André-Quillen cohomology of monoid algebras

Klaus Altmann Arne B. Sletsjøe

Abstract

We compute the André-Quillen (or Harrison) cohomology of an affine toric variety. The best results are obtained either in the general case for the first three cohomology groups, or in the case of isolated singularities for all cohomology groups, respectively.

1 Introduction

(1.1) Let k be a field of characteristic 0. For any finitely generated k-algebra A the so-called cotangent complex yielding the André-Quillen cohomology T_{A}^{n} = T^{n}(A, A; k) (n ≥ 0) may be defined. The first three of these A-modules are important for the deformation theory of A or its geometric equivalent SpecA: T_{A}^{1} equals the set of infinitesimal deformations, T_{A}^{0} describes their automorphisms, and T_{A}^{2} contains the obstructions for lifting infinitesimal deformations to larger base spaces. Apart from occurring in long exact sequences no meaning of the higher cohomology groups seems to be known when studying the deformation theory of closed subsets of SpecA. A very readable reference for the definition of André-Quillen cohomology and its relations to Hochschild and Harrison cohomology is Loday’s book [Lo]. For applications in deformation theory see for instance [Ld], [Pa], or the summary of the properties one has to know (without proofs) in the first section of [BC].

(1.2) For smooth k-algebras A, the higher T_{A}^{n} (i.e. n ≥ 1) vanish. For complete intersections the situation is still easy; T_{A}^{0} and T_{A}^{1} are well understood, and the remaining cohomology groups vanish. As far as we know, only few further examples exist where the cotangent complex or at least the André-Quillen cohomology groups are known. Palamodov has told us that he has computed (unpublished) the cotangent complex of an embedded point on a line; it turned out that the Poincaré series T (s) := ∑_{n≥0}(dim_{k} T^{n}) · s^{n} of this singularity is a rational function. It would be interesting to know whether this is always the case for isolated singularities. The result of the present paper is a spectral sequence converging to the Harrison (or André-Quillen) cohomology for affine toric varieties. In the case of an isolated singularity, this spectral sequence degenerates; this leads to a down to earth description of the modules T_{A}^{n}. Moreover, in the general case, the information is still sufficient to determine T_{A}^{0}, T_{A}^{1}, and T_{A}^{2}. That is, we use the methods of [Sl] to obtain straight formulas generalizing part of the results of [Al 1].

(1.3) The paper is organized as follows: We begin in §2 with fixing notation and recalling those facts of [Sl] that will be used in the following. The sections §3 and §4 contain the main theorems of the paper: First, we state our spectral sequence; then, in §4, we calculate some of the E_{1}-terms and show the vanishing of others. Finally, in §5, we present the resulting T_{A}^{n}-formulas promised before.

(1.4) Acknowledgements: The first author became interested in describing the algebra cohomology of monoid algebras during his visit at the Senter for høyere studier (Academy of Sciences) in Oslo. He liked the warm and stimulating atmosphere very much, and would like to thank those who made it possible for him to stay there. Moreover, we are grateful to R.-O. Buchweitz, J.A. Christophersen, and A. Laudal for many helpful discussions.

2 Inhomogeneous Harrison cohomology

(2.1) Let M, N be mutually dual, finitely generated, free Abelian groups; we denote by M_{IR}, N_{IR} the associated real vector spaces obtained via base change with IR. Assume we are given a rational, polyhedral cone σ = ⟨a^{1}, …, a^{m}⟩ ⊆ N_{IR} with apex in 0 and with a^{1}, …, a^{m} ∈ N denoting its primitive fundamental generators (i.e. none of the a^{i} is a proper multiple of an element of N). We define the dual cone σ^{∨} := {r ∈ M_{IR} | ⟨σ, r⟩ ≥ 0} ⊆ M_{IR} and denote by Λ := σ^{∨} ∩ M the resulting monoid of lattice points. The corresponding monoid algebra A := k[Λ] will be the object of the upcoming investigations. It is the ring of regular functions on the toric variety Y_{σ} = SpecA associated to σ. The ring A itself as well as most of its important modules (such as T_{A}^{n}) admit an M-(multi)grading. It is this grading which will make computations possible. For general facts concerning toric varieties see for instance [Fu] or [Od].

