Geodesic
Commutative Algebraic Groups up to Isogeny
Documenta mathematica, Tome 22 (2017), pp. 679-725
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Consider the abelian category Ck​ of commutative group schemes of finite type over a field k. By results of Serre and Oort, Ck​ 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 Ck​ by the full subcategory Fk​ of finite k-group schemes. We show that Ck​/Fk​ has homological dimension 1, and we determine its projective or injective objects. We also obtain structure results for Ck​/Fk​, which take a simpler form in positive characteristics.
DOI : 10.4171/dm/576
Classification : 14K02, 14L15, 18E35, 20G07
Mots-clés : homological dimension, commutative algebraic groups, isogeny category 
@article{10_4171_dm_576,
     author = {Michel Brion},
     title = {Commutative {Algebraic} {Groups} up to {Isogeny}},
     journal = {Documenta mathematica},
     pages = {679--725},
     year = {2017},
     volume = {22},
     doi = {10.4171/dm/576},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/576/}
}
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Michel Brion. Commutative Algebraic Groups up to Isogeny. Documenta mathematica, Tome 22 (2017), pp. 679-725. doi: 10.4171/dm/576

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