On the semicontinuity problem of fibers and global $F$-regularity

TL;DR

This paper investigates the semicontinuity of fiber properties in scheme morphisms, focusing on local and global aspects, including localization theorems for specific rings and the deformation of global $F$-regularity.

Contribution

It provides new localization theorems for seminormal and $F$-rational rings and explores the deformation of global $F$-regularity, advancing understanding of fiber property semicontinuity.

Findings

Localization theorems for seminormal rings

Localization theorems for $F$-rational rings

Deformation analysis of global $F$-regularity

Abstract

In this article, we discuss the semicontinuity problem of certain properties on fibers for a morphism of schemes. One aspect of this problem is local. Namely, we consider properties of schemes at the level of local rings, in which the main results are established by solving the lifting and localization problems for local rings. In particular, we obtain the localization theorems in the case of seminormal and $F$-rational rings, respectively. Another aspect of this problem is global, which is often related to the vanishing problem of certain higher direct image sheaves. As a test example, we consider the deformation of the global $F$-regularity.