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- W2045403733 abstract "be a minimal projective resolution. M is called bounded if the lengths of the kernels Ker di have a common upper bound, and M is called periodic if some Ker di is isomorphic to M. In [ 1, p. 7761, Alperin posed the following question: Is every bounded KG-module periodic? Alperin answered this question in the affirmative in case K is algebraic over its prime field [l, Theorem 11. Without any restriction on K, Eisenbud settled Alperin’s question in [6, Theorem 9.21. In Section 1 of the present paper, we look at an arbitrary QF ring R instead of a group algebra, and we give a condition which forces a bounded R-module M to be periodic. Before we will formulate our result, we make some preliminary remarks. Let M and X be modules over a ring R. With the Yoneda product for multiplication, the abelian group ExtX(M, M) = Oi,,, Extk(M, M) becomes a graded ring which we call the extension ring of M, and the abelian group Ext,*(M, X) becomes a graded right ExtR(M, M)-module. We recall some classical examples of extension rings. Let G be a finite group, and let K be a field on which G acts trivially. Then the cohomology ring H*(G, K) is the extension ring Ext,*(K, K). More generally, let M be a finitely generated KG-module. Then the ring Hom,(M, M) is a G-ring, and the cohomology ring H*(G, Hom,(M, M)) (with cup product) is canonically isomorphic to the extension ring ExtX(M, M) (with Yoneda product). These rings have been intensively studied; see Carlson [3] for details and for further references. Other well-known exampes of extension rings arise in the homological algebra of local rings, namely the extension" @default.
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- W2045403733 date "1986-07-01" @default.
- W2045403733 modified "2023-10-18" @default.
- W2045403733 title "Boundedness and periodicity of modules over QF rings" @default.
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- W2045403733 doi "https://doi.org/10.1016/0021-8693(86)90204-8" @default.
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