Derived equivalences as derived functors. (English) Zbl 0683.16030

Rickard, Jeremy

J. Lond. Math. Soc., II. Ser. 43, No. 1, 37-48 (1991).

In [J. Lond. Math. Soc., II. Ser. 39, No.3, 436-456 (1989; Zbl 0642.16034)], we proved that two algebras \(\Lambda\) and \(\Gamma\) are “derived equivalent”, meaning that the derived category of modules for \(\Lambda\) is equivalent to that for \(\Gamma\), precisely when \(\Gamma\) is isomorphic to the endomorphism ring of what we called a “tilting complex” for \(\Lambda\). The main result of this paper shows that, at least for sufficiently nice algebras (for example, algebras over a field), we can describe the equivalence of derived categories as a derived functor. This description is much easier to deal with than the rather complicated construction of [ibid.], although we need that construction in the proof.
Among other applications, we show that derived equivalences behave well with respect to the derived functors of Hom and tensor products, and we show that if a finite-dimensional algebra is derived equivalent to a symmetric algebra then it is symmetric itself.

Reviewer: J.Rickard

MSC:

16D90	Module categories in associative algebras
16P10	Finite rings and finite-dimensional associative algebras
18E30	Derived categories, triangulated categories (MSC2010)
16G30	Representations of orders, lattices, algebras over commutative rings

Keywords:

derived category of modules; tilting complex; equivalence of derived categories; derived functor; derived equivalences; finite-dimensional algebra; symmetric algebra

Citations:

Zbl 0672.16034; Zbl 0642.16034