
[0, 1, 0, 1] 1 0 0
[0, 1, 1, 0] 1 0 0
[1, 0, 0, 1] 1 0 0
[1, 0, 1, 0] 1 0 0
[0, 1, 1, 1] 0 1 0
[1, 0, 1, 1] 0 1 0
[1, 1, 0, 1] 0 1 0
[1, 1, 1, 0] 0 1 0
[1, 1, 1, 1] 0 0 1
2  4 0 0
3  0 4 0
4  0 0 1
Total  4 4 1

> Hilbert_function(Degrees=[[1,0,1,0],[1,0,0,1],[0,1,1,0],[0,1,0,1]]
> ,Components=0..10):

Component R R / I I
0  1 1 0
1  4 4 0
2  10 6 4
3  20 8 12
4  35 10 25
5  56 12 44
6  84 14 70
7  120 16 104
8  165 18 147
9  220 20 200
10  286 22 264

> interp([1,2],[4,6],X);

2 X + 2

Note that MAPLE has computed an incidence function different from the one we derived in Example 2.2.13. The command

> show_augmented_oriented_chain_complex():

allows to view the complex C̃(X; K). The ranks of the matrices with respect to the canonical bases are below the arrows, so one can check whether the conditions of Lemma 2.2.22 are fulfilled. The matrices appear by

> show_cc_matrices():

It is possible to create the subcomplex X_{≤b} for any degree b by the command

> subcomplex():

MAPLE always uses the restriction of ε to X_{≤b} as an incidence function on X_{≤b}. For instance, we consider the cell complexes X and Y of Example 3.1.21:

> create_cellcomplex(Degrees=[[2,1,0],[1,0,1],[0,1,2],[0,2,0]],Type=
> ["Simplicial",[[1,2,4],[2,3,4]]]):
> show_cellular_complex(grading="none",Polyring=[x,y,z]):

[ 0 − > R^{2} − > R^{5} − > R^{4} − > I − > 0 ]

I = (x^{2} y, x z, y z^{2}, y^{2})

> show_cc_matrices(Polyring=[x,y,z]):

[ x^{2} y x z y z^{2} y^{2} ]
[ \begin{matrix}−z & −y & 0 & 0 & 0 \\ x y & 0 & −y z & −y^{2} & 0 \\ 0 & 0 & x & 0 & −y \\ 0 & x^{2} & 0 & x z & z^{2} \end{matrix} ]

[ \begin{matrix}y & 0 \\ −z & 0 \\ 0 & y \\ x & −z \\ 0 & x \end{matrix} ]

> test_properties():

“The cellular complex is a minimal resolution of I.”

> b:=[1,2,2]:
> subcomplex(Degree=b):
> show_cellcomplex():

Cell  Dimension Degree
{}  −1
{v_{2}}  0  [1, 0, 1]
{v_{3}}  0  [0, 1, 2]
{v_{4}}  0  [0, 2, 0]
{v_{2}, v_{3}}  1  [1, 1, 2]
{v_{2}, v_{4}}  1  [1, 2, 1]
{v_{3}, v_{4}}  1  [0, 2, 2]
{v_{2}, v_{3}, v_{4}}  2  [1, 2, 2]

> show_augmented_oriented_chain_complex():

\left[{0 − > K − > K^{3} − > K^{3} − > K − > 0 \atop 1  2  1}\right]

> show_aocc_matrices():

[ 1 1 1 ]
[ \begin{matrix}−1 & −1 & 0 \\ 1 & 0 & −1 \\ 0 & 1 & 1 \end{matrix} ]

[ \begin{matrix}1 \\ −1 \\ 1 \end{matrix} ]

> create_cellcomplex(Degrees=[[2,1,0],[1,0,1],[0,1,2],[0,2,0]],Type=
> ["Simplicial",[[1,2,3],[1,3,4]]]):
> show_cellular_complex(grading="none",Polyring=[x,y,z]):

[ 0 − > R^{2} − > R^{5} − > R^{4} − > I − > 0 ]
I = (x^{2} y, x z, y z^{2}, y^{2})

> test_properties():

[1, 2, 1]

“The cellular complex is not a resolution of I.”

