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46 3. EXACT SEQUENCES OF MODULES Proof. The commutative diagram 0 0 N ∩ L N ⊕ L N + L x↦→(x,x) (x,y)↦→x−y M M ⊕ M M where the rows are short exact sequences, gives by 3.2.4 a five term long exact sequence 0 M/N ∩ L M/N ⊕ M/L M/N + L 3.2.8. Proposition (five lemma). Given a commutative diagram of homomorphisms M1 u1 M ′ 1 M2 u2 M ′ 2 M3 u3 M ′ 3 M4 u4 M ′ 4 M5 u5 M ′ 5 where the rows are exact sequences. If u1 is surjective, u2, u4 are isomorphism and u5 is injective, then u3 is an isomorphism. Proof. Let fi : Mi → Mi+1, f ′ i : M ′ i → M ′ i+1 . Split the given diagram in three as follows 0 0 0 0 0 M1 u1 Ker f ′ 2 Cok f2 u ′′ 3 Ker f ′ 4 Im f2 u ′ 3 Im f ′ 2 M2 u2 M ′ 2 M4 u4 M ′ 4 M3 u3 M ′ 3 Cok f1 u ′ 3 Im f ′ 2 Im f4 u5 M ′ 5 Cok f2 u ′′ 3 Cok f ′ 2 Note that Cok f1 Im f2 and Cok f ′ 2 Ker f ′ 4 . Now use 2.3.3 and the snake lemma to conclude that Ker u3 = 0 and Cok u3 = 0 and u3 is therefore an isomorphism. 3.2.9. Proposition (windmill lemma). Given homomorphism M f There is induced an eight term long exact sequence 0 Ker f Ker g ◦ f Cok f f g Ker g Cok g ◦ f 0 0 0 0 0 0 0 Cok g 0 g N 0 L .
Proof. Look at the two diagrams 0 0 3.2. THE SNAKE LEMMA 47 0 Ker g M f g◦f L 1 By the snake lemma the sequences 0 N M g L Ker f δ=f Cok f Ker g Ker g ◦ f f g N Ker g δ=g Cok g Cok g ◦ f 1 M g◦f L Cok f 0 are exact and overlap to give the windmill sequence. 3.2.10. Remark. The windmill is Ker g ◦ f Cok g ◦ f Cok f Ker g ◦ f Ker g f Ker f M N g◦f g Cok f 0 L Cok g Cok g ◦ f 3.2.11. Exercise. (1) Given a short exact sequence 0 M N Show that Ann(N) ⊂ Ann(M) ∩ Ann(L). (2) Give a short exact sequence 0 M N such that Ann(N) = Ann(M) ∩ Ann(L). (3) Given ideals I, J ⊂ R. Show that there is a short exact sequence. 0 R/I ∩ J R/I ⊕ R/J L L 0 0 0 0 0 R/I + J 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 + ·
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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
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Page 78 and 79: 78 6. FINITE MODULES Let the n × n
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Page 90 and 91: 90 7. MODULES OF FINITE LENGTH (5)
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Page 94 and 95: 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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