New York Journal of Mathematics

New York J. Math. 14 (2008) 325–378.

Koszul equivalences in A_{∞}-algebras

Di Ming Lu, John H. Palmieri, Quan Shui Wu and James J. Zhang

Abstract. We prove a version of Koszul duality and the induced derived equivalence for Adams connected A_{∞}-algebras that generalizes the classical Beilinson–Ginzburg–Soergel Koszul duality. As an immediate consequence, we give a version of the Bernšteĭn–Gel’fand–Gel’fand correspondence for Adams connected A_{∞}-algebras.

We give various applications. For example, a connected graded algebra A is Artin–Schelter regular if and only if its Ext-algebra Ext_{A}^{∗}(k, k) is Frobenius. This generalizes a result of Smith in the Koszul case. If A is Koszul and if both A and its Koszul dual A^{!} are noetherian satisfying a polynomial identity, then A is Gorenstein if and only if A^{!} is. The last statement implies that a certain Calabi–Yau property is preserved under Koszul duality.

Contents

Introduction 326

Part I. Koszul duality for algebras 329 1. Background on A_{∞}-algebras 329 2. The Koszul dual of an A_{∞}-algebra 336

Part II. Koszul duality for modules 343 3. Background on A_{∞}-modules 343 4. Adjunctions and equivalences 348

Received July 28, 2008. Mathematics Subject Classification. 16A03,16A62,16E65. Key words and phrases. A_{∞}-algebra, graded algebra, Artin–Schelter regular algebra,

Koszul duality, derived equivalence, Gorenstein property.

The first author is supported by the NSFC (project 10571152) of China and the NSF of Zhejiang Province (J20080154).

The third author is supported by the NSFC (key project 10331030) of China and Doctorate Foundation (No. 20060246003), Ministry of Education of China.

The fourth author is supported by the NSF of USA and the Royalty Research Fund of the University of Washington.

325^{ISSN 1076-9803/08}

5. Koszul equivalences and dualities 356 6. Minimal semifree resolutions 360 7. Towards classical Koszul duality 364 8. Some examples 366

Part III. Applications in ring theory 367 9. The Artin–Schelter condition 367 10. The BGG correspondence 374 References 376

Introduction

Koszul duality is an incredibly powerful tool used in many areas of mathematics. One aim of this paper is to unify some generalizations by using A_{∞}-algebras. Our version is comprehensive enough to recover the original version of Koszul duality and the induced derived equivalences due to Beilinson–Ginzburg–Soergel [7] and most of the generalizations in ring theory and algebraic geometry. Although we will restrict ourselves to Adams connected A_{∞}-algebras (a natural extension of a connected graded algebras — see Definition 2.1), we have set up a framework that will work for other classes of algebras arising from representation theory and algebraic geometry.

We fix a commutative field k and work throughout with vector spaces over k. We define A_{∞}-algebras over k in Definition 1.1.

Similar to [24, Section 11] we define the Koszul dual of an A_{∞}-algebra A to be the vector space dual of the bar construction of A — see Section 2 for details. This idea is not new and dates back at least to Beilinson–Ginzburg– Schechtman [6] for graded algebras. Keller also took this approach in [19] for differential graded algebras. Our first result is a generalization of [7, Theorem 2.10.2].

Theorem A. Let A be an augmented A_{∞}-algebra. Suppose that the Koszul dual of A is locally finite. Then the double Koszul dual of A is A_{∞}-isomorphic to A.

This is proved as Theorem 2.4. A special case of the above theorem was proved in [24, Theorem 11.2].

As in [7] we prove several versions of equivalences of derived categories induced by the Koszul duality. Let D^{∞}(A) be the derived category of right A_{∞}-modules over A. Let D_{per}^{∞}(A) (respectively, D_{fd}^{∞}(A)) denote the full triangulated subcategory of D^{∞}(A) generated by all perfect complexes (respectively, all right A_{∞}-modules whose homology is finite-dimensional) over A. The next result is a generalization of [7, Theorem 2.12.6].

Koszul equivalences in A_{∞}-algebras 327

Theorem B. Let A be an Adams connected A_{∞}-algebra and E its Koszul dual. If HE is finite-dimensional, then there is an equivalence of triangulated categories D_{per}^{∞}(A) ≅ D_{fd}^{∞}(E).

This is proved as Corollary 7.2(b)

Other equivalences of triangulated categories can be found in Sections 4 and 5. If A is either Artin–Schelter regular (Definitions 9.1(c) and 9.2(c)) or right noetherian with finite global dimension, then HE is finite-dimensional and hence Theorem B applies.

Koszul duality has many applications in ring theory, representation theory, algebraic geometry, and other areas. The next result is a generalization of the Bernšteĭn–Gel’fand–Gel’fand correspondence that follows from Theorem B. Let \underline{D_{fg}^{∞}}(A) be the stable derived category of A_{∞}-modules over A whose homology is finitely generated over HA, and let D^{∞}(proj A) be the derived category of the projective scheme of A. These categories are defined in Section 10, and the following theorem is part of Theorem 10.2.

Theorem C. Let A be an Adams connected A_{∞}-algebra that is noetherian Artin–Schelter regular. Let E be the Koszul dual of A. Then HE is finite-dimensional and there is an equivalence of triangulated categories D^{∞}(proj A) ≅ \underline{D_{fg}^{∞}}(E).

Applications of Koszul duality in ring theory are surprising and useful. We will mention a few results that are related to the Gorenstein property. In the rest of this introduction we let R be a connected graded associative algebra over a base field k.

Corollary D. Let R be a connected graded algebra. Then R is Artin– Schelter regular if and only if the Ext-algebra ⨁_{i∈ℤ} Ext_{R}^{i}(k_{R}, k_{R}) is Frobenius.

This result generalizes a theorem of Smith [32, Theorem 4.3 and Proposition 5.10] that was proved for Koszul algebras. It is proved in Section 9.3. Corollary D is a fundamental result and the project [25] was based on it.

The Gorenstein property plays an important role in commutative algebra and algebraic geometry. We prove that the Gorenstein property is preserved under Koszul duality; see Section 9.4 for details.

Corollary E. Let R be a Koszul algebra and let R^{!} be the Koszul dual of R in the sense of Beilinson–Ginzburg–Soergel [7]. If R and R^{!} are both noetherian having balanced dualizing complexes, then R is Gorenstein if and only if R^{!} is.

The technical hypothesis about the existence of balanced dualizing complexes can be checked when the rings are close to being commutative. For example, Corollary E holds when R and R^{!} are noetherian and satisfy a polynomial identity. This technical hypothesis is presented because we do not understand noncommutative rings well enough. We do not know any example in which the technical hypothesis is necessary; however, Corollary E does fail for nonnoetherian rings — for example, the free algebra R = k⟨x, y⟩ is Gorenstein, but R^{!} ≅ k⟨x, y⟩/(x^{2}, xy, yx, y^{2}) is not.

Note that Koszul duality preserves the Artin–Schelter condition (Proposition 9.3). Under the technical hypothesis of Corollary E the Artin–Schelter condition is equivalent to the Gorenstein property. Therefore Corollary E follows. We can restate Corollary E for A_{∞}-algebras in a way that may be useful for studying the Calabi–Yau property of the derived category D^{∞}(A) (see the discussion in Section 9.4).

The following is proved in Section 9.4.

Corollary F. Let A be an Adams connected commutative differential graded algebra such that RHom_{A}(k, A) is not quasi-isomorphic to zero. If the Extalgebra ⨁_{i∈ℤ} Ext_{A}^{i}(k_{A}, k_{A}) is noetherian, then A satisfies the Artin–Schelter condition.

The hypothesis on RHom_{A}(k, A) is a version of finite depth condition which is very mild in commutative ring theory and can be checked under some finiteness conditions. This is automatic if A is a finitely generated associative commutative algebra. As said before, the Artin–Schelter condition is equivalent to the Gorenstein property under appropriate hypotheses. Hence Corollary F relates the Gorenstein property of R with the noetherian property of R^{!} and partially explains why Corollary E holds, and at the same time it suggests that Corollary E should be a consequence of a more basic statement that relates the noetherian property of R with the Gorenstein property of R^{!}. On the other hand, we believe that there should be a version of Corollary E without using the noetherian property. As we commented for Corollary E, Corollary F should hold in a class of noncommutative rings R. Corollary F is also a variation of a result of Bøgvad and Halperin about commutative complete intersection rings [9].

This paper is part four of our A_{∞}-algebra project and is a sequel to [24, 25, 26]. Some results were announced in [24]. For example, Theorem B and Corollary D were stated in [24] without proof. We also give a proof of [24, Theorem 11.4] in Section 5.

The paper is divided into three parts: Koszul duality for algebras, Koszul duality for modules, and applications in ring theory.

Part I consists of Sections 1 and 2. Section 1 gives background material on A_{∞}-algebras and their morphisms. The reader may wish to skim it to see the conventions and notation used throughout the paper. Theorem A is proved in Section 2, and we use it to recover the classical Koszul duality of Beilinson, Ginzburg, and Soergel. We also discuss a few examples.

Part II consists of Sections 3–8. Section 3 gives background material on A_{∞}-modules; most of this is standard, but the results on opposites are new. Section 4 sets up a framework for proving equivalences of various derived categories of DG modules over a DG algebra. Section 5 uses this framework to prove DG and A_{∞} versions of the results of Beilinson, Ginzburg, and Soergel which establish equivalences between certain derived categories of modules over a Koszul algebra and over its Koszul dual. The point of Section 6 is a technical theorem which allows us, in Section 7, to rederive the classical results from the A_{∞}-algebra results. Theorem B is proved in Section 7, also. We discuss a couple of examples in Section 8.

Part III consists of Sections 9–10. Section 9 discusses Artin–Schelter regular algebras and Frobenius algebras, from the A_{∞}-algebra point of view, and includes proofs of Corollaries D, E, and F. Section 10 gives an A_{∞}- version of the BGG correspondence; Theorem C is proved there.

Part I. Koszul duality for algebras

1. Background on A_{∞}-algebras

In this section, we describe background material necessary for the rest of the paper. There are several subsections: grading conventions and related issues; A_{∞}-algebras and morphisms between them; the bar construction; and homotopy for morphisms of A_{∞}-algebras.

1.1. Conventions. Throughout we fix a commutative base field k. Unless otherwise stated, every chain complex, vector space, or algebra will be over k. The unadorned tensor product ⊗ is over k also.

Vector spaces (and the like) under consideration in this paper are bigraded, and for any bihomogeneous element a, we write

deg a = (deg_{1}(a), deg_{2}(a)) ∈ ℤ × G

for some abelian group G. The second grading is called the Adams grading. In the classical setting G is trivial, but in this paper we have G = ℤ; many of the abstract assertions in this paper hold for any abelian group G. If V is a bigraded vector space, then the degree (i, j) component of V is denoted by V_{j}^{i}. Usually we work with bihomogeneous elements, with the possibility of ignoring the second grading. All chain complexes will have a differential of degree (1, 0). The Koszul sign convention is in force throughout the paper, but one should ignore the second grading when using it: when interchanging elements of degree (i, s) and (j, t), multiply by (−1)^{ij}.

Given a bigraded vector space V, we write V^{♯} for its graded dual. Its suspension SV is the bigraded space with (SV)_{j}^{i} = V_{j}^{i+1}: suspension shifts the first grading down by one, and ignores the second grading. Write s : V → SV for the obvious map of degree −1. If V has a differential d_{V}, then define a differential d_{SV} on SV by d_{SV} (sv) = −sd_{V} (v). The Adams shift of V is ΣV with (ΣV)_{j}^{i} = V_{j+1}^{i}. If V has a differential, then there is no sign in the differential for ΣV: d_{ΣV} is just the shift of d_{V}.

If (M, d) and (N, d) are complexes, so are Hom_{k}(M, N) and M ⊗ N(= M ⊗_{k} N), with differentials given by

d(f) = df − (−1)^{deg_{1}(f)}fd, ∀ f ∈ Hom_{k}(M, N),
d(m ⊗ n) = dm ⊗ n + (−1)^{deg_{1}(m)}m ⊗ dn, ∀ m ⊗ n ∈ M ⊗ N,

respectively.

If C is a category, we write C(X, Y) for morphisms in C from X to Y. We reserve Hom to denote the chain complex with differential as in the previous paragraph.

1.2. A_{∞}-algebras and morphisms. In this paper, we will frequently work in the category of augmented A_{∞}-algebras; in this subsection and the next, we define the objects and morphisms of this category. Keller’s paper [20] provides a nice introduction to A_{∞}-algebras; it also has references for many of the results which we cite here and in later subsections. Lefèvre-Hasegawa’s thesis [23] provides more details for a lot of this; although it has not been published, it is available online. Another reference is [24] which contains some easy examples coming from ring theory. The following definition is originally due to Stasheff [34].

Definition 1.1. An A_{∞}-algebra over k is a ℤ × ℤ-graded vector space A endowed with a family of graded k-linear maps

m_{n} : A^{⊗n} → A, n ≥ 1,

of degree (2 − n, 0) satisfying the following Stasheff identities: for all n ≥ 1,

SI(n)  ∑(−1)^{r+st}m_{u}(1^{⊗r} ⊗ m_{s} ⊗ 1^{⊗t}) = 0,

where the sum runs over all decompositions n = r + s + t, with r, t ≥ 0 and s ≥ 1, and where u = r + 1 + t. Here 1 denotes the identity map of A. Note that when these formulas are applied to elements, additional signs appear due to the Koszul sign rule.

A DG (differential graded) algebra is an A_{∞}-algebra with m_{n} = 0 for all n ≥ 3.

The reader should perhaps be warned that there are several different sign conventions in the A_{∞}-algebra literature. We have chosen to follow Keller [20], who is following Getzler and Jones [14]. Stasheff [34] and Lefèvre-Hasegawa [23] use different signs: they have the sign (−1)^{rs+t} in SI(n), and this requires sign changes in other formulas (such as MI(n) below).

As remarked above, we work with bigraded spaces throughout, and this requires a (very mild) modification of the standard definitions: ordinarily, an A_{∞}-algebra is singly graded and deg m_{n} = 2−n; in our bigraded case, we have put deg m_{n} = (2−n, 0). Thus if one wants to work in the singly graded setting, one can just work with objects concentrated in degrees (∗, 0) = ℤ × {0}.

Definition 1.2. An A_{∞}-algebra A is strictly unital if A contains an element 1 which acts as a two-sided identity with respect to m_{2}, and for n ≠ 2, m_{n}(a_{1} ⊗ ⋯ ⊗ a_{n}) = 0 if a_{i} = 1 for some i.

In this paper we assume that A_{∞}-algebras (including DG algebras) are strictly unital.

Definition 1.3. A morphism of A_{∞}-algebras f : A → B is a family of k-linear graded maps

f_{n} : A^{⊗n} → B, n ≥ 1,

of degree (1 − n, 0) satisfying the following Stasheff morphism identities: for all n ≥ 1,

MI(n) ∑(−1)^{r+st}f_{u}(1^{⊗r}⊗m_{s}⊗1^{⊗t}) = ∑(−1)^{w}m_{q}(f_{i_{1}} ⊗f_{i_{2}} ⊗⋯⊗f_{i_{q}}),

where the first sum runs over all decompositions n = r + s + t with r, t ≥ 0, s ≥ 1, and where we put u = r + 1 + t; and the second sum runs over all 1 ≤ q ≤ n and all decompositions n = i_{1} + ⋯ + i_{q} with all i_{s} ≥ 1. The sign on the right-hand side is given by

w = (q − 1)(i_{1} − 1) + (q − 2)(i_{2} − 1) + ⋯ + 2(i_{q−2} − 1) + (i_{q−1} − 1).

When A_{∞}-algebras have a strict unit (as we usually assume), an A_{∞}- morphism between them is also required to be strictly unital, which means that it must satisfy these unital morphism conditions: f_{1}(1_{A}) = 1_{B} where 1_{A} and 1_{B} are strict units of A and B respectively, and f_{n}(a_{1} ⊗⋯⊗a_{n}) = 0 if some a_{i} = 1_{A} and n ≥ 2 (see [20, 3.5], [24, Section 4]).

As with A_{∞}-algebras, we have modified the grading on morphisms: we have changed the usual grading of deg f_{n} = 1−n to deg f_{n} = (1−n, 0). The composite of two morphisms is given by a formula similar to the morphism identities MI(n); see [20] or [23] for details.

Definition 1.4. A morphism f : A → B of A_{∞}-algebras is strict if f_{n} = 0 for n ≠ 1. The identity morphism is the strict morphism f with f_{1} = 1. A morphism f is a quasi-isomorphism or an A_{∞}-isomorphism if f_{1} is a quasi-isomorphism of chain complexes.

Note that quasi-isomorphisms of A_{∞}-algebras have inverses: a morphism is a quasi-isomorphism if and only if it is a homotopy equivalence — see Theorem 1.16 below.

We write Alg for the category of associative ℤ × ℤ-graded algebras with morphisms being the usual graded algebra morphisms, and we write Alg^{∞} for the category of A_{∞}-algebras with A_{∞}-morphisms.

Let A and B be associative algebras, and view them as A_{∞}-algebras with m_{n} = 0 when n ≠ 2. We point out that there may be nonstrict A_{∞}-algebra morphisms between them. That is, the function

Alg(A, B) → Alg^{∞}(A, B),

sending an algebra map to the corresponding strict A_{∞}-morphism, need not be a bijection. See Example 2.8 for an illustration of this.

The following theorem is important and useful.

Theorem 1.5 ([17]). Let A be an A_{∞}-algebra and let HA be its cohomology ring. There is an A_{∞}-algebra structure on HA with m_{1} = 0 and m_{2} equal to its usual associative product, and with the higher multiplications constructed from the A_{∞}-structure of A, such that there is a quasi-isomorphism of A_{∞}- algebras HA → A lifting the identity of HA.

This theorem was originally proved for ℤ-graded A_{∞}-algebras, and holds true in our ℤ × ℤ-setting.

1.3. Augmented A_{∞}-algebras. A strictly unital A_{∞}-algebra A comes equipped with a strict, strictly unital morphism η : k → A.

Definition 1.6. (a) A strictly unital A_{∞}-algebra A is augmented if there

is a strictly unital A_{∞}-algebra morphism ε : A → k so that ε ∘ η = 1_{k}.

