Divisors of a module and blow up

Abstract

In this paper we work with several divisors of a module $E \subseteq G \simeq R^e$ having rank $e$, such as the classical Fitting ideals of $E$ and of $G / E$, and the more recently introduced (generic) Bourbaki ideals $I(E)$ (Simis et al. (2003) $[19]$ ) or ideal norms $[[E]]_R$ (Villamayor (2006) [23]). We found several relations and equalities among them which allow to describe in some cases universal properties with respect to $E$ of their blow ups. As a byproduct we are also able to obtain lower bounds for the analytic spread $\ell\left(\bigwedge^e E\right)$, related with the algebraic local version of Zak's inequality as explained in Simis et al. (2002) [17].