Abelian Groups - László Fuchs [Springer]

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4 Direct Limits 59 We now move to three direct systems: A, B as above, and a third one, C D fC i .i 2 I/I j i g, all with the same directed index set I.IfˆW A ! B and ‰ W B ! C are homomorphisms between them such that the sequence 0 ! A i i !B i i !C i ! 0 is exact for each i 2 I, then we say that the sequence 0 ! A ˆ!B ‰ !C ! 0 (2.3) is exact. It is an important fact that direct limits of exact sequences is exact. More precisely, Theorem 4.6. Let A; B; C be direct systems over the same index set I, and ˆW A ! B and ‰ W B ! C homomorphisms between them. If the sequence (2.3) is exact, then the sequence ˆ ‰ 0 ! A D lim A i !B D lim B i !C D lim C i ! 0 !i !i !i of direct limits is likewise exact. Proof. Exactness at A and C is guaranteed by Lemma 4.5, so we prove exactness only at B . By Lemma 4.5, the diagram φ i 0 −−−−→ A i −−−−→ Bi ψ i −−−−→ Ci −−−−→ 0 π i ⏐ ⏐↓ ρ i ⏐ ⏐↓ ⏐ ⏐↓ σ i Φ 0 −−−−→ A ∗ ∗ −−−−→ B∗ Ψ ∗ −−−−→ C∗ −−−−→ 0 is commutative for all i 2 I.Ifa 2 A ,then i a i D a for some a i 2 A i ,so‰ ˆa D ‰ ˆ i a i D ‰ i i a i D i i i a i D 0. Nextletb 2 Ker ‰ .Forsomeb i 2 B i , we have i b i D b, whence i i b i D ‰ i b i D ‰ b D 0. There exists j 2 I with j i ib i D 0, thus j b j D j j i b i D 0. Since the top row in the diagram is exact, there is an a j 2 A j with j a j D b j . Setting a D j a j , we arrive at ˆa D ˆ j a j D j j a j D j j i b i D i b i D b,i.e.b 2 Im ˆ, and the bottom row is exact at B . ut Exercises (1) Show that lim Z.p n / D Z.p 1 /, using inclusion maps. !n (2) Let A n Š Z .n
60 2 Direct Sums and Direct Products (5) (a) Let A be the limit of the direct system A DfA i .i 2 I/I j i g,anda 2 A . There exist a j 2 I and an a j 2 A j such that j a j D a and o.a j / D o.a/. (b) Direct limit of torsion (torsion-free) groups is again torsion (torsion-free). (6) If G is finitely generated, and ˛ W G ! A (notations as above), then there exist a j 2 I and an ˛j W G ! A j such that ˛ D j˛j. (7) If A DfA i .i 2 I/I j i g and B DfB i .i 2 I/I j ig are direct systems of groups, then A ˚ B DfA i ˚ B i .i 2 I/I j i ˚ j ig is likewise a direct system whose direct limit is the direct sum of the direct limits of A and B. (8) What is wrong with the following argument? Because of Theorem 4.6, the sequence 0 ! Z.p 1 / ! Z.p 1 / ! Z.p 1 / ! 0 must be exact, since it can be obtained as the direct limit of the exact sequences 0 ! Z.p m / ! Z.p 2m / ! Z.p m / ! 0.m 2 N/. 5 Inverse Limits Inverse Systems Inverse limits are dual to direct limits: we just reverse the arrows. Assume fA i j i 2 Ig is a collection of groups, indexed by a poset I, andfor each pair i; j 2 I of indices with i j there is given a connecting homomorphism j i W A j ! A i such that (i) i i is the identity map of A i for all i 2 I;and (ii) if i j k in I, then j i k j D i k. In this case, A DfA i .i 2 I/I j i g is called an inverse system. Bytheinverse or projective limit, orsimplylimit, of this inverse system is meant a group A such that (a) there are maps i W A ! A i such that i D j i j for all i j; and (b) if G is any group with maps i W G ! A i .i 2 I/ subject to i D j i j for i j, then there is a unique map W G ! A satisfying i D i for all i 2 I. We write: A D lim A i ; and call the maps i W A ! A i canonical. i2I Theorem 5.1. An inverse system A DfA i .i 2 I/I j i g of groups has a limit, unique up to isomorphism. Proof. Consider the subgroup A in the direct product A D Q i2I A i that consists of all vectors a D .:::;a i ;:::/whose coordinates satisfy j i a j D a i for all i j. This is in fact a subgroup as is seen immediately. The projection maps i W a 7! a i satisfy i D j i j, so (a) holds for A . To verify (b), let G be a group as stated in (b), and for g 2 G define W g 7! .:::; i g;:::/ 2 Q i A i. Owing to the condition i D j i j,wehaveg 2 A . Clearly, W G ! A satisfies i D i for all i 2 I. If i D i 0 holds also
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Springer Monographs in Mathematics
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László Fuchs Abelian Groups 123
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Dedicated with love to my wife Shul
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viii Preface to relevant publicatio
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x Preface on abelian groups. The re
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xii Contents 4 Divisibility and Inj
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xiv Contents 7 Whitehead Groups If
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Table of Notations Set Theory ; : s
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Table of Notations xix N .p/: Nunk
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Table of Notations xxi Q A;;A : di
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2 1 Fundamentals is finite, countab
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4 1 Fundamentals Theorem 1.2. In a
