Projective modules over smooth, affine varieties over real closed fields

Abstract

Let X = Spec( A) be a smooth, affine variety of dimension n 2 over the field R of real numbers. Let P be a projective A-module of rank n such that its nth Chern class C n (P ) ∈ CH 0 (X) is zero.In this set-up, Bhatwadekar-Das-Mandal showed (amongst many other results) that P A ⊕ Q in the case that either n is odd or the topological space X(R) of real points of X does not have a compact, connected component. In this paper, we prove that similar results hold for smooth, affine varieties over an arbitrary real closed field R. The proof is algebraic and does not make use of Tarski's principle, nor of the earlier result for R.

Chow–Witt rings of classifying spaces for symplectic and special linear groups

“…Remark Note that Proposition also holds for real closed fields, using the comparison result [, Theorem 3] and the corresponding statement for Euler class groups from . More generally, it is conceivable that is torsion free if is smooth affine and oriented of dimension over a real pythagorean field.…”

“…The answer is also known to be positive over R and more generally over real closed fields, by [9,11,12]. In the context of Chow-Witt groups, we can reprove one of the main results of [12], namely the odd-rank case of [12, Theorem II].…”