Ann. scient. Éc. Norm. Sup., 4^{e} série, t. 27, 1994, p. 63 à 102.

DERIVING DG CATEGORIES

Bernhard Keller^{1}

14 July 1993

Abstract. We investigate the (unbounded) derived category of a differential Z-graded category (=DG category). As a first application, we deduce a ’triangulated analogue‘ (4.3) of a theorem of Freyd’s [5, Ex. 5.3 H] and Gabriel’s [6, Ch. V] characterizing module categories among abelian categories. After adapting some homological algebra we go on to prove a ’Morita theorem‘ (8.2) generalizing results of [19] and [20]. Finally, we develop a formalism for Koszul duality [1] in the context of DG augmented categories.

Summary

We give an account of the contents of this paper for the special case of DG algebras. Let k be a commutative ring and A a DG (k-)algebra, i.e. a Z-graded k-algebra

A = ∐_{p∈Z} A^{p}

endowed with a differential d of degree 1 such that

d(ab) = (da)b + (−1)^{p}a(db)

for all a ∈ A^{p}, b ∈ A. A DG (right) A-module is a Z-graded A-module M = ∐_{p∈Z} M^{p} endowed with a differential d of degree 1 such that

d(ma) = (dm)a + (−1)^{p}m(da)

for all m ∈ M^{p}, a ∈ A. A morphism of DG A-modules is a homogeneous morphism of degree 0 of the underlying graded A-modules commuting with the differentials. The DG A-modules form an abelian category CA. A morphism f : M → N of CA is null-homotopic if f = dr + rd for some homogeneous morphism r : M → N of degree -1 of the underlying graded A-modules. The homotopy category HA has the same objects as CA. Its morphisms are residue classes of morphisms of CA modulo null-homotopic morphisms. It is a triangulated [23] category (2.2). A quasi-isomorphism is a morphism of CA inducing isomorphisms in homology. The derived category DA is the localization [23] of HA with respect to the quasi-isomorphisms (4.1). It has infinite direct sums. Let H_{p}A be the smallest strictly (=closed under isomorphisms) full triangulated subcategory of HA containing A and closed under infinite direct sums. Each DG A-module M is quasi-isomorphic to a module pM ∈ H_{p}A. (3.1). The canonical projection HA → DA restricts to an equivalence H_{p}A → DA (4.1). This is classical [11, VI, 10.2] for right bounded modules over negative DG algebras (i.e. M^{p} = 0 for all p ≫ 0 and A^{p} = 0 for all p > 0).

1Supported by a grant of the Swiss National Foundation

The algebra A considered as a right DG A-module is small in DA, i.e. the functor (DA) (A,?) commutes with infinite direct sums. Moreover A is a generator of DA, i.e. DA coincides with its smallest strictly full triangulated subcategory containing A and closed under infinite direct sums. Now suppose that E is a Frobenius category [9] with infinite direct sums and that the associated stable category \underline{E} admits a small generator X. Then there is a DG algebra A and an S-equivalence G : \underline{E} → DA with GX →^{∼} A (4.3). This is an analogue of Freyd’s and Gabriel’s characterization of module categories among abelian categories [5, Ex. 5.3 H] [6, Ch. V]. It suggests that in the study of triangulated categories, categories of DG modules might take the rôle that module categories play in the theory of abelian categories.

Let B and C be DG algebras. A quasi-equivalence C → B is a B-C-bimodule (i.e. a right-B-left-C-bimodule) E containing an element e ∈ Z^{0} E such that the maps

B → E , b ↦ eb and C → E , c ↦ ce

induce isomorphisms in homology. For example, if we are given a quasi-isomorphism φ : C → B, we can take E =_{φ} B_{B} and e = 1. Suppose that A is a DG algebra which is flat as a k-module. There is an A-C-bimodule X such that

L(? ⊗_{C} X) : DC → DA , M ↦ (pM) ⊗_{C} X ,

is an equivalence iff C is quasi-equivalent to B = Hom (T, T) for some module T ∈ H_{p}A which is a small generator of DA (8.2). Here Hom (T, T) is the DG algebra whose nth component consists of the homogeneous graded morphisms f : T → T of degree n and whose differential maps f to d ∘ f − (−1)^{n}f ∘ d. It follows from ideas of Ravenel’s [18] that a DG A-module is small in DA iff it is contained in the smallest strictly full triangulated subcategory of DA containing A and closed under forming direct summands. We reproduce A. Neeman’s proof of this result [17, 2.2] in 5.3.

By applying suitable truncation functors to our DG algebras (9.1) we also generalize a result of [20] on realizing S-equivalences as derived functors (cf. also [13]).

Now suppose that k is a field. A DG augmented algebra is a DG algebra A endowed with a DG module \overline{A} whose homology is isomorphic to k viewed as a DG k-module concentrated in degree 0. There is a DG algebra A^{∗} and an A-A^{∗}-bimodule X such that L(X⊗_{A}?) : DA^{∗} → DA maps A^{∗} to \overline{A} and gives rise to an equivalence between the triangulated subcategories generated by A^{∗} and \overline{A} (10.2). We put \overline{A^{∗}} = RHom_{A}(X, DA), where DA = Hom_{k}(A, k). Then (A^{∗}, \overline{A^{∗}}) is a DG augmented algebra called the Koszul dual (cf. [1]) of (A, \overline{A}). It is unique up to a quasi-equivalence compatible with the augmentation. For example, if A = U(𝔊) for some Lie algebra 𝔊, then A^{∗} may be taken to be Hom_{k}(Λ𝔊, k) with the shuffle product and the usual derivation (6.5). Let A^{∨} = DDA. There is a canonical A^{∗∗}-A^{∨}-bimodule Y which in many cases gives rise to a quasiequivalence A^{∨} →^{∼} A^{∗∗} (10.3). We consider three special cases where A^{∨} is quasi-equivalent to A^{∗∗} and DA is related to DA^{∗} by a fully faithful embedding (10.5).

I am grateful to A. Neeman for pointing out Theorem 5.3 to me and calling my attention to his elegant proof in [17]. I thank the referee for his careful reading of the manuscript.

1. Graded categories and DG categories

1.1 Graded categories. Let k be a commutative ring. The tensor product over k will be denoted by ⊗. A graded category is a k-linear category A whose morphism spaces are Z-graded k-modules

A (A, B) = ∐_{p∈Z} A (A, B)^{p}

such that the composition maps

A (A, B) ⊗ A (B, C) → A (A, C)

are homogeneous of degree 0, ∀ A, B, C ∈ A. A simple example is the category Gra k of graded k-modules V = ∐_{p∈Z} V^{p} with

(Gra k) (V, W)^{p} = {f ∈ Hom_{k}(V, W) : f(V^{q}) ⊂ W^{p+q} , ∀ q}.

A graded category A is concentrated in degree 0 if A (A, B)^{p} = 0 for all p ≠ 0, A, B ∈ A. It is then completely determined by the k-linear category A^{0} having the same objects as A and the morphism spaces A^{0} (A, B) = A (A, B)^{0}.

If A and B are graded categories, a graded functor F : A → B is a k-linear functor whose associated maps

F (A, B) : A (A, B) → B (F A, F B)

are homogeneous of degree 0, ∀ A, B ∈ A.

Let A be a small graded category. The opposite graded category Aop has the same objects as A, its morphism spaces are Aop (A, B) = A (B, A), and the composition is given by

Aop (A, B)^{p} ⊗ Aop (B, C)^{q} → Aop (A, C)^{p+q} , g ⊗ f ↦ (−1)^{pq}fg.

A graded (right) A-module is a graded functor M : Aop → Gra k. For each A ∈ A we denote by A^{∧} the free A-module A (?, A). By definition

A^{∧}(f)g = (−1)^{pq}g ∘ f , ∀ f ∈ A (C, B)^{p} , ∀ g ∈ A (B, A)^{q}.

We define GA to be the category whose objects are graded A-modules and whose morphism spaces (GA) (M, N) consist of the morphisms of functors f : M → N such that fA : MA → NA is homogeneous of degree 0 for each A ∈ A.

If A is concentrated in degree 0, GA identifies with the category of sequences (M_{n})_{n∈Z} of A^{0}-modules (=k-linear contravariant functors from A^{0} to the category of k-modules).

We endow GA with the shift M ↦ M[1]: By definition,

(M[1]A)^{p} = (MA)^{p+1} and (M[1]a)(m) = (−1)^{pq}(Ma)(m)

for a ∈ A (B, A)^{p} and m ∈ (MA)^{q}. For a morphism f : M → N we put (f[1]A)^{p} = (fA)^{p+1}. The shift functor is clearly an autormorphism. Its nth iterate is denoted by M ↦ M[n], n ∈ Z.

The graded category Gra A has the same objects as GA and the morphisms spaces

(Gra A) (M, N) →^{∼} ∐_{p∈Z}(GA) (M, N[p]).

The composition of morphisms produced by f : M → N[q] and g : L → M[p] is given by f[p] ∘ g. We extend the shift functor to an automorphism of Gra A in the obvious way.

1.2 Differential graded categories. A differential graded category (=DG category) is a graded category A whose morphism spaces are endowed with differentials d (i.e. homogeneous maps d of degree 1 with d^{2} = 0) such that

d(fg) = (df)g + (−1)^{p}f(dg) , ∀ f ∈ A (B, C)^{p} , ∀ g ∈ A (A, B).

A simple example is the category Dif k of differential k-modules whose morphism spaces

(Dif k) (V, W) →^{∼} (Gra k) (V, W)

are endowed with the differential mapping (f^{p}) ∈ (Gra k) (V, W)^{n} to

(d ∘ f^{p} − (−1)^{n}f^{p+1} ∘ d).

If A and B are DG categories, a DG functor F : A → B is a graded functor such that F (df) = d(F f) for all morphisms f of A. A quasi-isomorphism F : A → B is a DG functor inducing a bijection obj A → obj B and quasi-isomorphisms A (A, B) → A (F A, F B) for all A, B ∈ A.

Let A be a small DG category. Its opposite Aop is the opposite graded category of A endowed with the same differential as A.

A DG (right) A-module is a DG functor M : Aop → Dif k. Denote by M| the underlying graded A-module of M. The objects of the DG category Dif A are the DG A-modules, its morphism spaces are the graded k-modules

(Dif A) (M, N) = (Gra A) (M| , N|),

endowed with the differential given by

df = d ∘ f − (−1)^{p}f ∘ d,

for each homogeneous f of degree p. One easily verifies that this is well defined.

If A is concentrated in degree 0, DG A-modules are in bijection with differential complexes of A^{0}-modules.

For each A ∈ A, the underlying graded module of the free module A^{∧} is the free graded module associated with A. The differential of A^{∧}(B) equals that of A (B, A). For each DG A-module M and each A ∈ A, the map

(Dif A) (A^{∧}, M) →^{∼} M(A) , f ↦ (fA)(1_{A}).

is an isomorphism of DG k-modules (’Yoneda-isomorphism‘).

We lift the shift functor from graded modules to DG modules by defining the differential of M[1] to be −d[1], where d : M → M[1] is the differential of M.

2. Homotopy categories

2.1 k-linear structures. Let A be a DG category. The category CA (resp. HA) has the same objects as Dif A. Its morphism spaces are

(CA) (M, N) = Z^{0} (Dif A) (M, N) resp. (HA) (M, N) = H^{0}(Dif A) (M, N).

Thus the morphisms of CA are homogeneous of degree 0 and commute with the differential. The morphisms of HA are residue classes \overline{f} of morphisms f of CA modulo null-homotopic morphisms, which by definition are of the form dr + rd for some morphism r : M → N[−1] of GA. We have a canonical projection functor CA → HA. Two DG modules are homotopy equivalent if they become isomorphic in HA. If A is concentrated in degree 0, CA (resp. HA) identifies with the category (resp. the homotopy category) of differential complexes of A^{0}-modules.

2.2 Exact and triangulated structures. We endow CA with an exact structure [16] by defining a conflation (=admissible short exact sequence [7, §9], [12, App. A]) to be a sequence

L →^{i} M →^{p} N

such that the underlying sequence of graded A-modules is split short exact.

We endow HA with the suspension functor S : HA → HA , M ↦ SM = M[1]. We define a triangle of HA to be an S-sequence [14] isomorphic to some

L →^{\overline{i}} M →^{\overline{p}} N →^{\overline{e}} SL ,

where (i, p) is a conflation and e = rds, where r and s are chosen homogeneous morphisms of degree 0 such that ps = 1_{N}, ri = 1_{L} and rs = 0.

