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

We introduce the module of derivations Θh,M attached to a given analytic map h:( Cⁿ,0)→ ( Cᵖ,0) and a submodule M⊆ Oₙᵖ and analyse several exact sequences related to Θh,M. Moreover, we obtain formulas for several numerical invariants associated to the pair $(h,M)$ and a given analytic map germ f:( Cⁿ,0)→ ( Cq,0). In particular, if X is an analytic subvariety of Cⁿ, we derive expressions for analytic invariants defined in terms of the module ΘX of logarithmic vector fields of X.