8. QUOTIENT DIVISIBLE MODULES

E. L. Lady

April 17, 1998

Now that we have seen how a p-divisible module can be understood by looking at d(Ŵ ⊗ G), it is tempting to wonder if this insight can be used globally. The following conjecture seems plausible: If φ: QG → QH and φ(d(Ŵ_{p} ⊗ G)) ⊆ d(Ŵ_{p} ⊗ H) for all primes p, then φ is a quasi-homomorphism from G to H.

From the Kurosh Theorem (Theorem 6.*) we see that this conjecture can be restated as a local-global principle: If φ: QG → QH is such that φ is a quasi-homomorphism from G_{p} to H_{p} for all prime ideals p, then φ must be a quasi-homomorphism from G to H.

In fact, over any dedekind domain with an infinite number of prime ideals this conjecture is false.

example 8.1. Let A be the rank-one module from Example 2.*, i.e. the submodule of Q generated by p^{−1} for all prime ideals ideals p. Let φ: QA → Q be the identity map on Q. Then φ is a quasi-homomorphism from A_{p} to W_{p} for all primes p since A_{p} ≈ W_{p} and QEnd W_{p} = Q. But φ is not a quasi-homomorphism from A to W since t(A) ≰ t(W) and thus Hom(A, W) = 0.

In [BP] Beaumont & Pierce defined a class of modules G which are precisely those for which the above conjecture is true. These modules are called quotient divisible modules. For the moment we will make a provisional definition:

definition 8.2. We will say that a finite rank torsion free module G has the localglobal quasi-homomorphism property if for all torsion free modules H and all φ: QG → QH, φ ∈ QHom(G, H) ⟺ φ ∈ QHom(G_{p}, H_{p}) for all primes p. (If we, as usual, identify QHom(G, H) with a submodule of Hom(QG, QH), then we have that G is quotient divisible if and only if QHom(G, H) = ⋂_{p} QHom(G_{p}, H_{p}).)

To motivate the definition of a quotient divisible module, think first about why one might expect all modules to have the local-global quasi-homomorphism property. The plausibility of the conjecture above arises from the equations G = ⋂_{p} G_{p} and H = ⋂_{p} H_{p}. When we consider Example 8.2 in that light, we realize that the reason it fails is that although A_{p} ≈ W_{p} for all p, the intersection ⋂ A_{p} is – as it were – skewed. So what we need to look for is a condition that guarantees that all the localizations G_{p} somehow “line up.”

To try to see what it would mean for the localizations G_{p} to “line up,” consider the following: let F be an essential projective submodule of G. Then (G/F)_{p} ≈ G_{p}/F_{p} is q-divisible for each prime q different from p. Furthermore, it follows readily from Proposition 1.* that (G_{p}/F_{p})[p] has finite dimension over W/p. We can thus easily conclude that (G/F)_{p} is the direct sum of a divisible module and a finite length p-primary module M, where M ⊆ (G/F)[p^{∞}]. If we now let F′ ⊇ F be the pre-image of M in G (i.e. M = F′/F), then F′ is finitely generated and thus projective, and G/F′ is divisible. One is thus tempted to say: “Without loss of generality, we may assume that (G/F)_{p} is divisible.” If true for every p, this would mean that G/F would be divisible. However this is fallacious, because in the example above, for instance, where A is the submodule of Q consisting of fractions with square-free denominators, there does not exist a finitely generated submodule F ⊆ A with A/F divisible. The fallacy arises because the submodule F′ chosen above depends on the prime p, and in general no single F′ will work for all primes p.

The existence of an essential finitely generated submodule F of G such that G/F is divisible is precisely the condition we are seeking that explains what it means for all the localizations G_{p} to “line up,” so that the intersection G = ⋂_{p} G_{p} is not “skewed.”

definition 8.3. A finite rank torsion free module G is called quotient divisible if there exists a finitely generated essential submodule F ⊆ G such that G/F is divisible.

