Tilting functors and stable equivalences for selfinjective algebras

Tachikawa, Hiroyuki; Wakamatsu, Takayoshi

doi:10.1016/0021-8693(87)90169-4

Cited by 28 publications

(13 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Further, by Theorem 3, A is stably equivalent to AI, and hence to TB Ã . Finally, by a known result due to H. Tachikawa and T. Wakamatsu [12] we conclude that TB Ã and TH are stably equivalent. Therefore, A is stably equivalent to TH, and this finishes the proof of Theorem 2.…”

Section: P R O O F First We Show That G Is a Functor For This We Hmentioning

confidence: 78%

“…There has been work connecting tilting theory and selfinjective algebras via trivial extension algebras (see [1], [5], [6], [12]). Recall that the trivial extension TB of an algebra B by its minimal injective cogenerator bimodule DB is the symmetric algebra whose additive structure is that of the group B È DB and whose multiplication is defined by aY xbY y abY ay xb for any aY b P B and any xY y P DB.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Stable equivalence of selfinjective algebras of tilted type

Skowroński

Yamagata

1998

Archiv der Mathematik

View full text Add to dashboard Cite

A finite dimensional K-algebra L is called selfinjective of tilted type if L is a quotient Bafn B , where B is the repetitive algebra of a tilted algebra B not of Dynkin type, n B is the Nakayama automorphism of B, and f is a positive automorphism of B. We prove that a selfinjective algebra A is stably equivalent to a selfinjective algebra L of tilted type if and only if A is socle equivalent to a selfinjective algebra L of tilted type.1. Introduction and the main results. Throughout the paper K will denote a fixed field. By an algebra we mean a finite dimensional associative K-algebra with an identity, which we shall assume to be basic and connected. For an algebra A, we shall denote by mod A the category of finitely generated right A-modules, and by mod A the stable module category of mod A. Recall that the objects of mod A are the objects of mod A without projective direct summands, and for any two objects M and N in mod A the space of morphisms from M to N in mod A is the quotient Hom A MY N Hom A MY NaPMY N, where PMY N is the subspace of Hom A MY N consisting of all A-homomorphisms which factorize through projective A-modules. Two algebras A and L are said to be stably equivalent if their stable module categories mod A and mod L are equivalent. We denote by D the standard duality Hom K ÀY K on mod A and by G A the Auslander±Reiten quiver of G A . Further, an algebra A is called selfinjective if A 9 DA in mod A, that is, the projective A-modules are injective. Moreover, A is called symmetric if A and DA are isomorphic as A-bimodules. If A is selfinjective then the left and right socle of A coincide, and we denote them by soc A. Two selfinjective algebras A and L are said to be socle equivalent if the factor algebras Aasoc A and Lasoc L are isomorphic.We are interested in the stable equivalence of selfinjective algebras whose Auslander± Reiten quiver has a prescribed form. There has been work connecting tilting theory and selfinjective algebras via trivial extension algebras (see [1], [5], [6], [12]). Recall that the trivial extension TB of an algebra B by its minimal injective cogenerator bimodule DB is the symmetric algebra whose additive structure is that of the group B È DB and whose multiplication is defined by

show abstract