3. QUASI-HOMOMORPHISMS

E. L. Lady

April 17, 1998

Sometimes when one looks at finite rank torsion free modules one sees things that are almost true but not quite. A module may almost be a direct sum of two submodules, but a few elements don’t fit. Or two modules may almost be isomorphic, but once again there are a few elements that get in the way.

Consider again Example 1.49. Using the concepts of Chapter 2 we can describe the construction as follows: We start with two rank-one modules A and B whose types t(A) and t(B) are incomparable (and also locally trivial, but that’s not relevant at the moment.) We then take an element w ≠ 0 ∈ W such that there exist elements a ∈ A, b ∈ B with a ∉ wA, b ∉ wB. We then construct a module H as the submodule of QA ⊕ QB generated by A ⊕ B and (a + b)/w. We can show that H is indecomposable. Indeed, the fact that (a + b)/w ∈ H but a/w ∉ A and b/w ∉ B prevents H from being the direct sum of A and B. On the other hand, no other non-trivial direct sum decomposition can work because if H = K ⊕ L with K, L ≠ 0 then K and L must be rank-one modules (since H has rank two) and by Proposition 2.* t(H) = {t(K), t(L), t(K) ∧ t(L)}. But t(H) = {t(A), t(B), t(A) ∧ t(B)}. Since t(A) and t(B) are incomparable, this can only work if t(K) = t(A), t(L) = t(B) or vice versa. But the only elements in H with type t(A) are those in A and the only elements with type t(B) are those in B, so this forces K = A and L = B, or vice versa, the case which has already been ruled out.

In this example, the single element (a + b)/w is the glue that prevents H from falling apart into the direct sum of A and B. Thus we have the sense that H is just barely indecomposable, and that the scantiness of the glue holding it together is in fact what makes it possible to play the trick that yields the non-unique direct decomposition in Example 1.49.

Contrast with this Example 1.45 as generalized in Example 2.28. Here we have three rank-one modules A, B, and C with mutually incomparable types and such that any pair have the same greatest lower bound, and we construct G as the submodule of Q generated by A(1, 0), B(0, 1) and C(1, 1), where C(1, 1), for instance, denotes {(c, c) | c ∈ C}. In contrast to the preceding example, the glue here is very strong, consisting of the infinite set of elements (c, c). (Or looked at another way, we have put an infinite number of different denominators under the element (1, 1).) In fact, it is really only a matter of point of view whether one thinks of G as being A ⊕ B held together with very strong glue consisting of all the elements (c, c) with c ∈ C, or whether G is based on A ⊕ C and the elements of B(0, 1) constitute the glue, or possibly G starts with B ⊕ C glued together with the elements of A(1, 0). In fact, the bonds holding G together are so strong that it only makes sense to think of it as a single unified whole. One should not expect to be able to use the module G to play dirty tricks with direct sum decompositions in the same way as was done in Example 1.49 using H.

Two modules with the same divisible hull which differ by only a finite number of elements, such as H and A ⊕ B do in Example 1.49, are called quasi-equal. (A more careful definition will be given below.) More generally, we say that G and H are quasi-isomorphic and we write G ∼ H if G is isomorphic to a module which is quasi-equal to H. The notion of quasi-isomorphism, invented by Bjarni Jónsson, is a very powerful tool for understanding direct sum decompositions and for gaining insight into finite rank torsion free modules in general. The following proposition shows that the definition of quasi-isomorphism just given is equivalent to the definition given in Chapter 2, where we said that two modules are quasi-isomorphic if each is isomorphic to a submodule of the other.

proposition 3.1. Let G and H be finite rank torsion free modules. The following conditions are equivalent:

(1) G is isomorphic to an essential submodule H′ of H such that H/H′ is finitely generated, i.e. G ∼ H.

(2) G is isomorphic to a submodule H′ of H such that H/H′ is not faithful.

(3) G is isomorphic to a submodule H′ of H such that H/H′ has finite length.

(4) There exist monomorphisms G → H and H → G.

(5) G is isomorphic to an essential submodule of H and H is a homomorphic image of G ⊕ F for some finitely generated W -module F.

proof: (3) ⇒ (1): Clear

(1) ⇒ (2): (1) implies that H/H′ is finitely generated and torsion, hence it is not faithful.

(2) ⇒ (3): To say that H/H′ is not faithful is to say that there exists w ≠ 0 ∈ W with w(H/H′) = 0, i.e. wH ⊆ H′. Then H/H′ is a homomorphic image of H/wH which has finite length by Proposition 1.22.

(2) ⇒ (4): (2) implies a monomorphism θ: G → H such that there exists w ≠ 0 ∈ W with wH ⊆ θ(G). Then we also get a monomorphism H → G by h ↦ θ^{−1}(wh).

(4) ⇒ (2): We may suppose that in fact G ⊆ H and there is a monomorphism from H to G or, equivalently, there exists a monomorphism η ∈ End H with η(H) ⊆ G. Since η is monic it is clearly not a left zero divisor in End H, so the left multiplication mapping φ ↦ ηφ is a one-to-one mapping from End H to itself. Thus by Proposition 1.26 η End H is an essential submodule of End H, so W ∩ η End H ≠ 0. I.e. there exists φ ∈ End H and w ≠ 0 ∈ W such that ηφ = w. Then wH = ηφ(H) ⊆ η(H) ⊆ G so that w(H/G) = 0 and H/G is not faithful.

(1) ⇒ (5): Suppose that H′ is an essential submodule of H such that there exists an isomorphism φ_{1} : G → H′ and that H is generated by H′ together with elements h_{1}, …, h_{n}. Then there is a map φ_{2} : W^{n} → H sending the ith basis element of W^{n} onto h_{i}. Then φ_{1} and φ_{2} induce a homomorphism φ from G ⊕ W^{n} onto the submodule of H generated by H′ and h_{1}, …, h_{n}, i.e. onto H.

(5) ⇒ (1): **** ✓

THE CATEGORY OF TORSION FREE MODULES UNDER QUASI-HOMOMORPHISMS. Jónsson developed the concept of quasi-isomorphism in order to provide an environment in which finite rank torsion free abelian groups would have the Krull-Schmidt Property, i.e. direct sum decompositions would be unique up to (quasi-)isomorphism. Jónsson defined his concept within a framework of lattice theory, but later Jim Reid, among others, showed how to look at quasi-isomorphism from a ring-theoretic point of view, which is more comfortable for most abelian group theorists. The presentation here essentially follows Reid’s.

In accord with our conventions, we denote by QHom(G, H) and QEnd G the divisible hulls of Hom(G, H) and End G. We have dim QHom(G, H) = rank Hom(G, H) and dim QEnd G = rank End G < ∞. Thus QEnd G is a finite dimensional Q-algebra. We then define the category of finite rank torsion free W -modules under quasi-homomorphisms as the category where the objects are finite rank torsion free modules and the morphisms from G to H are the elements of QHom(G, H). The maps in QHom(G, H) are called quasi-homomorphisms from G to H, and maps in QEnd G are called quasi-endomorphisms of G.

Recall that we usually make the identification Hom(G, H) ⊆ Hom_{Q}(QG, QH). We then have also QHom(G, H) ⊆ Hom_{Q}(QG, QH), and an instant’s thought then shows that

A quasi-homomorphism φ from G to H is a linear transformation φ: QG → QH such that wφ(G) ⊆ H for some w ≠ 0 ∈ W.

proposition 3.2. Two finite rank torsion free modules G and H are quasi-isomorphic if and only if they are isomorphic in the category of finite rank torsion free modules under quasi-homomorphisms. proof: ( ⇒ ): If G ∼ H then there exist monomorphisms φ: G → H and ψ: H → G. Then φ ∈ QHom(G, H) and ψ ∈ QHom(H, G). It suffices to see that φψ and ψφ have inverses in QEnd H and QEnd G. But in fact the proof of (4) ⇒ (2) in Proposition 3.1 shows that there exist η ∈ QEnd H and γ ∈ QEnd G such that ηφψ = w ≠ 0 ∈ W and γψφ = w′ ≠ 0 ∈ W, showing that φψ is invertible in QEnd H and ψφ invertible in QEndG. (See also Proposition 3.4 below.)

( ⇐ ): If there exist φ ∈ QHom(G, H) and ψ ∈ QHom(H, G) with ψφ = 1_{G}, φψ = 1_{H}, then φ and ψ must be monomorphisms, so G ∼ H by Proposition 3.1. ✓

We can restate Proposition 3.1 in another useful form. First, it will be useful to prove some properties that apply to any finite rank torsion free ring R. (Recall the convention that in this book except in Chapter 11 the word “ring” is used to mean W -algebra.) The following proposition is just a standard result in the theory of finite dimensional algebras once one notes that r is a left or right zero divisor in R if and only if the same is true in QR. In fact, if sr = 0 for some s ∈ QR, then there exists w ≠ 0 ∈ W with ws ∈ R and (ws)r = 0. proposition 3.3. Let R be a finite rank torsion free ring and r ∈ R. The following conditions are equivalent:

(1) r is not a left zero divisor in R.

