Modified proof of a local analogue of the Grothendieck conjecture

Victor Abrashkin

doi:10.5802/jtnb.703

Victor Abrashkin ¹

¹ Math Dept of Durham University Sci Laboratories, South Road DH7 7QR Durham, UK

Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 1-50.

Abstract (VO)
Résumé

A local analogue of the Grothendieck Conjecture is an equivalence between the category of complete discrete valuation fields $K$ with finite residue fields of characteristic $p \neq 0$ and the category of absolute Galois groups of fields $K$ together with their ramification filtrations. The case of characteristic 0 fields $K$ was studied by Mochizuki several years ago. Then the author of this paper proved it by a different method in the case $p > 2$ (but with no restrictions on the characteristic of $K$ ). In this paper we suggest a modified approach: it covers the case $p = 2$ , contains considerable technical simplifications and replaces the Galois group of $K$ by its maximal pro- $p$ -quotient. Special attention is paid to the procedure of recovering field isomorphisms coming from isomorphisms of Galois groups, which are compatible with the corresponding ramification filtrations.

L’analogue local de la conjecture de Grothendieck peut être formulé comme une équivalence entre la catégorie des corps $K$ complets pour une valuation discrete à corps résiduel fini de caractéristique $p \neq 0$ et la catégorie des groupes de Galois absolus des corps $K$ munis de la filtration de ramification. Le cas des corps de caractéristique $0$ a été étudié par Mochizuki il y a quelques années. Ensuite, l’auteur de cet article a établi, par une méthode différente l’analogue de la conjecture de Grothendieck dans le cas $p > 2$ (mais $K$ de caractéristique quelconque). Nous proposons ici une modification de cette approche qui inclut le cas $p = 2$ dans la preuve, contient des simplifications considérables et remplace le groupe de Galois par son pro- $p$ -quotient maximal. Une attention particulière est accordée au procédé de la reconstruction de l’isomorphisme de corps à partir d’un isomorphisme de groupe de Galois compatible avec les filtrations de ramification correspondantes.

MR Zbl

DOI : 10.5802/jtnb.703

Affiliations des auteurs :

Victor Abrashkin ¹

¹ Math Dept of Durham University Sci Laboratories, South Road DH7 7QR Durham, UK

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Victor Abrashkin. Modified proof of a local analogue of the Grothendieck conjecture. Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 1-50. doi : 10.5802/jtnb.703. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.703/

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Cité par 9 documents. Sources : Crossref, zbMATH