Syzygies of the Graded Algebra Relative to a Valuation

A. Campillo^{∗}.

D. Algebra. Fac. Ciencias. Prado de la Magdalena sn. 47005 Valladolid. SPAIN. Email: campillo@agt.uva.es.

C. Galindo^{∗}^{†}.

D. Matemáticas (ESTCE), UJI, Campus Riu Sec. 12071 Castellón. SPAIN. Email: galindo@nuvol.uji.es.

1 Introduction

Throughout this paper, v will denote a valuation of the quotient field F of a Noetherian local domain R. Also, we shall assume that v is centered at R. Denote by K the residue field of R and by S (:= v(R \ {0}) the value semigroup of v. For each m ∈ S, set P_{m}(P_{m}^{+}) := {f ∈ R|v(f) ≥ (>)m}. P_{m} and P_{m}^{+} are ideals of R and we call the graded algebra of R relative to v to the S-graded K-algebra

gr_{v}R = ⨁_{m∈S} P_{m}/P_{m}^{+}.

The analysis of the structure of gr_{v}R can be understood as an attempt to approach the problem of classifying valuations. This algebra is a tool which recently has been systematically used by Spivakovsky [8] and Teissier [9] for approaching local uniformization, a problem close to that of resolution of singularities of algebraic schemes. For additional properties of gr_{v}R, we refer to [7], [6], [3] and [5].

Teissier’s approach looks local uniformization by extending methods from toric geometry to infinite dimension. This paper keeps on with the same spirit by trying to establish analogies between gr_{v}R and affine toric varieties. Recall that such varieties are those given by algebras of finitely generated semigroups.

∗Supported by DGICYT PB-97-0471

†Supported by F. Bancaixa P1B-98-24

In the toric case, one studies the K-algebra K[S], where K is a field and S =< n_{1}, n_{2}, …, n_{h} > a cancellative commutative finitely generated semigroup such that S ∩ (−S) = 0. In the literature, there are two basic references for the study, the presentation and the syzygies of the (S-graded algebra) K[S]. Firstly, that given by Bayer and Sturmfels in their paper [1]. They consider the polynomial ring A = K[X_{1}, X_{2}, …, X_{h}] and regard K[S] as the quotient A/I, where I is the kernel of the natural epimorphism A → K[S]. Then, they obtain minimal free resolutions of K[S] as A-module from suitable resolutions of the A[L]-module M_{L} := K[N^{h} + L], L being a lattice of Z^{h}. In fact, L is the kernel of the natural map Z^{h} → G(S), where G(S) is the group spanned by S. Here, the point is that

I =< X^{a} − X^{b}|a − b ∈ L >,

where X^{c} = X_{1}^{c_{1}}X_{2}^{c_{2}} ⋯ X_{h}^{c_{h}}, whenever c = (c_{1}, c_{2}, …, c_{h}) ∈ N^{h}.

Secondly, one has the recent paper [4] by Campillo and Gimenez. Inspired by [2], they describe the successive construction of a minimal graded resolution of K[S] as A-module and prove that an explicit isomorphism can be given between the vector space of degree m syzygies and the vector space of augmented homology of the simplicial complex

△_{m} = {J ⊆ {1, 2, …, n}|m − ∑_{i∈J} n_{i} ∈ S}.

Using this fact, they obtain the syzygies for K[S] in combinatorial terms, namely by means of vector space complexes associated to directed graphs.

For the case of the graded algebra relative to a valuation, in [3] it is given an inductive method for computing a minimal system of generators M for gr_{v}R. This allows us to regard gr_{v}R as an A[v] module, where A[v] is a (possibly) infinitely dimensional polynomial ring whose variables correspond one to one to elements in M. So, it is natural try to adapt the ideas and techniques of the toric case to the algebra gr_{v}R. However, Bayer and Sturmfels techniques can not be adapted, as, for it, one would need that the kernel of the mapping A[v] → gr_{v}R were spanned by binomials such that the difference between their exponents were in a lattice, or at least to have a group structure for convenient choices of M. Because of the properties of the valuations, it is clear that it is not always true.

The purpose of this paper is to extend the techniques given in [4] to describe, in combinatorial terms, equations and syzygies for gr_{v}R. We will see how, in fact, main features in [4] can be adapted and extended for the case of graded algebras relative to valuations.

A first step towards our extension is already shown in [3] by making use of the facts that gr_{v}R is graded over the semigroup of values S of v and that S is a combinatorially finite semigroup (i.e., the number of decompositions of any element m ∈ S as a finite sum of others in S \ {0} is finite). This last property gives a finer condition than S ∩(−S) = 0, taking into account that, now, one is dealing with infinitely generated semigroups in general. Notice that the semigroup S can be seen as the infinite analogous of an object of toric geometry.

