Studies in Commutative Algebra

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The PI will work on several problems in commutative algebra that are centered around the homological conjectures, tight closure, and local cohomology. Some of these, such as the homological conjectures, are old problems for which recent advances have provided the hope of a solution; the problems on tight closure and finiteness properties of local cohomology modules are a continuation of the PI's long-term projects. Hochster's monomial conjecture is unresolved for rings that do not contain a field, such as those arising in number theory. Two approaches to this will be pursued: the first is a natural extension of Heitmann's work; another is via local cohomology theory. Brenner and Monsky recently proved that tight closure need not commute with localization. However, it appears likely that weak F-regularity---the property that all ideals of a ring are tightly closed---does localize. This will be approached via the notion of splinter rings. It is also proposed to attack Lyubeznik's conjecture that local cohomology modules of regular rings have finitely many associated prime ideals. This is now known in various cases, but remains unresolved for polynomial rings over the integers.This project is concerned with questions in commutative algebra. This is a field closely related to algebraic geometry: while algebraic geometry focuses on the geometry of solution sets of polynomial equations, the point of view in commutative algebra is to study the ring of polynomial functions on a solution set. Most of the questions that will be investigated may be viewed as questions about the existence of solutions for families of equations, and about the nature of the solution sets. Commutative algebra continues to develop a fascinating interaction with several branches of mathematics, and is becoming an increasingly valuable tool in engineering, coding theory, cryptography, and other applications of strategic interest.