Finite group torsors for the qfh topology
type: Item
instance of: scholarly article
title: Finite group torsors for the qfh topology
zbMATH Open document ID: 1050.14012
published in: Mathematische Zeitschrift
publication date: 2003-09-21T00:00:00Z
review text: Let \(S\) be a Noetherian scheme. The qfh topology for \(S\) is generated by the finite families of finite morphisms such that their sum is a universal topological epimorphism. This paper compares the qhf and étale torsors over \(S\) for certain group schemes. \textit{V. Voevodsky} [Sel. Math., New Ser. 2, No. 1, 111--153 (1996; Zbl 0871.14016)] showed that for each \(n\geq 0\) there are isomorphisms \(H_{et}^{n}( S,A) \cong H_{qfh}^{n}( S,A) \) for abelian schemes \(A\) which are locally constant for the étale topology. In the work under review, a non-abelian version of this result is obtained in the case \(n=1\). Specifically, it is shown that for \(G\) a finite étale group scheme over \(S\) that the canonical map \(H_{et}^{1}( S,G) \to H_{qfh}^{1}( S,G) \) is a bijection. The main result is equivalent to the statement that the qfh stack completion map \( BG\to BSt_{qfh}G\) is a local weak equivalence in the étale topology. First it is established that finite constant simplicial sheaves with finitely many simplices of each degree satisfy a qfh descent condition on strict local Hensel rings and that the inverse image of a qfh sheaf over such a ring can be computed using filtered colimits. By restricting to the case where \(G\) is constant and finite (which can be done since the problem is a local one in the étale topology) the author quickly proves the main result.
reviewed by: Alan Koch
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14F43
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 14L15
zbMATH DE Number: 1983501
zbMATH Keywords: finite group schemes
zbMATH Keywords: torsors
zbMATH Keywords: qfh topology
zbMATH Keywords: étale topology
author: J. F. Jardine
MaRDI profile type: Publication
full work available at URL: s00209-003-0526-y
OpenAlex ID: W2064869073
DOI: 10.1007/S00209-003-0526-Y
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description: scientific article; zbMATH DE number 1983501