aa r X i v : . [ m a t h . L O ] N ov Absolutely Indecomposable Modules
R¨udiger G¨obel and Saharon Shelah
Abstract
A module is called absolutely indecomposable if it is directly indecomposable in every genericextension of the universe. We want to show the existence of large abelian groups that are absolutelyindecomposable. This will follow from a more general result about R -modules over a large classof commutative rings R with endomorphism ring R which remains the same when passing to ageneric extension of the universe. It turns out that ‘large’ in this context has a precise meaning, namely being smaller than the first ω -Erd˝os cardinal defined below. We will first apply a result onlarge rigid valuated trees with a similar property established by Shelah [26] in 1982, and will provethe existence of related ‘ R ω -modules’ ( R -modules with countably many distinguished submodules)and finally pass to R -modules. The passage through R ω -modules has the great advantage that theproofs become very transparent essentially using a few ‘linear algebra’ arguments accessible alsofor graduate students. The result closes a gap in [12, 11], provides a good starting point for [16]and gives a new construction of indecomposable modules in general using a counting argument. There is a whole industry transporting symmetry properties from one category to another: For exampleconsider a tree or a graph (with extra properties if needed) together with its group of automorphisms.Then encode the tree or the graph into an object of your favored category in such a way that thebranches (or vertices) of the tree (of the graph) are recognized in the new structure. If the newcategory are abelian group argue by (infinite) divisibility, in case of groups and fields you use ofcourse infinite chains of roots (with legal primes) etc. Thus the automorphism group of the tree or thegraph is respected in the new category and by density arguments (or killing unwanted automorphismsby prediction arguments ‘on the way’) it happens that the automorphism group we start with becomes(modulo inessential maps: inner automorphisms in case of groups and Frobenius automorphisms incase of fields) the automorphism group of an object of the new category. For a few illustrating detailsthe reader may want to see papers by Heineken [22], Braun, G¨obel [2] (in case of groups), Corner,G¨obel [7] in case of modules (with group rings as the first category) or Fried and Kollar [14], Dugas,G¨obel [9] in case of fields and [10] for automorphism groups of geometric lattices. In this paper we alsoargue with symmetry properties of trees, but they are of a different kind. Given a cardinal λ which is This work is supported by the project No. I-706-54.6/2001 of the German-Israeli Foundation for Scientific Research& Development.This is GbSh880 in second author’s list of publications.AMS subject classification: primary: 13C05, 13C10, 13C13, 20K15, 20K25, 20K30; secondary: 03E05, 03E35. Keywords and phrases: absolutely indecomposable modules, generic extension, distinguished submodules, labelled trees,Erd˝os cardinal, rigid-like systems, automorphism groups. ω -)valuated trees based on this cardinal. This is a familyof λ subtrees T of size λ of the tree T λ = ω> λ of finite sequences of ordinals in λ together with avaluation map v : T −→ ω . Rigid means that there is no level preserving valuated homomorphismsbetween any two distinct members. (A tree homomorphism is valuated if the value of a branch is thesame as the value of its image.) Moreover this property is preserved if we change the universe, passingto a generic extension of the given universe (of set theory) we live in. The existence of such treeswas shown by Shelah [26]. These trees (used also in applied mathematics) were considered earlier inpapers by Nash-Williams, see [24] for example. We will encode them into free R ω -modules over anarbitrary not extraordinary large commutative ring R with 1 = 0. To be definite we can assume that R is the field Q of rationals or Z . Recall that R ω -modules are