On the vanishing of local cohomology of the absolute integral closure in positive characteristic

TL;DR

This paper extends the vanishing results of local cohomology for rings in positive characteristic, showing that their absolute integral closures form big Cohen-Macaulay algebras, generalizing previous results with a simpler proof.

Contribution

It generalizes the vanishing of local cohomology to rings that are images of Cohen-Macaulay rings, simplifying the proof of their absolute integral closures being Cohen-Macaulay.

Findings

All local cohomology below the dimension maps to zero in a finite extension.

Absolute integral closure of such rings is a big Cohen-Macaulay algebra.

Generalizes previous results to a broader class of rings.

Abstract

The aim of this paper is to extend the main result of C. Huneke and G. Lyubeznik in [Adv. Math. 210 (2007), 498--504] to the class of rings that are images of Cohen-Macaulay local rings. Namely, let $R$ be a local Noetherian domain of positive characteristic that is an image of a Cohen-Macaulay local ring. We prove that all local cohomology of $R$ (below the dimension) maps to zero in a finite extension of the ring. As a direct consequence we obtain that the absolute integral closure of $R$ is a big Cohen-Macaulay algebra. Since every excellent local ring is an image of a Cohen-Macaulay local ring, this result is a generalization of the main result of M. Hochster and Huneke in [Ann. of Math. 135 (1992), 45--79] with a simpler proof.