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48 3. EXACT SEQUENCES OF MODULES 3.3. Exactness of Hom 3.3.1. Proposition. Given an exact sequence 0 f M g N L and an R-module K. Then the following sequence is exact 0 HomR(K, M) HomR(K, N) HomR(K, L) Proof. Given h : K → M such that f ◦ h = 0 then h = 0 since f is injective. Given k : K → N such that g ◦ k = 0 then by 3.1.4 there is h : K → M such that f ◦ h = k. So the sequence is exact. 3.3.2. Proposition. Given a sequence 0 f M g N L Such that for any module K, the following sequence is exact 0 HomR(K, M) Then the original sequence is exact. HomR(K, N) HomR(K, L) Proof. For K = M the identity 1M is mapped to g ◦ f ◦ 1M = 0 so it is a 0sequence. Assume f(x) = 0. Take K = R then 1x is mapped to f ◦ 1x = 1 f(x) = 0. Therefore 1x = 0 and so x = 0. That insures that f is injective. Assume g(y) = 0. Take K = R then 1y is mapped to g ◦ 1y = 1 g(y) = 0. There exists a homomorphism h : R → M such that f ◦ h = 1y. By 2.5.12 h = 1x and therefore f(x) = y. The original sequence is now proven exact. 3.3.3. Proposition. Given an exact sequence M f g N L and a module K. Then the following sequence is exact 0 HomR(L, K) HomR(N, K) 0 HomR(M, K) Proof. Given h : L → K such that h ◦ g = 0 then h = 0 since g is surjective. Given k : N → K such that k ◦ f = 0 then by 3.1.5 there is h : L → K such that h ◦ g = k. So the sequence is exact. 3.3.4. Proposition. Given a sequence M f g N L such that for any module K, the following sequence is exact 0 HomR(L, K) Then the original sequence is exact. HomR(N, K) 0 HomR(M, K) Proof. For K = L the identity 1L is mapped to 1L ◦g ◦f = 0 so it is a 0-sequence. Take K = Cok g then pg : L → Cok g has pg ◦ g = 0, but by exactness 0 is the unique homomorphism satisfying this, so pg = 0. Therefore Cok g = 0 and g is surjective. Take K = Cok f, p : N → Cok f the projection. p ◦ f = 0 so by exactness there exists a unique q : L → Cok f such that q ◦ g = p. It follows that Ker g ⊂ Ker p = Im f. All together the original sequence is exact.
3.3.5. Proposition. Given a sequence The following are equivalent. 0 3.4. EXACTNESS OF TENSOR 49 f M g N (1) The sequence is split exact. (2) For any K the following sequence is exact 0 HomR(K, M) HomR(K, N) (3) For any K the following sequence is exact 0 HomR(L, K) HomR(N, K) L 0 HomR(K, L) HomR(M, K) If the conditions are true, then the sequences (2) and (3) are split exact. Proof. (1) ⇒ (2), (1) ⇒ (3) are clear by 3.1.14 giving that the sequences (2), (3) are split exact. (2) ⇒ (1): Let K = L, then there is a section to g. By 3.3.2 and 3.1.11 the original sequence is split exact. (3) ⇒ (1): Let K = M, then there is a retraction to f. By 3.3.4 and 3.1.11 the original sequence is split exact. 3.3.6. Exercise. (1) Show that the sequence 0 HomZ(Q/Z, Z) HomZ(Q/Z, Q) is exact, but the rightmost map is not surjective. (2) Show that the sequence 0 HomZ(Z/(n), Z) n HomZ(Z/(n), Z) is exact, but the rightmost map is not surjective. 3.4. Exactness of Tensor 3.4.1. Proposition. Given an exact sequence M f g N L 0 and an R-module K. Then the following sequence is exact M ⊗R K N ⊗R K L ⊗R K 0 0 HomZ(Q/Z, Q/Z) HomZ(Z/(n), Z/(n)) Proof. Let K ′ be any module. By 3.3.4 it is enough to see that the sequence 0 HomR(L ⊗R K, K ′ ) is exact. By 2.6.13 it amounts to see that the sequence 0 HomR(L, HomR(K, K ′ )) is exact. This follows from 3.3.3. 0 HomR(N ⊗R K, K ′ ) HomR(M ⊗R K, K ′ ) HomR(M, HomR(K, K ′ )) HomR(N, HomR(K, K ′ ))
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
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 70 and 71: 70 5. LOCALIZATION 5.4. Exactness a
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
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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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