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

Abstract.

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 $W \mathop\otimes\limits_{\mathcal{O}} V$ is also an indecomposable ${\mathcal{O}}$G-module. We relate the Green invariants of the ${\mathcal{O}}$G-modules V and $W \mathop\otimes\limits_{\mathcal{O}} V$. (This situation arises under important Morita equivalences.)