SOME APPLICATIONS OF KOSZUL DUALITY

IZURU MORI

Abstract. In this paper, we will discuss some applications of Koszul duality for a connected graded Koszul algebra A. Since A is an Artin-Schelter regular Koszul algebra if and only if its Koszul dual A^{!} is a Frobenius Koszul algebra, there is potential interaction between two research areas, Noncommutative Algebraic Geometry and the study of Frobenius algebras.

1. Introduction

Throughout, let A = ⊕_{i=0}^{∞}A_{i} be a finitely generated connected graded algebra over a field k. The augmentation ideal of A is denoted by 𝔪 = ⊕_{i=1}^{∞}A_{i}. We often view k as a graded left A-module by the identification k = A/𝔪. For a graded left A-module M = ⊕_{i=−∞}^{∞}M_{i} and an integer n ∈ ℤ, we define a graded left A-module M(n) by M(n) = M as an ungraded left A-module but M(n)_{i} = M_{n+i} for all i ∈ ℤ. A linear resolution of M is a resolution of the form

⋯ → ⨁ A(−2) → ⨁ A(−1) → ⨁ A → M → 0,

that is, a free resolution in which each differential is given by right multiplication of a matrix whose entries are all degree 1 elements (linear elements) of A. We say that A is Koszul if k has a linear resolution as a graded left A-module.

If A is a Koszul algebra, then its dual A^{!} is again a Koszul algebra. An important classical result is that there is a duality, known as the Koszul duality, between the categories of graded left modules having linear resolutions over A and A^{!}, respectively. Recently, it has been shown that the Koszul duality can be extended to a duality between the derived categories of finitely generated graded left modules over A and A^{!}, respectively. Through this extended Koszul duality, we are more able to translate results on A to those on A^{!}.

It is known that A is an Artin-Schelter regular Koszul algebra, which is one of the most important algebras in Noncommutative Algebraic Geometry, if and only if A^{!} is a Frobenius Koszul algebra. Further, the extended Koszul duality induces an equivalence between the derived category of a quantum projective space, which is a projective scheme associated to a noetherian Artin-Schelter regular Koszul algebra, and the graded stable category of the corresponding Frobenius Koszul algebra. So there is potential interaction between two research areas, Noncommutative Algebraic Geometry and the study of Frobenius algebras. In this paper, we will give some examples of such interaction.

This is an expository paper based on the results of the author and others. The detailed version of some of the results in this paper has been submitted for publication elsewhere.

2. Koszul Duality

Let L, M and N be graded left A-modules. The Yoneda product is a map

Ext_{A}^{i}(N, L) ⊗ Ext_{A}^{j}(M, N)→Ext_{A}^{i+j}(M, L)
[β] ⊗ [α]  ↦ [β ∘ α_{i}]

defined by the following diagram:

⋯ →F^{i+j}→ ⋯ →F^{j} → ⋯ →M→ 0
α_{i}↓  ^{α}↓
⋯ → G^{i} → ⋯ →N → 0
β↓
L

where

⋯ → F^{2} → F^{1} → F^{0} → M → 0, and
⋯ → G^{2} → G^{1} → G^{0} → N → 0

are free resolutions of M and N, respectively, and α_{i} is a lift of α.

We define a graded vector space E(M) over k by E(M)_{i} = Ext_{A}^{i}(M, k). Then E(k) has a structure of a connected graded algebra over k, and E(M) has a structure of a graded left E(k)-module by the Yoneda product.

Let A be a quadratic algebra over k, that is, A = T (V)/(R) where V is a finite dimensional vector space over k, T (V) is a tensor algebra over V, and R ⊂ V ⊗ V is a subvector space. We define the dual of A by A^{!} = T (V^{∗})/(R^{⊥}) where V^{∗} is the dual vector space of V and

R^{⊥} = {λ ∈ V^{∗} ⊗ V^{∗} | λ(r) = 0 for all r ∈ R}.

If A is a Koszul algebra, then it is known that A is quadratic and A^{!} ≅ E(k) as graded algebras. In this case, E(E(M)) has a graded left module structure over (A^{!})^{!} ≅ A. The following is an important classical result.

Theorem 2.1. [17] (Koszul duality) If A is a Koszul algebra, then A^{!} is also a Koszul algebra. In this case, if M is a graded left A-module having a linear resolution, then E(M) is a graded left A^{!}-module having a linear resolution, and

E(E(M)) ≅ M

as graded left A-modules. That is, E defines a duality

E : Lin A → Lin A^{!},

where Lin A is the category of graded left A-modules having linear resolutions.

