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