12 LEONID POSITSELSKI

f : F −→ M in the homotopy category of DG-modules over A such that the DG-

module F belongs to the minimal triangulated subcategory containing the DG-

module A and closed under infinite direct sums, while the cone of the morphism f

is an acyclic DG-module. When A is a DG-algebra over a field k, it suﬃces to

consider the bar-resolution of a DG-module M. It is a complex of DG-modules

over A, and its total DG-module formed by taking infinite direct sums provides the

desired DG-module F .

Let us give a detailed construction in the general case. Let M be a DG-module

over A. Choose a complex of free abelian groups M together with a surjective

morphism of complexes M −→ M such that the cohomology H(M ) is also a free

graded abelian group and the induced morphism of cohomology H(M ) −→ H(M)

is also surjective. For example, one can take M to be the graded abelian group

with the components freely generated by nonzero elements of the components of M,

endowed with the induced differential. Set F0 = A ⊗Z M ; then there is a natural

closed surjective morphism F0 −→ M of DG-modules over A and the induced

morphism of cohomology H(F0) −→ H(M) is also surjective. Let K be the kernel

of the morphism F0 −→ M (taken in the abelian category

Z0DG(A–mod)

of DG-

modules and closed morphisms between them). Applying the same construction

to the DG-module K in place of M, we obtain the DG-module F1, etc. Let F be

the total DG-module of the complex of DG-modules · · · −→ F1 −→ F0 formed by

taking infinite direct sums. One can easily check that the cone of the morphism

F −→ M is acyclic, since the complex · · · −→ H(F1) −→ H(F0) −→ H(M) −→ 0

is acyclic (it suﬃces to apply the result of [14] to the increasing filtration of the

total complex of · · · −→ F1 −→ F0 −→ M coming from the silly filtration of this

complex of complexes).

It remains to show that the DG-module F as an object of the homotopy cate-

gory can be obtained from the DG-module A by iterating the operations of shift,

cone, and infinite direct sum. Every DG-module Fn is a direct sum of shifts of

the DG-module A and shifts of the cone of the identity endomorphism of the

DG-module A. Denote by Xn the total DG-module of the finite complex of DG-

modules Fn −→ · · · −→ F0. Then we have F = lim

− →

Xn in the abelian category

Z0DG(A–mod). So there is an exact triple of DG-modules and closed morphisms

0 −→ Xn −→ Xn −→ F −→ 0. Since the embeddings Xn −→ Xn+1

split in DG(A–mod)#, the above exact triple also splits in this additive category.

Thus F is a cone of the morphism Xn −→ Xn in the triangulated category

Hot(A–mod).

1.5. Injective resolutions. A left DG-module M over a DG-ring A is said

to be injective if for any acyclic DG-module L over A the complex HomA(L, M) is

acyclic. The full triangulated subcategory of Hot(A–mod) formed by the injective

DG-modules is denoted by Hot(A–mod)inj.

For any right DG-module N over A and any complex of abelian groups V the

complex HomZ(N, V ) has a natural structure of left DG-module over A with the

graded A-module structure given by the formula (af)(n) = (−1)|a|(|f|+|n|)f(na).

The following Theorem provides another semiorthogonal decomposition of the

homotopy category Hot(A–mod) and another description of the derived category

D(A–mod).