This project is in the field of commutative algebra, especially Noetherian algebras and more especially polynomial rings over fields. The research in this proposal is directed at understanding the asymptotics of equations and their reduction modulo large prime numbers. Such asymptotics are captured through local cohomology, Hilbert-Kunz multiplicities, symbolic powers, and tight closure. The methods proposed are in part classical methods as well as those being developed by the proposer. The project also studies homological algebra, especially in terms of the graded Betti numbers and in terms of rings of finite Cohen-Macaulay type. A new asymptotic length function for local cohomology is proposed with applications to the homological conjectures, especially the monomial conjecture. Additional problems are proposed on the theory of liaison, especially regarding comparing the local and homogeneous versions, as well as understanding Gorenstein liaison classes.
Commutative algebra arose from the 19th century study of polynomial equations in many variables, and their solutions. The relationship between polynomial equations and geometry goes back at least to Descartes and the idea of coordinatizing the plane. Commutative algebra studies the solutions of such polynomial or power series equations by forming an algebraic object, called a ring, which consists of the 'generic' solutions. The algebraic properties of these generic solutions then give insight into the geometric and algebraic nature of the solutions. An important technique in this field has been to study such equations by reducing the coefficients modulo prime numbers for all large primes. A particular example of this has been the explosive development of the theory of tight closure over the last twenty years. Commutative algebra combines techniques from a number of other areas including combinatorics, topology, and analysis.
This grant was used to support students and myself in several projects in the field of commutative algebra. This field is the study of solutions of polynomial equations. These equations are typically polynomials in many variables, and we study their solutions by forming an algebraic structure, called a ring, in which the equations have a type of generic solution. One of the main tools used in this study is called "reduction to characteristic p", which consists a a method to study the equations over the integers with what is called mod p arithmetic. In this arithmetic, the number p (a prime number) is set to be zero, and all operations are then computed as if that number is zero. For example in mod 7 arithmetic, 6+6 = 5, since 6+6 = 7+5, but 7=0. Such a reductions turns out to be an extremely powerful tool due to the presence of the Frobenius map--a map taking elements to their pth powers. In particular, I studied several numerical invariants of singularities which arise from mod p arithmetic, and give very detailed information about the nature of the singularities. Here, a "singular" point of equations is a place where the behavior of solutions nearby are not smooth--don't have tangents. My field studies such singularities in great detail. I have disseminated my work through publications in refereed journals, as well as through giving many talks both in the United States and abroad, and through the organization of international conferences which bring experts together.