## [Cohen–Macaulay Property](/cohen-macaulay-property-0) Definition 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 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 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 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 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 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 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 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 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.