Set-theoretic methods in module theory

Univerzita Karlova v Praze Matematicko-fyzikální fakulta BAKALÁŘSKÁ PRÁCE Alexander Slávik Set-theoretic methods in module theory Katedra algebry Vedoucí bakalářské práce: prof. RNDr. Jan Trlifaj, CSc., DSc. Studijní program: Matematika Studijní obor: Obecná matematika Praha 2012

Děkuji svému vedoucímu prof. RNDr. Janu Trlifajovi, CSc., DSc. za cenné podněty, rady a připomínky při vedení bakalářské práce. Rovněž bych chtěl poděkovat své rodině za všestrannou podporu během mých studií.

Prohlašuji, že jsem tuto bakalářskou práci vypracoval(a) samostatně a výhradně s použitím citovaných pramenů, literatury a dalších odborných zdrojů. Beru na vědomí, že se na moji práci vztahují práva a povinnosti vyplývající ze zákona č. 121/2000 Sb., autorského zákona v platném znění, zejména skutečnost, že Univerzita Karlova v Praze má právo na uzavření licenční smlouvy o užití této práce jako školního díla podle 60 odst. 1 autorského zákona. V Praze dne 3. srpna 2012 Alexander Slávik

Název práce: Množinově-teoretické metody v teorii modulů Autor: Alexander Slávik Katedra: Katedra algebry Vedoucí bakalářské práce: prof. RNDr. Jan Trlifaj, CSc., DSc. Abstrakt: Třída modulů se nazývá dekonstruovatelná, pokud jde o třídu všech S-filtrovaných modulů pro nějakou množinu modulů S. Takovéto třdy nacházejí široké uplatnění v teorii aproximací modulů. V práci je dokázána dekonstruovatelnost třídy všech modulů majících C-resolventu a dekonstruovatelnost tříd všech modulů s omezenou C-resolventní dimenzí za předpokladu dekonstruovatelnosti třídy C. Dále jsou zkoumány lokálně F-volné moduly; je dokázána postačující podmínka na třídu F, aby byla třída všech lokálně F-volných modulů uzavřena na transfinitní extenze. Díky tomu lze zkonstruovat nové netriviální příklady nedekonstruovatelných tříd. Prezentovaná metoda zároveň poskytuje alternativní důkaz nedekonstruovatelnosti třídy všech plochých Mittag-Lefflerových modulů, nedávného výsledku D. Herbera a J. Trlifaje. Klíčová slova: Dekonstruovatelná třída, transfinitní extenze, filtrace, lokálně F volný modul Title: Set-theoretic methods in module theory Author: Alexander Slávik Department: Department of Algebra Supervisor: prof. RNDr. Jan Trlifaj, CSc., DSc. Abstract: A class of modules is called deconstructible if it coincides with the class of all S-filtered modules for some set of modules S. Such classes provide a convenient setting for construction of approximations. We prove that for any deconstructible class C the class of all modules possessing a C-resolution is deconstructible and the same holds for the classes of modules with bounded C-resolution dimension. Furthermore, we study the locally F-free modules; a sufficient condition on the class F is given for the class of all locally F-free modules to be closed under transfinite extensions. This enables us to show that there are many non-trivial examples of non-deconstructible classes, generalizing the recent result of D. Herbera and J. Trlifaj concerning the non-deconstructibility of the class of all flat Mittag-Leffler modules over a non-right perfect ring. Keywords: Deconstructible class, transfinite extension, filtration, locally F-free module 4

Contents Introduction........................... 7 1. Preliminaries.......................... 8 2. Deconstructibility for Filt(S)-resolved modules........... 10 3. Non-deconstructibility and locally F-free modules.......... 16 References............................ 23 5

A note about conventions Throughout the whole text, R denotes a ring (with a unit). All modules are right R-modules, unless explicitly stated otherwise. Symbols used in the text If the class S in any of the symbols below is a one-point set {N}, then the curly brackets may be omitted; e.g., Add(N) = Add({N}), N = {N} etc. Mod-R Add(S) Gen(S) Filt(S) S S S <κ S κ lim S ω the category of all R-modules the class of all direct summands in all direct sums of elements of S the class of all epimorphic images of direct sums of elements of S the class of all S-filtered modules the class of all the modules M such that Ext 1 R(S, M) = 0 for all S S the class of all the modules M such that Ext 1 R(M, S) = 0 for all S S the class of all < κ-presented modules from S the class of all κ-presented modules from S the class of all direct limits of countable direct systems of elements of S P n the class of all modules with projective dimension n i I M i the external direct sum of the system of modules (M i i I) i I M i the internal direct sum of the system of modules (M i i I) E(M) the injective envelope (hull) of M dim(r) the dimension of R R (p) the localization of R in p mspec(r) the set of all maximal ideals of R X the cardinality of X 6