(2.2) The following definitions are taken from [Sl], §2. We note that the original requirement that the monoids involved have no non-trivial subgroups is unnecessary for our purposes.

Definition: L ⊆ Λ is said to be monoid-like if for all elements λ_{1}, λ_{2} ∈ L the relation λ_{1} − λ_{2} ∈ Λ_{+} := Λ \ {0} implies λ_{1} − λ_{2} ∈ L. Moreover, a subset L_{0} ⊆ L of a monoid-like set is called full if (L_{0} + Λ) ∩ L = L_{0}.

For any subset P ⊆ Λ and n ≥ 1 we introduce S_{n}(P) := {(λ_{1}, …, λ_{n}) ∈ P^{n} | ∑_{v} λ_{v} ∈ P }. If L_{0} ⊂ L are as in the previous definition, then this gives rise to the following set:

C^{n}(L \ L_{0}, L; k) := {φ : S_{n}(L) → k | φ is shuffle invariant and vanishes on S_{n}(L \ L_{0})} .

(φ is said to be shuffle invariant if it vanishes on ( ∑_{π} sgn(π) π)(\underline{λ}) where π runs through all shuffles of the set {1, …, n}.) These k-vector spaces turn into a complex with the differential δ^{n} : C^{n−1}(L \ L_{0}, L; k) → C^{n}(L \ L_{0}, L; k) defined via

(δ^{n}φ)(λ_{1}, …, λ_{n}) := φ(λ_{2}, …, λ_{n})+∑_{v=1}^{n−1}(−1)^{v}φ(λ_{1}, …, λ_{v}+λ_{v+1}, …, λ_{n})+(−1)^{n}φ(λ_{1}, …, λ_{n−1}) .

Definition: The k-vector space HA^{n}(L \ L_{0}, L; k) := H^{n}(C^{•}(L \ L_{0}, L; k)) is called the n-th inhomogeneous Harrison cohomology of the pair (L, L_{0}).

(2.3) Theorem ([Sl]): Let R ∈ M. Then, defining Λ_{+} := Λ \ {0}, the homogeneous part of T_{A}^{n} in degree −R equals

T_{A}^{n}(−R) = HA^{n+1}(Λ_{+} \ (R + Λ), Λ_{+}; k) for n ≥ 0.

The proof of the theorem is spread throughout the first two sections of [Sl]: First, in (1.13), (1.14), the calculation of T_{A}^{n} has been reduced down to the monoid level. Then, Proposition (2.9) shows that the homogeneous pieces T_{A}^{n}(−R) equal the so-called graded Harrison cohomology groups Harr^{n+1,−R}(Λ, k[Λ]), and, finally, Theorem (2.10) states the above result.

3 The spectral sequence

(3.1) With the notation of (2.1) we define for any face τ ≤ σ and any degree R ∈ M the monoid-like set

K_{τ}^{R} := Λ_{+} ∩ (R − int τ^{∨}) .

These sets admit the following elementary properties:

(i) K_{0}^{R} = Λ_{+}, and K_{(i)}^{R} := K_{a^{i}}^{R} = {r ∈ Λ_{+} | ⟨a^{i}, r⟩ < ⟨a^{i}, R⟩} with i = 1, …, m.

(ii) For τ ≠ 0 the equality K_{τ}^{R} = ⋂_{a^{i}∈τ} K_{(i)}^{R} holds. Moreover, if σ is a top-dimensional cone, K_{σ}^{R} = Λ_{+} ∩ (R − int σ^{∨}) is a (diamond shaped) finite set.

(iii) Λ_{+} \ (R + Λ) = ⋃_{i=1}^{m} K_{(i)}^{R}.

