Abelian affine group schemes, plethories, and arithmetic topology

Sammanfattning: In Paper A we classify plethories over a field of characteristic zero. All plethories over characteristic zero fields are ``linear", in the sense that they are free plethories on a bialgebra. For the proof of this classification we need some facts from the theory of ring schemes where we extend previously known results. We also give a classification of plethories with trivial Verschiebung over a perfect field k of characteristic p>0.In Paper B we study tensor products of abelian affine group schemes over a perfect field k. We first prove that the tensor product G_1 \otimes G_2 of two abelian affine group schemes G_1,G_2 over a perfect field k exists. We then describe the multiplicative and unipotent part of the group scheme G_1 \otimes G_2. The multiplicative part is described in terms of Galois modules over the absolute Galois group of k. In characteristic zero the unipotent part of G_1 \otimes G_2 is the group scheme whose primitive elements are P(G_1) \otimes P(G_2). In positive characteristic, we give a formula for the tensor product in terms of Dieudonné theory. In Paper C we use ideas from homotopy theory to define new obstructions to solutions of embedding problems and compute the étale cohomology ring of the ring of integers of a totally imaginary number field with coefficients in Z/2Z. As an application of the obstruction-theoretical machinery, we give an infinite family of totally imaginary quadratic number fields such that Aut(PSL(2,q^2)), for q an odd prime power, cannot be realized as an unramified Galois group over K, but its maximal solvable quotient can. In Paper D we compute the étale cohomology ring of an arbitrary number field with coefficients in Z/nZ for n an arbitrary positive integer. This generalizes the computation in Paper C. As an application, we give a formula for an invariant defined by Minhyong Kim.