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Cotilting and a Hierarchy of Almost Cotorsion Groups
Journal of Algebra 224, 110᎐122 Ž2000. doi:10.1006rjabr.1999.8103, available online at http:rrwww.idealibrary.com on
Cotilting and a Hierarchy of Almost Cotorsion Groups1 Rudiger Gobel ¨ ¨ Fachbereich 6, Mathematik und Informatik, Uni¨ ersitat ¨ Essen, 45117 Essen, Deutschland E-mail: [email protected]
and Jan Trlifaj ˇ ´ Republika Katedra algebry MFF UK, Sokolo¨ ska ´ 83, 186 75 Praha 8, Ceska E-mail: [email protected] Communicated by Kent R. Fuller Received July 6, 1998
We describe the structure of all cotilting and partial cotilting groups, and of all cotilting torsion-free classes of groups. Under V s L, we also describe all tilting and partial tilting groups. As a by-product, we introduce new classes of groupsᎏcalled almost cotorsion groups. Unlike the well-known cotorsion groups, the almost cotorsion ones have a rather complex structure: we prove that any cotorsion free ring can be realized as an endomorphism ring of an almost cotorsion group. 䊚 2000 Academic Press
INTRODUCTION Starting with the paper of Miyashita w20x several key ideas of the classical tilting and cotilting theory of finite dimensional algebras have been extended to the setting of arbitrary associative rings. In order to obtain the results in full generality it is often necessary to work in the category Mod-R of all Žright R-. modules rather than mod-R, 1
Research supported by the German-Israeli Foundation for Scientific Research & Development Žproject no. G-0294-081.06r93. and by the Grant Agency of Czech Republic Žgrant no. 201r97r1162.. 110 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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the subcategory of all finitely generated modules. For example, the Bongartz idea of complementing a partial tilting module in a tilting one does not generalize in mod-R, but it does in Mod-R. More specifically, M g Mod-R is called a splitter if Ext R Ž M, M . s 0, w22x, w15x. Further, M g Mod-R is ‘‘ partial tilting’’ if M is a splitter such that proj.dim. M F 1. A ‘‘partial tilting’’ module M g mod-R is tilting provided that there is an exact sequence 0 ª R ª M⬘ ª M⬙ ª 0 where M⬘ and M⬙ are summands of a finite direct sum of copies of M. The classical Bongartz Lemma w3, 2.1x says that for any finite dimensional algebra R and any ‘‘partial tilting’’ module P g mod-R there is a module C g mod-R Žthe Bongartz complement of P, w17x. such that P [ C is a tilting module Žand Ž P [ C . H lmod-R s P H lmod-R, where M H s Ker Ext R Ž M, y. for M g Mod-R.. We call a module M g Mod-R partial tilting if GenŽ M . : M H and M H is a torsion class. Here, a class T : Mod-R is a torsion class in Mod-R if T is closed under arbitrary direct sums, homomorphic images and extensions, and GenŽ M . denotes the class of all modules generated by M, that is, of all homomorphic images of arbitrary direct sums of copies of M. Further, a module M g Mod-R is called tilting if GenŽ M . s M H . For finitely generated modules, the latter definition coincides with the definition of tilting given above, but partial tilting is a stronger notion than ‘‘partial tilting,’’ cf. w9, Sect. 1x. Even with our Žstronger. notion of partial tilting, the Bongartz Lemma does not generalize within mod-R: there exist finitely generated partial tilting modules over certain hereditary noetherian rings which are not summands of finitely generated tilting modules w9, Example 1.6x. Nevertheless, a generalization to Mod-R, for an arbitrary ring R, is available. By w9, Theorem 2.11x, for any partial tilting module P g Mod-R there is a module C g Mod-R such that P [ C is a tilting module and Ž P [ C . H s PH . This result has immediate consequences for the structure of tilting torsion classes of modules Žthat is, of the torsion classes T : Mod-R such that T s GenŽT . for a tilting module T, w2x.. By w9, Theorem 2.3x, a torsion class T is tilting iff T s P H for a splitter P g Mod-R. In the finite dimensional algebra case, the vector space duality provides dual results to the ones mentioned above. Surprisingly, the corresponding dual results hold true for arbitrary rings: A module M g Mod-R is called partial cotilting if CogŽ M . :H M and H M is a torsion-free class. Here, a class F : Mod-R is a torsion-free class in Mod-R if F is closed under arbitrary direct products, submodules and extensions. Further, H M s Ker Ext R Žy, M ., and CogŽ M . denotes the class of all modules cogenerated by M, that is, of all submodules of arbitrary direct products of copies of M. A module M g Mod-R is called
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cotilting if CogŽ M . sH M. A torsion-free class C : Mod-R is a cotilting torsion-free class provided that C s CogŽ C . for a cotilting module C, w7x, w8x. The dual version of the Bongartz Lemma reads as follows: for any partial cotilting module P g Mod-R there is a module D g Mod-R such that P [ D is a cotilting module and H Ž P [ D . sH P, w8, Theorem 2.11x. Similarly, a torsion-free class F is cotilting iff F sH P for a splitter P g Mod-R, w9, Corollary 2.12x. For any ring R, we have the following inclusions of subclasses in Mod-R: splitters = partial tilting modules = tilting modules = progenerators splitters = partial cotilting modules = cotilting modules
Ž 1.
= injective cogenerators. It is an interesting problem to describe these classes for particular rings R. In the present paper, we concentrate on the case when R s ⺪, the ring of all integers. In this case, splitters have recently been investigated in w15x and w16x. It is easy to see that the torsion part of any splitter A g Mod-Z is divisible, hence a direct summand in A. So we may restrict to the class R of all reduced torsion-free groups. There are two trivial examples of splitters: Ža. the torsion-free cotorsion groups Ža group A is cotorsion provided that ExtŽ⺡, A. s 0, where ⺡ is the additive group of all rationals., and Žb. the nuc-free groups Ža group A is nuc-free provided that A is free as an R-module, where R s nucŽ A. denotes the nucleus of A, so R s arb < a, b g ⺪, bA s A4 is the largest subring R of ⺡ such that A is an R-module.. In w15, Sect. 4x, examples of non-trivial splitters were constructed in R. These splitters are not 1-free, but their ranks and endomorphism rings can be prescribed w15, Example 5.5x. This indicates that no classification of non- 1-free splitters is possible. Moreover, though all 1-free splitters of cardinality 1 are nuc-free by w16, Main Theorem 1.5x, there exist nontrivial 1-free splitters of cardinality 2 1 w12, Corollary 8x. In Section 2 of this paper, we completely classify the Žpartial. cotilting groups, and the cotilting torsion-free classes of groups ŽTheorem 2.1, Corollary 2.2.. We also describe all Žpartial. tilting groups under V s L ŽTheorem 2.3.. As a by-product of our investigation, we discover classes of groups that exhibit similar homological properties to cotorsion groups, but have a rather complex structure. We call these groups almost cotorsion
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ŽDefinition 1.1.. In Theorem 1.7, we provide an explanation for the complexity: any cotorsion free ring appears as an endomorphism ring of an almost cotorsion group. Hence, unlike the cotorsion groups, the almost cotorsion ones do not appear to be classifiable. ˆ the pure injective hull of A, and by EŽ A. Let A g Mod-⺪. Denote by A the injective hull of A. Note that the notions of a cotorsion group and of a pure injective group coincide for torsion-free groups w14, Sect. 54x. A group A is cotorsion free provided that A has no non-zero cotorsion subgroups. A is -cotorsion free provided that imŽ ° M . is of finite rank for every ˆ A.. A is slender provided that reduced group M and every g HomŽ M, Ž1 n . s 0 for almost all n - , whenever g HomŽ⺪ , A. and Ž1 n < n . is the canonical basis of the free group ⺪ Ž .. It is well-known that slender « -cotorsion free, and cotorsion free « -cotorsion free, w10, Sect. 7x. The torsion part of a group A is denoted by T Ž A.. ⺠ denotes the set of all primes in ⺪. For p g ⺠, p denotes the group of all p-adic integers, and ⺪ p⬁ the Prufer p-group, so p ( EndŽ⺪ p⬁ .. Tp Ž A. denotes the p-tor¨ sion part of A, so T Ž A. s [p g ⺠ Tp Ž A.. We refer to w1x, w11x, and w14x for relevant properties of the notions defined above.
