RESOLUTIONS OF MONOMIAL IDEALS AND COHOMOLOGY OVER EXTERIOR ALGEBRAS

ANNETTA ARAMOVA, LUCHEZAR L. AVRAMOV, AND JÜRGEN HERZOG

Abstract. The paper studies the homology of finite modules over the exterior algebra E of a vector space V. To such a module M we associate an algebraic set V_{E}(M) ⊆ V, consisting of those v ∈ V that have a non-minimal annihilator in M. A cohomological description of its defining ideal leads, among other things, to complementary expressions for its dimension, linked by a ‘depth formula’. Explicit results are obtained for M = E/J, when J is generated by products of elements of a basis e_{1}, …, e_{n} of V. A (infinite) minimal free resolution of E/J is constructed from a (finite) minimal resolution of S/I, where I is the squarefree monomial ideal generated by ‘the same’ products of the variables in the polynomial ring S = K[x_{1}, …, x_{n}]. It is proved that V_{E}(E/J) is the union of the coordinate subspaces of V, spanned by subsets of { e_{1}, …, e_{n} } determined by the Betti numbers of S/I over S.

Introduction

Let V be a vector space with basis e_{1}, …, e_{n} over a field K, and let E = ⋀(V) be the exterior algebra over V. The standard basis elements e_{k_{1}} ∧ ⋯ ∧ e_{k_{s}} of E, k_{1} < ⋯ < k_{s}, are called monomials in E. An ideal J ⊆ E generated by monomials is called a monomial ideal. We study the (co)homological algebra of such ideals.

Along with J, we consider the corresponding squarefree monomial ideal I in the polynomial ring S = K[x_{1}, …, x_{n}]. Each S–module F_{i} in a minimal multigraded free resolution F of S/I can be written in the form

F_{i} = ⨁_{j=1}^{β_{i}} S(−a_{ij}) with uniquely determined a_{ij} ∈ ℕ^{n} .

A well known formula of Hochster [12] on the multigraded Betti numbers of squarefree monomial ideals shows that F is itself squarefree, in the sense that the coordinates of all shifts a_{ij} are equal to 0 or 1. Furthermore, there exist interesting non-minimal squarefree resolutions, for example the Taylor resolution [15].

Given any squarefree resolution F of the monomial ideal I ⊆ S, we choose a homogeneous basis B of F and construct a multigraded free resolution G of the monomial ideal J in the exterior algebra E. The resolution depends on B, but different choices of multihomogeneous bases lead to isomorphic complexes; if F is minimal, then so is G. The construction is given in Section 1.

Date: January 14, 1999.

1991 Mathematics Subject Classification. 13D02, 13D40, 16E10, 52B20.

Work on this paper started while the first and second author visited the third author; the hospitality of the University of Essen is gratefully acknowledged.

L.L.A. was partially supported by a grant from the National Science Foundation.

Section 2 contains applications. An explicit formula gives the multigraded Betti numbers of the monomial ideal J ⊆ E in terms of those of I. As a consequence, some interesting properties of J, like the linearity of its minimal resolution or the independence of its Betti numbers from the characteristic of the base field K, are seen to be equivalent to the corresponding properties of I. We also show that if I is a Gotzmann ideal in S, then J is a Gotzmann ideal in E. Our method yields exterior algebra analogues of the Taylor [15] and Eliahou-Kervaire [10] resolutions.

In Section 3 we associate with each finite E-module M an algebraic set V_{E}(M) ⊆ V. As for modular representations of finite groups, which provide the model, there are two constructions: in terms of the action of the graded ring Ext_{E}(K, K) on Ext_{E}(M, K), following Quillen [14], or in terms of the action of V on M, mimicking Carlson [7]. We prove that they yield the same result. Along with other properties of V_{E}(M), this parallels results over group algebras; techniques developed for that case have been successfully extended to other Hopf algebras, but they do not always apply here, because E is not a Hopf algebra (in the category of rings). Our approach is similar to that used in [4] to study modules over complete intersections, and takes advantage of the simple structure of Ext_{E}(K, K): by Cartan [8] it is the symmetric algebra of Hom_{K}(V, K). In particular, we prove that the dimension of V_{E}(M) is complementary to the (appropriately defined) depth of M over E.

When Δ is a simplicial complex and J = J_{Δ} is the ideal in E generated by all monomials e_{k_{1}} ∧⋯∧e_{k_{s}} such that { k_{1}, …, k_{s} } ∉ Δ, the K-algebra K⟨Δ⟩ = E/J_{Δ} is called the indicator algebra of Δ. It has proved to be important in the study of the f-vector of Δ; see for example [3]. The corresponding squarefree ideal I = I_{Δ} in S defines the more familiar Stanley-Reisner ring K[Δ] = S/I_{Δ}. In Section 4 we prove that V_{E}(K⟨Δ⟩) is an union of coordinate subspaces of V, determined by the supports of the shifts of a minimal free resolution of the Stanley-Reisner ring K[Δ] over S. This has consequences for the simplicial cohomology of Δ.

We are grateful to Ragnar-Olaf Buchweitz for several inspiring discussions.

1. The main construction

For the rest of the paper we fix some—mostly standard—notation.

A n-tuple (a_{1}, …, a_{n}) ∈ ℤ^{n} is squarefree if 0 ≤ a_{j} ≤ 1 for j = 1, …, n. For a = (a_{1}, …, a_{n}) ∈ ℤ^{n} we set |a| = a_{1} + ⋯ + a_{n}, and supp(a) = { j | a_{j} ≠ 0 }; by convention, supp(0) = [n], where [n] = { 1, …, n }. For an element u of an n-graded vector space M = ⨁_{a∈ℤ^{n}} M_{a}, the notation deg(u) = a is equivalent to u ∈ M_{a}; we set supp(deg(u)) = supp(u) and | deg(u)| = |u|. The decomposition M = ⨁_{j∈ℤ} M_{j}, where M_{j} = ⨁_{a∈ℤ^{n},|a|=j} M_{a}, turns M into a graded vector space.

Let S = K[x_{1}, …, x_{n}] be the polynomial ring on n commuting variables, and let E = K⟨e_{1}, …, e_{n}⟩ be the exterior algebra on n alternating variables. They are n-graded by deg(x_{j}) = deg(e_{j}) = ε_{j} = (0, …, 0, 1, 0 …, 0), with 1 in jth position. For σ ⊆ [n] we set x^{σ} = x_{k_{1}} ⋯ x_{k_{s}} and e_{σ} = e_{k_{1}} ∧ ⋯ ∧ e_{k_{s}}, where σ = { k_{1}, …, k_{s} } with k_{1} < ⋯ < k_{s}; we say that e_{σ} is a monomial in E. For a ∈ ℕ^{n} we set x^{a} = x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}} and e_{a} = e_{supp(a)} .

The following simple observation is used in many computations.

Observation 1.0. For monomials u, v ∈ E with supp(v) ⊆ supp(u) there exists a unique monomial u′ ∈ E such that vu′ = u; we then set v^{−1}u = u′. For monomials u, v, w, z ∈ E the equalities below hold whenever the left hand side is defined:

(v^{−1}u)w = v^{−1}(uw)  and  (z^{−1}v)(v^{−1}u) = z^{−1}u .

Construction 1.1. Let (F, θ) be a squarefree complex of n-graded S-modules, meaning that each F_{i} has a basis B_{i} with deg(f) squarefree for all f ∈ B_{i}.

Let P_{i} be a n-graded K-vector space with basis B_{i}, and set B = ⨆_{i} B_{i}. Let C_{j} be the n-graded right E-module with basis { y^{(a)} | a ∈ ℕ^{n} , deg (y^{(a)}) = a , |a| = j }. The tensor product C_{j} ⊗_{K} P_{i} becomes a right n-graded E–module, by

\begin{matrix}deg(y^{(a)} ⊗ f) = a + b; \\ (y^{(a)} ⊗ f)e = (−1)^{|b|}y^{(a)}e ⊗ f , \end{matrix} where b = deg(f) .

