Symmetry in the vanishing of Ext over stably symmetric algebras.
The paper under review deals with a question of M. Auslander, which can be stated in colloquial terms as follows: given a ring \(\Lambda\), is the property of asymptotic vanishing of the extension functors of an arbitrary pair of finitely generated modules \(M\) and \(N\) symmetric in \(M\) and \(N\)? Avramov and Buchweitz showed that the answer is ``yes'' for commutative complete intersections, but more recently Jorgensen and Sega found a commutative Gorenstein ring with the above property lacking the symmetry. The author's goal is to find classes of rings over which the property is symmetric. His motivation comes from noncommutative algebraic geometry where it is needed for defining an intersection multiplicity of modules. To this end he introduces the class of stably symmetric algebras. In short, an algebra is stably symmetric if some power of the syzygy autoequivalence is a Serre functor on the projective stabilization of the category of finitely generated \(\Lambda\)-modules. The terminology seems to be derived from the fact that every symmetric algebra (i.e., that is a Frobenius algebra with an identity Nakayama functor) is stably symmetric. With the additional assumptions that such an algebra is Noetherian, Gorenstein, and satisfies the Auslander condition (this is a certain finiteness condition on the non-vanishing of the extension functors) the author answers the original question in the positive. As a consequence, group rings of finite groups and exterior algebras on an odd number of letters all fall into the desired class. The author admits that he does not know any nontrivial examples of stably symmetric algebras which are not symmetric. The situation is different in the graded case and, when restricted to this class of algebras (and modules), all exterior algebras have the desired property. The paper also contains several interesting examples with explicit calculations.
categories of finitely generated modules
scientific article; zbMATH DE number 5143560
scientific article