Commutative algebra - Department of Mathematical Sciences - old ...

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