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

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.

💡 Research Summary

The paper investigates the necessity of the hypothesis “the order of the Galois group is coprime to the residue characteristic” in Paul Roberts’ theorem on the Cohen‑Macaulay property of integral closures. Roberts proved that if (R) is a regular local ring with fraction field (K) and (L/K) is a finite abelian Galois extension whose Galois group order is prime to the characteristic of the residue field of (R), then the integral closure (S) of (R) in (L) is Cohen‑Macaulay. The authors construct explicit counter‑examples showing that when this coprimality condition is removed, the conclusion can fail even in equal characteristic (p>0).

The construction uses modular invariant theory. Let (p) be any prime and consider the polynomial ring
(T = \mathbb{F}_{p}