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6 1 Fundamentals (8) A superfluous
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Page 32 and 33: 10 1 Fundamentals φ G ⏐ ↓ η 0
Page 34 and 35: 12 1 Fundamentals Lemma 2.5 (Snake
Page 36 and 37: 14 1 Fundamentals Exercises (1) Ass
Page 38 and 39: 16 1 Fundamentals non-zero subgroup
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Page 42 and 43: 20 1 Fundamentals 4 Sets Since the
Page 44 and 45: 22 1 Fundamentals It might be helpf
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Page 48 and 49: 26 1 Fundamentals of subgroups, ind
Page 50 and 51: 28 1 Fundamentals If is an infinit
Page 52 and 53: 30 1 Fundamentals For the proof of
Page 54 and 55: 32 1 Fundamentals C3. Identity. For
Page 56 and 57: 34 1 Fundamentals is exact in D; F
Page 58 and 59: 36 1 Fundamentals 7 Linear Topologi
Page 60 and 61: 38 1 Fundamentals Proof. (We need s
Page 62 and 63: 40 1 Fundamentals For two R-modules
Page 64 and 65: Chapter 2 Direct Sums and Direct Pr
Page 66 and 67: 1 Direct Sums and Direct Products 4
Page 72 and 73: 2 Direct Summands 51 Proof. If A D
Page 74 and 75: 2 Direct Summands 53 Exercises (1)
Page 76 and 77: 3 Pull-Back and Push-Out Diagrams 5
Page 78 and 79: 4 Direct Limits 57 In this case, A
Page 82 and 83: 5 Inverse Limits 61 for 0 W G ! A
Page 84 and 85: 5 Inverse Limits 63 for every i 2 I
Page 86 and 87: 6 Direct Products vs. Direct Sums 6
Page 88 and 89: 6 Direct Products vs. Direct Sums 6
Page 90 and 91: 7 Completeness in Linear Topologies
Page 96 and 97: Chapter 3 Direct Sums of Cyclic Gro
Page 98 and 99: 1 Freeness and Projectivity 77 Coro
Page 100 and 101: 1 Freeness and Projectivity 79 Let
Page 102 and 103: 2 Finite and Finitely Generated Gro
Page 108 and 109: 3 Factorization of Finite Groups 87
Page 110 and 111: 3 Factorization of Finite Groups 89
Page 112 and 113: 4 Linear Independence and Rank 91 E
Page 114 and 115: 4 Linear Independence and Rank 93 t
Page 116 and 117: 5 Direct Sums of Cyclic Groups 95 P
Page 118 and 119: 5 Direct Sums of Cyclic Groups 97 s
Page 120 and 121: 6 Equivalent Presentations 99 (5) (
Page 122 and 123: 6 Equivalent Presentations 101 Thes
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Page 126 and 127: 7 Chains of Free Groups 105 For Ded
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7 Chains of Free Groups 109 Hill’
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7 Chains of Free Groups 111 which i
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8 Almost Free Groups 113 (D) Let 0
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8 Almost Free Groups 115 Proof. (a)
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8 Almost Free Groups 117 be done by
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8 Almost Free Groups 119 with. In t
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9 Shelah’s Singular Compactness T
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10 Groups with Discrete Norm 123 Di
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10 Groups with Discrete Norm 125 Ch
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11 Quasi-Projectivity 127 11 Quasi-
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11 Quasi-Projectivity 129 for diffe
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132 4 Divisibility and Injectivity
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150 5 Purity and Basic Subgroups or
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152 5 Purity and Basic Subgroups Pr
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154 5 Purity and Basic Subgroups ge
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158 5 Purity and Basic Subgroups Th
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160 5 Purity and Basic Subgroups ˛
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166 5 Purity and Basic Subgroups 5
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168 5 Purity and Basic Subgroups Le
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172 5 Purity and Basic Subgroups F
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174 5 Purity and Basic Subgroups (H
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176 5 Purity and Basic Subgroups Ma
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180 5 Purity and Basic Subgroups So
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Chapter 6 Algebraically Compact Gro
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1 Algebraic Compactness 185 of the
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1 Algebraic Compactness 187 Na j ;
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1 Algebraic Compactness 189 Here jG
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2 Complete Groups 191 is a Cauchy s
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2 Complete Groups 193 A ⏐ μ A A
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3 The Structure of Algebraically Co
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3 The Structure of Algebraically Co
