CASTELNUOVO-MUMFORD REGULARITY FOR COMPLEXES AND RELATED TOPICS

KOHJI YANAGAWA (柳川浩二)

Abstract. Let A be a noetherian AS regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and gr A the category of finitely generated graded left A-modules. Following Jørgensen, we define the Castelnuovo-Mumford regularity reg(M^{•}) of a complex M^{•} ∈ D^{b}(gr A) in terms of the local cohomologies or the minimal projective resolution of M^{•}. Let A^{!} be the quadratic dual ring of A. Then A^{!} is selfinjective and Koszul (e.g., if A is the polynomial ring k[x_{1}, …, x_{d}], then A^{!} is the exterior algebra E = ⋀ ⟨y_{1}, …, y_{d}⟩). For the Koszul duality functor G : D^{b}(gr A) → D^{b}(gr A^{!}), we have reg(M^{•}) = max{ i | H^{i}(G(M^{•})) ≠ 0 }. As an application, we refine results of Martinez-Villa and Zacharia on weakly Koszul modules over A^{!} (especially, over E).

1. Introduction

Let A := ⨁_{i≥0} A_{i} be a noetherian AS regular Koszul quiver algebra over a field k. Such a quiver algebra (with relation) has been studied by Martinez-Villa and coworkers (c.f. [6, 9, 10, 11]). And a connected (i.e., A_{0} = k) AS regular algebra is very important in non-commutative algebraic geometry (c.f. [18]). If A is commutative and connected, it is a polynomial ring k[x_{1}, …, x_{d}] with deg x_{i} = 1 for each i.

Let Gr A (resp. Gr A^{op}) be the category of graded left (resp. right) A-modules, and gr A (resp. gr A^{op}) its full subcategory consisting of finitely generated modules. Set 𝔯 := ⨁_{i≥1} A_{i} to be the graded Jacobson radical. We have the left exact functor Γ_{𝔯} : Gr A → Gr A defined by Γ_{𝔯}(M) = { x ∈ M | 𝔯^{n}x = 0 for n ≫ 0 }, and its right derived functor RΓ_{𝔯} : D^{b}(Gr A) → D^{b}(Gr A). For M^{•} ∈ D^{b}(Gr A), the i^{th} cohomology of RΓ_{𝔯}(M^{•}) is denoted by H_{𝔯}^{i}(M^{•}). Similarly, we have the corresponding functors Γ_{𝔯^{op}}, RΓ_{𝔯^{op}}, and H_{𝔯^{op}}^{i} for graded right A-modules. When A is a polynomial ring k[x_{1}, …, x_{d}], H_{𝔯}^{i}(−) is known as the local cohomology module with support in the graded maximal ideal 𝔯.

We have a bounded cochain complex D^{•} of graded A-A bimodules which gives duality functors R Hom_{A}(−, D^{•}) : D^{b}(gr A) → D^{b}(gr A^{op}) and R Hom_{A^{op}}(−, D^{•}) : D^{b}(gr A^{op}) → D^{b}(gr A). These functors are quasi-inverse of each other. Moreover, we have “local duality theorem”

R\underline{Hom}_{A}(−, D^{•}) ≅ RΓ_{𝔯}(−)^{∨} and R\underline{Hom}_{A^{op}}(−, D^{•}) ≅ RΓ_{𝔯^{op}}(−)^{∨},

where (−)^{∨} stands for the graded k-dual. This is a quiver algebra version of [18].

For M^{•} ∈ D^{b}(gr A) and i, j ∈ ℤ, set β_{j}^{i}(M^{•}) := dim_{k} \underline{Ext}_{A}^{−i}(M^{•}, A/𝔯)_{−j}. Of course, β_{j}^{i}(−) measures the “size” of a minimal projective resolution. Using the above duality, we can generalize a well-know result of Eisenbud-Goto [5] concerning graded modules over a polynomial ring.

This note is basically a summary of [17] which has been accepted for publication in J. Pure Appl.

Algebra.

Definition-Theorem. (c.f. Jørgensen, [8]) For M^{•} ∈ D^{b}(gr A), we have

reg(M^{•}) := sup{ i + j | H_{𝔯}^{i}(M^{•})_{j} ≠ 0 } = sup{ i + j | β_{j}^{i}(M^{•}) ≠ 0 } < ∞.

We call this value the “Castelnuovo-Mumford regularity” of M^{•}.

For M^{•} ∈ D^{b}(gr A), set H(M^{•}) to be the complex such that H(M^{•})^{i} = H^{i}(M) for all i and the differential maps are zero. Then reg(H(M^{•})) ≥ reg(M^{•}). The difference reg(H(M^{•})) − reg(M^{•}) is a theme of the latter half of this note.