> b:=[1,2,1]:
> subcomplex(degree=b):
> show_cellcomplex():

Cell Dimension Degree
{}  −1
{v_{2}}  0  [1, 0, 1]
{v_{4}}  0  [0, 2, 0]

> show_incidence_function():

CELLULAR RESOLUTIONS OF MONOMIAL IDEALS 157

[ \begin{matrix}ε & \{v_{2}\} & \{v_{4}\} \\ \{\} & 1 & 1 \end{matrix} ]

> show_augmented_oriented_chain_complex():

\left[{0 − > K^{2} − > K − > 0 \atop 1}\right]

> show_aocc_matrices():

[ 1 1 ]

> Hilbert_function(Degrees=[[2,1,0],[1,0,1],[0,1,2],[0,2,0]],Compone
> nts=0..10):

Component R R / I I
0  1 1 0
1  3 3 0
2  6 4 2
3  10 2 8
4  15 2 13
5  21 2 19
6  28 2 26
7  36 2 34
8  45 2 43
9  55 2 53
10  66 2 64

The command

> generic():

checks whether a given monomial ideal I is generic. If so, Theorem 3.5.7 implies that the cellular complex 𝔽_{Δ_{I}} derived from the Scarf complex is a minimal free resolution of I. We look at Example 3.5.9:

> generic(Degrees=[[1,1,1],[4,3,0],[3,5,0],[0,4,3],[0,2,4],[2,0,2]])
> ;

true

> create_cellcomplex(Degrees=[[1,1,1],[4,3,0],[3,5,0],[0,4,3],[0,2,4
> ],[2,0,2]],Type="Scarf"):
> show_cellcomplex():

Cell  Dimension Degree
{}  −1
{v_{1}}  0  [1, 1, 1]
{v_{2}}  0  [4, 3, 0]
{v_{3}}  0  [3, 5, 0]
{v_{4}}  0  [0, 4, 3]
{v_{5}}  0  [0, 2, 4]
{v_{6}}  0  [2, 0, 2]
{v_{1}, v_{2}}  1  [4, 3, 1]
{v_{1}, v_{3}}  1  [3, 5, 1]
{v_{1}, v_{4}}  1  [1, 4, 3]
{v_{1}, v_{5}}  1  [1, 2, 4]
{v_{1}, v_{6}}  1  [2, 1, 2]
{v_{2}, v_{3}}  1  [4, 5, 0]
{v_{5}, v_{4}}  1  [0, 4, 4]
{v_{1}, v_{2}, v_{3}}  2  [4, 5, 1]
{v_{1}, v_{5}, v_{4}}  2  [1, 4, 4]

> show_cellular_complex(grading="none",Polyring=[x,y,z]):

[ 0 − > R^{2} − > R^{7} − > R^{6} − > I − > 0 ]
I = (x y z, x^{4} y^{3}, x^{3} y^{5}, y^{4} z^{3}, y^{2} z^{4}, x^{2} z^{2})

> test_properties():

“The cellular complex is a minimal resolution of I.”

One can reduce a given cell complex X by deleting a pair of cells of the same degree (see Section 3.6) by using

> delete_pair():

We consider Example 3.6.1.

> create_cellcomplex(Degrees=[[2,0,3],[3,0,2],[1,1,1],[0,2,0]],Type=
> "Taylor"):
> show_cellular_complex(grading="none",Polyring=[x,y,z]):

[ 0 − > R − > R^{4} − > R^{6} − > R^{4} − > I − > 0 ]
I = (x^{2} z^{3}, x^{3} z^{2}, x y z, y^{2})

> test_properties():

“The cellular complex is a resolution of I, but not minimal.”

> delete_pair([1,2,3,4],[1,2,4]):
> show_cellular_complex(grading="none",Polyring=[x,y,z]):

[ 0 − > R^{3} − > R^{6} − > R^{4} − > I − > 0 ]
I = (x^{2} z^{3}, x^{3} z^{2}, x y z, y^{2})

> test_properties():

“The cellular complex is a resolution of I, but not minimal.”

