BGG CORRESPONDENCE AND ARAMOVA-HERZOG’S THEORY ON EXTERIOR ALGEBRA

KOHJI YANAGAWA

This article is edited from my recent preprint [15] (some parts are abridged and some are extended). The formal version of [15] has been submitted to a journal.

Abstract. Let E = K⟨y_{1}, …, y_{n}⟩ be the exterior algebra. The distinguished pairs of a graded E-module N describe the growth of a minimal graded injective resolution of N. Römer gave a duality theorem between the distinguished pairs of N and those of its dual N^{∗}. In this paper, we show that under Bernstein-Gel’fand-Gel’fand correspondence, his duality is translated into a natural corollary of Serre duality for (complexes of) graded S = K[x_{1}, …, x_{n}]-modules.

1. Background

While the results of this article are purely ring theoretic, they have a combinatorial background. More precisely, they are related to the theory of Stanley-Reisner rings. This section is devoted to an explanation of this background, and independent from the latter sections in some sense. So one can skip this section.

Let S = K[x_{1}, …, x_{n}] be the polynomial ring over a field K. Set [n] := {1, …, n}. For an abstract simplicial complex Δ ⊂ 2^{[n]}, set I_{Δ} := ( ∏_{i∈F} x_{i} | F ⊂ [n], F ∉ Δ ) to be a monomial ideal of S. We call K[Δ] = S/I_{Δ} the Stanley-Reisner ring of Δ. This is a central concept of combinatorial commutative algebra. See [5, 12]. It is easy to see that Δ^{∨} := { F ⊂ [n] | [n] \ F ∉ Δ } is a simplicial complex again. We call Δ^{∨} the Alexander dual of Δ. It is known that H̃_{i}(|Δ|; K) ≅ H̃^{n−i−3}(|Δ^{∨}|; K) for all i, where H̃_{i}(|Δ|; K) is the ith reduced homology group of the geometric realization |Δ| of Δ with the coefficients in K. This isomorphism corresponds to the Alexander duality on the geometric realization | 2^{[n]} \ [n] |, which is homeomorphic to an (i − 2)dimensional sphere.

Let M be a finitely generated graded S-module. The ijth Betti number β_{i,j}(M) = dim_{K} Tor_{i}^{S}(K, M)_{j} of M is an important invariant. Following Bayer-Charalambous-Popescu [4], we say a Betti number β_{k,m}(M) ≠ 0 is extremal, if β_{i,j}(M) = 0 for all (i, j) ≠ (k, m) with i ≥ k and j − i ≥ m − k.

Theorem A (Bayer-Charalambous-Popescu, [4, Theorem 2.8]) Let Δ ⊂ 2^{{1,... ,n}} be a simplicial complex, and K[Δ] = S/I_{Δ} the Stanley-Reisner ring. Then β_{i,i+j}(K[Δ]) is extremal if and only if so is β_{j,i+j}(I_{Δ^{∨}}). Moreover, if this is the case, then β_{i,i+j}(K[Δ]) = β_{j,i+j}(I_{Δ^{∨}}).

We have β_{i,n}(K[Δ]) = dim_{K} H̃_{n−i−1}(|Δ|; K) =: h̃_{n−i−1}(|Δ|) by Hochster’s formula. If β_{i,n}(K[Δ]) ≠ 0 then it is always an extremal Betti numbers. The equality h̃_{n−i−1}(|Δ|) = β_{i,n}(K[Δ]) = β_{n−i,n}(I_{Δ^{∨}}) = β_{n−i+1,n}(K[Δ^{∨}]) = h̃_{i−2}(|Δ^{∨}|) induced by Theorem A corresponds to the usual Alexander duality. More generally, Theorem A gives an Alexander duality for (some) iterated Betti numbers (c.f. [8]).

Let E = K⟨y_{1}, …, y_{n}⟩ be the exterior algebra. To understand Theorem A, Aramova-Herzog [3] introduced distinguished pairs for a graded E-module N using its graded injective resolution. See Definition 2.7 below (our convention to describe these pairs is different from the original one in [3, 11]. Please be careful, when you refer these papers). Let K{Δ} = E/J_{Δ}, J_{Δ} := ( ∏_{i∈F} y_{i} | F ⊂ [n], F ∉ Δ ), be the exterior face ring of Δ (c.f. [2, 8]). Then [3, Corollary 9.6] states that (d, i) is a distinguished pair for K{Δ}^{∗} := Hom_{E}(K{Δ}, E) if and only if β_{d+i−n,d}(K[Δ]) is extremal. But Römer [11] proved that (d, i) is distinguished for N if and only if so is (d, 2n − d − i) for N^{∗}. Since k{Δ}^{∗} = J_{Δ^{∨}}, his result implies Theorem A (their arguments can also treat the value of β_{i,j}(K[Δ])).

