Full Record

Author | More, Ajinkya Ajay |

Title | Symbolic Powers and other Contractions of Ideals in Noetherian Rings. |

URL | http://hdl.handle.net/2027.42/94031 |

Publication Date | 2012 |

Date Accessioned | 2012-10-12 15:25:37 |

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 | 2019-06-03 |

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*…