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Symbolic Powers and other Contractions of Ideals in Noetherian Rings.

dc.contributor.authorMore, Ajinkya Ajayen_US
dc.date.accessioned2012-10-12T15:25:37Z
dc.date.availableNO_RESTRICTIONen_US
dc.date.available2012-10-12T15:25:37Z
dc.date.issued2012en_US
dc.date.submitted2012en_US
dc.identifier.urihttps://hdl.handle.net/2027.42/94031
dc.description.abstractThe 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 obtaineden_US
dc.language.isoen_USen_US
dc.subjectSymbolic Powers, Eisenbud-Mazur Conjecture, Regular Local Ring, Uniform Bounds, Contractionsen_US
dc.titleSymbolic Powers and other Contractions of Ideals in Noetherian Rings.en_US
dc.typeThesisen_US
dc.description.thesisdegreenamePhDen_US
dc.description.thesisdegreedisciplineMathematicsen_US
dc.description.thesisdegreegrantorUniversity of Michigan, Horace H. Rackham School of Graduate Studiesen_US
dc.contributor.committeememberHochster, Melvinen_US
dc.contributor.committeememberZhang, Junen_US
dc.contributor.committeememberZhang, Wenliangen_US
dc.contributor.committeememberSmith, Karen E.en_US
dc.contributor.committeememberZieve, Michael E.en_US
dc.subject.hlbsecondlevelMathematicsen_US
dc.subject.hlbtoplevelScienceen_US
dc.description.bitstreamurlhttp://deepblue.lib.umich.edu/bitstream/2027.42/94031/1/ajinkya_1.pdf
dc.owningcollnameDissertations and Theses (Ph.D. and Master's)


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