9. LOCALLY FREE MODULES

E. L. Lady

April 17, 1998

In the earlier chapter your author has tried to promote the point of view that the category of finite rank torsion free modules under quasi-homomorphisms is the appropriate environment for studying torsion free modules. For locally free modules, however, it turns out that quasi-homomorphisms do not play a very major role.

definition 9.1. G is locally free if G_{p} is a free W_{p}-module for all primes p. (Recall from Proposition 0.* that G_{p} is free over W_{p} ⟺ G_{p} is projective over W_{p} ⟺ G_{p} is finitely generated over W_{p}.)

proposition 9.2. (1) Submodules and homomorphic images of locally free modules are locally free.

(2) A rank-one module A is locally free if and only if t(A) is locally trivial.

(3) If G is locally free then IT(G), OT(G), and ET(G) are locally trivial.

(4) G is locally free if and only if G is t-bounded for some locally trivial type t.

(5) If H is locally free and H dominates G, then G is locally free.

(6) If G and H are locally free then so is G ⊗ H.

(7) If H is locally free then so is Hom(G, H).

(8) If W is semi-local then locally free modules are free.

proof: (1) If G_{p} is a free W_{p}-module then any submodule or homomorphic image of G_{p} is a free W_{p}-module.

(2) By Proposition 2.*.

(3) By Proposition 2.* IT(G) ≤ OT(G) ≤ ET(G). It follows immediately that if ET(G) is locally trivial then so are the other two. But ET(G) = t (⋀^{r} G), and ⋀^{r} G is a homomorphic image of G ⊗ ⋯ ⊗ G (r copies), hence is locally free by (6). Thus ET(G) is locally trivial by (2).

(4) ( ⇒ ): G is t-bounded, where t = OT(G), and by (3) if G is locally free then OT(G) is locally trivial.

( ⇐ ): If t = t(A) is locally trivial and G is t-bounded then by (2) A is locally free and by Proposition 4.* A dominates G. Thus by (3) G is locally free.

(5) If H dominates G then by Proposition 4.* H is a submodule of a direct sum of copies of G.

(6) If G_{p} and H_{p} are free W_{p}-modules then so is G_{p} ⊗ H_{p}.

7) If H_{p} is a free W_{p}-module then so is Hom(G_{p}, H_{p}).

8) If G is locally free then by 5) OT(G) is locally trivial. If W is semi-local then it follows from Proposition 2.* that OT(G) = t(W). Then G is t(W)-bounded so by Proposition 2.* G is projective. By Proposition 0.* G is free. ✓

proposition 9.3. A finite rank torsion free module G is locally free if and only if (∀p) p-rank G = rank G. proof:

As a special case of Proposition 4.* we have the following:

proposition 9.4. Let A be a locally free rank-one module. Let G and H be arbitrary finite rank torsion free modules.

(1) G ≈ Hom(A, A ⊗ G).

(2) Hom(A ⊗ G, A ⊗ H) ≈ Hom(G, H).

proof:

Restated, this proposition says the following:

proposition 9.5. If A is a rank-one locally free module and t = t(A) then the functor A ⊗ is an equivalence between the category of finite rank torsion free modules under homomorphisms and the category of t-saturated modules. An inverse to this functor is given by Hom(A, ). proof:

LOCALLY FREE MODULES AND QUOTIENT DIVISIBLE MODULES. The class of all finite rank torsion free modules can be thought of as a spectrum, where the quotient divisible modules are at one extreme and the locally free modules are at the other, and all other finite rank modules lie somewhere in between.

proposition 9.6. (1) If G is quotient divisible with no non-trivial projective summand and H is locally free, then Hom(G, H) = 0.

(2) If G is both quotient divisible and locally free then G is projective.

proof: 2) Let F be an essential free submodule of G. For every prime p, applying Lemma 3.13 to the free W_{p}-module G_{p} shows that G_{p} is quasi-equal to F_{p}. Since G and F are quotient divisible, Theorem 8.* shows that G and F are quasi-equal. Thus G is projective.

1) If φ ∈ Hom(G, H) then by Proposition 8.* φ(G) is quotient divisible and by Proposition 9.* it is locally free. Therefore by 2) φ(G) is projective, hence φ(G) is isomorphic to a summand of G. Since G has no non-trivial projective summands, φ(G) = 0 and therefore φ = 0. ✓ corollary 9.7. A locally free semi-prime ring is projective. proof: By Proposition 7.* a semi-prime ring is quotient divisible. ✓

Proposition 9.6 suggests that in the category of finite rank torsion free modules the classes of quotient divisible modules and of locally free modules play a role somewhat analogous to that of the torsion modules and the torsion free modules in the category of all W -modules. One should not get carried away with this analogy, however, and in particular one should not think that there is a legitimate torsion theory hiding somewhere in the woodwork here.

Recall from Proposition 3.* that if G is a finite rank torsion free module and φ: QG → QH then in order for φ to be a quasi-homomorphism from G to H two conditions must be satisfied:

(1) For all primes p, φ ∈ QHom(G_{p}, H_{p}).