Introduction A classic theorem of Kaplansky states that every projective module is a direct sum of countably generated projective modules. However, the classes of modules allowing decomposition into a direct sum of small modules are rather rare in general; for example, by another classic result of Faith and Walker, the existence of a cardinal κ, such that each injective module is a direct sum of κ-generated modules is equivalent to the ring R being right noetherian. On the other hand, there is a more general property called deconstructibility which appears to be ubiquitous. The point is in replacing direct sums by transfinite extension. Despite being a weaker condition, deconstructibility implies some other useful properties; e.g., each deconstructible class is precovering, each cotorsion pair generated by a deconstructible class is complete etc. In [1], Enochs et al. proved that for each n 0, the class P n of all modules of projective dimension n is deconstructible for each ring R. More in general, if (A, B) is a cotorsion pair generated by a set, then the class A is deconstructible [9]. The latter fact implies deconstructibility of many classes of modules studied in homological algebra. In particular, the class W = Z of all Whitehead groups is deconstructible under the additional assumption of Axiom of Constructibility, because it coincides with P 0 by [8]. However, by a surprising result in [4], the class W need not be deconstructible in other extensions of ZFC. Moreover, as recently shown in [6], non-deconstructibility occurs even in ZFC: the class of all flat Mittag-Leffler modules over each non-right perfect ring is not deconstructible. In Section 2 of this text, we extend the deconstructibility result from [1] mentioned above in a different direction: we prove that if C = Filt(S) for a set of modules S, then for each n 0, the class of all modules of C-resolution dimension n is deconstructible. Our key tool here is Hill s Lemma on transfinite extensions which enables us constructing filtrations compatible with the boundary maps. In Section 3, we present a different proof of the main result from [6] using trees on cardinals and their algebraic decoration, a technique as in [3]. This makes it possible to prove the result for a much larger class of modules, the locally F-free ones. In particular, we show that there are a number of other instances of non-deconstructibility in ZFC, e.g., for the classes of locally free modules coming from (infinite dimensional) tilting theory over Dedekind domains. 7

1. Preliminaries In this short section, we briefly recall the basic notions and propositions related to the topics discussed in the thesis. Proofs are omitted for the sake of brevity; we refer to [5] for further information. Definition 1.1. Let M be a module and µ an ordinal. A chain (an increasing sequence with respect to inclusion) H = (M α α µ) of submodules of M is continuous, if M α = β<α M α for each limit ordinal α µ. If M 0 = 0 and M µ = M, then H is a filtration of M. If C is a class of modules and H = (M α α µ) a filtration of M, then H is a C-filtration of M provided that for each α < µ, M α+1 /M α is isomorphic to an element of C. We say that M is C-filtered or a transfinite extension of elements of C then. The class of all C-filtered modules is denoted by Filt(C). Naturally, C is said to be closed under transfinite extensions if Filt(C) = C. A class C is deconstructible if there is a cardinal κ such that C = Filt(C <κ ), or, equivalently, it there is a set S such that C = Filt(S). Example 1.2. If C is the class (or a representative set) of all countably generated projective modules, then Filt(C) = P 0 by the Kaplansky Theorem [2, 26.2]. The class of all semiartinian modules is obtained as Filt(C) by taking C as the class of all simple modules. The following lemma states that the property of being a root of the functor Ext 1 R(, N) is preserved by transfinite extensions. Lemma 1.3 (Eklof Lemma, [5, 6.2]). Let N be a module. If M is a N-filtered module, then M N. Definition 1.4. Let A, B be classes of modules. A pair C = (A, B) is a cotorsion pair (or a cotorsion theory) if A = B and B = A. A cotorsion pair C is generated by a class D if C = ( (D ), D ). Notice that if S is a set of modules, then S = ( S S S), therefore any cotorsion pair generated by a set is also generated by a single module. There is an alternative description of the first component of a cotorsion pair provided it is generated by a set, cf. [5, 6.13]: Lemma 1.5. Let S be a set of modules and D the class of all the modules D such that there is a short exact sequence 0 F D S 0 with F free and S Filt(S). Then the class (S ) consists of all direct summands of modules in D. Finally, we present the Hill Lemma, which is a powerful tool when dealing with filtrations. It grants us the existence of a distributive, yet dense enough sublattice of submodules of a filtered module, which can be used for constructing new filtrations with various properties. 8

Lemma 1.6 (Hill Lemma, [5, 7.10]). Let κ be an infinite regular cardinal and C a set of < κ-presented modules. Let M be a module with a C filtration M = (M α α σ). Then there is a family F consisting of submodules of M such that (H1) M F. (H2) F is closed under arbitrary sums and intersections; it is a complete distributive sublattice of the lattice of all submodules of M. (H3) Let N, P F satisfy N P. Then the module P/N is C-filtered. Moreover, there is an ordinal τ σ and a continuous chain (F γ γ τ) of elements of F such that (F γ /N γ τ) is a C-filtration of P/N, and for each γ < τ there is a β < σ with F γ+1 /F γ isomorphic to M β+1 /M β. (H4) Let N F and X be a subset of M with X < κ. Then there is a P F, such that N X P and P/N is < κ-presented. Let us now recall the well-known characterization of the direct limits of countable direct systems. Lemma 1.7. Let (I, ) be a countable upper directed set and H = ( (F i i I), (f ij i j I) ) a direct system of modules. Then there is an increasing sequence (i k k < ω) of elements of I such that lim H = lim H, where H = ( (F ik k < ω), (f ik i l k l < ω) ). We ll now focus our attention to the tilting modules, which appear in the last section. Definition 1.8. A module T is called tilting if (T1) T has finite projective dimension, (T2) Ext i R(T, T (κ) ) = 0 for each 1 i < ω and each cardinal κ, (T3) there are r 0 and a long exact sequence 0 R T 0 T r 0 with T i Add(T ) for all i r. If n < ω, then T is called n-tilting provided it is tilting and T P n. Lemma 1.9. Let T be a 1-tilting module. Then (i) T = Gen(T ). (ii) if (A, B) is a cotorsion pair generated by T, then A B = Add(T ). 9