If A is a Koszul algebra, then we call A^{!} the Koszul dual of A.

Example 1. A free algebra

T = k⟨x_{1}, ⋯, x_{d}⟩

is a Koszul algebra where

T^{!} ≅ k⟨x_{1}, ⋯, x_{d}⟩/(x_{i}x_{j})_{i,j=1,··· ,d}.

Example 2. A polynomial algebra

S = k[x_{1}, ⋯, x_{d}] = k⟨x_{1}, ⋯, x_{d}⟩/(x_{i}x_{j} − x_{j}x_{i})_{i,j=1,··· ,d}

is a Koszul algebra where

S^{!} ≅ k⟨x_{1}, ⋯, x_{d}⟩/(x_{i}x_{j} + x_{j}x_{i}, x_{i}^{2})_{i,j=1,··· ,d}

is an exterior algebra.

Example 3. An algebra

A = k⟨x, y⟩/(y^{2})

is a Koszul algebra where

A^{!} ≅ k⟨x, y⟩/(x^{2}, xy, yx).

By this duality, we can translate some of the properties of A to those of A^{!}. For example, the following result is immediate.

Lemma 2.2. Let A be a Koszul algebra. Then A is regular, that is, A has finite global dimension, if and only if A^{!} is finite dimensional over k.

3. Extended Koszul Duality

The functor E above does not define a duality for the categories of arbitrary modules. So the above Koszul duality has limited applications. In order to overcome this limitation, we will extend the above Koszul duality to derived categories. If we assume the existence of a balanced dualizing complex defined below, then the duality behaves particularly well.

Definition 3.1. [20] A balanced dualizing complex of A is a complex D_{A} of graded A-A bimodules satisfying the following conditions, viewing D_{A} as a complex of graded left and right A-modules:

• D_{A} has finite injective dimension,

• h^{i}(D_{A}) are finitely generated for all i,

• Ext_{A}^{i}(D_{A}, D_{A}) ≅ \begin{cases}A & if i = 0 \\ 0 & if i ≠ 0 \end{cases} as graded A-A bimodules, and

• H_{𝔪}^{i}(D_{A}) ≅ \begin{cases}A^{∗} & if i = 0 \\ 0 & if i ≠ 0 \end{cases} as graded A-A bimodules,

where

H_{𝔪}^{i}(−) = lim_{n→∞} Ext_{A}^{i}(A/𝔪^{n}, −)

is the i-th local cohomology functor.

A balanced dualizing complex plays an essential role in Noncommutative Algebraic Geometry. Many noetherian algebras of importance have balanced dualizing complexes. For example, finite dimensional algebras, noetherian commutative algebras, and FBN algebras (including noetherian PI algebras) have balanced dualizing complexes (see [19]).

We will now extend the Koszul duality to derived categories as follows. Let grmod A be the category of finitely generated graded left A-modules and D^{b}(grmod A) the derived category of bounded complexes in grmod A.

Theorem 3.1. (Koszul duality) [4], [12] Let A be a Koszul algebra. If both A and A^{!} are noetherian and having balanced dualizing complexes, then there is a duality

Ē : D^{b}(grmod A) → D^{b}(grmod A^{!}).

We refer to [4] and [12] for the definition of Ē(X) for a complex X of graded left Amodules. If M is a finitely generated graded left A-module, then Ē(M) is a complex of left A^{!}-modules defined by

Ē(M)^{i} = A^{!} ⊗ M_{i}^{∗}(−i).

The following lemma says that the duality in the above theorem is in fact an extension of the classical Koszul duality.

Lemma 3.2. [12] Let A be a Koszul algebra and M a finitely generated graded left Amodule. Then M has a linear resolution if and only if Ē(M) ≅ E(M) in D^{b}(grmod A^{!}).

By this extended Koszul duality, we are more able to translate results on A to those on A^{!}. We will see such examples below.

4. Gorenstein Condition

One of the starting points of Noncommutative Algebraic Geometry was to classify Artin-Schelter regular algebras of global dimension 3, defined below.

Definition 4.1. [2] We say that A satisfies Gorenstein condition if, for some integer d, we have

Ext_{A}^{i}(k, A) ≅ \begin{cases}k & if i = d, \\ 0 & if i ≠ d. \end{cases}

We say that A is Artin-Schelter regular (AS-regular, for short) if A is regular and satisfying Gorenstein condition.

