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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 Retrieved
Date Indexed 2019-06-03
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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…x28;Uniform bounds on symbolic powers, chapter 4) Given a Noetherian complete local domain R, is there a positive integer k such that for any prime ideal P ⊂ R, P (kn) ⊆ P n for all positive integers n? 4. (General contractions of…

…Mazur conjecture Eisenbud and Mazur [EM97] studied symbolic powers in connection with the question of existence of non-trivial evolutions. Definition 1.1.1. Let R be a ring and S be a local R-algebra essentially of finite type. An evolution of…

…definition of symbolic powers. 3 Definition 1.1.2. Let R be a ring and I an ideal in R. For a positive integer n, the nth symbolic power of I is defined to be I (n) := {r ∈ R : r ∈ I n RP for all P such that P is a minimal prime of I}…

…let I be an ideal of height 2 having analytic spread 3. If Iis generically a complete intersection, unmixed and R(I) is normal and Cohen-Macaulay. Then mI n = mI n for all positive integers n. 10 1.3 Uniform bounds on symbolic powers of…

…results. In chapter 4 we explore the question of uniform bounds on symbolic powers of prime ideals. Finally in chapter 5 we raise some questions about contractions of powers of ideals from an overring and obtain some partial results to those questions for…

powers of ideals, chapter 5) Given an extension of Noetherian rings R ⊆ S and an ideal J in S what can be said about the behavior of In := J n ∩ R as n varies over positive integers? In particular, when is ⊕∞ i=0 In a Noetherian ring? 1.1 Eisenbud…

…prime ideals The question of equivalence of symbolic and adic topologies has generated considerable interest in the past two decades. For an unmixed ideal I in C[x1 , ..., xd ], Ein-Lazarsfeld-Smith (theorem 2.2, [ELS01])…

…ring R containing a field and ideal I of R, if h is the largest height of an associated prime ideal of I, then, I (hn) ⊆ I n for all positive integers n. In particular this implies that there is a uniform bound for the growth of symbolic

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