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

Mitsuyasu Hashimoto

Published 2009-08-19, updated 2010-11-24Version 3

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.