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Towards Cohomology of Real Closed Spaces

Tafari Clarke-James

Abstract

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 $\text{Sh}(\rce/X)$ 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}.

Towards Cohomology of Real Closed Spaces

Abstract

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 of my creation. Since can be defined without the use of -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}.

Paper Structure

This paper contains 19 sections, 44 theorems, 235 equations.

Table of Contents

  1. Sper(-) and Sections
  2. Orders=Cones
  3. Cones and Homomorphisms
  4. Introducing Pre-Geometry
  5. The Topologies of Sper(-)
  6. A Model-Theoretic Moment
  7. A Sheaf on Sper(-)
  8. Real Closed Rings are (co)Complete
  9. Properties of Real Closed Rings
  10. Real Closed Spaces I
  11. Objects
  12. Subspaces
  13. Morphisms
  14. A Deep Example: Semialgebraic Geometry & Real Closed Spaces
  15. The Real Étale Site
  16. ...and 4 more sections

Key Result

Theorem 4

If $R$ is a ring with positive cone of partial order $\alpha$, then $\mathop{\mathrm{supp}}\nolimits(\alpha)$ is an ideal in $R$.

Theorems & Definitions (129)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Example 6
  • Lemma 7
  • proof
  • ...and 119 more