A small Cohen-Macaulay module over a Noetherian local ring 
is a nonzero finitely
generated module 
with module depth equal to
the Krull dimension of 
. It is also called a finitely generated maximal Cohen-Macaulay
module. The word small distinguishes it from big
Cohen-Macaulay modules, which need not be finitely
generated.
The small Cohen-Macaulay module conjecture asks whether every complete local ring that is Noetherian and an integral
domain has such a module (Hochster 1975). A Cohen-Macaulay
ring supplies the example 
. Chen (2026) reported an obstruction and a proposed three-dimensional
counterexample from a completed section ring
of a complex projective surface.
The asserted nonexistence concerns all finitely generated modules, including those that do not arise from finite ring extensions. Ruling out Cohen-Macaulay rings finite over the integral domain alone would not suffice.
