The Grade Conjecture and asymptotic intersection multiplicity

commutative algebra;grade conjecture;characteristic p;frobenius;intersection multiplicity

Beder, Jesse

关键词: commutative algebra;    grade conjecture;    characteristic p;    frobenius;    intersection multiplicity;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/42274/Jesse_Beder.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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.