If A is noetherian and having a balanced dualizing complex (e.g. commutative), then A satisfies Gorenstein condition if and only if A has finite injective dimension as a left module over itself, so the above definition of Gorenstein condition agrees with the commutative one. Note that a noetherian regular algebra is AS-regular if and only if A has a balanced dualizing complex.

The first application of the extended Koszul duality is below.

Theorem 4.1. [7] [12] Let A be a Koszul algebra. Then A satisfies Gorenstein condition if and only if A^{!} satisfies Gorenstein condition.

Since a finite dimensional algebra satisfies Gorenstein condition if and only if it is Frobenius, we can recapture the following result as a corollary.

Corollary 4.2. [17] Let A be a Koszul algebra. Then A is AS-regular if and only if A^{!} is Frobenius.

5. Hilbert Series

In this section and the next section, we will discuss rationality of the Hilbert series and the Poincaré series, respectively, of a finitely generated graded left A-module M.

If dim_{k} M_{i} < ∞ for all i, then we define the Hilbert series of M by

H_{M}(t) = ∑_{i=−∞}^{∞}(dim_{k} M_{i})t^{i} ∈ ℤ[[t, t^{−1}]].

Let us calculate the Hilbert series of a few examples of graded algebras.

Example 4. If T = k⟨x_{1}, ⋯, x_{d}⟩ is a free algebra, then

H_{T} (t) = ∑_{i=0}^{∞} d^{i}t^{i} = \frac{1}{1 − dt} ∈ ℂ(t).

Example 5. If S = k[x_{1}, ⋯, x_{d}] is a polynomial algebra, then

H_{S}(t) = ∑_{i=0}^{∞} \binom{d − 1 + i}{d − 1}t^{i} = \frac{1}{(1 − t)^{d}} ∈ ℂ(t).

Example 6. If A = k⟨x, y⟩/(y^{2}), then

A_{0} = k.
A_{1} = kx + ky.
A_{2} = kx^{2} + kxy + kyx.
A_{3} = kx^{3} + kx^{2}y + kxyx + kyx^{2} + kyxy.
A_{4} = kx^{4} + kx^{3}y + kx^{2}yx + kxyx^{2} + kyx^{3} + kxyxy + kyx^{2}y + kyxyx.
⋯

If a_{i} = dim_{k} A_{i}, then, by induction,

a_{0} = 1, a_{1} = 2, a_{i} = a_{i−2} + a_{i−1} for i ≥ 2

(Fibonacci sequence). We can prove that

H_{A}(t) = ∑_{i=0}^{∞} a_{i}t^{i} = \frac{1 + t}{1 − t − t^{2}} ∈ ℂ(t).

There was a natural conjecture by V. E. Govorov (1972) that “if A is finitely presented, then H_{A}(t) ∈ ℂ(t)”. Although counterexamples were constructed by J. B. Shearer [16] (1980) and D. Anick [1] (1982), we will still expect that reasonably nice algebras, such as left noetherian algebras, have the following stronger property.

Definition 5.1. We say that A has the property (H) if H_{M}(t) ∈ ℂ(t) for all finitely generated graded left A-modules M.

In fact, many left noetherian algebras have the property (H).

Example 7. 1. Every finite dimensional algebra has the property (H).

2. Every quotient algebra of a left noetherian regular algebra has the property (H). In particular, every noetherian commutative algebra has the property (H).

3. (Stafford-Zhang [18], 1994) Every FBN algebra has the property (H). In particular, every noetherian PI algebra has the property (H).

6. Poincaré Series

If dim_{k} Ext_{A}^{i}(M, k) < ∞ for all i, then we define the Poincaré series of M by

P_{A}^{M}(t) = ∑_{i=0}^{∞} dim_{k} Ext_{A}^{i}(M, k)t^{i} ∈ ℤ[[t]].

There was a similar but more restrictive conjecture by Serre-Kaplansky (1965) that “if A is finite dimensional and commutative, then P_{A}^{k}(t) ∈ ℂ(t)”. Although a counterexample was constructed by D. Anick [1] (1982), finding a class of algebras having the following stronger property remains interesting.

Definition 6.1. We say that A has the property (P) if P_{A}^{M}(t) ∈ ℂ(t) for all finitely generated graded left A-modules M.

We know that the following classes of algebras have the property (P).

Example 8. 1. Every left noetherian regular algebra has the property (P).

2. (T. H. Gulliksen [6], 1974) Every commutative complete intersection algebra has the property (P).

Recently, Martínez and Zacharia proved the following result.