(3.2) Let us fix an element R ∈ M. The dependence of the sets K_{τ}^{R} on τ is a contravariant functor. This gives rise to the complexes C^{q}(KR_{•}; k) (q ≥ 1) defined as

C^{q}(K_{p}^{R}; k) := ⊕_{τ≤σ, dim τ=p} C^{q}(K_{τ}^{R}; k) (0 ≤ p ≤ dim σ)

with C^{q}(K_{τ}^{R}; k) := C^{q}(∅, K_{τ}^{R}; k) and the obvious differentials d^{p} : C^{q}(K_{p−1}^{R}; k) → C^{q}(K_{p}^{R}; k). (One has to use the maps C^{q}(K_{τ}^{R}; k) → C^{q}(K_{τ′}^{R}; k) for any pair τ ≤ τ′ of (p − 1)- and p-dimensional faces, respectively. The only problem might be the sign; it arises from comparison of the (pre-fixed) orientations of τ and τ′.) Our complex begins as

0 → C^{q}(Λ_{+}; k) → ⊕_{i=1}^{m}C^{q}(K_{(i)}^{R}; k) → ⊕_{⟨a^{i},a^{j}⟩≤σ}C^{q}(K_{(ij)}^{R}; k) → ⊕_{dim τ=3} C^{q}(K_{τ}^{R}; k) → · · · .

Lemma: The canonical k-linear map C^{q}(Λ_{+}\(R+Λ), Λ_{+}; k) → C^{q}(KR_{•}; k) is a quasiisomorphism, i.e. a resolution of the first vector space.

Proof: For an r ∈ Λ_{+} ⊆ M we define the k-vector space

V^{q}(r) := \{φ : \{\underline{λ} ∈ Λ_{+}^{q} |_{∑v}λ_{v} = r\} → k | φ is shuffle invariant\} .

Then, our complex splits into a direct product over r ∈ Λ_{+}. Its homogeneous factors equal

0 → V^{q}(r) → V^{q}(r)^{{i | r∈K_{(i)}^{R}}} → V^{q}(r)^{{τ≤σ | dim τ=2; r∈K_{τ}^{R}}} → V^{q}(r)^{{τ≤σ | dim τ=3; r∈K_{τ}^{R}}} → · · · .

On the other hand, denoting by H_{r,R}^{+} the halfspace H_{r,R}^{+} := {a ∈ N_{IR} | ⟨a, r⟩ < ⟨a, R⟩} ⊆ N_{IR}, the relation r ∈ K_{τ}^{R} is equivalent to τ \ {0} ⊆ H_{r,R}^{+}. Hence, the complex for computing the reduced cohomology of the topological space ⋃_{τ\{0}⊆H_{r,R}^{+}}(τ \ {0}) ⊆ σ equals

0 → k → k^{{i | r∈K_{(i)}^{R}}} → k^{{τ≤σ | dim τ=2; r∈K_{τ}^{R}}} → k^{{τ≤σ | dim τ=3; r∈K_{τ}^{R}}} → ⋯

if σ ∩ H_{r,R}^{+} ≠ ∅, i.e. r ∈ ⋃_{i} K_{(i)}^{R}; it is trivial otherwise. Since ⋃_{τ\{0}⊆H_{r,R}^{+}}(τ \ {0}) is con-

tractible, this complex is always exact. Thus, C^{q}(KR_{•}; k) = ∏_{r∈Λ_{+}} V^{q}(r)^{{τ≤σ | dim τ=•; r∈K_{τ}^{R}}} has ∏_{r∈Λ_{+}\(∪_{i}K_{(i)}^{R})} V^{q}(r) = C^{q}(Λ_{+} \ (R + Λ), Λ_{+}; k) as cohomology in 0, and it is exact elsewhere. □

(3.3) Combining the differentials d^{p} from (3.2) and δ^{q} from (2.2), we obtain a double complex

C^{•}(KR_{•}; k) (0 ≤ p ≤ dim σ; q ≥ 1).

Theorem: The André-Quillen cohomology of A = k[Λ] equals the cohomology of the total complex, that is

T_{A}^{n}(−R) = H^{n+1}(tot^{•}[C^{•}(KR_{•}; k)]) for n ≥ 0.

Moreover, given an element s ∈ Λ, the multiplication [·x^{s}] : T_{A}^{n}(−R) → T_{A}^{n}(−R + s) is given by the homomorphism C^{•}(KR_{•}; k) → C^{•}(KR−s_{•}; k) assigned to the inclusions K_{τ}^{R−s} ⊆ K_{τ}^{R}.