Lemma.

a) CA is a Frobenius category [9].

b) HA is a triangulated category [23].

Proof. a) Let F : CA → GA be the forgetful functor. For each N ∈ GA, let F_{ρ}N resp. F_{λ}N be the DG A-modules defined by

(F_{ρ}N)(A) = NA ⊕ (NA)[1] , d = [ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ] , (F_{ρ}N)(a) = [ \begin{matrix}Na & 0 \\ dNa & (−1)^{p}Na \end{matrix} ]
(F_{λ}N)(A) = (NA)[−1] ⊕ NA , d = [ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ] , (F_{λ}N)(a) = [ \begin{matrix}(−1)^{p}Na & 0 \\ (−1)^{p}dNa & Na \end{matrix} ] ,

where A ∈ Aop and a ∈ Aop (A, B)^{p}. For each M ∈ CA, define morphisms of DG A-modules ΦM = [1 d]^{t} : M → F_{ρ}F M and ΨM = [−d 1] : F_{λ}F M → M. We have bijections

(GA) (F M, N) →^{∼} (CA) (M, F_{ρ}N) , f ↦ (F_{ρ}f)(ΦM)
(GA) (N, F M) →^{∼} (CA) (F_{λ}N, M) , f ↦ (ΨM)(F_{λ}f).

Thus F_{ρ}N is injective and F_{λ}N is projective in CA for each N ∈ GA. Since ΦM and ΨM fit into conflations

M ⟶^{ΦM} F_{ρ}F M ⟶ M[1] , M[−1] ⟶ F_{λ}F M ⟶^{ΨM} M ,

we can conclude that CA has enough projectives and enough injectives. Moreover, M is itself projective (resp. injective) iff it is a direct summand of F_{ρ}F M (resp. of F_{λ}F M). Since F_{ρ}F M →^{∼} (F_{λ}F M)[1], we infer that M is projective iff it is injective. For later use, we introduce the notations P M = F_{ρ}F M and IM = F_{λ}F M.

b) HA identifies with the stable category associated with CA. Thus the assertion follows from [9, 9.4].

3. Resolution

3.1 P-resolutions. Let A be a DG category. Its homology category H^{∗}A is the graded category with the same objects as A and with the morphism spaces

(H^{∗}A) (A, B) = ∐_{n∈Z} H^{n}A (A, B).

We have a canonical functor H^{∗} : CA → Gra H^{∗}A defined by

(H^{∗}M)(A) = ∐_{n∈Z} H^{n}M(A).

It induces a functor

HA → GH^{∗}A

which will also be denoted by H^{∗}.

A DG module N is acyclic if H^{∗}N=0. A DG module Q is relatively projective (cf. [15, X, §10]) if, in CA, it is a direct summand of a direct sum of modules of the form A^{∧}[n], A ∈ A, n ∈ Z. A DG module has property (P) if it is homotopy equivalent to a DG module P admitting a filtration

0 = F_{−1} ⊂ F_{0} ⊂ F_{1} ⊂ … F_{p} ⊂ F_{p+1} … ⊂ P , p ∈ N

in CA such that (F1) P is the union of the F_{p}, p ∈ N,

(F2) the inclusion morphism F_{p−1} ⊂ F_{p} splits in GA, ∀ p ∈ N,

(F3) the subquotient F_{p}/F_{p−1} is isomorphic in CA to a relatively projective module, ∀ p ∈ N.

Note that (F1) and (F2) imply that the following sequence (∗) is split exact in GA and hence produces a triangle in HA

∐_{p∈N} F_{p} ⟶^{Φ} ∐_{q∈N} F_{q} ⟶^{can} P ;

here Φ has the components

F_{p} ⟶^{[1 −ι]^{t}} F_{p} ⊕ F_{p+1} ⟶^{can} ∐_{q∈N} F_{q} , ι = incl .

If A is concentrated in degree 0, a DG module P with (F1), (F2) and (F3) yields a complex of projective A^{0}-modules. Conversely a right bounded complex of projective A^{0}-modules gives rise to a DG module P with (F1), (F2) and (F3): Indeed, if P^{q} = 0 for q > 0, we can take F_{p} = ∐_{q>−p} P^{q}.

Theorem.

a) We have (HA) (P, N) = 0 for each acyclic N and each P with property (P).

b) For each M ∈ HA there is a triangle of HA

pM → M → aM → SpM,

where aM is acyclic and pM has property (P).

c) Let

… → \overline{Q_{n}} → \overline{Q_{n−1}} → … → \overline{Q_{1}} → \overline{Q_{0}} → H^{∗}M → 0

be a projective resolution of H^{∗}M in GH^{∗}A such that \overline{Q_{n}} →^{∼} H^{∗}Q_{n} for a relatively projective Q_{n} ∈ CA, ∀ n. Then pM is homotopy equivalent to a module P admitting a filtration F_{p} with (F1), (F2) and such that F_{p}/F_{p−1} →^{∼} Q_{p}[p] in CA, ∀ p.

We shall refer to pM as a P-resolution of M. If A is concentrated in degree 0, assertion c) implies that if M is a (possibly unbounded) complex of A^{0}-modules and Q_{∗}^{p} a given projective resolution of its pth homology, then M is quasi-isomorphic to a complex pM whose nth component is ∐_{p−q=n} Q_{q}^{p} .

We define H_{p}A to be the full subcategory of HA formed by the modules with property (P). Applying suitable Hom-functors to the triangle of b) and using a) we see that we have

(HA) (P, pM) →^{∼} (HA) (P, M) and (HA) (M, N) ←^{∼} (HA) (aM, N)

for all P ∈ H_{p}A and all acyclic N. In particular, if (HA) (M, N) = 0 for each acyclic N, we have 0 = (HA) (M, aM) ←^{∼} (HA) (aM, aM), so that aM = 0 and, by b), pM →^{∼} M. Hence a DG module M lies in H_{p}A iff (HA) (M, N) = 0 for each acyclic N. Therefore H_{p}A is a triangulated subcategory of HA. The inclusion H_{p}A ⊂ HA admits the right S-adjoint [14] M ↦ pM.

It follows from a) that each triangle

P → M → N → P [1],

where N is acyclic and P has property (P), is canonically isomorphic to the triangle of b). If (M_{i})_{i∈I} is a family of modules, we can apply this to the triangle

∐ pM_{i} → ∐ M_{i} → ∐ aM_{i} → ∐ pM_{i}[1]

to conclude that p and a commute with infinite direct sums.

Proof. a) The assertion holds for each P of the form A^{∧}[n], A ∈ A, n ∈ Z, since

(HA) (A^{∧}[n], N) = H^{0}(Dif A)(A^{∧}, N[−n]) = H^{−n}N(A) = 0

for each acyclic N. Hence it holds for relatively projective P. It also holds if F_{p} = P for p ≫ 0 since such a P lies in the triangulated subcategory generated by the relatively projectives. In the general case, we apply HA (?, N) to the triangle produced by the sequence (∗) and obtain an exact sequence

∏_{q∈Z}(HA) (F_{q}, N) ← (HA) (P, N) ← ∏_{p∈Z}(HA) (F_{p}[1], N).

Its outer terms vanish by the foregoing case.

b), c) Following [15, XII, 11] we endow CA with another exact structure: Its class of conflations E consists of the sequences

L → M → N

such that

0 → L(A)^{n} → M(A)^{n} → N(A)^{n} → 0
and 0 → H^{n}L(A) → H^{n}M(A) → H^{n}N(A) → 0

are short exact sequences of k-modules, for all A ∈ A, n ∈ Z. This is equivalent to requiring that

0 → L(A)^{n} → M(A)^{n} → N(A)^{n} → 0
and 0 → Z^{n} L(A) → Z^{n} M(A) → Z^{n} N(A) → 0

be short exact for all A ∈ A, n ∈ Z. The isomorphisms

(CA) (A^{∧}[−n], M) = Z^{0} (Dif A) (A^{∧}, M[n]) = Z^{n} M(A)
(CA) (P A^{∧}[−n], M) = M(A)^{n}

(2.2) show that if Q is relatively projective, then Q and P Q are E-projective. It is also clear that for each module M we may find an E-projective Q′ = Q ⊕ P Q′′ and a morphism p : Q′ → M inducing surjections

Q′(A)^{n} → M(A)^{n} and Z^{n} Q′(A) → Z^{n} M(A) , ∀ A ∈ A, ∀ n ∈ Z.

If K → Q′ is a kernel of p in CA, it is clear that K → Q′ → M is indeed a conflation. Thus, CA has enough E-projectives and we can inductively construct an E-resolution of M, i.e. an E-acyclic complex [12, 4.1]

… → Q′_{n} → Q′_{n−1} → … → Q′_{1} → Q′_{0} ⟶^{ε} M → 0

with E-projective Q′_{n} = Q_{n} ⊕ P Q′′_{n}, where Q_{n} and Q′′_{n} are relatively projective. Under the hypotheses of c), we can refine this construction as follows: The map

(CA) (Q, M) → (GH^{∗}A) (H^{∗}Q, H^{∗}M)

is clearly surjective if Q is of the form A^{∧}[n] for some A ∈ A, n ∈ Z. Hence it is surjective for relatively projective Q. We can therefore lift the given morphism \overline{Q_{0}} → H^{∗}M to a morphism p : Q_{0} → M of CA. Now we choose an E-projective P Q′′_{0}, with relatively projective Q′′_{0}, and a morphism q : P Q′′_{0} → M inducing epimorphisms

P Q′′_{0}(A)^{n} → M(A)^{n} , ∀ A ∈ A, ∀ n ∈ Z.

Then

Q′_{0} = Q_{0} ⊕ P Q′′_{0} ⟶^{[p q]} M

is the required deflation (=admissible epimorphism) with E-projective Q′_{0}. Observe that, since P Q′′_{0} is null-homotopic, Q′_{0} is homotopy equivalent to Q_{0}. Since H^{∗} : CA → GH^{∗}A carries Econflations to short exact sequences, we can successively lift the given resolution of H^{∗}M to an E-acyclic sequence

… → Q′_{n} → Q′_{n−1} → … → Q′_{1} → Q′_{0} ⟶^{ε} M → 0

such that Q′_{n} = Q_{n} ⊕ P Q′′_{n} for all n ∈ N. If

K = (… → K^{n} ⟶^{d_{K}^{n}} K^{n+1} → …) , n ∈ Z

is a differential complex over CA, its total module Tot K has the underlying graded module

∐_{n∈Z} K^{n}[−n]

and the differential

d = d_{K^{n}[−n]} + d_{K}^{n}.

Put

pM = Tot (… → Q′_{m} → Q′_{m−1} → … → Q′_{1} → Q′_{0} → 0 → 0 → …)

and

F′_{p} = Tot (… → 0 → 0 → Q′_{p} → Q′_{p−1} → … → Q′_{1} → Q′_{0} → 0 → 0 → …) , p ≥ 0.

Then pM with the filtration by the F′_{p} clearly satisfies (F1) and (F2), and F′_{p}/F′_{p−1} = Q′_{p}[p], ∀ p. By the lemma we will prove in 3.4, this implies that pM has property (P). The morphism ε : Q′_{0} → M induces a morphism φ : pM → M. It remains to be shown that H^{∗}φ is invertible or, equivalently, that

N = Tot (… → Q′_{m} → … → Q′_{1} → Q′_{0} → M → 0 → …)

is acyclic. This follows from the lemma we will prove in 3.3 applied to each N(A), A ∈ A.

3.2 I-resolutions. We record without proof the following ’dual‘ of 3.1. Fix an injective generator E of the category of k-modules. For each A ∈ A define the A-module A^{∨} by

B ↦ (Dif k) (A (A, B), E),

where E is viewed as a DG k-module concentrated in degree 0. A DG A-module is relatively injective if, in CA, it is a direct summand of a direct product of modules A^{∨}[n], A ∈ A, n ∈ Z. A DG module has property (I) if it is homotopy equivalent to a DG module I admitting a filtration

I = F_{0} ⊃ F_{1} ⊃ … ⊃ F_{p} ⊃ F_{p+1} ⊃ …, p ∈ N ,

such that (F1’) the canonical morphism I → lim_{⟵} I/F_{p} is invertible, (F2’) the inclusion morphism F_{p+1} ⊂ F_{p} splits in GA for all p ∈ N, (F3’) the subquotient F_{p}/F_{p+1} is isomorphic in CA to a relatively injective module, ∀ p ∈ N. By (F1’) and (F2’) the following sequence (∗′) is split exact in GA and hence produces a triangle in HA

I ⟶^{can} ∏_{p∈N} I/F_{p} ⟶^{Φ′} ∏_{q∈N} I/F_{q} ;

here Φ′ has the components

∏_{p∈N} I/F_{p} ⟶^{can} I/F_{q+1} ⊕ I/F_{q} ⟶^{[−π 1]} I/F_{q} ,

where π is the canonical projection I/F_{q+1} → I/F_{q}.