> delete_pair([2,3,4],[2,4]):
> show_cellular_complex(grading="none",Polyring=[x,y,z]):

[ 0 − > R^{2} − > R^{5} − > R^{4} − > I − > 0 ]

I = (x^{2} z^{3}, x^{3} z^{2}, x y z, y^{2})

> test_properties():

“The cellular complex is a resolution of I, but not minimal.”

> delete_pair([1,3,4],[1,4]):
> show_cellular_complex(grading="none",Polyring=[x,y,z]):

[ 0 − > R − > R^{4} − > R^{4} − > I − > 0 ]
I = (x^{2} z^{3}, x^{3} z^{2}, x y z, y^{2})

> show_cc_matrices(Polyring=[x,y,z]):

[ x^{2} z^{3} x^{3} z^{2} x y z y^{2} ]
[ \begin{matrix}−x & −y & 0 & 0 \\ z & 0 & −y & 0 \\ 0 & x z^{2} & x^{2} z & −y \\ 0 & 0 & 0 & x z \end{matrix} ]

[ \begin{matrix}y \\ −x \\ z \\ 0 \end{matrix} ]

> test_properties():

“The cellular complex is a minimal resolution of I.”

It is also possible to reduce a cell complex by deleting a cell together with two larger cells. We described this procedure in Section 3.6. It is the command

> delete():

For instance, it can be used for the hull complex in Example 3.6.2.

> create_cellcomplex(Degrees=[[2,0,0],[0,2,0],[0,0,2],[1,1,0],[1,0,1
> ],[0,1,1]],Type=["Simplicial",[[1,4,5],[2,4,6],[3,5,6],[4,5,6]]]):
> show_cellular_complex(grading="none",Polyring=[x,y,z]):

[ 0 − > R^{4} − > R^{9} − > R^{6} − > I − > 0 ]
I = (x^{2}, y^{2}, z^{2}, x y, x z, y z)

> test_properties():

“The cellular complex is a resolution of I, but not minimal.”

> delete([5,6]):
> show_cellular_complex(grading="none",Polyring=[x,y,z]):

[ 0 − > R^{3} − > R^{8} − > R^{6} − > I − > 0 ]
I = (x^{2}, y^{2}, z^{2}, x y, x z, y z)

> test_properties():

“The cellular complex is a minimal resolution of I.”

> Hilbert_function(Degrees=[[2,0,0],[0,2,0],[0,0,2],[1,1,0],[1,0,1],
> [0,1,1]],Components=0..10):

Component R R / I I
0  1 1 0
1  3 3 0
2  6 0 6
3  10 0 10
4  15 0 15
5  21 0 21
6  28 0 28
7  36 0 36
8  45 0 45
9  55 0 55
10  66 0 66

The exactness of a cellular complex 𝔽_{X} may depend on the choice of the field. Therefore, we have the optional argument char for the commands

> test_properties():
> show_augmented_oriented_chain_complex():
> Betti_numbers():

The default is char = 0. In this case MAPLE tests the acyclicity over K = ℝ. If char = p for a prime number p, MAPLE´ s computations are done over the finite field K = 𝔽_{p}. The choice of the field is relevant in Example 3.1.23.

> create_cellcomplex(Degrees=[[1,1,1,0,0,0],[1,1,0,0,0,1],[1,0,1,0,1
> ,0],[1,0,0,1,1,0],[1,0,0,1,0,1],[0,1,1,1,0,0],[0,1,0,1,1,0],[0,1,0
> ,0,1,1],[0,0,1,1,0,1],[0,0,1,0,1,1]],Type=[[1,2],[1,3],[1,6],[2,5]
> ,[2,8],[3,4],[3,10],[4,5],[4,7],[5,9],[6,7],[6,9],[7,8],[8,10],[9,
> 10],[1,2,3,8,10],[1,2,5,6,9],[1,3,4,6,7],[2,4,5,7,8],[3,4,5,9,10],
> [6,7,8,9,10]]):
> show_cellular_complex(grading="none",Polyring=[x,y,z,u,v,w]):

[ 0 − > R^{6} − > R^{15} − > R^{10} − > I − > 0 ]
I = (x y z, x y w, x z v, x u v, x u w, y z u, y u v, y v w, z u w, z v w)

> test_properties():

“The cellular complex is a minimal resolution of I.”

