On the Semi-Abelianness of Affine Group Schemes

TL;DR

This paper demonstrates that the category of affine group schemes over a field is semi-abelian by establishing coregularity and coexactness through Hopf algebra properties.

Contribution

It proves the semi-abelian nature of affine group schemes by linking Hopf algebra structures with categorical properties.

Findings

Category of commutative Hopf algebras is co-semi-abelian

Affine group schemes form a semi-abelian category

Coregularity and coexactness are established via Hopf algebra correspondences

Abstract

We prove that the category of commutative Hopf algebras over a field $k$ is co-semi-abelian. Consequently, the category of affine group $k$-schemes is semi-abelian. We establish coregularity by identifying the orthogonal factorization system of surjections and faithfully flat injections, and we deduce coexactness from Takeuchi's correspondence between normal Hopf ideals and Hopf subalgebras of commutative Hopf $k$-algebras.