Let G_{ℓ} be the residue module of ⨁_{ℓ=j+i} C_{j} ⊗_{K} P_{i} by the submodule generated by { y^{(a)} ⊗ f | supp(a) ⊈ supp(f) }, and write y^{(a)}f for the image of y^{(a)} ⊗ f in G_{ℓ}. Thus, G_{ℓ} is the n-graded right E-module with basis

Y_{ℓ} = \{y^{(a)}f | \begin{matrix}a ∈ ℕ^{n} , f ∈ B_{i} , supp(a) ⊆ supp(f) \\ ℓ = |a| + i , deg(y^{(a)}f) = a + deg(f) \end{matrix}\}

If in the complex (F, θ) the differential of f ∈ B_{i} has the form

θ(f) = ∑_{j: f_{j}∈B_{i−1}} λ_{j}x^{b−b_{j}}f_{j} with λ_{j} ∈ K , b = deg(f) , b_{j} = deg(f_{j}) ,

then define homomorphisms G_{ℓ} ⟶ G_{ℓ−1} of n-graded E-modules by

δ(y^{(a)}f) = (−1)^{|b|} ∑_{k∈supp(a)} y^{(a−ε_{k})}fe_{k}

ϑ(y^{(a)}f) = (−1)^{|a|} ∑_{j: f_{j}∈B_{i−1}} y^{(a)}f_{j}λ_{j}e_{b_{j}}^{−1}e_{b} .

and set ∂ = δ + ϑ: G_{ℓ} ⟶ G_{ℓ−1}.

Proposition 1.2. The preceding construction yields a complex (G, ∂) of right ngraded E-modules. If (G′, ∂′) is the complex obtained from homogeneous bases B′_{i} of F_{i}, then G′ ≅ G as complexes of n-graded E-modules.

Hochster’s formula [12] for the Betti numbers of a squarefree monomial ideal I ⊆ S shows that its minimal free resolution (F, θ) is squarefree. In that case, we can say more about the complex (G, ∂) described above.

Theorem 1.3. Let Σ be a set of subsets of [n], let I ⊆ S = K[x_{1}, …, x_{n}] be the ideal generated by the squarefree monomials { x^{σ} | σ ∈ Σ }, and let J ⊆ E = K⟨e_{1}, …, e_{n}⟩ be the ideal generated by the monomials { e_{σ} | σ ∈ Σ }.

If (F, θ) is a (minimal) free resolution of S/I over S, then the complex (G, ∂) of Construction 1.1 is a (minimal) free resolution of E/J over E.

Proof of the proposition. To show that ∂^{2} = 0 we establish equalities

δ^{2} = 0 ;  ϑ^{2} = 0 ;  δϑ = −ϑδ .

The first one comes from an easy direct computation.

Writing θ(f_{j}) = ∑_{k: g_{k}∈B_{i−2}} μ_{kj}x^{b_{j}−c_{k}}g_{k} ∈ F_{i−2}, we have

θ^{2}(f) = ∑_{j} λ_{j}x^{b−b_{j}}θ(f_{j}) = ∑_{j} λ_{j}x^{b−b_{j}} ∑_{k} μ_{kj}x^{b_{j}−c_{k}}g_{k}

= ∑_{k} ( ∑_{j} μ_{kj}λ_{j})x^{b−c_{k}}g_{k} = 0 .

Thus, ∑_{j} μ_{kj}λ_{j} = 0, so we get the second equality from:

ϑ^{2}(y^{(a)}f) = (−1)^{|a|} ∑_{j} ϑ(y^{(a)}f_{j})(λ_{j}e_{b_{j}}^{−1}e_{b})

= ∑_{j} ( ∑_{k} y^{(a)}g_{k}(μ_{kj}e_{c_{k}}^{−1}e_{b_{j}}))(λ_{j}e_{b_{j}}^{−1}e_{b})

= ∑_{k} y^{(a)}g_{k}( ∑_{j} μ_{kj}λ_{j}(e_{c_{k}}^{−1}e_{b_{j}})(e_{b_{j}}^{−1}e_{b}))

= ∑_{k} y^{(a)}g_{k}( ∑_{j} μ_{kj}λ_{j})e_{c_{k}}^{−1}e_{b} = 0 .

Note that if f ∈ B with deg(f) = b and e ∈ E with deg(e) = c, then

\begin{aligned}δ(y^{(a)}fe) & = δ(y^{(a)}f)e \\ ϑ(y^{(a)}fe) & = ϑ(y^{(a)}f)e \end{aligned} provided supp(a) ⊆ supp(b) + supp(c) .

When supp(a) ⊆ supp(b) these formulas hold by definition. If supp(a) ⊈ supp(b), then y^{(a)}f = 0, so we check that the right hand sides vanish. On the one hand, δ(y^{(a)}fe) = ± ∑_{k∈supp(a)} y^{(a−ε_{k})}fe_{k}e; if supp(a−ε_{k}) ⊈ supp(b), then y^{(a−ε_{k})}f = 0; otherwise, k ∈ supp(a) \ supp(f), hence k ∈ supp(c), so e_{k}e = 0. On the other hand, ϑ(y^{(a)}f) = ± ∑_{j} y^{(a)}g_{j}(λ_{j}e_{b_{j}}^{−1}e_{b}) with g_{j} ∈ B. Since supp(g_{j}) ⊆ supp(f), for all j we have supp(a) ⊈ supp(g_{j}), and hence y^{(a)}g_{j} = 0.

The third equality now results from the computation:

ϑ(δ(y^{(a)}f)) = (−1)^{|b|}ϑ( ∑_{k∈supp(a)} y^{(a−ε_{k})}fe_{k}) = (−1)^{|b|} ∑_{k∈supp(a)} ϑ(y^{(a−ε_{k})}f)e_{k}

= (−1)^{|b|+|a|−1} ∑_{k∈supp(a)} ( ∑_{j: f_{j}∈B_{i−1}} y^{(a−ε_{k})}f_{j}λ_{j}e_{b_{j}}^{−1}e_{b})e_{k}

= (−1)^{|a|−1} ∑_{j: f_{j}∈B_{i−1}} ( ∑_{k∈supp(a)}(−1)^{|b_{j}|}y^{(a−ε_{k})}f_{j}e_{k})λ_{j}e_{b_{j}}^{−1}e_{b}

= (−1)^{|a|−1} ∑_{j: f_{j}∈B_{i−1}} δ(y^{(a)}f_{j})λ_{j}e_{b_{j}}^{−1}e_{b}

= (−1)^{|a|−1}δ( ∑_{j: f_{j}∈B_{i−1}} y^{(a)}f_{j}λ_{j}e_{b_{j}}^{−1}e_{b}) = −δ(ϑ(y^{(a)}f)) .

When (G′, ∂′) is a complex obtained from a homogeneous basis B′ of F, write each f′ ∈ B′_{i} in the form f′ = ∑_{j: f_{j}∈B_{i}} λ_{j}x^{b′−b_{j}}f_{j} with b′ = deg(f′) and b_{j} = deg(f_{j}), and define homomorphisms of E-modules γ_{i}: G′_{i} ⟶ G_{i} by

γ_{i}(y^{(a)}f′) = ∑_{j: f_{j}∈B_{i}} y^{(a)}fλ_{j}e_{b_{j}}^{−1}e_{b′} .

Computations similar to (and more straightforward than) those above show that γ(ϑ′(y^{(a)}f′)) = ϑ(γ(y^{(a)}f′)) and γ(δ′(y^{(a)}f′)) = δ(γ(y^{(a)}f′)), so γ is a chain map. It is clearly bijective, so we have the desired isomorphism.

Proof of the theorem. Let (F, θ) be a n-graded free resolution of S/I over S, and let (G, ∂) be the complex obtained from it by Construction 1.1. To show that it is a resolution of E/J, we construct a K-linear chain homotopy χ such that

χ∂ + ∂χ = idG̃  (∗)

where G̃ is the complex obtained from G by replacing G_{0} with J.

Since F is exact, there is a homogeneous K-linear chain homotopy τ such that

τθ + θτ = idF̃

where F̃ is the complex obtained from F by replacing F_{0} with I.

Thus, for f ∈ B with deg(f) = b and σ ⊆ [n] such that supp(b)∩σ = ∅, we have

τ(fx^{σ}) = ∑_{k} μ_{k}x^{σ}x^{b−a_{k}}h_{k} where μ_{k} ∈ K , h_{k} ∈ B , a_{k} = deg(h_{k}) .

