Tame actions of group schemes: Integrals and slices
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 \(R\) and \(G\), can be characterized as actions with universal quotients such that after a faithfully flat base change the action is induced from an action of a diagonalizable group. This result extends the étale slice theorem of \textit{D. Luna} [Bull. Soc. Math. Fr., Suppl., Mém. 33, 81-105 (1973; Zbl 0286.14014)], and relates the authors' notion of tameness to that of \textit{A. Grothendieck} and \textit{J. P. Murre} [``The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme'', Lect. Notes Math. 208 (1971; Zbl 0216.33001)]: Locally a tame action looks like a cover which is tame with respect to a divisor with normal crossings. The paper concludes by obtaining an equivariant Euler characteristic homomorphism for sheaves equipped with a tame action by a finite flat group, generalizing papers by \textit{T. Chinburg} [Ann. Math., II. Ser. 139, No. 2, 443-490 (1994; Zbl 0828.14007)] and by \textit{T. Chinburg} and \textit{B. Erez} [Astérisque 209, 179-194 (1992; Zbl 0796.11051)] and applying that Euler characteristic map to reinterpret the class invariant homomorphism from the group of \(K\)-rational points of an elliptic curve \(E\) to the class group of \(A^D\): Here \(K\) is a number field with ring of integers \(R\), \(a \in \text{End}(E)\) and \(G= \text{Spec}(A)\) is the \(R\)-group scheme of \(a\)-torsion points on the Néron model of \(E\). For earlier descriptions of this map see \textit{P. Cassou-Noguès} and \textit{M. J. Taylor} [J. Théor. Nombres Bordx. 7, 307-331 (1995; Zbl 0852.11066)], \textit{M. J. Taylor} in: Group rings and class groups, Notes Talks DMV-Semin., Günzburg 1990, DMV Semin. 18, 153-210 (1992; Zbl 0811.11068)] and \textit{A. Agboola} [J. Théor. Nombres Bordx. 6, No. 2, 273-280 (1994; Zbl 0833.11055)]. These results are the first in a general program to study invariants attached to tame actions of group schemes.
scientific article; zbMATH DE number 908690
scientific article