Disentangling mappings defined on ICIS

We study germs of hypersurfaces $$(Y,0)\subset (\mathbb {C}^{n+1},0)$$ ( Y , 0 ) ⊂ ( C n + 1 , 0 ) that can be described as the image of $${\mathscr {A}}$$ A -finite mappings $$f:(X,S)\rightarrow (\mathbb {C}^{n+1},0)$$ f : ( X , S ) → ( C n + 1 , 0 ) defined on an icis ( X , S ) of dimension n . We extend the definition of the Jacobian module given by Fernández de Bobadilla, Nuño-Ballesteros and Peñafort-Sanchis when $$X=\mathbb {C}^n$$ X = C n , which controls the image Milnor number $$\mu _I(X,f)$$ μ I ( X , f ) . We apply these results to prove the case $$n=2$$ n = 2 of the generalised Mond conjecture, which states that $${\mu _I(X,f)\ge \text{ codim}_{\mathscr {A}_e}(X,f)}$$ μ I ( X , f ) ≥ codim A e ( X , f ) , with equality if ( Y , 0) is weighted homogeneous.