Theorem 6.1. [10] If A is a finite dimensional Koszul algebra such that A^{!} is left noetherian, then A has the property (P).

The second application of the extended Koszul duality is to generalize the above theorem.

Lemma 6.2. [12] Let A be a Koszul algebra and M a finitely generated graded left Amodule. If A is noetherian and having a balanced dualizing complex , then

P_{A}^{M}(t) = ∑_{i: finite} H_{h^{i}(Ē(M))}(t)t^{i}.

The following theorem is an immediate consequence of the above lemma.

Theorem 6.3. [12] Let A be a noetherian Koszul algebra having a balanced dualizing complex. If A^{!} has the property (H), then A has the property (P).

As we have seen, every finite dimensional algebra is noetherian and having a balanced dualizing complex, and every left noetherian regular algebra has the property (H), so the above theorem contains that of Martínez and Zacharia

7. Grothendieck Group

In the following two sections, we will apply the Koszul duality to the study of a quantum projective space and the graded stable category of a Frobenius Koszul algebra.

Let fdim A be the full subcategory of grmod A consisting of finite dimensional modules. Artin and Zhang [3] defined the noncommutative projective scheme associated to A to be the quotient category tails A = grmod A/ fdim A. The image of M ∈ grmod A in tails A is denoted by M̃. To understand this category, if A is left noetherian, then M̃ ≅ Ñ in tails A if and only if ⊕_{i=n}^{∞}M_{i} ≅ ⊕_{i=n}^{∞}N in grmod A for n ≫ 0. By Serre, if A is commutative and generated by degree 1 elements, then tails A is equivalent to the category of coherent modules over Proj A. If A is a noetherian AS-regular Koszul algebra, then we call tails A a quantum projective space.

Since ∼: grmod A → tails A is an exact functor, it induces a functor ∼: D^{b}(grmod A) → D^{b}(tails A).

Theorem 7.1. [13] Let A be a Koszul algebra such that both A and A^{!} are noetherian and having balanced dualizing complexes. Then, for X ∈ D^{b}(grmod A), X̃ ≅ 0 in D^{b}(tails A) if and only if Ē(X) has finite projective dimension. That is, Ē induces a duality

Ē : D^{b}(tails A) → D^{b}(grmod A^{!})/P(A^{!}),

where P(A^{!}) is the full subcategory of D^{b}(grmod A^{!}) consisting of complexes having finite projective dimension.

Now, suppose that A is a noetherian AS-regular Koszul algebra. Since A^{!} is Frobenius, D^{b}(grmod A^{!})/P(A^{!}) ≅ \underline{grmod}A^{!}, where \underline{grmod}A^{!} is the stable category of grmod A^{!}. Moreover, since the duality Hom_{A^{!}}(−, A^{!}) : grmod A^{!} → grmod(A^{!})^{op} induces a duality \underline{grmod}A^{!} → \underline{grmod}(A^{!})^{op}, where (A^{!})^{op} is the opposite graded algebra of A^{!}, we recapture the following classical result.

Theorem 7.2. [5] [9] If A is a noetherian AS-regular Koszul algebra, then there is an equivalence of categories

D^{b}(tails A) ≅ \underline{grmod}(A^{!})^{op}.

By this equivalence, we can translate results on a quantum projective space to those on the graded stable category of the corresponding Frobenius Koszul algebra, and vice versa. For example, let A and B be noetherian AS-regular Koszul algebras. Then the above theorem says that tails A and tails B are derived equivalent if and only if (A^{!})^{op} and (B^{!})^{op} are graded stable equivalent. Moreover, we can calculate the Grothendieck group.

Theorem 7.3. [14] [15] If A is a left noetherian regular Koszul algebra, then

K_{0}(tails A) ≅ ℤ[t]/(P_{A}^{k}(−t)).

The above theorem can be translated as follows.

Theorem 7.4. If A is a Frobenius Koszul algebra such that A^{!} is noetherian, then

K_{0}(\underline{grmod}A) ≅ ℤ[t]/(H_{A}(−t)).

Example 9. If A = k⟨x_{1}, ⋯, x_{d}⟩/(x_{i}x_{j} + x_{j}x_{i}, x_{i}^{2})_{i,j=1,··· ,d} is an exterior algebra, then H_{A}(t) = (1 + t)^{d}, so

K_{0}(\underline{grmod}A) ≅ ℤ[t]/((1 − t)^{d}).

