Integer torsion in local cohomology, and questions on tight closure theory

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Title Integer torsion in local cohomology, and questions on tight closure theory
Publication Type dissertation
School or College College of Science
Department Mathematics
Author Chan, Julian
Date 2011-05
Description Grothendieck's theory of local cohomology has applications to basic questions such as determining the minimal number of polynomial equations needed to define an algebraic set. These modules are typically not finitely generated, and a question of Huneke asks whether they have finitely many associated prime ideals. This was settled in the negative by Singh, who constructed a local cohomology module that has prime-torsion for each prime integer. We extend this work in Chapter 1 by showing that the module in question contains a copy of each finitely generated abelian group. Moreover, the module has a natural fine grading, and we are able to show that each finitely generated abelian group embeds into a single graded component. In Chapter 2 we study F-injectivity, i.e., the property that the Frobenius action on local cohomology modules is injective. We obtain an effective criterion to determine if a diagonal subalgebra of a bigraded hypersurface is F-injective; such subalgebras provide a rich source of examples of various F-singularities, and thus form a natural testing ground for various questions and conjectures related to classes of singularities defined via the Frobenius map. Chapter 3 is an investigation of the tight closure properties of rings of invariants of finite groups acting linearly on polynomial rings over fields of positive characteristic.
Type Text
Publisher University of Utah
Subject Integer torsion; Cohomology; Tight closure theory; Grothendieck's theory; Local cohomology; F-injectivity; Frobenius action
Dissertation Institution University of Utah
Dissertation Name Doctor of Philosophy
Language eng
Rights Management Copyright © Julian Chan 2011
Format application/pdf
Format Medium application/pdf
Format Extent 428,384 bytes
Identifier us-etd3,21994
Source Original housed in Marriott Library Special Collections, QA3.5 2011 .C43
ARK ark:/87278/s60g40wh
Setname ir_etd
ID 194539
Reference URL https://collections.lib.utah.edu/ark:/87278/s60g40wh
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