Derived categories and Morita theory
The authors develop a Morita theory for derived categories using tilting modules. Let A be an arbitrary ring, \({\mathcal A}=mod\)-A the category of right A-modules and T a right A-module, which is a tilting module; i.e., (i) T has a finite, finitely generated projective resolution of length f, (ii) \(Ext^ n_ A(T,T)=0\) for all positive integers n, (iii) A has a finite resolution \(0\to A\to T^ 0\to...\to T^ g\to 0\), with \(T^ i\in add(T)\). Let \(B=End_ A(T)\), and \({\mathcal B}=mod\)-B, \({\mathbb{F}}: {\mathcal A}\to {\mathcal B}\) is \(Hom_ A(T\),-) and \({\mathbb{G}}: {\mathcal B}\to {\mathcal A}\) is \(-\otimes_ BT\). \(D({\mathcal A})\) and \(D^ b({\mathcal A})\) is the derived category resp. the category of bounded complexes, similarly for \({\mathcal B}.\) The main result is Theorem: (a) The right derived functor \(R^{+,b}{\mathbb{F}}: D^ b({\mathcal A})\to D({\mathcal B})\) has image in \(D^ b({\mathcal B})\). In fact, \(\dim (R^{+,b})\leq f\). (b) Similarly \(L^{- ,b}{\mathbb{G}}: D^ b({\mathcal B})\to D({\mathcal A})\) has image in \(D^ b({\mathcal A})\) with \(\dim (L^{-,b})\leq g\). (c) The functors \(R^{+,b}{\mathbb{F}}\) and \(L^{-,b}{\mathbb{G}}\) induce mutually inverse equivalences between \(D^ b({\mathcal A})\) and \(D^ b({\mathcal B})\). (d) \(A\cong End_ B(T)\), and T satisfies the left analogues of (i), (ii) and (iii) for B. Moreover, the projective dimension of T as B-module is at most g, and in (iii) one may choose \(g=p\dim_ B(T).\) \textit{D. Happel} [Triangulated categories in representation theory of finite dimensional algebras (Preprint Jan. 85, to appear in Comment. Math. Helv.)] has obtained the equivalence between the derived categories in case A is a finite dimensional algebra of finite global dimension, using the notion of a tilting module only for \(f=g=1\). \textit{Y. Miyashita} [Math. Z. 193, 113-146 (1986; Zbl 0578.16015)] considers tilting modules over arbitrary rings - in the above generality, and also obtains (d) of the above theorem; however, he does not discuss derived categories, and hence obtains an induced equivalence only on parts of \({\mathcal A}\) and \({\mathcal B}\) resp. - cf. also the reviewer's paper [in Quaest. Math. 9, 365-391 (1986; see Zbl 0604.16003 above)]. The second main result of the authors is the following Theorem: Let \({\mathbb{F}}: {\mathcal A}\to {\mathcal B}\) be a left exact additive functor. Assume that \(R^{+,b}{\mathbb{F}}\) induces an equivalence of triangulated categories from \(D^ b({\mathcal A})\) to \(D^ b({\mathcal B})\), as well as an equivalence of the homotopy categories of bounded complexes over the finitely generated projective modules \(K^ b(P_ A)\) to \(K^ b(P_ B)\). Then \({\mathbb{F}}\cong Hom_ A(T,-),\) where T is a tilting module with \(B\cong End_ A(T)\).
scientific article; zbMATH DE number 3975153
scientific article