A note on the non-existence of small Cohen-Macaulay algebras

TL;DR

This paper constructs examples of certain algebraic structures in positive characteristic that cannot be extended to Cohen-Macaulay algebras, highlighting limitations in their existence.

Contribution

It introduces p-adic obstructions to demonstrate the non-existence of small Cohen-Macaulay algebras in specific cases.

Findings

Existence of rings without module-finite Cohen-Macaulay extensions

Contrast with Hochster-Huneke's result on unions of extensions

New obstructions in positive characteristic algebra

Abstract

By finding a p-adic obstruction, we construct many examples of positive characteristic complete noetherian local rings which do not admit any module-finite Cohen-Macaulay extension. These examples should be contrasted with a result of Hochster-Huneke: the union of all finite extensions is always Cohen-Macaulay.