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4 Pure-Injective Hulls 199 (2) The
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4 Pure-Injective Hulls 201 subgroup
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5 Locally Compact Groups 203 5 Loca
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5 Locally Compact Groups 205 Theore
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6 The Exchange Property 207 Evident
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6 The Exchange Property 209 In the
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6 The Exchange Property 211 to the
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Chapter 7 Homomorphism Groups Abstr
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1 Groups of Homomorphisms 215 (and
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1 Groups of Homomorphisms 217 is an
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1 Groups of Homomorphisms 219 F Not
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2 Algebraically Compact Homomorphis
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2 Algebraically Compact Homomorphis
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3 Small Homomorphisms 225 (3) The a
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3 Small Homomorphisms 227 completio
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Chapter 8 Tensor and Torsion Produc
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1 The Tensor Product 231 with the f
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1 The Tensor Product 233 for matchi
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1 The Tensor Product 235 finitely P
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2 The Torsion Product 237 (6) (a) I
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2 The Torsion Product 239 It is pre
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2 The Torsion Product 241 It remain
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3 Theorems on Tensor Products 243 T
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4 Theorems on Torsion Products 245
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5 Localization 251 (8) (Keef) If A
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5 Localization 253 Exercises (1) If
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256 9 Groups of Extensions and Coto
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300 10 Torsion Groups (A) If C is a
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302 10 Torsion Groups the indicator
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304 10 Torsion Groups 0 −−−
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306 10 Torsion Groups is tantamount
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308 10 Torsion Groups Proof. We hav
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310 10 Torsion Groups (3) Let A be
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312 10 Torsion Groups (C) B is a ba
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314 10 Torsion Groups pure-injectiv
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316 10 Torsion Groups Theorem 3.11
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318 10 Torsion Groups 4 More on Tor
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320 10 Torsion Groups (g) The large
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322 10 Torsion Groups that in torsi
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324 10 Torsion Groups subgroup of B
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326 10 Torsion Groups (c) Homomorph
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328 10 Torsion Groups then Ker con
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330 10 Torsion Groups A class more
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332 10 Torsion Groups with basic su
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334 10 Torsion Groups 8 Valuated Ve
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336 10 Torsion Groups V D A ` B for
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338 10 Torsion Groups Proof. Define
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340 10 Torsion Groups When Socles D
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342 10 Torsion Groups Hill [1] gave
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344 11 p-Groups with Elements of In
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410 12 Torsion-Free Groups is calle
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412 12 Torsion-Free Groups Proof. L
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414 12 Torsion-Free Groups If A is
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416 12 Torsion-Free Groups Exercise
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418 12 Torsion-Free Groups Converse
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420 12 Torsion-Free Groups fa X j a
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422 12 Torsion-Free Groups and 0 !