8. Serre Duality

Let M and N be graded left A-modules. The set of morphisms M → N in \underline{grmod}A is denoted by \underline{Hom}_{A}(M, N). The i-th stable cohomology is defined by

\underline{Ext}_{A}^{i}(M, N) = lim_{n≥max\{−i,0\}} \underline{Hom}_{A}(Ω^{n+i}M, Ω^{n}N),

where Ω^{n}M is the n-th syzygy of M. If σ : A → A is a graded algebra automorphism, then we define a graded left A-module ^{σ}M by ^{σ}M = M as a graded vector space over k but with the new action a · m = σ(a)m.

First, let A be a Frobenius algebra of Loewy length n. Then there exists a graded algebra automorphism ν : A → A, called the Nakayama automorphism, such that A^{∗} ≅ νA(n) as graded A-A bimodules.

Theorem 8.1. [8] Let A be a Frobenius algebra of Loewy length n, ν the Nakayama automorphism, and M, N finitely generated graded left A-modules. Then, for any integer i, there is a natural isomorphism

\underline{Ext}_{A}^{i}(M, N) ≅ \underline{Ext}_{A}^{−1−i}(N,^{ν}M(n))^{∗}.

Next, let A be a noetherian AS-regular Koszul algebra. Since A^{!} is Frobenius, we have the Nakayama automorphism ν : A^{!} → A^{!}, which canonically induces a graded algebra automorphism σ : A → A. The above theorem can be translated as follows.

Theorem 8.2. [8] (Serre duality) Let A be a noetherian AS-regular Koszul algebra of global dimension n, and M, N finitely generated graded left A-modules. Then, for any integer i, there is a natural isomorphism

Ext_{tails A}^{i}(M̃, Ñ) ≅ Ext_{tails A}^{n−1−i}(Ñ, { \atop ^{σ}M(−n)})^{∗}.

Now, we will rewrite the formulas in the above two theorems using a balanced dualizing complex D_{A} of A. First, let A be a Frobenius algebra of Loewy length n and ν the Nakayama automorphism. The functor

N (−) = Hom_{A}(−, A)^{∗} : grmod A → grmod A

is called the Nakayama equivalence. Since A is Frobenius,

D_{A} ≅ H_{𝔪}^{0}(A)^{∗} ≅ A^{∗} ≅ ^{ν}A(n)

as graded A-A-bimodules, so N can be written as

N (−) ≅ D_{A} ⊗_{A} − : grmod A → grmod A.

Hence we can rewrite the formula in Theorem 8.1 as

\underline{Ext}_{A}^{i}(M, N) ≅ \underline{Ext}_{A}^{−1−i}(N, N (M))^{∗} ≅ \underline{Ext}_{A}^{−1−i}(N, D_{A} ⊗_{A} M)^{∗}.

On the other hand, if A is a noetherian AS-regular algebra, then we can rewrite the formula in Theorem 8.2 as

Ext_{tails A}^{i}(M̃, Ñ) ≅ Ext_{tails A}^{−1−i}(Ñ, { \atop D_{A} ⊗_{A} M})^{∗}

by [21]. So these two formulas look strikingly alike.

9. A generalization of Nakayama equivalence

Let D_{A} be a balanced dualizing complex of A. By the previous section, it is interesting to view the functor D_{A} ⊗_{A} − as a generalization of the Nakayama equivalence. Unless A is Frobenius, it no longer induces an equivalence of categories of graded left A-modules. However, it induces an equivalence of the following derived categories.

Theorem 9.1. [11] (Foxby equivalence) If A is a noetherian algebra having a balanced dualizing complex D_{A}, then there is an equivalence of categories

D_{A} ⊗_{A}^{L} − : P(A) → I(A),

where I(A) is the full subcategory of D^{b}(grmod A) consisting of complexes having finite injective dimension.

Although we do not know how to generalize the Nakayama automorphism to an arbitrary algebra, the following may be an interesting idea. Let A be a noetherian algebra having a balanced dualizing complex D_{A}. Then it is known that A satisfies Gorenstein condition if and only if there exists a graded algebra automorphism σ : A → A such that D_{A} is isomorphic to a “shift” of ^{σ}A in the derived categories of graded A-A bimodules. We may consider σ as a generalization of the Nakayama automorphism because if A is a Frobenius algebra of Loewy length n and ν : A → A is the Nakayama automorphism, then we have seen that D_{A} ≅ ^{ν}A(n) as graded A-A bimodules.

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Department of Mathematics The University of Toledo Toledo, OH 43606-3390

  E-mail address: imori@utnet.utoledo.edu