We define a K-linear map χ on the K-basis of G̃ described in Construction 1.1 by

χ(y^{(a)}fe_{σ}) = \begin{cases}∑_{k} h_{k}μ_{k}e_{a_{k}}^{−1}(e_{b}e_{σ}) & if a = 0 and supp(b) ∩ σ = ∅  (1) \\ (−1)^{r+|b|}y^{ε_{s}}fe_{σ\\{s\}} & if a = 0 < min(supp(b) ∩ σ) = s  (2) \\ 0 & if a ≠ 0 and supp(b) ∩ σ = ∅  (3) \\ 0 & if 0 < min(a) < min(supp(b) ∩ σ)  (4) \\ (−1)^{r+|b|}y^{(a+ε_{s})}fe_{σ\\{s\}} & if min(a) ≥ min(supp(b) ∩ σ) = s  (5) \end{cases}

where b = supp(f) and r = |{ k ∈ σ | k < min(supp(b) ∩ σ) }|.

We establish (∗), by four separate computations. To simplify notation, we set

s(c) = supp(c) for c ∈ ℕ^{n}  and  u_{j} = λ_{j}e_{b_{j}}^{−1}e_{b} for j ∈ [n] .

(1) One has ∂(fe_{σ}) = ∑_{j} f_{j}(u_{j})e_{σ}. Since s(u_{j}) = s(b) \ s(b_{j}) for every j, we get

s(u_{j}) ∩ σ = ∅ and s(b_{j}) ∩ s(u_{j}e_{σ}) = s(b_{j}) ∩ σ = ∅ .

Write τ(f_{j}x^{σ}x^{b−b_{j}}) = ∑_{ℓ} g_{ℓ}ν_{ℓj}x^{σ}x^{b−c_{ℓ}} with g_{ℓ} ∈ B, ν_{ℓj} ∈ K and c_{ℓ} = deg(g_{ℓ}). As e_{b_{j}}u_{j}e_{σ} = λ_{j}e_{b}e_{σ}, one has χ(f_{j}u_{j}e_{σ}) = λ_{j} ∑_{ℓ} g_{ℓ}ν_{ℓj}e_{c_{ℓ}}^{−1}(e_{b}e_{σ}), therefore

χ(∂(fe_{σ})) = ∑_{ℓ} g_{ℓ}( ∑_{j} λ_{j}ν_{ℓj})e_{c_{ℓ}}^{−1}(e_{b}e_{σ}) .

On the other hand, if θ(h_{k}) = ∑_{ℓ} g_{ℓ}λ_{ℓk}x^{a_{k}−c_{ℓ}} with λ_{ℓk} ∈ K, then

∂(χ(fe_{σ})) = ∑_{k} ∑_{ℓ} g_{ℓ}μ_{k}λ_{ℓk}(e_{c_{ℓ}}^{−1}e_{a_{k}})(e_{a_{k}}^{−1}(e_{b}e_{σ})) = ∑_{ℓ} g_{ℓ}( ∑_{k} μ_{k}λ_{ℓk})e_{c_{ℓ}}^{−1}(e_{b}e_{σ}).

Since θτ + τθ = id_{F}, we see that there exists a ℓ_{0} such that g_{ℓ_{0}} = f, and

∑_{k} μ_{k}λ_{ℓk} + ∑_{j} λ_{j}ν_{ℓj} = \begin{cases}1 & if ℓ = ℓ_{0} ; \\ 0 & if ℓ ≠ ℓ_{0} . \end{cases}

This shows that ∂χ(fe_{σ}) + χ∂(fe_{σ}) = fe_{σ}, as desired.

(2) and (5). In either case, ∂χ(y^{(a)}fe_{σ}) is equal to

(−1)^{r} ∑_{k∈s(a+ε_{s})} y^{(a+ε_{s}−ε_{k})}fe_{k}e_{σ\{s}} +

(−1)^{r+|b|+|a|+1} ∑_{j: s(b_{j})⊇s(a+ε_{s})} y^{(a+ε_{s})}f_{j}u_{j}e_{σ\{s}} .

Note that y^{(a)}fe_{σ} appears above as a summand in the first sum for k = s. Now we compute χ(∂(y^{(a)}fe_{σ})). If s ∉ s(b_{j}) for some j, then s ∈ s(b) \ s(b_{j}) = s(u_{j}), therefore u_{j}e_{s} = 0, so that in ∂(y^{(a)}fe_{σ}) only the summands y^{(a)}f_{j}u_{j}e_{σ} with s ∈ s(b_{j}) remain. In this case min(s(b_{j}) ∩ s(u_{j}e_{σ})) = s, hence

χ(y^{(a)}f_{j}u_{j}e_{σ}) = (−1)^{r+|b_{j}|+|u_{j}|}y^{(a+ε_{s})}f_{j}u_{j}e_{σ\{s}} .

Since |u_{j}| + |b_{j}| = |b|, we see that the second sum in ∂χ(y^{(a)}fe_{σ}) appears in χ(∂(y^{(a)}fe_{σ})) with the opposite sign. If k ∈ s(a) and k ∉ σ, then k ≥ min(a) ≥ s, so min(s(b) ∩ (σ ∪ k)) = s. As min(a − ε_{k}) ≥ min(a) ≥ s, we get

(−1)^{|b|}χ(y^{(a−ε_{k})}fe_{k}e_{σ}) = (−1)^{r+1}y^{(a+ε_{s}−ε_{k})}fe_{k}e_{σ\{s}} .

The desired equality follows.

(3) For each j with s(a) ⊆ s(b_{j}), one has s(b_{j})∩σ = ∅, hence χ(y^{(a)}f_{j}u_{j}e_{σ}) = 0. Let k ∈ s(a), k ∉ σ and consider χ(y^{(a−ε_{k})}fe_{k}e_{σ}). We have now s(b)∩(σ ∪k)) = k. If k > min(a), then min(a − ε_{k}) = min(a), therefore χ(y^{(a−ε_{k})}fe_{k}e_{σ}) = 0. Let k = min(a). Then min(a − ε_{k}) ≥ k, hence (−1)^{|b|}χ(y^{(a−ε_{k})}fe_{k}e_{σ}) = y^{(a)}fe_{σ}. This proves the desired equality.

(4) For each j with s(a) ⊆ s(b_{j}), one has u_{j}e_{m} = 0 or min(s(b_{j})∩σ) = m, so that in both cases χ(y^{(a)}f_{j}u_{j}e_{σ}) = 0. Let k ∈ s(a), k ∉ σ and consider χ(y^{(a−ε_{k})}fe_{k}e_{σ}). If k > min(a), then min(a−ε_{k}) = min(a) < m, therefore min(a) < min(s(b)∩(σ∪k)) and by definition χ(y^{(a−ε_{k})}fe_{k}e_{σ}) = 0. Let k = min(a). Then min(s(b)∩(σ ∪k)) = k ≤ min(a − ε_{k}), therefore (−1)^{|b|}χ(y^{(a−ε_{k})}fe_{k}e_{σ}) = y^{(a)}fe_{σ}. This proves (∗).

2. Applications

Recall that each finite n-graded module M over A = E or A = S has a unique up to isomorphism minimal resolution by free n-graded A-modules, and homogeneous A-linear homomorphisms. The multigraded Betti number β_{ia}^{A}(M) is the number of basis elements of the i’th free module in such a resolution, that are homogeneous of degree a. The multigraded Poincaré series of M over A is defined by

P_{M}^{A}(t, u) = ∑_{i≥0} ∑_{a∈ℕ^{n}} β_{ia}^{A}(M)t^{i}u^{a} .

For the rest of this section, I is an ideal generated by squarefree monomials in S, and J denotes the corresponding monomial ideal in E.

Counting ranks in the resolution of Theorem 1.3 we get a new proof of [3, (6.4)].

Proposition 2.1. There is an equality of formal power series

P_{E/J}^{E}(t, u) = ∑_{i≥0} ∑_{a∈ℕ^{n}} β_{ia}^{S}(S/I)\frac{t^{i}u^{a}}{∏_{j∈supp(a)}(1 − tu_{j})} .  □

We record a couple of immediate consequences of this formula.