(2) r is not a right zero divisor in R.

(3) r is left invertible in QR.

(4) r is right invertible in QR.

proof:

(3) ⇒ (1): If rs = 0 and tr = 1 then s = trs = 0.

(4) ⇒ (2): Analogous.

(1) ⇒ (4): If r is not a left zero divisor in R then, as noted above, it is also not a left zero divisor in QR. Thus left multiplication by r is a monic linear transformation on the finite dimensional vector space QR, hence is an isomorphism, so rQR = QR. Thus 1 ∈ rQR, i.e. there exists s ∈ QR with rs = 1, so r is right invertible in QR.

(2) ⇒ (3): Analogous. ✓

proposition 3.4. Let φ ∈ End G. The following conditions are equivalent:

(1) φ is a monomorphism.

(2) φ is not a left zero divisor in End G.

(3) φ is not a right zero divisor in End G.

(4) φ is left invertible in QEnd G.

(5) φ is right invertible in QEnd G.

(6) G/φ(G) has finite length.

(7) φ is a quasi-automorphism, i.e. there exists ψ ∈ QEnd G with φψ = ψφ = 1_{G}.

proof: (1) ⇒ (2): If φ is monic then φα = 0 ⇒ α = 0.

(4) & (5) ⟺ (7): Clear.

(7) ⇒ (1): Clear.

(5) ⇒ (6): Let ψ ∈ QEnd G with φψ = 1 and let 0 ≠ w ∈ W be such that wψ(G) ⊆ G. Then wG = wφψ(G) = φ(wψ(G)) ⊆ φ(G) so that G/φ(G) is a homomorphic image of G/wG. But by Proposition 1.22 G/wG has finite length, hence so does G/φ(G).

(6) ⇒ (1): If length G/φ(G) < ∞ then G/φ(G) is torsion so φ(G) is an essential submodule of G and and φ is monic by Proposition 1.26.

The equivalence of (2) through (5) is a consequence of Proposition 3.3. ✓

proposition 3.5. If R is a finite rank torsion free ring and r ∈ R, then r is left invertible in R if and only if r is right invertible in R.

proof: Suppose that r is left invertible, say sr = 1 for s ∈ R. Then the proof of Proposition 3.4 shows that r has a two-sided inverse s′ in QR. Then s = s1 = srs′ = 1s′ = s′ so that rs = rs′ = 1 and s is in fact a two-sided inverse for

r. ✓

QUASI-CONCEPTS. This book is dedicated to the point of view that for many purposes the category of modules under quasi-homomorphisms is the appropriate environment for studying finite rank torsion free modules. It is also true that there are many times when one jumps back and forth between this category and the more traditional category very quickly, even within a single sentence. The way to deftly accomplish this without confusion is by judicious use of the prefix “quasi.”

(1) We say that two modules G and H are quasi-equal if QG = QH and the identity map on QG is a quasi-isomorphism from G to H and also from H to G. More prosaically, this means that there exist non-trivial elements w, w′ ∈ W such that wG ⊆ H and w′H ⊆ G.

(2) We say that a quasi-homomorphism φ: G → H is a quasi-surjection if H is quasi-equal to φ(G).

(3) By an (internal) quasi-direct decomposition of a module G we mean a pair of submodules H, K ⊆ QG such that G is quasi-equal to H ⊕ K. In this case, the associated projection QG → QH with kernel QK is in fact a quasi-homomorphism from G to H and is called the quasi-projection from G onto H.

(4) By a quasi-pure submodule of G we mean a module H ⊆ QG which is quasi-equal to a pure submodule of G.

(5) We say that a pair of quasi-homomorphisms H ⟶^{α} G ⟶^{β} K is quasi-exact if Ker β is quasi-equal to α(H), and we say that a short exact sequence of quasi-homomorphisms 0 → H ⟶^{α} G ⟶^{β} K → 0 is quasi-split if there exists a quasi-homomorphism σ: G → H with σα = 1_{H} or (equivalently) a quasi-homomorphism τ: K → G with βτ = 1_{K}. (See Proposition 3.16 below.)

(6) We say that G is strongly indecomposable if there is no non-trivial quasidirect decomposition of G. In other words, whenever G is quasi-equal to H ⊕ K then either H or K is trivial.

Note that isomorphic modules are quasi-isomorphic and strongly indecomposable modules are indecomposable.

example 3.6. The module H from Example 1.49, noted at the beginning of this chapter, where H is generated by A ⊕ B together with (a + b)/w, is merely indecomposable, not strongly indecomposable. On the other hand, the second example, G, generated by A(1, 0), B(0, 1) and C(1, 1), is strongly indecomposable.

One does not need a notion of “quasi-type” since invariance under quasi-isomorphisms is built into the very concept of type. In particular, one has the following result:

proposition 3.7. (1) If φ ∈ QHom(G, H) then for every g ∈ QG, t_{H}(φ(g)) ≥ t_{G}(g). (2) If H is a quasi-pure submodule of G then for every h ∈ QH, t_{H}(h) = t_{G}(h).

proof: (1) There exists w ≠ 0 ∈ W such that wφ ∈ Hom(G, H). Then t(φ(g)) = t(wφ(g)) ≥ t(g) by Proposition 2.21.

(2) If H is quasi-pure in G then H is quasi-equal to a pure submodule H′ of G. By Proposition 2.21 the type of an element h ∈ QH with respect to H′ is the same as its type with respect to G. Thus we may assume that G = H′, i.e. that H is quasi-equal to G. But this means that the identity map on QG belongs to both QHom(G, H) and QHom(H, G). Thus t_{G}(h) = t_{H}(h) for all h ∈ QH = QG by (1). ✓

The question arises as to whether it is permissible to use arrow notation to indicate quasi-homomorphisms. Your author is firmly committed to the belief that arrow notation is appropriate. Among other things, to disallow the use of arrows would be to disallow the use of commutative diagrams for quasi-homomorphisms, and this would definitely be a hardship. One the other hand, it has to be admitted that the indiscriminate use of arrows without indication as to what category the corresponding morphisms belong to can lead to a great deal of confusion. For this reason, in this book we have indicated quasi-homomorphisms by arrows only rarely, and only when there is a very clear indication that this is what we are doing.

As a further illustration of the idea that most of the statements one wants to make about finite rank torsion free modules can actually be made about quasi-equality classes, we have the following:

proposition 3.8. If G ∼ H then (1) rank G = rank H and for all prime ideals p, p-rank G = p-rank H.

(2) d(G) ≈ d(H).

(3) T(G) = T(H) and CT(G) = CT(H).

(4) End G and End H are quasi-isomorphic.

(5) QEnd G and QEnd H are isomorphic.

proof: (1) From Proposition 3.1 it is clear that if G ∼ H then QG ≈ QH, so rank G = rank H. Furthermore since G is isomorphic to an essential submodule of H by Proposition 1.24 p-rank G ≥ p-rank H. Likewise p-rank H ≥ p-rank G.

(2) By Proposition 3.1 we may assume that G ⊆ H and for some w ≠ 0 ∈ W, wH ⊆ G. Then d(wH) ⊆ d(G) ⊆ d(H). But it is easy to see that d(wH) = d(H).

(3) Since we may assume without loss of generality that G and H are quasi-equal, this follows immediately from Proposition 3.7.

(4) With the same assumption as in (2), let φ ∈ End H. Then wφ(H) ⊆ G so a fortiori wφ(G) ⊆ G. Thus w End H ⊆ End G. On the other hand, if ψ ∈ End G then wψ(H) = ψ(wH) ⊆ ψ(G) ⊆ G ⊆ H so that wψ ∈ End H. Thus w End G ⊆ End H so that End G and End H are quasi-equal.

(5) Immediate from (4) since quasi-isomorphic modules have isomorphic divisible

hulls. ✓

proposition 3.9. Suppose that G ∼ H.

(1) If φ is a monomorphism from H to G, then φ is a quasi-isomorphism.

(2) If H ⊆ G then G and H are quasi-equal. proof: (1) Let θ be a quasi-isomorphism from G to H. Then φθ is a monomorphism in QEnd G and so by Proposition 3.4 is is a quasi-automorphism of G. Thus φ is quasi-surjective and hence a quasi-isomorphism.

(2) If G ∼ H and H ⊆ G then apply (1) to 1_{QG} to see that G and H are quasi-equal. ✓

proposition 3.10. If G is strongly indecomposable and I is a fractional ideal, then IG is strongly indecomposable.

proof: If IG is quasi-equal to H ⊕ K then G is quasi-equal to

I^{−1}(H ⊕ K) = I^{−1}H ⊕ I^{−1}K. ✓

proposition 3.11. Let H ⊆ G. Then The following conditions are equivalent: (1) H is quasi-equal to G.

