Factorization of complete ideals in normal birational extensions in dimension two

Let (R,mR) be a two-dimensional regular local ring and let O be a normal birational extension of R. We relate the factorizations and semifactorization of complete mO-primary ideals in O to the factorizations of some complete mR-primary ideals in R. To this aim and given a complete mR-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.