On stable categories of group algebras

• Main Author: Poulton, Andrew
• Published: University of Bristol 2014
• Subjects: 512
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.658073

We study the stable category of a group algebra AG over a regular ring A, for a finite group G. We construct a right adjoint to the inclusion of the stable subcategory of A-projective AG-modules into the full stable category. We use this functor to study the stable category of VG-lattices, where V is a complete discrete valuation ring. We focus on HelIer lattices, the kernels of projective covers of torsion OGmodules. If k is the residue field of 0, we show that the Heller lattices of the simple kG-modules generate a dense sub category of the stable category laU-OG of OG-lattices. Turning to more general kG-modules, we show that the stable endomorphism ring of the Heller lattice of a kG-module M is isomorphic to the trivial extension algebra of the stable endomorphism ring of M when 0 is ramified, generalising a result due to S. Kawata. We conclude by discussing the structure of a connected component of the stable AuslanderReiten quiver containing the Heller lattice of an indecomposable kG-module. We also give necessary and sufficient conditions for the middle term of the almost split sequence ending in a virtually irreducible lattice to be indecomposable