A minimal S-graded resolution of gr_{v}R as A[v]-module exists since S is combinatorially finite. Moreover, as shown in [3], for each m ∈ S, there are K-vector space complexes C.(△_{m}, \underline{Q}_{m}), △_{m} being simplicial complexes given as above, in such a way that \underline{Q}_{m} provide ”weights” on the faces of △_{m}. These complexes satisfy the property that minimal systems of homogeneous generators for the successive syzygy modules of gr_{v}R can be derived from their homology.

The main part of this paper is devoted to give a method to compute the former homology. That method is developed in Section 3 and it works for any partition of certain subset Γ_{m} of the index set of M, although for it we need to enlarge the weights \underline{Q}_{m}. More explicitly, what we show is how such homology can be explicitly computed from bases of the homology of vector space complexes associated to directed graphs, which depend on the partition of Γ_{m}, and to weights which enlarge \underline{Q}_{m} or are quotients of these enlargements. Suitable partitions relate the algebras gr_{v}R and gr_{v}T, where T is a subring of R.

2 Linear weighted simplicial complexes

In this section, we introduce the concept of linear weighted simplicial complex and give, for valuations v of F centered at R, certain linear weighted simplicial complexes whose homology allows to describe minimal systems of homogeneous generators for the successive syzygy modules of gr_{v}R as A[v]-module. To study them, we also introduce the concepts of weighted enlargement of a linear weighted simplicial complex and of linear weighted relative simplicial complex, which will be used in the next section.

2.1 General definitions

Definition 1 A linear weighted simplicial complex is an assignation (△, \underline{Q}) where:

1. △ is a simplicial subcomplex of P (Z), P (Z) being the power set of a finite set Z, that is to say, △ is a finite collection of finite subsets of Z such that if J ∈ △, then P (J) ⊆ △. Each J is called a face of △ of dimension card(J) − 1.

2. For every L ∈ △, \underline{Q}(L) is a K-vector space. K being a fixed field.

3. For every inclusion L ⊂ J of elements in △ of codimension one, (i.e. cardJ = 1 + cardL), there exists a morphism of K-vector spaces l_{JL} : \underline{Q}(J) → \underline{Q}(L)

subjects to the condition

1. If L, D_{1}, D_{2}, J are in △ and one has the following inclusions of codimension one: L ⊂ D_{1}, L ⊂ D_{2}, D_{1} ⊂ J and D_{2} ⊂ J, then

l_{D_{1}L}l_{JD_{1}} + l_{D_{2}L}l_{JD_{2}} = 0.

To every linear assignation (△, \underline{Q}), one can associate a chain complex of vector spaces C.(△, \underline{Q}) as follows:

The r-chain vector space is C_{r}(△, \underline{Q}) = ⨁_{J∈△,card(J)=r+1} \underline{Q}(J), and the boundary ∂ : C_{r}(△, \underline{Q}) → C_{r−1}(△, \underline{Q}) is given by

∂(a) = ∑_{β∈J} l_{J,J\\{β\}}(a) ∈ ⨁_{β∈J} \underline{Q}(J \ \{β\}) ⊆ C_{r−1}(△, \underline{Q})

for a ∈ \underline{Q}(J).

If (△, \underline{Q}) is a linear weighted simplicial complex, then we can define a linear weighted relative simplicial complex as an assignation (△, △′, \underline{Q}) where △′ is a simplicial subcomplex of △. It is clear that we can associate to (△, △′, \underline{Q}) a complex of vector spaces C.(△, △′, \underline{Q}) where C_{r}(△, △′, \underline{Q}) = ⨁_{J∈△\△′,card(J)=r+1} \underline{Q}(J) and the boundary is defined as above. Thus, one gets the exact sequence of K-vector space complexes

0 → C.(△′, \underline{Q}) → C.(△, \underline{Q}) → C.(△, △′, \underline{Q}) → 0  (1)

and therefore the following homology long exact sequence:

⋯ → H_{r+1}(△, △′, \underline{Q}) → H_{r}(△′, \underline{Q}) → H_{r}(△, \underline{Q}) →  (2)
→ H_{r}(△, △′, \underline{Q}) → ⋯ → H_{−1}(△, \underline{Q}) → H_{−1}(△, △′, \underline{Q}) → 0.

Notice that, along this paper, we shall denote by H.(△, \underline{Q}) the homology of the vector space complex C.(△, \underline{Q}) which depends on the assignation (△, \underline{Q}) and by H.(△, △′, \underline{Q}) the homology in the relative case.

