aa r X i v : . [ m a t h . K T ] D ec HOMOLOGIES OF INVERSE LIMITS OF GROUPS
DANIL AKHTIAMOV
Abstract.
Let H n be the n -th group homology functor (with integer coeffcients)and let { G i } i ∈ N be any tower of groups such that all maps G i +1 → G i are surjective.In this work we study kernel and cokernel of the following natural map: H n (lim ←− G i ) → lim ←− H n ( G i )For n = 1 Barnea and Shelah [BS] proved that this map is surjective and its kernelis a cotorsion group for any such tower { G i } i ∈ N . We show that for n = 2 the kernelcan be non-cotorsion group even in the case when all G i are abelian and after it westudy these kernels and cokernels for towers of abelian groups in more detail. Introduction
It is well-known that H n commute with direct limits for any n ≥
0, where H n = H n ( − , Z ) s a functor from topological space or from groups to abelian groups.But in general it is not true for projective limits. Moreover, it is quite difficult to under-stand whether they commute or not in some concrete cases. Some of these cases are con-nected with difficult problems. For example, it is known that lim ←− H ( F/γ n ( F )) = 0 forfinitely generated free group F . Although it turned out difficult to understand whether H (lim ←− F/γ n ( F )) = 0 or not and it is still an open problem. And H (lim ←− F/γ n ( F )) = 0will imply that answer to the Strong Parafree Conjecture, which is also still open, isnegative [Hill, p. 294]. Also in some cases it is quite easy to see that kernel of thenatural map H n (lim ←− G i ) → lim ←− H n ( G i ) is non-zero, but it is hard to say somethingabout structure of this kernel. For example, let n = 1 and G i = F × i , where, again, F is a finitely generated free group. It is proven by Miasnikov and Kharlampovich [KM]that the kernel in this case contains 2-torsion but their prove is quite difficult and usesvery heavy machinery called ”Non-commutative Implicit Function Theorem”. And,additionally, question about p -torsions for prime p > F (lim ←− G i ) → lim ←− F ( G i ), where F is a functor, especially in the case F = H n . Itis well-known that for functor π n : T op ∗ → Ab and for a tower of connected pointedspaces X i , such that maps X i +1 → X i are fibrations there is the following short exactsequence, called Milnor sequence (see [GJ, VI Proposition 2.15]):0 → lim ←− π n +1 ( X i ) → π n (lim ←− X i ) → lim ←− π n ( X i ) → H n : Ch Z → Ab , where Ch Z is the category ofchain complexes of abelian groups and H n is n − th chain homology functor (see, for The author is supported by the grant of the Government of the Russian Federation for thestate support of scientific research carried out under the supervision of leading scientists, agreement14.W03.31.0030 dated 15.02.2018. instance, [W, p. 94, prop. 3.5.8]): Let C i be a tower of chain complexes (of abeliangroups) such that it satisfies degree-wise the Mittag-Leffler condition. Then there isthe following short exact sequence:0 → lim ←− H n +1 ( C i ) → H n (lim ←− C i ) → lim ←− H n ( C i ) → X i , such that maps X i +1 → X i are fibrations. (Un)fortunately, the answer is no for two reasons. First, the kernelmight be non-zero, while lim ←− H n +1 ( X i ) = 0. To see it let us consider X i = K ( F/γ i ( F )),where F is 2-generated free group. Using fibrant replacments, we can assume that allmaps X i +1 → X i are fibratons. Then, using Milnor sequences, we see that lim ←− X i = K (lim ←− F/γ n ( F ) , ←− H n +1 ( X i ) = 0for every n ≥ H ( F/γ n ( F )) =0 (and it implies that wedge of two circles is Z -bad space; actually wedge of two circlesis also Q -bad and Z /p -bad space for p >
2, but it was proven much more later byIvanov and Mkhailov in the papers [IM2], [IM3]). Second, the cokernel might be non-zero. Corresponding example actually was provided by Dwyer in [D] (Example 3.6).See Corollary 4 from my work for more details on this example.
Definition.
We call an abelian group A a cotorsion group if A = lim ←− B i forsome tower of abelian groups B i . Remark.
Usually people define cotorsion groups in a different way but thesedefinitions coincide because of [H] (Theorem 1). More detailed, there is the followingequivalent definition of cotorsion groups:
Definition.
We call an abelian group A a cotorsion group if Ext ( C, A )=0 forany torsion-free abelian group C .There is the following result which was proven by Shelah and Barnea ([BS], Corol-lary 0.0.9): Theorem.
