Class invariants: functional properties and application to kernel studies
type: Item
instance of: scholarly article
title: Class invariants: functional properties and application to kernel studies
zbMATH Open document ID: 1200.11084
author: Jean Gillibert
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2008-07-21T00:00:00Z
full work available at URL: https://eudml.org/doc/45201
full work available at URL: JTNB 2007 19 2 415 0.pdf
review text: The motivation of the present study goes back to \(E.\) Noether's well known ``normal basis'' theorem on the projectivity of the \({\mathcal O}_K[\Gamma]\)-module \({\mathcal O}_F\) for a tamely ramified Galois extension \(F/K\) of number fields, \(\Gamma =\;\text{Gal}\;(F/K).\) In the wildly ramified case, it is more convenient to replace the group algebra \({\mathcal O}_k [\Gamma]\) by a certain Hopf algebra \({\mathcal H}.\) If \(\Gamma\) is commutative, \({\mathcal H}\) is an \({\mathcal O}_k\)-Hopf order in \(K [\Gamma],\) and the original Noether problem boils down to studying the kernel of a certain canonical homomorphism \(\pi_0\;:\;H^1 (\text{Spec}\, {\mathcal O}_k, \;\text{Spec}\, {\mathcal H}^\ast) \to C\!\ell ({\mathcal H}), \, {\mathcal N} \to ({\mathcal N}) \ldotp ({\mathcal H}^\ast)^{-1}\) where \({\mathcal H}^\ast\) is the dual Hopf algebra of \({\mathcal H}.\) In this setting, \({\mathcal N}\) can be viewed as an \({\mathcal O}_K\)-order in \(F\) which is an \({\mathcal H}\)-Galois extension, and \({\mathcal N}\) ``admits a normal basis'' iff \(\pi_0 ({\mathcal N}) = 0,\) iff \({\mathcal N} \simeq {\mathcal H}^\ast.\) \noindent This morphism \(\pi_0\) has been generalized by Whitehead to a morphism \(\pi\) which ``measures'' the Galois structure of torsors : for a noetherian scheme \(S\) and a commutative \(S\)-group scheme \(G,\) finite and flat, Whitehead's morphism is a certain canonical \(\pi\;:\;H^1 (S, G) \to \;\text{Pic} (G^D),\) \(G^D\) being the Cartier dual of \(G,\) and a \(G\)-torsor ``admits a normal basis'' iff its image by \(\pi\) is null. To go further, consider an exact sequence \(0 \to {\mathcal N}(f) \to F_1 \displaystyle{\buildrel f \over\rightarrow} F_2 \to 0,\) where \(F_1\) and \(F_2\) are two abelian sheaves over \(S\) (for the \(fppf\) topology) such that the kernel \(N(f)\) can be represented by a finite flat \(S\)-group scheme. One can construct a morphism \(\psi_f\;:\;F_2(S) \displaystyle{\buildrel \delta \over\rightarrow…
reviewed by: Thong Nguyen Quang Do
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 14L15
zbMATH DE Number: 5302783
zbMATH Keywords: class-invariant homomorphism
MaRDI profile type: Publication
OpenAlex ID: W2963641460
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DOI: 10.5802/JTNB.595
description: scientific article; zbMATH DE number 5302783