Globalization of partial cohomology of groups
aa r X i v : . [ m a t h . R A ] J un GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS
MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIM ´ON
Abstract.
We study the relations between partial and global group cohomol-ogy. We show, in particular, that given a unital partial action of a group G ona ring A , such that A is a direct product of indecomposable rings, then anypartial n -cocycle with values in A is globalizable. Introduction
Given a partial action it is natural to ask whether there exists a global actionwhich restricts to the partial one. This question was first considered in the PhDThesis [1] (see also [2]) and independently in [45] and [39] for partial group actions,with subsequent developments in [3, 12, 17, 18, 21, 22, 24, 32, 33, 40, 44]. Moregenerally the problem was investigated for partial semigroup actions in [37, 38, 41,43], for partial groupoid actions in [10, 11, 36] and around partial Hopf (co)actionsin [5, 6, 7, 8, 14, 15, 16].Globalization results help one to use known facts on global actions in the studiesinvolving partial ones. Thus the first purely ring theoretic globalization fact [22,Theorem 4.5] stimulated intensive algebraic activity, permitting, in particular, todevelop a Galois Theory of commutative rings [25]. The latter, in its turn, inspiredthe definition and study of the concept of a partial action of a Hopf algebra in [13],which is based on globalizable partial group actions, and which became a startingpoint for interesting Hopf theoretic developments. Moreover, globalizable partialactions are more manageable, so that the great majority of ring theoretic studieson the subject deal with the globalizable case. Among the recent applications ofglobalization facts we mention their remarkable use to paradoxical decompositionsin [9] and to restriction semigroups in [41]. The reader is referred to the surveys[19, 20, 34] and to the recent book by R. Exel [31] for more information aboutpartial actions and their applications.In [30] R. Exel introduced the general concept of a continuous twisted partialaction of a locally compact group on a C ∗ -algebra and proved that any secondcountable C ∗ -algebraic bundle, which is regular in a certain sense, is isomorphicto the C ∗ -algebraic bundle constructed from a twisted partial group action. Thepurely algebraic version of this result was obtained in [23]. The concept involvesa twisting which satisfies a kind of 2-cocycle equality needed for an associativitypurpose. Thus, it was natural to work out a cohomology theory, encompassing suchtwistings, and this was done in [26]. The partial cohomology from [26] is strongly Mathematics Subject Classification.
Primary 20J06; Secondary 16W22, 18G60.
Key words and phrases.
Partial action, cohomology, globalization.This work was partially supported by CNPq of Brazil (Proc. 305975/2013-7), FAPESPof Brazil (Proc. 2012/01554-7, 2015/09162-9), MINECO (MTM2016-77445-P) and Fundaci´onS´eneca of Spain. related to H. Lausch’s cohomology of inverse semigroups [42] and nicely fits thetheory of partial projective group representations developed in [27], [28] and [29].The main globalization result from [24] says that if A is a (possibly infinite)product of indecomposable rings (blocks), then any unital twisted partial action α of a group G on A possesses an enveloping action, i. e. there exists a twisted globalaction β of G on a ring B such that A can be identified with a two-sided ideal in B , α is the restriction of β to A and B = P g ∈ G β g ( A ). Moreover, if B has an identityelement, then any two globalizations of α are equivalent in a natural sense. If A iscommutative, then α splits into two parts: a unital partial G -module structure on A (i. e. a unital partial action of G on A ) and a