The Grade Conjecture and asymptotic intersection multiplicity

Beder, Jesse

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https://hdl.handle.net/2142/42274 Copy

Description

Title

The Grade Conjecture and asymptotic intersection multiplicity

Author(s)

Beder, Jesse

Issue Date

2013-02-03T19:29:58Z

Director of Research (if dissertation) or Advisor (if thesis)

Dutta, Sankar P.

Doctoral Committee Chair(s)

Griffith, Phillip A.

Committee Member(s)

• Dutta, Sankar P.
• Schenck, Henry K.
• Haboush, William J.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(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.

Graduation Semester

2012-12

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http://hdl.handle.net/2142/42274

Copyright and License Information

Copyright 2012 Jesse Beder

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