2. Deconstructibility for Filt(S)-resolved modules The aim of this section is to establish a result similar to [1, 4.1] with the projective modules replaced by an arbitrary deconstructible class. In fact, Theorem 2.8 provides even more general result: the resolutions need not have finite length and their elements even need not belong to the same class. Let C = Filt(S) for a set S Mod-R. Then, by [5, 7.21], C is a precovering class (recall that a class D is precovering provided that every module M has a D-precover, that is, a homomorphism f : D M with D D such that the abelian group homomorphism Hom R (D, f): Hom R (D, D) Hom R (D, M) is surjective for each D D). Therefore, for any module M, we may obtain a chain complex of modules R : f i+2 f i+1 f i f 2 f 1 f 0 Ci+1 Ci C1 C0 M by taking f 0 : C 0 M a C-precover of M and f i : C i+1 C i a C-precover of Ker f i 1 for 1 i < ω. If the complex R is exact and f 0 is surjective, then we arrive at the concept of C-resolution, which is a subject of interest of relative homological algebra. Definition 2.1. Let C be a class of modules and M a module. A long exact sequence E : C i+1 C i C 1 C 0 M 0 with C i C for each i < ω is called a C-resolution of M. Let n < ω. If C n 0 and C i = 0 for n < i < ω, then E is said to have length n. The least n < ω such that M has a C-resolution with length n is called the C resolution dimension of M. Notice that if C is a class such that all C-precovers are surjective (which happens, for example, if P 0 C), then the complex R is necessarily exact, therefore all modules possess a C-resolution. We now turn our attention to basic observations concerning filtered modules and their number of generators. Lemma 2.2. Let M be a module and N its submodule. Assume that M/N is < κ-generated. Then there is a < κ-generated module G M such that M = N + G. Proof. Denote π the canonical map M M/N. Since M/N is < κ-generated, there is a set H M such that π(h) generates M/N and H < κ. Let G = H. For each m M, there are n < ω, r i R and h i H such that π(m) = n i=1 r ih i + N, that is, m = n i=1 r ih i + n for some n N, thus M = N + G. 10

Corollary 2.3. Let M be a module, N its submodule and κ an infinite cardinal. Assume that M is a union of a continuous chain (M α α < λ) with λ < cof(κ), M 0 = N and M α+1 /M α is < κ-generated for each 1 α < λ. Then M/N is < κ-generated. Proof. Assume first that N = 0. By Lemma 2.2, there are < κ-generated modules G α such that M α+1 = M α + G α for each α < κ. Therefore, M = α<λ G α. If H α is a generating set of G α with H α < κ, then α<λ H α is clearly a generating set of M, which has cardinality < κ, as it is a union of fewer then cof(κ) sets of cardinality < κ. If N 0, then M/N is a union of the continuous chain (M α /N α < λ) with the first element being trivial. Since (M α+1 /N)/(M α /N) = M α+1 /M α, the consecutive factors are < κ-generated and claim follows from the previous case. Definition 2.4. Let R be a ring and κ a cardinal. Then R is called right κ noetherian, provided that each right ideal I of R is κ-generated. The least infinite cardinal κ such that R is right κ-noetherian is the right dimension of R, denoted by dim(r). For example, if the ring R is noetherian, then dim(r) = ℵ 0. Lemma 2.5. Let κ be a cardinal such that κ dim(r). Then any submodule of a κ-generated module is κ-generated. Proof. Firstly observe that all submodules of cyclic modules are κ-generated, since every cyclic submodule is an epimorphic image of R, thus its every submodule is an epimorphic image of a right ideal. Further, if M is κ-generated module, then M is a union of a continuous chain (M α α < κ), where M α = β<α m βr, {m β β < κ} being a generating set of M. The third isomorphism theorem gives M α+1 /M α = (M α + m α+1 R)/M α = mα+1 R/(M α m α+1 R), thus the consecutive factors are cyclic. If N is a submodule of M, then for each α < κ, (N M α+1 )/(N M α ) embeds into M α+1 /M α via the monomorphism x + (N M α ) x + M α, therefore the continuous chain (N M α α < κ), the union of which is N, has consecutive factors κ-generated, and Lemma 2.3 applies. Corollary 2.6. Let κ be a cardinal such that κ dim(r). Then every κ generated module is κ-presented. Finally, we prove a homological-algebraic lemma, which enables us to build filtrations of whole resolutions, since the factor complexes of consecutive C resolutions will be C-resolutions as well. Lemma 2.7. Let R be an exact chain complex of modules f n 2 Rn 1 f n 1 Rn f n f n+1 Rn+1 11

with an exact subcomplex R of the form f n 2 R n 1 f n 1 R n f n R n+1 f n+1, that is, R n R n and f n = f n R n for each n Z. Then the factor complex, R = R/R, given by f n 2 Rn 1 /R n 1 f n 1 Rn /R n f n Rn+1 /R n+1 f n+1 with f n : R n R n+1 defined by x + R n f n (x) + R n+1, is exact, too. Proof. Firstly, observe that the maps f n are well-defined this is since f n (R n) R n+1 for each n Z. Let K n = Im f n 1 = Ker f n. Pick an n Z; the canonical decomposition of R into short exact sequences gives rise to a diagram 0 0 0 0 0 0 K n /K n R n 0 R n 1 /R n 1 R n /R n R n+1 /R n+1 0 0 K n K n+1 /K n+1 R n+1 0 0 R n 1 R n R n+1 0 0 K n R n K n+1 0 0 R n 1 R n R n+1 0 K n+1 R n+1 0 0 0 0 0 with all the maps defined using cosets in a natural way. It is routine to verify that the maps are well-defined and the diagram is commutative. Note that the bottom oblique row is exact, since Ker(f i R i) = Ker f i R i for i = n, n + 1. Thus, using The Nine Lemma on the middle three columns (which are clearly exact) and the oblique rows, we infer that the top oblique row is exact as well. The proof of the lemma is finished once we establish the exactness of the sequences R n 1 /R n 1 K n /K n R n 0 and 0 K n+1 /K n+1 R n+1 R n+1 /R n+1. The former holds because K n = Im f n 1, therefore every element of 0 12

