Factorization of complete ideals in normal birational extensions in dimension two

Let be a two-dimensional regular local ring and let be a normal birational extension of R. We relate the factorizations and semifactorization of complete -primary ideals in to the factorizations of some complete -primary ideals in R. To this aim and given a complete -primary ideal I R, we show that we can associate to each complete ideal sheaf with finite cosupport on X=BlI(R) a complete ideal in R from which it can be recovered.