1. ALMOST COTORSION GROUPS DEFINITION 1.1. Let be an infinite cardinal. Let T s - be the tree consisting of finite sequences of ordinals - . T is partially ordered by inclusion. Let Br s Br ŽT . be the set of all branches of T . Note that the elements of Br can be identified with -sequences of ordinals - . So < T < s and < Br < s . For each g Br and n - , define z n g ⺪ T by
Ž z n . s
Ž i q 1. ! Ž n q 1. !
if
᭚i - : ° i s & n F i,
Ž z n . s 0
otherwise.
Here, g HomŽ⺪ T , ⺪. denotes the th canonical projection for each g T . Then Ž z n < n - . form a divisibility chain modulo ⺪ ŽT ., that is z n y Ž n q 2 . z , nq1 s 1 ° n
᭙ g Br ᭙ n - ,
where 1 < g T 4 denotes the canonical basis of the free group ⺪ ŽT ..
Ž 2.
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Denote by ⺔ the subgroup of ⺪ T generated by
1 < g T 4 j z < g Br , n - 4 . Since the elements of Br are pairwise almost disjoint, we have ⺪ ŽT . ; ⺔ ; ⺪ T and ⺔r⺪ ŽT . (
[ž
gBr
Ý Ž z n q ⺪ ŽT . . ⺪
n-
/ (⺡
Ž B r .
.
In algebraic terms, ⺔ may be identified with a particular subgroup of ⺪ containing ⺪ Ž . and such that the factor by ⺪ Ž . is divisible and has rank . Let be a cardinal, G . Then T is naturally a subtree of T , and Br a subset of Br. Moreover, the canonical group embedding ⺪ T : ⺪T restricts to an embedding of ⺔ into ⺔ , and of ⺪ ŽT . into ⺪ ŽT.. Put D s Ker ExtŽ⺔ , y.. Then D = D 1 = ⭈⭈⭈ = D = Dq= ⭈⭈⭈ = C ,
Ž 3.
where C denotes the class of all cotorsion groups. We call the elements of D the -almost cotorsion groups. The following result is essentially due to Hunter w19x. It is the only fact about almost cotorsion groups needed in Section 2. LEMMA 1.2. F G D s C Proof. Define a strictly increasing sequence of cardinals, Ž ␣ < ␣ ) 0. by 1 s , ␣q1 s Ý n- ␣ n , where ␣ 0 s ␣ , ␣ , nq1 s 2 ␣ n , and ␣ s Ý  - ␣  for ␣ a limit ordinal. Note that ␣q1 s 2 ␣q1 for all ␣ G 0. It suffices to show that F␣ G 0 D ␣q1 s C . For this purpose, we prove for each s ␣q1 that ᭙A g Mod-⺪: Ž < A < F 2 & A g D . « A g C .
Ž 4.
Assume that ExtŽ⺔, A. s 0. An application of HomŽy, A. to the exact sequence 0 ª ⺪ ŽT. ª ⺔ ª ⺡ Ž B r. ª 0 shows that the connecting homomorphism Hom Ž ⺪ ŽT. , A . ª Ext Ž ⺡ Ž B r. , A . is onto. Assume that ExtŽ⺡, A . / 0. Since s 2 , we have
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LEMMA 1.3.
D l T s C l T for all G .
Proof. Let A be a reduced torsion group with ExtŽ⺔ , A. s 0. Denote by B the basic subgroup of A. Then ExtŽ⺔ , B . s 0, w14, Sect. 101, p. 193x. Assume that B has a direct summand, B⬘, such that B⬘ s [i- ⺪rk i ⺪ and either Ž1. k i s p n i Ž i - . for a prime p and for a strictly increasing sequence Ž n i < i - ., or Ž2. k i s pin i where pi Ž i - . are distinct primes. Then ExtŽ⺔ , B⬘. s 0 and < B⬘ < s , so Ž4. Žfor s 1 s . shows that B⬘ g C . Then B⬘ is bounded, a contradiction. It follows that B has no such direct summands B⬘, hence B is bounded. Since A is reduced and B is pure in A, w14, 27.5x gives A s B, and A is cotorsion. Nevertheless, the chain Ž3. does not stabilize. That is, D / C for each G . In order to prove this, we require a criterion for a reduced torsion-free group A to belong to D ᎏan Ext-lemma in the sense of ˆ can w15, Sect. 3x. ŽRecall the well-known fact that the elements of A be represented in the form Ý⬁ks 0 Ž k q 1.!a k where a k g A for all k, w14, Sect. 13x.. LEMMA 1.4 ŽExt-lemma.. Let G . Let A be a reduced torsion-free group. Then the following conditions are equi¨ alent: Ži. A g D ; ˆ there Žii. For each sequence Ž p < g Br . consisting of elements of A Ž < . is a sequence a g T consisting of elements of A such that for each g Br , p differs from Ý⬁ks0 Ž k q 1.!a ° k by an element of A.
ˆ ª ArA ˆ ª 0, and Proof. We have the exact sequence 0 ª A ª A ˆ ˆ is reduced cotorsion, w14, 41.8x. ArA is divisible torsion-free and A Applying the functor HomŽ⺔ , y. we get
.
Hom Ž⺔ , .
ˆ . Hom Ž ⺔ , ArA
6
ˆ Hom Ž ⺔ , A
ˆ ª Ext Ž ⺔ , A . ª Ext Ž ⺔ , A So Ži. is equivalent to
Ž i⬘ .
Hom Ž ⺔ , . is surjective.
. s 0.
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Further, Ži⬘. is equivalent to Ži⬙ ., where
ˆ . : f ° ⺪ ŽT . s 0 ᭚ g g Hom Ž ⺔ , Aˆ . : Ž i⬙ . ᭙ f g Hom Ž ⺔ , ArA g Ž ⺪ ŽT . . : A & f s g .
ˆ . then That Ži⬘. implies Ži⬙ . is clear. Conversely, if f ⬘ g HomŽ⺔ , ArA y1 ˆ we define g g HomŽ⺔ , A. by g Ž1 . g f ⬘Ž1 . for all g T . This is ˆ is pure injective. Let f s f ⬘ y g. possible since ⺪ ŽT . is pure in ⺔ and A ŽT . ˆ. such that f s g. Then f ° ⺪ s 0, so Ži⬙ . provides g g HomŽ⺔ , A Now, g ⬘ s g q g satisfies f ⬘ s g ⬘. ˆ . such that f ° ⺪ ŽT . s 0 corresponds Note that each f g HomŽ⺔ , ArA ŽT . ˆ ˆ .. In particular, Ž . 1-1 to f g Hom ⺔r⺪ , ArA s Hom ⺡ Ž⺔r⺪ ŽT ., ArA each such f is uniquely determined by the sequence Ž f Ž z 0 . < g Br . : ˆ ArA. ˆ. such that g Ž⺪ ŽT . . : A Since ⺪ ŽT . is pure in ⺔ , each g g HomŽ⺔ , A Ž Ž . < is uniquely determined by the sequence g 1 g T . : A. Moreover, Ž2. gives g Ž z n . s
⬁
Ž k q 1. !
ˆ g Ž 1 ° k . g A. Ý ksn Ž n q 1 . !
Since ⺪ ŽT . q Ý g B rz 0 ⺪ is essential in ⺔ , we infer that Ži⬙ . is equivalent to
ˆ ᭚Ž a < g T . : A ᭙ g Br : Žii⬘. ᭙Ž q < g Br . : ArA q s
⬁
žÝ
Ž k q 1 . !a ° k q A.
