1. DERIVED CATEGORY OF DG-MODULES 13
Theorem. (a) The category Hot(A–mod)inj is the minimal triangulated subcat-
egory of Hot(A–mod) containing the DG-module HomZ(A, Q/Z) and closed under
(b) The composition of functors Hot(A–mod)inj −→Hot(A–mod)−→D(A–mod)
is an equivalence of triangulated categories.
Proof. The proof is analogous to that of Theorem 1.4. Clearly, the cate-
gory Hot(A–mod)inj is closed under infinite products. It contains the DG-module
HomZ(A, Q/Z), since the complex HomA(L, HomZ(A, Q/Z)) HomZ(L, Q/Z) is
acyclic whenever the DG-module L is. To construct an injective resolution of a
DG-module M, one can embed in into a complex of injective abelian groups M so
that the cohomology H(M ) is also injective and H(M) also embeds into H(M ).
For example, one can take the components of M to be the products of Q/Z over all
nonzero homomorphisms of abelian groups from the components of M to Q/Z. Take
J0 = HomZ(A, M ) and consider the induced injective morphism of DG-modules
M −→ J0. Set K = J0/M, J−1 = HomZ(A, K ), etc., and J = Tot (J•). Then
the morphism of DG-modules M −→ J has an acyclic cone and the DG-module
J is isomorphic in Hot(A–mod) to a DG-module obtained from HomZ(A, Q/Z) by
iterating the operations of shift, cone, and infinite product.
1.6. Flat resolutions. A right DG-module N over a DG-ring A is said to be
flat if for any acyclic left DG-module M over A the complex N ⊗A M is acyclic. Flat
left DG-modules over A are defined in the analogous way. The full triangulated sub-
category of Hot(A–mod) formed by flat DG-modules is denoted by Hot(A–mod)fl.
We denote the thick subcategory of acyclic right A-modules by Acycl(mod–A) ⊂
Hot(mod–A). The quotient category Hot(mod–A)/Acycl(mod–A) is called the de-
rived category of right DG-modules over A and denoted by D(mod–A). The full
triangulated subcategory of flat right DG-modules is denoted by Hot(mod–A)fl ⊂
It follows from Theorems 1.4–1.5 and Lemma 1.3 that one can compute the
right derived functor ExtA(L, M) = HomD(A–mod)(L, M) for left DG-modules L
and M over a DG-ring A in terms of projective or injective resolutions. Namely,
one has ExtA(L, M) H(HomA(L, M)) whenever L is a projective DG-module or
M is an injective DG-module over A. The following Theorem allows to define a
left derived functor
M) for a right DG-module N and a left DG-module
M over A so that it could be computed in terms of flat resolutions.
Theorem. (a) The functor Hot(A–mod)fl/(Acycl(A–mod)∩Hot(A–mod)fl) −→
D(A–mod) induced by the embedding Hot(A–mod)fl −→ Hot(A–mod) is an equiva-
lence of triangulated categories.
(b) The functor Hot(mod–A)fl/(Acycl(mod–A) ∩ Hot(mod–A)fl) −→ D(mod–A)
induced by the embedding Hot(mod–A)fl −→ Hot(mod–A) is an equivalence of tri-
The proof of Theorem is based on the following Lemma.
Lemma. Let H be a triangulated category and A, F ⊂ H be full triangulated
subcategories. Then the natural functor F/A ∩ F −→ H/A is an equivalence of
triangulated categories whenever one of the following two conditions holds:
(a) for any object X ∈ H there exists an object F ∈ F together with a morphism
F −→ X in H such that a cone of that morphism belongs to A, or