2025/11/25 by Maithani, Aryaman, Singh, Anurag K., Sridhar, Prashanth
Mathematics · Computer Science · #Commutative Algebra and Its Applications #Coding theory and cryptography #Rings, Modules, and Algebras
paper · doi:10.48550/arxiv.2511.19800
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 p2 is not Cohen-Macaulay. Localizing at the homogeneous maximal ideal preserves the essential features of the construction.