(b) If A is an augmented A_{∞}-algebra with augmentation ε : A → k, its augmentation ideal is defined to be ker(ε_{1}).

(c) A morphism of augmented A_{∞}-algebras f : A → B must be strictly unital and must respect the augmentations: ε_{A} = ε_{B} ∘ f.

We write Alg_{aug}^{∞} for the resulting category of augmented A_{∞}-algebras.

Proposition 1.7 (Section 3.5 in [20]). The functor Alg_{aug}^{∞} → Alg^{∞} sending an augmented A_{∞}-algebra to its augmentation ideal is an equivalence of categories. The quasi-inverse sends an A_{∞}-algebra A to k ⊕ A with the apparent augmentation.

Using this equivalence, one can translate results and constructions for A_{∞}-algebras to the augmented case. The bar construction is an application of this.

1.4. The bar construction. The bar construction B(−) is of central importance in this paper, since we define the Koszul dual of A to be the vector space dual of its bar construction. In this subsection, we describe it. We also discuss the cobar construction Ω(−), the composite Ω(B(−)), and other related issues.

The following definition is a slight variant on that in [20, Section 3.6].

Definition 1.8. Let A be an augmented A_{∞}-algebra and let I be its augmentation ideal. The bar construction B_{aug}^{∞}A on A is a coaugmented differential graded (DG) coalgebra defined as follows: as a coaugmented coalgebra, it is the tensor coalgebra T (SI) on SI:

T (SI) = k ⊕ SI ⊕ (SI)^{⊗2} ⊕ (SI)^{⊗3} ⊕ · · · .

As is standard, we use bars rather than tensors, and we also conceal the suspension s, writing [a_{1}| ⋯ |a_{m}] for the element sa_{1} ⊗ ⋯ ⊗ sa_{m}, where a_{i} ∈ I for each i. The degree of this element is

deg[a_{1}| ⋯ |a_{m}] = (∑(−1 + deg_{1} a_{i}), ∑ deg_{2} a_{i}) .

The differential b on B_{aug}^{∞}A is the degree (1, 0) map given as follows: its component b_{m} : (SI)^{⊗m} → T (SI) is given by

(1.9) b_{m}([a_{1}| ⋯ |a_{m}])

= ∑_{j,n}(−1)^{w_{j,n}}[a_{1}| ⋯ |a_{j}|m_{n}(a_{j+1}, ⋯, a_{j+n})|a_{j+n+1}| ⋯ |a_{m}],

where m_{n} = (−1)^{n}m_{n} and

w_{j,n} = ∑_{1≤s≤j}(−1 + deg_{1} a_{s}) + ∑_{1≤t<n}(n − t)(−1 + deg_{1} a_{j+t}).

That is, its component mapping (SI)^{⊗m} to (SI)^{⊗u} is

∑ 1^{⊗j} ⊗ (s ∘ m_{n} ∘ (s^{−1})^{⊗n}) ⊗ 1^{⊗m−j−n},

where the sum is over pairs (j, n) with m ≥ j + n, and where u = m − n + 1. See [24, p. 115] for a discussion of the sign involved in m_{n}.

If A is an augmented DG algebra, then the above bar construction is the original bar construction and it is also denoted by BA.

Note that, with this definition, the bar construction of a bigraded algebra is again bigraded.

Remark 1.10. In [20, 3.6], Keller describes the bar construction in the nonaugmented situation. Aside from grading issues, the relation between his version and ours is as follows: if we write B^{∞} for Keller’s version, then B_{aug}^{∞} is the composite

Alg_{aug}^{∞} → Alg^{∞} ⟶^{B^{∞}} DGC → DGC_{coaug},

where the first arrow is the equivalence from Proposition 1.7, and the last arrow takes a coalgebra C to k ⊕ C, with the apparent coaugmentation.

The coderivation b encodes all of the higher multiplications of A into a single operation. Keller [20, 3.6] notes that if A and A′ are augmented A_{∞}-algebras, then there is a bijection between Hom sets

(1.11)  Alg_{aug}^{∞}(A, A′) ⟷ DGC_{coaug}(B_{aug}^{∞}A, B_{aug}^{∞}A′).

(Again, he is working with nonaugmented A_{∞}-algebras, but Proposition 1.7 allows us to translate his result to this setting.)

We briefly mention the cobar construction. In full generality, this would probably take a coaugmented A_{∞}-coalgebra as input, and produce an augmented DG algebra. We have no interest in working with A_{∞}-coalgebras, though, and we do not need this generality.

Definition 1.12. Given a coaugmented DG coalgebra C with coproduct Δ and differential b_{C}, the cobar construction ΩC on C is the augmented DG algebra which as an augmented algebra is the tensor algebra T (S^{−1}J) on the desuspension of the coaugmentation coideal J = cok(k → C). It is graded by putting

deg[x_{1}| ⋯ |x_{m}] = (∑(1 + deg_{1} x_{i}), ∑ deg_{2} x_{i}) .

Its differential is the sum d = d_{0} + d_{1} of the differentials

d_{0}[x_{1}| ⋯ |x_{m}] = − ∑_{i=1}^{m}(−1)^{n_{i}}[x_{1}| ⋯ |b_{C}(x_{i})| ⋯ |x_{m}],

and

d_{1}[x_{1}| ⋯ |x_{m}] = ∑_{i=1}^{m} ∑_{j=1}^{k_{i}}(−1)^{n_{i}+deg_{1} a_{ij}}[x_{1}| ⋯ |x_{i−1}|a_{ij}|b_{ij}| ⋯ |x_{m}]

where n_{i} = ∑_{j<i}(1 + deg_{1} x_{j}) and ∑_{j=1}^{k_{i}} a_{ij} ⊗ b_{ij} = \overline{Δ}(x_{i}). Here \overline{Δ} is the induced coproduct on J.

Definition 1.13 ([23, Section 2.3.4]). If A is an augmented A_{∞}-algebra, then its enveloping algebra UA is defined to be the DG algebra UA := Ω(B_{aug}^{∞}A).

Thus the enveloping algebra of an augmented A_{∞}-algebra is an augmented DG algebra.

Proposition 1.14 ([23, 1.3.3.6 and 2.3.4.3]). There is a natural quasi-iso-morphism of A_{∞}-algebras A → UA.

The map A → UA arises as follows: between the categories of DG coalgebras and DG algebras, the bar B and cobar Ω constructions are adjoint, with Ω the left adjoint, and thus for any DG coalgebra C, there is a map C → B(ΩC). In the case where C = B_{aug}^{∞}A, we get a map

B_{aug}^{∞}A → B(Ω(B_{aug}^{∞}A)) = B_{aug}^{∞}(Ω(B_{aug}^{∞}A)).

(One can view an augmented DG algebra R as an A_{∞}-algebra with all higher multiplications equal to zero. In this situation, the A_{∞}-bar construction B_{aug}^{∞}R reduces to the standard DG algebra bar construction B(R).) The bijection (1.11) says that this corresponds to a map

A → Ω(B_{aug}^{∞}A).

This is the map in Proposition 1.14. This proposition says that every augmented A_{∞}-algebra is quasi-isomorphic to an augmented DG algebra. A similar result is also true in the nonaugmented case, although we will not need this. The quasi-isomorphism between A and Ω(BA) is also a standard result in the case when A itself is an augmented DG algebra, although the natural map goes the other way in that setting; indeed, there is a chain homotopy equivalence ΩB(A) → A which is a map of DG algebras, but its inverse need not be an algebra map. See [13, Section 19], for example. One application of Proposition 1.14 is that in the DG case, there is a quasi-inverse in the category Alg_{aug}^{∞}.

Also from [13, Section 19], we have the following result.

Lemma 1.15 ([13, Section 19]). Let R be an augmented DG algebra. Assume that R is locally finite. Then there is a natural isomorphism Ω(R^{♯}) ≅ (B_{aug}^{∞}R)^{♯} = B^{♯}R.

In light of the lemma and Remark 1.10, we point out that if R is a locally finite augmented associative algebra, then the homology of the dual of its bar construction is isomorphic to Ext_{R}^{∗}(k, k) := ⨁_{i∈ℤ} Ext_{R}^{i}(k, k). This is also true when R is an A_{∞}-algebra; see [24, Lemma 11.1] and its proof. Thus by Theorem 1.5, there is a quasi-isomorphism of A_{∞}-algebras

Ext_{R}^{∗}(k, k) → B^{♯}R.

The A_{∞}-structure on Ext_{R}^{∗}(k, k) is studied in [26].

1.5. Homotopy. Earlier, we said that we work in the category of augmented A_{∞}-algebras. We also need the homotopy category of such algebras, and so we need to discuss the notion of homotopy between A_{∞}-algebra morphisms. See [20, 3.7] and [23, 1.2.1.7] for the following.

Let A and A′ be augmented A_{∞}-algebras, and suppose that f, g : A → A′ are morphisms of augmented A_{∞}-algebras. Let F and G denote the corresponding maps B_{aug}^{∞}A → B_{aug}^{∞}A′. Write b and b′ for the differentials on B_{aug}^{∞}A and B_{aug}^{∞}A′, respectively. Then f and g are homotopic, written f ≃ g, if there is a map H : B_{aug}^{∞}A → B_{aug}^{∞}A′ of degree −1 such that

ΔH = (F ⊗ H + H ⊗ G)Δ and F − G = b′ ∘ H + H ∘ b.

One can also express this in terms of a sequence of maps h_{n} : A^{⊗n} → A′ satisfying some identities, but we will not need this formulation. See [23, 1.2.1.7] for details (but note that he uses different sign conventions).

Two A_{∞}-algebras A and A′ are homotopy equivalent if there are morphisms f : A → A′ and g : A′ → A such that f ∘ g ≃ 1_{A′} and g ∘ f ≃ 1_{A}.

We will use the following theorem.

Theorem 1.16 ([18], [29], [23, 1.3.1.3]). (a) Homotopy is an equivalence

relation on the set of morphisms of A_{∞}-algebras A → A′.

(b) An A_{∞}-algebra morphism is a quasi-isomorphism if and only if it is a homotopy equivalence.

By part (a), we can define the homotopy category HoAlg_{aug}^{∞} to be the category of augmented A_{∞}-algebras in which the morphisms are homotopy classes of maps: that is,

HoAlg_{aug}^{∞}(A, A′) := (Alg_{aug}^{∞}(A, A′)/ ≃) .

By part (b), in this homotopy category, quasi-isomorphisms are isomorphisms.

2. The Koszul dual of an A_{∞}-algebra

Let A be an augmented A_{∞}-algebra. Its A_{∞}-Koszul dual, or Koszul dual for short, is defined to be E(A) := (B_{aug}^{∞}A)^{♯}. By [24, Section 11 and Lemma 11.1], E(A) is a DG algebra model of the A_{∞}-Ext-algebra ⨁_{i∈ℤ} Ext_{A}^{i}(k_{A}, k_{A}) where k_{A} is the trivial right A_{∞}-module over A, and where, by definition, Ext_{A}^{i}(M, N) = ⨁_{j∈ℤ} D^{∞}(A)(M, S^{i}Σ^{j}(N)) for any right A_{∞}-modules M and N. See Section 3 for the definitions of A_{∞}-modules and their derived categories.

In this section, we study some of the basic properties of A_{∞}-Koszul duality, we connect it to “classical” Koszul duality, and we discuss a few simple examples. The main result is Theorem A, restated as Theorem 2.4.

2.1. Finiteness and connectedness conditions. In this subsection, we introduce some technical conditions related to finite-dimensionality and connectivity of bigraded objects.

Definition 2.1. Let A be an augmented A_{∞}-algebra and let I be its augmentation ideal. We write I_{∗}^{i} for the direct sum I_{∗}^{i} = ⨁_{j} I_{j}^{i}, and similarly I_{j}^{∗} = ⨁_{i} I_{j}^{i}. We say that A is locally finite if each bihomogeneous piece A_{j}^{i} of A is finite-dimensional. We say that A is strongly locally finite if I satisfies the following:

(1) Each bihomogeneous piece I_{j}^{i} of I is finite-dimensional (i.e., A is locally finite).

(2) Either for all j ≤ 0, I_{j}^{∗} = 0; or for all j ≥ 0, I_{j}^{∗} = 0.

(3) Either for all j, there exists an m = m(j) so that for all i > m(j), I_{j}^{i} = 0; or for all j, there exists an m′ = m′(j) so that for all i < m′(j), I_{j}^{i} = 0.

We say that A is Adams connected if, with respect to the Adams grading, A is (either positively or negatively) connected graded and locally finite. That is:

• I_{j}^{∗} is finite-dimensional for all j.

• Either for all j ≤ 0, I_{j}^{∗} = 0, or for all j ≥ 0, I_{j}^{∗} = 0. We say that a DG algebra A is weakly Adams connected if:

• The DG bar construction B(A; A) ≅ B(A) ⊗ A is locally finite.

• The only simple DG A-modules are k and its shifts.

• A is an inverse limit of a family of finite-dimensional left DG Abimodules.

Lemma 2.2. We have the following implications.

(a) Adams connected ⇒ strongly locally finite ⇒ weakly Adams connected.

(b) A weakly Adams connected ⇒ E(A) locally finite.

(c) A strongly locally finite ⇒ E(A) strongly locally finite, and hence every iterated Koszul dual of A is strongly locally finite.

(d) A Adams connected ⇒ E(A) Adams connected, and hence every iterated Koszul dual of A is Adams connected.

Proof. The first implication in part (a) is clear. For the second implication, if A is strongly locally finite, then for connectivity reasons, k and its shifts will be the only simple modules. We defer the proof that B(A; A) is locally finite until after the proof of part (c).

For the inverse limit condition, we assume that I_{j}^{∗} = 0 when j ≤ 0, and that for each j, there is an m′(j) such that I_{j}^{i} = 0 when i < m′(j). The other cases are similar. We will describe a sequence of two-sided ideals J_{n} in A, n ≥ 1, so that A/J_{n} is finite-dimensional, and A = \underleftarrow{lim} A/J_{n}. Define J_{n} to be

J_{n} = I_{≥n}^{1} ⊕ I_{≥n+m′(1)}^{2} ⊕ I_{≥n+min(2m′(1),m′(2))}^{3} ⊕ ⋯
⊕ I_{≥n+min_{σ⊢n−1}(∑_{a∈σ} m′(a))}^{s} ⊕ ⋯ ⊕ I_{≥n+min(···)}^{n}
⊕ I_{∗}^{n+1} ⊕ I_{∗}^{n+2} ⊕ · · · .

The notation “σ ⊢ n − 1” means that σ partitions n − 1. The idea here is that, for example, if J contains all of the elements in bidegrees (1, ≥ n), and since A has elements in bidegrees (1, ≥ m′(1)), then for J to be an ideal, it should contain all of the elements in bidegrees (2, ≥ n + m′(1)).

Part (b) is clear: as graded vector spaces, B(A; A) and B(A) ⊗_{k} A are isomorphic; thus if B(A; A) is locally finite, so is B(A).

(c) Since E(A) is dual to the tensor coalgebra T (SI), we focus on T (SI). If we suppose that I_{j}^{∗} = 0 when j ≤ 0, then for all j < n, we have (I^{⊗n})_{j}^{∗} = 0. Shifting I by S does not change this: ((SI)^{⊗n})_{j}^{∗} will be zero if j < n. Therefore, T (SI) satisfies condition (2) of Definition 2.1. Dualizing, we see that E(A) satisfies the other version of condition (2): if J is its augmentation ideal, then J_{j}^{∗} = 0 when j ≥ 0. Similarly, if I_{j}^{∗} = 0 when j ≥ 0, then J_{j}^{∗} = 0 when j ≤ 0.

So if I satisfies condition (2), then so does J.

Now suppose that I satisfies (1) and this version of condition (3): for each j, there is an m′(j) such that I_{j}^{i} = 0 when i < m′(j). Then for fixed n and j:

• (I^{⊗n})_{j}^{i} is zero if i is small enough.

• (I^{⊗n})_{j}^{i} is finite-dimensional for all i. Therefore T (SI) satisfies condition (3). Furthermore, since for fixed j, (I^{⊗n})_{j}^{∗} is zero for all but finitely many values of n, we see that T (SI) satisfies condition (1). Dualizing, we see that the augmentation ideal J of E(A) satisfies (1) and (3), also (although J satisfies the “other version” of (3)). This completes the proof of part (c).

We return to part (a): if A is strongly locally finite, then by part (c), so is the bar construction B(A); more precisely, A and B(A) will satisfy the same version of condition (3). Hence it is easy to verify that their tensor product will be locally finite. This completes the proof of (a).

(d) A being Adams connected is equivalent to I satisfying (1), (2), and both versions of (3): for each j, there are numbers m(j) and m′(j) so that I_{j}^{i} = 0 unless m′(j) ≤ i ≤ m(j). By the proof of (c), this implies that E(A) satisfies the same conditions. □

Remark 2.3. One may interchange the roles of i and j in the definition of strong local finiteness, but the presence of the shift S in E(A) = T (SI) makes the situation asymmetric. Suppose that I satisfies the following: (1′) Each bihomogeneous piece I_{j}^{i} of I of A is finite-dimensional. (2′) Either for all i there exists an m = m(i) so that for all j ≥ m(i),

I_{j}^{i} = 0; or for all i there exists an m′ = m′(i) so that for all j ≤ m′,

I_{j}^{i} = 0.

(3′) Either for all i ≥ 1, I_{∗}^{i} = 0; or for all i ≤ 1, I_{∗}^{i} = 0. Then by imitating the proof of part (c) of the lemma, one can show that E(A) is locally finite; however, it may not satisfy (2′).

Suppose that I satisfies (1′), (2′), and the following: (3′′) Either for all i ≥ 0, I_{∗}^{i} = 0; or for all i ≤ 1, I_{∗}^{i} = 0. Then the same proof shows that the augmentation ideal of E(A) satisfies (1′), (2′), and (3′′) as well, and hence the same holds for every iterated Koszul dual of A.

2.2. A_{∞}-Koszul duality. Here is the main theorem of this section, which is a slight generalization of [24, Theorem 11.2]. This is Theorem A from the introduction.

Theorem 2.4. Suppose that A is an augmented A_{∞}-algebra with E(A) locally finite. Then there is a natural quasi-isomorphism of A_{∞}-algebras A ≅ E(E(A)).