(2) G/H is not faithful.

(3) G/H has finite length.

proof: (1) ⇔ Since H ⊆ G, H is quasi-equal to G if and only if there exists w ≠ 0 ∈ W such that wG ⊆ H, i.e. w(G/H) = 0.

(2) ⇔ (3) As seen in the proof of Proposition 3.1 (2) is equivalent to G/H having finite

length. ✓

proposition 3.12. (1) Let φ: QG → QH. Then φ is a quasi-homomorphism from G to H if and only if φ ∈ QHom(G_{p}, H_{p}) for all primes p and φ ∈ Hom(G_{p}, H_{p}) for all but finitely many p.

(2) Let QG = QH. Then G and H are quasi-equal if and only if G_{p} and H_{p} are quasi-equal for all primes p and G_{p} = H_{p} for all but finitely many p.

proof: (1) ( ⇒ ): If φ ∈ QHom(G, H) then for some w ≠ 0 ∈ W, wφ(G) ⊆ H. Then also wφ(G_{p}) ⊆ H_{p} for all p, so φ ∈ QHom(G_{p}, H_{p}) for all p. Furthermore for all p with w ∉ p, wH_{p} = H_{p} and so φ(G_{p}) ⊆ w^{−1}H_{p} = H_{p} so φ ∈ Hom(G_{p}, H_{p}). Since by Proposition 0.* there are only finitely many p with w ∈ p, it follows that φ ∈ Hom(G_{p}, H_{p}) for all but finitely many primes p.

( ⇐ ): For each p such that φ ∉ Hom(G_{p}, H_{p}) there exists w_{p} ≠ 0 ∈ W with w_{p}φ(G_{p}) ⊆ H_{p} because by assumption φ ∈ QHom(G_{p}, H_{p}). If there are only finitely many such p then we can multiply together the w_{p} to get a single w ≠ 0 ∈ W such that wφ(G_{p}) ⊆ H_{p} for all the exceptional p. Clearly this also holds for the non-exceptional p. Thus wφ(G) ⊆ H (see Proposition 0.*) so φ ∈ QHom(G, H).

(2) This follows immediately by applying (1) to the identity map on QG. ✓

The following triviality is used so pervasively in the remaining chapters that we have elevated it to the status of a lemma. lemma 3.13. Let R be a W -algebra and let M and N be R-modules which are torsion free as W -modules and such that M is finitely generated over R. Then

(1) QHom_{R}(M, N) = Hom_{R}(QM, QN).

(2) If N is a finitely generated essential R-submodule of QM then M and N are quasi-equal (as W -modules.)

(3) If R is a subring of S and S an essential subring of T such that T is finitely generated as an R-module, then S is quasi-equal to T.

proof: We need to show that Hom_{R}(QM, QN) ⊆ Q Hom_{R}(M, N), where the latter denotes the divisible hull of Hom_{R}(M, N) as a W -module. In fact, if φ: QM → QN is R-linear then there exists w ≠ 0 ∈ W such that wφ(m) ∈ N for some finite set of generators for M over R. Then wφ(M) ⊆ N so φ ∈ QHom_{R}(M, N).

In particular, if N is an R-submodule of QM with QN = QM, then this shows that 1_{QM} ∈ QHom_{R}(M, N). If N is also finitely generated as an R-module, then we also see that 1_{QM} is a quasi-homomorphism from N to M. This means that M and N are quasi-equal. ✓

The analog of Proposition 1.8 is valid for quasi-pure modules. In particular, we need to note the following:

proposition 3.14. If H ◁ G and H ⊆ K ⊆ G then K is a quasi-pure submodule of G if and only if K/H is a quasi-pure submodule of G/H.

proof: ( ⇒ ): If K is quasi-pure in G then K is quasi-equal to a pure submodule K′ of G. Then K′ = G ∩ QK′ = G ∩ QK so that H ⊆ K ⊆ K′, and there exists w ≠ 0 ∈ W such that wK′ ⊆ K. By Proposition 1.8 K′/H is pure in G/H. Since w(K′/H) ⊆ K/H ⊆ K′/H, it follows that K/H is quasi-equal to K′/H. Thus K/H is a quasi-pure submodule of G/H.

( ⇐ ): If K/H is quasi-pure in G/H, then K/H is quasi-equal to a pure submodule of G/H, which must have the form K′/H for some K′ such that H ⊆ K′ ⊆ G. Furthermore as in the preceding part of the proof K/H ⊆ K′/H, so that K ⊆ K′. By Proposition 1.8 K′ ◁ G. Now there exists w ≠ 0 ∈ W such that w(K′/H) ⊆ K/H. Then wK′ ⊆ K ⊆ K′, so K is quasi-equal to K′. Thus K is a quasi-pure submodule of

G. ✓

Another result we’ll need later is the following:

proposition 3.15. (1) If G = G_{1} + G_{2} and φ ∈ Hom(G, QH) for any H, then φ is a quasi-homomorphism from G to H if and only if the restrictions of φ to G_{1} and G_{2} are quasi-homomorphisms from G_{1} and G_{2} to H.

(2) If in addition QG_{1} = QG_{2} = QG then

QHom(G_{1} + G_{2}, H) = QHom(G_{1}, H) ∩ QHom(G_{2}, H).

proof: (1) If φ is a linear transformation from QG = QG_{1} + QG_{2} to QH and φ ∈ QHom(G, H) then for some w ≠ 0 ∈ W, wφ(G_{1} + G_{2}) ⊆ H and so wφ(G_{1}) ⊆ H and wφ(G_{2}) ⊆ H so that φ ∈ QHom(G_{1}, H) ∩ QHom(G_{2}, H). On the other hand if there exist w_{1}, w_{2} ≠ 0 ∈ W such that w_{1}φ(G_{1}) ⊆ H and w_{2}φ(G_{2}) ⊆ H, then w_{1}w_{2}φ(G_{1} + G_{2}) = w_{1}φ(G_{1}) + w_{2}φ(G_{2}) ⊆ H and φ ∈ QHom(G_{1} + G_{2}, H).

(2) This is simply a restatement of (1) since if QG_{1} = QG_{2} = QG then by our conventions QHom(G, H), QHom(G_{1}, H) and QHom(G_{2}, H) are all identified as submodules of Hom(QG, QH). ✓

QUASI-DIRECT DECOMPOSITIONS. We will not bother to formalize such obvious facts as the fact that the composition of two quasi-homomorphisms is a quasihomomorphism, or that if G is quasi-equal to G′ and H to H′ and φ ∈ QHom(G, H) then φ ∈ QHom(G′, H′). The point is that the “quasi” terminology works surprisingly well, and one can fearlessly take sentences which are known to be true for the usual category of modules under homomorphisms and change them into equally true sentences simply by inserting the word “quasi” throughout. For instance all the traditional theorems on direct sum decompositions have analogs for quasi-direct decompositions. The few proofs that follow should be enough to convince skeptics.

proposition 3.16. Let H and K be submodules of G. The following conditions are equivalent:

(1) G is quasi-equal to H ⊕ K.

(2) There exists a unique idempotent γ′ ∈ QEnd G such that H is quasi-equal to γ′(G) . and K is quasi-equal to Ker γ′.

(3) There exists γ ∈ End G and w ≠ 0 ∈ W such that γ^{2} = wγ and H is quasi-equal to γ(G) and K is quasi-equal to Ker γ.

proof: (1) ⇒ (2): If G is quasi-equal to H ⊕ K, then QG = QH ⊕ QK. Let γ be the unique idempotent linear transformation on QG with image QH and kernel QK. Let w, w′ ≠ 0 ∈ W be such that wG ⊆ H ⊕ K and w′(H ⊕ K) ⊆ G. Then wγ(G) = γ(wG) ⊆ γ(H ⊕ K) ⊆ H. Then ww′γ(G) ⊆ w′H ⊆ G, so γ ∈ QEnd G. Since also w′H = w′γ(H ⊕ K) ⊆ γ(G), γ(G) is quasi-equal to H. Now as an endomorphism of QG, γ has kernel QK. But since we are thinking of γ as a mapping with domain G, we have Ker γ = G ∩ QK ⊆ G. Then if w and w′ are as above, w′K ⊆ G ∩ QK = Ker γ and w(Ker γ) ⊆ QK ∩ (H ⊕ K) = K. Thus Ker γ is quasi-equal to K.

(2) ⇒ (3): If γ′ ∈ QEnd G is the idempotent above then let γ = wγ′, where w ≠ 0 ∈ W is chosen so that γ ∈ End G. Then γ^{2} = w^{2}γ′^{2} = wγ and H is quasi-equal to γ(G), K is quasi-equal to Ker γ.

