On invariant (co)homology of a group
Carlos Aquino, Rolando Jimenez, Martin Mijangos, Quitzeh Morales Meléndez
aa r X i v : . [ m a t h . K T ] J u l ON INVARIANT (CO)HOMOLOGY OF A GROUP
CARLOS AQUINO, ROLANDO JIMENEZ, MARTIN MIJANGOS AND QUITZEHMORALES MEL´ENDEZ
Abstract.
There are different notions of homology and cohomology that canbe defined for a group with an action of another group by group automor-phisms. In this paper we address three natural questions that arise in thiscontext. Namely, the relation of these notions with the usual (co)homology ofa semidirect product, the interpretation of the first homology group as somekind of abelianization and the classification of (invariant) group extensions.
Introduction
In [6] Knudson defined homology and cohomology groups for a group G withan action of another group Q by group automorphisms and computed homologygroups for some Z / • relate Knudson homology groups with the homology of the semidirect prod-uct G ⋊ Q as an application of Hochschild-Serre spectral sequence. Thisgeneralizes the spectral sequence in [5]; • give an interpretation of the first homology group as a so called weighed abelianization, a suitable group made up by orbits of the action. Therelation with classic group constructions is also adressed. And • define new cohomology groups and study the notion of invariant groupextensions related to them.The cohomology groups defined coincide with those of Knudson in some very specialcases. Some properties of the new groups are studied. The corresponding homologygroups do not coincide in general with those of Knudson.1. Knudson groups, invariant resolutions and semidirect product
Let Q and G be groups, and Q × G → G an action of Q on G by group auto-morphisms, i.e. q ( g g ) = q ( g ) q ( g ) , q (1) = 1for every q ∈ Q, g , g ∈ G . In this case, we call G a Q -group. Let A be an abeliangroup with trivial G - and Q -actions and denote by C ∗ ( G, A ) the complex C ∗ ( G ) ⊗ A The first three authors were partially supported by SEP-CONACyT Grant 284621.2010
Mathematics Subject Classification.
Primary: 55N25, 55T05; secondary: 18G40, 18G35.
Key words and phrases.
Cohomology of invariant group chains, Serre-Hochschild spectral se-quence, invariant group extensions. where C ∗ ( G ) is the bar complex. In this case, there is an induced action of Q on C ∗ ( G, A ) given by(1) q ([ g | · · · | g n ] ⊗ a ) = [ qg | · · · | qg n ] ⊗ a. According to [6], define the groups of homology and cohomology of invariant groupchains as H Q ∗ ( G, A ) = H ∗ ( C ( G, A ) Q )(2) H ∗ Q ( G, A ) = H ∗ ( Hom ( C ( G ) Q , A )) . (3)Since the differential of the chain complex C ∗ ( G, A ) is Q -equivariant, there is a welldefined action of Q on the homology groups H ∗ ( G, A ). The next theorem is provedin [6] when Q is a finite group. Definition 1.
Let A be an abelian group and n ∈ Z . We say that n is invertiblein A if the group homomorphism ψ : A −→ A given by ψ ( a ) = na for a ∈ A is anisomorphism. Theorem 1. ( [6] Proposition 3.3) Let G be a Q -group and let A be an abeliangroup with trivial G - and Q -actions. Assume that | Q | is invertible in A . Then thenatural map i ∗ : H Q ∗ ( G, A ) −→ H ∗ ( G, A ) Q is an isomorphism. The aim of this section is to prove the following:
Theorem 2.
Let Q be a finite group. Let G be a Q -group and let A be an abeliangroup with trivial G - and Q -actions. If | Q | is invertible in A , then H Qq ( G, A ) ∼ = H q ( G ⋊ Q, A ) . First we give some properties of the group homology in the case when the co-efficients are invertible. Then we will use Theorem 1 to relate the homology ofinvariant group chains to the homology of the semidirect product G ⋊ Q throughthe Hochschild-Serre spectral sequence.At the end of this section, we will show the usefulness of this theorem through anexample. Such an example is presented in [5] but here is treated in a more simpleway.Let Q be a finite group and A be a Q -module. Consider the homomorphism N : A −→ A , such that N ( a ) = P q ∈ Q qa . Since N satisfies N ( qa ) = N ( a ) and im N ⊆ A Q , the homomorphism ¯ N : A Q −→ A Q given by ¯ N (¯ a ) = P q ∈ Q qa is welldefined and is called the norm map. Proposition 1.
