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MODIFIED PROOF OF A LOCAL ANALOGUE OF THE GROTHENDIECK CONJECTURE
 

Summary: MODIFIED PROOF OF A LOCAL ANALOGUE
OF THE GROTHENDIECK CONJECTURE
Victor Abrashkin
Abstract. A local analogue of the Grothendieck Conjecture is an equivalence of
the category of complete discrete valuation fields K with finite residue fields of char-
acteristic p = 0 and the category of absolute Galois groups of fields K together with
their ramification filtrations. The case of characteristic 0 fields K was considered by
Mochizuki several years ago. Then the author proved it by different method if p > 2
(but char K = 0 or p). This paper represents 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 corresponding ramification filtrations.
R´esum´e. Un analogue local de la conjecture de Grothendieck est une ´equivalence
entre la cat´egorie des corps K complets pour une valuation discr`ete `a corps r´esiduels
finis de caract´eristique p = 0, et la cat´egorie des groupes galoisiens absolus de corps
K munis de la filtration de ramification. Le cas des corps de caract´eristique 0 a ´et´e
consid´er´e par Mochizuki il y a quelques ann´ees. Par la suite, le pr´esent auteur a
demontr´e l'´equivalence par une m´ethode diff´erente si p > 2 (mais char K = 0 or p).
Dans l'article pr´esent´e ici, une modification de l'approche pr´ec´edente est envisag´ee:

  

Source: Abrashkin, Victor - Department of Mathematical Sciences, University of Durham

 

Collections: Mathematics