1. DERIVED CATEGORY OF DG-MODULES 11
(b) C is a thick subcategory in H and the functor B −→ H/C is an equivalence
of triangulated categories;
(c) B and C generate H as a triangulated category, i. e., any object of H can
be obtained from objects of B and C by iterating the operations of shift
and cone;
(d) for any object X H there exists a distinguished triangle B −→ X −→
C −→ B[1] with B B and C C (and in this case for any morphism
X −→ X in H there exists a unique morphism between any distinguished
triangles of the above form for X and X , so this triangle is unique up
to a unique isomorphism and depends functorially on X);
(e) C is the full subcategory of H formed by all the objects C H such that
HomH(B, C) = 0 for all B B, and the embedding functor B −→ H has
a right adjoint functor (which can be then identified with the localization
functor H −→ H/C B);
(f) C is the full subcategory of H formed by all the objects C H such that
HomH(B, C) = 0 for all B B, B is a thick subcategory in H, and the
localization functor H −→ H/B has a right adjoint functor;
(g) B is the full subcategory of H formed by all the objects B H such that
HomH(B, C) = 0 for all C C, and the embedding functor C −→ H has
a left adjoint functor (which can be then identified with the localization
functor H −→ H/B C);
(h) B is the full subcategory of H formed by all the objects B H such that
HomH(B, C) = 0 for all C C, C is a thick subcategory in H, and the
localization functor H −→ H/C has a left adjoint functor.
1.4. Projective resolutions. A DG-module M is said to be acyclic if it is
acyclic as a complex of abelian groups, i. e., H(M) = 0. The thick subcategory of
the homotopy category Hot(A–mod) formed by the acyclic DG-modules is denoted
by Acycl(A–mod). The derived category of left DG-modules over A is defined as
the quotient category D(A–mod) = Hot(A–mod)/Acycl(A–mod).
A left DG-module L over a DG-ring A is called projective if for any acyclic left
DG-module M over A the complex HomA(L, M) is acyclic. The full triangulated
subcategory of Hot(A–mod) formed by the projective DG-modules is denoted by
Hot(A–mod)proj. The following Theorem says, in particular, that the homotopy
category H = Hot(A–mod) and its subcategories B = Hot(A–mod)proj and C =
Acycl(A–mod) satisfy the equivalent conditions of Lemma 1.3, and so describes the
derived category D(A–mod).
Theorem. (a) The category Hot(A–mod)proj is the minimal triangulated sub-
category of Hot(A–mod) containing the DG-module A and closed under infinite
direct sums.
(b) The composition of functors Hot(A–mod)proj −→Hot(A–mod)−→D(A–mod)
is an equivalence of triangulated categories.
Proof. First notice that the category Hot(A–mod)proj is closed under infinite
direct sums. It contains the DG-module A, since for any DG-module M over A
there is a natural isomorphism of complexes of abelian groups HomA(A, M) M.
According to Lemma 1.3, it remains to construct for any DG-module M a morphism
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