ks0
/
The latter is just a restatement of Žii.. Lemma 1.4 is based on computing Ext via Hom groups of a pure injective copresentation of A rather than dealing with short exact sequences. In fact, a similar approach, via Hom groups of the projective presentation 0 ª N ª B ª G⬘ ª 0 of the group G⬘ in w15, 3.2x, yields a very simple and short alternative proof of the Ext-lemma in w15, Sect. 3x. In order to prove that there are many non-cotorsion Ževen cotorsion free. torsion-free elements in D , we apply Shelah’s stationary black box principle Žcf. w13, 15x. and construct torsion-free elements of D with a prescribed endomorphism ring. First, we extend our notation as in w10, Sect. 1x:
DEFINITION 1.5. Let G and be cardinals such that s Žnote that this implies cf Ž . ) .. Let R be a ring such that < R < - . Let
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T s - and B s [ g T R Žthe free right R y module with the free basis T .. For ␣ - , we put T␣ s ␣ - and B␣ s [ g T␣ R. So B␣ : B : Bˆ and Bˆ␣ : Bˆ canonically. As in w10, 1.7x, each x g Bˆ can be represented in the form Ý g T c where c g Rˆ for all g T, and for each k - , c g Ž k q 1.!Rˆ for almost all g T. As in w10, 1.7x, we put x ° s c , and define the support of x by
w x x s g T < x ° / 04 ˆ we put w X x s Dx g X w x x. Further, Note that
-
4,
where ␦ : cf Ž . q 1 ª q 1 denotes a fixed continuous strictly increasing function such that ␦ Ž cf Ž .. s . We put 5 X 5 s sup5 x 5 < x g X 4 . If X : Bˆ and < X < F , then there is a minimal ␣ - such that X : Bˆ␣ ; then 5 X 5 s ␣. Since S s ␣ - < cf Ž ␣ . s 4 is a stationary subset of , we can split S in two stationary sets S1 , S2 ; such that S1 j S2 s S and S1 l S2 s ⭋. A triple T s Ž f, P, u. is called a trap provided that f : T ª T is a tree embedding, there is a subset Y : T such that < Y < s and P s [ g Y R, im f : P, w P x is a subtree of T, cf Ž5 P 5. s , 5 5 s 5 P 5 for each branch g Br Žim f ., and either
Ž a.
u s Ž p <  g Br . : Pˆ
and
5 u5 - 5 P 5
or
Ž b.
u g End Ž Pˆ. .
In the case Ža., T is called an Ext-trap, in the case Žb. an End-trap. Now, we formulate the principle needed: LEMMA 1.6 ŽStationary black box.. There is an ordinal * with < * < s and a sequence of traps, ŽŽ f␣ , P␣ , u␣ . < ␣ - *., such that Ži. ᭙ - ␣ - *: 5 P 5 F 5 P␣ 5, Žii. ᭙␣ /  - *: Br Žim f␣ . l Br Žim f . s ⭋, Žiii. ᭙␣ ,  - *:  q F ␣ « Br Žim f␣ . l Br Žw P x. s ⭋, Živ. ᭙␣ - *: 5 P␣ 5 g S1 iff Ž f␣ , P␣ , u␣ . is an Ext-trap, ˆ < X < F ᭙Ž p <  g Br . : Bˆ ᭚␣ - *: Žv. ᭙ X : B: 5 P␣ 5 g S1 & X : Pˆ␣ & 5 X 5 - 5 P␣ 5 & u␣ s Ž p <  g Br ., ˆ < X < F ᭙ g EndŽ Bˆ. ᭚␣ - *: Žvi. ᭙ X : B: 5 P␣ 5 g S2 & X : Pˆ␣ & 5 X 5 - 5 P␣ 5 & u␣ s ° P␣ .
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Proof. See w10, Appendixx, w13, 1.7x and w15, Theorem 4.8x.
THEOREM 1.7. Let G and s . Let R be a ring. Denote by Rq the additi¨ e group of R. Ži. Assume that < R < - and Rq is cotorsion free Ž free .. Then there exists a reduced torsion-free group A g D such that < A < s and EndŽ A. ( R. Moreo¨ er, A is cotorsion free Ž 1-free .. Žii. Assume that < R < - 2 and Rq is -cotorsion free Ž slender .. Then there exists a reduced torsion-free group A g D such that < A < s and EndŽ A. ( R [ FinŽ A.. Here, FinŽ A. denotes the ideal of EndŽ A. consisting of all endomorphisms with finite rank image. Moreo¨ er, A is -cotorsion free Ž slender .. Proof. We use the notation of 1.5. We construct a chain, Ž G␣ < ␣ - *., of pure R-submodules of Bˆ such that G␣ is cotorsion free Ž -cotorsion free, slender, 1-free., for all ␣ - *, as follows. First, G 0 s B, and G␣ s D - ␣ G for all limit ordinals ␣ - *. Assume ␣ s  q 1. If 5 P 5 g S2 , then Ž f , P , u . is an End-trap and we proceed as in w10, 3.2x. If 5 P 5 g S1 , then Ž f , P , u . is an Ext-trap. So u s Ž p < g Br . : Pˆ . As in w10, 3.2x, we get 5 p y y 5 s 5 P 5 for all g Br , where y s Ý⬁ks0 Ž k q 1.! f Ž ° k . g Pˆ . We enumerate the branches of T , so Br s ␥ < ␥ - 4 . Let G␣ 0 s G , and G␣␥ s D␦ - ␥ G␣ ␦ for all limit ordinals ␥ F . Further, let G␣ , ␥q1 be the purificaˆ for each ␥ - . As in w10, Lemma 6.2x tion of G␣␥ q Ž p␥ y y␥ . R in B, Žw10, Theorem 7.4Ža.x, and w10, Theorem 7.4Žb.x, respectively., we get that all G␣␥ Ž␥ F . are cotorsion free Ž -cotorsion free, slender, 1-free, respectively., and we put G␣ s G␣ . Let A s D␣ - * G␣ . Since B is reduced and torsion-free, so are Bˆ and ˆ s B. ˆ Since s , we have < A < s . By the construction, A. Clearly, A A is cotorsion free Ž -cotorsion free, slender, 1-free, respectively.. Also, EndŽ A. ( R [ FinŽ A. by the End-trap part of the construction Žcf. w10, 5.2 and 7.10x.. Moreover, by w10, 6.3x, we have FinŽ A. s 0 in the cotorsion free and free cases. It remains to show that condition Žii. of 1.4 is satisfied. ˆ By 1.6Žv., there exist  - * and an Ext-trap Let Ž p < g Br . : B. Ž f , P , u . such that u s Ž p < g Br . : Pˆ . By the construction, we have aX s p y Ý⬁ks0 Ž k q 1.! f Ž ° k . g A for each g Br . Since f Ž ° k . g T ; B ; A, we put a s f Ž . g A for each g T . It follows that condition Žii. of 1.4 holds true. Remark 1.8. For any A g Mod-⺪ and G , we have the following two implications Ext Ž ⺡, A . s 0 « Ext Ž ⺪ , A . s 0, Ž 5. Ext Ž ⺪ , A . s 0 « Ext Ž ⺔ , A . s 0,
Ž 6.
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By 1.3, both Ž5. and Ž6. can be reversed in the case when A is torsion. On the other hand, if A is torsion-free and ExtŽ⺪ , A. s 0, then A has no non-zero slender factor-groups, w8, Proposition 2.16x. By 1.7, for each G , there exist torsion-free slender groups A g D . It follows that the implication Ž6. cannot be reversed in general. Recently, Eklof has shown that in general Ž5. cannot be reversed either. Indeed, for each G , he has constructed an 1-free Žhence not cotorsion. group A such that ExtŽ⺪ , A. s 0 w12, Corollary 5x. Let A be a reduced torsion-free group such that A is almost cotorsion in the sense of 1.1. In view of 1.7, it may come as a surprise that A always has a non-trivial factor-group which is cotorsion: PROPOSITION 1.9. Let G and A g D . Assume A is reduced and torsion-free. Then there is a factor-group, B, of A such that B ( p for a prime p. Proof. By w22, Lemma 3.8x Žor w15, Lemma 7.4x., there is a pure subgroup A⬘ of A such that < B⬘ < F 2 and nucŽ B⬘. s nucŽ A., where B⬘ s ArA⬘. Clearly, B⬘ g D : D . By Ž4. Žfor s 1 s ., we infer that B⬘ is cotorsion. Since B⬘ is not divisible, B⬘ has a summand isomorphic to p for a prime p w14x. 2. COTILTING AND TILTING GROUPS In this section, we combine Lemma 1.2 with the results of w8x and w15x in order to describe the structure of arbitrary Žpartial. cotiliting and tilting groups. THEOREM 2.1. has the form
A group A g Mod-⺪ is partial cotilting if and only if A
[⺪ pgP
Ž ␣ p. p⬁
[ ⺡ Ž␥ . [
Ł
qgQ
⺓q ,
Ž 7.
