A Note on the Green Invariants in Finite Group Modular Representative Theory

摘要

As in Finite Group Modular Representation Theory, let ${mathcal{O}}$ be a commutative complete noetherian ring with an algebraically closed residue field k. Let G be a finite group and let N be a normal subgroup of G. First suppose that V is an indecomposable ${mathcal{O}}(G/N)$ -module, so that Inf G G/N (V) is an indecomposable ${mathcal{O}}$ G-module. We relate the Green invariants of V as an ${mathcal{O}}(G/N)$ -module to those of Inf G G/N (V) as an ${mathcal{O}}$ G-module. Secondly, let V and W be indecomposable ${mathcal{O}}$ G-modules. Assume that W is an endo-permutation lattice and that $W mathopotimeslimits_{mathcal{O}} V$ is also an indecomposable ${mathcal{O}}$ G-module. We relate the Green invariants of the ${mathcal{O}}$ G-modules V and $W mathopotimeslimits_{mathcal{O}} V$ . (This situation arises under important Morita equivalences.)