Five-term exact sequence for Kac cohomology
C\'esar Galindo, Yiby Morales

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.
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