(2) For all but finitely many p, φ ∈ Hom(G_{p}, H_{p}). Recall also from Theorem 8.* that if G is quotient divisible then condition (1) alone is sufficient. On the other hand, if G is locally free we now see that condition (2) alone suffices.

proposition 9.8. Let G be a locally free module, let H be any torsion free module, and let φ: QG → QH. Then φ ∈ QHom(G, H) if and only if φ ∈ Hom(G_{p}, H_{p}) for all but finitely many primes p. proof: ( ⇒ ): If G_{p} is a free W_{p}-module then by Proposition 3.* φ ∈ QHom(G_{p}, H_{p}). If G is locally free then this is true for all primes p so by Proposition 3.* φ ∈ QHom(G, H) for all but finitely many p. ✓

In parallel to the definition of a quotient divisible module we have the following characterization of locally free modules in terms of their Richman type:

proposition 9.9. Let F be an essential projective submodule of G. Then G is locally free if and only if G/F is p reduced. proof: ( ⇒ ): If G is locally free then for all p, G_{p}/F_{p} is a finitely generated W_{p}-module, thus is reduced. Since G/F is torsion, by Proposition 0.* G/F ≈ ⨁ G_{p}/F_{p}. Thus G/F is reduced.

( ⇐ ): If G_{p}/F_{p} is reduced then by Proposition 0.* it has finite length, so that G_{p} is quasi-equal to F_{p} and hence is a projective W_{p}-module. ✓

proposition 9.10. Let G be a finite rank torsion free module and let G_{0} be the quotient divisible core of G. Then there exists a locally free submodule G_{ℓ} of G such that G = G_{0} + G_{ℓ} and G_{0} ∩ G_{ℓ} is projective. proof: Let F be a projective submodule of G such that F ⊆ G_{0} and G_{0}/F = d(G/F) (see Proposition 8.*). Then G_{0}/F is a summand of G/F. Choose G_{ℓ} ⊆ G such that F ⊆ G_{ℓ} and G/F = G_{0}/F ⊕ G_{ℓ}/F. Then G = G_{0} + G_{ℓ} and G_{0} ∩ G_{ℓ} = F. Furthermore G_{ℓ}/F is a reduced torsion module, hence is p-reduced for all p. Hence by Proposition 9.9 G_{ℓ} is locally free. ✓

Unfortunately, one does not get any uniqueness properties for G_{ℓ}.

proposition 9.11. Let G be locally free and N = nil rad End G. Then N is a summand of End G and End G /N is projective. proof: It suffices to prove that End G /N is projective. In fact, it is a semi-prime ring, hence by Proposition 8.* is quotient divisible. By Proposition 9.* it is also locally free. Therefore it is projective. ✓

proposition 9.12. Let G be a locally free module and H a module with no non-trivial projective summand. Let N = nil rad End G. Let μ: G ⊗ H → G. Then μ(G ⊗ H) ⊆ NG. proof: Since Hom(G ⊗ H, G) ≈ Hom(H, Hom(G, G)) = Hom(H, End G), μ corresponds to a map μ̃: H → End G. Composing this with the quotient map End G → End G /N yields μ̄: H → End G /N. Since by Proposition 9.11 End G /N is projective, the image of μ̄ is projective and hence must be isomorphic to a summand of H. Since H has no non-trivial projective summand, μ̄ = 0 so that μ̃(H) ⊆ N. This says that for

g, ∈ G, h ∈ H, μ(g ⊗ h) = μ̃(h)(g) ∈ NG. ✓

WARFIELD DUALITY.

definition 9.13. Let A be a locally free rank-one module. Then for any finite rank torsion free G we define the Warfield dual of G with respect to A to be A^{∗}(G) = Hom(G, A). We also write A^{∗∗}(G) = A^{∗}(A^{∗}(G)) and let φ: G → A^{∗∗}(G) be the natural map defined by ρ(g)(φ) = φ(g).

theorem 9.14. Let A be a locally free rank-one module and t = t(A). Let G and H be any finite rank torsion free modules and let σ: A^{∗}(G) ⊗ G → A be defined by σ(γ ⊗ g) = γ(g). Then

(1) A^{∗}(G) is t-bounded.

(2) A^{∗}(G ⊗ H) ≈ Hom(G, A^{∗}(H)) ≈ Hom(H, A^{∗}(G)).

(3) If G is t-bounded then for any H the map θ: A^{∗}(G) ⊗ H → Hom(G, A ⊗ H) given by θ(γ ⊗ h)(g) = γ(g)h is an isomorphism.

(4) The natural map ρ: G → A^{∗∗}(G) is surjective.

(5) Ker ρ = G[t].

(6) If G is t-bounded and strongly indecomposable then A^{∗}(G) is strongly indecomposable.

(7) If G is t-bounded then OT(A^{∗}(G)) = [t: IT(G)].

proof: (1) By Proposition 4.* since A is t-bounded then A^{∗}(G) = Hom(G, A) is t-bounded.

(2) A^{∗}(G ⊗ H) = Hom(G ⊗ H, A) ≈ Hom(G, Hom(H, A)) = Hom(G, A^{∗}(H)) by Proposition 0.*.

(3) If G is t-bounded then by Proposition 4.* A dominates G so the result follows from Proposition 4.*.

(4) Since ρ factors through G/G[t] it suffices to deal with the case where G is t-bounded, hence dominated by A. It suffices to prove that in this case for each prime p the localized map G_{p} → A^{∗∗}(G)_{p} is an isomorphism. But since A dominates G, by Lemma 4.* A^{∗∗}(G)_{p} = Hom(Hom(G, A), A)_{p} ≈ Hom(Hom(G_{p}, A_{p}), A_{p}). Since G_{p} is a free W_{p}-module and A_{p} ≈ W_{p}, then result is then standard and easy.