> test_properties(char=2):

[1, 1, 1, 1, 1, 1]

“The cellular complex is not a resolution of I.”

> show_augmented_oriented_chain_complex():

\left[{0 − > K^{6} − > K^{15} − > K^{10} − > K − > 0 \atop 6  9  1}\right]

> show_augmented_oriented_chain_complex(char=2):

\left[{0 − > K^{6} − > K^{15} − > K^{10} − > K − > 0 \atop 5  9  1}\right]

   > Hilbert_function(Degrees=[[1,1,1,0,0,0],[1,1,0,0,0,1],[1,0,1,0,1,0
   > ],[1,0,0,1,1,0],[1,0,0,1,0,1],[0,1,1,1,0,0],[0,1,0,1,1,0],[0,1,0,0
   > ,1,1],[0,0,1,1,0,1],[0,0,1,0,1,1]],Components=0..10):

Component R R / I I
0  1  1  0
1  6  6  0
2  21 21 0
3  56 46 10
4  126 81 45
5  252 126 126
6  462 181 281
7  792 246 546
8  1287 321 966
9  2002 406 1596
10  3003 501 2502

   > interp([6,7,8,9,10],[181,246,321,406,501],X);

5 X^{2} + 1

There are also procedures which are helpful to compute the hull complex of a monomial ideal I. The command

   > create_hull_vertices():

generates the vertices

t^{a^{1}}, …, t^{a^{s}}

of the polyhedron P_{t} ⊂ ℝ^{n}. MAPLE always chooses t = (n + 1)! + 1.

   > system_of_hyperplanes():

shows the set of solutions of the system of linear equations

⟨v, t^{a^{j}}⟩ = 1, j ∈ J.

By setting values for the variables appearing in this set, the command

   > hyperplane():

yields a hyperplane H(v, 1). One can test if this hyperplane supports P_{t} by

   > test_hyperplane();

This command checks if the conditions of Proposition 3.3.6(b), which are

v ▷ 0,
⟨v, t^{a^{j}}⟩ = 1, j ∈ J,
⟨v, t^{a^{j}}⟩ > 1, j ∉ J,

are fulfilled. If this is the case,

F = conv {t^{a^{1}}, …, t^{a^{s}}}

is a bounded face of P_{t}. An arbitrary hyperplane can also directly be checked by

   > verify_hyperplane():

If one wants to prove that for a given set J ⊂ {1, …, s}, J ≠ ∅ the convex hull

F = conv {t^{a^{1}}, …, t^{a^{s}}}

is not a bounded face of P_{t}, first create the system of linear inequalities and linear equations obtained from

v ▷ 0,
⟨v, t^{a^{j}}⟩ = 1, j ∈ J,
⟨v, t^{a^{j}}⟩ > 1, j ∉ J.

This is done by

> create_sli():

The system can be shown by

> show_sli():

The command

> sli_addition():

adds a real constant on each side of an inequality, and

> sli_multiplication():

multiplies an inequality with a real constant. Two lines of the system can be added by

> sli_lineaddition():

If the set of solutions is already computed, one can also apply the command

> create_sli_p():

These commands shall be illustrated by Example 3.4.19:

> create_hull_vertices(Degrees=[[2,0,0],[0,2,0],[0,0,2],[1,1,0],[1,0
> ,1],[0,1,1]]):

[[625, 1, 1], [1, 625, 1], [1, 1, 625], [25, 25, 1], [25, 1, 25], [1, 25, 25]]

> system_of_hyperplanes([4,5]):

[\frac{1}{25} − \frac{26}{25} t_{1}, t_{1}, t_{1}]

> hyperplane([1/30]):