Let X i be an inverse system of pointed connected spaces, such thatall maps X i +1 → X i are Serre fibrations, π ( X i ) and π ( X i ) satisfies the Mittag-Lefflercondition. Then the natural map H (lim ←− X i ) → lim ←− H ( X i ) is surjective and its kernelis a cotorsion group. This result might give one a hope that, assuming towers π ( X i ) , π ( X i ) , . . . , π k ( X i )for large enough (maybe infinite) k satisfy Mittag-Leffler condition, we can ”fix” usualMilnor sequences and provide ”Milnor sequences for H n ”.(Un)fortunately, the answer is no already for n = 2. Moreover, it is false even forhomologies of abelian groups and it is proven in this work: Theorem 1.
There is an inverse system of abelian groups indexed by N suchthat all maps A i +1 → A i are epimorhisms and kernel of the natural map H (lim ←− A i ) → lim ←− H ( A i ) is not a cotorsion group. This theorem shows that things are very difficult even for abelian groups, so thefollowing result seems quite interesting:
OMOLOGIES OF INVERSE LIMITS OF GROUPS Theorem 2.
Let A i be an inverse system of torsion-free abelian groups indexedby N such that all maps A i +1 → A i are epimorhisms. Then for any n ∈ N the naturalmap H n (lim ←− A i ) → lim ←− H n ( A i ) is an embedding and its cokernel is a cotorsion group. Theorem 3.
Let A i be an inverse system of any abelian groups indexed by N such that all maps A i +1 → A i are epimorhisms. Then:(1) Cokernel of the natural map H (lim ←− A i ) → lim ←− H ( A i ) is a cotorsion group(2) Suppose, additionally, that torsion subgroup of A i is a group of bounded ex-ponent for any i . Then cokernel of the natural map H (lim ←− A i ) → lim ←− H ( A i ) is acotorsion group. In particular, it is true if all A i are finitely generated. Also in this work there are two results not about homologies but about anotherfunctors and inverse limits of abelian groups:
Statement 2.
Let B be any abelian group and B i be an inverse system of abeliangroups with surjective maps f i : B i +1 → B i between them. Then kernel of the map T or ( B, lim ←− B i ) → lim ←− T or ( B, B i ) is trivial. Corollary 6.
T or (lim ←− A i , lim ←− A i ) → lim ←− T or ( A i , A i ) is embedding for any inversesystem of abelian groups such that all maps A i +1 → A i are surjective. The author thanks Roman Mikhailov, Sergei O. Ivanov, Emmanuel Farjoun, FedorPavutnitsky and Saharon Shelah for fruitful discussions.2.
Homologies of inverse limits of groups
A goal of this chapter is to prove the following results:
Theorem 1.
There is an inverse system of abelian groups indexed by N suchthat all maps A i +1 → A i are epimorhisms and kernel of the natural map H (lim ←− A i ) → lim ←− H ( A i ) is not a cotorsion group. Theorem 2.
Let A i be an inverse system of torsion-free abelian groups indexedby N such that all maps A i +1 → A i are epimorhisms. Then for any n ∈ N the naturalmap H n (lim ←− A i ) → lim ←− H n ( A i ) is embedding and its cokernel is a cotorsion group. Theorem 3.
