Lefschetz number of the monodromy and resolution of singularities (Q1329628)

Let \(f(X,x) \to (\mathbb{C}, 0)\) be a germ of holomorphic function on an analytic space germ \(X\), not supposed to be smooth. We say that \(f\) is a smoothing if the singular locus of \(X\) is contained in \(H\), the zero locus of \(f\). Then the generic fibre of \(f\) is smooth and the monodromy acts on the cohomology of the Milnor fiber and the Lefschetz number \(\Lambda (f)\) is the alternating sum of the trace of these monodromies. The author proves that \(\Lambda (f)\) depends only on the residue class of \(f\) in \(m/{\mathcal I}\) where \(m\) is the maximal ideal and \({\mathcal I}\) is an ideal defined by using some resolution of \(X\). In fact \({\mathcal I}\) doesn't depend on this resolution. In particular, if the singularity of \(X\) is isolated, \({\mathcal I}\) contains \(m^ 2\). The Lefschetz number is computed when \(X\) is the cone over a smooth projective hypersurface, or the core over a hypersurface with isolated singularities.