Geodesic
Commutative Algebraic Groups up to Isogeny
Documenta mathematica, Tome 22 (2017), pp. 679-725.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Consider the abelian category $\Cal{C}_k$ of commutative group schemes of finite type over a field $k$. By results of Serre and Oort, $\Cal{C}_k$ 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 $\Cal{C}_k$ by the full subcategory $\Cal{F}_k$ of finite $k$-group schemes. We show that $\Cal{C}_k/\Cal{F}_k$ has homological dimension 1, and we determine its projective or injective objects. We also obtain structure results for $\Cal{C}_k/\Cal{F}_k$, which take a simpler form in positive characteristics.
Classification : 14L15, 14K02, 18E35, 20G07
Keywords: commutative algebraic groups, isogeny category, homological dimension 
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     author = {Brion, Michel},
     title = {Commutative {Algebraic} {Groups} up to {Isogeny}},
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     volume = {22},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2017__22__a33/}
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Brion, Michel. Commutative Algebraic Groups up to Isogeny. Documenta mathematica, Tome 22 (2017), pp. 679-725. http://geodesic.mathdoc.fr/item/DOCMA_2017__22__a33/