# Seminario di Algebra: “Kummer extensions of rings of formal power series, and perfectoid algebras”

## Lunedì 10 Aprile 2017, ore 14:15 - Aula 2BC30 - Yves André

ARGOMENTI: Seminari

Lunedì 10 Aprile 2017 alle ore 14:15 in Aula 2BC30, Yves André (Directeur de Recherche du CNRS at Paris 6 -Jussieu) terrà un seminario dal titolo “Kummer extensions of rings of formal power series, and perfectoid algebras”.

Abstract
We consider Kummer extensions of $A := Z_p[[T_1, … , T_n]]$, obtained by adjoining the $p^i$th roots of unity and of some element $a$ of $A$, then inverting $p$ and taking the $p$-integral elements. It is very difficult to describe such extensions. They are not always flat, but Hochster’s conjecture predicts that they are pure, i. e. $A$ is a direct summand (as $A$-module) of any such extension. We shall explain how one can prove this conjecture by going to the completed colimit in $i$ and using perfectoid theory via a deformation argument.