On the factoriality of some rings of complex Nash functions (Q1599938)

In this paper some algebraic properties of the ring of complex Nash functions of a factorial Nash subvariety are investigated. To be more precise, let \(X\) be a normal Nash subvariety of \(\mathbb C^n\). An affine Nash compact set \(K\) is defined as a compact set which can be identified with the maximal spectrum of the ring of Nash functions on \(K\), \({\mathcal N}_X(K)\). These compact sets can be characterized as holomorphically convex sets with respect to the analytic structure of \(X\) such that the inclusion map \({\mathcal N}_X(K) \to {\mathcal O}_X(K)\) is dense in the canonical \textbf{LF} topology of \({\mathcal O}_X(K)\). An Artin type approximation theorem is extended to Nash affine compact sets. The main result proved in this paper is the following: if \(K\) is a semianalytic Nash affine compact set of a normal Nash subvariety of \(\mathbb C^n\) then the map \(M \to M \otimes_{{\mathcal N}_X(K)} {\mathcal O}_X(K)\) induces an isomorphism \(\text{ Pic}({\mathcal N}_X(K)) \to \text{ Pic}({\mathcal O}_X(K))\). For a factorial Nash subvariety it follows that the ring \({\mathcal N}_X(K)\) is factorial if and only if \(H^2(K,{\mathbb Z}) = 0\), a result which is known in the analytic case [see the paper of \textit{H. G. Dales}, Proc. Am. Math. Soc. 44, 88-93 (1974; Zbl 0302.32018)].