Localizing a Cohen–Macaulay module

Let M be a finite Cohen–Macaulay module over a Noetherian local ring R. Then for every prime p of R, the localization M_p is a Cohen–Macaulay module over R_p (Stacks, Lemma 10.103.11 (0AAG)). This is what makes the two usual definitions of a Cohen–Macaulay ring agree: asking that R_m be Cohen–Macaulay at every maximal ideal m, or that R_p be Cohen–Macaulay at every prime p. Over a Noetherian local ring it also shows that a finite Cohen–Macaulay module satisfies every Serre condition (S_n) (Module.IsCohenMacaulay.satisfiesSerreS).

Proof

We follow Matsumura's argument, by induction on n = depth M_p. If n = 0, then p is an associated prime of M. A Cohen–Macaulay module has no embedded primes, so p is minimal in the support of M, and the support of M_p is the closed point alone; a module with zero-dimensional support is Cohen–Macaulay. If n > 0, then p is not an associated prime of M. No associated prime of M contains p either: being minimal in the support, it would equal p. Prime avoidance gives an M-regular x ∈ p. Then M ⧸ x M is Cohen–Macaulay over R, and its localization M_p ⧸ x M_p has depth n - 1, so it is Cohen–Macaulay over R_p by induction. Since x stays regular on M_p, the module M_p is Cohen–Macaulay.

Main results

• Module.IsCohenMacaulay.localizedModule: the localization of a finite Cohen–Macaulay module over a Noetherian local ring at a prime is Cohen–Macaulay.
• Module.isCohenMacaulay_localizedModule_of_mem_minimalPrimes: the localization of a finite module at a minimal prime of its support is Cohen–Macaulay.
• Module.isCohenMacaulay_iff_of_isLocalizedModule: whether M_p is Cohen–Macaulay can be tested on any localization map M → N at p, over any localization of the ring at p.

References

• Stacks Project, Lemma 10.103.11
• H. Matsumura, Commutative Ring Theory, Theorem 17.3 (iii).

theorem Module.isCohenMacaulay_localizedModule_of_mem_minimalPrimes {R : Type u} [CommRing R] [IsNoetherianRing R] {M : Type v} [AddCommGroup M] [Module R M] [Module.Finite R M] {p : Ideal R} [p.IsPrime] (hp : p ∈ (annihilator R M).minimalPrimes) : IsCohenMacaulay (Localization.AtPrime p) (LocalizedModule p.primeCompl M)

Let M be a finite module over a Noetherian ring and p a minimal prime of the support of M. Then M_p is a Cohen–Macaulay module over R_p: its support is the closed point alone.

theorem Module.isCohenMacaulay_iff_of_isLocalizedModule {R : Type u} [CommRing R] [IsNoetherianRing R] (p : Ideal R) [p.IsPrime] {B : Type v} [CommRing B] [IsLocalRing B] [Algebra R B] [IsLocalization.AtPrime B p] {M : Type w'} [AddCommGroup M] [Module R M] [Module.Finite R M] {N : Type w} [AddCommGroup N] [Module R N] [Module B N] [IsScalarTower R B N] [Small.{w, v} B] (f : M →ₗ[R] N) [IsLocalizedModule p.primeCompl f] : IsCohenMacaulay B N ↔ IsCohenMacaulay (Localization.AtPrime p) (LocalizedModule p.primeCompl M)

Being Cohen–Macaulay at a prime p does not depend on how the localization is presented. Let B be a localization of a Noetherian ring R at p, and let f : M → N exhibit the B-module N as the localization of a finite R-module M at p. Then N is Cohen–Macaulay over B exactly when LocalizedModule p.primeCompl M is Cohen–Macaulay over Localization.AtPrime p. For instance, this applies to the stalks of a quasi-coherent module on a scheme.

theorem Module.IsCohenMacaulay.localizedModule {R : Type u} [CommRing R] [IsLocalRing R] [IsNoetherianRing R] [Small.{v, u} R] {M : Type v} [AddCommGroup M] [Module R M] [Module.Finite R M] (h : IsCohenMacaulay R M) (p : Ideal R) [p.IsPrime] : IsCohenMacaulay (Localization.AtPrime p) (LocalizedModule p.primeCompl M)

The localization of a finite Cohen–Macaulay module over a Noetherian local ring at a prime p is a Cohen–Macaulay module over R_p. Stacks, Lemma 10.103.11 (0AAG).

theorem Module.IsCohenMacaulay.localization_atPrime {R : Type u} [CommRing R] [IsLocalRing R] [IsNoetherianRing R] (h : IsCohenMacaulay R R) (p : Ideal R) [p.IsPrime] : IsCohenMacaulay (Localization.AtPrime p) (Localization.AtPrime p)

A localization of a Cohen–Macaulay Noetherian local ring is Cohen–Macaulay as a ring.