Coherence of absolute integral closures

TL;DR

This paper proves that the absolute integral closure of certain noetherian local domains in equicharacteristic zero is not coherent in dimensions two and higher, extending known results and exploring implications for mixed characteristic cases.

Contribution

It establishes the non-coherence of absolute integral closures in equicharacteristic zero and extends related results to dimension three, providing new insights into mixed characteristic scenarios.

Findings

Absolute integral closure $R^{+}$ is not coherent for $ ext{dim}(R) extgreater=2$

Elementary proof of mixed characteristic case for $ ext{dim}(R)=2$

Extension of results to dimension 3 and implications for F-coherent rings

Abstract

We prove that the absolute integral closure $R^{+}$ of an equicharacteristic zero noetherian complete local domain $R$ is not coherent, provided $\dim(R)\geq 2$. As a corollary, we give an elementary proof of the mixed characteristic version of the result due to Asgharzadeh and extend it to dimension $3$. Furthermore, we apply the methods of Aberbach and Hochster used to prove the positive characteristic version of this result to study F-coherent rings and our work naturally suggests a mixed characteristic analogue of a result of Smith.