Bernstein-Gel’fand-Gel’fand correspondence (BGG correspondence, for short) is a well known theorem which states that the derived category D^{b}(gr S) of finitely generated graded S-modules is equivalent to the similar category D^{b}(gr E) for E. In this article, we show that BGG correspondence translates Römer’s duality into a natural consequence of Serre duality on D^{b}(gr S). A key point is that the duality functor (−)^{∗} = Hom_{E}(−, E) on D^{b}(gr E) corresponds to the duality functor R Hom_{S}(−, ω^{•}) on D^{b}(gr S), where ω^{•} is a dualizing complex of S.

The original paper [4] states Theorem A in the ℤ^{n}-graded context, while the arguments in [3, 11] are hard to work in this context. But, since BGG correspondence also holds for ℤ^{n}-graded modules, our method is powerful in this context too.

2. ℤ-graded case

Let W be an n-dimensional vector space over a field K, and S = ⨁_{i≥0} Sym_{i} W the polynomial ring. We regard S as a graded ring with S_{i} = Sym_{i} W. Let Gr S be the category of graded S-modules and their degree preserving S-homomorphisms, and gr S the full subcategory of Gr S consisting of finitely generated modules. Then there is an equivalence D^{b}(gr S) ≅ D_{gr S}^{b}(Gr S). (For derived categories, consult [7].) So we will freely identify these categories. For M = ⨁_{i∈ℤ} M_{i} ∈ Gr S and an integer j, M(j) denotes the shifted module with M(j)_{i} = M_{i+j}. For M^{•} ∈ D^{b}(Gr S), M^{•}[j] denotes the jth translation of M^{•}, that is, M^{•}[j] is the complex with M^{•}[j]^{i} = M^{i+j}. So, if M ∈ Gr S, M[j] is the cochain complex ⋯ → 0 → M → 0 → ⋯, where M sits in the (−j)th position. If M ∈ gr S and N ∈ Gr S, then Hom_{S}(M, N) has the structure of a graded S-module with Hom_{S}(M, N)_{i} = Hom_{Gr S}(M, N(i)).

Let ω^{•} ∈ D^{b}(gr S) be a minimal graded injective resolution of S(−n)[n]. That is, ω^{•} is a graded normalized dualizing complex of S. Then D_{S}(−) := Hom_{S}^{•}(−, ω^{•}) gives a duality functor from D^{b}(gr S) to itself. The ith cohomology of D_{S}(M^{•}) is Ext_{S}^{i}(M^{•}, ω^{•}). For M^{•} ∈ D^{b}(gr S) and i ∈ ℤ, set d_{i}(M^{•}) := dim_{S} H^{i}(M^{•}). Here the Krull dimension of the 0 module is −∞.

Definition 2.1. We say (d, i) ∈ ℕ × ℤ is a distinguished pair for a complex M^{•} ∈ D^{b}(gr S), if d = d_{i}(M^{•}) and d_{j}(M^{•}) < d + i − j for all j with j < i.

Example 2.2. Let M^{•} ∈ D^{b}(gr S) and d = d_{i}(M^{•}) ≥ 0. If d = max{d_{j}(M^{•}) | j ∈ ℤ}, then (d, i) is distinguished for M^{•}. Similarly, if i = min{j | H^{j}(M^{•}) ≠ 0}, then (d, i) is also distinguished. Thus M^{•} has several distinguished pairs in general.

In the sequel, deg_{S}(M) denotes the multiplicity (c.f. [5, Definition 4.1.5]) of a module M ∈ gr S, which is defined by the top term of the Hilbert polynomial of M.

Theorem 2.3. For M^{•} ∈ D^{b}(gr S), we have the following.

(1) A pair (d, i) is distinguished for M^{•} if and only if (d, −d − i) is distinguished for D_{S}(M^{•}).

(2) If (d, i) is a distinguished pair for M^{•}, then

deg_{S} H^{i}(M^{•}) = deg_{S} Ext_{S}^{−d−i}(M^{•}, ω^{•}).

Proof. (1) Since the statement is “symmetric”, it suffices to prove the direction ⇒.

From the double complex Hom_{S}^{•}(M^{•}, ω^{•}), we have a spectral sequence E_{2}^{p,q} = Ext_{S}^{p}(H^{−q}(M^{•}), ω^{•}) ⇒ Ext_{S}^{p+q}(M^{•}, ω^{•}). For simplicity, set e_{r}^{p,q} := dim_{S} E_{r}^{p,q}. By [5, §8.1, Theorem 8.1.1], we have

(2.1) e_{2}^{p,q} = dim_{S} Ext_{S}^{p}(H^{−q}(M^{•}), ω^{•}) = \begin{cases}−p & if p = −d_{−q}(M^{•}), \\ ≤ −p & if −d_{−q}(M^{•}) < p ≤ 0, \\ −∞ & otherwise. \end{cases}

(I) By (2.1), we have e_{2}^{−d,−i} = d. On the other hand, we have e_{2}^{p,q} < d for all (p, q) ≠ (−d, −i) with p + q = −d − i. In fact, the assertion follows from (2.1) if p > −d. So we may assume that p < −d and q = −d − i − p > −i. Since (d, i) is distinguished, we have d_{−q}(M^{•}) < d + i + q = −p. Thus E_{2}^{p,q} = 0 in this case. Anyway, we have e_{∞}^{p,q} < d for all (p, q) ≠ (−d, −i) with p + q = −d − i.