twisting which is a partial 2-cocycle w of G with values in the partial module A . In this case B is also commutative,and β splits into a global action of G on B (so we have a global G -module structureon B ) and a usual 2-cocycle of G with values in the group of units of the multiplieralgebra of B . The above mentioned results from [24] mean in this context thatgiven a unital G -module structure on A , for any 2-cocycle of G with values in A there exists a (usual) 2-cocycle u of G related to the global action on B such that w is the restriction of u . In this case we say that u is a globalization of w (seeDefinition 2.2). Moreover, if B has an identity element, then any two globalizationsof w are cohomologous.The purpose of the present article is to extend the results from [24] in the com-mutative case to arbitrary n -cocycles. The technical difficulties coming from [24]are being overcome by improvements and notations. In Section 1 we recall somenotions needed in the sequel. The main result of Section 2 is Theorem 2.4, inwhich we prove that given a unital partial G -module structure on a commutativering A , a partial n -cocycle w with values in A is globalizable if and only if w canbe extended to an n -cochain ˜ w of G with values in the unit group U ( A ) whichsatisfies a “more global” n -cocycle identity (15). This is the n -analogue of [24,Theorem 4.1] in the commutative setting. The technical part of our work is con-centrated in Section 3, in which we assume that A is a product of blocks, and thisassumption is maintained for the rest of the paper. Our goal is to construct a moremanageable partial n -cocycle w ′ which is cohomologous to w (see Theorem 3.12).In Section 4 we prove our main existence result Theorem 4.3. The defining formu-la for w ′ permits us to extend easily w ′ to an n -cochain f w ′ : G n → U ( A ) whichsatisfies our “more global” n -cocycle identity (see Lemma 4.2). Modifying f w ′ bya “co-boundary looking” function we define in (83) a function ˜ w : G n → U ( A )and show that ˜ w is a desired extension of w fitting Theorem 2.4, and permittingus to conclude that w is globalizable. The uniqueness of a globalization is treatedin Section 5. It turns out that it is possible to omit the assumption that the ring B under the global action has an identity element, imposed in [24] (with n = 2).More precisely, we prove in Theorem 5.3 that given a globalizable partial action α of G on a ring A , which is a product of blocks, and a partial n -cocycle w relatedto α , any two globalizations of w are cohomologous. More generally, arbitrary glo-balizations of cohomologous partial n -cocycles are also cohomologous. This resultsin Corollary 5.4 which establishes an isomorphism between the partial cohomologygroup H n ( G, A ) and the global one H n ( G, U ( M ( B ))) , where U ( M ( B )) stands forthe unit group of the multiplier ring M ( B ) of B . LOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 3 Background on globalization and cohomology of partial actions
In all what follows G will stand for an arbitrary group whose identity elementwill be denoted by 1 , and by a ring we shall mean an associative ring, which is notunital in general. Nevertheless, our main attention will be paid to partial actionson commutative and unital rings.In this section we recall a couple of concepts around partial actions. Definition 1.1. [22] Let A be a ring. A partial action α of G on A is a collectionof two-sided ideals D g ⊆ A ( g ∈ G ) and ring isomorphisms α g : D g − → D g suchthat (i) D = A and α is the identity automorphism of A ;(ii) α g ( D g − ∩ D h ) = D g ∩ D gh ;(iii) α g ◦ α h ( a ) = α gh ( a ) for each a ∈ D h − ∩ D h − g − .An equivalent form to state (i)–(iii) is as follows:(i) α = id A ;(iv) ∃ α h ( a ) , ∃ α g ◦ α h ( a ) ⇒ ∃ α gh ( a ) and