Commutative algebraic groups up to isogeny

TL;DR

This paper studies the structure of the category of commutative algebraic groups up to isogeny, revealing its homological dimension and characterizing its projective and injective objects, especially in positive characteristic.

Contribution

It establishes that the quotient category of commutative algebraic groups by finite groups has homological dimension one and provides structural insights, simplifying the understanding in positive characteristic.

Findings

Homological dimension of the quotient category is 1.

Characterization of projective and injective objects in the quotient category.

Simplified structure results in positive characteristic.

Abstract

Consider the abelian category $\mathcal{C}_k$ of commutative group schemes of finite type over a field $k$. By results of Serre and Oort, $\mathcal{C}_k$ has homological dimension $1$ (resp. $2$) if $k$ is algebraically closed of characteristic $0$ (resp. positive). In this article, we explore the abelian category of commutative algebraic groups up to isogeny, defined as the quotient of $\mathcal{C}_k$ by the full subcategory $\mathcal{F}_k$ of finite $k$-group schemes. We show that $\mathcal{C}_k/\mathcal{F}_k$ has homological dimension $1$, and we determine its projective or injective objects. We also obtain structure results for $\mathcal{C}_k/\mathcal{F}_k$, which take a simpler form in positive characteristics.