(5) Clear from (4) because a quasi-direct decomposition of A^{∗}(G) induces a quasi-direct decomposition of A^{∗∗}(G), and hence of G.

(6) The rank-one homomorphic images of A^{∗}(G) are of the form A^{∗}(B), where B is a pure submodule of G, and conversely. Thus OT(A^{∗}(G)) is the least upper bound of the set of types t(A^{∗}(B)) with t(B) ∈ T(G). But t(A^{∗}(B)) = t(Hom(B, A) = [t: t(B)]. ✓

proposition 9.15. If 0 → H → G → K → 0 is a short exact sequence of t(A)-bounded modules, then 0 → A^{∗}(K) → A^{∗}(G) → A^{∗}(H) → 0 is exact. proof: This is just Proposition 4.*. ✓

proposition 9.16. Let Γ: Hom(H, G) → A^{∗}(A^{∗}(G) ⊗ H)

be given by

Γ(ψ) = σ(1 ⊗ ψ): A^{∗}(G) ⊗ H → A.

If G is t-bounded then Γ is an isomorphism. proof: Let γ ∈ A^{∗}(G), h ∈ H, and ψ ∈ Hom(H, G). Then Γ(ψ)(γ ⊗ h) = σ(γ ⊗ ψ(h)) = ρ(ψ(h))(γ), so that Γ is the composition

Hom(H, G) → Hom(H, A^{∗∗}(G)) → A^{∗}(A^{∗}(G) ⊗ H),

where the first of these maps is induced by ρ and the second is the isomorphism given in (2). Hence if G is t-bounded then Γ is an isomorphism. ✓

proposition 9.17. Let G be locally free and let A = ⋀^{r} G, where r = rank G. Then A^{∗}(G) ≈ ⋀^{r−1} G.

proof: Any g_{1} ∧ ⋯ ∧ g_{r−1} ∈ ⋀^{r−1} G yields a map α: G → A by α(x) = g_{1} ∧ ⋯ ∧ g_{r−1} ∧ x ∈ ⋀^{r} G = A. This gives a map ⋀^{r−1} G → A^{∗}(G). Furthermore if g_{1} ∧ ⋯ ∧ g_{r−1} ≠ 0 then α ≠ 0 since g_{1} ∧ ⋯ ∧ g_{r−1} ∧ x ≠ 0 for any x not belonging to the pure submodule of G generated by g_{1}, …, g_{r}. Thus the map ⋀^{r−1} G → A^{∗}(G) is monic. The hard part is to see that it is surjective.

Let α′ ∈ A^{∗}(G) and let K = Ker α′. Let g_{1}, …, g_{r−1} be a linearly independant set in K and fix g′ ∈ G with g′ ∉ K. Then g_{1} ∧ ⋯ ∧ g_{r−1} ∧ g′ ≠ 0 ∈ A because g_{1}, …, g_{r−1}, g′ are linearly independant. Since rank A = 1 there exists u ∈ Q with α′(g′) = ug_{1} ∧ ⋯ ∧ g_{r−1} ∧ g′. Then the map α: x ↦ ug_{1} ∧ ⋯ ∧ g_{r−1} ∧ x is an element of QA^{∗}(G) and Ker α = K = Ker α′. It follows that α is a multiple of α′. But since α(g′) = α′(g′), it follows that α = α′.

Since we are given only that u ∈ Q, it remains to be shown that ug_{1} ∧ ⋯ ∧ g_{r−1} ∈ ⋀^{r−1} G. It will sufffice to prove that ug_{1} ∧ ⋯ ∧ g_{r−1} ∈ ⋀^{r−1} G_{p} for every p. By hypothesis G_{p} is a free W_{p}-module. Choose a basis k_{1}, …, k_{r−1}, g_{r} for G_{p} with k_{1}, …, k_{r−1} ∈ K. Then k_{1} ∧ ⋯ ∧ k_{r−1} is a generator for the cyclic module ⋀ K_{p}^{r−1} . Thus g_{1} ∧ ⋯ ∧ g_{r−1} = w′k_{1} ∧ ⋯ ∧ k_{r−1} for some w′ ∈ W. Then ug_{1} ∧ ⋯ ∧ g_{r−1} ∧ g_{r} = uw′k_{1} ∧ ⋯ ∧ k_{r−1} ∧ g_{r} ∈ ⋀^{r} G_{p}. Since k_{1}, …, k_{r−1}, g_{r} is a basis for ⋀^{r} G_{p} it follows that uw′ ∈ W so that ug_{1} ∧ ⋯ ∧ g_{r−1} = uw′k_{1} ∧ ⋯ ∧ k_{r−1} ∈ ⋀^{r−1} G_{p}. Since this is true for every prime p it follows that ug_{1} ∧ ⋯ ∧ g_{r−1} ∈ ⋀^{r−1} G. ✓

proposition 9.18. Let H be t(A)-bounded and φ: G → H. Then φ is a monomorphism mapping G onto an essential submodule of H if and only if A^{∗}(φ) is a monomorphism mapping A^{∗}(H) onto an essential submodule of A^{∗}(G).

proof: ( ⇒ ): We may assume that G ⊆ H and φ is the inclusion. Thus if γ ∈ A^{∗}(H) then A^{∗}(φ)(γ) is simply the restriction of γ to G. Thus since G is essential in H, if γ ≠ 0 then A^{∗}(φ)(γ) ≠ 0. Thus A^{∗}(φ) is monic. Now let γ ∈ A^{∗}(G). Then t(γ(H)) ≤ OT(H) ≤ t(A), so by Proposition 2.* for some w ≠ 0 ∈ W, wγ(H) ⊆ A. Thus wγ ∈ A^{∗}(H). Thus every element of A^{∗}(G) has a non-trivial multiple in A^{∗}(H), so A^{∗}(G) is an essential submodule of A^{∗}(H).

