1. DERIVED CATEGORY OF DG-MODULES 17 Theorem. One has Hi(A) = 0 for all i 0 if and only if any object X ∈ D can be included into a distinguished triangle Y −→ X −→ Z −→ Y [1] in D with Y ∈ D 1 and Z ∈ D 0 . Proof. “Only if”: suppose that the left DG-module X = A over A can be included into a distinguished triangle Y −→ X −→ Z −→ Y [1] in D with Y ∈ D 0 and Z ∈ D −1 (in the obvious notation). Then HomD(X, Z) H0(Z) = 0, hence X is a direct summand of Y in D, so Hi(X) = 0 for i 0. “If”: let M be a DG-module representing the object X. Let P be the direct sum of left DG-modules A[−i] over A taken over all the elements x ∈ Hi(M) for all i 1. There is a natural closed morphism of DG-modules P −→ M set M1 to be its cone. One has Hi(P ) = 0 for all i 0 and the maps Hi(P ) −→ Hi(M) are surjective for all i 1. Hence the maps Hi(M) −→ Hi(M1) are isomorphisms for all i 0, a monomorphism for i = 0, and zero maps for i 0. Repeating the same procedure for the DG-module M1, etc., we obtain a sequence of DG-module morphisms M −→ M1 −→ M2 −→ · · · , each morphism having the properties described above. Set N = lim − Mi then there is a closed morphism of DG-modules M −→ N over A such that the maps Hi(M) −→ Hi(N) are isomorphisms for all i 0 and a monomorphism for i = 0, and one has Hi(N) = 0 for all i 0. Set Q = cone(M → N)[−1] then one has Hi(Q) = 0 for all i 0. This provides the desired distinguished triangle. Remark 1. There is a much simpler proof of the above Theorem applicable in the case when Ai = 0 for all i 0. However, unlike in the situation of (the proof of) Theorem 1.8(b), it is not true that any DG-ring with zero cohomology in the negative degrees can be connected by a chain of quasi-isomorphisms with a DG-ring with zero components in the negative degrees. For a counterexample, consider the free associative algebra A over a field k generated by the elements x, y, and η in degree 0, z in degree 1, and ξ in degree −1, with the differential given by d(x) = d(y) = d(z) = 0, d(ξ) = xy, and d(η) = yz. One can check that Hi(A) = 0 for i 0. The nontrivial Massey product ξz − xη of the cohomology classes x, y, and z provides the obstruction. Remark 2. The construction of the above Theorem can be extended to arbi- trary DG-rings in the following way. Given a DG-ring A, denote by D 0 ⊂ D = D(A–mod) the full subcategory formed by all the DG-modules M over A such that Hi(M) = 0 for i 0 and by D 1 ⊂ D the minimal full subcategory of D containing the DG-modules A[−i] with i 1 and closed under infinite direct sums and the following operation of countably iterated extension. Given a sequence of objects Xi ∈ D 1 and a sequence of distinguished triangles Yi −→ Yi+1 −→ Xi+1 −→ Yi[1] in D with Y0 = X0, the homotopy colimit cone( i Yi → i Yi) of Yi should also belong to D 1 . Then the same construction as in the above proof provides for any X ∈ D a distinguished triangle Y −→ X −→ Z −→ Y [1] with Y ∈ D 1 and Z ∈ D 0 . This can be generalized even further in the spirit of Remark 1.8.3, by considering an arbitrary triangulated category D admitting infinite direct sums, and a set of compact objects C ⊂ D such that C[−1] ⊂ C in the role of the DG-modules A[−i] with i 1. The subcategory D 1 is then generated by C using the opera- tions of infinite direct sum and countably iterated extension, and the subcategory D 0 consists of all object X ∈ D such that HomD(C, X) = 0 for all C ∈ C. If one does not insist on the subcategories D 0 and D 1 being closed under shifts in

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