Divisors in global analytic sets
Let \(X\subset \mathbb{R}^{n}\) be a global analytic set i.e. \(X\) is the zero set of finitely many analytic functions on \(\mathbb{R}^{n}\). It is called irreducible if it does not admit proper global analytic subsets of the same dimension. A divisor in \(X\) is a formal sum \(Y=\sum_{i\in I}n_{i}Y_{i}\) with coefficients in \(\mathbb{Z}\) where \(\{Y_{i}\}_{i\in I}\) is a locally finite family of global irreducible analytic subsets of \(X\) such that \(\dim Y_{i}=\dim X-1\) and \(Y_{i}\cap \text{Reg}X\neq \emptyset \). The main results are: 1. if \(Y=\sum_{i\in I}Y_{i}\) is a reduced (i.e. \(n_{i}=1\)) divisor in \(X\) then there exists an analytic function \(h\) on \(X\) such that the multiplicity of \(h \) along \(Y_{i}\) is equal to 1 for all \(i\in I\) (the theorem does not assert that the zero set of \(h\) is equal to \(\bigcup Y_{i}\)), 2. Let additionally \(X\) be irreducible and pure dimensional, \(\dim X=q.\) Assume \(Y\) is a reduced divisor such that its ideal sheaf \(I(Y)\mathcal{O} _{X}\) of \(Y\) in the sheaf \(\mathcal{O}_{X}\) of germs of analytic function on \(X\) is locally principal. If \(X\setminus Y\) is not connected and the class \( [Y]\) of \(Y\) in the homology group of infinite chains \(H_{q-1}^{\infty }(X, \mathbb{Z}_{2})\) is equal to zero then \(Y\) is globally principal (i.e. \(Y= \text{div}(g)\) for some analytic function \(g\) on \(X\))\(.\)
scientific article; zbMATH DE number 5842811
scientific article