Local cohomology bounds and test ideals

TL;DR

This paper establishes conditions under which the finitistic test ideal equals the test ideal in prime characteristic rings and equates $F$-regularity notions for certain 4-dimensional rings, advancing understanding in algebraic geometry.

Contribution

It provides new sufficient conditions for the equality of test ideals and links $F$-regularity notions in specific ring classes, leveraging recent progress in the prime characteristic minimal model program.

Findings

Finitistic test ideal equals test ideal under certain conditions

Equivalence of $F$-regular and strongly $F$-regular for 4-dimensional rings

Advances in algebraic geometry via prime characteristic minimal model program

Abstract

We find sufficient conditions which imply equality of the finitistic test ideal and test ideal in rings of prime characteristic. Utilizing recent progress from the prime characteristic minimal model program we equate the notions of $F$-regular and strongly $F$-regular for 4-dimensional rings essentially of finite type over a field of prime characteristic $p>5$.