Let G be a group and A be a G -module.a) If n ∈ Z is invertible in A , then n is invertible in H ∗ ( G, A ) .b) If G is finite and | G | is invertible in A , then ¯ N : A G −→ A G is an isomor-phism. Proof. a) follows from functoriality of H ∗ ( G, A ) in the second coordinate. It isan exercise in [1, § | G | annihilates ker ¯ N and coker ¯ N , whichyields b) in the present situation. (cid:3) N INVARIANT (CO)HOMOLOGY OF A GROUP 3
From now on G will denote an arbitrary Q -group with Q a finite group, ϕ : Q −→ Aut ( G ) the group homomorphism inducing the action of Q on G , and A anabelian group with trivial G - and Q -actions.The Q -action on G induces a short exact sequence of groups1 −→ G −→ G ⋊ ϕ Q −→ Q −→ . We will identify the subgroup G × { e } of G ⋊ ϕ Q with G and { e } × Q with Q . Thereis an associated Hochschild-Serre spectral sequence E pq = H p ( Q, H q ( G, A )) ⇒ H p + q ( G ⋊ ϕ Q, A )(4)where the action of Q on H q ( G, A ) is the one given in Corollary III.8.2 of [1]. Recallthat the action of an element q ∈ Q on H ∗ ( G, A ) is given by the automorphism c ( q ) ∗ : H ∗ ( G, A ) −→ H ∗ ( G, A ) induced by the automorphism in the category ofpairs ([1] III.8) c ( q ) = ( α q , id A ) : ( G, A ) −→ ( G, A )( g, m ) ( qgq − , m )where α q : G −→ G is given by g qgq − and id A is the identity in A . Note inparticular that ϕ ( q ) = α q . Given a projective resolution F ǫ −→ Z on Z G in orderto compute this action at the chain level it is necessary to find an augmentationpreserving chain map of G -modules τ q : F −→ F such that τ q ( gx ) = α q ( g ) τ q ( x ) for g ∈ G and x ∈ F . In this way, we will have c ( q ) ∗ = ( τ q ⊗ G id A ) ∗ .Consider the bar resolution B ∗ ( G ) ǫ −→ Z and( τ q ) n : B n ( G ) −→ B n ( G )( g , g , ..., g n ) ( qg , qg , ..., qg n )for all n ≥ qg i represents the element ϕ ( q )( g i ). We can see that τ q is anaugmentation preserving map. Moreover, given g ∈ Gτ q ( g ( g , ..., g n )) = τ q ( gg , ...gg n ) for the action G on B n ( G )= ( q ( gg ) , ..., q ( gg n ))= ( qgqg , ...qgqg n ) since q is automorphism of G = qg ( qg , ..., qg n )= α q ( g ) τ q ( g , ..., g n ) . Hence c ( q ) ∗ = ( τ q ⊗ G id A ) ∗ and the action of Q on H ∗ ( G, A ) at the chain level isthe diagonal action on B ∗ ( G ) ⊗ G A . Since the action of G on A is trivial B ∗ ( G ) ⊗ G A ∼ = C ∗ ( G ) ⊗ A and the action turns out to be the diagonal action on C ∗ ( G ) ⊗ A which is the sameaction used in the definition of the homology of invariant group chains (1).The next corollary immediately follows from the previous section. Corollary 1. If | Q | is invertible in A , then the norm map H ∗ ( G, A ) Q −→ H ∗ ( G, A ) Q is an isomorphism. Proof.
By Proposition 1a | Q | is invertible in H ∗ ( G, A ) and the claim followsfrom Proposition 1b. (cid:3)
CARLOS AQUINO, ROLANDO JIMENEZ, MARTIN MIJANGOS AND QUITZEH MORALES
Theorem 3. If | Q | is invertible in A , then the E q term of the spectral sequence(4) is isomorphic to H Qq ( G, A ) . Proof.
We have that E q = H ( Q, H q ( G, A )) = H q ( G, A ) Q . By Corollary 1, H q ( G, A ) Q ∼ = H q ( G, A ) Q , and by Theorem 1, H q ( G, A ) Q ∼ = H Qq ( G, A ) . (cid:3) Proof of Theorem 2 . Invertibility of | Q | in A implies that | Q | is invert-ible in H ∗ ( G, A ) by Proposition 1a. By Corollary III.10.2 in [1] it follows that H p ( Q, H q ( G, A )) = 0 for all p >
0, thus E pq = H p ( Q, H q ( G, A )) = 0 if p ≥
1. Thismeans that the spectral sequence collapses at page two.Since this sequence converges to H ∗ ( G ⋊ ϕ Q, A ) and according to the previoustheorem E q ∼ = H Qq ( G, A ), then H Qq ( G, A ) ∼ = H q ( G ⋊ ϕ Q, A ) . (cid:3) Example.
We will compute the groups H Q ∗ ( G, A ) using the preceding Theoremfor Q = Z / h t i , G = Z /n and A = Z /m with m odd where n ≥ t ( g ) = g − , g ∈ G , and the action of G and Q on A is trivial. Under these conditions G ⋊ ϕ Q = D n and we just need to compute the homology groups of the dihedral groups. Wewill split the computations into two cases depending on the parity of n .Case 1. n odd.For this case we have that the cohomology groups H ∗ ( D n , Z ) are given by ([4]Theorem 5.3): H q ( D n , Z ) = Z q = 00 q ≡ mod Z / q ≡ mod q ≡ mod Z /n ⊕ Z / q ≡ mod , q > . Using the Universal Coefficient Theorem for cohomology we find that the homologygroups are(5) H q ( D n , Z ) = Z q = 0 Z / q ≡ mod q ≡ mod Z /n ⊕ Z / q ≡ mod q ≡ mod , q > BH q ( D n , B ) = B q = 0 B/ B q ≡ mod T ( B ) q ≡ mod B/ nB q ≡ mod T n ( B ) q ≡ mod , q > N INVARIANT (CO)HOMOLOGY OF A GROUP 5 where T n ( B ) denotes the set of all n -torsion elements. Using this for B = Z /m = A where m is odd we obtain H q ( D n , A ) = Z /m q = 00 q ≡ mod q ≡ mod Z /m ) / n ( Z /m ) = ( Z /m ) /n ( Z /m ) q ≡ mod T n ( Z /m ) q ≡ mod , q > . But ( Z /m ) /n ( Z /m ) ∼ = Z / ( m, n ) since ( Z /m ) /n ( Z /m ) −→ Z / ( m, n ) such that [ k ] ¯ k and Z / ( m, n ) −→ ( Z /m ) /n ( Z /m ) such that ¯ k [ k ] are well defined grouphomomorphisms and they are inverses of each other. On the other hand, it can beseen that { k m ( m,n ) | k = 0 , , ..., ( m, n ) − } ⊆ T n ( A ) . Moreover, if ¯ r ∈ T n ( A ) then there exists l ∈ Z such that nr = lm , which impliesthat r = lmn = lms ( m,n ) for some s ∈ Z . Since ( s, m ) = 1 and r ∈ Z , s | l and thus¯ r ∈ { k m ( m,n ) | k = 0 , , ..., ( m, n ) − } . Then { k m ( m,n ) | k = 0 , , ..., ( m, n ) − } ⊇ T n ( A ) . This implies that T n ( A ) = { k m ( m,n ) | k = 0 , , ..., ( m, n ) − } ∼ = Z / ( m, n ) . Summarizing we have H q ( D n , A ) = Z /m q = 0 Z / ( m, n ) q ≡ , mod , q >
00 otherwise . Case 2. n evenFor this case we have that cohomology groups H ∗ ( D n , Z ) are given by ([4]Theorem 5.2): H q ( D n , Z ) = Z q = 0( Z / q − q ≡ mod Z / q +22 q ≡ mod Z / q − q ≡ mod Z /n ⊕ ( Z / q q ≡ mod , q > . Using the Universal Coefficient Theorem for cohomology we find that the homologygroups H q ( D n , Z ) are: H q ( D n , Z ) = Z q = 0( Z / q +32 q ≡ mod Z / q q ≡ mod Z /n ⊕ ( Z / q +12 q ≡ mod Z / q q ≡ mod , q > . CARLOS AQUINO, ROLANDO JIMENEZ, MARTIN MIJANGOS AND QUITZEH MORALES
Using the Universal Coefficient Theorem for homology we have for any abeliangroup B : H q ( D n , B ) = B q = 0( B/ B ) q +32 ⊕ T ( B ) q − q ≡ mod B/ B ) q ⊕ T ( B ) q +22 q ≡ mod B/nB ⊕ ( B/ B ) q +12 ⊕ T ( B ) q − q ≡ mod B/ B ) q ⊕ T n ( B ) ⊕ T ( B ) q q ≡ mod , q > . In our case B = Z /m = A with odd m and we have H q ( D n , A ) = Z /m q = 00 q ≡ mod q ≡ mod Z /m ) /n ( Z /m ) q ≡ mod T n ( Z /m ) q ≡ mod , q > . Using the calculations made in the previous case we have H q ( D n , A ) = Z /m q = 0 Z / ( m, n ) q ≡ , mod , q >
00 otherwise . Summarizing both cases, for Q = Z / G = Z /n , A = Z /m where m is odd, t ( g ) = g − for every g ∈ G and A has trival G - and Q -actions, we have that thehomology groups of invariant group chains are: H Qq ( G, A ) = Z /m q = 0 Z / ( m, n ) q ≡ , mod , q >
00 otherwise . We can compare this result with Theorem 6 in [5].
Remark 1.
In general, homology of invariant group chains does not agree with thegroup homology of the semidirect product. For instance, consider Q = Z / h t i acting on G = Z /n with odd n by t ( g ) = g − , g ∈ G , and A = Z with trivial actionsof G and Q . On one hand we have ( [6] Example 5.1) H Z / p ( Z /n, Z ) = Z p = 0 Z /n p ≡ mod otherwise . On the other hand, the computations made in equation 5 show us that H Z / ∗ ( Z /n, Z ) = H ∗ ( Z /n ⋊ Z / , Z ) . The first homology group of invariant group chains
In this section we discuss the relations between Knudson first homology groupand two types of abelianization, the first one is and adhoc construction made tocoincide with this group, and the second is a natural construction given in termsof the semidirect product of groups and suitable commutants.
N INVARIANT (CO)HOMOLOGY OF A GROUP 7
Orbit group.
Let Q and G be groups, and Q × G → G an action of Q on G by group automorphisms, i.e. q ( g g ) = q ( g ) q ( g ) , q (1) = 1for every q ∈ Q, g , g ∈ G .Now, we define a new orbit set. For an element, g ∈ G consider its Q - orderedorbit : ( q ( g )) q ∈ Q . (6)These orbits form a set O ( G, Q ) ⊂ Q q ∈ Q ( G ) q , O ( G, Q ) = { ( q ( g )) q ∈ Q | g ∈ G } . (7)The group Q acts on O ( G, Q ) by the rule Q × O ( G, Q ) → O ( G, Q ); q · ( q ( g )) q ∈ Q = ( q ( q − ( g ))) q ∈ Q . The resulting orbit set O ′ ( G, Q ) = O ( G, Q ) /Q is in one-to-one correspondence withthe traditional orbit set G/Q , such correspondence is induced by the map O ( G, Q ) −→ G/Q ;( q ( g )) q ∈ Q
7→ { q ( g ) : q ∈ Q } . The coordinates of every element in O ′ ( G, Q ) depends only on the elements q ( g ) ∈ G for a set of representatives of the classes [ q ] ∈ Q/Q g , where Q g denotes the isotropygroup of the element g ∈ G . So, one can denote them as( q ( g )) [ q ] ∈ Q/Q g ∈ O ′ ( G, Q )and say that the classes are ordered up to the action of the group Q .Consider the product rule O ( G, Q ) × O ( G, Q ) −→ O ( G, Q )defined by multiplication coordinate by coordinate, i.e.