aa r X i v : . [ m a t h . K T ] J un (cid:7) mobile +995 599157836 EXTENSIONS AND (CO)HOMOLOGY OF Γ -GROUPS HVEDRI INASSARIDZE
Abstract.
This is a further investigation of our approach to groupactions in homological algebra in the settings of homology of Γ-simplicial groups, particularly of Γ-equivariant homology and coho-mology of Γ-groups. This approach could be called Γ-homologicalalgebra. The abstract kernel of non-abelian extensions of groups,its relation with the obstruction to the existence of non-abelian ex-tensions and with the second group cohomology are extended to thecase of non-abelian Γ-extensions of Γ-groups. We compute the ra-tional Γ-equivariant (co)homology groups of finite cyclic Γ-groups.The isomorphism of the group of n-fold Γ-equivariant extensions ofa Γ-group G by a G ⋊ Γ-module A with the (n+1)th Γ-equivariantgroup cohomology of G with coefficients in A is proven.We de-fine the Γ-equivariant Hochschild homology as the homology ofthe Γ- Hochschild complex involving the cyclic homology whenthe basic ring contains rational numbers and generalizing the Γ-equivariant(co)homology of Γ-groups when the action of the groupΓ on the Hochschild complex is induced by its action on the basicring. Important properties of the Γ-equivariant Hochschild homol-ogy related to Kahler differentials, Morita equivalence and derivedfunctors are established. Group (co)homology and Γ-equivariantgroup (co)homology of crossed Γ-modules are introduced and in-vestigated by using relevant derived functors. Relations of thesecond group cohomology with extensions of crossed Γ-modules,in particular with relative extensions of group epimorphisms inthe sense of Loday and with Γ-equivariant extensions of crossedΓ-modules are established. Long exact sequences of (co)homologyand Γ-equivariant (co)homology of crossed Γ-modules and the sixterms exact sequences for the integral homology and Γ-equivarianthomology of crossed Γ-modules are provided. Universal and Γ-equivariant universal central Γ-extensions of Γ-perfect crossed Γ-modules are constructed and Hopf formulas for the integral homol-ogy and Γ-equivariant integral homology of crossed Γ-modules areobtained. Finally, applications to algebraic K-theory, Galois the-ory of commutative rings and cohomological dimension of groupsare given.2010
Mathematics Subject Classification.
Key words and phrases. extensions of groups, homology of Γ-simplicial groups,Hochschild homology, symbol group, derived functors, Γ-equivariant group(co)homology, homology of crossed Γ-modules. introduction We continue the investigation of our approach to group actions inhomological algebra which we will call Γ-homological algebra. Theorigin of the equivariant study of group extensions theory in homo-logical algebra goes back to Whitehead paper [48]. Group actions onalgebraic and topological objects have many important applications inK-theory and homotopy theory([6,16,31,41]). Our goal is to continuethe development of extension theory in the category of Γ-groups andof relevant equivariant (co)homology theory that has been initiated in[27,28]. A different (co)homology theory of groups with operators wasprovided and investigated in [7-10], motivated by the graded categori-cal groups classification problem [8]. The introduction of Γ-equivariantchain complexes and their homology groups substantially contributeto the realization of our aim. Moreover this approach allows us topresent a version of equivariant Hochschild homology of any unital k-algebra A induced by the action of the group Γ on the k-algebra A.The Γ-equivariant Hochschild homology is closely related to Γ- equi-variant homology of groups [27]. This equivariant version differs of theequivariant Hochschild homology given in [38].We study extensions of Γ-groups that can be viewed as a part ofgroup actions in homological algebra, particularly of group actionson simplicial groups. Two important classes of Γ-group extensionsare considered. The first class is consisting of extensions having Γ-section map. The investigation of these extensions was initiated in[27] and called Γ-equivariant extensions of Γ-groups by Γ-equivariantG-modules. For the second class we deal with extensions of Γ-groupsendowed with a crossed Γ-module structure and called Γ-extensions ofcrossed Γ-modules (having Γ-section map). Our approach to extensionsof crossed modules substantially extends the class of relative extensionsof group epimorphisms introduced and investigated by Loday [33]. Itshould be noted that homology and cohomology of crossed modulesrelated to extensions of crossed modules was studied by many authors[3,7,12-13,15-17,23].The study of Γ-group extensions having Γ-section map is closelyrelated to the extension problem of group actions satisfying some con-ditions, in our case to lifting group actions that split for the givenextension of groups.Applications to algebraic K-theory, Galois theory of commutativerings and cohomological dimension of Γ-groups are given.The paper is divided into eight sections.Some notations that will be used throughout the paper:
XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 3 [Γ G ] denotes the set of elements γ gg − and Γ G is the normal sub-group of G generated by the set [Γ G ]. G Γ denotes the quotient group G/ Γ G .[ G, G ] Γ denotes the subgroup of the Γ-group G generated by thecommutant subgroup [G,G] and the elements of the form γ gg − , g ∈ G , γ ∈ Γ. It is called the Γ-commutant of the group G.[
G, H ] Γ denotes the subgroup of G generated by the elements x γ yx − y − ,where g ∈ G , y belongs to the normal subgroup H of G, and γ ∈ Γ. G ab Γ denotes the abelianization of the group G Γ .2. Preliminaries
In this section we recall some definitions and propositions given in[27] which will be used later. Moreover it is shown how equivariantversions of well known homological properties of groups are obtainedby using our approach to group actions in homological algebra.Let G Γ be the category whose objects are groups on which a fixedgroup Γ is acting, called Γ-groups, and morphisms are group homomor-phisms compatible with the action of Γ.Any exact sequence E of Γ-groups(2.1) E : 1 → A → B τ → G → β : G → X suchthat τ β = 1 G and β is compatible with the actions of Γ, that means β ( γ g ) = γ β ( g ) , g ∈ G, σ ∈ Γ. In addition if β is a homomorphism thenthe extension E is called split extension. Definition 2.1. (1) A Γ-equivariant G-module A is a G-module equippedwith a Γ-module structure and the actions of G and Γ are related toeach other by the equality σ ( g a ) = σ g ( σ a )for g ∈ G , σ ∈ Γ, a ∈ A .The category of Γ-equivariant G-modules is equivalent to the cate-gory of G ⋊ Γ-modules, where G ⋊ Γ is the semi-direct product of Gand Γ [9].If E is an extension with Γ-section map and A is a Γ-equivariant G-module then it is called Γ-equivariant extension of G by A. In addition ifX and G are Γ-equivariant G-modules then it is called proper sequenceof Γ-equivariant G-modules.
HVEDRI INASSARIDZE (2) A relatively free Γ-equivariant G-module F is a free G-modulewith basis a Γ-set and relatively projective Γ-equivariant G-modulesare retracts of relatively free Γ-equivariant G-modules.The class P of projective Γ-equivariant G-modules is a projectiveclass with respect to proper sequences of Γ-equivariant G-modules.For the cohomological description of the set E ( G, A ) of equivalenceclasses of Γ-equivariant extensions of G by A the Γ-equivariant ho-mology and cohomology of Γ-groups have been introduced as relative
T or P n and Ext n P , n ≥ Definition 2.2.
The Γ-equivariant homology and cohomology of Γ-groups are defined as follows H Γ n ( G, A ) =
T or P n ( Z , A ) , H n Γ ( G, A ) =
Ext n P ( Z , A ) , n ≥ , where the functors N and Hom are taken over the ring Z ( G ⋊ Γ)and the groups G and Γ are trivially acting on the abelian group Z ofintegers.Let(2.2) · ·· → B n → · · · → B → B → Z → Z , where B = Z ( G ) and B n , n > Z ( G )-module generated by [ g , g , ..., g n ], g i ∈ G . The action of Γon G naturally induces an action of Γ on the sequence (2.2) definedby γ ( mg ) = m γ g and γ ( g [ g , g , ..., g n ]) = γ g [ γ g , γ g , ..., γ g n ] for B and B n respectively, n ≥
1. Then the sequence 2.2 is the Γ-equivariantbar resolution of Z , the groups B n being relatively free Γ-equivariant G-modules. It follows that one has isomorphisms H Γ n ( G, A ) ∼ = H n ( B ∗ N G ⋉ Γ A )and H n Γ ( G, A ) ∼ = H n ( Hom G ⋉ Γ ( B ∗ , A ))For the Γ-equivariant cohomology of Γ-groups an alternative descrip-tion by cocycles is provided. To this end the group C n Γ ( G, A ) of Γ-maps, f : G n → A for n >
0, called Γ-cochains is considered. By using theclassical cobord operators δ n : C n Γ ( G, A ) → C n +1Γ ( G, A ), n >
0, onegets a cochain complex0 → C ( G, A ) → C ( G, A ) → C ( G, A ) → · · · → C n Γ ( G, A ) → · · · , where C ( G, A ) = A Γ , Kerδ = Der Γ ( G, A ) is the group of Γ-derivationsand the homology groups of the complex C ∗ Γ ( G, A ) are isomorphic tothe Γ-equivariant cohomology groups of the Γ-group G with coefficientsin the Γ-equivariant G-module A.Two Γ-equivariant extensions E and E’ of G by A are called equiva-lent if there is a morphism E → E ′ which is the identity on G and A. XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 5
We denote by E ( G, A ) the set of equivalence classes of Γ-equivariantextensions of G by A.
Theorem 2.3.
There is a bijection E ( G, A ) ∼ = H ( G, A ) . Remark . By using the Baer sum operation the set E ( G, A ) be-comes an abelian group and the bijection of Theorem 2.3 is an isomor-phism. This theorem will be extended to higher dimensions for n ≥ Definition 2.5.
A Γ-group is called Γ-free if it is a free group withbasis a Γ-set.Any free group F(G) generated by a Γ-group G becomes a Γ-freegroup by the following action of Γ: γ | g | = | γ g | , g ∈ G, γ ∈ Γ. Thedefining property of the Γ-free group F is that every Γ-map E f → G toa Γ-group G is uniquely extended to a Γ-homorphism F f ′ → G .Let F be the projective class of Γ-free groups in the category G Γ ofΓ-groups. Theorem 2.6.
There are isomorphisms H Γ n ( G, A ) ∼ = L F n − ( I ( G ) ⊗ G ⋉ Γ A ) , H n Γ ( G, A ) ∼ = R n − F Der Γ ( G, A ) for n ≥ , where I(G) is the kernel of the natural homomorphism Z ( G ) → Z of Γ -equivariant G-modules, Der(G,A) is the group of Γ -derivations, L F n − and R n − F denote respectively the left and right derivedfunctors with respect to the projective class F . We also recall some results on Γ-equivariant integral homology [27].These homology groups are simply denoted H Γ n ( G ) for H Γ n ( G, Z ), thegroups G and Γ acting trivially on Z . Theorem 2.7. (1)There is an isomorphism L F n ( G ab Γ ) ∼ = H Γ n +1 ( G ) for n ≥ .(2) There are exact sequences → Γ G/ [ G, G ] ∩ Γ G → H ( G ) → H Γ1 ( G ) → , · · · → H Γ n +1 ( G ) → L F n − U ( G ) → H n ( G ) → H Γ n ( G ) → L F n − U ( G ) → · · · → H Γ3 ( G ) → L F U ( G ) → H ( G ) → H Γ2 ( G ) → L F U ( G ) → H ( G ) → H Γ1 ( G ) → relating Γ -equivariant integral homology with the classical integralhomology of groups, where U is a covariant functor assigning to any Γ -group G the abelian group [ G, G ] Γ / [ G, G ] . HVEDRI INASSARIDZE
Theorem 2.8.
Let → A → B τ → G → be a short exact sequence of Γ -groups with Γ -section map and α : P → B a Γ -projective presentation of the Γ -group B. Then there is an exactsequence → V → H Γ2 ( B ) → H Γ2 ( G ) → A/ [ B, A ] Γ → H Γ1 ( B ) → H Γ1 ( G ) → , where V is the kernel of the Γ -homomorphism [ P, S ] Γ / [ P, R ] Γ → [ B, A ] Γ induced by α , R = Ker α and S = Ker τ α . Theorem 2.9.
If G is a Γ -group, then(1) H Γ1 ( G, A ) = G/ [ G, G ] Γ ⊗ A , G and Γ are trivially acting on A.(2) H Γ2 ( G ) is isomorphic to the group ( R ∩ [ P, P ] Γ ) / [ P, R ] Γ , where R = Kerα and α : P → G is a Γ -projective presentation of G (Hopfformula for the Γ -equivariant homology of groups. The Brown - Ellis formula is also obtained extending Hopf formulato higher Γ-equivariant homology of groups (see [28]).
Definition 2.10.
A Γ-subgroup L of a Γ-group G is called retract ofG if there is a Γ-homomorphism f : G → L sucht that its restrictionto L is the identity map. Theorem 2.11.
Let L be retract of a free Γ -group F. For any Γ -equivariant G-module A one has1) exact sequences → H Γ1 ( L, A ) → I ( L ) ⊗ Z( L ∝ Γ) A → A Γ → A L ⋉ Γ → and →→ A L ⋉ Γ → A Γ → Hom Z( L ⋉ Γ) ( I ( L ) , A ) → H ( L, A ) → ,2) H Γ n ( L, A ) = 0 and H n Γ ( L, A ) = 0 for n > . The consideration of retracts of Γ-free groups motivates the following
Question : Let G be a Γ-subgroup of a Γ-free group . Is G a Γ-freegroup? In other words we are asking whether the well known Nielsen -Schreier Theorem on free groups holds for Γ-free groupsFinally we provide an assertion establishing relation of Γ-equivariantcohomology of groups with well known equivariant cohomology of topo-logical spaces, namelyLet G be a Γ-group and X a topological space on which the groupsG and Γ are acting such that G is acting properly and γ ( g x ) = γ g ( γ x ). γ ∈ Γ , g ∈ G, x ∈ X. Theorem 2.12.
If X is either acyclic and Γ acts trivially on X or Xis Γ -contractible, then there is an isomorphism H n Γ ( G, A ) = H n Γ ( X/G, A ) for n ≥ , where G and Γ are trivially acting on the abelian group Aand H ∗ Γ ( X/G, A ) is the equivariant cohomology of the space X/G. XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 7 Extensions of Γ -groups We introduce an internal property of Γ-group extensions possessinga Γ-section map that will be used through out the paper.
Definition 3.1.
It will be said that the sequence 2.1 of Γ-groups pos-sesses the Γ-property if the restriction of τ on the subset [Γ X ] of X isinjective. Theorem 3.2.
The sequence 2.1 possesses the Γ -property iff it has a Γ -section map and Γ acts trivially on Kerτ.P roof.
