Homological dimension of isogeny categories of commutative algebraic groups

TL;DR

This paper calculates the homological dimension of isogeny categories of commutative algebraic groups over a field, relating it to the field's cohomological dimension at specific primes, thus extending previous results with a new approach.

Contribution

It provides a new method to determine the homological dimension of isogeny categories, generalizing earlier work by Serre, Oort, and Milne.

Findings

Homological dimension expressed in terms of field's cohomological dimension.

Generalization of previous results on algebraic groups.

Alternative approach to existing theorems.

Abstract

We determine the homological dimension of various isogeny categories of commutative algebraic groups over a field $k$, in terms of the cohomological dimension of $k$ at certain primes. This generalizes results of Serre, Oort and Milne, by an alternative approach.