Full Record

Author | Chen, Yuanyuan |

Title | Filtration Theorems and Bounding Generators of Symbolic Multi-powers |

URL | http://hdl.handle.net/2027.42/151674 |

Publication Date | 2019 |

Date Accessioned | 2019-10-01 18:28:14 |

Degree | PhD |

Discipline/Department | Mathematics |

Degree Level | doctoral |

University/Publisher | University of Michigan |

Abstract | We prove a very powerful generalization of the theorem on generic freeness that gives countable ascending filtrations, by prime cyclic A-modules A/P, of finitely generated algebras R over a Noetherian ring A and of finitely generated R-modules such that the number of primes P that occur is finite. Moreover, we can control, in a sense that we can make precise, the number of factors of the form A/P that occur. In the graded case, the number of occurrences of A/P up to a given degree is eventually polynomial. The degree is at most the number of generators of R over A. By multi-powers of a finite sequence of ideals we mean an intersection of powers of the ideals with exponents varying. Symbolic multi-powers are defined analogously using symbolic powers instead of powers. We use our filtration theorems to give new results bounding the number of generators of the multi-powers of a sequence of ideals and of the symbolic multi-powers as well under various conditions. This includes the case of ordinary symbolic powers of one ideal. Furthermore, we give new results bounding, by polynomials in the exponents, the number of generators of multiple Tor when each input module is the quotient of R by a power of an ideal. The ideals and exponents vary. The bound is given by a polynomial in the exponents. There are similar results for Ext when both of the input modules are quotients of R by a power of an ideal. Typically, the two ideals used are different, and the bound is a polynomial in two exponents. |

Subjects/Keywords | symbolic powers; filtration theorems; Mathematics; Science |

Contributors | Hochster, Mel (committee member); Tappenden, James P (committee member); Canton, Eric (committee member); Derksen, Harm (committee member); Smith, Karen E (committee member) |

Language | en |

Rights | Unrestricted |

Country of Publication | us |

Record ID | handle:2027.42/151674 |

Repository | umich |

Date Retrieved | 2019-10-31 |

Date Indexed | 2019-10-31 |

Grantor | University of Michigan, Horace H. Rackham School of Graduate Studies |

Issued Date | 2019-01-01 00:00:00 |

Note | [thesisdegreename] PHD; [thesisdegreediscipline] Mathematics; [thesisdegreegrantor] University of Michigan, Horace H. Rackham School of Graduate Studies; [bitstreamurl] https://deepblue.lib.umich.edu/bitstream/2027.42/151674/1/yych_1.pdf; |

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…at most the number of generators of R over A. By multi-*powers*
of a finite sequence of ideals we mean an intersection of *powers* of the ideals with exponents
varying. *Symbolic* multi-*powers* are defined analogously using *symbolic* *powers* instead of
*powers*…

…We use our filtration theorems to give new results bounding the number of generators
of the multi-*powers* of a sequence of ideals and of the *symbolic* multi-*powers* as well under
various conditions. This includes the case of ordinary *symbolic* *powers* of…

…Introduction
It appears to be very difficult to give a bound on the number of generators of a *symbolic*
power or of an intersection of *powers*. In this thesis, we will introduce a powerful tool to
give a bound for some particular cases.
We prove a very powerful…

…X Iknk , and of *symbolic* multi-*powers*,
pn1 q
I1
pnk q
X Ø Ø Ø X Ik
, under various conditions. This includes the case of ordinary *symbolic*
*powers* I pnq .
1
Furthermore, we give new results bounding, by polynomials in the ni , the number of
h…

…Although we do not study the containment problem for *symbolic* *powers* here, we do want
to point out that there is considerable recent literature on the existence of constants c such
that P pcnq is contained in P n for all n. P need not be prime, although…

…polynomial in h of degree at most r.
In the very last section, we use these Ļ-filtrations to give a bound on the number of
generators of an intersection of *powers* of two ideals or the ordinary *symbolic* *powers* I pnq
under particular restrictions that we will…

…calculate the number of generators of the *symbolic*
multi-*powers* of the intersection of prime monomial ideals, i.e., the intersection of *powers* of
these prime monomial ideals.
6
Definition 5.1.1. Suppose k Ä 1. Let Nkn be the number of non-negative integer…

…height of I. See definition 2.5.23.
I is usually an ideal of a ring R.
I e the extension of ideal I. See definition 2.7.8.
I c the contraction of ideal I. See definition 2.7.8.
I psq is the *symbolic* multi-*powers* of I. See definition 2.7.29.
I pnq is…