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

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 $G$.

💡 Research Summary

The paper investigates the topological and cohomological dimensions of the space of ℝ‑places of the rational function field K(x,y), where K is a totally Archimedean field (i.e., every ordering of K is Archimedean). An ℝ‑place of a field F is a homomorphism φ: F → ℝ∪{∞} that respects addition, multiplication, and sends 1 to 1; it either preserves or reverses the order. The collection M(F) of all ℝ‑places carries the topology of pointwise convergence, making it a Hausdorff, often compact, space.

The authors begin by recalling known results for the one‑variable case K(x). When K is totally Archimedean, M(K(x)) is homeomorphic to the real line ℝ; consequently its covering (Lebesgue) dimension dim M(K(x)) and its integral dimension dimℤ M(K(x)) both equal 1. Moreover, for any coefficient group G, the cohomological dimension dim_G M(K(x)) also equals 1.

The core of the paper extends the analysis to two variables. For any ℝ‑place φ of K(x,y), restriction to the subfield K(x) yields an ℝ‑place ψ∈M(K(x)). Conversely, given a pair (ψ, t) with ψ∈M(K(x)) and t∈ℝ, one can construct a unique ℝ‑place of K(x,y) that extends ψ and takes the value t on the second variable y. The authors prove that these restriction and extension maps are continuous inverses, establishing a topological homeomorphism

  M(K(x,y)) ≅ M(K(x)) × ℝ.

Since dim M(K(x)) = 1 and dim ℝ = 1, the covering dimension of the product is the sum, giving

  dim M(K(x,y)) = 2.

The same argument applies to the integral dimension, so dimℤ M(K(x,y)) = 2.

The more subtle part concerns cohomological dimension. For any Abelian group G that is 2‑divisible (i.e., every element is a double of some other element; examples include ℚ, ℚ/ℤ, and any ℚ‑vector space), the authors show that the Čech cohomology groups H^n(M(K(x,y)); G) vanish for n ≥ 2. The proof uses the Künneth formula for Čech cohomology applied to the product M(K(x)) × ℝ. Both factors have cohomological dimension 1, and the 2‑divisibility of G eliminates the Tor terms that could otherwise produce non‑trivial higher cohomology. Consequently,

  dim_G M(K(x,y)) = 1 for every 2‑divisible Abelian group G.

Thus the main theorem states:

  dim M(K(x,y)) = dimℤ M(K(x,y)) = 2,   dim_G M(K(x,y)) = 1 for all Abelian 2‑divisible G.

The paper concludes with several observations. First, the discrepancy between covering/integral dimension (2) and cohomological dimension (1) illustrates that ℝ‑place spaces can be non‑metrizable and possess exotic dimension behavior. Second, the methods suggest possible extensions: investigating fields K that are not totally Archimedean, or rational function fields in more than two variables, where one might expect the covering dimension to increase with the number of variables while cohomological dimension remains low for suitable coefficient groups. Finally, the authors note connections to real algebraic geometry, model theory, and the theory of ordered fields, indicating that the dimension theory of ℝ‑places provides a fertile interface between algebra, topology, and logic.