Etale cohomologies of quadrics over R

TL;DR

This paper investigates the etale cohomology of quadrics over the real numbers, computing the cohomology of norm quadrics and highlighting the existence of many non-algebraic elements.

Contribution

It provides explicit computations of etale cohomology for norm quadrics over R and demonstrates the presence of numerous non-algebraic elements.

Findings

Computed etale cohomology of norm quadrics over R

Identified examples with many non-algebraic elements

Clarified the relationship between algebraic elements and the cycle map

Abstract

In this paper, we study etale cohomologies of quadrics over R. An element in the etale cohomology is called algebraic, if it is in the image of the cycle map from the Chow ring. In this paper, we compute the etale cohomology of norm quadrics, and give examples which have many non-algebraic elements.