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

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

2009-08-19

arxiv.org/abs/0908.2703v3

Mathematics

Commutative Algebra

37 pages, updated the bibliography, and modified some errors

Scientific paper

10.1080/00927870903431241

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.

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