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Let  G  be the Klein four-group and letbe an arbitrary field of characteristic 2. A classification of indecomposable-modules is known. We calculate the relative cohomology groupsfor every indecomposable-module  N , where  χ  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 inare trivial.

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