(3) ⇒ (1): It will suffice to prove that following more general Lemma.

lemma 3.17. Let γ ∈ End G and let w ≠ 0 ∈ W be such that γ^{2} = wγ. Then wG ⊆ γ(G) ⊕ Ker γ ⊆ G. proof: Ker γ ∩ γ(G) = 0 since if x = γ(g) and γ(x) = 0 then wx = wγ(g) = γ^{2}(g) = γ(x) = 0 so x = 0. Now for any g ∈ G, wg = γ(g) + (wg − γ(g)), where γ(g) ∈ γ(G) and wg − γ(g) ∈ Ker γ since γ(wg − γ(g)) = wγ(g) − γ^{2}(g) = 0. Thus

wG ⊆ γ(G) ⊕ Ker γ ⊆ G. ✓

proposition 3.18. Let 0 → H ⟶^{α} G ⟶^{β} K → 0 be a short quasi-exact sequence. Then α(H) is a quasi-summand of G if and only if this sequence is quasi-split. More precisely, if we assume that α ∈ Hom(H, G) and β ∈ Hom(G, K) and 0 → H ⟶^{α} G ⟶^{β} K → 0 is actually exact, then for any w ≠ 0 ∈ W the following conditions are equivalent:

(1) There exists σ ∈ Hom(G, H) such that σα = w.

(2) There exists τ ∈ Hom(K, G) such that βτ = w.

(3) There exists L ⊆ G with α(H) ∩ L = 0 and wG ⊆ α(H) ⊕ L.

(4) There exists L ⊆ G such that α(H) ∩ L = 0 and β(L) is quasi-equal to K.

(5) α(H) is a direct summand of α(H) + wG.

proof: First note that if 0 → H ⟶^{α} G ⟶^{β} K → 0 is quasi-exact then there exists H′ quasi-equal to H and K′ quasi-equal to K such that α(H′) = Ker β ⊆ G and β(G) = K′, so that there is no loss of generality in assuming that α and β are honest homomorphisms and that the sequence is actually exact. Note also that α(H) is a quasi-summand of G if and only if (3) is true for some w ≠ 0 ∈ W, and that the sequence is quasi-split if and only if (1) and (2) are true for some w ≠ 0 ∈ W. Thus it suffices to prove the equivalence of (1) through (4).

(1) ⇒ (3): If there exists σ: G → H such that σα = w then let σ′ be defined on α(H) + wG by σ′(x) = σ(x/w) = σ(x)/w. Then σ′(α(h)) = σ(α(h))/w = h for h ∈ H, and in particular σ′(α(H)) ⊆ H. Also σ′(wG) = σ(G) ⊆ H. Thus σ′(α(H) + wG) ⊆ H so that σ′ ∈ Hom(α(H) + wG, H) and σ′α = 1_{H}, so σ′ splits α. Thus α(H) is a direct summand of α(H) + wG.

(2) ⇒ (3): Suppose that there exists τ: K → G with βτ = w. Now define β′ ∈ Hom(α(H) + wG, K) by β′(x) = β(x)/w. Then Ker β′ = Ker β = α(H) and β′(α(H) + wG) = β′(wG) ⊆ K. Since β′τ = 1_{K}, β′ is split and so α(H) is a direct summand of α(H) + wG.

(3) ⇔ (4): ****

(3) ⇒ (5): Given L ⊆ G with wG ⊆ α(H) ⊕ L ⊆ G, then α(H) ⊆ α(H) + wG ⊆ α(H) ⊕ L. Since α(H) is a direct summand of α(H) ⊕ L it follows that α(H) is a direct summand of α(H) + wG.

(5) ⇒ (1): If α(H) is a direct summand of α(H) + wG then there is is a splitting map σ′ ∈ Hom(α(H) + wG, H) with σ′α = 1_{H}. Let σ = wσ′. Then σ(G) = wσ′(G) = σ′(wG) ⊆ H so that σ ∈ Hom(G, H) and σα = w.

(4) ⇒ (2): If α(H) is a summand of α(H) + wG then there exists τ′: β(α(H) + wG) = wK → α(H) + wG such that βτ′ = 1_{wK}. Let τ = wτ′. Then τ(K) = wτ′(K) = τ′(wK) ⊆ G so that τ ∈ Hom(K, G) and βτ = w. ✓

example 3.19. Consider again Example 1.49, as reformulated at the beginning of this chapter. The module G is generated by A ⊕ B together with (a + b)/w for a certain w ∈ W, where a ∉ wA, b ∉ wB. The map γ: Q ⊕ Q → Q given by (x, y) ↦ wy maps G onto the submodule B′ of B generated by wB and b. Since wG ⊆ A ⊕ B it follows that the short exact sequence

0 → A → G → B′ → 0

is quasi-split. We can also notice that A + wG is generated by A and wB together with a + b. Thus b ∈ A + wG. But b generates B′ modulo wB so that A + wG = A ⊕ B′, so that A is a summand of A + wG, confirming the proposition. We have σ ∈ Hom(G, A) given by σ(a, b) = wa and τ ∈ Hom(B′, G) given by τ(b) = b and if we let α: A → G be the inclusion map then σα = w and γτ = w.

It turns out that any module H with rank two which is indecomposable but not strongly indecomposable looks pretty much like the H in Example 3.19.

proposition 3.20. Let H be an indecomposable module with rank H = 2 such that H is not strongly indecomposable. Then there exist pure rank-one submodules A, B of H and elements a ∈ A, b ∈ B and q ∈ Q such that H is generated by A ⊕ B together with q(a + b). Furthermore H/(A ⊕ B) ≈ A/IA ≈ B/IB ≈ W/I, where I is the ideal in W such that I^{−1} is the fractional ideal generated by 1 and q. If W is a principal ideal domain then we may choose q = 1/w for some w ∈ W. proof: Since H is not strongly indecomposable there exist non-trivial submodules A, B ⊆ H such that H is quasi-equal to A ⊕ B. Since rank H = 2, A and B must have rank one. We may assume that A and B are pure in H since we can replace them by their purifications.

Let α and β be the quasi-projections of H onto A and B, and let A′ = α(H), B′ = β(H). Then A ⊆ A′, B ⊆ B′ and A is quasi-equal to B′, B is quasi-equal to B′. By Proposition 2.2 A′/A ≈ W/I for some ideal I and A = IA′. Now α induces a surjection ᾱ: H → A′/A and we claim that Ker ᾱ = A ⊕ B. In fact, let h = a + b ∈ H with a ∈ QA, b ∈ QB. Then the assumption that A ◁ H yields h ∈ Ker ᾱ ⟺ α(h) = a ∈ A ⟺ a ∈ H ⟺ b = h − a ∈ H ⟺ h ∈ A ⊕ B. Thus H/(A ⊕ B) ≈ A′/A ≈ W/I and likewise H/(A ⊕ B) ≈ B′/B, so that B′/B ≈ W/I. It follows that B = IB′. Now 1 ∈ I^{−1} (since I ⊆ W) so by Proposition 0.* I^{−1} is generated by 1 together with some q ∈ I. If I is principal, generated by w ∈ W, then we may choose q = 1/w. Then A ⊕ B ⊆ H ⊆ A′ ⊕ B′ = I^{−1}(A ⊕ B) = (A ⊕ B) + q(A ⊕ B). Since H/(A ⊕ B) is cyclic, H is generated by A ⊕ B together with some single element h ∈ q(A ⊕ B), so h = q(a + b) for some a ∈ A, b ∈ B. ✓

Another familiar theorem about direct sum decompositions which carries over to quasi-direct compositions is the following:

proposition 3.21. Let H ⊆ K ⊆ G.

(1) If H is a quasi-summand of G then H is a quasi-summand of K. In fact, if G is quasi-equal to H ⊕ L then K is quasi-equal to H ⊕ (L ∩ K).

(2) If H ◁ G and K is a quasi-summand of G then K/H is a quasi-summand of G/H.

(3) If H is a quasi-summand of G then H is quasi-pure in G. proof: (1) If H is a quasi-summand of G then by Proposition 3.16 H is quasiequal to γ(G) for some idempotent γ ∈ QEnd G. Since γ(QG) = QH ⊆ QK, γ restricts to a linear endomorphism γ′ defined on QK. Now because γ is idempotent, H = γ(H) ⊆ γ(K) ⊆ γ(G). But H is quasi-equal to γ(G), so it follows that H is equal to γ(K) = γ′(K). Thus γ′ ∈ QEnd K. Thus γ′ is an idempotent quasi-endomorphism of K. By Proposition 3.16 K is quasi-equal to H ⊕ Ker γ′. But Ker γ′ = K ∩ Ker γ, which is quasi-equal to K ∩ L since Ker γ is quasi-equal L.

(2) A quasi-splitting map G/K → G induces a quasi-splitting map (G/H)/(K/H) = G/K → G/H.

