Unramified Cohomology of Quadrics in Characteristic Two

TL;DR

This paper investigates the unramified cohomology groups of smooth projective quadrics over fields of characteristic two, extending known results from characteristic not two and providing a complete description of certain cohomological maps.

Contribution

It completely determines the kernel and cokernel of the natural map from field cohomology to unramified cohomology for quadrics in characteristic two, extending prior characteristic not two results.

Findings

Complete description of kernel and cokernel of cohomology maps

Extension of known results to characteristic two

Advancement in understanding unramified cohomology of quadrics

Abstract

Let $F$ be a field of characteristic 2 and let $X$ be a smooth projective quadric of dimension $\ge 1$ over $F$. We study the unramified cohomology groups with 2-primary torsion coefficients of $X$ in degrees 2 and 3. We determine completely the kernel and the cokernel of the natural map from the cohomology of $F$ to the unramified cohomology of $X$. This extends the results in characteristic different from 2 obtained by Kahn, Rost and Sujatha in the nineteen-nineties.