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).
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