The relative Heller operator and relative cohomology for the Klein 4-group

TL;DR

This paper computes the relative cohomology groups for indecomposable modules over the Klein Four-group with respect to proper subgroups, extending previous work to non-trivial coefficients and analyzing cup product structures.

Contribution

It extends the classification of relative cohomology for the Klein Four-group to include non-trivial coefficients and examines the triviality of cup products in positive degrees.

Findings

Calculated all relative cohomology groups for indecomposable modules

Extended previous cohomology results to non-trivial coefficients

Proved cup products in positive degrees are trivial

Abstract

Let $G$ be the Klein Four-group and let $k$ be an arbitrary field of characteristic 2. A classification of indecomposable $kG$-modules is known. We calculate the relative cohomology groups $H_\{chi}^i(G,N)$ for every indecomposable $kG$-module $N$, where $\{chi}$ is the set of proper subgroups in $G$. This extends work of Pamuk and Yalcin to cohomology with non-trivial coefficients. We also show that all cup products in strictly positive degree in $H_\{chi}^*(G,k)$ are trivial.