( q ( g )) q ∈ Q ( q ( g )) q ∈ Q = ( q ( g ) q ( g )) q ∈ Q . (8)This product is well defined, associative, it has the neutral element( q (1)) q ∈ Q = (1) q ∈ Q (9)and the inverse of the element ( q ( g )) q ∈ Q ∈ O ( G, Q ) is the element( q ( g − )) q ∈ Q = ( q ( g ) − ) q ∈ Q . (10)Indeed, ( q ( g )) q ∈ Q ( q ( g ) − ) q ∈ Q =( q ( g ) q ( g ) − ) q ∈ Q = (1) q ∈ Q . In fact, the group O ( G, Q ) is isomorphic to the group G . This new expressionof the same object is a way to take into account the action of the group Q and tohave a well-defined product of Q -orbits as a whole.If all isotropy groups of elements g ∈ G are finite, then the coordinates of anelement ( q ( g )) q ∈ Q ∈ O ( G, Q ) repeat as many times as the order of any isotropygroup Q q ( g ) . In the case the whole group Q is finite, one may define the map(11) O ( G, Q ) −→ C ( G ); ( q ( g )) q ∈ Q X q ∈ Q q ( g ) = | Q g | X [ q ] ∈ Q/Q g q ( g ) CARLOS AQUINO, ROLANDO JIMENEZ, MARTIN MIJANGOS AND QUITZEH MORALES to the free abelian group generated by G as a set. The image of this map lies inthe fixed point subgroup C ( G ) Q but does not generate the whole group, which isgenerated by the elements of the form P [ q ] ∈ Q/Q g q ( g ).One may also define the map O ′ ( G, Q ) −→ C ( G ); ( q ( g )) [ q ] ∈ Q/Q g X [ q ] ∈ Q/Q g q ( g ) . Then one obtains a map O ( G, Q ) −→ C ( G ) Q as the composition O ( G, Q ) −→ O ′ ( G, Q ) −→ C ( G ) Q whose image generates C ( G ) Q . However, the set O ′ ( G, Q ) does not have a groupstructure that could make this into a group homomorphism.One may ask instead for a minimal relation subgroup R ( G, Q ) in C ( G ) Q makingthe map (11) into a group homomorphism, i.e. such that the composition O ( G, Q ) −→ C ( G ) Q −→ C ( G ) Q /R ( G, Q )is a homomorphism.For this, note that the map (11) sends the product g g to | Q g g | X [ q ] ∈ Q/Q g g q [ g g ] . So, the element(12) | Q g g | X [ q ] ∈ Q/Q g g q [ g g ] − | Q g | X [ q ] ∈ Q/Q g q [ g ] − | Q g | X [ q ] ∈ Q/Q g q [ g ]must belong to R ( G, Q ). However, there are different pairs of elements g and g (having different orbits) that may give the same product g = g g . Therefore, therelation (12) might not be minimal. Note that the integers | Q g g | , | Q g | , | Q g | have a common divisor, namely | Q g ∩ Q g | . Therefore, the relation (12) is generatedby(13) | Q g g || Q g ∩ Q g | P [ q ] ∈ Q/Q g g q [ g g ] −− | Q g || Q g ∩ Q g | P [ q ] ∈ Q/Q g q [ g ] − | Q g || Q g ∩ Q g | P [ q ] ∈ Q/Q g q [ g ] . This justifies the following definition.
Definition 2.
The weighed orbit abelianization O ( G, Q ) wab of the action of a (fi-nite) group Q on a group G by group automorphisms is the biggest abelian groupgenerated by the Q -orbits P q ∈ Q q ( g ) , g ∈ G , such that the map O ( G, Q ) −→ O ( G, Q ) wab ; ( q ( g )) q ∈ Q X q ∈ Q q ( g )(14) is a group homomorphism. The resulting group, which is by definition a quotient of C ( G ) Q is called weighedbecause of the relations given in terms of the orders of isotropy groups Q g .By construction, one has the following. N INVARIANT (CO)HOMOLOGY OF A GROUP 9
Theorem 4.
Let Q be a finite group, G be a group, and Q × G → G be an actionof Q on G by group automorphisms. Denote by H Q ( G, Z ) the first homology groupof Q -invariant chains on the group G (see [6] ). Then, there is an isomorphism H Q ( G, Z ) ∼ = O ( G, Q ) wab . (15) Proof.
It was shown in [5, Theorem 2] the following. H Q ( G, Z ) = Z nP ¯ q ∈ Q/Q [ g ] q [ g ] g ∈ G o Z n a P ¯ q ∈ Q/Q [ g q [ g ] − b P ¯ q ∈ Q/Q [ g g q [ g g ] + c P ¯ q ∈ Q/Q [ g q [ g ] g , g ∈ G o , where a = | Q [ g || Q [ g ∩ Q [ g | , b = | Q [ g g || Q [ g ∩ Q [ g | and c = | Q [ g || Q [ g ∩ Q [ g | . (cid:3) Relations between G and H Q ( G, Z ) . In this section we compare the group H Q ( G, Z ) with a construction of an abelian group made of orbits of the action of Q on G . This construction is a natural extension of the usual notion of abelianiza-tion of a group, which is made of orbits of the group G acting on itself by groupconjugation.The map G → C ( G ) Q sending each element g ∈ G to the corresponding gen-erator P [ q ] ∈ Q/Q g q [ g ] does not induce a homomorphism between the groups G and H Q ( G, Z ). Instead, the (not necessarily surjective) norm map N : G → C ( G )given by sending each element g ∈ G to the sum of the elements on its orbit P [ q ] ∈ Q q [ g ] = | Q g | P [ q ] ∈ Q/Q g q [ g ] does induce a homomorphism, because theproduct g g is sent to | Q g g | X [ q ] ∈ Q/Q g g q [ g g ] == | Q g | X [ q ] ∈ Q/Q g q [ g ]+ | Q g | X [ q ] ∈ Q/Q g q [ g ] . However, this map sends each of the elements q ( g ) , q ∈ Q of the orbit of g to thesame element in H Q ( G, Z ). So, one might seek for a group defined in terms of G , Q and the action Q × G → G identifying all the elements in the same orbit andbeing abelian.A version of this in the case of inner automorphisms is the well known abelian-ization G ab of the group G . This is because this group is generated by conjugacyclasses of its elements.In the semidirect product G ⋊ Q induced by the action φ : Q × G → G of Q on G , consider the commutator subgroup [ G, Q ]. The generators of this subgrouphave the form [ g, q ] = g ( qg − q − ) = gφ ( q, g − ) = gq ( g − ). Denote by [ G, Q ] G ⋊ Q the normal closure of [ G, Q ] in G ⋊ Q . Then, in the quotient G ⋊ Q/ [ G, Q ] G ⋊ Q one has that g = q ( g ) for every q ∈ Q, g ∈ G . Definition 3.
