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

Commutative algebraic groups up to isogeny