α g ◦ α h ( a ) = α gh ( a ), where a ∈ A .Partial actions can be obtained as restrictions of global ones as follows. Let β be a global action of G on a ring B and A a two-sided ideal in B . Then setting D g = A∩ β g ( A ) and denoting by α g the restriction of β g to D g − , we readily see that α = { α g : D g − → D g | g ∈ G } is a partial action of G on A , called the restriction of β to A , and α is said to be an admissible restriction of β if B = P g ∈ G β g ( A ).Clearly, if B 6 = P g ∈ G β g ( A ), then replacing B by P g ∈ G β g ( A ), the partial action α can be viewed as an admissible restriction. Partial actions isomorphic to restrictionsof global ones are called globalizable . The notion of an isomorphism of partial actionis defined as follows. Definition 1.2 (see p. 17 from [2] and Definition 4 from [28]) . Let A and A ′ berings and α = { α g : D g − → D g | g ∈ G } , α ′ = { α ′ g : D ′ g − → D ′ g | g ∈ G } bepartial actions of G on A and A ′ , respectively. A morphism ( A , α ) → ( A ′ , α ′ ) ofpartial actions is a ring homomorphism ϕ : A → A ′ such that for any g ∈ G and a ∈ D g − the next two conditions are satisfied:(i) ϕ ( D g ) ⊆ D ′ g ;(ii) ϕ ( α g ( a )) = α ′ g ( ϕ ( a )).We say that a morphism ϕ : ( A , α ) → ( A ′ , α ′ ) of partial actions is an isomorphism if ϕ : A → A ′ is an isomorphism of rings and ϕ ( D g ) = D ′ g for each g ∈ G .By [22, Theorem 4.5] a partial action α on a unital ring A is globalizable exactlywhen each ideal D g is a unital ring, i. e. D g is generated by an idempotent which iscentral in A , and which will be denoted by 1 g . In order to guarantee the uniquenessof a globalization one considers the next. Definition 1.3 (Definition 4.2 from [22]) . A global action β of G on a ring B issaid to be an enveloping action for the partial action α of G on a ring A if α isisomorphic to an admissible restriction of β .By the above mentioned Theorem 4.5 from [22], an enveloping action β for aglobalizable partial action of G on a unital ring A is unique up to an isomorphism. This was called equivalence in [22, Definition 4.1].
MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIM ´ON
Denote by F = F ( G, A ) the ring of functions from G to A , i. e. F is the Cartesianproduct of copies of A indexed by the elements of G . Note that by the proof ofTheorem 4.5 from [22], the ring under the global action is a subring B of F , andconsequently B is commutative if and only of so too is A .Every ring is a semigroup with respect to multiplication, and if in Definition 1.1we assume that A is a (multiplicative) semigroup and the α g are isomorphisms ofsemigroups, then we obtain the concept of a partial action of G on a semigroup(see [27]). Furthermore, the concept of a morphism of partial actions on semigroupsis obtained from Definition 1.2 by assuming that ϕ : A → A ′ is a homomorphismof semigroups.Partial cohomology was defined in [26] as follows. Let α = { α g : D g − → D g | g ∈ G } be a partial action of G on a commutative monoid A . Assume that each ideal D g is unital, i. e. D g is generated by an idempotent 1 g = 1 A g , which is central in A .In this case we shall say that α is a unital partial action. Then D g ∩ D h = D g D h ,for all g, h ∈ G, so the properties (ii) and (iii) from Definition 1.1 can be replacedby (ii’) α g ( D g − D h ) = D g D gh ;(iii’) α g ◦ α h = α gh on D h − D h − g − .Note also that (iii’) implies a more general equality α x ( D x − D y . . . D y n ) = D x D xy . . . D xy n , (1)for any x, y , . . . , y n ∈ G , which easily follows by observing that D x − D y . . . D y n = D x − D y . . . D x − D y n . Definition 1.4 (see [26]) . A commutative monoid A with a unital partial action α of G on A will be called a (unital) partial G -module . A morphism of (unital)partial G -modules ϕ : ( A , α ) → ( A ′ , α ′ ) is a morphism of partial actions such thatits restriction on each D g is a homomorphism of monoids D g → D ′ g , g ∈ G .Following [26], the category of (unital) partial G -modules and their morphismsis denoted by pMod ( G ). Definition 1.5 (see [26]) . Let
