Towards Cohomology of Real Closed Spaces

TL;DR

This paper revitalizes the theory of real closed spaces and rings by establishing foundational results, constructing an equivalence of topoi, and suggesting applications of motivic cohomology to deepen understanding of recent developments.

Contribution

It develops the fundamentals of real closed rings and spaces with minimal machinery, proves their closure properties, and constructs an explicit equivalence of topoi for computational purposes.

Findings

Real closed rings are closed under limits and colimits.

An equivalence of topoi between Scheiderer's sheaves and a new sheaf category is established.

The theory of real closed spaces becomes more accessible and computable.

Abstract

It was shown by Claus Scheiderer prior to 1994 that real closed spaces have \'{e}tale cohomology. Following Scheiderer, study of real closed spaces fell out of fashion and o-minimal geometry became the focus for those at the intersection of model theory and geometry. I decided to breathe new life into the theory of real closed rings and spaces, as studied by Schwartz in 1989. In Section 1, I build the fundamentals of the theory using as little machinery as possible, and presented them as clearly as I could. Hidden gems include a full proof that real closed rings are closed under limits and colimits. In Section 2, I give an introduction to the category of real closed spaces in the first half. In the second half, I construct an equivalence of topoi between Scheiderer's sheaves on the real \'{e}tale site, and sheaves on a real \'{e}tale site $\rce/X$ of my creation. Since $\text{Sh}(\rce/X)$ can be defined without the use of $G$-topoi, the equivalence of topoi renders Scheiderer's theory computable. I end with a discussion of how one might use motivic cohomology to better understand recent results of Annette Huber in \cite{no_deRham_huber}.