Almost-split sequences and group rings
Almost split sequences are discussed in the category of RG-lattices, i.e., finitely generated R-free RG-modules, where R is either a field of characteristic \(p>0\) or a complete discrete valuation ring of characteristic zero with residue characteristic p, and G is a finite group. In particular, applications to tensor products of such modules are given. Some information is developed about splitting trace lattices, which are those lattices X such that the RG-lattice \(End_ R(X)\) has trivial direct summand. Using these, some facts are obtained about direct summands of tensor products. In the case where \(R=K\) is a field, let P be the projective cover of the trivial simple module, the almost split sequence for P/soc P has the form \[ 0\to \Omega (K)\to P\oplus E\to P/soc P\to 0. \] The authors present a new proof of the criterion of \textit{P. Webb} [Math. Z. 179, 97-121 (1982; Zbl 0479.20008)] for E to be indecomposable. Some results on indecomposable KG-modules are also given, where \(p=2\) and G has dihedral Sylow 2-subgroups.
scientific article; zbMATH DE number 3955191