Corollary 2.2. (1) The multigraded Betti numbers of I are independent of the characteristic of the field K if and only if this is true for J.

(2) The ideal I has a linear free resolution over S if and only if the ideal J has a linear free resolution over E. □

An important class of ideals in S with linear resolution are the Gotzmann ideals. Recall that an ideal L ⊆ A, where A = S or A = E, is called Gotzmann if it

is generated by elements of the same degree, say d, and its span in degree d + 1 is the smallest possible: rank_{K} L_{d+1} ≤ rank_{K} L′_{d+1} holds for all graded ideals L′ ⊆ A with rank_{K} L′_{d} = rank_{K} L_{d}. It is a widely open question which monomial ideals are Gotzmann. From a combinatorial point of view, it is particularly interesting for ideals generated by squarefree monomials.

Proposition 2.3. If the ideal I ⊆ S is Gotzmann, then so is the ideal J ⊆ E.

Note that the converse may fail: J = (e_{1} ∧ e_{2} ∧ e_{3}, e_{1} ∧ e_{2} ∧ e_{4}, e_{1} ∧ e_{3} ∧ e_{4}) ⊆ E is a Gotzmann ideal, but I = (x_{1}x_{2}x_{3}, x_{1}x_{2}x_{4}, x_{1}x_{3}x_{4}) ⊆ S is not.

Proof. Let J′ ⊆ E be an ideal generated in degree d, with rank_{K} J′_{d} = rank_{k} J_{d}.

The algebraic Kruskal-Katona Theorem [3, (4.4)] yields a monomial ideal J^{lex} generated in degree d, with rank_{K} J_{d}^{lex} = rank_{K} J′_{d} and rank_{K} J_{d+1}^{lex} ≤ rank_{K} J′_{d+1}. For the squarefree monomial ideal I′ ⊆ S corresponding to J^{lex}, we have

rank_{K} J_{d+1} = nβ_{0d}(J) − β_{1d+1}(J)
= nβ_{0d}(I) − (β_{1d+1}(I) + dβ_{0d}(I))
= rank_{K} I_{d+1} − d rank_{K} I_{d}
≤ rank_{K} I′_{d+1} − d rank_{K} I′_{d}
= nβ_{0d}(I′) − (β_{1d+1}(I′) + dβ_{0d}(I′))
= nβ_{0d}(J^{lex}) − β_{1d+1}(J^{lex})
= rank_{K} J_{d+1}^{lex}

where the inequality is the Gotzmann hypothesis on I, the second and penultimate equalities come from Proposition 2.1, the rest are read off from the corresponding minimal resolutions. Altogether, we get rank_{K} J_{d+1} ≤ rank_{K} J′_{d+1}, as desired.

Applying Theorem 1.3 to the Taylor resolution of monomial ideals in polynomial rings, cf. [15] or [9, p. 439], we obtain an analogue over exterior algebras.

For a set of monomials { u_{1}, …, u_{m} } and a subset τ ⊆ [m] = { 1, …, m }, we denote u_{τ} the least common multiple of the monomials { u_{j} | j ∈ τ }.

Proposition 2.4. Let J ⊆ E be an ideal generated by a set { u_{1}, …, u_{m} } of monomials. The right E-modules T_{i} with basis

{ y^{(a)}f_{τ} | a ∈ ℕ^{n} , |a| + |τ| = i , τ ⊆ [m] , supp(a) ⊆ supp(u_{τ}) }

where deg(y^{(a)}f_{τ}) = a + deg u_{τ}, and the E-linear maps defined by

∂(y^{(a)}f_{τ}) = (−1)^{|u_{τ} |} ∑_{k∈supp(a)} y^{(a−ε_{k})}f_{τ} e_{k} +

∑_{j: supp(u_{τ\{j}})⊇supp(a)}(−1)^{r_{j}+|a|}y^{(a)}f_{τ\{j}}u_{τ\{j}}^{−1}u_{τ}

where r_{j} = |{ t ∈ τ | t < j }|, form a n-graded resolution of E/J . □

Example 2.5. When J = (e_{1}, …, e_{n}) the proposition provides a minimal n-graded resolution of K = E/(e_{1}, …, e_{n}) over E. Another one is the Cartan resolution (C, ∂), where C_{i} has a basis { w^{(c)} | c ∈ ℕ^{n}, |c| = i }, and

d(w^{(c)}) = ∑_{k∈supp(c)} w^{(c−ε_{k})}e_{k} .

To get an isomorphism of complexes γ: C ⟶ T, note that each c ∈ ℕ^{n} can be written uniquely as c = a + b with supp(c) = supp(b) and b squarefree, and set

γ(w^{(c)}) = (−1)^{|b|(|a|+(|b|−1)/2)}y^{(a)}f_{supp(b)} .

Our last application is to stable ideals, a notion extended in [3] from polynomial rings to exterior algebras: setting max(e_{σ}) = max{ i | i ∈ σ }, call a monomial ideal J ⊆ E stable if e_{j}e_{σ\{m}} ∈ J for each e_{σ} ∈ J and each j < m = max(e_{σ}).

For a monomial ideal J ⊆ E, we denote G(J) the uniquely defined minimal generating set of J consisting of monomials. As in [10], it is easily seen that each monomial u′ ∈ J has a unique decomposition u′ = uw with u ∈ G(J) and max(u) < min(w). Applying Theorem 1.3 to the resolution of squarefree stable ideals in S given in [2], we get a resolution for stable monomial ideals in E.

Proposition 2.6. If J ⊆ E is a stable ideal, then E/J has a minimal resolution (G, ∂) by n-graded free E-modules G_{ℓ} with basis

\{y^{(a)}f_{σ,u} | \begin{matrix}a ∈ ℕ^{n} , σ ⊆ [n] , u ∈ G(J) \\ supp(a) ⊆ σ ∪ supp(u) , σ ∩ supp(u) = ∅ , max(σ) < max(u) \\ i = |a| + |σ| + 1 , deg(y^{(a)}f_{σ,u}) = a + deg(e_{σ}) + deg(u) \end{matrix}\}

and differentials ∂_{ℓ}: G_{ℓ} ⟶ G_{ℓ−1} given by

∂(y^{(a)}f_{σ,u}) = (−1)^{|u|+|σ|} ∑_{ℓ∈supp(a)} y^{(a−ε_{ℓ})}f_{σ,u}e_{ℓ} +

(−1)^{|a|} ∑_{j∈σ} ((−1)^{|σ|}y^{(a)}f_{σ\{j},u}e_{j} + (−1)^{(|σ|−1)|w_{j}|}f_{σ\{j},u_{j}}w_{j})

where u_{j} ∈ G(J) is determined from the unique decomposition ue_{j} = u_{j}w_{j} described above, and y^{(b)}f_{ρ,v} = 0 if max(ρ) > max(v) or supp(b) ⊈ ρ ∪ supp(v). □

The preceding result was originally proved by different means in [3, (2.1)].

3. Cohomology

We study right modules over the exterior algebra E. Since the ideal (V) ⊂ E is nilpotent, each (finite) E-module M has a unique up to isomorphism minimal free resolution F by (finite) free E-modules. The rank β_{i}^{E}(M) of the free E-module F_{i} is known as the i’th Betti number of M over E. The size of F is measured by the complexity of M over E, that is introduced as follows:

cx_{E} M = inf{ c ∈ ℤ | β_{i}^{E}(M) ≤ αi^{c−1} for some α ∈ ℝ and all i ≥ 1 } .

For each v ∈ V = E_{1}, the equality v^{2} = 0 implies Mv ⊆ Ann_{M}(v). We say that v is M-regular if equality holds, or, equivalently, if the infinite complex of K-spaces

(M, ρ^{v}): … ⟶ M ⟶^{ρ^{v}} M ⟶^{ρ^{v}} M ⟶ …  where  ρ^{v}(y) = yv

has trivial homology H_{∗}(M, ρ^{v}). Otherwise, we say that v is M-singular.

