zbMATH — the first resource for mathematics

Stable equivalence for self-injective algebras and a generalization of tilting modules. (English) Zbl 0726.16009
Let T(A) denote the trivial extension \(A\ltimes DA\) of an Artin algebra A by the minimal injective cogenerator DA. The main results state that for an equivalence S: mod-T(A)\(=mod\)-T(B) with two additional conditions there is a generalized tilting bimodule \({}_ BT_ A\) where A and B are Artin algebras. Several interesting properties of generalized tilting modules are also obtained as well as a construction of a stably equivalent functor.

16G10 Representations of associative Artinian rings
16D90 Module categories in associative algebras
16D50 Injective modules, self-injective associative rings
Full Text: DOI
[1] Auslander, M; Reiten, I, Stable equivalence of Artin algebras, (), 8-70
[2] Auslander, M; Reiten, I, Representation theory of Artin algebras, III, Comm. algebra, 3, 239-294, (1975) · Zbl 0331.16027
[3] Auslander, M; Reiten, I, Representation theory of Artin algebras, V, Comm. algebra, 5, 519-554, (1977) · Zbl 0396.16008
[4] Fossum, R.M; Griffith, P.A; Reiten, I, Trivial extensions of abelian categories, () · Zbl 0255.16014
[5] Happel, D, On the derived category of a finite-dimensional algebra, Comment. math. helv., 62, 339-389, (1987) · Zbl 0626.16008
[6] Happel, D; Ringel, C.M, Tilted algebras, Trans. amer. math. soc., 274, 399-443, (1982) · Zbl 0503.16024
[7] Miyashita, Y, Tilting modules of finite projective dimension, Math. Z., 193, 113-146, (1986) · Zbl 0578.16015
[8] \scJ. Rickard, Equivalences of derived categories of modules, preprint. · Zbl 1033.20005
[9] Tachikawa, H; Wakamatsu, T, Tilting functors and stable equivalences for selfinjective algebras, J. algebra, 109, 138-165, (1987) · Zbl 0616.16012
[10] Tachikawa, H; Wakamatsu, T, Applications of reflection functors for self-injective algebras, (), 308-327
[11] \scT. Wakamatsu, On modules with trivial self-extensions, preprint. · Zbl 0646.16025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.