[\frac{2}{375}, \frac{1}{30}, \frac{1}{30}, 1]

> test_hyperplane();

true

> system_of_hyperplanes([1,4,5]):

[\frac{1}{675}, \frac{1}{27}, \frac{1}{27}]

> hyperplane([]):

[\frac{1}{675}, \frac{1}{27}, \frac{1}{27}, 1]

> test_hyperplane();

true

> system_of_hyperplanes([1,2]):

[ t_{1}, t_{1}, −626 t_{1} + 1]

CELLULAR RESOLUTIONS OF MONOMIAL IDEALS 163

> hyperplane([1/1000]):

[\frac{1}{1000}, \frac{1}{1000}, \frac{187}{500}, 1]

> test_hyperplane();

“Vertex 4 is below the hyperplane.” false

> create_sli():
> show_sli():

[ \begin{matrix}1 . & t_{1} & > & 0 \\ 2 . & t_{1} & > & 0 \\ 3 . & −626 t_{1} + 1 & > & 0 \\ 4 . & 1 & = & 1 \\ 5 . & 1 & = & 1 \\ 6 . & −391248 t_{1} + 625 & > & 1 \\ 7 . & −576 t_{1} + 1 & > & 1 \\ 8 . & −15624 t_{1} + 25 & > & 1 \\ 9 . & −15624 t_{1} + 25 & > & 1 \end{matrix} ]

> sli_addition(7,-1):
> sli_multiplication(7,-1/576):
> show_sli():

[ \begin{matrix}1 . & t_{1} & > & 0 \\ 2 . & t_{1} & > & 0 \\ 3 . & −626 t_{1} + 1 & > & 0 \\ 4 . & 1 & = & 1 \\ 5 . & 1 & = & 1 \\ 6 . & −391248 t_{1} + 625 & > & 1 \\ 7 . & t_{1} & < & 0 \\ 8 . & −15624 t_{1} + 25 & > & 1 \\ 9 . & −15624 t_{1} + 25 & > & 1 \end{matrix} ]

> create_sli_p(hyperplanes=vector([1/651, 26/651-_t[1],_t[1]]),verti
> ces=[1,6]):
> sli_lineaddition(7,8):
> show_sli():

[ \begin{matrix}1 . & \frac{1}{651} & > & 0 \\ 2 . & \frac{26}{651} − t_{1} & > & 0 \\ 3 . & t_{1} & > & 0 \\ 4 . & 1 & = & 1 \\ 5 . & \frac{5417}{217} − 624 t_{1} & > & 1 \\ 6 . & \frac{9}{217} + 624 t_{1} & > & 1 \\ 7 . & \frac{225}{217} − 24 t_{1} & > & 1 \\ 8 . & \frac{242}{217} & > & 2 \\ 9 . & 1 & = & 1 \end{matrix} ]

Example 3.4.20 gives another illustration:

> create_hull_vertices(Degrees=[[1,0,1,0],[1,0,0,1],[0,1,1,0],[0,1,0
> ,1]]):

[[121, 1, 121, 1], [121, 1, 1, 121], [1, 121, 121, 1], [1, 121, 1, 121]]

> verify_hyperplane(hyperplane=[439/61000,439/61000,1/1000,1/1000,1]
> ,vertices=[1,2,3,4]);

true

> system_of_hyperplanes([1,4]):

[\frac{1}{122} − t_{1}, \frac{1}{122} − t_{2}, t_{1}, t_{2}]

> hyperplane([1,1]):

[\frac{−121}{122}, \frac{−121}{122}, 1, 1, 1]

> test_hyperplane();

“There are non positive coefficients.” “Vertex 2 is in the hyperplane.” “Vertex 3 is in the hyperplane.”