( ⇐ ): If A^{∗}φ is a monomorphism mapping A^{∗}(H) onto an essential submodule of A^{∗}(G), then by the previous paragraph A^{∗∗}(φ) is a monomorphism mapping A^{∗∗}(G) onto an essential submodule of A^{∗∗}(H). Using the natural isomorphism ρ between A^{∗∗} and the identity functor, we conclude that φ is a monomorphism mapping G onto a pure submodule of H. ✓

proposition 9.19. Let G_{1} and G_{2} be t-bounded modules with the same divisible hull. Then A^{∗}(G_{1} + G_{2}) = A^{∗}(G_{1}) ∩ A^{∗}(G_{2}) and A^{∗}(G_{1} ∩ G_{2}) = A^{∗}(G_{1}) + A^{∗}(G_{2}).

proof: Because G_{1} and G_{2} are t-bounded, A^{∗}(G_{1}) and A^{∗}(G_{2}) are essential submodules of Hom(QG_{1}, Q) = Hom(QG_{2}, Q). Now if φ: QG_{1} → Q then φ(G_{1} + G_{2}) ⊆ A ⟺ φ(G_{1}) ⊆ A & φ(G_{2}) ⊆ A. Thus A^{∗}(G_{1} + G_{2}) = A^{∗}(G_{1} ∩ A^{∗}(G_{2}). And from this we derive

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

Stated another way, Proposition 9.19 says that the functor A^{∗} affords a lattice antiisomorphism between the set of t-bounded essential submodules of any finite dimensional Q-space V and the set of t-bounded essential submodules of Hom(V, Q). Combining this with any isomorphism from Hom(V, Q) to V will give a lattice anti-isomorphism from the lattice of t-bounded essential submodules of V onto itself.

The following corollary may be compared to Proposition 8.*. There seems to be a hint here that ET(G) may play the same role for locally free modules that the family of numbers {p-rank G}_{p∈Spec W} plays for quotient divisible modules. corollary 9.20. If G is an essential submodule of a locally free module H and ET(G) = ET(H), then G is quasi-equal to H. proof: Let A = ⋀^{r} G and t = t(A) = ET(G) = ET(H), so that A ∼ ⋀^{r} H. By Proposition 9.19 ⋀^{r−1} G ≈ A^{∗}(G) and ⋀^{r−1} H ∼ A^{∗}(H). Now the inclusion map G ↪ H induces a monomorphism ⋀^{r} G → ⋀^{r} H, and thus a monomorphism A^{∗}(G) → A^{∗}(H). On the other hand, by Proposition 9.19, applying the functor A^{∗} to the inclusion yields a monomorphism A^{∗}(H) → A^{∗}(G). It thus follows that A^{∗}(G) and A^{∗}(H) are quasi-isomorphic. Thus G and H are quasi-isomorphic, and since G ⊆ H they must be quasi-equal. ✓

proposition 9.21. Let A be a locally free rank-one module, t = t(A), and let G be a t-bounded module. Let δ: A → Hom(G, A ⊗ G) be defined by δ(a)(g) = a ⊗ g and let θ: A^{∗}(G) ⊗ G → Hom(G, A ⊗ G) be defined as in Proposition 9.2. Then there is an isomorphism

Δ: Hom(G, H) ≈ Hom(A, A^{∗}(G) ⊗ H)

given by Δ(φ) = (1 ⊗ φ)θ^{−1}δ. proof: By the naturality of θ, Δ(φ) = θ^{−1}(1 ⊗ φ)_{∗}δ and so Δ is simply the composition Hom(G, H) ≈ Hom(A ⊗ G, A ⊗ H) ≈ Hom(A, Hom(G, A ⊗ H)) ≈

Hom(A, A^{∗}(G) ⊗ H). ✓

There is an intriguing parallel here to Theorem 8.* for Arnold duality. Theorem 8.* implies that if G is quotient divisible then QHom(G, H) ≈ Hom(Q, A(G) ⊗ H).

corollary 9.22. If G is t-bounded then A ⊗ Hom(G, H) ≈ (A^{∗}(G) ⊗ H)(t). proof: By Proposition 9.21 and Proposition 4.* A ⊗ Hom(G, H) ≈

A ⊗ Hom(A, A^{∗}(G) ⊗ H) ≈ (A^{∗}(G) ⊗ H)(t). ✓

corollary 9.23. If G is t-bounded then (G ⊗ H)(t) ≈ A ⊗ Hom(A^{∗}(G), H). proof: Since G is t-bounded, (G ⊗ H)(t) ≈ (A^{∗∗}(G) ⊗ H)(t) ≈ A ⊗ Hom(A^{∗}(G), H) by Proposition 9.22. ✓

MODULES WITH NON-TRIVIAL TRACE.