Let A i be an inverse system of any abelian groups indexed by N such that all maps A i +1 → A i are epimorhisms. Then:(1) Cokernel of the natural map H (lim ←− A i ) → lim ←− H ( A i ) is a cotorsion group(2) Suppose, additionally, that torsion subgroup of A i is a group of bounded ex-ponent for any i . Then cokernel of the natural map H (lim ←− A i ) → lim ←− H ( A i ) is acotorsion group. In particular, it is true if all A i are finitely generated.Proof of Theorem 1. Let us consider A ′ i,p = Z i ⊕ ( ∞ M i +1 Z /p Z ) , A ′ i := M p ∈ P A ′ i,p , B := M p ∈ P Z /p Z , A i := A ′ i × B We are going to define maps ψ i : A ′ i → A ′ i − in the most natural way. Let us denoteby e i,p , . . . , e ii,p elements of basis of the free abelian summand of A ′ i,p and let us denoteby e i +1 i,p , e i +2 i,p , . . . elements of basis of the Z /p Z -vector space, which is the second directsummand of A ′ i,p . Now let us define ψ i ( e ji,p ) := e ji − ,p . It is obvious that there is unique DANIL AKHTIAMOV ψ i with such properties. And, finally, let φ i : A i → A i − be the map defined by thematrix (cid:20) ψ i Id B (cid:21) . Using Kunneth formula and usig that all A ′ i and B are abelian, we have: H (lim ←− A i ) = H (lim ←− ( A ′ i × B )) = H ((lim ←− A ′ i ) × B ) = H (lim ←− A ′ i ) ⊕ H ( B ) ⊕ ((lim ←− A ′ i ) ⊗ B )In the similar way we can get:lim ←− H ( A i ) = lim ←− H (( A ′ i × B )) = lim ←− H ( A ′ i ) ⊕ H ( B ) ⊕ ( A ′ i ⊗ B ) = lim ←− H ( A ′ i ) ⊕ H ( B ) ⊕ lim ←− ( A ′ i ⊗ B )Thus our map H (lim ←− A i ) → lim ←− H ( A i ) is a map from H (lim ←− A ′ i ) ⊕ H ( B ) ⊕ ((lim ←− A ′ i ) ⊗ B ) to lim ←− H ( A ′ i ) ⊕ H ( B ) ⊕ lim ←− ( A ′ i ⊗ B ). Analyzing maps from the Kunnethformula, we see that this map is given by a diagonal matrix, the corresponding maps H (lim ←− A ′ i ) → lim ←− H ( A ′ i ) and (lim ←− A ′ i ) ⊗ B → lim ←− ( A ′ i ⊗ B ) coincide with obvious mapswhich come from definition of inverse limit and the corresponding map H ( B ) → H ( B )is isomorphism. Thus kernel of the natural map (lim ←− A ′ i ) ⊗ B → lim ←− ( A ′ i ⊗ B ) is directsummand of kernel of the natural map H (lim ←− A i ) → lim ←− H ( A i ). So it is enough toprove that kernel of the natural map (lim ←− A ′ i ) ⊗ B → lim ←− ( A ′ i ⊗ B ) is not a cotorsiongroup and now we will do it. Let us note that the map (lim ←− A ′ i ) ⊗ B → lim ←− ( A ′ i ⊗ B )can be decomposed in the following way:(lim ←− A ′ i ) ⊗ B = M p ∈ P (lim ←− A ′ i ) ⊗ Z /p Z → M p ∈ P lim ←− ( A ′ i ⊗ Z /p Z ) → lim ←− ( M p ∈ P A ′ i ⊗ Z /p Z ) = lim ←− ( A ′ i ⊗ B )It is easy to see that the map M p ∈ P lim ←− ( A ′ i ⊗ Z /p Z ) → lim ←− ( M p ∈ P A ′ i ⊗ Z /p Z ) is injective.Then kernel of the map (lim ←− A ′ i ) ⊗ B → lim ←− ( A ′ i ⊗ B ) equals M p ∈ P Ker [(lim ←− A ′ i ) ⊗ Z /p Z → lim ←− ( A ′ i ⊗ Z /p Z )]. It is proven at [IM] (Corollary 2.5) that Ker [(lim ←− A ′ i ) ⊗ Z /p Z → lim ←− ( A ′ i ⊗ Z /p Z )] ∼ = lim ←− ( T or ( A ′ i , Z /p Z )). Let us note that T or ( A ′ i , Z /p Z ) = ∞ M i +1 Z /p Z .So lim ←− ( T or ( A ′ i , Z /p Z )) ∼ = lim ←− ( L ∞ i +1 Z /p Z ), which is, obviously, p -torsion abeliangroup and which is nonzero because of [MP] (p. 330, Proposition A.20). Then, fi-nally, we have that kernel of the map (lim ←− A ′ i ) ⊗ B → lim ←− ( A ′ i ⊗ B ) is not a cotorsiongroup because of [B] (Theorem 8.5). Q.E.D.We will need the following lemmas and statements in order to prove Theorem 2and Theorem 3. All inverse systems supposed to be indexed by N . Statement 1.
Let C be any category with projective limits and let B i be a tower ofobjects from C . Let F : C → Ab be a functor such that lim ←− n Coker [ F (lim ←− B i ) → F ( B n )] =0 . Then cokernel of the natural map F (lim ←− B i ) → lim ←− F ( B i ) is a cotorsion group. This statement was proved in the third version of Barnea’s and Shelah’s preprint[BS] in the case when functor F preserves surjection. However, their proof was quitecomplicated. Our proof of the statement is straightforward. OMOLOGIES OF INVERSE LIMITS OF GROUPS Proof.
