Student Commutative Algebra Seminar

When analyzing the singularities of algebraic varieties over a field of positive characteristics, Kunz's theorem plays a crucial role in determining the regularity of the ring. For instance, suppose R is a Noetherian local ring over an F-finite field. Then, the theorem states that the ring R is regular if and only if R^(1/p^e) is a free R-module for some (and for all) e>0. This suggests that the ring R is not free if R^(1/p^e) is not free and that investigating the extent of R^(1/p^e) being close to a free module measures the singularity of the ring. Under this point of view, we will define F-split and strongly F-regular rings in this talk and discuss several examples of strongly F-regular rings. Furthermore, we will discuss Fedder's criterion, which detects when the ring becomes F-split. Speaker(s): Seungsu Lee (University of Utah)