Example of noetherian local ring without Maximal Cohen-Macaulay modules.

Question

A finitely generated module $M$ with $\mathrm{depth}_{R}(M)= \dim R$ is called Maximal Cohen-Macaulay module. I want to find an example of noetherian local ring without Maximal Cohen-Macaulay modules. Can somebody give an example or some references?

Answer 1

A domain that admits a maximal Cohen-Macaulay module must be universally catenary (see the stacks project or Cohen-Macaulay modules by Hochster), so one can look to examples of local domains that are not universally catenary. Such examples were constructed by Nagata; see the stacks project here for one such example of a local domain that is not even catenary. There is also a famous example of Nagata of a local domain that is catenary but not universally catenary, which can be used instead (see Section 3 here).

The only known examples have the types of pathologies the examples mentioned above do, and I should point out there is a famous open question due to Hochster as to whether every complete local domain admits a maximal Cohen-Macaulay module (termed a small Cohen-Macaulay module in the literature). Actually, Hochster conjectured originally in the 1970's that this question has an affirmative answer, but later conjectured in the 2000's the answer to be negative (see Page 3 here for an overview). These days the answer is largely thought to be negative, but this remains unproven.