Hochschild (co)homology of the second kind I

We define and study the Hochschild (co)homology of the second kind (known also as the Borel-Moore Hochschild homology and the compactly supported Hochschild cohomology) for curved DG-categories. An isomorphism between the Hochschild (co)homology of the second kind of a CDG-category B and the same of the DG-category C of right CDG-modules over B, projective and finitely generated as graded B-modules, is constructed. Sufficient conditions for an isomorphism of the two kinds of Hochschild (co)homology of a DG-category are formulated in terms of the two kinds of derived categories of DG-modules over it. In particular, a kind of"resolution of the diagonal"condition for the diagonal CDG-bimodule B over a CDG-category B guarantees an isomorphism of the two kinds of Hochschild (co)homology of the corresponding DG-category C. Several classes of examples are discussed.

Introduction CDG-algebras (where "C" stands for "curved") were introduced in connection with nonhomogeneous Koszul duality in [13]. Several years earlier, (what we would now call) A ∞ -algebras with curvature were considered in [3] as natural generalizations of the conventional A ∞ -algebras. In fact, [3] appears to be the first paper where the Hochschild (and even cyclic) homology of curved algebras was discussed.
Recently, the interest to these algebras was rekindled by their connection with the categories of matrix factorizations [18,2,12,1,22]. In these studies, beginnings of the theory of Hochschild (co)homology for CDG-algebras have emerged. The aim of the present paper is to work out the foundations of the theory on the basis of the general formalism of derived categories of the second kind as developed in the second author's paper [15]. The terminology, and the notion of a differential derived functor of the second kind, which is relevant here, go back to the classical paper [5].
The subtle but crucial difference between the differential derived functors of the first and the second kind lies in the way one constructs the totalizations of bicomplexes: one can take either direct sums or direct products along the diagonals. The construction of the differential Tor and Ext of the first kind, which looks generally more natural at the first glance, leads to trivial functors in the case of a CDG-algebra with nonzero curvature over a field. So does the (familiar) definition of Hochschild (co)homology of the first kind.
On the other hand, with a CDG-algebra B one can associate the DG-category C of right CDG-modules over B, projective and finitely generated as graded B-modules. For the DG-category C, the Hochschild (co)homology of the first kind makes perfect sense. The main problem that we address in this paper is the problem of comparison between the Hochschild (co)homology of the first kind of the DG-category C and the Hochschild (co)homology of the second kind of the original CDG-algebra B (defined using the differential Tor/Ext of the second kind).
We proceed in two steps: first, compare the Hochschild (co)homology of the second kind for B and C, and then deal with the two kinds of Hochschild (co)homology of C. The first step is relatively easy: our construction of an isomorphism works, at least, for any CDG-algebra B over a field k (see Section 2.6). However, a trivial counterexample shows that the two kinds of Hochschild (co)homology of C are not isomorphic in general (see Section 4.9). There are natural maps between the two kinds of Hochschild (co)homology, though. A sufficient condition for these maps to be isomorphisms is formulated in terms of the derived categories of the second kind of CDG-bimodules over B. In the maximal generality that we have been able to attain, this is a kind of "resolution of the diagonal" condition for the CDG-bimodule B over B (see Theorems 3.5.C-D and Corollaries 4.6.B, 4.7.B, and 4.8).
Let us say a few more words about the first step. There is no obvious map between the Hochschild complexes of B and C, so one cannot directly compare their cohomology. Instead, we construct a third complex (both in the homological and the cohomological versions) endowed with natural maps from/to these two complexes, and show that these maps are quasi-isomorphisms. To obtain the intermediate complex, we embed both B and C into a certain larger differential category.
The idea of these embeddings goes back to A. Schwarz's work [19]. The starting observation is that a CDG-algebra is not a CDG-module over itself in any natural way (even though it is naturally a CDG-bimodule over itself). It was suggested in [19], however, that one can relax the conditions on differential modules over CDG-algebras (called "Q-algebras" in [19]) thereby making the modules carry their own curvature endomorphisms. In recognition of A. Schwarz's vision, we partly borrow his terminology by calling such modules QDG-modules.
Any CDG-algebra is naturally both a left and a right QDG-module over itself. While CDG-modules form a DG-category, QDG-modules form a CDG-category. Both a CDG-algebra B (considered as a CDG-category with a single object) and the DG-category C of CDG-modules over it embed naturally into the CDG-category D of QDG-modules over B, so the Hochschild complex of D provides an intermediate object for comparison between the Hochschild complexes of B and C. Now let us turn to the second step. The (conventional) derived category of DG-modules over a DG-algebra is defined as the localization of the homotopy category of DG-modules by the class of quasi-isomorphisms, or equivalently, by the thick subcategory of acyclic DG-modules. This does not make sense for CDG-modules, since their differentials have nonzero squares, so their cohomology cannot be defined. Indeed, the subcategory of acyclic DG-modules is not even invariant under CDG-isomorphisms between DG-algebras [15,Examples 9.4].
The definition of the derived categories of the second kind, various species of which are called the coderived, the contraderived, the absolute derived, and the complete derived categories, for DG-and CDG-modules are not based on any notion of cohomology of a differential module. Rather, the classes of coacyclic, contraacyclic, absolutely acyclic, and completely acyclic CDG-modules are built up starting from short exact sequences of CDG-modules (with closed morphisms between them).
For reasons related to the behavior of tensor products with respect to infinite direct sums and products of vector spaces, the derived categories and functors of the second kind work better for coalgebras than for algebras, even though one is forced to use them for algebras if one is interested in curved algebras and modules. (For derived categories and functors of the first kind, it is the other way.) That is why one has to impose additional conditions like finiteness of homological dimension, Noetherianness, etc., on the underlying graded algebras of one's CDG-algebras in order to make the derived categories of the second kind well-behaved and the relation between them and the derived functors of the second kind working properly. We did our best to make such additional conditions as weak as possible in this paper, but the price of generality is technical complexity.
Unlike the Tor and Ext, the Hochschild (co)homology is essentially an invariant of a pair (a field or commutative ring, an algebra over it). It is not preserved when the ground field or ring is changed. In this paper, we always work over an arbitrary commutative ring k, or a commutative ring of finite homological dimension, as needed. The only exceptions are some examples depending on the Koszul duality results from [15], which are established only over a field. Working over a commutative ring involves all kinds of k-flatness or k-projectivity conditions that need to be imposed on the algebras and modules, both in order to define the Hochschild (co)homology and to compute various (co)homology theories in terms of standard complexes.
Recent studies of the categories of matrix factorizations and of the associated CDG-algebras showed the importance of developing the relevant homological algebra using only Z/2-grading (as opposed to the conventional Z-grading). In this paper we work with CDG-algebras and CDG-categories graded by an arbitrary abelian group Γ endowed with some additional data that is needed to define Γ-graded complexes and perform operations with them. The behavior of our (co)homology theories with respect to a replacement of the grading group Γ is discussed in detail (see Section 2.5).
We exhibit several classes of examples of DG-algebras and DG-categories for which the two kinds of Tor, Ext, and Hochschild (co)homology coincide. These examples roughly correspond to the classes of DG-algebras for which the derived categories of the first and second kind are known to coincide [15,Section 9.4]. In particular, one of these classes is that of the DG-categories that are cofibrant with respect to G. Tabuada's model category structure (see Section 4.4).
Examples of CDG-algebras B such that the two kinds of Tor and Ext for the corresponding DG-category C of CDG-modules over B, finitely generated and projective as graded B-modules, are known to coincide are fewer; and examples when we can show that the two kinds of Hochschild (co)homology for this DG-category C coincide are fewer still. Among the former are all the CDG-rings B whose underlying graded rings are Noetherian of finite homological dimension (see Section 4.7). In the latter class we have some CDG-algebras over fields admitting Koszul filtrations of finite homological dimension (see Section 4.6), curved commutative local algebras describing germs of isolated hypersurface singularities (due to the results of [2]), and curved commutative smooth algebras over perfect fields with the curvature function having no other critical values but zero (due to the recent results of [8]; see Section 4.8).
Our discussion of the Hochschild (co)homology of the DG-categories of matrix factorizations is finished in Section 4.10, where we show that the Hochschild (co)homology of the second kind of the DG-category of matrix factorizations over a smooth affine variety over an algebraically closed field of characteristic zero is isomorphic to the direct sum of the Hochschild (co)homology of the first kind of the similar DG-categories corresponding to all the critical values of the potential.
We are grateful to Anton Kapustin, Ed Segal, Daniel Pomerleano, Kevin Lin, and Junwu Tu for helpful conversations. A. P. is partially supported by the NSF grant DMS-1001364. L. P. is partially supported by a grant from P. Deligne 2004 Balzan prize and an RFBR grant.

CDG-Categories and QDG-Functors
This section is written in the language of CDG-categories. Expositions in the generality of CDG-rings, which might be somewhat more accessible to an inexperienced reader, can be found in [13,15,19]. For a discussion of DG-categories, we refer to [6], [21], and [15, Section 1.2].
1.1. Grading group. Let Γ be an abelian group endowed with a symmetric bilinear form σ : Γ × Γ −→ Z/2 and a fixed element 1 ∈ Γ such that σ(1, 1) = 1 mod 2. We will use Γ as the group of values for the gradings of our complexes. The differentials will raise the degree by 1, and signs like (−1) σ(a,b) will appear in the sign rules.
For example, in the simplest cases one may have Γ = Z, 1 = 1, and σ(a, b) = ab mod 2 for a, b ∈ Γ, or, alternatively, Γ = Z/2, 1 = 1 mod 2, and σ(a, b) = ab. One can also take Γ to be any additive subgroup of Q, containing Z and consisting of fractions with odd denominators, 1 = 1, and σ(a, b) = ab mod 2. Of course, it is also possible that Γ = Z d for any finite or infinite d, etc. When working over a commutative ring k containing the field F 2 , we will not need the form σ, and so Γ = Q or Γ = 0 become admissible choices as well.
From now on, we will assume a grading group data (Γ, σ, 1) to be fixed. When appropriate, we will identify the integers with their images under the natural map Z −→ Γ sending 1 to 1 without presuming this map to be injective, and denote σ(a, b) simply by ab for a, b ∈ Γ. So we will write simply 1 instead of 1, etc. This map Z −→ Γ will be also used when constructing the total complexes of polycomplexes some of whose gradings are indexed by the integers and the other ones by elements of the group Γ. Conversely, to any a ∈ Γ one assigns the class σ(1, a) ∈ Z/2, which we will denote simply by a in the appropriate contexts.

CDG-categories.
A CDG-category C is a category whose sets of morphisms Hom C (X, Y ) are Γ-graded abelian groups (i. e., C is a Γ-graded category) endowed with homogeneous endomorphisms d : Hom C (X, Y ) −→ Hom C (X, Y ) of degree 1 and fixed elements h X ∈ Hom C (X, X) of degree 2 for all objects X, Y ∈ C. The endomorphisms d are called the differentials and the elements h X are called the curvature elements. The following equations have to be satisfied: d(f g) = d(f )g + (−1) |f | f d(g) for any composable homogeneous morphisms f and g in C of the degrees |f | and |g| ∈ Γ, d 2 (f ) = h Y f − f h X for any morphism f : X −→ Y in C, and d(h X ) = 0 for any object X ∈ C.
The simplest example of a CDG-category is the category Pre(A) of precomplexes over an additive category A. The objects of Pre(A) are Γ-graded objects X in A endowed with an endomorphism d X : X −→ X of degree 1. The Γ-graded abelian group of morphisms Hom Pre(A) (X, Y ) is the group of homogeneous morphisms X −→ Y of Γ-graded objects. The differentials d : Hom(X, Y ) −→ Hom(X, Y ) are given by the rule d(f ) = d Y f − (−1) |f | f d X , and the curvature elements are h X = d 2 X . In particular, when A = Ab is the category of abelian groups, we obtain the CDG-category of precomplexes of abelian groups Pre(Ab).
A CDG-category with a single object is another name for a CDG-ring. A CDG-ring (B, d, h) is a Γ-graded ring B endowed with an odd derivation d of degree 1 and a curvature element h ∈ B 2 such that d 2 (b) = [h, b] for any b ∈ B and d(h) = 0.
An isomorphism between objects X and Y of a CDG-category C is (an element of) a pair of morphisms i : X −→ Y and j : Y −→ X of degree 0 such that ji = id X , ij = id Y , and d(i) = 0 = d(j); any one of the latter two equations implies the other one. It also follows that jh Y i = h X .
Let X be an object of a CDG-category C and τ ∈ Hom C (X, X) be its homogeneous endomorphism of degree 1. An object Y ∈ C is called the twist of an object X with an endomorphism τ (the notation: Y = X(τ )) if homogeneous morphisms i : X −→ Y and j : Y −→ X of degree 0 are given such that ji = id X , ij = id Y , and jd(i) = τ .
In this case one has jh Y i = h X + dτ + τ 2 . For any object X ∈ C and an element n ∈ Γ, an object Y ∈ C is called the shift of X with the grading n (the notation: Y = X[n]) if homogeneous morphisms i : X −→ Y and j : Y −→ X of the degrees n and −n, respectively, are given such that ji = id X , ij = id Y , and d(i) = 0 = d(j). In this case one has jh Y i = h X .
An object X ∈ C is called the direct sum of a family of objects X α ∈ C if homogeneous morphisms i α : X α −→ X of degree 0 are given such that the induced map Hom C (X, Y ) −→ α Hom C (X α , Y ) is an isomorphism of Γ-graded abelian groups for any object Y ∈ C, and di α = 0. In this case one has h X i α = i α h Xα , so the endomorphism h X corresponds to the family of morphisms i α h Xα under the above isomorphism for Y = X. The (direct) product of a family of object is defined in the dual way. An object X is the direct sum of a finite family of objects X α ∈ C if and only if it is their direct product. Of course, the notions of a shift and a direct sum/product of objects make sense in a (nondifferential) Γ-graded category, too; one just drops the conditions involving d and h.
Twists, shifts, direct sums, and products of objects of a CDG-category are unique up to a unique isomorphism whenever they exist.
A DG-category is a CDG-category in which all the curvature elements are zero.
The opposite CDG-category to a CDG-category C is constructed as follows. The class of objects of C op coincides with the class of objects of C. For any objects X, Y ∈ C the Γ-graded abelian group Hom C op (X op , Y op ) is identified with Hom C (Y, X), and the differential d op on this group coincides with d. The composition of morphisms in C op differs from that in C by the sign rule, f op g op = (−1) |f ||g| (gf ) op . Finally, the curvature elements in C op are h X op = −h X . In particular, this defines the CDG-ring Now let k be a commutative ring. A k-linear CDG-category is a CDG-category whose Γ-graded abelian groups of morphisms are endowed with Γ-graded k-module structures so that the compositions are k-bilinear and the differentials are k-linear.
The tensor product C ⊗ k D of two k-linear CDG-categories C and D is constructed as follows. The objects of C ⊗ k D are pairs (X ′ , X ′′ ) of objects X ′ ∈ C and X ′′ ∈ D. The Γ-graded k-module of morphisms 1.3. QDG-functors. Let C and D be CDG-categories. A covariant CDG-functor F : C −→ D is a homogeneous additive functor between the Γ-graded categories C and D, endowed with fixed elements a X ∈ Hom D (F (X), F (X)) of degree 1 for all objects X ∈ C such that F (Covariant or contravariant) CDG-functors C −→ D are objects of the DG-category of CDG-functors. The Γ-graded abelian group of morphisms between covariant CDG-functors F and G is the Γ-graded group of homogeneous morphisms, with the sign rule, between F and G considered as functors between Γ-graded categories. More precisely, a morphism f : F −→ G of degree n ∈ Γ is a collection of morphisms f X : F (X) −→ G(X) of degree n in D for all objects X ∈ C such that f Y F (g) = (−1) n|g| G(g)f X for any morphism g : X −→ Y in C. The differential d on the Γ-graded group Hom(F, G) of morphisms between CDG-functors F = (F, a) and A (covariant or contravariant) QDG-functor F between CDG-categories C and D is the same set of data as a CDG-functor satisfying the same equations, except for the equation connecting F (h X ) with h F (X) , which is omitted. QDG-functors C −→ D are objects of the CDG-category of QDG-functors. The Γ-graded abelian group of morphisms between QDG-functors and the differential on it are defined exactly in the same way as in the CDG-functor case. The curvature element of a QDG-functor for any object X ∈ C. A CDG-functor or QDG-functor F = (F, a) : C −→ D is said to be strict if a X = 0 for all objects X ∈ C. The identity CDG-functor Id C of a CDG-category C is the strict CDG-functor (Id C , 0). The composition of strict QDG-functors is a strict QDG-functor, and the composition of (strict) CDG-functors is a (strict) CDG-functor.
Two CDG-functors F : C −→ D and G : D −→ C between CDG-categories C and D are called mutually inverse equivalences of CDG-categories if they are equivalences of the Γ-graded categories such that the adjunction isomorphisms i : GF −→ Id C and j : F G −→ Id D are closed morphisms of CDG-functors, i. e., d(i) = 0 = d(j) (any one of the two equations implies the other one). A CDG-functor F : C −→ D is an equivalence if and only if it is fully faithful as a functor between Γ-graded categories and any object Y ∈ D is a twist of an object F (X) for some X ∈ C.
An equivalence (F, G) between CDG-categories C and D is called a strict equivalence if the CDG-functors F and G are strict. A strict CDG-functor F : C −→ D is a strict equivalence if and only if it is fully faithful as a functor between Γ-graded categories and any object Y ∈ D is isomorphic to an object F (X) for some X ∈ C.
A strict CDG-functor between DG-categories is called a DG-functor. An equivalence of DG-categories is their strict equivalence as CDG-categories.
If all objects of the category D admit twists with all of their endomorphisms of degree 1, then the embedding of the DG-category of strict CDG-functors C −→ D into the DG-category of all CDG-functors is an equivalence of DG-categories, and the embedding of the CDG-category of strict QDG-functors C −→ D into the CDG-category of all QDG-functors is a strict equivalence of CDG-categories.
A QDG-functor between k-linear CDG-categories is k-linear if its action on the Γ-graded k-modules of morphisms in the CDG-categories is k-linear. Given three k-linear CDG-categories C, D, E, the functor of composition of k-linear QDG-functors C −→ D and D −→ E is a strict k-linear CDG-functor on the tensor product of the k-linear CDG-categories of QDG-functors. The composition (on either side) with a fixed CDG-functor is a strict CDG-functor between the CDG-categories of QDG-functors, and the composition with a fixed QDG-functor is a strict QDG-functor between such CDG-categories.
Given two k-linear QDG-functors The tensor product of strict QDG-functors is a strict QDG-functor, and the tensor product of (strict) CDG-functors is a (strict) CDG-functor.

