Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Semi-Abelianness of Affine Group Schemes

David Forsman

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.

On the Semi-Abelianness of Affine Group Schemes

Abstract

We prove that the category of commutative Hopf algebras over a field is co-semi-abelian. Consequently, the category of affine group -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 -algebras.
Paper Structure (6 sections, 7 theorems, 1 equation)

This paper contains 6 sections, 7 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of Commutative Hopf Algebras
  4. Coregularity
  5. Coexactness
  6. Conclusion

Key Result

Lemma 2.2

Let $C$ be a pointed, regular, and protomodular category. Then the following are equivalent:

Theorems & Definitions (18)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Theorem 3.3: Faithful Flatness
  • proof
  • Theorem 3.4: Coregularity
  • proof
  • ...and 8 more