Ž p g P . and  q Ž q g Q . where P and Q are two disjoint sets of primes, ␣ p $ are non-zero cardinals, ␥ is a cardinal, and ⺓ q s Žq q . Ž q g Q .. Proof. Assume A is partial cotilting. If p g ⺠ satisfies Tp Ž A. / 0, then ArpA ( ExtŽ⺪rp⺪, A. s 0, as A is a splitter. It follows that T Ž A. is divisible, so A s T Ž A. [ B for torsion-free group B. If B / 0, then ExtŽ A , A. s 0 implies ExtŽ⺪ , A. s 0, hence A g D , for all G . By 1.2, A is cotorsion, hence of the form Ž7. for some P, Q : ⺠. If p g P l Q, then ExtŽ⺪ p⬁ , ⺓ p . ( HomŽ⺪ p⬁ , EŽ⺓ p .r⺓ p . is a non-zero direct summand of ExtŽ A, A., a contradiction. So P l Q s ⭋.
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Next, we prove the ‘‘if’’ part: for each X : ⺠, let NX s G g Mod⺪ < Tp Ž G . s 0 ᭙ p g X 4 . Then NX is a torsion-free class. Assume A is of the form Ž7.. Then H A s Fq g Q H ⺓ q . In particular, for each B g Mod-⺪, we have B gH A iff T Ž B . gH A iff EŽT Ž B .. gH A. The latter is equivalent to B g NQ . Assume ␥ ) 0. Then CogŽ A. s CogŽ[p g P ⺪ p⬁ [ ⺡. s N⺠ _ P . If ␥ s 0 and P / ⭋, then ⺡ g CogŽ[p g P ⺪ p⬁ ., so again CogŽ A. s N⺠ _ P . If ␥ s 0 and P s ⭋, then CogŽ A. s CogŽŁ q g Q ⺓ q . is a subclass of the class of all reduced torsion free groups. Anyway, we get CogŽ A. :H A s NQ , so A is partial cotilting. COROLLARY 2.2. Ži. Let A g Mod-⺪. Then A is cotilting if and only if A is of the form Ž7. with Q s ⺠ _ P, and either ␥ ) 0, or P / ⭋. Žii. Let F be a torsion-free class in Mod-⺪. Then F is cotilting if and only if there is Q : ⺠ such that F s A g Mod-⺪ < Tq Ž A. s 0 ᭙q g Q4 . Proof. From 2.1, we get NQ sH A s CogŽ A. s N⺠ _ P , and ␥ ) 0 in case P s ⭋. By w8, Lemma 2.6Ža.x, CogŽ A. is a torsion-free class for any group A of the form Ž7.. Nevertheless, by Žthe proof of. 2.1, CogŽ A. is a cotilting torsion free class iff ␥ ) 0 or P / ⭋. For any ring R, there is a well-known symmetry: if TR is a finitely generated tilting module, then so is S T for S s EndŽTR .. Moreover, S TR is faithfully balanced, and the bimodule S TR is called a tilting bimodule. Also, TR induces a category equivalence between GenŽTR . s T H and Ker TorS Žy, T ., generalizing the Morita equivalence w2, 4, 5, 18, 20x. A similar symmetry is not available for cotilting modules. Call S CR a cotilting bimodule provided that S CR is a faithfully balanced bimodule such that both S C and CR are cotilting modules. Then S CR induces a particular kind of duality between subcategories of Mod-R and S-Mod generalizing the Morita duality w6x. Nevertheless, in general, CR cotilting does not imply that S CR is a cotilting bimodule. Indeed, taking already the trivial case in 2.2Ži., so P s ⺠, ␥ s 0, and ␣ p s 1 for all p g P, we get S s EndŽ A ⺪ . s Ł p g P p s EndŽS A., hence S A ⺪ is not balanced. Next, we characterize all Žpartial. tilting groups assuming the axiom of constructibility Ž V s L.: THEOREM 2.3 Ž V s L.. if A has the form
A group A g Mod-⺪ is partial tilting if and only
[⺪ pgP
Ž ␣ p. p⬁
[ R ŽP . ,
Ž 8.
where P : ⺠, ␣ p Ž p g P . are non-zero cardinals,  is a cardinal, and R P is the subring of ⺡ generated by ⺪ j py1 < p g P 4 .
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Proof. Assume A is partial tilting. Since A is a splitter, the torsion part of A is divisible, hence splits off. So there is a subset P of ⺠ such that A ( [p g P ⺪ Žp␣⬁ p . [ ⺡ Ž ␥ . [ C, where ␣ p Ž p g P . are non-zero cardinals, ␥ is a cardinal, and C is a reduced torsion-free splitter. Let R s nucŽ C .. By w15, Theorem 2.5x Žor by w22x., py1 < p g P 4 : R. Since ExtŽ C, C Ž . . s 0 and V s L, w15, Theorem 7.5x implies that C is a free right R-module. So C ( R Ž  . for a cardinal  . If ␥ / 0, then C is cotorsion and torsion-free, hence pure-injective. If C / 0, then R is a pure-injective group, so R s ⺡ and C is not reduced, a contradiction. This proves that C s 0 and A is divisible. For each p g ⺠ and each group G we have pG s G iff HomŽ⺪ p⬁ , EŽ G .rG. s 0 iff ExtŽ⺪ p⬁ , G . s 0. So ⺪ p⬁ H coincides with the Žtorsion. class of all p-divisible groups. On the other hand, ⺡ H is just the class of all cotorsion groups. If q g ⺠ _ P, then ⺪ q n g AH for all n - , but [n- ⺪ q n f Q H , w14, Corollary 54.4x. Since AH is a torsion class, we infer that A has the form Ž8. for P s ⺠ Žand R P s ⺡.. Assume ␥ s 0. We will show that qy1 f R for all q g ⺠ _ P. Indeed, assume qy1 g R for some q g ⺠. Since ⺪ q n is cotorsion for all n - , we get ⺪ q n g R H for all n - . On the other hand, a direct computation shows that [n- ⺪ q n f R H . Since AH is a torsion class, we infer that some ⺪ q n f AH , hence q g P. This proves that R is the subring of ⺡ generated by ⺪ j py1 < p g P 4 . So A has the form Ž8. for the given P. Conversely, assume A is of the form Ž8.. If  / 0, then GenŽ A. s ModR P is the class of all groups which are p-divisible for all p g P. If  s 0, then GenŽ A. is the class of all divisible torsion groups D such that Tq Ž D . s 0 whenever q g ⺠ _ P. Since R Py and ⺪-extensions of R P-modules coincide, we have Mod-R P : R PH . So if  / 0, then GenŽ A. s AH s Mod-R P , and A is a tilting group. If  s 0, then GenŽ A. ; AH s Mod-R P , so A is partial tilting, but not tilting. COROLLARY 2.4 Ž V s L.. Ži. Let A g Mod-⺪. Then A is tilting if and only if A has the form Ž8. with  / 0. Žii. Let T be a torsion class in Mod-⺪. Then T is tilting if and only if there is P : ⺠ such that T s A g Mod y ⺪ < pA s A ᭙ p g P 4 .
REFERENCES 1. F. W. Anderson and K. R. Fuller, ‘‘Rings and Categories of Modules,’’ 2nd ed., Springer, New York, 1992.