R -modules with countably many ( ω )distinguished submodules and free means that the module and its distinguished submodules and factormodules are free as well. Such creatures are considered in Brenner, Butler, Corner (see [3, 4, 5, 1])and G¨obel, May [18] for arbitrary commutative rings and an account about the advanced theory incase of fields can be seen in [27] and in the references given there. We will show the existence of free R ω -modules with endomorphism algebra R by transporting the absolute rigid trees into the categoryof R ω -modules. It turns out that the passage through R ω -modules makes the anticipated proofs verytransparent. Moreover our main result on R ω -modules with distinguished submodules is only a fewsteps away from the desired result on R -modules if R has enough primes (like Z ).The corollary on the existence of large absolutely (fully) rigid abelian groups replaces the earlierunsuccessful approach in [12] and [11, Chapter XV]: Let R = 0 be any fixed countable ring. Thenby Corollary 4.2 there exists an absolutely rigid R ω -module of size λ (or an absolute family of size λ of non-trivial R -modules with only the zero-homomorphism between distinct member) iff λ < κ ( ω ).The same holds if R ω -modules are replaced by abelian groups. Thus as a byproduct we present a newconstruction of large, absolutely indecomposable abelian groups, not using stationary sets as [25, 7].So, if we restrict to the problem on the existence of large absolute indecomposable abelian groupsaddressed in [12, 11], then it follows from the above (realizing for example Z as the endomorphismring in Corollary 4.2) that from λ < κ ( ω ) follows the existence of such abelian groups. The conversedirection would need a strengthening of the Theorem 2.2 from [12] now showing the existence ofnon-trivial idempotents. (The second author believes that this guess might be true.)It is also a different matter how to replace R ω -modules by R -modules or R -modules and theendomorphism algebra R by a general not extraordinary large prescribed R -algebra A . This willfollow from [16], a paper which had to wait for Theorem 4.1 in place of [12]. ω –Erd˝os car-dinal We first describe the result on trees we want to apply by encoding them into modules with distinguishedsubmodules.Let κ ( ω ) denote the first ω - Erd˝os cardinal . This is defined as the smallest cardinal κ such that κ → ( ω ) <ω , i.e. for every function f from the finite subsets of κ to 2 there exist an infinite subset X ⊂ κ and a function g : ω → f ( Y ) = g ( | Y | ) for all finite subsets Y of X . The cardinal2 ( ω ) is strongly inaccessible; see Jech [23, p. 392]. Thus κ ( ω ) is a large cardinal. We should alsoemphasize that κ ( ω ) may not exist in every universe.If λ < κ ( ω ), then let T λ = ω> λ = { f : n −→ λ : with n < ω and n = Dom f } be the tree of all finite sequences f (of length or level lg( f ) = n ) in λ . Since n = { , . . . n − } asordinal, we also write f = f (0) ∧ f (1) ∧ . . . ∧ f ( n − g = f ↾ m for any m ≤ n we obtainall initial segments of f . We will write g ✁ f . Thus g ≤ f ⇐⇒ g ⊆ f as graphs ⇐⇒ g ✁ f. A subtree T of T λ is a subset which is closed under initial segments and a homomorphism between twosubtrees of T λ is a map that preserves levels and initial segments. (Note that a homomorphism doesnot need to be injective or preserve (cid:2) .) The tree T is valuated if with the tree we have a valuationmap v : T −→ ω . In the following a tree will always come with a valuation and Hom( T , T ) denotesthe valuated homomorphisms between subtrees T and T , i.e if v i is the valuation of T i ( i = 1 , ϕ is such a valuated homomorphism , then v ( ηϕ ) = v ( η ) for all η ∈ T . Shelah [26] showed theexistence of an absolutely rigid family of 2 λ valuated subtrees of T λ . Theorem 2.1 If λ < κ ( ω ) is infinite and T λ = ω> λ , then there is a family T α ( α ∈ λ ) of valuatedsubtrees of T λ (of size λ ) such that for α, β ∈ λ and in any generic extension of the universe thefollowing holds. Hom( T α , T β ) = ∅ = ⇒ α = β. Proof.