The set V_{E}(M) ⊆ V of M-singular elements is called the rank variety of M. If M = ⨁_{a∈ℤ} M_{a} is graded, regularity can also be introduced by the vanishing

of the cohomology H^{∗}(M, v) of the finite complex of K-vector spaces

(M, v): … ⟶ M_{a−1} ⟶^{ρ_{a−1}^{v}} M_{a} ⟶^{ρ_{a}^{v}} M_{a+1} ⟶ …

Recall that when M and N are graded E-modules, their graded tensor product M⊗_{K}^{gr}N and homomorphism space Hom_{K}^{gr}(N, M) have diagonal actions:

\begin{matrix}(x ⊗ y)e_{σ} = ∑_{τ⊆σ}(−1)^{k|τ|} sgn_{σ\τ}^{τ} xe_{τ} ⊗ ye_{σ\τ} \\ (γe_{σ})(y) = ∑_{τ⊆σ}(−1)^{|τ|(k+(|τ|+1)/2)} sgn_{σ\τ}^{τ} γ(ye_{τ})e_{σ\τ} \end{matrix} for y ∈ N_{k} and σ ⊆ [n]

where sgn_{σ\τ}^{τ} is the sign of the permutation (τ, σ \ τ); that these are (graded) E-modules follows from the fact that E is a super Hopf algebra.

The properties of V_{E}(M) are similar to those of the varieties of modular representations, but proofs are simpler, compare the account by Benson [5].

Theorem 3.1. If the field K is algebraically closed, then the rank varieties of finite E–modules M, N satisfy the following properties.

(1) V_{E}(M) is a cone (that is, a homogeneous algebraic subset) in V.

(2) dim V_{E}(M) = cx_{E} M and 2^{n−cx_{E} M} divides rank_{K} M .

(3) V_{E}(M) = {0} if and only if M is free.

(4) V_{E}(M) = V_{E}(N) if M is a syzygy of N.

(5) If M ⊆ N, then each one of the three varieties V_{E}(M), V_{E}(N), V_{E}(N/M), is contained in the union of the other two.

(6) V_{E}(M ⊕ N) = V_{E}(M) ∪ V_{E}(N) .

(7) V_{E}(M⊗_{K}^{gr}N) = V_{E}(M) ∩ V_{E}(N) = V_{E}(Hom_{K}^{gr}(N, M)) if M, N are graded.

(8) Each cone in V is the rank variety of some graded E–module.

As over commutative rings, the notion of regularity can be extended to sequences. Elements v_{1}, …, v_{r} ∈ V form an M-regular sequence if v_{i} is (M/M(v_{1}, …, v_{i−1}))- regular for 1 ≤ i ≤ r, in other words, if yv_{i} ∈ M(v_{1}, …, v_{i−1}) implies that y ∈ M(v_{1}, …, v_{i}) for 1 ≤ i ≤ r. It is clear that each M-regular sequence can be extended to a maximal one. The supremum of the lengths of M-regular sequences is called the depth of M over E, and denoted depth_{E} M.

Parts of the preceding theorem depend on a depth-formula for modules over exterior algebras, that is similar to the extension of the classical Auslander-Buchsbaum equality to modules over complete intersections, obtained in [4].

Theorem 3.2. If the field K is infinite and M is a finite E-module, then each maximal M-regular sequence has depth_{E} M elements, and

depth_{E} M + cx_{E} M = n .

Examples 3.3. (1) If rank_{K} M is odd, then cx_{E} M = n.

Indeed, if depth_{E} M > 0, then taking an M-regular v ∈ V we get rank_{K} M = rank_{K}(Ann_{M}(v)) + rank_{K}(Mv) = 2 rank_{K}(Mv), so rank_{K} M is even.

(2) The depth equality fails when K is finite and n ≥ 2.

Indeed, if v ∈ V \ {0} then E ⟶^{λ_{v}} E ⟶^{λ_{v}} E with λ_{v}(e) = ve is an exact complex of E–modules, so cx_{E}(E/(v)) = 1, and hence M = ⨁_{v∈V} E/(v) has complexity 1; on the other hand, it is clear that V_{E}(M) = V, hence depth_{R} M = 0.

To start on the proofs, we record some simple facts on regularity.

Remarks 3.4. Let M be an E-module.

(1) When v^{2} = 0, any K[v]-module is a direct sum of copies of K[v] and K[v]/(v). Thus, v ∈ V = E_{1} is regular if and only if M is free over the subalgebra K[v] ⊆ E.

(2) For v ∈ V, let π: E ⟶ E/(v) and ρ: M ⟶ M/Mv be the canonical homomorphisms. If v is M-regular, then they induce isomorphisms

Ext_{π}^{i}(ρ, K): Ext_{E/(v)}^{i}(M/Mv, K) ≅ Ext_{E}^{i}(M, K) Tor_{i}^{π}(ρ, K): Tor_{i}^{E}(M, K) ≅ Tor_{i}^{E/(v)}(M/Mv, K) for i ≥ 0 .

Indeed, M is free over K[v] by (2), so if G is a free resolution of M over E, then G/Gv is a free resolution of M/Mv over E/(v). Thus, Ext_{π}^{∗}(ρ, K) and Tor_{∗}^{π}(ρ, K) are the maps induced in homology by the isomorphisms of complexes Hom_{E/(v)}(G/Gv, K) ≅ Hom_{E}(G, K) and G⊗_{E} K ≅ (G/Gv)⊗_{E/(v)} K, respectively.

(3) Regularity of a sequence v = v_{1}, …, v_{d} ∈ V is detected by its Cartan complex C(v; M), defined by C_{i}(v, M) = ⨁_{a∈ℕ^{n},|a|=i} w^{(a)}M with w^{(a)}M ≅ M for each a ∈ ℕ^{n} and ∂(w^{(a)}u) = ∑_{ℓ∈supp(a)} w^{(a−ε_{ℓ})}ue_{ℓ} for u ∈ M.

We set H(v; M) = H(C(v; M)), and note that the following are equivalent:

(i) v is M-regular.

(ii) M is a free module over K[v_{1}, …, v_{d}] .

(iii) H_{1}(v; M) = 0 .

(iv) H_{i}(v; M) = 0 for i ≥ 1 .

Indeed, let E′ be an exterior algebra on alternating variables e′_{1}, …, e′_{d}, and let φ: E′ ⟶ E be the homomorphism of K-algebras with φ(e′_{i}) = v_{i} for i = 1, …, r. If C′ is the Cartan resolution of the right E′–module K, cf. Remark 2.5, then C(v; M) = C′ ⊗_{E} M, so H_{i}(v; M) = Tor_{i}^{E′}(K, M). Thus, (i) ⟹ (iv) by iterated use of (2). If (iii) holds, then Tor_{1}^{E′}(K, M) = 0; computing Tor from a minimal free resolution of M over E′ we see that M′ is free over E′; it follows that φ is an isomorphism, so (iii) ⟹ (ii) holds. Finally, (ii) ⟹ (i) is trivial.

(4) By (3), each permutation of an M-regular sequence is itself M-regular.

To study the geometry of V_{E}(M) we use product structures in cohomology. We recall the basics, referring for details to MacLane [13] or Bourbaki [6].

Construction 3.5. For E-modules M, L, N and i, j ∈ ℤ, composition pairings

Ext_{E}^{j}(L, N) × Ext_{E}^{i}(M, L) ⟶ Ext_{E}^{i+j}(M, N)

are introduced as follows. Let C and G be E-free resolutions of L and M, respectively, and represent elements in Ext_{E}^{i}(M, L) and Ext_{E}^{j}(L, N) by E-linear homomorphisms κ: G_{i} ⟶ L with κ∂_{i+1} = 0 and ξ: C_{j} ⟶ N with ξ∂_{j+1} = 0. Choosing a lifting of κ to an E-linear chain map κ̃: G ⟶ C of degree −i, define the product cl(ξ) cl(κ) to be the class of the composition ξκ̃_{i+j}: G_{i+j} ⟶ N.

The pairings are K-bilinear, associative, and natural (hence, independent of the choices made above). They make Ext_{E}^{∗}(K, K) = ⨁_{i=0}^{∞} Ext_{E}^{i}(K, K) into a graded algebra, and Ext_{E}^{∗}(M, K) = ⨁_{i=0}^{∞} Ext_{E}^{i}(M, K) into a graded left module over it.

