A finiteness condition on regular local overrings of a local domain

Abstract

The local factorization theorem of Zariski and Abhyankar implies that between a given pair of 2 2 -dimensional regular local rings, S ⊇ R S \supseteq R , having the same quotient field, every chain of regular local rings must be finite. It is shown in this paper that this property extends to every such pair of regular local rings, regardless of dimension. An example is given to show that this does not hold if "regular" is replaced by "Cohen-Macaulay," by "normal," or by "rational singularity." More generally, it is shown that the set R ( R ) \mathcal {R}(R) of n n dimensional regular local rings birationally dominating a given n n -dimensional local domain, R R , and ordered by containment, satisfies the descending chain condition. An example is given to show that if R R is regular the two examples of minimal elements of R ( R ) \mathcal {R}(R) given by J. Sally do not exhaust the set of minimal elements of R ( R ) \mathcal {R}(R) .