QDG-modules.
A left QDG-module over a small CDG-category C is a strict covariant QDG-functor C −→ Pre(Ab). Analogously, a right QDG-module over C is a strict contravariant QDG-functor C op −→ Pre(Ab). (Left or right) CDG-modules over a CDG-category C are similarly defined in terms of strict CDG-functors with values in the CDG-category Pre(Ab). The CDG-categories of left and right QDG-modules over C are denoted by C-mod qdg and mod qdg -C; the DG-categories of left and right CDG-modules over C are denoted by C-mod cdg and mod cdg -C. Since the CDG-category Pre(Ab) admits arbitrary twists, one obtains (strictly) equivalent (C)DG-categories by considering not necessarily strict QDG-or CDG-functors.
Given a CDG-ring or CDG-category C, we will denote by C # the underlying Γ-graded ring or category. For a QDG-module M over C, we similarly denote by M # the underlying Γ-graded C # -module (i. e., homogeneous additive functor from C # to the Γ-graded category of Γ-graded abelian groups) of M.
If k is a commutative ring and C is a k-linear CDG-category, then any QDG-functor C −→ Pre(Ab) can be lifted to a k-linear QDG-functor C −→ Pre(k-mod) in a unique way, where k-mod denotes the abelian category of k-modules. So the CDG-category C-mod qdg can be also described as the CDG-category of (strict) k-linear QDG-functors C −→ Pre(k-mod). Notice that another notation for the CDG-category Pre(k-mod) is k-mod qdg , where k is considered as a CDG-ring concentrated in degree 0 with the trivial differential and curvature, while k-mod cdg is a notation for the DG-category of complexes of k-modules.
Let C be a small k-linear CDG-category, N be a right QDG-module over C, and M be a left QDG-module. The tensor product N # ⊗ C # M # is a Γ-graded k-module defined as the quotient module of the direct sum of N(X) ⊗ k M(X) over all objects X ∈ C by the sum of the images of the maps The tensor product over C is a strict CDG-functor and its restriction to the DG-subcategories of CDG-modules is a DG-functor A QDG-functor between CDG-categories F : C −→ D induces a strict QDG-functor of inverse image (restriction of scalars) F * : D-mod qdg −→ C-mod qdg . Here we use the natural strict equivalence between the CDG-categories of arbitrary and strict QDG-functors C −→ Pre(Ab). When F is a CDG-functor, the functor F * is a strict CDG-functor, and it restricts to a DG-functor D-mod cdg −→ C-mod cdg . For any right QDG-module N and left QDG-module M over a k-linear CDG-category D and a k-linear CDG-functor F : C −→ D there is a natural map of precomplexes of k-modules F * (N) ⊗ C F * (M) −→ N ⊗ D M, commuting with the differentials.
For any CDG-category B there is a natural strict CDG-functor B −→ mod qdg -B assigning to an object X ∈ B the right QDG-module R X : Y −→ Hom B (Y, X) over B.
Here the differential on R X (Y ) coincides with the differential on Hom B (Y, X).
A CDG-module over a DG-category is called a DG-module. The DG-categories of left and right DG-modules over a small DG-category C are denoted by C-mod dg and mod dg -C. In particular, k-mod dg is yet another notation for the DG-category of complexes of k-modules for a commutative ring k. If C is a k-linear DG-category, then the objects of C-mod dg can be viewed as DG-functors C −→ k-mod dg , and the objects of mod dg -C can be viewed as DG-functors C op −→ k-mod dg .
Given left QDG-modules M ′ and M ′′ over k-linear CDG-categories B ′ and B ′′ , their tensor product M ′ ⊗ k M ′′ is the QDG-module over B ′ ⊗ k B ′′ defined as the composition of the tensor product of strict QDG-functors M ′ ⊗ M ′′ : B ′ ⊗ k B ′′ −→ Pre(k-mod) ⊗ k Pre(k-mod) with the strict CDG-functor of tensor product of precomplexes ⊗ k : Pre(k-mod) ⊗ k Pre(k-mod) −→ Pre(k-mod). The latter functor assigns to two precomplexes of k-modules their tensor product as Γ-graded k-modules, endowed with the differential defined by the usual formula. The tensor product of CDG-modules is a CDG-module.
1.5. Pseudo-equivalences. Let us call a homogeneous additive functor F # : C # −→ D # between Γ-graded additive categories C # and D # a pseudo-equivalence if F is fully faithful and any object Y ∈ D # can be obtained from objects F (X), X ∈ C # , using the operations of finite direct sum, shift, and passage to a direct summand.
A CDG-functor between CDG-categories F : C −→ D is called a pseudoequivalence if it is fully faithful as a functor between the Γ-graded categories and any object Y ∈ D can be obtained from objects F (X), X ∈ C, using the operations of finite direct sum, shift, twist, and passage to a direct summand.
The category of (left or right) Γ-graded modules over a small Γ-graded category C # is abelian. Let us call a right Γ-graded module N over C # (finitely generated ) free if it is a (finite) direct sum of representable modules R X , where X ∈ C # . A Γ-graded module P over C # is a projective object in the abelian category of Γ-graded modules if and only if it is a direct summand of a free Γ-graded module. A Γ-graded module P is a compact projective object (i. e., a projective object representing a covariant functor preserving infinite direct sums on the category of modules) if and only if it is a direct summand of a finitely generated free Γ-graded module. In this case, a Γ-graded modules P is said to be finitely generated projective.
Given a CDG-category B, denote by mod cdg fgp -B and mod qdg fgp -B the DG-category of right CDG-modules and the CDG-category of right QDG-modules over B, respectively, which are finitely generated projective as Γ-graded modules. The representable QDG-modules R X are obviously objects of mod qdg fgp -B, so there is a strict CDG-functor R : B −→ mod qdg fgp -B. There is also the strict CDG-functor of tautological embedding I : mod cdg fgp -B −→ mod qdg fgp -B. Lemma A. The CDG-functors R and I are pseudo-equivalences.
Proof. First of all notice that any two objects of a CDG-category C that are isomorphic in the Γ-graded category C # are each other's twists. In particular, so are any two QDG-modules over a CDG-category B that are isomorphic as Γ-graded B # -modules. Hence in order to prove that R is a pseudo-equivalence, it suffices to show that any (finitely generated) projective Γ-graded right B # -module P admits a QDG-module structure. Indeed, if there is a Γ-graded right B # -module Q such that the Γ-graded module P ⊕ Q admits a differential d making it a QDG-module, and ι : P −→ P ⊕Q and π : P ⊕Q −→ P are the embedding of and the projection onto the direct summand P in P ⊕ Q, then the differential πdι on P makes it a QDG-module.
To prove that I is a pseudo-equivalence, it suffices to show that the Γ-graded B # -module P ⊕ P [−1] admits a CDG-module structure for any (finitely generated) projective right B # -module P . Define the right CDG-module Q over B with the group Q(X) consisting of formal expressions of the form p ′ + d(p ′′ ), p ′ , p ′′ ∈ P (X), with P (X) embedded into Q(X) as the set of all expressions p+d(0). The differential d on Q(X), being restricted to P (X), maps p + d(0) to 0 + d(p) and B acts on P ⊂ Q as it acts on P . The action of B is extended from P to Q in the unique way making the Leibniz rule satisfied, and the differential d is extended from P to Q in the unique way making the equation on d 2 hold (see [15, proof of Theorem 3.6] for explicit formulas). There is a natural exact sequence of Γ-graded B # -modules Proof. First of all, it is obvious that if F : C # −→ D # is a pseudo-equivalence of Γ-graded categories, then the induced functor of restriction of scalars between the categories of (left or right) Γ-graded modules over D and C is an equivalence of Γ-graded categories. These equivalences transform the functor of tensor product of Γ-graded modules over D into the functor of tensor product of Γ-graded modules over C. Thus, it remains to check that any QDG-module over C can be extended to a QDG-module over D. And this is also straightforward.
More generally, one can see that the assertions of Lemma B hold for any CDG-functor F : C −→ D that is a pseudo-equivalence as a Γ-graded functor C # −→ D # . The assertions of both Lemmas A and B remain valid if one replaces finitely generated projective modules with finitely generated free ones.
Lemma C. If a CDG-functor F : C −→ D is a pseudo-equivalence of CDG-categories, then so is the CDG-functor F op :

Ext and Tor of the Second Kind
This section contains an exposition of the classical theory of the two kinds of differential derived functors, largely following [5], except that we deal with CDG-categories rather than DG-(co)algebras. The classical theory allows to establish an isomorphism between the Hochschild (co)homology of the second kind of a CDG-category B and the DG-category C of right CDG-modules over B that are finitely generated and projective as graded B-modules. We also construct a natural map between the two kinds of Hochschild (co)homology of any DG-category C linear over a field k.

2.1.
Ext and Tor of the first kind. Given a DG-category D, denote by Z 0 (D) the category whose objects are the objects of D and whose morphisms are the closed (i. e., annihilated by the differential) morphisms of degree 0 in D. Let H 0 (D) denote the category whose objects are the objects of D and whose morphisms are the elements of the cohomology groups of degree 0 of the complexes of morphisms in D. The categories Z 0 (D) and H 0 (D) have preadditive category structures (i. e., the abelian group structures on the sets of morphisms). In addition, these categories are endowed with the shift functors X −→ X[n] for all n ∈ Γ, provided that shifts of all objects exist in D (see 1.2). Finally, let H(D) denote the Γ-graded category whose objects are the objects of D and whose morphisms are the Γ-graded groups of cohomology of the complexes of morphisms in D.
Let k be a commutative ring and C be a small k-linear DG-category. Let us endow the additive categories Z 0 (C-mod dg ) and Z 0 (mod dg -C) with the following exact category structures. In other words, for any object X ∈ C the sequence M ′ (X) −→ M(X) −→ M ′′ (X) must be a short exact sequence of complexes of k-modules whose Γ-graded cohomology modules also form a short exact sequence (i. e., the boundary maps vanish).
Denote the additive category Z 0 (k-mod dg ) of Γ-graded complexes of k-modules with its exact category structure defined above by Com ex (k-mod). Let d denote the differentials on objects of Com ex (k-mod). We will be interested in the derived categories D − (Com ex (k-mod)) and D + (Com ex (k-mod)) of complexes, bounded from above or below, over the exact category Com ex (k-mod). The differential acting between the terms of a complex over Com ex (k-mod) will be denoted by ∂.
The objects of D − (Com ex (k-mod)) can be viewed as bicomplexes with one grading by the integers bounded from above and the other grading by elements of the group Γ. (The differential d preserves the grading by the integers, while changing the Γ-valued grading; and the differential ∂ raises the grading by the integers by 1, while preserving the Γ-valued grading.) To any such bicomplex, one can assign its Γ-graded total complex, constructed by taking infinite direct sums along the diagonals. This defines a triangulated functor from D − (Com ex (k-mod)) to the unbounded derived category of Γ-graded complexes of k-modules, Analogously, the objects of D + (Com ex (k-mod)) can be viewed as bicomplexes with one grading by the integers bounded from below and the other grading by elements of the group Γ. To any such bicomplex, one can assign its Γ-graded total complex, constructed by taking infinite products along the diagonals. This defines a triangulated functor Tot ⊓ : D + (Com ex (k-mod)) −−→ D(k-mod). Any complex over Com ex (k-mod) bounded from above (resp., below) that becomes exact (with respect to the differential ∂) after passing to the cohomology of the Γ-graded complexes of k-modules (with respect to the differential d) is annihilated by the functor Tot ⊕ (resp., Tot ⊓ ).
Remark. The latter assertion does not hold for the total complexes of unbounded complexes over Com ex (k-mod), constructed by taking infinite direct sums or products along the diagonals. That is the reason why we define the functors Tot ⊕ and Tot ⊓ for bounded complexes only. The assertion holds, however, for the functor of "Laurent totalization" of unbounded complexes, which coincides with Tot ⊕ for complexes bounded from above and with Tot ⊓ for complexes bounded from below. See [5] and the introduction to [15] (cf. Remark 2.2). Now consider the functor of two arguments (see 1.4) We would like to construct its left derived functor ). For this purpose, notice that both exact categories Z 0 (mod dg -C) and Z 0 (C-mod dg ) have enough projective objects. Specifically, for any object X ∈ C the representable DG-module R X ∈ Z 0 (mod dg -C) is projective, and so is the the cone of the identity endomorphism of R X (taken in the DG-category mod dg -C). Any object of Z 0 (mod dg -C) is the image of an admissible epimorphism acting from an (infinite) direct sum of shifts of objects of the above two types.
Given a right DG-module N and a left DG-module M over C, choose a left projective resolution Q • of N and a left projective resolution P • of M in the exact categories Z 0 (mod dg -C) and Z 0 (C-mod dg ). When substituted as one of the arguments of the functor ⊗ C , any projective object of one of the exact categories of DG-modules makes this functor an exact functor from the other exact category of DG-modules to the exact category Com ex (k-mod). This allows to define N ⊗ L C M ∈ D − (Com ex (k-mod)) as the object represented either by the complex Q • ⊗ C M, or by the complex N ⊗ C P • , or by the total complex of the bicomplex Q • ⊗ C P • .
Analogously, consider the functor of two arguments assigning to any two left DG-modules over C the complex of morphisms between them as DG-functors C −→ k-mod dg . We would like to construct its right derived functor Notice that the exact category Z 0 (C-mod dg ) has enough injective objects. For any projective object Q ∈ Z 0 (mod dg -C) and an injective k-module I, the object Hom k (Q, I) ∈ Z 0 (C-mod dg ) is injective, and any injective object in the exact category Z 0 (C-mod dg ) is a direct summand of an object of this type. To prove these assertions, it suffices to check that for any DG-module M ∈ Z 0 (mod dg -C), any object X ∈ C, and any element of M(X) or H(M)(X) there is a DG-module Q as above and a closed morphism of DG-modules M −→ Hom k (Q, I) that is injective on the chosen element. Given left DG-modules L and M over C, choose a left projective resolution P • of L and a right injective resolution J • of M in the exact category Z 0 (C-mod dg ). Substituting a projective object as the first argument or an injective object as the second argument of the functor Hom C , one obtains an exact functor from the exact category of DG-modules in the other argument to the exact category Com ex (k-mod). This allows to define RHom C (L, M) ∈ D + (Com ex (k-mod)) as the object represented either by the complex Hom C (P • , M), or by the complex Hom C (L, J • ), or by the total complex of the bicomplex Hom C (P • , J • ).
Composing the derived functor ⊗ L C with the functor Tot ⊕ , we obtain the derived functor Tor C : Z 0 (mod dg -C) × Z 0 (C-mod dg ) −−→ D(k-mod). Similarly, composing the derived functor RHom C with the functor Tot ⊓ , we obtain the derived functor One can compute the derived functors Tor C and Ext C using resolutions of a more general type than above. Specifically, let N and M be a left and a right DG-module over C. Let · · · −→ F 2 −→ F 1 −→ F 0 −→ M be a complex of left DG-modules over C and (closed morphisms between them) such that the complex of Γ-graded Assume that the DG-modules F i are h-flat (homotopy flat), i. e., for any i 0 and any right DG-module R over C such that H(R) = 0 one has H(R ⊗ C F i ) = 0. Let Q • be a left projective resolution of the DG-module N in the exact category of right DG-modules over C. Then the natural maps Tot ⊕ (Q Analogously, let L and M be left DG-modules over C. Let · · · −→ P 2 −→ P 1 −→ P 0 −→ L be a complex of left DG-modules over C which becomes exact after passing to the Γ-graded cohomology modules. Assume that the DG-modules P i are h-projective (homotopy projective), i. e., for any i 0 and any left DG-module R over C such that H(R) = 0 one has H(Hom C (P i , R)) = 0. Then the complex of k-modules Tot ⊓ (Hom C (P • , M)) represents the object Ext C (L, M) in D(k-mod). Similarly, let M −→ J 0 −→ J 1 −→ J 2 −→ · · · be a complex of left DG-modules over C which becomes exact after passing to the cohomology modules. Assume that the DG-modules J i are h-injective, i. e., for any i 0 and any left DG-module R over C such that H(R) = 0 one has H(Hom C (R, J i )) = 0. Then the complex of k-modules Tot ⊓ (Hom C (L, J • )) represents the object Ext C (L, M).
In particular, it follows that the functors Tor C and Ext C transform quasiisomorphisms of DG-modules (i. e., morphisms of DG-modules inducing isomorphisms of the Γ-graded cohomology modules) in any of their arguments into isomorphisms in D(k-mod).
Furthermore, consider the case when the complex of morphisms between any two objects of C is an h-flat complex of k-modules. Then for any left DG-module M over C such that the complex of k-modules M(X) is h-flat for any object X ∈ C, the bar-construction where we use the simplifying notation C(X, Y ) = Hom C (Y, X) for any objects X, Y ∈ C, defines a left resolution of the DG-module M which consists of h-flat DG-modules over C and remains exact after passing to the cohomology modules. Thus, for any right DG-module N over C the total complex of the bar-complex constructed by taking infinite direct sums along the diagonals, represents the object Tor C (N, M) in D(k-mod). The h-flatness condition on the DG-module M can be replaced with the similar condition on the DG-module N.
Analogously, assume that the complex of morphisms between any two objects of C is an h-projective complex of k-modules. Let L and M be left DG-modules over C such that either the complex of k-modules L(X) is h-projective for any object X ∈ C or the complex of k-modules M(X) is h-injective for any object X ∈ C. Then the total complex of the cobar-complex

