Scaffolds and integral Hopf Galois module structure on purely inseparable extensions

Abstract

Let $p$ be prime. Let $L/K$ be a finite, totally ramified, purely inseparable extension of local fields, $\left[L:K\right]=p^n,n\ge 2.$ It is known that $L/K$ is Hopf Galois for numerous Hopf algebras $H,$ each of which can act on the extension in numerous ways. For a certain collection of such $H$ we construct "Hopf Galois scaffolds" which allow us to obtain a Hopf analogue to the Normal Basis Theorem for $L/K.$ The existence of a scaffold structure depends on the chosen action of $H$ on $L.$ We apply the theory of scaffolds to describe when the fractional ideals of $L$ are free over their associated orders in $H.$

16 pages. This version better describes the numerous ways that $H$ can act on $L$, and fixes ambiguities as to which action is being considered

References in corpus (1)

• Hopf Galois Structures on Primitive Purely Inseparable Extensions

Cited by in corpus (1)

• Hopf Galois Structures on Primitive Purely Inseparable Extensions