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2. I. Assem, Tilting theoryᎏan introduction, Topics Algebra 26, No. I Ž1990., 127᎐180. 3. K. Bongartz, Tilted algebras, in ‘‘Proceedings of the ICRA III, Puebla 1980,’’ Lecture Notes in Mathematics, Vol. 903, pp. 26᎐38, Springer, Berlin, 1981. 4. S. Brenner and M. Butler, Generalizations of the Bernstein ᎐Gelfand᎐Ponomarev reflexion functors, in ‘‘Proceedings of the ICRA II, Ottawa 1979,’’ Lecture Notes in Mathematics, Vol. 832, pp. 103᎐169, Springer, Berlin, 1979. 5. R. R. Colby and K. R. Fuller, Tilting, cotilting and serially tilted rings, Comm. Algebra 18 Ž1990., 1589᎐1615. 6. R. Colpi, Cotilting bimodules and their dualities, in ‘‘Proceedings of the Euroconf. Murcia ’98,’’ to appear. 7. R. Colpi, G. D’Este, and A. Tonolo, Quasi-tilting modules and counter equivalences, J. Algebra 191 Ž1997., 461᎐494. 8. R. Colpi, A. Tonolo, and J. Trlifaj, Partial cotilting modules and the lattices induced by them, Comm. Algebra 25 Ž1997., 3225᎐3237. 9. R. Colpi and J. Trlifaj, Tilting modules and tilting torsion theories, J. Algebra 178 Ž1995., 614᎐634. 10. A. L. S. Corner and R. Gobel, Prescribing endomorphism algebras, a unified treatment, ¨ Proc. London. Math. Soc. 50 Ž1985., 447᎐479. 11. P. C. Eklof and A. H. Mekler, ‘‘Almost Free Modules,’’ North-Holland, New York, 1990. 12. P. C. Eklof and J. Trlifaj, How to make Ext vanish, preprint. 13. B. Franzen and R. Gobel, Prescribing endomorphism algebras, Rend. Sem. Mat. Uni¨ . ¨ Pado¨ a 80 Ž1988., 215᎐241. 14. L. Fuchs, ‘‘Infinite Abelian Groups,’’ Vols. 1 and 2, Academic, New York, 1970 and 1973. 15. R. Gobel ¨ and S. Shelah, Cotorsion theories and splitters, Trans. Amer. Math. Soc., to appear. 16. R. Gobel ¨ and S. Shelah, Almost free splitters, Colloq. Math. 81 Ž1999., 193᎐221. 17. D. Happel, Selforthogonal modules, in ‘‘Abelian groups and Modules,’’ pp. 257᎐276, Kluwer, Dordrecht, 1995. 18. D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 Ž1982., 399᎐443. 19. R. H. Hunter, Balanced subgroups of abelian groups. Trans. Amer. Math. Soc. 215 Ž1976., 81᎐98. 20. Y. Miyashita, Tilting modules of finite projective dimension, Math. Z. 193 Ž1986., 113᎐146. 21. L. Salce, Cotorsion theories for abelian groups, Symposia Math. 23 Ž1979., 11᎐32. 22. P. Schultz, Self-splitting groups, Univ. Western Australia, Perth, 1980, preprint. 23. S. Shelah, Existence of rigid-like families of abelian p-groups, in ‘‘Model Theory and Algebra,’’ Lecture Notes in Mathematics, Vol. 498, pp. 384᎐402, Springer, Berlin, 1975.
Cotilting and a Hierarchy of Almost Cotorsion Groups1 Rudiger Gobel ¨ ¨ Fachbereich 6, Mathematik und Informatik, Uni¨ ersitat ¨ Essen, 45117 Essen, Deutschland E-mail: [email protected]
and Jan Trlifaj ˇ ´ Republika Katedra algebry MFF UK, Sokolo¨ ska ´ 83, 186 75 Praha 8, Ceska E-mail: [email protected] Communicated by Kent R. Fuller Received July 6, 1998
We describe the structure of all cotilting and partial cotilting groups, and of all cotilting torsion-free classes of groups. Under V s L, we also describe all tilting and partial tilting groups. As a by-product, we introduce new classes of groupsᎏcalled almost cotorsion groups. Unlike the well-known cotorsion groups, the almost cotorsion ones have a rather complex structure: we prove that any cotorsion free ring can be realized as an endomorphism ring of an almost cotorsion group. 䊚 2000 Academic Press
INTRODUCTION Starting with the paper of Miyashita w20x several key ideas of the classical tilting and cotilting theory of finite dimensional algebras have been extended to the setting of arbitrary associative rings. In order to obtain the results in full generality it is often necessary to work in the category Mod-R of all Žright R-. modules rather than mod-R, 1
Research supported by the German-Israeli Foundation for Scientific Research & Development Žproject no. G-0294-081.06r93. and by the Grant Agency of Czech Republic Žgrant no. 201r97r1162.. 110 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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the subcategory of all finitely generated modules. For example, the Bongartz idea of complementing a partial tilting module in a tilting one does not generalize in mod-R, but it does in Mod-R. More specifically, M g Mod-R is called a splitter if Ext R Ž M, M . s 0, w22x, w15x. Further, M g Mod-R is ‘‘ partial tilting’’ if M is a splitter such that proj.dim. M F 1. A ‘‘partial tilting’’ module M g mod-R is tilting provided that there is an exact sequence 0 ª R ª M⬘ ª M⬙ ª 0 where M⬘ and M⬙ are summands of a finite direct sum of copies of M. The classical Bongartz Lemma w3, 2.1x says that for any finite dimensional algebra R and any ‘‘partial tilting’’ module P g mod-R there is a module C g mod-R Žthe Bongartz complement of P, w17x. such that P [ C is a tilting module Žand Ž P [ C . H lmod-R s P H lmod-R, where M H s Ker Ext R Ž M, y. for M g Mod-R.. We call a module M g Mod-R partial tilting if GenŽ M . : M H and M H is a torsion class. Here, a class T : Mod-R is a torsion class in Mod-R if T is closed under arbitrary direct sums, homomorphic images and extensions, and GenŽ M . denotes the class of all modules generated by M, that is, of all homomorphic images of arbitrary direct sums of copies of M. Further, a module M g Mod-R is called tilting if GenŽ M . s M H . For finitely generated modules, the latter definition coincides with the definition of tilting given above, but partial tilting is a stronger notion than ‘‘partial tilting,’’ cf. w9, Sect. 1x. Even with our Žstronger. notion of partial tilting, the Bongartz Lemma does not generalize within mod-R: there exist finitely generated partial tilting modules over certain hereditary noetherian rings which are not summands of finitely generated tilting modules w9, Example 1.6x. Nevertheless, a generalization to Mod-R, for an arbitrary ring R, is available. By w9, Theorem 2.11x, for any partial tilting module P g Mod-R there is a module C g Mod-R such that P [ C is a tilting module and Ž P [ C . H s PH . This result has immediate consequences for the structure of tilting torsion classes of modules Žthat is, of the torsion classes T : Mod-R such that T s GenŽT . for a tilting module T, w2x.. By w9, Theorem 2.3x, a torsion class T is tilting iff T s P H for a splitter P g Mod-R. In the finite dimensional algebra case, the vector space duality provides dual results to the ones mentioned above. Surprisingly, the corresponding dual results hold true for arbitrary rings: A module M g Mod-R is called partial cotilting if CogŽ M . :H M and H M is a torsion-free class. Here, a class F : Mod-R is a torsion-free class in Mod-R if F is closed under arbitrary direct products, submodules and extensions. Further, H M s Ker Ext R Žy, M ., and CogŽ M . denotes the class of all modules cogenerated by M, that is, of all submodules of arbitrary direct products of copies of M. A module M g Mod-R is called
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cotilting if CogŽ M . sH M. A torsion-free class C : Mod-R is a cotilting torsion-free class provided that C s CogŽ C . for a cotilting module C, w7x, w8x. The dual version of the Bongartz Lemma reads as follows: for any partial cotilting module P g Mod-R there is a module D g Mod-R such that P [ D is a cotilting module and H Ž P [ D . sH P, w8, Theorem 2.11x. Similarly, a torsion-free class F is cotilting iff F sH P for a splitter P g Mod-R, w9, Corollary 2.12x. For any ring R, we have the following inclusions of subclasses in Mod-R: splitters = partial tilting modules = tilting modules = progenerators splitters = partial cotilting modules = cotilting modules
Ž 1.