(II) Since d_{i−j+1}(M^{•}) < d+j−1 < d+j for all j ≥ 2, we have that E_{2}^{−d−j,−i+j−1} = 0. So we have E_{r}^{−d−j,−i+j−1} = 0 for all r ≥ 2. Next we will show that d = e_{2}^{−d,−i} = e_{3}^{−d,−i} = ⋯ = e_{r}^{−d,−i} by induction on r. Recall that E_{r+1}^{−d,−i} is the cohomology of

E_{r}^{−d−r,−i+r−1} → E_{r}^{−d,−i} → E_{r}^{−d+r,−i−r+1}.

But we have seen that E_{r}^{−d−r,−i+r−1} = 0. Moreover, e_{r}^{−d+r,−i−r+1} ≤ e_{2}^{−d+r,−i−r+1} ≤ d − r < d by (2.1), and e_{r}^{−d,−i} = d by the induction hypothesis. Thus e_{r+1}^{−d,−i} = d. Hence e_{∞}^{−d,−i} = d. From this fact and (I), we have that dim_{S} Ext_{S}^{−d−i}(M^{•}, ω^{•}) = d.

(III) Finally, we will show that Ext_{S}^{−d−i−j}(M^{•}, ω^{•}) < d + j for all j > 0. To see this, it suffices to show that e_{2}^{p,q} < d + j for all j > 0 and all (p, q) with p + q = −d − i − j. If p > −d − j, the assertion is clear. If p = −d − j, then q = −i and d_{−q}(M^{•}) = d < −p. So E_{2}^{p,q} = 0 in this case. Hence we may assume that p < −d − j and −q = d + i + j + p < i. Since (d, i) is distinguished, d_{−q}(M^{•}) < d + (i + q) = −j − p < −p. So we have E_{2}^{p,q} = 0 in this case too.

(2) Since deg_{S} E_{r}^{−d,−i} = deg_{S} E_{r+1}^{−d,−i} for all r ≥ 2 by the argument in (II) of the proof of (1), we have deg_{S} E_{2}^{−d,−i} = deg_{S} E_{∞}^{−d,−i}. Hence

deg_{S} Ext_{S}^{−d−i}(M^{•}, ω^{•}) = deg_{S} E_{∞}^{−d,−i} = deg_{S} E_{2}^{−d,−i} = deg_{S} Ext_{S}^{−d}(H^{i}(M^{•}), ω^{•}), where the first equality follows from (I) and (II). But, since dim(H^{i}(M^{•})) = d, we have deg_{S} Ext_{S}^{−d}(H^{i}(M), ω^{•}) = deg_{S} H^{i}(M^{•}).

Remark 2.4. For the above theorem, only (2.1) and the fact that inj. dim_{S} ω^{•} < ∞ are essential. So the theorem holds in much wider contexts.

(1) Theorem 2.3 (1) also holds for a noetherian local commutative ring R admitting a dualizing complex. The part (2) also holds for R, if we replace deg_{S}(−) by l_{R_{𝔭}}(− ⊗_{R} R_{𝔭}) for a prime ideal 𝔭 ⊂ R with dim R/𝔭 = d.

(2) Let A be an associative ring with 1. For a left (or right) A-module M, set j(M) := min{ i | Ext_{A}^{i}(M, A) ≠ 0 }. We say A is Auslander Gorenstein if A is left and right noetherian, inj. dim_{A}A = inj. dim A_{A} < ∞, and satisfies the following condition: For every finitely generated left (or right) A-module M and for all i ≥ 0, we have j(N) ≥ i for all submodule N ⊂ Ext_{A}^{i}(M, A).

Familiar examples of Auslander Gorenstein rings include commutative Gorenstein local rings (in this case, j(M) = dim A − dim M), Weyl algebras, and universal enveloping algebras of finite dimensional Lie algebras. See [1] for further information.

If A is Auslander Gorenstein, then −j(M) is an exact dimension function. If we use this “dimension” to define distinguished pairs for objects in D^{b}(mod_{A}) or D^{b}(mod_{A^{op}}), Theorem 2.3 also holds for the duality functor R Hom_{A}(−, A) between D^{b}(mod_{A}) and D^{b}(mod_{A^{op}}). More generally, the theorem holds for rings with Auslander dualizing complexes (see [17]).

Next, we assume that A = ⨁_{i≥0} A_{i} is a graded K(≅ A_{0})-algebra satisfying the following conditions.

(a) There is a polynomial f(t) ∈ ℚ[t] such that f(i) = dim_{K} A_{i} for i ≫ 0.

(b) A is Auslander regular (i.e., A is Auslander Gorenstein and gl. dim A < ∞).