Let A^{!} be the quadratic dual ring of A. Then A^{!} is finite dimensional, Koszul and selfinjective by [10]. (e.g., If A is the polynomial ring k[x_{1}, …, x_{d}], then A^{!} is the exterior algebra ⋀ ⟨y_{1}, …, y_{d}⟩). The Koszul duality functors F : D^{b}(gr A^{!}) → D^{b}(gr A) and G : D^{b}(gr A) → D^{b}(gr A^{!}) give an equivalence D^{b}(gr A) ≅ D^{b}(gr A^{!}) (c.f. [2]). We have

reg(M^{•}) = max{ i | H^{i}(G(M^{•})) ≠ 0 }.

For N ∈ gr A^{!} and n ∈ ℤ, N_{⟨n⟩} denotes the submodule of N generated by the degree n component N_{n}. We say N is weakly Koszul, if N_{⟨n⟩} has an n-linear projective resolution (i.e., β_{j}^{i}(N_{⟨n⟩}) ≠ 0 ⇒ i + j = n) for all n. Martinez-Villa and Zacharia [11] proved that the i^{th} syzygy Ω_{i}(N) of N ∈ gr A^{!} is weakly Koszul for i ≫ 0. Of course, the same is true for N ∈ gr (A^{!})^{op}. Set lpd(N) := min{ i ∈ ℕ | Ω_{i}(N) is weakly Koszul }.

Theorem. Let N ∈ gr A^{!}, and N′ := \underline{Hom}_{A^{!}}(N, A^{!}) ∈ gr (A^{!})^{op} its dual. Then

lpd(N′) = reg( H ∘ F(N) ).

If A^{!} is the exterior algebra E = ⋀ ⟨y_{1}, …, y_{d}⟩, we have an upper bound of lpd(N) depending only on max{ dim_{k} N_{i} | i ∈ ℤ } and d. This bound gives huge numbers, and must be very far from optimal. On the other hand, we have lpd(E/J) ≤ min{1, d − 2} for a monomial ideal J of E. This slightly improves a result of Herzog and Römer.

2. Preliminaries

First, we sketch basic properties of an algebra of a quiver with relations.

Let Q be a finite quiver. That is, Q = (Q_{0}, Q_{1}) is a finite oriented graph, where Q_{0} is the set of vertices and Q_{1} is the set of arrows. The path algebra kQ is a positively graded algebra with grading given by the lengths of paths. Let J be the graded Jacobson radical of kQ (i.e., the ideal generated by all arrows). If I ⊂ J^{2} is a graded ideal, we say A = kQ/I is a graded quiver algebra. Of course, A = ⨁_{i≥0} A_{i} is a graded ring. The subalgebra A_{0} is a product of copies of the field k, one copy for each element of Q_{0}. If A_{0} = k (i.e., Q has only one vertex), we say A is connected. If a graded algebra R = ⨁_{i≥0} R_{i} with R_{0} = k is generated by R_{1} as a k-algebra and dim_{k} R_{1} < ∞, then it can be regarded as a graded quiver algebra. Set 𝔯 := ⨁_{i≥1} A_{i}. Unless otherwise specified, we assume that A is left and right noetherian throughout this note.

Let Gr A (resp. Gr A^{op}) be the category of graded left (resp. right) A-modules, and gr A (resp. gr A^{op}) its full subcategory consisting of finitely generated modules. Since A is noetherian, gr A and gr A^{op} are abelian categories. In the sequel, we will define several concepts for Gr A, but the corresponding concepts for Gr A^{op} can be defined in the similar way.

The n^{th} shift M(n) of M = ⨁_{i∈ℤ} M_{i} ∈ Gr A is defined by M(n)_{i} = M_{n+i}. Set ι(M) := inf{ i | M_{i} ≠ 0 }.

For v ∈ Q_{0}, we have the idempotent e_{v} of A associated with v. Note that 1 = ∑_{v∈Q_{0}} e_{v}. Set P_{v} := Ae_{v} and_{v}P := e_{v}A. Then we have_{A}A = ⨁_{v∈Q_{0}} P_{v} and A_{A} = ⨁_{v∈Q_{0}}(_{v}P). Each P_{v} and _{v}P are indecomposable projectives. Conversely, any indecomposable projective in Gr A (resp. Gr A^{op}) is isomorphic to P_{v} (resp. _{v}P) for some v ∈ Q_{0} up to degree shifting. Set k_{v} := P_{v}/(𝔯P_{v}) and _{v}k := _{v}P/(_{v}P 𝔯). Each k_{v} and _{v}k are simple.

Let C^{b}(Gr A) be the category of bounded cochain complexes in Gr A, and D^{b}(Gr A) its derived category. For a complex M^{•} and an integer p, let M^{•}[p] be the p^{th} translation of M^{•}. That is, M^{•}[p] is a complex with M^{i}[p] = M^{i+p}. A module M can be regarded as a complex ⋯ → 0 → M → 0 → ⋯ with M at the 0^{th} term.