On the other hand, assume that (△, \underline{Q}) and (△, \underline{Q′}) are two linear weighted simplicial complexes associated to the same simplicial complex △. (△, \underline{Q′}) is called a weighted enlargement of (△, \underline{Q}) if for each J ∈ △, \underline{Q}(J) is a vector subspace of \underline{Q′}(J) and the morphisms l′_{JL}, associated to (△, \underline{Q′}), satisfy that the restriction of l′_{JL} to \underline{Q}(J) equals l_{JL} for each inclusion of codimension one L ⊂ J of elements in △. In this case, we can consider the linear weighted simplicial complex (△, \underline{Q′}/\underline{Q}) where (\underline{Q′}/\underline{Q})(J) stands for the quotient vector space of \underline{Q′}(J) with respect to \underline{Q}(J) for each J ∈ △ and, for every inclusion of codimension one L ⊂ J of elements in △, the morphism (\underline{Q′}/\underline{Q})(J) → (\underline{Q′}/\underline{Q})(L) is the one induced by l′_{JL}. Therefore, the exact sequence of K-vector space complexes

0 → C.(△, \underline{Q}) → C.(△, \underline{Q′}) → C.(△, \underline{Q′}/\underline{Q}) → 0

and the associated homology long exact sequence

⋯ → H_{r+1}(△, \underline{Q′}/\underline{Q}) → H_{r}(△, \underline{Q}) → H_{r}(△, \underline{Q′}) →
→ H_{r}(△, \underline{Q′}/\underline{Q}) → ⋯ → H_{−1}(△, \underline{Q′}) → H_{−1}(△, \underline{Q′}/\underline{Q}) → 0

hold.

2.2 Linear weighted complexes associated to valuations

Let v be a valuation of F centered at R and S its value semigroup. Set M = {x_{γ}}_{γ∈Γ} a minimal system of homogeneous generators of the graded algebra of R relative to v, gr_{v}R. In [3], we give a method for computing a set M as above. The set Γ can be viewed as a (possibly infinite) set of pairs γ = (β, i), where i is a positive integer and β an element in S called the degree of γ and denoted by deg(γ). If γ = (β, i), then x_{γ} ∈ P_{β}/P_{β}^{+} and the number of elements in Γ of a fixed degree is finite since the dimension as Kvector space of P_{β}/P_{β}^{+} so is. Consider A[v] := K[{X_{γ}}_{γ∈Γ}], where {X_{γ}}_{γ∈Γ} is a set of algebraically independent indeterminates. Give to X_{γ} the degree deg(γ) ∈ S, then A[v] is an S-graded algebra and the epimorphism of graded K-algebras ϕ_{0} : A[v] → gr_{v}R, given by X_{γ} → x_{γ}, allows to regard gr_{v}R as an A[v]-module. Homogeneous generators for its successive syzygy modules can be obtained from the homology of the linear weighted simplicial complexes (△_{m}, \underline{Q}_{m}) which we define next (see [3]).

For each m ∈ S, set Γ_{m} = {γ ∈ Γ|m−deg(γ) ∈ S}. Γ_{m} is a finite set since S is a combinatorially finite semigroup. Then, △_{m} := {J ⊆ Γ_{m}|m−β_{J} ∈ S}, where β_{J} = ∑_{γ∈J} deg(γ); \underline{Q}_{m}(J) := (gr_{v}R)_{m−β_{J}}, where (gr_{v}R)_{α} = P_{α}/P_{α}^{+} denotes the homogeneous component of degree α of gr_{v}R, and for J ∈ △_{m} and L = J\\{γ\}, l_{JL}^{m} : \underline{Q}_{m}(J) ⟶ \underline{Q}_{m}(L) is defined so l_{JL}^{m}(a) = (−1)^{η_{J} (γ)}x_{γ}a, η_{J}(γ) being the number of the place that γ has among the elements in J with respect to a previously fixed total ordering on Γ.

With the aim of studying the homology of the K-vector space complex associated to (△_{m}, \underline{Q}_{m}), we give a weighted enlargement (△_{m}, \underline{Q′}_{m}) of (△_{m}, \underline{Q}_{m}). If J ∈ △_{m}, then \underline{Q′}_{m}(J) = (T_{m}^{−1}gr_{v}R)_{m−β_{J}}, where T_{m} denotes the set of finite products of elements in {x_{γ}|γ ∈ Γ_{m}} which is a multiplicative system in gr_{v}R, T_{m}^{−1}gr_{v}R the quotient ring of gr_{v}R with respect to T_{m} and (T_{m}^{−1}gr_{v}R)_{α} the degree α homogeneous component of the former quotient ring. The vector space morphism l_{JL}^{m} : \underline{Q′}_{m}(J) ⟶ \underline{Q′}_{m}(L) will have the same definition as above.