(c) A is Cohen-Macaulay with respect to Gel’fand-Kirillov dimension (c.f. [1]). Then a finitely generated graded A-module M has the Hilbert polynomial, and

we can define the multiplicity deg_{A}(M). If we use Gel’fand-Kirillov dimension to define distinguished pairs, both (1) and (2) of Theorem 2.3 hold for A. So the under the additional assumption that A is Koszul, it might be interesting to generalize Corollary 2.8 below and related results to the quadratic dual ring A^{!}.

Let V be the dual vector space of W, and E = ⋀ V the exterior algebra. We regard E as a negatively graded ring with E_{−i} = ⋀^{i} V (this is the opposite convention from [3, 11]). Let gr E be the category of finitely generated graded E-modules and their degree preserving E-homomorphisms. Here “E-module” means a left and right module N with ea = (−1)^{(deg e)(deg a)}ae for all homogeneous e ∈ E and a ∈ N.

Let {x_{1}, …, x_{n}} be a basis of W, and {y_{1}, …, y_{n}} its dual basis of V. Set L(N^{•}) = ⨁_{i∈ℤ} S ⊗_{K} N^{i} and L(N^{•})^{m} = ⨁_{i−j=m} S ⊗_{K} N_{j}^{i}. The differential defined by

L(N^{•})^{m} ⊃ S ⊗_{K} N_{j}^{i} ∋ 1 ⊗ z ↦ ∑_{1≤l≤n} x_{l} ⊗ y_{l}z + (−1)^{m}(1 ⊗ δ^{i}(z)) ∈ L(N^{•})^{m+1}

makes L(N^{•}) a cochain complex of free S-modules. Here δ^{i} is the ith differential map of N^{•}. Moreover, L gives a functor from D^{b}(gr E) to D^{b}(gr S). Similarly, we have a functor R : D^{b}(gr S) → D^{b}(gr E). The following is a crucial result.

Theorem 2.5 (BGG correspondence, c.f.[6]). The functors L and R give a category equivalence D^{b}(gr S) ≅ D^{b}(gr E).

For N ∈ gr E, then N^{∗} := Hom_{E}(N, E) ≅ Hom_{K}(N, K)(n) is a graded E-module again. (−)^{∗} gives an exact duality functor on gr E, and it can be extended to the duality functor D_{E} on D^{b}(gr E).

Proposition 2.6. For N^{•} ∈ D^{b}(gr E), we have

D_{S} ∘ L(N^{•}) ≅ L ∘ D_{E}(N^{•})(−2n)[2n].

Proof. Since L(N^{•}) consists of free S-modules, we have

D_{S} ∘ L(N^{•}) ≅ Hom_{S}^{•}(L(N^{•}), S(−n)[n]).

It is easy to see that

Hom_{S}^{m}(L(N^{•}), S(−n)[n]) ≅ ⨁_{j−i=m+n} S(−n) ⊗_{K} (N_{j}^{i})^{∨},

where (−)^{∨} means the graded K-dual. On the other hand,

L ∘ D_{E}(N^{•})^{m} = ⨁_{i−j=m} S ⊗_{K} D_{E}(N^{•})_{j}^{i} = ⨁_{i−j=m} S(n) ⊗_{K} (N_{−n−j}^{−i})^{∨}
= ⨁_{j−i=m−n} S(n) ⊗_{K} (N_{j}^{i})^{∨}.

So we can easily construct a quasi-isomorphism D_{S}∘L(N^{•}) → L∘D_{E}(N^{•})(−2n)[2n].

For N^{•} ∈ D^{b}(gr E), we have H^{i}(L(N^{•}))_{j} ≅ Ext_{E}^{j+i}(K, N^{•})_{j} by [6, Theorem 3.7]. So the Laurent series P_{i}(t) = ∑_{j∈ℤ}(dim_{K} Ext_{E}^{j+i}(K, N^{•})_{j})·t^{j} is the Hilbert series of the finitely generated graded S-module H^{i}(L(N^{•})). If H^{i}(L(N^{•})) ≠ 0, there exists a Laurent polynomial Q_{i}(t) ∈ ℤ[t, t^{−1}] such that

P_{i}(t) = \frac{Q_{i}(t)}{(1 − t)^{d}} ,

where d = d_{i}(L(N^{•})) = dim_{S} H^{i}(L(N^{•})). Set e_{i}(N^{•}) := Q_{i}(1) = deg_{S} H^{i}(L(N^{•})).

Definition 2.7. Let N^{•} ∈ D^{b}(gr E). We say (d, i) ∈ ℕ × ℤ is a distinguished pair for N^{•} if and only if it is distinguished for L(N^{•}) (in the sense of Definition 2.1).

Note that a distinguished pair (d, i) for N^{•} and e_{i}(N^{•}) concern the growth of the “(−i)-linear strand” of a minimal graded injective resolution of N^{•}. The above definition is (essentially) same as that of [3, 11]. But our convention to describe these pairs is different from the original one.

