Definable $mathcal{C}$$^{r}$ sheaf on o-minimal spectrum (Model theoretic aspects of the notion of independence and dimension)

Abstract

Consider an o-minimal expansion of the real field ℝ~ and a definable $mathcal{C}$$^{r}$ submanifold M of ℝm, where r is a nonnegative integer. Let L be the first-order language of ℝ~. The o-minimal spectrum M~ of M is the set of all complete m-types of the first-order theory Thℝ(ℝ~) which imply a formula defining M. A stalk of the sheaf of definable $mathcal{C}$$^{r}$ functions on M~ at a point α ∈ M~ is a local ring. Its residue field is naturally an L-structure. We show that the residue field is a minimal elementary extension of the o-minimal structure ℝ containing $mathcal{C}$$^{r}$df(M)/supp(α) and satisfying that, for any a⁻ ∈ ($mathcal{C}$$^{r}$df(M))n and any formula φ(x⁻), the extension satisfies the sentence φ(a⁻) if and only if the definable subset of M defined by φ(a⁻) is an element of a. Here, the notation $mathcal{C}$$^{r}$df(M) denotes the ring of all definable $mathcal{C}$$^{r}$ functions on M.