Affine group schemes over symmetric monoidal categories
type: Item
instance of: scholarly article
title: Affine group schemes over symmetric monoidal categories
zbMATH Open document ID: 1248.14050
author: Abhishek Banerjee
published in: Pacific Journal of Mathematics
publication date: 2012-05-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/1506.04890
full work available at URL: p02.xhtml
reviewed by: Emre Coskun
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 18D10
zbMATH DE Number: 6029225
zbMATH Keywords: group schemes
zbMATH Keywords: symmetric monoidal categories
MaRDI profile type: Publication
OpenAlex ID: W2092894287
arXiv ID: 1506.04890
DOI: 10.2140/PJM.2012.255.25
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review text: Let \(S\) be a scheme. A group scheme \(G\) over \(S\) is said to have finite order \(r\) over \(S\) if \(G\) can be written as \(\text{Spec}(A)\), where \(A\) is a sheaf of \(\mathcal{O}_S\)-algebras which is locally free of finite rank \(r\). In [\textit{J. Tate} and \textit{F. Oort}, Ann. Sci. Ec. Norm. Super. (4), No. 3, 1--21 (1970; Zbl 0195.50801)], the authors presented the following result, due to Deligne: A commutative group scheme \(G\) over \(S\) of order \(r\) is annihilated by \(r\). In the paper under review, the author proves a similar result for group schemes over symmetric monoidal categories. Namely, let \((C, \otimes, 1)\) be an abelian, closed, \(\mathbb{C}\)-linear symmetric monoidal category and let \(G\) be an affine commutative group scheme over \(C\) free and of finite rank \(r\). Then \(G\) is annihilated by \(r\). Here, an affine group scheme over \(C\) is a covariant functor from the category of algebras in \(C\) to the category of groups, with the usual compatibility conditions.NEWLINENEWLINEIt must be said here that the subject of relative algebraic geometry over a symmetric monoidal category is has been developed in the papers [\textit{P. Deligne}, in: The Grothendieck Festschrift, Collect. Artic. in honor of the 60th birthday of A. Grothendieck. Vol II, Prog. Math. No. 87, 111--195 (1990; Zbl 0727.14010); \textit{M. Hakim}, Topos anneles et schemas relatifs. Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 64. Berlin-Heidelberg-New York: Springer-Verlag. (1972; Zbl 0246.14004); \textit{B. Toen} and \textit{M. Vaquie}, J. K-Theory 3, No. 3, 437--500 (2009; Zbl 1177.14022)]. The current paper contributes another result to this theory.
description: scientific article; zbMATH DE number 6029225