If A is weakly Adams connected, then essentially by definition (or see Lemma 2.2), E(A) is locally finite. Hence by Lemma 2.2, E(A) is locally finite if A is Adams connected or strongly locally finite. The summary of the theorem’s proof is that the double Koszul dual is the enveloping algebra UA of A (see Definition 1.13 and Proposition 1.14).

Proof. By definition, the double Koszul dual E(E(A)) is (B_{aug}^{∞}((B_{aug}^{∞}A)^{♯}))^{♯}. Apply Lemma 1.15 to the DG algebra E(A) = (B_{aug}^{∞}A)^{♯}, which is locally finite. Then there are natural DG algebra isomorphisms

E(E(A)) = (B_{aug}^{∞}((B_{aug}^{∞}A)^{♯}))^{♯} ≅ Ω(((B_{aug}^{∞}A)^{♯})^{♯}) ≅ Ω(B_{aug}^{∞}A).

Proposition 1.14 gives a natural A_{∞}-isomorphism A ⟶^{≅} Ω(B_{aug}^{∞}A). □

Koszul duality E(−) is a contravariant functor from A_{∞}-algebras to DG algebras, and since one can view a DG algebra as being an A_{∞}-algebra, there are functions

Alg_{aug}^{∞}(A, A′) ⟶^{E(−)} DGA_{aug}(E(A′), E(A)) ⟶ Alg_{aug}^{∞}(E(A′), E(A)),

for augmented A_{∞}-algebras A and A′.

Corollary 2.5. Suppose that A and A′ are augmented A_{∞}-algebras with E(A), E(E(A)), E(A′), and E(E(A′)) locally finite.

(a) Then E(−) gives a bijection

Alg_{aug}^{∞}(A, A′) ⟶^{∼} DGA_{aug}(E(A′), E(A)).

(b) The composite

Alg_{aug}^{∞}(A, A′) ⟶^{∼} DGA_{aug}(E(A′), E(A)) → Alg_{aug}^{∞}(E(A′), E(A))

induces a bijection

HoAlg_{aug}^{∞}(A, A′) ⟶^{∼} HoAlg_{aug}^{∞}(E(A′), E(A)).

(c) Hence every A_{∞}-algebra map f : E(A′) → E(A) is homotopic to a DG algebra map.

Proof. (a) From (1.11) we have a bijection

Alg_{aug}^{∞}(A, A′) ⟶^{∼} DGC_{coaug}(B_{aug}^{∞}A, B_{aug}^{∞}A′).

We are assuming that B_{aug}^{∞}A and B_{aug}^{∞}A′ are locally finite, so the vector space duality maps

DGC_{coaug}(B_{aug}^{∞}A, B_{aug}^{∞}A′) → DGA_{aug}((B_{aug}^{∞}A′)^{♯}, (B_{aug}^{∞}A)^{♯})
→ DGC_{coaug}(((B_{aug}^{∞}A)^{♯})^{♯}, ((B_{aug}^{∞}A′)^{♯})^{♯})

are bijections. Therefore so is

Alg_{aug}^{∞}(A, A′) ⟶^{∼} DGA_{aug}(E(A′), E(A)).

(b, c) The naturality of the quasi-isomorphism (= homotopy equivalence) in Theorem 2.4 says that the function

HoAlg_{aug}^{∞}(A, A′) ⟶^{E(E(−))} HoAlg_{aug}^{∞}(E(E(A)), E(E(A′)))
f ⟼ E(E(f))

is a bijection. That is, the composite

HoAlg_{aug}^{∞}(A, A′) → HoAlg_{aug}^{∞}(E(A′), E(A))
→ HoAlg_{aug}^{∞}(E(E(A)), E(E(A′)))

is a bijection. The first map here is induced by

Alg_{aug}^{∞}(A, A′) ⟶^{∼} DGA_{aug}(E(A′), E(A)) ⟶^{i} Alg_{aug}^{∞}(E(A′), E(A)),

and the second by

Alg_{aug}^{∞}(E(A′), E(A)) ⟶^{∼} DGA_{aug}(E(E(A)), E(E(A′)))

⟶^{j} Alg_{aug}^{∞}(E(E(A)), E(E(A′))).

Since both of the functions i and j are inclusions, the functions Ho i and Ho j must be bijections. This proves (b) and (c). □

2.3. Classical Koszul duality. Classically, a Koszul algebra is a connected graded associative algebra R which is locally finite, is generated in degree 1, has quadratic relations, and has its ith graded Ext-group Ext_{R}^{i}(k, k) concentrated in degree −i for each i; see [7, Theorem 2.10.1] or [32, Theorem 5.9(6)], for example. (In those papers, Ext_{R}^{i}(k, k) is actually required to be concentrated in degree i, but that is the result of different grading conventions.) Its (classical) Koszul dual, also denoted by R^{!}, is Ext_{R}^{∗}(k, k). One can show that if R is a Koszul algebra, then so is R^{!} — see [7, 2.9.1], for example.

A standard example is an exterior algebra R = Λ(x_{1}, …, x_{n}) on generators x_{i} each in degree 1; then its Koszul dual R^{!} is the polynomial algebra k[y_{1}, …, y_{n}], with each y_{i} in degree (1, −1).

We want all of our algebras to be bigraded, though, and we want the double Koszul dual to be isomorphic, as a bigraded algebra, to the original algebra. Thus we might grade Λ(x_{1}, …, x_{n}) by putting each x_{i} in degree (0, 1), in which case R^{!} = k[y_{1}, …, y_{n}]. The grading for R^{!} is given as follows: y_{i} is represented in the dual of the bar construction for R by the dual of [x_{i}], and since deg[x_{i}] = (−1, 1), y_{i} has degree (1, −1). The double Koszul dual is exterior on classes dual to [y_{i}] in the bar construction on R^{!}, each of which therefore has degree (0, 1).

Note that these are graded in such a way that there are no possible nonzero higher multiplications m_{n} on them. This absence of higher multiplications is typical for a Koszul algebra, as Keller [20, 3.3] and the authors [24, Section 11] point out. Conversely, if we grade our algebras in such a way that there are no possible higher multiplications, we can recover classical Koszul duality.

Definition 2.6. Fix a pair of integers (a, b) with b ≠ 0. A bigraded associative algebra A is an (a, b)-generated Koszul algebra if it satisfies these conditions:

(a) A_{0,0} = k.

(b) A is locally finite.

(c) A is generated in bidegree (a, b).

(d) The relations in A are generated in bidegree (2a, 2b).

(e) For each i, the graded vector space Ext_{A}^{i}(k, k) is concentrated in degree (i(a − 1), −ib).

In fact, conditions (c) and (d) should follow from condition (e): one should be able to imitate the proofs of [7, 2.3.1 and 2.3.2].

If A is a bigraded associative algebra, the classical Koszul dual of A, denoted by A^{!}, is defined to be HE(A) — the homology of the A_{∞}-Koszul dual E(A) = (B_{aug}^{∞}A)^{♯}. Forgetting grading issues, A^{!} is isomorphic to Ext_{A}^{∗}(k, k). In classical ring theory, we often consider the classical Koszul dual as an associative algebra — an A_{∞}-algebra with m_{n} = 0 if n ≠ 2.

Corollary 2.7. Fix a pair of integers (a, b) with b ≠ 0. If A is an (a, b)- generated Koszul algebra, then E(A) and E(E(A)) are quasi-isomorphic to associative algebras A^{!} and (A^{!})^{!}, respectively, and there is an isomorphism of bigraded algebras A ≅ (A^{!})^{!}.

This result is known [7] so we only give a sketch of proof.

Sketch of proof of Corollary 2.7. By Theorem 1.5, the A_{∞}-Koszul dual E(A) is quasi-isomorphic to A^{!} with some possible higher multiplications. We need to show, among other things, that in this case, the higher multiplications on A^{!} are zero.

Since b ≠ 0, both E(A) and E(E(A)) will be locally finite.

The bidegree (deg_{1} x, deg_{2} x) of a nonzero nonunit element x ∈ Ext_{A}^{∗}(k, k) satisfies

(∗)  \frac{deg_{1} x}{deg_{2} x} = −\frac{a − 1}{b},

and this fraction makes sense since b ≠ 0. The same is true for any tensor product of such elements. Since the higher multiplication m_{n} has degree (2 − n, 0), one can see that if n ≠ 2, the bidegree of m_{n}(x_{1} ⊗ ⋯ ⊗ x_{n}) will not satisfy (∗), and so will be zero. Thus there is no nonzero higher multiplications on A^{!} which is compatible with the bigrading. This implies that the A_{∞}-algebra E(A) is quasi-isomorphic to the associative algebra A^{!}.

Now we claim that A^{!} is (1 − a, −b)-generated Koszul. There is an obvious equivalence of categories between ℤ-graded algebras and ℤ × ℤ-graded algebras concentrated in degrees (na, nb) for n ∈ ℤ; under that equivalence, (a, b)-generated Koszul algebras correspond to Koszul algebras in the sense of [7]. Koszul duality takes ℤ-graded algebras generated in degree 1 to ℤ × ℤ-graded algebras generated in degree (1, −1). It takes ℤ × ℤ-graded algebras generated in degree (a, b) to ℤ × ℤ-graded algebras generated in degree (1 − a, −b). The proof of [7, 2.9.1] carries over to show that since A is (a, b)-generated Koszul, its dual A^{!} is (1 − a, −b)-generated Koszul.

Since A^{!} is Koszul, its Koszul dual (A^{!})^{!} is associative (or there is no nonzero higher multiplications on (A^{!})^{!}, by the first part of the proof). A similar grading argument shows that any morphism f : A → (A^{!})^{!} of A_{∞}- algebras must be strict; thus the isomorphism of A_{∞}-algebras A → (A^{!})^{!} is just an isomorphism of associative algebras. □

2.4. Examples: exterior and polynomial algebras. In this subsection, we consider some simple examples involving exterior algebras and polynomial algebras. The first example shows that in the classical setting, it is crucial that a Koszul algebra be generated in a single degree.

Example 2.8. Assume that the ground field k has characteristic 2, and consider the exterior algebra Λ = Λ(x_{1}, x_{2}) with deg x_{i} = (0, i): this is not a classical Koszul algebra, nor is it an (a, b)-generated Koszul algebra, since there are generators in multiple degrees. The same goes for Λ⊗Λ. The Extalgebra for Λ is the polynomial algebra Λ^{!} = k[y_{1}, y_{2}] with deg y_{i} = (1, −i). Similarly, the Ext-algebra for Λ ⊗ Λ is isomorphic to Λ^{!} ⊗ Λ^{!}. Although it is true that (Λ^{!})^{!} ≅ Λ, there are naturality problems. In particular, the map

Alg(Λ, Λ ⊗ Λ) → Alg(Λ^{!} ⊗ Λ^{!}, Λ^{!})

is not injective: one can show that the following two maps induce the same map on Ext:

f : x_{1} ↦ x_{1} ⊗ 1 + 1 ⊗ x_{1},  g : x_{1} ↦ x_{1} ⊗ 1 + 1 ⊗ x_{1},
f : x_{2} ↦ x_{2} ⊗ 1 + 1 ⊗ x_{2},  g : x_{2} ↦ x_{2} ⊗ 1 + 1 ⊗ x_{2} + x_{1} ⊗ x_{1}.

(Indeed, any coassociative coproduct on Λ induces the Yoneda product on Ext_{Λ}^{∗}(k, k).) This shows the importance of the requirement that Koszul algebras be generated in a single degree.

Now, Theorem 2.4 and Corollary 2.5 apply here. The A_{∞}-version of the Koszul dual of Λ is quasi-isomorphic to Λ^{!}: E(Λ) ≅ Λ^{!} = k[y_{1}, y_{2}], where the A_{∞}-structure on Λ^{!} is given by m_{n} = 0 when n ≠ 2. Corollary 2.5 says that there is a bijection

HoAlg_{aug}^{∞}(Λ, Λ ⊗ Λ) ⟶^{∼} HoAlg_{aug}^{∞}(Λ^{!} ⊗ Λ^{!}, Λ^{!}).

This fixes the flaw above; the two (strict) maps f and g correspond to A_{∞}- algebra morphisms E(f) and E(g), and while E(f)_{1} = E(g)_{1}, the morphisms must differ in some higher component. (Even though the algebras involved here have A_{∞}-structures with zero higher multiplications, there are nonstrict A_{∞}-algebra morphisms between them. Also, k[y_{1}, y_{2}] is quasi-isomorphic, not equal, to the A_{∞}-Koszul dual of Λ, so Corollary 2.5(c), which says that every nonstrict map on Koszul duals is homotopic to a strict one, does not apply here.)

Example 2.9. Let A = Λ(x) with deg x = (a, 0); Corollary 2.7 does not apply in this case. The Koszul dual is E(A) = k[y] with deg y = (1 − a, 0) and with m_{n} = 0 for n ≠ 2. Assume that a ≠ 0, 1; then E(A) is locally finite, as is E(E(A)) by Remark 2.3. Thus Corollary 2.5 says that there is a bijection

HoAlg_{aug}^{∞}(Λ(x), Λ(x)) ⟶^{∼} HoAlg_{aug}^{∞}(k[y], k[y]).

For degree reasons, every map Λ(x) → Λ(x) must be strict, so each map is given by the image of x: x ⟼ cx for any scalar c ∈ k.

On the other hand, degree reasons do not rule out nonstrict maps k[y] → k[y]. Strict maps will correspond to those from Λ(x) to itself, with the map given by the scalar c corresponding to the map y ↦ cy. Thus, as pointed out in Corollary 2.5(c), if there are any nonstrict maps, then they are homotopic to strict ones. In particular, one can see that if two A_{∞}-algebra maps f, g : k[y] → k[y] are homotopic, then f_{1} = g_{1}. So if f = (f_{1}, f_{2}, …) is such a map, then it will be homotopic to the strict map (f_{1}, 0, 0, …). None of this is immediately clear from the morphism and homotopy identities, so Koszul duality, in the form of Corollary 2.5, gives some insight into A_{∞}-maps from k[y] to itself.

Example 2.10. Now consider A = Λ(x) with deg x = (1, 0). As in the previous example, every map Λ(x) → Λ(x) must be strict. In this case, B_{aug}^{∞}A is the vector space spanned by the classes [x| ⋯ |x], all of which are in bidegree (0, 0). Since B_{aug}^{∞}A is not locally finite, Theorem 2.4 does not apply, and the Koszul dual E(A) ends up being the power series ring k[[y]] instead of the polynomial ring k[y]. Consider the composite

Alg_{aug}^{∞}(A, A) ⟶^{∼} DGC_{coaug}(B_{aug}^{∞}A, B_{aug}^{∞}A) → DGA^{aug}(E(A), E(A)).

The first map is a bijection by (1.11), but the second map is not, essentially since the map is given by vector space duality and the vector spaces involved are not finite-dimensional. Since strict A_{∞}-maps are homotopic if and only if they are equal, we get a proper inclusion

HoAlg_{aug}^{∞}(A, A) ↪^{≠} HoAlg_{aug}^{∞}(E(A), E(A)).

Thus Corollary 2.5 fails here.

Example 2.11. We also mention the case when A = Λ(x) with deg x = (a, b) with b ≠ 0. In this case, E(A) = k[y] with deg y = (1 − a, −b). Also, A_{∞}-morphisms Λ(x) → Λ(x) and k[y] → k[y] must be strict, and it is easy to show that the strict maps are in bijection, as Corollary 2.5 says they should be. Indeed in this case, classical Koszul duality (Corollary 2.7) applies, since Λ(x) is (a, b)-generated Koszul with b ≠ 0.

Part II. Koszul duality for modules

3. Background on A_{∞}-modules

Koszul duality relates not just to algebras, but also to modules over them. In this section, we briefly review the relevant categories of modules over an A_{∞}-algebra. See Keller [20, 4.2] for a few more details, keeping in mind that since he is not working in the augmented setting, a little translation is required, especially in regards to the bar construction. The paper [24, Section 6] also has some relevant information, as does Lefèvre-Hasegawa’s thesis [23].

There are several subsections here: the definition of A_{∞}-module; the bar construction; derived categories; and several sections about “opposites.”

3.1. A_{∞}-modules. Let A be an A_{∞}-algebra. A bigraded vector space M is a right A_{∞}-module over A if there are graded maps

m_{n} : M ⊗ A^{⊗n−1} → M, n ≥ 1,

of degree (2 − n, 0) satisfying the Stasheff identities SI(n), interpreted appropriately. Similarly, a bigraded vector space N is a left A_{∞}-module over A if there are graded maps

m_{n} : A^{⊗n−1} ⊗ N → N, n ≥ 1,

of degree (2 − n, 0) satisfying the Stasheff identities SI(n), interpreted appropriately.

Morphisms of right A_{∞}-modules are defined in a similar way: a morphism f : M → M′ of right A_{∞}-modules over A is a sequence of graded maps

f_{n} : M ⊗ A^{⊗n−1} → M′, n ≥ 1,

of degree (1 − n, 0) satisfying the Stasheff morphism identities MI(n). Morphisms of left A_{∞}-modules are defined analogously, and so are homotopies in both the right and left module settings.

Now suppose that A is an augmented A_{∞}-algebra. A right A_{∞}-module M over A is strictly unital if for all x ∈ M and for all a_{i} ∈ A, m_{2}(x ⊗ 1) = x and

m_{n}(x ⊗ a_{2} ⊗ ⋯ ⊗ a_{n}) = 0, for all n ≥ 3

if a_{i} = 1 for some i. A morphism f of such is strictly unital if for all n ≥ 2, we have f_{n}(x ⊗ a_{2} ⊗ ⋯ ⊗ a_{n}) = 0 if a_{i} = 1 for some i.

Given an augmented A_{∞}-algebra A, let Mod^{∞}(A) denote the category of strictly unital right A_{∞}-modules with strictly unital morphisms over A.

Suppose A is an augmented A_{∞}-algebra. The morphism ε : A → k makes the vector space k into a left A_{∞}-module over A. It is called the trivial left A_{∞}-module over A and is denoted by _{A}k. The trivial right A_{∞}-module over A is defined similarly, and is denoted by k_{A}.

3.2. The bar construction for modules. The bar construction is as useful for A_{∞}-modules as it is for A_{∞}-algebras: recall that the bar construction on A is B_{aug}^{∞}A = T (SI). A strictly unital right A_{∞}-module structure on a bigraded vector space M gives a comodule differential on the right B_{aug}^{∞}Acomodule

B_{aug}^{∞}(M; A) := SM ⊗ T (SI),

as in Definition 1.8. Also, morphisms of right modules M → M′ are in bijection with morphisms of right DG comodules B_{aug}^{∞}(M; A) → B_{aug}^{∞}(M′; A) as in (1.11), and the notion of homotopy translates as well.

