Abelian étale coverings currents on graphs and semistable reduction of curves
type: Item
instance of: scholarly article
title: Abelian étale coverings currents on graphs and semistable reduction of curves
zbMATH Open document ID: 0869.14010
DOI: 10.1007/BF02567516
author: Mohamed Saïdi
published in: Manuscripta Mathematica
publication date: 1996-06-05T00:00:00Z
full work available at URL: https://eudml.org/doc/156161
review text: Let \(R\) be a discrete valuation ring which is strictly henselian. Let \(\Xi\) be a semi-stable curve over \(R\) with generic fiber \(\Xi_\eta\) smooth and geometrically connected. The main result is the following (theorem 4.3 (a)): let \(\overline K\), resp. \(k\), be an algebraic closure of \(K=\text{Quot}(R)\), resp. the residue field of \(R\), then, for a positive integer \(n\) prime to the characteristic of \(k\), one has the exact sequence \[ 0\to H^1_{et} (\Xi\times_Rk, \mathbb{Z}/n \mathbb{Z}) \to H^1_{et} (\Xi \times_R \overline K, \mathbb{Z}/n \mathbb{Z})\to {\mathcal C} (\Gamma, \mathbb{Z}/n \mathbb{Z}) \to 0 \] where \({\mathcal C} (\Gamma, \mathbb{Z}/n \mathbb{Z})\) are the currents (also called harmonic cochains or harmonic cocycles) defined on the intersection graph \(\Gamma\) of \(\Xi \times_Rk\) and with values in \(\mathbb{Z}/n \mathbb{Z}\). The exactness of this sequence is implies, among other properties, by the fact that every \(\mu_n\)-torsor over \(\Xi\times_R \overline K\) is defined over \(\text{Quot} (R')\), where \(R'\) is a finite extension of \(R\) (theorem 4.2 (b)). This result has been announced by \textit{M. Saïdi} in C. R. Acad. Sci., Paris, Sér. I 316, No. 12, 1299-1302 (1993; Zbl 0788.14023). As applications (of results and of arguments of the proofs), the author gives descriptions of semi-stable models for cyclic étale coverings of \(\Xi_\eta\) of degrees prime to the characteristic of \(k\).
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 866134
zbMATH Keywords: discrete valuation ring
zbMATH Keywords: currents
zbMATH Keywords: cyclic étale coverings
reviewed by: Marc Reversat
MaRDI profile type: Publication
cites work: Stable reduction and uniformization of abelian varieties. I
cites work: Néron Models
cites work: The irreducibility of the space of curves of a given genus
cites work: Uniformisation des variétés abéliennes
cites work: Q2551490
cites work: Vanishing cycles, ramification of valuations, and class field theory. (With the collaboration of Takeshi Saito.)
cites work: Q3136243
cites work: Q4305700
cites work: On the deformation of Artin-Schreier to Kummer
cites work: Etale coverings of a Mumford curve
cites work: Discrete groups, Mumford curves and theta functions
Recommended article: Q3136243
Recommended article: Q4388173
Recommended article: Q4323689
Recommended article: Q4505327
Recommended article: Maximal wild monodromy in unequal characteristic
Recommended article: Étale cohomology of a DM curve-stack with coefficients in \(\mathbb G _m\)
Recommended article: Q4703012
Recommended article: On the maximal unramified quotients of \(p\)-adic étale cohomology groups and logarithmic Hodge--Witt sheaves
Recommended article: \(p\)-adic étale cohomology
Recommended article: Q4630741
description: scientific article; zbMATH DE number 866134