1. Derived Category of DG-Modules

1.1. DG-rings and DG-modules. A DG-ring A = (A, d) is a pair consisting

of an associative graded ring A =

i∈Z

Ai

and an odd derivation d: A −→ A of

degree 1 such that

d2

= 0. In other words, it is supposed that

d(Ai)

⊂

Ai+1

and d(ab) = d(a)b +

(−1)|a|ad(b)

for a, b ∈ A, where |a| denotes the degree of a

homogeneous element, i. e., a ∈ A|a|.

A left DG-module (M, dM ) over a DG-ring A is a graded left A-module M =

i∈Z

M i endowed with a differential dM : M −→ M of degree 1 compatible with

the derivation of A and such that dM

2

= 0. The compatibility means that the

equation dM (ax) = d(a)x +

(−1)|a|adM

(x) holds for all a ∈ A and x ∈ M.

A right DG-module (N, dN ) over A is a graded right A-module N endowed

with a differential dN of degree 1 satisfying the equations dN (xa) = dN (x)a +

(−1)|x|xd(a)

and dN

2

= 0, where x ∈ N

|x|.

Let L and M be left DG-modules over A. The complex of homomorphisms

HomA(L, M) from L to M over A is constructed as follows. The component

HomA(L,

i

M) consists of all homogeneous maps f : L −→ M of degree i such that

f(ax) = (−)i|a|af(x) for all a ∈ A and x ∈ L. The differential in the com-

plex HomA(L, M) is given by the formula d(f)(x) = dM (f(x)) −

(−1)|f|f(dL(x)).

Clearly, one has

d2(f)

= 0; for any composable morphisms of left DG-modules f

and g one has d(fg) = d(f)g +

(−1)|f|fd(g).

For any two right DG-modules R and N over A, the complex of homomor-

phisms HomA(R, N) is defined by the same formulas as above and satisfies the

same properties, with the only difference that a homogeneous map f : R −→ N

belonging to HomA(R, N) must satisfy the equation f(xa) = f(x)a for a ∈ A and

x ∈ R.

Let N be a right DG-module and M be a left DG-module over A. The tensor

product complex N ⊗A M is defined as the graded quotient group of the graded

abelian group N ⊗Z M by the relations xa ⊗ y = x ⊗ ay for x ∈ N, a ∈ A, y ∈ M,

endowed with the differential given by the formula d(x ⊗ y) = d(x) ⊗ y

+(−1)|x|x

⊗

d(y). For any two right DG-modules R and N and any two left DG-modules L and

M the natural map of complexes HomA(R, N) ⊗Z HomA(L, M) −→ HomZ(R ⊗A

L, N ⊗A M) is defined by the formula (f ⊗ g)(x ⊗ y) =

(−1)|g||x|f(x)

⊗ g(y). Here

Z is considered as a DG-ring concentrated in degree 0.

For any DG-ring A, its cohomology H(A) = Hd(A), defined as the quotient

of the kernel of d by its image, has a natural structure of graded ring. For a left

DG-module M over A, its cohomology H(M) is a graded module over H(A); for a

right DG-module N, its cohomology H(N) is a right graded module over H(A).

A DG-algebra A over a commutative ring k is a DG-ring endowed with DG-

ring homomorphism k −→

A0

whose image is contained in the center of the algebra

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