1. DERIVED CATEGORY OF DG-MODULES 15

the DG-module structure on HomA(B, M) is defined so that HomA(B, M) −→

HomZ(B, M) is a closed injective morphism of DG-modules).

The functor Rf obviously maps acyclic DG-modules to acyclic DG-modules,

and so induces a functor D(B–mod) −→ D(A–mod), which we will denote by IRf .

The functor Ef has a left derived functor LEf obtained by restricting Ef to either

of the full subcategories Hot(A–mod)proj or Hot(A–mod)fl ⊂ Hot(A–mod) and com-

posing it with the localization functor Hot(B–mod) −→ D(B–mod). The functor

Ef has a right derived functor REf obtained by restricting Ef to the full subcate-

gory Hot(A–mod)inj ⊂ Hot(A–mod) and composing it with the localization functor

Hot(B–mod) −→ D(B–mod). The functor LEf is left adjoint to the functor IRf

and the functor REf is right adjoint to the functor IRf .

Theorem. The functors IRf , LEf , REf are equivalences of triangulated

categories if and only if the morphism f induces an isomorphism H(A) H(B).

Proof. Morphisms in D(A–mod) between shifts of the DG-module A recover

the cohomology H(A) and analogously for the DG-algebra B, so the “only if”

assertion follows from the isomorphism LEf (A) B. To prove the “if” part,

we will show that the adjunction morphisms LEf (IRf (N)) −→ N and M −→

IRf (LEf (M)) are isomorphisms for any left DG-modules M over A and N over B.

The former morphism is represented by the composition B⊗A G −→ B⊗A N −→ N

for any flat DG-module G over A endowed with a quasi-isomorphism G −→ N of

DG-modules over A. This composition is a quasi-isomorphism, since the morphisms

B⊗A G ←− A⊗A G −→ A⊗A N N are quasi-isomorphisms. The latter morphism

is represented by the fraction M ←− F −→ B ⊗A F for any flat DG-module F

over A endowed with a quasi-isomorphism F −→ M of DG-modules over A. The

morphism F A ⊗A F −→ B ⊗A F is a quasi-isomorphism.

1.8. DG-module t-structure. An object Y of a triangulated category D is

called an extension of objects Z and X if there is a distinguished triangle X −→

Y −→ Z −→ X[1]. Let D = D(A–mod) denote the derived category of left DG-

modules over a DG-ring A. Let D

0

⊂ D denote the full subcategory formed by all

DG-modules M over A such that

Hi(M)

= 0 for i 0 and D

0

⊂ D denote the

minimal full subcategory of D(A–mod) containing the DG-modules A[i] for i 0

and closed under extensions and infinite direct sums.

Theorem. (a) The pair of subcategories (D 0, D 0) defines a t-structure [3]

on the derived category D(A–mod).

(b) The subcategory D 0 ⊂ D coincides with the full subcategory formed by all

DG-modules M over A such such that Hi(M) = 0 for i 0 if and only if Hi(A) = 0

for all i 0.

Proof. Part (a): clearly, one has D

0[1]

⊂ D

0,

D

0[−1]

⊂ D

0,

and

HomD(D

0,

D

0[−1])

= 0. It remains to construct for any DG-module M over A a

closed morphism of DG-modules F −→ M inducing a monomorphism on

H1

and an

isomorphism on

Hi

for all i 0 such that F can be obtained from the DG-modules

A[i] with i 0 by iterated extensions and infinite direct sums in the homotopy

category of DG-modules. This construction is similar to that of the proof of Theo-

rem 1.4, with the following changes. One chooses a surjective morphism M −→ M

onto M from a complex of free abelian groups M with free abelian groups of co-

homology so that Hi(M ) = 0 for i 0 and the maps Hi(M ) −→ Hi(M) are