Symmetry in the vanishing of Ext over stably symmetric algebras.
type: Item
instance of: scholarly article
title: Symmetry in the vanishing of Ext over stably symmetric algebras.
zbMATH Open document ID: 1134.16003
author: Izuru Mori
published in: Journal of Algebra
publication date: 2007-04-16T00:00:00Z
review text: 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.
reviewed by: Alex Martsinkovsky
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16S37
Mathematics Subject Classification ID: 16S38
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 5143560
zbMATH Keywords: stably symmetric algebras
zbMATH Keywords: Frobenius algebras
zbMATH Keywords: Koszul algebras
zbMATH Keywords: Sklyanin algebras
zbMATH Keywords: Tate-Vogel cohomology
zbMATH Keywords: Auslander condition
zbMATH Keywords: Gorenstein rings
zbMATH Keywords: categories of finitely generated modules
zbMATH Keywords: stable module categories
zbMATH Keywords: Serre functors
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2005.07.009
OpenAlex ID: W2031917605
cites work: Noncommutative projective schemes
cites work: Support varieties and cohomology over complete intersections
cites work: Homological dimensions of unbounded complexes
cites work: ABSOLUTE, RELATIVE, AND TATE COHOMOLOGY OF MODULES OF FINITE GORENSTEIN DIMENSION
cites work: The homological theory of contravariantly finite subcategories:auslander-buchweitz contexts, gorenstein categories and (co-)stabilization
cites work: On the vanishing of group cohomology
cites work: Reconstruction of a variety from the derived category and groups of autoequivalences
cites work: Q4143433
cites work: Symmetry in the vanishing of Ext over Gorenstein rings
cites work: Nonvanishing cohomology and classes of Gorenstein rings
cites work: Q3378989
cites work: Cohomology of tails, Tate-Vogel cohomology, and noncommutative Serre duality over Koszul quiver algebras.
cites work: Serre's vanishing conjecture for Ext-groups.
cites work: Rationality of the Poincaré series for Koszul algebras.
cites work: Riemann-Roch like theorem for triangulated categories
cites work: Co-Point Modules Over Frobenius Koszul Algebras
cites work: Bézout's theorem for non-commutative projective spaces
cites work: LOCAL LIMITATIONS OF THE Ext FUNCTOR DO NOT EXIST
cites work: Q4875159
cites work: Dualizing complexes over noncommutative graded algebras
cites work: Serre duality for noncommutative projective schemes
DOI: 10.1016/J.JALGEBRA.2005.07.009
Recommended article: Vanishing of self-extensions over symmetric algebras.
Recommended article: Symmetry in the vanishing of Ext over Gorenstein rings
Recommended article: A note on symmetry in the vanishing of Ext
Recommended article: The vanishing of self-extensions over \(n\)-symmetric algebras of quasitilted type.
Recommended article: Q5470670
Recommended article: Symmetric algebras stably equivalent to the trivial extensions of tubular algebras
Recommended article: Symmetric Hochschild extension algebras
Recommended article: SYMMETRY IN ALGEBRAIC ALGEBRAS WITH POSITIVE INVOLUTION
Recommended article: Generalized symmetrization in enveloping algebras
Recommended article: Q4694096
description: scientific article; zbMATH DE number 5143560