Finally if Σ_{m} denotes the power set of Γ_{m}, then it is clear that the linear weighted simplicial complexes (Σ_{m}, \underline{Q′}_{m}) and (Σ_{m}, \underline{Q′}_{m}/\underline{Q}_{m}) make sense taking into account that if J ∈ Σ_{m}, then (\underline{Q′}_{m}/\underline{Q}_{m})(J) will be the quotient vector space \underline{Q′}_{m}(J)/\underline{Q}_{m}(J) whenever J ∈ △_{m}, and \underline{Q′}_{m}(J) otherwise.

3 Linear weighted graphic homologies

The concept of graphic homology was introduced in [4] by Campillo and Gimenez in order to study the syzygies of affine toric varieties. In this section, we generalize this concept by defining when the homology of a complex of vector spaces is what we shall call linear weighted graphic. We shall prove that an example of this type of homology is that of the K-vector space complex associated to (△_{m}, \underline{Q}_{m}) and so, we shall get the main goal of this type of homologies H.(C), which is that bases for the spaces H_{r}(C) can be explicitly reached from the homology of vector space complexes associated to suitable graphs and linear weights.

3.1 General definitions

Let S be a combinatorially finite semigroup. Assume that there exist a set Γ and a map deg : Γ → S such that deg(Γ) is a finite system of generators of S. Consider Λ a subset of Γ and B a subset of S. Then, we shall call the directed graph of S associated to the pair (Λ, B) to the directed graph G_{ΛB}(S), whose vertex set is formed by those elements m of the group spanned by S such that there exists a subset L of Λ satisfying m − β_{L} ∈ B, β_{L} = ∑_{γ∈L} deg(γ), and such that two vertices m and m′ are joined (m with m′) whenever m′ = m + deg(λ) for some λ ∈ Λ. In what follows, we shall denote G_{ΛB} instead of G_{ΛB}(S) whenever it does not cause confusion.

Let m be a vertex of G_{ΛB}, denote by Ω_{ΛB}(m) the set of subsets L of Λ such that m − β_{L} ∈ B. Assume that (Σ, \underline{R}) is a linear weighted simplicial complex, such that Ω_{ΛB}(m) ⊆ Σ ⊆ P (Λ). Then, we can state the following

Proposition 1 Let S, Γ, Λ, B, m be as before. Assume that for every b ∈ B and for every λ, λ′ ∈ Λ such that b + deg(λ) + deg(λ′) ∈ B, it is satisfied b + deg(λ) ∈ B and b + deg(λ′) ∈ B. Then, the family of vector spaces {C_{r}(G_{ΛB}(m)}_{r≥−1},

C_{r}(G_{ΛB}(m)) = ⨁_{L∈Ω_{ΛB}(m),card(L)=r+1} \underline{R}(L)

together with the vector space morphisms C_{r}(G_{ΛB}(m)) → C_{r−1}(G_{ΛB}(m)) induced by the projection of the boundaries of C.(Σ, \underline{R}) are a chain complex of vector spaces denoted by C.(G_{ΛB}(m), \underline{R}).

Proof. It suffices to show that if L ∈ Ω_{ΛB}(m) and L \ {λ, λ′} ∈ Ω_{ΛB}(m), λ, λ′ ∈ L, λ ≠ λ′, then L \ {λ} and L \ {λ′} are in Ω_{ΛB}(m). Indeed, suppose that L = {λ_{1}, λ_{2}, …, λ_{r−1}, λ, λ′}, then m − ∑_{i=1}^{r−1} deg(λ_{i}) ∈ B. Also m − deg(λ) − deg(λ′) − ∑_{i=1}^{r−1} deg(λ_{i}) = b ∈ B. So, b + deg(λ) + deg(λ′) ∈ B and b + deg(λ) and also b + deg(λ′) are in B as desired.

Definition 2 A directed graph G_{ΛB}(S) which satisfies the condition given in the last proposition is said to be a chain graph.

Definition 3 The homology H.(C) of a vector space complex C. is defined to be linear weighted purely graphic if there exist a chain graph G_{ΛB}(S), a vertex m of G_{ΛB}(S) and a linear weighted simplicial complex (Σ, \underline{R}), Ω_{ΛB}(m) ⊆ Σ ⊆ P (Λ), such that

H_{r}(C) ≅ H_{r}(G_{ΛB}(m), \underline{R}),

for all r ≥ 1.

Definition 4 The homology H.(C) of a vector space complex C. is said to be linear weighted graphic if bases of the vector spaces H_{r}(C) can be explicitly computed from bases of finitely many linear weighted purely graphic homologies by explicitly given exact sequences.

From now on, v will be a valuation of F centered at R and our conditions and notations will be those of the Introduction and 2.2.