PROPERTIES OF QUOTIENT DIVISIBLE MODULES. Recall that according to Proposition 2.*, if F and F′ area finitely generated essential submodules of a finite rank torsion free module G then the quotients G/F and G/F′ are torsion modules differing only by a finite length summand. One can thus associate with G an equivalence class of torsion modules G/F which is called the Richman type of G. Obviously G is quotient divisible if and only if its Richman type contains a divisible module.

proposition 8.4. (1) Direct sums, pure submodules and torsion free homomorphic images of quotient divisible modules are quotient divisible.

(2) If G is quotient divisible and H is quasi-isomorphic to G, then H is quotient divisible.

(3) Projective modules are quotient divisible.

(4) If W is semi-local then every finite rank torsion free W -module is quotient divisible.

(5) If R is a quasi-separable semi-prime ring then R is a quotient divisible W -module.

(6) If R is a splitting ring (not necessarily with finite rank) then every R-split module is quotient divisible. proof: (1) It is clear that the direct sum of two quotient divisible modules is quotient divisible. Now let H ◁ G and let F be a finitely generated essential submodule of G such that G/F is divisible. Then H ∩ F is a finitely generated submodule of H and H/H ∩ F is a pure submodule (???) of the divisible module G/F hence is divisible. Thus H is quotient divisible. Furthermore (F + H)/H is an essential submodule of G/H and (G/H)/((F + H)/H) ≈ G/(F + H), which is a homomorphic image of the divisible module G/F, hence is divisible. Thus G/H is quotient divisible.

(2) We may suppose G ⊆ H. Then by Proposition 3.* H/G has finite length. Let F be a finitely generated submodule of G such that G/F is torsion divisible. Then the divisible module G/F must be a summand of H/F: H/F ≈ G/F ⊕ H/G. If we now let F′ ⊇ F be such that F′/F is the component of the above direct sum isomorphic to H/G, then F′ is finitely generated (both F and F′/F are) and H/F′ ≈ G/F, hence H/F′ is divisible. Thus H is quotient divisible.

(3) Clear. Choose F = H.

(4) For each prime p, choose a finitely generated submodule F_{p} of G such that G_{p}/F_{p} is divisible. (Note quite right!) Let F be the submodule of G generated by all the F_{p}.

(5) By Proposition 7.* and Proposition 7.* it suffices to consider the case when R is an integral domain D. Since QD is separable over W, the integral closure of D in its quotient field QD is a finitely generated D-module, hence is quasi-equal to D. Thus there is no loss of generality in supposing D integrally closed. Now let W′ be the integral closure of W in QD. By the separability of QD, W′ is a finitely generated essential W -submodule of D. Furthermore W′ is a dedekind domain and D is a rank-one W′-module as well as a subring of QD. By Proposition 2.*, then, D/W′ is a divisible W′-module, hence a fortiori divisible as a W -module. Therefore D is quotient divisible. ✓

corollary 8.5. If G and H are quotient divisible submodules of a Q-vector space V , then G ∩ H and G + H are also quotient divisible. proof: G ∩ H and G + H are isomorphic to a pure submodule and a homomorphic

image of G ⊕ H. ✓

Among modules of arbitrary finite rank, the quotient divisible ones are special in the same way that the rank-one modules with idempotent type are special among the class of all rank-one modules:

proposition 8.6. (1) A rank-one module A is quotient divisible if and only if t(A) is idempotent.

(2) If G is quotient divisible then T(G) and IT(G) contain only idempotent types.

(3) A Butler module G is quotient divisible if and only if T(G) contains only idempotent types. proof: (1) ( ⇒ ): If t(A) is idempotent then by Proposition 2.*, A is quasi-equal to a subring of Q. By Proposition 8.4, a subring of Q is quotient divisible.

( ⇐ ): We may suppose W ⊆ A. If A is quotient divisible then A is quasi-equal to A′, where A′/W is divisible. Hence wlog assume A/W divisible. ****

(2) If G is quotient divisible then by Proposition 8.4, every pure rank-one submodule of G is quotient divisible and hence by (1) is idempotent. Thus T(G) contains only idempotent types.