K n /K n R n has the form f n 1 (x) + K n R n for some x R n 1, and x + R n 1 is the desired preimage. To verify the latter, observe that the kernel of the map x + K n+1 R n+1 x + R n+1 is {x + K n+1 R n+1 x R n+1} which has zero intersection with K n+1 /K n+1 R n+1. We are now ready to prove the main result. Theorem 2.8. Let κ be a cardinal such that κ dim(r), M a module and S 1, S 2,... sets of κ-presented modules. Assume that there is a long exact sequence R : f n+1 f n f 2 f 1 f 0 Dn D1 D0 M 0, with D i Filt(S i ) for i < ω. Then there is a filtration (M α α λ) of M such that for every α < λ, there is a long exact sequence R α : f α,n+1 Dα,n fα,n fα,2 Dα,1 fα,1 Dα,0 fα,0 Mα+1 /M α 0 with D α,n Filt(S n ) κ and M α+1 /M α κ-presented. Proof. Since κ dim(r), Corollary 2.6 ensures that we need not distingush between the notions of κ-presented and κ-generated. Let λ = κ + ϱ, where ϱ is the minimal number of generators of M, and let {m α α < λ} be a generating set of M. We will inductively construct a continuous chain of long exact sequences (R α α λ) of the form R α : fn+1 D α,n+1 f n D α,n f 2 D α,2 f 1 D α,1 f 0 D α,0 Dα,n Dα,1 Dα,0 Mα 0 with D α,n Filt(S n ) and R α+1 /R α = R α. Denote by F i the family of submodules of D i obtained from an S i -filtration of D i using the Hill Lemma 1.6; we shall pick the elements of chains (D α,i α λ) from these families. Put M 0 = 0 and D 0,i = 0 for every i < ω as well. Assume that M α and R α are already constructed and M α M. Let γ < λ be the least index such that m γ / M α (this ensures that M = α<λ M α). We will now construct modules D i,j and D i,j, i j < ω (for simplicity, we allow i > j if j = 0) in such way that (1) D i,j, D i,j F i, (2) D i,0 = D i,0 = D α,i, (3) D i,j D i,j D i,j+1 D i,j+1, (4) D i,t/d i,j ( denotes either or ) is κ-presented, whenever the necessary inclusion holds (i.e., t > j or t = j if the arrows are oriented / ), (5) f i+1 (D i+1,j) Ker(f i D i,j), (6) f i+1 (D i+1,j) Ker(f i D i,j), (7) f(d 0,1) m γ. Notice that to satisfy the property (4), it suffices to ensure that the consecutive factors (in view of the property (3)) are κ-presented. Property (7) ensures that M λ = M. 13

To begin the construction, property (H4) from the Hill Lemma gives the module D 0,1 D 0,0 {d} with D 0,1/D 0,0 κ-presented, where d D 0 satisfies f 0 (d) = m γ. Now, since Ker(f 0 D 0,0) = Ker(f 0 D 0,1) D 0,0, the third isomorphism theorem gives Ker(f 0 D 0,1)/ Ker(f 0 D 0,0) = (Ker(f 0 D 0,1) + D 0,0)/D 0,0 D 0,1/D 0,0. As the last module is κ-generated, the middle one (and therefore the first one) is κ-generated as well by Lemma 2.5. Let G D0,1 be a set satisfying G κ and Ker(f 0 D0,0) + G = Ker(f 0 D0,1). Then, because of exactness of R α at D α,0, there is a set H D 1 such that H κ and f 1 (H) = G. Hill s property (H4) yields the module D1,1 D1,0 H with D1,1/D 1,0 being κ-presented. Clearly f 1 (D1,1) Ker(f 0 D0,1). The procedure further proceeds in two alternating steps, the step and the step. The -step. Assume that we have already constructed all required modules with the second index j. Let D 0,j+1 = D 0,j. In the same fashion as above, we construct a module D 1,j+1 F 1 satisfying f 1 (D 1,j+1) Ker(f 0 D 0,j+1) and having D 1,j+1/D 1,j κ-presented using the fact that D 0,j+1/D 0,j is κ- presented and f 1 (D 1,j) f 1 (D 1,j) Ker(f 0 D 0,j) from induction hypothesis. Next, we obtain the module D 2,j+1 with analogous properties, then D 3,j+1 etc. Finally, as D j+1,j is not defined for j > 0, D j+1,j+1 has to be constructed in the same fashion, but directly from D j+1,0. This step ensures that (5) holds. The -step. Assume that we have already constructed all required modules with the second index < j and the -modules with the second index = j. Let Dj,j = Dj,j. There is a set G D j such that Dj,0 + G = Dj,j and G κ. Hill s property (H4) gives the module Dj 1,j Dj 1,j f(g) with Dj 1,j/D j 1,j κ-presented. As the inclusion f(dj,0) Dj 1,j holds clearly, this implies f j (Dj,j) Dj 1,j. Similarly, we construct Dj 2,j satisfying f j 1 (Dj 1,j) Dj 2,j, then Dj 3,j etc., up to D0,j. This step ensures the property (6). Let D α+1,i = i j<ω D i,j and M α+1 = f 0 (D α+1,0 ). Since D α+1,i is a union of countable chain with consecutive factors κ-presented, D α+1,i /D α,i is κ- presented as well by Corollary 2.3, and D α+1,i F i by (H2). We claim that the modules D α+1,i, M α+1 together with the restrictions of homomorphisms f n form an exact sequence R α+1. Firstly, since f i (Di,j) Di 1,j (property (6)), inclusion f i (D α+1,i ) D α+1,i 1 holds, and, because of the exactness of R, f i (D α+1,i ) Ker(f i 1 D α+1,i 1 ). Moreover, properties (3) and (5) together give Ker(f i 1 D i 1,j 1) Ker(f i 1 D i 1,j) f i (D i,j) f i (D i,j), 14