2.2.
Ext and Tor of the second kind: general case. Let B be a small k-linear CDG-category. Then the categories Z 0 (B-mod cdg ) and Z 0 (mod cdg -B) of (left and right) CDG-modules over B and closed morphisms of degree 0 between them are abelian. In particular, consider the abelian category Z 0 (k-mod cdg ) of Γ-graded complexes of k-modules and denote it by Com ab (k-mod). We will be interested in the derived categories D − (Com ab (k-mod)) and D + (Com ab (k-mod)) of complexes, bounded from above or below, over the abelian category Com ab (k-mod). The objects of D − (Com ab (k-mod)) can be viewed as bicomplexes with one grading by the integers bounded from above and the other grading by elements of the group Γ. To any such bicomplex, one can assign its Γ-graded total complex, constructed by taking infinite products along the diagonals. This defines a triangulated functor Analogously, the objects of D + (Com ab (k-mod)) can be viewed as bicomplexes with one grading by the integers bounded from below and the other grading by elements of the group Γ. To any such bicomplex, one can assign its Γ-graded total complex, constructed by taking infinite direct sums along the diagonals. This defines a triangulated functor Remark. The functors of total complexes of unbounded complexes over Com ab (k-mod), constructed by taking infinite direct sums or infinite products along the diagonals, are not well-defined on the derived category D(Com ab (k-mod)). The procedure of "Laurent totalization" of unbounded complexes, which coincides with Tot ⊓ for complexes bounded from above and with Tot ⊕ for complexes bounded from below, defines a functor on D(Com ab (k-mod)), though. Notice that this Laurent totalization is different from the one discussed in Remark 2.1 (the chosen direction along the diagonals is opposite in the two cases). Now consider the functor of two arguments (see 1.4) We would like to construct its left derived functor . Notice that the abelian categories Z 0 (mod cdg -B) and Z 0 (B-mod cdg ) have enough projective objects. More precisely, for any projective left Γ-graded module P over B # the corresponding freely generated CDG-module Q, as constructed in the proof of Lemma 1.5.A, is a projective object of Z 0 (B-mod cdg ). Any projective object in Z 0 (B-mod cdg ) is a direct summand of an object of this type. For any projective object Let us call a left Γ-graded B # -module P # flat if the functor of tensor product with P # over B # is exact on the abelian category of right Γ-graded B # -modules. Given a left CDG-module P over B, if the left B # -module P # is flat, then the functor of tensor product with P is exact as a functor In view of the above remarks, we can define N ⊗ L B M ∈ D − (Com ab (k-mod)) as the object represented either by the complex Q • ⊗ B M, or by the complex N ⊗ B P • , or by the total complex of the bicomplex Q • ⊗ B P • .
Analogously, consider the functor of two arguments assigning to any two left CDG-modules over B the complex of morphisms between them as strict CDG-functors B −→ k-mod cdg . We would like to construct its right derived functor Notice that the abelian category Z 0 (B-mod cdg ) has enough injective objects. For any injective object J in Z 0 (B-mod cdg ), the underlying left Γ-graded B # -module J # is injective. One can construct these injective CDG-modules as the duals to projective (or flat) right CDG-modules (see the discussion of injective DG-modules in 2.1) or obtain them as the CDG-modules cofreely cogenerated by injective Γ-graded Composing the derived functor ⊗ L B with the functor Tot ⊓ , we obtain the derived functor . Similarly, composing the derived functor RHom B with the functor Tot ⊕ , we obtain the derived functor . The derived functors Tor B,II and Ext II B are called the Tor and Ext of the second kind of CDG-modules over B.
Notice that the derived functors ⊗ L B and RHom B assign distinghuished triangles to short exact sequences of CDG-modules in any argument, hence so do the derived functors Tor B,II and Ext II B . Given a k-linear CDG-functor F : B −→ C, a right CDG-module N over C, and a left CDG-module M over C, there is a natural morphism in D(k-mod). Analogously, given a k-linear CDG-functor F : B −→ C and left CDG-modules L and M over C, there is a natural morphism If the functor F # : B # −→ C # is a pseudo-equivalence of Γ-graded categories, then these natural morphisms are isomorphisms for any CDG-modules L, M, and N. Now let C be a small k-linear DG-category. Then the identity functors from the exact categories Z 0 (mod dg -C) and Z 0 (C-mod dg ) to the abelian categories Z 0 (mod cdg -C) and Z 0 (C-mod cdg ) are exact, so any resolution in Z 0 (mod dg -C) or Z 0 (C-mod dg ) is also a resolution in Z 0 (mod cdg -C) or Z 0 (C-mod cdg ). Besides, any DG-module that is projective or injective in the exact category Z 0 (mod dg -C) or Z 0 (C-mod dg ) is also projective or injective as a Γ-graded C # -module. It follows that there are natural morphisms and in D(k-mod) for any DG-modules L, M, and N over C.

2.3.
Flat/projective case. Let B be a small k-linear CDG-category, N a right CDG-module over B, and M a left CDG-module over B. Consider the Γ-graded complex of k-modules Bar ⊓ (N, B, M) constructed in the following way. As a Γ-graded k-module, Bar ⊓ (N, B, M) is obtained by totalizing a bigraded k-module with one grading by elements of the group Γ and the other grading by nonpositive integers, the totalizing being performed by taking infinite products along the diagonals. The component of degree −i ∈ Z of that bigraded module is the Γ-graded k-module where, as in 2.1, we use the simplifying notation B(X, Y ) = Hom B (Y, X).
The differential on Bar ⊓ (N, B, M) is the sum of the three components ∂, d, and δ given by the formulas where the products b j b j+1 denote the composition of morphisms in B and the products nb 1 and b i m denote the action of morphisms in B on the CDG-modules, Proof. Choose a left resolution Q • of the right CDG-module N and a left resolution P • of the left CDG-module M such that the Γ-graded B # -modules P # j and Q # j are flat. Consider the tricomplex Bar ⊓ (Q • , B, P • ) and construct its Γ-graded total complex by taking infinite products along the diagonals. Then this total complex maps naturally to both the complex Bar ⊓ (N, B, M) and the total complex Tot ⊓ (Q • ⊗ B P • ) of the tricomplex Q • ⊗ B P • , constructed also by taking infinite products along the diagonals. These morphisms of Γ-graded complexes are both quasi-isomorphisms. Cf. the proof of Proposition 2.4.A below, where some additional details can be found.
Let F : B −→ C be a k-linear CDG-functor, N be a right CDG-module over C, and M be a left CDG-module over C. Then there is a natural morphism of complexes of k-modules F * : Bar ⊓ (F * N, B, F * M) −→ Bar ⊓ (N, C, M) given by the rule The image of an arbitrary element in Bar ⊓ (F * N, B, F * M) is constructed as the sum of the images of (the infinite number of) its bihomogeneous components, the sum being convergent bidegree-wise in Bar ⊓ (N, C, M).
Suppose the CDG-categories B and C satisfy the assumptions of Proposition A, and so does one of the CDG-modules N and M. Then the morphism of bar-complexes F * represents the morphism (7) of the objects Tor in D(k-mod). Now let L and M be left CDG-modules over B. Consider the Γ-graded complex of k-modules Cob ⊕ (L, B, M) constructed as follows. As a Γ-graded k-module, Cob ⊕ (L, B, M) is obtained by totalizing a bigraded k-module with one grading by elements of the group Γ and the other grading by nonnegative integers, the totalizing being done by taking infinite direct sums along the diagonals. The component of degree i ∈ Z of that bigraded module is the Γ-graded k-module The differential on Cob ⊕ (L, B, M) is the sum of the three components ∂, d, and δ given by the formulas Suppose the CDG-categories B and C satisfy the assumptions of Proposition B, and so does one of the CDG-modules L and M. Then the morphism of cobar-complexes F * represents the morphism (8) of the objects Ext in D(k-mod).
Denote by Bar ⊕ (N, B, M) the Γ-graded complex of k-modules constructed in the same way as Bar ⊓ (N, B, M), except that the totalization is being done by taking infinite direct sums along the diagonals. Similarly, denote by Cob ⊓ (L, B, M) the Γ-graded complex of k-modules constructed in the same way as Cob ⊕ (L, B, M) except that the totalization is being done by taking infinite products along the diagonals.
Assume that C is a small DG-category in which the complex of morphisms between any two objects is an h-flat complex of flat k-modules, and either a right DG-module N or a left DG-module M over C is such that all the complexes of k-modules N(X) or M(X) are h-flat complexes of flat k-modules. Then the natural map Analogously, assume that the complex of morphisms between any two objects in a DG-category C is an h-projective complex of projective k-modules, and either a left DG-module L over C is such that all the complexes of k-modules L(X) are h-projective complexes of projective k-modules, or a left DG-module M over C is such that all the complexes of k-modules M(X) are h-injective complexes of injective  (11) and (13).
Proposition C. Let B be a small k-linear CDG-category. Assume that the maps k −→ Hom B (X, X) corresponding to the curvature elements h X ∈ Hom B (X, X) admit k-linear retractions Hom B (X, X) −→ k, i. e., they are embeddings of k-module direct summands. In particular, this holds when k is a field and all the elements h X are nonzero. Then for any CDG-modules L, M and N the complexes Bar ⊕ (N, B, M) and Cob ⊓ (L, B, M) are acyclic.
Proof. This follows from the fact that the differentials δ on the bigraded bar-and cobar-complexes are acyclic.
2.4. Hochschild (co)homology. Let B be a small k-linear CDG-category. Consider the CDG-category B ⊗ k B op ; since it is naturally isomorphic to its opposite CDG-category, there is no need to distinguish between the left and the right CDG-modules over it. Furthermore, there is a natural (left) CDG-module over the CDG-category B ⊗ k B op assigning to an object (X, Y op ) ∈ B ⊗ k B op the precomplex of k-modules B(X, Y ) = Hom B (Y, X). By an abuse of notation, we will denote this CDG-module (as well as the corresponding right CDG-module) simply by B. Assume that the Γ-graded k-modules B # (X, Y ) are flat for all objects X, Y ∈ B.
The Remark. We define the Hochschild (co)homology of the second kind for CDG-categories B satisfying the above flatness assumption only, even though our definition makes sense without this requirement. In fact, this assumption is never used in this paper (except in the discussion of explicit complexes below in this section, which requires a stronger projectivity assumption in the cohomology case anyway). However, we believe that our definition is not the right one without the flatness assumption, since one is not supposed to use underived nonexact functors when defining (co)homology theories. So to define the Hochschild (co)homology of the second kind in the general case one would need to replace a CDG-category B with a CDG-category, equivalent to it in some sense and satisfying the flatness requirement. We do not know how such a replacement could look like. The analogue of this procedure for Hochschild (co)homology of the first kind is well-known (in this case it suffices to replace a DG-category C with a quasi-equivalent DG-category with h-flat complexes of morphisms; see below). There are smaller and much more important complexes computing the Hochschild (co)homology, namely, the Hochschild complexes. The homological Hochschild complex of the second kind Hoch ⊓ • (B, M) is constructed in the following way. As a Γ-graded k-module, Hoch ⊓ • (B, M) is obtained by taking infinite products along the diagonals of a bigraded k-module with one grading by elements of the group Γ and the other grading by nonpositive integers. The component of degree −i ∈ Z of that bigraded k-module is the Γ-graded k-module is the sum of the three components ∂, d, and δ given by the formulas Proof. Choose a left resolution P • of the CDG-module M such that the Γ-graded B # ⊗ k B #op -modules P # j are flat. Consider the bicomplex Hoch ⊓ • (B, P • ) and construct its total complex by taking infinite products along the diagonals. This total complex maps naturally to both the complex Hoch ⊓ • (B, M) and the total complex of the bicomplex B ⊗ B⊗ k B op P • , constructed by taking infinite products along the diagonals. These morphisms of Γ-graded complexes are both quasi-isomorphisms.
Indeed, the morphism Hoch ⊓ The latter assertion follows from the similar statement for the bigraded Hochschild complex of the Γ-graded B # ⊗ k B #op -module P # with the differential ∂.
There is a natural morphism of complexes of k-modules F * : where the value of ρ in the exponent is given by the formula (12). The image of an arbitrary element in Hoch ⊓ • (B, F * M) is constructed as the sum of the images of (the infinite number of) its bihomogeneous components, the sum being convergent bidegree-wise in Hoch • (C, M).
The morphism F * of Hochschild complexes computes the map of Hochschild homology (16) HH II * (B, F * M) −−→ HH II * (C, M) obtained by passing to the homology in the morphism of Tor objects (7) for the The cohomological Hochschild complex of the second kind Hoch ⊕,• (B, M) is constructed as follows. As a Γ-graded k-module, Hoch ⊓ • (B, M) is obtained by taking infinite direct sums along the diagonals of a bigraded k-module with one grading by elements of the group Γ and the other grading by nonnegative integers. The component of degree i ∈ Z of that bigraded k-module is the Γ-graded k-module . The differential on Hoch ⊕,• (B, M) is the sum of the three components ∂, d, and δ given by the formulas where the value of λ in the exponent is given by the formula (14). Suppose the CDG-categories B and C satisfy the assumptions of Proposition B. Then the morphism F * of Hochschild complexes computes the map of Hochschild cohomology (19) HH play an important role when B is a DG-category, but apparently not otherwise, as we will see below.
Let C be a small k-linear DG-category. Assume that the complexes of k-modules C(X, Y ) are h-flat for all objects X, Y ∈ C. The (conventional) Hochschild homology (of the first kind) HH * (C, M) of a k-linear DG-category C with coefficients in a DG-module M over C ⊗ k C op is the homology of the object Tor C⊗ k C op (C, M) ∈ D(k-mod). In particular, the Hochschild homology of the DG-module M = C over C is called simply the Hochschild homology of C and denoted by HH * (C, C) = HH * (C).
The (conventional) Hochschild cohomology (of the first kind) HH * (C, M) of a k-linear DG-category C with coefficients in a DG-module M over C ⊗ k C op is the cohomology of the object Ext C⊗ k C op (C, M) ∈ D(k-mod). In particular, the Hochschild cohomology of the DG-module M = C over C is called simply the Hochschild cohomology of C and denoted by HH * (C, C) = HH * (C).
Let F : C −→ D be a k-linear DG-functor between DG-categories whose complexes of morphisms are h-flat complexes of k-modules. Then for any DG-module M over D ⊗ k D op passing to the homology in the morphism of Tor objects (3) for the DG-functor F ⊗ F op provides a natural map of Γ-graded k-modules (20) HH Composing this map with the map induced by the closed morphism C −→ F * D of DG-modules over C ⊗ k C op , we obtain a natural map Passing to the cohomology in the morphism of Ext objects (4) for the DG-functor F ⊗ F op provides a natural map When the complexes C(X, Y ) are h-flat complexes of flat k-modules for all objects X, Y ∈ C, both the Hochschild (co)homology of the first and the second kind are defined for any DG-module M over C ⊗ k C op . In this case, there are natural morphisms of Γ-graded k-modules

