Coherence of absolute integral closures

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.