Regularity and intersections of bracket powers

TL;DR

This paper characterizes regular rings in prime characteristic by the commutation of ideal intersections with bracket powers, highlighting the importance of reducedness and connecting to Frobenius map flatness and content theory.

Contribution

It establishes a new characterization of regular rings via intersection and bracket powers, linking it to Frobenius flatness and content theory in prime characteristic.

Findings

Regular rings are characterized by intersection-bracket power commutation.

Reducedness is necessary for the equivalence.

Connections to Frobenius flatness and Ohm-Rush content theory.

Abstract

Among reduced Noetherian prime characteristic commutative rings, we prove that a regular ring is precisely one where finite intersection of ideals commutes with taking bracket powers. However, reducedness is essential for this equivalence. Connections are made with Ohm-Rush content theory, intersection-flatness of the Frobenius map, and various flatness criteria.