Proposition 3.6. There is a natural in V isomorphism of graded K-algebras

Ext_{E}^{∗}(K, K) ≅ Sym_{K}^{∗}(V^{∨}) where V^{∨} = Hom_{K}(V, K) .

If M is a finite E-module, then the Ext_{E}^{∗}(K, K)-module Ext_{E}^{∗}(M, K) is finite.

Proof. Cartan’s resolution (C, ∂) of K over E, cf. Example 2.5, is minimal, so

Ext^{i}(K, K) = H^{i} ( Hom_{E}(C, K)) = Hom_{E} ( ⨁_{a∈ℕ^{n},|a|=i} Ew^{(a)}, K) .

The homomorphisms of E-modules { χ^{a}: C_{i} ⟶ K | a ∈ ℕ^{n}, |a| = i }, such that χ^{a}(w^{(b)}) = 1 for b = a and χ^{a}(w^{(b)}) = 0 for b ∈ ℕ^{n} with |b| = i and b ≠ a form a K-basis of Hom_{E}(C, K). The E-linear maps

χ_{i+j}^{a}: C_{i+j} ⟶ C_{j} defined by χ_{i+j}^{a}(w^{(b)}) = \begin{cases}w^{(b−a)} & if b − a ∈ ℕ^{n}; \\ 0 & otherwise , \end{cases}

define a lifting of χ^{a} to a chain map C ⟶ C. This means that χ^{a}χ^{b} = χ^{a+b} for all b ∈ ℕ^{n}, so Ext_{E}^{∗}(K, K) is the polynomial ring on χ_{1} = χ^{ε_{1}}, …, χ_{n} = χ^{ε_{n}}.

To see that the Ext_{E}^{∗}(K, K)-module Ext_{E}^{∗}(M, K) is finite we argue by induction on q = max{ r | ME_{r} ≠ 0 }. If q = 1, then M ≅ K^{s} for some s and the assertion is clear. If q > 1, then M′ = M(V) ≠ 0, so the exact sequence 0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0 of E-modules yields an exact sequence of Ext_{E}^{∗}(K, K)-modules

Ext_{E}^{∗}(M′, K) ⟶ Ext_{E}^{∗}(M, K) ⟶ Ext_{E}^{∗}(M′′, K)  (3.6.1)

in which those on the outside are noetherian by the induction hypothesis.

Remark 3.7. If χ_{1}, …, χ_{n} is the basis of V^{∨} dual to the basis e_{1}, …, e_{n} of V, then we identify Ext_{E}^{∗}(K, K) with the graded polynomial ring S = K[χ_{1}, …, χ_{n}] in which each χ_{i} has degree 1; the elements of S act as functions on V.

Applied to the S-module Ext_{E}^{∗}(M, K), the Hilbert-Serre theorem yields:

Corollary 3.8. The Krull dimension of the S-module Ext_{E}^{∗}(M, K) is equal to cx_{E} M, and there exists a polynomial p_{M}(t) ∈ ℤ[t] with p_{M}(1) > 0, such that

P_{M}^{E}(t) = \frac{p_{M}(t)}{(1 − t)^{c}}  with  c = cx_{E} M .  □

Now we give a basic cohomological description of the rank variety.

Theorem 3.9. If K is algebraically closed and M is a finite E-module, then

V_{E}(M) = { v ∈ V | ξ(v) = 0 for all ξ ∈ Ann_{S} ( Ext_{E}^{∗}(M, K)) } .

Proof. Let I = Ann_{S} ( Ext_{E}^{∗}(M, K)). For v ∈ V, set V^{v} = Ker (V^{∨} ⟶ (vK)^{∨}), and let P_{v} denote the homogeneous prime ideal (V^{v}) of S. By the Nullstellensatz, we have to prove that I ⊆ P_{v} if and only if v is M-singular.

If v is singular, then by Remark 3.4.1 we have an isomorphism of K[v]-modules M ≅ K[v]^{p} ⊕ K^{q} with q > 0. The inclusion ι: K[v] ↪ E induces a diagram

Ext_{E}^{∗}(K, K) ⊗_{K} Ext_{E}^{∗}(M, K) ⟶ Ext_{E}^{∗}(M, K)
Ext_{ι}^{∗}(K,K)⊗Ext_{ι}^{∗}(M,K)↓  ↓^{Ext_{ι}^{∗}(M,K)}
Ext_{K[v]}^{∗}(K, K) ⊗_{K} Ext_{K[v]}^{∗}(M, K) ⟶ Ext_{K[v]}^{∗}(M, K) .

It commutates by naturality of composition products, so Ext_{ι}^{∗}(K, K)(I) annihilates

Ext_{K[v]}^{∗}(M, K) ≅ K^{p} ⊕ Ext_{K[v]}^{∗}(K, K)^{q} .

It is then equal to 0, that is, I ⊆ Ker Ext_{ι}^{∗}(K, K) = P_{v}.

If v is regular, then π: E ⟶ E/(v) and ρ: M ⟶ M/Mv induce a diagram

Ext_{E}^{∗}(K, K) ⊗_{K} Ext_{E}^{∗}(M, K)  ⟶  Ext_{E}^{∗}(M, K)
Ext_{π}^{∗}(K,K)⊗ Ext_{π}^{∗}(ρ,K)↑  ↑^{Ext_{π}^{∗}(ρ,K)}
Ext_{E/(v)}^{∗}(K, K) ⊗_{K} Ext_{E/(v)}^{∗}(M/Mv, K) ⟶ Ext_{E/(v)}^{∗}(M/Mv, K) .

It is commutative by naturality, and Ext_{π}^{∗}(ρ, K) is an isomorphism by Remark 3.4.2. As Ext_{E/(v)}^{∗}(M/Mv, K) is a finite Ext_{E/(v)}^{∗}(K, K)-module by Proposition 3.6, we conclude that so is Ext_{E}^{∗}(M, K). It follows that the composition

Ext_{E/(v)}^{∗}(K, K) ⟶^{Ext_{π}^{∗}(K,K)} Ext_{E}^{∗}(K, K) = S ⟶ S/I

is a finite homomorphism of rings. Assuming that P_{v} ⊆ I, we conclude that

Sym_{K}^{∗}[V^{v}] ≅ Ext_{E/(v)}^{∗}(K, K) ⟶ S/P_{v} = Ext_{K[v]}^{∗}(K, K) ≅ Sym_{K}^{∗}[(Kv)^{∨}]

is a finite homomorphism: this is absurd, since it maps V^{v} to 0.

Proof of Theorem 3.2. Let v = v_{1}, …, v_{d} be an arbitrary maximal M-regular sequence in V. We want to prove that depth_{E} M = d and cx_{E} M = n − d.

We first assume that K is algebraically closed; the elements in a regular sequence being K-linearly independent, we have d ≤ n, so we can induce on d. An equality d = 0 means that each element of V is M-singular, that is, depth_{E} M = 0; on the other hand, Theorem 3.9 yields cx_{R} M = dim V_{E}(M) = dim V = n.

If d > 0, then the images of v_{2}, …, v_{d} in E/(v_{1}) form a maximal (M/Mv_{1})- regular sequence. The induction hypothesis yields depth_{E}(M/Mv_{1}) = d − 1 and

cx_{E/(v_{1})}(M/Mv_{1}) = (n − 1) − (d − 1) = n − d .

As cx_{E/(v_{1})}(M/Mv_{1}) = cx_{E} M by Remark 3.4.2, we are done.

Let now K be an arbitrary infinite field. Taking an algebraic closure K̄ of K, we consider the finite module M̄ = M ⊗_{K} K̄ over the exterior algebra Ē = E ⊗_{K} K̄ of the K̄-vector space V̄ = V ⊗_{K} K̄. Due to the flatness of Ē over E, we see that (considered as a sequence in V̄) any M-regular sequence in V is M̄-regular, and that βĒ ; i(M̄) = β_{i}^{E}(M) for each i. This yields

depth_{E} M ≤ depthĒ M̄ = d  and  cx_{E} M = cxĒ M̄ = n − d .

Assuming that the M̄-regular sequence v is not maximal, we can find in V̄ /K̄v an element v that is (M̄/M̄(v))-regular. As the set of regular elements is Zariski-open and K is infinite, we can even pick v in V/(v), and get an M-regular sequence v, v. This is absurd, so v is a maximal M̄-regular sequence and we have

d ≤ depth_{E} M ≤ depthĒ M̄ = d .

