An introduction to the almost purity theorem

Specialeforsvar ved Corvin Paul

Titel: An introduction to the almost purity theorem

Abstract: In the 60's Tate discovered that computing certain Galois cohomology groups of a local field becomes tractable after adjoining lots of $p$-th roots. In his PhD thesis Scholze introduced perfectoid fields which are a class of fields containing lots of $p$-th roots. Furthermore, he proved the tilting equivalence that relates a perfectoid field with an associated characteristic p perfectoid field. In addition, he proved a generalisation of Faltings almost purity theorem. The almost purity theorem relates étale extensions of a perfectoid field $K$ with almost étale extensions over the ring of integers $K^\circ$. We give an exposition of the ingredients used to state the almost purity theorem and prove the theorem in characteristic $p$. Using the almost purity theorem we give a new proof of Tate's computations.

Vejleder: Dustin Clausen
Censor: Pieter Beelen