Galois structure of de Rham cohomology of tame covers of schemes
type: Item
instance of: scholarly article
title: Galois structure of de Rham cohomology of tame covers of schemes
zbMATH Open document ID: 0828.14007
DOI: 10.2307/2946586
author: Ted Chinburg
published in: Annals of Mathematics. Second Series
publication date: 1996-01-08T00:00:00Z
review text: The object of this paper is to generalize to tame covers of schemes the theory of the Galois module structure of tamely ramified rings of integers. To motivate this generalization, we recall two theorems of Noether and Taylor, respectively. Let \(N/K\) be a finite Galois extension of number fields with group \(G = \text{Gal} (N/K)\). Let \({\mathfrak O}_N\) be the ring of integers of \(N\). Noether proved that \({\mathfrak O}_N\) is a projective \(\mathbb{Z} [G]\)-module if and only if \(N/K\) is at most tamely ramified: we suppose from now on that is the case. Then \({\mathfrak O}_N\) defines a class \(({\mathfrak O}_N)\) in the Grothendieck group \(K_0 (\mathbb{Z} [G])\) of all finitely generated projective \(\mathbb{Z} [G]\)-modules. The class group \(\text{Cl} (\mathbb{Z} [G])\) of \(\mathbb{Z} [G]\) is defined to be the quotient of \(K_0 (\mathbb{Z} [G])\) by the subgroup generated by the class of \(\mathbb{Z} [G]\). Let \(({\mathfrak O}_N)^{\text{stab}}\) be the image of \(({\mathfrak O}_N)\) in \(\text{Cl} (\mathbb{Z} [G])\). In Invent. Math. 63, 41-79 (1981; Zbl 0469.12003), \textit{M. J. Taylor} proved Fröhlich's conjecture that \(({\mathfrak O}_N)^{\text{stab}}\) is equal to another invariant \(W_{N/K}\) in \(\text{Cl} (\mathbb{Z} [G])\) which Cassou-Noguès had defined by means of the root-numbers of symplectic representations of the Galois group \(G\). To generalize the results above, let \(G\) be an abstract finite group. We consider tame \(G\)-covers \(f : X \mapsto Y\) of schemes of finite type over a Noetherian ring \(A\) with respect to a divisor \(D\) on \(Y\) having normal crossings. Let \(\text{CT} (A[G])\) be the Grothendieck group of all finitely generated \(A[G]\)-modules which are cohomologically trivial as \(\mathbb{Z}[G]\)- modules. The ``forgetful'' homomorphism \(K_0 (\mathbb{Z} [G]) \mapsto \text{CT} (\mathbb{Z} [G])\) is an isomorphism. The image of \(({\mathfrak O}_N)\) under this isomorphism is \(\Psi (X/Y)\). -- Our main results conce…
Mathematics Subject Classification ID: 14E20
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 591417
zbMATH Keywords: tame covers of schemes
zbMATH Keywords: Galois module structure
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2946586
OpenAlex ID: W2078931011
description: scientific article; zbMATH DE number 591417