3.2 The semigroups S_{m}

Let m be an element in the value semigroup S of v. Let S̄_{m} = {m′ ∈ S|m − m′ ∈ S}. The set S̄_{m} is finite since S is combinatorially finite. Let S_{m} be the subsemigroup of S spanned by S̄_{m}. It is obvious that Γ_{m} = {γ ∈ Γ|deg(γ) ∈ S̄_{m}}.

Proposition 2 With the above notations, the following equality

△_{m} = {J ⊆ Γ_{m}|m − β_{J} ∈ S_{m}}

holds.

Proof. Let J = {γ_{1}, γ_{2}, …, γ_{d}} be in △_{m}, then m = s + ∑_{i=1}^{d} deg(γ_{i}) where s ∈ S. γ_{j} ∈ Γ_{m} for each j = 1, 2, …, d, as m − deg(γ_{j}) = s + ∑_{i≠j} deg(γ_{i}). Moreover, m − β_{J} = s, s ∈ S, and then m − s = β_{J}, which implies s ∈ S̄_{m} and m − β_{J} ∈ S_{m}. This concludes the proof since the reverse inclusion is trivial.

Proposition 3 For each m ∈ S, the semigroup S_{m} is generated by the degrees of the elements in Γ_{m}.

Proof. Denote by deg(Γ_{m}) the set of degrees of the elements in Γ_{m}, then deg(Γ_{m}) ⊆ S̄_{m} and < deg(Γ_{m}) >⊆ S_{m}, where < A > stands for the semigroup of S generated by A, A ⊆ S.

Conversely, let t be an element in S_{m}, then t = ∑_{i=1}^{d} n_{i}t_{i}, where n_{i} ∈ Z^{+} (the set of positive integers) and t_{i} ∈ S̄_{m} (1 ≤ i ≤ d). So, for each i, we can write m = t_{i} + s, s ∈ S. On the other hand, M = {x_{γ}}_{γ∈Γ} generates gr_{v}R and since t_{i} ∈ S, we get t_{i} = ∑_{j=1}^{e} k_{j}deg(γ_{j}), k_{j} ∈ Z^{+} and γ_{j} ∈ Γ (1 ≤ j ≤ e). Now, for every j = 1, 2, …, e, one has

m − deg(γ_{j}) = s + (k_{j} − 1)deg(γ_{j}) + ∑_{l≠j} k_{l}deg(γ_{l}),

and therefore deg(γ_{j}) ∈ S̄_{m}. Thus, γ_{j} ∈ Γ_{m} and t ∈< deg(Γ_{m}) >.

3.3 The homology H.(△_{m}, \underline{Q}_{m})

We want to prove that the homology H.(△_{m}, \underline{Q}_{m}) is linear weighted graphic. To do it, we shall use the enlarged weights \underline{Q′}_{m} defined in 2.2, and their corresponding quotient weights (\underline{Q′}_{m}/\underline{Q}_{m}). In the sequel, both weights will be denoted by \underline{R}_{m} for simplicity’s sake. Throughout the rest of this subsection, we shall use several sets which depend on two elements in S denoted by α and m. This is not strictly necessary for the development of the present section, but it will be useful in the last remark.

Let Γ_{α} = Λ_{α} ∪ Π_{α} be a partition of Γ_{α} for a fixed element α in the semigroup of values S of v. Such partitions arise in practice when one considers a subring T of R and since gr_{v}T ⊆ gr_{v}R, one takes Π_{α} to be the index set of the degree α elements of a minimal system of generators for gr_{v}T.

We shall call the Apery set of Γ_{α} relative to Π_{α} to the set

A(α) = {a ∈ S_{α}|a − deg(e) ∉ S_{α} for all e ∈ Π_{α}}.

If no confusion arises, A(α) will be denoted simply by A, and analogously Λ_{α} will be denoted by Λ and Π_{α} by Π.

Let m be an element in S_{α}. For our reasoning, we can assume, without loss of generality, that m is a vertex of G_{ΛA}(S_{α}). Consider the sets:

K_{αm} := {L ⊆ Γ_{α}|L ∩ Π ≠ ∅ and m − β_{L} ∈ S_{α}} ∪ {L ⊆ Λ|m − β_{L} ∈ S_{α} \ A}

and

△_{αm} := {L ⊆ Γ_{α}|m − β_{L} ∈ S_{α}}.

