Abelian extensions of equicharacteristic regular rings need not be Cohen-Macaulay

TL;DR

This paper demonstrates that in equicharacteristic regular rings, abelian extensions may not preserve the Cohen-Macaulay property when the extension degree shares a prime factor with the residue field characteristic, contrary to prior results.

Contribution

It provides explicit counterexamples showing that the Cohen-Macaulay property can fail in abelian extensions of equicharacteristic regular rings when the degree is divisible by the characteristic.

Findings

Constructed polynomial rings over fields of characteristic p with non-Cohen-Macaulay integral closures

Showed that localization preserves the non-Cohen-Macaulay property

Extended the understanding of when Cohen-Macaulayness is preserved in extensions

Abstract

By a theorem of Roberts, the integral closure of a regular local ring in a finite abelian extension of its fraction field is Cohen-Macaulay, provided that the degree of the extension is coprime to the characteristic of the residue field. We show that the result need not hold in the absence of this requirement on the characteristic: for each positive prime integer $p$, we construct polynomial rings over fields of characteristic $p$, whose integral closure in an elementary abelian extension of order $p^2$ is not Cohen-Macaulay. Localizing at the homogeneous maximal ideal preserves the essential features of the construction.