2.5.
Change of grading group. Let us first introduce some terminology that will be used throughout the rest of the paper. A Γ-graded module N # over a Γ-graded category B # is said to have flat dimension d if d is the minimal length of a left flat resolution of N # in the abelian category of Γ-graded B # -modules, or equivalently, the functor of tensor product with N # over B # has the homological dimension d.
Projective and injective dimensions of Γ-graded B # -modules are defined in the similar way. The left homological dimension of a Γ-graded category B # is the homological dimension of the abelian category of Γ-graded left B # -modules, and the weak homological dimension of B # is the homological dimension of the functor of tensor product of Γ-graded modules over B # .
The functor φ * has a left adjoint functor φ ! and a right adjoint functor φ * . The former assigns to a Γ-graded k-module V the Γ ′ -graded k-module V ′ constructed by taking the direct sums of the grading components of V over all the preimages in Γ of a given element n ′ ∈ Γ ′ , while the latter involves taking direct products over the preimages of n ′ in Γ.
All three functors φ ! , φ * , and φ * are exact. Besides, they transform (pre)complexes of k-modules to (pre)complexes of k-modules and commute with passing to the cohomology of the complexes of k-modules. So they induce triangulated functors between the derived categories D Γ (k-mod) and D Γ ′ (k-mod) of Γ-graded and Γ ′ -graded complexes of k-modules.
Given a Γ-graded k-linear CDG-category B, one can apply the functor φ ! to all its precomplexes of morphisms, obtaining a Γ ′ -graded k-linear CDG-category φ ! B. To a (left or right) CDG-module M ′ over φ ! B one can assign a CDG-module φ * M ′ over B, and to a CDG-module M over B one can assign CDG-modules φ ! M and φ * M over φ ! B.
The functors φ ! , φ * , and φ * are compatible with the functors of tensor product and Hom of CDG-modules in the following sense. For any left CDG-modules L, M and right CDG-module N over B there are natural isomorphisms For any left CDG-modules L ′ and M ′ over φ ! B there are natural isomorphisms It follows from the isomorphisms (26) that the functors φ ! preserve all the flatness and projectivity properties of CDG-and DG-modules considered above in this paper, while the functors φ * preserve the injectivity properties. Furthermore, the functors φ ! commute with the functors Tot ⊕ , while the functors φ * commute with the functors Tot ⊓ . Therefore, in view of the isomorphisms (25), for any Γ-graded DG-category C and DG-modules L, M, and N over it there are natural isomorphisms Furthermore, the functor φ ! preserves tensor products of k-linear (C)DG-categories. Thus, assuming that the complexes of morphisms in the DG-category C are h-flat complexes of k-modules, for any DG-module M over C ⊗ k C op there are natural isomorphisms of Hochschild (co)homology In particular, there is an isomorphism and a natural morphism The latter morphism is an isomorphism when the kernel of the map φ : Γ −→ Γ ′ is finite (so the functors φ ! and φ * are isomorphic). The analogous results for (co)homology theories of the second kind hold under more restrictive conditions, since the functor φ ! does not commute with Tot ⊓ in general, nor does the functor φ * commute with Tot ⊕ . However, there are morphisms of functors Hence for any Γ-graded CDG-category B and CDG-modules L, M, and N over it there are natural morphisms in D Γ ′ (k-mod). The morphisms (31-32) are always isomorphisms when the kernel of the map φ : Γ −→ Γ ′ is finite. They are also isomorphisms when the derived functors in question can be computed using finite resolutions (cf. 3.3). So the morphism (31) is an isomorphism whenever one of the Γ-graded B # -modules N # and M # has finite flat dimension. The morphism (32) is an isomorphism whenever either the Γ-graded B # -module L # has finite projective dimension, or the Γ-graded B # -module has finite injective dimension.
Thus, assuming that the Γ-graded k-modules of morphisms in the category B # are flat, for any CDG-module M over B⊗ k B op there are natural morphisms of Hochschild (co)homology which are always isomorphims when the kernel of the map φ : Γ −→ Γ ′ is finite. The map (33) is an isomorphism whenever one of the Γ-graded B # ⊗ k B #op -modules B # and M # has finite flat dimension. The map (34) is an isomorphism whenever either the Γ-graded B # ⊗ k B #op -module B # has finite projective dimension, or the Γ-graded In particular, there is a natural map , which is an isomorphism when either the kernel of the map φ : Γ −→ Γ ′ is finite, or the Γ-graded B # ⊗ k B #op -module B # has finite flat dimension. There are also natural maps which are both isomorphisms when the kernel of the map φ : Γ −→ Γ ′ is finite. Given a Γ ′ -graded k-linear CDG-category B ′ , one can apply the functor φ * to all of its precomplexes of morphisms, obtaining a Γ-graded k-linear CDG-category φ * B ′ . To a (left or right) CDG-module M ′ over B ′ one can assign a CDG-module φ * M ′ over φ * B ′ . Assume that the map φ : Γ −→ Γ ′ is surjective. Then the functors φ * : Z 0 (B ′ -mod cdg ) −→ Z 0 (φ * B ′ -mod cdg ) and Z 0 (mod cdg -B ′ ) −→ Z 0 (mod cdg -φ * B ′ ) are equivalences of abelian categories. For any left CDG-modules L ′ , M ′ and right CDG-module N ′ over B ′ there are natural isomorphisms Furthermore, the functor φ * commutes with the functors Tot ⊕ and Tot ⊓ when applied to polycomplexes with one grading by elements of the group Γ ′ and the remaining gradings by the integers. Therefore, there are natural isomorphisms in D Γ (k-mod), and similar isomorphisms for the Tor and Ext of the first kind over a k-linear DG-category C.
There is a natural strict CDG-functor φ * B ′ ⊗ k φ * B ′op −→ φ * (B ′ ⊗ k B ′op ). So, assuming that the Γ ′ -graded k-modules of morphisms in the category B ′# are flat, for any CDG-module M ′ over B ′ ⊗ k B ′op there are natural maps and, in particular, . One can see that the maps (40-42) are isomorphisms whenever the kernel Γ ′′ of the map φ : Γ −→ Γ ′ is finite and its order |Γ ′′ | is invertible in k. Indeed, the CDG-category φ * B ′ ⊗ k φ * B ′op is linear over the group ring k[Γ ′′ ] of the abelian group Γ ′′ , and the CDG- The same assertions apply to Hochschild (co)homology of the first kind of a Γ ′ -graded DG-category C whose complexes of morphisms are h-flat complexes of k-modules.
2.6. DG-category of CDG-modules. Let B be a small k-linear CDG-category such that the Γ-graded k-modules B # (X, Y ) are flat for all objects X, Y ∈ B. Denote by C = mod cdg fgp -B the DG-category of right CDG-modules over B, projective and finitely generated as Γ-graded B # -modules, and by D = mod qdg fgp -B the CDG-category of right QDG-modules over B satisfying the same condition. The results below also apply to finitely generated free modules in place of finitely generated projective ones.
There are strict k-linear CDG-functors R : B −→ D and I : C −→ D, and moreover, these CDG-functors are pseudo-equivalences of CDG-categories (see 1.5). Strictly speaking, the categories C and D as we have defined them are only essentially small rather than small, i. e., they are strictly equivalent to small CDG-categories. So from now on we will tacitly assume that C and D have been replaced with their small full subcategories containing at least one object in every isomorphism class and such that the functors R and I are still defined.
The pseudo-equivalences R and I induce equivalences between the DG-categories of (left or right) CDG-modules over the CDG-categories B, C, and D. Let N be a right CDG-module and L, M be left CDG-modules over B; denote by N C , N D , L C , L D , etc. the corresponding CDG-modules over C and D (which are defined uniquely up to a unique isomorphism). By the results of 2.2 (see (7)(8)), the CDG-functors R and I induce isomorphisms This is a generalization of [18,Theorem 2.14].
When the ring k has finite weak homological dimension, any Γ-graded complex of flat k-modules is flat. So if the Γ-graded k-modules of morphisms in the category B # , and hence also in the category C # , are flat, then the complexes of morphisms in the DG-category C are h-flat. Thus, both the Hochschild (co)homology of the first and the second kind are defined for the DG-category C, and therefore the natural maps between the Hochschild (co)homology of the first and second kind of the DG-category C with coefficients in any DG-module over C ⊗ k C op are defined.

Derived Categories of the Second Kind
In this section we interpret, under certain homological dimension assumptions, the Ext and Tor of the second kind over a CDG-category in terms of the derived categories of the second kind of CDG-modules over it. This allows to obtain sufficient conditions for an isomorphism of the Hochschild (co)homology of the first and second kind for a DG-category, and in particular, for the DG-category C of CDG-modules over a CDG-category B, projective and finitely generated as Γ-graded B # -modules.
3.1. Conventional derived category. Given a DG-category D, the additive category H 0 (D) has a natural triangulated category structure provided that a zero object and all shift and cones exist in D. In particular, for any small DG-category C the categories H 0 (C-mod dg ) and H 0 (mod dg -C) are triangulated. These are called the homotopy categories of (left and right) DG-modules over C.
A (left or right) DG-module M over C is said to be acyclic if the complexes M(X) are acyclic for all objects X ∈ C, i. e., H(M) = 0. Acyclic DG-modules form thick subcategories, closed under both infinite directs sums and infinite products, in the homotopy categories of DG-modules. The quotient categories by these thick subcategories are called the (conventional) derived categories (of the first kind) of DG-modules over C and denoted by D(C-mod dg ) and D(mod dg -C).
The full subcategory of h-projective DG-modules H 0 (C-mod dg ) prj ⊂ H 0 (C-mod dg ) is a triangulated subcategory whose functor to D(C-mod dg ) is an equivalence of categories [6], and the same applies to the full subcategory of h-injective DG-modules H 0 (C-mod dg ) inj ⊂ H 0 (C-mod dg ). To prove these results, one notices first of all that any projective object in the exact category Z 0 (C-mod dg ) is an h-projective DG-module, and similarly for injectives (see 2.1 for the discussion of this exact category and its projective/injective objects). Let P • be a left projective resolution of a DG-module M in Z 0 (C-mod); then the total DG-module of P • , constructed by taking infinite direct sums along the diagonals, is an h-projective DG-module quasi-isomorphic to M. Similarly, if J • is a right injective resolution of a DG-module M in Z 0 (C-mod dg ), then the total DG-module of J • , constructed by taking infinite products along the diagonals, is an h-injective DG-module quasi-isomorphic to M [15, Section 1].
Furthermore, the full subcategory of h-flat DG-modules H 0 (C-mod dg ) fl ⊂ H 0 (C-mod dg ) is a triangulated subcategory whose quotient category by its intersection with thick subcategory of acyclic DG-modules is equivalent to D(C-mod dg ). This follows from the above result for h-projective DG-modules and the fact that any h-flat DG-module is h-projective. The same applies to the full subcategory of h-flat right DG-modules H 0 (mod dg -C) fl ⊂ H 0 (mod dg -C).
Let k be a commutative ring and C be a small k-linear DG-category. Restricting the triangulated functor of two arguments (see (2)) to the full subcategory of h-projective DG-modules in the first argument, one obtains a functor that factors through the derived category in the second argument, providing the derived functor Alternatively, restricting the functor Hom C to the full subcategory of h-injective DG-modules in the second argument, one obtains a functor that factors through the derived category in the first argument, leading to the same derived functor Ext C . The composition of this derived functor with the localization functor Z 0 (C-mod dg ) −→ D(C-mod dg ) is isomorphic to the derived functor Ext C constructed in 2.1. For any left DG-modules L and M over C there is a natural isomorphism Analogously, restricting the triangulated functor of two arguments (see (1)) to the full subcategory of h-flat DG-modules in the first argument one obtains a functor that factors through the Cartesian product of the derived categories, providing the derived functor The same derived functor can be obtained by restricting the functor ⊗ C to the full subcategory of h-flat DG-modules in the second argument. Up to composing with the localization functors Z 0 (mod dg -C) −→ D(mod dg -C) and Z 0 (C-mod dg ) −→ D(C-mod dg ), this is the same derived functor Tor C that was constructed in 2.1.

