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

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–André 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.

Key Words:

• F-Injective
• F-Pure
• Strongly F-regular

2000 Mathematics Subject Classification:

• Primary 13A35
• Secondary 14L30

ACKNOWLEDGMENTS

The author thanks Professor K.-i. Watanabe for communicating the author his result (see Remark 2.20). Special thanks are also due to Professor A. Singh and Professor K.-i. Yoshida for valuable advice. The author is grateful to Professor K. Schwede and Professor F. Enescu for giving valuable comments to the former version of this article.

Notes

Communicated by A. Singh.