The orbit group of the action of the group Q on G , denoted by G//Q ,is the image of G in ( G ⋊ Q ) / [ G, Q ] G ⋊ Q under the composition (16) G ֒ → G ⋊ Q −→ ( G ⋊ Q ) / [ G, Q ] G ⋊ Q . Denote by p : G → G//Q the map (16) onto its image.This group has the following universal property: if ϕ : G → H is a grouphomomorphism such that ϕ ( q ( g )) = ϕ ( g ) for any q ∈ Q and g ∈ G , then thereis a unique homomorphism ψ : G//Q → H such that the following diagram iscommutative: G ϕ " " ❋❋❋❋❋❋❋❋❋ p (cid:15) (cid:15) G//Q ψ / / H, i.e. ψ ◦ p = ϕ .This means that there is an induced map G//Q → H Q ( G, Z ) commuting withthe map N : G → H Q ( G, Z ). As H Q ( G, Z ) is an abelian group, this factors througha homomorphism ( G//Q ) ab → H Q ( G, Z ) sending the element ¯ g ∈ ( G//Q ) ab to theelement | Q g | [ g ] Q ∈ H Q ( G, Z ).By the previous discussion, we have the following. Theorem 5.
There is a homomorphism (17) (
G//Q ) ab −→ H Q ( G, Z ) from the orbit group abelianization to the first homology group of invariant groupchains commuting with the norm map, i.e. such that the diagram G ¯ N (cid:30) (cid:30) ❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁ p (cid:15) (cid:15) G//Q (cid:15) (cid:15) ( G//Q ) ab / / H Q ( G, Z ) , is commutative. (cid:3) As it was discussed also, this homomorphism is not in general surjective, becauseit is induced by the (non surjective) norm map. It might not be injective also: usingthe relation (13), it is easy to show that the order n of an element g ∈ G annihilatesthe corresponding element [ g ] Q ∈ H Q ( G, Z ). Therefore, if n divides | Q g | , thenthe image of such element would be zero.3. Invariant cohomology, invariant group extensions with abeliankernel and free actions
In this section we propose a new definition of invariant cohomology that gen-eralizes the usual cohomology of a group, this cohomology is an invariant of the Q -group G that provides algebraic information. We also define their correspondinghomology that turns out to be a generalization of the homology defined in [6]. For N INVARIANT (CO)HOMOLOGY OF A GROUP 11 this, we introduce the category Q - G M od that turns out to be equivalent to thecategory of modules over Z ( G ⋊ Q ). Definition 4.
Let G be a Q -group. Let M be, simultaneously, a G -module and a Q -module. We say that M is a Q - G module if q ( gm ) = q ( g ) qm (18) for g ∈ G, q ∈ Q, m ∈ M . We denote by Q - G M od the category whose objects are Q - G modules and morphisms are the functions f : M → N such that f is both G -linear and Q -linear. Proposition 2.
The category Q - G M od is equivalent to the category of modulesover Z ( G ⋊ Q ) . Proof.