= injective cogenerators. It is an interesting problem to describe these classes for particular rings R. In the present paper, we concentrate on the case when R s ⺪, the ring of all integers. In this case, splitters have recently been investigated in w15x and w16x. It is easy to see that the torsion part of any splitter A g Mod-Z is divisible, hence a direct summand in A. So we may restrict to the class R of all reduced torsion-free groups. There are two trivial examples of splitters: Ža. the torsion-free cotorsion groups Ža group A is cotorsion provided that ExtŽ⺡, A. s 0, where ⺡ is the additive group of all rationals., and Žb. the nuc-free groups Ža group A is nuc-free provided that A is free as an R-module, where R s nucŽ A. denotes the nucleus of A, so R s arb < a, b g ⺪, bA s A4 is the largest subring R of ⺡ such that A is an R-module.. In w15, Sect. 4x, examples of non-trivial splitters were constructed in R. These splitters are not 1-free, but their ranks and endomorphism rings can be prescribed w15, Example 5.5x. This indicates that no classification of non- 1-free splitters is possible. Moreover, though all 1-free splitters of cardinality 1 are nuc-free by w16, Main Theorem 1.5x, there exist nontrivial 1-free splitters of cardinality 2 1 w12, Corollary 8x. In Section 2 of this paper, we completely classify the Žpartial. cotilting groups, and the cotilting torsion-free classes of groups ŽTheorem 2.1, Corollary 2.2.. We also describe all Žpartial. tilting groups under V s L ŽTheorem 2.3.. As a by-product of our investigation, we discover classes of groups that exhibit similar homological properties to cotorsion groups, but have a rather complex structure. We call these groups almost cotorsion
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ŽDefinition 1.1.. In Theorem 1.7, we provide an explanation for the complexity: any cotorsion free ring appears as an endomorphism ring of an almost cotorsion group. Hence, unlike the cotorsion groups, the almost cotorsion ones do not appear to be classifiable. ˆ the pure injective hull of A, and by EŽ A. Let A g Mod-⺪. Denote by A the injective hull of A. Note that the notions of a cotorsion group and of a pure injective group coincide for torsion-free groups w14, Sect. 54x. A group A is cotorsion free provided that A has no non-zero cotorsion subgroups. A is -cotorsion free provided that imŽ ° M . is of finite rank for every ˆ A.. A is slender provided that reduced group M and every g HomŽ M, Ž1 n . s 0 for almost all n - , whenever g HomŽ⺪ , A. and Ž1 n < n . is the canonical basis of the free group ⺪ Ž .. It is well-known that slender « -cotorsion free, and cotorsion free « -cotorsion free, w10, Sect. 7x. The torsion part of a group A is denoted by T Ž A.. ⺠ denotes the set of all primes in ⺪. For p g ⺠, p denotes the group of all p-adic integers, and ⺪ p⬁ the Prufer p-group, so p ( EndŽ⺪ p⬁ .. Tp Ž A. denotes the p-tor¨ sion part of A, so T Ž A. s [p g ⺠ Tp Ž A.. We refer to w1x, w11x, and w14x for relevant properties of the notions defined above.
1. ALMOST COTORSION GROUPS DEFINITION 1.1. Let be an infinite cardinal. Let T s - be the tree consisting of finite sequences of ordinals - . T is partially ordered by inclusion. Let Br s Br ŽT . be the set of all branches of T . Note that the elements of Br can be identified with -sequences of ordinals - . So < T < s and < Br < s . For each g Br and n - , define z n g ⺪ T by
Ž z n . s
Ž i q 1. ! Ž n q 1. !
if
᭚i - : ° i s & n F i,
Ž z n . s 0
otherwise.
Here, g HomŽ⺪ T , ⺪. denotes the th canonical projection for each g T . Then Ž z n < n - . form a divisibility chain modulo ⺪ ŽT ., that is z n y Ž n q 2 . z , nq1 s 1 ° n
᭙ g Br ᭙ n - ,
where 1 < g T 4 denotes the canonical basis of the free group ⺪ ŽT ..
Ž 2.
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Denote by ⺔ the subgroup of ⺪ T generated by
1 < g T 4 j z < g Br , n - 4 . Since the elements of Br are pairwise almost disjoint, we have ⺪ ŽT . ; ⺔ ; ⺪ T and ⺔r⺪ ŽT . (
[ž
gBr
Ý Ž z n q ⺪ ŽT . . ⺪
n-
/ (⺡
Ž B r .
.
In algebraic terms, ⺔ may be identified with a particular subgroup of ⺪ containing ⺪ Ž . and such that the factor by ⺪ Ž . is divisible and has rank . Let be a cardinal, G . Then T is naturally a subtree of T , and Br a subset of Br. Moreover, the canonical group embedding ⺪ T : ⺪T restricts to an embedding of ⺔ into ⺔ , and of ⺪ ŽT . into ⺪ ŽT.. Put D s Ker ExtŽ⺔ , y.. Then D = D 1 = ⭈⭈⭈ = D = Dq= ⭈⭈⭈ = C ,
Ž 3.
where C denotes the class of all cotorsion groups. We call the elements of D the -almost cotorsion groups. The following result is essentially due to Hunter w19x. It is the only fact about almost cotorsion groups needed in Section 2. LEMMA 1.2. F G D s C Proof. Define a strictly increasing sequence of cardinals, Ž ␣ < ␣ ) 0. by 1 s , ␣q1 s Ý n- ␣ n , where ␣ 0 s ␣ , ␣ , nq1 s 2 ␣ n , and ␣ s Ý  - ␣  for ␣ a limit ordinal. Note that ␣q1 s 2 ␣q1 for all ␣ G 0. It suffices to show that F␣ G 0 D ␣q1 s C . For this purpose, we prove for each s ␣q1 that ᭙A g Mod-⺪: Ž < A < F 2 & A g D . « A g C .
Ž 4.
Assume that ExtŽ⺔, A. s 0. An application of HomŽy, A. to the exact sequence 0 ª ⺪ ŽT. ª ⺔ ª ⺡ Ž B r. ª 0 shows that the connecting homomorphism Hom Ž ⺪ ŽT. , A . ª Ext Ž ⺡ Ž B r. , A . is onto. Assume that ExtŽ⺡, A . / 0. Since s 2 , we have
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LEMMA 1.3.
D l T s C l T for all G .
Proof. Let A be a reduced torsion group with ExtŽ⺔ , A. s 0. Denote by B the basic subgroup of A. Then ExtŽ⺔ , B . s 0, w14, Sect. 101, p. 193x. Assume that B has a direct summand, B⬘, such that B⬘ s [i- ⺪rk i ⺪ and either Ž1. k i s p n i Ž i - . for a prime p and for a strictly increasing sequence Ž n i < i - ., or Ž2. k i s pin i where pi Ž i - . are distinct primes. Then ExtŽ⺔ , B⬘. s 0 and < B⬘ < s , so Ž4. Žfor s 1 s . shows that B⬘ g C . Then B⬘ is bounded, a contradiction. It follows that B has no such direct summands B⬘, hence B is bounded. Since A is reduced and B is pure in A, w14, 27.5x gives A s B, and A is cotorsion. Nevertheless, the chain Ž3. does not stabilize. That is, D / C for each G . In order to prove this, we require a criterion for a reduced torsion-free group A to belong to D ᎏan Ext-lemma in the sense of ˆ can w15, Sect. 3x. ŽRecall the well-known fact that the elements of A be represented in the form Ý⬁ks 0 Ž k q 1.!a k where a k g A for all k, w14, Sect. 13x.. LEMMA 1.4 ŽExt-lemma.. Let G . Let A be a reduced torsion-free group. Then the following conditions are equi¨ alent: Ži. A g D ; ˆ there Žii. For each sequence Ž p < g Br . consisting of elements of A Ž < . is a sequence a g T consisting of elements of A such that for each g Br , p differs from Ý⬁ks0 Ž k q 1.!a ° k by an element of A.
ˆ ª ArA ˆ ª 0, and Proof. We have the exact sequence 0 ª A ª A ˆ ˆ is reduced cotorsion, w14, 41.8x. ArA is divisible torsion-free and A Applying the functor HomŽ⺔ , y. we get
.
Hom Ž⺔ , .
ˆ . Hom Ž ⺔ , ArA
6
ˆ Hom Ž ⺔ , A
ˆ ª Ext Ž ⺔ , A . ª Ext Ž ⺔ , A So Ži. is equivalent to
Ž i⬘ .
Hom Ž ⺔ , . is surjective.
. s 0.
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Further, Ži⬘. is equivalent to Ži⬙ ., where
ˆ . : f ° ⺪ ŽT . s 0 ᭚ g g Hom Ž ⺔ , Aˆ . : Ž i⬙ . ᭙ f g Hom Ž ⺔ , ArA g Ž ⺪ ŽT . . : A & f s g .
ˆ . then That Ži⬘. implies Ži⬙ . is clear. Conversely, if f ⬘ g HomŽ⺔ , ArA y1 ˆ we define g g HomŽ⺔ , A. by g Ž1 . g f ⬘Ž1 . for all g T . This is ˆ is pure injective. Let f s f ⬘ y g. possible since ⺪ ŽT . is pure in ⺔ and A ŽT . ˆ. such that f s g. Then f ° ⺪ s 0, so Ži⬙ . provides g g HomŽ⺔ , A Now, g ⬘ s g q g satisfies f ⬘ s g ⬘. ˆ . such that f ° ⺪ ŽT . s 0 corresponds Note that each f g HomŽ⺔ , ArA ŽT . ˆ ˆ .. In particular, Ž . 1-1 to f g Hom ⺔r⺪ , ArA s Hom ⺡ Ž⺔r⺪ ŽT ., ArA each such f is uniquely determined by the sequence Ž f Ž z 0 . < g Br . : ˆ ArA. ˆ. such that g Ž⺪ ŽT . . : A Since ⺪ ŽT . is pure in ⺔ , each g g HomŽ⺔ , A Ž Ž . < is uniquely determined by the sequence g 1 g T . : A. Moreover, Ž2. gives g Ž z n . s
⬁
Ž k q 1. !
