$F$-regularity, test elements, and smooth base change

The Smooth Base Change Theorem

F-regularity relative to modules

In this paper we will generalize  some of known results on the tight closure of an ideal to the tight closure of an ideal relative to a module .

Deformation of F-purity and F-regularity

For a Noetherian local domain (R, m, K), it is an open question whether strong F–regularity deforms. We provide an affirmative answer to this question when the canonical module satisfies certain additional assumptions. The techniques used here involve passing to a Gorenstein ring, using an anti– canonical cover.

F -pure Homomorphisms, Strong F -regularity, and F -injectivity

We discuss Matijevic–Roberts type theorem on strong F -regularity, F -purity, and Cohen–Macaulay F -injective (CMFI for short) property. Related to this problem, we also discuss the base change problem and the openness of loci of these properties. In particular, we define the notion of F -purity of homomorphisms using Radu–André homomorphisms, and prove basic properties of it. We also discuss a...

F-regularity Does Not Deform

We show that the property of F-regularity does not deform, and thereby settle a longstanding open question in the theory of tight closure. Specifically, we construct a three dimensional N-graded domain R which is not F-regular (or even F-pure), but has a quotient R=tR which is F-regular. Examples are constructed over fields of characteristic p > 0, as well as over fields of characteristic zero.