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 =
and an odd derivation d: A −→ A of
degree 1 such that
= 0. In other words, it is supposed that

and d(ab) = d(a)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 =
M i endowed with a differential dM : M −→ M of degree 1 compatible with
the derivation of A and such that dM
= 0. The compatibility means that the
equation dM (ax) = d(a)x +
(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 +
and dN
= 0, where x N
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
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))
Clearly, one has
= 0; for any composable morphisms of left DG-modules f
and g one has d(fg) = d(f)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

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) =
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 −→
whose image is contained in the center of the algebra
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