Tight Closure, Local Cohomology, and Related Questions

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The proposed research problems stem from long-standing questions and conjectures in commutative algebra. These are related to the tight closure theory of Hochster and Huneke, to the homological conjectures, and to the theory of local cohomology. The PI will pursue an approach to Hochster's monomial conjecture which lies at the intersection of these three topics. This conjecture is unresolved for rings which do not contain a field, such as those which arise in number theory. The proposed approach involves annihilating the elements of obstruction local cohomology modules by elements of arbitrarily low valuation. This idea has proved remarkably strong in the work of Heitmann, where he settled the monomial conjecture for rings of dimension up to three. Obtaining a description of such annihilators of local cohomology is a vast program, and the proposed research will focus on some concrete initial cases. In joint work with Uli Walther, the PI will work on Lyubeznik's conjecture that local cohomology modules of regular rings have finitely many associated prime ideals. This is now known in various cases due to the work of Huneke-Sharp and Lyubeznik, but remains unresolved for polynomial rings over the integers.Commutative algebra is a field closely related to algebraic geometry: while algebraic geometry focuses on the geometry of solutions sets of polynomial equations, in commutative algebra the main objects of study are functions on these solution sets. Most of the questions which will be investigated in the proposed research are 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.