Let T be the functor T : Q - G M od → G ⋊ Q M od , M M ′ , f f ′ where M is a Q - G module, f is a morphism of Q - G modules and M ′ and f ′ aredefined as follows: as a set M ′ is defined as the underlying set of M and the ac-tion of an element ( g, q ) ∈ G ⋊ Q over m ∈ M is given by ( g, q ) m = g ( qm ) and f ′ ( m ) = f ( m ). It is easy to see that f ′ is a morphism in the category G ⋊ Q M od .Let L be the functor L : G ⋊ Q M od → Q - G M od , M ¯ M , f ¯ f where M is a G ⋊ Q module and f is a morphism of G ⋊ Q modules. As a set ¯ M is defined asthe underlying set of M , an element q ∈ Q acts on m ∈ M by the rule qm = (1 , q ) m and g ∈ G acts by gm = ( g, m . The morphism ¯ f is defined by ¯ f ( m ) = f ( m ) forall m ∈ M . It is easy to see that ¯ f is a morphism in the category Q - G M od .Thus, T defines an isomorphism of categories with inverse L . (cid:3) In this way, one says that a Q - G -module M is a free Q - G -module, if and onlyif T ( M ) is a free G ⋊ Q -module. The following proposition is a characterization ofsuch Q - G modules: Proposition 3. A Q - G module M is free if and only if M admits a Z G -basis where Q acts freely. Proof. If M admits a Z G -basis where Q acts freely, then one can write M asthe sum M ∼ = L i ∈ I ( Z G ) i with Q acting freely on the set of indices I . For each i ∈ I , we have a Z ( G ⋊ Q )-isomorphism: M q ∈ Q ( Z G ) qi ∼ = Z ( G ⋊ Q )mapping the element g ∈ ( Z G ) qi to the element ( g, q ) ∈ Z ( G ⋊ Q ). If E is a set ofrepresentatives of the quotient I/Q then one has M ∼ = M j ∈ E M q ∈ Q ( Z G ) qj ∼ = M j ∈ E Z ( G ⋊ Q ) j . Conversely, if M is a free Q - G module, then M is a free Z ( G ⋊ Q )-module, so M ∼ = M i ∈ I Z ( G ⋊ Q ) i ∼ = M ( q,i ) ∈ Q × I ( Z G ) ( q,i ) and the action of Q on Q × I given by q ′ ( q, i ) = ( q ′ q, i ) is free. (cid:3) Let G be a Q -group and M a Q - G module. There are natural actions: Q × Hom G ( B n ( G ) , M ) → Hom G ( B n ( G ) , M ) q · f ([ q | · · · | g n ]) = qf ([ q − g | · · · | q − g n ]) Q × B n ( G ) ⊗ G M → B n ( G ) ⊗ G Mq ([ q | · · · | g n ] ⊗ m ) = [ qg | · · · | qg n ] ⊗ qm and the differential induced in Hom G ( B ( G ) , M ) and B ( G ) ⊗ G M by the bar res-olution are Q -equivariant. We define the homology and cohomology of invariantsas: HH Qn ( G, M ) = H n (( B ( G ) ⊗ G M ) Q )(19) HH nQ ( G, M ) = H n ( Hom G ( B ( G ) , M ) Q )(20) Remark 2. If Q acts trivially on G and on M , then HH nQ ( G, M ) = H n ( G, M ) and HH Qn ( G.M ) = H n ( G, M ) . In this way, these invariants are an immediategeneralization of the usual homology and cohomology of the group G . The following propositions show that under certain conditions, this cohomologycoincides with other invariants.
Proposition 4. If Q is a finite group and | Q | is invertible in M , then HH nQ ( G, M ) ∼ = H n ( G, M ) Q . Proof.
The action of Q on Hom G ( B ( G ) , M ) induces a well-defined action of Q on H n ( G, M ) Q × H n ( G, M ) → H n ( G, M ) , ( q, f ) q · f The natural inclusion i : Hom G ( B n ( G ) , M ) Q → Hom G ( B n ( G ) , M ) induces a ho-momorphism: HH nQ ( G, M ) → H n ( G, M ) Q with inverse given by: H n ( G, M ) Q → HH nQ ( G, M ) , f | Q | X q ∈ Q q · f (cid:3) Proposition 5. If Q is a finite group, M is a trivial Q - G module and | Q | isinvertible in M , then HH nQ ( G, M ) ∼ = H nQ ( G, M ) . Proof.
It is easy to see that if M is Q - G trivial, then HH nQ ( G, M ) ∼ = H n ( Hom ( C ( G ) , M ) Q )We define α : Hom ( C n ( G ) Q , M ) → Hom ( C n ( G ) , M ) Q by α ( f )([ g | · · · | g n ] ⊗ n ) = 1 | Q | f ( X q ∈ Q q [ g | · · · | g n ] ⊗ n ) . This homomorphism is an isomorphism with inverse β : Hom ( C n ( G ) , M ) Q → Hom ( C n ( G ) Q , M ) , β ( f ) = f | C n ( G ) Q N INVARIANT (CO)HOMOLOGY OF A GROUP 13
Also, one has α ( δ ( f ))([ g | · · · | g n +1 ] ⊗ n ) = 1 | Q | δf ( X q ∈ Q q [ g | · · · | g n +1 ] ⊗ n )= 1 | Q | f ( X q ∈ Q ([ qg | · · · | qg n ] ⊗ n + X ( − i [ qg | · · · | qg i qg i +1 | · · · | qg n +1 ] ⊗ n +( − n +1 [ qg | · · · | qg n ] ⊗ n ))= α ( f )([ g | · · · | g n +1 ] ⊗ n ) + X i ≤ n ( − i α ( f )([ g | · · · | g i g i +1 | · · · | g n +1 ] ⊗ n )+( − n +1 α ( f )([ g | · · · | g n ] ⊗ n ) = d ( α ( f ))([ g | · · · | g n +1 ] ⊗ n )In this way, the diagram: Hom ( C n ( G ) Q , M ) δ / / α (cid:15) (cid:15) Hom ( C n +1 ( G ) Q , M ) α (cid:15) (cid:15) Hom ( C n ( G ) , M ) Q d / / Hom ( C n +1 ( G ) , M ) Q is commutative and we obtain an isomorphism: HH nQ ( G, M ) ∼ = H n ( Hom ( C ( G ) , M ) Q ) ∼ = H nQ ( G, M ) (cid:3) Low dimensional cohomology and group extensions with abelian ker-nel.
Here we generalize classical results on low dimensional cohomology of groupsfor the theory we have defined.In dimension 0, by definition we have HH Q ( G, M ) = ker ( M Q → Hom G ( B ( G ) , M ) Q ) = M Q ∩ M G but M G admits a Q -module structure, so we can write HH Q ( G, M ) = ( M G ) Q Next we describe HH Q ( G, M ) in terms of invariant derivations.
