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

More documents

Recommendations

Info

30 2. MODULES 2.4.11. Proposition. Let F be a free module with basis yα. For a module M and a family of elements xα ∈ M there is a unique homomorphism g : F → M such that g(yα) = xα given by g( aαyα) = aαxα. Proof. The basis yα ∈ F gives the isomorphism 2.4.7 f : α R → F . The family 1xα : R → M gives a homomorphism g ′ : R → M by 2.4.3. Then g = g ′ ◦ f −1 . 2.4.12. Corollary. Let M be an R-module and M R the free module with basis ex indexed by x ∈ M. The homomorphism R → M, axex ↦→ ax x M is surjective identifying M as a factor module of a free module in a natural way. 2.4.13. Definition. A module is indecomposable if it is not isomorphic to a direct sum of two nonzero submodules, otherwise decomposable. 2.4.14. Example. Q is an indecomposable Z-module. Namely if m1 n1 nonzero numbers in two submodules, then n1m2 m1 n1 ber in the intersection. α = n2m1 m2 n2 , m2 n2 are is a nonzero num- 2.4.15. Exercise. (1) Show that if a ring is decomposable as a module, then it is the product of two nonzero rings. (2) Let Mα be a finite family of modules. Show that Mα = Mα, (3) Let Nα ⊂ Mα be a family of submodules modules. Show that Mα/ Nα Mα/Nα and that Mα/ Nα Mα/Nα 2.5. Homomorphism modules 2.5.1. Lemma. Let R be a ring and f, g : M → N homomorphisms. (1) (f + g)(x) = f(x) + g(x) is a homomorphism. (2) If a ∈ R, then (af)(x) = af(x) is a homomorphism. Proof. Calculate according to 2.1.1. (1) (f + g)(x + y) = f(x + y) + g(x + y) = f(x) + f(y) + g(x) + g(y) = (f + g)(x) + (f + g)(y) and (f + g)(ax) = f(ax) + g(ax) = a(f(x) + g(x)) = a(f + g)(x). (2) (af)(x + y) = af(x + y) = af(x) + af(y) = (af)(x) + (af)(y) and (af)(bx) = af(bx) = abf(x) = baf(x) = b(af)(x). The last calculation uses that R is commutative 1.1.2 (4). 2.5.2. Definition. Let R be a ring and M, N modules. By 2.5.1, the homomorphism module HomR(M, N) is the additive group of all homomorphism with scalar multiplication R × HomR(M, N) → HomR(M, N), (a, f) ↦→ af = [x ↦→ af(x)] 2.5.3. Definition. Let a ∈ R be a ring and f : M → M ′ , g : N → N ′ , h, k : M ′ → N homomorphisms of modules. By 2.5.1 (1) (h + k) ◦ f = h ◦ f + k ◦ f. (2) (ah) ◦ f = a(h ◦ f). (3) g ◦ (h + k) = g ◦ h + g ◦ k. (4) g ◦ (ah) = a(g ◦ h).
There is induced a homomorphism of R-modules. 2.5. HOMOMORPHISM MODULES 31 Hom(f, g) : HomR(M ′ , N) → HomR(M, N ′ ) (h : M ′ → N) ↦→ (g ◦ h ◦ f : M → N ′ ) 2.5.4. Definition. Let R, S be a rings. A functor is a construction T , which to R-modules M, N associates S-modules T (M), T (N) and a homomorphism such that HomR(M, N) → HomS(T (M), T (N)), f ↦→ T (f) (1) T (1M) = 1 T (M) (2) T (g ◦ f) = T (g) ◦ T (f) In case the homomorphism goes and HomR(M, N) → HomS(T (N), T (M)), f ↦→ T (f) (1) T (1M) = 1 T (M) (2) T (g ◦ f) = T (f) ◦ T (g) the functor is contravariant. Clearly functors transform isomorphisms into isomorphism. Given functors T, T ′ a natural homomorphism is a family νM : T (M) → T ′ (M) of homomorphisms, such that for each f : M → N the following diagram commutes T (M) T (f) νN T (N) In the contravariant case the diagram is T (M) T (f) T (N) νM T ′ (M) νM T ′ (f) T ′ (N) T ′ (M) T ′ (f) νN T ′ (N) A natural isomorphism is a natural homomorphism such that each νM is an isomorphism. 2.5.5. Proposition. Let R be a ring. (1) The construction is a functor. (2) The construction is a contravariant functor N ↦→ HomR(M, N), g ↦→ Hom(1M, g) M ↦→ HomR(M, N), f ↦→ Hom(f, 1N)
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 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
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.
Page 115:
10.3. DEDEKIND DOMAINS 115 10.3.10.
Page 118 and 119:
118 BIBLIOGRAPHY
Page 120 and 121:
120 INDEX finite type rings, 95 fin
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?