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
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