Filtrations associated with singularities

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