Theorem.

a) We have (HA) (N, I) = 0 for each acyclic N and each I with property (I).

b) For each M ∈ HA there is a triangle of HA

a′M → M → iM → Sa′M ,

where a′M is acyclic and iM has property (I). c) Let

0 → H^{∗}M → \overline{J_{0}} → \overline{J_{1}} → … → \overline{J_{n}} → \overline{J_{n+1}} → …

be an injective resolution of H^{∗}M in GH^{∗}A such that \overline{J_{n}} →^{∼} H^{∗}J_{n} for a relatively injective J_{n} ∈ CA, ∀ n. Then iM is homotopy equivalent to a module I admitting a decreasing filtration F_{p} with (F1’) and (F2’) and such that F_{p}/F_{p+1} →^{∼} J_{p}[−p] in CA for all p ∈ N.

3.3 Acyclic total complexes. Let

N = ∐_{p,q∈Z} N^{pq}

be a bigraded abelian group with commuting differentials d_{I} and d_{II} of bidegree (1, 0) and (0, 1), respectively. Let Tot N and T̂ot N be the differential graded groups with components

(Tot N)^{n} = ∐_{p+q=n} N^{pq} resp. (T̂ot N)^{n} = ∏_{p+q=n} N^{pq} , n ∈ Z ,

and the differential given by

dt = d_{I}t + (−1)^{p}d_{II}t , t ∈ N^{pq}.

For r ∈ Z denote by N^{∗r} (resp. B^{∗r}, Z^{∗r}, H^{∗r}) the differential graded groups with components

N^{nr} (resp. Im d_{II}^{n,r−1}, Ker d_{II}^{nr}, Ker d_{II}^{nr}/Im d_{II}^{n,r−1} ) , n ∈ Z ,

and the differential induced by d_{I}.

Lemma. If N^{∗r} and H^{∗r} are acyclic for all r ∈ Z, then Tot N and T̂ot N are acyclic.

Proof. If N^{∗r} is acyclic for all r ∈ Z, the same holds for the B^{∗r}. Thus if N^{∗r} and H^{∗r} are acyclic for all r ∈ Z, then so are the Z^{∗r}. To prove that Tot N is acyclic we consider the differential bigraded subgroups N_{m} ⊂ N, m ≥ 1, with N_{m}^{∗r} = 0 for r ∉ [−m, m], N_{m}^{∗r} = N^{∗r} for r ∈ [−m, m − 1], and N_{m}^{∗m} = Z^{∗m}. Clearly each Tot N_{m} admits a finite filtration with acyclic subquotients and hence is acyclic. Since we have

Tot N ←^{∼} Tot lim_{⟶} N_{m} ←^{∼} lim_{⟶} Tot N_{m} ,

the assertion follows. Similarly, to prove that T̂ot N is acyclic, we consider the quotients Q_{m} of N, m ≥ 1, with Q_{m}^{∗r} = 0 for r ∉ [−m, m], Q_{m}^{∗r} = N_{m}^{∗r} for r ∈ [−m + 1, m] and Q_{m}^{∗,−m} = B^{∗,−m+1}. As above, each T̂ot Q_{m} is acyclic and we have

T̂ot N →^{∼} T̂ot lim_{⟵} Q_{m} →^{∼} lim_{⟵} T̂ot Q_{m}.

Moreover for each m ≥ 1, the components of the canonical morphism

p_{m} : T̂ot Q_{m+1} → T̂ot Q_{m}

are surjective. Therefore, p_{m} also induces surjections onto the groups B^{n}T̂ot Q_{m} = Z^{n}T̂ot Q_{m}, n ∈ Z. By the Mittag-Leffler-criterion [8, 0_{III}, 13.1], T̂ot N is acyclic.

3.4 Adjusting limits. Let P′ be a DG A-module and

F′_{0} ⊂ F′_{1} ⊂ … ⊂ F′_{p} ⊂ … ⊂ P′

a filtration satisfying (F1) and (F2). Suppose that for each p ≥ 1 a DG module Q_{p} and a homotopy equivalence F′_{p}/F′_{p−1} →^{∼} Q_{p} are given.

Lemma. The DG module P′ is homotopy equivalent to a DG module P admitting a filtration F_{p} satisfying (F1) and (F2) and such that F_{p}/F_{p−1} is isomorphic to Q_{p} in CA, ∀ p.

Proof. We will inductively construct a sequence

F_{0} ⊂ F_{1} ⊂ … ⊂ F_{p} ⊂ …

and a sequence of homotopy equivalences \overline{f_{p}} : F′_{p} → F_{p} such that the squares

F′_{p} → F′_{p+1}
\overline{f_{p}} ↓  ↓ \overline{f_{p+1}}
F_{p} → F_{p+1}

are commutative (in HA), the sequence F_{p} satisfies (F2) and F_{p}/F_{p−1} →^{∼} Q_{p} in CA, ∀ p. Of course, we put F_{0} = Q_{0} and let \overline{f_{0}} : F′_{0} → F_{0} be the given homotopy equivalence. Suppose that the construction has been completed for all p < n. We have

Ext_{CA}(F′_{n}/F′_{n−1}, F′_{n−1}) →^{∼} Ext_{CA}(Q_{n}, F_{n−1}) ,

where Ext_{CA} denotes classes of extensions in the exact category CA (2.2). We choose a conflation

F_{n−1} → F_{n} → Q_{n}

whose class corresponds to that of the given extension of F′_{n}/F′_{n−1} by F′_{n−1}. Then we have a commutative diagram

F′_{n−1} → F′_{n} → F′_{n}/F_{n−1} → F′_{n−1}[1]
\overline{f_{n−1}} ↓  ↓  ↓ \overline{f_{n−1}}[1]
F_{n−1} → F_{n} →  Q_{n}  → F_{n−1}[1]

We choose \overline{f_{n}} so as to fit into the diagram. Now let P be the union of the F_{p}. Using the sequence (∗) of 3.1 we get triangles

∐_{p∈Z} F′_{p} ⟶^{\overline{Φ}} ∐_{q∈Z} F′_{q} ⟶ P′ ⟶ S ∐_{p∈Z} F′_{p}

∐_{p∈Z} F_{p} ⟶^{\overline{Φ}} ∐_{q∈Z} F_{q} ⟶ P ⟶ S ∐_{p∈Z} F_{p}.

The \overline{f_{p}} yield a commutative square

∐_{p∈Z} F′_{p} ⟶^{\overline{Φ}} ∐_{q∈Z} F′_{q}
\overline{a} ↓  ↓ \overline{b}
∐_{p∈Z} F_{p} ⟶^{\overline{Φ}} ∐_{q∈Z} F_{q}

where \overline{a} and \overline{b} are homotopy equivalences. Using axiom TR3 [23, Ch. I, §1] and the five lemma we see that P is homotopy equivalent to P′.

4. Derived categories and stable categories

4.1 Derived categories. Let A be a small DG category. Let Σ be the class of quasiisomorphisms of HA (i.e. morphisms \overline{s} such that H^{∗}\overline{s} is invertible). By definition [11, Ch. VI, 10] the derived category of A is the localization DA = (HA)[Σ^{−1}] [23]. It follows from theorem 3.1 that the canonical functor HA → DA induces an equivalence H_{p}A → DA. If A is concentrated in degree 0, DA identifies with the unbounded derived category of the category of A^{0}-modules. As in the case of the derived category of an exact category, one constructs [7, 12.3] a functor which completes the images in DA of pointwise short exact sequences of CA into triangles.

Since (infinite) direct sums of acyclic modules are acyclic, DA has direct sums, and the canonical functors CA → HA → DA commute with direct sums.

4.2 Small objects and generators. Let A be a small DG category and T a k-linear triangulated category with infinite direct sums. An object X ∈ T is small if T (X,?) commutes with (infinite) direct sums. By the five lemma, if two vertices of a triangle of T are small, then so is the third one. Each A^{∧} is small in DA. Indeed, let (M_{i})_{i∈I} be a family of modules and A ∈ A. Then

(DA)(A^{∧}, ∐_{i∈I} M_{i}) →^{∼} H^{0} ∐ M_{i}(A) →^{∼} ∐ H^{0}M_{i}(A) →^{∼} ∐_{i∈I}(DA)(A^{∧}, M_{i}).

Let H_{p}^{b}A be the smallest strictly (=closed under ismorphisms) full triangulated subcategory of H_{p}A containing the A^{∧}, A ∈ A.

A set X ⊂ T is a set of generators if T coincides with its smallest strictly full triangulated subcategory containing X and closed under direct sums. It follows from the sequence (∗) of 3.1 that the A^{∧}, A ∈ A, form a set of generators for DA.

Let F, F′ : DA → T be two k-linear S-functors commuting with direct sums and μ : F → F′ a morphism of S-functors [14].

Lemma.

a) The restriction of F to H_{p}^{b}A is fully faithful iff F induces bijections

(DA) (A^{∧}, B^{∧}[n]) → T (F A^{∧}, F B^{∧}[n])

for all A, B ∈ A, n ∈ Z. b) F is fully faithful if F | H_{p}^{b}A is fully faithful and F A^{∧} is small for each A ∈ A.

c) F is an equivalence iff F | H_{p}^{b}A is fully faithful and the F A^{∧}, A ∈ A, form a set of small generators for T .

d) The morphism μ : F → F′ is invertible iff μA^{∧} is invertible for each A ∈ A.

Proof. a) results from ’devissage‘ (cf. e.g. [9, 10.10]).

b) Let A ∈ A. By the five lemma, the modules M such that the map

(DA) (A^{∧}, M) → T (F A^{∧}, F M)

is bijective form a strictly full triangulated subcategory of DA. It contains all the generators B^{∧}, B ∈ A, and is closed under infinite direct sums (since both, A^{∧} and F A^{∧}, are small and F commutes with infinite direct sums). This subcategory therefore coincides with DA. The same argument shows that for fixed M ∈ DA, the map

(DA) (L, M) → T (F L, F M)

is bijective for each L ∈ DA.

c) is now clear.

d) The DG modules M with invertible μM form a strictly full triangulated subcategory of DA which moreover is closed under infinite direct sums. This subcategory equals DA iff it contains the A^{∧}, A ∈ A, as these form a set of generators for DA.

4.3 Stable categories. Let E be a k-linear Frobenius category [9] with (infinite) direct sums. Since E has enough injectives, it is clear that direct sums of conflations (=admissible short exact sequences) of E are conflations. Moreover, direct sums of injectives (=projectives in E) are injective. In particular, the associated stable category \underline{E} is a triangulated category with infinite direct sums. Suppose that \underline{E} admits a set of small generators X ⊂ \underline{E}.

Theorem. (cf. [5, Ex. 5.3 H]) There is a DG category A and an S-equivalence G : \underline{E} → DA giving rise to an equivalence between X ⊂ \underline{E} and the full subcategory of DA formed by the free modules A^{∧}, A ∈ A.

Proof. Let Ẽ be the category of acyclic [14, 1.5] differential complexes

P = (… → P^{n} →^{d} P^{n−1} → …) , n ∈ Z

with projective components P^{n} ∈ E. Endow Ẽ with the pointwise split short exact sequences. Then Ẽ is a Frobenius category and it is easy to see that the functor P ↦ Z^{0} P induces an S-equivalence

G_{1} : ˜ \underline{E} → \underline{E}.

For each X ∈ X, choose X̃ ∈ Ẽ with Z^{0} X̃ →^{∼} X. Let A be the DG category whose objects are the X̃ and whose morphism spaces are

A (X̃, Ỹ) →^{∼} Hom (X̃, Ỹ) ,

where for P, Q ∈ Ẽ, the DG k-module Hom (P, Q) has the components

∏_{p∈Z} E (P^{p}, Q^{n+p}) , n ∈ Z ,

and the differential given by d(f^{p}) = (d ∘ f^{p} − (−1)^{n}f^{p+1} ∘ d). Note that

˜\underline{E} (P, S^{n}Q) →^{∼} H^{n}Hom (P, Q).

