Five-term exact sequence for Kac cohomology

TL;DR

This paper develops a method using relative group cohomology to compute Kac cohomology for finite group pairs, providing a practical tool for analyzing abelian extensions of finite-dimensional Hopf algebras.

Contribution

It introduces a new approach to compute Kac cohomology via relative cohomology and resolutions, including a spectral sequence and five-term exact sequence.

Findings

Derived the first two pages of a spectral sequence converging to Kac cohomology

Established a five-term exact sequence for Kac cohomology

Demonstrated the method's usefulness through examples

Abstract

We use relative group cohomologies to compute the Kac cohomology of matched pairs of finite groups. This cohomology naturally appears in the theory of abelian extensions of finite dimensional Hopf algebras. We prove that Kac cohomology can be computed using relative cohomology and relatively projective resolutions. This allows us to use other resolutions, besides the bar resolution, for computations. We compute, in terms of relative cohomology, the first two pages of a spectral sequence which converges to the Kac cohomology and its associated five-term exact sequence. Through several examples, we show the usefulness of the five-term exact sequence in computing groups of abelian extensions.