Divisors of a module and blow up
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 $I(E)$ (A. Simis, B. Ulrich, and W. Vasconcelos [18]) or ideal norms $[[E]]_R$ (O. Villamayor [22]). 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 $\ell(\bigwedge^{e}E)$ related with the algebraic local version of Zak's inequality as explained in A. Simis, K. Smith and B. Ulrich [16].