Let denote by Φ i := Ker ( F (lim ←− B i ) → F ( B i )) and by Ψ i := Im ( F (lim ←− B i ) → F ( B i )). Then we have the following exact sequences:0 → Φ i → F (lim ←− B i ) → Ψ i → → Ψ i → F ( B i ) → Coker [ F (lim ←− B i ) → F ( B i )] → . From the second sequence we get by the assumption lim ←− n Coker [ F (lim ←− B i ) → F ( B n )] =0 lim ←− Ψ i = lim ←− F ( B i ). From the first we deduce a new exact sequence:0 → lim ←− Φ i → F (lim ←− B i ) → lim ←− Ψ i → lim ←− Φ i → → lim ←− Φ i → F (lim ←− B i ) → lim ←− F ( B i ) → lim ←− Φ i → ←− of any inverse system of abelian groups is a cotorsiongroup by [H] (Theorem 1). Q.E.D. Corollary 1.
Let B i be an inverse system of groups (respectively abelian groups)and any maps between them. Let F : Grp → Ab (respectively F : Ab → Ab ) be a functorsuch that lim ←− n Coker [ F (lim ←− B i ) → F ( B n )] = 0 . Then kernel of the map F (lim ←− B i ) → lim ←− F ( B i ) equals lim ←− Φ i , where Φ n = Ker [ F (lim ←− B i ) → F ( B n )] . Corollary 2.
Cokernel of the map Λ n (lim ←− B i ) → lim ←− Λ n ( B i ) is a cotorsion groupfor any inverse system B i , such that B i +1 → B i are epimorphisms.Proof. Since B i +1 → B i are epimorphisms, the maps Λ n (lim ←− B i ) → Λ n ( B i ) = 0are also epimorphisms (it easily follows from constructive description of projectivelimits in the category of groups). Then Λ n (lim ←− B i ) → Λ n ( B i ) is epimorphism, becauseΛ n is right-exact. Then Coker [Λ n (lim ←− B i ) → Λ n ( B i )] = 0 and we are done because ofStatement 1. Corollary 3.
Cokernel of the map H (lim ←− B i ) → lim ←− H ( B i ) is a cotorsion groupfor any inverse system of abelian groups B i , such that B i +1 → B i are epimorphisms.Proof. It is well-known [Breen, section 6] that H is naturally isomorphic to Λ in the category of abelian groups, so the corollary follows from Corollary 2. Definition.
Let G be any group. Let γ ( G ) := G and γ i +1 ( G ) := [ G, γ i ( G )]. Series γ ( G ) := G are called lower central series of a group G . Let denote b G := lim ←− G/γ i ( G ). Corollary 4.
Cokernel of the map H ( b G ) → lim ←− H ( G/γ i ( G )) is a cotorsiongroup for any group G .Proof . It is clear that the map H ( G ) → H ( G/γ i ( G )) factors through the map H ( b G ) → H ( G/γ i ( G )). Then it is enough to prove that lim ←− Coker [ H ( G ) → H ( G/γ i ( G )] =0. Let note that maps between groups Coker [ H ( G ) → H ( G/γ i ( G )] are zero. Really,using that H ( G ) = H ( G/γ i ( G )), we see from the 5-term exact sequence [Brown, p.47,exercise 6a] that these cokernels are equal to γ i ( G ) ∩ [ G,G ][ γ i ( G ) ,G ] = γ i ( G ) γ i +1 ( G ) . Q.E.D.Following statement is not really necessary for a proof of the Theorems, but it isinteresting by itself and makes clearer what is happening. DANIL AKHTIAMOV
Statement 2.
Let B be any abelian group and B i be an inverse system of abeliangroups with surjective maps f i : B i +1 → B i between them. Then kernel of the map T or ( B, lim ←− B i ) → lim ←− T or ( B, B i ) is trivial.Proof. It is proven in [IM] (Proposition 2.6) that for any free resolution P • of B there are the following exact sequences(take Λ = Z and use a fact which states thatring Z has a global dimension one):0 → H (lim ←− P • ⊗ B i ) → lim ←− T or ( B, B i ) → H (lim ←− P • ⊗ B i ) is isomorphic to lim ←− T or ( B, B i ).Let take P • be a minimal resolution, i.e. such that P s = 0 when s >
1. It ispossible because global dimension of Z equals 1.Let us note that the maps P s ⊗ lim ←− B i → lim ←− ( P s ⊗ B i ) are embeddings becauseof Lemma 1 ( s = 0 , C • is defined from this sequence):0 → P • ⊗ (lim ←− B i ) → lim ←− ( P • ⊗ B i ) → C • → P s =0 for s >
1, note that C s = 0 for s >
1. Thus we have the followingexact sequence: 0 → H ( P • ⊗ (lim ←− B i )) → H (lim ←− ( P • ⊗ B i ))But H ( P • ⊗ (lim ←− B i )) = T or ( B, lim ←− B i ) by definition and we already have gotthat H (lim ←− P • ⊗ B i ) = lim ←− T or ( B, B i ) so we are done. Q.E.D. Corollary 5.