3.2.
Derived categories of the second kind. Let B be a small CDG-category. As in 3.1, the homotopy categories of CDG-modules H 0 (B-mod cdg ) and H 0 (mod cdg -B) over B are naturally triangulated. Given a short exact sequence 0 −→ K ′ −→ K −→ K ′′ −→ 0 in the abelian category Z 0 (B-mod cdg ), one can consider it as a finite complex of closed morphisms in the DG-category B-mod cdg and take the corresponding total object in B-mod cdg [15, Section 1.2].
A left CDG-module over B is called absolutely acyclic if it belongs to the minimal thick subcategory of H 0 (B-mod cdg ) containing the total CDG-modules of exact triples of CDG-modules. The quotient category of H 0 (B-mod cdg ) by the thick subcategory of absolutely acyclic CDG-modules is called the absolute derived category of left CDG-modules over B and denoted by D abs (B-mod cdg ) [15, Section 3.3].
A left CDG-module over B is called coacyclic if it belongs to the minimal triangulated subcategory of H 0 (B-mod cdg ) containing the total CDG-modules of exact triples of CDG-modules and closed under infinite direct sums. The quotient category of H 0 (B-mod cdg ) by the thick subcategory of coacyclic CDG-modules is called the coderived category of left CDG-modules over B and denoted by D co (B-mod cdg ).
The definition of a contraacyclic CDG-module is dual to the previous one. A left CDG-module over B is called contraacyclic if it belongs to the minimal triangulated subcategory of H 0 (B-mod cdg ) containing the total CDG-modules of exact triples of CDG-modules and closed under infinite products. The quotient category of H 0 (B-mod cdg ) by the thick subcategory of contraacyclic CDG-modules is called the contraderived category of left CDG-modules over B and denoted by D ctr (B-mod cdg ).
Coacyclic, contraacyclic, and absolutely acyclic right CDG-modules are defined in the analogous way. The corresponding exotic derived (quotient) categories are denoted by D co (mod cdg -B), D ctr (mod cdg -B), and D abs (mod cdg -B).
We will use the similar notation D co (C-mod dg ), D ctr (C-mod dg ), etc., in the particular case of the coderived, contraderived, and absolutely derived categories of DG-modules over a small DG-category C. Notice that any coacyclic or contraacyclic DG-module is acyclic. The converse is not true [15,Examples 3.3].
Furthermore, given an exact subcategory in the abelian category of Γ-graded B # -modules, one can define the class of absolutely acyclic CDG-modules with respect to this exact subcategory (or the DG-category of CDG-modules whose underlying Γ-graded modules belong to this exact subcategory). For this purpose, one considers exact triples of CDG-modules whose underlying Γ-graded modules belong to the exact subcategory, takes their total CDG-modules, and uses them to generate a thick subcategory of the homotopy category of all CDG-modules whose underlying Γ-graded modules belong to the exact subcategory. When the exact subcategory is closed under infinite direct sums (resp., infinite products), the class of coacyclic (resp., contraacyclic) CDG-modules with respect to this exact subcategory is defined. Taking the quotient category, one obtains the coderived, contraderived, or absolute derived category of CDG-modules with the given restriction on the underlying Γ-graded modules.
We will be particularly interested in the coderived and absolute derived categories of CDG-modules over B whose underlying Γ-graded B # -modules are flat or have finite flat dimension (see 2.5 for the terminology). Denote the DG-categories of right CDG-modules over B with such restrictions on the underlying Γ-graded modules by mod cdg fl -B and mod cdg ffd -B, and their absolute derived categories by D abs (mod cdg fl -B) and D abs (mod cdg ffd -B). The coderived category of mod cdg fl -B, defined as explained above, is denoted by D co (mod cdg fl -B). The definition of the coderived category D co (mod cdg ffd -B) requires a little more care because the class of modules of finite flat dimension is not closed under infinite direct sums; only the classes of modules of flat dimension not exceeding a fixed number d are. Let us call a CDG-module N over B d-flat if its underlying Γ-graded B # -module M # has flat dimension not greater than d. Define an object N ∈ H 0 (mod cdg ffd -B) to be coacyclic with respect to mod cdg ffd -B if there exists an integer d 0 such that the CDG-module N is coacyclic with respect to the DG-category of d-flat CDG-modules over B. The coderived category D co (mod cdg ffd -B) is the quotient category of the homotopy category H 0 (mod cdg ffd -B) by the thick subcategory of CDG-modules coacyclic with respect to mod cdg ffd -B. Similarly, let B-mod cdg prj , B-mod cdg fpd , B-mod cdg inj , and B-mod cdg fid denote the DG-categories of left CDG-modules over B whose underlying Γ-graded B # -modules are projective, of finite projective dimension, injective, and of finite injective dimension, respectively. The notation for the homotopy categories and exotic derived categories of these DG-categories is similar to the above. The definition of the coderived category D co (B-mod cdg fpd ) and the contraderived category D ctr (B-mod cdg fid ) involves the same subtle point as discussed above. It is dealt with in the same way, i. e., the class of CDG-modules coacyclic with respect to B-mod cdg fpd or contraacyclic with respect to B-mod cdg fid is defined as the union of the classes of CDG-modules coacyclic or contraacyclic with respect to the category of modules of the projective or injective dimension bounded by a fixed integer. Proof. The first equivalence in part (b) is easy to prove. By [15, Theorem 3.5(b)], CDG-modules that are projective as Γ-graded modules are semiorthogonal to any contraacyclic CDG-modules in H 0 (B-mod cdg ). The construction of [15, proof of Theorem 3.6] shows that any object of H 0 (B-mod cdg fpd ) is a cone of a morphism from a CDG-module that is absolutely acyclic with respect to B-mod cdg fpd to an object of H 0 (B-mod cdg prj ). It follows that the functor H 0 (B-mod cdg prj ) −→ D abs (B-mod cdg fpd ) is an equivalence of triangulated categories. Moreover, any object of H 0 (B-mod cdg fpd ) that is contraacyclic with respect to B-mod cdg is absolutely acyclic with respect to B-mod cdg fpd . To prove the second equivalence in part (b), it suffices to show that any object of H 0 (B-mod cdg prj ) that is coacyclic with respect to B-mod cdg fpd is coacyclic with respect to B-mod cdg prj (as any object of the latter kind is clearly contractible). The proof of this is analogous to the proof of part (a) below. It follows that any CDG-module coacyclic with respect to B-mod cdg fpd is absolutely acyclic with respect to B-mod cdg fpd . The proof of part (c) is analogous to the proof of part (b) up to the duality.
To prove part (a), notice that the same construction from [15, proof of Theorem 3.6] allows to present any object of H 0 (mod cdg ffd -B) as a cone of a morphism from a CDG-module that is absolutely acyclic with respect to mod cdg ffd -B to an object of H 0 (mod cdg fl -B). By [15,Lemma 1.6], it remains to show that any object of H 0 (mod cdg fl -B) that is coacyclic (absolutely acyclic) with respect to mod cdg ffd -B is coacyclic (absolutely acyclic) with respect to mod cdg fl -B. We follow the idea of the proof of [14,Theorem 7.2.2]. Given an integer d 0, let us call a d-flat right CDG-module N over B d-coacyclic if it is coacyclic with respect to the exact category of d-flat CDG-modules over B. We will show that for any d-coacyclic CDG-module N there exists an (d − 1)-coacyclic CDG-module L together with a surjective closed morphism of CDG-modules L −→ N whose kernel K is also (d − 1)-coacyclic. It will follow that any (d − 1)-flat d-coacyclic CDG-module N is (d − 1)-coacyclic, since the total CDG-module of the exact triple K −→ L −→ M is (d − 1)-coacyclic, as is the cone of the morphism K −→ L. By induction we will conclude that any 0-flat d-coacyclic CDG-module is 0-coacyclic. The argument for absolutely acyclic CDG-modules will be similar.
To prove that a d-coacyclic CDG-module can be presented as a quotient of a (d−1)coacyclic CDG-module by a (d −1)-coacyclic CDG-submodule, we will first construct such a presentation for totalizations of exact triples of d-flat CDG-modules, and then check that the class of d-flat CDG-modules presentable in this form is stable under taking cones and homotopy equivalences. Proof. Choose projective objects P ′ and P ′′′ in the abelian category of CDG-modules Z 0 (mod cdg -B) (see 2.2) such that there are surjective morphisms P ′ −→ N ′ and P ′′′ −→ N ′′′ . Then there exists a surjective morphism from the exact triple CDG-modules P ′ −→ P ′′ = P ′ ⊕ P ′′′ −→ P ′′′ onto the exact triple N ′ −→ N ′′ −→ N ′′′ . Let K ′ −→ K ′′ −→ K ′′′ be the kernel of this morphism of exact triples; then the CDG-modules P (i) are 0-flat, while the CDG-modules K (i) are (d − 1)-flat. Therefore, the total CDG-module P of the exact triple P ′ −→ P ′′ −→ P ′′′ is 0-coacyclic (in fact, 0-flat and contractible), while the total CDG-module K of the exact triple Proof. Denote by L ′′′ the CDG-module L ′ ⊕ L ′′ ; then there is the embedding of a direct summand L ′ −→ L ′′′ and the surjective closed morphism of CDG-modules L ′′′ −→ N ′′ whose components are the composition L ′ −→ N ′ −→ N ′′ and the surjective morphism L ′′ −→ N ′′ . These two morphisms form a commutative square with the morphisms L ′ −→ N ′ and N ′ −→ N ′′ . The kernel K ′′′ of the morphism L ′′′ −→ N ′′ is the middle term of an exact triple of CDG-modules K ′′ −→ K ′′′ −→ L ′ . Since the CDG-modules K ′′ and L ′ are (d − 1)-coacyclic, the CDG-module K ′′′ is (d − 1)-coacyclic, too. Set L = cone(L ′ → L ′′′ ) and K = cone(K ′ → K ′′′ ).
To prove part (b), notice that the above morphisms of CDG-modules L ′ −→ L ′′′ and K ′ −→ K ′′′ are injective; denote their cokernels by L 0 and K 0 . Then the CDG-module L 0 ≃ L ′′ is (d − 1)-coacyclic. In the assumptions of part (b), the CDG-module K 0 is the kernel of the surjective morphism L 0 −→ N 0 , so it is (d − 1)flat. Hence it follows from the exact triple Lemma C. For any contractible d-flat CDG-module N there exists an exact triple K −→ P −→ N with with a 0-flat contractible CDG-module P and a (d − 1)-flat contractible CDG-module K.
Proof. It is easy to see using the explicit description of projective objects in Z 0 (mod cdg -B) given in 2.2 that any projective CDG-module is contractible. Let p : P −→ N be a surjectuve morphism onto N from a projective CDG-module P . Let t : N −→ N be a contracting homotopy for N and θ : P −→ P be a contracting homotopy for P . Then pθ − tp : P −→ N is a closed morphism of CDG-modules of degree −1. Since P is projective and p is surjective, there exists a closed morphism b : P −→ P of degree −1 such that pθ − tp = pb. Hence θ − b is another contracting homotopy for P making a commutative square with the contracting homotopy t and the morphism p. It follows that the restriction of θ − b on the kernel K of the morphism p is a contracting homotopy for the CDG-module K. Analogously, consider the functor (see (5)) and restrict it to the subcategory H 0 (mod cdg fl -B) in the first argument. This restriction factors through the Cartesian product D co (mod cdg fl -B) × D co (B-mod cdg ). Indeed, the tensor product of a CDG-module that is flat as a Γ-graded module with a coacyclic CDG-module is clearly acyclic, as is the tensor product of a CDG-module coacyclic with respect to mod cdg fl -B with any CDG-module over B. Taking into account Theorem 3.2(a), we obtain a left derived functor 3.4. Comparison of the two theories. Let C be a small k-linear DG-category. Recall (see 3.1) the notation H 0 (C-mod dg ) prj for the homotopy category of h-projective left DG-modules over C. As in 3.2, let C-mod dg prj and H 0 (C-mod dg prj ) denote the DG-category of left DG-modules over C whose underlying Γ-graded C # -modules are projective, and its homotopy category. Finally, denote by H 0 (C-mod dg prj ) prj the full triangulated subcategory in H 0 (C-mod dg prj ) formed by the h-projective left DG-modules over C whose underlying Γ-graded C # -modules are projective. The functors are equivalences of triangulated categories. Moreover, for any left DG-module L over C there exists a DG-module P ∈ H 0 (C-mod dg prj ) prj together with a quasi-isomorphism P −→ L of DG-modules over C (see [6] or [15,Section 1]).
The equivalence of categories H 0 (C-mod dg prj ) prj −→ D(C-mod dg ) factors as the following composition , where the middle arrow is also an equivalence of categories (by Theorem 3.2(b)). Besides, there is the localization functor D ctr (C-mod dg ) −→ D(C-mod dg ). This allows to construct a natural morphism (49) Ext II C (L, M) −−→ Ext C (L, M) in D(k-mod) for any objects L ∈ D co (C-mod dg fpd ) and M ∈ D ctr (C-mod dg ). Specifically, for a given DG-module L choose a DG-module F ∈ H 0 (C-mod dg prj ) and a closed morphism F −→ L with a cone coacyclic with respect to C-mod dg fpd . Next, for the DG-module F choose a DG-module P ∈ H 0 (C-mod dg prj ) prj together with a quasi-isomorphism P −→ F . Then the complex Hom C (F, M) represents the object Ext II C (L, M), the complex Hom C (P, M) represents the object Ext C (L, M), and the morphism Hom C (F, M) −→ Hom C (P, M), induced by the morphism P −→ F , represents the desired morphism (49). This morphism does not depend on the choices of the objects F and P .
To see that the comparison morphism (49) coincides with the morphism (10) constructed in 2.2, choose a projective resolution P • of the object L in the exact category Z 0 (C-mod dg ). Then both the resolution P • and its finite canonical truncation τ −d P • for d large enough are resolutions of L that can be used to compute Ext II C (L, M) by the procedure of 2.2, while the whole resolution P • can also be used to compute Ext C (L, M) by the procedure of 2.1. Set F to be the total DG-module of the finite complex of DG-modules τ −d P • and P the total DG-module of the complex of DG-modules P • , constructed by taking infinite direct sums along the diagonals.
Then the morphism of complexes Hom C (F, M) −→ Hom C (P, M) represents both the morphisms (10) and (49) in D(k-mod).
Analogously, denote by H 0 (C-mod dg inj ) inj the full triangulated subcategory in H 0 (C-mod dg inj ) formed by the h-injective left DG-modules over C whose underlying Γ-graded C # -modules are injective. Here, as above, the notation H 0 (C-mod dg ) inj for the category of h-injective DG-modules comes from 3.1, while the notation C-mod dg inj and H 0 (C-mod dg inj ) for the categories of DG-modules whose underlying Γ-graded modules are injective is similar to that in 3.2. The functors are equivalences of triangulated categories; moreover, for any left DG-module M over C there exists a DG-module J ∈ H 0 (C-mod dg inj ) inj together with a quasi-isomorphism M −→ J of CDG-modules over C.
The equivalence of categories H 0 (C-mod dg inj ) inj −→ D(C-mod dg ) factors as the following composition , where the middle arrow is also an equivalence of categories (by Theorem 3.2(c)). Besides, there is the localization functor D co (C-mod dg ) −→ D(C-mod dg ). This allows to construct a natural morphism (50) Ext II C (L, M) −−→ Ext C (L, M) in D(k-mod) for any objects L ∈ D co (C-mod dg ) and M ∈ D ctr (C-mod dg fid ). Specifically, for a given DG-module M choose a DG-module I ∈ H 0 (C-mod dg inj ) and a closed morphism M −→ I with a cone contraacyclic with respect to C-mod dg fid . Next, for the DG-module I choose a DG-module J ∈ H 0 (C-mod dg inj ) inj together with a quasi-isomorphism I −→ J. Then the complex Hom C (L, I) represents the object Ext II C (L, M), the complex Hom C (L, J) represents the object Ext C (L, M), and the morphism Hom C (L, I) −→ Hom C (L, J) represents the desired morphism (50). This comparison morphism agrees with the comparison morphism (10) from 2.2 where the former is defined.
Finally, denote by H 0 (mod dg fl -C) fl the full triangulated subcategory in H 0 (mod dg fl -C) formed by h-flat right DG-modules over C whose underlying Γ-graded C # -modules are flat. As above, H 0 (mod dg -C) fl is the homotopy category of h-flat right DG-modules over C, while mod dg fl -C and H 0 (mod dg fl -C) denote the DG-category of right DG-modules whose underlying Γ-graded C # -modules are flat, and its homotopy category.
The functors between the quotient categories of H 0 (mod dg fl -C) fl and H 0 (mod dg -C) fl by their intersections with the thick subcategory of acyclic DG-modules and the derived category D(mod dg -C) are equivalences of triangulated categories. Moreover, for any right DG-module N over C there exists a DG-module Q ∈ H 0 (mod dg fl -C) fl together with a quasi-isomorphism of DG-modules Q −→ N [15, Section 1.6].
The localization functor H 0 (mod dg fl -C) fl −→ D(mod dg -C) factors into the composition . (the middle arrow being described by Theorem 3.2(a)). There is also the localization functor D co (C-mod dg ) −→ D(C-mod dg ). This allows to construct a natural morphism in D(k-mod) for any objects N ∈ D co (mod dg ffd -C) and M ∈ D co (C-mod dg ) in the same way as above.
Specifically, for a given DG-module N choose a DG-module F ∈ H 0 (mod dg fl -C) and a closed morphism F −→ N with a cone coacyclic with respect to mod dg ffd -C. Next, for the DG-module F choose a DG-module Q ∈ H 0 (mod dg fl -C) fl together with a quasi-isomorphism Q −→ F . Then the complex F ⊗ C M represents the object Tor C,II (N, M), the complex Q ⊗ C M represents the object Tor C (N, M), and the morphism Q ⊗ C M −→ F ⊗ C M represents the desired morphism (51). This comparison morphism agrees with the morphism (9) from 2.2 where the former is defined.
Proposition. (a) The natural morphism Tor C (N, M) −→ Tor C,II (N, M) is an isomorphism whenever the Γ-graded C # -module N # has finite flat dimension and there exists a closed morphism Q −→ N into N from a DG-module Q ∈ H 0 (mod dg fl -C) fl with a cone that is coacyclic with respect to mod dg ffd -C. (b) The natural morphism Ext II C (L, M) −→ Ext C (L, M) is an isomorphism whenever the Γ-graded C # -module L # has finite projective dimension and the object L ∈ H 0 (C-mod dg fpd ) belongs to the triangulated subcategory generated by H 0 (C-mod dg prj ) prj and the subcategory of objects coacyclic with respect to C-mod dg fpd . Equivalently, the latter conclusion holds whenever the Γ-graded C # -module L # has finite projective dimension and the object L ∈ D ctr (C-mod dg ) belongs to the image of the functor H 0 (C-mod dg ) prj −→ D ctr (C-mod dg ).
(c) The natural morphism Ext II C (L, M) −→ Ext C (L, M) is also an isomorphism if the Γ-graded C # -module M # has finite injective dimension and the object M ∈ H 0 (C-mod dg fid ) belongs to the triangulated subcategory generated by H 0 (C-mod dg inj ) inj and the subcategory of objects contraacyclic with respect to C-mod dg fid . Equivalently, the latter conclusion holds whenever the Γ-graded C # -module M # has finite injective dimension and the object M ∈ D co (C-mod dg ) belongs to the image of the functor H 0 (C-mod dg ) inj −→ D co (C-mod dg ).
Notice that the equivalence of categories H 0 (C-mod dg ) prj ≃ D(C-mod dg ) identifies the functor H 0 (C-mod dg ) prj −→ D ctr (C-mod dg ) with the functor left adjoint to the localization functor D ctr (C-mod dg ) −→ D(C-mod dg ). Analogously, the equivalence of categories H 0 (C-mod dg ) inj ≃ D(C-mod dg ) identifies the functor H 0 (C-mod dg ) inj −→ D co (C-mod dg ) with the functor right adjoint to the localization functor D co (C-mod dg ) −→ D(C-mod dg ).
Before we prove the proposition, let us introduce some more notation. The triangulated category of (C)DG-modules coacyclic (resp., contraacyclic) with respect to a given DG-category of (C)DG-modules D will be denoted by Ac co (D) (resp., Ac ctr (D)). So Ac co (D) and Ac ctr (D) are triangulated subcategories of H 0 (D). Similarly, Ac abs (D) denotes the triangulated subcategory of absolutely acyclic (C)DG-modules. Finally, given a DG-category C, we denote by Ac(C-mod dg ) and Ac(mod dg -C) the full subcategories of acyclic DG-modules in the homotopy categories H 0 (C-mod dg ) and H 0 (mod dg -C).
Proof. Part (a) follows immediately from the above construction of the morphism (51). To prove the first assertion of part (b), notice that any morphism from an object of H 0 (C-mod dg prj ) prj to an object of Ac co (C-mod dg fpd ) vanishes in H 0 (C-mod dg ). In fact, any morphism from an object of H 0 (C-mod dg prj ) to an object of Ac co (C-mod dg fpd ) vanishes, and any morphism from an object of H 0 (C-mod dg ) prj to an object of Ac(C-mod dg ) vanishes in the homotopy category. By the standard properties of semiorthogonal decompositions (see, e. g., [15,Lemma 1.3]), it follows that any object L in the triangulated subcategory generated by H 0 (C-mod dg prj ) prj and Ac co (C-mod dg fpd ) in H 0 (C-mod dg fpd ) admits a closed morphism P −→ L from an object P ∈ H 0 (C-mod dg prj ) prj with a cone in Ac co (C-mod dg fpd ). To prove the equivalence of the two conditions in part (b), notice that, by the same semiorthogonality lemma, a DG-module L ∈ C-mod dg fpd belongs to the triangulated subcategory generated by H 0 (C-mod dg prj ) prj and Ac co (C-mod dg fpd ) in H 0 (C-mod dg fpd ) if and only if, as an object of D co (C-mod dg fpd ), it belongs to the image of H 0 (C-mod dg prj ) prj in D co (C-mod dg fpd ). Then use the concluding remarks in 3.2. Part (c) is similar to part (b) up to the duality.
In particular, if the left homological dimension of the Γ-graded category C # is finite (see 2.5 for the terminology), then the classes of coacyclic, contraacyclic, and absolutely acyclic left DG-modules over C coincide [15,  Analogously, if the weak homological dimension of the Γ-graded category C # is finite and the category H 0 (mod dg fl -C) coincides with its full subcategory H 0 (mod dg fl -C) fl , then the natural morphism Tor C (N, M) −→ Tor C,II (N, M) is an isomorphism for any DG-modules N and M over C. This follows from part (a) of Proposition.