(3) ( ⇒ ): By (2).

( ⇐ ): If G is a Butler module then by Proposition 5.* G is isomorphic to a homomorphic image of a module of the form ⨁ A_{i}, where the A_{i} are rank-one modules with t(A_{i}) ∈ T(G). If T(G) contains only idempotent types, then by (1) each A_{i} is quotient divisible. Hence by Proposition 8.4, G is quotient divisible. ✓

example 8.7. There exist modules G such that T(G) and CT(G) contain only idempotent types but G is not quotient divisible.

proposition 8.8. If G is a quotient divisible module and H is a submodule of G with rank H = rank G and p-rank H ≤ p-rank G for all primes p then H is quasi-equal to G. proof: Since H is essential in G the hypothesis implies that p-rank H = p-rank G for all p. ****

proposition 8.9. A quotient divisible module G is projective if and only if p-rank G = rank G for all primes p. proof: ( ⇒ ): By Proposition 1.*.

( ⇐ ): Let F be an essential free submodule of G. Then p-rank F = rank F = rank G = p-rank G for all primes p. Therefore by Proposition 8.8, G is quasi-equal to F. Therefore G is projective by Proposition 3.*. ✓

proposition 8.10. (1) If G and H are quotient divisible then so is G ⊗ H.

(2) If H is quotient divisible then so is Hom(G, H).

proof: (1) Let F_{1} and F_{2} be essential free submodules of G and H respectively such that G/F_{1} and H/F_{2} are divisible. Then F_{1} ⊗ F_{2} is an essential submodule of G ⊗ H and (G ⊗ H)/(F_{1} ⊗ F_{2}) is a pure submodule (????) of G/F_{1} ⊗ H ⊗ F_{2} , which is divisible.

(2) Let F be an essential free submodule of G. Then G/F is torsion, so Hom(G/F, H) = 0. Therefore there is an exact sequence

Hom(G/F, H) → Hom(G, H) → Hom(F, H).

????

We can now fulfill a promise from Chapter 7: proposition 8.11. Let F be a finite separable extension of Q and G a finite rank torsion free module. Then G is isomorphic to an essential subring of F if and only if it satisfies the following three conditions:

(1) rank G = [F: Q].

(2) QEnd G contains a field isomorphic to F.

(3) G is quotient divisible.

proof: ( ⇒ ): It was shown in Theorem 7.* that if G is an integral domain with quotient field F then conditions (1) and (2) are satisfied. Furthermore G is quotient divisible by Proposition 8.4.

( ⇐ ): **** ✓

THE LOCAL GLOBAL PRINCIPLE.

proposition 8.12. Any G with the local-global quasi-homomorphism property (Definition 8.*) must be quotient divisible.

proof: In fact, for any G let F be a finitely generated essential submodule of G and let G′ ⊇ F be such that G′/F = d(G/F). Then clearly G′ is quotient divisible. Furthermore (G/G′)_{p} has finite length for each prime p (as follows from Proposition *), and thus for each p by Proposition 3.* the identity map 1_{QG} is a quasi-homomorphism from G_{p} to G′_{p} inverse to the inclusion map G′_{p} ⊆ G_{p} Thus if G satisfies the local-global quasihomomorphism property then G is quasi-isomorphic to G′. Hence by Proposition 8.* G is quotient divisible. ✓

We will now show that G has the local-global quasi-homomorphism property if and only if G is quotient divisible. First observe the following:

proposition 8.13. Let F be an essential finitely generated submodule of G. Suppose that G has the following property with respect to F:

Whenever H is a torsion free W -module and φ ∈ QHom(G_{p}, H_{p}) for some prime p and φ(F) ⊆ H, then φ ∈ Hom(G_{p}, H_{p}).