Denote by B Γ( τ ) the normal subgroup of B generated by theelements γ gg − such that τ ( γ gg − ) = 1 , γ ∈ Γ , g ∈ G. It is evident that B Γ( τ ) is a Γ-subgroup of B and the canonical map δ : B → B/B Γ( τ ) isa Γ-homomorphism.Let α : G → B be a section map for the Γ-homomorphism τ , thenthe sequence E Γ : 1 → A/B Γ( τ ) → B/B Γ( τ ) τ ′ → G → τ ′ is induced by τ . In effect, for that it suffices to show that if σ g = γ g then σ ( δα ( g )) = γ ( δα ( g )), σ, γ ∈ Γ , g ∈ G . One has τ ( σ α ( G )) = σ ( τ α ( g )) = σ g and τ ( γ α ( G )) = γ ( τ α ( g )) = γ g . Thus τ ( σ α ( G )) = τ ( γ α ( G )) and there-fore τ ( γ − σ α ( g )) = γ − τ ( σ α ( g )) = γ − τ ( τ α ( g )) = g. It follows that γ − σ α ( g ) .α ( g ) − ∈ B Γ( τ ) implying the equality δ ( γ − σ α ( g )) = δ ( α ( g ))and finally the required equality.Now assume the sequence E possesses the Γ-property. Then B Γ( τ ) = 1implying the isomorphism of the sequences E ∼ = E Γ . Conversely let thesequence E satisfies the conditions of the theorem. That means it hasa Γ-section map α : G → B and the group Γ trivially acts on Kerτ.
Let τ ( γ b · b − ) = 1 for some b ∈ B, γ ∈ Γ . This yields the equality γ ( ατ ( b )) = ατ ( b ) . By using the equality b = ατ ( b ) · c, c ∈ A one gets γ b = γ ( ατ ( b )) · γ c = ατ ( b ) · c = b. This completes the proof.
Corollary 3.3.
The sequence E Γ possesses the Γ -property and everyits section map is a Γ -section map. Definition 3.4.
A Γ-group G is called Γ-perfect if G = [ G, G ] Γ orequivalently, if H Γ1 ( G ) = 0 [33,27].ExampleLet F(G) be the Γ-free group generated by the Γ-group G. The shortexact sequence of Γ-groups1 → R → F ( G ) τ → G → , HVEDRI INASSARIDZE where τ ( | g | ) = g , has a Γ-section map α : G → F ( G ), sending anyelement g to | g | . Then the short sequence of Γ-groups(3.1) 0 → R/ [ F ( G ) , R ] Γ → F ( G ) / [ F ( G ) , R ] Γ τ ′ → G → ηα andΓ is trivially acting on R/ [ F ( G ) , R ] Γ , where τ ′ is induced by τ and η : F ( G ) → F ( G ) / [ F ( G ) , R ] Γ is the canonical Γ-homorphism. Thereforeby Theorem 3.2 the sequence (3.1) has the Γ-property.If the Γ-group G is Γ-perfect the sequence (3.1) yields the followingΓ-extension of G(3.2)0 → R ∩ [ F ( G ) , F ( G )] Γ / [ F ( G ) , R ] Γ → [ F ( G ) , F ( G )] Γ / [ F ( G ) , R ] Γ τ ′′ → G → . Since the group [ F ( G ) , F ( G )] Γ / [ F ( G ) , R ] Γ is a Γ-subgroup of F ( G ) / [ F ( G ) , R ] Γ ,the sequence (3.2) also has the Γ-property and therefore it is a Γ-extension of G with Γ-section map. The sequence (3.2) is the universalcentral Γ-equivariant extension of the Γ-perfect group G and the group R ∩ [ F ( G ) , F ( G )] Γ / [ F ( G ) , R ] Γ is isomorphic to H Γ2 ( G ) [27]. As we seein this example the Γ-property has been used substantially.Now we continue our investigation of Γ-extensions of Γ-groups byconsidering the non-abelian case. Let(3.3) 1 → J σ → X τ → G → θ : X → AutJ implying the Γ-homorphism ψ : G → AutJ/InJ , where Γ isassuming trivially acting on AutJ and InJ denotes the group of innerautomorphisms of J.
Definition 3.5.
The triple (
G, J, ψ ) is called abstract kernel of thenon-abelian extension (3.3) of Γ-groups.
Theorem 3.6.
1) For any abstract kernel ( G, J, ψ ) there is a cor-rectly defined element,called obstruction for ( G, J, ψ ) , and belongingto H ( G, C ) . The abstract kernel ( G, J, ψ ) possesses an extension iff Obs ( G, J, ψ ) = 0 .2) If there exists an extension of the Γ -group G with abstract ker-nel ( G, J, ψ ) , then the set of equivalence classes of extensions with Γ -property of G by J is bijective with H ( G, C ) , where C is the center ofJ. P roof.
We will follow the classical proof (when the action of Γ on Gis trivial).
XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 9
First of all it should be noted that for a given extension (3.3) ofthe abstract kernel (
G, J, ψ ) a section map α : G → X with α (1) = 1induces by conjugation an automorphism ϕ ( x ) ∈ ψ ( x ) , x ∈ G of thegroup J and maps f, µ : G × G → J satisfying the well known equalities α ( x ) + l = ϕ ( x )( l ) + α ( x ), l ∈ J, x ∈ G,α ( x ) + α ( y ) = f ( x, y ) + α ( xy ) , x, y ∈ G, (1) ϕ ( x )( f ( x, y )) + f ( x, yz ) = f ( x, y ) + f ( xy, z ) , z ∈ G, (2) ϕ ( x ) ϕ ( y ) = µ ( f ( x, y )) ϕ ( xy ) . Taking into account the action of Γ one has γ ( α ( x )+ γ l = γ ( ϕ ( x )( l ))+ γ ( α ( x )). Therefore α ( γ ( x )) + l = γ ϕ ( x )( l ) + α ( γ x ). On the other hand α ( γ x ) + l = ϕ ( γ x )( l ) + α ( γ x ) . Thus ϕ ( γ x )( l ) = γ ϕ ( x )( l ) showing that ϕ is a Γ-map. Similarly it can be proved that the maps f and µ areΓ-maps.Conversely, for given Γ-maps ϕ : G → Aut ( J ) , f, µ : G × G → J satisfying the equalities (1) and (2), and ϕ (1) = 1 , f (1 , g ) = f ( x,
1) =0 , x ∈ J, g ∈ G, one can construct an extension of the Γ-group G withΓ-property by considering the set B ( J, ϕ, f, G ) of couples ( x, g ) , x ∈ J, g ∈ G, and defining the group structure on it as follows:( x, g ) + ( x , g ) = ( x + ϕ ( g ) x + f ( g, g ) , gg ), and Γ is acting on B ( J, ϕ, f, G ) componentwise γ ( x, g ) = ( x, γ g ) , γ ∈ Γ.This Γ-extensionis called semi-direct product Γ-extension of the Γ-group G by J. It isevident that it has the Γ-property.For any Γ-extension of G by J with Γ-property and abstract kernel(
G, J, ψ ) the arising Γ-maps ϕ , f and ψ should satisfy the equalities (1)and (2).That is not the case in general and the obstruction is definedby the equality(3) ϕ ( x )( f ( x, y )) + f ( x, yz ) = k ( x, y, z ) + f ( x, y ) + f ( xy, z ) , where k(x,y,z) is an element of the center C of J which is the kernel of µ. It is evident that k : G × G × G → C is a Γ-map and it can be provedsimilarly to the classical case that it is 3-th Γ-cocycle of the chaincomplex C ∗ Γ ( G, C ) . The Γ-cocycle k is called obstruction Ob ( G, J, ψ )for the abstract kernel (
G, J, ψ ) . Finally, any Γ-extension of the abstract kernel (
G, J, ψ ) is equivalentto a semi-direct product Γ-extension and the set of semi-direct productΓ-extensions of G by J is bijective to H ( G, C ). The proof of theses twoassertions completely follows the well known case when Γ acts triviallyon G and J and it is left to the reader.As noted in Preliminaries the bijection of the set of Γ-equivariantextensions of G by A with the second Γ-equivariant homology groupof the Γ-group G could be extended to higher dimensions. For that weneed the notion of n-fold Γ-equivariant extension of G by A which is defined as a long exact sequence of Γ-groups:0 → A → B α → B → ... → B n α n → G → , where 0 → A → B → Imα → Imα i − → B i → Imα i →→ ≤ i ≤ n −
1, are proper sequences of Γ-equivariant G-modules and0 → Imα n − → B n → G → e is a Γ-equivariant extension of G by Imα n − .Let E n Γ ( G, A ), n ≥
1, denotes the class of equivalence classes of n-fold Γ-equivariant extensions of G by A. It becomes an abelian groupby using the Baer sum operation for all n ≥ Theorem 3.7.
There is an isomorphism E n Γ ( G, A ) ∼ = H n +1Γ ( G, A ) for n ≥ .P roof. We need the following property of the relative derived func-tors
Ext ∗ Γ ( A, B ) of
Hom G ∝ Γ ( A, B ) defined in the category of Γ-equivarianrG-modules by using relatively projective Γ-resolutions with respect tothe class P of proper sequences.For any proper sequence of Γ-equivariant G-modules 0 → A → D → B → → Ext ( B, L ) → Ext ( D, L ) → Ext ( A, L )) θ → Ext ( B, L ) → ... → Ext n Γ ( A, L )) θ → Ext n +1Γ ( B, L ) → ..., → Ext ( L, A ) → Ext ( L, D ) → Ext ( L, B )) δ → Ext ( L, A ) → ... → Ext n Γ ( L, B )) δ → Ext n +1Γ ( L, A ) → .... The connected sequence of contravariant functors (
Ext n Γ ( − , A ) , θ n , n =1 , , ... ) is the right universal sequence of contravariant functors (orright satellite of of the functor Ext ( − , L )) relative to the class P of proper sequences of Γ-equivariant G-modules [26]. In effect, let( U n , µ n , n = 1 , , ... ) be a connected sequence of contravariant func-tors related to the class P and f : Ext ( − , A ) → U ( − ) be a mor-phism of functors. It will be shown that there exists a unique exten-sion of the morphism f to f n : Ext n Γ ( − , A ) → U n ( − ) , n = 2 , , ... compatible with the connecting homomorphisms. To define the ex-tension f : Ext ( J, A ) → U ( J ) consider the short exact sequence0 → L → P → J →
0, where P is a relatively projective Γ-equivariantG-module, implying the isomorphism θ : Ext ( L, A ) → Ext ( J, A ).Then the homomorphism f is given by µ f ( θ ) − . It is easily checkedthat f is correctly defined, compatible with the connecting homomor-phisms θ and δ , and it is the unique extension of f . The extensionof f n to f n +1 for n > XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 11
It will be shown now that the connected sequence of contravariantfunctors ( E n Γ ( − , A ) , σ n , n = 1 , , ... ) in the category of Γ-equivariant G-modules is also the right universal sequence of contravariant functorsrelative to the class P of proper sequences of Γ-equivariant G-modules.For the proper sequence E = 0 → J → B → L → δ n : E n Γ ( J, A ) → E n +1Γ ( L, A ) is given by σ n ([ E n ]) =[ E n ⊗ E ], where E n = 0 → A → X → X → ... → X n → J → E n ⊗ E is an n+1- fold extension of L by A obtained by splicing then-fold extension E n with the proper sequence E.Let ϕ : E ( − , A ) → U ( − ) be a morphism of functors. Its uniquelydefined extension ϕ : E ( − , A ) → U ( − ) is realized as follows: for[ E ] ∈ Ext ( J, A ), E = 0 → A → X α → X → J →
0, by usingconnecting homomorphisms with respect to the proper sequence 0 → Imα → X → J → ϕ ([ E ]) = µ ϕ ([ E ]), where E =0 → A → X → Imα →
0. The extension of ϕ n to ϕ n +1 for n > f : Ext ( − , A ) → E ( − , A ) im-plies the isomorphism of these two right universal sequences of con-travariant functors and this yields the isomorphism Ext n Γ ( B, A ) ∼ = E n Γ ( B, A ) for n ≥ h : E ( G, − ) → E ( Z , − ).If we consider the functors E n Γ ( B, A ) with respect to the second vari-able then the sequence E n Γ ( Z , − ) , η n , n ≥
2, and the sequence E n Γ ( G, − ) , λ n , n ≥
1, are right universal sequences of covariant functors on the categoryof Γ-equivariant G-modules with respect to the class P of proper se-quences of Γ-equivariant G-modules, the connecting homomorphisms η n and λ n being defined and the universality shown similarly to the pre-vious sequence ( E n Γ ( − , A ) , σ n , n = 1 , , ... ). Therefore the isomorphism h : E ( G, − ) → E ( Z , − ) induces isomorphisms h n : E n Γ ( G, − ) → E n +1Γ ( Z , − ) for all n ≥
1. It remains to apply the isomorphism E n +1Γ ( Z , − ) → Ext n +1Γ ( Z , − ). This completes the proof.4. Some computations
It is reasonable to ask for the computation of the Γ-equivariant(co)homology of groups introduced in [27]. As noted above Γ-equivarianthomology and cohomology groups of retracts of Γ-free groups are triv-ial for n >
