COMMUTATIVE ALGEBRA

YOSHIFUMI TSUCHIMOTO

10. Minimal free resolution of finitely generated modules over local rings

Proposition 10.1. Let A be a local ring with the residue field k. Let M, N be A-modules. We assume N is a finite module. Then an Amodule homomorphism f : M → N is surjective if and only if f ⊗ 1_{k} : M ⊗_{A} k → N ⊗_{A} k is surjective.

Definition 10.2. Let

0 ← M ←^{ϵ} F_{0} ←^{d_{1}} F_{1} ←^{d_{2}} F_{2} ←^{d_{3}} …

be a resolution of M. We say that it is minimal free resolution if the following conditions are satisfied.

(1) each F_{i} is a finite free A-module.

(2) \overline{d_{i}} = 0.

(3) \overline{ϵ} : L_{0} ⊗_{A} k → M ⊗ k is an isomorphism.

Lemma 10.3. A finite module M over a Noetherian local ring has a minimal free resolution.

Definition 10.4. Let M be an A-module over a ring A. We define the projective dimension of M (proj dim M) to be the minimal length of the projective resolution of M.

Proposition 10.5. Let A be a local ring with the residue field k. Then:

(1) If we have a minimal resolution 0 ← M ← F_{0} ← F_{1} ← F_{2} ← … of an A-module M, then we have Tor_{i}^{A}(M, k) = rank(F_{k}).

(2) proj dim(M) = sup{i| Tor_{i}^{A}(M, k) ≠ 0} ≤ proj dim_{A} k.

Definition 10.6. We define the global dimension of a ring A by

gl dim(A) = sup{proj dim M|M ∈ Ob(A -modules)}

Lemma 10.7. Let A be a local ring. Let k be its residue field. Then we have

gl dim(A) = proj dim(k).

Theorem 10.8 (Serre). Let A be a Noetherian local ring. Then:

A is regular ⟺ gl dim A = dim A ⟺ gl dim A < ∞.

Theorem 10.9 (Serre). A localization of a regular local ring at a prime ideal is also a regular local ring.

Definition 10.10. A Noetherian ring is said to be regular if its localilzation at every prime is a regular local ring.