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

102 9. PRIMARY DECOMPOSITION
9.1.5. Proposition. Let R be a ring and M a module.
(1) M = 0 if and only if Supp(M) = ∅.
(2) For any module
Supp(M) ⊂ V (Ann(M))
.
(3) If M is finite, then
Supp(M) = V (Ann(M))
Proof. (1) See 5.4.1. (2) If MP = 0 then for u /∈ P there is x ∈ M such that
ux = 0. So Ann(M) ⊂ P . (3) Let x1, . . . , xn generate M. If Ann(M) =
∩ Ann(xi) ⊂ P then some Ann(xi) ⊂ P , so xi
1 = 0 in MP .
9.1.6. Proposition. Let N1, . . . , Nk ⊂ M be submodules such that Supp(M/Ni)∩
Supp(M/Nj) = ∅, i = j. Then the homomorphism
M/ ∩i Ni →
M/Ni
is an isomorphism.
Proof. By induction on k it is enough to treat the case k = 2. By 9.1.4 the support
of the cokernel is empty.
9.1.7. Proposition. Let R be a ring and M, N modules.
(1)
Supp(M ⊗R N) ⊂ Supp(M) ∩ Supp(N)
(2) If M, N are finite, then
Supp(M ⊗R N) = Supp(M) ∩ Supp(N)
Proof. There is an isomorphism (M ⊗R N) MP ⊗RP NP . (1) This is clear. (2)
This follows from 6.4.3.
9.1.8. Corollary. Let φ : R → S be a ring homomorphism and M an R-module.
(1) For the change of rings module
(2) If M is finite, then
Supp(M ⊗R S) ⊂ φ ∗−1 (Supp(M))
Supp(M ⊗R S) = φ ∗−1 (Supp(M))
9.1.9. Corollary. Let R be a ring, I an ideal in R and M a finite R-module. Then
Supp(M/IM) = Supp(M) ∩ V (I)
9.1.10. Corollary. Let R be a ring, U a multiplicative subset and M a finite Rmodule.
Then
Supp(U −1 M) = Supp(M) ∩ Spec(U −1 R)
9.1.11. Proposition. Let (R, P ) → (S, Q) be a local homomorphism and M a
finite R-module. If Supp(M) = ∅ then Supp(M ⊗R S) = ∅.
Proof. The homomorphism R → S is faithfully flat 5.5.8.
9.1.12. Proposition. Let M be a finite R-module and P ∈ Supp(M). Then there
is a nonzero homomorphism M → R/P .
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