On the Galois and Flat Cohomology of Unipotent Algebraic Groups over Local and Global Function Fields. II

Abstract

Let k be a field of positive characteristic p with degree of imperfection 1 and a nontrivial discrete valuation. We show that the Galois and flat cohomology of unipotent k-groups over k of dimension $<\mathit{p}-1$ are finite (in fact, trivial) if and only if they are k-split. We give some examples to show that the bound $\mathit{p}-1$ is best possible and present several applications to local-global principles.