Similarly, one has a bar construction for left A_{∞}-modules, defined by

B_{aug}^{∞}(A; N) := T (SI) ⊗ SN,

with the same formula for the differential.

3.3. Derived categories. Let D^{∞}(A) be the derived category associated to the module category Mod^{∞}(A). By [20, 4.2] (see also [23, 2.4.2]), this derived category is the same as the homotopy category of Mod^{∞}(A): in the homotopy category for A_{∞}-modules, quasi-isomorphisms have already been inverted. This statement is the module version of Theorem 1.16.

Given a DG algebra R, write Mod R for the category of unital DG right R-modules, and write D(R) for its derived category. A good reference for the derived category D(R) is [19]; also see [21, 22]. See [23, 2.4.3 and 4.1.3] and [24, 7.2 and 7.3] for the following.

Proposition 3.1. (a) Suppose that A and B are augmented A_{∞}-algebras. If f : A → B is an A_{∞}-isomorphism, then the induced functor

f^{∗} : D^{∞}(B) → D^{∞}(A)

is a triangulated equivalence.

(b) Suppose that R is an augmented DG algebra. Then the inclusion Mod R ↪ Mod^{∞}(R) induces a triangulated equivalence

D(R) → D^{∞}(R).

(c) Hence if A is an augmented A_{∞}-algebra and R is an augmented DG algebra with an A_{∞}-isomorphism A → R, there is a triangulated equivalence F : D(R) → D^{∞}(A). Under this equivalence, F (k_{R}) ≅ k_{A} and F (R) ≅ A.

Hence in the category D^{∞}(A) one can perform many of the usual constructions, by first working in the derived category D(UA) of its enveloping algebra and then applying the equivalence of categories D(UA) → D^{∞}(A). See Section 5 for an application of this idea. We note that the Adams shift is an automorphism of D^{∞}(A).

3.4. The opposite of a DG algebra. If A = (A, m_{1}, m_{2}) is a DG algebra with differential m_{1} and multiplication m_{2}, we define the opposite algebra of A to be (A^{op}, m_{1}^{op}, m_{2}^{op}), where A^{op} = A, m_{1}^{op} = −m_{1}, and

m_{2}^{op}(a ⊗ b) = (−1)^{(deg a)(deg b)}m_{2}(b ⊗ a).

That is, m_{2}^{op} = m_{2} ∘ τ, where τ is the twist function, which interchanges tensor factors at the expense of the appropriate Koszul sign. One can verify that this is a DG algebra: m_{1}^{op} and m_{2}^{op} satisfy a Leibniz formula. (Choosing m_{1}^{op} = m_{1} also works, but is not compatible with the bar construction: see below.)

If f : A → A′ is a map of DG algebras, define f^{op} : A^{op} → (A′)^{op} by f^{op} = f. Then f^{op} is also a DG algebra map, so op defines an automorphism of the category of DG algebras, and it is clearly its own inverse.

Dually, given a DG coalgebra C = (C, d, Δ), we define its opposite coalgebra to be (C^{op}, d^{op}, Δ^{op}), where C^{op} = C, d^{op} = −d, and Δ^{op} = τ ∘ Δ. With these definitions, for any DG algebra A there is an isomorphism

B(A^{op}) ⟶^{Φ} B(A)^{op},
[a_{1}| … |a_{m}] ⟼ (−1)^{∑_{i<j}(−1+deg a_{i})(−1+deg a_{j})}[a_{m}| … |a_{1}],

of DG coalgebras. Note that, had we defined the differential m_{1}^{op} in A^{op} by m_{1}^{op} = m_{1}, this map Φ would not be compatible with the differentials, and so would not be a map of DG coalgebras.

If F : (C, d, Δ) → (C′, d′, Δ′) is a morphism of DG coalgebras, then define F^{op} to be equal to F. One can check that F^{op} : C^{op} → (C′)^{op} is also a map of DG coalgebras, making op into a functor op : DGC → DGC. As above, it is an automorphism which is its own inverse.

The definition of homotopy in Subsection 1.5 works in general for DG coalgebra morphisms: if F and G are DG coalgebra morphisms C → C′, then a homotopy from F to G is a map H : C → C′ of degree −1 such that

ΔH = (F ⊗ H + H ⊗ G)Δ, F − G = d′ ∘ H + H ∘ d.

We write H = H(F → G) to indicate the “direction” of the homotopy. One can check that, in this situation, the map H also defines a homotopy from G^{op} to F^{op}, as maps C^{op} → (C′)^{op}. Therefore, we may define H^{op}(G^{op} → F^{op}) to be H(F → G). As a consequence, op induces an automorphism on the homotopy category of DG coalgebras.

3.5. The opposite of an A_{∞}-algebra. Now we define a functor

op : Alg^{∞} → Alg^{∞}

which generalizes the opposite functor on the category of DG algebras. Given an A_{∞}-algebra (A, m_{1}, m_{2}, m_{3}, …), define (A^{op}, m_{1}^{op}, m_{2}^{op}, m_{3}^{op}, …) as follows: as a bigraded vector space, A^{op} is the same as A. The map m_{n}^{op} : (A^{op})^{⊗n} → A^{op} is defined by m_{n}^{op} = (−1)^{ε(n)}m_{n} ∘ (twist), where “twist” is the map which reverses the factors in a tensor product, with the appropriate Koszul sign, and

ε(n) = \begin{cases}1, & if n ≡ 0, 1 (mod 4), \\ 0, & if n ≡ 2, 3 (mod 4). \end{cases}

Equivalently, since only the parity of ε(n) is important, ε(n) = \binom{n+2}{2}, or ε(n) = \binom{n}{2} + 1. Thus when applied to elements,

m_{n}^{op}(a_{1} ⊗ ⋯ ⊗ a_{n}) = (−1)^{ε(n)+∑_{i<j}(deg a_{i})(deg a_{j})}m_{n}(a_{n} ⊗ ⋯ ⊗ a_{1}).

Lemma 3.2. The function ε satisfies the following additivity formula: for any q ≥ 1 and any i_{s} ≥ 1, s = 1, 2, …, q,

∑_{1≤s≤q} ε(i_{s})+ε ( ∑_{1≤s≤q} i_{s} − q + 1)+ ∑_{1≤s<t≤q}(i_{s} −1)(i_{t} −1) ≡ q +1 (mod 2).

Proof. The q = 1 case is trivial, the q = 2 case may be established by (for example) considering the different congruence classes of i_{1} mod 4, and for larger q, one can use a simple induction argument. □

Lemma 3.3. (A^{op}, m_{1}^{op}, m_{2}^{op}, m_{3}^{op}, …) is an A_{∞}-algebra.

Proof. We need to check that (A^{op}, m_{1}^{op}, m_{2}^{op}, m_{3}^{op}, …) satisfies the Stasheff identities. This is a tedious, but straightforward, verification, which we leave to the reader. The q = 2 case of Lemma 3.2 is useful. □

We also need to specify what happens to morphisms. Given a morphism f : A → B, we define f^{op} : A^{op} → B^{op} by defining

f_{n}^{op} : (A^{op})^{⊗n} → B^{op}

to be f_{n}^{op} = (−1)^{1+ε(n)}f_{n} ∘ (twist); that is,

f_{n}^{op}(a_{1} ⊗ ⋯ ⊗ a_{n}) = (−1)^{1+ε(n)+∑_{i<j} deg a_{i} deg a_{j}}f_{n}(a_{n} ⊗ ⋯ ⊗ a_{1}).

Lemma 3.4. The family f^{op} = (f_{n}^{op}) is a morphism of A_{∞}-algebras.

Proof. This is another tedious verification. Lemma 3.2 is used here. □

To complete this circle of ideas, we should consider the bar construction. That is, consider the following diagram of functors:

\xymatrix{Alg^{∞} \ar[d]^{B(−)} \ar[rr]^{op}_{∼} &  & Alg^{∞} \ar[d]^{B(−)} \\ DGC \ar[r]^{op}_{∼} & DGC \ar@{.>}[r]^{∼?} & DGC}

The horizontal arrows are equivalences of categories. The vertical arrows are fully faithful embeddings. Starting with an A_{∞}-algebra A in the upper left corner, mapping down and then to the right gives B(A)^{op}, while mapping to the right and then down gives B(A^{op}). It would be nice if these two DG coalgebras agreed, and indeed they do. The proof of the following is left to the reader.

Lemma 3.5. For any A_{∞}-algebra A, the map

Φ : B(A^{op}) ⟶ B(A)^{op},
[a_{1}| … |a_{m}] ⟼ (−1)^{∑_{i<j}(−1+deg a_{i})(−1+deg a_{j})}[a_{m}| … |a_{1}]

is an isomorphism of DG coalgebras.

Note that B(A)^{op} is the opposite coalgebra to B(A), as defined in Subsection 3.4.

This result gives us a second way to prove Lemma 3.4, that the definition f_{n}^{op} = (−1)^{1+ε(n)}f_{n} ∘ (twist) defines a morphism of A_{∞}-algebras: one just has to check that if the A_{∞}-algebra morphism f : A → A′ corresponds to the DG coalgebra morphism B(f) : B(A) → B(A′), then the composite

B(A)^{op} ⟶_{≅}^{Φ^{−1}} B(A^{op}) ⟶^{B(f^{op})} B((A′)^{op}) ⟶_{≅}^{Φ} B(A′)^{op}

is equal to B(f)^{op} = B(f). This is straightforward.

Once we know that the bar construction works well with opposites, we can define the opposite of a homotopy between A_{∞}-algebra maps in terms of the bar construction. Thus op defines an automorphism on the homotopy category of A_{∞}-algebras.

3.6. The opposite of an A_{∞}-module. Since modules over an A_{∞}-alge-bra are defined using exactly the same identities SI(n) as for A_{∞}-algebras, and since morphisms between modules satisfy only slight variants on the identities MI(n), essentially the same proofs show that the opposite of a right A-module is a left A^{op}-module, etc. That is, there are equivalences of categories

(left A_{∞}-modules over A) ⟶^{op} Mod^{∞}(A^{op}),
D^{∞}(left A_{∞}-modules over A) ⟶^{op} D^{∞}(A^{op}).

So whenever left A_{∞}-modules arise, we may easily convert them to right A_{∞}-modules, and vice versa.

4. Adjunctions and equivalences

This section lays more groundwork: generalities for establishing equivalences between categories via Auslander and Bass classes, results about derived functors for DG modules, and ⊗-Hom adjointness. Two of the main results of the section are Propositions 4.10 and 4.11, which describe when certain subcategories of derived categories of DG-modules are equivalent.

4.1. Auslander and Bass classes. Let C and D be two categories. Let F : C → D be left adjoint to a functor G : D → C. Then there are natural transformations

η : 1_{C} → GF,
ε : F G → 1_{D}.

We define two full subcategories as follows. The Auslander class associated to (F, G) is the subcategory of C whose objects are

{M | η_{M} : M → GF (M) is an isomorphism}.

The Auslander class is denoted by 𝔸. The Bass class associated to (F, G) is the subcategory of D whose objects are

{N | ε_{N} : F G(N) → N is an isomorphism}.

The Bass class is denoted by 𝔹. These definitions are abstractions of ideas of Avramov and Foxby [2, Section 3]. The following lemma is proved by imitating [2, Theorem 3.2].

Lemma 4.1. Let (F, G) be a pair of adjoint functors between C and D.

(a) The functors F and G restrict to an equivalence of categories between 𝔸 and 𝔹.

(b) If C and D are additive and F and G are additive functors, then 𝔸 and 𝔹 are additive subcategories. If C and D are triangulated and F and G are triangulated functors, then 𝔸 and 𝔹 are triangulated subcategories.

4.2. Derived functors over a DG algebra. The derived category and derived functors over a DG algebra are well-understood constructions nowadays. See [33], [19], and [13], for example. We review some details in this subsection. As with A_{∞}-algebras and modules, every DG module in this paper is ℤ × ℤ-graded.

Let R be a DG algebra and let M be a DG R-module. Then M is called acyclic if HM = 0; it is called free if it is isomorphic to a direct sum of shifts of R; and it is called semifree if there is a sequence of DG submodules

0 = M_{−1} ⊂ M_{0} ⊂ ⋯ ⊂ M_{n} ⊂ ⋯

such that M = ⋃_{n} M_{n} and that each M_{n}/M_{n−1} is free on a basis of cycles. Semifree modules are a replacement for free complexes over an associative algebra.

Notation. If R is a DG algebra and M and N are DG R-modules, we write Hom_{R}(M, N) for the DG k-module whose degree n elements are degree n R-module maps M → N, ignoring the differential; see Subsection 1.1 for the formula for the differential in Hom_{R}(M, N). Similarly, End_{R}(M) means the complex Hom_{R}(M, M).

In DG homological algebra, K-projective and K-injective DG modules are used to define derived functors. A DG R-module M is called K-projective if the functor Hom_{R}(M, −) preserves quasi-isomorphisms, or equivalently, Hom_{R}(M, −) maps acyclic DG R-modules to acyclic DG k-modules. For example, a semifree DG R-module is always K-projective. A DG R-module M is called K-flat if the functor M ⊗_{R} − preserves quasi-isomorphisms; every K-projective DG R-module is K-flat. A DG R-module N is called Kinjective if the functor Hom_{R}(−, N) preserves quasi-isomorphisms, or equivalently, Hom_{R}(−, N) maps acyclic DG R-modules to acyclic DG k-modules.

Given a DG R-module M, a map f : L → M is called a semifree (or Kprojective or K-flat, respectively) resolution of M if f : L → M is a quasiisomorphism and L is semifree (or K-projective or K-flat, respectively).

Similarly, a K-injective resolution of M is a quasi-isomorphism M → L where L is K-injective. In all of these cases, we will also abuse notation slightly and refer to L itself as the resolution, omitting mention of the map f.

The right derived functor of Hom_{R}(−, −) is RHom_{R}(−, −), defined by

RHom_{R}(M, N) := Hom_{R}(P, N) or RHom_{R}(M, N) := Hom_{R}(M, I)

where P is a K-projective resolution of M and I is a K-injective resolution of N. The left derived functor of − ⊗_{R} − is − ⊗_{R}^{L} −, defined by

M ⊗_{R}^{L} N := M ⊗_{R} Q or M ⊗_{R}^{L} N := S ⊗_{R} N

where S is a K-flat resolution of M and Q is a K-flat resolution of N.

4.3. Tensor-Hom and Hom-Hom adjunctions. We discuss ⊗-Hom and Hom-Hom adjointness, both basic and derived. These are well-known, at least in the case of modules over an associative algebra. The DG case may not be as familiar, so we provide some details. Here is the basic version.

Lemma 4.2. Let A, B, C, and D be DG algebras.

(a) [Hom-Hom adjointness] Let _{A}L_{C}, _{D}M_{B} and _{A}N_{B} be DG bimodules. Then

Hom_{A^{op}}(_{A}L_{C}, Hom_{B}(_{D}M_{B},_{A}N_{B})) ≅ Hom_{B}(_{D}M_{B}, Hom_{A^{op}}(_{A}L_{C},_{A}N_{B}))

as DG (C, D)-bimodules.

(b) [⊗-Hom adjointness] Let _{D}L_{B}, _{B}M_{A} and _{C}N_{A} be DG bimodules. Then

Hom_{A}(_{D}L ⊗_{B} M_{A},_{C}N_{A}) ≅ Hom_{B}(_{D}L_{B}, Hom_{A}(_{B}M_{A},_{C}N_{A}))

as DG (C, D)-bimodules.

The isomorphism in part (a) gives a pair of adjoint functors

A-Mod ⇄ (Mod B)^{op},

namely Hom_{A^{op}}(−,_{A}N_{B}) (left adjoint) and Hom_{B}(−,_{A}N_{B}) (right adjoint). This explains the label, “Hom-Hom adjointness.”

Proof. (a) The desired isomorphism

ϕ : Hom_{A^{op}}(_{A}L_{C}, Hom_{B}(_{D}M_{B},_{A}N_{B})) → Hom_{B}(_{D}M_{B}, Hom_{A^{op}}(_{A}L_{C},_{A}N_{B}))

is defined by the following rule. Let f ∈ Hom_{A^{op}}(_{A}L_{C}, Hom_{B}(_{D}M_{B},_{A}N_{B})), and l ∈ L and m ∈ M, write f(l) ∈ Hom_{B}(_{D}M_{B},_{A}N_{B}) and f(l)(m) ∈ N; then ϕ(f) : M → Hom_{A^{op}}(_{A}L_{C},_{A}N_{B}) is determined by

ϕ(f)(m)(l) = (−1)^{|l||m|}f(l)(m)

for all l ∈ L, m ∈ M. It is straightforward to check that ϕ is an isomorphism of DG bimodules.

The above construction is given in the unpublished manuscript [3, (3.4.2), p. 27]. The isomorphism ϕ is called the swap isomorphism [3, Sect. 3.4].

(b) This is standard and a proof is given in [3, (3.4.3), p. 28]. A non-DG version is in [31, Theorem 2.11, p. 37]. □

To get derived versions of these, we need information about bimodules, semifree resolutions, K-projectives, etc.

Lemma 4.3. Let A and B be DG algebras. Let M and L be DG (B, A)- bimodules, or equivalently, DG B^{op} ⊗ A-modules.

(a) Let N be a DG A-module. If there is a sequence of DG submodules

0 = N_{−1} ⊂ N_{0} ⊂ ⋯ ⊂ N_{n} ⊂ ⋯

such that N = ⋃_{n} N_{n}, C_{n} := N_{n}/N_{n−1} is K-projective, and the underlying graded module C_{n} is projective, then N is K-projective.

(b) If M is semifree (B, A)-bimodule, then it is K-projective over A. As a consequence, if M → L is a semifree resolution of L, then restricted to the right-hand side, it is a K-projective resolution of L_{A}.

(c) If M is K-injective (B, A)-bimodule, then it is K-injective over A. As a consequence, if L → M is a K-injective resolution of L, then restricted to the right-hand side, it is a K-injective resolution of L_{A}.