1. Here we provide an attempt to the investigation of thisproblem for finite cyclic Γ-groups.
It is well known that for the computation of the (co)homology offinite cyclic groups the following free resolution of Z is used:(4.1) ... D → Z ( Z m ) N → Z ( Z m ) D → Z ( Z m ) N → Z ( Z m ) D → Z ( Z m ) ǫ → Z → , where Z m is a finite cyclic group of order m and generator t, D = t - 1and N = 1 + t + t + ... + t m − . Assume now that a group Γ is actingon Z m with trivial action on Z . Similarly to the action of Γ on thebar resolution of Z it induces an action of Γ on the resolution (4.1) of Z . It is easily checked that the homomorphism D is compatible withtrivial action of Γ only. This case is not interesting, since it is reducedto the usual (co)homology of groups. Therefore the resolution (4.1) isunsuitable for the computation of Γ-equivariant (co)homology of finitecyclic Γ-groups.By slightly changing the value of D we are able to compute therational Γ-equivariant (co)homology of the finite cyclic Γ-group Z m .Let B ∗ ⊗ Q → Q be the bar resolution of the field Q of rationalnumbers obtained by tensoring the bar resolution B ∗ → Z of Z by Q , where B ⊗ Q ∼ = Q ( G ), and B n , n ≥ , is the free Q ( G )- modulegenerated by the elements [ g , g , ..., g n ] , g i ∈ G .The rational (co)homology of groups with coefficients in Z ( G )-modulesis defined as follows: Definition 4.1. Q H n ( G, A ) = H n (( B ∗ ⊗ Q ) ⊗ Z ( G ) A )) and Q H n ( G, A ) =
Hom Z ( G ) ( B ∗ ⊗ Q , A ) for n ≥ Z ( G )-module AIt is easily checked that so defined rational homology and cohomol-ogy of groups don’t depend on the projective Q ( G )-resolution of Q andthere are isomorphisms Q H n ( G, A ) ∼ = H n ( G, A ⊗ Q ) and Q H n ( G, A ) = H n ( G, Hom ( Q , A )), n ≥
0. In particular taking into account the iso-morphisms Q ⊗ Q ∼ = Q and Hom ( Q , Q ) ∼ = Q one has the isomorphisms Q H n ( G, Q ) ∼ = H n ( G, Q ) and Q H n ( G, Q ) ∼ = H n ( G, Q ), n ≥
0, for thetrivial Z ( G )-module Q .Based on this definition of rational (co)homology of groups we pro-vide the following definition of rational Γ-equivariant homology andcohomology of Γ-groups. Let Γ be a group acting on the group G andtrivially acting on Q implying the action of Γ on the bar resolution B ∗ ⊗ Q . Definition 4.2. Q H Γ n ( G, A ) = H n (( B ∗ ⊗ Q ) ⊗ Z ( G ⋊ Γ) A )) and Q H n Γ ( G, A ) =
Hom Z ( G ⋊ Γ) ( B ∗ ⊗ Q , A ) for n ≥ Z ( G ⋊ Γ)-moduleA We recall that any t i is a generator of Z m iff (i,m)= 1 and the totalnumber of generators is equal to the number of integers coprime with XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 13 m and less than m which is called the Euler ϕ -function. Any auto-morphism of Z m maps a generator to a generator and therefore it hasthe form t → t k , 1 ≤ k < m , where (k,m)= 1, k and m not havingnon-trivial commun divisors.Since Γ is not trivially acting, that means there is γ ∈ Γ such that γ t = t k , k >
1. For the computation of rational (co)homology ofZ m = { , t, t , ..., t m − } , t m = 1, we will use the following sequence offree Q(Z m )-modules over Q:(4.2) ... D → Q ( Z ) N → Q ( Z ) D → Q ( Z ) N → Q ( Z ) D → Q ( Z ) ǫ → Q → , where D = t m − + t m − , ..., t, − ( m −
1) and N = 1 + t + t + ... + t m − . Itwill be shown that the sequence (4.2) is a projective Q(Z m )-resolutionof Q . It is evident that homomorphims D and N are compatible withthe action of Γ and DN = 0, ND = 0, ǫ D = 0.For f ( t ) = a + a t + a t + ... + a m − t m − ∈ Q ( Z m ) one has D ( f ( t )) = − ( m − a + a + a + ... + a m − + ( a − ( m − a + a + ... + a m − ) t + ... + ( a + a + ... + a m − − ( m − a m − + a m − ) t m − +( a + a + ... + a m − − ( m − a m − ) t m − .If D ( f ( t )) = 0 this yields a system of m equalities: a + a + ... + a i − − ( m − a i + a i +1 + .. + a m − = 0 , i = 0 , , ..., m −
1, implying the equalities a = a = a = ... = a m − = a m − and therefore N ( a ) = f ( t ).If N(f(t)) = 0, then a + a + ... + a m − + a m − = 0. Assume there is ϕ ( t ) = x + x t + x t + ... + x m − t m − such that D ( ϕ ( t )) = f ( t ). Thisyields the equalities − ( m − x + x + x + ... + x m − = a ,x − ( m − x + x + ... + x m − = a ,....................................................x + x − x + ... + x m − − ( m − x m − = a m − .Since Σ i a i = 0, this system of m linear equations has infinitely manysolutions in the field Q of rational numbers, for every b ∈ Q the solutionhas the form x i = ( a m − a i ) /m + b, i = 0 , , ..., m − x m − = b . Thesame holds for the case ǫ ( f ( t )) = 0. It follows that the sequence (4.2)is exact.It remains to show that 0 → KerD → Q ( Z m ) → ImD → → KerN → Q(Z m ) → ImD → Q ( Z m )-modules having Γ-section map. Since ImN = KerD, any element ofImN has the form f ( t ) = a + at + at + ... + at m − and Γ acts triviallyon f(t). Therefor the map α : ImN → Q(Z m ), α ( f ( t )) = a is a Γ-section map for the sequence 0 → KerN → Q ( Z m ) → ImD →
0. Forthe sequence 0 → KerD → Q ( Z m ) → ImD → f ( t ) ∈ ImD satisfies the equality Σ i a i = 0, since ImD = KerN, and consider the element ϕ ( t ) = − /m ( f ( t )). It will be shown that D ( ϕ ( t )) = f ( t ). Ineffect one has D ( ϕ ( t )) =( − ( m − − /m ( a )) + a + a + ... + a m − )+( − /m ( a ) − ( m − − /m ( a )) − /m ( a ) − ... − /m ( a m − )) t ++ ... + ( − /m ( a ) − /m ( a ) − /m ( a ) − ... − /m ( a m − ) − ( m − − /m ( a m − ))) t m − = a − /m ( a + a + a + ... + a m − )+( a − ( − /m ( a + a + a + ... + a m − )) t ++ ... + ( a m − − ( − /m ( a + a − a + ... + a m − + a m − ) t m − = f ( t ).Therefore the map α : ImD → Q(Z m ) sending f(t) to -1/m(f(t)) isa Γ-section map.We have proven that the sequence (4.2) is a Q ( Z m )-equivariant pro-jective resolution of the trivial Q ( Z m )- module Q , where Q ( Z m ) is arelatively free Q (Γ)-equivariant Z ( Z m ⋊ Γ)-module. Therefore the ho-mology groups of the complex ... D ∗ → Q ( Z m ) ⊗ Z (Z m ⋊ Γ) A N ∗ → Q ( Z m ) ⊗ Z ( Z m ⋊ Γ) A D ∗ → Q ( Z m ) ⊗ Z ( Z m ⋊ Γ) A N ∗ → Q ( Z m ) ⊗ Z ( Z m ⋊ Γ) A D ∗ → Q ( Z m ) ⊗ Z ( Z m ⋊ Γ) A → Q H n (Z m , A ) of the cyclicgroup Z m with coefficients in the Z ((Z m ⋊ Γ)-module A. If one usesthe isomorphisms Q ( Z m ) ⊗ Z ( Z m ⋊ Γ) A ∼ = ( Q ⊗ Z ( Z m )) ⊗ Z (Z m ⋊ Γ) A ∼ = Q ⊗ ( Z ( Z m ) ⊗ Z ( Z m ⋊ Γ) A ) ∼ = Q ⊗ A Γ and the fact that every element of Q ⊗ A Γ can be written in the form q ⊗ [ a ], q ∈ Q , [ a ] ∈ A Γ . we havefinally obtained Theorem 4.3.
Let Γ be a group not trivially acting on the finite cyclicgroup Z m . Then for any Γ -equivariant Z ( Z m ⋊ Γ) -module A one has Q H Γ0 ( Z m , A ) = Q ⊗ A Γ Q H Γ2 n − ( Z m , A ) = ( q ⊗ [ at m − + at m − + ... + at ] = q ⊗ [( m − a ]) /ImN ∗ Q H Γ2 n ( Z m , A ) = ( q ⊗ [ at m − + at m − + ... + at + a ] = 0) /ImD ∗ ,for n > and the homomorphisms D ∗ and N ∗ are induced by D andN respectively. For the rational Γ-equivariant cohomology Q H n (Z m , A ) of the cyclicgroup Z m with coefficients in the Z (( Z m ⋊ Γ)-module A one uses thecomplex0 → Hom Z (Z m ⋊ Γ) ( Q (Z m ) , A ) D ∗ → Hom Z (Z m ⋊ Γ) ( Q ( Z m ) , A ) N ∗ → Hom Z (Z m ⋊ Γ) ( Q ( Z m ) , A ) D ∗ → Hom Z (Z m ⋊ Γ) ( Q ( Z m ) , A ) N ∗ → Hom Z ( Z m ⋊ Γ) ( Q ( Z m ) , A ) D ∗ → ... Theorem 4.4.
Let Γ be a group not trivially acting on the finite cyclicgroup Z m . Then for any Γ -equivariant Z ( Z m ⋊ Γ) -module A one has Q H ( Z m , A ) = Hom ( Q , A Γ ) Q H n − ( Z m , A ) = KerN ∗ /ImD ∗ XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 15 Q H n Γ ( Z m , A ) = KerD ∗ /ImN ∗ ,for n > and the homomorphisms D ∗ and N ∗ are induced by D andN respectively. One sees that the rational Γ-equivariant (co)homology of finite cyclicΓ-groups is of period 2 for n > -derived functors and Γ -equivariant Hochschildhomology Let Γ be a group acting on the ring Λ with unit and let A be a leftΛ-module on which Γ is acting such that γ ( λa ) = γ λ γ a, γ ∈ Γ , λ ∈ Λ , a ∈ A. Then A will be called Γ-equivariant left Λ-module. Denote by Γ A the Λ-submodule of A generated by the elements γ a − a , a ∈ A , γ ∈ Γ,and by A Γ the quotient of A by Γ A . Definition 5.1.
It will be said a group Γ is acting on a chain complex L ∗ of left Λ-modules ... → L n δ n → L n − δ n − → ... if Γ acts on each L n becoming Γ-equivariant Λ-module and δ n satisfiesthe following condition(1) δ n ( γ l n − l n ) ∈ Γ L n − , l n ∈ L n , γ ∈ Γ . In particular condition (1) is satisfied if δ n is compatible withe theaction of Γ. Definition 5.2.
The homology groups H Γ n ( L ∗ ), n ∈ Z , of the chaincomplex L ∗ are defined as the homology groups of the quotient chaincomplex of L ∗ : L Γ ∗ = ... → ( L n ) Γ δ ′ n → ( L n − ) Γ δ ′ n − → ... The groups H Γ n ( L ∗ ) are called Γ-equivariant homology groups of L ∗ .The consideration of the chain complex L Γ ∗ is motivated by the fol-lowing important cases.Case 1 - Relation with cyclic homology.
It will be said that a group Γ is acting on a left Λ-module M if it isacting on Λ and M such that γ ( λm ) = γ λ γ m . In that case M is calledΓ-equivariant Λ-module.It will be said that a group Γ is acting on unital κ -algebra A if itis acting on A and κ such that γ ( ka ) = γ k γ a , k ∈ κ, a ∈ A, γ ∈ Γ.The group Γ acts on A-bimodule M if it is acting on the κ -algebra Aand on M such that γ (( am ) a ′ ) − ( γ a γ m ) γ a ′ = γ a ( γ m γ a ′ ). In that case M is called Γ-equivariant A-bimodule or equivalently Γ-equivariant A e -module, where A e is the enveloping algebra of A, A e = A ⊗ A op , A op being the opposite algebra of A.Let(5.1) C ∗ ( A, M ) = ... → M ⊗ A ⊗ n b → M ⊗ A ⊗ n − b → ... → M ⊗ A b → M be the Hochschild complex, where the κ -module M ⊗ A ⊗ n is in degreen and the tensor product is taken over κ . Definition 5.3.
Let Γ be a group acting on the Hochschild complex C ∗ ( A, M ). Then H Γ ∗ ( C ∗ ( A, M )) is called Γ-equivariant Hochschild ho-mology of the κ -algebra A with coefficients in Γ-equivariant A e - mod-ule M. If the action of Γ is induced by its actions on A and M, then H Γ ∗ ( C ∗ ( A, M )) will be denoted H Γ ∗ ( A, M ) and HH Γ ∗ ( A ) for M = A.Let M = A and assume the group Z of integers acts trivially on κ and A. Define the action of Z on A ⊗ n +1 via the composition of thenatural homomorphism Z → Z / ( n + 1) Z ∼ = Z n +1 with the action of Z n +1 on A ⊗ n +1 given by t n ( a , ..., a n ) = ( − n ( a n , a , ..., a n − ) on the generator ( a , ..., a n ) of A ⊗ n +1 , n ≥
1. Then A ⊗ n +1 becomes Z -equivariant κ -module and it iswell known that the κ -homomorphism b satisfies condition (1) of Defi-nition 5.1. Therefore Z is acting on the chain complex C ∗ ( A, A ) whichis just the Connes complex and the homology groups H Z n ( C ∗ ( A, A )), n (cid:23)
0, are Connes homology groups of the κ -algebra A. It is wellknown that they are isomorphic to cyclic homology groups of A when κ contains the group Q of rational numbers.We conclude that the cyclic homology of the algebra A over κ con-taining Q is Z -equivariant Hochschild homology H Z ∗ ( C ∗ ( A, A )) .Case 2 -
Relation with Γ- equivariant homology of groups Let G be a group on which the group Γ is acting and consider thebar Z ( G )-resolution of Z : B ∗ ( Z ) = ... → B n → B n − → ... → B → B → Z → B = Z ( G ) and B n is the free Z ( G ) - module generated by[ g , ..., g n ], g i ∈ G, n ≥
1. The action of the group Γ on B ∗ ( Z ) is givenby γ ( g [ g , ..., g n ]) = γ g [ γ g , ..., γ g n ] , n ≥ Z .The action of Γ can be extended to the integral homology complexof G: C ∗ ( G ) = ... → C n → C n − → ... → C → C → C = Z , C n is the free abelian group generated by [ g , ..., g n ], g i ∈ G, n ≥ γ ([ g , ..., g n ]) = [ γ g , ..., γ g n ] , n ≥ XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 17
According to [27] the n-th Γ-equivariant integral homology H Γ n ( G ) ofG is the n-th homology group of the chain complex B ∗ ( Z ) ⊗ Z ( G ⋊ Γ) Z .It is easily checked that H Γ n ( G ) is isomorphic to the n-th Γ-equivarianthomology H Γ n ( C ∗ ( G )) of the chain complex C ∗ ( G ). In effect, Let F =Σ i Z ( G ) a i be a Z ( G ⋊ Γ)- module free as Z ( G )-module and the setof generators Y = { a i } be a Γ-set. As mentioned above it is calledrelatively free Z ( G ⋊ Γ)- module. Then the following isomorphismholds: F ⊗ Z ( G ⋊ Γ) Z ∼ = (Σ i Z a i ) Γ ,implying the needed isomorphism.Besides the group action on the Hochschild complex C ∗ ( A, A ) ofthe case 1 it is interesting to consider the action of the group Γ on C ∗ ( A, A ) induced by the action of Γ on the κ -algebra A, γ ( a , ..., a n ) =( γ a , ..., γ a n ) on the generators ( a , ..., a n ) of A ⊗ n , n ≥
1. Under thisaction of Γ the κ -homomorphism b is compatible and the κ -module A ⊗ n is Γ-equivariant. We mean particularly the case when A = Z ( G )and the action of Γ on A is induced by the action of Γ on the group G.It is clear in this case one has isomorphisms H Γ n ( C ∗ ( Z ( G ) , Z ( G ))) ∼ = H Γ n ( C ∗ ( G )) ∼ = H Γ n ( G ) , n ≥ κ -algebras for Q ⊂ κ and theΓ-equivariant integral homology of groups.In what follows it always will be assumed that the Γ-equivariantHochschild homology of the κ -algebra A is defined by the action of Γon A. In order to describe the Γ-equivariant Hochschild homology interms of derived functors the notion of Γ-equivariant derived functorswill be introduced.Let A ΓΛ be the category of Γ-equivariant left Λ-modules. A morphismof the category A ΓΛ is a Λ-homomorphism f : M → M ′ such that f ( γ m ) = γ f ( m ) , m ∈ M, γ ∈ Γ. As mentioned in Preliminaries, ifΛ = Z ( G ) the category A ΓΛ is equivalent to the category of G ⋊ Γ-modules. It is evident if Λ = Z and Γ acts trivially on Z the category A ΓΛ is equivalent to the category of Z (Γ)-modules. A Γ-equivariantΛ-module free as Λ-module with basis a Γ-set is called relatively freeΓ-equivariant Λ-module. A retract of a relatively free Γ-equivariantΛ-module is called relatively projective Γ-equivariant Λ-module. Anyshort exact sequence of Γ-equivariant Λ-modules having a Γ-sectionmap is called proper exact sequence of Γ-equivariant Λ-modules.A long exact sequence of Γ-equivariant Λ-modules P ∗ ( M ) = ... → P n α n → P n − → ... → P α → P τ → M → P n is relativelyprojective Γ-equivariant Λ-module and sequences 0 → Kerτ → P τ → M →
0, 0 → Kerα n → P n → Imα n → , n ≥
1, are proper sequences.It is obvious there is a natural action of Γ on the chain complex P ∗ ( M ). Definition 5.4.
