Abelian class groups of reductive group schemes
type: Item
instance of: scholarly article
title: Abelian class groups of reductive group schemes
zbMATH Open document ID: 1278.14064
author: Cristian D. Gonzales-Aviles
published in: Israel Journal of Mathematics
publication date: 2013-11-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/1108.3264
review text: Let \(K\) be a number field or a function field in one variable over a finite field \(k\). Let \(S\) be a nonempty open subscheme of \(\mathrm{Spec}(\mathcal{O}_K)\) when \(K\) is a number field or a nonempty open affine subscheme of the smooth projective irreducible curve over \(k\) with function field \(K\) when \(K\) is a function field. For each open subscheme \(U\) of \(S\), let \(A_S(U)\) be the ring of \(U\)-integral adeles of \(S\) and let \(A_S = \varinjlim_U A_S(U)\). Suppose that \(G\) is an affine group scheme of finite type over \(S\) with smooth generic fiber, consider the class set of \(G\) \[ C(G) = G(A_S(S)) \backslash G(A_S) / G(K). \] In general, \(C(G)\) is just a set and carries no additioanl structure. In this paper, the author uses the idea of ``abelian group approximation'' as in the theory of quadratic forms to study \(C(G)\) when \(G\) is a reductive \(S\)-group scheme. Let \(S_{\mathrm{fl}}\) be the small fppf site over \(S\) and \(H^i_{\mathrm{ab}}(S_{\mathrm{fl}},G)\) be the abelian cohomology groups of \(G\), define \[ C_{\mathrm{ab}}(G) = \mathrm{Ker}(H^1_{\mathrm{ab}}(S_{\mathrm{fl}},G) \to H^1_{\mathrm{ab}}(K_{\mathrm{fl}},G)). \] \vskip 0.1in The first main result of the paper is Theorem 1: Assume that \(K\) has no primes. Then there exist a natural right action of \(H_{\mathrm{ab}}^0(S_{\mathrm{fl}},G)\) on \(H^1(S_{\mathrm{\'{e}t}},\tilde{G})\) where \(\tilde{G}\) is the simply connected central cover of the derived group \(G\), and a canonical exact sequence of pointed sets \[ 1 \to H^1(S_{\text{ét}},\tilde{G})/H_{\mathrm{ab}}^0(S_{\mathrm{fl}},G) \to C(G) \to C_{\mathrm{ab}}(G) \to 1, \] where the first nontrivial map is injective. When \(K\) has no real primes and the simply-connected central cover of \(G_K\) satisfies the strong approximation property with respect to the set of primes of \(K\) which do not correspond to a closed point of \(S\), there exists a canonical isomorphism between \(C(G)\) and \(C_{\mathrm{ab}}(G)\). T…
reviewed by: Xiao Xiao
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 14F20
zbMATH DE Number: 6221771
zbMATH Keywords: reductive group scheme
zbMATH Keywords: abelian class group
MaRDI profile type: Publication
OpenAlex ID: W2022023853
arXiv ID: 1108.3264
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DOI: 10.1007/S11856-012-0147-4
description: scientific article; zbMATH DE number 6221771