An Isaac Newton Institute Programme

Model Theory and Applications to Algebra and Analysis

Galois cohomology for surgical fields

14th April 2005

Author: Martin-Pizarro, Amador (Berlin)

Abstract

In 1995, Pillay and Poizat introduced the notion of a surgical structure (translated from the french chirurgicale), a structure such that to each definable set there was an element of a poset attached to it (denoted h by its dimension) in such a way that if there was a partition of a set X into finite pieces and each of these pieces could be sent via some finite-to-one map to another set Y, then dim(X)is bounded above by dim(Y). Moreover, an accumulation character was required on this assignment, i.e, given a definable equivalence relation on a definable set, there were only finitely many classes of dimension the dimension of the ambient set.

Under these weak assumptions, they proved that a field interpreted in such a structure is perfect and has small absolute Galois group.

I will show how these techniques can be extended to consider certain Galois cohomological groups relative to the field, and discuss their geometrical meaning.

The talk is intended to be self contained and for a general audience in Model Theory.