Ramification theory in arithmetic geometry and existence of slices

Sophie Marques, November 19th, 2013

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.