The result is a consequence of the Main Theorem 5.3 in [26, p. 208]. The family of rigidtrees is constructed in [26, p. 214, Theorem 5.7] and the proof, that the trees are rigid, follows fromTheorem 5.8 using the Conclusion 2.14 in [26]. In Shelah’s notation κ ( ω ) is the first beautiful cardinal > ℵ .This property of rigid families of valuated trees in Theorem 2.1 fails, if we choose λ ≥ κ ( ω ). Infact the following result from [12] on rigid families of R -modules reflects this. Theorem 2.2 (Eklof-Shelah [12]) Let λ be a cardinal ≥ κ ( ω ) and R any ring with 1.(i) If { M ν | ν < λ } is a family of non-zero left R -modules, then there are distinct ordinals µ, ν < λ ,such that in some generic extension V [ G ] of the universe V , there is an injective homomorphism φ : M µ → M ν .(ii) If M is an R -module of cardinality λ , then there exists a generic extension V [ G ] of the universe V , such that M has an endomorphism that is not multiplication by an element of R .Thus κ ( ω ) is the precise border line for Theorem 3.1 and we can not expect absolute results onendomorphism rings and rigid families of abelian groups above κ ( ω ), see Corollary 4.2.Combining Theorem 2.2 with our main result this also conversely shows that the implication ofTheorem 2.1 fails whenever λ ≥ κ ( ω ), i.e. there is a generic extension V [ G ] of the universe V andthere are distinct ordinals α, β ∈ λ with Hom( T α , T β ) = ∅ .3 The main construction
Let R = 0 be a commutative ring. As we shall write endomorphisms on the right, it will be convenientto view R -modules as left R -modules. Next we define a free R -module F of rank λ over a suitableindexing set (obviously) used to encode trees T α from Theorem 2.1 into the structure when turningthe free R -module F into an R ω -module module F with ω distinguished submodules.We enumerate a subfamily of λ valuated trees from Theorem 2.1 by the indexing set I = ω> ( ω> λ ).Thus T η with valuation map v η : T η −→ ω ( η ∈ I )without repetition. Next define inductively subsets S n ⊆ n ( ω> λ ) such that the following holds.(0) S = {⊥} (1) If S n is defined, then S n +1 = { η ∧ h ν i : η ∈ S n , ⊥ 6 = ν ∈ T η } .Let S = S n ∈ ω S n and also let η ∧ h⊥i = η for ⊥ ∈ T η .Put S nk = { η ∧ h ν i ∈ S : lg η = n, lg ν = k } ⊆ S n +1 . Here ν = ν ∧ . . . ∧ ν k − with ν i ∈ λ is asequence of ordinals and η = η ∧ . . . ∧ η n − with η i ∈ T η ∧ ... ∧ η i − a sequence of branches from specialtrees. Moreover write T kη = { ν ∈ T η : lg ν = k } ⊆ T η and T kη = { ν ∈ T η : v η ( ν ) = k } ( η ∈ I ) . Now we define the free R -modules:(i) F = L η ∈ S Re η (ii) F nk = L η ∈ S n L ν ∈ T kη R ( e η ∧ h ν ↾ k − i − e η ∧ h ν i )(iii) F nk = L η ∈ S n ( L ν ∈ T kη Re η ∧ h ν i )(iv) F kn = L η ∈ S n ( L ν ∈ T kη Re η ∧ h ν i )(v) F = h R ( e η − e η ′ ) : η, η ′ ∈ S i and F = Re ⊥ .We note that F = L ⊥ 6 = η ∈ S R ( e ⊥ − e η ) and F = F ⊕ F .Next we define R ω -modules. These are R -modules with ω distinguished submodules. We enumeratethe distinguished submodules by a particular well-ordered, countable indexing set W = h , i ∧ L ∧ L ∧ L with L i a copy of ω × ω ( i = 1 , , . We view W as an ordinal. Then an R ω -module X is an R -module X with a family of submodules X i ( i ∈ W ). We will also say that X is a free R ω -module if X, X i , X/X i ( i ∈ W ) are free R -modules.In particular 4 = ( F, F , F , F nk , F pq , F rs : ( nk ) ∈ L , ( pq ) ∈ L ) , ( rs ) ∈ L ) is a free R ω − module. (3.1)If X , Y are R ω -modules, then ϕ is an R ω -homomorphism ( ϕ ∈ Hom R ( X , Y )) if ϕ ∈ Hom R ( X, Y )and X i ϕ ⊆ Y i for all i ∈ W , where Y = ( Y, Y i : i ∈ W ). We also write Hom R ( X , X ) = End R X .We want to show the following Theorem 3.1
Let R be a commutative ring with = 0 and | R | , λ < κ ( ω ) . A free R -module F ofrank λ can be made into a free R ω -module F = ( F, F i : i ∈ W ) such that End R F = R holds in anygeneric extension of the given universe. Note that the size of R and the rank λ can be arbitrary < κ ( ω ); in particular R = Z / Z . If λ isfinite, then we can choose directly a suitable finite family of F i s with the required endomorphism ring.Otherwise λ is infinite and we can apply Theorem 2.1. So we choose F = ( F, F i : i ∈ W ) as in (3.1)depending on the valuated trees from Theorem 2.1. Then clearly it remains to show End R F = R .We first show the following crucial Lemma 3.2
Let ϕ ∈ End R F with F as in (3.1) and F = L η ∈ S Re η . If η ∈ S , then e η ϕ ∈ Re η . Proof.