(3) If G is quasi-equal to H ⊕ K and w, w′ ≠ 0 ∈ W are such that wG ⊆ H ⊕ K and w′(H ⊕ K) ⊆ G, then w(G ∩ QH) ⊆ H and w′H ⊆ G ∩ QH, so that H is quasi-equal to the pure submodule G ∩ QH of G. ✓

JÓNSSON’s THEOREM. The brilliant idea of looking at the category of finite rank torsion free W -modules under quasi-isomorphisms is that we are able to work in a category where the homomorphism groups are all finite dimensional vector spaces. Since this is what really makes most of the classical theorems about modules over finite dimensional algebras work, we can expect most of these theorems to carry over into our category. In particular, we get the Jónsson’s analog of the Krull-Schmidt Theorem:

theorem 3.22. The representation of a module as a quasi-direct sum of strongly indecomposable modules is unique up to quasi-isomorphism.

proof: We can simply follow the standard Azamaya proof, once we have established the following:

proposition 3.23. If G is strongly indecomposable, then QEnd G is a local ring.

proof: The proof is still classical. Namely, a finite dimensional algebra without proper idempotents is a local ring. (In fact if QN = nil rad QEnd G then (QEnd G)/QN can have no proper idempotents, since idempotents modulo QN lift to idempotents in QEnd G by Proposition 1.32. Thus (QEnd G)/QN is a semi-simple ring without idempotents, hence is a skew field, in particular has no proper left ideals. It follows that QN is the unique maximal left ideal in QEnd G.)

It thus suffices to see that if G is strongly indecomposable then QEnd G has no proper idempotents, which is equivalent to saying that if QEnd G has a proper idempotent then G has a non-trivial quasi-decomposition. But we have already proved this as Proposition 3.16. ✓

For those who non-commutative ring theory is weak, an alternative proof of Proposition 3.23 occurs further below (Corollary 3.26).

proof of Jónsson’s Theorem. We will actually prove a stronger result: theorem 3.24. Let G be quasi-equal to G_{1} ⊕ ⋯ ⊕ G_{s} and also to H_{1} ⊕ ⋯ ⊕ H_{t}, where the G_{i} and H_{j} are strongly indecomposable. Then s = t and the H_{j} can be renumbered such that for each i, the summands G_{i} and H_{i} are interchangeable. In other words if for each i = 1, …, s we let X_{i} denote either G_{i} or H_{i}, then G is quasi-equal to X_{1} ⊕ ⋯ ⊕ X_{s}. proof: We may assume wlog that G = G_{1} ⊕ … G_{s}. Let γ_{1} be the projection of G onto G_{1} with kernel equal to G_{2} ⊕ ⋯ ⊕ G_{s} and for j = 1, …, t let η_{j} be the quasi-idempotent in QEnd G with η_{j}(G) quasi-equal to H_{j}. Let φ_{j} ∈ QEnd G_{1} be the restriction of γ_{1}η_{j} to G_{1} . Then φ_{1} + ⋯ + φ_{t} = 1 ∈ QEnd G_{1} since ∑ η_{j} = 1 ∈ QEnd G. Since QEnd G_{1} is a local ring, at least one of the φ_{i}, say φ_{1} , must be invertible (otherwise all φ_{i} ∈ nil rad QEnd G which implies the contradiction 1 ∈ nil rad QEnd G). By renumbering the H_{j} we may suppose that φ_{1} is invertible, i.e. γ_{1}η_{1} restricts to an quasi-automorphism of G_{1} . By Proposition 3.18 this means that η_{1} restricts to a split quasi-monomorphism from G_{1} onto a quasi-summand of H_{1} . But by assumption H_{1} is strongly indecomposable, so that η_{1} must thus restrict to a quasi-isomorphism from G_{1} to H_{1} and γ_{1} to a quasi-isomorphism from H_{1} to G_{1} . Thus G_{1} is quasi-equal to γ_{1}(H_{1}), so G is quasi-equal to γ_{1}(H_{1}) ⊕ G_{2} ⊕ ⋯ ⊕ G_{s}.

We now claim that γ_{1}(H_{1}) ⊕ G_{2} ⋯ ⊕ G_{s} is quasi-equal to H_{1} ⊕ G_{2} ⊕ ⋯ ⊕ G_{s}. First note that since γ_{1} restricted to H_{1} is monic, then H_{1} ∩ (G_{2} ⊕ ⋯ ⊕ G_{s}) = H_{1} ∩ Ker γ_{1} = 0, so that the sum H_{1} + G_{2} ⊕ ⋯ ⊕ G_{s} is in fact direct. Now let w ≠ 0 ∈ W be such that wH_{1} ⊆ G = G_{1} ⊕ ⋯ ⊕ G_{s}. If g_{1} ∈ γ_{1}(H_{1}) then there exists h_{1} ∈ H_{1} with g_{1} = γ_{1}(h_{1}), and what this means is that wg_{1} = wh_{1} +x for some x ∈ G_{2} ⊕⋯⊕G_{s}. Thus wγ_{1}(H_{1}) ⊆ wH_{1} ⊕G_{2} ⊕⋯⊕G_{s} and so wγ_{1}(H_{1})⊕G_{2} ⊕⋯⊕G_{s} ⊆ wH_{1} ⊕G_{2} ⊕⋯⊕G_{s} ⊆ G. Since we have seen that G is quasi-equal to γ_{1}(H_{1}) ⊕ G_{2} ⊕ ⋯ ⊕ G_{s}, this shows that G is quasi-equal to H_{1} ⊕ G_{2} ⊕ ⋯ ⊕ G_{s}, so that H_{1} and G_{1} are interchangeable up to quasi-equality.

We now have G_{1} ⊕ G_{2} ⊕ ⋯ ⊕ G_{s} quasi-equal to G_{1} ⊕ H_{2} ⊕ ⋯ ⊕ H_{t}. Applying the same argument again to G_{2} shows that G_{2} is interchangeable with either G_{1} or H_{i} for some i = 2, …, t. But clearly G_{2} and G_{1} are not interchangeable. Thus we may renumber the H_{i} so that G_{2} and H_{2} are interchangeable. Continuing in this fashion, we obtain the

theorem. ✓

MORE STOLEN THEOREMS. There are other theorems we can steal from the classical theory of finite dimensional algebras. The following result is known in the classical theory as Fitting’s Lemma.

theorem 3.25. Let γ ∈ End G. Then G is quasi-equal to H ⊕ K, where H and K are invariant under γ and the restriction of γ to H is a quasi-automorphism and γ^{n}(K) = 0 for some n ≥ 1. proof: The ascending chain of pure submodules Ker γ ⊆ Ker γ^{2} ⊆ … must become stationary at some n ≤ rank G. Let K = Ker γ^{n} and let H = γ^{n}(G). Then clearly H and K are invariant under γ and by definition γ^{n}(K) = 0. Furthermore, H ∩ Ker γ = 0: in fact, if h ∈ H ∩ Ker γ then (∃g) h = γ^{n}(g) and γ(h) = γ^{n+1}(g) = 0 so g ∈ Ker γ^{n+1} = Ker γ^{n} so h = γ^{n}(g) = 0. Thus the restriction of γ to H is monic, hence is a quasi-automorphism, as required. Now let θ ∈ QEnd H be an inverse to the restriction of γ^{n}. Then θγ^{n} ∈ QHom(G, H) and θγ^{n} restricts to the identity on H, hence θγ^{n} is a quasi-projection and G is quasi-equal to H ⊕ Ker(θγ^{n}). But

Ker θγ^{n} = Ker γ^{n} = K. ✓

From this we get an extremely useful corollary. (In fact your author ventures that he has probably got more mileage out of the following result than from any other theorem in the subject.)

corollary 3.26. G is strongly indecomposable if and only if every (quasi-) endomorphism of G is either monic or belongs to nil rad QEnd G. In particular, this shows (again) that QEnd G is a local ring. proof: ( ⇐ ): If every endomorphism of G is either monic or belongs to the nil radical of QEnd G then QEnd G can have no proper idempotents, because if γ is a proper idempotent then (∃w ≠ 0 ∈ W) wγ ∈ End G and wγ is neither monic nor nilpotent.

( ⇒ ): If γ ∈ End G, apply Proposition 3.25. Since G has no proper quasi-direct decomposition, either γ is a quasi-automorphism or G = Ker γ^{n} so that γ is nilpotent. Furthermore if γ is not a quasi-automorphism then by Propositioin 3.* it is not monic, and therefore for every β ∈ QEnd G, βγ is not a quasi-automorphism, and therefore must also be nilpotent. Thus the left ideal generated by γ contains only nilpotent elements, hence γ ∈ nil rad QEnd G.

