Central simple algebras of index p^n in characteristic p

Abstract: Let k be a field of characteristic p>0. In this talk, we shall study p-algebras; that is, central simple k-algebras of degree a power of p. By a theorem of Teichmüller, every such algebra (say of exponent e and index d) is Brauer equivalent to a tensor product of cyclic algebras of degree e. Teichmüller moreover proved in the 30's that the number of cyclic algebras appearing in the tensor product can be bounded by d!(d!-1) (twenty years ago, this result was improved by Mammone and Mammone-Merkurjev in some particular cases). We shall improve this bound down to d-1. Key tools are a careful study of the Frobenius morphism for Severi-Brauer varieties, as well as a structure theorem for some commutative unipotent group schemes.