Spherical Algebras and their Auslander—Yoneda AlgebrasOberseminar Algebra und Darstellungstheorie

by Jan Thomm (Universität zu Köln), Marvin Plogmann (Universität zu Köln)

Thursday Sep 24, 2026, 4:00 PM → 6:00 PM Europe/Berlin

Endenicher Allee 60/1-008-Seminarraum (Mathezentrum)

Considering a hereditary algebra A and a module M over A, Ext^n(M,A)=0 unless n is equal to the projective dimension of M. Generalising this observation to algebras of finite global dimension, Auslander and Bridger define a module M to be spherical if Ext^n(M,A)=0 unless n is the projective dimension of M (or n=0). Motivated by this, we call an algebra spherical if every indecomposable module is spherical. In this talk, we will discuss structural properties of the module category of a spherical algebra and give examples beyond the hereditary case. We will then show that representation-finite bispherical algebras, that is, algebras that are spherical with respect to both left and right modules, give rise to Auslander—Gorenstein algebras: If M is an additive generator of the module category, then its Yoneda algebra Ext^*(M,M) is Auslander—Gorenstein. In fact, an appropriate version of this property characterizes bispherical algebras, even in the representation-infinite case. Finally, we will discuss the Auslander—Reiten bijection for the Auslander—Gorenstein algebras arising in this way.