Local and global monomialization of morphisms
by
Dale Cutkosky
University of Missouri, Columbia

Suppose that S and R are local rings, essentially of finite type over a field k, and such that S is an extension of R. The structure of such an extension is extremely complicated, even when R and S are regular.

Suppose that V is a valuation ring of the quotient field K of S, such that V dominates S. Then we can ask if there are regular local rings R' and S' such that R' is obtained from R by a sequence of monoidal transforms, S' is obtained from S by a sequence of monoidal transforms, S' dominates R' and the extension S' of R' has an especially good structure.

We show that it is possible to obtain a diagram of mappings as above making the extension S' of R' a monomial mapping whenever the quotient field of S is a finite extension of the quotient field of R, and the characteristic of k is 0.

We also discuss more global theorems of this type.

Date received: July 10, 1999