Prime Characteristic Rings, Birational Morphisms, and Valuations

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Commutative algebra is the local theory of algebraic geometry, and algebraic geometry is the study of geometric objects defined by polynomial equations. The study of algebraic geometry stems from Descartes’ introduction of the coordinate plane to better understand the geometry of objects defined by polynomial equations. For example, the polynomial equation y=x^2 defines a parabola in the coordinate plane and the polynomial equation x^2+y^2=1 defines a circle. In contrast, the graph of the equation y^2 = x^3 has a sharp point, a type of singularity. Geometric objects defined by polynomial equations may admit singularities. The local study of these singularities is a necessity to understanding the geometry of the object. The main research goals of this project concern itself with the study of singularities when the defining equations have coefficients in a positive characteristic numbering system. Singularities of prime characteristic rings are studied, classified, and understood through descriptive behavior of the Frobenius endomorphism. This project focuses on the study of F-pure and F-regular singularities, numerical invariants designed to relate rings among these singularity classes, and test ideals. A particular emphasis is to determine sufficient conditions that imply equality of the finitistic and big test ideal of a local F-pure ring. The purpose of doing so is to investigate the weak-implies-strong conjecture from tight closure theory, to establish unifying behavior of F-regular rings, and to relate the numerical invariant F-signature with the behavior of valuation rings centered over an F-regular ring.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.