Corollary 2.8 (c.f. [11, Theorem 3.8]). Let N^{•} ∈ D^{b}(gr E). A pair (d, i) is distinguished for N^{•} if and only if (d, 2n − d − i) is distinguished for D_{E}(N^{•}). If this is the case, we have e_{i}(N^{•}) = e_{2n−d−i}(D_{E}(N^{•})).

Proof. For the first statement, it suffices to prove the direction ⇒. By Theorem 2.3, (d, −d − i) is a distinguished pair for D_{S} ∘ L(N^{•}) ≅ L ∘ D_{E}(N^{•})(−2n)[2n]. For a complex M^{•} ∈ D^{b}(gr S), we have H^{j}(M^{•}(−2n)[2n]) = H^{2n+j}(M^{•})(−2n) and d_{j}(M^{•}(−2n)[2n]) = d_{2n+j}(M^{•}). Thus (d, 2n−d−i) is distinguished for L∘D_{E}(N^{•}). The last equality follows from Theorem 2.3 (2).

For a module N ∈ gr E, d_{i}(L(N)) can be 0 quite often. But we have the following.

Proposition 2.9. Assume that a module N ∈ gr E does not have a free summand. If (d, i) is a distinguished pair for N, then we have d > 0.

Proof. Let 0 → N → I^{0} → I^{1} → ⋯ (resp. · · · → I^{−1} → I^{0} → N → 0) be a minimal injective (resp. projective) resolution of N. For j ≥ 0, set Ω_{j}(N) := (ker(I^{j} → I^{j+1}))[−j]. Obviously, 0 → Ω_{j}(N) → I^{j} → I^{j+1} → ⋯ is a minimal injective resolution. On the other hand, since N does not have a free summand, ⋯ → I^{−1} → I^{0} → ⋯ → I^{j−1} → Ω_{j}(N) → 0 is a minimal projective resolution. If d_{i}(L(N)) > 0, then d_{i}(L(Ω_{j}(N))) = d_{i}(L(N)) for all j ≥ 0. But, if d_{i}(L(N)) = 0, then d_{i}(L(Ω_{j}(N))) = −∞ for j ≫ 0. On the other hand, since a minimal injective resolution of N^{∗} is the dual of a minimal projective resolution of N, we have d_{i}(L(N^{∗})) = d_{i}(L(Ω_{j}(N)^{∗})) for all i and all j ≥ 0. So N^{∗} and Ω_{j}(N)^{∗} have the same distinguished pairs. For a contradiction, we assume that (0, i) is a distinguished pair for N. Then (0, 2n − i) is distinguished for N^{∗} and Ω_{j}(N)^{∗}. So (0, i) is distinguished for Ω_{j}(N) for all j ≥ 0. This contradicts the above observation.

We say a distinguished pair (d, i) is positive, if d > 0. Since [3, 11] study a distinguished pair for a module, they only treat a positive one.

Remark 2.10. When N^{•} is a module, Corollary 2.8 was proved in [11, Theorem 3.8]. On the other hand, for positive distinguished pairs, we can prove the corollary from [11, Theorem 3.8] directly: Let I^{•} be an injective resolution of N^{•} and P^{•} a projective resolution of I^{•}. From the quasi-isomorphism f : P^{•} → I^{•}, we have the exact complex (T^{•}, ∂^{•}) := cone(f). Then N := ker ∂_{0} (resp. N^{∗}) has the same positive distinguished pairs as N^{•} (resp. D_{E}(N^{•})).

A variant of BGG correspondence gives an equivalence \underline{gr}E ≅ D^{b}(Coh(ℙ^{n−1})) of triangulated categories, where \underline{gr} E is the stable category, and Coh(ℙ^{n−1}) is the category of coherent sheaves on ℙ^{n−1} = Proj S. More precisely, the composition of the functor L : gr E → D^{b}(gr S) and the natural functor D^{b}(gr S) → D^{b}(Coh(ℙ^{n−1})) induces this equivalence. Note that the functor gr S ∋ M → M̃ ∈ Coh(ℙ^{n−1}) ignores modules of finite length. Hence if d_{i}(M^{•}) = 0 then H^{i}(M̃^{•}) = 0.

In the rest of this section, we assume that K is algebraically closed. Let N ∈ gr E. Following [2], we say v ∈ E_{−1} = V is N-regular if Ann_{N}(v) = vN. It is easy to see that v is N-regular if and only if it is N^{∗}-regular. We say V_{E}(N) = {v ∈ V | v is not N-regular} is the rank variety of N (see [2]). [2, Theorem 3.1] states that V_{E}(N) is an algebraic subset of V = Spec S, and dim V_{E}(V) = max{ d_{i}(L(N)) | i ∈ ℤ }. By the above remark, V_{E}(N) = V_{E}(N^{∗}). We can refine this observation using the grading of N.

