Modified proof of a local analogue of the Grothendieck conjecture

Details Modified proof of a local analogue of the Grothendieck conjecture Modified proof of a local analogue of the Grothendieck conjecture

2009-07-17

arxiv.org/abs/0907.3035v1

Mathematics

Number Theory

42 pages; see also http://www.maths.dur.ac.uk/~dma0va/

Scientific paper

A local analogue of the Grothendieck Conjecture is an equivalence of the category of complete discrete valuation fields $K$ with finite residue fields of characteristic $p\ne 0$ 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 $\operatorname{char}K=0$ 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 the corresponding ramification filtrations.

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