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32 2. MODULES Proof. By 2.5.3 the construction on homomorphisms are homomorphisms. Given also homomorphisms f ′ : M ′ → M ′′ and g ′ : N ′ → N ′′ . Then and Hom(1M, g ′ ◦ g) = Hom(1M, g ′ ) ◦ Hom(1M, g) Hom(f ′ ◦ f, 1N) = Hom(f, 1N) ◦ Hom(f ′ , 1N) showing the conditions on compositions. 2.5.6. Corollary. Let a ∈ R give scalar multiplications aM, aN. (1) Hom(aM, 1N) = Hom(1M, aN) : HomR(M, N) → HomR(M, N) f ↦→ af is scalar multiplication a HomR(M,N). (2) The map R → HomR(M, M), a ↦→ aM is a homomorphism. 2.5.7. Example. Let R = R1 × R2 be the product ring. The constructions in 2.2.9 is: (1) A functor which to a pair of an R1-module and an R2-module associates an R-module.(2) A functor which to an R-module associates a pair of an R1-module and an R2-module. 2.5.8. Proposition. Let R be a ring and Mα a family of modules. For any module N there are natural isomorphisms (1) HomR( α Mα, N) α HomR(Mα, N) (2) HomR(N, α Mα) α HomR(N, Mα) Proof. This is 2.4.3 reformulated. (1) A homomorphism g : α Mα → N is uniquely determined by the family gα = g ◦ iα : Mα → N. (2) A homomorphism f : N → α Mα is uniquely determined by the family fα = pα ◦f : N → Mα. 2.5.9. Lemma. Let R be a ring and M, N modules. For x ∈ M there is a homomorphism HomR(M, N) → N, f ↦→ f(x). Proof. Calculate according to 2.1.1 (f + g) ↦→ (f + g)(x) = f(x) + g(x) and (af) ↦→ (af)(x) = af(x). 2.5.10. Definition. The natural homomorphism 2.5.9 is the evaluation at x. evx : HomR(M, N) → N, f ↦→ f(x) 2.5.11. Lemma. There is a natural homomorphism M → HomR(HomR(M, N), N), x ↦→ evx Proof. Calculate according to 2.1.1 evx+y(f) = f(x + y) = f(x) + f(y) = evx(f) + evy(f). and evax(f) = f(ax) = af(x) = aevx(f) to see that the map is a homomorphism. 2.5.12. Proposition. Let R be a ring and M a module. The evaluation ev1 : HomR(R, M) M, f ↦→ f(1) is a natural isomorphism. x ↦→ 1x 2.3.11 is the inverse. Proof. Calculate the composite ev1(x ↦→ 1x) = 1x(1) = x and 1 f(1)(a) = af(1) = f(a) proving the claims.
2.6. TENSOR PRODUCT MODULES 33 2.5.13. Definition. Let R be a ring and M a module. The dual module is M ∨ = HomR(M, R) If f : M → N is a homomorphism, then the dual homomorphism is f ∨ = Hom(f, 1R) : N ∨ → M ∨ This construction is a contravariant functor. 2.5.14. Lemma. There is a natural homomorphism where evx(f) = f(x) 2.5.10. M → M ∨∨ = HomR(HomR(M, R), R), x ↦→ evx Proof. This is a special case of 2.5.11 2.5.15. Definition. A module M is a reflexive module if the homomorphism 2.1.14, is an isomorphism. 2.5.16. Example. Let R be a ring. M → M ∨∨ (1) The module R is reflexive. (2) If (a) = R, (0) in a domain, then R/(a) is not reflexive. 2.5.17. Exercise. (1) Show that HomZ(Q, Z) = 0. (2) Calculate HomZ(Z/(m), Z/(n)) = 0 for integers m, n. (3) Let I ⊂ R be an ideal and M a module. Show that HomR(R/I, M) = {x ∈ M|I ⊂ Ann(x)}. (4) Let R be a ring. Show that a free module with a finite basis is a reflexive module. (5) If (n) ⊂ (m) ⊂ Z, then show that (m)/(n) is a reflexive Z/(n)-module. 2.6. Tensor product modules 2.6.1. Definition. Let R be a ring and M, N modules. The tensor product module M ⊗R N = F/F ′ is the factor module F/F ′ , where F = ⊕M×NR is the free module with basis (x, y) = e (x,y) 2.4.6 and F ′ is the submodule generated by all elements of form (x1 + x2, y) − (x1, y) − (x2, y), (x, y1 + y2) − (x, y1) − (x, y2) (ax, y) − a(x, y), (x, ay) − a(x, y) The projection of the basis element (x, y) onto M ⊗R N is x ⊗ y = (x, y) + F ′ . 2.6.2. Remark. The relations are interpreted. (1) There are identities in M ⊗R N (x1 + x2) ⊗ y = x1 ⊗ y + x2 ⊗ y, x ⊗ (y1 + y2) = x ⊗ y1 + x ⊗ y2 (2) The map ax ⊗ y = a(x ⊗ y) = x ⊗ ay ⊗ : M × N → M ⊗R N, (x, y) ↦→ x ⊗ y has partial maps x ↦→ x⊗y : M → M ⊗R N and y ↦→ x⊗y : N → M ⊗R N that are all homomorphisms.
Page 1: ELEMENTARY COMMUTATIVE ALGEBRA LECT
Page 5 and 6: Contents Prerequisites 7 1. A dicti
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82 6. FINITE MODULES 6.5.9. Corolla
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84 6. FINITE MODULES (1) E is an in
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86 7. MODULES OF FINITE LENGTH 7.2.
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88 7. MODULES OF FINITE LENGTH Proo
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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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98 8. NOETHERIAN MODULES 8.5. Fract
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9 Primary decomposition 9.1. Suppor
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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 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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