Recall that S can be seen as the Yoneda algebra Ext_{E}^{∗}(K, K), and Ext_{E}^{∗}(K, N) has the S-module structure by the Yoneda product. By the same argument as [2, Theorem 3.9], we have that

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

But [Ext_{E}^{∗+i}(K, N)]_{∗} := ⨁_{j∈ℤ} Ext_{E}^{j+i}(K, N)_{j} is an S-module which is isomorphic to H^{i}(L(N)) (see the proof of [6, Proposition 2.3]. Note that Ext_{E}^{i}(K, K) = [Ext_{E}^{i}(K, K)]_{i} for all i), and we have Ext_{E}^{∗}(K, N) ≅ ⨁_{j∈ℤ}[Ext_{E}^{∗+i}(K, N)]_{∗}. Set

V_{E}^{i}(N) = { v ∈ V | ξ(v) = 0 for all ξ ∈ Ann_{S}( [Ext_{E}^{∗+i}(K, N)]_{∗}) }.

We have V_{E}(N) = ⋃_{i} V_{E}^{i}(N) and d_{i}(L(N)) = dim V_{E}^{i}(N). For an algebraic set X ⊂ Spec S of dimension d, set Top(X) to be the union of the all irreducible components of X of dimensions d.

Proposition 2.11. If (d, i) is a distinguished pair for N ∈ gr E, then we have Top(V_{E}^{i}(N)) = Top(V_{E}^{2n−d−i}(N^{∗})).

Proof. By the proof of Theorem 2.3, Ann_{S}( H^{i}(L(N)) ) has the same top dimensional components as Ann_{S}( H^{−d−i}(D_{S} ∘ L(N)) ).

In the above situation, we have V_{E}^{i}(N) ≠ V_{E}^{2n−d−i}(N^{∗}) in general.

3. Squarefree case

In this section, we regard S = K[x_{1}, …, x_{n}] as an ℕ^{n}-graded ring with deg x_{i} = (0, …, 0, 1, 0, …, 0) where 1 is in the ith position. Similarly, E = K⟨y_{1}, …, y_{n}⟩ is a −ℕ^{n}-graded ring with deg y_{i} = − deg x_{i}. Let *gr S (resp. *gr E) be the category of finitely generated ℤ^{n}-graded S-modules (resp. E-modules). The functors L and R defining the BGG correspondence D^{b}(gr S) ≅ D^{b}(gr E) also work in the ℤ^{n}-graded context. That is, the functors L : D^{b}(*gr E) → D^{b}(*gr S) and R : D^{b}(*gr S) → D^{b}(*gr E) are defined by the same way as the ℤ-graded case, and they give an equivalence D^{b}(*gr S) ≅ D^{b}(*gr E), see [14, Theorem 4.1]. Note that the dualizing complex ω^{•} of S is ℤ^{n}-graded, and D_{S}(−) = Hom_{S}^{•}(−, ω^{•}) is also a duality functor on D^{b}(*gr S). Similarly, D_{E}(−) = Hom_{E}(−, E) is a duality functor on D^{b}(*gr E). As Proposition 2.6, for N^{•} ∈ D^{b}(*gr E), we have D_{S} ∘L(N^{•}) ≅ L∘D_{E}(N^{•})(−2)[2n] in D^{b}(*gr S). Here we set j := (j, j, …, j) ∈ ℕ^{n} for j ∈ ℤ.

For a = (a_{1}, …, a_{n}) ∈ ℤ^{n}, set supp(a) := {i | a_{i} > 0} ⊂ [n] := {1, …, n} and |a| = ∑_{i=1}^{n} a_{i}. We say a ∈ ℤ^{n} is squarefree if a_{i} = 0, 1 for all i ∈ [n]. When a ∈ ℤ^{n} is squarefree, we sometimes identify a with supp(a).

Definition 3.1 ([13]). We say a ℤ^{n}-graded S-module M is squarefree, if the following conditions are satisfied.

(a) M is ℕ^{n}-graded (i.e., M_{a} = 0 if a ∉ ℕ^{n}) and finitely generated.

(b) The multiplication map M_{a} ∋ y ↦ (∏ x_{i}^{b_{i}}) · y ∈ M_{a+b} is bijective for all a, b ∈ ℕ^{n} with supp(a + b) = supp(a).

For a simplicial complex Δ ⊂ 2^{[n]}, the Stanley-Reisner ideal I_{Δ} := ( ∏_{i∈F} x_{i} | F ∉ Δ ) and the Stanley-Reisner ring K[Δ] := S/I_{Δ} are squarefree modules. Note that if M is squarefree then M_{a} ≅ M_{F} as K-vector spaces for all a ∈ ℕ^{n} with supp(a) = F. Let Sq_{S} be the full subcategory of *gr S consisting of squarefree modules. In *gr S, Sq_{S} is closed under kernels, cokernels and extensions ([13, Lemma 2.3]), and we have that D^{b}(Sq_{S}) ≅ D_{Sq_{S}}^{b}(*gr S). If M^{•} ∈ D^{b}(Sq_{S}), then D_{S}(M^{•}) ∈ D_{Sq_{S}}^{b}(*gr S) (see [14]). So D_{S} gives a duality functor on D^{b}(Sq_{S}).

Remark 3.2. This remark is an advertisement for my recent paper [16], and quite independent from other parts of this article.

