The graded algebra of a valuation over a two dimensional local Notherian domain
by
Carlos Galindo
Department of Mathematics. Universidad Jaume I. 12071 Castellon. Spain

Let R be a two-dimensional local Noetherian domain. In this talk, we shall give an explicit computation of the graded algebra gr_vR, relative to a valuation v of the quotient field of R centered at R.

We study gr_vR by viewing it as a homomorphic image of a polinomial ring A[v] via an epimorphism of K-algebras f : A[v] --> gr_vR, K is the residue field of R. If we consider no assumption on the dimension of R, then we can compute the number of generators of a minimal generating set of the i-syzygy module of gr_vR as A[v]-module, whenever the kernel of the morphism f, I₀, is spanned by a regular set. In our case (dim R = 2), we are able to give, in an explicit manner, a minimal set of generators for I₀. This set is regular, so we can compute the number of generators of the syzygy modules of gr_vR.

Date received: May 5, 1999