1. Let T be an additive covariant functor from A ΓΛ tothe category A Γ Z . The left Γ-equivariant derived functors L Γ n T, n ≥ L Γ n T ( M ) = H Γ n ( T P ∗ ( M )).2. Let T be an additive contravariant functor from A ΓΛ to the cat-egory A Γ Z . The right Γ-equivariant derived functors R n Γ T, n ≥
0, of Tare defined as R n Γ T ( M ) = H Γ n ( T P ∗ ( M )).It is easily checked that these derived functors are correctly definedand they don’t depend of the Γ-projetive resolution of M.Consider the action of Γ on the tensor product M ⊗ Λ L of Γ-equivariantΛ-modules M and L induced by the action of Γ on the couples (m,l), γ ( m, l ) =( γ m, γ l ), m ∈ M, l ∈ L .Then M ⊗ Λ L becomes a Γ-equivariant abeliangroup or equivalently Z (Γ)-module. The left Γ- derived functors ofthe functor − ⊗ Λ L : A ΓΛ → A Γ Z will be denoted T or Λ , Γ n ( − , L ) , n ≥ Z ( G ), we recover the functors T or P n defined in [27], where P is the projective class of proper sequences of Γ-equivariant Z ( G )-modules and L is a trivial Γ-equivariant Z ( G )-module, in particular T or Z ( G ) , Γ n ( Z , L ) ∼ = H Γ n ( G, L ), n ≥
0, if Γ acts on Λ trivially.The right Γ-derived functors of the functor
Hom Λ ( − , L ) , n ≥
0, willbe denoted
Ext n Λ , Γ ( − , L ) , n ≥
0. If Λ = Z ( G ), we recover the functors Ext n P defined in [27] and Ext n Z ( G ) , Γ ( Z , L ) ∼ = H n Γ ( G, L ) , n ≥ Definition 5.5.
If the group Γ is acting on the κ -algebra A, let [ A, A ] Γ denote the κ -submodule of A generated by the elements { γ a − a, aa ′ − a ′ a } , a, a ′ ∈ A, γ ∈ Γ. It will be called the Γ-additive commutator ofA.Let A be a commutative unital κ -algebra and Ω ( A ) be the A-moduleof K¨ a hler differentials generated by the κ -linear symbols da, a ∈ A withdefining relation d ( ab ) = adb + bda, a, b ∈ A . One defines the action ofΓ on Ω ( A ) as follows: γ ( da ) = d ( γ a )) , a ∈ A, γ ∈ Γ.The Γ-equivariant A-module Ω ( A ) of K¨ a hler differentials is definedas (Ω ( A )) Γ . Theorem 5.6.
Let Γ be a group acting on unital κ -algebra A. Thenone has1. HH Γ0 ( A ) = A/ [ A, A ] Γ ,
2. If A is commutative, HH Γ1 ( A ) ∼ = Ω ( A ) ,
3. If A is relatively projective Γ -equivariant κ -module, H Γ n ( A, M ) =
T or A e , Γ n ( A, M ) for every Γ -equivariant A e -bimodule M,4. Morita equivalence for Γ -equivariant Hochschild homology. Theinclusion maps A → M r ( A ) , M → M r ( M ) , induce the isomorphism XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 19 H Γ n ( A, M ) → H Γ n (M r ( A ) , M r ( M )) , r ≥ , n ≥ , where M r ( A ) and M r ( M ) are the κ -algebra of r-matrices over A and M respectively. P roof
1) Taking into account the homomorphism b is a Γ-homomorphismthe first equality is straightforward. If A is commutative, HH Γ0 ( A ) = A Γ .2) It is well known that the maps HH ( A ) → Ω ( A ) and Ω ( A ) → HH ( A ) sending respectively the class of a ⊗ a ′ to ada’ and ada’ tothe class of a ⊗ a ′ are inverse to each other. It suffices to remark that γ ( a ⊗ a ′ ) − a ⊗ a ′ = γ a ⊗ γ a ′ − a ⊗ a ′ is sending to γ ad γ a ′ - ada’ = γ ( ada ′ )- ada’ and conversely, γ ( ada ′ ) - ada’ is sending to γ ( a ⊗ a ′ ) − a ⊗ a ′ .3) Consider the Hochschild bar complex C bar ∗ ( A ) = ... → A ⊗ n +1 b ′ → A ⊗ n b ′ → ... b ′ → A ⊗ ,where A ⊗ n +1 is in degree n-1, n ≥
1, and b ′ = P n − i =0 ( − d i , d i are differentials of the Hochschild complex C ∗ ( A, A ) and the groupΓ is acting on the Hochschild bar complex as it is defined for theHochschild complex. It is evident every A ⊗ n , n ≥
2, is relatively projec-tive Γ-equivariant κ -module, since A possesses this property. Moreover A ⊗ n +2 , n ≥
0, is relatively projective Γ-equivariant left A e -module withthe action( α, β )( a , ..., a n +1 ) = ( αa , a , ..., a n , a n +1 β ). The contracting homo-topy s : A ⊗ n → A ⊗ n +1 , s ( a , ..., a n ) = (1 , a , ..., a n ), is a Γ-map satisfy-ing the equality b’s + sb’ = id.Therefore the chain complex C bar + ∗ ( A ) = C bar ∗ ( A ) b → A , b ( a ⊗ a ′ ) = aa ′ , a, a ′ ∈ A , is a Γ-projective resolution of the Γ-equivariant A e -module A. Upon tensoring this Γ-projective resolution with a Γ-equivariant A e -module M one obtains the Hochschild complex because of the iso-morphism M ⊗ A e A ⊗ n +2 ∼ = M ⊗ κ A ⊗ n . This implies the equalities H Γ n ( A, M ) = H Γ n ( C bar + ∗ ( A ) ⊗ A e M ) = Γ T or A e n ( A, M ).4) The action of the group Γ on A induces an action on M r ( M )given by γ [ m ij ] = [ γ m ij ] , γ ∈ Γ , m ij ∈ M , making the group GL r ( M )of invertible r-matrices a Γ-group. This action is compatible with thenatural inclusions M r ( M ) → M r+1 ( M ) and GL r ( M ) → GL r +1 ( M ) in-ducing the action of Γ on M( M ) = lim → r M r ( M ) and on GL( M ) = lim → r GL r ( M ) respectively.It also induces an action of Γ on the trace map tr: M r ( M ) → M , tr ([ m ij ]) = Σ ri m ii , given by ( γ tr )([ m ij ]) = Σ ri ( γ m ii ) , γ ∈ Γ. The tracemap is extended to tr: M r ( M ) ⊗ M r ( A ⊗ n ) → M ⊗ A . By identifyingM r ( M ) with M r ( κ ) ⊗ M any element of M r ( M ) is a sum of elements u i v i with v i ∈ M r ( κ ) and u i ∈ M , and the trace map takes the formtr ( v a ⊗ ... ⊗ v n a n ) = tr ( v ...v n ) a ⊗ ... ⊗ a n , v i ∈ M n ( κ ) , a ∈ M and a j ∈ A, j ≥
1. The action of Γ on the extended trace map is given by ( γ tr )( v a ⊗ ... ⊗ v n a n ) = tr ( γ v ... γ v n ) γ a ⊗ ... ⊗ γ a n , γ ∈ Γ. It isevident that the extended trace map is a Γ-map taking into accountthat tr ( γ ( v ...v n )) = tr ( γ v ... γ v n ).Thus the extended trace map yields a morphism of chain complexes tr ∗ : C ∗ (M r (A) , M r (M)) → C ∗ ( A, M ) compatible with the action ofthe group Γ and therefore a morphism tr Γ ∗ : C Γ ∗ (M r (A) , M r (M)) → C Γ ∗ ( A, M ). On the other hand one has a morphism inc Γ ∗ : C Γ ∗ ( A, M ) → C Γ ∗ (M r (A) , M r (M)) induced by the inclusion maps A → M r ( A ), M → M r ( M ). It is immediate that tr Γ ∗ inc Γ ∗ = id . It is well known that inc Γ ∗ tr Γ ∗ is homotopic to id and the homotopy h = Σ i ( − i h i is definedby the formula h i ( a , ..., a n ) = Σ e ij ( a jk ) ⊗ e ( a km ) ⊗ ... ⊗ e ( a ipq ) ⊗ e q (1) ⊗ a i +1 ⊗ ... ⊗ a n ,where the sum is extended over all possible sets of indices (j,k,m,...,p,q), a is in M r (M), other a i are in M r (A) and the e ij denoted elementarymatrices. Accordingly to this homotopy formula one has the equalities γ ( h i ( a , ..., a n )) = Σ γ e ij ( a jk ) ⊗ γ e ( a km ) ⊗ ... ⊗ γ e ( a ipq ) ⊗ γ e q (1) ⊗ γ a i +1 ⊗ ... ⊗ γ a n = h i ( γ a , ..., γ a n ),showing the homotopy h is compatible with the action of Γ and itinduces the homotopy of inc Γ ∗ tr Γ ∗ to the identity. This completes theproof of the theorem which extends well known results on Hochschildhomology for Γ acting trivially on A.Besides Λ = Z ( G ), the case Λ = Z (Γ acting trivially on Z ) is alsointeresting. One means the consideration of the right Γ-equivariant de-rived functors Ext n Λ , Γ ( − , M ) of the contravariant functor Hom ΓΛ ( − , M )from the category A ΓΛ of Γ-equivariant left Λ-modules to the cate-gory of abelian groups, where Hom ΓΛ ( L, M ) is the abelian group ofΛ-homorphisms f : L → M compatible with the action of Γ and as-suming that Γ is trivially acting on Hom ΓΛ ( L, M ).6.
Extensions of crossed Γ -modules In this section the investigation of extensions of Γ-groups is contin-ued for the class of Γ-groups endowed with a crossed Γ-module struc-ture. These extensions are called Γ-extensions of crossed Γ-modules.As noted in the Introduction the extension theory of crossed moduleshas been treated by many mathematicians. Our approach to extensiontheory of crossed modules substantially extends the class of relativeextensions of epimorphisms of groups introduced and investigated byLoday [33].
XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 21
A crossed Γ-module (
G, µ ) is a pair consisting of a Γ-group G and a Γ-homomorphism µ : G → Γ (Γ acting on itsel by conjugation) satisfyingthe Peiffer identity: µ ( g ) g ′ = gg ′ g − , g, g ′ ∈ G. A homomorphism from a crossed Γ-module (
G, µ ) to a crossed Γ-module ( G ′ , µ ′ ) is a Γ-homorphism f : G → G ′ such that µ ′ f = µ .Denote by CrΓ the category of crossed Γ-modules. A crossed Γ-module( G, µ ) will be called trivial if µ is the trivial map, µ ( g ) = e, g ∈ G .There is an obvious equivalence between the category of trivial crossedΓ-modules and the category of Γ-modules.A crossed Γ-module ( G, µ ) will be called elementary crossed Γ-moduleif µ is injective. It is equivalent to the inclusion crossed Γ-module G µ → Γ, where G = µ ( G ) is a normal subgroup of Γ . If (
G, µ ) is acrossed Γ-module and A is a Γ-module satisfying the following property: γ a = a , for a ∈ A, γ ∈ Imµ , then A will be called crossed equivariantΓ-module. In particular
Kerµ is a crossed equivariant Γ-module. It isobvious any crossed equivariant Γ-module is a Γ /Imµ -module.
Definition 6.1.