lemma 9.24. Let G be t-bounded and let δ: A → Hom(G, A ⊗ G), σ: A^{∗}(G) ⊗ G → A, and θ: A^{∗}(G) ⊗ G → Hom(G, A ⊗ G) be defined as in Theorem 9.14 and Proposition 9.21. Let φ ∈ End G. Then Trace φ = σ(1 ⊗ φ)θδ(1) ∈ W. proof: Since A dominates G, Proposition 1.* shows that it suffices to prove the result locally. Hence no generality is lost in supposing G free and A = W. Let g_{1}, …, g_{r} be a basis for G and let γ_{1}, …, γ_{r} ∈ Hom(G, Q) be the dual basis. Then in fact γ_{i} ∈ A^{∗}(G) for all i since γ_{i}(G) ⊆ W = A. Now θ^{−1}δ(1) = ∑ γ_{i} ⊗ g_{i} and for φ ∈ End G, σ(1 ⊗ φ)θ^{−1}δ(1) = ∑ γ_{i}(φ(g_{i})) = Trace φ. (In fact γ_{i}(φ(g_{i})) is simply the i^{th} diagonal entry in the matrix for φ.) Furthermore since σ(1 ⊗ φ)θ^{−1}δ ∈ End A ≈ W, then

Trace φ = σ(1 ⊗ φ)θ^{−1}δ(1) ∈ W. ✓

We say that G has non-trivial trace if there exists φ ∈ End G such that Trace φ ≠ 0. It is well known, of course, that if φ = 1 ∈ End G then Trace φ = rank G ∈ W. Thus if char W ≠ 0 then every module has non-trivial trace.

proposition 9.25. Let G be t-bounded. The following conditions are equivalent: (1) G has non-trivial trace.

(2) σ: A^{∗}(G) ⊗ G → A is a quasi-split surjection.

(3) δ: A → Hom(G, A ⊗ G) is a quasi-split monomorphism.

proof: 1) ⇒ 2) & 3): By Lemma 9.24 if Trace φ ≠ 0 then σ(1 ⊗ φ)θ^{−1}δ ≠ 0 ∈ End A. Since all non-trivial endomorphisms of A are quasi-automorphisms (because QEnd A = Q), it follows that δ and σ split.

(2) ⇒ (1): Let η ∈ QHom(A, A^{∗}(G) ⊗ G) be a quasi-splitting for σ. Then for some w ≠ 0 ∈ W, wη ∈ Hom(A, A^{∗}(G) ⊗ G). By Proposition 9.21 wη = (1 ⊗ φ)θ^{−1}δ for some φ ∈ End G. Then by Lemma 9.24 Trace φ = wση(1) = w ≠ 0.

(3) ⇒ (1): Analogous. ✓

proposition 9.26. Let G be a t-bounded strongly indecomposable module. Let N = nil rad End G and Z = Center(End G /N). Then G has non-trivial trace if and only if QZ is separable over Q and (rank G) /[QZ: Q] is not a multiple of char W.

proof: Let D = QEnd G /N. Since G is strongly indecomposable, D is a skew field by Proposition 3.*. Let X_{i} = N^{i}QG/N^{i+1}QG for i = 1, 2, …, m, where N^{m} = 0. Let X = ⨁ X_{i}. Then X is a D-module and is canonically isomorphic to QG as a Q-vector space. For φ QEnd G let φ′ be the image of φ in D. Since X is a D-space, left multiplication by φ′ yields a scalar linear transformation φ̂ on X. Furthermore if we identify X and QG by using the obvious map, then φ also becomes a linear transformation on X and φ − φ̂ is nilpotent, so Trace φ = Trace φ̂. Then Trace φ̂ = (dim_{D} X) Trace_{D/Q} φ′ = (dim_{X} D) Trace_{QZ/Q}(Trace_{D/QZ} φ′), where by Trace_{D/Q} φ′ is meant, of course, the usual field-theoretic trace of φ′ ∈ D.

( ⇐ ): If QZ is separable over Q then there exists φ′ ∈ QZ ⊆ D with Trace_{QZ/Q} φ′ ≠ 0. If furthermore dim_{QZ} X = (rank G)/[Z: Q] is not a multiple of char W, then Trace_{X} φ̂ = (dim_{QZ} X) Trace_{QZ/Q} φ′ ≠ 0. Thus Trace φ ≠ 0 for any pre-image φ ∈ QEnd G for φ′.

( ⇒ ): Now if QZ is not separable over Q then it is known from field theory that Trace_{QZ/Q} φ′ = 0 for all φ′. And if dim_{D} X is a multiple of char W then clearly Trace φ̂ = 0 for all scalar linear transformations φ̂ on the D-space X. And finally if [D: QZ] is a multiple of char W it is known from the theory of simple algebras

[Bourbaki] that Trace_{D/QZ} φ′ = 0 for all φ′ ∈ D. Thus Trace φ = Trace φ̂ = 0 for all φ if either QZ is not separable over Q or (rank G)/[QZ: Q] = [D: QZ] dim_{D} X is a multiple

of char W. ✓

corollary 9.27. Let G be locally free with non-trivial trace.

(1) For any H the natural map δ_{H}: H → Hom(G, H ⊗ G) given by δ_{H}(h)(g) = h ⊗ g is a quasi-split monomorphism and the adjointness isomorphism Hom(X ⊗ G, H) ≈ Hom(X, Hom(G, H) takes quasi-split monomorphisms to quasi-split monomorphisms.