Note that this shows that nil rad QEnd G is a maximal left ideal (and necessarily the unique one), since any element not in the nilradical is invertible in QEnd G. Thus QEnd G is a local ring. ✓

corollary 3.27. If G is strongly indecomposable and has no nilpotent endomorphisms, then rank(End G) divides rank G. proof: By the Corollary 3.26 every non-trivial endomorphism of G is monic, hence invertible in QEnd G, and it follows that QEnd G is a skew field. Now QG is a vector space over this skew field, hence rank G = dim QG must be a multiple of dim QEnd G = rank End G. ✓

corollary 3.28. Let G be strongly indecomposable and N = nil rad End G. Then N = {φ ∈ End G | φ(G) ⊆ NG}. proof: Clearly φ ∈ N ⇒ φ(G) ⊆ NG. Conversely, by Proposition 1.33 rank NG < rank G. Thus if φ(G) ⊆ NG then φ is not monic, and for all γ ∈ End G, γφ is not monic. Thus by Corollary 3.26 γφ is nilpotent. Thus the left ideal in End G generated by φ is nil and it follows by Proposition 1.30 that φ ∈ N. ✓

The following corollary to Fitting’s Lemma would be trivial if the hypothesis were that G be p-reduced, but is surprising since we require only the much weaker assumption that G has no p-divisible quasi-summand. It is also surprising that the hypothesis only imposes a condition relevant to a single prime p. proposition 3.29. Let p be a prime ideal and G a module with no p-divisible quasi-summand. Let φ ∈ End G. If φ preserves p-heights then φ is monic. proof: Assume that φ preserves p-heights and apply Proposition 3.25 to get a quasi-decomposition of G as H ⊕ K where φ restricts to a quasi-automorphism of H and K = Ker φ^{n} for some n. Since φ preserves p-heights, if g ∈ Ker φ then ht_{p}(g) = ht_{p}(0) = ∞. By induction we see that ht_{p}(g) = ∞ for all g ∈ Ker φ^{n} = K, so K is p-divisible. By assumption, then, K = 0, so φ is a quasi-automorphism. ✓

MURLEY MODULES. We noted in Chapter Two that rank-one modules are the atoms of the theory of finite rank torsion free modules. If we restrict attention to homomorphisms of reduced modules, though, we find that indecomposable modules G with the property that p-rank G ≤ 1 for every prime ideal p have many of the properties of rank-one modules.

definition 3.30. A Murley module is a reduced torsion free module G such that p-rank G ≤ 1 for all prime ideals p. A cohesive module is a reduced finite rank torsion free module G such that every proper homomorphic image of G is divisible. A Murley ring is a torsion free W -algebra R such that R is a Murley module and a cohesive ring is a torsion free ring which is a cohesive module.

These definitions differ from the ones in [Arnold], [Murley], and [Dubois] by the added restriction that G be reduced. This avoids certain nuisance special cases. We will not assume that Murley modules have finite rank, but we will require that a Murley module cannot be written as an infinite direct sum of non-trivial summands.

proposition 3.31. G is a cohesive module if and only if G is a Murley module and for all primes p, G is either p-divisible or p-reduced. proof: ( ⇐ ): If H is a pure submodule of G and p is a prime then by assumption either G is p-divisible, so p-rank H = p-rank G = 0, or G is p-reduced and p-rank G = 1, so H is also p-reduced and 1 ≤ p-rank H ≤ p-rank G = 1. In either case p-rank G/H = 0 so G/H is p-divisible. Since this is true for all p, G/H is divisible.

( ⇒ ): If on the one hand p-rank G > 1 for some p then G must contain a pure rank-one module A which is not p-divisible (otherwise G would be p-divisible, contradicting the assumption that p-rank G > 1). Then p-rank A = 1 < p-rank G so that p-rank G/A ≠ 0 and G/A is a proper homomorphic image of G which is not divisible. And if there exists p with 0 ≠ p^{−∞}G ⊊ G then G/p^{∞}G is not divisible since it is not p-divisible. ✓

proposition 3.32. G is a Murley module if and only if G is isomorphic to a pure submodule of ∏_{p} Ŵ_{p}.

proof: ( ⇐ ): If G ◁ ∏ Ŵ_{p} then G is reduced and for all p, p-rank G ≤ p-rank ∏_{p} Ŵ_{p} = 1.

( ⇒ ): Suppose that G is a Murley module. By Proposition 1.*, for each prime p the completion Ĝ_{p} is a free Ŵ_{p}-module with the same p-rank as p-rank G, namely 1 or 0. Thus Ĝ_{p} = 0 or Ĝ_{p} ≈ Ŵ_{p} for all p. Now the canonical maps θ_{p}: G → Ĝ_{p} induce a map θ: G → ∏ Ĝ_{p}. Furthermore Ker θ = ⋂ Ker θ_{p} = ⋂ p^{∞}G = d(G) = 0, so that θ is monic. Finally, for each prime p, θ_{p}(G_{p}) ◁ Ĝ_{p} and it follows that θ(G) is p-pure in ∏_{p} Ĝ_{p}. Since this is true for every p, θ(G) is a pure submodule of ∏_{p} Ĝ_{p} ≈ ∏_{p} Ŵ_{p}. ✓

proposition 3.33. G is a Murley module if and only if every finite length homomorphic image of G is cyclic or trivial. proof: ( ⇐ ): If every homomorphic image of G is cyclic, then G/pG is a cyclic W/p-module, hence is simple or trivial. Thus p-rank G ≤ 1.

( ⇒ ): Let G be a Murley module and suppose by way of contradiction that there exists a non-trivial homomorphic image G/H which is not cyclic. Then by Proposition 0.* some p-primary component of G/H is not cyclic. This p-primary component is a homomorphic image of G/H and thus a homomorphic image of G. Therefore wlog we may suppose that G/H is p-primary and not cyclic. But then there is a homomorphic image G/H′ of G/H such that p(G/H′) = 0 and G/H′ is not cyclic. Then pG ⊆ H′, so G/pG is not cyclic. It follows that p-rank G = length G/pG > 1, a contradiction. ✓

Recall that the reduced tensor product of G and H is defined to be G ⋆ H = (G ⊗ H)/d(G ⊗ H).

proposition 3.34. (1) A pure submodule of a Murley module is a Murley module. (2) If G and H are Murley modules then so are G ⋆ H and Hom(G, H).

(3) If G and H are cohesive then so is Hom(G, H).

proof: (1) If H ◁ G and G is a Murley module, then d(H) ⊆ d(G) = 0 so H is reduced. Furthermore for every prime p, p-rank H ≤ p-rank G ≤ 1. Thus H is a Murley module.

(2) If H is reduced then by Proposition 1.34 so is Hom(G, H). Furthermore by Proposition 1.36 p-rank Hom(G, H) ≤ (p-rank G)(p-rank H) ≤ 1. Likewise G ⋆ H has p-rank at most one, and by definition G ⋆ H is reduced.

(3) By Proposition 1.34 if H is p-reduced or p-divisible then so is Hom(G, H). Since Hom(G, H) is Murley module by (1), the result follows from Proposition 3.27. ✓

Proposition 3.34 is stated in terms of G ⋆ H rather than G ⊗ H because G ⊗ H need not be reduced even when G and H are, as was seen in Example 2.36.

In general, the determination of d(G ⊗ H) is a non-trivial problem. We will address this issue in Chapter 8.

The following Proposition shows a very important way that Murley modules are like rank-one modules. proposition 3.35. G is a Murley module if and only if whenever H is quasi-equal to G then H = IG for some fractional ideal I. If H ⊆ G then we may choose I ⊆ W.

proof: ( ⇐ ): Suppose that G has the indicated property and let p be a prime. We must show that p-rank G ≤ 1, i.e. that G/pG is a simple module. Now a submodule of G/pG has the form H/pG where pG ⊆ H ⊆ G. By hypothesis H = IG for some fractional ideal I. Now recall from Proposition 2.11 that if I_{1} and I_{2} are submodules of Q then (I_{1} ∩ I_{2})G = I_{1}G ∩ I_{2}G and (I_{1} + I_{2})G = I_{1}G + I_{2}G. From this we see that (I ∩ W)G = IG ∩ G = IG = H, so we lose no generality in supposing I ⊆ W. Furthermore we have (I + p)G = IG + pG = H + pG = H so we may as well suppose p ⊆ I ⊆ W. Since p is a maximal ideal, either I = p and H = pG, or I = W and H = G. Thus G/pG has no proper submodules so p-rank G = length G/pG = 1.

( ⇒ ): If H is quasi-equal to G then wH ⊆ G for some w ≠ 0 ∈ W, and if wH = IG then H = w^{−1}IG. It thus suffices to consider the case where H ⊆ G and H is quasi-equal to G. Then by Proposition 3.11 G/H is a finite length homomorphic image of G, hence by Proposition 3.33 G/H ≈ W/I for some ideal I. Then I(G/H) = 0 so IG ⊆ H ⊆ G. But by Proposition 3.33 again G/IG ≈ W/J for some ideal J and since I(W/J) = 0 we get I ⊆ J. Thus length G/IG = length W/J ≤ length W/I = length G/H ≤ length G/IG since IG ⊆ H.

