The behavior of differential, quadratic and bilinear forms under purely inseparable field extensions

TL;DR

This paper investigates how differential, quadratic, and bilinear forms behave under purely inseparable field extensions, providing generators for kernels of restriction maps and characterizing their structure in characteristic p.

Contribution

It introduces a system of generators for the kernels of restriction maps of differential and quadratic forms in purely inseparable extensions, extending understanding of form behavior in characteristic p.

Findings

Generated kernels of differential forms under purely inseparable extensions.

Determined the quadratic Witt kernel in characteristic 2.

Extended results to bilinear Witt kernels for modular extensions.

Abstract

Let $F$ be a field of characteristic $p$ and let $E/F$ be a purely inseparable field extension. We study the group $H_p^{n+1}(F)$ of classes of differential forms under the restriction map $H_p^{n+1}(F)\to H_p^{n+1}(E)$ and give a system of generators of the kernel $H_p^{n+1}(E/F)$. In the case $p=2$, we use this to determine the kernel $W_q(E/F)$ of the restriction map $W_q(F) \to W_q(E)$ between the group of nonsingular quadratic forms over $F$ and over $E$. We also deduce the corresponding result for the bilinear Witt kernel $W(E'/F)$ of the restriction map $W(F) \to W(E')$, where $E'/F$ denotes a modular purely inseparable field extension.