It follows that depth_{E} M = d and depth_{E} M + cx_{E} M = n, as desired.

Lemma 3.10. For each ξ ∈ Ext_{E}^{i}(K, K) there is a graded E-module L_{ξ} such that

V_{E}(L_{ξ}) = { v ∈ V | ξ(v) = 0 } .

Proof. In the Cartan resolution C of K over E, set D_{i} = ∂_{i}(C_{i}), let ξ̄: D_{i} ⟶ K be the E-linear map that corresponds to ξ under the isomorphisms

Ext^{i}(K, K) = Hom_{E}(C_{i}, K) ≅ Hom_{E}(D_{i}, K)

and set L_{ξ} = Ker ξ̄. The exact sequence of E-modules

0 ⟶ L_{ξ} ⟶ D_{i} ⟶ K ⟶ 0

induces an exact sequence of graded modules over S = Ext^{∗}(K, K),

S ⟶^{ξ̄^{∗}} Ext_{E}^{∗}(D_{i}, K) ⟶ Ext_{E}^{∗}(L_{ξ}, K) ⟶^{ð} S(1) ⟶^{ξ̄^{∗}(1)} Ext_{E}^{∗}(D_{i}, K)(1)

where ξ̄^{∗} = Ext_{E}^{∗}(ξ̄, K) maps 1 ∈ S^{0} to ξ ∈ Ext_{E}^{i}(D_{i}, K) = S^{i}. Thus, ξ̄^{∗} and ξ̄^{∗}(1) are injective, yielding Ext_{E}^{∗}(L_{ξ}, K) ≅ S^{≥i}(i)/Sξ. As \sqrt{S^{≥i}(i)/Sξ} = \sqrt{Sξ}, we conclude from Theorem 3.9 that V_{E}(L_{ξ}) has the desired form.

Proof of Theorem 3.1. (1) Note that rank_{K}(Mv) ≤ rank_{K} ( Ann_{M}(v)) for each v ∈ V, and the inequality is strict precisely when v is M-singular. Setting m = rank_{K} M, we rewrite the inequality as rank_{K}(ρ^{v}) < m − rank_{K}(ρ^{v}), that is, as rank_{K}(ρ^{v}) < m/2. Thus, V_{E}(M) is the zero-set of the minors of order ⌊m/2⌋ of a matrix representing multiplication by a generic element of V. Clearly, v ∈ V_{E}(M) implies λv ∈ V_{E}(M) for each λ ∈ K, so the variety is homogeneous.

(2) Let cx_{E} M = c. By Corollary 3.8 and elementary dimension theory, the number c is equal to the Krull dimension of the ring S/ Ann_{S} ( Ext_{E}^{∗}(M, K)), which is the dimension of the variety V_{E}(M).

Theorem 3.2 yields an M-regular sequence v_{1}, …, v_{n−c} in V, so M is free over E′ = K[v_{1}, …, v_{n−c}] by Remark 3.4, so rank_{K} M = 2^{n−c} rank_{E′} M.

(3) If V_{E}(M) = {0}, then cx_{E} M = 0, so the preceding argument works with r = n, and shows that M is free over K[v_{1}, …, v_{n}] = E. Conversely, if M is free over E the non-zero elements of V are obviously M-regular, hence V_{E}(M) = {0}.

(5) An exact sequence of E-modules 0 ⟶ M ⟶ N ⟶ M/N ⟶ 0 induces an exact sequence of complexes of vector spaces

0 ⟶ (M, ρ^{v}) ⟶ (N, ρ^{v}) ⟶ (M/N, ρ^{v}) ⟶ 0

and hence an exact sequence of homology spaces

H_{∗}(M, ρ^{v}) ⟶ H_{∗}(N, ρ^{v}) ⟶ H_{∗}(M/N, ρ^{v}) ⟶ H_{∗}(M, ρ^{v}) ⟶ H_{∗}(N, ρ^{v})

which implies the desired assertions follow immediately.

(4) It suffices to consider the case when M and N appear in an exact sequence 0 ⟶ M ⟶ P ⟶ N ⟶ 0 with a free E-module P. By (5) and (3) we then have

V_{E}(M) ⊆ V_{E}(N) ∪ V_{E}(P) = V_{E}(N) ⊆ V_{E}(M) ∪ V_{E}(P) = V_{E}(M) .

(6) follows immediately from the definitions.

(7) Recall that v ∈ V acts on M⊗_{K}^{gr}M by the formula (x ⊗ y)v = x ⊗ yv + (−1)^{k}xv ⊗ y, when y ∈ N_{k}. This means that x ⊗ y ↦ y ⊗ x is an isomorphism

(M⊗_{K}^{gr}N, v) ≅ (N, v) ⊗_{K} (M, v)

where the tensor product on the right hand side is one of complexes of K-vector spaces. The Künneth formula then gives an isomorphism of graded vector spaces

H^{∗}(M⊗_{K}^{gr}N, v) ≅ H^{∗}(N, v) ⊗_{K} H^{∗}(M, v)

from which we get V_{E}(M⊗_{K}^{gr}N) = V_{E}(M) ∩ V_{E}(N).

A similar argument yields H^{∗}(Hom_{K}^{gr}(N, M), v) ≅ Hom_{K}(H^{∗}(N, v), H^{∗}(M, v)), establishing the equality V_{E}(Hom_{K}^{gr}(N, M)) = V_{E}(M) ∩ V_{E}(N).

(8) Given a cone W ⊆ V, pick homogeneous polynomials ξ_{1}, …, ξ_{s} ∈ S that define it, and note that W = V_{E}(L_{ξ_{1}}⊗_{K}^{gr} ⋯ ⊗_{K}^{gr}L_{ξ_{s}}) by (7) and Lemma 3.10.

4. Simplicial complexes

For σ ⊆ [n], let Kσ denote the coordinate subspace spanned by { e_{j} | j ∈ σ }. In a n-graded situation, we refine some results of the preceding section as follows.

Proposition 4.1. Let M be a finite n-graded E–module.

(1) Ext_{E}^{∗}(M, K) is a finite (1 + n)-graded left module over the polynomial ring S = K[χ_{1}, …, χ_{n}], in which χ_{i} has (1 + n)-degree (1, ε_{i}).

(2) There exists a polynomial p_{M}(t, u_{1}, …, u_{n}) ∈ ℤ[t, u_{1}, …, u_{n}] such that

P_{M}^{E}(t, u_{1}, …, u_{n}) = \frac{p_{M}(t, u_{1}, …, u_{n})}{∏_{j=1}^{n}(1 − tu_{j})} ;

if M_{a} = 0, then no monomial t^{i}u^{a} appears in p_{M}(t, u_{1}, …, u_{n}).

(3) The variety V_{E}(M) is a union of coordinate subspaces of V.

(4) Each union of coordinate subspaces is the variety of a n-graded E–module.

Proof. (1) Take a n-graded free resolution G of M, and let Ext_{E}^{ia}(M, K) consist of those elements of Ext_{E}^{i}(M, K) = H^{i} Hom(G, K) that can be represented by a homomorphism κ: G_{i} ⟶ K, such that κ(G_{ib}) = 0 when a ≠ b ∈ ℤ^{n}. Performing Construction 3.5 with this G and the n-graded Cartan resolution C of K, cf. Example 2.5, and using n-homogeneous maps, one gets bilinear pairings

Ext_{E}^{jb}(K, K) × Ext_{E}^{ia}(M, K) ⟶ Ext_{E}^{i+j,a+b}(M, K) for all i, j ∈ ℤ; a, b ∈ ℤ^{n} .

They make Ext_{E}^{∗}(M, K) into a (1 + n)-graded left module over Ext_{E}^{∗}(K, K), and the identification Ext_{E}^{∗}(K, K) = S of Remark 3.7 is compatible with this grading.

(2) The expression for P_{M}^{R}(t, u_{1}, …, u_{n}) comes from (1), by the multigraded version of the Hilbert-Serre theorem. The assertion on the monomials in the numerator is obvious when M ≅ ⨁_{i=1}^{s} K(a_{i}) with a_{i} ∈ ℤ^{n}. As (3.6.1) is an exact sequence of (1 + n)-graded vector spaces, we conclude by induction on rank_{K} M.