3.5.
Comparison for DG-category of CDG-modules. Let B be a small k-linear CDG-category and C = mod cdg fgp -B the DG-category of right CDG-modules over B, projective and finitely generated as Γ-graded B # -modules. The results below also apply, mutatis mutandis, to finitely generated free modules in place of finitely generated projective ones.
The DG-categories of (left or right) CDG-modules over B and DG-modules over C are naturally equivalent; let M C denote the DG-module over C corresponding to a CDG-module M over B (see 1.5 and 2.6). Denote by B-mod cdg fgp the DG-category of left CDG-modules over B, finitely generated and projective as Γ-graded B # -modules. Let k ∨ be an injective cogenerator of the abelian category of k-modules; for example, one can take k ∨ = k when k is a field, or k ∨ = Hom Z (k, Q/Z) for any ring k.
Recall that an object X of a triangulated category T with infinite direct sums is called compact if the functor Hom T (X, −) preserves infinite direct sums. A set of compact objects S ⊂ T generates T as a triangulated category with infinite direct sums if and only if the vanishing of all morphisms X −→ Y [ * ] in T for all X ∈ S implies vanishing of an object Y ∈ T [9, Theorem 2.1(2)].
Theorem A. (a) If the Γ-graded category B # has finite weak homological dimension and the image of the functor H 0 (mod cdg fgp -B) −→ D co (mod cdg -B) generates D co (mod cdg -B) as a triangulated category with infinite direct sums, then for any right DG-module N C and left DG-module M C over C the natural morphism The same conclusion holds if the Γ-graded category B # has finite weak homological dimension and all objects of H 0 (mod cdg fl -B) can be obtained from objects of H 0 (mod cdg fgp -B) using the operations of shift, cone, filtered inductive limit, and passage to a homotopy equivalent CDG-module over B.
(b) If the Γ-graded category B # has finite left homological dimension and the image of the functor H 0 (B-mod cdg fgp ) −→ D co (B-mod cdg ) generates D co (B-mod cdg ) as a triangulated category with infinite direct sums, then for any left DG-modules L C and M C over C the natural morphism Ext II C (L C , M C ) −→ Ext C (L C , M C ) is an isomorphism. Proof. The proof is based on the results of 3.4. The DG-category B-mod cdg fgp is equivalent to the DG-category C op ; the equivalence assigns to a right CDG-module F the left CDG-module G = Hom B op (F, B) and to a left CDG-module B the right CDG- module F = Hom B (G, B) over B. Given a left CDG-module M over B, the corresponding left DG-module M C over C assigns to a CDG-module F ∈ mod cdg fgp -B the complex of k-modules F ⊗ B M ≃ Hom B (G, M). Given a right CDG-module N over B, the corresponding right DG-module N C over C assigns to a CDG-module F ∈ mod cdg fgp -B the complex of k-modules Hom B op (F, N) ≃ N ⊗ B G. The categories of (left or right) Γ-graded modules over the Γ-graded categories B # and C # are also equivalent. Γ-graded modules corresponding to each other under these equivalences have equal flat, projective, and injective dimensions. So the (weak, left, or right) homological dimensions of the Γ-graded categories B # and C # are equal. The equivalence between the DG-categories of (left or right) CDG-modules over B and DG-modules over C preserves the classes of coacyclic, acyclic, and absolutely acyclic (C)DG-modules. Given a left CDG-module M over B, the DG-module M C is acyclic if and only if the complex F ⊗ B M ≃ Hom B (G, M) is acyclic for any CDG-modules F ∈ mod cdg fgp -B and G ∈ B-mod cdg fgp (related to each other as above); similarly for a right CDG-module N over B.
For any small CDG-category B the functor H 0 (B-mod cdg fgp ) −→ D co (B-mod cdg ) is fully faithful, and the objects in its image are compact in the coderived category [7]. Thus, the classes of acyclic and coacyclic left DG-modules over C coincide if and only if D co (B-mod cdg ) is generated by H 0 (B-mod cdg fgp ) as a triangulated category with infinite direct sums. Now parts (a) and (b) follow from Proposition 3.4(a-b); see also the concluding remarks in 3.4. For details related to the proof of the second assertion of part (a), see the last paragraph of the proof of Theorem B below.
The next, more technical result is a generalization of Theorem A to the case of Γ-graded categories B # of infinite homological dimension.
Let us denote by T i ⊕ ⊂ T (resp., T i ⊓ ⊂ T ) the minimal triangulated subcategory of a triangulated category T containing subcategories T i and closed under infinite direct sums (resp., infinite products). Given a class of CDG-modules E ⊂ Z 0 (mod cdg -B), we denote by E ∪ ⊂ Z 0 (mod cdg -B) the full subcategory of all CDG-modules that can be obtained from the objects of E using the operations of shift, cone, filtered inductive limit, and passage to a homotopy equivalent CDG-module.  Indeed, a DG-module over any small DG-category C is h-projective if and only if it belongs to the minimal triangulated subcategory of H 0 (C-mod dg ) containing the representable DG-modules and closed under infinite direct sums [6,15]. Representable left DG-modules over C correspond to the objects of B-mod cdg fgp under the equivalence between the DG-categories C-mod dg and B-mod cdg (see the proof of Theorem A). It follows that the DG-module L C ∈ H 0 (C-mod dg fpd ) belongs to the triangulated subcategory generated by H 0 (C-mod dg fpd ) fpd and the objects coacyclic with respect to C-mod dg fpd if and only if a CDG-module L over B belongs to the minimal triangulated subcategory of H 0 (B-mod cdg fpd ) containing H 0 (B-mod cdg fgp ) and all objects coacyclic with respect to B-mod cdg fpd and closed under infinite direct sums. Similarly, a left DG-module over a k-linear DG-category C is h-injective if and only if it belongs to the minimal triangulated subcategory of H 0 (C-mod dg ) containing the DG-modules Hom k (R X , k ∨ ), where R X are the representable right DG-modules over C. Representable right DG-modules over C correspond to the objects of mod cdg fgp -B under the equivalence between the DG-categories mod dg -C nad mod cdg -B. So the DG-module M C ∈ H 0 (C-mod dg fid ) belongs to the subcategory generated by H 0 (C-mod dg fid ) fid and the objects contraacyclic with respect to C-mod dg fid if and only if a CDG-module M over B belongs to the minimal triangulated subcategory of H 0 (B-mod cdg fid ) containing Ac ctr (B-mod cdg fid ) and all CDG-modules Hom k (F, k ∨ ) for F ∈ H 0 (mod cdg fgp -B), and closed inder infinite products. Finally, a right DG-module over a DG-category C is h-flat whenever it can be obtained from the representable right DG-modules using the operations of shift, cone, filtered inductive limit, and passage to a homotopy equivalent DG-module (we do not know whether the converse is true). Indeed, the class of h-flat DG-modules is closed under shifts, cones, filtered inductive limits, and homotopy equivalences, since these operations commute with the tensor product of DG-modules over C and preserve acyclicity of complexes of k-modules. Thus, if a right CDG-module Q over B can be obtained from objects of mod cdg fgp -B using the operations of shift, cone, filtered inductive limit, and passage to a homotopy equivalent CDG-module, then the corresponding DG-module Q C over C is h-flat.
The equivalence of the two conditions both in (b) and in (c) follows from the same semiorthogonality arguments as in the proof of Proposition 3.4. Now assume that the commutative ring k has finite weak homological dimension and the Γ-graded k-modules B # (X, Y ) are flat for all objects X, Y ∈ B. Recall that the DG-categories of left and right CDG-modules over B ⊗ k B op are naturally isomorphic. To any left CDG-module L and right CDG-module N over B one can assign the (left) CDG-module L ⊗ k N over the CDG-category B ⊗ k B op . The same conclusion holds if the Γ-graded category B # ⊗ k B #op has finite weak homological dimension and all objects of H 0 (B ⊗ k B op -mod cdg fl ) can be obtained from objects in the image of (52) using the operations of shift, cone, filtered inductive limit, and passage to a homotopy equivalent CDG-module over B ⊗ k B op .
(b) If the Γ-graded category B # ⊗ k B #op has finite left homological dimension and the image of the functor (52) generates D co (B ⊗ k B op -mod cdg ) as a triangulated category with infinite direct sums, then the natural map HH II, * (C, M C ) −→ HH * (C, M C ) is an isomorphism for any DG-module M C over the DG-category C ⊗ k C op .
Proof. This is a particular case of the next Theorem D.
Theorem D. (a) Suppose that the Γ-graded B # ⊗ k B #op -module B # has finite flat dimension and there exists a CDG-module and a closed morphism Q −→ B of CDG-modules over B ⊗ k B op with a cone in Ac co (mod cdg ffd -B ⊗ k B op ). Then the natural map HH * (C, M C ) −→ HH II * (C, M C ) is an isomorphism for any DG-module M C over C ⊗ k C op .
(b) Suppose that the Γ-graded B # ⊗ k B #op -module B # has finite projective dimension and the CDG-module B over B ⊗ k B op belongs to Equivalently, the same conclusion holds if the the Γ-graded B # ⊗ k B #op -module B # has finite projective dimension and the object B ∈ D co (B ⊗ k B op -mod cdg fpd ) belongs to the minimal triangulated subcategory of D co (B ⊗ k B op -mod cdg fpd ), containing the CDG-modules G ⊗ k F , where F ∈ H 0 (mod cdg fgp -B) and G ∈ H 0 (B-mod cdg fgp ), and closed under infinite direct sums.
Proof. It suffices to notice that CDG-modules G ⊗ F over B ⊗ k B op correspond precisely to representable DG-modules over C ⊗ k C op under the equivalence of DG-categories B ⊗ k B op -mod cdg ≃ C ⊗ k C op -mod dg . The rest of the argument is similar to the proof of Theorem B and based on Proposition 3.4(a-b).
3.6. Derived tensor product functor. The following discussion is relevant in connection with the role that the external tensor products of CDG-modules play in the above Theorems 3.5.C-D.
Let k be a commutative ring of finite weak homological dimension, and let B ′ and B ′′ be k-linear CDG-categories such that the Γ-graded k-modules of morphisms in the categories B ′# and B ′′# are flat. Consider the functor of tensor product We would like to construct its left derived functor . Denote by B ′ -mod cdg k-fl the DG-category of left CDG-modules M ′ over B ′ for which all the Γ-graded k-modules M ′# (X) are flat, and similarly for CDG-modules over B ′′ . Notice that the natural functor from the quotient category of H 0 (B ′ -mod cdg k-fl ) by its intersection with Ac co (B ′ -mod cdg ) to the coderived category D co (B ′ -mod cdg ) is an equivalence of triangulated categories. Indeed, the construction of [15,proof of Theorem 3.6] shows that for any left CDG-module M ′ over B ′ there exists a closed morphism F ′ −→ M ′ , where F ′ ∈ H 0 (B ′ -mod cdg k-fl ), with a coacyclic cone. So it remains to use [15,Lemma 3.6].
Restrict the above functor ⊗ k to the subcategory H 0 (B ′ -mod cdg k-fl ) in the first argument. Clearly, this restriction factors through the coderived category D co (B ′′ -mod cdg ) in the second argument. Let us show that it also factors through the coderived category D co (B ′ -mod cdg ) in the first argument (cf. [14,Lemma 2.7]). Indeed, let M ′ be an object of H 0 (B ′ -mod cdg k-fl ) ∩ Ac co (B ′ -mod cdg ) and M ′′ be a left CDG-module over B ′′ . Choose a CDG-module F ′′ ∈ H 0 (B ′′ -mod cdg k-fl ) such that there is a closed morphism F ′′ −→ M ′′ with a coacyclic cone. Then the CDG-module M ′ ⊗ k F ′′ is coacyclic, since M ′ is coacyclic and F ′′ is k-flat; at the same time, the cone of the morphism M ′ ⊗ k F ′′ −→ M ′ ⊗ k M ′′ is coacyclic, the cone of the morphism F ′′ −→ M ′′ is coacyclic and M ′ is k-flat. Thus, the CDG-module M ′ ⊗ k M ′′ is also coacyclic.
We have constructed the desired derived functor ⊗ L k . Clearly, the same derived functor can be obtained by restricting the functor ⊗ k to the subcategory H 0 (B ′′ -mod cdg k-fl ) in the second argument. Analogously, one can construct a derived functor , or the similar functor with modules of finite projective dimension replaced by those of finite flat dimension. All one has to do is to restrict the functor ⊗ k to the homotopy category of CDG-modules whose underlying Γ-graded modules satisfy both conditions of k-flatness and finiteness of the projective dimension over B ′ or B ′′ .
In these situations one does not even need the condition that the weak homological dimension of k is finite. However, one has to use the fact that the tensor product over k preserves finitness of projective/flat dimensions, provided that at least one of the Γ-graded modules being multiplied is k-flat.

