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Author
Title Uniform Symbolic Topologies in Non-Regular Rings
URL
Publication Date
Date Accessioned
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
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;

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…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…

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…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]…

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