Towards Cohomology of Real Closed Spaces

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}.