Morita theory for derived categories. (English) Zbl 0642.16034

Rickard, Jeremy

J. Lond. Math. Soc., II. Ser. 39, No. 3, 436-456 (1989).

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)

Keywords:

equivalences of derived categories; bounded complexes of modules; endomorphism rings; tilting complexes; tilting modules

Citations:

Zbl 0604.16025; Zbl 0626.16008