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⊗AG −→ B⊗AN −→ 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⊗AG ←− A⊗AG −→ A⊗AN 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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