which yields the inclusion f i (D α+1,n ) Ker(f i 1 D α+1,i 1 ) and, therefore, the exactness of R α+1. To show that M α+1 /M α is κ-generated, consider the following diagram with both rows exact: 0 D α,0 D α+1,0 D α+1,0 /D α,0 0 f 0 D α,0 f 0 D α+1,0 g 0 M α M α+1 M α+1 /M α 0 π The map g : D α+1,0 /D α,0 M α+1 /M α is defined by formula x + D α,0 f 0 (x) + M α ; it is well-defined, because x y D α,0 implies f 0 (x y) M α, which means f 0 (x)+m α = f 0 (y)+m α. The diagram is easily verified to be commutative. Since both f 0 D α+1,0 and π are epimorphisms, g must be epic as well. The module M α+1 /M α is thus a homomorphic image of a κ-generated module D α+1,0 /D α,0, hence it is κ-generated itself. For limit ordinal α λ, put D α,i = β<α D β,i and M α = β<α M β and let the morphisms in R α be the corresponding restrictions. Such a construction clearly yields R α exact, and by (H2), we infer that D α,i F i for i < ω. Since all the complexes R α are exact, the factor complexes R α = R α+1 /R α are exact by Lemma 2.7, therefore the resolutions R α have all the desired properties. Notice that in the proof of Theorem 2.8, the sequence R λ = β<λ R β does not have to be equal to R. Corollary 2.9. Let C be a deconstructible class of modules. (i) The class of all modules possessing any C-resolution is deconstructible. (ii) The class of all modules with C-resolution dimension n is deconstructible for each n < ω; in particular, the classes P n are deconstructible. Proof. Let S Mod-R be a set such that C = Filt(S). The case (i) is obtained by taking S i = S in the Theorem 2.8, the case (ii) by taking S i = S for i n and S i = {0} for n < i < ω. As the class P 0 is deconstructible by the theorem of Kaplansky, the statement for P n is just a special case of (ii). 15

3. Non-deconstructibility and locally F-free modules In contrast with the preceding part, the aim of this section is to establish the existence of subclasses of Mod-R, which are not deconstructible, i.e., not of the form Filt(T) for a set T Mod-R. Our attention is focused on the classes of so called locally F-free modules, which seem to provide various examples of such behavior. This setting covers the case of flat Mittag-Leffler modules, the class of which has been recently proved not to be deconstructible unless R is a right perfect ring, cf. [6]. Throughout this section, F is a subclass of Mod-R and C the class of modules isomorphic to a countable direct sum of elements of F. Definition 3.1. Let M be a module. M is called F-free if it is isomorphic to a direct sum of elements of F. M is called locally F-free if there is a system S of submodules of M satisfying the properties (S1) S C, (S2) for every countable subset C M, there is an S S such that C S, (S3) 0 S and S is closed under unions of countable chains. The set S is said to witness the local F-freeness of M. Clearly, each F-free module is locally F-free, and the converse hold for all countably generated modules. We will denote by L the class of all locally F-free modules and by lim F the ω class of all countable direct limits of the modules from F. Example 3.2. (i) If F is the class of all countably generated projective modules, then F-free = projective, lim F is the class of all countably presented flat modules, and ω L is the class of all flat Mittag-Leffler (= ℵ 1 -projective) modules, see [6]. (ii) If F is the class of all countably generated pure-projective modules, then L is the class of all Mittag-Leffler modules, cf. [5, 3.14]. The following lemma provides sufficient conditions for the class L to be closed under transfinite extensions. It is a vital tool for proving the non-deconstructibility of L in particular cases, which will become clear in the light of Theorem 3.8. Lemma 3.3. Assume that Ext 1 R(F, G) = 0 whenever F F and G C and F consists of countably presented modules. Then the class L is closed under transfinite extensions. Proof. Let M be a module possessing an L-filtration (M α α λ). By induction on α, we will construct sets S α witnessing local F-freeness of M α in such a way that S γ S δ for γ δ λ and the following condition ( ) is satisfied: Whenever γ δ λ and S S δ, then S M γ S γ ; moreover, S = (S M γ ) P, where P C. ( ) 16

