Commutative Algebraic Groups up to Isogeny

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.