Fregularity

Fregularity is a property studied in commutative algebra and algebraic geometry, defined for Noetherian rings (often local rings) of prime characteristic p > 0. It sits in the framework of tight closure theory and Frobenius endomorphisms, providing a way to measure how mild a ring’s singularities are in positive characteristic. In many texts the terms “strongly F-regular” and “F-regular” are used, with the former emphasizing a stronger technical condition; in practice they are closely related and often taken to coincide.

A common formulation is that a Noetherian ring R of characteristic p > 0 is strongly F-regular if, for every nonzero element c in R, there exists an integer e ≥ 0 such that the R-linear map R → F^e(R) sending 1 to c and extended via Frobenius splits as an R-module homomorphism. Equivalently, strong F-regularity implies that every ideal of R is tightly closed. This tight closure perspective links F-regularity to a finiteness of certain closure operations and to the behavior of singularities under Frobenius.

Key consequences of F-regularity include favorable singularity properties: strongly F-regular rings are typically Cohen–Macaulay and normal, and they exhibit good behavior under localization. The concept provides a positive characteristic analog to certain mild singularities in characteristic zero, and reductions to characteristic p are used to study how singularities behave across characteristics.

Examples and scope: all regular rings of characteristic p are strongly F-regular, and a wide class of singularities in positive characteristic are known to be F-regular. The theory connects to broader themes such as F-purity, tight closure, and the study of singularities across characteristics. See also tight closure, F-purity, and Cohen–Macaulay properties.