16 LEONID POSITSELSKI surjective for all i 0. Then for F0 = A ⊗Z M and K = ker(F0 M) one chooses a surjective morphism K −→ K onto K from a complex of free abelian groups K with free abelian groups of cohomology so that Hi(K ) = 0 for i 1 and the maps Hi(K ) −→ Hi(K) are surjective for all i 1 in order to put F1 = A ⊗Z K , etc. The DG-module F is constructed as the total DG-module of the complex · · · −→ F1 −→ F0 formed by taking infinite direct sums. The “only if” assertion in part (b) is clear. To prove “if”, replace A with its quasi-isomorphic DG-subring τ 0 A with the components 0 A)i = Ai for i 0, 0 A)0 = ker(A0 A1), and 0 A)i = 0 for i 0 then notice that the canonical filtrations on DG-modules over τ 0 A considered as complexes of abelian groups are compatible with the action of the ring τ 0 A. Remark 1. The t-structure described in part (a) of Theorem can well be degenerate, though it is clearly nondegenerate under the assumptions of part (b). Namely, one can have i D 0 [i] = 0. For example, take A = k[x] to be the graded algebra of polynomials with one generator x of degree 1 over a field k and endow it with the zero differential. Then the graded A-module k[x, x−1] considered as a DG-module with zero differential belongs to the above intersection, since it can be presented as the inductive limit of the DG-modules x−jk[x]. Moreover, take A = k[x, x−1], where deg x = 1 and d(x) = 0 then D 0 = 0 and D 0 = D. Remark 2. One might wish to define a dual version of the above t-structure on D(A–mod) where D 0 would be the minimal full subcategory of D containing the DG-modules HomZ(A, Q/Z)[i] for i 0 and closed under extensions and infinite products, while D 0 would consist of all DG-modules M with Hi(M) = 0 for i 0. The dual version of the above proof does not seem to work in this case, however, because of a problem related to nonexactness of the countable inverse limit. Remark 3. The above construction of the DG-module t-structure can be gen- eralized in the following way (cf. [51]). Let D be a triangulated category with infinite direct sums. An object C D is said to be compact if the functor HomD(C, −) preserves infinite direct sums. Let C D be a subset of objects of D consisting of compact objects and such that C[1] C. Let D 0 be the full subcategory of D formed by all objects X such that HomD(C, X[−1]) = 0 for all C C, and let D 0 be the minimal full subcategory of D containing C and closed under extensions and infinite direct sums. Then (D 0 , D 0 ) is a t-structure on D. Indeed, let X be an object of D. Consider the natural map into X from the direct sum of objects from C indexed by morphisms from objects of C to X let X1 be the cone of this map. Applying the same construction to the object X1 in place of X, we obtain the object X2, etc. Let Y be the homotopy inductive limit of Xn, i. e., the cone of the natural map n Xn −→ n Xn. Then Y D 0 [−1] and cone(X Y )[−1] D 0 . 1.9. Silly filtrations. Let A be a DG-ring and D = D(A–mod) denote the derived category of left DG-modules over it. Denote by D 0 D the full subcat- egory formed by all the DG-modules M such that Hi(M) = 0 for i 0 and by D 1 D the full subcategory of all the DG-modules M such that Hi(M) = 0 for i 0. We refer the reader to [49, Introduction and Appendices B–C] for the general discussion of silly filtrations. For the purposes of the present exposition it suffices to say that the directions of arrows in the distinguished triangle in the next Theorem are opposite to the ones in distinguished triangles related to t-structures.
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