K_{αm} is a simplicial subcomplex of △_{αm}. The linear weighted complexes (△_{αm}, \underline{R}_{m}) and (K_{αm}, \underline{R}_{m}) make sense, and the same happens for the linear weighted relative simplicial complex (△_{αm}, K_{αm}, \underline{R}_{m}) whose associated vector space complex C.(△_{αm}, K_{αm}, \underline{R}_{m}) is exactly the complex C.(G_{ΛA}(m), \underline{R}_{m}). This complex will be denoted C.(A_{m}, \underline{R}_{m}) for short. As a consequence, we get the following homology long exact sequence:

⋯ → H_{r+1}(A_{m}, \underline{R}_{m}) → H_{r}(K_{αm}, \underline{R}_{m}) → H_{r}(△_{αm}, \underline{R}_{m}) →  (3)

→ H_{r}(A_{m}, \underline{R}_{m}) → ⋯ → H_{−1}(△_{αm}, \underline{R}_{m}) → H_{−1}(A_{m}, \underline{R}_{m}) → 0.

Since the homology H.(A_{m}, \underline{R}_{m}) is linear weighted purely graphic, our interest lies in studying the homology H.(K_{αm}, \underline{R}_{m}). In order to do it, we shall extend the construction in [4] for the toric situation by considering the linear weighted simplicial subcomplex of (Σ_{α} ∩ Σ_{m}, \underline{R}_{m}), (\overline{K}_{αm}, \underline{R}_{m}), given by

\overline{K}_{αm} = K_{αm} ∪ \{I ∪ J ⊆ Σ_{α}|I ⊆ Λ, J ⊆ Π, card(J) ≥ 2,
m − β_{I} − β_{J} ∉ S_{α} but m − β_{I} − deg(e) ∈ S_{α} for all e ∈ J}.

Proposition 4 (\overline{K}_{αm}, \underline{R}_{m}) is indeed a linear weighted simplicial complex which is acyclic.

Proof. To start with, we shall show that \overline{K}_{αm} is a simplicial complex which proves that (\overline{K}_{αm}, \underline{R}_{m}) is a linear weighted simplicial complex. In fact, let L = I ∪ J ∈ \overline{K}_{αm} be and take L′ = I′ ∪ J′ such that I′ ⊆ I ⊆ Λ and J′ ⊆ J ⊆ Π. If L ∈ K_{αm}, then L′ ∈ \overline{K}_{αm} since K_{αm} is a simplicial complex. Suppose now that L ∈ \overline{K}_{αm} \ K_{αm}, if m − β_{L′} ∉ S_{α} then L′ ∈ \overline{K}_{αm} \ K_{αm} as m − β_{I′} − deg(e) ∈ S_{α} for all e ∈ J′ and to finish L′ ∈ K_{αm} whenever m − β_{L′} ∈ S_{α}.

Now let us see that the complex is acyclic. Assume that c is a t-cycle in the complex C.(\overline{K}_{αm}, \underline{R}_{m}), c = ∑_{L∈Ω⊆\overline{K}_{αm}} v_{L}, v_{L} ∈ \underline{R}_{m}(L). If there exists some L ∈ Ω such that L ⊆ Λ, then m − β_{L} ∈ S_{α} \ A and therefore m − β_{L∪\{γ_{L}\}} ∈ S_{α} for some γ_{L} ∈ Π. Writing c_{1} = c − ∂(∑_{L∈Ω,L⊆Λ} \frac{v_{L}}{x_{γ_{L}}}), we obtain another cycle which is homologous to c. Notice that, whenever the weight \underline{R}_{m} is \underline{Q′}_{m}/\underline{Q}_{m}, \frac{v_{L}}{x_{γ_{L}}} means the class of the division by x_{γ_{L}} of some representative of v_{L}. So, there is no loss of generality in assuming that c = ∑_{L∈Ω} v_{L} where L ∩ Π ≠ ∅ for all L.

Let us fix a total ordering on Λ and Π and define the following total ordering on Σ_{α}: L ≤ L′; L, L′ ∈ Σ_{α} if, and only if, they satisfy either card(L ∩ Π) < card(L′ ∩ Π), or card(L ∩ Π) = card(L′ ∩ Π) and L ∩ Π is less than L′ ∩ Π for the lexicographical ordering induced by that of Π, or L ∩ Π = L′ ∩ Π and L ∩ Λ is less than L′ ∩ Λ for the lexicographical ordering induced by the one of Λ. Let L be the smallest set of Ω under this ordering and L \ {γ} the smallest one-codimensional face of L. c is a cycle, therefore we can find L′ ∈ Ω such that L \ {γ} is a one-codimensional face of L′. By minimality, L′ = L \ {γ} ∪ {γ′}, γ′ ≠ γ and γ′ ∈ Π. It is clear that \frac{v_{L}}{x_{γ′}} ∈ C_{t+1}(\overline{K}_{αm}, \underline{R}_{m}). Set c_{1} := c − ∂(\frac{v_{L}}{x_{γ′}}) = ∑_{K∈Ω_{1}} w_{K}, then c_{1} = 0 concludes the proof and otherwise the smallest set of Ω_{1} is strictly greater than L. Repeating the procedure we conclude the proof, since we can get only finitely many sets Ω_{i} in this way.