Proof. (a) First consider the sequence

0 → N_{0} → N_{1} → C_{1} → 0

of DG A-modules. This is a split (hence exact) sequence after omitting the differentials, as the underlying graded module C_{1} is projective. Let X be an acyclic DG A-module. Then we have an exact sequence

0 → Hom_{A}(C_{1}, X) → Hom_{A}(N_{1}, X) → Hom_{A}(N_{0}, X) → 0.

If the two ends Hom_{A}(C_{1}, X) and Hom_{A}(N_{0}, X) are acyclic, so is the middle term Hom_{A}(N_{1}, X). This shows that if C_{0}(= N_{0}) and C_{1} are K-projective, so is N_{1}. By induction on n we see that N_{n} is K-projective and projective for all n. Since every sequence

0 → N_{n−1} → N_{n} → C_{n} → 0

splits, the map Hom_{A}(N_{n}, X) → Hom_{A}(N_{n−1}, X) is surjective. This means that the inverse system {Hom_{A}(N_{n}, X)}_{n} satisfies Mittag-Leffler condition. Since each Hom_{A}(N_{n}, X) is acyclic,

Hom_{A}(N, X) = \underleftarrow{lim} Hom_{A}(N_{n}, X)

is acyclic by [37, Theorem 3.5.8].

(b) The second assertion follows from the first one.

By part (a) and the definition of a semifree module, we may assume M is free. Since a free module is a direct sum of shifts of B^{op} ⊗ A, we may assume M is a copy of B^{op} ⊗ A. Therefore M is free over A, and hence K-projective over A.

(c) The second assertion follows from the first one.

Let N_{A} be an acyclic DG A-module. By ⊗-Hom adjointness (Lemma 4.2(b)), we have

Hom_{A}(N,_{B}M_{A}) = Hom_{A}(N, Hom_{B^{op}⊗A}(B^{op} ⊗ A,_{B}M_{A}))
≅ Hom_{B^{op}⊗A}(N ⊗_{A} (B^{op} ⊗ A),_{B}M_{A})
≅ Hom_{B^{op}⊗A}(N ⊗ B^{op},_{B}M_{A}).

Since N is acyclic, so is N ⊗ B^{op}. Since _{B}M_{A} is K-injective, the above formula implies that Hom_{A}(N, M_{A}) is acyclic. Hence M_{A} is K-injective. □

We can combine the previous two lemmas to get derived Hom-Hom and ⊗-Hom adjointness.

Lemma 4.4 (Derived Hom-Hom adjointness). Let A, B, C, and D be DG algebras. Let _{A}L_{B}, _{B}M_{C} and _{A}N_{D} be DG bimodules.

(a) There is an isomorphism of complexes

RHom_{A^{op}}(_{A}L_{C}, RHom_{B}(_{D}M_{B},_{A}N_{B})) ≅ RHom_{B}(_{D}M_{B}, RHom_{A^{op}}(_{A}L_{C},_{A}N_{B}))

in D(C^{op} ⊗ D).

(b) There is an isomorphism of k-vector spaces

D(A^{op})(_{A}L, RHom_{B}(M_{B},_{A}N_{B})) ≅ D(B)(M_{B}, RHom_{A^{op}}(_{A}L,_{A}N_{B})).

Proof. (a) This follows from Lemmas 4.2 and 4.3, and by taking semifree resolutions of the DG bimodules M and L.

(b) This follows from (a) by taking H^{0}. □

Lemma 4.5 (Derived ⊗-Hom adjointness). Let A, B, C and D be DG algebras. Let _{D}L_{B}, _{B}M_{A} and _{C}N_{A} be DG bimodules.

(a) There is an isomorphism of complexes

RHom_{A}(_{D}L ⊗_{B}^{L} M_{A},_{C}N_{A}) ≅ RHom_{B}(_{D}L_{B}, RHom_{A}(_{B}M_{A},_{C}N_{A}))

in D(C^{op} ⊗ D).

(b) There is an isomorphism of k-vector spaces

D(A)(L ⊗_{B}^{L} M_{A}, N_{A}) ≅ D(B)(L_{B}, RHom_{A}(_{B}M_{A}, N_{A})).

Proof. (a) This follows from Lemmas 4.2 and 4.3, and by taking semifree resolutions of the DG bimodules M and L and a K-injective resolution of the bimodule N.

(b) This follows from (a) by taking H^{0}. □

4.4. Balanced bimodules and equivalences. In this subsection we prove Propositions 4.10 and 4.11, the main results in this section; these establish a framework for proving the derived equivalences in Section 5.

Let A and E be two DG algebras. A DG A-module M is a DG (E, A)- bimodule if and only if there is a map of DG algebras E → Hom_{A}(M, M).

Definition 4.6. Let B be a DG (E, A)-bimodule. We call B left balanced if there is a quasi-isomorphism B → N of DG (E, A)-bimodules such that N is K-injective over E^{op} ⊗ A and the canonical map E → Hom_{A}(N, N) is a quasi-isomorphism of DG algebras. The right balanced property is defined in a similar way, in terms of the map A^{op} → Hom_{E^{op}}(N, N).

Lemma 4.7. Let B be a DG (E, A)-bimodule. Then the following conditions are equivalent:

(i) B is left balanced. (ii) If P → B is a quasi-isomorphism of DG (E, A)-bimodules with P_{A}

being K-projective, then the canonical map E → Hom_{A}(P, P) is a quasi-isomorphism of DG algebras.

(iii) If B → I is a quasi-isomorphism of DG (E, A)-bimodules with I_{A} being K-injective, then the canonical map E → Hom_{A}(I, I) is a quasiisomorphism of DG algebras.

Proof. (i) ⇔ (ii). Suppose that there is a quasi-isomorphism B → N where N is K-injective over E^{op} ⊗ A. (By a result of Spaltenstein [33, Corollary 3.9], there always is such a map.) By Lemma 4.3(c), N is K-injective over A. The quasi-isomorphism f : P → B → N induces two maps

E ⟶^{i_{N}} Hom_{A}(N, N) ⟶^{f^{∗}} Hom_{A}(P, N)

and

E ⟶^{i_{P}} Hom_{A}(P, P) ⟶^{f_{∗}} Hom_{A}(P, N)

of (E, E)-bimodules. Since N is K-injective over A and P is K-projective over A, both g and h are quasi-isomorphisms. It is easy to see that f^{∗}i_{N} = f_{∗}i_{P}: they both map e ∈ E to ef = fe ∈ Hom_{A}(P, N). Therefore i_{N} is a quasi-isomorphism if and only if i_{P} is.

(ii) ⇔ (iii). This proof is similar. □

By the above lemma, we can construct plenty of left balanced bimodules. For example, let M be a K-injective (or K-projective) DG A-module and let E = Hom_{A}(M, M). It follows from the lemma that M becomes a left balanced DG (E, A)-bimodule with its natural left E-module structure.

We now recall a few definitions.

Definition 4.8. An object M in an additive category C with infinite direct sums is called small if C(M, −) commutes with arbitrary direct sums.

Let A be an A_{∞}-algebra and M be a right A_{∞}-module over A. Let triang_{A}^{∞}(M) denote the triangulated subcategory of D^{∞}(A) generated by M and its Adams shifts. (Recall that every subcategory in this paper is full.) Let thick_{A}^{∞}(M) be the thick subcategory generated by triang_{A}^{∞}(M): the smallest triangulated subcategory, closed under summands, which contains triang_{A}^{∞}(M). Similarly, if R is a DG algebra and N is a DG R-module, let triang_{R}(N) denote the triangulated subcategory of D(R) generated by N and its shifts, and let thick_{R}(N) be the thick subcategory generated by triang_{R}(N). Let D_{per}^{∞}(A) = thick_{A}^{∞}(A), and let D_{per}(R) = thick_{R}(R). We call objects in D_{per}^{∞}(A) and D_{per}(R) perfect complexes. Let loc_{A}^{∞}(M) denote the localizing subcategory (= triangulated and closed under arbitrary direct sums) generated by triang_{A}^{∞}(M), and similarly for loc_{R}(N). We can also define each of these with M or N replaced by a collection of modules. It is well-known that loc_{R}(R) = D(R).

Lemma 4.9. (a) If R is a DG algebra and N is a right DG R-module, then N is small in D(R) if and only if RHom_{R}(N, −) commutes with arbitrary colimits, and if and only if N ∈ D_{per}(R).

(b) If A is an A_{∞}-algebra and M is a right A_{∞}-module over A, then M is small in D^{∞}(A) if and only if M ∈ D_{per}^{∞}(A).

Proof. (a) The equivalence that N is small if and only if N is in D_{per}(R) is somewhat standard; see Keller [19, 5.3], for example. If RHom_{R}(N, −) commutes with arbitrary colimits, then since homology also commutes with colimits, D(R)(N, −) = H RHom_{R}(N, −) does as well, and so N is small. Finally, RHom_{R}(R, −) commutes with arbitrary colimits, and hence so does RHom_{R}(N, −) for any object N in D_{per}(R) = thick_{R}(R).

(b) This follows from part (a) and Propositions 1.14 and 3.1(c). □

Let B be a DG (E, A)-bimodule. By Lemma 4.5(b),

F_{B} := − ⊗_{E}^{L} B : D(E) → D(A) and
G_{B} := RHom_{A}(B, −) : D(A) → D(E)

form a pair of adjoint functors. Let 𝔸_{B} and 𝔹_{B} be the Auslander and Bass classes associated to the pair (F_{B}, G_{B}). By Lemma 4.1, (F_{B}, G_{B}) induces a triangulated equivalence between 𝔸_{B} and 𝔹_{B}.

The next two results are precursors of the derived equivalences in the next section.

Proposition 4.10. Let A and E be DG algebras, and suppose that B is a left balanced DG (E, A)-module. Define adjoint functors F = F_{B} = − ⊗_{E}^{L} B and G = G_{B} = RHom_{A}(B, −), as above.

(a) Then F_{B} and G_{B} induce an equivalence of categories 𝔸_{B} ≅ 𝔹_{B}. Furthermore, E_{E} ∈ 𝔸_{B}, B_{A} ∈ 𝔹_{B}, and F_{B}(E_{E}) = B_{A}.

(b) There is an equivalence of triangulated categories

triang_{E}(E) ≅ triang_{A}(B).

(c) There is an equivalence of triangulated categories

D_{per}(E) = thick_{E}(E) ≅ thick_{A}(B).

(d) If B_{A} is small, then there is an equivalence of triangulated categories

D(E) ≅ loc_{A}(B).

Proof. (a) The first assertion is Lemma 4.1. Without loss of generality, we assume that B is K-injective over E^{op} ⊗ A. Since B is left balanced,

E → Hom_{A}(B, B) ≅ RHom_{A}(B, B) = RHom_{A}(B, E ⊗_{E}^{L} B) = GF (E)

is a quasi-isomorphism of DG A-modules. Hence E ∈ 𝔸_{B}. Clearly F (E) = B and G(B) = E. Consequently, B ∈ 𝔹_{B}.

(b,c) These follow from (a).

(d) By definition, F commutes with arbitrary colimits. If B is small, G commutes with arbitrary colimits. In this case, 𝔸_{B} and 𝔹_{B} have arbitrary colimits. Since E ∈ 𝔸_{B}, 𝔸_{B} = D(E). Since F is an equivalence, 𝔹_{B} = loc_{A}(B). □

Now we consider two other functors F̃^{B} = RHom_{E^{op}}(−, B) : D(E^{op}) → D(A) and G̃^{B} = RHom_{A}(−, B) : D(A) → D(E^{op}). Both of them are contravariant; however, if we view them as F^{B} : D(E^{op}) → D(A)^{op} and G^{B} : D(A)^{op} → D(E^{op}), then they become covariant. By Lemma 4.4(a), (F^{B}, G^{B}) is an adjoint pair. Let 𝔸^{B} and 𝔹^{B} be the Auslander and Bass classes associated to the pair (F^{B}, G^{B}).

Proposition 4.11. Let A and E be DG algebras, and suppose that B is a left balanced DG (E, A)-module. Define adjoint functors F = F^{B} and G = G^{B}, as above.

(a) Then F^{B} and G^{B} induce an equivalence of categories 𝔸^{B} ≅ 𝔹^{B}. Furthermore, _{E}E ∈ 𝔹^{B}, B_{A} ∈ 𝔸^{B}, and F^{B}(_{E}E) = B_{A}.

(b) If B is also right balanced, then A_{A} ∈ 𝔸^{B}, _{E}B ∈ 𝔹^{B}, and F^{B}(_{E}B) = A_{A}.

(c) There is an equivalence of triangulated categories

triang_{E^{op}}(E) ≅ triang_{A}(B)^{op}.

If B is also right balanced, then

triang_{E^{op}}(E, B) ≅ triang_{A}(A, B)^{op}.

(d) There is an equivalence of triangulated categories

D_{per}(E^{op}) = thick_{E^{op}}(E) ≅ thick_{A}(B)^{op}.

If B is also right balanced, then

thick_{E^{op}}(E, B) ≅ thick_{A}(A, B)^{op}.

(e) If _{E}B is small and B is right balanced, then there is an equivalence of triangulated categories

D_{per}(E^{op}) ≅ thick_{A}(A, B)^{op} = thick_{A}(B)^{op}.

As a consequence, A ∈ thick_{A}(B)^{op}.

Proof. (a) The first assertion follows from Lemma 4.1. We may assume that B is K-injective over E^{op} ⊗ A. Since B is left balanced,

G^{B}F^{B}(_{E}E) = RHom_{A}(RHom_{E^{op}}(E, B), B) = RHom_{A}(B, B) ⟵ E

is a quasi-isomorphism. This shows that E ∈ 𝔹^{B}. Since F^{B}(E) = B, we have B ∈ 𝔸^{B}.

(b) This is the right-hand version of (a).

(c,d,e) These follow from (a,b). □

Proposition 4.10 also implies the following easy fact.

Corollary 4.12. Let M be an object in D^{∞}(A) for some A_{∞}-algebra A. Then thick_{A}(M) is triangulated equivalent to D_{per}(E) for some DG algebra E.

Proof. By Proposition 3.1 we may assume that A is a DG algebra, and then we may replace D^{∞}(A) by D(A). Hence we may assume that M is a right DG A-module.

Let B_{A} be a K-projective resolution of M and let E = End_{A}(B_{A}). Then B is a left balanced (E, A)-bimodule. Note that M ≅ B in D(A). The assertion follows from Proposition 4.10(c). □

5. Koszul equivalences and dualities

In the setting of classical Koszul duality [7], there is an equivalence between certain subcategories of the derived categories of a Koszul algebra A and of its Koszul dual; the subcategories consist of objects satisfying certain finiteness conditions. In this section, we explore the analogous results for non-Koszul algebras, DG algebras, and A_{∞}-algebras. The main results are Theorems 5.4 and 5.5 in the DG setting, and Theorems 5.7 and 5.8 in the A_{∞} setting.

5.1. Koszul equivalence and duality in the DG case. Let A be an augmented DG algebra and let E(A) = (B_{aug}^{∞}A)^{♯} be its Koszul dual, as defined in Section 2.2. The usual bar construction B(A; A) [13, p. 269], where the second A is viewed as a DG A-bimodule, agrees with the A_{∞}- module version B_{aug}^{∞}(A; A) from Section 3.2. By [13, Proposition 19.2(b)], B(A; A) is a semifree resolution of the right A-module k. Thus to define derived functors we may replace k_{A} with B(A; A)_{A}.

The following lemma can be viewed as a dual version of [13, Proposition 19.2].

Lemma 5.1. Let B = B(A; A) and let E = E(A).

(a) The natural embedding i : _{E}k → _{E}B is a quasi-isomorphism of left DG E-modules.

(b) If A is weakly Adams connected (Definition 2.1), then B is a Kinjective DG left E-module.

A left DG E-module is called semiinjective if it is an injective left graded E-module and a K-injective left DG E-module.

Proof. (a) By [13, Proposition 19.2(a)], the augmentations in BA and A define a quasi-isomorphism ε ⊗ ε : B_{A} → k_{A} of right DG A-modules. The map i is a quasi-isomorphism because the composition k ⟶^{i} B ⟶^{ε⊗ε} k is the identity. It is easy to see that the map i is a left DG E-module homomorphism.

(b) Let E^{♯} be the E-bimodule Hom_{k}(E, k). It follows from the adjunction formula in Lemma 4.2(b) that E^{♯} is semiinjective as a left and a right DG E-module. If V is a finite-dimensional k-vector space, then E^{♯} ⊗ V ≅ Hom_{k}(E, V) and this is also semiinjective as a left and a right DG E-module.

Since B(A; A) is locally finite, E = (BA)^{♯} is locally finite and E^{♯} = (BA^{♯})^{♯} ≅ BA. Hence B(A;_{A}k) ≅ BA is a semiinjective left DG E-module. By induction one can easily show that if M is a finite-dimensional left DG A-module, then the bar construction B(A; M) is a semiinjective left DG E-module. Since A is weakly Adams connected, A = \underleftarrow{lim} N_{n} where the {N_{n}}_{n≥0} are finite-dimensional left DG A-modules. Since each N_{n} is finitedimensional, we may further assume that the map N_{n} → N_{n−1} is surjective for all n. By the assertion just proved, B(A; N_{n}) is a semiinjective left DG E-module for each n, as is B(A; N_{n}/N_{n−1}).

Since A = \underleftarrow{lim} N_{n} and since B(A; A) is locally finite,

B(A; A) = \underleftarrow{lim} B(A; N_{n}).

A result of Spaltenstein [33, Corollary 2.5] says that such an inverse limit of K-injectives is again K-injective, and this finishes the proof. (Spaltenstein’s result is for inverse limits of K-injectives in the category of chain complexes over an abelian category, but the proof is formal enough that it extends to the category of DG modules over a DG algebra.) □

Remark 5.2. By the above lemma, _{E}B is isomorphic to _{E}k in D(E^{op}). By [13, Proposition 19.2(a)], B_{A} is isomorphic to k_{A} in D(A). However, B is not isomorphic to k in D(E^{op} ⊗ A) in general.

Lemma 5.3. Let B be the right DG A-module B(A; A) and let C = End_{A}(B).

(a) B is a left balanced (C, A)-bimodule.

(b) If E := E(A) is locally finite, then B is a left balanced (E, A)-bimodule via the natural isomorphism E → C. As a consequence,

HE ≅ H RHom_{A}(k, k).