Examples
The purpose of this section is mainly to illustrate the results of Section 3. Examples of DG-categories C for which the two kinds of Hochschild (co)homology are known to coincide are exhibited in 4.1-4.5. Examples of CDG-algebras B such that the two kinds of Hochschild (co)homology can be shown to coincide for the DG-category of CDG-modules C = mod cdg fgp -B are considered in 4.6-4.8. Counterexamples are discussed in 4.9 and 4.10. Hochschild (co)homology of matrix factorizations are considered in 4.8-4.10.
4.1. DG-category with zero differentials. Let C be a small k-linear DG-category such that the differentials in the complexes C(X, Y ) vanish for all objects X, Y ∈ C.
Proposition. (a) If N is a right DG-module over C such that the differentials in the complexes N(X) vanish for all objects X ∈ C and the Γ-graded C # -module N # has finite flat dimension, then the natural morphism Tor C (N, M) −→ Tor C,II (N, M) is an isomorphism for any left DG-module M over C.
(b) If L be a left DG-module over C such that the differentials in the complexes L(X) vanish for all objects X ∈ C and the Γ-graded C # -module L # has finite projective dimension, then the natural morphism Ext II C (L, M) −→ Ext C (L, M) is an isomorphism for any left DG-module M over C.
(c) If M is a left DG-module over C such that the differentials in the complexes M(X) vanish for all objects X ∈ C and the Γ-graded C # -module M # has finite injective dimension, then the natural morphism Ext II C (L, M) −→ Ext C (L, M) is an isomorphism for any left DG-module L over C.
Proof. To prove part (a), notice that a finite flat left resolution P • of the Γ-graded C # -module N # , with every term of it endowed with a zero differential, can be used to compute both kinds of derived functor Tor that we are interested in. The proofs of parts (b) and (c) are similar. Proof. Any DG-module over a DG-category with vanishing differentials is an extension of two DG-modules with vanishing differentials. Indeed, the kernel and image of the differential d on such a DG-module is a DG-submodule. So it remains to use the fact that both kinds of functors Ext and Tor assign distinguished triangles to short exact sequences of DG-modules in any argument, together with the preceding proposition. Part (b) also follows from the fact that the classes of acyclic and absolutely acyclic left DG-modules over C coincide in its assumptions; see [7].
Corollary B. Let C be a DG-category such that the complexes C(X, Y ) are complexes of flat k-modules with zero differentials for all objects X, Y ∈ C.
(a) If the Γ-graded C # ⊗ k C #op -module C # has finite flat dimension, then the natural morphism of Hochschild homology HH * (C, M) −→ HH II * (C, M) is an isomorphism for any DG-module M over C ⊗ k C op .
(b) If the Γ-graded C # ⊗ k C #op -module C # has finite projective dimension, then the natural morphism of Hochschild cohomology HH II, * (C, M) −→ HH * (C, M) is an isomorphism for any DG-module M over C ⊗ k C op .
Proof. This follows directly from Proposition.

4.2.
Nonpositive DG-category. Assume that our grading group Γ is isomorphic to Z and the isomorphism identifies 1 with 1 (see 1.1).
Let C be a small k-linear DG-category. Assume that the complexes of k-modules C(X, Y ) are concentrated in nonpositive degrees for all objects X, Y ∈ C. Let us call a (left or right) DG-module M over C bounded above if all the complexes of k-modules M(X) are bounded above uniformly, i. e., there exists an integer n such that the complexes M(X) are concentrated in the degree n for all X. DG-modules bounded below are defined in the similar way. Proof. Parts (a-c) follow from the corresponding parts of Proposition 3.4.
To prove part (b), let us choose a finite left resolution P • of the DG-module L in the abelian category Z 0 (C-mod dg ) such that the DG-modules P i are bounded above and their underlying graded C # -modules are projective. Then the total DG-module P of P • maps into L with a cone absolutely acyclic with respect to C-mod dg fpd , so it suffices to show that P is h-projective. Indeed, any left DG-module P over C that is bounded above and projective as a graded C # -module is h-projective. To prove the latter assertion, one can construct by induction in n an increasing filtration of P by DG-submodules such that the associated quotient DG-modules are direct summands of direct sums of representable DG-modules shifted by the degree determined by the number of the filtration component.
The proof of part (c) is similar up to duality, and to prove part (a) one has to show that a right DG-module Q over C that is bounded above and flat as a graded C # -module is h-flat. This can be done, e. g., by using (the graded version of) the Govorov-Lazard flat module theorem to construct a filtration similar to the one in the projective case, except that the associated quotient DG-modules are filtered inductive limits of direct sums of (appropriately shifted) representable DG-modules. Now assume that the complexes C(X, Y ) are complexes of flat k-modules concentrated in nonpositive cohomological degrees.
Corollary. (a) For any DG-module M over C ⊗ k C op bounded above, the natural morphism HH * (C, M) −→ HH II * (C, M) is an isomorphism. If the graded C # ⊗ k C #op -module C # has finite flat dimension, then the latter morphism is an isomorphism for any DG-module M.
(b) For any DG-module M over C ⊗ k C op bounded below, the natural morphism HH II, * (C, M) −→ HH * (C, M) is an isomorphism. If the graded C # ⊗ k C #op -module C # has finite projective dimension, then the latter morphism is an isomorphism for any DG-module M.

Proof. Apply Propositions A and B(a-b) to the DG-category
So the map HH * (C) −→ HH II * (C) is an isomorphism under our assumptions on C. The map HH II, * (C) −→ HH * (C) is an isomorphism provided that either the DG-module C over C ⊗ k C op is bounded below [1,Proposition 3.15], or the graded C # ⊗ k C #op -module C # has finite projective dimension.

4.3.
Strongly positive DG-category. As in 4.2, we assume that the grading group Γ is isomorphic to Z and the isomorphism identifies 1 with 1.
Let k be a field and C be a k-linear DG-category such that the complexes of k-vector spaces C(X, Y ) are concentrated in nonnegative degrees for all objects X, Y ∈ C, the component C 1 (X, Y ) vanishes for all X and Y , the component C 0 (X, Y ) vanishes for all nonisomorphic X and Y , and the k-algebra C 0 (X, X) is semisimple for all X. Here a noncommutative ring is called (classically) semisimple if the abelian category of (left or right) modules over it is semisimple. We keep the terminology from 4.2 related to bounded DG-modules. Proof. The proof uses the construction of the natural morphisms (9-10) from 2.2. To prove part (a), one can compute both kinds of Tor in question using the reduced bar-resolution of the DG-module N over C relative to C 0 , i. e., Here C 0 is considered as a DG-category with complexes of morphisms concentrated in degree 0 and endowed with zero differentials, C/C 0 is a DG-module over C 0 ⊗ k C 0op , and C is a DG-module over C 0 ⊗ k C op . The semisimplicity condition on C 0 guarantees projectivity of right DG-modules of the form R⊗ C 0 C as objects of the exact category Z 0 (mod dg -C) for all right DG-modules R over C 0 . Due to the positivity/boundedness conditions on C, N, and M, there is no difference between the two kinds of totalizations of the resulting bar-bicomplex. The proof of part (b) is similar. Proof. The proof is similar to that of Proposition 4.2.B. E. g., in part (b) the key is to show that any DG-module over C that is bounded below and projective as a graded C # -module is h-projective. One constructs an increasing filtration similar to that in 4.2 with the only difference that the associated quotient DG-modules are projective objects of the exact category Z 0 (C-mod dg ).
Corollary. (a) For any DG-module M over C ⊗ k C op bounded below the natural morphism HH * (C, M) −→ HH II * (C, M) is an isomorphism. If the graded C # ⊗ k C #op -module C # has finite flat dimension, then the latter morphism is an isomorphism for and DG-module M.
(b) For any DG-module M over C ⊗ k C op bounded above, the natural morphism HH II, * (C, M) −→ HH * (C, M) is an isomorphism. If the graded C # ⊗ k C #op -module C # has finite projective dimension, then the latter morphism is an isomorphism for any DG-module M.
So the map HH * (C) −→ HH II * (C) is an isomorphism under our assumptions on C. The map HH II, * (C) −→ HH * (C) is an isomorphism provided that either the DG-module C over C ⊗ k C op is bounded above, or the graded C # ⊗ k C #op -module C # has finite projective dimension.

