Symmetry in the vanishing of Ext over stably symmetric algebras
Symmetry in the vanishing of Ext over stably symmetric algebras
复制标题
DOI：
10.1016/j.jalgebra.2005.07.009
复制
发表时间：
2007-04
影响因子：
0.9
通讯作者：
I. Mori
中科院分区：
文献类型：
--
作者：
I. Mori
域k上的Frobenius代数称为对称的，如果Nakayama自同构是内自同构。稳定对称代数是对称k-代数的推广。本文研究了这类代数R的Ext消失的对称性，即对所有的n-生成R-模M和N，ExtRi（M，N）=0当且仅当ExtRi（N，M）=0.利用Serre对偶证明了一类Noether稳定对称Gorenstein代数，如有限群的群代数和n为奇数时的外代数Λ（kn），具有这种对称性.利用Koszul对偶证明了每个外代数Λ（kn），无论n是偶数还是奇数，都具有分次模的对称性.
A Frobenius algebra over a field k is called symmetric if the Nakayama automorphism is an inner automorphism. A stably symmetric algebra is defined to be a generalization of a symmetric k-algebra. In this paper we will study symmetry in the vanishing of Ext for such algebras R, namely, for all finitely generated R-modules M and N, ExtRi(M,N)=0 for all i≫0 if and only if ExtRi(N,M)=0 for all i≫0. We show that a certain class of noetherian stably symmetric Gorenstein algebras, such as the group algebra of a finite group and the exterior algebra Λ(kn) when n is odd, have this symmetry using Serre duality. We also show that every exterior algebra Λ(kn), whether n is even or odd, has this symmetry for graded modules using Koszul duality.