Symbolic Powers and other Contractions of Ideals in Noetherian Rings.

[摘要] The results in this thesis are motivated by the following four questions:1. (Eisenbud-Mazur conjecture): Given a regular local ring (R,m) containing a field of characteristic zero and an unmixed ideal I in R, the second symbolic power is contained in the ideal mI.2. (Integral closedness of mI) Given a regular local ring (R,m) and a radical ideal I in R, whenis mI integrally closed?3. (Uniform bounds on symbolic powers) Given a complete local domain R, is there a constant k such that for any prime ideal P in R, the kn’th symbolic power of P is contained in its n’th ordinary power, for all positive integers n.4. (General contractions of powers of ideals) Given an extension of Noetherian rings R contained in S and an ideal J in S what can be said about the behavior of the ideals obtained by contraction of various powers of J?It is shown that if I is an ideal generated by a single binomial and several monomials in a polynomial ring over a field where m is the homogeneous maximal ideal, then, mI is integrally closed. The Eisenbud-Mazur conjecture is shown to hold for the case of certain prime ideals in certain subrings of a formal power series ring over a field. Some computational results using Macaulay2 are discussed. For a Noetherian complete local domain (R,m), it is shown that there exists a numerical function f such that for any prime ideal P in R, the f(n)’th symbolic power of P is contained in its n’th ordinary power. Suppose R contained in S is a module-finite extension of domains and R is normal, while S is regular,equicharacteristic, then, under mild conditions on R and S, it is shown that there exists a positive integer c such that for any prime ideal P in R, the cn’th symbolic power of P is contained in the n’th ordinary power of P. Two questions are raised about the behavior of contractions of powers ideals from a polynomial ring in one indeterminate to its coefficient ring and some partial results are obtained

[发布机构] University of Michigan

[效力级别] Eisenbud-Mazur Conjecture

[关键词] Symbolic Powers;Eisenbud-Mazur Conjecture;Regular Local Ring;Uniform Bounds;Contractions;Mathematics;Science;Mathematics