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424 12 Torsion-Free Groups (ii) If
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426 12 Torsion-Free Groups number o
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428 12 Torsion-Free Groups along B
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430 12 Torsion-Free Groups whenever
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432 12 Torsion-Free Groups If we ha
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434 12 Torsion-Free Groups Then we
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436 12 Torsion-Free Groups (ii) For
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438 12 Torsion-Free Groups (7) (de
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440 12 Torsion-Free Groups it is ro
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442 12 Torsion-Free Groups 1 for `
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444 12 Torsion-Free Groups A D B i
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446 12 Torsion-Free Groups (5) For
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448 12 Torsion-Free Groups Proof. L
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450 12 Torsion-Free Groups local (T
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452 12 Torsion-Free Groups stable r
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454 12 Torsion-Free Groups (3) (War
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456 12 Torsion-Free Groups write x
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458 12 Torsion-Free Groups For thos
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460 12 Torsion-Free Groups A C 1
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462 12 Torsion-Free Groups ˛./.˛
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464 12 Torsion-Free Groups so the s
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466 12 Torsion-Free Groups followed
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468 12 Torsion-Free Groups (i) ther
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470 12 Torsion-Free Groups 11 Duali
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472 12 Torsion-Free Groups Lemma 11
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474 12 Torsion-Free Groups rk p D.A
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476 12 Torsion-Free Groups is proj(
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478 12 Torsion-Free Groups division
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Chapter 13 Torsion-Free Groups of I
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1 Direct Decompositions of Infinite
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2 Slender Groups 489 (6) A torsion-
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2 Slender Groups 491 Example 2.5. T
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2 Slender Groups 493 Proof. From Th
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2 Slender Groups 495 F Notes. The r
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3 Characterizations of Slender Grou
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3 Characterizations of Slender Grou
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4 Separable Groups 501 Exercises (1
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4 Separable Groups 503 where the G
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4 Separable Groups 505 In the follo
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4 Separable Groups 507 c C g D h 2
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5 Vector Groups 509 (7) Show that t
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5 Vector Groups 511 Isomorphism of
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6 Powers ofZ of Measurable Cardinal
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7 Whitehead Groups If V = L 519 (7)
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7 Whitehead Groups If V = L 521 Pro
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7 Whitehead Groups If V = L 523 Sˇ
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8 Whitehead Groups Under Martin’s
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8 Whitehead Groups Under Martin’s
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Chapter 14 Butler Groups Abstract T
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1 Finite Rank Butler Groups 531 and
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1 Finite Rank Butler Groups 533 Pro
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1 Finite Rank Butler Groups 535 Let
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2 Prebalanced and Decent Subgroups
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2 Prebalanced and Decent Subgroups
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3 The Torsion Extension Property 54
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4 Countable Butler Groups 547 (ii)
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5 B 1 -andB 2 -Groups 549 (F) Homog
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6 Solid Subgroups 551 understand th
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6 Solid Subgroups 553 fa n j n
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6 Solid Subgroups 555 Again changin
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7 Solid Chains 557 (4) A corank 1 s
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7 Solid Chains 559 B C1 =B admits
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7 Solid Chains 561 Lemma 7.6 (Bican
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8 Butler Groups of Uncountable Rank
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9 More on Infinite Butler Groups 56
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9 More on Infinite Butler Groups 57
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Chapter 15 Mixed Groups Abstract Mi
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1 Splitting Mixed Groups 575 For ev
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1 Splitting Mixed Groups 577 so A m
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2 Baer Groups are Free 579 (4) (Opp
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2 Baer Groups are Free 581 We note
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2 Baer Groups are Free 583 One does