ˆ g Ž 1 ° k . g A. Ý ksn Ž n q 1 . !
Since ⺪ ŽT . q Ý g B rz 0 ⺪ is essential in ⺔ , we infer that Ži⬙ . is equivalent to
ˆ ᭚Ž a < g T . : A ᭙ g Br : Žii⬘. ᭙Ž q < g Br . : ArA q s
⬁
žÝ
Ž k q 1 . !a ° k q A.
ks0
/
The latter is just a restatement of Žii.. Lemma 1.4 is based on computing Ext via Hom groups of a pure injective copresentation of A rather than dealing with short exact sequences. In fact, a similar approach, via Hom groups of the projective presentation 0 ª N ª B ª G⬘ ª 0 of the group G⬘ in w15, 3.2x, yields a very simple and short alternative proof of the Ext-lemma in w15, Sect. 3x. In order to prove that there are many non-cotorsion Ževen cotorsion free. torsion-free elements in D , we apply Shelah’s stationary black box principle Žcf. w13, 15x. and construct torsion-free elements of D with a prescribed endomorphism ring. First, we extend our notation as in w10, Sect. 1x:
DEFINITION 1.5. Let G and be cardinals such that s Žnote that this implies cf Ž . ) .. Let R be a ring such that < R < - . Let
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T s - and B s [ g T R Žthe free right R y module with the free basis T .. For ␣ - , we put T␣ s ␣ - and B␣ s [ g T␣ R. So B␣ : B : Bˆ and Bˆ␣ : Bˆ canonically. As in w10, 1.7x, each x g Bˆ can be represented in the form Ý g T c where c g Rˆ for all g T, and for each k - , c g Ž k q 1.!Rˆ for almost all g T. As in w10, 1.7x, we put x ° s c , and define the support of x by
w x x s g T < x ° / 04 ˆ we put w X x s Dx g X w x x. Further, Note that
-
4,
where ␦ : cf Ž . q 1 ª q 1 denotes a fixed continuous strictly increasing function such that ␦ Ž cf Ž .. s . We put 5 X 5 s sup5 x 5 < x g X 4 . If X : Bˆ and < X < F , then there is a minimal ␣ - such that X : Bˆ␣ ; then 5 X 5 s ␣. Since S s ␣ - < cf Ž ␣ . s 4 is a stationary subset of , we can split S in two stationary sets S1 , S2 ; such that S1 j S2 s S and S1 l S2 s ⭋. A triple T s Ž f, P, u. is called a trap provided that f : T ª T is a tree embedding, there is a subset Y : T such that < Y < s and P s [ g Y R, im f : P, w P x is a subtree of T, cf Ž5 P 5. s , 5 5 s 5 P 5 for each branch g Br Žim f ., and either
Ž a.
u s Ž p <  g Br . : Pˆ
and
5 u5 - 5 P 5
or
Ž b.
u g End Ž Pˆ. .
In the case Ža., T is called an Ext-trap, in the case Žb. an End-trap. Now, we formulate the principle needed: LEMMA 1.6 ŽStationary black box.. There is an ordinal * with < * < s and a sequence of traps, ŽŽ f␣ , P␣ , u␣ . < ␣ - *., such that Ži. ᭙ - ␣ - *: 5 P 5 F 5 P␣ 5, Žii. ᭙␣ /  - *: Br Žim f␣ . l Br Žim f . s ⭋, Žiii. ᭙␣ ,  - *:  q F ␣ « Br Žim f␣ . l Br Žw P x. s ⭋, Živ. ᭙␣ - *: 5 P␣ 5 g S1 iff Ž f␣ , P␣ , u␣ . is an Ext-trap, ˆ < X < F ᭙Ž p <  g Br . : Bˆ ᭚␣ - *: Žv. ᭙ X : B: 5 P␣ 5 g S1 & X : Pˆ␣ & 5 X 5 - 5 P␣ 5 & u␣ s Ž p <  g Br ., ˆ < X < F ᭙ g EndŽ Bˆ. ᭚␣ - *: Žvi. ᭙ X : B: 5 P␣ 5 g S2 & X : Pˆ␣ & 5 X 5 - 5 P␣ 5 & u␣ s ° P␣ .
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Proof. See w10, Appendixx, w13, 1.7x and w15, Theorem 4.8x.
THEOREM 1.7. Let G and s . Let R be a ring. Denote by Rq the additi¨ e group of R. Ži. Assume that < R < - and Rq is cotorsion free Ž free .. Then there exists a reduced torsion-free group A g D such that < A < s and EndŽ A. ( R. Moreo¨ er, A is cotorsion free Ž 1-free .. Žii. Assume that < R < - 2 and Rq is -cotorsion free Ž slender .. Then there exists a reduced torsion-free group A g D such that < A < s and EndŽ A. ( R [ FinŽ A.. Here, FinŽ A. denotes the ideal of EndŽ A. consisting of all endomorphisms with finite rank image. Moreo¨ er, A is -cotorsion free Ž slender .. Proof. We use the notation of 1.5. We construct a chain, Ž G␣ < ␣ - *., of pure R-submodules of Bˆ such that G␣ is cotorsion free Ž -cotorsion free, slender, 1-free., for all ␣ - *, as follows. First, G 0 s B, and G␣ s D - ␣ G for all limit ordinals ␣ - *. Assume ␣ s  q 1. If 5 P 5 g S2 , then Ž f , P , u . is an End-trap and we proceed as in w10, 3.2x. If 5 P 5 g S1 , then Ž f , P , u . is an Ext-trap. So u s Ž p < g Br . : Pˆ . As in w10, 3.2x, we get 5 p y y 5 s 5 P 5 for all g Br , where y s Ý⬁ks0 Ž k q 1.! f Ž ° k . g Pˆ . We enumerate the branches of T , so Br s ␥ < ␥ - 4 . Let G␣ 0 s G , and G␣␥ s D␦ - ␥ G␣ ␦ for all limit ordinals ␥ F . Further, let G␣ , ␥q1 be the purificaˆ for each ␥ - . As in w10, Lemma 6.2x tion of G␣␥ q Ž p␥ y y␥ . R in B, Žw10, Theorem 7.4Ža.x, and w10, Theorem 7.4Žb.x, respectively., we get that all G␣␥ Ž␥ F . are cotorsion free Ž -cotorsion free, slender, 1-free, respectively., and we put G␣ s G␣ . Let A s D␣ - * G␣ . Since B is reduced and torsion-free, so are Bˆ and ˆ s B. ˆ Since s , we have < A < s . By the construction, A. Clearly, A A is cotorsion free Ž -cotorsion free, slender, 1-free, respectively.. Also, EndŽ A. ( R [ FinŽ A. by the End-trap part of the construction Žcf. w10, 5.2 and 7.10x.. Moreover, by w10, 6.3x, we have FinŽ A. s 0 in the cotorsion free and free cases. It remains to show that condition Žii. of 1.4 is satisfied. ˆ By 1.6Žv., there exist  - * and an Ext-trap Let Ž p < g Br . : B. Ž f , P , u . such that u s Ž p < g Br . : Pˆ . By the construction, we have aX s p y Ý⬁ks0 Ž k q 1.! f Ž ° k . g A for each g Br . Since f Ž ° k . g T ; B ; A, we put a s f Ž . g A for each g T . It follows that condition Žii. of 1.4 holds true. Remark 1.8. For any A g Mod-⺪ and G , we have the following two implications Ext Ž ⺡, A . s 0 « Ext Ž ⺪ , A . s 0, Ž 5. Ext Ž ⺪ , A . s 0 « Ext Ž ⺔ , A . s 0,
Ž 6.