For M, N ∈ Gr A, set \underline{Hom}_{A}(M, N) := ⨁_{i∈ℤ} Hom_{Gr A}(M, N(i)) to be a graded kvector space with \underline{Hom}_{A}(M, N)_{i} = Hom_{Gr A}(M, N(i)). Similarly, we can also define \underline{Hom}_{A}^{•}(M^{•}, N^{•}), R\underline{Hom}_{A}(M^{•}, N^{•}), and \underline{Ext}_{A}^{i}(M^{•}, N^{•}) for M^{•}, N^{•} ∈ D^{b}(Gr A).

If V is a k-vector space, V^{∗} denotes the dual space Hom_{k}(V, k). For M ∈ Gr A (resp. M ∈ Gr A^{op}), M^{∨} := ⨁_{i∈ℤ}(M_{i})^{∗} has a graded right (resp. left) A-module structure given by (fa)(x) = f(ax) (resp. (af)(x) = f(xa)) and (M^{∨})_{i} = (M_{−i})^{∗}. If W is a graded A-A bimodule, then so is W^{∨}. Note that I_{v} := (_{v}P)^{∨} (resp. _{v}I := (P_{v})^{∨}) is injective in Gr A (resp. Gr A^{op}). Moreover, I_{v} and _{v}I are graded injective hulls of k_{v} and _{v}k respectively. In particular, the A-A bimodule A^{∨} is injective both in Gr A and in Gr A^{op}.

Let W be a graded A-A-bimodule (we mainly concern the cases W = A or W = A^{∨}). If M ∈ Gr A, we can regard \underline{Hom}_{A}(M, W) as a graded right A-module by (fa)(x) = f(x)a. We have a natural isomorphism \underline{Hom}_{A}(M, A^{∨}) ≅ M^{∨}. We can also define R\underline{Hom}_{A}(M^{•}, W) ∈ D^{b}(Gr A^{op}) and \underline{Ext}_{A}^{i}(M^{•}, W) ∈ Gr A^{op} for M^{•} ∈ D^{b}(Gr A).

Let P^{•} be a bounded complex in gr A such that each P^{i} is projective. We say P^{•} is minimal if ∂(P^{i}) ⊂ 𝔯P^{i+1} for all i. Any complex M^{•} ∈ C^{b}(gr A) has a minimal projective resolution, which is unique up to isomorphism. We denote a graded module A/𝔯 by A_{0}. Set β_{j}^{i}(M^{•}) := dim_{k} \underline{Ext}_{A}^{−i}(M^{•}, A_{0})_{−j}. Let P^{•} be a minimal projective resolution of M^{•}, and P^{i} := ⨁_{l=1}^{m} T^{i, l} an indecomposable decomposition. Then we have

β_{j}^{i}(M^{•}) = #{ l | T^{i, l}(j) ≅ P_{v} for some v }.

Definition 1. Let A be a graded quiver algebra. We say A is Artin-Schelter regular (AS-regular, for short), if

• A has finite global dimension d.

• \underline{Ext}_{A}^{i}(k_{v}, A) = \underline{Ext}_{A^{op}}^{i}(_{v}k, A) = 0 for all i ≠ d and all v ∈ Q_{0}.

• There are a permutation δ on Q_{0} and an integer n_{v} for each v ∈ Q_{0} such that \underline{Ext}_{A}^{d}(k_{v}, A) ≅ _{δ(v)}k(n_{v}) (equivalently, \underline{Ext}_{A^{op}}^{d}(_{v}k, A) ≅ k_{δ^{−1}(v)}(n_{v}) ) for all v.

Remark 2. The AS regularity is a very important concept in non-commutative algebraic geometry (see for example [18]). But there many authors assume the connectedness of A. We also remark that Martinez-Villa and coworkers called the rings in Definition 1 generalized Auslander regular algebras.

Definition 3. For an integer l ∈ ℤ, we say M^{•} ∈ gr A has an l-linear (projective) resolution, if β_{j}^{i}(M^{•}) = 0 for all i, j with i + j ≠ l. If M^{•} has an l-linear resolution for some l, we say M^{•} has a linear resolution.

Definition 4. We say A is Koszul, if the graded left A-module A_{0} has a linear resolution. (Note that A_{0} ≅ ⨁_{v∈Q_{0}} k_{v}.)

In the above definition, we can regard A_{0} as a right A-module (we get the equivalent definition). The next fact is easy to prove.

Lemma 5. If A is AS-regular, Koszul, and has global dimension d, then \underline{Ext}_{A}^{d}(k_{v}, A) ≅ δ(v)k(d) and \underline{Ext}_{A^{op}}^{d}(_{v}k, A) ≅ k_{δ^{−1}(v)}(d) for all v ∈ Q_{0}. Here δ is the permutation of Q_{0} given in Definition 1.

In the rest of this paper, A is always a noetherian AS-regular Koszul quiver algebra of global dimension d.

Example 6. (1) A polynomial ring k[x_{1}, …, x_{d}] is clearly a noetherian AS-regular Koszul (quiver) algebra of global dimension d.

