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38 2. MODULES (1) The change of ring functor maps an R-module M to the R/I-module M/IM. The natural isomorphism 2.7.6 is HomR(M, N) Hom R/I(M/IM, N) for any R/I-module N. (2) The induced module functor maps an R-module M to the R/I-module {x ∈ M|I ⊂ Ann(x)}. The natural isomorphism 2.7.10 is HomR(N, M) Hom R/I(N, HomR(R/I, M)) for any R/I-module N. 2.7.12. Definition. Let R → S, S ′ be ring homomorphisms. The tensor product ring over R is S ⊗R S ′ with multiplication given by (b ⊗ b ′ )(c ⊗ c ′ ) = bc ⊗ b ′ c ′ extended by linearity. R → S ⊗R S ′ , r ↦→ r ⊗ 1 = 1 ⊗ r is the natural ring homomorphism. 2.7.13. Proposition. Let φ, φ ′ : R → S, S ′ and ψ, ψ ′ : S, S ′ → T give a commutative diagram of ring homomorphisms, ψ ◦ φ = ψ ′ ◦ φ ′ . Then b ⊗ b ′ ↦→ ψ(b)ψ ′ (b ′ ) is the unique homomorphism making the following diagram commutative. Proof. This is clear by 2.6.3. R S ′ S S ⊗R S ′ T 2.7.14. Example. Let R → S be a ring homomorphism. Then is an isomorphism. R[X] ⊗R S S[X] 2.7.15. Exercise. (1) Show that the change of rings of a free R-module is a free Smodule. (2) Let φ : R → S be a ring homomorphism. Show that the change of rings of a scalar multiplication a : M → M on an R-module is a scalar multiplication φ(a) : M ⊗R S → M ⊗R S. (3) Show that the change of rings of the composition of two homomorphisms is the composition of the change of rings of each homomorphism. (4) Show the isomorphism R[X] ⊗R R[Y ] R[X, Y ]
3 Exact sequences of modules 3.1. Exact sequences 3.1.1. Definition. Let f : M → N and g : N → L be homomorphisms of modules. The sequence of homomorphisms is a M f g N L (1) 0-sequence: g ◦ f = 0 or equivalently Im f ⊂ Ker g (2) exact sequence: Im f = Ker g For a sequence of more homomorphisms the conditions should be satisfied for every consecutive composition. E.g. The sequence M f g N h L K is a 0-sequence if g ◦ f = 0 and h ◦ g = 0. The sequence is exact if Im f = Ker g and Im g = Ker h. 3.1.2. Remark. An interpretation of 2.3.3 gives: (1) The sequence is exact if and only if f is injective. (2) The sequence 0 M f f M N is exact if and only if f is surjective. (3) The sequence 0 f M N is exact if and only if f is an isomorphism. 3.1.3. Proposition. (1) For a homomorphism f : M → N the sequence 0 Ker f f M N N 0 0 Cok f is exact. (2) For scalar multiplication with a ∈ R on M the sequence 0 is exact. Ker aM aM M M 39 M/aM 0 0
Page 1: ELEMENTARY COMMUTATIVE ALGEBRA LECT
Page 5 and 6: Contents Prerequisites 7 1. A dicti
Page 7: Prerequisites The basic notions fro
Page 10 and 11: 10 1. A DICTIONARY ON RINGS AND IDE
Page 22 and 23: 22 2. MODULES (5) If R is a field,
Page 24 and 25: 24 2. MODULES (1) IM = {a1y1 + ·
Page 26 and 27: 26 2. MODULES 2.3.6. Corollary. Let
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Page 30 and 31: 30 2. MODULES 2.4.11. Proposition.
Page 32 and 33: 32 2. MODULES Proof. By 2.5.3 the c
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Page 36 and 37: 36 2. MODULES 2.6.12. Example. Let
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Page 66 and 67: 66 5. LOCALIZATION (2) For a prime
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Page 72 and 73: 72 5. LOCALIZATION (1) φ is flat.
Page 74 and 75: 74 6. FINITE MODULES 6.1.7. Corolla
Page 76 and 77: 76 6. FINITE MODULES (4) Let A be a
Page 78 and 79: 78 6. FINITE MODULES Let the n × n
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Page 82 and 83: 82 6. FINITE MODULES 6.5.9. Corolla
Page 84 and 85: 84 6. FINITE MODULES (1) E is an in
Page 86 and 87: 86 7. MODULES OF FINITE LENGTH 7.2.
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88 7. MODULES OF FINITE LENGTH Proo
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90 7. MODULES OF FINITE LENGTH (5)
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92 7. MODULES OF FINITE LENGTH 7.5.
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94 8. NOETHERIAN MODULES Proof. Use
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96 8. NOETHERIAN MODULES 8.3.2. Cor
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98 8. NOETHERIAN MODULES 8.5. Fract
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9 Primary decomposition 9.1. Suppor
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9.2. ASS OF MODULES 103 Proof. The
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9.2. ASS OF MODULES 105 9.2.12. Def
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9.4. DECOMPOSITION OF MODULES 107 9
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9.5. DECOMPOSITION OF IDEALS 109 9.
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10 Dedekind rings 10.1. Principal i
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10.3. DEDEKIND DOMAINS 113 10.2.4.
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10.3. DEDEKIND DOMAINS 115 10.3.10.
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118 BIBLIOGRAPHY
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120 INDEX finite type rings, 95 fin
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Commutative algebra - Department of Mathematical Sciences - old ...

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