Central extensions of some linear cycle sets
aa r X i v : . [ m a t h . K T ] M a r CENTRAL EXTENSIONS OF SOME LINEAR CYCLE SETS
JORGE A. GUCCIONE AND JUAN J. GUCCIONE
Abstract.
For each member A of a family of linear cycle sets whose underlying abelian group is cyclicof order a power of a prime number, we compute all the central extensions of A by an arbitrary abeliangroup. Contents
Introduction A cycle set , as defined in [21], is a set A endowed with a binary operation · , such that the left translations a b · a are bijective and the identities ( a · b ) · ( a · c ) = ( b · a ) · ( b · c ) are satisfied. In [21] it was proved that non-degenerate cycle sets (i.e., with invertible squaring map a a · a )are in bijective correspondence with non-degenerate involutive set-theoretic solutions of the Yang-Baxterequation, whose study was started by Etingof, Schedler, and Soloviev in [10]. These solutions are connectedwith many domains of algebra: Garside structures, Hopf-Galois theory, affine torsors, Artin-Schelter regularrings, groups of I -type, left symmetric algebras, etcetera (see, for instance [4, 5, 7–9, 11–16, 18, 22, 23]). A linear cycle set is a cycle set ( A, · ) endowed with an abelian group operation + satisfying the identities a · ( b + c ) = a · b + a · c and ( a + b ) · c = ( a · b ) · ( a · c ) . The interest in these structures is due to the fact that they are equivalent to brace structures, and sothey are strongly related with the non-degenerated involutive set theoretic solutions of the Yang-Baxterequations. For instance, the structure group of a non-degenerate solution [10] is a brace in a natural way.Motivated by the problem of the classify braces, in [2] the authors point out the importance of to developa extensions theory of braces (or equivalently, of linear cycle sets). This was made out by Bachiller in [1],using the language of braces; by Ben David and Ginosar in [3], using the language of bijective -cocycles(other avatar of linear cycle sets); and by Lebed and Vendramin in [19], using the language of linear cyclesets. In the last approach the authors introduce cohomology theories, RH ∗ N ( A , Γ) and H ∗ N ( A , Γ) , in orderto classify the central cycle type extensions and the central extensions of a linear cocycle A = ( A, + , · ) by an abelian group Γ , respectively. These cohomologies are defined by using explicit cochain complexes (cid:0) RC ∗ N ( A , Γ) , ∂ ∗ (cid:1) and (cid:0) C ∗ N ( A , Γ) , ∂ ∗ (cid:1) . This allows to use homological methods in order to studied them.To be something more precise, when A is a group ( A, +) endowed with the trivial linear cycle set structure a · b := b , one can use resolutions, satellite functors, simplicial methods, etcetera, to make calculations andto obtain theoretical results about RH ∗ N ( A , Γ) and H ∗ N ( A , Γ) ; and it is reasonable to expect that, under Mathematics Subject Classification.
Key words and phrases.
Cycle sets, Yang Baxter equation, extensions, cohomology. right circumstances, these calculations and results can be extended to more general types linear cycle sets.For example, this occurs if the necessary hypotheses to apply the Perturbation Lemma are satisfied.Let p ∈ N be a prime number and let ν, η ∈ N be such that < ν ≤ η ≤ ν . Let u := p ν , v := p η , t := p η − ν , u ′ := p ν − η and let A be the linear cycle set ( Z /v Z ; · ) , where ı · := (1 − uı ) . Let Γ be anadditive abelian group. The main result of this paper are the following: Theorem A.
Assume that u = v . For each γ, γ ∈ Γ such that vγ = 0 , let ξ γ , ξ γ : Z v Z × Z v Z → Γ be themaps defined by ξ γ ( ı , ı ) := ξ γ ([ g ı ⊗ g ı ]) and ξ γ ( ı , ı ) := ξ γ ( g ı ⊗ g ı ) , where ξ γ and ξ γ are as above of Proposition 3.15. The following facts hold: (1) Γ × Z v Z is a linear cycle set via ( c, ı ) + ( c ′ , ı ′ ) := (cid:0) c + c ′ + ξ γ ( ı, ı ′ ) , ı + ı ′ (cid:1) and ( c, ı ) · ( c ′ , ı ′ ) := (cid:0) c ′ + ξ γ ( ı, ı ′ ) , ı · ı ′ (cid:1) . Following [19] we denoted this linear cycle set by Γ ⊕ ξ γ ,ξ γ Z v Z . Moreover ⊕ ξ γ ,ξ γ Z v Z Z v Z , ι π where ι and π are the evident maps, is a central extension of ( Z /v Z ; · ) by Γ in the sense of [19, Def-inition 5.5] . (2) The extension associated with ( ξ γ , ξ γ ) and ( ξ γ ′ , ξ γ ′ ) are equivalent if and only if γ ′ = γ and vγ ′ = vγ ; and each central extension of ( Z /v Z ; · ) by Γ , is equivalent to one of these. Theorem B.
Assume that < u < v . For each γ, γ ∈ Γ such that vγ = uγ , let ξ γ , ξ γ ,γ : Z v Z × Z v Z → Γ be the maps defined by ξ γ ( ı , ı ) := ξ γ ([ g ı ⊗ g ı ]) and ξ γ ,γ ( tı + , ı ) := ξ γ ,γ ( g ı ⊗ g tı + ) , where ξ γ and ξ γ ,γ are as above of Proposition 3.17, ≤ ı < u and ≤ < t . The following facts hold: (1) Γ × Z v Z is a linear cycle set via ( c, ı ) + ( c ′ , ı ′ ) := (cid:0) c + c ′ + ξ γ ( ı, ı ′ ) , ı + ı ′ (cid:1) and ( c, ı ) · ( c ′ , ı ′ ) := (cid:0) c ′ + ξ γ ,γ ( ı, ı ′ ) , ı · ı ′ (cid:1) . Following [19] we denoted this linear cycle set by Γ ⊕ ξ γ ,γ ,ξ γ Z v Z . Moreover ⊕ ξ γ ,γ ,ξ γ Z v Z Z v Z , ι π where ι and π are the evident maps, is a central extension of ( Z /v Z ; · ) by Γ in the sense of [19, Def-inition 5.5] . (2) The extension associated with ( ξ γ , ξ γ ,γ ) and ( ξ γ ′ , ξ γ ′ ,γ ′ ) are equivalent if and only if γ − γ ′ ∈ u Γ and t ( γ − γ ′ ) = γ − γ ′ ; and each central extension of ( Z /v Z ; · ) by Γ , is equivalent to one of these. Theorem C.
Assume that u = 2 and v = 4 . For each γ, γ , γ ′ ∈ Γ such that γ = 2 γ and γ ′ = 0 , let ξ γ , ξ γ ,γ ′ ,γ : Z Z × Z v Z → Γ be the maps defined by ξ γ ( ı , ı ) := ξ γ ([ g ı ⊗ g ı ]) and ξ γ ,γ ′ ,γ (2 ı + , ı ) := ξ γ ,γ ′ ,γ ( g ı ⊗ g ı + ) , where ξ γ and ξ γ ,γ ′ ,γ are as above of Proposition 3.19, ≤ ı, < . The following facts hold: (1) Γ × Z v Z is a linear cycle set via ( c, ı ) + ( c ′ , ı ′ ) := (cid:0) c + c ′ + ξ γ ( ı, ı ′ ) , ı + ı ′ (cid:1) and ( c, ı ) · ( c ′ , ı ′ ) := (cid:0) c ′ + ξ γ ,γ ′ ,γ ( ı, ı ′ ) , ı · ı ′ (cid:1) . Following [19] we denoted this linear cycle set by Γ ⊕ ξ γ ,γ ′ ,γ ,ξ γ Z v Z . Moreover ⊕ ξ γ ,γ ′ ,γ ,ξ γ Z v Z Z v Z , ι π where ι and π are the evident maps, is a central extension of ( Z /v Z ; · ) by Γ in the sense of [19, Def-inition 5.5] . (2) The extension associated with ( ξ γ , ξ γ ,γ ′ ,γ ) and ( ξ γ , ξ γ ,γ ′ ,γ ) are equivalent if and only if γ − γ ∈ , γ − γ = 2( γ − γ ) and γ ′ = γ ′ .; and each central extension of ( Z /v Z ; · ) by Γ , is equivalentto one of these. ENTRAL EXTENSIONS OF SOME LINEAR CYCLE SETS 3
In order to prove these results we first compute the normalized full linear cycle set cohomology H N ( A , Γ) .This is done in Theorems 3.14, 3.16 and 3.18. In this paper we work with the category of Z -algebras, all the maps are Z -linear, ⊗ means ⊗ Z and Hom means
Hom Z . Moreover, for each Z -algebra F , we set D := D/ Z . Given a Z -algebra D we call S . ( D ) the simplicial complex of D -bimodules with objects S n ( D ) := D ⊗ n +2 ,face maps µ i : S n ( D ) → S n − ( D ) ( i = 0 , . . . , n ) and degeneracy maps ǫ i : S n ( D ) → S n +1 ( D ) ( i = 0 , . . . , n ),defined by: µ i ( x ⊗ · · · ⊗ x n +1 ) := x ⊗ · · · ⊗ x i x i +1 ⊗ · · · ⊗ x n +1 ,ǫ i ( x ⊗ · · · ⊗ x n +1 ) := x ⊗ · · · ⊗ x i ⊗ ⊗ x i +1 ⊗ · · · ⊗ x n +1 . The chain complex associated with S . ( D ) is the Hochschild resolution ( D ⊗∗ +2 , b ′∗ ) of D , and the chaincomplex ( D ⊗ D ⊗∗ ⊗ D, b ′∗ ) , obtained dividing ( D ⊗∗ +2 , b ′∗ ) by the subcomplex generated by images ofthe degeneracy maps is the normalized Hochschild resolution of D . Let D e := D ⊗ D op be the envelopingalgebra of D . Each D -bimodule M is a left D e -module via ( d ⊗ d ′ ) m := dmd ′ and a right D e -module via m ( d ⊗ d ′ ) := d ′ md . Conversely, each left (right) D e -module is a D -bimodule in an evident way. Clearly,having a morphism of D -bimodules is the same as having a morphism of left (right) D e -modules.Let Υ be the family of all the epimorphism of D -bimodules which split as left D -module maps. Wesay that a D -bimodule X is Υ -relative projective if for each f : Y → Y in Υ and each a D -bimodulemap g : X → Y , there exists a D -bimodule map h : X → Y such that g = f h . It is well known thata D -bimodule X is Υ -relative projective if and only if there exists a left D -module X ′ such that X is adirect sum of D e ⊗ D X ′ . A complex of D -bimodules ( X ∗ , d ∗ ) is a Υ -relative projective resolution of D ifeach X n is Υ -relative projective, and there exists a D -bimodule morphism d : X → D such that D X X X X X · · · , d d d d d d is contractil as a left D -module complex. The complex ( D ⊗ D ⊗∗ ⊗ D, b ′∗ ) is an Υ -relative projective res-olution of D . Let µ D : D ⊗ D → D be the multiplication map. A contracting homotopy of D D ⊗ D D ⊗ D ⊗ D D ⊗ D ⊗ ⊗ D · · · , µ D b ′ b ′ b ′ as a complex of left D -modules, is the degree map ξ ∗ , given by ξ n +1 ( x ) := ( − n +1 x ⊗ for x ∈ D ⊗ D ⊗ n ⊗ D . Using relative projective resolutions, a theory of relative derived functors can be developed,which is similar to the standard one (see [20]). Thus, we can define the Hochschild homology of D withcoefficients in a D-bimodule M as the tor relative to the family of epimorphisms Υ . So, the Hochschildhomology H ∗ ( D, M ) , of D with coefficients in M , is the homology of M ⊗ D e ( D ⊗ D ⊗∗ ⊗ D, b ′∗ ) . There arecanonical identifications ̥ n : M ⊗ D ⊗ n → M ⊗ D e ⊗ (cid:0) D ⊗ D ⊗ n ⊗ D (cid:1) , given by ̥ n ( m ⊗ x ) := m ⊗ D e (1 ⊗ x ⊗ .Using them we obtain that (cid:0) M ⊗ D ⊗∗ , b ∗ (cid:1) ≃ M ⊗ D e (cid:0) D ⊗ D ⊗∗ ⊗ D, b ′∗ (cid:1) , where b n ( m ⊗ x ⊗ · · · ⊗ x n ) := mx ⊗ x ⊗ · · · ⊗ x n + n − X i =1 ( − ı m ⊗ x ⊗ · · · ⊗ x ı − ⊗ x ı x ı +1 ⊗ x ı +2 ⊗ · · · ⊗ x n + ( − n x n m ⊗ x ⊗ · · · ⊗ x n − . If D is a projective Z -module (which is what occurs in this work), then D ≃ Z ⊕ D and the D -bimodules D ⊗ n +2 and D ⊗ D ⊗ n ⊗ D are projective. Therefore, in this case, the Hochschild homology can be definedusing the usual functor tor. The main purpose of our comment on relative derived functors above is tomake subsection 2.1 and the reference [17] more understandable. JORGE A. GUCCIONE AND JUAN J. GUCCIONE
