Spherical modules and the Auslander--Gorenstein condition for Auslander--Yoneda algebras

Abstract: For a finite dimensional algebra $A$ of finite global dimension we study the Auslander--Yoneda algebra defined as the Yoneda algebra of the direct sum of all indecomposable $A$-modules. We show that the Auslander--Yoneda algebra is an Auslander--Gorenstein algebra if and only if every indecomposable left and right $A$-module is spherical in the sense of Auslander and Bridger. This motivates the study of spherical algebras defined by the condition that every indecomposable module is spherical. We characterize spherical algebras by a certain natural pair of subcategories being a split torsion pair. Moreover, we prove that representation-finite algebras which are spherical are directed and give a full classification of spherical Nakayama algebras. Furthermore, we show that replicated algebras of hereditary algebras are spherical. As a final application of the new notion of spherical algebras, we give a negative answer to a question of Venjakob on Auslander regular algebras in general, but show that there is a positive answer when assuming that every indecomposable left $A$-module is spherical.