It is clear that the composition of the exact functor

Ẽ → CA , P ↦ (X̃ ↦ Hom (X̃, P))

with the canonical projection CA → DA vanishes on projectives of Ẽ (=null-homotopic complexes in Ẽ ) and hence induces an S-functor

G_{2} : ˜ \underline{E} → DA.

For X̃ ∈ X̃ the module G_{2}X̃ is isomorphic to X̃^{∧}, the free module associated with X̃ ∈ A. If P_{i}, i ∈ I, is a family in Ẽ and X̃ ∈ X̃, the nth homology of the morphism

∐ Hom (X̃, P_{i}) → Hom (X̃, ∐ P_{i})

identifies with

∐ \underline{Ẽ}(X̃, S^{n}P_{i}) → \underline{Ẽ}(X̃, ∐ S^{n}P_{i}) ,

which is bijective since X̃ is small in Ẽ. Hence G_{2} commutes with direct sums. We have already seen that G_{2} induces bijections

\underline{Ẽ}(X̃, S^{n}Ỹ) →^{∼} H^{n}Hom (X̃, Ỹ) →^{∼} H^{n}A (X̃, Ỹ) →^{∼} (DA) (G_{2}X̃, S^{n}G_{2}Ỹ) , X̃, Ỹ ∈ X̃ , n ∈ Z.

By the argument of 4.2 b), we conclude that G_{2} is fully faithful. The essential image of G_{2} contains the generators A^{∧}, A ∈ A, of DA. So G_{2} is essentially surjective. We let G be the composition of G_{2} with an S-quasi-inverse of G_{1}.

5. Small objects

Let A be a small DG category. Each free module A^{∧}, A ∈ A, is small in DA, and so are the objects of the smallest strictly full triangulated subcategory of DA containing the A^{∧}, A ∈ A, and closed under forming direct summands. Ravenel’s ideas [18] imply that this subcategory coincides with the full subcategory of small objects of DA. In 5.3, we give A. Neeman’s proof [17, 2.2] of Ravenel’s result.

5.1 Homotopy limits and small objects. Let T be a triangulated category with (infinite) sums. Let

X_{0} →^{f_{0}} X_{1} →^{f_{1}} … → X_{p} →^{f_{p}} X_{p+1} → …, p ∈ N

be a sequence of morphisms of T . Let there be given a homotopy limit of the sequence, i.e. an object X with morphisms ψ_{p} : X_{p} → X fitting into a triangle

∐ X_{p} →^{Φ} ∐ X_{q} →^{Ψ} X → S ∐ X_{p} ,

where Φ is defined as in 3.1 and Ψ has the components ψ_{q}. Note that a homotopy limit is unique up to non-unique isomorphism.

Let M ∈ T be small. Then T (M,?) commutes with direct sums and thus transforms the above triangle into the long exact sequence

… → ∐ T (M, X_{p}) →^{Φ_{∗}} ∐ T (M, X_{q}) →^{Ψ_{∗}} T (M, X) → …

It is easy to see that (SΦ)_{∗} is injective. We therefore have an isomorphism

lim_{⟶} T (M, X_{p}) →^{∼} Cok Φ_{∗} →^{∼} T (M, X).

5.2 Brown’s representability theorem. Keep the hypotheses of 5.1 and assume that T admits a set of small generators X. For completeness we include a proof of the following

Theorem. [3] A cohomological functor F : T → (Ab)op is representable iff it commutes with direct sums.

Remark. More precisely, the proof will show that each such F is represented by the homotopy limit of a sequence

X_{0} →^{f_{0}} X_{1} → … → X_{p} →^{f_{p}} X_{p+1} → …, p ∈ N ,

where X_{0} as well as the cone (=third corner of a triangle) over each f_{p} is an (infinite) sum of objects S^{n}X, X ∈ X, n ∈ Z. In particular, each M ∈ T is the homotopy limit of such a sequence, as we see by taking F = T (?, M).

Proof. We have to prove that the condition is sufficient. Let X^{+} be the class of direct sums of objects S^{n}X, n ∈ Z, X ∈ X. For each M ∈ T put M^{∧} = T (M,?). Since X is a set, there is an X_{0} ∈ X^{+} and a morphism π_{0} : X_{0}^{∧} → F inducing a surjection

X_{0}^{∧}(S^{n}X) → F S^{n}X

for all X ∈ X, n ∈ Z. We will inductively construct a sequence

X_{0} →^{f_{0}} X_{1} → … → X_{p} →^{f_{p}} X_{p+1} → …, p ∈ N ,

and morphisms π_{p+1} : X_{p+1}^{∧} → F such that π_{p+1}f_{p}^{∧} = π_{p}. Suppose that for some p ≥ 0 we have constructed X_{p} and π_{p}. Choose Z_{p} ∈ X^{+} admitting a morphism ρ_{p} : Z_{p} → X_{p} which induces a surjection

Z_{p}^{∧}(S^{n}X) → Ker π_{p}(S^{n}X)

for all X ∈ X, n ∈ Z. Define X_{p+1} by the triangle

Z_{p} →^{ρ_{p}} X_{p} →^{f_{p}} X_{p+1} → SZ_{p}.

Since we have an exact sequence

F Z_{p} ←^{F ρ_{p}} F X_{p} ← F X_{p+1}

and by definition π_{p}ρ_{p}^{∧} = 0, we can choose π_{p+1} : X_{p+1}^{∧} → F such that π_{p+1}f_{p}^{∧} = π_{p}. Define X_{∞} by the triangle

∐_{p∈N} X_{p} →^{Φ} ∐_{q∈N} X_{q} →^{Ψ} X_{∞} → S ∐_{p∈N} X_{p} ,

where Φ has the components

X_{p} ⟶^{[1 −f_{p}]^{t}} X_{p} ⊕ X_{p+1} ⟶^{can} ∐_{q∈N} X_{q}.

Since F : T → (Ab)op commutes with direct sums, it takes sums of T to products of Ab. Thus we have an exact sequence

∏_{p∈N} F X_{p} ← ∏_{q∈N} F X_{q} ← F X_{∞} ,

which shows that there is a morphism π_{∞} : X_{∞}^{∧} → F such that π_{∞}Ψ_{q}^{∧} = π_{q}^{∧} for all q ∈ N. By an easy diagram chase we see that π_{∞} induces an isomorphism

T (S^{n}X, X_{∞}) → F S^{n}X

for all X ∈ X, n ∈ Z. Since X generates T , we can conclude that π_{∞} is an isomorphism.

5.3 Small objects. Keep the hypotheses of 5.2. If U and V are classes of objects of T , we denote by U ∗ V the class of objects X occuring in a triangle

U → X → V → SU

with U ∈ U, V ∈ V. The octahedral axiom implies that the operation ∗ is associative. The objects of X ∗X ∗…∗X (n factors) are called extensions of length n of objects of X. The following theorem and its proof can be found in [17, 2.2].

Theorem. [18] [17] Each small object of T is a direct summand of an extension of objects S^{n}X, X ∈ X, n ∈ Z.

Remarks. a) We will of course apply the theorem to the case where T is the derived category of a DG algebra A and where X consists of the free modules A^{∧}, A ∈ A.

b) One can adapt the proof of [19, 6.3] to show that, if A is a negative DG category, i.e. A (A, B)^{n} = 0 for all n > 0, A, B ∈ A, then each small object of DA is an extension of DA-direct summands of finite sums of free modules A^{∧}, A ∈ A.

Proof. [17] Let M be a small object of T . Choose a sequence

X_{0} →^{f_{0}} X_{1} → … → X_{p} →^{f_{p}} X_{p+1} → …, p ∈ N ,

as in remark 5.2. By 5.1 we have an isomorphism

lim_{⟶} T (M, X_{p}) →^{∼} T (M, M).

In particular, the identity of M factors through some X_{p}, which means that M is a direct summand of X_{p}. Now X_{p} is an extension of sums of objects S^{n}X, X ∈ X, n ∈ Z. So we can apply the following lemma to Z′ = 0 and Z = X_{p} to obtain the commutative square

\begin{aligned}M′ & → & M \\ ↓ & & ↓ \\ 0 & → & X_{p} , \end{aligned}

where the cone on the first line is an extension M′′ of objects S^{n}X, X ∈ X, n ∈ Z. Since M → X_{p} is a (split) monomorphism, the morphism M′ → M vanishes and thus M is a direct summand of M′′.

Lemma. [17, 2.3] Let M ∈ T be small and let c : Z′ → Z be a morphism whose mapping cone is an extension of (infinite) sums of objects S^{n}X, X ∈ X, n ∈ Z. Then each diagram

\begin{aligned}& & M \\ & & ↓ \\ Z′ & →^{c} & Z \end{aligned}

may be completed to a commutative square

\begin{matrix}M′ & →^{b} & M \\ ↓ & & ↓ \\ Z′ & →^{c} & Z \end{matrix}

such that the cone over b is an extension of objects S^{n}X, X ∈ X, n ∈ Z.

Proof. By assumption the cone Z′′ over c is an extension of sums of objects S^{n}X, X ∈ X, n ∈ Z. We proceed by induction on the length l of Z′′. If we have l = 1, then Z′′ is itself a sum of objects S^{n}X, X ∈ X, n ∈ Z. By the smallness of Y, the composition M → Z → Z′′ factors through a finite subsum M′′ ⊂ Z′′. We find the required square by completing

M → M′′
↓  ↓
Z′ →^{c} Z → Z′′ → SZ′

to a morphism of triangles

M′ →^{b} M → M′′ → SM′
↓ ↓ ↓ ↓
Z′ →^{c} Z → Z′′ → SZ′.

If we have l > 1, then Z′′ occurs in a triangle

Z′′_{0} → Z′′ → Z′′_{1} → SZ′′_{0}

where both, Z′′_{0} and Z′′_{1}, are of length < l. By forming an octahedron over

Z → Z′′ → Z′′_{1}

we see that c is the composition of two morphisms c_{0} and c_{1} whose cones are Z′′_{0} and Z′′_{1}. By the induction hypothesis we have a commutative diagram

M′ →^{b_{0}} M_{1} →^{b_{1}} M
↓ ↓ ↓
Z′ →^{c_{0}} Z_{1} →^{c_{1}} Z ,

where the cones of b_{0} and b_{1} are extensions of objects of X. By the octahedral axiom the same holds for b = b_{1}b_{0}.

6. Standard functors

6.1 Hom and tensor. Let A and B be small DG categories. The tensor product A ⊗ B is the DG category whose objects are the pairs (A, B) of objects A ∈ A, B ∈ B, and whose morphism spaces are

(A ⊗ B) ((A, B), (A′, B′)) →^{∼} A (A, A′) ⊗ B (B, B′).

The composition of A ⊗ B is given by the formula

(f′ ⊗ g′)(f ⊗ g) = (−1)^{pq}f′f ⊗ g′g

for f ∈ A (A, A′)^{p} and g′ ∈ B (B′, B′′)^{q}.

Let X be an A-B-bimodule, i.e. a module over A ⊗ Bop. It gives rise to a pair of adjoint DG functors

\begin{matrix}Dif A \\ T_{X} ↑↓ H_{X} \\ Dif B \end{matrix}

which are defined as follows

(H_{X}M)(B) = (Dif A) (X(?, B), M)
(T_{X}N)(A) = Cok (∐_{B,C∈B} NC ⊗ B (B, C) ⊗ X(A, B) →^{ν} ∐_{B∈B} NB ⊗ X(A, B)) ,

where ν(n ⊗ f ⊗ x) = (Nn)(f) ⊗ x − n ⊗ X(A, f)(x). Observe that for each B ∈ B we have

T_{X}B^{∧} →^{∼} X(?, B) since

(Dif A) (T_{X}B^{∧}, M) = (Dif B) (B^{∧}, H_{X}M) = (H_{X}M)(B) = (Dif A) (X(?, B), M)

for each M ∈ Dif A. For brevity, we put X^{B} = X(?, B).

The functors H_{X} and T_{X} induce a pair of adjoint functors between HA and HB which will also be denoted by H_{X} and T_{X}. We denote by LT_{X} the left derived functor of T_{X}, i.e. the composition

DB → H_{p}B ⟶^{T_{X}} HA → DA , N ↦ T_{X}pN.

Observe that LT_{X} commutes with direct sums since p and T_{X} do.