A linear cycle set A := ( A ; · ) is an abelian additive group A , endowed with a binary operation · such thatthe left translations a a ′ · a are permutations of A and the following conditions are fulfilled ( a · a ′ ) · ( a · a ′′ ) = ( a ′ · a ) · ( a ′ · a ′′ ) , [1.1] a · ( a ′ + a ′′ ) = a · a ′ + a · a ′′ , [1.2] ( a + a ′ ) · a ′′ = ( a · a ′ ) · ( a · a ′′ ) . [1.3]We will use multiplicative notation. Let G A := { X a : a ∈ A } , endowed with the group structure given by X a X a ′ := X a + a ′ . We set X a · X a ′ := X a · a ′ and X a · a ′ := a · a ′ . In [19, Section 4] the authors introduce theories of (co)homology, H N ∗ ( A , Γ) and H ∗ N ( A , Γ) , that we recallnow. For each s ≥ , we let sh( D ⊗ s ) denote the subgroup of D ⊗ s generated by the shuffles X σ ∈ sh l,s − l sg( σ ) d σ − (1) ⊗ · · · ⊗ d σ − (1) , taken for all ≤ l < s and d k ∈ D . Here sh l,s − l is the subset of all the permutations σ of s elementssatisfying σ (1) < · · · < σ ( l ) and σ ( l + 1) < · · · < σ ( s ) . For each r ≥ and s ≥ , let b C Nrs ( A , Z ) := M ( s ) ⊗ D ⊗ r , where M ( s ) := D ⊗ s sh( D ⊗ s ) . Given g , . . . , g s ∈ G A , we let [ g ⊗ · · · ⊗ g s ] denote the class of g ⊗ · · · ⊗ g s in M ( s ) . Consider the double complex (cid:0) b C N ∗∗ ( A , Z ) , ∂ h ∗∗ , ∂ v ∗∗ (cid:1) , where ∂ h rs ([ g ⊗ · · · ⊗ g s ] ⊗ g s +1 ⊗ · · · ⊗ g r + s ) := [ g s +1 · g ⊗ · · · ⊗ g s +1 · g s ] ⊗ g s +1 · g s +2 ⊗ · · · ⊗ g s +1 · g r + s + r + s − X = s +1 ( − − s [ g ⊗ · · · ⊗ g s ] ⊗ g s +1 ⊗ · · · ⊗ g − ⊗ g g +1 ⊗ g +2 ⊗ · · · ⊗ g r + s + ( − r [ g ⊗ · · · ⊗ g s ] ⊗ g s +1 ⊗ · · · ⊗ g r + s − and ∂ v rs ([ g ⊗ · · · ⊗ g s ] ⊗ g s +1 ⊗ · · · ⊗ g r + s ) := ( − r +1 [ g ⊗ · · · ⊗ g s ] ⊗ g s +1 ⊗ · · · ⊗ g r + s + s − X =1 ( − + r +1 [ g ⊗ · · · ⊗ g − ⊗ g g +1 ⊗ g +2 ⊗ · · · ⊗ g s ] ⊗ g s +1 ⊗ · · · ⊗ g r + s + ( − r + s +1 [ g ⊗ · · · ⊗ g s − ] ⊗ g s +1 ⊗ · · · ⊗ g r + s . Recall that the total complex of (cid:0) b C N ∗∗ ( A , Z ) , ∂ h ∗∗ , ∂ v ∗∗ (cid:1) is the chain complex (cid:0) b C N ∗ ( A , Z ) , ∂ ∗ (cid:1) , where b C Nn ( A , Z ) := M r + s = n b C Nrs ( A , Z ) and ∂ n | b C Nrs ( A , Z ) := ∂ h rs + ∂ v rs . Let Γ be an abelian additive group. The normalized full homology groups and the normalized full cohomol-ogy groups of A with coefficients in Γ are the homology groups of b C N ∗ ( A , Γ) := ( b C N ∗ ( A , Z ) , ∂ ∗ ) ⊗ Γ and thecohomology groups of b C ∗ N ( A , Γ) := Hom (cid:0) ( b C N ∗ ( A , Z ) , ∂ ∗ ) , Γ (cid:1) , respectively. We let b H N ∗ ( A , Γ) and b H ∗ N ( A , Γ) denote the full normalized homology and the full normalized cohomology, of A with coefficients in Γ . Remark . The complex b C N ∗ ( A , Γ) is not the complex (cid:0) C N ∗ ( A , Γ) , ∂ ∗ (cid:1) introduced in [19, Definition 4.2],but they are isomorphic via the maps Ξ rs : b C Nrs ( A , Γ) → C Nrs ( A , Γ) , given by Ξ rs ([ X a ⊗ · · · ⊗ X a s ] ⊗ X a s +1 ⊗ · · · ⊗ X a r + s ) := ( a s +1 , . . . , a r + s , a , . . . , a s ) . Similarly, b C ∗ N ( A , Γ) ≃ (cid:0) C ∗ N ( A , Γ) , ∂ ∗ (cid:1) , and so b H N ∗ ( A , Γ) = H N ∗ ( A , Γ) and b H ∗ N ( A , Γ) = H ∗ N ( A , Γ) . Next, we recall the perturbation lemma. We present the version given in [6].A special deformation retract ( Y ∗ , ∂ ∗ ) ( X ∗ , d ∗ ) p ∗ ı ∗ X ∗ X ∗ +1 , h ∗ +1 [1.4]consists of the following: ENTRAL EXTENSIONS OF SOME LINEAR CYCLE SETS 5 (1) Chain complexes ( Y, ∂ ) , ( X, d ) and morphisms ı , p between them, such that pı = id .(2) A homotopy h from ıp to id , such that hı = 0 , ph = 0 and hh = 0 .A perturbation of [1.4] is a map δ ∗ : X ∗ → X ∗− such that ( d + δ ) = 0 . We call it small if id − δ h isinvertible. In this case we write A := (id − δ h ) − δ and we consider the diagram ( Y ∗ , ∂ ∗ ) ( X ∗ , d ∗ + δ ∗ ) p ∗ ı ∗ X ∗ X ∗ +1 , h ∗ +1 [1.5]where ∂ := ∂ + pAi , i := i + hAi , p := p + pAh and h := h + hAh .In all the cases considered in this paper the morphism δ h is locally nilpotent (in other words, for all x ∈ X ∗ there exists n ∈ N such that ( δ h ) n ( x ) = 0 ). Consequently, (id − δ h ) − = P ∞ n =0 ( δ h ) n . Theorem 1.2 ([6]) . If δ is a small perturbation of [1.4] , then the diagram [1.5] is a special deformationretract. Proposition 1.3.
Consider morphisms of double complexes ( Y ∗∗ , ∂ h ∗∗ , ∂ v ∗∗ ) ( X ∗∗ , d h ∗∗ , d v ∗∗ ) , p ∗∗ ı ∗∗ [1.6] such that p ∗∗ ı ∗∗ = id . Assume that in each row s we have a special deformation retract ( Y ∗ s , ∂ h ∗ s ) ( X ∗ s , d h ∗ s ) p ∗ s ı ∗ s X ∗ s X ∗ +1 ,s , h ∗ +1 ,s [1.7] endowed with a small perturbation δ h ∗ s : X ∗ s → X ∗− ,s . Let A ∗∗ := (id − δ h ∗∗ h ∗∗ ) − δ h ∗∗ and consider thediagram ( Y ∗∗ , ∂ h ∗∗ , ∂ v ∗∗ ) ( X ∗∗ , d h ∗∗ + δ h ∗∗ , d v ∗∗ ) p ∗∗ ı ∗∗ X ∗∗ X ∗ +1 , ∗ , h ∗ +1 , ∗ , [1.8] where ∂ h := ∂ h + pAı , ı := ı + hAı , p := p + pAh and h := h + hAh . The following facts hold: (1) The maps ı ∗∗ and p ∗∗ are morphisms of double complexes such that p ∗∗ ı ∗∗ = id . (2) For each row s , the map h ∗ +1 ,s is a homotopy from ı ∗ s p ∗ s to id .Proof. Let ( Y ∗ , ∂ t ∗ ) and ( X ∗ , d t ∗ ) be the total chain complexes of ( Y ∗∗ , ∂ h ∗∗ , ∂ v ∗∗ ) and ( X ∗∗ , d h ∗∗ , d v ∗∗ ) , respec-tively. We have an homotopy equivalence data ( Y ∗ , ∂ t ∗ ) ( X ∗ , d t ∗ ) p t ∗ ı t ∗ X ∗ X ∗ +1 , h t ∗ +1 [1.9]where ı t ∗ , p t ∗ an h t ∗ are given by ı tn := L r + s = n ı rs , p tn := L r + s = n p rs and h tn +1 := L r + s = n +1 h rs . Considerthe small perturbation δ ∗ : X ∗ → X ∗− , given by δ n := L r + s = n δ rs . The result follows immediately byapplying the perturbation lemma to this case. (cid:3) Let D be the Z -algebra of a finite cyclic group. In this section we construct a chain complex suitable forour purposes, giving the Hochschild homology of the Z -algebra D with coefficients is an abelian group M ,which is considered as a D -bimodule via the trivial actions. This complex will be the complex ( Y ∗ , ∂ ∗ ) in aspecial deformation retract as in [1.4], in which ( X ∗ , d ∗ ) will be the normalized Hochschild chain complexof D with coefficients in M . It is natural to try to use the minimal resolution of D in order to construct ( Y ∗ , ∂ ∗ ) , but this does not work because, in this case, the perturbation is not small. So we are forced touse a more involved complex. Let v, t ∈ N with t < v . Assume that t | v and set u := v/t . Consider the cycle groups C v := h g i , C u := h x i and C t := h y i of order v , u and t , respectively. The group algebra D := Z [ C v ] is isomorphic to the crossedproduct E := B ⋊ ζ Z [ C t ] , in which B := Z [ C u ] , C t acts trivially on B and ζ is the cocycle given by ζ ( y , y ′ ) := ( if + ′ < t,x otherwise, JORGE A. GUCCIONE AND JUAN J. GUCCIONE where ≤ , ′ < t . We recall that E is a free Z -module with basis { x ı w y : 0 ≤ ı < u and ≤ < t } andthat the multiplication map of E is given by x ı w y x ı ′ w y ′ = x ı + ı ′ ζ ( y , y ′ ) w y + ′ , where ≤ , ′ < t . Themap f : E → D , defined by f ( x ı w y ) := g tı + , where ≤ < t , is an algebra isomorphism.For all α, β ≥ , let Y β := E ⊗ Z [ C t ] and X αβ := E ⊗ E . The groups X αβ are E -bimodules in an obviousmanner and the groups Y β are E -bimodules via the left regular action and the right action ( x ı w y ⊗ y k ) x h w y l := x ı + h ζ ( y k , y l ) w y ⊗ y k + l , where ≤ k, l < t . Consider the diagram of E -bimodules and E -bimodule maps... Y X X · · · Y X X · · · Y X X · · · , ∂ ∂ ∂ υ d d υ d d υ d d where v β ( w ⊗ w ) := w ⊗ and ∂ β − ( w ⊗
1) := w ⊗ y − w y ⊗ , d α − ,β ( w ⊗ w ) := w ⊗ xw − xw ⊗ w ,∂ β ( w ⊗
1) := t − X h =0 x u − w y h ⊗ y t − h − , d α,β ( w ⊗ w ) := u − X ı =0 x ı w ⊗ x u − ı − w . Clearly, the column and the rows of this diagram are chain complexes.
Proposition 2.1.
