2024/10/25 by Tafari Clarke-James, Clarke-James, Tafari
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Advanced Topics in Algebra #Advanced Banach Space Theory
paper · pdf · doi:10.48550/arxiv.2410.19929
It was shown by Claus Scheiderer prior to 1994 that real closed spaces have étale 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 étale site, and sheaves on a real étale site \rce/X of my creation. Since 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 \citenodeRhamhuber.