Let (1) 0 → ( A, σ → ( X, η ) τ → ( G, µ ) → e be a sequence of crossed Γ-modules such that the induced sequence0 → A σ → X τ → G → e is an exact sequence of Γ-groups. Then the sequence (1) will be calledΓ-extension of the crossed Γ-module ( G, µ ) by the crossed equivariantΓ-module A. In that case η ( X ) acts trivially on A and σ ( A ) belongs tothe center of X.If in addition there is a Γ-map α : ( G, µ ) → ( X, η ) such that thecomposite τ α is the identity map, then it will be called Γ-extensionwith Γ-section map or Γ-equivariant extension of the crossed Γ-module(
G, µ ).Two Γ-extensions of (
G, µ ) by the crossed Γ-module (A,1)0 → ( A, σ → ( X , η ) τ → ( G, µ ) → e ,and0 → ( A, σ → ( X , η ) τ → ( G, µ ) → e are called isomorphic if there is a Γ-homomorphism ϑ : ( X , η ) → ( X , η ) inducing the identity map on (A,1) and τ ϑ = τ .Denote by E (( G, µ ) , ( A, E Γ (( G, µ ) , ( A, E ( − , ( A, E Γ ( − , ( A, E (( G, µ ) , − ), E Γ (( G, µ ) , − ) on the category of crossed equivariant Γ-modules to the category of sets are determined in a standard way. In particular, forthe case of E Γ (( G, µ ) , ( A, E = ( A, σ → ( X, η ) τ → ( G, µ )] ∈ E Γ (( G, µ ) , ( A, α and f : ( G ′ , µ ′ ) → ( G, µ ) be a Γ-homomorphism. Bytaking the fiber product D = { ( x, g ′ ) } , τ ( x ) = f ( g ′ ) , x ∈ X, g ′ ∈ G ′ ofthe diagram X σ → G f ← G ′ one obtains the Γ-extension E ′ = ( A, σ ′ → ( D, δ ) p → ( G ′ , µ ′ ), where σ ′ ( a ) = ( σ ( a ) , e ) , p ( x, g ′ ) = g ′ , δ ( x, g ′ ) = µ ( x ). The Γ-section map α ′ : ( G ′ , µ ′ ) → ( D, δ ) is given by α ′ ( g ′ ) =( αf ( g ′ ) , g ′ ). This defines the contravariant functor E (( A, , f ) : E Γ (( G, µ ) , ( A, → E Γ (( G ′ , µ ′ ) , ( A, E (( A, , f )([ E ]) = [ E ′ ]. To define the covariantfunctor E (( G, µ ) , ( h )) : E Γ (( G, µ ) , ( A, → E Γ (( G, µ ) , ( A ′ , h : ( A, → ( A ′ ,
1) is a Γ-homomorphism, take the direct product( A ′ × X, η ′ ), η ′ ( a ′ , x ) = η ( x ), and the Cokernel ( β , η ′′ ) of the injection β : A → ( A ′ ⊗ X, η ′ ), β ( a ) = ( − h ( a ) , σ ( a )), η ′′ ([( a ′ , x )]) = η ′ ( x ). Thisdefines a Γ-extension E ′′ = ( A ′ , σ ′′ → ( Cokernelβ, η ′′ ) τ ′′ → ( G, µ ), where σ ′′ ( a ′ ) = [( a ′ , τ ′′ [( a ′ , x )] = η ( x ), with Γ-section map α ′′ : ( G, µ ) → ( Cokernelβ, η ′′ ) , α ′′ = h ′ α , where h ′ ( x ) = [(0 , x )]., and therefore thecovariant functor E (( G, µ ) , ( h ))([ E ]) = E ′′ .To define (co)homology and Γ-equivariant (co)homology of crossedΓ-modules two important classes will be defined in the category CrΓ ofcrossed Γ-modules.The objects of the first class P Γ of crossed Γ-modules are constructedas follows. Let ( G, µ ) be an arbitrary crossed Γ-module and takethe free group F (Γ × G ) generated by the couples ( γ, g ), γ ∈ Γ, g ∈ G . There is an action of Γ on F (Γ × G ) given by γ ′ ( γ, g ) =( γ ′ γ, g ), γ, γ ′ ∈ Γ , g ∈ G , and a Γ-homomorphism η : F (Γ × G ) → G, η ( γ, g ) = γ g , inducing a Γ-homomorphism µη : F (Γ × G ) → Γ.Consider the normal subgroup of F (Γ × G ) generated by the elements µη ( x ) x ′ x − x ′− for x, x ′ ∈ F (Γ × G ). Let F ( G,µ ) denotes the quotient Γ-group F (Γ × G ) / { µη ( x ) x ′ x − x ′− } . Since ( G, µ ) is a crossed Γ-module,the Γ-homomorphism η sends the normal subgroup { µη ( x ) x ′ x − x ′− } to the unit. This yields a Γ-homomorphism η ′ : F ( G,µ ) → G and acrossed Γ-module ( F ( G,µ ) , µη ′ ) which is called free crossed Γ-modulegenerated by ( G, µ ) implying the canonical surjective homomorphism η ′ : ( F ( G,µ ) , µη ′ ) → ( G, µ ). This construction was used by Loday toshow the existence of the universal central relative extension of a groupepimorphism [33]. The objects of the class P Γ are retracts of freecrossed Γ-modules and are called projective crossed Γ-modules.The construction of the second class P Γ − e of crossed Γ-modules isrealized similarly. Consider the free group F(G) generated by theelements g , g ∈ G . There is an action of Γ on F(G) given by XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 23 γ | g | = | γ g | , g ∈ G, γ ∈ Γ, and let ϕ : F ( G ) → G be the canoni-cal Γ-homomorphism, ϕ ( | g | ) = g , having a Γ-section map σ : G → F ( G ) , σ ( g ) = | g | , g ∈ G . This yields a crossed Γ-module ( F ( G ) , µϕ )and a homomorphism ϕ : ( F ( G ) , µϕ ) → ( G, µ ) having a Γ-sectionmap. The quotient F Γ( G,µ ) = F ( G ) / { µϕ ( x ) x ′ x − x ′− } , x, x ′ ∈ F ( G )provides a crossed Γ-module ( F Γ( G,µ ) , µϕ ′ ) induced by ϕ which will becalled Γ-equivariant free crossed Γ-module. and the canonical surjec-tion ( F Γ( G,µ ) , µϕ ′ ) → ( G, µ ) having a Γ-section map. The class F Γ − e is consisting of all Γ-equivarianr free crossed Γ-modules. The objectsof the class P Γ − e are retracts of free crossed Γ-modules and are calledΓ-equivariant projective crossed Γ-modules. Proposition 6.2.
The classes P Γ and P Γ − e are projective classes inthe category CrΓ of crossed Γ -modules. P roof . To prove the class P Γ is projective it suffices to show thatfor any surjective homomorphism f : ( G ′ , µ ′ ) → ( G, µ ) of crossed Γ-modules and any homomorphism h : F ( L,ν ) → ( G, µ ), where F ( L,ν ) is afree crossed Γ-module, there is a homomorphism h ′ : F ( L,ν ) → ( G ′ , µ ′ )that fh’ = h. For every element (e,l) of Γ × L, l ∈ L , choose an elementg’ of G’ such that f(g’) = h([(e,l)]) and define the Γ-map Γ × L → G ′ sending ( γ, l ) to γ g ′ which induces the required homomorphism h’.For the class P Γ − e it suffices to show that for any surjective ho-momorphism f : ( G ′ , µ ′ ) → ( G, µ ) of crossed Γ-modules having aΓ-section map and any homomorphism h : F Γ( L,ν ) → ( G, µ ), where F Γ( L,ν ) is a Γ-equivariant free crossed Γ-module, there is a homomor-phism h ′ : F Γ( L,ν ) → ( G ′ , µ ′ ) that fh’ = h. It is easily checked thath’ can be defined as h ′ ([ x ]) = σh ([ x ]) where x ∈ F ( L ) an σ is thesection-map of f. This completes the proof of the Proposition.There is a Γ-homomorphism F (Γ × G ) → F ( G ) sending the gen-erator | ( γ, g ) | to | γ g | inducing Γ-homomorphism F ( G,µ ) → F Γ( G,µ ) andtherefore a natural morphism ω : P Γ → P Γ − e from the projective class P Γ to the projective class P Γ − e .For the cohomological interpretation of the abelian group of Γ-extensionsof crossed Γ-modules the right derived functors R n ¯ P T, n ≥
0, of a con-travariant functor T from the category A with finite inverse limits tothe category of abelian groups with respect to a projective class ¯ P will be defined. The case of left derived functors of a covariant func-tor to the category of abelian groups or to the category of groups wasconsidered in [47] and [24,26] respectively.To this aim let us recall some definitions given in [26]. Definition 6.3.
A ¯ P -projective resolution of an object A of the cate-gory A is a pseudo-simplicial projective object over A, P ∗ → A , whichis ¯ P -exact and ¯ P -epimorphic.Since the category A contains finite inverse limits, every object Aadmits a ¯ P -projective resolution which is unique up to simplicial ho-motopy. Definition 6.4.
The right derived functors R n ¯ P T of the contravariantfunctor T with respect to the projective class ¯ P are given by R n ¯ P T ( A ) = H n T ( P ∗ ) , n ≥ . It is obvious that the category CrΓ of crossed Γ-modules is a categorywith finite inverse limits. Denote by
Hom Γ (( G, µ ) , ( A, Hom Γ ( − , ( A, Hom Γ ( − , ( A, P and P Γ respectively. Namely Definition 6.5.
The n-th cohomology and Γ-equivariant cohomologyof the crossed Γ-module (
G, µ ) with coefficients in a Γ-module A aregiven by H n P Γ (( G, µ ) , A ) = R n − P Γ Hom Γ (( G, µ ) , ( A, H n P Γ − e (( G, µ ) , A ) = R n − P Γ − e Hom Γ (( G, µ ) , ( A, n ≥ Theorem 6.6.
One has H P Γ (( G, µ ) , A ) = H P Γ − e (( G, µ ) , A ) ∼ = Hom Γ (( G, µ ) , ( A, . H P Γ (( G, µ ) , A ) ∼ = E (( G, µ ) , ( A, and H P Γ − e (( G, µ ) , A ) ∼ = E Γ (( G, µ ) , ( A, , where A is crossed equivariant Γ -module. H P Γ (( Imµ, σ ) , A ) ∼ = E xt (Γ /Imµ, Γ , A ) ,where E xt (Γ /Imµ, Γ , A ) is the abelian group of relative extensions of (Γ /Imµ, Γ) defined by Loday [33] and ( Imµ, σ, ) is the induced inclusioncrossed Γ -module, σ : Imµ ֒ → Γ .4) The short exact sequence of Γ -modules (6.1) 0 → A ′ f ′ → A f → A ′′ → induces a long exact cohomology sequence for the Γ -module ( G, µ ) ifthe Γ -modules of (6.1) are crossed equivariant: XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 25 → Hom Γ (( G, µ ) , A ′ ) → Hom Γ (( G, µ ) , A )) → Hom Γ (( G, µ ) , A ′′ )) → H P Γ (( G, µ ) , A ′ ) → H P Γ (( G, µ ) , A ) → H P Γ (( G, µ ) , A ′′ ) → H P Γ (( G, µ ) , A ′ ) → ... → H n P Γ (( G, µ ) , A ′ ) → H n P Γ (( G, µ ) , A ) → H P Γ (( G, µ ) , A ′′ ) → H n +1 P Γ (( G, µ ) , A ′ ) → ... .If the sequence (6.1) of Γ -modules possesses the Γ -property then itinduces the long cohomology sequence → Hom Γ (( G, µ ) , A ′ ) → Hom Γ (( G, µ ) , A )) → Hom Γ (( G, µ ) , A ′′ )) → H P Γ − e (( G, µ ) , A ′ ) → H P Γ − e (( G, µ ) , A ) → H P Γ − e (( G, µ ) , A ′′ ) → H P Γ − e (( G, µ ) , A ′ ) → ... → H n P Γ − e (( G, µ ) , A ′ ) → H n P Γ − e (( G, µ ) , A ) → H P Γ − e (( G, µ ) , A ′′ ) → H n +1 P Γ − e (( G, µ ) , A ′ ) → ... . P roof . 1) Obvious.2) Consider the canonical P Γ -projective resolution of ( G, µ ) ... τ → ( K , l ) λ ⇛ λ P (( K ,l ) ,µ ) τ → ( K , l ) λ ⇒ λ P (( G,µ ) ,µη ′ ) τ → ( G, µ ),where ( K , l ) is the simplicial kernel of τ and ( K , l ) is the simpli-cial kernel of ( τ λ , τ λ ). Let f : P ( K n,l ) → A be a Γ-homomorphismsuch that P i =0 τ λ i f = o . Therefore P i =0 λ i f = o and this implies aΓ-homomorphism f : ( K , l ) → A given by f ( x ) = f ( y ) for y = τ ( x ) , x ∈ ( K , l ). Correctness follows from the fact if y = τ ( x ) thenthe triple ( y y − , y y − , y y − ) belongs to ( K , l ) implying f ( y y − ) =0. By the same argument one has f ( x, x ) = 0 for any x ∈ P (( G,µ ) ,µη ′ ) .Now by using the diagram( K , l ) λ ⇒ λ P (( G,µ ) ,µη ′ ) τ → ( G, µ ),and the Γ-homomorphism f : ( K , l ) → A the following crossed Γ-extension of ( G, µ ) is constructed. Take the direct product A × P ( G,µ ) with component wise action of Γ on it. One obtains the crossed Γ-module ( A × P ( G,µ ) , ¯ µ ) where ¯ µ ([ a, x ]) = µη ′ ( x ), x ∈ P ( G,µ ) , a ∈ A . Byintroducing on A × P ( G,µ ) the following equivalence relation:( a, x ) ∼ ( b, y ) if τ ( x ) = τ ( y ) and a · f ( x, y ) = b , this yields thecrossed Γ-module ( A × P ( G,µ ) / ∼ , α ), α ([ a, x ]) = µη ′ ( x ), and the men-tioned crossed Γ-extension of ( G, µ ): E = 0 → ( A, σ → ( A × P ( G,µ ) / ∼ , α ) β → ( G, µ ) → e , where σ ( a ) = [( a, e )] , β ([ a, x ]) = τ ( x ). This allows to define correctly ahomomorphism ϑ : H P (( G, µ ) , A ) → E (( G, µ ) , ( A, f ] to[ E ]. Conversely for any crossed Γ-extension E of ( G, µ ): E = 0 → ( A, → ( X, η ) → ( G, µ ) → e the Γ-homomorphism τ :( P ( G,µ ) , µη ′ ) → ( G, µ ) induces a Γ-homomorphism h ′ : P ( G,µ ) , µη ′ ) → ( X, η ) and its composite with the homomorphism (
X, η ) → ( G, µ ) isequal to τ . Therefore this implies a homomorphism g ′ : K , l ) → ( A,
1) such that the composite of g’ with the homomorphism ( A, → ( X, η ) is equal to γ ( γ ) − . The Γ-homomorphism g = g ′ τ : P (( G,µ ) ,µη ′ ) → A satisfies the equality P i =0 τ λ i g = 0 and implies the homomorphism ϑ ′ : E (( G, µ ) , ( A, → H P Γ (( G, µ ) , A ) sending [ E ] to [ g ] such that thehomomorphisms ϑ and ϑ ′ are inverse to each other.3) First let us recall the definition of relative extensions of groupepimorphisms. Definition 6.7. [33]Let ν : N → Q be an epimorphism of groups. A relative extensionof (Q,N) is given by an exact sequence of groups1 → L λ → M µ → N ν → Q → η of N on M such that ( M, µ ) is a crossed N-module.It is evident that every relative extension (
M, µ ) of (Q,N) inducesthe N-extension of (
Imµ, σ )0 → ( L, → ( M, µ ) → (( Imµ, σ ) → σ : Imµ ֒ → N is the inclusion crossed N-module and one gets amap E xt ( Q, N, L ) → E (( Imµ, σ ) , ( L, Imµ, σ )0 → ( L, → ( X, µ ) → ( Imµ, σ ) → N/Imµ, N )1 → L → X → N → N/Imµ → E (( Imµ, σ ) , ( L, → E xt ( N/Imµ, N, L ) ∼ = E xt ( Q, N, L ) which is the inverse of the map E xt ( Q, N, L ) → E (( Imµ, σ ) , ( L, → A ′ f ′ → A f → A ′′ → Hom Γ (( P ( G,µ ) , µη ′ ) , ( A, → Hom Γ (( P ( G,µ ) , µη ′ ) , ( A ′′ , h : P ( G,µ ) → A ′′ be a Γ-homomorphism and τ : P (Γ × G ) → P ( G,µ ) be the canonical surjection. For hτ ( e, g ) , g ∈ G , take a ∈ A such that f ( a ) = hτ ( e, g ). This yields the Γ-homomorphism ¯ h : P (Γ × G ) → A given by ¯ h ( γ, g ) = γ a , γ ∈ Γ , g ∈ G such that f ¯ h = hτ . It is clearthat any element of the subgroup { µη ( x ) x ′ x − x ′− } goes to the unitby ¯ h , since A is a crossed equivariant Γ-module and one obtains aΓ-homomorphism h ′ : ( P ( G,µ ) → A induced by ¯ h such that f h ′ = h . XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 27
If the sequence (6.1) possesses the Γ-property, then the Γ-homomorphism h : P Γ( G,µ ) → A ′′ induces the Γ-homomorphism h ′′ : P Γ( G,µ ) → A ′′ which implies the Γ-homomorphism h ′ : P Γ( G,µ ) → A given by h ′ ( x ) = σh ′′ ( x ), x ∈ G , where σ is the Γ-section map of (6.1). One gets f h ′ ( µη ( x ) x ′ x − x ′− ) = f ( µη ( h ′ ( x )( h ′ ( x ) − ) = 1. The Γ-property of(6.1) implies the equality h ′ ( µη ( x ) x − ) = 1. Therefore the subgroup { µη ( x ) x ′ x − x ′− } goes to unit by the Γ-homomorphism h’. It is evidentthat the Γ-homomorphism ¯ h : P Γ( G,µ ) → A induced by h’ satisfies theequality f ¯ h = h. This completes the proof of the theorem.7.