Each one of the rows of the above diagram is contractible as a ( E, Z ) -bimodule complex.A contracting homotopy σ β : Y β → X β and σ α +1 ,β : X αβ → X α +1 ,β for α ≥ , of the β -th row, is givenby σ β ( x ı w y ⊗ y h ) := x ı w y ⊗ w y h ,σ α − ,β ( x ı w y ⊗ x k w y h ) := k − X l =0 x ı + l w y ⊗ x k − l − w y h ,σ α,β ( x ı w y ⊗ x k w y h ) := δ k,u − x ı w y ⊗ w y h , where ≤ k < u and δ k,u − is the delta of Kronecker.Proof. We must check that v β σ β = id Y β , σ β v β + d β σ β = id X β and σ α,β d α,β + d α +1 ,β σ α +1 ,β id X αβ . [2.1]A direct computation shows that v β σ β ( w ⊗ y ) = v β (cid:0) w ⊗ w y (cid:1) = w ⊗ y ,σ β v β ( w ⊗ x ı w y ) = σ β (cid:0) x ı w ⊗ y (cid:1) = x ı w ⊗ w y ,d α +1 ,β σ α +1 ,β ( w ⊗ x ı w y ) = ı − X l =0 d α +1 ,β (cid:0) x l w ⊗ x ı − l − w y (cid:1) = w ⊗ x ı w y − x ı w ⊗ w y ,d α,β σ α,β ( w ⊗ x ı w y ) = ( ≤ ı < u − , P u − l =0 x l w ⊗ x u − l − w y j ı = u − , σ α +1 ,β d α +1 ,β ( w ⊗ x ı w y ) = ( w ⊗ x ı w y if ≤ ı < u − , − P u − l =0 x l +1 w ⊗ x u − l − w y if ı = u − , σ α,β d α,β ( w ⊗ x ı w y ) = u − X l =0 σ α,β (cid:0) x l w ⊗ x u − l − ı w y (cid:1) = x ı w ⊗ w y . Equalities [2.1] follows immediately from these facts. (cid:3)
ENTRAL EXTENSIONS OF SOME LINEAR CYCLE SETS 7
Proposition 2.2.
The complex of E -bimodules E Y Y Y Y Y Y · · · , e µ ∂ ∂ ∂ ∂ ∂ ∂ where e µ is the E -bimodule morphism given by e µ ( w ⊗
1) := w , is contractible as a left E -module complex.A contracting homotopy σ − : E → Y and σ − β +1 : Y β → Y β +1 for β ≥ , is given by σ − ( x ı w y ) := x ı w y ⊗ ,σ − β ( x ı w y ⊗ y k ) := δ t − ,k x ı +1 w y ⊗ ,σ − β +1 ( x ı w y ⊗ y k ) := k − X l =0 x ı ζ ( y , y l ) w y + l ⊗ y k − l − , where ≤ k < t .Proof. A direct computation shows that e µσ − ( w ) = e µ (cid:0) w ⊗ (cid:1) = w ,σ − e µ ( w ⊗ y ) = σ − (cid:0) w y (cid:1) = w y ⊗ ,∂ β +1 σ − β +1 ( w ⊗ y ) = − X l =0 ∂ β +1 (cid:0) w y l ⊗ y − l − (cid:1) = w ⊗ y − w y ⊗ ,∂ β σ − β ( w ⊗ y ) = ∂ β (cid:0) δ t − , xw ⊗ (cid:1) = δ t − , t − X l =0 w y l ⊗ y t − l − ,σ − β +1 ∂ β +1 ( w ⊗ y ) = σ − β +1 (cid:0) w ⊗ y +1 − w y ⊗ y (cid:1) = ( w ⊗ y if < t − , − P t − l =0 w y l +1 ⊗ y t − l − if = t − , σ − β ∂ β ( w ⊗ y ) = t − X l =0 σ − β (cid:0) x u − ζ ( y t − l − , y ) w y l ⊗ y t − l − (cid:1) = w y ⊗ . The result follows easily from these facts. (cid:3)
For α ≥ and ≤ l ≤ β , we define E -bimodule maps d lαβ : X αβ → X α + l − ,β − l , recursively by: d l ( z ) := − σ ∂ υ ( z ) if l = 1 and α = 0 , − σ d d ( z ) if l = 1 and α > , − P l − =1 σ d l − d ( z ) if < l and α = 0 , − P l − =0 σ d l − d ( z ) if < l and α > , [2.2]for z ∈ E ⊗ Z w . Theorem 2.3.
Let Υ be the family of all the epimorphism of E -bimodules which split as left E -modulemaps. The chain complex E X X X X X · · · , µ E d d d d d where µ E is the multiplication map, X n := L α + β = n X αβ and d n is the E -bimodule morphism defined by d n ( x ) := n X l =1 d l n ( x ) if x ∈ X n , n − α X l =0 d lα,n − α ( x ) if x ∈ X α,n − α with α > ,is a Υ -relative projective resolution of E .Proof. This is an immediate consequence of [17, Corollary A2]. (cid:3)
Remark . In the previous definition and in the rest of this work we identify each X rs with its imageinside X α + β . JORGE A. GUCCIONE AND JUAN J. GUCCIONE
In order to carry out our computations we also need to give an explicit contracting homotopy of thisresolution. For this we define left E -module maps σ ll,β − l : Y β −→ X l,β − l and σ lα + l +1 ,β − l : X αβ −→ X α + l +1 ,β − l , recursively by σ lα + l +1 ,β − l := − P l − ı =0 σ d l − ı σ ı ( < l ≤ β and α ≥ − ). Proposition 2.5.
A contracting homotopy σ : E → X and σ n +1 : X n → X n +1 ( n ≥ ), of the resolutionintroduced in Theorem 2.3, is given by σ ( x ) := σ σ − ( x ) and σ n +1 ( x ) := − n +1 X l =0 σ ll,n − l +1 σ − n +1 υ n ( x ) + n X l =0 σ ll +1 ,n − l ( x ) if x ∈ X n , n − α X l =0 σ lα + l +1 ,n − α − l ( x ) if x ∈ X α,n − α with α > .Proof. This is a direct consequence of [17, Corollary A2]. (cid:3)
The following theorem gives a closed expression of the homomorphisms d lαβ that appear in the relativeprojective resolution of E , obtained above. Theorem 2.6.
The maps d l vanish for l > . Moreover d α, β +1 ( w ⊗ w ) = ( − α (cid:0) w y ⊗ w − w ⊗ w y (cid:1) , d α,β ( w ⊗ w ) = − x u − w ⊗ w ,d α, β ( w ⊗ w ) = ( − α +1 t − X h =0 x u − w y h ⊗ w y t − h − , d α +1 ,β ( w ⊗ w ) = 0 . Proof.
We sketch the proof. We first prove the formula for d αβ by induction on α . By equality [2.2], wehave d , β +1 ( w ⊗ w ) = − σ , β ∂ β +1 υ β +1 ( w ⊗ w ) = w y ⊗ w − w ⊗ w y and d , β ( w ⊗ w ) = − σ , β − ∂ β υ β ( w ⊗ w ) = − t − X h =0 x u − w y h ⊗ w y t − h − , which proves the case α = 0 . Assume the formula is true for α . Then, d α +1 , β +1 ( w ⊗ w ) = − σ α +1 , β d α, β +1 d α +1 , β +1 ( w ⊗ w ) = ( − α +1 ( w y ⊗ w − w ⊗ w y ) and d α +1 , β ( w ⊗ w ) = − σ α +1 , β − d α, β d α +1 , β ( w ⊗ w ) = ( − α t − X h =0 x u − w y h ⊗ w y t − h − , as desired. We next prove the formula for d αβ . For α = 0 , we have d β ( w ⊗ w ) = − σ ,β − d ,β − d β ( w ⊗ w ) = − x u − w ⊗ w . Assume the formula is true for α . Then, d α +1 ,β ( w ⊗ w ) = − σ α +2 ,β − (cid:0) d αβ d α +1 ,β + d α +1 ,β − d α +1 ,β (cid:1) ( w ⊗ w ) = ( if α is even, − x u − w ⊗ w if α is odd,as desired. Finally, since σ α +1 ,β − d α +1 ,β − d αβ ( w ⊗ w ) = σ α +1 ,β − d α,β − d αβ ( w ⊗ w ) = σ α +2 ,β − d α +1 ,β − d αβ ( w ⊗ w ) = 0 , from equality [2.2] it follows that d l = 0 for l > . (cid:3) Proposition 2.7.
The homotopy σ found in Proposition 2.5 satisfies σ n +1 ( x ) = − σ ,n +1 σ − n +1 υ n ( x ) + n X l =0 σ ll +1 ,n − l ( x ) for all x ∈ X n . ENTRAL EXTENSIONS OF SOME LINEAR CYCLE SETS 9
Proof.
By the definitions of σ , υ and σ − , it suffices to prove that σ ll, β − l (cid:0) x ı w y ⊗ (cid:1) = σ ll, β +1 − l (cid:0) x ı w y ⊗ y k (cid:1) = 0 for all l ≥ and ≤ k < t − . [2.3]By the definition of σ l and Theorem 2.6, for this it sufficient to consider the cases l = 1 and l = 2 .Moreover, since the σ l ’s are left E -linear, we can assume that ı = = 0 . For l = 1 , we have σ l, β − l (cid:0) w ⊗ (cid:1) = − σ , β − d , β σ , β ( w ⊗
1) = t − X h =0 σ , β − (cid:0) x u − w y h ⊗ w y t − h − (cid:1) = 0 and σ l, β (cid:0) w ⊗ y k (cid:1) = − σ , β d , β +1 σ , β +1 ( w ⊗ y k ) = σ , β ( w ⊗ w y k +1 − w y ⊗ w (cid:1) = 0 . Therefore, σ , β − (cid:0) w ⊗ (cid:1) = − σ , β − d , β σ , β ( w ⊗
1) = σ , β − ( x u − w ⊗ w ) = 0 and σ , β − (cid:0) w ⊗ y k (cid:1) = − σ , β − d , β − σ , β − ( w ⊗ y k ) = σ , β − ( x u − w ⊗ w y k ) = 0 , which finishes the proof. (cid:3) Remark . Let ≤ < t . A direct computation shows that σ σ − υ ( w ⊗ x ı w y ) = (P − l =0 x ı w y l ⊗ w y − l − if w ⊗ x ı w y ∈ X , β , δ t − , x ı +1 w ⊗ w if w ⊗ x ı w y ∈ X , β +1 . Proposition 2.9.