Lemma.

a) LT_{X} is an equivalence iff the morphisms B (B, C) → (Dif A) (X^{B}, X^{C}) induce isomorphisms in homology, ∀ B, C ∈ B, and the X^{B}, B ∈ B, form a set of small generators for DA.

b) A morphism X → X′ of A-B-bimodules is a quasi-isomorphism iff the induced morphism LT_{X} → LT_{X′} is invertible.

c) Suppose that X has property (P) over A ⊗ Bop. If A is k-flat, then T_{X} preserves acyclicity. If B is k-projective, then T_{X} preserves property (P). If k is a field then T_{X}N has property (P) for each DG B-module N.

Proof. a) follows from 4.2 c), and b) from 4.2 d). It suffices to prove c) for the case where X = (A′, B′)^{∧} for some (A′, B′) ∈ A ⊗ Bop. Then we have T_{X}N = N(B′) ⊗_{k} A (A′,?). So the first two assertions are clear. To prove the last one, we fix an acyclic DG A-module M and observe that

(Dif A) (T_{X}N, M) →^{∼} (Dif k) (N(B′), M(A′)).

Since k is a field, M(A′) is even null-homotopic. Hence we have (HA) (T_{X}N, M) = 0, and the assertion follows from 3.1.

Example. Let F : B → A be a DG functor and put X(A, B) = A (A, F B) for A ∈ A, B ∈ B. Then clearly X^{B} = (F B)^{∧}. Hence LT_{X} is an equivalence iff H^{∗}F : H^{∗}A → H^{∗}B is an equivalence.

6.2 Right projective bimodules. We keep the assumptions of 6.1 and assume in addition that X^{B} has property (P) for each B ∈ B. Since

(H_{X}M)(B) = (Dif A) (X^{B}, M) ,

it follows from theorem 3.1 that H_{X}M is acyclic for each acyclic M. The induced functor DA → DB will be denoted by RH_{X}. We have

(HA) (T_{X}P, M) = (HB) (P, H_{X}M) = 0

whenever P has property (P) and M is acyclic. By 3.1 we conclude that T_{X} preserves property (P). Using this we see that

(DA) (LT_{X}N, M) = (HA) (T_{X}pN, M) = (HB) (pN, H_{X}M) = (DB) (N, RH_{X}M) ,

i.e. that RH_{X} is a right adjoint of LT_{X}.

Now define a B-A-module X^{⊤} by

X^{⊤}(B, A) = (Dif A) (X^{B}, A^{∧}).

For each M ∈ Dif A, we have a canonical morphism T_{X^{⊤}}M → H_{X}M.

Lemma.

a) The morphism LT_{X^{⊤}}M → RH_{X}M is invertible for all M ∈ H_{p}^{b}A. It is invertible for all M iff the X^{B} are small in DA, ∀ B ∈ B.

b) If LT_{X} : DB → DA is an equivalence, its quasi-inverse is isomorphic to LT_{X^{⊤}}.

Proof. a) The morphism is clearly invertible for free M. By ’devissage‘ it is invertible for M ∈ H_{p}^{b}A. Since H_{X} commutes with infinite direct sums iff the X^{B} are small, the second assertion follows from 4.2 d).

b) If LT_{X} is an equivalence then so is RH_{X}. In particular, RH_{X} commutes with direct sums. The assertion now follows from a) and 4.2 d).

Example. Keep the notations of example 6.1. If LT_{X} is an equivalence, a quasi-inverse is given by LT_{X^{⊤}}.

6.3 Flat targets. We keep the assumptions of 6.1 and assume in addition that A is k-flat, i.e. A (A, B) is a flat k-module, ∀ A, B ∈ A. Let pX be a P-resolution of X over A ⊗ Bop. Note that for B ∈ B the A-module (pX)^{B} need not have property (P) (unless B (B′, B) is projective over k for each B′ ∈ B). In particular, the canonical morphism p(X^{B}) → (pX)_{B} of HA need not be a quasi-isomorphism.

Lemma.

a) We have LT_{X}N →^{∼} T_{pX}N for each N ∈ DB.

b) Let C be another DG category and Y a B-C-bimodule. We have LT_{X}LT_{Y} →^{∼} LT_{Z}, where Z = T_{pX}Y.

Proof. a) By 6.1 b) we have LT_{pX} →^{∼} LT_{X}. So we only have to show that LT_{pX}N →^{∼} T_{pX}N for each N ∈ DB. It is enough to check that T_{pX}N is acyclic for each acyclic N. Now T_{pX}N inherits from pX a complete filtration which splits in GA and has subquotients T_{Q}N, where Q is relative projective. So it is enough to show that T_{F} N is acyclic for each F = (A′, B′)^{∧}, (A′, B′) ∈ Aop ⊗ B. But

(T_{F} N)(A) →^{∼} A (A, A′) ⊗ N(B′).

b) follows from a) and the fact that T_{pX}T_{Y} →^{∼} T_{Z} as functors Dif C → Dif A.

6.4 Tensor functors and DG functors. Let A and B be small DG categories. Let F : Dif B → Dif A be an arbitrary DG functor. Its left derived functor is the composition

DB → H_{p}B →^{F} HA → DA , N ↦ F pN

Let X be the bimodule X(A, B) = (F B^{∧})(A) = (Dif A) (A^{∧}, F B^{∧}). For each B-module N, the canonical morphism

NB →^{∼} (Dif B) (B^{∧}, N) → (Dif A) (F B^{∧}, F N) = (Dif A) (X(?, B), F N) = (H_{X}F N)(B)

comes from a natural morphism N → H_{X}F N. By adjunction, we obtain T_{X}N → F N. The induced morphism

LT_{X}N → LF N

is clearly invertible for N = B^{∧}[n], B ∈ B, n ∈ Z. This implies the first assertion of the following lemma. The second one follows from lemma 4.2.

Lemma. The canonical morphism

LT_{X}N → LF N

is invertible for each N ∈ H_{p}^{b}B. It is invertible for all N ∈ DB iff LF commutes with direct sums.

6.5 Example: Lie algebra cohomology. Let R be a k-algebra with 1 and L a (k, R)-Lie algebra [21, §2], i.e. L is a Lie algebra over R, and R is endowed with a left L-module structure such that

[X, rY] = (Xr)Y + r[X, Y]

for all X, Y ∈ L, r ∈ R. In addition, we assume that L is projective as an R-module. For example this holds for the (R, C^{∞}(M))-Lie algebra formed by the C^{∞}-vector fields on a C^{∞}-manifold M [21, §4]. Let the Lie algebra Z be the semi-direct product of L by R and let A be the ’universal algebra of differential operators generated by R and L‘: A is an associative k-algebra endowed with a k-linear morphism ι : Z → A which is universal for the properties

ι([U, V]) = [ι(U), ι(V)] and ι(rU) = ι(r)ι(U)

for all U, V ∈ Z, r ∈ R. The canonical Z-action on R uniquely extends to an A-module structure. Let ε denote the map A → R, a ↦ a.1.

Let E be the graded exterior R-algebra over L and let X be the differential complex with components X^{n} = A ⊗_{R} E^{−n} and the differential [21, §4]

d(a ⊗ X_{1} ∧ … ∧ X_{n}) = ∑_{i=1}^{n}(−1)^{i−1}aX_{i} ⊗ X_{1} ∧ … X̂_{i} … ∧ X_{n}

+ ∑_{j<k}(−1)^{j+k}a ⊗ [X_{j}, X_{k}] ∧ X_{1} ∧ … X̂_{j} … X̂_{k} … ∧ X_{n}.

The complex X together with the augmentation ε : X^{0} → R is a projective resolution of the left A-module R [21, §4]. The corresponding quasi-isomorphism X → R will also be denoted by ε.

Let B be the DG R-module (Dif Aop) (X, R). We will freely make use of the identifications

B = (Dif Aop) (X, R) = Hom_{A}(A ⊗_{R} E, R) = Hom_{R}(E, R).

Endowed with the ’shuffle product‘ B becomes a DG algebra [10, §9] : Recall that for f ∈ B^{p}, g ∈ B^{q}, and n = p + q, one puts

(fg)(X_{1} ∧ … ∧ X_{n}) = ∑ σ_{ij}f(X_{i_{1}}, …, X_{i_{p}}) g(X_{j_{1}}, …, X_{j_{q}}) ,

where σ_{ij} is the parity of the permutation

1 ↦ i_{1} , …, p ↦ i_{p} , p + 1 ↦ j_{1} , …, p + q ↦ j_{q} ,

and the sum ranges over all tuples i, j with i_{1} < … < i_{p}, j_{1} < … < j_{q} and {1, …, p + q} = {i_{1}, …, i_{p}} ∪ {j_{1}, …, j_{q}}.

Let f ∈ B^{p}. We define a DG left B-module structure on X by putting f.(a⊗X_{1} ∧…∧X_{n}) = 0 for p > n and, with the same notations as for the shuffle product,

f(a ⊗ X_{1} ∧ … ∧ X_{n}) = ∑ σ_{ij}a ⊗ f(X_{i_{1}}, …, X_{i_{p}}) X_{j_{1}} ∧ … ∧ X_{j_{q}}

for p < n and p + q = n. It is clear that the actions of A and B on X commute among each other and agree on R so that X becomes an Aop-B-bimodule. Note that X| Aop has property (P) (3.1).

Lemma.

a) The functors LT_{X} : DB → DAop and RH_{X} induce quasi-inverse S-equivalences between H_{p}^{b}B and the full triangulated subcategory of DAop generated by R.

b) If L is finitely generated over R, then LT_{X} : DB → DAop is fully faithful and RH_{X} ←^{∼} LT_{X^{⊤}}.

Proof. a) By 4.2 a) we have to check that the morphism of complexes

λ : B → (Dif Aop) (X, X)

mapping f to left multiplication by f is a quasi-isomorphism. By definition the composition of λ with

ε_{∗} : (Dif Aop) (X, X) → (Dif Aop) (X, R)

is the identity. Since ε : X → R is a quasi-isomorphism and X has property (P), ε_{∗} is a quasiisomorphism. Hence so is λ.

b) If L is finitely generated, X| Aop is a bounded complex of finitely generated projective A-modules. In particular, X is small in DAop. The assertion now follows from 4.2 b) and 6.2 a).

6.6 Example: Bar resolution. Let A be a small DG category. Let Ỹ be the bar resolution [4, IX, §6] of A, i.e. the complex of A-A-bimodules with Ỹ(A, B)^{n} = 0 for n > 0 and

Ỹ^{−n}(A, C) = ∐_{B_{0},...,B_{n}} A (B_{0}, C) ⊗ A (B_{1}, B_{0}) ⊗ … ⊗ A (B_{n}, B_{n−1}) ⊗ A (A, B_{n}) , n ≥ 0

endowed with the differential d of degree 1 with

d(a_{0} ⊗ a_{1} ⊗ … ⊗ a_{n} ⊗ a_{n+1}) = ∑_{i=0}^{n}(−1)^{i}a_{0} ⊗ … ⊗ a_{i}a_{i+1} ⊗ … ⊗ a_{n+1}

Let Y be the total module of Ỹ (cf. the proof of 3.1). Define I to be the A-A-bimodule I(A, B) = A (A, B). By [4, IX, §6] we have a quasi-isomorphism ε : Y → I induced by the composition map

∐_{B_{0}} A (B_{0}, C) ⊗ A (A, B_{0}) → A (A, C).

The maps

Ỹ^{−n} → ∐_{p+q=n} Ỹ^{−p} ⊗ Ỹ^{−q}

given by

a_{0} ⊗ … ⊗ a_{n+1} ↦ (a_{0} ⊗ … ⊗ a_{p} ⊗ 1 ⊗ 1 ⊗ a_{p+1} ⊗ … ⊗ a_{n+1})

yield a morphism

Δ : Y → Y ∘ Y,

where by definition ? ∘ Y = T_{Y}. We have commutative diagrams

\begin{matrix}Y & ⟶^{Δ} & Y ∘ Y \\ ∥ & & ↓ Y ∘ ε \\ Y & ⟶^{can} & Y ∘ I \end{matrix} \begin{matrix}Y & ⟶^{Δ} & Y ∘ Y \\ ∥ & & ↓ ε ∘ Y \\ Y & ⟶^{can} & I ∘ Y \end{matrix}  \begin{matrix}Y & ⟶^{Δ} & Y ∘ Y & ⟶^{Y ∘Δ} & Y ∘ (Y ∘ Y) \\ ∥ & & & & ↓ can \\ Y & ⟶^{Δ} & Y ∘ Y & ⟶^{Δ∘Y} & (Y ∘ Y) ∘ Y. \end{matrix}

Now let B be a set of DG A-modules. The above diagrams ensure that we can make B into a DG category by requiring that

B (L, M) →^{∼} (Dif A) (Y ∘ L, M) ,

that the identity 1_{L}^{B} corresponds to the composition

Y ∘ L ⟶^{ε∘L} I ∘ L \underrightarrow{can} L ,

and that the composition of two morphisms of B coming from g : Y ∘ L → M and f : Y ∘ M → N is given by the composition

Y ∘ L ⟶^{Δ∘L} (Y ∘ Y) ∘ L ⟶^{can} Y ∘ (Y ∘ L) ⟶^{Y ∘g} Y ∘ M →^{f} N.