Then G has the local-global quasi-homomorphism property.

proof: Let φ: QG → QH and suppose φ ∈ QHom(G_{p}, H_{p}) for all primes p. Since F is finitely generated, there exists 0 ≠ w ∈ W such that wφ(F) ⊆ H. The hypothesis then yields wφ ∈ Hom(G_{p}, H_{p}) for all p, so that wφ(G) ⊆ ⋂ H_{p} = H. Thus wφ ∈ Hom(G, H), so that φ ∈ QHom(G, H). This shows that G has the local-global homomorphism property. ✓

Now the clincher: proposition 8.14. If F is a submodule of G such that G/F is divisible, and φ ∈ QHom(G, H) for some torsion free H, then φ ∈ Hom(G, H) ⟺ φ(F) ⊆ H. proof: ( ⇒ ): Clear.

( ⇐ ): If φ(F) ⊆ H then φ induces a homomorphism φ̄: G/F → QH/H. If also φ ∈ QHom(G, H) then wφ(G) ⊆ H for some w ≠ 0 ∈ W, so that wφ̄(G/F) = 0. But φ̄(G/F) is divisible since G/F is divisible. Since a non-trivial divisible module is faithful, this means φ̄(G/F) = 0, i.e. φ(G) ⊆ H. Thus φ ∈ Hom(G, H). ✓

To summarize:

theorem 8.15. G has the local-global quasi-homomorphism property if and only if G is quotient divisible.

proposition 8.16. If G and H are quotient divisible modules with rank G = rank H and p-rank G = p-rank H for all primes p and φ: G → H is a monomorphism, then φ is in fact a quasi-isomorphism. proof:

Your author finds Proposition 8.14 intruiguing, especially when G has a free submodule F with G/F divisible, as is the case, for instance, when G is quotient divisible and W is a principal ideal domain. In this case, let f_{1}, …, f_{r} be a set of generators for F. Then these are constitute a finite set of test elements, as it were, for a quasi-homomorphism. Namely, according to Proposition 8.14 if φ ∈ QHom(G, H) then φ(G) ⊆ H if and only if φ(f_{i}) ∈ H for all i.

From this we get an interesting corollary. In general, given finite rank torsion free modules G and H it is fairly difficult to get much information about Hom(G, H). When G is quotient divisible, we have the following result:

proposition 8.17. If G is quotient divisible and rank G = r, then for any torsion free H, Hom(G, H) is isomorphic to a pure submodule of H^{r−1} ⊕ 𝔞H, for some ideal 𝔞 of W. If there exists a free submodule F of G with G/F divisible (in particular, if W is a principal ideal domain), then Hom(G, H) is isomorphic to a pure submodule of H^{r}. proof: Let F be a projective submodule of G such that G/F is divisible. According to Proposition 8.14 Hom(G, H) = QHom(G, H) ∩ Hom(F, H), where we have identified QHom(G, H) as a subspace of QHom(F, H) using the exact sequence

Hom(G/F, H) = 0 → Hom(G, H) → Hom(F, H).

Thus Hom(G, H) is isomorphic to a pure submodule of Hom(F, H). By Proposition 0.* F = F_{1} ⊕ I where F_{1} is free and I is an ideal of W. Then by Proposition 1.*

Hom(F, H) ≈ Hom(F_{1}, H) ⊕ Hom(I, H) ≈ H^{r−1} ⊕ I^{−1}H.

But I^{−1} ≈ 𝔞 for some ideal 𝔞 ⊆ W. If F is free, then Hom(F, H) ≈ H^{r}. ✓ proposition 8.18. Suppose that there exists a free submodule F ⊆ G such that G/F is divisible. Let Hom(G, H) be embedded as a pure submodule of H^{r} as in Proposition 8.17. Let φ ∈ Hom(G, H) correspond to ⟨h_{1}, …, h_{r}⟩ ∈ H^{r} under this embedding. Then φ is monic if and only if h_{i} ≠ 0 for all i. Furthermore if H′ is the submodule of H generated by the set {h_{1}, …, h_{r}}, then the reduced part of H/H′ is isomorphic to H/φ(G).

