The dimension of the space of R-places of certain rational function fields

TL;DR

This paper determines the topological and cohomological dimensions of the space of real places of rational function fields in two variables over a totally Archimedean field, revealing a nuanced dimensional structure.

Contribution

It establishes the exact covering, integral, and cohomological dimensions of the space of R-places for certain rational function fields, a novel result in this area.

Findings

The space $M(K(x,y))$ has covering and integral dimensions equal to 2.

The cohomological dimension of $M(K(x,y))$ is 1 for any Abelian 2-divisible group.

The results clarify the topological complexity of spaces of real places in rational function fields.

Abstract

We prove that the space $M(K(x,y))$ of $\mathbb R$-places of the field $K(x,y)$ of rational functions of two variables with coefficients in a totally Archimedean field $K$ has covering and integral dimensions $\dim M(K(x,y))=\dim_\IZ M(K(x,y))=2$ and the cohomological dimension $\dim_G M(K(x,y))=1$ for any Abelian 2-divisible coefficient group $G$.