Commutative algebraic groups up to isogeny
type: Item
instance of: scholarly article
title: Commutative algebraic groups up to isogeny
zbMATH Open document ID: 1378.14040
author: Michel Brion
published in: Documenta Mathematica
publication date: 2017-05-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1602.00222
full work available at URL: 20.html
review text: Let \(k\) be an arbitrary field and let \(\mathcal{G}_k\) be the category with objects the commutative group schemes of finite type over \(k\) and with morphisms the homomorphisms of \(k\)-group schemes. By a result of Grothendieck \(\mathcal{G}_k\) is an abelian category. Moreover, by results of Serre and Oort, \(\mathcal{G}_k\) has homological dimension \(1\) (resp. \(2\)) if \(k\) is algebraically closed of characteristic \(0\) (resp. positive). However, Milne proved that the homological dimension of \(\mathcal{G}_k\) can be arbitrary large if \(k\) is non-algebraically closed. In this article, the author studies the quotient category \(\mathcal{G}_k/\mathcal{F}_k\), where \(\mathcal{F}_k\) is the Serre subcategory of commutative finite group schemes. Then \(\mathcal{G}_k/\mathcal{F}_k\) is obtained from \(\mathcal{G}_k\) by inverting all isogenies (i.e. all morphisms with finite kernels and cokernels). The full subcategory \({\mathcal{C}_k}\) of \(\mathcal{G}_k/\mathcal{F}_k\) with objects the smooth connected commutative group schemes of finite type over \(k\) turns out to be equivalent to \(\mathcal{G}_k/\mathcal{F}_k\) (Lemma 3.1). Therefore the author works mainly with \({\mathcal{C}}_k\) instead of \(\mathcal{G}_k/\mathcal{F}_k\) as the morphisms in \({\mathcal{C}}_k\) admit a simpler description. Let \({\mathcal{A}}_k\) / \({\mathcal{T}}_k\) / \({\mathcal{U}}_k\) / \({\mathcal{L}}_k\) / \({\mathcal{S}}_k\) denote the full subcategories of \({\mathcal{C}}_k\) with objects abelian varieties / tori / unipotent algebraic groups / linear algebraic groups / semi-abelian varieties. The main results of the article are the following: {\parindent=6mm \begin{itemize}\item[{\(\bullet\)}] The homological dimension of \({\mathcal{C}}_k\) is \(1\) (\S 4.3). \item[{\(\bullet\)}] The base change under any purely inseparable field extension \(k'\) of \(k\) yields an equivalence of categories \({\mathcal{C}}_k \to {\mathcal{C}}_{k'}\) (Theorem 3.11). \item[{\(\bullet\)}] The…
reviewed by: Ronan Terpereau
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 14K02
Mathematics Subject Classification ID: 18E35
Mathematics Subject Classification ID: 20G07
zbMATH DE Number: 6713329
zbMATH Keywords: commutative algebraic groups
zbMATH Keywords: isogeny category
zbMATH Keywords: homological dimension
MaRDI profile type: Publication
arXiv ID: 1602.00222
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description: scientific article