Analytic spread modulo an element and symbolic Rees algebras (Q584324)

This article concerns the behavior of the symbolic powers of prime ideals of dimension one in regular local rings; the motivating problem is to find conditions for the symbolic Rees algebra \(S=\oplus P^{(n)}t^ n \) to be noetherian. The main result is that if T is a d-dimensional local domain (regular or complete takes care of the technical hypotheses) with an infinite residue field, P is a prime of height \(d-1,\) and there is an element \(f\in P^{(m)}\) such that \({\mathfrak q}=P/fT\) has analytic spread \(d-2\) (in other words, \(\ell ({\mathfrak q})=ht({\mathfrak q}))\) and reduction number at most \(m-1\), then S is noetherian. Examples of height two primes in \(K[[U,V,W]]\) are given (almost complete intersections that are the defining ideals of one-dimensional semigroup curves) for which the theorem implies S is noetherian. From these, the author constructs examples of primes in hypersurface domains with \(\ell({\mathfrak q})=ht({\mathfrak q})\).