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 Title: The Grade Conjecture and asymptotic intersection multiplicity Author(s): Beder, Jesse Director of Research: Dutta, Sankar P. Doctoral Committee Chair(s): Griffith, Phillip A. Doctoral Committee Member(s): Dutta, Sankar P.; Schenck, Henry K.; Haboush, William J. Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): commutative algebra grade conjecture characteristic p frobenius intersection multiplicity Abstract: In this thesis, we study Peskine and Szpiro's Grade Conjecture and its connection with asymptotic intersection multiplicity $\chi_\infty$. Given an $A$-module $M$ of finite projective dimension and a system of parameters $x_1, \ldots, x_r$ for $M$, we show, under certain assumptions on $M$, that $\chi_\infty(M, A/\underline{x}) > 0$. We also give a necessary and sufficient condition on $M$ for the existence of a system of parameters $\underline{x}$ with $\chi_\infty(M, A/\underline{x}) > 0$. We then prove that if the Grade Conjecture holds for a given module $M$, then there is a system of parameters $\underline{x}$ such that $\chi_\infty(M, A/\underline{x}) > 0$. We also prove the Grade Conjecture for complete equidimensional local rings in any characteristic. Issue Date: 2013-02-03 URI: http://hdl.handle.net/2142/42274 Rights Information: Copyright 2012 Jesse Beder Date Available in IDEALS: 2013-02-03 Date Deposited: 2012-12
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