Definition 5. A Q -derivation of G in M is a Q -equivariant map d : G → M suchthat d ( g g ) = d ( g ) + g d ( g ) , we denote the set of Q -derivations by Der Q ( G, M ) .For each element m ∈ M Q we can define an inner Q -derivation d m : G → M , d m ( g ) = ( g − m , we denote the set of inner Q -derivations by IDer Q ( G, M ) = { d m | m ∈ M Q } . In the sequence: M Q ∂ / / Hom G ( B ( G ) , M ) Q ∂ / / Hom G ( B ( G ) , M ) Q ∂ and ∂ are given by: ∂ ( m )([ g ]) = ( g − m = d m ( g )( ∂ f )([ g | g ]) = g f ([ g ]) − f ([ g g ]) + f ([ g ])so, Im ( ∂ ) = IDer Q ( G, M ) and ker ( ∂ ) = Der Q ( G, M ) in this way, HH Q ( G, M ) =
Der Q ( G, M ) /IDer Q ( G, M ) Now we discuss group extensions in the present context. In the classical work[3] it is shown that equivalent classes of group extensions(21) 0 → K → G → H → , where K is a commutative group and given a (right) action H × K −→ K by groupautomorphisms, are classified by the second cohomology group H ( H, K ).In the sequel, for simplicity, in (21) the group K is identified with its image in G and the group H is identified with the quotient G/K
For a general group extension (21), as usual, we take a normalized section s : H −→ G , i.e. j ◦ s = Id H , s ( e ) = e , where e denotes the neutral element. Now,consider the equation(22) x s · y s = f ( x, y ) · ( x · y ) s , where f ( x, y ) ∈ K , and x s denotes the image of the element x ∈ H under thesection s . This gives a so called set of factors f : H × H −→ K . In order to clarifythe change in equation (22) due to a change of representatives in the (right) cosetsin G , we have the following:For elements a, b ∈ K , the product ( a · x s ) · ( b · y s ) can be rewritten in the form(23) ( a · x s ) · ( b · y s ) = a · T x ( b ) · f ( x, y ) · ( x · y ) s , where T x : K −→ K is the (left) inner automorphism given by conjugation by theelement x s ∈ G , i.e.(24) T x ( a ) = x s ax − s , which does not depend on the representative x s ∈ G of the class x ∈ H becausethe conjugation action of the abelian group K on itself is trivial.Let G be a Q -group. Assume that this action leaves the group K invariant, i.e. QK ⊂ K . Then K is also a Q -group. Moreover, one has q ( hx ) = q ( h ) q ( x ), whichmeans that the quotient H is a Q -group with the action Q × H → H defined by q ( Kg ) = Kq ( g ).We assume that the action of the group Q commutes with the section: q ( x s ) =( q ( x )) s .One should check how the action of Q on K interacts with its H -module struc-ture: q ( T x ( a )) = q ( x s ) q ( a ) q ( x s ) − = ( q ( x )) s q ( a )( q ( x )) − s = T q ( x ) ( q ( a )) . If one writes T x ( a ) = xa , then one has q ( xa ) = q ( x ) q ( a ). For such extensions, thefactor set f : H × H −→ K is Q -equivariant with respect to the diagonal action Q × ( H × H ) → H × H . Indeed, by applying the automorphism q ∈ Q on bothsides of the equation (22) one obtains(25) q ( x s ) · q ( y s ) = qf ( x, y ) · q (( x · y ) s ) . Then, applying the same equation to ( q ( x )) s and to ( q ( y )) s one has(26) q ( x ) s · q ( y ) s = f ( q ( x ) , q ( y )) · ( q ( x ) · q ( y )) s . But, by assumption, q ( x s ) = ( q ( x )) s , q ( y s ) = ( q ( y )) s . So, the right side of equations(25) and (26) coincide and one can cancel the factor q (( x · y ) s ) = ( q ( x · y )) s = ( q ( x ) · q ( y )) s on the left side of these equations. N INVARIANT (CO)HOMOLOGY OF A GROUP 15
In this way, we have the following:
Theorem 6.
Let G , H be Q -groups and let K be a Q - H module. Then the set ofequivalence classes of Q -equivariant extensions (27) 0 → K → G → H → , that admit a normalized Q -equivariant section s : H → G is in one-to-one corre-spondence with the elements of the group HH Q ( H, K ) . Proof.
The previous arguments show that such extensions with Q -linear sec-tions are defined by Q -linear set of factors. The usual arguments for classifyinggroup extensions follow. (cid:3) It is clear that a Q -linear set of factors restricts to a homomorphism on fixedpoints, because the subgroup C • ( H ) Q ⊂ C • ( H ) is Q -invariant. However, it is nottrue in general that every homomorphism f ∈ Hom( C • ( H ) Q , K ) is the restrictionof some ˜ f ∈ Hom Q ( C • ( H ) , K ). This is because, for elements x ∈ C • ( H ) Q onehas q ˜ f ( x ) = ˜ f ( qx ) = ˜ f ( x ), and this means that the image of C • ( H ) Q under ˜ f iscontained in the subgroup K Q of invariants of the Q -module K , i.e. one has a map˜ f | C • ( H ) Q : C • ( H ) Q −→ K Q . The set of all such restrictions may not coincide with Hom( C • ( H ) Q , K ) if K is nota trivial Q -module. Corollary 2. If M is a Q - G module with trivial actions and | Q | is invertible in M , then the group H Q ( G, M ) classifies Q -equivariant extensions → M → E → G → inducing a trivial action of G on M . Free actions.