T or (lim ←− A i , lim ←− A i ) → lim ←− T or ( A i , A i ) is an embedding for anyinverse system of abelian groups such that all maps A i +1 → A i are epimorphic.Proof. Let consider following commutative diagramm with ψ being isomorphism: T or (lim ←− A i , lim ←− A j ) lim ←− T or ( A i , A i )lim ←− T or ( A i , lim ←− A j ) lim ←− lim ←− T or ( A i , A j ) ϕf ψg But f is monomorphism because of Statement 3 and g is monomorphism becauseof Statement 3 and left exactness of lim ←− . Q.E.D. Statement 3.
Let B be a cotorsion group and A be another abelian group. Letsuppose that there are i : A → B and π : B → A , such that πi = nId A for n ≥ .Then A is a cotorsion group.Proof. Since
Ext ( Q , B ) = 0, a map πi = nId A induces zero endomorphism of Ext ( Q , A ). Let consider the following exact sequences(we denote n -torsion subgroupof A by A n ): 0 → A n → A → nA → → nA → A → A/nA → OMOLOGIES OF INVERSE LIMITS OF GROUPS It is known that any n -torsion group is a cotorsion group (see [B], Theorem 8.5).Thus the maps A → nA and nA → A induce isomorphisms on Ext ( Q , − ), andthen so do nId A . But it induces zero endomorphism of Ext ( Q , A ), so we are done.Q.E.D. Proof of Theorem 3.
We already proved the first point of this theorem (Corollary2), so let us prove second point of the theorem. It is known [Breen, section 6] thatfor any abelian group A there is the following short exact sequence (here L Λ is firstderived functor of functor Λ ):0 → Λ ( A ) → H ( A ) → L Λ ( A ) → A = lim ←− A i we have the following sequence:0 → Λ (lim ←− A i ) → H (lim ←− A i ) → L Λ (lim ←− A i ) → (lim ←− A i +1 ) → Λ (lim ←− A i ) is an epimorphism for any i , lim ←− Λ (lim ←− A i ) = 0and we have the following sequence:0 → lim ←− Λ ( A i ) → lim ←− H ( A i ) → lim ←− L Λ ( A i ) → Ker [ L Λ (lim ←− A i ) → lim ←− L Λ ( A i )] ⊆ Ker [ T or (lim ←− A i , lim ←− A i ) → lim ←− T or ( A i , A i )] =0 (here we used Corollary 5). Then, using Snake Lemma, we have the following shortexact sequence:0 → Coker [Λ (lim ←− A i → lim ←− Λ ( A i )] → Coker [ H (lim ←− A i ) → lim ←− H ( A i )] → Coker [ L Λ (lim ←− A i ) → lim ←− L Λ ( A i )] → Coker [Λ (lim ←− A i → lim ←− Λ ( A i )] is acotorsion group. Then it is enough to prove that Coker [ L Λ (lim ←− A i ) → lim ←− L Λ ( A i )]is a cotorsion group.It is known [Breen, sections 4 and 5] that for any abelian group A there are maps L Λ ( A ) → T or ( A, A ) and
T or ( A, A ) → L Λ ( A ), such that their composition equalsto 2 Id L Λ ( A ) . Then they induce natural maps Coker [ L Λ (lim ←− A i ) → lim ←− L Λ ( A i )] → Coker [ T or (lim ←− A i , lim ←− A i ) → lim ←− T or ( A i , A i )] and Coker [ L Λ (lim ←− A i ) → lim ←− L Λ ( A i )] → Coker [ T or (lim ←− A i , lim ←− A i ) → lim ←− T or ( A i , A i )], such that their composition is mul-tiplication by two. Then, using Statement 3, we see that it is enough to provethat Coker [ T or (lim ←− A i , lim ←− A i ) → lim ←− T or ( A i , A i )] is a cotorsion group. But if tor-sion subgroups of A i are groups of bounded exponent, then T or ( A i , A i ) are torsiongroups of bounded exponent. Hence they are cotorsion groups [B, Theorem 8.5], andlim ←− T or ( A i , A i ) is a cotorsion groups, because it is an inverse limit of cotorsion groups(it is obvious from our second definition of cotorsion groups that an inverse limitof cotorsion groups is itself a cotorsion group). Thus Coker [ T or (lim ←− A i , lim ←− A i ) → lim ←− T or ( A i , A i )] is a cotorsion group as an image of a cotorsion group and we are done.Q.E.D.We need the following statement in order to prove Theorem 2: DANIL AKHTIAMOV
Statement 4.
