Abstract
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.
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