Studies on Integrality of Ideals

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

This project deals with questions in commutative algebra. While the questions address a variety of topics, there is an underlying common thread that unifies them all: the theory of integral closure. The PI, together with her collaborators, first plans to develop methods that produce the integral closure of an ideal. In joint work with Bernd Ulrich the proposer has established that linkage is a method for capturing integral elements over a complete intersection. Now the investigator wants to extend this same procedure to Gorenstein linkage and residual intersections. Second, the PI intends to clarify the connection between the core of an ideal and the adjoint of Lipman (or the multiplier ideal of Ein and Lazarsfeld), find an explicit formula for the core, and compute the core of monomial ideals (a question posed by Eisenbud and Sturmfels). The investigator intends to use linkage theory, residual intersection theory and Groebner basis theory. Third, the investigator would like to study the special fiber ring. This object is very important from a geometric point of view because it encodes algebraic information on the special fiber of the blowup. The focus here is on finding conditions that force the fiber to be unmixed or even Cohen-Macaulay. Last, the PI plans to study the Hilbert function of zero-dimensional normal ideals (an ideal is normal if all its powers are integrally closed). One of the proposer's goals is to show that all the Hilbert coefficients of such ideals are non-negative, while another goal is to characterize the vanishing of the Hilbert coefficients in terms of the Cohen-Macaulayness of the associated graded ring of the ideal itself or of one of its powers. The proposed research is concerned with problems in commutative algebra. On a basic level commutative algebra is about techniques for solving systems of polynomial equations. Commutative algebra had a revolutionary growth in the past fifty years as it provided the tools for understanding many problems in pure and applied mathematics. In many applied problems, polynomial equations and hence commutative algebra play a crucial role. Applied areas where commutative algebra results have been used in the past include operations research, computer science, robotics, control theory, coding theory and cryptography to mention a few.