(2) Let k⟨x_{1}, …, x_{d}⟩ be the free associative algebra, and (q_{i,j}) a d×d matrix with entries in k satisfying q_{i,j}q_{j,i} = q_{i,i} = 1 for all i, j. Then A = k⟨x_{1}, …, x_{n}⟩/( x_{j}x_{i} − q_{i,j}x_{i}x_{j} | 1 ≤ i, j ≤ d) is a noetherian AS-regular Koszul algebra with global dimension d. This fact must be well-known to the specialist, but we will sketch a proof here. Since x_{1}, ..., x_{d} ∈ A_{1} form a regular normalizing sequence with k = A/(x_{1}, …, x_{d}), A is a noetherian ring with a balanced dualizing complex by [12, Lemma 7.3]. We can construct a free resolution of k = A/𝔯, which is a “q-analog” of the Koszul complex of a polynomial ring k[x_{1}, …, x_{d}]. So A is Koszul and has global dimension d. Since A has finite global dimension and admits a balanced dualizing complex, it is AS-regular (c.f. [12, Remark 3.6 (3)]).

(3) For examples of non-connected AS regular algebras, see [6].

For M ∈ Gr A, set

Γ_{𝔯}(M) = lim_{→} \underline{Hom}_{A}(A/𝔯^{n}, M) = \{ x ∈ M | A_{n} x = 0 for n ≫ 0 \} ∈ Gr A.

Then Γ_{𝔯}(−) gives a left exact functor from Gr A to itself. So we have a right derived functor RΓ_{𝔯} : D^{b}(Gr A) → D^{b}(Gr A). For M^{•} ∈ D^{b}(Gr A), H_{𝔯}^{i}(M^{•}) denotes the i^{th} cohomology of RΓ_{𝔯}(M^{•}). Similarly, we can define RΓ_{𝔯^{op}} and H_{𝔯^{op}}^{i} for D^{b}(Gr A^{op}) in the same way. If M is an A-A bimodule, H_{𝔯}^{i}(M) and H_{𝔯^{op}}^{i}(M) are also.

Since A is AS regular, we have RΓ_{𝔯}(A) ≅ A^{∨}(d)[−d] in D^{b}(gr A). By the same argument as [18, Proposition 4.4], we also have RΓ_{𝔯}(A) ≅ A^{∨}(d)[−d] in D^{b}(gr A^{op}). It does not mean that H_{𝔯}^{d}(A) ≅ A^{∨}(d) as A-A bimodules. But there is an A-A bimodule L such that L ⊗_{A} H_{𝔯}^{d}(A) ≅ A^{∨}(d) as A-A bimodules. Here the underlying graded additive group of L is A, but the bimodule structure is give by A × L × A ∋ (a, l, b) ↦ ϕ(a)lb ∈ A = L for a (fixed) graded k-algebra automorphism ϕ of A. In particular, L ≅ A as left A-modules and as right A-modules (separately). If A is commutative, then ϕ is the identity map.

Set L′ ≅ \underline{Hom}_{A}(L, A) and D^{•} := L′(−d)[d]. Note that D^{•} belongs both D^{b}(gr A) and D^{b}(gr A^{op}). We have H_{𝔯}^{i}(D^{•}) = H_{𝔯^{op}}^{i}(D^{•}) = 0 for all i ≠ 0 and H_{𝔯}^{0}(D^{•}) ≅ H_{𝔯^{op}}^{0}(D^{•}) ≅ A^{∨} as A-A bimodules by the same argument as [18, §4]. Thus (an injective resolution of) D^{•} is a balanced dualizing complex of A in the sense of [18]. It is easy to check that R\underline{Hom}_{A}(−, D^{•}) and R\underline{Hom}_{A^{op}}(−, D^{•}) give duality functors between D^{b}(gr A) and D^{b}(gr A^{op}), which are quasi-inverse of each other.

Theorem 7 (c.f. Yekutieli [18] and Martinez-Villa [9]). For M^{•} ∈ D^{b}(gr A), we have

RΓ_{𝔯}(M^{•})^{∨} ≅ R\underline{Hom}_{A}(M^{•}, D^{•}). In particular, (H_{𝔯}^{i}(M^{•})_{j})^{∗} ≅ \underline{Ext}_{A}^{−i}(M^{•}, D^{•})_{−j}.

The above result was proved by Yekutieli in the connected case. (In some sense, Martinez-Villa proved a more general result than ours, but he did not concern complexes.) The proof of [18, Theorem 4.18] also works in our case.

Definition 8 (Jørgensen, [8]). For M^{•} ∈ D^{b}(gr A), we say

reg(M^{•}) := sup{ i + j | H_{𝔯}^{i}(M^{•})_{j} ≠ 0 }

is the Castelnuovo-Mumford regularity of M^{•}.