Proof: The first part is a straightforward consequence of Theorem (2.3) and the previous lemma – just use the corresponding spectral sequence of the double complex. For the A-module structure of T_{A}^{n}, one easily observes from the proofs in [Sl] that the multiplication with x^{s} arises from the complex homomorphism C^{•}(Λ_{+} \ (R + Λ), Λ_{+}; k) → C^{•}(Λ_{+} \ ([R − s] + Λ), Λ_{+}; k) provided by the inclusion R + Λ ⊆ [R − s] + Λ. □

(3.4) Corollary: There is a spectral sequence E_{1}^{p,q} = ⊕_{dim τ=p}HA^{q}(K_{τ}^{R}; k) ⟹ T_{A}^{p+q−1}(−R).

Proof: This is the other spectral sequence associated to the double complex C^{•}(KR_{•}; k). □

4 The E_{1}-level

(4.1) Definition: Let K ⊆ M be an arbitrary subset of the lattice M. A function f : K → k is called quasilinear if f(r) + f(s) = f(r + s) for any r and s with r, s, r + s ∈ K. The vector space of quasilinear functions is denoted by \overline{Hom} (K, k).

Recalling the differential δ^{2} : C^{1}(K_{τ}^{R}) → C^{2}(K_{τ}^{R}) from (2.2) shows that the E_{1}^{•,1}-summands HA^{1}(K_{τ}^{R}; k) equal \overline{Hom} (K_{τ}^{R}, k).

(4.2) The orbits of the torus acting on Y_{σ} are parametrized by the faces of σ; the singular locus of Y_{σ} is the disjoint union of some of these orbits. We call a face τ ≤ σ smooth if our toric variety is smooth along orb(τ). It is one of the basic facts that smooth faces are characterized by being generated from a part of a ZZ-basis of N. In particular, 0 and the one-dimensional faces are always smooth.

Proposition: If τ ≤ σ is a smooth face, then the injections Hom_{k}(span_{k}K_{τ}^{R}, k) ↪ \overline{Hom} (K_{τ}^{R}, k) are even isomorphisms. Moreover, span_{k}K_{τ}^{R} = ⋂_{a^{i}∈τ} span_{k}K_{(i)}^{R}, and the latter vector spaces equal span_{k}K_{(i)}^{R} = M_{k}, (a^{i})^{⊥}, or 0 if ⟨a^{i}, R⟩ ≥ 2, = 1, or ≤ 0, respectively.

Proof: Let f : K_{τ}^{R} → k be quasilinear; it suffices to extend f to a ZZ-linear map defined on span_{Z Z}K_{τ}^{R} ⊆ M. If R was non-positive on any of the generators of τ, then K_{τ}^{R} would be empty anyway. Hence, if (w.l.o.g.) τ = ⟨a^{1}, …, a^{k}⟩, we may assume that ⟨a^{i}, R⟩ ≥ 2 for i = 1, …, l and ⟨a^{j}, R⟩ = 1 for j = l + 1, …, k. K_{τ}^{R} contains the easy part τ^{⊥} ∩Λ_{+}; it is no problem at all to extend f_{|τ^{⊥}∩Λ_{+}} to a ZZ-linear function defined on τ^{⊥} ∩ M. In general, we have to show that for elements s^{v} ∈ K_{τ}^{R} the value ∑_{v} f(s^{v}) only depends on s := ∑_{v} s^{v}, not on the summands themselves. (Then, f(s) may be defined as this value.) By smoothness of τ there exist elements r^{1}, …, r^{l} ∈ K_{τ}^{R} such that ⟨a^{i}, r^{j}⟩ = δ_{ij} for 1 ≤ i ≤ k and 1 ≤ j ≤ l. Hence, quasilinearity of f implies

f(s^{v}) = ∑_{i=1}^{l}⟨a^{i}, s^{v}⟩f(r^{i}) + f(p^{v}) with p^{v} := s^{v} −_{∑i}⟨a^{i}, s^{v}⟩r^{i} ∈ τ^{⊥} ∩ M .

Summing up yields

∑_{v} f(s^{v}) = ∑_{i=1}^{l}⟨a^{i}, s⟩f(r^{i}) + ∑_{v} f(p^{v}) = ∑_{i=1}^{l}⟨a^{i}, s⟩f(r^{i}) + f(s −_{∑i}⟨a^{i}, s⟩r^{i}) .