Homology of crossed Γ -modules To define the homology of crossed Γ-modules the left derived functors L P n T , n ∈
0, of a covariant functor T : A → Gr from the category A with finite inverse limits to the category Gr of groups will be used[24,26]. Definition 7.1. [26]Let P ∗ → A be a pseudo-simplicial projective resolution over theobject A of the category A : P ∗ = ... τ → P λ ⇛ λ P λ ⇒ λ P τ → A ,and consider the chain complex ( L ∗ T ( P ∗ ) , d ∗ ), where L n T ( P ∗ ) = T ( P n ) ∩ KerT ( λ n ) ∩ ... ∩ KerT ( λ nn − ) , n ≥
0, and d n : L n T ( P ∗ ) → L n − T ( P ∗ ) is the restriction of T ( λ nn ) on L n T ( P ∗ ).The n-th homology group of the chain complex ( L ∗ T ( P ∗ ) , d ∗ ) definesthe n-th left derived functor L P Γ n T of T with respect to the projectiveclass P Γ .If the values of the functor T belong to the category of abelian groupsone can also use the definition of Tierney - Vogel [47] by consideringthe homology groups of the chain complex J T ( P ∗ ): J T ( P ∗ ) = { T ( P n ) , δ n , n ≥ } , where δ n = n P i =0 ( − i T ( δ ni ).The natural homomorphism LT ( P ∗ ) → J T ( P ∗ ) induces an isomor-phism of their homology groups and the proof of this assertion is com-pletely similar to the proof for simplicial groups [35]. Definition 7.2.
The n-th homology group of the crossed Γ-module(
G, µ ) with coefficients in the Γ-module A and with respect to theprojective class P Γ is given by H P Γ n (( G, µ ) , A ) = L P Γ n − ( I ( G ) ⊗ G ⋊ Γ A ), n ≥ The Γ-equivariant n-th homology group of the crossed Γ-module(
G, µ ) with coefficients in the Γ-module A and with respect to theprojective class P Γ − e is defined as H P Γ − e n (( G, µ ) , A ) = L P Γ − e n − ( I ( G ) ⊗ G ⋊ Γ A ), n ≥ Proposition 7.3.
One has1) H P Γ n (( G, µ ) , A ) = 0 for n ≥ and H P Γ (( G, µ ) , A ) = I ( G ) ⊗ G ∝ Γ A ,if ( G, µ ) is a projective crossed Γ -module.2)If Γ acts trivially on A, then H P Γ (( G, µ ) , A ) = G/ [ G, G ] Γ ⊗ A.P roof . 1) Let (
P, µ ) be a projective crossed Γ-module and (( Y ∗ , δ ∗ ) , τ, ( P, µ ))be a P Γ -projective resolution of ( P, µ ). Since P is projective, thereis a Γ-homomorphism h : P → Y such that τ h = 1 and inducingthe left contractibility h n : Y n → Y n +1 , n ≥
0. It follows that theabelian augmented pseudo-simplicial group ( I ( Y ∗ ) ⊗ G ⋊ Γ A ) , I ( τ ) ⊗ G ⋊ Γ A ) , I ( P ) ⊗ G ⋊ Γ A ) is left contractive and therefore aspherical. This alsoyields L P Γ ( I ( G ) ⊗ G ⋊ Γ A ) = I ( G ) ⊗ G ⋊ Γ A .2) First it will be shown that every exact sequence of Γ-groups G τ → G τ → G → e .induces the exact sequence H Γ1 ( G ) H Γ1 ( τ ) → H Γ1 ( G ) H Γ1 ( τ ) → H Γ1 ( G ) → → Γ G/ [ G, G ] ∩ Γ G → H ( G ) → H Γ1 ( G ) → (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) Kerξ (cid:15) (cid:15) / / Γ G / [ G , G ] ∩ Γ G (cid:15) (cid:15) ξ / / Γ G/ [ G, G ] ∩ Γ G (cid:15) (cid:15) / / (cid:15) (cid:15) / / KerH ( τ ) σ (cid:15) (cid:15) / / H ( G ) (cid:15) (cid:15) H ( τ ) / / H ( G ) (cid:15) (cid:15) / / KerH Γ1 ( τ ) / / H Γ1 ( G ) (cid:15) (cid:15) H Γ1 ( τ ) / / H Γ1 ( G ) (cid:15) (cid:15) / /
00 0showing that the homomorphism σ is surjective. Consider now thecommutative diagram XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 29 H ( G ) H ( τ ) (cid:15) (cid:15) δ / / KerH ( τ ) σ (cid:15) (cid:15) H Γ1 ( G ) δ ′ / / KerH Γ1 ( τ ) . By Stallings - Stammbach exact homology sequence [44] one con-cludes that δ is surjective implying the surjection of δ ′ and therefore theexactness of the required sequence. The isomorphism I ( G ) ⊗ G ∝ Γ A ∼ = H Γ1 ( G ) ⊗ A [27] gives us the exactness of the sequence I ( G ) ⊗ G ⋊ Γ A → I ( G ) ⊗ G ⋊ Γ A → I ( G ) ⊗ G ⋊ Γ A → P ∗ , µ ∗ ) , τ, ( G, µ )) is a P Γ -projectiveresolution of the crossed Γ-module ( G, µ ) then π ( I ( P ∗ ) ⊗ P ∗ ⋊ Γ A ) isisomorphic to I ( G ) ⊗ G ⋊ Γ A ∼ = G/ [ G, G ] Γ ⊗ A . This completes the proofof the proposition. Remark . The results of Prop.7.3 also hold for the Γ-equivarianthomology H P Γ − e ∗ (( G, µ ) , A ).8. Central Γ -extensions of crossed Γ -modules In what follows H P Γ n (( G, µ ) , Z ) and H P Γ − e n (( G, µ ) , Z ) are denoted H P Γ n ( G, µ ) and H P Γ − e n ( G, µ ) respectively.
Definition 8.1.
The Γ-extension of the crossed Γ-module (
G, µ ) by aΓ-module A0 → ( A, σ → ( U, η ) τ → ( G, µ ) → e is called central if Γ acts trivially on A. It is called universal if forany central Γ-extension ( Y, δ ) of (
G, µ )0 → ( B, → ( Y, δ ) → ( G, µ ) → e there is a unique Γ- homomorphism ( U, η ) → ( Y, δ ) over (
G, µ ).The Γ-extension of the crossed Γ-module (
G, µ ) by a Γ-module Awith Γ-section map0 → ( A, σ → ( U, η ) τ → ( G, µ ) → e is called Γ-equivariant central if Γ acts trivially on A. It is calledΓ-equivariant universal if for any central crossed Γ-extension ( Y, δ ) of(
G, µ ) with Γ-section map0 → ( B, → ( Y, δ ) → ( G, µ ) → e there is a unique Γ- homomorphism ( U, η ) → ( Y, δ ) over (
G, µ ).For the construction of the universal Γ-extension and the Γ-equivariantuniversal Γ-extension of the crossed Γ-module (
G, µ ) the projectiveclasses P Γ and P Γ − e will be used respectively. Consider the above mentioned free group F (Γ , G ) with Γ-action γ ′ ( γ, g ) = ( γ ′ γ, g ) and the Γ-homomorphism η ′ : F (Γ , G ) → G givenby η ′ ( γ, g ) = γ g . Denote by R the kernel of η ′ and by P the quo-tient of the free crossed Γ-module F ( G,µ ) by the normal subgroup gen-erated by the elements [ γ r · r − ] , r ∈ R, γ ∈ Γ. This yields a crossedΓ-module (
P, τ ), where τ is induced by the canonical homomorphism η ′ : ( F ( G,µ ) , µη ′ ) → ( G, µ ).For the Γ-equivarianr case take the free group F(G) with Γ-action γ | g | = | γ g | , g ∈ G. and the kernel L of the canonical homomorphism ϕ : F ( G ) → G . Let P Γ be the quotient of the Γ-equivariant free crossedΓ-module F Γ( G,µ ) by the normal subgroup generated by the elements[ γ l · l − ], l ∈ L, γ ∈ Γ. This yields a crossed Γ-module ( P Γ , τ Γ ), where τ Γ is induced by the canonical surjection F Γ( G,µ ) , µϕ ′ ) → ( G, µ ). Definition 8.2.
The crossed Γ-module (
G, µ ) is called Γ-perfect if H P Γ ( G, µ ) = H P Γ − e ( G, µ ) = 0 . Theorem 8.3.
1) A central Γ -extension ( U, η ) of a crossed Γ -module ( G, µ ) is universal if and only if it is Γ -perfect and every central Γ -extension of ( U, η ) splits.2) If ( G, µ ) is Γ -perfect then the Γ -extension → ( R ′ , → ([ P, P ] / [ P, R ] Γ , ¯ η ) τ ′ → ( G, µ ) → e is the universal Γ -extension of ( G, µ ) , where τ ′ is induced by τ and ¯ η is induced by µη ′ .3) A central Γ -equivariant extension ( U, η ) of a crossed Γ -module ( G, µ ) is Γ -equivariant universal if and only if it is Γ -perfect and everycentral Γ -equivariant extension of ( U, η ) splits.4) If ( G, µ ) is Γ -perfect then the Γ -extension → ( L ′ , → ([ P Γ , P Γ ] / [ P Γ , L ] Γ , ¯ ϕ ) τ ′ → ( G, µ ) → e is the Γ -equivariant universal Γ -extension of ( G, µ ) , where τ ′ is in-duced by τ and ¯ ϕ is induced by µϕ . P roof . 1)The way follows to the classical proof for central groupextensions [37] and that has been also realized for central Γ-equivariantgroup extensions [27]. Let E = 0 → ( C, α → ( U, η ) β → ( G, µ ) → e be the universal Γ-extension of ( G, µ ). If (
U, η ) is not perfect, thereis two distinct morphisms f , f : ( U, η ) → ( U/ [ U, U ] Γ , η ′ ) from E to F = 0 → ( U/ [ U, U ] Γ , σ → ( U/ [ U, U ] Γ , η ′ ) τ → ( G, µ ) → e over ( G, µ ),where f ( x ) = (1 , β ( x )), f ( x ) = ( ψ ( x ) , β ( x )), x ∈ U , and ψ : U → U/ [ U, U ] Γ is the canonical homomorphism. That is in contradictionwith the universality of E. XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 31
Let F = 0 → ( D, σ → ( W, δ ) ω → ( U, η ) → e be a central Γ-extension of ( U, η ). The sequence of Γ-modules0 → ( Kerβω, → ( W, δ ) → ( G, µ ) → e is a central Γ-extensionof ( G, µ ). The universality of E yields a homomorphism f ′ : ( U, η ) → ( W, β ) over (
G, µ ) and the composite ωf ′ is also a homomorphism over( G, µ ) implying the equality ωf ′ = 1 and the splitting of F.2)Let0 → ( D, → ( X, δ ) → ω → ( G, µ ) → G, µ ). Then there is a homomorphism f : ( P, τ ) → ( X, δ ) of crossed Γ-modules over (
G, µ ) such that thediagram ([
P, P ] Γ / [ P, R ] Γ , η ′ ) τ ′ / / f ′ (cid:15) (cid:15) ( G, µ ) k (cid:15) (cid:15) ( X, δ ) ω / / ( G, µ ) ., where τ ′ , η ′ and f ′ are induced by τ , µ and f respectively. Thehomomorphism τ ′ is surjective, since ( G, µ ) is Γ-perfect and there-fore τ ′ is a central Γ-extension of ( G, µ ). Every Γ- homomorphism[
P, P ] Γ / [ P, R ] Γ → Kerω is trivial implying the uniqueness of f ′ over( G, µ ).3)and 4) We omit the proof since it goes along the same lines as theproof of 1) and 2) respectively by replacing in particular the crossedΓ-module (
P, τ ) by the crossed Γ-module ( P Γ , τ Γ ). Conclusion
A central Γ-module has universal Γ-extension if and onlyif it is Γ-perfect. A central Γ-equivariant Γ-module has Γ-equivariantuniversal Γ-extension if and only if it is Γ-perfect.
Theorem 8.4.
1) Let → ( A, → ( B, δ ) ϑ → ( G, µ ) → be a Γ -extension of ( G, µ ) and τ : ( F, η ) → ( B, δ ) be a free presenta-tion of ( B, δ ) . Then there is an exact sequence → U → H P Γ ( B, δ ) → H P Γ ( G, µ ) ρ → A/ [ B, A ] Γ → H P Γ ( B, δ ) → H P Γ ( G, µ ) → ,where U is the kernel of [ F, S ] Γ / [ F, R ] Γ → [ B, A ] Γ , R = Kerτ and S = Kerθτ .2) If → ( A, → ( B, δ ) ϑ → ( G, µ ) → is a Γ -equivariant extension of ( G, µ ) and τ Γ : ( F Γ , η Γ ) → ( B, δ ) be a Γ -equivariant free presentation of ( B, δ ) . Then there is an exactsequence → U Γ → H P Γ − e ( B, δ ) → H P Γ − e ( G, µ ) ρ → A/ [ B, A ] Γ → H P Γ − e ( B, δ ) → H P Γ − e ( G, µ ) → ,where U Γ is the kernel of [ F Γ , S Γ ] Γ / [ F Γ , R Γ ] Γ → [ B, A ] Γ , R Γ = Kerτ Γ and S = Kerθτ Γ . P roof . 1) Let ( F ∗ ( G ) , η ∗ ) → ( G, µ ) and ( F ∗ ( B ) , β ∗ ) → ( B, δ ) be F -projective resolutions of ( G, µ ) and (
B, δ ) respectively induced by τ and θτ . The short exact sequence of augmented pseudo-simplicialgroups([ F ∗ , F ∗ ] Γ → [ G, G ] Γ ) → ( F ∗ → G ) → ( F ∗ / [ F ∗ , F ∗ ] Γ → G/ [ G, G ] Γ )yields the short exact sequence0 → H P Γ ( G, µ ) → π ([ F ∗ , F ∗ ] Γ ) → [ G, G ] Γ → (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / Kerα ′ (cid:15) (cid:15) / / Kerα (cid:15) (cid:15) κ / / A (cid:15) (cid:15) / / A/κ ( Kerα ) (cid:15) (cid:15) / / (cid:15) (cid:15) / / / / H P Γ ( B, δ ) α ′ (cid:15) (cid:15) / / π ([ F B ∗ , F B ∗ ] Γ ) α (cid:15) (cid:15) / / B γ (cid:15) (cid:15) / / H P Γ ( B, δ ) (cid:15) (cid:15) / / / / H P Γ ( G, µ ) / / π ([ F G ∗ , F G ∗ ] Γ ) (cid:15) (cid:15) / / G (cid:15) (cid:15) / / H P Γ ( G, µ ) (cid:15) (cid:15) / /
01 1 0It is easily checked that
Kerα = [
F, S ] Γ / [ F, R ] Γ . Therefore κ ( Kerα ) =[
B, A ] Γ and Kerκ is isomorphic to
Kerα ′ . The connecting homomor-phism ρ is defined in a natural way. This completes the proof of thetheorem.As a result of Theorem 8.4 one obtains the Hopf formula for thecrossed Γ-module homology and for the Γ-equivariant crossed Γ-modulehomology. Namely XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 33
Corollary 8.5.
