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Title Symbolic Powers and other Contractions of Ideals in Noetherian Rings.
URL
Publication Date
Date Accessioned
Degree PhD
Discipline/Department Mathematics
Degree Level doctoral
University/Publisher University of Michigan
Abstract 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
Subjects/Keywords Symbolic Powers, Eisenbud-Mazur Conjecture, Regular Local Ring, Uniform Bounds, Contractions; Mathematics; Science
Contributors Hochster, Melvin (committee member); Zhang, Jun (committee member); Zhang, Wenliang (committee member); Smith, Karen E. (committee member); Zieve, Michael E. (committee member)
Language en
Rights Unrestricted
Country of Publication us
Record ID handle:2027.42/94031
Repository umich
Date Indexed 2020-09-09
Grantor University of Michigan, Horace H. Rackham School of Graduate Studies
Issued Date 2012-01-01 00:00:00
Note [thesisdegreename] PHD; [thesisdegreediscipline] Mathematics; [thesisdegreegrantor] University of Michigan, Horace H. Rackham School of Graduate Studies; [bitstreamurl] http://deepblue.lib.umich.edu/bitstream/2027.42/94031/1/ajinkya_1.pdf;

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