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

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70 5. LOCALIZATION 5.4. Exactness and localization 5.4.1. Proposition. Let R be a ring and M a module. The following conditions are equivalent. (1) M = 0. (2) MP = 0 for all prime ideals P . (3) MP = 0 for all maximal ideals P . Proof. (1) ⇒ (2) ⇒ (3) is clear. (3) ⇒ (1): Let 0 = x ∈ M be given. Then Ann(x) ⊂ P is contained in a maximal ideal, 5.1.2. Clearly 0 = x 1 ∈ MP contradicts (3). 5.4.2. Corollary. Let R be a ring and f : M → N a homomorphism. The following conditions are equivalent. (1) f is injective. (2) fP is injective for all prime ideals P . (3) fP is injective for all maximal ideals P . Proof. Use 5.4.1 on Ker f. 5.4.3. Corollary. Let R be a ring and f : M → N a homomorphism. The following conditions are equivalent. (1) f is surjective. (2) fP is surjective for all prime ideals P . (3) fP is surjective for all maximal ideals P . Proof. Use 5.4.1 on Cok f. 5.4.4. Corollary. Let R be a ring and f : M → N a homomorphism. The following conditions are equivalent. (1) f is an isomorphism. (2) fP is an isomorphism for all prime ideals P . (3) fP is an isomorphism for all maximal ideals P . 5.4.5. Corollary. Let R be a ring and 0 f M g N a sequence of homomorphisms. The following conditions are equivalent. (1) The sequence is short exact. (2) The sequence 0 MP fP NP is short exact for all prime ideals P . (3) The sequence 0 MP fP NP is short exact for all maximal ideals P . L gP LP gP LP 5.4.6. Corollary. Let R be a ring and F a module. The following conditions are equivalent. (1) F is flat. (2) FP is flat for all prime ideals P . 0 0 0
(3) FP is flat for all maximal ideals P . 5.5. FLAT RING HOMOMORPHISMS 71 Proof. Let 0 → M → N. Use 5.3.7 and 5.4.2 on M ⊗R F → N ⊗R F . 5.4.7. Proposition. Let R be a ring and M a module. Then there is an exact sequence 0 → M → P maximal Proof. Let 0 = x ∈ M be given. Then Ann(x) ⊂ P is contained in a maximal ideal, 5.1.2. Clearly 0 = x 1 ∈ MP . 5.4.8. Corollary. Let R be a ring. Then there is an injective ring homomorphism R → P maximal 5.4.9. Corollary. Let R be a ring. The following conditions are equivalent. (1) R is reduced. (2) RP is reduced for all prime ideals P . (3) RP is reduced for all maximal ideals P . Proof. Use 5.1.8 and 5.4.10. 5.4.10. Exercise. (1) Let R be a ring and 0 f M g N a split exact sequence. Show that the localized sequence is split exact for all prime ideals P . RP MP L 5.5. Flat ring homomorphisms 5.5.1. Definition. A flat R-module F is faithfully flat if for any M = 0 the tensor product M ⊗R F = 0. 5.5.2. Example. A nonzero free module is faithfully flat. 5.5.3. Lemma. Let F be a faithfully flat R-module and M → N a homomorphism. If M ⊗R F → N ⊗R F is injective, then M → N is injective. Proof. Let f : M → N. Then 0 → Ker f ⊗R F → M ⊗R F → N ⊗R F is exact. It follows, that Ker f = 0. 5.5.4. Definition. A ring homomorphism R → S is a flat ring homomorphism if S is a flat R-module and a faithfully flat ring homomorphism if S is faithfully flat. 5.5.5. Proposition. Let R → S be a faithfully flat ring homomorphism. for any ideal I ⊂ R the extended contracted returns I, i.e. I = IS ∩ R Proof. Tensor the homomorphism R/I → R/IS∩R with S. The induced R/I⊗R S → R/IS ∩ R ⊗R S is canonically isomorphic to the identity S/IS → S/IS. By 5.5.3 R/I → R/IS ∩ R is injective giving I = IS ∩ R. 5.5.6. Corollary. A faithfully flat ring homomorphism R → S is injective. 5.5.7. Proposition. Let φ : R → S be a ring homomorphism. The following conditions are equivalent 0
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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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Page 20 and 21: 20 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
Page 28 and 29: 28 2. MODULES (2) Give an example o
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
Page 34 and 35: 34 2. MODULES (3) The formation of
Page 36 and 37: 36 2. MODULES 2.6.12. Example. Let
Page 38 and 39: 38 2. MODULES (1) The change of rin
Page 40 and 41: 40 3. EXACT SEQUENCES OF MODULES 3.
Page 42 and 43: 42 3. EXACT SEQUENCES OF MODULES Fo
Page 44 and 45: 44 3. EXACT SEQUENCES OF MODULES an
Page 46 and 47: 46 3. EXACT SEQUENCES OF MODULES Pr
Page 58 and 59: 58 4. FRACTION CONSTRUCTIONS 4.1.4.
Page 60 and 61: 60 4. FRACTION CONSTRUCTIONS 4.2.7.
Page 62 and 63: 62 4. FRACTION CONSTRUCTIONS and ge
Page 64 and 65: 64 4. FRACTION CONSTRUCTIONS 4.6. T
Page 66 and 67: 66 5. LOCALIZATION (2) For a prime
Page 68 and 69: 68 5. LOCALIZATION (2) Any local ri
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
Page 80 and 81: 80 6. FINITE MODULES 6.5. Finite Pr
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.
Page 88 and 89: 88 7. MODULES OF FINITE LENGTH Proo
Page 90 and 91: 90 7. MODULES OF FINITE LENGTH (5)
Page 92 and 93: 92 7. MODULES OF FINITE LENGTH 7.5.
Page 94 and 95: 94 8. NOETHERIAN MODULES Proof. Use
Page 96 and 97: 96 8. NOETHERIAN MODULES 8.3.2. Cor
Page 98 and 99: 98 8. NOETHERIAN MODULES 8.5. Fract
Page 101 and 102: 9 Primary decomposition 9.1. Suppor
Page 103 and 104: 9.2. ASS OF MODULES 103 Proof. The
Page 105 and 106: 9.2. ASS OF MODULES 105 9.2.12. Def
Page 107 and 108: 9.4. DECOMPOSITION OF MODULES 107 9
Page 109 and 110: 9.5. DECOMPOSITION OF IDEALS 109 9.
Page 111 and 112: 10 Dedekind rings 10.1. Principal i
Page 113 and 114: 10.3. DEDEKIND DOMAINS 113 10.2.4.
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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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