F-coherent rings with applications to tight closure theory

Abstract

The aim of this paper is to introduce a new class of Noetherian rings of positive characteristic in terms of perfect closures and study their basic properties. If the perfect closure of a Noetherian ring is coherent, we call it an $F$-coherent ring. Some interesting applications are given in connection with tight closure theory. In particular, we discuss relationships between $F$-coherent rings and $F$-pure, $F$-regular, and $F$-injective rings. The final section discusses how the coherent property effects the behavior of tight closure for general perfect rings.

10 pages, comments are welcome

References in corpus (3)

• On seminormal monoid rings
• Tight closure does not commute with localization
• Homological properties of the perfect and absolute integral closures of Noetherian domains

Cited by in corpus (6)

• On the Witt vectors of perfect rings in positive characteristic
• Coherence of absolute integral closures
• A variant of perfectoid Abhyankar's lemma and almost Cohen-Macaulay algebras
• Homological aspects of perfect algebras
• Remarks on the Small Cohen-Macaulay conjecture and new instances of maximal Cohen-Macaulay modules
• On the Gorensetein $(n,d)$-Flat and Gorensetein $(n,d)$-Injective Modules