By Theorem 7 and the fact that R\underline{Hom}_{A}(M^{•}, D^{•}) ∈ D^{b}(gr A^{op}), we have reg(M^{•}) < ∞ for all M^{•} ∈ D^{b}(gr A).

Theorem 9 (Jørgensen, [8]). If M^{•} ∈ C^{b}(gr A), then

reg(M^{•}) = max{ i + j | β_{j}^{i}(M^{•}) ≠ 0 }. (2.1)

When A is a polynomial ring and M^{•} is a module, the above theorem is a fundamental result obtained by Eisenbud and Goto [5]. In the non-commutative case, under the assumption that A is connected but not necessarily regular, this has been proved by Jørgensen [8]. (If A is not regular, we have reg(A) > 0 in many cases. So one has to assume that reg A = 0 there.) In our case (i.e., A is AS-regular), we have a much simpler proof. So we will give it here. This proof is also different from one given in [5].

Proof. Set Q^{•} := \underline{Hom}_{A}^{•}(P^{•}, L′(−d)[d] ). Here P^{•} is a minimal projective resolution of M^{•}, and L′ is the A-A bimodule given in the construction of the dualizing complex D^{•}. Note that \underline{Hom}_{A}(P_{v}, L′) ≅ _{δ^{−1}(v)}P for all v ∈ Q_{0}. Let s be the right hand side of (2.1), and l the minimal integer with the property that β_{s−l}^{l}(M^{•}) ≠ 0. Then ι(Q^{−d−l}) = l − s + d, and ι(Q^{−d−l+1}) ≥ l − s + d. Since Q^{•} is a minimal complex, we have

0 ≠ H^{−d−l}(Q^{•})_{l−s+d} = \underline{Ext}_{A}^{−d−l}(M^{•}, D^{•})_{l−s+d} = (H_{𝔯}^{d+l}(M^{•})_{−l+s−d})^{∗}.

Thus reg(M^{•}) ≥ s. The opposite inequality can be proved similarly and more easily.

3. Koszul duality

In this section, we study the relation between the Castelnuovo-Mumford regularity of complexes and the Koszul duality. For precise information of this duality, see [2, §2].

Recall that A = kQ/I is a graded quiver algebra over a finite quiver Q. Let Q^{op} be the opposite quiver of Q. That is, Q_{0}^{op} = Q_{0} and there is a bijection from Q_{1} to Q_{1}^{op} which sends an arrow α : v → u in Q_{1} to the arrow α^{op} : u → v in Q_{1}^{op}. Consider the bilinear form ⟨−, −⟩ : (kQ)_{2} × (kQ^{op})_{2} → A_{0} defined by

⟨αβ, γ^{op}δ^{op}⟩ = \begin{cases}e_{v} & if α = δ and β = γ, \\ 0 & otherwise \end{cases}

for all α, β, γ, δ ∈ Q_{1}. Here v ∈ Q_{0} is the vertex with β ∈ Ae_{v}. Let I^{⊥} ⊂ kQ^{op} be the ideal generated by { y ∈ (kQ^{op})_{2} | ⟨x, y⟩ = 0 for all x ∈ I_{2} }. We say kQ^{op}/I^{⊥} is the quadratic dual ring of A, and denote it by A^{!}. Clearly, (A^{!})_{0} = A_{0}. Since A is Koszul, so is A^{!}. Since A is AS regular, A^{!} is a finite dimensional selfinjective algebra with A = ⨁_{i=0}^{d} A_{i} by [10, Theorem 5.1]. If A is a polynomial ring, then A^{!} is the exterior algebra ⋀(A_{1})^{∗}.

Let V be a finitely generated left A_{0}-module. Then Hom_{A_{0}}(A^{!}, V) is a graded left A^{!}-module with (af)(a′) = f(a′a) and Hom_{A_{0}}(A^{!}, V)_{i} = Hom_{A_{0}}((A^{!})_{−i}, V). Since A^{!} is selfinjective, we have Hom_{A_{0}}(A^{!}, A_{0}) ≅ A^{!}(d). Hence Hom_{A_{0}}(A^{!}, V) is a projective (and injective) left A^{!}-module for all V. If V has degree i (e.g., V = M_{i} for some M ∈ gr A), then we set Hom_{A_{0}}(A^{!}, V)_{j} = Hom_{A_{0}}(A_{−j−i}^{!}, V).

For M^{•} ∈ C^{b}(gr A), let G(M^{•}) := Hom_{A_{0}}(A^{!}, M^{•}) ∈ C^{b}(gr A^{!}) be the total complex of the double complex with G(M^{•})^{i,j} = Hom_{A_{0}}(A^{!}, M_{j}^{i}) whose vertical and horizontal differentials d′ and d′′ are defined by

d′(f)(x) = ∑_{α∈Q_{1}} αf(α^{op}x), d′′(f)(x) = ∂_{M^{•}}(f(x))

for f ∈ Hom_{A_{0}}(A^{!}, M_{j}^{i}) and x ∈ A^{!}. The grading of G(M^{•}) is given by

G(M^{•})_{q}^{p} := ⨁_{p=i+j, q=−l−j} Hom_{A_{0}}((A^{!})_{l}, M_{j}^{i}).

