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.