Commutative algebra - Department of Mathematical Sciences - old ...
Commutative algebra - Department of Mathematical Sciences - old ...
Commutative algebra - Department of Mathematical Sciences - old ...
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9<br />
Primary decomposition<br />
9.1. Support <strong>of</strong> modules<br />
9.1.1. Definition. Let R be a ring and M a module.<br />
(1) The set <strong>of</strong> prime ideals is the spectrum and denoted Spec(R).<br />
(2) For a ring homomorphism φ : R → S restriction defines a map<br />
(3) For a subset B ⊂ R<br />
is a subset <strong>of</strong> the spectrum.<br />
(4) The support <strong>of</strong> M is<br />
φ ∗ : Spec(S) → Spec(R)<br />
Q ↦→ φ −1 (Q)<br />
V (B) = {P ∈ Spec(R)|MP = 0}<br />
Supp(M) = {P ∈ Spec(R)|MP = 0}<br />
(5) A minimal prime ideal in Supp(M) is a minimal prime <strong>of</strong> M.<br />
9.1.2. Proposition. Let R be a ring.<br />
(1) Let I ⊂ R be an ideal. Then<br />
and is identified with Spec(R/I).<br />
(2) Let U be a multiplicative subset.<br />
Supp(R/I) = V (I)<br />
Supp(U −1 R) = {P ∈ Spec(R)|P ∩ U = ∅}<br />
and is identified with Spec(U −1 R).<br />
Pro<strong>of</strong>. This is a restatement <strong>of</strong> 1.3.5, 5.1.5.<br />
9.1.3. Proposition. Let 0 → N → M → L → 0 be a short exact sequence <strong>of</strong><br />
modules. Then<br />
Supp(M) = Supp(N) ∪ Supp(L)<br />
Pro<strong>of</strong>. This follows from 5.4.5.<br />
9.1.4. Corollary. (1) Let N ⊂ M be a submodule. Then<br />
Supp(M) = Supp(N) ∪ Supp(M/N)<br />
(2) Given submodules N, L ⊂ M. Then<br />
Supp(M/N ∩ L) ∪ Supp(M/N + L) = Supp(M/N) ∪ Supp(M/L)<br />
and<br />
Supp(M/N + L) ⊂ Supp(M/N) ∩ Supp(M/L)<br />
Pro<strong>of</strong>. (2) Use the sequence 3.2.7.<br />
101