Tame actions of group schemes: Integrals and slices
type: Item
instance of: scholarly article
title: Tame actions of group schemes: Integrals and slices
zbMATH Open document ID: 0907.14021
DOI: 10.1215/S0012-7094-96-08212-5
author: Ted Chinburg
author: Boas Erez
author: Georgios Pappas
author: Martin J. Taylor
published in: Duke Mathematical Journal
publication date: 1998-06-24T00:00:00Z
review text: Let \(S= \text{Spec} (R)\) be an affine scheme, let \(G= \text{Spec} (A)\) be an affine group scheme and \(X= \text{Spec} (B)\) be an affine \(S\)-scheme. An action \((X,G)\) of \(G\) on \(X\) corresponding to a coaction \(B \to B \otimes_R A\) is tame if there is an \(A\)-comodule map \(\alpha: A \to B\) such that \(\alpha(1_A)= 1_B\), i. e. a total integral in the sense of \textit{Y. Doi} [Commun. Algebra 13, 2137-2159 (1985; Zbl 0576.16004)]. The main part of this groundbreaking paper is devoted to three results. 1. Let \(I\) be the module of integrals of \(A^D\), the dual of \(A\), and \(B^A\) the ring of (co)invariants of \(B\). If \(A\) and \(B\) are locally free over \(R\), then the action \((X,G)\) is tame iff \(IB= B^A\), that is, \(B\) is a tame \(A^D\)-module algebra in the sense of \textit{L. N. Childs} and \textit{S. Hurley} [Trans. Am. Math. Soc. 298, 763-778 (1986; Zbl 0609.16005)]. Thus for \(B, R\) rings of integers of a Galois extension \(L/K\) of number fields with Galois group \(\Gamma\) and \(A= R\Gamma^D\), \((G, X)\) is tame iff the trace map \(B \to R\) is surjective, iff \(L/K\) is tamely ramified. Thus the authors' notion of tame action greatly extends the classical number-theoretic notion of tameness. 2. A tame action \((X, G)\) always admits a universal quotient in the category of schemes, and that quotient is affine. As an example shows, quotients of actions by finite groups are not always universal. 3. If \(R\) is Noetherian and \(G\) is commutative, finite and flat, then a tame action \((X, G)\) admits flat slices. That is, at each \(y \in \text{Spec} (X/G)\) there is a flat morphism \(Y' \to Y\) containing \(y\) in its image and a closed subgroup \(H\) of \(G\) stabilizing some point \(x\) of \(X\) over \(y\), and a \(Y'\)-scheme \(Z\) with an \(H\)-action, so that \(Y'= Z/H_{(Y')}\), \(H_{(Y')}\) is diagonalizable, and \((X_{(Y')}, G_{(Y')})\) is induced from \((Z, H_{(Y')})\). In fact, tame actions, with the same hypotheses on \(R\)…
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14L24
Mathematics Subject Classification ID: 19L47
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 55S91
zbMATH DE Number: 908690
zbMATH Keywords: universal quotient
zbMATH Keywords: equivariant Euler characteristic
zbMATH Keywords: class invariant map
zbMATH Keywords: slice theorem
zbMATH Keywords: divisor with normal crossings
zbMATH Keywords: induced action
zbMATH Keywords: total integral
zbMATH Keywords: tame actions of group schemes
reviewed by: Lindsay N. Childs
MaRDI profile type: Publication
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Tame actions of group schemes: Integrals and slices (full work available at URL)
full work available at URL: s0012-7094-96-08212-5

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OpenAlex ID: W1980585661

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