Thus H = IG. ✓

Recall that we say that a submodule H of G is fully invariant if φ(H) ⊆ H for all φ ∈ End H.

proposition 3.36. A Murley module G is a direct sum of strongly indecomposable modules. In particular, an indecomposable Murley module is strongly indecomposable. Furthermore every summand of G is fully invariant.

proof: We know that any finite rank torsion free module G is quasi-equal to a direct sum H_{1} ⊕ ⋯ ⊕ H_{n}, where the H_{i} are strongly indecomposable. If G is a Murley module then by Proposition 3.35 G = IH_{1} ⊕ ⋯ ⊕ IH_{n} for some fractional ideal I. Since the IH_{k} are strongly indecomposable by Proposition 3.10, the result is established.

Now if G = H ⊕ K and if φ: H → K then φ(H)_{∗} is p-divisible by every prime such that K is p-divisible, and also by every prime such that H is p-divisible. But for every prime p either H or K is p-divisible since p-rank(H ⊕ K) ≤ 1. Thus φ(H)_{∗} is divisible, and since G is reduced φ(H)_{∗} = 0, so φ = 0. Thus Hom(H, K) = 0 and likewise Hom(K, H) = 0. It follows immediately that H and K are fully invariant in G. ✓

Direct sum decompositions involving fully invariant summands are extremely nice. They are discussed at the beginning of Chapter 11.

corollary 3.37. If G is a Murley module and G = ⨁ G_{i}, then End G ≈ ∏ End G_{i}.

proof: Immediate since the G_{i} are fully invariant submodules of G. ✓ corollary 3.38. The class of Murley modules has the Krull-Schmidt Property. In fact, if G is a Murley module and G = G_{1} ⊕ ⋯ ⊕ G_{s} = H_{1} ⊕ ⋯ ⊕ H_{t}, where the G_{i} and H_{i} are indecomposable, then s = t and the H_{i} can be renumbered so that for each i, G_{i} and H_{i} are identical submodules of G.

proof: This follows from the fact that the G_{i} and H_{i} are fully invariant submodules of G. If γ_{1} is the projection of G onto G_{1} then for each i, γ_{1}(H_{i}) ⊆ H_{i} because H_{i} is fully invariant. It follows that γ_{1} restricts to a projection from H_{i} onto H_{i} ∩ G_{1} . Since H_{i} is indecomposable, either H_{i} = G_{1} or H_{i} = 0. We can then renumber the H_{i} so that H_{1} = G_{1} and complete the proof by induction. ✓

proposition 3.39. Let H be a p-local Murley module and G a reduced module containing H as an essential submodule. Then H is quasi-equal to G.

proof: G/H is a proper submodule of QG/H because G is reduced. But since H is a p-local Murley module, by Proposition 1.24 QG/H = QH/H ≈ W (p^{∞}). Therefore all proper submodules of QG/H are cyclic, so G/H is cyclic and H is quasi-equal to

G. ✓

MURLEY RINGS.

proposition 3.40. Let R be a Murley ring and S a subring of QR quasi-equal to R. Then R = S.

proof: By Proposition 3.35 S = IR for some fractional ideal I ⊆ Q. Thus S is an R-module. Since 1 ∈ S, it follows that R ⊆ S. Analogously, S ⊆ R. ✓

proposition 3.41. Let R and S be finite rank torsion free Murley rings such that p-rank R = p-rank S for all p and such that R is an essential subring of S. Then R = S.

proof: It suffices to prove that R_{p} = S_{p} for all p. Therefore we may assume that W is local. It now follows from Proposition 3.39 that R is quasi-equal to S and therefore R = S by Proposition 3.40. ✓

lemma 3.42. If R is a Murley ring and L, M are R-modules which are torsion free as W -modules then

(1) If R is not p-divisible then pR is a prime ideal in R and (pR) -rank M = p-rank M.

(2) p-rank(L ⊗_{R} M) = (p-rank L)(p-rank M).

(3) If furthermore L if finitely generated over R and M is reduced (as a W -module) then L ⊗_{R} M ≈ L ⋆ M.

(4) If M is a reduced R-module then R ⋆ M ≈ M. proof: (1) First note that if R is not p-divisible then p-rank R = 1 and so R/pR ≈ W/p and hence R/pR is a simple W -module and thus is a fortiori simple as an R-module. Thus pR is a maximal ideal of R. Now (pR) -rank M = length_{R} M/pRM = length_{R} M/pM = dim_{R/pR} M/pM = dim_{W/p} M/pM = p-rank M since R/pR ≈ W/p.

(2) If X is an R-module then p-rank X = length X/pX = length R/pR ⊗_{R} X. (The length here is computed over W.) Also R/pR ⊗_{R} R/pR ≈ (R/pR)/p(R/pR) ≈ R/pR. From this it follows that p-rank L ⊗_{R} M = length R/pR ⊗_{R} L ⊗_{R} M = length L/pL ⊗_{R} M/pM. On the other hand, by Proposition 1.36 (p-rank L)(p-rank M) = p-rank L ⊗ M = length L/pL ⊗_{W} M/pM. Thus it suffices to see that L/pL ⊗_{R} M/pM ≈ L/pL ⊗_{W} M/pM. Now if p-rank R = 0 then the R-modules L and M are p-divisible and all the p-ranks are trivial. Otherwise p-rank R = 1. Since L/pL and M/pM are vector spaces over R/pR it suffices to see that (R/pR) ⊗_{R} (R/pR) ≈ (R/pR) ⊗_{W} (R/pR), in other words that (R/pR) ⊗_{W} (R/pR) ≈ R/pR. But this is clear since R/pR ≈ W/pW.

(3) If L is finitely generated over R and M is reduced, then L ⊗_{R} M is reduced. This is clear is L is free, or even if L is projective, and this case will provide as much as we need to establish the propositions which follow. Once these are established and we know that R is a finite product of dedekind domains, it will follow that any finitely generated torsion free L is projective so that this special case is in fact not special at all.

Assuming then that L ⊗_{R} M is reduced, the surjection L ⊗ M → L ⊗_{R} M induces a surjection μ: L ⋆ M → L ⊗_{R} M. Since p-rank(L ⋆ M) = (p-rank L)(p-rank M) = p-rank L ⊗_{R} M, Ker μ has p-rank 0 for all p. Thus Ker μ is divisible, and since L ⋆ M is reduced Ker μ = 0 and μ is an isomorphism.

(4): By (2) R ⋆ M ≈ R ⊗_{R} M ≈ M. ✓

proposition 3.43. Let R be a Murley ring.

(1) R ⋆ R ≈ R.

(2) End R ≈ R.

(3) R is commutative.

(4) R has trivial nil radical.

(5) If L and M are R-modules and M is reduced, then every homomorphism L → M is R-linear.

proof: (1) Immediate from Lemma 3.42.

(5) If φ: L → M then φ induces an R-linear map R⋆L → R⋆M. But by Lemma 3.42 R ⋆ L and R ⋆ M are just L and M.

(2) By (5) End R = End_{R} R ≈ R.

(3) If r ∈ R then there is a map φ ∈ End R defined by φ(x) = rx. By (5) φ is R-linear. Thus for all x, r ∈ R, rx = φ(x) = φ(x · 1) = xφ(1) = xr.

(4) It suffices to show that if r ∈ R with r^{2} = 0 then r = 0. If r^{2} = 0 then the multiplication map x ↦ xr gives a short exact sequence

0 → ann(r) ⟶^{⊆} R → rR → 0

so that for any p, p-rank R = p-rank rR + p-rank ann(r). Let I = (rR)_{∗}. Now if R is p-divisible then I is also p-divisible. Likewise if rR is p-divisible then the pure ideal I is also p-divisible. And if rR is not p-divisible then p-rank rR = p-rank R = 1 and p-rank ann(r) = 0. But rR ⊆ ann(r) since r^{2} = 0 and so I = (rR)_{∗} ⊆ ann(r) since ann(r) is pure in R, and so in this case also p-rank I = 0. Thus I is p-divisible for every prime p, so I is divisible. Since R is reduced, I = 0. Thus rR = 0, i.e. r = 0. ✓

proposition 3.44. A Murley ring has no proper idempotents if and only if it is strongly indecomposable as a W -module. proof: ( ⇐ ): If r is an idempotent in R then left multiplication by r is an idempotent endomorphism of R, so R is not indecomposable, hence certainly not strongly indecomposable.

