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19 Free Abelian Groups 19 Free abelian groups The main properties of free abelian groups are that they are projective and that every subgroup of a free abelian group is free abelian. We have already proved the dual properties for divisible groups: Every divisible group is injec- tive and every quotient of a divisible group is divisible. After going through the usual proofs I’ll use diagrams to explain why these statements are in fact equivalent. Definition 19.1. A (left) R-module P is called projective if for any R-module epimorphism p : B ³ C, any morphism1 f : P ! C lifts to B in the sense that there is a morphism g : P ! B so that pg = f: B 9g ¡¡ ¡ p ¡ f ? PC- Proposition 19.2. Every projective module is isomorphic to a direct sum- mand of a free module. Proof. Suppose that P is a projective R-module. Let F be a free module which maps onto P (e.g., take the free module with one generator for every element of P ). Then the projection map p : F ! P has a section s: F 9s ¡¡ ¡ p ¡ ? PP- idP The image of s is a direct summand of F isomorphic to P . Theorem 19.3. Every free R-module is projective. Proof. A morphism on a free module R[X] with basis X is uniquely deter- mined by its values on the elements of X which are arbitrary, i.e., we have the adjunction relation: HomR(R[X];B) = MorSets(X; F B) FB is the underlying set of B (with the rest of the structure “forgotten”). Any lifting to B of the set function f : X ! C extends uniquely to a morphism g : R[X] ! B lifting f. The cardinality of the set X is called the rank of the free R-module R[X]. 1I use the word morphism to mean morphism in the appropriate category. In this case it means R-module homomorphism. 1 Theorem 19.4. The rank of a free abelian group is well-defined. In other words, Z[X] »= Z[Y ] iff X; Y have the same cardinality. Proof. The rank of a free abelian group F is the dimension over Z=p of F=pF which is well-defined. Theorem 19.5. Every subgroup of a free abelian group is free abelian with smaller or equal rank. Remark 19.6. The proof uses only the fact that Z is a PID so the theorem holds for free modules over any PID. Proof. Let H be a subgroup of the free abelian group G = Z[X]. By the Axiom of Choice we can find a well-ordering of the set X. Then every nonzero element h 2 H can be expanded as a linear combination and we can use the well-ordering to pick out the leading term ¸(h) = n®x® if h = n®x® + n¯x¯ + ¢ ¢ ¢ where x® > x¯ > xγ > ¢ ¢ ¢ (and n® 6= 0). Let Y be the set of all x 2 X a multiple of which occurs as the leading term of some element of H. For each y 2 Y let I(y) = f0g [ fn 2 Z j ny = ¸(h) for some h 2 Hg This is an ideal. [If ny = ¸(h) then kny = ¸(kh), if my = ¸(h0) then 0 (n ¡ m)y = ¸(h ¡ h ) unless n = m.] Let ny 2 I(y) be a generator of this ideal. Then there is an h 2 H so that ¸(h) = nyy. Call it h(y). [This uses the Axiom of Choice.] We claim that Z = fh(y) j y 2 Y g is a free basis for the group H. It is clear that Z is linearly independent over Z since the leading terms of each of its elements are multiples of distinct elements of X. So it is enough to show that Z generates H. Suppose not. Then take an element h 2 H which is not in hZi so that ¸(h) = mx where x 2 X is minimal. Then x 2 Y and m 2 I(x) = (nx). So m = knx and kh(x) also has leading term mx. Consequently, h ¡ kh(x) (which also lies in H but not in hZi) has a smaller leading term than h contradicting the minimality of x. Finally, we note that the basis Z for H has the same number of elements as the subset Y of X so the rank of H is less than or equal to the rank of G = Z[X]. Corollary 19.7. PID’s have the property that projective modules are the same as free modules so every submodule of a projective module is projective. Proof. A projective module is always isomorphic to a submodule of a free module by Proposition 19.2. By Theorem 19.5 above, these are free. 2 Hereditary rings Now I want to explain that the properties of divisible groups and projec- tive (i.e. free) groups that we have proved separately are in fact logically equivalent. Theorem 19.8. For any ring R the following are equivalent. 1. Every submodule of a projective module is projective. 2. Every quotient module of an injective module is injective. Definition 19.9. A ring having these properties is called hereditary. We also say that it has global dimension · 1. [It has global dimension 0 if every module is projective. (Equivalently, every module is injective.) Rings of g.dim.0 are called semisimple.] The proof of this theorem uses the following general fact: Lemma 19.10. For any ring R we have the following. 1. Every module is a quotient of a projective module. 2. Every module embeds in an injective module. Proof of Theorem 19.8. Since (2) is obvious for abelian groups but we had to work hard to prove (1) we will show (2) implies (1). The “dual” argument will show that (1) implies (2). ⊆ B - I 9g ¡ ¡ p q ¡ ? ¡ f ? ⊆ HC- - I=A Suppose that H is a submodule of a projective module P and we have the lifting problem in the above diagram. (p : B ! C is onto and we need to lift f : H ! C up to B.) Using the lemma, embed B into an injective module I. Let A = ker p ⊆ B ⊆ I. Then C embeds in the quotient I=A which is injective by assumption (2). By definition of injectivity, the morphism H ! I=A extends to a mor- phism h : P ! I=A. Since P is projective, this lifts to a morphism h˜ : P ! I. But then h˜ sends H into B since B = q¡1(C) and qh˜jH = f. Consequently, g = h˜jH is a lifting of f. Proof that every group embeds in a divisible group. First of all note that ev- ery cyclic subgroup hgi · G embeds in a divisible group: Z embeds in Q and Z=n »= ©Z=pk embeds in a direct sum of Z=p1’s. Extending each of these homomorphisms to all of G we get an infinite collection of maps fg : G ! Dg where fg(g) 6= 0. Their product gives an embedding Q Q fg : G ½ g2G Dg: 3.
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