We then have a canonical A-B-bimodule X(A, L) := (Y ∘L)(A), where the action of g : Y ∘L → M is given by the composition

Y ∘ L ⟶^{Δ∘L} (Y ∘ Y) ∘ L ⟶^{can} Y ∘ (Y ∘ Y) ⟶^{Y ∘g} Y ∘ M.

Now suppose that k is a field. Then each Ỹ^{n} is relatively projective over A ⊗ Aop. Since Y admits the filtration F^{p} = ∐_{n≥−p} Ỹ^{n}, it has property (P) over A ⊗ Aop. Using 6.1 b) and c) we infer that the composition η

Y ∘ M ⟶^{ε∘M} I ∘ M ⟶^{can} M

is a P-resolution for each DG A-module M. Therefore the morphism

η_{∗} : (Dif A) (Y ∘ L, Y ∘ M) → (Dif A) (Y ∘ L, M) , L, M ∈ B ,

is a quasi-isomorphism. And so is the canonical morphism

B (L, M) → (Dif A) (X^{L}, X^{M}) = (Dif A) (Y ∘ L, Y ∘ M)

since it has η_{∗} as a left inverse. Using 4.2 we infer the

Lemma.

a) The restriction of LT_{X} to H_{p}^{b}B is fully faithful.

b) If each L ∈ B is small in DA, then LT_{X} is fully faithful.

c) LT_{X} is an equivalence iff the objects of B form a set of small generators for DA.

7. Quasi-functors and lifts

7.1 Quasi-functors. Let A and B be small DG categories. Denote by \underline{A} the full subcategory of DA whose objects are the A^{∧}, A ∈ A, and by Z\underline{A} the full subcategory whose objects are the A^{∧}[n], n ∈ Z, A ∈ A. Note that we have

(Z\underline{A}) (A^{∧}[n], B^{∧}[m]) = H^{m−n}A (A, B)

for all A, B ∈ A, n, m ∈ Z.

Let X be an A-B-bimodule. By definition, X is a quasi-functor B → A if it satisfies the conditions of the following lemma. Note that in this case LT_{X} gives rise to a functor Z\underline{B} → Z\underline{A} and hence to a functor H^{∗}B → H^{∗}A.

Lemma. The following are equivalent

i) LT_{X} gives rise to a functor \underline{B} → \underline{A}.

ii) For each B ∈ B the functor (DA) (?, X^{B}) is representable by an object of \underline{A}.

iii) For each B ∈ B there is an A ∈ A and an element x_{B} ∈ Z^{0}X(A, B) such that for each A′ ∈ A the morphism

A (A′, A) → X(A′, B) , f ↦ X(f, B)(x_{B})

induces isomorphisms in homology.

Proof. Exercise.

Suppose for example that A and B are concentrated in degree 0. Then A^{0} is equivalent to \underline{A}. Thus by i), a quasi-functor X yields a functor F^{0} : B^{0} → A^{0}; hence a functor F : B → A. It is easy to see that in D(A ⊗ Bop), X is isomorphic to the bimodule (A, B) ↦ A (A, F B).

7.2 Quasi-equivalences. Keep the hypotheses of 7.1. By definition, X is a quasi-equivalence if the conditions of the following lemma hold. In this case B is quasi-equivalent to A.

Lemma. The following are equivalent

i) LT_{X} is an equivalence giving rise to an equivalence \underline{B} → \underline{A}.

ii) LT_{X} gives rise to equivalences Z\underline{B} → Z\underline{A} and \underline{B} → \underline{A}.

iii) There is a subset D ⊂ A × B projecting onto A as well as onto B, and for each (A, B) ∈ D there is an element x_{AB} ∈ Z^{0}X(A, B) such that the morphisms

A (A′, A) → X(A′, B) , f ↦ X(f, B)(x_{AB})
B (B, B′) → X(A, B′) , g ↦ X(A, g)(x_{AB})

induce isomorphisms in homology for each A′ ∈ A, B′ ∈ B.

Proof. Exercise.

Example. Each DG functor F : B → A inducing an equivalence H^{∗}F : H^{∗}B → H^{∗}A yields a quasi-equivalence X(A, B) = A (A, F B). If A and B are concentrated in degree 0, each quasiequivalence comes from an equivalence F : B → A.

Remark. If k is a field, ’quasi-equivalence‘ is an equivalence relation (6.1 c and 6.2 b imply reflexivity; 6.3 b implies transitivity).

7.3 Lifts. Let A be a small DG category. Let U ⊂ DA be a full small subcategory and ZU ⊂ DA the full subcategory whose objects are the U[n], U ∈ U, n ∈ Z. A lift of U is a DG category B together with an A-B-bimodule X such that LT_{X} gives rise to equivalences Z\underline{B} →^{∼} ZU

and B →^{∼} U.

Z\underline{B} ⟶^{∼} ZU
↓  ↓
DB ⟶^{LT_{X}} DA.

Examples. With the notations of 6.5, (B, X) is a lift of U = {R}. — If k is a field, any U ⊂ DA may be lifted using the bar resolution of 6.6.

The definition of a lift implies in particular that LT_{X} induces an equivalence from H_{p}^{b}B onto the triangulated subcategory of DA generated by U (4.2 a). If X^{B} has property (P) for each B ∈ B, a quasi-inverse is induced by RH_{X}. Indeed, if M ∈ H_{p}^{b}B, we have

(DB) (S^{n}B^{∧}, RH_{X}LT_{X}M) →^{∼} (DA) (LT_{X}S^{n}B^{∧}, LT_{X}M) ←^{∼} (DB) (S^{n}B^{∧}, M)

since LT_{X} is fully faithful on H_{p}^{b}B. This means that RH_{X}LT_{X}M ← M is invertible.

We see from 6.1 that LT_{X} is itself an equivalence iff the objects of U form a system of small generators for DA.

If U is given, we can always construct a standard lift by taking B to be the full subcategory of Dif A formed by chosen objects pU, U ∈ U, and X to be the bimodule

(A, pU) ↦ (pU)(A) , pU ∈ B, A ∈ A.

Now let (B, X) be any lift of U such that X^{B} has property (P) for each B ∈ B. Let C be a DG category and F : Dif C → Dif A a DG functor such that LF : DC → DA induces a functor \underline{C} → U.

DC ← \underline{C}
↓ LF  ↓
DB ⟶^{LT_{X}} DA ← U
↑  ↑
\underline{B} → U

Lemma. Put Y (B, C) = (H_{X}F C^{∧})(B).

a) LT_{Y} induces a functor \underline{C} → \underline{B} ; hence Y is a quasi-functor. It is a quasi-equivalence if LF induces an equivalence Z\underline{C} → ZU.

b) There is a canonical morphism

LT_{X}LT_{Y} M → LF M ,

which is invertible for M ∈ H_{p}^{b}C. It is invertible for arbitrary M ∈ DC iff LF commutes with direct sums.

c) If (C, Z) is a lift of U and F = T_{Z}, then Y is a quasi-equivalence C → B and we have LT_{X}LT_{Y} →^{∼} LT_{Z}. If moreover Z_{C} has property (P) for each C ∈ C, then RH_{Y} RH_{X} →^{∼} RH_{Z}.

Remark. In 10.3 we will need the following fact. Suppose that F, T_{X} and T_{Y} all preserve acyclicity so that their derived functors are isomorphic to the functors induced by them. Then the morphism of b) is produced by the composition

T_{X}T_{Y} →^{T_{X}α} T_{X}H_{X}F →^{ΦF} F

which is even defined as a morphism of DG functors. Here α : T_{Y} → H_{X}F denotes the canonical morphism constructed in 6.4, and Φ the adjunction morphism.

Proof. a) Consider the functor G = H_{X} ∘ F : Dif C → Dif B. We have LG = RH_{X}LF. So LG induces a functor \underline{C} → \underline{B}. By definition we have Y (B, C) = (GC^{∧})(B). Hence we have a morphism T_{Y} → G such that LT_{Y} M → LGM is invertible for each M ∈ H_{p}^{b}C (6.4). So LT_{Y} induces a functor \underline{C} → \underline{B}. We have morphisms

LT_{X}LT_{Y} → LT_{X}LG = LT_{X}RH_{X}LF → LF

which are invertible on H_{p}^{b}C. Thus LT_{X} induces an equivalence Z\underline{C} → Z\underline{B} iff LF induces an equivalence Z\underline{C} → ZU. The second assertion now follows from 7.2.

b) follows from the proof of a) and 4.2 d).

The first two assertions of c) are immediate form a) and b). The last assertion is clear since if LT_{Y} is an equivalence and LT_{X}LT_{Y} →^{∼} LT_{Z}, then RH_{Y} RH_{X} is right adjoint to LT_{Z}.

7.4 On the unicity of lifts. Keep the hypotheses of 7.3 and assume in addition that A is k-flat. Since X^{B} has property (P), ∀ B ∈ B, we have a well defined pair of adjoint functors

H_{X}^{!} : D(A ⊗ Cop) → D(B ⊗ Cop) , Z ↦ H_{X}Z
T_{X}^{!} : D(B ⊗ Cop) → D(A ⊗ Cop) , Y ↦ T_{X}pY

Lemma. For each Y ∈ D(B ⊗ Cop) we have

LT_{X}LT_{Y} →^{∼} LT_{Z} ,

where Z = T_{X}^{!}Y. Moreover T_{X}^{!} induces an equivalence between the full subcategories

\{Y : LT_{Y} gives rise to a functor \underline{C} → \underline{B}\} ⊂ D(B ⊗ Cop)
and \{Z : LT_{Z} gives rise to a functor \underline{C} → U\} ⊂ D(A ⊗ Cop).

Proof. We have T_{X}pY ←^{∼} T_{pX}pY by 6.1 b) and T_{pX}pY →^{∼} T_{pX}Y by the k-flatness of A (6.3 a). So we have T_{X}^{!}Y →^{∼} T_{pX}Y. By 6.3 b) this implies the first assertion. Since LT_{X} gives rise to a functor \underline{B} → U, we infer that T_{X}^{!} induces indeed a functor between the given subcategories. Suppose that LT_{Y} gives rise to a functor \underline{C} → \underline{B}. We have to show that the canonical morphism pY → H_{X}T_{X}pY of H(B ⊗ Cop) is a quasi-isomorphism. But we have already seen that H_{X}T_{X}pY →^{∼} H_{X}T_{pX}Y, and on the other hand, for each B ∈ B, we have

(pY)_{B} →^{∼} Y_{B} →^{∼} H_{X}T_{X}p(Y_{B}) →^{∼} H_{X}T_{pX}Y_{B} ,

where we use 6.3 a) for the third isomorphism and the fact that Y_{B} ∈ U for the second one. Now suppose that LT_{Z} gives rise to a functor \underline{C} → \underline{A}. We have to show that the canonical morphism T_{X}p(H_{X}Z) → Z of D(A ⊗ Cop) is invertible. As above we have T_{X}p(H_{X}Z) →^{∼} T_{pX}H_{X}Z and

Z_{C} ←^{∼} T_{X}pH_{X}Z_{C} ←^{∼} T_{pX}H_{X}Z_{C} ,

where we use Z_{C} ∈ U for the first isomorphism and 6.3 a) for the second one.

8. Application: Derived equivalences

8.1 Arbitrary targets. Let A and C be small DG categories.

Theorem. Assertion i) implies ii), and ii) implies iii).

i) There is a DG functor H : Dif C → Dif A such that LH : DC → DA is an equivalence.

ii) C is quasi-equivalent to a full DG subcategory B of Dif A whose objects have property (P) and form a set of small generators for DA.

iii) There are a DG category B and DG functors

Dif C →^{G} Dif B →^{F} Dif A

such that LG and LF are equivalences.

