Projective modules over smooth, affine varieties over real closed fields
This article proves results in the spirit of some of the following articles. \textit{M.~P.~Murthy}'s paper [Ann. Math. (2) 140, No. 2, 405--434 (1994; Zbl 0839.13007)], \textit{S. M. Bhatwadekar, M. K. Das} and \textit{S. Mandal} [Invent. Math. 166, No. 1, 151--184 (2006; Zbl 1107.13013)], the authors [J. Pure Appl. Algebra 213, No. 10, 1936--1944 (2009; Zbl 1171.13007)]. The main theorem in the paper is the following: Let \(R\) be a real closed field and let \(X=\mathrm{Spec}(A)\) be a smooth affine variety of dimension \(n\geq 2\) over \(R\). Let \(X(R)\) denote the \(R\)-rational points of \(X\) and let \(K\) denote the module of \(n\)-forms. Let \(P\) be a projective module over \(A\) of rank \(n\) with determinant \(L\). Asuume that \(c_n(P)=0\), the \(n\)-th Chern class in the Chow group of \(A\). Then \(P\cong A\oplus Q\) in the follwing cases. 1) \(X(R)\) has no closed and bounded semialgebraically connected component; 2) For every closed and bounded semialgebraically connected component \(W\) of \(X(R)\), \(L_{|W}\) is not isomorphic to \(K_{|W}\); 3) \(n\) is odd.
scientific article; zbMATH DE number 5696922