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.