Finally, the second claim follows by ⋂_{a^{i}∈τ} span_{k}K_{(i)}^{R} = ⋂_{j=l+1}^{k}(a^{j})^{⊥} = span_{k}(τ^{⊥}; r^{1}, …, r^{l}) = span_{k}K_{τ}^{R}. □

(4.3) We turn to the remaining part of the first level and show the vanishing of E_{1}^{p,≥2} if Y_{σ} is smooth in codimension p:

Theorem: If τ ≤ σ is a smooth face, then HA^{q}(K_{τ}^{R}; k) = 0 for q ≥ 2.

Proof: We proceed by induction on dim τ, i.e. we may assume that the vanishing holds for all proper faces of τ. Let r(τ) be an arbitrary element of int(σ^{∨} ∩ τ^{⊥}) ∩ M, i.e. τ = σ ∩ [r(τ)]^{⊥}. Then, via R_{g} := R − g · r(τ) with g ∈ ZZ, one obtains an infinite (if τ ≠ σ) series of degrees admitting the following two properties:

(i) K_{τ}^{R_{g}} = K_{τ}^{R} for any g ∈ ZZ (since R_{g} = R on τ), and

(ii) K_{τ′}^{R_{g}} ≠ ∅ implies τ′ ≤ τ for any face τ′ ≤ σ and g ≫ 0 (since ⟨a^{j}, R_{g}⟩ ≤ 0 if a^{j} ∉ τ).

In particular, in degree −R_{g} with g ≫ 0, the first level of our spectral sequence is shaped as follows:

• For p < dim τ only HA^{q}(K_{τ′}^{R}; k) with τ′ ≤ τ appear as summands of E_{1}^{p,q}. By the induction hypothesis they even vanish for q ≥ 2,

• for p = dim τ it follows that E_{1}^{p,q} = HA^{q}(K_{τ}^{R}; k), and

• all vector spaces E_{1}^{p,q} vanish beyond the [p = dim τ]–line.

Hence, the differentials d_{r} : E_{r}^{p,q} → E_{r}^{p+r,q−r+1} are trivial for r ≥ 1, q ≥ 2, and we obtain

T_{A}^{q+dim τ−1}(−R_{g}) = HA^{q}(K_{τ}^{R}; k) for g ≫ 0 .

Moreover, under this identification, the multiplication

[·x^{r(τ)}] : T_{A}^{q+dim τ−1}(−R_{g}) → T_{A}^{q+dim τ−1}(−R_{g+1})

is just the identity map. On the other hand, we may restrict T_{σ}^{n} := T_{A}^{n} onto the smooth, open subset Y_{τ} := Spec k[τ^{∨} ∩ M] ⊆ Y_{σ}. Since k[τ^{∨} ∩ M] equals the localization of k[σ^{∨} ∩ M] by the element x^{r(τ)}, we obtain

T_{σ}^{n} ⊗_{k[σ^{∨}∩M]} k[σ^{∨} ∩ M]_{x^{r(τ)}} = T_{τ}^{n} = 0 for n ≥ 1.

In particular, any element of T_{A}^{q+dim τ−1}(−R_{g}) ⊆ T_{A}^{q+dim τ−1} will be killed by some power of x^{r(τ)}; but this means HA^{q}(K_{τ}^{R}; k) = 0. □

(4.4) Corollary: E_{1}^{0,q} = E_{1}^{1,q} = 0 for q ≥ 2.

5 Applications

(5.1) The main ingredient for describing the André-Quillen cohomology of Y_{σ} will be the complex \overline{Hom} (KR_{•}, k) = (E_{1}^{•,1}, d_{1}) which is built from the vector spaces \overline{Hom} (K_{τ}^{R}, k) as C^{q}(KR_{•}; k) was from C^{q}(K_{τ}^{R}; k) in (3.2). It contains (span_{k}KR_{•})^{∗} as a subcomplex.

(5.2) First, we discuss the case of an isolated singularity Y_{σ}. Here, our spectral sequence degenerates completely.