proof: ****

remark 8.19. It would seem that in most cases, a quotient divisible module G would have a free submodule F such that G/F is divisible. For if F is a finitely generated essential submodule of G with G/F divisible then F is a direct sum of ideals. Consider a rank-one summand I of F. If I is not a pure submodule of G then since I is isomorphic to an ideal of W it would seem plausible that there would exist a cyclic module I′ with I ⊆ I′ ⊆ I_{∗}. If one can do this for each rank-one summand for each rank-one summand I of F, then let F′ be the direct sum of these cyclic modules I′. Then F′ is free and F ⊆ F′, so that G/F′ is a homomorphic images of G/F and hence is divisible.

Clearly there are cases where this will not work, however. For instance if G is projective but not free and F is an essential free submodule of G then G/F is non-trivial with finite length, hence not divisible.

corollary 8.20. If G is quotient divisible and A is a rank-one module and t = t(A), then Hom(G, A) is a t-projective module.

proof: By Proposition 8.17 Hom(G, A) is quasi-isomorphic to a pure submodule of the t-projective module A^{r}. By Proposition 4.* quasi-pure submodules of t-projective modules are t-projective. ✓

When t is idempotent, by Proposition 4.* Corollary 8.20 is true without any hypothesis on G. However for an arbitrary type t, and especially when t is locally trivial, this result is in surprising contrast to what we will later see in Proposition 9.*.

EXISTENCE THEOREM.

theorem 8.21. Let U be a finite dimensional Q-space. For every prime p of W, let D_{p} be a QŴ_{p}-subspace of Ŵ_{p} ⊗ U. Then there exists a unique (up to quasi-equality) quotient divisible W -module G such that

(1) QG = U.

(2) For every prime p, d(Ŵ_{p} ⊗ G) = D_{p}. proof: Let F be an essential free submodule of U. For each prime ideal p, choose a pure submodule F_{p} of F such that F_{p} ∩ D_{p} = 0 and (Ŵ_{p} ⊗ F_{p}) ⊕ D_{p} is an essential Ŵ_{p}-submodule of Ŵ_{p} ⊗ U. (How do we know this can be done?) Let G_{p} = U ∩ ((Ŵ_{p} ⊗ F_{p}) ⊕ D_{p}). (The subscript p here does not denote localization.) Let G_{p} = ⋂_{p} G_{p}. Then QG = U and for each prime p, d(Ŵ_{p} ⊗ G) = D_{p} since ****. Furthermore, we claim that G/F is divisible, which will show that G is quotient divisible. In fact, it suffices to prove that for each prime p, Ŵ_{p} ⊗ (G/F) is divisible (why?). But Ŵ_{p} ⊗ (G/F) ≈ (Ŵ_{p} ⊗ G)/(Ŵ_{p} ⊗ G) ≈ D_{p}, a divisible module. Thus G/F is divisible and since F is an essential free submodule of G, this shows that G is quasi-divisible.

Now if G′ is another quotient divisible module with QG′ = U and d(Ŵ_{p} ⊗ G) = D_{p} for all p, and if φ = 1_{U}, then by the Local Global Homomorphism Principle φ is a quasi-homomorphism both from G to G′ and from G′ to G. Thus G′ is quasi-equal to

G. ✓

THE QUOTIENT DIVISIBLE CORE.

proposition 8.22. (1) If G is a finite rank torsion free module then there exists an essential quotient divisible submodule G_{0} of G.

(2) G_{0} is unique up to quasi-equality.

(3) G/G_{0} is a reduced torsion module. proof: (1) Let F be an essential free submodule of G and let G_{0} be the inverse image in G of the divisible submodule of G/F. Thus G_{0}/F is divisible so G_{0} is quotient divisible.

(3) G/G_{0} ≈ (G/F)/(G_{0}/F) and the latter is a reduced torsion module since G_{0}/F is the maximal divisible summand of the torsion module G/F.

