Potent Elements and Tight Closure in Artinian Modules.

Tight Closure;Artinian Module;Potent Element;Isolated Singularity;Local Cohomology;Finitistic Tight Closure;Mathematics;Science;Mathematics

Stubbs, JoeTappenden, Jamie ;

University of Michigan

关键词: Tight Closure;    Artinian Module;    Potent Element;    Isolated Singularity;    Local Cohomology;    Finitistic Tight Closure;    Mathematics;    Science;    Mathematics;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/60783/jstubbsz_1.pdf?sequence=1&isAllowed=y

瑞士|英语

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

【 摘 要 】

In this thesis we develop the theory of potent elements, a new method for studying the question of whether tight closure equals finitistic tight closure for the zero submodule of an arbitrary Artinian module.G. Lyubeznik and K. E. Smith conjectured that the two types of tight closure agree for Artinian modules over an excellent local ring.Our main results simultaneously unify and generalize results in the literature and are strong enough to settle the conjecture of Lyubeznik and Smith;;s in new cases.For example, our results establish that tight closure equals finitistic tight closure in every $Z$-graded Artinian module over a polynomial ring over an isolated singularity.Along the way we prove a difficult uniform annihilator theorem for certain $Ext$ modules which is of independent interest. We also develop the theory of $u$-split complexes which we expect to have further applications.