8 LEONID POSITSELSKI

A, where k is considered as a DG-ring concentrated in degree 0; equivalently, a

DG-algebra is a complex of k-modules with a k-linear DG-ring structure.

Remark. One can consider DG-algebras and DG-modules graded by an abelian

group Γ different from Z, provided that Γ is endowed with a parity homomorphism

Γ −→ Z/2 and an odd element 1 ∈ Γ, so that the differentials would have degree 1.

In particular, one can take Γ = Z/2, that is have gradings reduced to parities, or

consider fractional gradings by using some subgroup of Q consisting of rationals

with odd denominators in the role of Γ. Even more generally, one can replace the

parity function with a symmetric bilinear form σ : Γ × Γ −→ Z/2, to be used in

the super sign rule in place of the product of parities; one just has to assume that

σ(1, 1) = 1 mod 2. All the most important results of this paper remain valid in

such settings. The only exceptions are the results of subsections 3.4 and 4.3, where

we consider bounded grading.

1.2. DG-categories. A DG-category is a category whose sets of morphisms

are complexes and compositions are biadditive maps compatible with the gradings

and the differentials. In other words, a DG-category DG consists of a class of ob-

jects, complexes of abelian groups HomDG(X, Y ), called the complexes of morphisms

from X to Y , defined for any two objects X and Y , and morphisms of complexes

HomDG(Y, Z) ⊗Z HomDG(X, Y ) −→ HomDG(X, Z), called the composition maps,

defined for any three objects X, Y , and Z. The compositions must be associative

and unit elements idX ∈ HomDG(X, X) must exist; the equations d(idX ) = 0 then

hold automatically.

For example, left DG-modules over a DG-ring A form a DG-category, which

we will denote by DG(A–mod). The DG-category of right DG-modules over A will

be denoted by DG(mod–A).

A covariant DG-functor DG −→ DG consists of a map between the classes of

objects and (closed) morphisms between the complexes of morphisms compatible

with the compositions. A contravariant DG-functor is defined in the same way,

except that one has to take into account the natural isomorphism of complexes

V ⊗ W W ⊗ V for complexes of abelian groups V and W that is given by

the formula v ⊗ w −→

(−1)|v||w|w

⊗ v. (Covariant or contravariant) DG-functors

between DG and DG form a DG-category themselves. The complex of morphisms

between DG-functors F and G is a subcomplex of the product of the complexes

of morphisms from F (X) to G(X) in DG taken over all objects X ∈ DG ; the

desired subcomplex is formed by all the systems of morphisms compatible with all

morphisms X −→ Y in DG .

For example, a DG-ring A can be considered as a DG-category with a sin-

gle object; covariant DG-functors from this DG-category to the DG-category of

complexes of abelian groups are left DG-modules over A, while contravariant DG-

functors between the same DG-categories can be identified with right DG-modules

over A.

A closed morphism f : X −→ Y in a DG-category DG is an element of

HomDG(X,

0

Y ) such that d(f) = 0. The category whose objects are the objects

of DG and whose morphisms are closed morphisms in DG is denoted by Z0(DG).

An object Y is called the product of a family of objects Xα (notation: Y =

α

Xα) if a closed isomorphism of contravariant DG-functors HomDG(−,Y )