Let ≤ ı < u and ≤ < t . For all α ≥ and β ≥ , the following equalities hold: σ α +2 , β ( w ⊗ x ı w y ) = ( − α +1 δ u − ,ı δ t − , w ⊗ w ,σ α +2 , β − ( w ⊗ x ı w y ) = ( − α +1 δ u − ,ı − X l =0 x u − w y l ⊗ w y − l − . Moreover, σ lα + l +1 ,β − l = 0 for all l ≥ , α ≥ and β ≥ l .Proof. We sketch the proof. By the discussion above Proposition 2.5, we have σ α +2 , β ( w ⊗ x ı w y ) = − σ α +2 , β d α +1 , β +1 σ α +1 , β +1 ( w ⊗ x ı w y ) = ( − α +1 δ u − ,ı δ t − , w ⊗ w , and σ α +2 , β − ( w ⊗ x ı w y ) = − σ α +2 , β − d α +1 , β σ α +1 , β ( w ⊗ x ı w y ) = ( − α +1 δ u − ,ı − X l =0 x u − w y l ⊗ w y − l − , which proves the statement for σ . Our next purpose is to prove that σ α +3 ,β − = 0 . We assert that σ α +3 ,β − d α +1 ,β σ α +1 ,β = 0 . In fact, if α is even this follows from the fact that d α +1 ,β = 0 , while if α isodd, then the assertion is also true, because σ α +3 ,β − d α +1 ,β σ α +1 ,β ( w ⊗ x ı w y ) = − δ u − ,ı σ α +3 ,β − (cid:0) x u − w ⊗ w y (cid:1) = 0 . Thus, σ α +3 ,β − ( w ⊗ x ı w y ) = − σ α +3 ,β − d α +2 ,β − σ α +2 ,β − ( w ⊗ x ı w y ) = 0 , as desired. Since, moreover d = 0 , in order to prove that σ α +4 ,β − = 0 it suffices to check that the equality σ α +4 ,β − d α +2 ,β − σ α +2 ,β − = 0 holds. If α is odd this follows from the fact that d α +2 ,β − = 0 ; while, if α is even, then a direct computation proves that we also have σ α +4 ,β − d α +2 ,β − σ α +2 ,β − ( w ⊗ x ı w y ) = 0 .The proof that σ lα + l +1 ,β − l = 0 for l ≥ , follows easily by induction. (cid:3) Let ( E ⊗ E ⊗∗ ⊗ E, b ′∗ ) be the normalized Hochschild resolution of the Z -algebra E . It is easy to see thatthere exist unique morphisms of E -bimodule chain complexes φ ∗ : ( X ∗ , d ∗ ) −→ ( E ⊗ E ⊗∗ ⊗ E, b ′∗ ) and ϕ ∗ : ( E ⊗ E ⊗∗ ⊗ E, b ′∗ ) −→ ( X ∗ , d ∗ ) , such that- φ = ϕ = id E ⊗ E ,- ϕ n +1 ( w ⊗ x ⊗ w ) = σ n +1 ϕ n b ′ n +1 ( w ⊗ x ⊗ w ) for all n ≥ and x ∈ E ⊗ n +1 , - the restriction of φ n +1 to X ı,n +1 − ı satisfies φ n +1 ( w ⊗ w ) = ξ n +1 φ n d n +1 ( w ⊗ w ) , where ξ n +1 is as in subsection 1.1. Proposition 2.10. ϕ ∗ φ ∗ = id and φ ∗ ϕ ∗ is homotopically equivalent to the identity map. A homotopy isthe one degree map ω ∗ +1 : φ ∗ ϕ ∗ → id , recursively defined by ω := 0 and ω n +1 ( y ) := ξ n +1 ( φ n ϕ n − id − ω n b ′ n )( y ) for n ≥ and y ∈ E ⊗ E ⊗ n ⊗ Z w . [2.4] Moreover, ϕ ∗ ω ∗ = 0 , ω ∗ +1 φ ∗ = 0 and ω ∗ +1 ω ∗ +1 = 0 .Proof. We prove the first two assertions by induction. Clearly ϕ φ = id . Assume that ϕ n φ n = id . Sincethe image of ξ n +1 is included in E ⊗ E ⊗ n +1 ⊗ Z w , we have ϕ n +1 φ n +1 ( y ) = σ n +1 ϕ n b ′ n +1 ξ n +1 φ n d n +1 ( y )= σ n +1 ϕ n φ n d n +1 ( y ) − σ n +1 ϕ n ξ n b ′ n φ n d n +1 ( y )= id n +1 ( y ) − d n +2 σ n +2 ( y ) , for y ∈ X ı,n +1 − ı ∩ ( E ⊗ Z w ) . Consequently, to conclude that ϕ n +1 φ n +1 = id it suffices to check that σ n +2 ( w ⊗ w ) = 0 , which follows easily from Proposition 2.7. Next we prove the second assertion. Clearly φ ϕ − id = 0 = b ′ ω . Let U n := φ n ϕ n − id and T n := U n − ω n b ′ n . Assuming that b ′ n ω n + ω n − b ′ n − = U n − ,we get that b ′ n +1 ω n +1 ( y ) + ω n b ′ n ( y ) = b ′ n +1 ξ n +1 T n ( y ) + ω n b ′ n ( y )= T n ( y ) − ξ n b ′ n T n ( y ) + ω n b ′ n ( y )= U n ( y ) − ξ n b ′ n U n ( y ) + ξ n b ′ n ω n b ′ n ( y )= U n ( y ) − ξ n U n − b ′ n ( y ) + ξ n b ′ n ω n b ′ n ( y ) , = U n ( y ) − ξ n U n − b ′ n ( y ) + ξ n U n − b ′ n ( y ) − ξ n ω n − b ′ n − b ′ n ( y )= U n ( y ) , for y ∈ E ⊗ E ⊗ n ⊗ Z , where the first equality holds by identity [2.4]; the second one, since ξ is a contractinghomotopy; the third one, by the definition of T n ; the fourth one, since U ∗ is a morphism; and the fifth one,by the assumption. (cid:3) For each α, β, n ∈ N such that α + β = n , we let ϕ αβn : E ⊗ E ⊗ n ⊗ E → X αβ denote the unique mapsuch that ϕ n = P α + β = n ϕ αβn . Remark . A direct computation using the definitions of φ ∗ and ϕ ∗ , shows that φ ( w ⊗ w ) = w ⊗ w y ⊗ w on X ,φ ( w ⊗ w ) = w ⊗ xw ⊗ w on X ,φ ( w ⊗ w ) = − t − X h =0 x u − w y h ⊗ w y ⊗ w y t − h − ⊗ w on X , φ ( w ⊗ w ) = w ⊗ w y ⊗ xw ⊗ w + w ⊗ xw ⊗ w y ⊗ w on X , φ ( w ⊗ w ) = u − X h =0 x h w ⊗ xw ⊗ x u − h − w ⊗ w on X ,ϕ ( w ⊗ x ı w y ⊗ w ) = − X h =0 x ı w y h ⊗ w y − h − and ϕ ( w ⊗ x ı w y ⊗ w ) = − ı − X h =0 x h w ⊗ x ı − h − w y , where ≤ ı < u and ≤ < t . Remark . A direct computation shows that ω ( w ⊗ x ı w y ⊗ w ) = − X h =0 x ı w y h ⊗ w y ⊗ w y − h − ⊗ w − ı − X h =0 x h w ⊗ xw ⊗ x ı − h − w y ⊗ w , where ≤ ı < u and ≤ < t . ENTRAL EXTENSIONS OF SOME LINEAR CYCLE SETS 11
Let v , u , t , D and E be as in Section 2.1 and let D := D/ Z . Recall that the map f : E → D , given by f ( x ı w y ) := g tı + , where ≤ < t , is an algebra isomorphism. Here we obtain a chain complex givingthe Hochschild homology of D with coefficients in a commutative group M , considered as a D -bimodulevia the trivial actions (that is g ı m = mg ı = m ). We are interested in the cases M := D ⊗ s / sh( D ⊗ s ) with s ∈ N .For each α, β ∈ N , let M αβ be a copy of M . Let d lαβ : M αβ → M α + l − ,β − l ( α, β ≥ , ≤ l ≤ min(2 , β ) and α + l > ) be the morphisms defined by: d α − ,β ( m ) := 0 , d α, β − ( m ) := 0 , d α,β ( m ) := − m,d α,β ( m ) := um, d α, β ( m ) := ( − α +1 tm, d α +1 ,β ( m ) := 0 . [2.5]By Theorem 2.6, tensoring M over D e with ( X ∗ , d ∗ ) and using the identifications θ αβ : M αβ → M ⊗ D e X αβ ,given by θ αβ ( m ) := m ⊗ w ⊗ w , we obtain the chain complex X ( M ) X ( M ) X ( M ) X ( M ) X ( M ) X ( M ) · · · , d d d d d d where X n ( M ) := L α + β = n M αβ and d n is the E -bimodule morphism defined by d n ( m ) := min( n, X l =1 d l n ( m ) if m ∈ M n , min( n − α, X l =0 d lα,n − α ( m ) if m ∈ M α,n − α with α > . [2.6]Let (cid:0) M ⊗ D ⊗∗ , b ∗ (cid:1) be the normalized Hochschild chain complex of the Z -algebra D with coefficients in M . Recall that there is a canonical identification (cid:0) M ⊗ D ⊗∗ , b ∗ (cid:1) ≃ M ⊗ D e (cid:0) D ⊗ D ⊗∗ ⊗ D, b ′∗ (cid:1) . Let φ ∗ : ( X ∗ ( M ) , d ∗ ) −→ (cid:0) M ⊗ D ⊗∗ , b ∗ (cid:1) and ϕ ∗ : (cid:0) M ⊗ D ⊗∗ , b ∗ (cid:1) −→ ( X ∗ ( M ) , d ∗ ) [2.7]be the morphisms of chain complexes induced by φ ∗ and ϕ ∗ , respectively. By definition φ = ϕ = id M .Moreover, by Proposition 2.10 we know that ϕ ∗ φ ∗ = id and φ ∗ ϕ ∗ is homotopically equivalent to theidentity map. More precisely, a homotopy ω ∗ +1 , from φ ∗ ϕ ∗ to id , is the family of maps (cid:16) ω n +1 : M ⊗ D ⊗ n −→ M ⊗ D ⊗ n +1 (cid:17) n ≥ , [2.8]induced by (cid:0) ω n +1 : E ⊗ E ⊗ n ⊗ E −→ E ⊗ E ⊗ n +1 ⊗ E (cid:1) n ≥ . By Proposition 2.10 we also know that ω = 0 , ϕ ∗ ω ∗ = 0 , ω ∗ +1 φ ∗ = 0 and ω ∗ +1 ω ∗ +1 = 0 .For each α, β, n ∈ N such that α + β = n , we let ϕ αβn : M ⊗ D ⊗ n → M αβ denote the unique map suchthat ϕ n = P α + β = n ϕ αβn .In Section 3 we will use the following result with M := D ⊗ s / sh( D ⊗ s ) . Proposition 2.13.
The following assertions hold: (1)
For each α, β ≥ , there exists x αβ ∈ D ⊗ α + β such that φ α + β ( m ) = m ⊗ x αβ , for all m ∈ M αβ . (2) For each α, β ≥ , there exists a map ψ αβα + β : D ⊗ α + β → Z such that ϕ αβα + β ( m ⊗ g ı ⊗ · · · ⊗ g ı α + β ) = ψ αβα + β ( g ı ⊗ · · · ⊗ g ı α + β ) m for all m ∈ M αβ . (3) For each n ≥ , there exists a map b ω n +1 : D ⊗ n → D ⊗ n +1 such that ω n +1 ( m ⊗ g ı ⊗ · · · ⊗ g ı n ) = m ⊗ b ω n +1 ( g ı ⊗ · · · ⊗ g ı r + s ) for all m ∈ M αβ .Proof. All the assertions follow from the fact that the left and right actions of D on M are trivial. (cid:3) Remark . By Remark 2.11, we have φ ( m ) = m ⊗ g on M , φ ( m ) = m ⊗ g t on M , φ ( m ) = − t − X l =1 m ⊗ g ⊗ g l on M , φ ( m ) = m ⊗ g ⊗ g t + m ⊗ g t ⊗ g on M , φ ( m ) = u − X l =1 m ⊗ g t ⊗ g tl on M , ϕ ( m ⊗ g tı + ) = m,ϕ ( m ⊗ g tı + ) = − ım, where ≤ ı < u and ≤ < t . Remark . By Remark 2.12, we have ω ( m ⊗ g tı + ) = − X l =1 m ⊗ g ⊗ g l − ı − X l =0 m ⊗ g t ⊗ g tl + , where ≤ ı < u and ≤ < t . Let p ∈ N be a prime number and let ν, η ∈ N be such that < ν ≤ η ≤ ν . Let v := p η , u := p ν , t := p η − ν and u ′ := p ν − η . Note that u ′ t = u and ut = v . Consider the linear cycle set A := ( Z /v Z ; · ) ,where ı · := (1 − uı ) . In this section we compute the cohomologies H N ( A , Γ) and H N ( A , Γ) of A with coefficients in an arbitrary abelian group Γ . Then, using the last result we prove Theorems A, Band C of the introduction. Let C v := h g i be the multiplicative cyclic group of order v , endowed with thebinary operation g ı · g := g ı · . Let D := Z [ C v ] and D := D/ Z . Let sh( D ⊗ s ) be as in subsection 1.2.1.For each r ≥ and s ≥ , let M ( s ) := D ⊗ s / sh( D ⊗ s ) and let U rs := X r ( M ( s )) = L α + β = r M ( s ) αβ ,where each M ( s ) αβ is a copy of M ( s ) . Consider the double complex ( U ∗∗ , d h ∗∗ , d v ∗∗ ) , where the s -row ( U ∗ s , d h ∗ s ) is the complex ( X ∗ ( M ( s )) , d ∗ ) , introduced in Subsection 2.2, and d v rs := L α + β = r d v αβs , in which d v αβs : M ( s ) αβ → M ( s − αβ is the map defined by d v αβs ([ g ı ⊗ · · · ⊗ g ı s ]) := ( − r +1 [ g ı ⊗ · · · ⊗ g ı s ]+ s − X j =1 ( − j + r +1 [ g ı ⊗ · · · ⊗ g ı j − ⊗ g ı j + ı j +1 ⊗ g ı j +2 ⊗ · · · ⊗ g ı s ]+ ( − r + s +1 [ g ı ⊗ · · · ⊗ g ı s − ] , [3.1]where [ g ı ⊗ · · · ⊗ g ı s ] , etcetera, are as in Subsection 1.2.1. Let A tr be the group Z /v Z , endowed with thetrivial structure of linear cycle set. For each s ≥ , let φ ∗ s : ( U ∗ s , d h ∗ s ) −→ (cid:0) b C N ∗ s ( A tr ) , ∂ h ∗ s (cid:1) and ϕ ∗ s : (cid:0) b C N ∗ s ( A tr ) , ∂ h ∗ s (cid:1) −→ ( U ∗ s , d h ∗ s ) be the maps φ ∗ and ϕ ∗ introduced in [2.7], with M replaced by M ( s ) . By items (1) and (2) of Proposi-tion 2.13, in the diagram ( U ∗∗ , d h ∗∗ , d v ∗∗ ) (cid:0) b C N ∗ s ( A tr ) , ∂ h ∗∗ , ∂ v ∗∗ (cid:1) ϕ ∗∗ φ ∗∗ [3.2]the maps φ ∗∗ and ϕ ∗∗ are morphisms of double complexes. Moreover, we know that ϕ ∗∗ φ ∗∗ = id , and thatin each row s , we have a special deformation retract ( U ∗ s , d h ∗ s ) (cid:0) b C N ∗ s ( A tr ) , ∂ h ∗ s ) ϕ ∗ s φ ∗ s b C N ∗ s ( A tr ) b C N ∗ +1 ,s ( A tr ) , ω ∗ +1 ,s [3.3]where ( ω n +1 ,s ) n ≥ is the family of maps ( ω n +1 ) n ≥ , introduce in [2.8], with M replaced by M ( s ) (see Sub-section 2.2). For each s ∈ N , we have a perturbation δ ∗ s : M ( s ) ⊗ D ⊗∗ → M ( s ) ⊗ D ⊗∗− , where δ ( g ı ⊗ g ı ) = g ı · g ı − g ı = g (1 − uı ) ı − g ı ,δ ( g ı ⊗ g ı ⊗ g ı ) := g ı · g ı ⊗ g ı · g ı − g ı ⊗ g ı = g (1 − uı ) ı ⊗ g (1 − uı ) ı − g ı ⊗ g ı ,δ ([ g ı ⊗ g ı ] ⊗ g ı ) := [ g ı · g ı ⊗ g ı · g ı ] − [ g ı ⊗ g ı ] = [ g (1 − uı ) ı ⊗ g (1 − uı ) ı ] − [ g ı ⊗ g ı ] ,δ ns = 0 for ns / ∈ { , , } . [3.4] ENTRAL EXTENSIONS OF SOME LINEAR CYCLE SETS 13
In order to carry out our computations we are going to apply Proposition 1.3 to this data. For this, we firstmust prove that δ ∗ is small. Since ω = ω = 0 , the unique non-trivial point is that δ ω is nilpotent.But, by Remark 2.15 and the fact that g · g ı = g (1 − u ) ı and g · g l = g (1 − u ) l = g t ( u − u ′ l )+ l , we have δ ω ( g ı ⊗ g tı + ) = − X l =1 g · g i ⊗ g · g l − − X l =1 g ı ⊗ g l = − X l =1 g ı g − uı ⊗ g t ( u − u ′ l )+ l − − X l =1 g i ⊗ g l , [3.5]where ≤ < t . Using this it is easy to see that ( δ ω ) t − = 0 . Remark . The chain double complex (cid:0) b C N ∗∗ ( A tr , Z ) , b ∂ h ∗∗ , ∂ v ∗∗ (cid:1) , obtained by applying the perturbation [3.4]to (cid:0) b C N ∗∗ ( A tr , Z ) , ∂ h ∗∗ , ∂ v ∗∗ (cid:1) , only coincides with (cid:0) b C N ∗∗ ( A , Z ) , ∂ h ∗∗ , ∂ v ∗∗ (cid:1) for ∗∗ = 01 , ∗∗ = 11 , ∗∗ = 21 , ∗∗ = 02 , ∗∗ = 12 and ∗∗ = 03 . Thus, the chain double complex U ( A ) := ( U ∗∗ , b d h ∗∗ , b d v ∗∗ ) , obtained by applyingProposition 1.3 to the above data, will be useful only to compute the full (co)homology of A in degrees and . Note that b d v ∗∗ = d v ∗∗ .For each α, β, r ∈ N and s ∈ N such that α + β = r , we let ϕ αβrs : M ( s ) ⊗ D ⊗ r → M ( s ) αβ denote theunique map such that ϕ rs = P α + β = r ϕ αβrs . Clearly ϕ αβrs is the map ϕ αβr introduced above Proposition 2.13,with M replaced by M ( s ) . A direct computation using equality [3.5] and Remark 2.14 shows that ϕ δ ω ( g ı ⊗ g tı + ) = j − X l =1 l ( g ı g − uı − g ı ) = (cid:18) (cid:19) g ı ( g − uı − and ϕ δ ω ( g ı ⊗ g tı + ) = − − X l =1 ( u − u ′ l ) g ı g − uı = u ′ (cid:18)(cid:18) j (cid:19) − (cid:18) j − (cid:19) t (cid:19) g ı g − uı , where ≤ < t . An inductive argument using these equalities, [3.5] and the fact that g u = 1 shows that, ϕ ( δ ω ) s ( g ı ⊗ g tı + ) = (cid:18) s + 1 (cid:19) g ı (cid:0) g − uı − (cid:1) s [3.6]and ϕ ( δ ω ) s ( g ı ⊗ g tı + ) = u ′ (cid:18)(cid:18) s + 1 (cid:19) − (cid:18) − s (cid:19) t (cid:19) g ı g − uı ( g − uı − s − , [3.7]for all ≤ < t and s ∈ N . Let U ( A ) T be the subcomplex U ( A ) T := U U U U U U , b d v03 b d v02 b d v12 b d h12 b d h11 b d h21 of U ( A ) . Recall that U rs = L α,β ≥ α + β = r M ( s ) αβ . Theorem 3.2.
The chain double complex U ( A ) T is a partial total complex of the diagram D ( A ) T := M (1) M (2) M (3) M (1) M (1) M (2) M (2) M (1) M (1) M (1) , b d h1021 b d h2021 b d h1012 b d v012 b d v003 b d h1011 b d h1111 b d v002 b d v102 b d h0201 where b d v ∗∗∗ = d v ∗∗∗ (see formula [3.1] ), and the other not zero maps are given by: b d h0201 ( g ı ) := ug ı , b d h1011 ( g ı ) := g ı ( g − uı − , b d h1111 ( g ı ) := g ı (1 − g − uı ) , b d h1021 ( g ı ) := − g ı t − X s =0 g − suı , b d h1012 ([ g ı ⊗ g ı ]) := [ g (1 − u ) ı ⊗ g (1 − u ) ı ] − [ g ı ⊗ g ı ] , b d h2021 ( g ı ) := − g ı + u ′ g ı t − X s =1 sg − uıs ! . Proof.
In order to prove this theorem we will apply Proposition 1.3 to the data consisting of diagrams [3.2]and [3.3]. We begin by computing the first row of U ( A ) T . Since ω = 0 and ϕ = id M (1) we know that b d h11 = d h11 + δ φ . Moreover, by Remark 2.14 and the definition of δ , we have δ φ ( g ı ) = 0 on M (1) and δ φ ( g ı ) = δ ( g ı ⊗ g ) = g (1 − u ) ı − g ı on M (1) .Consequently, by equalities [2.5] and [2.6], b d h11 ( g ı ) = 0 = b d h0101 ( g ı ) on M (1) and b d h11 ( g ı ) = g ı ( g − uı −
1) = b d h1011 ( g ı ) on M (1) .We next compute b d h21 . By Remark 2.14 and the definition of δ , we have δ φ ( g ı ) = 0 on M (1) , [3.8] δ φ ( g ı ) = g (1 − u ) ı ⊗ g (1 − u ) t − g ı ⊗ g t = g ı (cid:0) g − uı − (cid:1) ⊗ g t on M (1) , [3.9] δ φ ( g ı ) = t − X l =1 g ı ⊗ g l − t − X l =1 g ı g − uı ⊗ g t ( u − u ′ l )+ l on M (1) . [3.10]Hence, again by Remark 2.14, ϕ δ φ ( g ı ) = 0 on M (1) , [3.11] ϕ δ φ ( g ı ) = g ı (1 − g − uı ) on M (1) , [3.12] ϕ δ φ ( g ı ) = − (cid:18) t (cid:19) g ı (cid:0) g − uı − (cid:1) on M (1) , [3.13] ϕ δ φ ( g ı ) = u ′ (cid:18) t (cid:19) g ı g − uı on M (1) . [3.14]By equalities [2.5], [2.6] and [3.8], we obtain that b d h21 ( g ı ) = d h21 ( g ı ) = b d h0201 ( g ı ) on M (1) . More-over, by equalities [3.6], [3.7] and [3.9], ϕ (cid:0) δ ω (cid:1) s δ φ ( g ı ) = 0 on M (1) and ϕ (cid:0) δ ω (cid:1) s δ φ ( g ı ) = 0 on M (1) for all s ≥ . ENTRAL EXTENSIONS OF SOME LINEAR CYCLE SETS 15
Hence, by equalities [2.5], [2.6], [3.11] and [3.12], b d h21 ( g ı ) = d h21 ( g ı ) + ϕ δ φ ( g ı ) + ϕ δ φ ( g ı ) = b d h1111 ( g ı ) on M (1) .We now compute b d h21 on M (1) . Using again equalities [3.6] and [3.7], we obtain that ϕ (cid:0) δ ω (cid:1) s (cid:18) t − X l =1 g ı ⊗ g l (cid:19) = t − X l =1 (cid:18) ls + 1 (cid:19) g ı (cid:0) g − uı − (cid:1) s = (cid:18) ts + 2 (cid:19) g ı (cid:0) g − uı − (cid:1) s ,ϕ (cid:0) δ ω (cid:1) s (cid:18) t − X l =1 g ı g − uı ⊗ g t ( u − u ′ l )+ l (cid:19) = (cid:18) ts + 2 (cid:19) g ı g − uı (cid:0) g − uı − (cid:1) s ,ϕ (cid:0) δ ω (cid:1) s (cid:18) t − X l =1 g ı ⊗ g l (cid:19) = − u ′ ( s + 1) (cid:18) ts + 2 (cid:19) g ı g − uı ( g − uı − s − ,ϕ (cid:0) δ ω (cid:1) s (cid:18) t − X l =1 g ı g − uı ⊗ g t ( u − u ′ l )+ l (cid:19) = − u ′ ( s + 1) (cid:18) ts + 2 (cid:19) g ı g − uı ( g − uı − s − , for all s ≥ . So, by equality [3.10], ϕ (cid:0) δ ω (cid:1) s δ φ ( g ı ) = − (cid:18) ts + 2 (cid:19) g ı (cid:0) g − uı − (cid:1) s +1 on M (1) , [3.15] ϕ (cid:0) δ ω (cid:1) s δ φ ( g ı ) = u ′ ( s + 1) (cid:18) ts + 2 (cid:19) g ı g − uı (cid:0) g − uı − (cid:1) s on M (1) , [3.16]for all s ≥ . Thus, by equalities [3.13] and [3.15], t − X s =0 ϕ (cid:0) δ ω (cid:1) s δ φ ( g ı ) = − t − X s =1 (cid:18) ts + 1 (cid:19) g ı (cid:0) g − uı − (cid:1) s = − g ı − t + t − X s =1 g − suı ! , and, similarly, by equalities [3.14] and [3.16], t − X s =0 ϕ (cid:0) δ ω (cid:1) s δ φ ( g ı ) = t − X s =0 u ′ ( s + 1) (cid:18) ts + 2 (cid:19) g ı g − uı (cid:0) g − uı − (cid:1) s = u ′ g ı g − uı t − X s =0 ( s + 1) g − uıs ! . Consequently, by [2.5] and [2.6], we have b d h21 ( g ı ) = b d h1021 ( g ı ) + b d h2021 ( g ı ) . We now compute the second rowof U ( A ) T . Since ω = 0 and ϕ = id M (2) , we know that b d h12 = d h12 + δ φ . Moreover, by Remark 2.14and the definition of δ , δ φ ([ g ı ⊗ g ı ]) = 0 on M (2) and δ φ ([ g ı ⊗ g ı ]) = [ g (1 − u ) ı ⊗ g (1 − u ) ı ] − [ g ı ⊗ g ı ] on M (2) .Therefore, by equalities [2.5] and [2.6], b d h12 ([ g ı ⊗ g ı ]) = b d h0102 ([ g ı ⊗ g ı ]) on M (2) and b d h12 ([ g ı ⊗ g ı ]) = b d h1012 ([ g ı ⊗ g ı ]) on M (2) ,as desired. (cid:3) Remark . Let (cid:0) b C N ∗ ( A , Z ) , ∂ ∗ (cid:1) be as in subsection 1.2.1, let Tot( U ( A )) be the total complex of U ( A ) andlet b ϕ ∗∗ : (cid:0) b C N ∗∗ ( A , Z ) , ∂ h ∗∗ , ∂ v ∗∗ (cid:1) −→ U ( A ) be the map obtained by applying Proposition 1.3 to the specialdeformation retracts [3.3], endowed with the perturbations δ ∗ s given in [3.4]. By that proposition, themap b ϕ ∗ : (cid:0) b C N ∗ ( A , Z ) , ∂ ∗ (cid:1) → Tot( U ( A )) , induced by b ϕ ∗∗ , is an homotopy equivalence. Since ω = 0 and ω = 0 , we have b ϕ = ϕ = id M (1) and b ϕ = ϕ = id M (2) . On the other hand, by Remark 2.14 andequalities [3.6] and [3.7], for each ≤ ı < u and ≤ < t , we have b ϕ ( g ı ⊗ g tı + ) = b ϕ ( g ı ⊗ g tı + ) + b ϕ ( g ı ⊗ g tı + ) , where b ϕ : M (1) ⊗ D −→ M (1) and b ϕ : M (1) ⊗ D −→ M (1) are the maps given by b ϕ ( g ı ⊗ g tı + ) := − X s =0 (cid:18) s + 1 (cid:19) g ı (cid:0) g − uı − (cid:1) s [3.17]and b ϕ ( g ı ⊗ g tı + ) := − ıg ı + − X s =1 u ′ (cid:18)(cid:18) s + 1 (cid:19) − (cid:18) − s (cid:19) t (cid:19) g ı g − uı ( g − uı − s − . [3.18] Proposition 3.4.