(2) Now suppose G_{1} is also an essential quotient divisible submodule of G. Then by Corollary 8.5 G_{0} ∩ G_{1} is quotient divisible. Therefore wlog we may suppose that G_{0} ⊆ G_{1} (Explain!). Now let F be a finitely generated submodule of G_{0} such that G_{0}/F is torsion divisible. Then by Proposition 2.* G_{1}/F is isomorphic to the direct sum of a divisible torsion module and a finite length module T. Then T is not faithful, so there exists w ≠ 0 ∈ W with wT = 0. I.e. wG_{1}/F ⊆ G_{0}/F. Since F ⊆ G_{0} ⊆ G_{1} it follows that wG_{1} ⊆ G_{0} . Thus G_{1} is quasi-equal to G_{0} . ✓

definition 8.23. The quotient divisible core of G is defined to be an essential quotient divisible submodule of G.

Proposition 8.23 thus says that every finite rank torsion free G has a quotient divisible core, and the quotient divisible core is unique up to quasi-equality.

In analogy to the construction of the category of finite rank torsion free modules under quasi-homomorphisms, it is tempting to construct a category C in the following way:

The objects of C are finite rank torsion free modules. For two modules G and H, Hom_{C}(G, H) is defined to be the set of linear transformations φ: QG → QH such that

(∀p) φ_{p} ∈ QHom(G_{p}, H_{p}).

Unfortunately (or fortunately, for those who wish to discourage the proliferation of categories), the category C doesn’t really give us anything new:

proposition 8.24. If φ ∈ Hom(QG, QH) then φ is a morphism in C if and only if φ restricts to a quasi-homomorphism from G_{0} to H_{0} , where G_{0} and H_{0} are the quotient divisible cores of G and H.

proof:

QUOTIENT DIVISIBLE MURLEY MODULES.

proposition 8.25. An indecomposable Murley module is quotient divisible if and only if its inner type is idempotent.

proof:

Although we have seen that only quotient divisible modules have the local-global quasi-homomorphism property, for Murley modules there is an analogous, slightly weaker, property:

proposition 8.26. Two Murley modules G and H are quasi-equal if and only if QG = QH, IT(G) = IT(H), and d(Ŵ_{p} ⊗ G) = d(Ŵ_{p} ⊗ H) for all primes p.

proposition 8.27. If G is a Murley module then End G is a direct sum of quotient divisible strongly homogeneous modules. In particular, if G is indecomposable then End G is strongly homogeneous.

proposition 8.28. Let G be a Murley module. The following conditions are equivalent:

(1) G is strongly homogeneous.

(2) The quotient divisible core of G is strongly homogeneous and isomorphic to End G.

(3) G is a rank-one torsion free (End G)-module.

ARNOLD DUALITY (GLOBAL CASE). theorem 8.29. There exists a contravariant functor on the category of finite rank torsion free W -modules under quasi-homomorphisms such that for every G the following are true:

(1) QAG = Hom(G, Q).

(2) If γ: G → K then A(γ) = γ^{∗} = Hom(γ, Q).

(3) rank AG = rank G.

(4) For every prime p, p-rank AG = rank G − p-rank G.

(5) AG is divisible if and only if G is locally free.

(6) A^{2}G is naturally quasi-isomorphic to the quotient divisible core of G.

(7) For any torsion free W -module H, d(G ⊗ H) ≈ QHom(AG, H).

proof: For each prime p, the localization G_{p} of G is Ŵ_{p}-split, and therefore it has an Arnold dual A_{p}G_{p} within the category of p-local modules. Furthermore QA_{p}G_{p} = Hom(G_{p}, Q) = Hom(G, Q) and if γ: G → K then γ induces A_{p}(γ_{p}): A_{p}K_{p} → A_{p}G_{p} and A_{p}(γ_{p}) = γ_{p}^{∗} = γ^{∗} = Hom(γ, Q). By Theorem 8.21, there exists up to quasi-equality a unique quotient divisible module AG such that for every prime p, (AG)_{p} = A_{p}G_{p} and QAG = QA_{p}G_{p} = Hom(G, Q). Furthermore rank AG = rank A_{p}G_{p} = rank G and for each prime p, p-rank AG = p-rank A_{p}G_{p} = rank G − p-rank G. And if γ: G → K then γ^{∗} ∈ QHom((AH)_{p}, (AG)_{p}) for every prime p, so by the Local-Global Homomorphism Principle γ^{∗} ∈ QHom(AK, AG). We define Aγ = γ^{∗}. Clearly this makes A a functor from the category of finite rank torsion free W -modules under quasihomomorphisms into itself. By construction, assertions (1) and (2) hold and (3) is true since rank AG = rank QAG = rank Hom(G, Q) = rank G.