> create_sli():
> sli_lineaddition(6,7):
> show_sli():

[ \begin{matrix}1 . & \frac{1}{122} − t_{1} & > & 0 \\ 2 . & \frac{1}{122} − t_{2} & > & 0 \\ 3 . & t_{1} & > & 0 \\ 4 . & t_{2} & > & 0 \\ 5 . & 1 & = & 1 \\ 6 . & 1 − 120 t_{1} + 120 t_{2} & > & 1 \\ 7 . & 2 & > & 2 \\ 8 . & 1 & = & 1 \end{matrix} ]

It is helpful to have available elementary operations of monomial ideals. Use

   > minimal_generators():

in order to get the system of minimal generators of a monomial ideal. For example, the monomial ideal

I = (x^{3}, xy, y^{4}, x^{2}, xy^{2}) ⊂ R = K[x, y]

can also be written as

I = (xy, y^{4}, x^{2}),

which is a representation of I using its set of minimal generators.

   > minimal_generators([[3,0],[1,1],[0,4],[2,0],[1,2]]);

[[1, 1], [0, 4], [2, 0]]

The sum of two ideals will be computed by the command

   > ideal_addition():

and the product by

   > ideal_multiplication():

One gets the n-th power of an ideal by

   > ideal_power():

Use

   > ideal_intersection():

to obtain the intersection of two ideals. For example, for the two ideals

I = (x^{3}, xy, y^{4}),
J = (x^{2}, xy^{2})

in the polynomial ring R = K[x, y] we have

I + J = (xy, y^{4}, x^{2}),
I · J = (x^{5}, x^{3}y, x^{2}y^{3}, xy^{6}),
J^{3} = (x^{6}, x^{5}y^{2}, x^{4}y^{4}, x^{3}y^{6}),
I ∩ J = (x^{3}, x^{2}y, xy^{2}).

   > A:=[[3,0],[1,1],[0,4]];

A := [[3, 0], [1, 1], [0, 4]]

   > B:=[[2,0],[1,2]];

B := [[2, 0], [1, 2]]

  > C:=ideal_addition(A,B);

C := [[1, 1], [0, 4], [2, 0]]

  > C:=ideal_multiplication(A,B);

C := [[5, 0], [3, 1], [2, 3], [1, 6]]

  > C:=ideal_power(B,3);

C := [[6, 0], [5, 2], [4, 4], [3, 6]]

  > C:=ideal_intersection(A,B);

C := [[3, 0], [2, 1], [1, 2]]

References

[1] Günter Ewald. Combinatorial Convexity and Algebraic Geometry. Springer, New York, 1996.

[2] Branko Grünbaum. Convex Polytopes. Interscience Publ., London, 1967.

[3] Winfried Bruns; Jürgen Herzog. Cohen Macaulay rings. Cambridge University Press, Cambridge, 1993.

[4] Winfried Bruns; Jürgen Herzog. On multigraded resolutions. J. Symbolic Computation, 118:245–257, 1995.

[5] William S. Massey. Singular Homology Theory. Springer, New York, 1980.

[6] James R. Munkres. Elements of Algebraic Topology. Addison-Wesley, California, 1984.

[7] Edwin H. Spanier. Algebraic Topology. Springer, New York, 1981.

[8] Dave Bayer; Bernd Sturmfels. Cellular resolutions of monomial modules. J. reine angew. Math., 502:123–140, 1998.

[9] Winfried Bruns; Udo Vetter. Determinantal Rings. Chapman and Hall, London, 1974.

[10] Ezra Miller; Bernd Sturmfels; Kohji Yanagawa. Generic and Cogeneric Monomial Ideals. J. Symbolic Computation, 29:691–708, 2000.

[11] A. Björner; M. Las Vergnas; B. Sturmfels; N. White; G. Ziegler. Oriented matroids. Cambridge University Press, Cambridge, 1993.

Erklärung

Ich versichere, daß ich die Arbeit ohne fremde Hilfe und ohne Benutzung anderer als der angegebenen Quellen angefertigt habe und daß die Arbeit in gleicher oder ähnlicher Form noch keiner anderen Prüfungsbehörde vorgelegen hat und von dieser als Teil einer Prüfungsleistung angenommen wurde. Alle Ausführungen, die wörtlich oder sinngemäß übernommen wurden, sind als solche gekennzeichnet.

Paderborn, den 10.06.2002