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 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 2020-09-01 Date Indexed 2020-09-09 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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