Similarly, for a complex N^{•} ∈ C^{b}(gr A^{!}), we can define a new complex F(N^{•}) := A⊗_{A_{0}} N^{•} ∈ C^{b}(gr A) as the total complex of the double complex with F(N^{•})^{i,j} = A⊗_{A_{0}} N_{j}^{i} whose vertical and horizontal differentials d′ and d′′ are defined by

d′(a ⊗ x) = ∑_{α∈Q_{1}} aα ⊗ α^{op}x, d′′(a ⊗ x) = a ⊗ ∂_{N^{•}}(x)

for a ⊗ x ∈ A ⊗_{A_{0}} N^{i}. The gradings of F(N^{•}) is given by

F(N^{•})_{q}^{p} := ⨁_{p=i+j, q=l−j} A_{l} ⊗_{A_{0}} N_{j}^{i}.

Clearly, each term of F(N^{•}) is a projective A-module. For a module N ∈ gr A^{!}, F(N) is a minimal complex. Hence we have

β_{j}^{i}(F(N)) = \begin{cases}dim_{k} N_{i} & if i + j = 0, \\ 0 & otherwise. \end{cases}

The operations F and G define functors F : D^{b}(gr A^{!}) → D^{b}(gr A) and G : D^{b}(gr A) → D^{b}(gr A^{!}), and they give an equivalence D^{b}(gr A) ≅ D^{b}(gr A^{!}) of triangulated categories. This equivalence is called the Koszul duality. When A is a polynomial ring, this equivalence is called Bernstein-Gel’fand-Gel’fand correspondence. See, for example, [4].

Proposition 10 (c.f. [4, Proposition 2.3]). In the above situation, we have

β_{j}^{i}(M^{•}) = dim_{k} H^{i+j}(G(M^{•}))_{−j}.

Proof.

\underline{Ext}_{A^{!}}^{i}(A_{0}, N^{•})_{j} ≅ Hom_{D^{b}(gr A^{!})}(A_{0}, N^{•}[i](j))
≅ Hom_{D^{b}(gr A)}( F(A_{0}), F(N^{•}[i](j)) )
≅ Hom_{D^{b}(gr A)}( A, F(N^{•})[i + j](−j) )
≅ H^{i+j}(F(N^{•}))_{−j}.

The next result immediately follows from Theorem 9 and Proposition 10.

Corollary 11. reg(M^{•}) = max{ i | H^{i}(G(M^{•})) ≠ 0 }.

For M^{•} ∈ D^{b}(gr A), set H(M^{•}) to be the complex such that H(M^{•})^{i} = H^{i}(M) for all i and all differential maps are zero. By a spectral sequence argument, we see that

reg(H(M^{•})) ≥ reg(M^{•}). (3.1)

In the next section, we will see that the difference reg(M^{•})−reg(H(M^{•})) can be arbitrary large. For N^{•} ∈ D^{b}(gr A^{!}), we can define H(N^{•}) is the same way.

We can refine Proposition 10 using the notion of linear strands of projective resolutions, which was introduced by Eisenbud et. al. ([4, §3]). Let P^{•} be a minimal projective resolution of M^{•} ∈ D^{b}(gr A). Consider the decomposition P^{i} := ⨁_{j∈ℤ} P^{i,j} such that any indecomposable summand of P^{i,j} is isomorphic to a summand of A(−j). For an integer l, we define the l-linear strand proj. lin_{l}(M^{•}) of a projective resolution of M^{•} as follows: The term proj. lin_{l}(M^{•})^{i} of cohomological degree i is P^{i,l−i} and the differential P^{i,l−i} → P^{i+1,l−i−1} is the corresponding component of the differential P^{i} → P^{i+1} of P^{•}. So the differential of proj. lin_{l}(M^{•}) is represented by a matrix whose entries are elements in A_{1}. Set proj. lin(M^{•}) := ⨁_{l∈ℤ} proj. lin_{l}(M^{•}). Clearly, β_{j}^{i}(M^{•}) = β_{j}^{i}(proj. lin(M^{•})) for all i, j.

Proposition 12 (c.f. [4, Corollary 3.6]). For N^{•} ∈ D^{b}(gr A^{!}), we have

proj. lin_{l}(F(N^{•})) = F(H^{l}(N^{•}))[−l], in particular, proj. lin(F(N^{•})) = F(H(N^{•})).

4. Weakly Koszul Modules

Let B be a noetherian Koszul algebra with the graded Jacobson radical 𝔯. For M ∈ gr B and i ∈ ℤ, M_{⟨i⟩} denotes the submodule of M generated by its degree i component M_{i}. The next result naturally appears in the study of Koszul algebras, and might be a folk-theorem (see [17] for further information).

