A property of a Noetherian local ring or a module meaning that its depth equals its Krull dimension (locally), i.e., it attains maximal possible depth relative to its dimension.

Principle

Maximal non‑vanishing of regular sequences: a Cohen–Macaulay object has the largest possible length of a regular sequence, so homological obstructions are minimized.

Demonstration

Regular local rings and polynomial rings are Cohen–Macaulay. For example, k[x1,...,xn] and its localizations have depth n equal to their Krull dimension n.

Misapplication

Mistaking Cohen–Macaulay for regularity or freeness: Cohen–Macaulay guarantees depth = dimension but does not imply the ring is regular (regularity requires finite global dimension).

Consequence

Cohen–Macaulay rings enjoy favorable homological and geometric properties: nicer behavior of depth under localization, unmixedness of ideals, well‑behaved multiplicities and duality theories.

Reversal

A non Cohen–Macaulay ring has depth strictly less than its dimension; this leads to more complicated Ext patterns, embedded associated primes and pathological local cohomology.

Boundary

Most commonly used for Noetherian local or graded rings and finitely generated modules; outside Noetherian or without a notion of depth the condition may be undefined or require modification.

Semantic Tension

Tension with Serre's S_k conditions and with Gorenstein: Cohen–Macaulay equals Serre S_dim but additional conditions (e.g. canonical module simplicity) distinguish Gorenstein from Cohen–Macaulay.

Synthesis

Cohen–Macaulayness identifies rings or modules whose depth meets dimension, signaling that regular sequences of maximal length exist and yielding tractable homological and geometric behavior.