Let B be an (n − 1)-simplex which is the geometric realization of 2^{[n]}. From a squarefree S-module M, we can construct a k-sheaf M^{+} on B (we are thinking classical topology on B). For example, k[Δ]^{+} ≅ j_{∗}\underline{k}_{|Δ|}, where \underline{k}_{|Δ|} is the constant sheaf on |Δ| and j : |Δ| → B is the embedding map. We have an isomorphism H^{i}(B, M^{+}) ≅ [H_{𝔪}^{i+1}(M)]_{0} for all i ≥ 1, and an exact sequence 0 → [H_{𝔪}^{0}(M)]_{0} → M_{0} → H^{0}(B, M^{+}) → [H_{𝔪}^{1}(M)]_{0} → 0, where H_{𝔪}^{i}(−) is the local cohomology module with support in the maximal ideal 𝔪 := (x_{1}, …, x_{n}). We have that ω_{K[Δ]}^{•} := Hom_{S}(K[Δ], ω^{•}) ∈ D^{b}(Sq_{S}), and j^{∗}(ω_{K[Δ]}^{•})^{+} is Verdier’s dualizing complex of |Δ| (c.f. [9]). Moreover, Serre duality for R Hom_{K[Δ]}(−, ω_{K[Δ]}^{•}) corresponds to the Poincaré-Verdier duality on |Δ| in our context. See [16] for detail.

Definition 3.3 (Römer [11]). A ℤ^{n}-graded E-module N = ⨁_{a∈ℤ^{n}} N_{a} is squarefree if N is finitely generated and N = ⨁_{F ⊂[n]} N_{−F}.

For example, any monomial ideal of E is a squarefree module. Let Sq_{E} be the full subcategory of *gr E consisting of squarefree E-modules. Then we have the functors S : Sq_{E} → Sq_{S} and E : Sq_{S} → Sq_{E} giving an equivalence Sq_{S} ≅ Sq_{E}. Here S(N)_{F} = N_{−F} for N ∈ Sq_{E}, and the multiplication map S(N)_{F} ∋ z ↦ x_{i}z ∈ S(N)_{F ∪{i}} for i ∉ F is given by S(N)_{F} = N_{−F} ∋ z ↦ (−1)^{α(i,F)}y_{i}z ∈ N_{−(F ∪{i})} = S(N)_{F ∪{i}}, where α(i, F) = #{ j ∈ F | j < i }. See [11] for further information. Of course, S and E can be extended to the functors between D^{b}(Sq_{S}) and D^{b}(Sq_{E}).

If N ∈ Sq_{E}, then N^{∗} = Hom_{E}(N, E) is squarefree again. So (−)^{∗} gives the duality functor D_{E} on D^{b}(Sq_{E}). We have the Alexander duality functor A := S ∘ D_{E} ∘ E on Sq_{S} (or D^{b}(Sq_{S})). For example, A(K[Δ]) = I_{Δ^{∨}}, where Δ^{∨} is the Alexander dual complex of Δ (see §1). In general, we have A(H^{i}(M^{•}))_{F} = (H^{−i}(M^{•})_{[n]\F})^{∨}.

Remark 3.4. Let Λ := ⟨ e_{F, G} | F, G ⊂ [n], F ⊂ G ⟩ be the incidence algebra of the Boolean lattice 2^{[n]} with the coefficients in K, and mod_{Λ} the category of finitely generated right Λ-modules. Then we have a functor Ψ : mod_{Λ} → Sq_{S} giving an equivalence mod_{Λ} ≅ Sq_{S}. See [14, Proposition 2.2]. Since inj. dim Λ < ∞, R Hom_{Λ}(−, Λ[n]) gives a duality functor from D^{b}(mod_{Λ}) to D^{b}(mod_{Λ^{op}}). But the map defined by e_{F, G} ↦ e_{G^{c}, F^{c}} gives a ring isomorphism Λ^{op} ≅ Λ, where F^{c} := [n] \ F. Under this isomorphism R Hom_{Λ}(−, Λ[n]) gives the duality functor D_{Λ} from D^{b}(mod_{Λ}) to itself. Then we have D_{S} ≅ Ψ ∘ D_{Λ} ∘ Ψ^{−1}. Similarly, the duality A on Sq_{S} corresponds to the duality Hom_{k}(−, k) on mod_{Λ}. See [14, Remark 3.3].

An associated prime ideal of M ∈ *gr S is of the form P_{F} = (x_{i} | i ∉ F) for some F ⊂ [n]. Let M be a squarefree module. A monomial prime ideal P_{F} is a minimal prime of M if and only if F is a maximal element of the set {G ⊂ [n] | M_{G} ≠ 0}. The following is a squarefree version of Definition 2.1.

Definition 3.5. We say (F, i) ∈ 2^{[n]} × ℤ is a distinguished pair for a complex M^{•} ∈ D^{b}(Sq_{S}), if P_{F} is a minimal prime of H^{i}(M^{•}) and H^{j}(M^{•})_{G} = 0 for all j with j < i and G ⊃ F with |G| < |F | + i − j.

