Studies in Commutative Algebra

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This research project concerns the study of questions in the theory of commutative algebra. This is a field that deals with the study of algebraic varieties: geometric objects given as the solutions of a system of polynomial equations. For example, the solution set of a single polynomial equation in two variables can be geometrically realized as a curve in the plane (e.g., a parabola y=x^2). One important aspect is to understand the local picture of these solution sets, which has applications in many sciences and engineering. For example, the parabola is smooth (meaning that locally it looks like a line) while the curve defined by y^2=x^3 is smooth except at the origin, where it is a cusp. The singular or non-smooth points of an algebraic variety have rich and subtle local structure, and detailing their properties is a crucial part of many investigations. The projects that will be explored are focused on the local properties of algebraic varieties, such as their local intersection numbers and the multiplicities (which is a measure of how bad the singular points are). Several of the questions under study are longstanding and of fundamental importance. This project investigates several longstanding open questions in commutative algebra from new perspectives. One is to attack Serre's conjecture on positivity of intersection multiplicities and Hochster's direct summand conjecture using a new notion called lim Cohen-Macaulay sequences of modules, whose existence will establish both conjectures. Another is to attack Lech's conjecture on Hilbert-Samuel multiplicities using advances in positive characteristic methods. The investigator recently settled this conjecture in dimension three in equal characteristic (the conjecture was previously known only in dimension less than or equal to two). The goal here is to improve the methods and to seek solutions in higher dimension. Other research projects include studying singularities in positive characteristic, especially investigating their behavior under deformation, under passing to a generic linkage, and their connections with F-module and D-module theory. A crucial technique to be employed is the Frobenius structures on local cohomology modules.