Let η ∈ S be fixed and recall that T kη = T η ∩ k λ . We consider its successors η ∧ h ν i in S with ⊥ 6 = ν ∈ T η and let lg η = n, lg ν = k . Thus η ∧ h ν i ∈ S nk and ν ∈ T kη . If ϕ ∈ End R F , then we claim e η ∧ h ν i ϕ = X l 5e will apply the two displayed equations and suppose for contradiction that e η ϕ / ∈ Re η . Hence e η ϕ = P l From Lemma 3.2 follows e ⊥ ϕ = re ⊥ , e η ϕ = r η e η for some r, r η ∈ R andall ⊥ 6 = η ∈ S . Moreover ( e ⊥ − e η ) ∈ F , and therefore ( e ⊥ − e η ) ϕ ∈ F and ( e ⊥ − e η ) ϕ = re ⊥ − r η e η ∈ R ( e ⊥ − e η ) by support (in the direct sum). Hence re ⊥ − r η e η = r ′ ( e ⊥ − e η ) for some r ′ ∈ R and r = r ′ , r η = r ′ implies r η = r for all η ∈ S . Thus ϕ = r ∈ R . We want to strengthen Theorem 3.1 showing the existence of fully rigid systems of R ω -modules on λ .This is a family F U ( U ⊆ λ ) of R ω -modules such that the following holds. Hom R ( F U , F V ) = (cid:26) R if U ⊆ V U V R -algebras A as endomorphism algebras End R F = A which are also absolute, see Fuchs, G¨obel [16]. We first extend the well-ordered indexing set W for F by one more element and let W ′ := h , , i ∧ L ∧ L ∧ L with L i ∼ = ω × ω. Hence W ′ and W are both order-isomorphic to ω × ω but W ′ has virtually one more element than W added at place 2 to the definition of F . This allows us to replace F from Theorem 3.1 by F U wherethe new place is F := F U := M e ∈ U eR for any U ⊆ S. From Theorem 3.1 follows Hom R ( F U . F V ) ⊆ R for any U, V ⊆ S. Clearly Hom R ( F U . F V ) = R if U ⊆ V . On the other hand, if u ∈ U \ V , then e u ϕ = re u by thedisplayed formula. But re u ∈ F V only if r = 0. Hence Hom R ( F U . F V ) = 0 whenever U V . Finallynote that | S | = λ . We established the existence of fully rigid systems. Theorem 4.1 If R is any commutative ring with = 0 and λ, | R | < κ ( ω ) , then there is a fully rigidsystem F U ( U ⊆ λ ) of free R ω -modules with the following properties.(i) F is free of rank λ and F U = ( F, F , F , F U , F i : i ∈ L ∧ L ∧ L ) , thus only F = F U dependson U .(ii) The family F U ( U ⊆ λ ) is absolute, i.e. if the given universe is replaced by a generic extension,then the family is still fully rigid. The last theorem and a result from [12] (see Theorem 2.2) immediately characterize the first ω -Erd˝os cardinal. For clarity we restrict ourself to countable rings R . Corollary 4.2 Let R by any countable commutative ring. Then the following conditions for a cardinal λ are equivalent.(i) There is an absolute R ω -module X of size λ with End R M = R .(ii) There is a fully rigid family F U ( U ⊆ λ ) of free R ω -modules.(iii) There is a family of R ω -modules of size λ with only the zero-homomorphism between two distinctmembers.