(4) For any prime p, p-rank AG = p-rank(AG)_{p} = p-rank A_{p}(G_{p}) = rank G_{p} − p-rank G_{p} = rank G − p-rank G by Proposition 6.**.

(6) Let ρ: QG → QA^{2}G = Hom(Hom(G, Q), Q) be given by ρ(g)(φ) = φ(g). By Proposition 6.**, for every prime p, ρ is a quasi-isomorphism from G_{p} to A_{p}^{2}(G_{p}) = (A^{2}G)_{p}. Thus by Proposition 8.*, ρ is a quasi-isomorphism from the quotient divisible core of G to the quotient divisible core of A^{2}G. But by construction A^{2}G is quotient divisible. Therefore A^{2}G is quasi-isomorphic to the quotient divisible core of G.

(5) G is locally free if and only if G_{p} is a free W_{p}-module for each prime p. By Proposition 6.* this is true if and only if (AG)_{p} = A_{p}G_{p} is divisible for each prime p, which is equivalent to the statement that AG is divisible.

(7) Note that d(G ⊗ H) = ⋂_{p} d(G_{p} ⊗ H_{p}), where the intersection is taken over all primes p. And by Proposition 6.*, for each prime p there is an isomorphism σ_{p}: d(G_{p} ⊗ H_{p}) ⟶^{≈} QHom((AG)_{p}, H_{p}). Finally, since AG is quotient divisible, by Proposition 8.** QHom(AG, H) = ⋂_{p} QHom((AG)_{p}, H) ≈ ⋂_{p} d(G_{p} ⊗ H_{p}) =

d(G ⊗ H). ✓

proposition 8.30. Let W′ be a finite rank quasi-separable integral extension of W.

(1) If L is a W′-module then the Arnold dual of L as a W′-module is the same as its Arnold dual as a W -module.

(2) If G is a W -module then the Arnold Dual of W′ ⊗ G as a W′-module is W′ ⊗ AG.

corollary 8.31. Let D be an integrally closed quasi-separable integral domain and let D′ be the unique integrally closed subring of QD such that D ∩ D′ is the integral closure of W in QD and D + D′ = QD. Then D′ is quasi-isomorphic to the Arnold dual of D. proof: Since D is integrally closed it is a W′-module, where W′ is the integral closure of W in the field QD. By Proposition 8.30, in computing AD we may replace W by W′ as the ground ring. Thus there is no loss of generality in supposing that rank D = 1. Therefore rank AD = 1 and by Proposition 8.6, t(AD) is idempotent. Therefore we may choose AD to be a subring of QW′. Furthermore, by Theorem 6.*, for each prime p, p-rank AD = 1 if and only if p-rank D = 0. Thus by Proposition 2.* AD is the unique subring of Q such that D ∩ AD = W′ and D + AD = QW′ = QD. ✓

proposition 8.32. (1) Let H be an essential submodule of G and let γ: H ↪ G be the inclusion map. Then Aγ: AG → AH is a monomorphism embedding AG as an essential submodule of AH.

(2) If G_{1} and G_{2} are essential submodules of a Q-vector space V then A(G_{1} + G_{2}) = AG_{1} ∩ AG_{2} and A(G_{1} ∩ G_{2}) = AG_{1} + AG_{2} .

proof: (1) If φ ∈ QAG = Hom(G, Q) then Aγ(φ) = φγ is simply the restriction of φ to H. Since H is essential in G, φ ≠ 0 ⇒ Aγ(φ) ≠ 0. Thus Aγ is monic. Since rank AG = rank AH, Aγ thus embeds AG as an essential submodule of AH.

