Commutative algebraic groups up to isogeny

Michel Brion

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arXiv · 1602.00222

Commutative algebraic groups up to isogeny

Michel Brion

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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.

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BibTeX RIS

Michel Brion. 2016-01-31. Commutative algebraic groups up to isogeny. https://arxiv.org/abs/1602.00222

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