A ∈ pMod ( G ) and n be a positive integer. An n -cochain of G with values in A is a function f : G n → A , such that f ( x , . . . , x n ) isan invertible element of the ideal D ( x ,...,x n ) = D x D x x . . . D x ...x n . By a -cochain we shall mean an invertible element of A , i. e. a ∈ U ( A ), where U ( A ) stands forthe group of invertible elements of A .Denote the set of n -cochains by C n ( G, A ). It is an abelian group under thepointwise multiplication. Indeed, its identity is e n which is the n -cochain definedby e n ( x , . . . , x n ) = 1 ( x ,...,x n ) := 1 x x x . . . x ...x n , and the inverse of f ∈ C n ( G, A ) is f − ( x , . . . , x n ) = f ( x , . . . , x n ) − , where f ( x , . . . , x n ) − means the inverse of f ( x , . . . , x n ) in D ( x ,...,x n ) .The multiplicative form of the classical coboundary homomorphism now can beadapted to our context by replacing the global action by a partial one, and takinginverse elements in the corresponding ideals, as follows. Definition 1.6 (see [26]) . Let ( A , α ) ∈ pMod ( G ) and n be a positive integer. Forany f ∈ C n ( G, A ) and x , . . . , x n +1 ∈ G define( δ n f )( x , . . . , x n +1 ) = α x (1 x − f ( x , . . . , x n +1 )) LOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 5 n Y i =1 f ( x , . . . , x i x i +1 , . . . , x n +1 ) ( − i f ( x , . . . , x n ) ( − n +1 . (2)If n = 0 and a is an invertible element of A , we set( δ a )( x ) = α x (1 x − a ) a − . (3)According to Proposition 1.5 from [26] the coboundary map δ n is a homomor-phism C n ( G, A ) → C n +1 ( G, A ) of abelian groups, such that δ n +1 δ n f = e n +2 (4)for any f ∈ C n ( G, A ). As in the classical case one defines the abelian groups of par-tial n -cocycles, n -coboundaries and n -cohomologies of G with values in A by setting Z n ( G, A ) = ker δ n , B n ( G, A ) = im δ n − and H n ( G, A ) = ker δ n / im δ n − , n ≥ H ( G, A ) = Z ( G, A ) = ker δ ). Then two partial n -cocycles which represent thesame element of H n ( G, A ) are called cohomologous .Taking n = 0, we see that H ( G, A ) = Z ( G, A ) = { a ∈ U ( A ) | ∀ x ∈ G : α x (1 x − a ) = 1 x a } ,B ( G, A ) = { f ∈ C ( G, A ) | ∃ a ∈ U ( A ) : f ( x ) = α x (1 x − a ) a − } . Notice that H ( G, A ) is exactly the subgroup of α -invariants of U ( A ), as defined(for the case of rings) in [25, p. 79]. In order to relate partial cohomology to twistedpartial actions, consider the cases n = 1 and n = 2. In the first case we have( δ f )( x, y ) = α x (1 x − f ( y )) f ( xy ) − f ( x )with f ∈ C ( G, A ), so that Z ( G, A ) = { f ∈ C ( G, A ) | ∀ x, y ∈ G : α x (1 x − f ( y )) f ( x ) = 1 x f ( xy ) } ,B ( G, A ) = { f ∈ C ( G, A ) | ∃ g ∈ C ( G, A ) : f ( x, y ) = α x (1 x − g ( y )) g ( xy ) − g ( x ) } , and for n = 2( δ f )( x, y, z ) = α x (1 x − f ( y, z )) f ( xy, z ) − f ( x, yz ) f ( x, y ) − , with f ∈ C ( G, A ), and Z ( G, A ) = { f ∈ C ( G, A ) | ∀ x, y, z ∈ G : α x (1 x − f ( y, z )) f ( x, yz ) = f ( x, y ) f ( xy, z ) } . Now, a unital twisted partial action (see [23, Def. 2.1]) of G on a commutativering A splits into two parts: a unital partial action of G on A , and a twistingwhich, in our terminology, is a 2-cocycle with values in the partial G -module A .Furthermore, the concept of equivalent unital twisted partial actions from [24, Def.6.1] is exactly the notion of equivalence of partial 2-cocycles.We shall use multipliers in order to define globalization of partial cocycles, andfor this purpose we remind the reader that the multiplier ring of M ( A ) of anassociative