Remark. It is straightforward that the complex C.(\overline{K}_{αm}, K_{αm}, \underline{R}_{m}) makes sense. Thus (\overline{K}_{αm}, K_{αm}, \underline{R}_{m}) is a linear weighted relative simplicial complex and its associated homology long exact sequence (2) and the former proposition prove

H_{r+1}(\overline{K}_{αm}, K_{αm}, \underline{R}_{m}) ≅ H_{r}(K_{αm}, \underline{R}_{m}).

This isomorphism has an explicit expression and we note that its inverse can be obtained in an explicit manner following the previous proof.

Now, we are going to describe a vector space complex whose homology is isomorphic to that of C.(\overline{K}_{αm}, K_{αm}, \underline{R}_{m}). This one works in a similar way to the vector space complexes associated to linear weighted simplicial complexes. Let

D_{α} := {d ∈ S_{α}|∃J ⊆ Π such that card(J) ≥ 2, d − β_{J} ∉ S_{α}
and d − deg(e) ∈ S_{α}, ∀e ∈ J }.

The sets J which guarantee that d ∈ D_{α} are called associated to d in D_{α}. Let also, for m ∈ S_{α},

D_{αm} := {I∪J ∈ Σ_{α}|I ⊆ Λ and J a subset of Π associated to m − β_{I} ∈ D_{α}}.

Put C_{r}(D_{αm}, \underline{R}_{m}) = ⨁_{L∈D_{αm},card(L)=r+2} \underline{R}_{m}(L). (D_{αm}, \underline{R}_{m}) is not a linear weighted simplicial complex, however, with the boundary induced by that of C.(Σ_{m}, \underline{R}_{m}), it follows from the argument in the proof of Proposition 1 that C.(D_{αm}, \underline{R}_{m}) is a vector space complex, and the following homology isomorphism

H_{r+1}(\overline{K}_{αm}, K_{αm}, \underline{R}_{m}) ≅ H_{r}(D_{αm}, \underline{R}_{m}), r ≥ −1,  (4)

holds.

Keeping the notations of this subsection, we are ready to state and prove the following

Proposition 5 The homology H.(△_{αm}, \underline{R}_{m}) is linear weighted graphic.

Proof. The sequence (3), Proposition 4, the last remark and (4) allow to state the long exact sequence

⋯ → H_{r+1}(A_{m}, \underline{R}_{m}) →^{μ} H_{r}(D_{αm}, \underline{R}_{m}) →^{ρ} H_{r}(△_{αm}, \underline{R}_{m}) →^{ρ′}  (5)

→ H_{r}(A_{m}, \underline{R}_{m}) →^{μ′} H_{r−1}(D_{αm}, \underline{R}_{m}) → · · · .

Since ρ and ρ′ are explicitly known and we have explicitly described μ and μ′, we can compute a basis of H_{r}(△_{αm}, \underline{R}_{m}) from bases of the other four vector spaces appearing in (5). As a result, the proof will be completed after showing that the homology H_{r}(D_{αm}, \underline{R}_{m}) is graphic.

Indeed, define for each j such that 1 ≤ j ≤ card(Π) the following subset of \overline{K}_{αm},

K_{αm}(j) := K_{αm} ∪ \{L ∈ \overline{K}_{αm} \ K_{αm}|L = I ∪ J and card(J) ≤ j\}.

If 1 ≤ i < j < k ≤ card(Π), then we can write the following exact sequence of vector space complexes associated to linear weighted relative simplicial complexes:

0 → C.(K_{αm}(j), K_{αm}(i), \underline{R}_{m}) → C.(K_{αm}(k), K_{αm}(i), \underline{R}_{m}) →
→ C.(K_{αm}(k), K_{αm}(j), \underline{R}_{m}) → 0.

Taking sequences as above for (i, j, k) equal to (1, 2, 3), (1, 3, 4), …, (1, card(Π) − 1, card(Π)), we conclude by considering their associated homology long exact sequences, that we are able to compute bases for

H.(K_{αm}(card(Π)), K_{αm}(1), \underline{R}_{m}) = H.(D_{αm}, \underline{R}_{m})

from bases of H.(K_{αm}(j), K_{αm}(j − 1), \underline{R}_{m}), 2 ≤ j ≤ card(Π). Finally writing, for J ⊆ Π fixed,

D_{α}^{J} = {d ∈ S_{α}|J is associated to d in D_{α}},

we obtain

H_{r}(K_{αm}(j), K_{αm}(j − 1), \underline{R}_{m}) ≅ ⨁_{J⊆Π,card(J)=j} H_{r−j}(G_{ΛD_{α}^{J}}(m), \underline{R}_{m−β_{J}}),

which ends the proof.