Let A i be an inverse system of torsion-free abelian groups indexedby N such that all maps A i +1 → A i are epimorphisms and let B be any torsion-freeabelian group. Then the natural map B ⊗ lim ←− A i → lim ←− ( B ⊗ A i ) is an embedding.Proof. First let prove the statement for B = Q . Let us consider the followingshort exact sequence: 0 → Z → Q → Q / Z → − ⊗ A i for this sequence we get:0 → T or ( Q / Z , A i ) → A i → Q ⊗ A i → Q / Z ⊗ A i → T or ( Q / Z , A ) = t ( A ) for any abelian group A , we have:0 → t ( A i ) → A i → Q ⊗ A i → Q / Z ⊗ A i → ←− , we have:0 → lim ←− t ( A i ) → lim ←− A i → lim ←− ( Q ⊗ A i )Let us apply the exact functor Q ⊗ − for this sequence:0 → Q ⊗ lim ←− t ( A i ) → Q ⊗ lim ←− A i → Q ⊗ lim ←− ( Q ⊗ A i )Let us note that Q ⊗ lim ←− ( Q ⊗ A i ) ∼ = lim ←− ( Q ⊗ A i ), because lim ←− ( Q ⊗ A i ) is a Q -vectorspace. It means that we got the following:0 → Q ⊗ lim ←− t ( A i ) → Q ⊗ lim ←− A i → lim ←− ( Q ⊗ A i )But in the our case t ( A i ) = 0 for any i . So we have:0 → Q ⊗ lim ←− A i → lim ←− ( Q ⊗ A i )Now let us prove the statement in the case B = Q ⊕ I , where I is any cardinal.Using the sequence for B = Q we get:0 → ( Q ⊗ lim ←− t ( A i )) ⊕ I → ( Q ⊗ lim ←− A i ) ⊕ I → (lim ←− ( Q ⊗ A i )) ⊕ I Since it is obvious that the map (lim ←− ( Q ⊗ A i )) ⊕ I → lim ←− (( Q ⊗ A i ) ⊕ I ) is injective, wehave: 0 → ( Q ⊗ lim ←− t ( A i )) ⊕ I → ( Q ⊗ lim ←− A i ) ⊕ I → lim ←− (( Q ⊗ A i ) ⊕ I )But in the our case t ( A i ) = 0 for any i . So we have the exactness of the followingsequence: 0 → ( Q ⊗ lim ←− A i ) ⊕ I → lim ←− (( Q ⊗ A i ) ⊕ I )Since the functor −⊗ A commutes with direct sums for any A , we proved the statementin the case when B is any Q -vector space.Now let us prove the statement for any torsion-free B . Let Q B be the injectivehull of B . Since B is torsion-free, Q B = Q ⊕ I for some cardinal I . Let us consider thefollowing short exact sequence( B ′ is defined from this sequence):0 → B → Q B → B ′ → ←− A i is torsion-free and lim ←− B ⊗ A i = 0 because the maps B ⊗ A i +1 → B ⊗ A i are epimorphisms, it gives us two sequences:0 → B ⊗ lim ←− A i → Q B ⊗ lim ←− A i → B ′ ⊗ lim ←− A i → OMOLOGIES OF INVERSE LIMITS OF GROUPS → lim ←− ( B ⊗ A i ) → lim ←− ( Q B ⊗ A i ) → lim ←− ( B ′ ⊗ A i ) → → Ker [ B ⊗ lim ←− A i → lim ←− ( B ⊗ A i )] → Ker [ Q B ⊗ lim ←− A i → lim ←− ( Q B ⊗ A i )]Since Q B is a Q -vector space, we already proved that Ker [ Q B ⊗ lim ←− A i → lim ←− ( Q B ⊗ A i )] =0. Then Ker [ B ⊗ lim ←− A i → lim ←− ( B ⊗ A i )] = 0 and we are done. Q.E.D. Proof of Theorem 2.