(2) If H dominates G then the natural map σ: Hom(G, H) ⊗ G → H given by σ_{H}(ψ ⊗ g) = ψ(g) is a quasi-split surjection and the adjointness isomorphism Hom(X, Hom(G, H)) ≈ Hom(X ⊗ G, H) takes split quasi-surjections to split quasi-surjections.

proof: Choose A so that t(A) = OT(G). First we show that δ_{H} and σ_{H} are quasi-split. By Lemma 9.4 to see that δ_{H} is quasi-split monic it suffices to show that A ⊗ δ_{H}: A ⊗ H → A ⊗ Hom(G, H ⊗ G) is quasi-split monic, and for this it suffices to show that the composition

δ_{A⊗H}: A ⊗ H → A ⊗ Hom(G, H ⊗ G) → Hom(G, A ⊗ H ⊗ G)

is quasi-split monic. Hence, replacing H with A ⊗ H we may assume wlog that H dominates A and G.

With this assumption, the map θ_{H}: Hom(G, H) ⊗ G → Hom(G, G ⊗ H) given by θ_{H}(φ ⊗ g)(g′) = g ⊗ φ(g′) is an isomorphism (c.f. Lemma 4.*) and if h ∈ H then there exists η ∈ QHom(A, H) with η(1) = h. Now apply the naturality of δ, σ, and θ to see that if φ ∈ End G then

σ_{H}(1 ⊗ φ)θ_{H}^{−1}δ_{H}(h) = σ_{H}(1 ⊗ φ)θ_{H}^{−1}δ_{H}η(1)
= ησ_{A}(1 ⊗ φ)θ_{A}^{−1}δ_{A}(1) = η(Trace φ) = (Trace φ)h.

Thus clearly if there exists φ ∈ End G with Trace φ ≠ 0 then

σ_{H}(1 ⊗ φ)θ_{H}^{−1}δ_{H} = Trace φ ≠ 0 ∈ W ⊆ End H

and since multiplication by Trace φ is a quasi-automorphism in End H it follows that δ_{H} and σ_{H} are quasi-split as required.

Now if φ ∈ Hom(X ⊗ G, H) then its adjoint in Hom(X, Hom(G, H)) can be factored as

φ_{∗}δ_{X}: X → Hom(G, X ⊗ G) → Hom(G, H).

Since δ_{X} is quasi-split monic, if φ is quasi-split monic then so is its adjoint. Likewise the adjoint for ψ ∈ Hom(X, Hom(G, H)) factors as σ_{H}(ψ⊗1): X⊗G → Hom(G, H)⊗G → H, which is a split quasi-surjection if ψ is. ✓

Whereas examples of failure of cancellation in direct sum decompositions are fairly surprising and need to be somewhat carefully contrived, for tensor products cancellation is not something one even expects. For instance if G = Q then from an isomorphism G ⊗ H ≈ G ⊗ H′ we can conclude only that rank H = rank H′. On the other hand, Proposition 9.4 shows that if A is a locally free rank-one module then we do get cancellation: A ⊗ H ≈ A ⊗ H′ implies H ≈ H′. One of the themes of this chapter is that properties which hold for rank-one locally free modules often are valid in somewhat weakened form for locally free modules in general. Thus one might even be rash enough to conjecture that if G is locally free then G ⊗ H ∼ G ⊗ H′ implies H ∼ H′. Your author will acknowledge that he has invested a considerable amount of time in investigating this unlikely conjecture. In fact the greater part of the material in this chapter was developed pursuant to that investigation. Unfortunately, the closest he has come to success is the following corollary:

corollary 9.28. If G is locally free with non-trivial trace, then for any finite rank torsion free H there exist, up to quasi-isomorphism, at most finitely many H′ such that G ⊗ H ∼ G ⊗ H′.

proof: If G ⊗ H ∼ G ⊗ H′ then by Corollary 9.27 H′ is isomorphic to a quasi-summand of Hom(G, H′ ⊗ G) ∼ Hom(G, H ⊗ G). By Jónsson’s Theorem (Theorem 3.24) Hom(G, H ⊗ G) has, up to quasi-isomorphism, only finitely many quasi-summands. ✓

proposition 9.29. Let G be t-bounded and for any H let Δ: Hom(G, H) → Hom(A, A^{∗}(G) ⊗ H) and Γ: Hom(H, G) → Hom(A^{∗}(G) ⊗ H, A) be the isomorphisms from Theorem 9.14 and Proposition 9.21. Let φ ∈ Hom(G, H) and ψ ∈ Hom(H, G).

(1) If G has non-trivial trace and φ is a quasi-split monomorphism then so is Δ(φ).

(2) If G has non-trivial trace and ψ is a quasi-split quasi-surjection, then so is Γ(ψ).

(3) If G is strongly indecomposable and Δ(φ) is a quasi-split monomorphism then so is φ.

(4) If G is strongly indecomposable and Γ(ψ) is a quasi-split quasi-surjection then so is ψ.

proof: Since Γ(ψ) = σ(1 ⊗ ψ) and Δ(φ) = (1 ⊗ φ)θ^{−1}δ, Lemma 9.24 shows that

Γ(ψ)Δ(φ) = Trace ψφ.

