Abelian extensions of equicharacteristic regular rings need not be Cohen-Macaulay
Aryaman Maithani, Anurag K. Singh, Prashanth Sridhar
TL;DR
Roberts' theorem guarantees that the integral closure of a regular ring in a finite abelian extension is Cohen-Macaulay when the extension degree is coprime to the residue characteristic; this paper shows that in equal characteristic such guarantees can fail for elementary abelian extensions of order $p^2$. It constructs explicit invariant-ring-based counterexamples in characteristic $p$ by taking $R=T^G$ with $T$ a polynomial ring over $\mathbb{F}_p$ and $G$ an abelian $p$-group, then letting $H$ be a suitable subgroup so that $S=T^H$ is the integral closure of $R$ in a finite abelian extension with Galois group $G/H$ and $S$ is not Cohen-Macaulay (via Kemper's bireflection criterion). The results yield non-Cohen-Macaulay $S$ with arbitrarily large Cohen-Macaulay defect, preserved under localization at the homogeneous maximal ideal, and extend to towers with depth growing as $d+2$ and defect $d-2$. This work links non-Cohen-Macaulay behavior in equal-characteristic invariant rings to Roberts-type questions and illustrates the natural occurrence of such phenomena in modular invariant theory.
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.