Vanishing of self-extensions over symmetric algebras

Abstract

Abstract. We study self-extensions of modules over symmetric artin algebras. We show that non-projective modules with eventually vanishing self-extensions must lie in AR components of stable type ZA∞. Moreover, the degree of the highest non-vanishing self-extension of these modules is determined by their quasilength. This has implications for the Auslander-Reiten Conjecture and the Extension Conjecture.

‘Ext-Finite Modules for Weakly Symmetric Algebras With Radical Cube Zero’

“…There is also the 'generalized Auslander-Reiten condition', GARC, which has been introduced in [1] in the context of homological conjectures, which has attracted a lot of interest, see for example [6,7,13,8,9]. The condition GARC for a ring R states:…”

Ext-Finite Modules for Weakly Symmetric Algebras With Radical Cube Zero

“…There is also the 'generalized Auslander-Reiten condition', GARC, which has been introduced in [1] in the context of homological conjectures, which has attracted a lot of interest, see for example [6,7,13,8,9]. The condition GARC for a ring R states:…”

“…There is also the 'generalised Auslander-Reiten condition', GARC, which has been introduced in [2] in the context of homological conjectures, and which has attracted a c 2016 Australian Mathematical Publishing Association Inc. 1446-7887/2016 $16.00 lot of interest; see for example [6][7][8][9]13]. The condition GARC for a ring R is stated as follows.…”

The vanishing of self-extensions over n-symmetric algebras of quasitilted type

“…In particular, a weakly symmetric radical cube zero algebra does not satisfy (GARC) precisely when it is tame and does not satisfy (Fg) [E]. We also mention that the condition (GARC) has been studied for symmetric algebras in [DP2]. However, the study of n-symmetric algebras requires completely different methods.…”