Proof. i) implies ii): By 6.4 we have LH →^{∼} LT_{Z} for some A-C-bimodule Z. So (C, Z) is a lift of U = {LHC^{∧} : C ∈ C}. Take B to be a standard lift of U. The assertion then follows from 7.3 c) and 4.2 c).

ii) implies iii): By 7.2 we have an equivalence LT_{X} : DC → DB and by 7.3 an equivalence LF : DB → DA.

8.2 Flat targets. Let A and C be small DG categories and assume that A is k-flat.

Theorem. The following are equivalent

i) There is an A-C-bimodule X such that LT_{X} : DC → DA is an equivalence.

ii) C is quasi-equivalent to a full DG subcategory B of Dif A whose objects have property (P) and form a set of small generators for DA.

Proof. i) implies ii) by 8.1. Conversely, ii) implies i) by 8.1 iii), 6.4 and 6.3 b).

Remark. Recall from section 5 that a DG module is small in DA iff it is contained in the smallest strictly full triangulated subcategory of DA containing the free modules and closed under forming direct summands.

9. Application: Stalk categories

9.1 Modules over H^{0}A. Let A be a small DG category. Let H^{0}A (resp. τ^{≤0}A) be the DG category with the same objects as A and with the morphism spaces

(H^{0}A) (A, B) = H^{0}A (A, B) , A, B ∈ A,

viewed as DG k-modules concentrated in degree 0 (resp.

(τ^{≤0}A) (A, B) = τ^{≤0}A (A, B) , A, B ∈ A,

where τ^{≤0}K denotes the subcomplex C of K with C^{n} = 0 for n > 0, C^{0} = Z^{0}K, and C^{n} = K^{n} for n < 0). We have the obvious functors

H^{0}A ⟵^{π} τ^{≤0}A ⟶^{ι} A.

As in example 6.1, they yield functors

DH^{0}A ⟵^{LT_{X}} Dτ^{≤0}A ⟶^{LT_{Y}} DA ,

where X(A, B) = (H^{0}A) (A, πB) and Y (A, B) = A (A, ιB). The functor LT_{X} is an equivalence iff A satisfies the ’Toda-condition‘ (cf. [22])

H^{n}A (A, B) = 0 , ∀ n < 0 , ∀ A, B ∈ A.

In this case (example 6.2), we have a canonical functor from DH^{0}A to DA given simply by the composition

DH^{0}A ⟶^{LT_{X^{⊤}}} Dτ^{≤0}A ⟶^{LT_{Y}} DA.

If A is k-flat, this simplifies to

DH^{0}A ⟶^{LT_{Z}} DA ,

where Z is the A-H^{0}A-bimodule T_{pY} X^{⊤} (6.3 b).

9.2 Equivalences. Let B be a small k-linear category. We identify B with a DG category concentrated in degree 0. Let A be an arbitrary small DG category.

Theorem. (cf. [19], [12]) The following are equivalent

i) There are DG categories A_{1}, A_{2} and DG functors

Dif B →^{F_{3}} Dif A_{2} →^{F_{2}} Dif A_{1} →^{F_{1}} Dif A

such that LF_{1}, LF_{2} and LF_{3} are equivalences.

ii) There is an S-equivalence DB →^{∼} DA.

iii) B is equivalent to a full subcategory U of DA whose objects form a set of small generators and satisfy (DA) (U, V [n]) = 0 for all n ≠ 0, U, V ∈ U.

Remark. We refer to [19, 6.4] for more precise information in the case where A and B are rings.

Proof. By 4.2 c), ii) implies iii). To prove that iii) implies i), let A_{1} be a full subcategory of Dif A consisting of chosen objects pU, U ∈ U. Let F_{1} = T_{X} where X(A, A_{1}) = A_{1}(A). By 6.1, LF_{1} is an equivalence. By the assumption on U we have H^{n}A_{1} (A, B) = 0 for n ≠ 0 and arbitrary A, B ∈ A_{1}, and H^{0}A_{1} is equivalent to B. Now the assertion is clear from 9.1.

Using 6.3 b) and 6.4 we find the

Corollary. (cf. [20]) If A is k-flat, the following are equivalent

i) There is an A-B-bimodule X such that LT_{X} : DB → DA is an equivalence.

ii) There is an S-equivalence DB → DA.

iii) B is equivalent to a full subcategory U of DA whose objects form a set of small generators and satisfy (DA) (U, V [n]) = 0 for all n ≠ 0, U, V ∈ U.

Remark. We refer to [20] for more precise information in the case where A and B are rings. A straightforward construction of the bimodule in this case is given in [13].

10. Application: Koszul duality for DGA categories

10.1 Preliminaries. Suppose that k is a field. Define the functor D : Dif k → Dif k by

DM = (Dif k) (M, k) ,

where k is viewed as a DG k-module concentrated in degree 0. Let A be a DG k-category. For each A ∈ A we define the A-module A^{∨} by

A^{∨}(B) = DA (A, B) , B ∈ A.

For each DG module M and each A ∈ A we have a canonical isomorphism of DG k-modules

(Dif A) (M, A^{∨}) →^{∼} DM(A)
φ ↦ (m ↦ ((φA)(m))(1_{A})).

In particular, we have a canonical morphism

A (A, B) → DDA (A, B) →^{∼} DA^{∨}(B) →^{∼} (Dif A) (A^{∨}, B^{∨}) ,

which is a quasi-isomorphism if dim H^{n}A (A, B) < ∞ for each n ∈ Z. So in this case the full subcategory A^{∨} of Dif A formed by the A^{∨}, A ∈ A, is quasi-equivalent to A.

Fix A ∈ A. To compute (DA) (?, A^{∨}), we first remark that if N is acyclic, we have

(HA) (N, A^{∨}) = H^{0}DN(A) = 0.

Therefore

(DA) (M, A^{∨}) →^{∼} (HA) (pM, A^{∨}) ←^{∼} (HA) (M, A^{∨}) →^{∼} H^{0}DM(A) ,

and in particular H^{n}A^{∨} (A^{∨}, B^{∨}) →^{∼} (DA) (A^{∨}, B^{∨}[n]). So if we define the A-A^{∨}-bimodule X_{∨} by (A, B^{∨}) ↦ B^{∨}(A), then (A^{∨}, X_{∨}) is a lift (7.3) of {A^{∨} : A ∈ A} ⊂ DA.

10.2 The Koszul dual. Suppose from now on that A is an augmented DG category (=DGA category) i.e.

a) Distinct objects of A are non-isomorphic.

b) For each A ∈ A a DG module \overline{A} is given such that H^{0}\overline{A}(A) →^{∼} k and H^{n}\overline{A}(B) = 0 whenever n ≠ 0 or B ≠ A.

Now let (A^{∗}, X) be a lift (7.3) of \{\overline{A} : A ∈ A\} ⊂ DA. After deleting some objects from A^{∗} we may (and will) assume that we have a bijection A ↦ A^{∗} between the objects of A and those of A^{∗} such that LT_{X}A^{∗∧} →^{∼} \overline{A} for each A ∈ A. By 6.3 a) we also may (and will) assume that X has property (P) as a bimodule. Since k is a field, this implies in particular that X(?, A^{∗}) has property (P) for each A^{∗} ∈ A^{∗} (6.1 c). Hence the functors H_{X} and T_{X} both preserve acyclicity and induce a pair of adjoint functors between DA^{∗} and DA, which will also be denoted by T_{X} and H_{X}.

We make A^{∗} into an augmented DG category by putting

\overline{A^{∗}} = H_{X}A^{∨}.

This is a good definition since indeed

H^{n}\overline{A^{∗}}(B^{∗}) →^{∼} (DA^{∗}) (B^{∗∧}, \overline{A^{∗}}[n]) →^{∼} (DA^{∗}) (B^{∗∧}, H_{X}A^{∨}[n])
→^{∼} (DA) (T_{X}B^{∗∧}, A^{∨}[n]) →^{∼} (DA) (\overline{B}, A^{∨}[n])
→^{∼} H^{n}D\overline{B}(A).

We define A^{∗} with the \overline{A^{∗}}, A^{∗} ∈ A^{∗}, to be the Koszul dual of the DGA category A (cf. [1]). We sum up our notations in the diagram

\begin{matrix}\overline{A} & DA & A^{∨} \\ ↑ & T_{X} ↑↓ H_{X} & ↓ \\ A^{∗∧} & DA^{∗} & \overline{A^{∗}}. \end{matrix}

If B is another DGA category, a quasi-functor Y : B → A is compatible with the augmentations

if H_{Y} \overline{A} →^{∼} \overline{B} whenever T_{Y} B^{∧} →^{∼} A^{∧}.

By 7.3 c) the Koszul dual is determined by the above construction up to a quasi-equivalence compatible with the augmentation, i.e. if X′ and A^{∗′} result from different choices made in the construction, there is an A^{∗′}-A^{∗}-bimodule Y having property (P) such that T_{Y} : DA^{∗} → DA^{∗′}

satisfies T_{X′}T_{Y} →^{∼} T_{X}, T_{Y} A^{∗∧} →^{∼} A^{∗′∧} and

H_{Y} \overline{A^{∗′}} →^{∼} H_{Y} H_{X′}A^{∨} →^{∼} H_{X}A^{∨} →^{∼} \overline{A^{∗}}

for each A ∈ A.

The Koszul dual defined in [2] is quasi-equivalent to the full subcategory of Dif A^{∗} formed by the A^{∗∧}[n(A)], where n : A → Z is a given ’weight function‘ for A. Note that the morphism spaces of this category simply identify with the shifted spaces

A^{∗} (A^{∗}, B^{∗})[n(B) − n(A)] , A, B ∈ A.

Examples. a) Let 𝔊 be a k-Lie algebra and U(𝔊) its universal enveloping algebra. In the notations of 6.5 (with R = k), the Koszul dual of A = U(𝔊) is quasi-equivalent to B.

b) Let V be a finite-dimensional k-vector space, DV its dual over k, ΛDV the exterior algebra on DV, and SV the graded symmetric algebra on V. View A = ΛDV as a DG algebra concentrated in degree 0, and B = SV as a DG algebra with the components B^{n} = S^{n}V and vanishing differential. Define (commuting) right and left A-actions on ΛV by

v^{∗}.(v_{1} ∧ … ∧ v_{n}) = ∑_{i=1}^{n}(−1)^{i+1}v^{∗}(v_{i})v_{1} ∧ … v̂_{i} … ∧ v_{n}
(v_{1} ∧ … ∧ v_{n}).v^{∗} = ∑_{i=1}^{n}(−1)^{n+i}v^{∗}(v_{i})v_{1} ∧ … v̂_{i} … ∧ v_{n}.

Endow the graded A-B-bimodule X = SV ⊗ ΛV with the differential

d : X^{p} → X^{p+1} , x ↦ (−1)^{p} ∑_{i=1}^{n}(v_{i} ⊗ v_{i}^{∗})x ,

where the v_{i}, 1 ≤ i ≤ n, form a basis of V and (v_{i}^{∗}) is the dual basis. Then (B, X) is a lift of the trivial A-module k. Hence the Koszul dual of A is quasi-equivalent to B.

c) Let V be a finite-dimensional k-vector space, I ⊂ Z an interval and A_{I} the DG category concentrated in degree 0 whose objects are the i ∈ I and whose morphism spaces are the

A_{I} (i, j) = S^{j−i}V

concentrated in degree 0. For each i ∈ I let \overline{ı} be the DG A_{I}-module concentrated in degree 0 with \overline{ı}(j) = k for i = j and \overline{ı}(j) = 0 for i ≠ j. Let B_{I} be the DG category whose objects are the symbols i^{∗}, i ∈ I and whose morphism spaces are the stalk complexes

B_{I} (i^{∗}, j^{∗}) = (Λ^{i−j}DV)[j − i].

Let X_{I} be the A_{I}-B_{I}-bimodule given by

X_{I}(i, j^{∗})^{n} = Λ^{−n}V ⊗ S^{n+j−i}V

endowed with the differential given by left multiplication by ∑_{i=1}^{n} v_{i}^{∗} ⊗ v_{i}, where the v_{i}, 1 ≤ i ≤ n, form a basis of V and (v_{i}^{∗}) is the dual basis. Then (B_{I}, X_{I}) is a lift of \{\overline{ı} : i ∈ I\} ⊂ DA_{I}. So the Koszul dual of A_{I} is quasi-equivalent to B_{I}. Clearly, the modules i^{∨}, i ∈ I, are the unions of their finite-dimensional submodules and the functor i^{∧} ↦ i^{∨} is an equivalence. It therefore follows from the lemma on the ’symmetric‘ case (10.5) that the Koszul dual of B_{I} is quasi-equivalent to A_{I}.

