Modules of derivations, logarithmic ideals and singularities of maps on analytic varieties

We introduce the module of derivations $\Theta_{h,M}$ attached to a given analytic map $h:(\mathbb C^n,0)\to (\mathbb C^p,0)$ and a submodule $M\subseteq \mathcal O_n^p$ and analyse several exact sequences related to $\Theta_{h,M}$. Moreover, we obtain formulas for several numerical invariants associated to the pair $(h,M)$ and a given analytic map germ $f:(\mathbb C^n,0)\to (\mathbb C^q,0)$. In particular, if $X$ is an analytic subvariety of $\mathbb C^n$, we derive expressions for analytic invariants defined in terms of the module $\Theta_X$ of logarithmic vector fields of $X$.