(c) If A is weakly Adams connected, then B is a right balanced (E, A)- bimodule.

Proof. (a) Since B is semifree over A, the assertion follows from Lemma 4.7.

(b) The first assertion follows from Lemma 4.7 and the fact that E → End_{A}(B) is a quasi-isomorphism [13, Ex 4, p. 272].

Since B_{A} is a K-projective resolution of k_{A},

HE ≅ H End_{A}(B) ≅ H RHom_{A}(k, k).

(c) By Lemma 5.1(b), B is a K-injective left DG E-module. To show that B is right balanced, we must show that the canonical map ϕ : A^{op} → End_{E^{op}}(B) is a quasi-isomorphism. This canonical map sends a ∈ A to the endomorphism y ↦ ya. Since _{E}B is K-injective, H End_{E^{op}}(B) ≅ H RHom_{E^{op}}(k, k). By part (b),

H RHom_{E^{op}}(k, k) ≅ H(E(E))^{op} ≅ HA^{op},

where the last isomorphism follows from Theorem 2.4. Therefore, since HA is locally finite, it suffices to show that Hϕ is injective. Let a ∈ A^{op} be a cocycle such that ϕ(a) = 0 in H End_{E^{op}}(B). Then a ≠ 1 and there is an f ∈ End_{E^{op}}(B) such that ϕ(a) = d(f). Applying this equation to x = [ ] ⊗ 1 ∈ B = B(A; A), we obtain

[ ] ⊗ a = ϕ(a)(x) = d(f)(x)
= (d ∘ f − (−1)^{deg_{1} f}f ∘ d)(x)
= d ∘ f([ ] ⊗ 1) ± f ∘ d([ ] ⊗ 1).

Since f is a left E-module homomorphism, f([ ] ⊗ 1) = [ ] ⊗ w for some w ∈ A. By definition, d([ ] ⊗ 1) = 0. Therefore

[ ] ⊗ a = d ∘ f([ ] ⊗ 1) = d([ ] ⊗ w) = [ ] ⊗ dw,

and hence a = dw as required. □

Here is a version of [7, 1.2.6] for DG algebras.

Theorem 5.4. Let A be an augmented DG algebra and let E = E(A) be the Koszul dual of A. Assume that E is locally finite. The functors RHom_{A}(k, −) and − ⊗_{E}^{L} k induce the following equivalences.

(a) The category triang_{A}(k) is triangulated equivalent to triang_{E}(E).

(b) The category thick_{A}(k) is triangulated equivalent to D_{per}(E).

(c) Suppose that k_{A} is small in D(A). Then loc_{A}(k) is triangulated equivalent to D(E).

Proof. Note that B = B(A; A) ≅ k_{A} as a right DG A-module. Then the assertions follow from Proposition 4.10 and Lemma 5.3(b). □

Theorem 5.5. Let A be an augmented DG algebra and let E = E(A) be the Koszul dual of A. Assume that A is weakly Adams connected. The functors RHom_{A}(−, k) and RHom_{E^{op}}(−, k) induce the following equivalences.

(a) triang_{A}(k, A)^{op} is triangulated equivalent to triang_{E^{op}}(E, k).

(b) thick_{A}(k, A)^{op} is triangulated equivalent to thick_{E^{op}}(E, k).

(c) Suppose that k_{A} is small in D(A). Then

D_{per}(A)^{op} = thick_{A}(k, A)^{op} ≅ thick_{E^{op}}(E, k) = thick_{E^{op}}(k).

Proof. This follows from Proposition 4.11 and Lemmas 5.1 and 5.3. □

Corollary 5.6. Let A be a weakly Adams connected augmented DG algebra and let E = E(A) be its Koszul dual. If k_{A} is small in D(A), then HE is finite-dimensional.

Proof. By Theorem 5.5(c), E is in the thick subcategory generated by k, and every object in thick_{E^{op}}(k) has finite-dimensional homology. □

See Corollary 6.2 for a related result for DG algebras, and Corollaries 5.9 and 7.2 for similar results about A_{∞}-algebras.

5.2. Koszul equivalence and duality in the A_{∞} case. Now suppose that A is an A_{∞}-algebra. By Proposition 1.14, A is quasi-isomorphic to the DG algebra UA, so by Proposition 3.1, the derived category D^{∞}(A) is equivalent to D(UA). We can use this to prove the following, which is a version of [7, 1.2.6] for A_{∞}-algebras.

Theorem 5.7. Let A be an augmented A_{∞}-algebra and let E = E(A) be the Koszul dual of A. Assume that A is strongly locally finite (Definition 2.1).

(a) The category triang_{A}^{∞}(k) is triangulated equivalent to triang_{E}^{∞}(E).

(b) The category thick_{A}^{∞}(k) is triangulated equivalent to D_{per}^{∞}(E).

(c) Suppose that k_{A} is small in D^{∞}(A). Then loc_{A}^{∞}(k) is triangulated equivalent to D^{∞}(E).

Proof. We can replace A by UA and E by E(UA) = E(E(E(A))). The assertions follow from Proposition 3.1 and Theorem 5.4. □

Similarly, Proposition 3.1 combined with 5.5 give the following.

Theorem 5.8. Let A be an augmented A_{∞}-algebra and let E = E(A) be the Koszul dual of A. Assume that A is strongly locally finite.

(a) triang_{A}^{∞}(k, A)^{op} is triangulated equivalent to triang_{E^{op}}^{∞}(E, k).

(b) thick_{A}^{∞}(k, A)^{op} is triangulated equivalent to thick_{E^{op}}^{∞}(E, k).

(c) Suppose that k_{A} is small in D^{∞}(A). Then

D_{per}^{∞}(A)^{op} = thick_{A}^{∞}(k, A)^{op} ≅ thick_{E^{op}}^{∞}(E, k) = thick_{E^{op}}^{∞}(k).

Just as Theorem 5.5 implied Corollary 5.6, this result implies the following.

Corollary 5.9. Let A and E be as in Theorem 5.8. If k_{A} is small in D^{∞}(A), then HE is finite-dimensional.

See Corollary 7.2 for the converse of Corollary 5.9.

Corollary 5.10. Via the equivalence in Theorem 5.8, there is an isomorphism of k-vector spaces

H_{j}^{i}(RHom_{A}(k, A)) ≅ H_{−j}^{−i}(RHom_{E^{op}}(k, E))

for all i, j.

Proof. Again we may assume that A and E are DG algebras. The functor G^{B} in Proposition 4.11 is defined as G^{B} = RHom_{A}(−, B) ≅ RHom_{A}(−, k), and changes S to S^{−1} and Σ to Σ^{−1}; hence the assertion follows from the fact G^{B}(k_{A}) = _{E}E and G^{B}(A_{A}) = _{E}k. □

Remark 5.11. One might hope to prove Theorems 5.7 and 5.8 directly, working in the category D^{∞}(A) rather than D(UA). Keller [20, 6.3] and Lefèvre-Hasegawa [23, 4.1.1] have described the appropriate functors: A_{∞}-

versions of RHom_{A}(−, −) and −⊗_{E}^{L} −, which they write as Hom^{∞}• A(−, −) and

−⊗^{∞}_{E}−. Although we fully expect these to satisfy all of the required properties

(such as adjointness), it was easier to use the more standard results in the DG setting.

6. Minimal semifree resolutions

In this short section we consider the existence of a minimal semifree resolution of a DG module over a DG algebra. The main result is Theorem 6.1; this result is needed in several places. There are similar results in the literature — see, for example [4, Section 1.11] or [11, Lemma A.3] — but they require that A be connected graded with respect to the first (non-Adams) grading. We need to use this in other situations, though, so we include a detailed statement and proof.

We say that A is positively connected graded in the second (Adams) grading if A_{<0}^{∗} = 0 and A_{0}^{∗} = k; negatively connected graded in the Adams grading is defined similarly. Write 𝔪 for the augmentation ideal of A; then a semifree resolution F → M of a module M is called minimal if d_{F} (F) ⊂ F 𝔪.

Theorem 6.1. Let A be a DG algebra and let M be a right DG A-module.

(a) Assume that A is positively connected graded in the second grading and that HM_{≤n}^{∗} = 0 for some n, or that A is negatively connected graded in the second grading and that HM_{≥n}^{∗} = 0 for some n. Then M has a minimal semifree resolution L → M with L_{≤n}^{∗} = 0 (respectively L_{≥n}^{∗} = 0).

(b) Assume further that HA and HM are both bounded on the same side with respect to the first grading: assume that for each j, there is an m so that (HA)_{j}^{≤m} = 0 and (HM)_{j}^{≤m} = 0, or (HA)_{j}^{≥m} = 0 and (HM)_{j}^{≥m} = 0. If HA and HM are locally finite (respectively, locally finite with respect to the second grading), then so is L.

Proof. Without loss of generality, we assume that A_{<0}^{∗} = 0 and HM_{≤n}^{∗} = 0 for some n. After an Adams shift we may further assume that n = −1, that is, HM = HM_{≥0}^{∗}. We will construct a sequence of right DG A-modules {L_{u}}_{u≥0} with the following properties:

(1) 0 = L_{−1} ⊂ L_{0} ⊂ ⋯ ⊂ L_{u} ⊂ · · ·.

(2) L_{u}/L_{u−1} is a free DG A-module generated by a cycles of Adams-degree u.

(3) L_{u} ⊗_{A} k has a trivial differential; that is, d_{L_{u}}(L_{u}) ⊂ L_{u}𝔪.

(4) There is a morphism of DG A-modules ε_{u} : L_{u} → M such that the kernel and coker of H(ε_{u}) have Adams degree at least u + 1.

If A and M satisfy the hypotheses in part (b), then each L_{u} will also satisfy (5) L_{u} is locally finite (respectively, locally finite with respect to the second

grading). Let L_{−1} = 0. We proceed to construct L_{u} inductively for u ≥ 0, so

suppose that {L_{−1}, L_{0}, ⋯, L_{u}} have been constructed and satisfy (1)–(4), and if relevant, (5).

Consider the map H(ε_{u}) : HL_{u} → HM; let C be its cokernel and let K be its kernel. We will focus on the parts of these in Adams degree u + 1. Choose an embedding i of C_{u+1} into the cycles in M, and let P_{u} be the free DG A-module C_{u+1}⊗A on C_{u+1}, equipped with a map f : P_{u} → M, sending x⊗1 to i(x) for each x ∈ C_{u+1}. Since A is positively connected graded in the Adams grading, the map f induces an isomorphism in homology in Adams degrees up to and including u+1. Similarly, let Q_{u} be the free DG A-module on K_{u+1}, mapping to L_{u} by a map g̃ inducing a homology isomorphism in degrees less than or equal to u + 1. Then let L_{u+1} be the mapping cone of

Q_{u} ⟶^{g} L_{u} ⊕ P_{u},

where g maps Q_{u} to the first summand by the map g̃. Since Q_{u} is free and since the composite (ε_{u} + f)g induces the zero map on homology, this composite is null-homotopic. Therefore there is a map ε_{u+1} : L_{u+1} → M. In more detail, since L_{u+1} is the mapping cone of g : Q_{u} → J_{u} := L_{u} ⊕ P_{u}, it may be written as L_{u+1} = S(Q_{u}) ⊕ J_{u}, with differential given by

d(q, l) = (−d_{Q_{u}}(q), g(q) + d_{J_{u}}(l)).

The null-homotopy gives an A-module homomorphism θ : Q_{u} → M of degree (−1, 0) such that

θd_{Q_{u}} + d_{M}θ = δ_{u}g

where δ_{u} = ε_{u} + f. The A-module homomorphism ε_{u+1} : L_{u+1} → M is defined by

ε_{u+1}(q, l) = θ(q) + δ_{u}(l) ∀q ∈ Q_{u} and l ∈ J_{u}.

One can check that ε_{u+1} commutes with the differentials and hence is a morphism of DG A-modules. The morphism ε_{u+1} is an extension of ε_{u} + f, hence ε_{u} is the restriction of ε_{u+1} to L_{u}.

Now we claim that the kernel and cokernel of H(ε_{u+1}) are in Adams degree at least u + 2. There is a long exact sequence in homology

⋯ → H_{j}^{i}(Q_{u}) ⟶^{H(g)} H_{j}^{i}(L_{u} ⊕ P_{u}) → H_{j}^{i}(L_{u+1}) ⟶^{δ} H_{j}^{i+1}(Q_{u}) → · · · .

In Adams degrees less than u + 1, Q_{u} and P_{u} are zero, so L_{u} and L_{u+1} are isomorphic. In Adams degree u + 1, H^{∗}(Q_{u}) maps isomorphically to K_{u}, a vector subspace of H(L_{u}⊕P_{u}), so the boundary map δ is zero, and the above long exact sequence becomes short exact. Indeed, there is a commutative diagram, where the rows are short exact:

\xymatrix{0 \ar[r] & H^{i}(Q_{u})_{≤u+1} \ar[r] \ar[d] & H^{i}(L_{u} ⊕ P_{u})_{≤u+1} \ar[r] \ar[d] & H^{i}(L_{u+1})_{≤u+1} \ar[r] \ar[d]^{H(ε_{u+1})} & 0 \\ 0 \ar[r] & 0 \ar[r] & H^{i}(M)_{≤u+1} \ar[r] & H^{i}(M)_{≤u+1} \ar[r] & 0}

The snake lemma immediately shows that H(ε_{u+1}) has zero kernel and zero cokernel in Adams degree ≤ u + 1, as desired. This verifies property (4) for L_{u+1}.

With property (4), and the fact that L_{i}/L_{i−1} has a basis of cycles in Adams degree i, (1) and (2) are easy to see. To see (3) we use induction on u. It follows from the construction and induction that d_{L_{u}}(L_{u}) ⊂ 𝔪L_{u} + L_{u−1}. Since the semibasis of L_{u−1} has Adams degree no more than u − 1 and the semibasis of P_{u} ⊕ Q_{u} has Adams degree u, we see that d_{L_{u}}(L_{u}) ⊂ 𝔪L_{u} + 𝔪L_{u−1} = 𝔪L_{u}.

Let L be the direct limit \underrightarrow{lim} L_{u}. Then L is semifree and there is a map ϕ : L → M such that the kernel and cokernel of H(ϕ) are zero. Such an L is a semifree resolution of M. Property (3) implies that L is minimal.

If the hypotheses of part (2) are satisfied, then the construction of L_{u} shows that (5) holds. Since L_{j}^{i} = (L_{u})_{j}^{i} for u ≫ 0, L is also locally finite (respectively, locally finite with respect to the Adams grading). □

We are often interested in the complex RHom_{A}(k, k) or in its homology, namely, the Ext-algebra, Ext_{A}^{∗}(k, k). As noted in Section 4.2, to compute this, we replace k by a K-projective resolution P, and then RHom_{A}(k, k) = Hom_{A}(P, k). Since semifree implies K-projective, we can use a minimal semifree resolution, as in the theorem. The construction of L gives the following: for each u, there is a short exact sequence of DG A-modules

0 → L_{u−1} → L_{u} → L_{u}/L_{u−1} → 0.

where L_{u}/L_{u−1} is a free DG A-module, and this leads to a short exact sequence

0 → Hom_{A}(L_{u}/L_{u−1}, k) → Hom_{A}(L_{u}, k) → Hom_{A}(L_{u−1}, k) → 0.

We see that Hom_{A}(L_{u}/L_{u−1}, k)_{i} = 0 when i < u, and therefore

Hom_{A}(L_{u}, k)_{u} ≅ Hom_{A}(L_{u}/L_{u−1}, k)_{u} ≅ (B^{♯})_{−u},

where B is a graded basis for the free DG module L_{u}/L_{u−1}. Note that B is concentrated in degrees (∗, u), so its graded dual B^{♯} is in degrees (∗, −u). Again by the short exact sequence, by induction on u, Hom_{A}(L_{u−1}, k)_{i} = 0 when i ≥ u, so we see that

RHom_{A}(k, k)_{u} ≅ Hom_{A}(L, k)_{u} ≅ Hom_{A}(L_{u}, k)_{u} ≅ (B^{♯})_{−u}.

Furthermore, since L_{u}/L_{u−1} is free on a basis of cycles, or alternatively because the resolution L is minimal, we see that

Ext_{A}(k, k)_{u} ≅ (B^{♯})_{−u}.

This leads to the following corollary; see Corollary 5.6 for a related result.

Corollary 6.2. Let A be a DG algebra which is connected graded, either positively or negatively, in the second grading, and let E = E(A) be its Koszul dual.

(a) Then HE = Ext_{A}^{∗}(k, k) is finite-dimensional if and only if k_{A} is small in D(A).

(b) If HE is finite-dimensional (or if k_{A} is small in D(A)), then HA is Adams connected.

Proof. (a) If HE = Ext_{A}^{∗}(k, k) is finite-dimensional, then by the above computation, the minimal semifree resolution L of k is built from finitely many free pieces, and so L is a perfect complex: it is in thick_{A}(k). Therefore k_{A} is small in D(A).

Conversely, if k_{A} is small, then it is isomorphic in D(A) to a perfect complex, and we claim that if X is a perfect complex, then H RHom_{A}(X, k) is finite-dimensional as a vector space: this is true if X = A, and therefore it is true for every object in the thick subcategory generated by A. Therefore k_{A} small implies that HE is finite-dimensional.

(b) Now suppose that HE is finite-dimensional. Without loss of generality, suppose that A is positively graded connected in the second grading. We claim that for each i, (HA)_{i}^{∗} is finite-dimensional.

Note that in the construction of the minimal semifree resolution for k, the first term L_{0} is equal to A, and the map L_{0} → k is the augmentation. Consider the short exact sequence

0 → L_{u−1} → L_{u} → L_{u}/L_{u−1} → 0,

for u ≥ 1. Since L_{u}/L_{u−1} is free on finitely many classes in Adams degree u, then in Adams degree i, H(L_{u}/L_{u−1})_{i} is isomorphic to a finite sum of copies of HA_{i−u}. Therefore if i ≤ u, then this is finite-dimensional. Therefore when i ≤ u, H(L_{u−1})_{i} is finite-dimensional if and only if H(L_{u})_{i} is. For i fixed and u sufficiently large, H(L_{u})_{i} stabilizes and gives H(L)_{i}. But HL ≅ k, since L is a semifree resolution of k. Thus H(L_{0})_{i} = (HA)_{i} is finite-dimensional for each i, as desired. □

7. Towards classical Koszul duality

In this section we recover the classical version of the Koszul duality given by Beilinson–Ginzburg–Soergel [7]. First we give some useful results about A_{∞}-algebras with finite-dimensional cohomology, and then we use these to recover classical Koszul duality, in Theorem 7.5.