The successor case. Assume that we have already constructed S α. By assumption, M α+1 /M α L; denote S α the set witnessing local F-freeness of M α+1 /M α. Put S α+1 = { S M α+1 S M α S α & π(s) S α }, where π denotes the projection M α+1 M α+1 /M α. The inclusion S α S α+1 is clear, and, by induction hypothesis, S β S α+1 for β α + 1. We shall verify that S α+1 defined in this way satisfies conditions ( ) and (S1) (S3). By assumption, Ext 1 R(π(S), S M α ) = 0 for all S S α+1, therefore S = (S M α ) π(s) C, which shows that S α+1 satisfies the condition (S1). The previous paragraph also proves ( ) for case γ = α, δ = α + 1 by taking P = π(s) such that S = (S M α ) P. If γ α, then S M γ = (S M α ) M γ. Since M α satisfies ( ) and S M α S α, we conclude that S M γ S γ. Moreover, there is a Q C such that S M α = (S M γ ) Q, so S = (S M γ ) Q P, which completes the proof of ( ). In order to prove the condition (S2), consider a countable subset C of M α+1. Then there exists S S α such that π( C ) S. As S is a countable direct sum of countably presented modules, it is countable presented, therefore there is a countably generated module T M α+1 such that π(t ) = S and C T. Moreover, T M α = Ker(π T ) is countably generated as well, so there is an S S α satisfying T M α S. The exact sequence 0 S T +S S 0 splits by assumption, therefore T + S S α+1. Since C T + S, we have established (S2). Finally, consider a chain S 0 S 1 S i S i+1 of elements of S α+1 ; let S = i<ω S i. Since S i M α S α for all i < ω and S M α = i<ω (S i M α ), we conclude that S M α S α. Similarly, π(s) = i<ω π(s i) S α as π(s i ) S α for all i < ω, thus S S α+1 and condition (S3) is satisfied. The limit case. Let α λ be a limit ordinal and assume that the systems S β for β < α are already constructed. Put S α = { i<ω S i S i S αi & S i S i+1 for all i < ω and some α i < α }. Again, the inclusions S β S α for β α are clear. To establish (S1), consider S = i<ω S i S α. We may clearly assume that the sequence (α i i < ω) is increasing. Let S i = j<ω (S j M αi ) S αi ; the condition ( ) ensures that S j M αi S αi for all i, j < ω and by (S3), S i S αi as well. It is easy to see that S = i<ω S i and S i M αj = S j for all j i < ω. Therefore, by ( ), we obtain P i C such that S i+1 = S i P i for all i < ω, which implies S i = S 0 j<i P j. We conclude that S = S 0 i<ω P i, therefore S C. Let C = {c i i < ω} be a countable subset of M α and α i < α ordinals such that c i M αi. Again, we may assume that the sequence (α i i < ω) is increasing. We ll inductively construct a chain of modules S 0 S 1 S i S i+1 such that {c 1, c 2..., c i } S i and S i S αi for all i < ω. Let S 0 S α0 be such that c 0 S 0. Assuming we have already constructed M 0, M 1,..., M i, let S i+1 S αi+1 be such that D i {c i+1 } S i+1, where D i M αi ( M αi+1 ) is a countable set such that D i = S i. Clearly S i S i+1 and C i<ω S i S α, which proves the condition (S2). To see that ( ) holds, consider γ < α and S = i<ω S i S α, where (α i i < ω) is an increasing sequence. We may w.l.o.g. assume that γ α 0 and S i = S i, 17

using the notation above. Then S 0 = (S 0 M γ ) Q for some Q C and by the arguments above, S = S 0 P i = (S 0 M γ ) Q i<ω and Q i<ω P i C. For verification of (S3), let S 0 S 1 S i S i+1 be a chain of modules from S α, where S i = j<ω S ij, S ij S αij, (α ij j < ω) being an increasing sequence of ordinals < α for every i < ω. Put β = sup i,j<ω α ij. If β < α, then S ij S β for all i, j ω, whence S i S β for all i < ω, therefore i<ω S i S β S α. If β = α, then clearly cof α = ω; let (β i i < ω) be an increasing sequence of ordinals < α with sup i<ω β i = α. Then, by ( ), S i M βi S βi, therefore i<ω S i = i<ω (S i M βi ) S α. The main idea used in proving non-deconstructibility of locally F-free modules is that the property of being a root of Ext 1 R(, M) for a module M holds for all countable direct limits of elements of F, provided it holds for locally F-free modules. This result may be obtained by constructing a module X, the structure of which is closely related to the one of an infinite tree. This method is sometimes called decorating a tree, with the branch-submodules being called patterns. Definition 3.4. Let κ be an infinite cardinal. A tree on κ is a set T κ = {τ : n κ n < ω} partially ordered by inclusion. A length of τ T κ is defined as l(τ) = dom(τ). A branch of a tree is a maximal linearly ordered subset. The set of all branches will be denoted Br. Clearly, each ν Br(T κ ) can be identified with a (countable) sequence of ordinals < κ, so Br(T κ ) = {ν : ω κ}. Also note that T κ = κ and Br(T κ ) = κ ω. Lemma 3.5. Let N be the direct limit of a chain system g 0 g 1 g 2 g i g i+1 H : F 0 F1 F2 Fi+1, with F i F and g i Hom(F i, F i+1 ) for i < ω. Let κ be an infinite cardinal and denote D = τ T κ F τ, P = τ T κ F τ, where F τ = F l(τ). Then there is a locally F-free module X such that D X P and N (Br(Tκ)) = X/D. Proof. We begin the construction with defining particular branch-submodules of X. For ν Br(T κ ), i < ω and x F i define x νi P by x if τ = ν i, π τ (x νi ) = g j 1... g i (x) if τ = ν j, i < j < ω, 0 otherwise, i<ω P i 18