1) If τ : ( F, η ) → ( G, µ ) is a free presentation of ( G, µ ) then H P Γ ( G, µ ) ∼ = R ∩ [ F, F ] Γ / [ F, R ] Γ , R = Kerτ .2)If τ Γ : ( F Γ , η Γ ) → ( G, µ ) is a Γ -equivariant free presentation of ( G, µ ) then H P Γ − e ( G, µ ) ∼ = R Γ ∩ [ F Γ , F Γ ] Γ / [ F Γ , R Γ ] Γ , R Γ = Kerτ Γ . P roof . By Theorem 8.4 the Γ-extension of (
G, µ ):( R, σ → ( F, η ) → ( G, µ )induces the exact sequence H P Γ ( F, η ) → H P Γ ( G, µ ) → R/ [ F, R ] Γ σ ′ → F/ [ F, F ] Γ → G/ [ G, G ] Γ →
0, where σ ′ is induced by σ and H P Γ ( F, η ) = 0. since (
F, η ) is afree crossed Γ-module. Finally one gets H P Γ ( G, µ ) ∼ = Kerσ ′ = R ∩ [ F, F ] Γ / [ F, R ] Γ .The proof for the case 2) of Theorwm 8.4 and Corollary 8.5 is com-pletely similar and it is omitted.9. Applications to algebraic K-theory, Galois theory ofcommutative rings and cohomological dimension ofgroups
The first application deals with the connection of the Γ-equivarianthomology of groups and the homology of crossed Γ-modules with therelative algebraic K-functor K ( f ), where f : Λ → Λ ′ is a surjectivehomomorphism of rings with unit. For this purpose we recall the defi-nition of K ( f ). Definition 9.1.
The relative Steinberg group St ( f ) of the surjectivehomomorphism f is the quotient of the free group F ( St (Λ) × Y ) by theminimal St (Λ)-equivarient normal subgroup satisfying the relations( A ) y uij y vij = y u + vij , ( B ) x λij y v = y vij , ( B ) x λij y vkl = y vkl , j = k, i = l, ( B ) x λij y vjk = y λvik y vjk , i = k, ( B ′ ) x λij y vki = y − vλkj y vki , j = k, ( C ) x λij · t = y vij ty − uij , t ∈ F ( St (Λ) × Y ) , where Y is the set of { y uij } , i,j are positive integers and u belongs tothe kernel I of the homomorphism f [33].One defines the homomorphism ϕ f : St ( f ) → E (Λ , I ) ⊂ GL ( I )given by ϕ f ( x · y uij ) = ϕ Λ ( x ) e uij ϕ Λ ( x ) − , where the homomorphism ϕ Λ : ST (Λ) → E (Λ) is sending the generator x λij of St (Λ) to e λij . The group E (Λ , I ) is the direct limit of { E n (Λ , I ) } , n → ∞ , and E n (Λ , I ) is thenormal subgroup of E n (Λ) of elementary n-matrices generated by I-elementary matrices of the form I n + ve ij , v ∈ I and i = j . The group E (Λ) is acting on E (Λ , I ) by conjugation and it is well known that E (Λ , I ) is E (Λ)-perfect. Definition 9.2. [33] K ( f ) = Kerϕ f and Cokerϕ f = K ( f ).The groups K ( f ) and K ( f ) are also noted K (Λ , I )and K (Λ , I )respectively [37,41,25].Denote D the fiber product Λ × Λ ′ Λ with projections p : D → Λand p : D → Λ. Let St(I) be the kernel of St ( p ) and let C ( I ) =[ St ( p ) , St ( p )]. There is a homomorphism µ : St ( I ) /C ( I ) → St (Λ)induced by St ( p ) on St(I). In [33] it is shown that the set of relationsdefining the group St(I) given by Swan [46] is equivalent to the set ofrelations ( A , B , B , B , B ′ ) implying the isomorphisms θ : St ( f ) → St ( I ) /C ( I ), K ( f ) ∼ = → K ( I ) /C ( I ), and the sequence(9.1) 0 → Ker ( µθ ) → St ( f ) µθ → St (Λ) St ( f ) → St (Λ ′ ) → St (Λ ′ ) , St (Λ)).Accordingly to results of [33] the following short exact sequence isprovided(9.2) 0 → K ( f ) → St ( f ) ϕ f → E (Λ , I ) → , where ϕ f = ϕ Λ µϑ (see also [30]). Theorem 9.3.
There is an exact sequence → [ P, S ] St (Λ) / [ P, R ] St (Λ) → H P St ( λ ) ( St ( f ) /St (Λ)( ϕ f )) → H P St ( λ ) ( E (Λ , I )) →→ K ( f ) /St (Λ)( ϕ f )) → , where α : P → St ( f ) is a St (Λ)-projective presentation of St(f), R = Kerα and S = Kerϕ f α.P roof . Consider the normal subgroup of St(f) generated by theelements γ x · x − , x ∈ St ( f ) , γ ∈ St (Λ), such that ϕ f ( γ x · x − ) = 1 . .This subgroup is denoted St (Λ)( ϕ f ). By Corollary 3.3 this yields theexact sequence0 → K ( f ) /St (Λ)( ϕ f )) → St ( f ) /St (Λ)( ϕ f )) → E (Λ , I ) → St (Λ)-equivariant extension of E (Λ , I ) havinga St (Λ)-section map. The group St(f)is St (Λ)-perfect [33] implying St ( f ) /St (Λ)( ϕ f )) is also St (Λ)-perfect and therefore H P St ( λ ) ( St ( f ) /St (Λ)( ϕ f ))) = XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 35
0. It remains to apply Theorem 2.8 to get the required exact sequence.This completes the proof.
Remark . The exact sequence of Theorem 9.3 can be replaced bythe exact sequence0 → [ P, S ] St (Λ) / [ P, R ] St (Λ) → H P St ( λ ) ( St ( f ) /St (Λ)( ϕ f )) → ˜ H P St ( λ ) ( E (Λ , I )) → K ( f ) → H P St ( λ ) ( E (Λ , I )) is the fiber product H P St ( λ ) ( E (Λ , I )) × K ( f ) /St (Λ)( ϕ f ) K ( f ).We are now going to establish the relation of the homology of crossedΓ-modules with the relative algebraic K-functor K ( f ).The short exact sequence (9.2) induces the following E (Λ)-extensionof the inclusion crossed E (Λ)-module ( E (Λ , I ) , σ ), σ : E (Λ , I ) ֒ → E (Λ):0 → ( K ( f ) , → ( St ( f ) , ϕ f ) → ( E (Λ , I ) , σ ) → . Take the quotient St’(f) of St(f) by the normal subgroup generatedby the elements γ x · x − , γ ∈ Kerϕ Λ , x ∈ St ( f ) , implying the shortexact sequence0 → K ( f ) → St ′ ( f ) ϕ ′ f → E (Λ , I ) → E (Λ)-modules, E (Λ) is trivially acting on K ( f ) and its action onSt’(f) is realized via the homomorphism ϕ Λ .Finally one obtains a central E (Λ)-extension of the inclusion E (Λ)-module ( E (Λ , I ) , σ ), σ : E (Λ , I ) ֒ → E (Λ),(9.3) 0 → ( K ( f ) , → ( St ′ ( f ) , ϕ ′ f ) → ( E (Λ , I ) , σ ) → , where ϕ ′ f is induced by ϕ Λ µθ .The sequence(9.4)0 → ( K ( f ) /E (Λ)( ϕ ′ f ) , → ( St ′ ( f ) /E (Λ)( ϕ ′ f ) , σϕ ′ f ) → ( E (Λ , I ) , σ ) → E (Λ)-equivariant extension of the inclusion crossed E (Λ)-module( E (Λ , I ) , σ ). It is evident that the crossed E (Λ)-module ( St ′ ( f ) , σϕ ′ f ) is E (Λ)-perfect and therefore the crossed ( E (Λ)-module ( St ′ ( f ) /E (Λ)( ϕ ′ f ) , σϕ ′ f )is also E (Λ)-perfect implying H P E (Λ) − e ( St ′ ( f ) /E (Λ)( ϕ ′ f ) , σϕ ′ f )) = 0 . Theorem 9.5.
1) The sequence of St (Λ) − modules → ( Ker ( µθ ) , → ( St ( f ) , µθ ) → ( KerSt ( f ) , σ ) → is the universal St (Λ )-extension of the inclusion St (Λ) -module ( KerSt ( f ) , σ ) and there is an isomorphism H P St ( λ ) ( KerSt ( f ) , σ ) ∼ = Ker ( µθ ) .2) The sequence (9.4) is the E (Λ) -equivariant extension of the inclu-sion crossed E (Λ) -module ( E (Λ , I ) , σ ) and there is an isomorphism H P E (Λ) − e ( E (Λ , I ) , σ ) ∼ = → K ( f ) /E (Λ)( ϕ ′ f ) .P roof . 1) As noted above in [33] it is proven that the group St(f)is St (Λ)-perfect. Therefore the crossed St (Λ)-module ( St ( f, µθ ) is also St (Λ)-perfect and H P St ( λ ) ( St ( f ) , µθ ) = 0. Since the sequence (9.1) isthe universal relative extension of ( St (Λ ′ ) , St (Λ)) it follows that everycentral St (Λ)-extension of the crossed St (Λ)-module ( St ( f ) , µθ ) splitsimplying H P St ( λ ) ( St ( f ) , µθ ) = 0. It remains to apply the first part ofTheorem 8.4 to get the required isomorphism.2) It is evident that the crossed E (Λ)-module ( St ′ ( f ) , ϕ ′ f ) is E (Λ)-perfect. Therefore the crossed E (Λ)-module ( St ′ ( f ) /E (Λ)( ϕ ′ f ) , σϕ ′ f ) isalso E (Λ)-perfect and H P E ( λ ) − e ( St ′ ( f ) /E (Λ)( ϕ ′ f ) , σϕ ′ f ) = 0.Let 0 → ( A, → ( U ′ , η ′ ) β ′ → ( St ′ ( f ) /E (Λ)( ϕ ′ f ) , σϕ ′ f ) → E (Λ)-equivariant extension of ( St ′ ( f ) /E (Λ)( ϕ ′ f ) , σϕ ′ f ).Consider the fiber product D ′ q ′ (cid:15) (cid:15) q ′ / / ( St ( f ′ ) , ϕ ′ f ) g ′ (cid:15) (cid:15) ( U ′ , η ′ ) β ′ / / St ′ ( f ) /E (Λ)( ϕ ′ f ) , σϕ ′ f ) . , where g ′ is induced by the natural homomorphism St ( f ′ ) → St ( f ′ ) /E (Λ)( ϕ ′ f ).The sequence 0 → ( A, → D ′ q ′ → ( St ( f ′ ) , ϕ ′ f ) is E (Λ)-equivariant ex-tension of ( St ′ ( f ) , ϕ ′ f ) which becomes a St (Λ)-equivariant extension of( St ′ ( f ) , ϕ ′ f ) via the homomorphism ϕ Λ : St (Λ) → E (Λ).Now take the fiber product D q (cid:15) (cid:15) q / / ( St ( f ) , ϕ f ) g (cid:15) (cid:15) D ′ q ′ / / ( St ( f ′ ) , ϕ ′ f ) . The St (Λ)-equivariant extension D q → ( St ( f ) , ϕ f ) is St (Λ)-splittingand therefore it E (Λ)-splits too. Let σ : ( St ( f ) , ϕ f ) → D be the split-ting homomorphism implying the homomorphism q σ : ( St ( f ) , µθ ) → D ′ of crossed E (Λ)-modules such that q ′ q σ : ( St ( f ) , µθ ) = g . For γ x · x − ∈ Kerϕ Λ one has g ( γ x · x − ) = 1 implying the equality q ′ ( γ q σ ( x ) · q σ ( x ) − ) = 1 . By Theorem 3.2 the E (Λ)-equivariant XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 37 extension D ′ q ′ → ( St ( f ′ ) , ϕ ′ f ) has the E-property implying the equal-ities q σ ( γ x · x − ) = γ q σ ( x ) · q σ ( x ) − = 1 . Therefore the homo-morphism q σ sends to the unit the normal subgroup of St(f) gener-ated by the elements γ x · x − ∈ Kerϕ Λ , inducing E (Λ)-homomorphism σ ′ : ( St ′ ( f ) , ϕ ′ f ) → D ′ such that q ′ σ ′ = 1 . Starting with the first diagram and with the splitting homomor-phism σ ′ it is easily shown by the same line of argumentation as forthe previous case that the E (Λ)-equivariant extension β ′ : ( U ′ , η ′ ) → ( St ′ ( f ) /E (Λ)( ϕ ′ f ) , σϕ ′ f ) splits. It follows H P E ( λ ) − e ( St ′ ( f ) /E (Λ)( ϕ ′ f ) , σϕ ′ f ) =0. It remains to apply the second part of Theorem 8.4. This completesthe proof of the theorem.The second application concerns the investigation of the relationshipbetween the equivariant symbol group of non commutative local ringsand the Milnor algebraic K-functor K by using the Γ-homology ofgroups and the homology of crossed Γ-modules that will extend thewell known Matsumoto’s theorem for fields [34].Let A be a unital ring and Sym(A) be the symbol group of the ringA generated by the elements { u, v } , u, v ∈ A ∗ , satisfying the followingrelations( S ) { u, − u } = 1 , u = 1 , − u ∈ A ∗ , ( S ) { uu ′ , v } = { u, v }{ u ′ , v } ,( S ) { u, vv ′ } = { u, v }{ u, v ′ } .where A ∗ denotes the multiplicative group of invertible elements ofthe ring A [ ]. By Matsumoto’s theorem the groups Sym(A) and K ( A )are isomorphic when A is a field [34].For our purpose it is necessary to introduce the notion of equivariantsymbol group. Definition 9.6.