Now if φ is quasi-split monic then we can choose ψ ∈ Hom(H, G) so that ψφ is any desired endomorphism of G. Hence if G has non-trivial trace we can choose ψ so that Γ(ψ)Δ(φ) = Trace ψφ ≠ 0 ∈ End A ≈ W, and we conclude that Δ(φ) is quasi-split surjective. Analogously we see that if ψ ∈ Hom(H, G) is a split quasi-surjective then Γ(ψ) is as well.

Conversely, if Δ(φ) is quasi-split monic then by Proposition 9.* the splitting map has the form Γ(ψ) for some ψ ∈ QHom(H, G), and so Trace ψφ = Δ(φ)Γ(ψ) ≠ 0. Since nilpotent linear transformation have trivial trace, this means that ψφ is not nilpotent. Thus if G is strongly indecomposable then by Proposition 3.* ψφ is a quasi-automorphism of G so that φ is also quasi-split monic. Likewise if Γ(ψ) is a quasi-split surjection then ψ is as well. ✓ QUASI-SUMMAND OF TENSOR PRODUCTS. In Chapter 5 we saw that if G and H are quotient divisible Butler modules then G ⊗ H will often have rank-one quasi-summands. In fact, if one looks at the sequence G, G ⊗ G, G ⊗ G ⊗ G, … then eventually a rank-one quasi-summand is inevitable.

For locally free modules the situation is surprisingly different. We have seen in Proposition 9.25 that if G is t(A)-bounded with non-trivial trace then the rank-one module A will be a quasi-summand of A^{∗}(G) ⊗ G. It now turns out that if G and H strongly indecomposable then this is the only case where G ⊗ H can have a rank-one quasi-summand.

theorem 9.30. Let G and H be strongly indecomposable modules. Suppose that G ⊗ H has a locally free rank-one quasi-summand A. Let t = t(A). Then

(1) G and H are locally free and t-bounded and have non-trivial trace.

(2) H ∼ A^{∗}(G).

(3) t(A) = IT(G)OT(H).

(4) Let N = nil rad End G. Then a maximal completely decomposable quasi-summand of G ⊗ H has rank equal to rank(End G) − rank N.

proof: Since a split quasi-surjection G ⊗ H → A must vanish on G[t] ⊗ H and by Proposition 9.14 G/G[t] ≈ A^{∗∗}(G), such a split quasi-surjection induces a map A^{∗∗}(G) ⊗ H → A which must also be a split quasi-surjection. By Jónsson’s Theorem (Theorem 3.24), A is then a quasi-summand of A^{∗}(K) ⊗ H for some strongly indecomposable quasi-summand K of A^{∗}(G). Since K is t-bounded, Proposition 9.29 yields a split quasi-surjection from H to K. Since H is by assumption strongly indecomposable, thus H ∼ K, so that in particular H is t-bounded and hence by Lemma 9.2 is locally free and by the proof of Proposition 9.29 has non-trivial trace. By symmetry, G is also locally free, t-bounded, and has non-trivial trace. Thus by Proposition 9.14 A^{∗}(G) is strongly indecomposable. Since K is a quasi-summand of A^{∗}(G) we conclude that H ∼ K ∼ A^{∗}(G).

Furthermore OT(H) = OT(A^{∗}(G)) = [t: IT(G)] and so IT(G)OT(H) = t. Finally this shows that a maximal completely decomposable submodule if G ⊗ H is t-homogeneous and thus corresponds to a maximal linearly independent set of quasi-split monomorphisms from A to G ⊗ H ∼ G ⊗ A^{∗}(G). Hence by Proposition 9.29 and the fact that G is strongly indecomposable it corresponds to a maximal linearly independent set of quasi-automorphisms of G. Thus the rank of such a maximal completely decomposable quasi-summand equals rank(End G /N) = rank End G − rank N, where N = nil rad End G. ✓

remark 9.31. Call two strongly indecomposable modules G and H similar if there exist rank-one locally free modules A_{1} and A_{2} such that A_{1} ⊗ G_{1} ∼ A_{2} ⊗ G_{2} . If G is locally free then the Warfield dual of G defines a similarity class independent of A, since if OT(G) = t(A_{1}) and t(A) ≥ t(A_{1}) then A^{∗}(G) ≈ A^{∗}(A_{1}) ⊗ A_{1}^{∗}(G) and A^{∗}(A_{1}) is a locally free rank-one module. Thus Theorem 9.17 shows that if G is locally free and strongly indecomposable with non-trivial trace then there exists a strongly indecomposable module H such that G ⊗ H has a rank-one quasi-summand, and that H is unique up to similarity. By Proposition 9.* we can choose H = ⋀^{r−1} G, where r = rank G.

lemma 9.32. Let G be a strongly indecomposable module and η: Hom(G ⊗ H, A) be a split quasi-surjection. Let ψ ∈ Hom(G, A^{∗}(H)) be adjoint to η. Then G is t-bounded with non-trivial trace and ψ is a split quasi-monomorphism. proof: Since A is a quasi-summand of G ⊗ H_{1} for some strongly indecomposable quasisummand H_{1} of H, Theorem 9.31 shows that G is t-bounded with non-trivial trace, so that ρ: G → A^{∗∗}(G) is an isomorphism and A is isomorphic to a quasi-summand of A^{∗∗}(G) ⊗ H. Since A^{∗}(G) is strongly indecomposable Proposition 9.29 implies that the map ψ_{1} ∈ Hom(H, A^{∗}(G)) adjoint to η is a split quasi-surjection. Thus A^{∗}(ψ_{1}) is a split quasi-monomorphism. But for g ∈ G, h ∈ H,

