On an analog of Serre’s conjectures, Galois cohomology and defining equation of unipotent algebraic groups

Abstract

In this note we establish the validity, in the case of unipotent group schemes over non-perfect fields, of an analog of Serre’s conjectures for algebraic groups, which relates properties of Galois (or flat) cohomology of unipotent group schemes to finite extensions of non-perfect fields. We also establish an interesting property of Russell’s defining equations of connected smooth one-dimensional unipotent groups over a field $k$.