LOCAL FACTORIZATION OF NONSINGULAR BIRATIONAL MORPHISMS IN DIMENSION GREATER THAN TWO (REGULAR LOCAL RINGS, DESCENDING CHAIN CONDITION, RATIONAL SINGULARITY)

The Local Factorization Theorem of Zariski and Abhyankar implies that between a given pair of 2-dimensional regular local rings, S (GREATERTHEQ) R, having the same quotient field, every chain of regular local rings must be finite in length. It is shown that this property extends to every such pair of regular local rings, regardless of dimension. Examples are given to show that this does not hold if regular is weakened to various statements, including Gorenstein , rational singularity , and normal . More generally, it is shown that the set of n-dimensional regular local rings birationally containing an arbitrary integral domain must satisfy the descending chain condition. Some conditions which imply a uniform bound on the lengths of certain chains between two fixed n-dimensional regular local rings, as above, are given. Finally, a new class, containing infinitely many minimal regular local overrings containing a fixed regular local ring, is presented