[figure: q = 1; q = 2; q = 3; ⋮; p = 0; \overline{Hom} (K_{0}^{R}, k); p = 1; \overline{Hom} (K_{1}^{R}, k); p = dim σ; ⋮; …; …; HA^{q}(K_{σ}^{R}; k); 0; 0; d_{r}]

Theorem: Let Y_{σ} be an isolated singularity. Then, the André-Quillen cohomology in degree −R equals

T_{A}^{n}(−R) = \{ \begin{matrix}H^{n}(\overline{Hom} (KR_{•}, k)) = H^{n}((span_{k}KR_{•})^{∗}) & for 0 ≤ n ≤ dim σ − 1 \\ H^{dim σ}(\overline{Hom} (KR_{•}, k)) & for n = dim σ \\ HA^{n−dim σ+1}(K_{σ}^{R}; k) & for n ≥ dim σ + 1 . \end{matrix}

Proof: The first level of the spectral sequence is non-trivial only in E_{1}^{p,1} with 0 ≤ p ≤ dim σ and E_{1}^{dim σ,q} with q ≥ 1, respectively. Moreover, the complexes \overline{Hom} (KR_{•}, k) and (span_{k}KR_{•})^{∗} are equal up to position p = dim σ − 1. □

(5.3) In the general case, we still have enough information to determine the deformation relevant modules T_{A}^{0}, T_{A}^{1}, and T_{A}^{2}. Under the additional hypothesis of smoothness in codimension two, these results have already been obtained in [Al 1] with a different proof.

Theorem: Let σ be an arbitrary rational, polyhedral cone with apex in 0. Then, for every R ∈ M,

T_{A}^{n}(−R) = H^{n}(\overline{Hom} (KR_{•}, k)) for n = 0, 1, 2.

Moreover, for n = 0, 1, this vector space equals H^{n}((span_{k}KR_{•})^{∗}), too.

Proof: This is a direct consequence of Corollary (4.4). □

(5.4) Since the complex (span_{k}KR_{•})^{∗} is much easier to handle than \overline{Hom} (KR_{•}, k), it pays to look for sufficient conditions for T_{A}^{2} = H^{2}((span_{k}KR_{•})^{∗}) to hold.

Proposition: If Y_{σ} is Gorenstein in codimension two, i.e. for every two-dimensional face ⟨a^{i}, a^{j}⟩ ≤ σ there is an element r(i, j) ∈ M such that ⟨a^{i}, r(i, j)⟩ = ⟨a^{j}, r(i, j)⟩ = 1, then

T_{A}^{2} = H^{2}((span_{k}KR_{•})^{∗}).

Proof: For edges ⟨a^{i}, a^{j}⟩ ≤ σ we have to show that (span_{k}K_{(ij)}^{R})^{∗} ↪ \overline{Hom} (K_{(ij)}^{R}, k) is an isomorphism. We will adapt the proof of (4.2). It may be assumed that ⟨a^{i}, R⟩; ⟨a^{j}, R⟩ ≥ 1 and r(i, j) ∈ K_{(ij)}^{R}. Let d := | det(a^{i}, a^{j})| ∈ ZZ; it is the smallest positive value of a^{i} possible on elements of Λ ∩ (a^{j})^{⊥}. We choose an r^{i} ∈ Λ ∩ (a^{j})^{⊥} with ⟨a^{i}, r^{i}⟩ = d; together with r(i, j) it will play the same role as r^{1}, …, r^{l} did in (4.2). Case 1: ⟨a^{i}, R⟩ > d and ⟨a^{i}, R⟩ ≥ ⟨a^{j}, R⟩ (in particular, r^{i} ∈ K_{(ij)}^{R}): Then, elements s^{v} or s (cf. (4.2)) may be represented as

s = ⟨a^{j}, s⟩ · r(i, j) + \frac{⟨a^{i}, R⟩ − ⟨a^{j}, R⟩}{d} r^{i} + [(a^{i}, a^{j})^{⊥} − elements] .