10.3 The double dual. The composition of H_{X} with the functor T_{X_{∨}} : DA^{∨} → DA of 10.1 induces a functor \underline{A^{∨}} → \{\overline{A^{∗}} : A ∈ A\} ⊂ DA^{∗}. Thus (7.3 a), we have a quasi-functor Y : A^{∨} → A^{∗∗}, which is a quasi-equivalence iff the restriction of H_{X} : DA → DA^{∗} to the subcategory formed by the A^{∨}[n], A ∈ A, n ∈ Z, is fully faithful.

DA^{∨}  \underrightarrow{T_{X_{∨}}}  DA
T_{Y} ↓  T_{X} ↑↓ H_{X}
DA^{∗∗}  \overrightarrow{T_{X_{∗}}}  DA^{∗}.

We endow A^{∨} with the augmentation defined by

\overline{A^{∨}}(B^{∨}) = D(Dif A) (\overline{A}, B^{∨}) →^{∼} DD\overline{A}(B).

Lemma. The quasi-functor Y : A^{∨} → A^{∗∗} is compatible with the augmentations.

Proof. Let (A^{∗∗}, X_{∗}) be the chosen lift for the \overline{A^{∗}}, A ∈ A. Recall that we assume that X_{∗} has property (P) as a bimodule. Fix A ∈ A. We have to show that \overline{A^{∨}} →^{∼} H_{Y} \overline{A^{∗∗}}. By definition H_{Y} \overline{A^{∗∗}} = H_{Y} H_{X_{∗}}A^{∗∨}. We will show that H_{Y} H_{X_{∗}}A^{∗∨} →^{∼} \overline{A^{∨}} by explicitly exhibiting a quasiisomorphism. For short we write ^{∨}(?,?) for (Dif A^{∨})(?,?), . . .. We have the following series of morphisms of DG k-modules, functorial in B^{∨} ∈ A^{∨}

(H_{Y} H_{X_{∗}}A^{∗∨})(B^{∨}) →^{∼}  ^{∨}(B^{∨∧}, H_{Y} H_{X_{∗}}A^{∗∨}) →^{∼} ^{∗}(T_{X_{∗}}T_{Y} B^{∨∧}, A^{∗∨})
→^{∼} D ^{∗}(A^{∗∧}, T_{X_{∗}}T_{Y} B^{∨∧}) ← D ^{∗}(A^{∗∧}, H_{X}T_{X_{∨}}B^{∨∧}).

The last arrow is induced by the morphism

T_{X_{∗}}T_{Y} → H_{X}T_{X_{∨}}

of DG functors Dif A^{∨} → Dif A^{∗} exhibited in remark 7.3. It is a quasi-isomorphism since B^{∨∧} ∈ H_{p}^{b}A^{∨} (7.3 b). We continue the series of morphisms:

D ^{∗}(A^{∗∧}, H_{X}T_{X_{∨}}B^{∨∧}) →^{∼} D (T_{X}A^{∗∧}, T_{X_{∨}}B^{∨∧})
→^{∼} D (T_{X}A^{∗∧}, B^{∨})

since by construction T_{X_{∨}}B^{∨∧} →^{∼} X_{∨}(?, B^{∨}) →^{∼} B^{∨} in Dif A. Now since T_{X}A^{∗∧} is quasi-equivalent to \overline{A}, we have a quasi-isomorphism

D (T_{X}A^{∗∧}, B^{∨}) ← D (\overline{A}, B^{∨}).

By definition the last term is isomorphic to \overline{A^{∨}}(B^{∨}).

10.4 Properties of A^{∗}. Let M be a DG A-module and n ∈ N. By definition we have sdim M ≤ n (resp. pdim M ≤ n, resp. idim M ≤ n) if there is a sequence

0 = M_{−1} → M_{0} → M_{1} → M_{2} → … → M_{n} = M

of morphisms of DA such that in each triangle

M_{i−1} → M_{i} → Q_{i} → M_{i−1}[1] , 0 ≤ i ≤ n ,

the module Q_{i} is isomorphic to a finite direct sum of modules of the form \overline{A}[n] (resp. A^{∧}[n], resp. A^{∨}[n]), A ∈ A, n ∈ Z. The (possibly infinite) numbers sdim M, pdim M and idim M are referred to as the semi-simple, the projective, and the injective dimension of M, respectively.

Let ν : Dif A → Dif A be the functor defined by

(νM)(A) = D(Dif A) (M, A^{∧}).

For example, we have νA^{∧} = A^{∨} by the definition of A^{∨} for each A ∈ A. We have a natural transformation

D(Dif A) (M, N) → (Dif A) (N, νM)

which is defined as follows: Given a linear form φ on (Dif A) (M, N) and an f ∈ (Dif A) (A^{∧}, N) ←^{∼} N(A), the associated linear form on (Dif A) (M, A^{∧}) maps g to φ(fg). Clearly this is an isomorphism for M = B^{∧}[n], B ∈ A, n ∈ Z, and therefore a quasi-isomorphism for M ∈ H_{p}^{b}A.

Lemma.

a) If sdim M < ∞ and pdim M < ∞ then H_{X}LνM →^{∼} (Lν)H_{X}M in DA^{∗}.

b) For each A ∈ A we have

1) pdim \overline{A^{∗}} ≤ sdim A^{∨} 2) sdim A^{∗∧} ≤ idim \overline{A}
3) idim \overline{A^{∗}} ≤ sdim A^{∧}  4) sdim A^{∗∨} ≤ pdim \overline{A}

Proof. a) Since sdim M < ∞, we have T_{X}M ∈ H_{p}^{b}A^{∗} and M →^{∼} T_{X}N for N →^{∼} H_{X}M. We assume that N (and hence T_{X}N) has property (P). We have to show that H_{X}νT_{X}N →^{∼} νN. We write ^{∗}(?,?) and (?,?) instead of (Dif A^{∗}) (?,?) and (Dif A) (?,?). We have the following series of quasi-isomorphisms functorial in A^{∗} ∈ A^{∗}

(H_{X}νT_{X}N)(A^{∗}) → ^{∗}(A^{∗∧}, H_{X}νT_{X}N) → (T_{X}A^{∗∧}, νT_{X}N).

Since T_{X}N ∈ H_{p}^{b}A and N ∈ H_{p}^{b}A^{∗}, we also have the following quasi-isomorphisms:

(T_{X}A^{∗∧}, νT_{X}N) → D(T_{X}N, T_{X}A^{∗∧}) → D ^{∗}(N, A^{∗∧}) = (νN)(A^{∗}).

b) Assertions 1) and 2) are trivial since H_{X} \overline{B} →^{∼} B^{∗∧}, B ∈ A, and H_{X}A^{∨} = \overline{A}, A ∈ A. For 3) we use that

\overline{A^{∗}} →^{∼} H_{X}A^{∨} →^{∼} H_{X}νA^{∧} →^{∼} (Lν)H_{X}A^{∧}

if sdim A^{∧} < ∞, and B^{∗∨} = (Lν)H_{X} \overline{B} for each B ∈ A. For 4) we use that A^{∗∨} →^{∼} LνH_{X} \overline{A} →^{∼} H_{X}Lν\overline{A} if pdim \overline{A} < ∞ and \overline{B^{∗}} = H_{X}LνB^{∧} for each B ∈ A.

10.5 Three special cases. We consider three cases where A^{∨} is quasi-equivalent to A^{∗∗}, and there is a fully faithful embedding relating DA and DA^{∗}.

Lemma. (The ’finite‘ case) Suppose that pdim \overline{A} < ∞ and sdim A^{∧} < ∞ for all A ∈ A. a) sdim A^{∗∨} < ∞ and idim \overline{A^{∗}} < ∞ for all A^{∗} ∈ A^{∗}.

b) T_{X} and H_{X} are quasi-inverse equivalences between DA^{∗} and DA.

c) We have quasi-equivalences A →^{∼} A^{∨} →^{∼} A^{∗∗}.

Examples. a) The category B_{I} of 10.2 c) for finite I.

b) Let Λ be a finite-dimensional k-algebra of finite global dimension all of whose simple modules are one-dimensional. We take A to be the k-linear category formed by chosen representatives of the indecomposable projective A-modules and for each A ∈ A we take \overline{A} to be the head of A.

Proof. a) holds by 10.4 b).

b) Since pdim \overline{A} < ∞, we have \overline{A} ∈ H_{p}^{b}A for each A ∈ A. Moreover, since sdim B^{∧} < ∞, the triangulated subcategory generated by the \overline{A} contains each B^{∧}, B ∈ A. Hence the \overline{A}, A ∈ A, form a system of small generators for DA and the assertion follows from 6.1 a) and 6.2.

c) Since H_{X} is fully faithful, A^{∨} is quasi-equivalent to A^{∗∗} (10.3). Since sdim A^{∧} < ∞ for all A ∈ A, we have

∞ > dim H^{n}A^{∧}(B) = dim H^{n}A (A, B)

for all A, B ∈ A so that A → A^{∨} is a quasi-equivalence (example 7.2).

Lemma. (The ’exterior‘ case) Suppose that sdim A^{∧} < ∞ and sdim A^{∨} < ∞ for all A ∈ A.

a) pdim \overline{A^{∗}} < ∞ and idim \overline{A^{∗}} < ∞ for each A^{∗} ∈ A^{∗}.

b) T_{X} and H_{X} induce quasi-inverse equivalences between H_{p}^{b}A^{∗} and the smallest full triangulated subcategory of DA containing the \overline{A}, A ∈ A.

c) T_{X^{⊤}} : DA → DA^{∗} is fully faithful.

d) We have quasi-equivalences A →^{∼} A^{∨} →^{∼} A^{∗∗}.

Remark. Part b) yields theorem 16 of [2].

Examples. a) Example 10.2 b).

b) The category B_{I} of example 10.2 c).

c) If Λ is a finite-dimensional algebra of arbitrary global dimension with one-dimensional simples, we can proceed as in example b) of the ’finite case‘.

Proof. a) holds by 10.4 b). By the definition of ’lift‘ (7.3) we have b).

c) Let T be the full triangulated subcategory of DA generated by the \overline{A}, A ∈ A. The restriction of H_{X} to T is fully faithful (7.3). Since H_{p}^{b}A is contained in T , H_{X} is fully faithful on H_{p}^{b}A, and H_{X}A^{∧} lies in H_{p}^{b}A^{∗} for each A ∈ A. In particular, H_{X}A^{∧} is small for each A ∈ A. Since T_{X^{⊤}} agrees with H_{X} on H_{p}^{b}A (6.2 a), the assertion follows from 4.2 b).

d) Since the A^{∨}, A ∈ A, lie in T , A^{∨} is quasi-equivalent to A^{∗∗}. Since the A^{∧}, A ∈ A, lie in T , we have

∞ > dim H^{n}A^{∧}(B) = dim H^{n}A (B, A)

for all A, B ∈ A, so that A → A^{∨} is a quasi-equivalence (example 7.1).

Lemma. (The ’symmetric‘ case) Suppose that pdim \overline{A} < ∞ and idim \overline{A} < ∞ for all A ∈ A.

a) sdim A^{∗∧} < ∞ and sdimA^{∗∨} < ∞ for all A^{∗} ∈ A^{∗}.

b) T_{X} and H_{X} induce quasi-inverse equivalences between H_{p}^{b}A^{∗} and the smallest full triangulated subcategory of DA containing the \overline{A}, A ∈ A.

c) T_{X} : DA^{∗} → DA is fully faithful.

d) We have a quasi-equivalence A^{∨} → A^{∗∗} if each B^{∨}, B ∈ A, lies in the smallest triangulated subcategory of DA closed under direct sums and containing the \overline{A}, A ∈ A.

Examples. In example 10.2 a), we have pdim \overline{A} < ∞ and idim \overline{A} < ∞ if 𝔊 is finite-dimensional. This also holds in 10.2 c). For 10.2 c) the assumption of d) is satisfied as well.

Proof. a) holds by 10.4 b). By the definition of ’lift‘ (7.3) we have b).

c) and d): By 4.2 b), T_{X} is fully faithful. So T_{X} induces an equivalence onto its image, which is precisely the smallest strictly full triangulated subcategory containing the \overline{A}, A ∈ A, and closed under direct sums. A quasi-inverse is induced by H_{X}. Thus the restriction of H_{X} to the subcategory of DA formed by the B^{∨}, B ∈ A, is fully faithful. Now d) follows by 10.3.

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