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Morita theory for derived categories. (English) Zbl 0642.16034
A necessary and sufficient condition is given for the equivalence of the derived categories \(D^ b\)(Mod-\(\Lambda)\) and \(D^ b\)(Mod-\(\Gamma)\) of bounded complexes of modules for two rings \(\Lambda\) and \(\Gamma\). The condition is that \(\Gamma\) should be the endomorphism ring of what we call a tilting complex for \(\Lambda\). This generalises the result that an equivalence exists when \(\Gamma\) is the endomorphism ring of a tilting module for \(\Lambda\), which is due to E. Cline, B. Parshall and L. Scott [J. Algebra 104, 397-409 (1986; Zbl 0604.16025)] itself generalizing previous results of D. Happel [Comment. Math. Helv. 62, 339-389 (1987; Zbl 0626.16008)]. Equivalences of other categories such as \(D^-\)(Mod-\(\Lambda)\) and \(D^-\)(Mod-\(\Gamma)\) are also discussed.
Reviewer: J.Rickard

MSC:
16D90 Module categories in associative algebras
16B50 Category-theoretic methods and results in associative algebras (except as in 16D90)
16Gxx Representation theory of associative rings and algebras
16P10 Finite rings and finite-dimensional associative algebras
18E30 Derived categories, triangulated categories (MSC2010)
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