A vanishing theorem for finitely supported ideals in regular local rings
A cohomological vanishing property is proved for finitely supported ideals in an arbitrary d-dimensional regular local ring. (Such vanishing implies some refined Briancon-Skoda-type results, not otherwise known in mixed char- acteristic.) It follows that the adjoint e I of a finitely supported ideal I has order ord e I = sup(ordI +1 −d, 0), and that taking adjoints of finitely supported ideals commutes with taking strict transforms at infinitely near points. In particular, e I is also finitely supported.
On Complete Ideals in Regular Local Rings
Publisher Summary This chapter presents an approach to unique factorization of complete ideals in two-dimensional regular local rings, based on a decomposition theorem that is valid in all dimensions. The theory of complete ideals in two-dimensional regular local rings was founded by Zariski. Zariski's work was motivated by the birational theory of linear systems on smooth surfaces. Zariski first raised the question of higher-dimensional generalizations, but not much has happened in this respect during the intervering fifty years. The chapter presents the assumption that R is a noetherian local domain, with maximal ideal m and fraction field K , then a prime divisor of R is a valuation ν of K whose valuation ring R ν dominates R such that the transcendence degree of the field R ν /m ν over R / m is as large as possible, that is, dim R – 1.
On Hilbert's thirteenth problem and related questions
Hilbert's thirteenth problem involves the study of solutions of algebraic equations. The object is to obtain a complexity estimate for an algebraic function. As of now, the problem remains open. There are only a few partial algebraic results in this connection, but at the same time the problem has stimulated a series of studies in the theory of functions with their subsequent applications. The most brilliant result in this cycle is Kolmogorov's theorem on superpositions of continuous functions.