(2) It suffices to prove this in the local case. Thus we may suppose that G_{1} and G_{2} are R-split for some splitting ring R. Now QAG_{1} = QAG_{2} = Hom(QG_{1}, Q) and likewise (R ⊗ AG_{1})^{⋆} = (R ⊗ AG_{2})^{⋆}. And by Proposition 3.* QHom(G_{1} + G_{2}, R) = QHom(G_{1}, R) ∩ QHom(G_{2}, R). Thus by Proposition 8.* A(G_{1} + G_{2}) = AG_{1} ∩ AG_{2} . And

A(G_{1} ∩ G_{2}) = A(A^{2}G_{1} ∩ A^{2}G_{2}) = A(A(AG_{1} + AG_{2})) = AG_{1} + AG_{2}. ✓

remark. There are two ways in which one might want to improve on Arnold duality.

(1) One might look for a different way of defining A so that AG is determined up to isomorphism rather than only up to quasi-equality.

(2) One might try to redefine A so that AG is only quotient divisible when G is and A^{2}G ∼ G for all finite rank torsion free G. Unfortunately, though, neither of these is possible. proof: (1) If A were defined on the category of finite rank torsion free modules under homomorphisms and A^{2}G ≈ G for quotient divisible modules, then Q = End Q ≈ End AQ = End W = W, a contradiction.

(2) Let B be a rank-one module with t(B) locally trivial. Then p-rank B = rank B = 1 for all p, so p-rank AB = rank B − p-rank B = 0 for all p. Thus AB is a divisible rank-one module, so AB = Q and thus A^{2}B = AQ = W ≉ B. ✓

GENERALIZED BUTLER MODULES.

theorem 8.33. Let G be a finite rank torsion free W -module. The following conditions are equivalent:

(1) G is R-split for some finite rank quasi-separable splitting ring R.

(2) G is isomorphic to a pure submodule of a direct sum of quasi-separable integral domains.

(3) G is a homomorphic image of a finite direct sum of quasi-separable integral domains.

(4) There exists a separable extension Q′ of Q such that W′ ⊗ G is a quotient divisible Butler W′-module, where W′ is the integral closure of W in Q′.

proof: (1) ⇒ (2): If G is R-split then G is isomorphic to a pure submodule of R^{k} for some k. But R is a finite product of quasi-separable domains.

(2) ⇒ (1): If G is isomorphic to a pure submodule of ⨁ D_{i}, it follows from Theorem 7.* and Lemma 7.* that there exists a quasi-separable splitting ring R such that all the D_{i} are R-split. Then G is R-split.

(1) ⇒ (3): If G is R-split then so is its Arnold dual A(G). Apply Arnold duality to the pure embedding A(G) → ⨁ D_{i} to get a homomorphism φ: ⨁ A(D_{i}) → G such that Image(φ) is quasi-equal to G. By Proposition 7.* the A(D_{i}) are quasi-separable domains. Since G/ Image(φ) is finitely generated, we can adjoint a finite number of copies of W to ⨁ A(D_{i}) and make φ surjective.

(3) ⇒ (4): By Proposition 7.* we can find a finite integral quasi-separable extension W′ of W such that the W′ ⊗ D_{i} are all direct sums of rank-one W′-modules of idempotent type. Since W′ ⊗ G is a homomorphic image of ⨁ W′ ⊗ D_{i} it is thus a Butler W′-module. By Proposition 6.* it is quotient divisible.

(4) ⇒ (2): If W′ ⊗ G is a quotient divisible Butler W′-module then W′ ⊗ G is isomorphic to a pure submodule of a direct sum of rank-one W′-modules of idempotent type, i.e. a direct sum of subrings of Q′. But G is a pure submodule of W′ ⊗ G. ✓