Proposition 13. In the above situation, the following are equivalent.

(1) M_{⟨i⟩} has a linear projective resolution for all i.

(2) H^{i}(proj. lin(M)) = 0 for all i ≠ 0.

(3) gr_{𝔯} M := ⨁_{i=0}^{∞} 𝔯^{i−1}M/𝔯^{i}M has a linear resolution as a B (≅ gr_{𝔯} B)-modules.

Definition 14 (Martinez-Villa et.al., c.f. [11]). We say M ∈ gr B is weakly Koszul, if it satisfies the equivalent conditions of Proposition 13.

If M ∈ gr B has a linear resolution, then it is weakly Koszul. Moreover, if M is weakly Koszul, then the i^{th} syzygy Ω_{i}(M) is also for all i ≥ 1.

Let A be a noetherian AS-regular Koszul quiver algebra of global dimension d, and A^{!} its quadratic dual, as in the previous sections.

Theorem 15 (Martinez-Villa and Zacharia, [11]). If N ∈ gr A^{!} (or N ∈ gr (A^{!})^{op}), then the i^{th} syzygy Ω_{i}(N) is weakly Koszul for i ≫ 0.

Definition 16 (Herzog et. al., [7, 14]). For 0 ≠ N ∈ gr A^{!} (or N ∈ gr (A^{!})^{op}), set

lpd(N) := inf{ i ∈ ℕ | Ω_{i}(N) is weakly Koszul }.

Remark 17. Herzog and Iyengar ([7]) studied the invariant lpd over noetherian commutative Koszul algebras. Among other things, they proved that lpd(M) is always finite over some “nice” rings (e.g., graded complete intersections which are Koszul).

The next result follows from Corollary 11 and Proposition 12.

Theorem 18. Let N ∈ gr A^{!}, and N′ := Hom_{A^{!}}(N, A^{!}) ∈ gr (A^{!})^{op} its dual. Then we have

lpd(N′) = reg( H ∘ F(N) )
= max{ reg(H^{i}(F(N))) + i | i ∈ ℤ }.

Note that reg(H ∘ F(N)) ≥ reg(F(N)) = max{ i | H^{i}(G ∘ F(N)) ≠ 0 } = 0 by the inequality (3.1) and Corollary 11.

If lpd(N) ≥ 1 for some N ∈ gr A^{!}, then sup{ lpd(L) | L ∈ gr A^{!} } = ∞. In fact, if Ω_{−i}(N) is the i^{th} cosyzygy of N (since A^{!} is selfinjective, we can consider cosyzygies), then lpd(Ω_{−i}(N)) > i. But when A is the polynomial ring S = k[x_{1}, …, x_{d}] and A^{!} is the exterior algebra E = ⋀ ⟨y_{1}, …, y_{d}⟩, we have an upper bound of lpd(N) for N ∈ gr E depending only on max{ dim_{k} N_{i} | i ∈ ℤ } and d. But before stating this, we recall a result on a upper bound of reg(M) for M ∈ gr S.

Theorem 19 (Brodmann and Lashgari, [3]). Let S == k[x_{1}, …, x_{d}] be a polynomial ring. Assume that a graded submodule M ⊂ S^{⊕n} is generated by elements whose degrees are at most δ. Then we have reg(M) ≤ n^{d!}(2δ)^{(d−1)!}.

When n = 1 (i.e., when M is an ideal), the above bound is given by Bayer and Mumford [1], and sharper than it seems. In fact, for each m ∈ ℕ, there is an ideal I ⊂ k[x_{1}, …, x_{10m+1}] which is generated by elements of degree at most four but satisfies reg(I) ≥ 2^{2^{m}} + 1. For our study on lpd(N), the case when δ = 1 (but n is general) is essential. When n = δ = 1, we have reg(M) = 1 in the situation of Theorem 19. So I believe that the bound can be largely improved (at least) when δ = 1.

Theorem 20. Let E = ⋀ ⟨y_{1}, …, y_{d}⟩ be an exterior algebra, and N ∈ gr E. Set n := max{ dim_{k} N_{i} | i ∈ ℤ }. Then lpd(N) ≤ n^{d!}2^{(d−1)!}.

Proof. Set L := N′ ∈ gr E. (For graded E-modules, we do not have to distinguish left modules form right ones.) By Theorem 18, it suffices to prove reg(H^{i}(F(L))) + i ≤ n^{d!}2^{(d−1)!} for each i. We may assume that i = 0. Note that H^{0}(F(L)) is the cohomology of the sequence

S ⊗_{k} L_{−1} ⟶^{∂_{−1}} S ⊗_{k} L_{0} ⟶^{∂_{0}} S ⊗_{k} L_{1}.