(A^{∗}(ψ_{1})ρ)(g)(h) = (A^{∗}(ψ_{1})(ρ(g)))(h) = ρ(g)(ψ_{1}(h)) = ψ_{1}(h)(g) = η(g ⊗ h) = ψ(g)(h)

so that ψ = A^{∗}(ψ_{1})ρ. Thus ψ is a split quasi-monomorphism. ✓

theorem 9.33. Let K be a locally free module with non-trivial trace. Let G be a strongly indecomposable module and H a module such that K is quasi-isomorphic to a quasi-summand of G ⊗ H. Then G is likewise locally free with non-trivial trace and is isomorphic to a quasi-summand of Hom(H, K). More specifically, the adjointness isomorphism Hom(G ⊗ H, K) ≈ Hom(G, Hom(H, K)) takes split quasi-surjections to split quasi-monomorph isms. proof: Choose A so that t(A) = OT(K). Then Proposition 9.14 yields the following isomorphisms:

Hom(G ⊗ H, K) ≈ A^{∗}(G ⊗ H ⊗ A^{∗}(K))
≈ Hom(G, A^{∗}(H ⊗ A^{∗}(K)) ≈ Hom(G, Hom(H, K)),

and it is easily seen that the composition of these is the adjointness isomorphism in question. Proposition 9.29 and Lemma 9.32 then show that split quasi-surjections are taken to split quasi-monomorphisms and that G is t-bounded, hence locally free, with non-trivial trace. ✓

corollary 9.34. If K is locally free with non-trivial trace then for any H there exist up to quasi-isomorphism at most finitely many strongly indecomposable modules G such that K is isomorphic to a quasi-summand of G ⊗ H. If H has non-trivial trace and K is strongly indecomposable and dominates H, then there is at least one such G. proof: By Theorem 9.33 the possible strongly indecomposable modules G such that K is isomorphic to a quasi-summand of G ⊗ H lie among the quasi-summands of Hom(H, K). Thus by Jónsson’s Theorem there are only finitely many such G.

Now by Corollary 9.27 if H has non-trivial trace and K dominates H then there is a quasi-split map Hom(H, K) ⊗ H → K. Thus by Jónsson’s Theorem if K is strongly indecomposable then there exists at least one strongly indecomposable quasi-summand G of Hom(H, K) such that K is isomorphic to a quasi-summand of G ⊗ H. ✓ corollary 9.35. Let G be a quasi-direct sum of strongly indecomposable modules with non-trivial trace. Let H be a locally free module with non-trivial trace dominating G. Then the map ρ_{1} : G → Hom(Hom(G, H), H) given by ρ_{1}(g)(φ) = φ(g) is a quasi-split monomorphism. proof: It suffices to prove that theorem for each strongly indecomposable quasisummand in some quasi-direct decomposition of G, hence no generality is lost in supposing G strongly indecomposable. By Corollary 9.27 σ: G ⊗ Hom(G, H) → H is a quasi-split surjection. By Theorem 9.34 the adjoint map σ′: G → Hom(Hom(G, H), H) is a quasi-split monomorphism. But σ′ = ρ_{1} : in fact, for any g ∈ G, φ ∈ Hom(G, H),

σ′(g)(φ) = φ(g ⊗ φ) = φ(g) = ρ_{1}(g)(φ). ✓

corollary 9.36. Let G be strongly indecomposable and let G ⊗ H = K ⊕ K′, where K ≠ 0 and K is locally free with non-trivial trace. Let G_{1} be a submodule of G.

(1) If K ⊆ G_{1} ⊗ H then G_{1} is quasi-equal to G.

(2) If G_{1} ⊗ H ⊆ K′ then G_{1} = 0.

proof: 1) Let K ⊆ G_{1} ⊗ H. Then by Proposition 3.* K is a quasi-summand of G_{1} ⊗ H. If we suppose, without loss of generality, that K is strongly indecomposable, then by Jónsson’s Theorem K is a quasi-summand of G_{2} ⊗ H for some strongly indecomposable quasi-summand G_{2} of G_{1} . Furthermore if μ: G ⊗ H → K is a split quasi-surjection then the restriction of μ to G_{2} ⊗ H → K is also a quasi-split surjection. Call this restriction μ_{2} . Since K has non-trivial trace, Theorem 9.33 shows that the adjoint map μ′_{2} : G_{2} → Hom(H, K) is a quasi-split monomorphism. But μ_{2} is simply the restriction to G_{2} of the map μ′: G → Hom(H, K) adjoint to μ. Thus we have

G_{2} ⟶^{⊆} G ⟶^{μ′} Hom(H, K)

and the composition of these two maps is a quasi-split monomorphism. By Proposition 3.* the inclusion G_{2} ↪ G is thus also a quasi-split monomorphism, i.e. G_{2} is a quasi-summand of G. Since G is strongly indecomposable, we see that G_{2} = G. But G_{2} ⊆ G_{1} ⊆ G, so G_{1} = G.

2) Let ψ: G ⊗ K → K be the quasi-projection. By Theorem 9.33 the adjoint map ψ′: G → Hom(H, K) is monic. Since ψ′(G_{1}) = 0 it follows that G_{1} = 0. ✓