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3 Valuated Groups. Height-Matrices
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4 Nice, Isotype, and Balanced Subgr
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5 Mixed Groups of Torsion-Free Rank
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5 Mixed Groups of Torsion-Free Rank
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6 Simply Presented Mixed Groups 597
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7 Warfield Groups 599 Exercises (1)
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7 Warfield Groups 601 (i) A=F is to
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7 Warfield Groups 603 Lemma 7.7 (St
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8 The Categories WALK and WARF 605
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8 The Categories WALK and WARF 607
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9 Projective Properties of Warfield
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9 Projective Properties of Warfield
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Chapter 16 Endomorphism Rings Abstr
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1 Endomorphism Rings 615 Lemma 1.5.
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1 Endomorphism Rings 617 For a subg
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1 Endomorphism Rings 619 Before sta
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1 Endomorphism Rings 621 Recently,
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2 Endomorphism Rings of p-Groups 62
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3 Endomorphism Rings of Torsion-Fre
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4 Endomorphism Rings of Special Gro
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5 Special Endomorphism Rings 637 se
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5 Special Endomorphism Rings 639 Pr
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5 Special Endomorphism Rings 641 Re
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6 Groups as Modules Over Their Endo
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7 Groups with Prescribed Endomorphi
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7 Groups with Prescribed Endomorphi
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656 17 Automorphism Groups Example
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658 17 Automorphism Groups linear g
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660 17 Automorphism Groups Exercise
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662 17 Automorphism Groups Accordin
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664 17 Automorphism Groups (2) If A
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666 17 Automorphism Groups (b) If G
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668 17 Automorphism Groups Then a 7
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670 17 Automorphism Groups for all
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Chapter 18 Groups in Rings and in F
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1 Additive Groups of Rings and Modu
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2 Rings on Groups 677 (11) For a ri
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2 Rings on Groups 679 is a subring,
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2 Rings on Groups 681 Beaumont-Lawv
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3 Additive Groups of Noetherian Rin
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4 Additive Groups of Artinian Rings
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4 Additive Groups of Artinian Rings
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5 Additive Groups of Regular Rings
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5 Additive Groups of Regular Rings
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6 E-Rings 693 (2) A group is isomor
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6 E-Rings 695 (F) Every endomorphis
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7 Groups of Units in Commutative Ri
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7 Groups of Units in Commutative Ri
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8 Multiplicative Groups of Fields 7
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References Books D.M. Arnold — [A
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References 709 D.M. Arnold, R. Hunt
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References 711 R. Burkhardt — [1]
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References 713 M. Dugas — [1] Fas
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References 715 A.A. Fomin — [1] T
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References 717 R. Göbel, S. Shelah
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References 719 P. Hill, M. Lane, C.
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References 721 A. Kertész — [1]
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References 723 A. Mader — [1] On
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References 725 L.G. Nongxa — [1]
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References 727 endomorphism rings.
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References 729 G.M. Tsukerman — [
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Author Index A Abel, N.H., vii Albr
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Author Index 733 Grinshpon, S.Ya.,
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Author Index 735 O’Neill, J.D., 3
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Subject Index A Abelian group, 1 Ab
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Subject Index 739 homomorphism, 37,
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Subject Index 741 Full rational gro
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Subject Index 743 Łoś-Eda theorem
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Subject Index 745 Rank 1 torsion-fr
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Subject Index 747 vector space, 334
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Abelian Groups - László Fuchs [Springer]

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