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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

…*filtration* *theorems* to give new results bounding the number of generators
of the multi-powers of a… …LIST OF FIGURES
Figure
3.1
Commutative diagram of factors of an ω 2 -*filtration*… …the
*filtration*.
In the graded case, the number of occurrences of A{p up to a given… …standard
graded algebras over a field. We use these *theorems* to give new results bounding the… …Definition 3.1.1. Let M be a R-module. We define recursively the notion of an ω r -*filtration*
2…

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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 August 06, 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. 06 Aug 2020.

Vancouver:

Chen Y. Filtration Theorems and Bounding Generators of Symbolic Multi-powers. [Internet] [Doctoral dissertation]. University of Michigan; 2019. [cited 2020 Aug 06]. 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