The Cartier Core Map and F-graded Systems

Abstract

This dissertation studies singularities in positive characteristic rings and the operators that define these singularities. One approach we take is via the obstructions to strong F-regularity: given a commutative Noetherian F-finite ring R of prime characteristic and a Cartier algebra D, we define a self-map, called the Cartier core, on the Frobenius split locus of the pair (R,D) by sending a point P to the splitting prime of (R_P,D_P). We prove the Cartier core map is continuous, containment preserving, and fixes the D-compatible ideals. We show the Cartier core map can be extended to arbitrary ideals J, where it outputs the largest D-compatible ideal contained in J in the case that the pair (R,D) is Frobenius split. The other approach we take is by studying F-graded systems of ideals in R, which are sequences of ideals giving rise to Cartier algebras on R. We identify how properties of these systems (or modifications of these systems) affect the singularity properties of the corresponding Cartier algebra. In particular, we show that in a regular local ring for a special class of such systems called p-families, strong F-regularity and F-splitting are the same. Further, we make use of this and a new operation we introduce called p-stabilization to get a criterion that in a regular local ring, a system is strongly F-regular exactly when its p-stabilization is F-split. Finally, we associate a combinatorial object to systems built out of monomial ideals in such a way that encapsulates the behavior of the p-stabilization.