Commutative Algebraic Groups up to Isogeny

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.

2010 Mathematics Subject Classification: 14K02, 14L15, 18E35, 20G07.

Keywords and Phrases: commutative algebraic groups, isogeny category, homological dimension.

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