Adic Finiteness: Bounding Homology and Applications

TL;DR

This paper introduces adic finiteness conditions to establish amplitude inequalities and characterizes regular local rings in prime characteristic via derived local cohomology and Frobenius endomorphism.

Contribution

It extends amplitude inequalities by replacing finite generation with adic finiteness and provides a new criterion for regularity of local rings using derived local cohomology.

Findings

Adic finiteness replaces finite generation in amplitude inequalities.

A local ring is regular iff derived local cohomology has finite flat dimension under Frobenius.

New connections between homological properties and ring regularity in prime characteristic.

Abstract

We prove a versions of amplitude inequalities of Iversen, Foxby and Iyengar, and Frankild and Sather-Wagstaff that replace finite generation conditions with adic finiteness conditions. As an application, we prove that a local ring $R$ of prime characteristic is regular if and only if for some proper ideal $\mathfrak b$ the derived local cohomology complex $\mathbf{R}\Gamma_{\mathfrak{b}}(R)$ has finite flat dimension when viewed through some positive power of the Frobenius endomorphism.