Theorem 3.6. Let M^{•} ∈ D^{b}(Sq_{S}). A pair (F, i) is distinguished for M^{•} if and only if (F, −|F | − i) is distinguished for D_{S}(M^{•}). If this is the case, dim_{K} H^{i}(M^{•})_{F} = dim_{K} H^{−|F |−i}(D_{S}(M^{•}))_{F}.

Proof. Note that the spectral sequence E_{2}^{p,q} = Ext_{S}^{p}(H^{−q}(M^{•}), ω^{•}) ⇒ Ext_{S}^{p+q}(M^{•}, ω^{•}) is ℤ^{n}-graded, and E_{r}^{p,q} is squarefree for all p, q and r ≥ 2. So we can prove the theorem in a similar way to Theorem 2.3. See [15] for detail.

If N^{•} ∈ D^{b}(Sq_{E}), then it is easy to see that L(N^{•})(−1) ∈ D^{b}(Sq_{S}). So L(−) := L(−)(−1) gives a functor from D^{b}(Sq_{E}) to D^{b}(Sq_{S}). Moreover, we have L ≅ A ∘ D_{S} ∘ S by [14, Proposition 4.3].

Definition 3.7. Let N^{•} ∈ D^{b}(Sq_{E}). We say (F, i) is a distinguished pair for N^{•} if it is a distinguished pair for L(N^{•}) ∈ D^{b}(Sq_{S}) in the sense of Definition 3.5.

The next result can be proved by the same way as Corollary 2.8 using Theorem 3.6.

Proposition 3.8. Let N^{•} ∈ D^{b}(Sq_{E}). A pair (F, i) is distinguished for N^{•} if and only if (F, 2n − |F | − i) is distinguished for D_{E}(N^{•}). If this is the case, we have dim_{K} H^{i}(L(N^{•}))_{F} = dim_{K} H^{2n−|F |−i}(L ∘ D_{E}(N^{•}))_{F}.

If M^{•} ∈ D^{b}(*gr S), then Tor_{i}^{S}(K, M^{•}) := H^{−i}(K ⊗_{E} P^{•}) is a ℤ^{n}-graded module, where P^{•} is a graded free resolution of M^{•}. Set β_{i,a}(M^{•}) := dim_{K} Tor_{i}^{S}(K, M^{•})_{a} for a ∈ ℤ^{n}. We say β_{i,a}(M^{•}) is the (i, a)th Betti number of M^{•}. If M^{•} ∈ D^{b}(Sq_{S}) and β_{i,a}(M^{•}) ≠ 0, then a is squarefree (see [14]). Now we back to extremal Betti numbers mentioned in §1.

Definition 3.9 ([4]). A Betti number β_{i,F} (M^{•}) ≠ 0 is extremal if β_{j,G}(M^{•}) = 0 for all (j, G) ≠ (i, F) with j ≥ i, G ⊃ F, and |G| − j > |F | − i.

Proposition 3.10 (c.f. [3]). Let M^{•} ∈ D^{b}(Sq_{S}) and N^{•} := E(M^{•}) ∈ D^{b}(Sq_{E}). A pair (F, i) is distinguished for D_{E}(N^{•}) if and only if β_{i+|F |−n,F} (M^{•}) is an extremal Betti number. If this is the case, then we have β_{i+|F |−n,F} (M^{•}) = H^{i}(L ∘ D_{E}(N^{•}))_{F}.

Proof. For j ∈ ℤ and G ⊂ [n], we have the following.

β_{j,G}(M^{•}) = dim_{K}[H^{|G|−j−n}(D_{S} ∘ A(M^{•}))]_{[n]\G} (by [14, Corollary 3.6])
= dim_{K}[H^{n+j−|G|}(A ∘ D_{S} ∘ A(M^{•}))]_{G}
= dim_{K}[H^{n+j−|G|}(L ∘ E ∘ A(M^{•}))]_{G}
= dim_{K}[H^{n+j−|G|}(L ∘ D_{E}(N^{•}))]_{G}.

The assertion easily follows from this equality.

Corollary 3.11 (c.f. [4, 11, 10]). Let M^{•} ∈ D^{b}(Sq_{S}). A Betti number β_{i,F} (M^{•}) is extremal if and only if so is β_{|F |−i.F} (A(M^{•})). If this is the case, β_{i,F} (M^{•}) = β_{|F |−i.F} (A(M^{•})).

Proof. If β_{i,F} (M^{•}) is extremal, then (F, n + i − |F |) is a distinguished pair for D_{E} ∘ E(M^{•}) by Proposition 3.10. By Proposition 3.8, (F, n−i) is a distinguished pair for E(M^{•}) ≅ D_{E}∘E∘A(M^{•}). So β_{|F |−i,F} (A(M^{•})) is extremal. The converse implication can be proved by the same way. The last equality follows from Proposition 3.8.

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Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan

yanagawa@math.sci.osaka-u.ac.jp