and put Y νi = {x νi x F i }. Y νi is easily seen to be a submodule of P. Furthermore, Y νi = Fi, where the isomorphism is given by the formula x νi x. The system of modules {X νi i < ω} is readily seen to be independent. Thus, we may define X ν = i<ω Y νi. Observe that X ν = i<ω F i. On the other hand, X ν F ν i for every i < ω, since for every x F i, x νi (g i (x)) ν,i+1 X ν is precisely the sequence with x at ν i and zeros everywhere else. This implies that X ν = ( i<j F ) ( ν i j i<ω Y νi) for every j < ω. Finally, put X = ν Br(T X κ) ν. It is clear that D X. Let X ν = (X ν + D)/D. To prove that N (Br(Tκ)) = X/D, we will first show that N = X ν. For each i < ω, define f i : F i X ν by x x νi + D. Firstly, for every i < j < ω, f i = f j g j 1... g i, since the elements x νi and ( g j 1... g i (x) ) νj differ only at finitely many indices namely ν i, ν i + 1,..., ν j 1 for every x F i. If f i (x) = 0 for some i < ω and x F i, then the sequence x νi is zero almost everywhere, whence g j (x) = 0 for some j i. Finally, i<ω Im f i = X ν, therefore ( Xν, (f i i < ω) ) is a direct limit of the chain (F); in other words, N = X ν. Since every element of X ν has non-zero components only at the indexes {ν i i < ω} and intersection of any two such sets for different νs is finite, we infer that the system of modules { X ν ν Br(T κ )} is independent. Therefore, X/D = ν Br(T X κ) ν = N (Br(T κ)). To see that X is locally F-free, define a system of submodules of X S = { X S S Br(T κ ) & S ω }, where X S = ν S X ν. We shall prove that S witnesses local F-freeness of X. In order to verify (S1), consider a countable set S = {ν i i < ω} Br(T κ ). Then X S = i<ω X S i, where S i = {ν 0, ν 1,..., ν i }. We will show that, for every i < ω, X Si+1 = X Si k j<ω Y ν i+1,j, where k = min { n < ω (ν i+1 n) / {ν m m < ω & ν S i } }, that is, k is the level of T κ, where the branch ν i+1 leaves the subtree consisting of branches from S i (such k exists, since S i is finite). This holds, since X νi+1 = ( j<k F ) ( ν i+1 j k j<ω Y ) ν i+1,j and XSi X νi+1 = j<k F ν i+1 j. Therefore, X Si+1 = X Si k j<ω Y ν i+1,j, where the second summand is isomorphic to k j<ω F j, which is an element of C. Therefore, X S is a countable direct sum of elements of C, whence an element of C itself. If C = {x i i < ω} is a countable subset of X, then for every i < ω there is n(i) < ω and x 0 i, x 1 i,..., x n(i) i such that x i = x 0 i + x 1 i + + x n(i) i and x j i X ν i,j for some ν i,j Br(T κ ). Then C X S, where S = {ν i,j i, j < ω}, which proves the condition (S2). Finally, if X S0 X S1... X Si X Si+1 is a chain of elements of S, then i<ω X S i = i<ω X S i = ) i<ω ( ν S i X ν = ν i<ω S X i ν S. The trees of main interest will be those satisfying Br(T κ ) = 2 κ. The following purely set-theoretic lemma grants their existence. Lemma 3.6. For every cardinal λ, there is a cardinal κ such that κ λ and κ ω = 2 κ. 19

Proof. We ll define a sequence of sets (S i i < ω) in the following fashion: S 0 = λ, S i+1 = P(S i ) for all i < ω, and put S = i<ω S i, κ = S. Let T be a set of all (countable) sequences of elements of S clearly T = κ ω. To show that κ ω 2 κ, consider a function f : P(S) T defined by formula Z (Z S i i < ω) for every Z P(S). Notice that since Z S i S i, Z S i P(S i ) = S i+1 S, the sequence f(z) is indeed an element of T. If z Z 1 \ Z 2 for Z 1, Z 2 P(S), then z Z 1 S j, but z / Z 2 S j, where j < ω is such that z S j, which means that the sequences f(z 1 ), f(z 2 ) have different j-th term, thus f(z 1 ) f(z 2 ), which shows that f is one-to-one. The reverse inequality, κ ω 2 κ, is true for every infinite cardinal κ, since 2 κ = κ κ κ ω. Lemma 3.7 (Hunter s counting trick). Let M be a module such that L M. Then lim ω F M. Proof. Lemma 3.6 ensures the existence of an infinite cardinal κ such that 2 κ = κ ω, M κ and F i κ for each i < ω. Take N lim F; by Lemma 1.7, we ω may w.l.o.g. assume that N is a direct limit of a chain H as in Lemma 3.5. The same lemma then gives a locally F-free module X and an exact sequence 0 D X N (Br(Tκ)) 0. By assumption, Ext 1 R(X, M) = 0, therefore applying the functor Hom R (, M) yields an exact sequence 0 Hom R (N (Br(Tκ)), M) Hom R (X, M) Hom R (D, M) Ext 1 R(N (Br(Tκ)), M) 0. Assume that Ext 1 R(N, M) 0. Then, since Br(T κ ) = κ ω = 2 κ, we have Ext 1 R(N (Br(Tκ)), M) = Ext 1 R (N, M) 2κ 2 2κ, whereas Hom R (D, M) M Fτ (κ κ ) κ = 2 κ < 2 2κ, τ T κ which is a contradiction with the surjectivity of the connecting homomorphism Hom R (D, M) Ext 1 R(N (Br(Tκ)), M). In the following, D will denote the class of all direct summands of the modules M that fit into an exact sequence 0 P M C 0 where P is a free module and C C. Lemma 3.8. Assume that L is closed under transfinite extensions, and there exists a countably generated module C (lim F) \ D. Then L is not deconstructible. ω Proof. Assume there is a cardinal κ such that L = Filt L <κ. Consider the cotorsion pair generated by L <κ, that is, ( ((L <κ ) ), L κ ). By the Eklof Lemma 1.3, (L <κ ) = L, so C ((L <κ ) ) by Lemma 3.7. By Lemma 1.5, C is isomorphic to a direct summand in a module E of the form 0 P E L 0 where P is a free module and L L. 20

