# Algebraic Geometry Seminar

#### Ramification Theory in Arithmetic Geometry and Existence of Slices

**Speaker:**
Sophie Marques, Courant Institute of Mathematical Sciences

**Location:**
Warren Weaver Hall 317

**Date:**
Tuesday, November 19, 2013, 3:30 p.m.

**Synopsis:**

In algebraic number theory, unramified extensions are completely understood since they have a simple structure. Ramification makes the understanding of an extension complicated. However, when the ramification is tame, the added complexity is small, and we can still describe them explicitly. One can extend the notion of ramification to actions involving affine group schemes. We will mainly focus on tame ramification in this context. We will discuss two definitions of tameness and how they are related. We will see how the fundamental results of ramification theory in algebraic number theory can be translated, and in which cases were we able to prove them for actions involving group schemes. We get for instance a result describing the structure of the inertia group for a tame action. This permits us to induce a tame action by an action of some fppf lifting of an inertia group. Finally, we will discuss some possible candidates to define higher ramification groups.