Full Record

Author | Walker, Robert |

Title | Uniform Symbolic Topologies in Non-Regular Rings |

URL | http://hdl.handle.net/2027.42/149907 |

Publication Date | 2019 |

Date Accessioned | 2019-07-08 19:44:06 |

Degree | PhD |

Discipline/Department | Mathematics |

Degree Level | doctoral |

University/Publisher | University of Michigan |

Abstract | When does a Noetherian commutative ring R have uniform symbolic topologies (USTP) on primes – read, when does there exist an integer D>0 such that the symbolic power P^{(Dr)} lies in P^r for all prime ideals P in R and all r >0? Groundbreaking work of Ein – Lazarsfeld – Smith, as extended by Hochster and Huneke, and by Ma and Schwede in turn, provides a beautiful answer in the setting of finite-dimensional excellent regular rings. Their work shows that there exists a D depending only on the Krull dimension: in other words, the exact same D works for all regular rings as stated of a fixed dimension. Referring to this last observation, we say in the thesis that the class of excellent regular rings enjoys class solidarity relative to the uniform symbolic topology property (USTP class solidarity), a strong form of uniformity. In contrast, this thesis shows that for certain classes of non-regular rings including rational surface singularities and select normal toric rings, a uniform bound D does exist but depends on the ring, not just its dimension. In particular, for rational double point surface singularities over the field C of complex numbers, we show that USTP solidarity is plainly impossible. It is natural to sleuth for analogues of the Improved Ein – Lazarsfeld – Smith Theorem where the ring R is non-regular, or where the above ideal containments can be improved using a linear function whose growth rate is slower. This thesis lies in the overlap of these research directions, working with Noetherian domains. |

Subjects/Keywords | Symbolic Powers of Ideals in Noetherian Integral Domains; Rationally Singular Combinatorially Defined Algebras; Weil divisor class groups of Noetherian normal integral domains; Mathematics; Science |

Contributors | Smith, Karen E (committee member); Jacobson, Daniel (committee member); Hochster, Mel (committee member); Jeffries, Jack (committee member); Koch, Sarah Colleen (committee member); Speyer, David E (committee member) |

Language | en |

Rights | Unrestricted |

Country of Publication | us |

Record ID | handle:2027.42/149907 |

Repository | umich |

Date Retrieved | 2019-08-20 |

Date Indexed | 2019-08-21 |

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/2027.42/149907/1/robmarsw_1.pdf; |

Sample Search Hits | Sample Images | Cited Works

…seminar talk.
1.2
A Highlight Reel Backdrop to the Dissertation Problem
In this chapter, all rings are nonzero Noetherian commutative with identity.
This thesis is focused on comparing the asymptotic growth of *symbolic* *powers* of
ideals in Noetherian…

…43].
We investigate two collections of ideals, namely, the regular and *symbolic* *powers* of
a fixed ideal, invoking geometric, combinatorial, or algebraic considerations.
To clarify, suppose we fix an ideal I in a Noetherian commutative ring R, say…

…instance, in the polynomial ring R = R[x, y] in
two real variables, if I = (x, y)R, then I 2 = (x2 , xy, y 2 )R. Meanwhile, the *symbolic*
*powers* of I are a family of ideals {I (N ) } in R indexed by positive…

…Z/2Z.
The reader should not infer from the above example that computation of *symbolic*
*powers* is easy. Indeed, *symbolic* *powers* are difficult to understand algebraically –
it is generally hard to find generating sets for them. They are more readily…

…a radical ideal I in S, and Z = Zeros(I) ⊆ X
the zero locus of I in X. The Zariski – Nagata theorem says the *symbolic*- and
differential *powers* of I coincide (see [17, Thm. 3.14], [18], and [59, Cor. 2.9]…

…radical ideal I. Indeed, all
prime ideals are maximal, and one can show that (*symbolic*) *powers* of distinct
maximal ideals are comaximal, so ideal intersections and ideal products coincide.
The radical ideal I is a finite product of maximal ideals…

…observation to make
is that given comaximal ideals I, J in any Noetherian commutative ring R,
the *symbolic* *powers* I (a) and J (b) are comaximal for all a, b ∈ Z≥0 , and hence
I (a) ∩ J (b) = I (a) J (b…

…an ideal in a Noetherian commutative ring R, its regular *powers* {I N } and
its *symbolic* *powers* {I (N ) } each form a graded sequence of ideals. As discussed in
papers such as [13, 14, 16, 37, 40, 41, 47, 50, 51]…