( ⇒ ): Suppose that R_{1} and R_{2} are W -submodules of R such that R = R_{1} ⊕ R_{2} . By Proposition 3.36 R_{1} and R_{2} are fully invariant in R. In particular, they are closed under multiplication by elements of R, i.e. they are ideals. Thus R is the ring-theoretic product R_{1} × R_{2} . Since R has no proper idempotents, it follows that R_{1} = 0 or R_{2} = 0. Thus R is an indecomposable W -module. By Proposition 3.36 R is strongly indecomposable. ✓

proposition 3.45. Let R be a Murley ring with no non-trivial idempotents. If R has infinite rank assume also that that QR is a field. Then R is a dedekind domain and every ideal I ⊆ R has the form 𝔞R, where 𝔞 = W ∩ I. proof: By Proposition 3.43 R is commutative with trivial nil radical. Thus if rank R < ∞ then QR is a finite dimensional commutative Q-algebra with trivial radical, hence a product of fields. By Proposition 3.44 R is strongly indecomposable, therefore by Proposition 3.16 QEnd R has no proper idempotents. Then QR has no proper idempotents, since QR can be identified as a subring of QEnd R. Therefore QR is in fact a field.

Now let I be a non-trivial ideal in R. Since QR is a field, IQR = QR. But QR is the localization of R with respect to the non-zero elements of W, so from the fact that 1 ∈ IQR we conclude that in IQR, 1 = i/s for some i ∈ I, s ≠ 0 ∈ W. Thus i ∈ W ∩ I so W ∩ I ≠ 0. If w ≠ 0 ∈ I ∩ W then wR ⊆ IR = I, showing that I is quasi-equal to R. Then by Proposition 3.35 I = 𝔞R for some ideal 𝔞 of W. Then 𝔞 ⊆ I ∩ W so that I = 𝔞R ⊆ (I ∩ W)R ⊆ IR = I, so that we may choose 𝔞 = I ∩ W.

Now if 𝔞R is a non-trivial ideal in R, where 𝔞 ⊆ W, then 𝔞^{−1}R is an R-submodule of QR and 𝔞^{−1}R𝔞R = 𝔞^{−1}𝔞R = R. Thus every non-trivial ideal in R is invertible, so R is a dedekind domain. ✓

theorem 3.46. A finite rank Murley ring is a finite product of dedekind domains. In particular, a Murley ring is indecomposable if and only if it is a dedekind domain. proof: By Corollary 3.36 R is the direct sum of strongly indecomposable modules R_{1} , …,R_{n} and these are fully invariant in R. But this implies that they are ideals in R, so R is the ring-theoretic product of R_{1}, …, R_{n}. These R_{i} must then all be indecomposable Murley rings and therefore by Proposition 3.44 and Proposition 3.45 they are dedekind domains. ✓

THE TORSION SUBMODULE OF EXT. Finally, for those who know about Ext, we give the following important result due to Carol Walker.

theorem 3.47. Let E: 0 → H → G → K → 0 be a short exact sequence. Then this sequence is quasi-split if and only if the corresponding element of Ext(K, H) lies in the torsion submodule. proof: For w ≠ 0 ∈ W multiplication by w on K is monic, hence multiplication by w on Ext(K, H) is surjective since Ext( , H) is right exact. I.e. Ext(K, H) is divisible. It follows that Q ⊗ Ext(K, H) equals the quotient of Ext(K, H) modulo its torsion submodule. Now since Q is flat the short exact sequence E: 0 → H → G → K → 0 leads to the long exact sequence 0 → QHom(K, H) → QHom(K, G) → QHom(K, K) → Q ⊗ Ext(K, H) → Q ⊗ Ext(K, G) → Q ⊗ Ext(K, K) → 0. Now if the original sequence is quasi-split, then QHom(K, G) → QHom(K, K) is surjective, and conversely if this map is surjective then 1_{K} ∈ QHom(K, K) lifts to a map in QHom(K, G), which is equivalent to saying that the original sequence quasi-splits. Thus we see that the original sequence quasi-splits if and only if the image of 1_{K} ∈ QHom(K, K) in Q ⊗ Ext(K, H) is trivial, i.e. if the connecting homomorphism takes 1_{K} into the torsion submodule of Ext(K, H). But it is well known in homological algebra [see Fuchs I] that the image of 1_{K} in Ext(K, H) is the element corresponding to the original sequence E. ✓

The following proposition gives an alternative, more explicit proof of the above theorem.

proposition 3.48. Let H ◁ G and let E be the element of Ext(G/H, H) represented by the short exact sequence 0 → H → G ⟶^{γ} G/H → 0. Let w ≠ 0 ∈ W. Then wE = 0 if and only if H is a direct summand of H + wG. Furthermore this is equivalent to the existence of X ⊆ G with X ∩ H = 0 such that wG ⊆ H ⊕ X ⊆ G (which implies, of course, that E is quasi-split). proof: Multiplication by w gives a map G/H → G/H which induces the map Ext(H, G/H) → Ext(H, G/H) taking E to wE. But G/H ⟶^{w} G/H factors as G/H ⟶^{w} w(G/H) ⟶^{⊆} G/H where the first map is an isomorphism, so the extension wE is isomorphic to the extension E′ which is the pull-back of E under the inclusion w(G/H) ↪ G/H. This pull-back is shown in the following diagram:

E′: 0 ⟶ H ⟶ H + wG ⟶ w(G/H) ⟶ 0
∥  ^{⊆}↓  ^{⊆}↓
E: 0 ⟶ H ⟶^{⊆} G ⟶ G/H ⟶ 0 .

(In fact, if the right hand square is to be a pull-back, then the middle module in the top sequence must be simply the inverse image of w(G/H) in G, i.e. H + wG, as shown.)

Then wE = 0 if and only if the top sequence splits, i.e. if and only if H is a direct summand of H + wG.

By Proposition 3.18 this is equivalent to the existence of X ⊆ G with H +wG = H ⊕X

and wG ⊆ H + wG = H ⊕ X ⊆ G. ✓

restatement. If H ◁ G then the element of Ext(K, H) corresponding to the short exact sequence 0 → H → G → G/H → 0 is annihilated by an element w ≠ 0 ∈ W if and only if there exists X ⊆ G with X ∩ H = 0 such that G is generated by H ⊕ X together with a family {(h_{i} + k_{i})/w}_{i∈I} with w ≠ 0 ∈ W and h_{i} ∈ H, k_{i} ∈ X. Furthermore in this case there is no loss of generality in assuming that the index set I is finite.

We can (unhistorically) think of the motivation for inventing the category of finite rank torsion free modules under quasi-homomorphisms as follows: Finite rank torsion free modules are too complicated to be understood simply by looking at them. Attempts to understand these objects will consist of finding ways to associate with a module G various other objects which are sufficiently simpler than G that we have some reasonable chance of comprehending them, and yet still provide useful information about the original module G. For instance, we can associate to G various numbers, such as its rank and p-rank. Or sets, such as T(G) and CT(G).

A very natural approach is to find functors from the category of finite rank torsion free W -modules into nicer categories. For instance one has a functor into the category of finite dimensional vector spaces by associating to each G its divisible hull QG. This takes us into a category that has extremely nice properties, but the price we pay is that the only information QG provides about G is its rank.

The category of finite length modules over a finite dimensional Q-algebra Λ is a lot nicer than the category of finite rank torsion free W -modules, even though it still has its complications. Alas, finding a useful functor into this category is not feasible. But if we examine what it is that makes this category nice, it is largely the fact that the homomorphism groups between two finite length Λ-modules are finite dimensional vector spaces, and in particular that the endomorphism ring of a finite length module is a finite dimensional algebra. This is what makes the Krull-Schmidt-Azumaya Theorem applicable to the category.

The category of finite rank torsion free W -modules under quasi-homomorphisms gives us a category which has the nice properties of the category of finite length modules over a finite dimensional algebra, and at the same time is sufficiently complicated that much of the important information about a finite rank torsion free module persists when we look at it in this category.

It is clear that if G is quasi-equal to G′ and H quasi-equal to H′, then QHom(G, H) = QHom(G′, H′). For this reason, there is a lot to be said for thinking of the objects in the category of finite rank torsion free modules under quasihomomorphisms as being not modules, but quasi-equality classes of modules. Certainly this makes sense if one remembers that from the point of view of category theory objects exist merely as place-holders for morphisms.

This way of thinking can lead to some valuable insights. For instance in the category of modules under quasi-homomorphisms W is a simple object: its only subobjects are the ideals, and these are all quasi-equal to W. Furthermore if W is a local ring, then in the category of modules under quasi-homomorphisms it can be seen that all modules have finite length. In fact, since any finite rank module has a filtration where the factors are rank-one modules, it suffices to see that rank-one modules have finite length. But from the point of view of quasi-equality there are only two submodules of Q, namely W and Q. This insight was what led your author to suspect that some of the work which Maurice Auslander and Idun Reiten were doing for abelian categories where all objects have finite length might be applicable to the study of finite rank torsion free modules.

On the other hand, purely as a matter of style it is certainly more convenient to talk about modules than to talk about quasi-equality classes, and so that is the language in which the discussion in this book will take place.