(iv) λ < κ ( ω ) with κ ( ω ) the first ω - Erd˝os cardinal. We note, that the last theorem can also be applied to vector spaces (and ω in (i), (ii) and (iii) canbe replaced by 4 or 5 as demonstrated in [16]) 7 Passing to R -modules We will restrict ourself to only one application of Theorem 4.1. A forthcoming paper by Fuchs, G¨obel[16] will exploit Theorem 4.1 and new results will be obtained in two directions. Firstly the numberof primes needed in Corollary 5.1 will be reduced to four (which is minimal), moreover R -algebras A will be realized as End R M = A in order to give more absolute results. These applications wereobtained earlier but had to wait for publication until it became possible to replace certain results in[12] by Theorem 4.1. Corollary 5.1 Let R be a domain with infinitely many comaximal primes. If λ, | R | < κ ( ω ) , thenthere is an absolute fully rigid family M U ( U ⊆ λ ) of torsion-free R -modules M U of size λ . Thus thefollowing holds in any generic extension of the given universe of set theory. Hom R ( M U , M V ) = (cid:26) R if U ⊆ V if U V Proof. Let p i ( i ∈ W ′ ) be a countable family of comaximal primes of R and choose F U =( F, F , F , F U , F i : i ∈ L ∧ L ∧ L ) from Theorem 4.1. We will now construct R -modules M U with F ⊆ M U ⊆ Q ⊗ F where Q denotes the quotient field of R . Also, if X ⊆ F , then we denote by p −∞ X := [ n ∈ ω p − n X ⊆ Q ⊗ F. Now let M U := h p −∞ i F i , p −∞ F U : i ∈ W i . Thus F ⊆ M U ⊆ Q ⊗ F because F + F = F and Q ⊗ F U := ( Q ⊗ F, Q ⊗ F , Q ⊗ F , Q ⊗ F U , Q ⊗ F i : i ∈ L ∧ L ∧ L ) satisfies End Q ( Q ⊗ F U ) = Q by Theorem 4.1. Consider now any ϕ ∈ End R M U . Theprimes ensure that p −∞ i F i ϕ ⊆ p −∞ i F i for all i ∈ W ′ and ϕ extends uniquely to an endomorphism(also called) ϕ ∈ End Q ( Q ⊗ F U ). It follows that ϕ = q ∈ Q , thus ϕ is scalar multiplication by q onthe right. It remains to show that ( ϕ =) q ∈ R and possibly ϕ = 0.Now we recall that the family of primes, in particular p and p are comaximal, thus p −∞ R ∩ p −∞ R = R . Choose any e η ∈ F . Then e η ϕ ∈ p −∞ Re η , hence q ∈ p −∞ R . Similarly, e ⊥ ϕ ∈ p −∞ Re ⊥ ,thus also q ∈ p −∞ R and q ∈ p −∞ R ∩ p −∞ R = R as required. 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Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories , Alge-bra, Logic and Applications, Vol. 4, Gordon & Breach Science Publishers, London, 1992.Address of the authors:R¨udiger G¨obelFachbereich 6, MathematikUniversit¨at Duisburg EssenD 45117 Essen, Germany e-mail: [email protected] Saharon ShelahInstitute of Mathematics,Hebrew University, Jerusalem, Israeland Rutgers University, New Brunswick, NJ, U.S.A e-mail: [email protected]: [email protected]