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

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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 . i
9.2. ASS OF MODULES 103 Proof. The module HomR(M, R/P )P HomRP (MP , k(P )) is nonzero 7.1.4. 9.1.13. Proposition. Let R be a noetherian ring and M a finite module. M has finite length if and only if Supp(M) consists only of maximal ideals. Proof. This is 8.6.7. 9.1.14. Proposition. Let R be a noetherian ring and F a finite module. If R has only finitely many maximal ideals and FP is free of rank n, then F is free of rank n. Proof. If R is artinian, this is clear from 7.4.5. If P1, . . . , Pk are the maximal ideals, then choose x1, . . . , xn ∈ F giving a basis for F/P1 · · · PnF . The homomorphism R n → F, ei ↦→ xi is an isomorphism 6.4.6 and 8.1.5. 9.1.15. Exercise. (1) 9.2. Ass of modules 9.2.1. Definition. Let M be an R-module. A prime ideal P ⊂ R is an associated prime ideal of M if P = Ann(x) for some x ∈ M. The set of prime ideals associated to M is Ass(M). 9.2.2. Proposition. Let P ⊂ R be a prime ideal. (1) P ∈ Ass(M) if and only if there is an injective homomorphism R/P → M. (2) Ass(M) ⊂ Supp(M) (3) Let 0 = N ⊂ R/P be a nonzero submodule, then Proof. This is clear from the definition. Ass(N) = {P } 9.2.3. Proposition. Let 0 → N → M → L → 0 be a short exact sequence of modules. Then Ass(N) ⊂ Ass(M) ⊂ Ass(N) ∪ Ass(L) Proof. The left inclusion is trivial. Next let P ∈ Ass(M) such that P /∈ Ass(N). Choose a submodule L of M such that L R/P . Then Ass(L ∩ N) ⊂ Ass(L) ∩ Ass(N). It follows that L ∩ N = 0, and therefore M/N contains a submodule isomorphic to R/P . 9.2.4. Corollary. Let 0 → N → M → L → 0 be a split exact sequence of modules. Ass(M) = Ass(N) ∪ Ass(L) 9.2.5. Corollary. (1) Let N ⊂ M be a submodule. Then (2) Let M, N be modules. Then Ass(N) ⊂ Ass(M) ⊂ Ass(N) ∪ Ass(M/N) Ass(M ⊕ N) = Ass(M) ∪ Ass(N) (3) Given submodules N, L ⊂ M. Then Ass(M/N ∩ L) ⊂ Ass(M/N) ∪ Ass(M/L) ⊂ Ass(M/N ∩ L) ∪ Ass(M/N + L) Proof. (3) Use the sequence 3.2.7.
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ELEMENTARY COMMUTATIVE ALGEBRA LECT
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Contents Prerequisites 7 1. A dicti
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Prerequisites The basic notions fro
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10 1. A DICTIONARY ON RINGS AND IDE
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22 2. MODULES (5) If R is a field,
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24 2. MODULES (1) IM = {a1y1 + ·
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26 2. MODULES 2.3.6. Corollary. Let
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28 2. MODULES (2) Give an example o
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30 2. MODULES 2.4.11. Proposition.
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32 2. MODULES Proof. By 2.5.3 the c
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34 2. MODULES (3) The formation of
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36 2. MODULES 2.6.12. Example. Let
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38 2. MODULES (1) The change of rin
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40 3. EXACT SEQUENCES OF MODULES 3.
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42 3. EXACT SEQUENCES OF MODULES Fo
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44 3. EXACT SEQUENCES OF MODULES an
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