For a ring A with unit the equivariant symbol group
Sym A ∗ ( A ) is defined as the group ( Sym ( A )) A ∗ .The symbol group Sym(A) becomes A ∗ -group by the action u { v, w } = { uvu − , uwu − } . Therefore the equivariant symbol group of an unitalcommutative ring coincides with its symbol group.Now assume the ring A is a non commutative local ring such that A/Rad ( A ) = F . Consider the group U(A) generated by the elements h u, v i , u, v ∈ A ∗ , A ∗ , satisfying the following relations( U ) h u, − u i = 1 , u = 1 , − u ∈ A ∗ , ( U ) h uv, w i = u h v, w ih u, w i ,( U ) h u, vw ih v, wu ih w, uv i = 1,where u h v, w i = h uvu − , uwu − i [21]. The group U(A) becomes A ∗ -group with respect to this action andresults of [21] show us that there is a surjective A ∗ -homomorphism U ( A ) → Sym ( A ). In addition there is a short exact sequence of A ∗ -groups relating U(A) with K ( A ) [21]:(9.5) 0 → K ( A ) → U ( A ) τ → [ A ∗ , A ∗ ] → , where A ∗ acts trivially on K ( A ) and by conjugation on [ A ∗ , A ∗ ],and τ ( h u, v i ) = [ u, v ]. Moreover the sequence (9.5) induces a central A ∗ -extension of the inclusion crossed A ∗ -module ([ A ∗ , A ∗ ] , i ):(9.6) 0 → ( K ( A ) , → ( U ( A ) , τ ) iτ → ([ A ∗ , A ∗ ] , i ) → . We will need the corresponding A ∗ -equivariant versions of these twosequences. Namely,(9.7) 0 → ( K ( A )) /A ∗ ( τ ) → U ′ ( A ) τ ′ → [ A ∗ , A ∗ ] → , (9.8) 0 → ( K ( A )) /A ∗ ( τ ) , → ( U ′ ( A ) , iτ ′ ) τ ′ → ([ A ∗ , A ∗ ] , i ) → , where U ′ ( A ) = ( U ( A )) /A ∗ ( τ ) and τ ′ is induced by τ .The subgroup A ∗ ( τ ) of K ( A ) is generated by the elements h γ, x i such that γx = xγ , where γ ∈ A ∗ , x ∈ [ A ∗ , A ∗ ]. In effect, let Q h u i , v i i be an element of U(A). One has Q γ h u i , v i i = Q h γ, [ u i , v i ] ih u i , v i i = h γ, Q [ u i , v i ] i Q h u i , v i i . Thus γ ( Q h u i , v i i ) · ( Q h u i , v i i ) − = h γ, Q [ u i , v i ] i and the equality τ ( γ ( Q h u i , v i i ) · ( Q h u i , v i i ) − ) = 1 implies τ ( γ ( Q h u i , v i i ) · ( Q h u i , v i i ) − ) = [ γ Q h u i , v i ] = 1.By this way the defining relations for the group U’(A) have been alsoprovided as follows:( U ) h u, − u i = 1 , u = 1 , − u ∈ A ∗ , ( U ) h uv, w i = u h v, w ih u, w i ,( U ) h u, vw ih v, wu ih w, uv i = 1,( U ) uv = vu, u ∈ A ∗ , v ∈ [ A ∗ , A ∗ ]. Theorem 9.7.
There is an exact sequence → [ P, S ] A ∗ / [ P, R ] A ∗ → H A ∗ ( U ′ ( A )) → H A ∗ ([ A ∗ , A ∗ ]) → K ( A ) /A ∗ ( τ ) → Sym A ∗ ( A ) → [ A ∗ , A ∗ ] / [ A ∗ , A ∗ ] A ∗ → , where α : P → U ′ ( A ) is A ∗ -projective presentation of U’(A), R = Kerα and S = Kerτ ′ α . P roof . First of all it will be proved that the groups H A ∗ ( U ′ ( A )) and Sym A ∗ ( A ) are isomorphic. Consider the system of relations ( S , S , S ′ )which is equivalent to the system ( S , S , S ) of defining relations for the XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 39 symbol group Sym(A)[21]. It is easily checked that the system of rela-tions (( S , S , S ′ , L , L ) is equivalent to the system ( U , U , U , M , M ),where ( L : w { u, v } = { u, v } , ( L ) : { u, v }{ u ′ , v ′ } = { u ′ , v ′ }{ u, v } ,and( M ) : w h u, v i = h u, v i , ( M ) : h u, v ih u ′ , v ′ i = h u ′ , v ′ ih u, v i . For thegroup U(A) the relation h u, v ih u ′ , v ′ i = [ u,v ] h u ′ , v ′ ih u, v i holds [21], im-plying the group U ( A ) A ∗ defined by the system of relations (( U , U , U , M )is abelian. Therefore the group Sym ( A ) A ∗ is also abelian and this yieldsthe following sequence of equivalences of defining systems:( S , S , S , L ) ≈ ( S , S , S ′ , L ) ≈ (( S , S , S ′ , L , L ) ≈ ( U , U , U , M , M ) ≈ ( U , U , U , M )showing the isomorphism of groups Sym ( A ) A ∗ and U ( A ) A ∗ . Finallythis induces the following isomorphisms H A ∗ ( U ′ ( A )) ∼ = U ′ ( A ) A ∗ ∼ = U ( A ) A ∗ ∼ = Sym ( A ) A ∗ . It remains to apply Theorem 2.8 for the se-quence (9.7). This completes the proof of the theorem. Corollary 9.8. (1) As a consequence of this theorem there is an exactsequence → [ P, S ] A ∗ / [ P, R ] A ∗ → H A ∗ ( U ′ ( A )) → ˜ H A ∗ ([ A ∗ , A ∗ ]) → K ( A ) → Sym A ∗ ( A ) → [ A ∗ , A ∗ ] / [ A ∗ , A ∗ ] A ∗ → , and(2) if [ A ∗ , A ∗ ] is quasi-perfect then the sequence → [ P, S ] A ∗ / [ P, R ] A ∗ → H A ∗ ( U ′ ( A )) → ˜ H A ∗ ([ A ∗ , A ∗ ]) → K ( A ) → Sym A ∗ ( A ) → is exact. where ˜ H A ∗ ([ A ∗ , A ∗ ]) is the fiber product H A ∗ ([ A ∗ , A ∗ )]) × K ( A ) /A ∗ ( τ ) K ( A ).The sequence of Corollary 9.8,(2) generalizes the exact sequence H ( U ( D )) → H ([ D ∗ , D ∗ ]) → K ( D ) → Sym ( D ) → D ∗ , D ∗ ]is perfect, and the sequence of Corollary 9.8,(1) can be considered asan abelian version of the exact sequence of Guin [21]( A ∗ ) ab ⊗ Z K ( A ) → ¯ H ( A ∗ , U ( A )) → ¯ H ( A ∗ , [ A ∗ , A ∗ ]) → K ( A ) → Sym ( A ) → [ A ∗ , A ∗ ] / [ A ∗ , [ A ∗ , A ∗ ]] → H is the first non abelian homomolgy of groups with coeffi-cients in crossed modules. Remark . Guin’s low dimensional non-abelian group homoloy withcoefficients in crossed modules is closely related to integral homology ofcrossed modules. Let (
A, δ ) be a crossed G-module and ¯ H ( G, A ) , ¯ H ( G, A )denote Guin’s group homology with coefficients in the crossed G -module( A, δ ). In [5] it is shown that there is a group homomorphism ϕ : G N A → A , ϕ ( g ⊗ a ) = g a · a − , where G N A is the non-abelian ten-sor product of Brown-Loday, and ( G N A, ϕ ) is a crossed A-module.Then ¯ H ( G, A ) = cokerϕ and ¯ H ( G, A ) =
Kerϕ . It is easily seen thatthere is an isomorphism of the abelianization ¯ H ab ( G, A ) with H P ( A, δ )and there is an exact sequence H P ( G N A, ϕ ) → H P (Γ A, σ ) → ¯ H ( G, A ) → G N A/ ([ G N A, G N A ]) A → Γ A/ ([Γ A, Γ A ]) A → → ( ¯ H ( G, A ) , → ( G N A, ϕ ) → (Γ A, σ ) → C p n be the cyclic group of order p n and S n be the splitting ring of the polynomial x p n − S ∗ n is the group of invertibleelements and µ p n is the group of n-roots of 1 in the splitting ring S n , N B ( R, C p n ) is the set of isomorphism classes of Galois extensions withnormal basis of R with Galois group C p n and Γ n is the Galois group of S n . Theorem 9.10.
There are bijections1.
N B ( R, C p n ) ∼ = Ext Z , Γ n ( µ p n , S ∗ n ) , where Γ n is trivially acting on Z .2. N B ( R, C p n ) ∼ = H n ( µ p n , S ∗ n ) , where S ∗ n is a trivial µ p n -module. P roof . First of all it is necessary to show that one has a bijection of
Ext , Γ ( L, M ) with E , ΛΓ ( L, M ) which is the set of isomorphism classesof short exact sequences 0 → M → X → L → ... → F n α n → F n − α n − → ... α → F α → F ( L ) τ → L → F ( L ) is the free Λ-module with basis | l | , l ∈ L , τ is thenatural surjective homomorphism and the action of Γ on F ( L ) isgiven by γ | l | = | γ l | , l ∈ L , F = F ( Kerτ ) and by induction on nthe relatively free Γ-equivariant Λ-module F n , n > , is defined as F ( Kerα n − ), the homomorphism α n is induced by the natural homo-morphism F ( Kerα n − ) → Kerα n − . Then Ext n Λ , Γ ( L, M ) , n ≥
0, isthe n-th homology group of the chain complex0 → Hom ΓΛ ( F ( L ) , M ) → Hom ΓΛ ( F , M ) → ... → Hom ΓΛ ( F n − , M ) → Hom ΓΛ ( F n , M ) → ... . XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 41
It is evident
Ext , Γ ( L, M ) ∼ = Hom ΓΛ ( L, M ). Let E : 0 → M → X → L → σ : L → X , [ E ] ∈ E , ΛΓ ( L, M ). The homomorphism τ induces a Λ-homorphism F ( L ) → X compatible with the action of Γsending | l | to σ ( l ). Let f denote its restriction on Kerτ and let [ f α ] bethe class of the homomorphism f α : F → M . Therefore one obtains amap from E , ΛΓ ( L, M ) to
Ext , Γ ( L, M ) sending [E] to [ f α ]. Conversely,if [ g ] ∈ Ext , Γ ( L, M ), then g induces a Λ-homomorphism g ′ : Kerτ → M compatible with the action of Γ. Similarly to the classical case(when Γ is acting trivially on Λ) by using the homomorphism g’ onecan construct a short exact sequence of Γ-equivariant Λ-modules with Γ-section map as follows. Take the sum M ⊕ F ( L )which is a Γ-equivariantΛ-module with componentwise action of Γ and its quotient Y by thesubmodule generated by the elements ( − g ′ ( x ) , ϑ ( x )), x ∈ Kerτ , where ϑ : Kerτ → F ( L ) is the inclusion map. One gets the needed shortexact sequence E g : 0 → M β → Y η → L → β ( m ) = [( m, η ([ m, x ]) = τ ( x ), and the Γ-section map is δ ( l ) = [(0 , | l | )]. Thus one obtains a map Ext , Γ ( L, M ) → E , ΛΓ ( L, M )sending [g] to [ E g ]. It is easily checked that both maps induced by[ E ] [ f α ] and [ g ] [ E g ] respectively are inverse to each other.In [29] there is the following formula of G.Janelidze N B ( R, C p n ) = Ext S n ( Hom ( J, U n ( R ))) , U ( R n )), where J = C p n , U ( R n ) = S ∗ n , U n ( R ) = µ p [ n ] and S n is the category of Γ n -sets.As noted by C.Greither [20] this beautiful formula allowed us toestablish the bijection of RN B ( R, C p n ) with the set of isomorphismclasses of short exact sequences 0 → S ∗ n → X → µ p n → n -modules having Γ n -section map. This completes the proof of the firstbijection.For the second bijection it suffices to remark that suppose 0 → M → X → L → Z , then the groupX is abelian and the considered sequence is a short exact sequenceof Γ-equivariant L-modules with Γ-section map. In that case by [27,Theorem 20] this implies the bijection E , Z Γ ( L, M ) ∼ = H ( L, M ). Thiscompletes the proof of the theorem.Finally, the relation of Γ-equivariant cohomology of Γ-groups withequivariant dimensions of groups with operators will be established,particularly with the equivariant cohomological dimension of Γ-groups.
Recently in [19] the important and well known theorems of Eilenberg-Ganea [14] and Stallings-Swan [43,45] relating the cohomological dimen-sion, the geometric dimension and the Lusternik-Schnirelmann cate-gory have been extended to the setting of Γ-groups as follows:1) Equivariant Eilenberg - Ganea Theorem:Let G be a Γ-group, where Γ is finite. Then the chain of inequalities cd Γ ( G ) ≤ cat Γ ( G ) ≤ gd Γ ( G ) ≤ sup { , cd Γ ( G ) } is satisfied. Furthermore, if cd Γ ( G ) = 2 then cat Γ ( G ) = 2.2) Equivariant Stallings - Swan Theorem:Let G be a Γ-group, where Γ is finite. The following equalities areequivalent: (1) gd Γ ( G ) = 1 , (2) cat Γ ( G ) = 1 , (3) cd Γ ( G ) = 1 , (4) G is a non-trivial free Γ-group.For this purpose the equivariant version of these three quantitieshave been provided and the equivariant group cohomology has beenintroduced that is the generalization of the Γ-equivariant cohomologyof Γ-groups allowing a wider class of coefficients. It is defined as thegroup cohomology H ∗ ( O G ( G ⋊ Γ) , M ) = Ext ∗ O G ( G ⋊ Γ) ( Z , M ), where G denotes the family of subgroups of G ⋊ Γ which are conjugate to asubgroup of Γ and O G ( G ⋊ Γ) is the orbit category whose objects are theΓ-sets ( G ⋊ Γ) /H for H ∈ G and morphisms are Γ-maps. An O G ( G ⋊ Γ)-module is a contravariant functor from the category O G ( G ⋊ Γ) to thecategory of abelian groups and Z is the constant functor with value Z . The equivariant cohomological dimension cd Γ ( G ) of a Γ-group G isdefined as the least dimension d such that H d +1 ( O G ( G ⋊ Γ) , M ) = 0for all O G ( G ⋊ Γ)-modules M.It would be natural to introduce another algebraic cohomological di-mension cd Γ ( G ) of a Γ-group G based on the Γ-equivariant cohomologyof Γ-groups as follows: Definition 9.11.
The cohomological dimension cd Γ ( G ) of the Γ- groupG is the least dimension d such that H d +1Γ ( G, M ) = 0 for all ( G ⋊ Γ)-modules M.In [19, Remark 9.1] it is notified that the Γ-equivariant cohomologyof Γ-groups is the relative group cohomology in the sense of Hochschild[22] and Adamson [1] (see also Benson [4]) enhancing the interest tothis equivariant group cohomology. The Γ-equivariant group cohomol-ogy H ∗ Γ ( G, M ) is isomorphic to H ∗ ( G ⋊ Γ) , Γ; M ) and there is an iso-morphism H ∗ ( G ⋊ Γ) , Γ; M ) ∼ = H ∗ ( O G ( G ⋊ Γ) , M − ), where M − is a O G ( G ⋊ Γ)-module induced by M [39].
XTENSIONS AND (CO)HOMOLOGY OF Γ-GROUPS 43
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A. Razmadze Mathematical Institute of Tbilisi State University, 6,Tamarashvili Str., Tbilisi 0179, Georgia.
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