Height and coheight of primes in a Cohen–Macaulay local ring

In a Noetherian local ring R that is Cohen–Macaulay over itself, every prime p satisfies ht p + dim (R ⧸ p) = dim R (Stacks, Lemma 10.103.10 (0AAF)). The inequality ≤ holds in every ring (Ideal.height_add_ringKrullDim_quotient_le); the Cohen–Macaulay hypothesis gives the reverse one. Without it the equation fails: in a local ring whose spectrum is a plane meeting a line at a point, a minimal prime p cutting out the line has ht p + dim (R ⧸ p) = 1 < 2.

The proof chooses an R-regular sequence r₁, …, rₖ in p of maximal length. Its length is at most ht p, and by maximality every element of p is a zero divisor on N = R ⧸ (r₁, …, rₖ). By prime avoidance p lies in an associated prime of N. Now N is again Cohen–Macaulay, of dimension dim R - k, so it has no embedded primes, and p is itself associated to N. By unmixedness dim (R ⧸ p) = dim N = dim R - k ≥ dim R - ht p.

This is the dimension formula that makes Cohen–Macaulay rings catenary: applied in the localizations of a Cohen–Macaulay ring, it says that heights add along chains of primes.

Main results

• Module.IsCohenMacaulay.height_add_ringKrullDim_quotient: for a Noetherian local ring R that is Cohen–Macaulay over itself and a prime p, ht p + dim (R ⧸ p) = dim R (Stacks, Lemma 10.103.10 (0AAF)).

References

• Stacks Project, Lemma 10.103.10.
• H. Matsumura, Commutative Ring Theory, Theorem 17.4 (i).

theorem Module.IsCohenMacaulay.height_add_ringKrullDim_quotient {R : Type u} [CommRing R] [IsLocalRing R] [IsNoetherianRing R] (h : IsCohenMacaulay R R) (p : Ideal R) [p.IsPrime] : ↑p.height + ringKrullDim (R ⧸ p) = ringKrullDim R

The dimension formula in a Cohen–Macaulay local ring. For a Noetherian local ring R that is Cohen–Macaulay over itself and a prime p of R, ht p + dim (R ⧸ p) = dim R. Stacks, Lemma 10.103.10 (0AAF); H. Matsumura, Commutative Ring Theory, Theorem 17.4 (i).