Realisation functors in tilting theory (Q1745316)

Derived Morita theory for rings is a fundamental piece in the study of derived categories. The authors consider the problem of the equivalences of derived categories in the more general setting of abelian categories. The main result of the paper establishes that if there exists a restrictable triangle equivalence \(\Phi :D({\mathcal B})\to D({\mathcal A})\), where \( {\mathcal A}, {\mathcal B}\) are abelian categories such that \(D({\mathcal A})\) has set-indexed coproducts (resp. set-indexed products) and \({\mathcal B}\) has a projecive generator (resp. injective cogenerator), then there is a bounded tilting (resp. cotilting) object \(M\) in \(D( {\mathcal A})\) such that its associated heart \({\mathcal H}_M\) is equivalent to \({\mathcal B}\). Moreover, this last condition implies that there is a triangle equivalence between the two bounded derived categories \(D^b({\mathcal B})\) and \(D^b({\mathcal A})\), the two statements are equivalent if \(\mathcal B\) has a projective generator and \(\mathcal A\) is the category of right modules over a ring. This last equivalence extend Rickard's derived Morita theory for rings [\textit{J. Rickard}, J. Lond. Math. Soc., II. Ser. 39, No. 3, 436--456 (1989; Zbl 0642.16034)]. Since Grothendieck categories satisfies the preceding conditions their result can be considered a derived Morita theorem for Grothendieck categories. Filtered derived categories and realization functors, studied in Section 3, and (co)silting objects in triangulated categories, Section 4, are the principal technical ingredients of the paper. Thus, it is shown that for projective algebras over commutative rings the realization functor associated to a compact tilting object \(T\) in \(D(A)\) such that \(\mathrm{End}_{D(A)}(T)\cong B\) is an equivalence of standard type, i.e. naturally equivalent to \(-\overset{\mathbf{L}}{\otimes}_B X\) with \(X\) a complex of \(B-A\) bimodules. In last section, after presenting some conditions for a recollement of abelian categories to induce a recollement of unbounded derived categories, a characterization of recollement derived categories equivalent to the derived version of a recollement of abelian categories is given. Moreover, any recollement of \(D(A)\), where \(A\) is a finite dimensional hereditary algebra over a field, by derived module categories is equivalent to a stratifying one, i.e. induced by a homological ring epimorphism.