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By 1.3, both Ž5. and Ž6. can be reversed in the case when A is torsion. On the other hand, if A is torsion-free and ExtŽ⺪ , A. s 0, then A has no non-zero slender factor-groups, w8, Proposition 2.16x. By 1.7, for each G , there exist torsion-free slender groups A g D . It follows that the implication Ž6. cannot be reversed in general. Recently, Eklof has shown that in general Ž5. cannot be reversed either. Indeed, for each G , he has constructed an 1-free Žhence not cotorsion. group A such that ExtŽ⺪ , A. s 0 w12, Corollary 5x. Let A be a reduced torsion-free group such that A is almost cotorsion in the sense of 1.1. In view of 1.7, it may come as a surprise that A always has a non-trivial factor-group which is cotorsion: PROPOSITION 1.9. Let G and A g D . Assume A is reduced and torsion-free. Then there is a factor-group, B, of A such that B ( p for a prime p. Proof. By w22, Lemma 3.8x Žor w15, Lemma 7.4x., there is a pure subgroup A⬘ of A such that < B⬘ < F 2 and nucŽ B⬘. s nucŽ A., where B⬘ s ArA⬘. Clearly, B⬘ g D : D . By Ž4. Žfor s 1 s ., we infer that B⬘ is cotorsion. Since B⬘ is not divisible, B⬘ has a summand isomorphic to p for a prime p w14x. 2. COTILTING AND TILTING GROUPS In this section, we combine Lemma 1.2 with the results of w8x and w15x in order to describe the structure of arbitrary Žpartial. cotiliting and tilting groups. THEOREM 2.1. has the form
A group A g Mod-⺪ is partial cotilting if and only if A
[⺪ pgP
Ž ␣ p. p⬁
[ ⺡ Ž␥ . [
Ł
qgQ
⺓q ,
Ž 7.
Ž p g P . and  q Ž q g Q . where P and Q are two disjoint sets of primes, ␣ p $ are non-zero cardinals, ␥ is a cardinal, and ⺓ q s Žq q . Ž q g Q .. Proof. Assume A is partial cotilting. If p g ⺠ satisfies Tp Ž A. / 0, then ArpA ( ExtŽ⺪rp⺪, A. s 0, as A is a splitter. It follows that T Ž A. is divisible, so A s T Ž A. [ B for torsion-free group B. If B / 0, then ExtŽ A , A. s 0 implies ExtŽ⺪ , A. s 0, hence A g D , for all G . By 1.2, A is cotorsion, hence of the form Ž7. for some P, Q : ⺠. If p g P l Q, then ExtŽ⺪ p⬁ , ⺓ p . ( HomŽ⺪ p⬁ , EŽ⺓ p .r⺓ p . is a non-zero direct summand of ExtŽ A, A., a contradiction. So P l Q s ⭋.
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Next, we prove the ‘‘if’’ part: for each X : ⺠, let NX s G g Mod⺪ < Tp Ž G . s 0 ᭙ p g X 4 . Then NX is a torsion-free class. Assume A is of the form Ž7.. Then H A s Fq g Q H ⺓ q . In particular, for each B g Mod-⺪, we have B gH A iff T Ž B . gH A iff EŽT Ž B .. gH A. The latter is equivalent to B g NQ . Assume ␥ ) 0. Then CogŽ A. s CogŽ[p g P ⺪ p⬁ [ ⺡. s N⺠ _ P . If ␥ s 0 and P / ⭋, then ⺡ g CogŽ[p g P ⺪ p⬁ ., so again CogŽ A. s N⺠ _ P . If ␥ s 0 and P s ⭋, then CogŽ A. s CogŽŁ q g Q ⺓ q . is a subclass of the class of all reduced torsion free groups. Anyway, we get CogŽ A. :H A s NQ , so A is partial cotilting. COROLLARY 2.2. Ži. Let A g Mod-⺪. Then A is cotilting if and only if A is of the form Ž7. with Q s ⺠ _ P, and either ␥ ) 0, or P / ⭋. Žii. Let F be a torsion-free class in Mod-⺪. Then F is cotilting if and only if there is Q : ⺠ such that F s A g Mod-⺪ < Tq Ž A. s 0 ᭙q g Q4 . Proof. From 2.1, we get NQ sH A s CogŽ A. s N⺠ _ P , and ␥ ) 0 in case P s ⭋. By w8, Lemma 2.6Ža.x, CogŽ A. is a torsion-free class for any group A of the form Ž7.. Nevertheless, by Žthe proof of. 2.1, CogŽ A. is a cotilting torsion free class iff ␥ ) 0 or P / ⭋. For any ring R, there is a well-known symmetry: if TR is a finitely generated tilting module, then so is S T for S s EndŽTR .. Moreover, S TR is faithfully balanced, and the bimodule S TR is called a tilting bimodule. Also, TR induces a category equivalence between GenŽTR . s T H and Ker TorS Žy, T ., generalizing the Morita equivalence w2, 4, 5, 18, 20x. A similar symmetry is not available for cotilting modules. Call S CR a cotilting bimodule provided that S CR is a faithfully balanced bimodule such that both S C and CR are cotilting modules. Then S CR induces a particular kind of duality between subcategories of Mod-R and S-Mod generalizing the Morita duality w6x. Nevertheless, in general, CR cotilting does not imply that S CR is a cotilting bimodule. Indeed, taking already the trivial case in 2.2Ži., so P s ⺠, ␥ s 0, and ␣ p s 1 for all p g P, we get S s EndŽ A ⺪ . s Ł p g P p s EndŽS A., hence S A ⺪ is not balanced. Next, we characterize all Žpartial. tilting groups assuming the axiom of constructibility Ž V s L.: THEOREM 2.3 Ž V s L.. if A has the form
A group A g Mod-⺪ is partial tilting if and only
[⺪ pgP
Ž ␣ p. p⬁
[ R ŽP . ,
Ž 8.
where P : ⺠, ␣ p Ž p g P . are non-zero cardinals,  is a cardinal, and R P is the subring of ⺡ generated by ⺪ j py1 < p g P 4 .
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121
Proof. Assume A is partial tilting. Since A is a splitter, the torsion part of A is divisible, hence splits off. So there is a subset P of ⺠ such that A ( [p g P ⺪ Žp␣⬁ p . [ ⺡ Ž ␥ . [ C, where ␣ p Ž p g P . are non-zero cardinals, ␥ is a cardinal, and C is a reduced torsion-free splitter. Let R s nucŽ C .. By w15, Theorem 2.5x Žor by w22x., py1 < p g P 4 : R. Since ExtŽ C, C Ž . . s 0 and V s L, w15, Theorem 7.5x implies that C is a free right R-module. So C ( R Ž  . for a cardinal  . If ␥ / 0, then C is cotorsion and torsion-free, hence pure-injective. If C / 0, then R is a pure-injective group, so R s ⺡ and C is not reduced, a contradiction. This proves that C s 0 and A is divisible. For each p g ⺠ and each group G we have pG s G iff HomŽ⺪ p⬁ , EŽ G .rG. s 0 iff ExtŽ⺪ p⬁ , G . s 0. So ⺪ p⬁ H coincides with the Žtorsion. class of all p-divisible groups. On the other hand, ⺡ H is just the class of all cotorsion groups. If q g ⺠ _ P, then ⺪ q n g AH for all n - , but [n- ⺪ q n f Q H , w14, Corollary 54.4x. Since AH is a torsion class, we infer that A has the form Ž8. for P s ⺠ Žand R P s ⺡.. Assume ␥ s 0. We will show that qy1 f R for all q g ⺠ _ P. Indeed, assume qy1 g R for some q g ⺠. Since ⺪ q n is cotorsion for all n - , we get ⺪ q n g R H for all n - . On the other hand, a direct computation shows that [n- ⺪ q n f R H . Since AH is a torsion class, we infer that some ⺪ q n f AH , hence q g P. This proves that R is the subring of ⺡ generated by ⺪ j py1 < p g P 4 . So A has the form Ž8. for the given P. Conversely, assume A is of the form Ž8.. If  / 0, then GenŽ A. s ModR P is the class of all groups which are p-divisible for all p g P. If  s 0, then GenŽ A. is the class of all divisible torsion groups D such that Tq Ž D . s 0 whenever q g ⺠ _ P. Since R Py and ⺪-extensions of R P-modules coincide, we have Mod-R P : R PH . So if  / 0, then GenŽ A. s AH s Mod-R P , and A is a tilting group. If  s 0, then GenŽ A. ; AH s Mod-R P , so A is partial tilting, but not tilting. COROLLARY 2.4 Ž V s L.. Ži. Let A g Mod-⺪. Then A is tilting if and only if A has the form Ž8. with  / 0. Žii. Let T be a torsion class in Mod-⺪. Then T is tilting if and only if there is P : ⺠ such that T s A g Mod y ⺪ < pA s A ᭙ p g P 4 .
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