The difference ⟨a^{i}, R⟩ − ⟨a^{j}, R⟩ is always divisible by d, i.e. the coefficients are integers. Case 2: ⟨a^{i}, R⟩; ⟨a^{j}, R⟩ ≤ d: This implies K_{(ij)}^{R} ⊆ (a^{i}, a^{j})^{⊥} +ZZ ·r(i, j) = (a^{i} −a^{j})^{⊥}. In particular, we may use the representation s = ⟨a^{i}, s⟩ · r(i, j) + [(a^{i}, a^{j})^{⊥} − elements] . □

Corollary: Let Y_{σ} be a three-dimensional, toric Gorenstein singularity, i.e. σ = ⟨a^{1}, …, a^{m}⟩ with a^{m+1} := a^{1} being the cone over a lattice polygon embedded in height one. Then

T_{A}^{2}(−R)^{∗} = ⋂_{i=1}^{m}(span_{k}K_{(i,i+1)}^{R})/span_{k}(⋂_{i=1}^{m}K_{(i,i+1)}^{R}) .

(5.5) Finally, we would like to mention an alternative to the complexes (span_{k}KR_{•})^{∗} and \overline{Hom} (KR_{•}, k). Let E ⊆ Λ_{+} be the (finite) set of non-splittable elements in Λ_{+}; it is the minimal generator set of the monoid Λ = σ^{∨} ∩ M and gives rise to a canonical surjection π : ZZ^{E} → M. The relations among E-elements are gathered in the ZZ-module L(E) := ker π. Every q ∈ L(E) splits into a difference q = q^{+} − q^{−} with q^{+}, q^{−} ∈ IN^{E} and ∑_{v} q_{v}^{+} q_{v}^{−} = 0. We denote by q̄ ∈ Λ the image q̄:= π(q^{+}) = π(q^{−}).

Definition: With E_{τ}^{R} := E ∩ K_{τ}^{R}, we define L(E_{τ}^{R}) := L(E) ∩ ZZ^{E_{τ}^{R}} and \overline{L}(E_{τ}^{R}) ⊆ L(E_{τ}^{R}) to be the submodule generated by the relations q ∈ L(E) such that q̄ ∈ K_{τ}^{R}. (Notice that with E_{0}^{R} = E the notation differs slightly from that in [Al 1].)

As usual (cf. (3.2)), we may construct complexes from these finitely generated, free abelian groups. They fit into the following commutative diagram with exact rows:

[figure: 0; (span_{k}KR_{•})^{∗}; k^{E_{•}^{R}}; Hom_{Z Z}(L(ER_{•}), k); 0; 0; \overline{Hom} (KR_{•}, k); k^{E_{•}^{R}}; Hom_{Z Z}(\overline{L}(ER_{•}), k); 0; ∼]

Proposition: The k-duals of T_{A}^{n}(−R) equal

T_{A}^{1}(−R)^{∗} = L_{k}(⋃_{i}E_{(i)}^{R})/∑_{i}L_{k}(E_{(i)}^{R})

and

T_{A}^{2}(−R)^{∗} = ker ( ⊕_{i} L_{k}(E_{(i)}^{R}) → L(E))/ im ( ⊕_{ij} \overline{L}_{k}(E_{(ij)}^{R}) → ⊕_{i}L_{k}(E_{(i)}^{R})) .

Proof: As in the proof of Lemma (3.2), one obtains that the complex k^{E_{•}^{R}} has no cohomology except H^{0} = k^{E\∪_{i}E_{(i)}^{R}}. Hence, the long exact sequence for the second row in the above diagram yields isomorphisms

H^{p−1}(Hom_{Z Z}(\overline{L}(ER_{•}), k)) ⟶^{∼} H^{p}(\overline{Hom} (KR_{•}, k)) for p ≥ 2.

Taking a closer look at the first terms shows that the same result is true for p = 1 if E_{0}^{R} is replaced by ⋃_{i} E_{(i)}^{R}. Finally, we know that \overline{L}(E_{τ}^{R}) = L(E_{τ}^{R}) if dim τ ≤ 1. □

(5.6) Remark: The vector space T_{A}^{1}(−R) has also a convex-geometric interpretation; it is related to the set of Minkowski summands of the cross cut σ ∩ [R = 1]. For details, we refer to [Al 2].

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Klaus Altmann Institut für reine Mathematik der Humboldt-Universität zu Berlin Ziegelstr. 13A D-10099 Berlin, Germany E-mail: altmann@mathematik.hu-berlin.de Arne B. Sletsjøe Department of Mathematics University of Oslo Pb. 1053 Blindern N-0316 Oslo, Norway E-mail: arnebs@math.uio.no