Let consider following commutative diagram with ψ beingisomorphism.lim ←− A i ⊗ lim ←− A j lim ←− ( A i ⊗ A i )lim ←− ( A i ⊗ lim ←− A j ) lim ←− lim ←− ( A i ⊗ A j ) ϕf ψg Then we see that f is embedding because of Statement 4 and g is embeddgingbecause of Statement 4 and left-exactness of lim ←− . This implies that φ is also embedding.Then, since there is a natural embedding Λ ( A ) → A ⊗ A and H is naturally isomorphicto Λ [Breen, section 6] for abelian groups, we proved the theorem for n = 2.Now let us note that H n is naturally isomorphic to Λ n on the category of torsion-free abelian groups also for any n > n .We can assume that n ≥
3. Let us note that the map (lim ←− A i ) ⊗ n → lim ←− ( A i ) ⊗ n may be decomposed up to isomorphism in the following way:(lim ←− A i ) ⊗ n ∼ = lim ←− A i ⊗ (lim ←− A j ) ⊗ ( n − → lim ←− A i ⊗ lim ←− ( A j ) ⊗ ( n − → lim ←− i lim ←− j A i ⊗ ( A j ) ⊗ ( n − ∼ =lim ←− ( A i ) ⊗ n . But all maps in the decomposition are monic because of inductional as-sumption, left-exactness of lim ←− and exactness of − ⊗ B for torsion-free B . Then themap (lim ←− A i ) ⊗ n → lim ←− ( A i ) ⊗ n is also monic, and then so is H n (lim ←− A i ) → lim ←− H n ( A i ).Q.E.D. 3. Applications to topology
Theorem 4.
Let X i be an inverse system of pointed connected spaces, such thatall maps X i +1 → X i are Serre fibrations, all π ( X i ) are abelian, all maps π ( X i +1 ) → π ( X i ) are epimorphisms and π ( X i ) i ∈ N satisfy the Mittag-Leffler condition. Then:(1) Cokernel of the natural map H (lim ←− X i ) → lim ←− H ( X i ) is a cotorsion group.(2) Suppose, additionally, that all π ( X i ) are torsion-free. Then kernel of thenatural map H (lim ←− X i ) → lim ←− H ( X i ) is a cotorsion group.(3) Suppose that condition (2) is satisfied and, additionally, π ( X i ) i ∈ N satisfy theMittag-Leffler condition. Then the natural map H (lim ←− X i ) → lim ←− H ( X i ) is embedding.Proof. Let us denote Π ( X ) := Im [ π ( X ) → H ( X )]. It is obvious that Π is afunctor. Then we have the following sequence:0 → Π ( X ) → H ( X ) → H ( π ( X )) → Then we get the following sequences, feeding X = lim ←− X i and X = X i :0 → Π ( X i ) → H ( X i ) → H ( π ( X i )) → → lim ←− Π ( X i ) → lim ←− H ( X i ) → lim ←− H ( π ( X i )) → lim ←− Π ( X i ) → lim ←− H ( X i ) → lim ←− H ( π ( X i )) → π ( X i ) → Π ( X i ) is epimorphism, lim ←− π ( X i ) → lim ←− Π ( X i ) it is well-known(e.g. see [Brown, p. 42, Theorem 5.2]) that for any space X there is the following exactsequence: π ( X ) → H ( X ) → H ( π ( X )) →
0. also epimorphism. Then lim ←− π ( X i ) = lim ←− Π ( X i ) = 0, because π ( X i ) satisfy theMittag-Leffler condition. So we have the following sequence:0 → lim ←− Π ( X i ) → lim ←− H ( X i ) → lim ←− H ( π ( X i )) → → Π (lim ←− X i ) → H (lim ←− X i ) → H ( π (lim ←− X i )) → X ii ∈ N is a tower of pointed fibrations [GJ,VI Proposition 2.15]:0 → lim ←− π ( X i ) → π (lim ←− X i ) → lim ←− π ( X i ) → π ( X i ) satisfy the Mittag-Leffler condition, we have: π (lim ←− X i ) ∼ = lim ←− π ( X i )Finally, we have two sequences:0 → lim ←− Π ( X i ) → lim ←− H ( X i ) → lim ←− H ( π ( X i )) → → Π (lim ←− X i ) → H (lim ←− X i ) → H (lim ←− π ( X i )) →
0. Let us note that
Coker [lim ←− Π ( X i ) → Π (lim ←− X i )] = 0 and Ker [lim ←− Π ( X i ) → Π (lim ←− X i )] ∼ = lim ←− π ( X i ) . It follows from the following sequence:0 → lim ←− π ( X i ) → π (lim ←− X i ) → lim ←− π ( X i ) → → lim ←− π ( X i ) → Ker [lim ←− H ( X i ) → H (lim ←− X i )] → Ker [lim ←− H ( π ( X i )) → H (lim ←− π ( X i ))] → Coker [ H (lim ←− X i ) → lim ←− H ( X i )] ∼ = Coker [ H (lim ←− π ( X i )) → lim ←− H (lim ←− ( X i ))Then point (1) of the Theorem follows from point (1) of Theorem 1. Let us provepoints (2) and (3) of the Theorem. It follows from Theorem 2 for n = 2 that Ker [lim ←− H ( π ( X i )) → H (lim ←− π ( X i ))] = 0. Then we have: Ker [lim ←− H ( X i ) → H (lim ←− X i )] ∼ = lim ←− π ( X i )So we proved point (3) and point (2). Q.E.D. OMOLOGIES OF INVERSE LIMITS OF GROUPS Theorem 5.
