On the non-existence of small Cohen–Macaulay algebras

By finding a p  -adic obstruction, we construct many examples of complete noetherian local normal FpFp-algebras R   such that no module-finite extension R↪SR↪S is Cohen–Macaulay. These examples should be contrasted with a result of Hochster–Huneke: the directed union of all such extensions is always Cohen–Macaulay.