(3) The annihilator of the multigraded S-module Ext_{E}^{∗}(M, K) being a monomial ideal in χ_{1}, …, χ_{n}, its radical is an intersection of prime ideals generated by subsets of { χ_{1}, …, χ_{n} }. The desired assertion follows from Theorem 3.9.

(4) Note that ⋂_{i=1}^{s} V_{E}(Kσ_{i}) = V_{E}( ⨁_{i=1}^{s} E/(Kσ_{i})).

Theorem 4.2. If J is a monomial ideal in E and I is the corresponding squarefree monomial ideal in S, then

V_{E}(E/J) = ⋃_{a∈Σ} K supp(a)

where Σ is the set of shifts of a minimal free resolution of S/I over S, and so

cx_{E}(E/J) = max{ |a| | a ∈ Σ } .

The proof of the theorem is deferred to the end of the section.

Let Δ be a simplicial complex with n vertices, and set K⟨Δ⟩ = E/J, where J is generated by { e_{σ} | σ ∉ Δ }. We give a combinatorial interpretation of the complex

(K⟨Δ⟩, v): 0 ⟶ K⟨Δ⟩_{1} ⟶^{ρ^{v}} K⟨Δ⟩_{2} ⟶^{ρ^{v}} …

For a subset ρ ⊆ [n], we denote Δ_{ρ} the restriction of Δ to ρ, that is, the simplicial complex with faces σ ∈ Δ such that σ ⊆ ρ. Furthermore, for a face σ ∈ Δ we introduce the link of σ in Δ_{ρ} as the simplicial complex

lk_{Δ_{ρ}} σ = ⟨τ ∈ Δ_{ρ} | τ ∪ σ ∈ Δ⟩.

For v ∈ V, v = ∑_{i=1}^{n} λ_{i}e_{i}, we call supp(v) = { i | λ_{i} ≠ 0 } the support of v.

Now the cohomology of (K⟨Δ⟩, v) can be interpreted as follows:

Proposition 4.3. The complex (K⟨Δ⟩, v) only depends on ρ = supp(v), namely, it is isomorphic to (K⟨Δ⟩, v_{ρ}) with v_{ρ} = ∑_{j∈ρ} e_{j}. Furthermore,

H^{i}(K⟨Δ⟩, v) ≅ ⨁_{σ∈Δ,σ⊆[n]\ρ} H̃^{i−1}(lk_{Δ_{ρ}} σ; K)

where H̃^{∗}( ; K) denotes reduced simplicial cohomology with coefficients in K.

Proof. The map φ: V ⟶ V given by φ(e_{j}) = λ_{j}^{−1}e_{j} for j ∈ ρ and φ(e_{j}) = e_{j} for j ∉ ρ extends to an isomorphism of K-algebras φ: K⟨Δ⟩ ⟶ K⟨Δ⟩, with φ(v) = v_{ρ}.

As a K⟨Δ_{ρ}⟩-module the algebra K⟨Δ⟩ decomposes as follows:

K⟨Δ⟩ = ⨁_{σ∈Δ,σ⊆[n]\ρ} e_{σ} · K⟨Δ_{ρ}⟩.

Now note that e_{σ}K⟨Δ_{ρ}⟩ ≅ K⟨lk_{Δ_{ρ}} σ⟩, and that (K⟨lk_{Δ_{ρ}} σ⟩, v) is isomorphic to the augmented oriented cochain complex of lk_{Δ_{ρ}} σ with values in K.

By a theorem of Hochster [12], ρ ⊆ [n] is the support of a shift of the resolution of k[Δ] if and only if H̃(Δ_{ρ}; K) ≠ 0, so Theorem 4.2 and Proposition 4.3 yield

Corollary 4.4. Let Δ be a simplicial complex with n vertices. For a subset σ ⊆ [n] and a field K the following conditions are equivalent:

(i) There exists ρ ⊆ [n] with σ ⊆ ρ such that H̃(Δ_{ρ}; K) ≠ 0.

(ii) There exists τ ∈ Δ with τ ∩ σ = ∅, such that H̃(lk_{Δ_{σ}} τ; K) ≠ 0. □

We single out a special case: For any simplicial complex Δ with H̃^{∗}(Δ; k) ≠ 0 and any subset σ of the vertex set of Δ, there is a face τ of Δ such that H̃(lk_{Δ_{τ}} σ; K) ≠ 0.

Proof of Theorem 4.2. Let F be a minimal free resolution of S/I over S, let G be the minimal free resolution of E/J over E of Theorem 1.3, and let Y_{ℓ} be the basis of G_{ℓ} from Construction 1.1. A homogeneous K-basis of Hom_{E}(G_{ℓ}, K) = Ext_{E}^{ℓ}(E/J, K) is given by { κ_{f}^{a} | κ_{f}^{a}(y^{(a)}f) = 1 and κ_{f}^{a}(Y_{ℓ} \ {y^{(a)}f}) = 0 }.

In the Cartan resolution C of K over E, cf. Example 2.5, set 1 = w^{(0)} and w_{j} = w^{(ε_{j})}. Fixing a homomorphisms κ_{f}^{a}: G_{ℓ} ⟶ K, with f ∈ B_{i} and deg(f) = b, we note that a lifting of κ_{f}^{a} to a chain map κ_{f}^{a}: G ⟶ C can be started by

(κ_{f}^{a})_{ℓ}(y^{(a′)}f′) = \begin{cases}1 & when a = a′ and f = f′; \\ 0 & otherwise ; \end{cases}

(κ_{f}^{a})_{ℓ+1}(y^{(a′)}f′) = \begin{cases}(−1)^{|b|}w_{j} when a = a′ + ε_{j} , j ∈ supp(b) , and f = f′; \\ (−1)^{|a|}w_{j}λ_{f′f}e_{j}^{−1}e_{b}^{−1}e_{b′}  when a = a′, j ∈ supp(b′ − b) , and θ(f′) = ∑_{g∈B_{i}} λ_{f′g}x^{b′−c}g with b′ = deg(f′) , c = deg(g) ; \\ 0 otherwise . \end{cases}

These cases are disjoint because b′ is squarefree, so by Construction 3.5 we have

χ_{j}κ_{f}^{a} = \begin{cases}(−1)^{|b|}κ_{f}^{a+ε_{j}} & for j ∈ supp(f) ; \\ (−1)^{|a|} ∑_{f′∈B_{i+1} : b′=b+ε_{j}} λ_{f′f}κ_{f′}^{a} & for j ∈ null(f) = [n] \ supp(f) . \end{cases}

Ordering the subsets of [n] by inclusion, we set B[0] = ∅ and

B[p] = { f ∈ B \ B[p − 1] | supp(f) is maximal in B \ B[p − 1] } for p ≥ 1 .

The multiplication table shows that the K-span of { κ_{f}^{a} | supp(f) ∈ ⋃_{p≤q} B[p] } is a submodule M[q] of M = Ext_{E}^{∗}(M, K) over S = K[χ_{1}, …, χ_{n}], such that

\frac{M[q]}{M[q − 1]} ≅ ⨁_{f∈B[q]} S\overline{κ}_{f}^{0}  and  Ann_{S}(\overline{κ}_{f}^{0}) = ( null(f)) .

From the finite filtration 0 = M[0] ⊆ ⋯ ⊆ M[n] = M we then get

\sqrt{Ann_{S} M} = \sqrt{⋂_{q=1}^{n} Ann_{S} \frac{M[q]}{M[q − 1]}} = ⋂_{q=1}^{n} \sqrt{Ann_{S} \frac{M[q]}{M[q − 1]}} = ⋂_{f∈B} ( null(f)) .

The desired result now follows from Theorem 3.9.

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Institute of Mathematics, Bulgarian Academy of Sciences Sofia 1113, Bulgaria

E-mail address: algebra@bgearn.acad.bg

Department of Mathematics, Purdue University West Lafayette, Indiana 47907, USA

E-mail address: avramov@math.purdue.edu

FB 6 Mathematik und Informatik, Universität-GHS-Essen Postfach 103764, Essen 45117, Germany

E-mail address: mat300@uni-essen.de