$F$-pure homomorphisms, strong $F$-regularity, and $F$-injectivity

TL;DR

This paper explores properties of $F$-purity, strong $F$-regularity, and $F$-injectivity in algebraic rings, establishing their relationships, base change behaviors, and defining new notions using homomorphisms.

Contribution

It introduces the notion of $F$-purity of homomorphisms, compares strong and very strong $F$-regularity, and proves their equivalence in certain classes of rings.

Findings

$F$-purity of homomorphisms defined using Radu-Andre homomorphisms.

Strong and very strong $F$-regularity coincide in local, $F$-finite, and essentially finite-type rings.

$F$-pure base change of strong $F$-regularity proved.

Abstract

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-Andre homomorphisms, and prove basic properties of it. We also discuss a strong version of strong $F$-regularity (very strong $F$-regularity), and compare these two versions of strong $F$-regularity. As a result, strong $F$-regularity and very strong $F$-regularity agree for local rings, $F$-finite rings, and essentially finite-type algebras over an excellent local rings. We prove the $F$-pure base change of strong $F$-regularity.