4.4.
Cofibrant DG-category. A small k-linear DG-category is called cofibrant if it is a retract (in the category of DG-categories and functors between them) of a DG-category k x n,α of the following form. As a Γ-graded category, k x n,α is freely generated by a set of homogeneous morphisms x n,α , where n runs over nonnegative integers and α belongs to some set of indices. This means that the morphisms in k x n,α are the formal k-linear combinations of formal compositions of the morphisms x n,α . It is additionally required that the element dx n,α belongs to the class of morphisms multiplicatively and additively generated by the morphisms x m,β with m < n. The cofibrant DG-categories are exactly (up to the zero object issue) the cofibrant objects in the model category structure constructed in [20] (see also [21]).
The following lemmas will be used in conjunction with the results of 3.4 in order to prove comparison results for the two kinds of Ext, Tor, and Hochschild (co)homology for cofibrant DG-categories.
Lemma A. Let D be a DG-category of the form k x n,α as above.
(a) If a right DG-module N over D is such that all the complexes of k-modules N(X) are h-flat complexes of flat k-modules, then there exists a closed morphism Q −→ N, where Q ∈ H 0 (mod dg fl -D) fl , with a cone absolutely acyclic with respect to mod dg ffd -D. (b) If a left DG-module L over D is such that all the complexes of k-modules L(X) are h-projective complexes of projective k-modules, then there exists a closed morphism P −→ L, where L ∈ H 0 (D-mod dg prj ) prj , with a cone absolutely acyclic with respect to D-mod dg fpd . (c) If a left DG-module M over D is such that all the complexes of k-modules L(X) are h-injective complexes of injective k-modules, then there is a closed morphism M −→ J, where J ∈ H 0 (D-mod dg inj ) inj , with a cone absolutely acyclic with respect to D-mod dg fid . Lemma B. (a) If C is a cofibrant k-linear DG-category and the ring k has a finite weak homological dimension, then the weak homological dimension of the Γ-graded category C # is also finite. Moreover, the categories H 0 (mod dg fl -C) and H 0 (mod dg fl -C) fl coincide in this case.
(b) If C is a cofibrant k-linear DG-category and the ring k has finite homological dimension, then the left homological dimension of the Γ-graded category C # is finite. Moreover, the classes of acyclic and absolutely acyclic left DG-modules over C coincide in this case.
Proof of Lemmas A and B. Let us first prove parts (b) of both lemmas. The following arguments generalize the proof of [15,Theorem 9.4] to the DG-category case. For any objects X, Y ∈ D denote by V (X, Y ) the free Γ-graded k-module spanned by those elements x n,α that belong to D(X, Y ). Consider the short exact sequence of Γ-graded D # -modules The middle and right term are endowed with DG-module structures, so the left term also acquires such a structure. There is a natural increasing filtration on the left term induced by the filtration of V related to the indexes n of the generators x n,α . It is a filtration by DG-submodules and the differentials on the associated quotient modules are the differentials on the tensor product induced by the differentials on the factors D and L (as is the differential on the middle term).
It follows that whenever all the complexes of k-modules L(X) are coacyclic (absolutely acyclic), both the middle and the left terms of the exact sequence are coacyclic (absolutely acyclic) DG-modules, so L(X) is also a coacyclic (absolutely acyclic) DG-module. In particular, if the homological dimension of k is finite and L is acyclic, then it is absolutely acyclic. Furthermore, when all the complexes L(X) are h-projective complexes of projective k-modules, both the middle and the left terms belong to H 0 (D-mod dg prj ) prj . So it suffices to take the cone of the left arrow as the DG-module P .
It also follows from the same exact sequence considered as an exact sequence of Γ-graded D # -modules that the Γ-graded D # -module L # has the projective dimension at most 1 whenever all L # (X) are projective Γ-graded k-modules. Since for any projective Γ-graded D # -module F # the Γ-graded k-modules F # (X) are projective, the left homological dimension of D # can exceed the homological dimension of k by at most 1.
Since Ext and Tor over Γ-graded categories are functorial with respect to Γ-graded functors, the (weak, left, or right) homological dimension of a retract C # of a Γ-graded category D # does not exceed that of D # . To prove the second assertion of Lemma B(b) for a retract C of a DG-category D as above, consider DG-functors I : C −→ D and Π : D −→ C such that ΠI = Id C . Let M be an acyclic DG-module over C; then the DG-module Π * M over D is acyclic, hence absolutely acyclic, and it follows that M = I * Π * M is also absolutely acyclic.
It remains to prove the second assertion of Lemma B(a). If the underlying Γ-graded D # -module of a right DG-module N over D is flat, then the above exact sequence remains exact after taking the tensor product with N over D. Besides, the Γ-graded k-modules N # (X) are flat, since the Γ-graded k-modules D # (X, Y ) are. If the weak homological dimension of k is finite, it follows that the complexes of k-modules N(X) are h-flat. Now if the complexes of k-modules L(X) are acyclic, then the tensor products of the left and the middle terms with N over D are acyclic, hence the complex N ⊗ D L is also acyclic.
Finally, let us deduce the same assertion for a retract C of the DG-category D. For this purpose, notice that for any DG-functor F : C −→ D the functor F * : H 0 (mod dg -D) −→ H 0 (mod dg -C) has a left adjoint functor F ! given by the rule F ! (N) = N ⊗ C D. In other words, the DG-module F ! (N) assigns the complex of k-modules N ⊗ C F * S X to an object X ∈ D, where S X is the left (covariant) representable DG-module over D corresponding to X. The functor F ! transforms objects of H 0 (mod dg fl -C) to objects of H 0 (mod dg fl -D) and h-flat DG-modules to h-flat DG-modules, since for any right DG-module N over C and left DG-module M over D one has F ! N ⊗ D M ≃ N ⊗ C F * M. Now if (I, Π) is our retraction and N ∈ H 0 (mod dg fl -C), then I ! N ∈ H 0 (mod dg fl -D), hence I ! N is h-flat, and it follows that N = Π ! I ! N is also h-flat. Proof. Since the morphisms (9-10) are functorial with respect to DG-functors F : C −→ D, i. e., make commutative squares with the morphisms (3-4) and (7)(8), it suffices to prove the statements of Corollary for a DG-category D = k x n,α . Now the first assertions in both (a) and (b) follow from Lemma A and Proposition 3.4, while the second ones follow from Lemma B and the concluding remarks in 3.4.
Lemma D. Let D be a DG-category of the form k x n,α . Then the Γ-graded D # ⊗ k D #op -module D # has projective dimension at most 1. There exists an h-projective DG-module P over D ⊗ k D op and a closed morphism of DG-modules P −→ D with a cone absolutely acyclic with respect to D ⊗ k D op -mod dg fpd . Proof. It suffices to consider the short exact sequence and argue as above.
Corollary E. Let C be a cofibrant k-linear DG-category. Then for any DG-module M over C ⊗ k C op , the natural morphisms of Hochschild (co)homology HH * (C, M) −→ HH II * (C, M) and HH II, * (C, M) −→ HH * (C, M) are isomorphisms. Proof. The assertions for a DG-category D = k x n,α follow from Lemma D and Proposition 3.4(a-b). To deduce the same results for a retract C of a DG-category D, use the fact that the comparison morphisms (23) make commutative squares with the morphisms (16), (19) and (20), (22). 4.5. DG-algebra with Koszul filtration. Let A be a DG-algebra over a field k endowed with an increasing filtration F i A, i 0, such that Assume that the associated graded algebra gr F A is Koszul (in the grading i induced by the filtration F ) and has finite homological dimension (here we use the Koszulity condition without the assumption of finite-dimensionality of the components of gr F A, see e. g. [17]). Then one can assign to A a coaugmented CDG-coalgebra C endowed with a finite decreasing filtration G [15, Section 6.8] (cf. 4.6 below).
Corollary. Assume that the coaugmented coalgebra C is conilpotent (see [17] or Proof. The (left or right) homological dimension of the graded algebra A # is finite, since one can compute the spaces Ext over it using the nonhomogeneous Koszul resolution. By [15,Corollary 6.8.2], the classes of acyclic and absolutely acyclic DG-modules over A coincide. Hence the first two assertions follow from the concluding remarks in 3.4. To prove the last assertion, notice that the DG-algebra A ⊗ k A op is endowed with the induced filtration having the same properties as required above of the filtration on A; the corresponding CDG-coalgebra is naturally identified with C ⊗ k C op . Since C is conilpotent, so is C ⊗ k C op . Thus, the classes of acyclic and absolutely acyclic DG-modules over A ⊗ k A op coincide, too. 4.6. CDG-algebra with Koszul filtration. Let B = (B, d, h) be a CDG-algebra over a field k endowed with an increasing filtration F i B, i 0, such that F 0 B = k, Assume that the associated graded algebra gr F B is Koszul and has finite homological dimension. Then one can assign to the filtered CDG-algebra (B, F ) a CDG-coalgebra C endowed with a finite decreasing filtration G [15, Section 6.8].
Let C = mod cdg fgp -B be the DG-category of right CDG-modules over B, projective and finitely generated as Γ-graded B # -modules. All the results below will also hold for finitely generated free modules in place of finitely generated projective ones. Proof. The homological dimension of the graded algebra B # is finite (see 4.5). By [15, Corollary 6.8.1], the coderived category D co (B-mod cdg ) is generated by H 0 (B-mod cdg fgp ) as a triangulated category with infinite direct sums. Thus, the assertions of the corollary follow from Theorem 3.5.A.
Let C ss denote the maximal cosemisimple Γ-graded subcoalgebra of the Γ-graded coalgebra C [15,Section 5.5]. Assume that the differential d and the curvature linear function h on C annihilate C ss , and the tensor product coalgebra C ss ⊗ k C ss op is cosemisimple. The latter condition always holds when the field k is perfect and the grading group Γ contains no torsion of the order equal to the characteristic of k. Proof. The CDG-algebra B ⊗ k B op is endowed with the induced filtration having the same properties; the corresponding CDG-coalgebra is naturally identified with C ⊗ k C op . The coderived category of CDG-modules D co (B-mod cdg ) is equivalent to the coderived category of CDG-comodules D co (C-comod cdg ) [15,Theorem 6.8]. This equivalence transforms the functor of tensor product into the similar functor of tensor product ⊗ k : D co (C-comod cdg ) × D co (C op -comod cdg ) −−→ D co (C ⊗ k C op -comod cdg ).
When the coalgebra C ss ⊗ k C ss op is cosemisimple, any DG-comodule over it (considered as a DG-coalgebra with zero differential) can be obtained from tensor products of DG-comodules over C ss and C ssop using the operations of cone and passage to a direct summand. The coderived category D co (C ⊗ k C op -comod cdg ) of CDG-comodules over C ⊗ k C op is generated by DG-comodules over C ss ⊗ k C ss op as a triangulated category with infinite direct sums, since the coalgebra without counit (C⊗ k C op )/(C ss ⊗ k C ss op ) is conilpotent [15,Section 5.5]. Therefore, the conditions of Theorem 3.5.C are satisfied for the CDG-algebra B.

4.7.
Noetherian CDG-ring. Let B be a CDG-algebra over a commutative ring k and C = mod cdg fgp -B the DG-category of right CDG-modules over B, projective and finitely generated as Γ-graded B # -modules.
Corollary A. Assume that the Γ-graded ring B # is graded left Noetherian and has finite left homological dimension. Then Proof. Notice that for a left Noetherian (graded) ring the weak and left homological dimensions coincide. Whenever the graded ring B # is left Noetherian, the coderived category D co (B-mod cdg ) is compactly generated by CDG-modules whose underlying Γ-graded modules are finitely generated (a result of D. Arinkin, [15,Theorem 3.11.2]). Assuming additionally that the left homological dimension of B # is finite, it follows easily that D co (B-mod cdg ) is compactly generated by H 0 (B-mod cdg fgp ). (See the beginning of 3.5 for a brief discussion of compact generation.) It remains to apply Theorem 3.5.A(a-b) to deduce the assertions of the corollary.
Before formulating our next result, let us define yet another exotic derived category of CDG-modules. Given a small CDG-category D, the complete derived category D cmp (D-mod cdg ) of left CDG-modules over D is the quotient category of the homotopy category H 0 (D-mod cdg ) by its minimal triangulated subcategory, containing Ac abs (D-mod cdg ) and closed under both infinite direct sums and products. CDG-modules belonging to the latter subcategory are called completely acyclic. Now assume that the ring k has finite weak homological dimension and the Γ-graded k-module B # is flat. Assume further that the Γ-graded ring B # is both left and right Noetherian of finite homological dimension, the Γ-graded ring B # ⊗ k B #op is graded Noetherian and the Γ-graded module B # over B # ⊗ k B #op has finite projective dimension.  [15,Theorem 3.11.2] and the discussion in 3.6, the triangulated subcategory with infinite direct sums in D cmp (B ⊗ k B op -mod cdg ) generated by the CDG-modules L ⊗ k N with L and N as above coincides with the triangulated subcategory with infinite direct sums generated by the CDG-modules G ⊗ k F with G ∈ H 0 (B-mod cdg fgp ) and F ∈ H 0 (mod cdg fgp -B). The construction from [15,proof of Theorem 3.6] shows that there exists a closed morphism from a CDG-module P ∈ H 0 (B ⊗ k B op -mod cdg fgp ) into the CDG-module B with the cone absolutely acyclic with respect to B ⊗ k B op -mod cdg fpd . The triangulated subcategory with infinite direct sums generated by H 0 (B ⊗ k B op -mod cdg fgp ) in H 0 (B ⊗ k B op ) is semiorthogonal to all completely acyclic CDG-modules, and maps fully faithfully to D co (B ⊗ k B op -mod cdg fpd ) and to D cmp (B ⊗ k B op -mod cdg ) [7]. So the condition that the object P is generated by the objects G ⊗ k F can be equivalently checked in any of these triangulated categories. Notice that since the objects of H 0 (B ⊗ k B op -mod cdg fgp ) are compact in these triangulated categories, it does not matter whether to generate P from G ⊗ K F using shift, cones, and infinite direct sums, or shift, cones, and passages to direct summands only.
One can drop the assumption that the Γ-graded ring B # ⊗ k B #op is graded Noetherian by replacing the complete derived category D cmp (B ⊗ k B op -mod cdg ) with the coderived category D co (B ⊗ k B op -mod cdg fpd ) in the formulation of Corollary B. Notice also that when the left homological dimension of B # ⊗ k B #op is finite, all the exotic derived categories D cmp (B ⊗ k B op -mod cdg ), D abs (B ⊗ k B op -mod cdg ), D co (B ⊗ k B op -mod cdg fpd ), etc. coincide [15, Theorem 3.6(a)]. 4.8. Matrix factorizations. Set Γ = Z/2. Let R be a commutative regular local ring; suppose that R is also an algebra of essentially finite type over its residue field k. Let w ∈ R be a noninvertible element whose zero locus has an isolated singularity at the closed point of the spectrum of R. Consider the CDG-algebra (B, d, h) over k, where B is the algebra R placed in degree 0, d = 0, and h = −w. Let C = mod cdg fgp -B be the corresponding DG-category of right CDG-modules; its objects are conventionally called the matrix factorizations of w.
The computations in [18] and [2] show that the two kinds of Hochschild (co)homology for the k-linear DG-category C are isomorphic. The somewhat stronger assertion that the natural maps HH * (C, M) −→ HH II * (C, M) and HH II, * (C, M) −→ HH * (C, M) are isomorphisms for any DG-module M over C ⊗ k C op follows from our Corollary 4.7.B. Indeed, according to [2, Theorem 4.1 and the discussion in Section 6.1] the assumption of the corollary is satisfied in this case.
More generally, let X be a smooth affine variety over a field k and R be the k-algebra of regular functions on X. Let w ∈ R be such a function; consider the CDG-algebra (B, d, h) constructed from R and w as above. Let C = mod cdg fgp -B be the DG-category of right CDG-modules over B, projective and finitely generated as Γ-graded B # -modules.
Corollary A. Assume that the morphism w : X \ {w = 0} −→ A 1 k from the open complement of the zero locus of w in X to the affine line is smooth. Assume moreover that either (a) there exists a smooth closed subscheme Z ⊂ X such that w : X \ Z −→ A 1 k is a smooth morphism and w| Z = 0, or (b) the field k is perfect. Then the natural maps HH * (C, M) −→ HH II * (C, M) and HH II, * (C, M) −→ HH * (C, M) are isomorphisms for any DG-module M over C ⊗ k C op .
Proof. The proof is based on Corollary 4.7.B, Orlov's theorem connecting matrix factorizations with the triangulated categories of singularities [10], and some observations from the paper [8]. We will show that all objects of H 0 (B ⊗ k B op -mod cdg fgp ) can be obtained from the objects G⊗ k F with G ∈ H 0 (B-mod cdg fgp ) and F ∈ H 0 (mod cdg fgp -B) using the operations of cone and passage to a direct summand. By Orlov's theorem, the triangulated categories H 0 (B-mod cdg fgp ) and H 0 (mod cdg fgp -B) can be identified with the triangulated category D b Sing (X 0 ) of singularities of the zero locus X 0 ⊂ X of the function w. Similarly, the triangulated category H 0 (B ⊗ k B op -mod cdg fgp ) is identified with the triangulated category D b Sing (Y 0 ) of singularities of the zero locus Y 0 ⊂ X × k X of the function w × 1 − 1 × w on the Cartesian product X × k X. Sing Sing (Y 0 ) induced by the composition of the external tensor product of coherent sheaves on two copies of X 0 and the direct image under the closed embedding X 0 × k X 0 −→ Y 0 .
Proof. Rather than checking the assertion of the lemma for Orlov's cokernel functor Σ : H 0 (B-mod cdg fgp ) −→ D b Sing (X 0 ), one can use the construction of the inverse functor Υ : D b Sing (X 0 ) −→ H 0 (B-mod cdg fgp ) given in [16], for which the desired compatibility is easy to establish. Alternatively, it suffices to use the result of [8,Lemma 2.18].
Let D abs (B-mod cdg fg ) denote the absolute derived category of left CDG-modules over B whose underlying Γ-graded B # -modules are finitely generated; the notation D abs (mod cdg fg -B) for right CDG-modules will have the similar meaning. Then the external tensor product of finitely generated CDG-modules induces a functor D abs (B-mod cdg fg ) × D abs (mod cdg fg -B) −→ D abs (B ⊗ k B op -mod cdg fg ); furthermore, the natural functor H 0 (B-mod cdg fgp ) −→ D abs (B-mod cdg fg ) is an equivalence of categories, since B # is Noetherian of finite homological dimension.
Let M ∈ H 0 (B-mod cdg fgp ); the direct image of the corresponding coherent sheaf Σ(M) on X 0 under the closed embedding X 0 −→ X can be viewed as an object of D abs (B-mod cdg fg ). It is clear from the above-mentioned lemma from [8] that this object is naturally isomorphic to the image of M in D abs (B-mod cdg fg ). Let N ∈ H 0 (mod cdg fgp -B); then the coherent sheaf Σ(N) on X 0 , viewed as an object of D abs (mod cdg fg -B), is isomorphic to N. Since the external tensor product is well-defined on the absolute derived categories of finitely generated CDG-modules, it follows that the coherent sheaf Σ(M) ⊠ k Σ(N) on X 0 × k X 0 , viewed as an object of D abs (B ⊗ k B op -mod cdg fg ), is isomorphic to M ⊗ k N.
differential d and curvature h of the CDG-coalgebra C in Corollary 4.6.B, nor can one drop the condition on the critical values of the potential w in Corollary 4.8.A. Set Γ = Z/2 and (B, d, h) = (k, 0, 1). So B is the k-algebra k placed in the grading 0 mod 2 and endowed with the zero differential and a nonzero curvature element. Then any CDG-module over B is contractible, any one of the two components of the differential of a CDG-module being its contracting homotopy. So the DG-category C is quasi-equivalent to zero, hence HH * (C) = 0 = HH * (C).
On the other hand, the CDG-algebra B ⊗ k B op is simply the Z/2-graded k-algebra k with the zero differential and curvature. The CDG-module B over B ⊗ k B op is the Z/2-graded k-module k concentrated in degree 0 mod 2. So Tor Therefore, we have natural isomorphisms (53) HH II * (B (c) ) ≃ HH II * (B) and HH II, * (B (c) ) ≃ HH II, * (B), and consequently similar isomorphisms for the Hochschild (co)homology of the second kind of the DG-categories C = mod cdg fgp -B and C (c) = mod cdg fgp -B (c) . Hochschild (co)homology of the first kind of the DG-categories C and C (c) are not isomorphic in general (and in fact can be entirely unrelated, as the following example illustrates).
Let k be an algebraically closed field of characteristic 0, X be a smooth affine variety over k, and w be a regular function on X. Let B be the CDG-algebra associated with X and w as in 4.8. The function w has a finite number of critical values c i ∈ k. When c is not a critical value, the Hochschild (co)homology of the first kind HH * (C (c) ) and HH * (C (c) ) vanish, since the category H 0 (C (c) ) does. We have natural maps HH * (C (c i ) ) −−→ HH II * (C (c i ) ) ≃ HH II * (C) and HH II, * (C) ≃ HH II, * (C (c i ) ) −−→ HH * (C (c i ) ).