2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76711Suppose that X to Y is a generically finite map of nonsingular varieties over a field of characteristic zero, and v is a valuation of the function field of X. We prove that it is possible to perform a sequence of monoidal transforms X' to X and Y' to Y so that X' to Y' is a monomial mapping at the center of v. We deduce from this that a birational morphism of nonsingular varieties can be factored along a valuation by a sequence of blowups and blowdowns with nonsingular centers.145 pages In the revision a new chapter (chapter 6) has been added which proves a stronger local factorization Theorem. The introduction has also been modified. There are no other changes from the original March '98 submissionAlgebraic GeometryCommutative Algebra14E, 13Btext