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1.
Chen, Yuanyuan.
Filtration Theorems and Bounding Generators of *Symbolic* Multi-* powers*.

Degree: PhD, Mathematics, 2019, University of Michigan

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

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.
*Advisors/Committee Members: Hochster, Mel (committee member), Tappenden, James P (committee member), Canton, Eric (committee member), Derksen, Harm (committee member), Smith, Karen E (committee member).*

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

…*Symbolic* multi-*powers* are defined analogously using *symbolic* *powers* instead of
*powers*. We use our… …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… …*symbolic*
power or of an intersection of *powers*. In this thesis, we will introduce a powerful tool… …number
of generators of the multi-*powers* of ideals, i.e., I1n1 X ¨ ¨ ¨ X Iknk , and of *symbolic*…

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Chen, Y. (2019). Filtration Theorems and Bounding Generators of Symbolic Multi-powers. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/151674

Chicago Manual of Style (16^{th} Edition):

Chen, Yuanyuan. “Filtration Theorems and Bounding Generators of Symbolic Multi-powers.” 2019. Doctoral Dissertation, University of Michigan. Accessed September 29, 2020. http://hdl.handle.net/2027.42/151674.

MLA Handbook (7^{th} Edition):

Chen, Yuanyuan. “Filtration Theorems and Bounding Generators of Symbolic Multi-powers.” 2019. Web. 29 Sep 2020.

Vancouver:

Chen Y. Filtration Theorems and Bounding Generators of Symbolic Multi-powers. [Internet] [Doctoral dissertation]. University of Michigan; 2019. [cited 2020 Sep 29]. Available from: http://hdl.handle.net/2027.42/151674.

Council of Science Editors:

Chen Y. Filtration Theorems and Bounding Generators of Symbolic Multi-powers. [Doctoral Dissertation]. University of Michigan; 2019. Available from: http://hdl.handle.net/2027.42/151674

2.
Walker, Robert.
Uniform *Symbolic* Topologies in Non-Regular Rings.

Degree: PhD, Mathematics, 2019, University of Michigan

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

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.
*Advisors/Committee Members: 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).*

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

…We investigate two collections of ideals, namely, the regular and *symbolic* *powers* of
a… …the *symbolic*
*powers* of I are a family of ideals {I (N ) } in R indexed… …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… …Nagata theorem says the *symbolic*- and
differential *powers* of I coincide (see [17, Thm…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Walker, R. (2019). Uniform Symbolic Topologies in Non-Regular Rings. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/149907

Chicago Manual of Style (16^{th} Edition):

Walker, Robert. “Uniform Symbolic Topologies in Non-Regular Rings.” 2019. Doctoral Dissertation, University of Michigan. Accessed September 29, 2020. http://hdl.handle.net/2027.42/149907.

MLA Handbook (7^{th} Edition):

Walker, Robert. “Uniform Symbolic Topologies in Non-Regular Rings.” 2019. Web. 29 Sep 2020.

Vancouver:

Walker R. Uniform Symbolic Topologies in Non-Regular Rings. [Internet] [Doctoral dissertation]. University of Michigan; 2019. [cited 2020 Sep 29]. Available from: http://hdl.handle.net/2027.42/149907.

Council of Science Editors:

Walker R. Uniform Symbolic Topologies in Non-Regular Rings. [Doctoral Dissertation]. University of Michigan; 2019. Available from: http://hdl.handle.net/2027.42/149907