7.1. Finite-dimensional A_{∞}-algebras. Let A be an A_{∞}-algebra. Let D_{fd}^{∞}(A) denote the thick subcategory of D^{∞}(A) generated by all A_{∞}-modules M over A such that HM is finite-dimensional.

Lemma 7.1. Let A be a strongly locally finite A_{∞}-algebra. For parts (b) and (c), assume that A is Adams connected and that HA is finite-dimensional.

(a) thick_{A}^{∞}(k) = D_{fd}^{∞}(A).

(b) A is quasi-isomorphic to a finite-dimensional Adams connected DG algebra.

(c) thick_{A}^{∞}(k) = D_{fd}^{∞}(A) ⊇ D_{per}^{∞}(A) and loc_{A}^{∞}(k) = D^{∞}(A).

Proof. (a) Clearly thick_{A}^{∞}(k) ⊂ D_{fd}^{∞}(A). To show the converse, we may replace A by the DG algebra UA ≅ E(E(A)) (Proposition1.14). Since A is strongly locally finite, so is E(E(A)), by Lemma 2.2. So we may assume that A is a strongly locally finite DG algebra. In this case every 1-dimensional right DG A-module M is isomorphic to a shift of the trivial module k. As a consequence, M ∈ thick_{A}^{∞}(k). Induction shows that M ∈ thick_{A}^{∞}(k) if M is finite-dimensional. If M is a right DG A-module with HM being finite-dimensional, then the minimal semifree resolution L of M is Adams locally finite by Theorem 6.1. Thus M is quasi-isomorphic to a finite-dimensional right DG A-module by truncation: replace L by ⨁_{−N≤s≤N} L_{s}^{∗} for N sufficiently large. Therefore M is in thick_{A}^{∞}(k). This shows that thick_{A}^{∞}(k) = D_{fd}^{∞}(A).

(b) By Theorem 2.4, A is quasi-isomorphic to B := E(E(A)). Since A is Adams connected, so is B. Since HA ≅ HB, H(B_{≥n}^{∗}) = 0 for some n. Hence B is quasi-isomorphic to C := B/B_{≥n}^{∗}. Therefore A is quasi-isomorphic to the Adams connected finite-dimensional DG algebra C.

(c) By part (b) we may assume that A is finite-dimensional, which implies that A is in D_{fd}^{∞}(A) = thick_{A}^{∞}(k) ⊆ loc_{A}^{∞}(k). Therefore

loc_{A}^{∞}(A) = D^{∞}(A) ⊆ loc_{A}^{∞}(k) ⊆ D^{∞}(A).

This proves the last statement. □

Corollary 7.2. Let A be an augmented A_{∞}-algebra and let E = E(A).

(a) Suppose that A is strongly locally finite. k_{A} is small in D^{∞}(A) if and only if HE is finite-dimensional.

(b) Suppose that A is Adams connected. If k_{A} is small in D^{∞}(A) (or if HE is finite-dimensional), then D_{per}^{∞}(A) ≅ D_{fd}^{∞}(E).

Proof. (a) If k_{A} is small, then by Corollary 5.9, HE is finite-dimensional.

Conversely, suppose that HE is finite-dimensional. By Proposition 1.14, A is quasi-isomorphic to the augmented DG algebra UA ≅ E(E(A)), so by Proposition 3.1, k_{A} is small in D^{∞}(A) if and only if k_{UA} is small in D(UA). According to Corollary 6.2, k_{UA} is small if and only if HE(UA) is finitedimensional. Since A is strongly locally finite, so is E, and we have a quasiisomorphism E(E(E)) ≅ E. This means that HE(UA) ≅ HE(A) = HE.

(b) This follows from Theorem 5.7(b) (switching A and E) and Lemma 7.1(a) (for E). (Since A is Adams connected, then by Lemma 2.2, E is Adams connected and strongly locally finite, so these results hold for E.) □

Part (b) is Theorem B from the introduction.

7.2. Twisting the grading. To recover the classical derived equivalences induced by Koszul duality, we need to twist the grading of E(A). Recall that if A is Adams connected and is concentrated in degrees {0} × ℕ, then E(A) is in degrees {(n, −n)|n ≥ 0}. The aim of this section is to change the grading of E(A), to put it in degrees {0} × ℕ.

Let A be a DG algebra with zero differential (there are some problems if d ≠ 0). Let A be the DG algebra with zero differential such that:

(a) A = A as ungraded associative algebras.

(b) A_{j}^{i} = A_{−j}^{i+j} (so A_{j}^{i} = A_{−j}^{i+j}). Note that the sign before j is not essential; for example, we can define a different A by A_{j}^{i} = A_{j}^{i−j} or A_{j}^{i} = A_{−j}^{i−j} or A_{j}^{i} = A_{+j}^{i+j}. Since A is a DG algebra with zero differential, so is A. If A is Koszul, then A^{!} = HE(A) = HE(A) which lives in degrees {0} × ℤ.

Remark 7.3. If A is a DG algebra with nonzero differential d, then we don’t know how to make A into a DG algebra naturally. That is, we don’t see a good way of defining a differential for A.

For any right DG A-module M, we define a corresponding right DG Amodule M by:

(a) M = M as ungraded right A-modules.

(b) M_{j}^{i} = M_{−j}^{i+j}.

One can easily check that M with the differential of M is a right DG Amodule.

The following is routine.

Lemma 7.4. The assignment M ↦ M defines an equivalence between Mod A and Mod A which induces a triangulated equivalence between D(A) and D(A). If further A is finite-dimensional, then D_{fd}(A) ≅ D_{fd}(A).

We now reprove [7, Theorem 2.12.6] (in a slightly more general setting). Let D_{fg}^{∞}(A) denote the thick subcategory of D^{∞}(A) generated by all A_{∞}- modules M over A such that HM is finitely generated over HA. If A is a DG algebra, D_{fg}(A) is defined in a similar way. Note that D_{fg}^{∞}(A) = D_{fd}^{∞}(A) if and only if HA is finite-dimensional.

Theorem 7.5. Suppose A is a connected graded finite-dimensional Koszul algebra. Let A^{!} = HE(A).

(a) D(A) ≅ loc_{A^{!}}(k).

(b) If A^{!} is right noetherian, then D_{fg}(A) = D_{fd}(A) ≅ D_{per}(A^{!}) = D_{fg}(A^{!}).

Proof. Note that A^{!} is quasi-isomorphic to E(A). It follows from Lemma 7.4 that we can replace A^{!} by E := E(A)

(a) This follows from Theorem 5.4(b) (switching A and E).

(b) By Corollary 7.2(b) (again switching A and E), D_{per}(E) ≅ D_{fd}(A). Since A is finite-dimensional, D_{fd}(A) = D_{fg}(A). Note that A^{!} is concentrated in degrees {0} × ℕ. Since A^{!} is right noetherian and has finite global dimension (because A is finite-dimensional Koszul), it is clear that D_{per}(A^{!}) = D_{fg}(A^{!}). □

8. Some examples

Example 8.1. Let Λ = Λ(x_{1}, x_{2}) with deg x_{i} = (0, i). As noted in Example 2.8, this is not a classical Koszul algebra; nonetheless, it satisfies many of the same derived equivalences as Koszul algebras. Let Λ^{!} denote the Ext algebra Ext_{Λ}^{∗}(k, k) ≅ k[y_{1}, y_{2}], where deg y_{i} = (1, −i); this is an A_{∞}-algebra with no higher multiplications. Then E(Λ) is quasi-isomorphic to the associative algebra Λ^{!}, so there is a triangulated equivalence D^{∞}(E(Λ)) ≅ D(Λ^{!}). Similarly, we have D^{∞}(Λ) ≅ D(Λ). Since the homology of E(Λ^{!}) ≅ E(E(Λ)) is isomorphic to Λ, which is finite-dimensional, the trivial module k_{Λ^{!}} is small in D(Λ^{!}). So Theorems 5.7 and 5.8 give triangulated equivalences

thick_{Λ}(k) ≅ D_{per}(Λ^{!}),  thick_{Λ}(k, Λ)^{op} ≅ thick_{Λ^{!}}(Λ^{!}, k),
thick_{Λ^{!}}(k) ≅ D_{per}(Λ),  D_{per}(Λ^{!})^{op} ≅ thick_{Λ}(k),
loc_{Λ^{!}}(k) ≅ D(Λ).

Compare to the classical Koszul equivalences of Theorem 7.5: part (b) of that theorem is the first of these equivalences, while part (a) of the theorem is the last of these.

Slightly more generally, the results of Theorem 7.5 hold for exterior algebras on finitely many generators, as long as they are graded so as to be Adams connected: Λ(x_{1}, …, x_{n}), graded by setting deg x_{i} = (a_{i}, b_{i}) with each b_{i} positive, or each b_{i} negative.

Example 8.2. We fix an integer p ≥ 3 and define two A_{∞}-algebras, B(0) and B(p), each with m_{1} = 0. As associative algebras, they are both isomorphic to Λ(y) ⊗ k[z] with deg y = (1, −1) and deg z = (2, −p). The algebra B(0) has no higher multiplications, while B(p) has a single nonzero higher multiplication m_{p}. This map m_{p} satisfies m_{p}(y^{⊗p}) = z; more generally, m_{p}(a_{1} ⊗ ⋯ ⊗ a_{p}) is zero unless each a_{i} has the form a_{i} = yz^{j_{i}} for some j_{i} ≥ 0, and

m_{p}(yz^{j_{1}} ⊗ ⋯ ⊗ yz^{j_{p}}) = z^{1+∑ j_{i}}.

See [24, Example 3.5] for more on B(p).

We claim that k_{B(0)} is not small in D^{∞}(B(0)) ≅ D(B(0)), while k_{B(p)} is small in D^{∞}(B(p)). The Ext algebra Ext_{B(0)}^{∗}(k, k) is isomorphic to the associative algebra k[u] ⊗ Λ(v) with deg u = (0, 1) and deg v = (1, p). In particular, this algebra is not finite-dimensional, so by Corollary 7.2, k_{B(0)} is not small in D(B(0)). In contrast, by [24, Proposition 12.6], the Koszul dual of B(p) is A_{∞}-isomorphic to the associative algebra A(p) = k[x]/(x^{p}), where deg x = (0, 1). Since A(p) is finite-dimensional, k_{B(p)} is small in D^{∞}(B(p)). This verifies the claim.

Part III. Applications in ring theory

9. The Artin–Schelter condition

In this section we prove Corollaries D, E and F. We start by discussing Artin–Schelter regularity, for both associative algebras and A_{∞}-algebras. The Eilenberg–Moore spectral sequence is a useful tool for connecting results about modules over HA to modules over A, if A is a DG algebra or an A_{∞}- algebra. Then we discuss Frobenius algebras and prove Corollary D, and we discuss dualizing complexes and prove Corollary E. At the end of the section, we prove Corollary F.

9.1. Artin–Schelter regularity.

Definition 9.1. Let R be a connected graded algebra.

(a) R is called Gorenstein if injdim R_{R} = injdim_{R}R < ∞.

(b) R is called Artin–Schelter Gorenstein if R is Gorenstein of injective dimension d and there is an integer l such that

Ext_{R}^{i}(k, R) ≅ Ext_{R^{op}}^{i}(k, R) ≅ \begin{cases}0 & i ≠ d, \\ Σ^{l}(k) & i = d. \end{cases}

(c) R is called Artin–Schelter regular if R is Artin–Schelter Gorenstein and has global dimension d.

Artin–Schelter regular algebras have been used in many ways in noncommutative algebraic geometry.

Now we consider analogues for A_{∞}-algebras. When A is an unbounded A_{∞}-algebra, there is no good definition of global or injective dimension, so we only consider a version of condition (b) in Definition 9.1.

Definition 9.2. Let A be an augmented A_{∞}-algebra and let k be the trivial module.

(a) We say A satisfies the right Artin–Schelter condition if there are integers l and d such that

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

If further k_{A} is small in D^{∞}(A), then A is called right A_{∞}-Artin– Schelter regular, or just right Artin–Schelter regular, if the context is clear.

(b) We say A satisfies the left Artin–Schelter condition if there are integers l and d such that

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

If further _{A}k is small in D^{∞}(A^{op}), then A is called left A_{∞}-Artin– Schelter regular (or just left Artin–Schelter regular).

(c) We say A satisfies the Artin–Schelter condition if the conditions in parts (a) and (b) hold for the same pair of integers (l, d). If further k_{A} is small in D^{∞}(A), then A is called A_{∞}-Artin–Schelter regular (or just Artin–Schelter regular).

(d) Finally, if A is A_{∞}-Artin–Schelter regular, we say that A is noetherian if D_{fg}^{∞}(A) = D_{per}^{∞}(A).

Suppose that R is a connected graded algebra. If R is Artin–Schelter regular as an associative algebra, then R is A_{∞}-Artin–Schelter regular. Conversely, if R is A_{∞}-Artin–Schelter regular, then one can use Corollary 6.2 to show that R is Artin–Schelter regular.

Also, if R is a connected graded algebra, then the cohomology of every object in D_{fg}(R) is bounded (with respect to the homological grading), and we see that D_{fg}(R) = D_{per}(R) if and only if R is noetherian of finite global dimension; this motivates our definition of “noetherian” for A_{∞}-Artin–Schelter regular algebras.

It is easy to see that A satisfies the left A_{∞}-Artin–Schelter condition if and only if A^{op} satisfies the right A_{∞}-Artin–Schelter condition. We conjecture that the left A_{∞}-Artin–Schelter condition is equivalent to the right A_{∞}-Artin–Schelter condition. We verify this, with some connectedness and finiteness assumptions, in Theorems 9.8 and 9.11.

Proposition 9.3. Let A be an augmented A_{∞}-algebra and let E = E(A) be the Koszul dual of A.

(a) Assume that A is strongly locally finite. Then A satisfies the right Artin–Schelter condition if and only if E satisfies the left Artin–Schelter condition.

(b) Assume that E is strongly locally finite. Then A satisfies the left Artin– Schelter condition if and only if E satisfies the right Artin–Schelter condition.

Proof. (a) By Corollary 5.10 A satisfies the right Artin–Schelter condition if and only if E satisfies the left Artin–Schelter condition.

(b) Switch A and E and use the fact that A → E(E(A)) is a quasiisomorphism: the assertion follows from part (a). □

9.2. The Eilenberg–Moore spectral sequence. In this subsection we recall the Eilenberg–Moore spectral sequence [22, Theorem III.4.7] and [13, p. 280]. This helps to translate homological results for modules over HA to similar results for modules over A.

Suppose that A is a DG algebra and that M and N are right DG Amodules. Because of the ℤ × ℤ-grading on A, M, and N, Ext_{HA}^{∗}(HM, HN) is ℤ^{3}-graded, and we incorporate the gradings into the notation as follows:

Ext_{HA}^{p}(HM, HN)_{s}^{q}.

On the other hand, Ext_{A}^{∗}(M, N) is ℤ^{2}-graded, since it is defined to be the homology of RHom_{A}(M, N). That is,

Ext_{A}^{i}(M, N)_{j} := D(A)(M, S^{i}Σ^{j}N) ≅ H(RHom_{A}(M, S^{i}Σ^{j}N)).

Because of this, each E_{r}-page of the Eilenberg–Moore spectral sequence is ℤ^{3}-graded, while the abutment is ℤ^{2}-graded.

Theorem 9.4 ([22, Theorem III.4.7]). Let A be a DG algebra, and let M and N be right DG A-modules. Then there is a spectral sequence of the form

(E_{2})_{s}^{p,q} ≅ Ext_{HA}^{p}(HM, HN)_{s}^{q} ⇒ Ext_{A}^{p+q}(M, N)_{s},

natural in M and N. All differentials preserve the lower (Adams) grading s.

This is a spectral sequence of cohomological type, with differential in the E_{r}-page as follows:

d_{r} : (E_{r})_{s}^{p,q} → (E_{r})_{s}^{p+r,q−r+1}.

Ignoring the Adams grading, the E_{2}-term is concentrated in the right halfplane (i.e., p ≥ 0), and it converges strongly if for each (p, q), d_{r} : E_{r}^{p,q} → E_{r}^{p+r,q−r+1} is nonzero for only finitely many values of r.

There is also a Tor version of this spectral sequence, which we do not use. See [22, Theorem III.4.7] and [13, p. 280] for more details.

Note that the above theorem also holds for A_{∞}-algebras; see [22, Theorem V.7.3]. Another way of obtaining an A_{∞}-version of this spectral sequence is to use the derived equivalence between D^{∞}(A) and D(UA).

Corollary 9.5. Let A be an A_{∞}-algebra. If HA satisfies the left (respectively, right) Artin–Schelter condition, then so does A.

Proof. We may assume that A is a DG algebra. Let M = k and N = A. Note that Hk = k. Since HA satisfies the left Artin–Schelter condition, ⨁_{p,q} Ext_{HA}^{p}(HM, HN)^{q} is 1-dimensional. By Theorem 9.4, ⨁_{n} Ext_{A}^{n}(k, A) is 1-dimensional. Therefore A satisfies the left Artin–Schelter condition. □

One naive question is if the converse of Corollary 9.5 holds.

9.3. Frobenius A_{∞}-algebras. In this subsection we define Frobenius DG algebras and Frobenius A_{∞}-algebras and then prove Corollary D.

Definition 9.6. An augmented DG algebra A is called left Frobenius (respectively, right Frobenius) if:

(a) HA is finite-dimensional.

(b) There is a quasi-isomorphism of left (respectively, right) DG A-modules α : S^{l}Σ^{d}(A) → A^{♯} for some integers l and d.

An augmented DG algebra A is called Frobenius if it is both left and right Frobenius.

Lemma 9.7. Suppose A is a DG algebra such that there is a quasi-isomor-phism of left DG A-modules α : S^{l}Σ^{d}(A) → A^{♯} for some integers l and d.

(a) There is a quasi-isomorphism of right DG A-modules β : S^{l}Σ^{d}(A) → A^{♯} for the same integers l and d.

(b) If A is connected graded with respect to some grading which is compatible with the ℤ^{2}-grading — see below — then HA is finite-dimensional.