Since im(∂_{0})(−1) is a submodule of S^{⊕ dim_{k} L_{1}} generated by elements of degree 1, we have reg(im(∂_{0})) < n^{d!}2^{(d−1)!} by Theorem 19. Consider the short exact sequence 0 → ker(∂_{0}) → S ⊗_{k} L_{0} → im(∂_{0}) → 0. Since reg(S ⊗_{k} L_{0}) = 0, we have reg(ker(∂_{0})) ≤ n^{d!}2^{(d−1)!}. Similarly, we have reg(im(∂_{−1})) ≤ n^{d!}2^{(d−1)!} by Theorem 19. By the short exact sequence 0 → im(∂_{−1}) → ker(∂_{0}) → H^{0}(F(L)) → 0, we have reg(H^{0}(F(L))) ≤ n^{d!}2^{(d−1)!}.

In a special case, there is much more reasonable bound for lpd(N).

Definition 21 (Römer, [13]). We say an integer vector a = (a_{1}, …, a_{d}) ∈ ℤ^{d} is squarefree, if a_{i} = 0, 1 for all i. Let N = ⨁_{a∈ℤ^{d}} N_{a} be a finitely generated ℤ^{d}-graded modules over the exterior algebra E = ⋀ ⟨y_{1}, …, y_{d}⟩. We say N is squarefree, if N_{a} ≠ 0 implies that a is squarefree.

This concept naturally appears in the study of combinatorial commutative algebra (c.f. [14, 16]). For example, all monomial ideals of E are squarefree.

Proposition 22 (Herzog and Römer, [14]). If N is a squarefree E-module, then we have lpd(N) ≤ d − 1.

In [17], we describe lpd(N) for a squarefree E-module N in terms of combinatorial commutative algebra. We will show it below in the case when N is a monomial ideal. We also remark that there is a squarefree E-module N with lpd(N) = d − 1.

Set [d] := {1, …, d}. Let Δ ⊂ 2^{[d]} be an (abstract) simplicial complex (i.e., F ∈ Δ and G ⊂ F imply G ∈ Δ). It is easy to see that Δ^{∨} := { F ⊂ [d] | [d] \ F ∉ Δ } is a simplicial complex again. We also have Δ^{∨∨} = Δ. Set J_{Δ} = ( ∏_{i∈F} y_{i} | F ⊂ [d], F ∉ Δ ) to be a monomial ideal of E. Any monomial ideal of E is given in this way. Similarly, set I_{Δ} = ( ∏_{i∈F} x_{i} | F ⊂ [d], F ∉ Δ ) to be a monomial ideal of S, and call it the Stanley-Reisner ideal of Δ. Any squarefree monomial ideal of S is given in this way.

Proposition 23. For a simplicial complex Δ ⊂ 2^{[d]}, we have

(4.1) lpd(J_{Δ}) = max{ i − depth_{S}( Ext_{S}^{d−i}(S/I_{Δ^{∨}}, S) ) | 0 ≤ i ≤ d }.

Here we set the depth of the 0 module to be +∞.

If Ext_{S}^{d−i}(S/I_{Δ^{∨}}, S) ≠ 0, then we have i − depth_{S}( Ext_{S}^{d−i}(S/I_{Δ^{∨}}, S) ) ≥ 0. One might think the right side of the equality (4.1) is strange. But the right side of (4.1) equals 0 if and only if S/I_{Δ^{∨}} is sequentially Cohen-Macaulay (see [15]). In this sense, lpd(J_{Δ}) measures “how is S/I_{Δ^{∨}} far from sequentially Cohen-Macaulay?”.

Corollary 24 (Römer, [13]). For a simplicial complex Δ ⊂ 2^{[d]}, the following are equivalent.

(1) J_{Δ} ⊂ E is weakly Koszul.

(2) I_{Δ} ⊂ S is weakly Koszul.

(3) S/I_{Δ^{∨}} is sequentially Cohen-Macaulay.

We remark that there are many examples of Stanley-Reisner ideals I_{Δ} ⊂ S which are weakly Koszul (dually, Stanley-Reisner rings S/I_{Δ} which are sequentially Cohen-Macaulay).

Corollary 25. If d ≥ 3, then we have lpd(E/J_{Δ}) ≤ d − 2.

In this moment, I have no idea whether the above bound is (nearly) sharp for large d.

References

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[12] I. Mori, Rationality of the Poincare series for Koszul algebras, J. Algebra 276 (2004), 602–624.

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[14] T. Römer, On minimal graded free resolutions, Thesis, University of Essen, 2001.

[15] R. Stanley, Combinatorics and commutative algebra, 2nd ed. Progress in Math., 41. Birkhäuser 1996.

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[17] K. Yanagawa, Castelnuovo-Mumford regularity for complexes and weakly Koszul modules, J. Pure Appl. Algebra, in press.

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Department of Mathematics

Graduate School of Science

Osaka University

Toyonaka, Osaka 560-0043, JAPAN

  E-mail address: yanagawa@math.sci.osaka-u.ac.jp