Denote π the map E L. Since C is countably generated, π(c) is countably generated as well, and therefore contained in some submodule D C, as L is locally F-free. As C is a direct summand in E, it is a direct summand in π 1 (D), which fits into the exact sequence 0 P π 1 (D) D 0. We infer that C D, a contradiction. Theorem 3.9. Let T be a tilting module which is a direct sum of countably presented modules, T = i I T i. Let F be a representative set of {T i i I}. Assume there exists a countably generated module C (lim F) \ D. Then L is ω not deconstructible. Proof. Since T is tilting, Ext 1 R(F, G) = 0 for all F F and G C. The claim now follows from Lemmas 3.3 and 3.8. The module described in the definition below seem to be a good candidate for the role of the module C in the hypotheses of Theorem 3.9. Definition 3.10. Let R be a non-right perfect ring and (a i i < ω) a sequence of non-zero elements of R such that Ra 0 Ra 1 a 0 Ra n... a 0 Ra n+1... a 0. The Bass module (with coefficients a i ) is the direct limit of a chain R a 0 R a 1 a n 1 R an R a n+1 ( a i stands for the R R homomorphism x xa i ). Let C be a Bass module. As C has a free presentation 0 R (ω) ϱ R (ω) C 0, where ϱ is defined by 1 i 1 i 1 i+1 a i, 1 i being the i-th element of the canonical basis of R (ω) for i < ω, it is countably presented. Furthermore, its projective dimension is always 1, cf. [2, 28.2]. Corollary 3.11. (i) Let R be a non-right perfect ring. Assume that F consists of countably generated projective modules and R F. Then the class L is not deconstructible. In particular, the class of all locally free modules and the class of all flat Mittag-Leffler modules are not deconstructible. (ii) Let R be a Dedekind domain with the quotient field Q R, and P be a nonempty set of maximal ideals in R such that mspec(r)\p is a countable set. Let F = { p P R (p)} {E(R/q) q mspec(r) \ P }. Then the class L is not deconstructible. Proof. (i) If F consists of countably generated projective modules, then C does as well. Moreover, for projective modules the notions of countably generated and countably presented coincide, and clearly Ext 1 R(F, G) = 0 for each F F and G C. Finally, note that D = C in this case. If N is any Bass module, then N may not be a direct summand in an element of D, as that would imply N 21

being projective. We infer that the N (lim ω F) \ D, therefore the class L is not deconstructible. The choice F = {R} corresponds with the case of locally free modules. The case of flat Mittag-Leffler modules is obtained if F is the class of all countably generated projectives, cf. [6]. (ii) As in the case (i), the countably presented modules coincide with the countably generated ones, since R is noetherian. Let R P = p P R (p). Then R P is a subring of Q containing R, so R P is a Dedekind domain by [7, 12.1 and 12.5]. Take p P. Since every nonzero ideal factors uniquely as a product of prime ideals in Dedekind domains, p 2 p; let x be a (non-zero) element of p\p 2. Denote by C the Bass R P module corresponding to the choice of a i = x for all i < ω. Then C is isomorphic to an R P -submodule of Q, and C lim F as an R-module. ω Let T = R P q mspec(r)\p E(R/q). Then T is a 1-tilting module (cf. [5, 14.30]), and by Lemma 1.5, T generates a cotorsion pair (A, B) such that D A. All the modules E(R/q), q mspec(r)\p are countably generated; moreover, from the exact sequence 0 R R P q mspec(r)\p E(R/q) 0 and the assumption on P, we infer that R P is countably generated as well. The claim will thus follow from Theorem 3.9 once we prove that C / D. Assume C D. As C Gen(T ), C B by Lemma 1.9. Therefore C A B = Add(T ), again by Lemma 1.9. Assume that C is a direct summand in M N, where M, N are modules such that M = R (µ) P, N = q mspec(r)\p E(R/q)(νq) with µ, ν q cardinals for each q mspec(r) \ P (in fact, we may assume µ, ν q ω, as C is countably generated). Let π M be the canonical projection M N M. Since C is a torsionfree module, the restriction π M C is injective, therefore C is isomorphic to a submodule of M. However, Cx = C, while M contains no non-zero submodule U such that Ux = U, because n<ω xn R P = 0 by the Krull Intersection Theorem. Remark. In the case (ii) of Corollary 3.11, a tilting module is obtained also for P =, namely T = Q Q/R. However, if F = {Q} {E(R/q) q mspec(r)}, then L is the class of all divisible (= injective) modules, which is deconstructible by the Faith-Walker Theorem, since R is noetherian. 22

References [1] S. T. Aldrich, E. Enochs, O. Jenda, L. Oyonarte, Envelopes and covers by modules of finite injective and projective dimensions, J. Algebra 242 (2001), 447 459. [2] F. W. Anderson, K. R. Fuller, Rings and Categories of Modules, 2nd ed., Graduate Texts in Mathematics, Springer-Verlag, New York 1992. [3] S. Bazzoni, J. Šťovíček, Flat Mittag-Leffler modules over countable rings, Proc. Amer. Math. Soc. 140 (2012), 1527 1533 [4] P. C. Eklof, S. Shelah, On the existence of precovers, Illinois J. Math. 47 (2003), 173 188. [5] R. Göbel, J. Trlifaj, Approximations and Endomorphism Algebras of Modules, 2nd revised and expanded edition, Vol. 1 Approximations, GEM 41, W. de Gruyter, Berlin 2012. [6] D. Herbera, J. Trlifaj, Almost free modules and Mittag-Leffler conditions, Advances in Math. 229 (2012), 3436 3467. [7] H. Matsumura, Commutative Ring Theory, CSAM 8, Cambridge Univ. Press, Cambridge 1989. [8] S. Shelah, A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals, Israel J. Math. 21 (1975), 319 349. [9] J. Šťovíček, J. Trlifaj, Generalized Hill Lemma, Kaplansky theorem for cotorsion pairs and some applications, Rocky Mountain J. Math. 39 (2009), 305 324. 23