Now, as a clear consequence, we get our main result:

Theorem 1 For each m in S, the homology H.(△_{m}, \underline{Q}_{m}) of the K- vector space complex associated to the linear weighted simplicial complex C.(△_{m}, \underline{Q}_{m}), given in Section 2, is linear weighted graphic.

Remark. We can determine sets C_{r} such that H_{r}(△_{m}, \underline{Q}_{m}) = 0 whenever m ∉ C_{r}. For −1 ≤ i ≤ card(Λ), consider the simplicial subcomplexes of (\overline{K}_{αm}, \underline{R}_{m}) given by

M_{αm}(i) := K_{αm} ∪ \{L ∈ \overline{K}_{αm} \ K_{αm}|L = I ∪ J and card(I) ≤ i\}.

Note that whenever r ≥ −1, we have the following isomorphism

H_{r}(\overline{K}_{αm}, K_{αm}, \underline{R}_{m}) ≅ H_{r}(M_{αm}(card(Λ)), M_{αm}(−1), \underline{R}_{m}).

Set Σ_{αm} = {I ⊆ Λ|m−β_{I} ∈ S_{α}} and, for d ∈ S_{α}, Θ_{d} = {J ⊆ Π|d−β_{J} ∈ S_{α}}. Then, the following isomorphism of homologies holds:

H_{r+1}(M_{αm}(i), M_{αm}(i − 1), \underline{R}_{m}) ≅ ⨁_{I∈Σ_{αm},card(I)=i} H_{r−i}(Θ_{m−β_{I}} , \underline{R}_{m−β_{I}}).

Indeed, it follows from the following straightforward equality,

H_{r+1}(M_{αm}(i), M_{αm}(i − 1), \underline{R}_{m}) = ⨁ H_{r+1−i}(P (Π_{m−β_{I}}), Θ_{m−β_{I}} , \underline{R}_{m−β_{I}}),

where the sum is over all cardinality i sets I in Σ_{αm}, Π_{m−β_{I}} being the set {e ∈ Π|m − β_{I} − deg(e) ∈ S_{α}} and by bearing in mind that the linear weighted simplicial complex (P (Π_{m−β_{I}}), \underline{R}_{m−β_{I}}) is acyclic. As a consequence, H_{r}(D_{αm}, \underline{R}_{m}) = 0 (r ≥ 0) when H_{r−card(I)}(Θ_{m−β_{I}} , \underline{R}_{m−β_{I}}) = 0 for all I ⊆ Λ such that card(I) ≤ r and m − β_{I} ∈ S_{α}.

Now, for l ≥ 0, set D_{α}(\underline{R}, l) = \{d ∈ S_{α}|H_{l}(Θ_{d}, \underline{R}_{d}) ≠ 0\} and

C_{r}(\underline{R}, α) := \{m ∈ S_{α}|m = a + β_{I}; a ∈ A(α), I ⊆ Λ_{α} and card(I) = r + 1\}∪

\{m ∈ S_{α}|∃I ⊆ Λ_{α}, card(I) = i ≤ r with m − β_{I} = d ∈ D_{α}(\underline{R}, r − i)\}.

Pick, m ∈ S, r ≥ 1 and assume that for some α in S, m ∈ S_{α}, Γ_{m} = Γ_{α} and m ∉ C_{r}(\underline{R}, α). Then, the homology H_{r}(△_{m}, \underline{R}_{m}) vanishes.

Notice that, C_{r}(\underline{R}, α) depends on the partition Λ_{α} ∪ Π_{α} of Γ_{α} and on the weights \underline{R}_{d}, d ∈ S_{α}, which are either \underline{Q′}_{d} or \underline{Q′}_{d}/\underline{Q}_{d}. Hence to deduce H_{r}(△_{m}, \underline{Q}_{m}) = 0, we need that m ∉ C_{r}(\underline{Q′}, α) and m ∉ C_{r+1}(\underline{Q′}/\underline{Q}, α), where C_{r}(\underline{Q′}, α) and C_{r}(\underline{Q′}/\underline{Q}, α) are defined as C_{r}(\underline{R}, α) but particularizing the weights \underline{R}_{d} to \underline{Q′}_{d} or \underline{Q′}_{d}/\underline{Q}_{d}. Therefore, if we set C_{r}(α) = C_{r}(\underline{Q′}, α) ∪ C_{r+1}(\underline{Q′}/\underline{Q}, α), then H_{r}(△_{m}, \underline{Q}_{m}) vanishes whenever m ∉ C_{r}(α).

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