Reduction of the group scheme of p-power roots of unity
and its torsors
Abstract : After the full development of Galois theory in algebra and its extension to the geometric setting, it has become a natural point of view to study rings and varieties via their torsors. When these objects are defined over a local or a global field, the question of their reduction at a prime is posed. We will discuss the problem of finding models of reduction for torsors, with an eye especially open for the models of the Galois groups. In the first lecture, I will introduce the general problem of reduction of torsors and report on some situations where nice models can be found. In the second lecture, I will recall the essentials of the work of Sekiguchi and Suwa on the unification of Kummer and Artin-Schreier-Witt theories, and use it to construct some models of the group schemes of p-power roots of unity. In the third lecture, I will relate this construction to the classification of finite flat group schemes due to Breuil and Kisin.