Boundedness and periodicity of modules over QF rings
Let KG be the group algebra of a finite group with coefficients in a field K. For a finitely generated KG-module M, let \(...\to P_ 1\to^{d_ 1}P_ 0\to^{d_ 0}M\to 0\) be a minimal projective resolution. M is called bounded if the lengths of the kernels ker \(d_ i\) have a common upper bound, and M is called periodic if some ker \(d_ i\) is isomorphic to M. In case K is algebraic over its prime field then every bounded KG-module is periodic. In this paper an arbitrary QF ring R is considered instead of a group algebra and a condition is given which forces a bounded module \(M_ R\) to be periodic. 1. Let \(M_ R\) be a bounded module over a QF ring R. Let the ring \(Ext_ R^*(M;M)\) be right noetherian, and let the right \(Ext^*_ R(M;M)\)-module \(Ext^*_ R(M;N)\) be noetherian for every simple module \(N_ R\). Then M is periodic. For a QF algebra R the assumptions on M in 1. are sufficient and necessary. In particular we have: 2. For a bounded module \(M_ R\) over a QF algebra R, the following conditions are equivalent: (i) M is periodic; (ii) The ring \(Ext^*_ R(M;M)\) is right noetherian, and the right \(Ext^*_ R(M;M)\)-module \(Ext^*_ R(M;N)\) is noetherian for every simple module \(N_ R\); (iii) For every finitely generated module \(X_ R\), the right \(Ext^*_ R(M;M)\)-module \(Ext^*_ R(M;X)\) is noetherian.
scientific article; zbMATH DE number 3955135