Projective modules over smooth, affine varieties over Archimedean real closed fields (Q1025068)

Let \(\mathbb{R}\) be an Archimedean real closed field. Let \(X=\text{Spec}\, A\) be a smooth affine variety of dimension \(n\geq 2\) over \(\mathbb{R}\). Let \(X(\mathbb{R})\) denote the \(\mathbb{R}\)-rational points of \(X\) and let \(K\) denote the module of \(n\)-forms, \(\Omega^n_{A/\mathbb{R}}\). Let \(P\) be a projective module of rank \(n\) over \(A\) and denote by \(L\) the determinant \(\wedge^n P\). Assume that \(c_n(P)\), the nth Chern class of \(P\) in \(CH_0(X)\) is zero. Then the authors prove that \(P\) has a free direct summand in several cases. For example, one of the cases is when for every closed and bounded semi-algebraically connected component \(W\) of \(X(\mathbb{R})\), the restrictions of \(L,K\) to \(W\) are not isomorphic. The case when \(\mathbb{R}=\mathbb{R}\) was dealt with in [\textit{S. M. Bhatwadekar, M. K. Das, S. Mandal}, Invent. Math. 166, No. 1, 151--184 (2006; Zbl 1107.13013)].