Divisors in global analytic sets

Abstract

We prove that any divisor $Y$ of a global analytic set $X\subset R^n$ has a generic equation, that is, there is an analytic function vanishing on $Y$ with multiplicity one along each irreducible component of $Y$. We also prove that there are functions with arbitrary multiplicities along $Y$. The main result states that if $X$ is pure dimensional, $Y$ is locally principal, $X\backslash Y$ is not connected and $Y$ represents the zero class in $H_{q–1}^{\infty }(X,Z_2)$ then the divisor $Y$ is globally principal.