Filtrations associated with singularities

We fix a complex analytic normal singularity germ (X, o) of dimension 2 and a (not necessarily irreducible) reduced Weil divisor (S, o) (X, o). The embedded resolution of the pair determines a multi-index filtration of the local ring Oₗ, ₎, which measures the embedded geometry of the pair. Furthermore, from the (induced) resolution of (S, o) we also consider a multi-index filtration associated with (S, o). This latter one can be lifted to a filtration of Oₗ, ₎ too. The main result proves that the second filtration of Oₗ, ₎ can be realized as a `limit' filtration of the first one (if we blow up certain centers sufficiently many times).