SUPPORT OF NOETHERIAN MODULES OVER COMMUTATIVE RINGS

subcategory noethA which is formed by all noetherian A-modules. An A-module is locally noetherian if all its finitely generated submodules are noetherian. We denote by NoethA the category of locally noetherian A-modules. Note that the ring A is noetherian precisely when every A-module is locally noetherian. The spectrum SpecA of A is the set of prime ideals of A. A subset of SpecA is Zariski closed if it is of the form V (a) = {p ∈ SpecA | a ⊆ p} for some ideal a of A. The support of an A-module M is by definition suppM = {p ∈ SpecA |Mp 6 = 0}. Lemma 1. Let a be an ideal of A. Then suppA/a = V (a). Proof. Fix p ∈ SpecA and let S = A \ p. Recall that for any A-module M, an element x/s in S−1M = Mp is zero iff there exists t ∈ S such that tx = 0. Thus we have (A/a)p = 0 iff there exists t ∈ S with t(1 + a) = t+ a = 0 iff a 6 ⊆ p. Lemma 2. Let 0 → M ′ → M → M ′ ′ → 0 be an exact sequence of A-modules. Then suppM = suppM ′ ∪ suppM ′′. Proof. For each prime ideal p, the sequence 0→M ′p →Mp →M ′′p → 0 is exact. Lemma 3. Let M = iMi be an A-module, written as a sum of submodules Mi. Then suppM = i suppMi. Proof. The assertion is clear if the sum iMi is direct, since