Let Y i be a sequence of spaces. Then cokernel of the map H k ( ∞ Y i =1 Y i ) → lim ←− n H k ( n Y i =1 Y i ) is a cotorsion group for every k .Proof. Since composition of the natural maps H k ( n Y i =1 Y i ) → H k ( ∞ Y i =1 Y i ) and H k ( ∞ Y i =1 Y i ) → H k ( n Y i =1 Y i ) is the identity map, the map H k ( ∞ Y i =1 Y i ) → H k ( n Y i =1 Y i ) is sur-jective. So we are done because of Statement 1. Q.E.D. Remark.
It is easy to see from this from this proof that the same fact holds for anycategory which has infinite products instead of category of spaces and for and functor F from this category to Ab instead of H k . But I formulated it in this way because itseems more natural in this section.We will also prove the following Theorem, which does not follow from our previousresults but follows from Shelah’s and Barnea’s. It is connected with Theorem 3, so Ifound quite natural to formulate and prove it here. This Theorem is a generalization ofShelah’s and Barnea’s [Corollary 0.0.9. BS]. Actually we show that it is not necessaryto assume that π ( X i ) satisfies the Mittag-Leffler conditon. Theorem 6.
Let X i be an inverse system of pointed connected spaces, suchthat all maps X i +1 → X i are Serre fibrations and π ( X i ) satisfies the Mittag-Lefflercondition. Then the natural map H (lim ←− X i ) → lim ←− H ( X i ) is surjective and its kernelis a cotorsion group.Proof. Let us consider the following exact sequence:0 → lim ←− π ( X i ) → π (lim ←− X i ) → lim ←− π ( X i ) → ←− X i is also connected and we have: H (lim ←− X i ) ∼ = π (lim ←− X i ) ab It follows that:
Ker [ H (lim ←− X i ) → lim ←− H ( X i )] = Ker [( π (lim ←− X i )) ab → lim ←− ( π ( X i )) ab ]and Coker [ H (lim ←− X i ) → lim ←− H ( X i )] = Coker [( π (lim ←− X i )) ab → lim ←− ( π ( X i )) ab ]. Alsowe have the following sequence:0 → lim ←− π ( X i ) → π (lim ←− X i ) → lim ←− π ( X i ) → ←− π ( X i ) → ( π (lim ←− X i )) ab → (lim ←− π ( X i )) ab → ←− π ( X i ) is a cotorsion group by the Theorem of Huber(Theorem 1, [H])and any quotient group of a cotorsion group is itself a cotorsion group, Ker [( π (lim ←− X i )) ab → (lim ←− π ( X i )) ab ] is a cotorsion group. Now consider the map (lim ←− π ( X i )) ab → lim ←− ( π ( X i )) ab .It is surjective and its kernel is a cotorsion group by Theorem 0.0.1 of Shelah and Barneafrom [BS]. Now note that we can decompose the map ( π (lim ←− X i )) ab → lim ←− ( π ( X i )) ab in the following way:( π (lim ←− X i )) ab → (lim ←− π ( X i )) ab → lim ←− ( π ( X i )) ab Then the map is surjective as a composition of two surjective maps and its kernel isan extension of a cotorsion group by a cotorsion group. Thus it is obvious from oursecond definition of cotorsion groups that the kernel itself is a cotorsion group . Q.E.D.4.