(c) If HA is finite-dimensional, then HA is Frobenius as an associative algebra.

(d) A satisfies the left Artin–Schelter condition and RHom_{A}(k, A) is quasiisomorphic to S^{−l}Σ^{−d}(k).

The compatibility requirement for the grading in part (b) means that there should be numbers a and b so that the nth graded piece is equal to ⨁_{ai+bj=n} A_{j}^{i}.

Proof. (a) Let β = S^{l}Σ^{d}(α^{♯}).

(b) Since S^{l}Σ^{d}(A) → A^{♯} is an quasi-isomorphism,

(HA)_{m−d}^{n−l} ≅ (HA^{♯})_{−m}^{−n} ≅ ((HA)_{m}^{n})^{♯}

for all n, m. This implies that HA is locally finite. If A is connected graded with respect to some compatible grading, then so is HA. Then the above formula implies that HA is finite-dimensional.

(c) The quasi-isomorphism α gives rise to an isomorphism

H(α) : S^{l}Σ^{d}(HA) → (HA)^{♯}.

If HA is finite-dimensional, then HA is Frobenius.

(d) Since A → S^{−l}Σ^{−d}A^{♯} is a K-injective resolution of A, we can compute RHom_{A}(k, A) by Hom_{A}(k, S^{−l}Σ^{−d}(A^{♯})), which is S^{−l}Σ^{−d}(k). □

By Lemma 9.7 above, A is left Frobenius if and only if it is right Frobenius. So we can omit both “left” and “right” before Frobenius. It is therefore easy to see that A is Frobenius if and only if A^{op} is. We show that the Frobenius property is a homological property.

Theorem 9.8. Let A be an Adams connected DG algebra such that HA is finite-dimensional. Then the following are equivalent.

(a) A is Frobenius.

(b) HA is Frobenius.

(c) A satisfies the left Artin–Schelter condition.

(d) A satisfies the right Artin–Schelter condition.

Proof. By Lemma 7.1, we may assume that A is finite-dimensional. (a) ⇒ (b): Use Lemma 9.7(c).

(b) ⇒ (a): Since HA is Frobenius, there is an isomorphism of right HAmodules f : S^{l}Σ^{d}(HA) → HA^{♯}. Pick x ∈ ZA^{♯} so that the class of x generates a submodule of HA^{♯} that is isomorphic to S^{l}Σ^{d}(HA). Hence the map a → xa is a quasi-isomorphism A → A^{♯}.

(a) ⇒ (d): By vector space duality,

RHom_{A^{op}}(k, A) ≅ RHom_{A}(A^{#}, k) ≅ RHom_{A}(S^{l}Σ^{d}(A), k) = S^{−l}Σ^{−d}(k).

Hence A satisfies the right Artin–Schelter condition.

(d) ⇒ (a): Suppose RHom_{A^{op}}(k, A) ≅ S^{−l}Σ^{−d}(k) by the right Artin– Schelter condition. Since HA is locally finite, by vector space duality, we obtain that RHom_{A}(A^{#}, k) is quasi-isomorphic to S^{−l}Σ^{−d}(k). By Theorem 6.1 A^{♯} has a minimal semifree resolution, say P → A^{♯}. Since P is minimal, it has a semifree basis equal to (⨁_{i} Ext_{A}^{i}(A^{#}, k))^{#} = Σ^{d}(k). Hence P ≅ S^{l}Σ^{d}(A) for some l and d. Thus S^{l}Σ^{d}(A) → A^{♯} is a quasi-isomorphism and A is left Frobenius. By Lemma 9.7, A is Frobenius.

Thus we have proved that (a), (b) and (d) are equivalent. By left-right symmetry, (a), (b) and (c) are equivalent. □

We obtain an immediate consequence.

Corollary 9.9. Let f : A → B be a quasi-isomorphism of Adams connected DG algebras. Assume that HA ≅ HB is finite-dimensional. Then A is Frobenius if and only if B is.

Since finite-dimensional Hopf algebras are Frobenius, every finite-dimen-sional DG Hopf algebra is Frobenius.

Suggested by the DG case, we make the following definition.

Definition 9.10. An A_{∞}-algebra A is called Frobenius if HA is finite-di-mensional and there is a quasi-isomorphism of right A_{∞}-modules S^{l}Σ^{d}(A) → A^{♯} for some l and d.

The following is similar to Theorem 9.8 and the proof is omitted.

Theorem 9.11. Let A be an Adams connected A_{∞}-algebra such that HA is finite-dimensional. The following are equivalent.

(a) A is Frobenius.

(b) HA is Frobenius.

(c) A satisfies the left Artin–Schelter condition.

(d) A satisfies the right Artin–Schelter condition.

Now we are ready to prove Corollary D from the introduction. We restate it for the reader’s convenience.

Corollary D. Let R be a connected graded algebra. Then R is Artin– Schelter regular if and only if the Ext-algebra ⨁_{i∈ℤ} Ext_{R}^{i}(k_{R}, k_{R}) is Frobenius.

Proof. Note that R is right Artin–Schelter regular if and only if k_{R} is small and R satisfies the right A_{∞}-Artin–Schelter condition [35, Proposition 3.1]. Now we assume, temporarily, that R is Adams connected; by Lemma 2.2, this means that E(R) is, as well. Then the above conditions are in turn equivalent to: HE(R) = ⨁_{i} Ext^{i}(k_{R}, k_{R}) is finite-dimensional and E(R) satisfies the left Artin–Schelter condition. By Theorem 9.11, this is equivalent to HE(R) being Frobenius.

Now, we use Corollary 6.2 to justify the Adams connected assumption: if R is Artin–Schelter regular, then k_{R} is small, which implies that R = HR is Adams connected. Conversely, if HE(R) is Frobenius, then it is finitedimensional, which also implies that R is Adams connected. □

This proof in fact shows that (for associative algebras) left Artin–Shelter regularity is equivalent to right Artin–Shelter regularity.

9.4. Dualizing complexes and the Gorenstein property. The balanced dualizing complex over a graded ring B was introduced by Yekutieli [38]. We refer to [38] for the definition and basic properties. Various noetherian graded rings have balanced dualizing complexes; see [36, 39].

Lemma 9.12. Suppose R is a noetherian connected graded ring with a balanced dualizing complex. Then R satisfies the right Artin–Schelter condition if and only if R is Gorenstein. In this case, R will be Artin–Schelter Gorenstein.

Proof. Let B be the balanced dualizing complex over R. Then the functor F := RHom_{R}(−, B) induces an equivalence D_{fg}(R) ≅ D_{fg}(R^{op})^{op} and satisfies F (k_{R}) = _{R}k. By the right Artin–Schelter condition,

RHom_{R}(k, R) ≅ S^{l}Σ^{d}(k)

for some l and d. Applying the duality functor F, we have

RHom_{R^{op}}(B, k) = RHom_{R^{op}}(F (R), F (k)) ≅ RHom_{R}(k, R) ≅ S^{l}Σ^{d}(k).

Therefore _{R}B is quasi-isomorphic to S^{−l}Σ^{−d}(R). Since _{R}B has finite injective dimension by definition, _{R}R has finite injective dimension. Also it follows from _{R}B ≅ S^{−l}Σ^{−d}(R) that B_{R} ≅ S^{−l}Σ^{−d}(R) by the fact that R^{op} = RHom_{R^{op}}(B, B). So since B_{R} has finite injective dimension, so does R_{R}.

For the converse, note that both H RHom_{R}(k, R) and H RHom_{R^{op}}(k, R) are finite-dimensional since the existence of B implies that R satisfies the χ-condition. The assertion follows from the proof of [40, Lemma 1.1]. □

Now we restate and prove Corollary E.

Corollary E. Let R be a Koszul algebra and let R^{!} be the Koszul dual of R. If R and R^{!} are both noetherian having balanced dualizing complexes, then R is Gorenstein if and only if R^{!} is.

Proof. By Lemma 9.12 the Artin–Schelter condition is equivalent to the Gorenstein property. The assertion follows from Proposition 9.3. □

We say a connected graded algebra A has enough normal elements if every nonsimple graded prime factor ring A/P contains a homogeneous normal element of positive degree. A noetherian graded ring satisfying a polynomial identity has enough normal elements.

Corollary 9.13. Let R be a Koszul algebra and let R^{!} the Koszul dual of R. If R and R^{!} are both noetherian having enough normal elements, then R is (Artin–Schelter) Gorenstein if and only if R^{!} is.

Proof. By [39, Theorem 5.13], R and R^{!} have balanced dualizing complexes. By [40, Proposition 2.3(2)], under the hypothesis, the Artin–Schelter Gorenstein property is equivalent to the Gorenstein property. The assertion follows from Corollary E. □

Let R^{e} = R ⊗ R^{op}. Following the work of Van den Bergh [36], Ginzburg [15] and Etingof–Ginzburg [10], an associative algebra R is called twisted Calabi–Yau if

Ext_{R^{e}}^{i}(R, R^{e}) ≅ \begin{cases}^{ϕ}R^{1} & if i = d \\ 0 & if i ≠ d \end{cases}

for some d (note that we do not require R to have finite Hochschild dimension). If the above equation holds for ϕ = Id_{R}, then A is called Calabi–Yau. If R is connected graded, then ^{ϕ}R^{1} should be replaced by Σ^{l}(^{ϕ}R^{1}) for some integer l. It follows from Van den Bergh’s result [36, Proposition 8.2] that if R is connected graded noetherian and Artin–Schelter Gorenstein, then R is twisted Calabi–Yau. It is easy to see that if R has finite global dimension and R^{e} is noetherian, then Artin–Schelter regularity is equivalent to the twisted Calabi–Yau property. It is conjectured that the Artin–Schelter Gorenstein property is equivalent to the twisted Calabi–Yau property for all connected graded noetherian rings.

We end this section with a proof of Corollary F.

Corollary F. Let A be an Adams connected commutative differential graded algebra such that RHom_{A}(k, A) is not quasi-isomorphic to zero. If the Extalgebra ⨁_{i∈ℤ} Ext_{A}^{i}(k_{A}, k_{A}) is noetherian, then A satisfies the Artin–Schelter condition.

Proof. Since A is commutative, its Ext algebra H = HE(A) = Ext_{A}^{∗}(k, k) is a graded Hopf algebra which is graded cocommutative [12, p. 545]. By hypothesis, H is noetherian, so it satisfies [12, (1.1)]. Since RHom_{A}(k, A) ≠ 0, Corollary 5.10 implies that RHom_{E^{op}}(k, E) ≠ 0 where E = E(A). By Theorem 9.4 RHom_{H^{op}}(k, H) ≠ 0. Since H^{op} ≅ H, we have RHom_{H}(k, H) ≠ 0 which says that H has finite depth. By [12, Theorem C], the noetherian property of H implies that H is elliptic, and elliptic Hopf algebras are classified in [12, Theorem B]. It is well-known that the Hopf algebras in [12, Theorem B] are Artin–Schelter Gorenstein. By Corollary 9.5, E(A) satisfies the Artin–Schelter condition, and therefore by Proposition 9.3, A does as well. □

10. The BGG correspondence

The classical Bernšteĭn–Gel’fand–Gel’fand (BGG) correspondence states that the derived category of coherent sheaves over ℙ^{n} is equivalent to the stable derived category over the exterior algebra of (n + 1)-variables [8, Theorem 2]. Some generalizations of this were obtained by Baranovsky [5], He–Wu [16], Mori [27] and so on. In this section we prove a version of the BGG correspondence in the A_{∞}-algebra setting, as a simple application of Koszul duality.

If R is a right noetherian ring, then the stable bounded derived category over R, denoted by \underline{D_{fg}^{b}}(R), is defined to be the Verdier quotient D_{fg}(R)/D_{per}(R). With R concentrated in degrees {0} × ℤ, every complex in D_{fg}(R) is bounded. When R is a finite-dimensional Frobenius algebra, then the stable module category over R is equivalent to the stable bounded derived category over R [30].

Recall from Sections 7.1 and 7.2, respectively, that

D_{fd}^{∞}(A) = thick_{A}^{∞}(M | M ∈ Mod A, dim_{k} HM < ∞),
D_{fg}^{∞}(A) = thick_{A}^{∞}(M | M ∈ Mod A, HM finitely generated over HA).

If HA is finite-dimensional, then by Lemma 7.1(a), D_{fd}^{∞}(A) = thick_{A}^{∞}(k), which is also equal to thick_{A}^{∞}(k, A). Modelled on the definition of \underline{D_{fg}^{b}}(R), we define the stable derived category of an A_{∞}-algebra A to be

\underline{D_{fg}^{∞}}(A) = D_{fg}^{∞}(A)/D_{per}^{∞}(A)

where the right-hand side of the equation is a Verdier quotient. Boundedness of complexes does not make sense here since A itself may not be bounded, but for a weakly Adams connected A_{∞}-algebra we have the following variation. The small stable derived category of a weakly Adams connected A_{∞}-algebra A is defined to be

\underline{D_{sm}^{∞}}(A) = thick_{A}^{∞}(k, A)/D_{per}^{∞}(A)

where the right-hand side of the equation is a Verdier quotient. If HA is finite-dimensional, then

\underline{D_{sm}^{∞}}(A) = D_{fd}^{∞}(A)/D_{per}^{∞}(A) = \underline{D_{fg}^{∞}}(A).

In general, \underline{D_{sm}^{∞}}(A) ⊂ \underline{D_{fg}^{∞}}(A). It is arguable whether or not \underline{D_{sm}^{∞}}(A) is a good definition. One reason we use the above definition is to make the BGG correspondence easy to prove.

It is easy to see that \underline{D_{sm}^{∞}}(A) = 0 if and only if k_{A} is small (see Definition 4.8 and Lemma 4.9(b)). In this case we call A regular. This is consistent with terminology for associative algebras: Orlov called the triangulated category \underline{D}_{fg}^{b}(R) the derived category of the singularity of R [28].

Given any connected graded ring R, we define the projective scheme over R to be the quotient category

Proj R := Mod R/Tor R

where Tor R is the Serre localizing subcategory generated by all finite-dimen-sional graded right R-modules [1]. When R is right noetherian, we denote its noetherian subcategory by proj R. The bounded derived category of proj R is D^{b}(proj A) which is modelled by the derived category of coherence sheaves over a projective scheme. When A is an A_{∞}-algebra, we can define the derived category directly without using Proj A. The derived category of projective schemes over A is defined to be

D^{∞}(Proj A) = D^{∞}(A)/loc_{A}^{∞}(k);

the derived category of finite projective schemes over A is defined to be

D^{∞}(proj A) = D_{fg}^{∞}(A)/thick_{A}^{∞}(k);

and the derived category of small projective schemes over A is defined to be

D_{sm}^{∞}(proj A) = thick_{A}^{∞}(k, A)/thick_{A}^{∞}(k).

If A is right noetherian and regular (e.g., A is a commutative polynomial ring), then D^{∞}(proj A) = D_{sm}^{∞}(proj A) and this is equivalent to the derived category of the (noncommutative) projective scheme proj A [1].

Lemma 10.1. Let A be an A_{∞}-algebra satisfying the Artin–Schelter condition. Then \underline{D_{sm}^{∞}}(A)^{op} ≅ \underline{D_{sm}^{∞}}(A^{op}).

Proof. First of all we may assume A is a DG algebra. Let B be the (A, A)-bimodule A. Clearly B is a balanced (A, A)-bimodule. The equivalences given in Proposition 4.10 are trivial, but Proposition 4.11 is not trivial. By Proposition 4.11(a), 𝔸^{B} ≅ 𝔹^{B} and F^{B}(_{A}A) = A_{A} ∈ 𝔸^{B}. By the Artin–Schelter condition, F^{B}(_{A}k) = RHom_{A^{op}}(k, A) = S^{l}Σ^{d}(k) and G^{B}(S^{l}Σ^{d}(k)) = S^{−l}Σ^{−d}(G^{B}(k)) = k. So _{A}k ∈ 𝔸^{B} and F^{B}(_{A}k) = S^{l}Σ^{d}(k). This implies that F^{B} induces an equivalence between thick_{A^{op}}^{∞}(k, A)^{op} and thick_{A}^{∞}(k, A) which sends _{A}A to A_{A}. The assertion follows from the definition of \underline{D_{sm}^{∞}}(A). □

The next theorem is a version of the BGG correspondence.

Theorem 10.2. Let A be a strongly locally finite A_{∞}-algebra and let E be its Koszul dual.

(a) There is an equivalence of triangulated categories

D_{sm}^{∞}(proj A)^{op} ≅ \underline{D_{sm}^{∞}}(E^{op}).

(b) If A is Adams connected graded noetherian right Artin–Schelter regular, then there is an equivalence of triangulated categories

D^{∞}(proj A) ≅ \underline{D_{fg}^{∞}}(E).

Proof. (a) By Theorem 5.8(b) there is an equivalence of triangulated categories

thick_{A}^{∞}(k, A)^{op} ≅ thick_{E^{op}}^{∞}(k, E)

which maps k_{A} to E. Therefore we have

D_{sm}^{∞}(proj A)^{op} = (thick_{A}^{∞}(k, A)/thick_{A}^{∞}(k))^{op}
≅ thick_{E^{op}}^{∞}(k, E)/thick_{E^{op}}^{∞}(E) = \underline{D_{sm}^{∞}}(E^{op}).

(b) Since A is AS regular, Corollary D says that E is Frobenius and hence HE is finite-dimensional. By Lemma 10.1, \underline{D_{sm}^{∞}}(E^{op}) ≅ \underline{D_{sm}^{∞}}(E)^{op}. Since HE is finite-dimensional, \underline{D_{sm}^{∞}}(E) = \underline{D_{fg}^{∞}}(E). Since A is connected graded Artin–Schelter regular, k_{A} is small, so

thick_{A}^{∞}(k, A) = thick_{A}^{∞}(A) = D_{per}^{∞}(A).

By the definition of noetherian (Definition 9.2), D_{fg}^{∞}(A) = D_{per}^{∞}(A). Therefore D^{∞}(proj A) = D_{sm}^{∞}(proj A). The assertion follows from part (a). □

Theorem C is part (b) of the above theorem.

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Department of Mathematics, Zhejiang University, Hangzhou 310027, China dmlu@zju.edu.cn

Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195, USA palmieri@math.washington.edu zhang@math.washington.edu

Institute of Mathematics, Fudan University, Shanghai, 200433, China qswu@fudan.edu.cn

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