One of the greatest difficulties in the study of invariant co-homology lies in its very definition, since the bar resolution B ( G ) → Z is not aprojective resolution in the category Q - G M od . In this section, we analyze thecase when the action of Q on G is “free” in order to replace the bar resolution witha projective resolution of the augmentation ideal. At the end of the paragraph wewill analize some examples. Definition 6.
An action of a group Q on a group G is free if Q g = { } for each g ∈ G, g = 1 . In this case, we say that G is a free Q -group. Remark 3. B ( G ) = Z G is a free Z G -module where any Z G -basis consists of asingle element. In this way, if | Q | > , Q can not act freely on this basis. Therefore, B ( G ) is not a free Q - G module. Proposition 6. If G is a free Q -group, then B n ( G ) is a free Q - G module for each n > . Proof.
The set { [ g | · · · | g n ] | g i ∈ G, g i = 1 } is a Z G -base and Q acts freelyon it. Thus B n ( G ) is a free Q - G module for each n > (cid:3) If G is a free Q -group, we can consider the restriction of the bar resolution tothe augmentation ideal and we obtain a projective resolution: · · · / / B ( G ) d / / B G d / / I G / / of the augmentation ideal I G in the category Q - G M od . In addition, the naturalmorphism: η : Hom G ( I G , M ) → Der ( G, M ) , f d f , where d f ( g ) = f ( g − Q -module isomorphism. Then, Ext Q - G ( I G , M ) = Hom G ( I G , M ) Q ∼ = Der Q ( G, M )and, therefore, we obtain the following expression for HH nQ ( G, M ) by applying thederived functor of
Hom Q - G ( − , M ) of the Q - G module I G : HH nQ ( G, M ) = ( M G ) Q n = 0 Ext Q - G ( I G , M ) /IDer Q ( G, M ) n = 1 Ext n − Q - G ( I G , M ) n ≥ A is a Q - G module with trivial actions, then IDer Q ( G, A ) = 0 and we obtain: HH nQ ( G, A ) =
A n = 0
Ext n − Q - G ( I G , M ) n ≥ Remark 4.
If the action of Q on G is free and M is a Q - G module, then a Q -equivariant extension: → M → E → G → always admits a normalized Q -equivariant section s : G → E . In this case, intheorem 6, we can consider all extensions and not only those that have a normalized Q -equivariant section. Example.
Let Q be a group and let S be a set such that Q acts freely on S . Thenthe free group generated by S , F ( S ) is a Q -group with action induced by the actionof Q on S . Then I F ( S ) is a free Z F ( S )-module with basis S − { s − | s ∈ S } in addition, Q acts freely on this basis. So I F ( S ) is a projective Q - F ( S ) module.Then, 0 / / I F ( S ) 1 / / I F ( S ) / / I F ( S ) in Q - F ( S ) M od therefore, we have: HH nQ ( F ( S ) , Z ) = Z n = 0 Der Q ( F ( S ) , Z ) n = 10 n ≥ HH Q ( F ( S ) , Z ) = 0, all the extensions that induce the trivial action on Z are Q -equivalent to the extension of the direct product:0 → Z → Z × F ( S ) → F ( S ) → Example.
Consider the groups Q = Z / h s | s i and G = Z = h t i with Z / Z by s : Z → Z , t t −
1N INVARIANT (CO)HOMOLOGY OF A GROUP 17 in this way, Z is a free Z / I Z is the free Z G -modulegenerated by the element t −
1. It is easy to see that I Z is not a free Z / Z module.Consider the following projective resolution for that ideal: · · · / / Z G ( Z / d / / Z G ( Z / d / / Z G ( Z / ǫ / / I Z / / Z G ( Z /
2) is the free Z / Z module generated by Z / ǫ : Z G ( Z / → I Z , x + ys ( x − t − y )( t − d i ( x + ys ) = x (1 + ts ) + yt − (1 + ts ) , i = 2 k − x (1 − ts ) − yt − (1 − ts ) , i = 2 k with x, y ∈ Z . When applying the functor Hom G ( − , Z ) Q to the resolution (we areconsidering the coefficients module Z as a Z / Z module with trivial actions) weobtain Hom G ( I Z , Z ) Q = Der Q ( Z , Z ) = 0 and Hom G ( Z G ( Z / , Z ) Q ∼ = Z . In thisway, we have the following cochain complex:0 / / Z / / Z / / Z / / Z / / · · · therefore, Ext n Z / - Z ( I Z , Z ) = n = 2 k Z / n = 2 k − HH n Z / ( Z , Z ) = Z n = 00 n = 2 k − Z / n = 2 k ≥ HH Z / ( Z , Z ) = Z /
2, there are only two Z / → Z → E → Z → Z . Acknowledgements
The authors express their gratitude to the anonymous referee for the carefulrevision and useful comments to improve this work.
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Instituto de Matem´aticas, Unidad Oaxaca, Universidad Nacional Aut´onoma de M´exico,Le´on 2, 68000 Oaxaca de Ju´arez, Oaxaca, M´exico
E-mail address : [email protected] Instituto de Matem´aticas, Unidad Oaxaca, Universidad Nacional Aut´onoma de M´exico,Le´on 2, 68000 Oaxaca de Ju´arez, Oaxaca, M´exico
E-mail address : [email protected] Instituto de Matem´aticas, Unidad Oaxaca, Universidad Nacional Aut´onoma de M´exico,Le´on 2, 68000 Oaxaca de Ju´arez, Oaxaca, M´exico
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