For all ≤ ı < u and ≤ < t , the following identities hold: b ϕ ( g ı ⊗ g tı + ) = g ı − X l =0 g − ulı and b ϕ ( g ı ⊗ g tı + ) = − ıg ı + u ′ g ı − X l =1 ( − l − t ) g − ulı . Proof.
Both formulas follow by induction on using [3.17], [3.18] and that (cid:0) s +1 (cid:1) = (cid:0) − s (cid:1) + (cid:0) − s +1 (cid:1) . (cid:3) Here we will use freely the notations introduced in subsection 1.2.1. Let A be as at the beginning of thissection and let Γ be an additive abelian group. In Theorems 3.14. 3.16 and 3.18, we compute H N ( A , Γ) .Moreover, we obtain a family of -cocycles of (cid:0) C ∗ N ( A , Γ) , ∂ ∗ (cid:1) that applies surjectively on H N ( A , Γ) , andwe determine when two of these cocycles are cohomologous (see Propositions 3.15, 3.17 and 3.19). By[19, Theorem 5.8] this classify the central extensions of A by Γ . We use this fact in order to proveTheorems A, B and C.For each l ∈ N and α, β ∈ N we define Γ( l ) αβ := Hom( M ( l ) αβ , Γ) . There are obvious identifications Γ(1) αβ = X ≤ ı We have H N ( A , Γ) = b H N ( A , Γ) ≃ Γ u .Proof. By Remarks 1.1 and 3.1 we know that H N ( A , Γ) = b H N ( A , Γ) = H ( U ( A , Γ)) . By definition b d (cid:16)X γ ı g ı (cid:17) = X ı, ( γ ı + − γ ı − γ ) g ı ⊗ g and b d (cid:16)X γ ı g ı (cid:17) = X ı ( γ ı − uı − γ ı ) . From the first equality we get ker( b d ) = { X γ ı g ı : γ ı = ıγ and vγ = γ v = 0 } . Consequently, − uγ = (1 − u ) γ − γ = γ − u − γ = 0 , and so H ( U ( A , Γ)) = ker( b d ) ∩ ker( b d ) = { X γ ı g ı : γ ı = ıγ and uγ = 0 } , which is clearly isomorphic to Γ u . (cid:3) ENTRAL EXTENSIONS OF SOME LINEAR CYCLE SETS 17 Our next purpose is to compute b H N ( A , Γ) in the cases above mentioned. Lemma 3.7. P γ ı g ı ⊗ g ∈ ker( b d ) if and only if γ ı = P ı + − k = γ k − P ı − k =1 γ k for ≤ ı, < v .Proof. Assume that P γ ı g ı ⊗ g ∈ ker( b d ) . Then, for all a ≤ b , we have b d (cid:16)X γ ı g ı ⊗ g (cid:17) ( g ⊗ g a ⊗ g b ) = − γ ab + γ a +1 ,b − γ ,a + b + γ a . Thus, γ a +1 ,b = γ ab + γ ,a + b − γ a . An inductive argument using this fact proves that the statement is truewhen a ≤ b . For a > b , we have γ ab = γ ba = X a ≤ k b and b < ≤ v − ı + b , otherwise, [3.19]is the unique P γ ı g ı ⊗ g ∈ ker (cid:0) b d (cid:1) with γ b = γ and γ = 0 for = b . Note that X γ ı g ı ⊗ g = v − X b =1 f b ( γ b ) for each X γ ı g ı ⊗ g ∈ ker (cid:0) b d (cid:1) . Remark . A direct computation shows that b d ( γg ı ) = − v − X =1 = ı γ ( g ı ⊗ g + g ⊗ g ı ) − γg ı ⊗ g ı + v − X a,b =1 a + b ≡ ı (mod v ) γg a ⊗ g b . Thus, by Remark 3.8, b d ( γg ı ) = ( − f ( γ ) − f ( γ ) − · · · − f v − ( γ ) if ı = 1 , f ı − ( γ ) − f ı ( γ ) if ı = 1 .Consequently, since the f ı ( γ ) ’s generates ker (cid:0) b d (cid:1) , we have ker (cid:0) b d (cid:1) / Im (cid:0) b d (cid:1) = Γ /v Γ . Lemma 3.10. Let γ ∈ Γ . If u = v , then b d ( f ( γ )) = 0 . Otherwise b d ( f ( γ )) = − γg ⊗ g − u ′ X k =1 γg ⊗ g kt − u ′ − X k = u ′ +1 γg ⊗ g kt +1 − t − X h =2 ( h +1) u ′ − X k = hu ′ γg ⊗ g kt + h + v − X ı =2 v − X =1 (cid:0) Λ( γ, (1 − u ) ı, (1 − u ) − Λ( γ, ı (cid:1) g ı ⊗ g . [3.20] Proof. By definition b d ( f ( γ )) = X(cid:0) Λ( γ, (1 − u ) ı, (1 − u ) − Λ( γ, ı (cid:1) g ı ⊗ g . We will use [3.19] in order to compute Λ( γ, − u, (1 − u ) . In order to carry out this task, for each < < v we need to find k such that ≤ kv + (1 − u ) < v . But this happens if and only if ( k − v < ( u − ≤ kv ,and it is evident that such a k there exists and it is unique. Moreover, ≤ k < u and ( u − = kv .In fact, if k ≥ u , then kv + (1 − u ) > uv + (1 − u ) v ≥ v , while if ( u − 1) = kv , then v | , because gcd( u − , v ) = 1 . By equality [3.19], for all , k , such that < < v and < kv + (1 − u ) < v , we have Λ( γ, v − u +1 ,kv + (1 − u ) = γ if u = v and < kv + (1 − u ) ≤ , − γ if u < v and < kv + (1 − u ) ≤ u , otherwise. Consequently, Λ( γ, v − u +1 ,kv + (1 − u ) = γ if and only if u = v and k = = 1 . From this and Re-mark 3.8 it follows easily that if u = v , then b d ( f ( γ )) = 0 . Assume now that u < v . We nextpurpose is to determine when Λ( γ, v − u +1 ,kv + (1 − u ) = − γ . Note that < kv + (1 − u ) ≤ u ⇔ kv − u ≤ ( u − < kv − ⇔ kt + kt − uu − ≤ < kt + kt − uu − . Thus, Λ( γ, v − u +1 ,kv + (1 − u ) = − γ if and only if = kt + (cid:24) kt − uu − (cid:25) . Write k = hu ′ + l , where h = 0 and ≤ l ≤ u ′ , or h = 1 and ≤ l < u ′ , or < h < t and ≤ l < u ′ . Then kt − uu − hu ′ + l ) t − uu − hu + lt − uu − h − u − − h + lt + h − u − h − lt + h − u − . We claim that (cid:6) kt − uu − (cid:7) = h . To check this we must prove that < lt + h − < l . Assume first that h = 0 .Then lt − > , because t > and l > ; while lt − < u ′ t − u − , because < l < u ′ . Assume now that ≤ h < t . Then lt + h − > , because l > or h > ; while lt + h − ≤ ( u ′ − t + h − u − t + h − < u − .Summarizing, for all , k , such that < < v and < kv + (1 − u ) < v , we have Λ( γ, v − u +1 ,kv + (1 − u ) = − γ if and only if = kt if ≤ k ≤ u ′ , kt + 1 if u ′ + 1 ≤ k < u ′ , kt + h if hu ′ ≤ k < ( h + 1) u ′ with < h < t .Using this fact it is easy to see that equality [3.20] holds. (cid:3) Remark . Let B, Z ⊆ Γ(2) ⊕ Γ(1) ⊕ Γ(1) be the -coboundaries and the -cocycles of U ( A , Γ) respectively, and let Z ′ ⊆ Z be the subgroup of cocycles z = (cid:0)X γ ı g ı ⊗ g , X γ ı g ı , X γ ′ ı g ı (cid:1) , such that P γ ı g ı ⊗ g = f ( γ ) for some γ ∈ Γ . By Remark 3.9 we know that H ( U ( A , Γ)) = Z ′ /B ∩ Z ′ .Moreover, since f ( γ ) ∈ ker (cid:0) b d (cid:1) , a triple z = (cid:0) f ( γ ) , P γ ı g ı , P γ ′ ı g ı (cid:1) is in Z ′ if and only if(1) b d (cid:0)P γ ′ ı g ı (cid:1) = 0 , b d (cid:0)P γ ′ ı g ı (cid:1) = 0 and b d (cid:0)P γ ′ ı g ı (cid:1) = 0 ,(2) b d (cid:0)P γ ′ ı g ı (cid:1) = − b d (cid:0)P γ ı g ı (cid:1) ,(3) b d (cid:0)P γ ı g ı (cid:1) = − b d (cid:0) f ( γ ) (cid:1) .Clearly the first condition is satisfied if and only if uγ ′ ı = 0 for all ı , γ ′ ı = γ ′ ı − ıu for all ı , and γ ′ ı + = γ ′ ı + γ ′ for all ı, . But this happens if and only if γ ′ ı = ıγ ′ for all ı and uγ ′ = 0 . [3.21]On the other hand, item (3) says that X ( γ ı + γ − γ ı + ) g ı ⊗ g = b d (cid:16)X γ ı g ı (cid:17) = − b d (cid:0) f ( γ ) (cid:1) , which, by Lemma 3.10, implies that(4) If t = 1 (or, equivalently, u = v ), then γ ı = ıγ . Moreover vγ = γ v = 0 .(5) If < t = u (or, equivalently, u ′ = 1 ), then γ kt + l = ( kt + l ) γ − ( k + 1) γ if k = 0 and ≤ l ≤ t , ( kt + l ) γ − ( k + 1) γ if k = 1 and ≤ l ≤ t + 2 , ( kt + l ) γ − ( k + 1) γ if ≤ k < t − and k < l ≤ t + k + 1 , [3.22]and vγ − uγ = γ v = 0 .(6) If < t < u (or equivalently, < u ′ < u ), then γ kt + l = ( kt + l ) γ − ( k + 1) γ if k = 0 and ≤ l ≤ t , ( kt + l ) γ − ( k + 1) γ if ≤ k < u ′ and ≤ l ≤ t , ( kt + l ) γ − ( k + 1) γ if k = u ′ and ≤ l ≤ t + 1 , ( kt + l ) γ − ( k + 1) γ if u ′ < k ≤ u ′ − and ≤ l ≤ t + 1 , ( kt + l ) γ − ( k + 1) γ if ≤ h < t , k = hu ′ − and h ≤ l ≤ t + h , ( kt + l ) γ − ( k + 1) γ if ≤ h < t , hu ′ ≤ k ≤ ( h + 1) u ′ − and h < l ≤ t + h , [3.23] ENTRAL EXTENSIONS OF SOME LINEAR CYCLE SETS 19 and vγ − uγ = γ v = 0 (note that, if u ′ = 2 , then the fourth line in [3.23] is empty; while, if t = 2 ,then the last two lines are empty).Conversely under these conditions, item (3) holds. Summarizing, items (1) and (3) are satisfied if andonly if equality [3.21] and conditions (4), (5) or (6) are fulfilled, depending on the case. Finally, by thedefinition of b d and equality [3.21], b d (cid:16)X γ ′ ı g ı (cid:17) = X ı − γ ′ ı + u ′ t − X s =1 sγ ′ ı − ıus ! g ı = X ı (cid:18) u ′ (cid:18) t (cid:19) − (cid:19) γ ′ ı g ı = ( − P γ ′ ı g ı if t = 2 , u − P γ ′ ı g ı if t = 2 . [3.24] Lemma 3.12. Let P γ ı g ı ∈ Γ(1) and γ ∈ Γ . If < u < v = u and condition (5) holds, or < u < v < u and condition (6) holds, then b d (cid:16)X γ ı g ı (cid:17) = − η − ν − X =0 X { ı : v ( ı )= } (cid:18) ı + u ′ p (cid:18) t ( )2 (cid:19)(cid:19) ( tγ − γ ) g ı − X { ı : t | ı } ı (cid:0) tγ − γ (cid:1) g ı , [3.25] where v ( ı ) := max { l ≥ p l | ı } and t ( ) := t/p .Proof. Assume first that v < u . By Remark 3.11 we know that vγ − uγ = 0 and equality [3.23] issatisfied. A direct computation shows that this equality can be written as γ kt + l = ( kt + l ) γ − ( k + 1) γ if k = 0 and ≤ l < t , ( kt + l ) γ − kγ if ≤ k ≤ u ′ and l = 0 , ( kt + l ) γ − ( k + 1) γ if ≤ k ≤ u ′ and ≤ l < t , ( kt + l ) γ − kγ if u ′ < k < u ′ and ≤ l ≤ , ( kt + l ) γ − ( k + 1) γ if u ′ < k < u ′ and ≤ l < t , ( kt + l ) γ − kγ if ≤ h < t , hu ′ ≤ k < ( h + 1) u ′ and ≤ l ≤ h , ( kt + l ) γ − ( k + 1) γ if ≤ h < t , hu ′ ≤ k < ( h + 1) u ′ and h < l < t . [3.26]Write ı := p v ( ı ) ı ′ . Clearly t − X s =0 γ ı − suı = t − X s =0 γ ı − sup v ( ı ) ı ′ = t − X s =0 γ ı + sup v ( ı ) = p v ( ı ) t ( v ( ı )) − X s =0 γ ı + sup v ( ı ) . Consequently, b d (cid:16)X γ ı g ı (cid:17) = − X ı (cid:18) t − X s =0 γ ı − suı (cid:19) g ı = η − ν − X =0 X { ı : v ( ı )= } p t ( ) − X s =0 γ ı + sup + X { ı : t | ı } tγ ı g ı . By equality [3.26], if t | ı , then tγ ı = tıγ − ıγ . So, in order to finish the proof of equality [3.25], we onlymust check that t ( ) − X s =0 γ ı + sup = (cid:18) ıp + u ′ p (cid:18) t ( )2 (cid:19)(cid:19) ( tγ − γ ) for all ı such that v ( ı ) = . [3.27]We divided the proof of this in five cases. In the first four we use equality [3.26] and that u = tu ′ .1) If ı = 1 , then := v ( ı ) = 0 , and so t ( ) − X s =0 γ ı + sup = t − X s =0 (cid:0) su (cid:1) γ − u ′ + 1 + t − X s =2 su ′ ! γ = (cid:18) u ′ (cid:18) t (cid:19)(cid:19) (cid:0) tγ − γ (cid:1) . 2) If < ı < t , then t ( ) − X s =0 γ ı + sup = t ( ) − X s =0 (cid:0) ı + sup (cid:1) γ − ıp − X s =0 ( su ′ p + 1) + t ( ) − X s = ı su ′ p γ = (cid:18) ıp + u ′ p (cid:18) t ( )2 (cid:19)(cid:19) (cid:0) tγ − γ (cid:1) . 3) If t < ı < u , then ı = tq + ¯ ı with < q < u ′ and < ¯ ı < t , which implies that t ( ) − X s =0 γ ı + sup = t ( ) − X s =0 γ ¯ ı +( su ′ p + q ) t = t ( ) − X s =0 ( ı + su ′ p t ) γ − ¯ ıp − X s =0 ( su ′ p + q + 1) + t ( ) − X s = ¯ ıp ( su ′ p + q ) γ. Thus p − X s =0 γ ı + sup = (cid:18) t ıp + u ′ p t (cid:18) t ( )2 (cid:19)(cid:19) γ − (cid:18) ¯ ıp + t ( ) q + u ′ p (cid:18) t ( )2 (cid:19)(cid:19) γ = (cid:18) ıp + u ′ p (cid:18) t ( )2 (cid:19)(cid:19) ( tγ − γ ) . 4) If ςu < ı < ( ς +1) u , where < ς < p , then ı = tq + ¯ ı with ςu ′ ≤ q < ( ς +1) u ′ and < ¯ ı < t . So, t ( ) − X s =0 γ ı + sup = t ( ) − X s =0 γ ¯ ı +( su ′ p + q ) t = t ( ) − X s =0 ( ı + su ′ p t ) γ − ⌈ ¯ ı − ςp ⌉ − X s =0 ( su ′ p + q + 1) + t ( ) − X s = ⌈ ¯ ı − ςp ⌉ ( su ′ p + q ) γ. Since l ¯ ı − ςp m = ¯ ıp , this implies that t ( ) − X s =0 γ ı + sup = (cid:18) t ıp + u ′ p t (cid:18) t ( )2 (cid:19)(cid:19) γ − (cid:18) ¯ ıp + t ( ) q + u ′ p (cid:18) t ( )2 (cid:19)(cid:19) γ = (cid:18) ıp + u ′ p (cid:18) t ( )2 (cid:19)(cid:19) ( tγ − γ ) . 5) If p u < ı , then by the previous cases we have t ( ) − X s =0 γ ı + sup = t ( ) − X s =0 γ ¯ ı + sup = (cid:18) ¯ ıp + u ′ p (cid:18) t ( )2 (cid:19)(cid:19) ( tγ − γ ) = (cid:18) ıp + u ′ p (cid:18) t ( )2 (cid:19)(cid:19) ( tγ − γ ) , where ≤ ¯ ı < p u is the remainder of the integer division of ı by p u (the last equality follows from thefact that vγ − uγ = 0 ).Assume now that v = u . By Remark 3.11 we know that vγ − uγ = 0 and equality [3.22] is satisfied.A direct computation shows that this equality can be written as γ kt + l = ( kt + l ) γ − ( k + 1) γ if k = 0 and ≤ l < t , ( kt + l ) γ − kγ if k = 1 and l = 0 , ( kt + l ) γ − ( k + 1) γ if k = 1 and ≤ l < t , ( kt + l ) γ − kγ if < k < t and ≤ l ≤ k , ( kt + l ) γ − ( k + 1) γ if < k < t and k < l < t . [3.28]In the case v = u the proof of equality [3.25] follows the same pattern than in the case v < u , but usingequality [3.28] instead of [3.26]. We leave the details to the reader. (cid:3) Lemma 3.13. Let P γ ı g ı ∈ Γ(1) , P γ ′ ı g ı ∈ Γ(1) and γ ∈ Γ . Assume the hypothesis of Lemma 3.12holds and that u > . Then equality [3.21] and condition (2) are satisfied if and only if γ ′ ı = − ı ( tγ − γ ) for all ı .Proof. By Lemma 3.12 we have b d (cid:16)X γ ı g ı (cid:17) = − η − ν − X =0 X { ı : v ( ı )= } (cid:18) ı + u ′ p (cid:18) t ( )2 (cid:19)(cid:19) ( tγ − γ ) g ı − X { ı : t | ı } ı (cid:0) tγ − γ (cid:1) g ı . [3.29]Assume first that p is odd. Then u | u ′ p (cid:0) t ( )2 (cid:1) for all ≤ < η − ν , and thus u ′ p (cid:0) t ( )2 (cid:1) ( tγ − γ ) = 0 ,since u ( tγ − γ ) = vγ − uγ = 0 . Consequently, by equalities [3.24] and [3.29], condition (2) holds if andonly if γ ′ ı = − ı ( tγ − γ ) for all ı (note that these γ ′ ı ’s satisfy condition [3.21]). Assume now that p = 2 and ν > . Since | t , we have u ′ (cid:0) t (cid:1) = u ( t − ≡ u (mod u ) . Consequently, if condition (2) is true, then (cid:16) u − (cid:17) γ ′ = (cid:18) u ′ (cid:18) t (cid:19)(cid:19) ( tγ − γ ) = (cid:16) u (cid:17) ( tγ − γ ) . [3.30]Since uγ ′ = u ( tγ − γ ) = 0 , this implies that − γ ′ = 2( tγ − γ ) , and so − u γ ′ = u ( tγ − γ ) , because | u .Adding this equality to [3.30], we obtain that − γ ′ = (1 + u )( tγ − γ ) = tγ − γ . By condition [3.21] thisimplies that γ ′ ı = − ı ( tγ − γ ) for all ı (note that these γ ′ ı ’s satisfy condition [3.21]). Conversely assume that γ ′ ı = − ı ( tγ − γ ) for all ı . By equalities [3.24] and [3.29], in order to prove that condition (2) is satisfied,we must check that (cid:16) u − (cid:17) γ ′ ı = (cid:18) ı + u ′ (cid:18) t ( )2 (cid:19)(cid:19) ( tγ − γ ) , [3.31]for all ı such that := v ( ı ) ∈ { , . . . , η − ν − } . If > , then u ′ (cid:18) t ( )2 (cid:19) ( tγ − γ ) = u ′ t t ( ) − tγ − γ ) = 2 − u ( t ( ) − tγ − γ ) = 0 , ENTRAL EXTENSIONS OF SOME LINEAR CYCLE SETS 21 and so, since | ı , we have (cid:16) u − (cid:17) γ ′ ı = (cid:16) − u (cid:17) ı ( tγ − γ ) = ı ( tγ − γ ) = (cid:18) ı + u ′ (cid:18) t ( )2 (cid:19)(cid:19) ( tγ − γ ) , as desired. Assume then that = 0 . Hence u ′ (cid:18) t ( )2 (cid:19) = u ′ (cid:18) t (cid:19) = u t − ≡ − u u ) , where the last equality holds since | t . Since, moreover u ( ı − γ ′ = 0 , we have u γ ′ ı = u ıγ ′ = u γ ′ , andso (cid:18) ı + u ′ (cid:18) t ( )2 (cid:19)(cid:19) ( tγ − γ ) = (cid:16) ı − u (cid:17) ( tγ − γ ) = − γ ′ ı + u γ ′ = (cid:16) − u (cid:17) γ ′ ı , which finishes the proof. (cid:3) Let A be as at the beginning of this Section, let Γ be an additive abelian group and let B and Z be thegroups of -coboundaries and -cocycles of U ( A , Γ) , respectively. Recall that Γ r := { γ ∈ Γ : rγ = 0 } , foreach natural number r .In the following result we set z ( γ , γ ′ , γ ) := ( f ( γ ) , P ıγ g ı , P ıγ ′ g ı ) , where γ, γ ∈ Γ . Theorem 3.14. If u = v , then H N ( A , Γ) = b H N ( A , Γ) = H ( U ( A , Γ)) ≃ Γ v ⊕ Γ v Γ . [3.32] Moreover, the set Z := { ( z ( γ , − γ , γ )) : γ ∈ Γ and γ ∈ Γ u } , is a subgroup of Z , that applies surjectivelyon H ( U ( A , Γ)) and B ∩ Z = { ( f ( γ ) , , 0) : γ ∈ v Γ } . Finally the map Θ : Z → Γ ⊕ Γ u , defined by Θ ( z ( γ , − γ , γ )) := ( γ , γ ) is an isomorphism that induces the isomorphism in [3.32] .Proof. By Remarks 1.1 and 3.1 we have H N ( A , Γ) = b H N ( A , Γ) = H ( U ( A , Γ)) . Let Z ′ be as in Remark 3.11.Thus Z ′ = { z ( γ , γ ′ , γ ) : vγ = uγ ′ = 0 and condition (2) is satisfied } . Since t = 1 , by the definition of b d and equality [3.24], condition (2) holds if and only if γ = γ ′ . Hence Z ′ = Z . Clearly the map Θ : Z → Γ v ⊕ Γ , defined by Θ ( z ( γ , − γ , γ )) := ( γ , γ ) is an isomorphism. Wenow compute B ∩ Z = n(cid:16) b d ( x ) , b d ( x ) , (cid:17) : x ∈ Γ(1) and b d ( x ) = f ( γ ) for some γ ∈ Γ o . Write x = P γ ′′ ı g ı . By the definition of b d , we have b d ( x ) = v − X =1 ( γ ′′ +1 − γ ′′ − γ ′′ ) g ⊗ g − ( γ ′′ + γ ′′ v − ) g ⊗ g v − + v − X ı =2 v − X =1 ( γ ′′ ı + − γ ′′ − γ ′′ ı ) g ı ⊗ g . So, by Remark 3.8, we get that b d ( x ) = f ( γ ) if and only if γ ′′ − γ ′′ = γ , γ ′′ ı +1 = γ ′′ + γ ′′ ı for < ı < v − ,and γ ′′ v − = − γ ′′ (or, equivalently, if and only if γ = − vγ ′′ and γ ′′ ı = − ( v − ı ) γ ′′ for < ı < v ). Moreover,since u = v , we have b d ( x ) = 0 . Consequently B ∩ Z = { ( f ( − vγ ′′ ) , , 0) : γ ′′ ∈ Γ } . Thus the map Θ induces an isomorphism H ( U ( A , Γ)) ≃ Γ v ⊕ Γ v Γ . (cid:3) Assume that we are under the hypothesis of Theorem 3.14. For each γ ∈ Γ and γ ∈ Γ v , let ξ γ : M (2) → Γ and ξ γ : M (1) ⊗ D → Γ be the maps defined by ξ γ ([ g ı ⊗ g ı ]) := γ if ı = ı = 1 , − γ if ı , ı ≥ and ı + ı ≤ v + 1 , otherwise, and ξ γ ( g ı ⊗ g ı ) := − ı ı γ . Proposition 3.15. The map ( ξ γ , ξ γ ) : M (2) ⊕ ( M (1) ⊗ D ) → Γ is a -cocycle of b C ∗ N ( A , Γ) . Moreover,each -cocycle of b C ∗ N ( A , Γ) is cohomologous to a ( ξ γ , ξ γ ) and two -cocycles ( ξ γ , ξ γ ) and ( ξ γ ′ , ξ γ ′ ) arecohomologous if and only if γ ′ = γ and vγ ′ = vγ .Proof. By Remark 3.3, Theorem 3.14 and the discussion above Remark 3.5, it suffices to check that (cid:0) ξ γ , ξ γ (cid:1) = (cid:16) f ( γ ) , X γ ı g ı , − X γ ı g ı (cid:17) [ b ϕ ] , where ( f ( γ ) , P γ ı g ı , − P γ ı g ı ) is as in the statement of Theorem 3.14. But this follows by that theorem,equality [3.19] and Proposition 3.4 with t = 1 . (cid:3) Proof of Theorem A. This follows from Remark 1.1, Proposition 3.15 and [19, Theorem 5.8]. (cid:3) In the following result for each γ, γ ∈ Γ , we set z ( γ , γ ) := ( f ( γ ) , P γ ı g ı , − P ı ( tγ − γ ) g ı ) , where the γ i ’s with i ≥ are as in [3.23] if v < u , and the γ i ’s with i ≥ are as in [3.22] if v = u . Theorem 3.16. If < u < v ≤ u , then H N ( A , Γ) = b H N ( A , Γ) = H ( U ( A , Γ)) ≃ Γ u Γ ⊕ Γ u . [3.33] Moreover the set Z := { z ( γ , γ ) : vγ = uγ } is a subgroup of Z that applies surjectively on H ( U ( A , Γ)) and B ∩ Z = { z ( uγ, vγ ) : γ ∈ Γ } . Finally the map Θ : Z → Γ ⊕ Γ u , defined by Θ( z ( γ , γ )) := ( γ , tγ − γ ) is an isomorphism that induces the isomorphism in [3.33] .Proof. Assume first that v < u . By Remarks 1.1 and 3.1 we have H N ( A , Γ) = b H N ( A , Γ) = H ( U ( A , Γ)) .Let Z ′ be as in Remark 3.11. We have Z ′ = n(cid:16) f ( γ ) , X γ ı g ı , X γ ′ ı g ı (cid:17) : γ ′ ı = ıγ ′ , uγ ′ = 0 , vγ = uγ and condition (2) and equality [3.23] hold o . By Lemma 3.13 we have Z ′ = Z . Clearly, the map Θ : Z → Γ ⊕ Γ u , given by Θ( z ( γ , γ )) := ( γ , tγ − γ ) ,is an isomorphism. We now compute B ∩ Z = n(cid:16) b d ( x ) , b d ( x ) , (cid:17) : x ∈ Γ(1) and b d ( x ) = f ( γ ) for some γ ∈ Γ o . Write x = P γ ′′ ı g ı . By the definition of b d , we have b d ( x ) = v − X =1 ( γ ′′ +1 − γ ′′ − γ ′′ ) g ⊗ g − ( γ ′′ + γ ′′ v − ) g ⊗ g v − + v − X ı =2 v − X =1 ( γ ′′ ı + − γ ′′ − γ ′′ ı ) g ı ⊗ g . So, by Remark 3.8, we get that b d ( x ) = f ( γ ) if and only if γ ′′ − γ ′′ = γ , γ ′′ ı +1 = γ ′′ + γ ′′ ı for < ı < v − ,and γ ′′ v − = − γ ′′ (or, equivalently, if and only if γ = − vγ ′′ and γ ′′ ı = ( ı − v ) γ ′′ for < ı < v ). Hence, b d ( x ) = b d (cid:16)X γ ′′ ı g ı (cid:17) = X(cid:0) γ ′′ (1 − u ) ı − γ ′′ ı (cid:1) g ı = − uγ ′′ g + X ı ≥ (cid:0) γ ′′ (1 − u ) ı − γ ′′ ı (cid:1) g ı . Thus, B ∩ Z = { z ( uγ, vγ ) : γ ∈ Γ } , and so the map Θ induces an isomorphism H ( U ( A , Γ)) ≃ Γ u Γ ⊕ Γ u .The case v = u follows in the same way. The unique difference is that, in the characterization of Z ′ we must use equality [3.22] instead of [3.23]. (cid:3) Assume that we are under the hypothesis of Theorem 3.16. For each γ, γ ∈ Γ such that vγ = uγ , let ξ γ : M (2) → Γ and ξ γ ,γ : M (1) ⊗ D → Γ be the maps defined by ξ γ ([ g ı ⊗ g ı ]) := γ if ı = ı = 1 , − γ if ı , ı ≥ and ı + ı ≤ v + 1 , otherwise,and ξ γ ,γ ( g ı ⊗ g tı + ) := ı (cid:18) ı − u ′ (cid:18) j (cid:19)(cid:19) ( tγ − γ ) + − X l =0 γ ı − ulı , where ≤ ı < u , ≤ < t and the γ r ’s are as in equality [3.23] if v = u and they are in equality [3.23]if v < u (take into account that if r < or r ≥ v , then to apply equalities [3.22] and [3.23], in order tocompute explicitly the map γ r in function of γ and γ , it is necessary to replace r by the remainder of theinteger division of r by v ). Proposition 3.17. The map ( ξ γ , ξ γ ,γ ) : M (2) ⊕ ( M (1) ⊗ D ) → Γ is a -cocycle of b C N ∗ ( A , Γ) . Moreovereach -cocycle of b C N ∗ ( A , Γ) is cohomologous to a ( ξ γ , ξ γ ,γ ) and two -cocycles ( ξ γ , ξ γ ,γ ) and ( ξ γ ′ , ξ γ ′ ,γ ′ ) are cohomologous if and only if γ − γ ′ ∈ u Γ and t ( γ − γ ′ ) = γ − γ ′ .Proof. By Remark 3.3 and Theorem 3.16 it suffices to check that ( ξ γ , ξ γ ,γ ) = z ( γ , γ )[ b ϕ ] , (cid:0) ξ γ , ξ γ ,γ (cid:1) = (cid:16) f ( γ ) , X γ ı g ı , − X ı ( tγ − γ ) g ı (cid:17) id M (2) b ϕ b ϕ . where z ( γ , γ ) is as in the statement of Theorem 3.16. But this follows by that theorem, equality [3.19],the fact that u ( tγ − γ ) = 0 , and Proposition 3.4. (cid:3) ENTRAL EXTENSIONS OF SOME LINEAR CYCLE SETS 23 Proof of Theorem B. This follows from Remark 1.1, Proposition 3.17 and [19, Theorem 5.8]. (cid:3) In the following result we set z ( γ , γ ′ , γ ) := ( f ( γ ) , P γ ı g ı , P γ ′ ı g ı ) , where γ, γ , γ ′ ∈ Γ , the γ i ’s with i ≥ are as in [3.22] and γ ′ ı = ıγ ′ for ı ≥ . Theorem 3.18. If v = u = 4 , then H N ( A , Γ) = b H N ( A , Γ) = H ( U ( A , Γ)) ≃ Γ2Γ ⊕ Γ ⊕ Γ . [3.34] Moreover, the subgroup Z := { z ( γ , γ ′ , γ ) : 2 γ ′ = 0 and γ = 2 γ } of Z that surjectively on H ( U ( A , Γ)) , Z ∩ B = { z (2 γ, , γ ) : γ ∈ Γ } and the map Θ : Z → Γ ⊕ Γ ⊕ Γ , defined by Θ( z ( γ , γ ′ , γ )) := ( γ , γ + γ, γ ′ ) ,is an isomorphism that induces the isomorphism in [3.34] .Proof. By Remarks 1.1 and 3.1 we have H N ( A , Γ) = b H N ( A , Γ) = H ( U ( A , Γ)) . Let Z ′ be as in Remark 3.11.Then Z ′ = n(cid:16) f ( γ ) , X γ ı g ı , X γ ′ ı g ı (cid:17) : γ ′ ı = ıγ ′ , uγ ′ = 0 , vγ = uγ and condition (2) and equality [3.22] hold o . By the fact that γ − γ = 0 and equalities [3.24] and [3.25], we have b d (cid:16)X γ g (cid:17) = − γ − γ ) g − γ − γ ) g − γ − γ ) g = 0 = b d (cid:16)X γ ′ g (cid:17) , which shows in particular that condition (2) is fulfilled. Hence Z ′ = Z . Clearly the map Θ : Z → Γ ⊕ Γ ⊕ Γ ,defined by Θ( z ( γ , γ ′ , γ )) := ( γ , γ + γ, γ ′ ) , is an isomorphism. We now compute the group Z ∩ B . Let x := γ ′′ g + γ ′′ g + γ ′′ g ∈ Γ(1) . Arguing as above we get that b d ( x ) = f ( γ ) if and only if γ ′′ = − γ ′′ , γ ′′ = − γ ′′ and γ = − γ ′′ . Hence, b d (cid:0) γ ′′ g + γ ′′ g + γ ′′ g (cid:1) = ( γ ′′ − γ ′′ ) g + ( γ ′′ − γ ′′ ) g + ( γ ′′ − γ ′′ ) g = − γ ′′ g + 2 γ ′′ g , and so Z ∩ B = { z (2 γ, , γ ) : γ ∈ Γ } . Thus, Θ induces an isomorphism H ( U ( A , Γ)) ≃ Γ2Γ ⊕ Γ ⊕ Γ . (cid:3) Assume that we are under the hypothesis of Theorem 3.18. For each γ ′ ∈ Γ and γ, γ ∈ Γ such that γ = 2 γ , let ξ γ : M (2) → Γ be as above of Proposition 3.17 and ξ γ ,γ ′ ,γ : M (1) ⊗ D → Γ be the mapdefined by ξ γ ,γ ′ ,γ ( g ı ⊗ g ı + ) := − X l =0 γ ı − ulı − ı (cid:18) ı − (cid:18) (cid:19)(cid:19) γ ′ = ı = 0 or ı = = 0 , − γ ′ if ı = 1 , = 0 and ı ∈ { , } , if ı = 1 , = 0 and ı = 2 , γ − ıγ ′ if = 1 and ı = 1 , γ − γ if = 1 and ı = 2 , − γ − ıγ ′ if = 1 and ı = 3 . Proposition 3.19. The map ( ξ γ , ξ γ ,γ ′ ,γ ) : M (2) ⊕ ( M (1) ⊗ D ) → Γ is a -cocycle of b C ∗ N ( A , Γ) . More-over each -cocycle of b C ∗ N ( A , Γ) is cohomologous to a ( ξ γ , ξ γ ,γ ′ ,γ ) and two -cocycles ( ξ γ , ξ γ ,γ ′ ,γ ) and ( ξ γ , ξ γ ,γ ′ ,γ ) are cohomologous if and only if γ − γ ∈ , γ − γ = 2( γ − γ ) and γ ′ = γ ′ .Proof. Mimic the proof of Proposition 3.17. (cid:3) Proof of Theorem C. This follows from Remark 1.1, Proposition 3.19 and [19, Theorem 5.8]. 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Email address : [email protected] Departamento de Matemática, Facultad de Ciencias Exactas y Naturales-UBA, Pabellón 1-Ciudad Uni-versitaria, Intendente Guiraldes 2160 (C1428EGA) Buenos Aires, Argentina.Instituto Argentino de Matemática-CONICET, Saavedra 15 3er piso, (C1083ACA) Buenos Aires, Ar-gentina. Email address ::