non-necessarily unital ring A is the set M ( A ) = { ( R, L ) ∈ End ( A A ) × End ( A A ) : ( aR ) b = a ( Lb ) for all a, b ∈ A} with component-wise addition and multiplication (see [4] or [22] for more details).Here we use the right-hand side notation for homomorphisms of left A -modules,whereas for homomorphisms of right modules the usual notation is used. Thus given R : A A → A A , L : A A → A A and a ∈ A , we write a aR and a La . For a MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIM ´ON multiplier u = ( R, L ) ∈ M ( A ) and an element a ∈ A we set au = aR and ua = La ,so that the associativity equality ( au ) b = a ( ub ) always holds with a, b ∈ A .Notice that eu = ue (5)for any u ∈ M ( A ) and any central idempotent e ∈ A . For eu = ( e ) u = e ( eu ) = ( eu ) e = e ( ue ) = ( ue ) e = ue. Any a ∈ A determines a multiplier u a by setting u a b = ab and bu a = ba , b ∈ A , so that a u a gives the canonical homomorphism A → M ( A ), whichis an isomorphism if A has 1 A (in this case the inverse isomorphism is given by M ( A ) ∋ u u A = 1 A u ∈ A ). According to [22] a ring A is said to be non-degenerate if the canonical map A → M ( A ) is injective. This is guaranteed if A isleft (or right) s -unital , i. e. for any a ∈ A one has a ∈ A a (respectively, a ∈ a A ).Furthermore, given a ring isomorphism φ : A → A ′ , the map M ( A ) ∋ u φuφ − ∈ M ( A ′ ), where φuφ − = ( φ − Rφ, φLφ − ), u = ( R, L ), is an isomorphismof rings. In particular, an automorphism φ of A gives rise to an automorphism u φuφ − (6)of M ( A ).We shall also use the next. Remark 1.7. If A is a commutative idempotent ring, then M ( A ) is also commu-tative. Proof.
Indeed, for arbitrary u, v ∈ M ( A ) and a, b ∈ A we have that ( uv )( ab ) = u ( v ( ab )) = u ( v ( a ) b ) = u ( bv ( a )) = u ( b ) v ( a ) = v ( a ) u ( b ) = v ( au ( b )) = v ( u ( b ) a ) = v ( u ( ba )) = ( vu )( ab ). Similarly, ( ab )( uv ) = ( ab )( vu ) for all a, b ∈ A , and conse-quently, uv = vu , showing that M ( A ) is commutative. (cid:3) The notion of a globalization of a partial cocycle and itsrelation with an extendibility property
In this section we introduce the concept of a globalization of a partial n -cocyclewith values in a commutative unital ring A and show that a partial n -cocycle w isglobalizable, provided that an extendibility property for w holds. We start with ageneral auxiliary result which does not involve partial actions.Let G be a group and A a commutative unital ring. For f ∈ F = F ( G, A )denote by f | t the value f ( t ) and define β x : F → F by β x ( f ) | t = f ( x − t ) , (7)where x, t ∈ G . Then β is a global action of G on F which was used in [22] to dealwith the globalization problem for partial actions on unital rings.Let e w : G n → U ( A ) be a function, i. e. e w is an element of the group C n ( G, U ( A ))of global (classical) n -cochains of G with values in U ( A ). Define u : G n → U ( F ) by u ( x , . . . , x n ) | t = e w ( t − , x , . . . , x n − ) ( − n e w ( t − x , x , . . . , x n ) n − Y i =1 e w ( t − , x , . . . , x i x i +1 , . . . , x n ) ( − i , n > . (8)We proceed with a technical fact which will be used in the main result of thissection. LOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 7
Lemma 2.1.
The n -cochain u is an n -cocycle with respect to the action β of G on U ( F ) , i. e. u ∈ Z n ( G, U ( F )) .Proof. We need to show that the function β x ( u ( x , . . . , x n +1 )) n Y i =1 u ( x , . . . , x i x i +1 , . . . , x n +1 ) ( − i u ( x , . . . , x n ) ( − n +1 (9)is identity, i. e. it equals 1 F for any x , . . . , x n +1 ∈ G . Evaluating (9) at t andusing (7), we get u ( x , . . . , x n +1 ) | x − t n Y i =1 u ( x , . . . , x i x i +1 , . . . , x n +1 ) ( − i | t u ( x , . . . , x n ) ( − n +1 | t . (10)Denote by ˜ δ n : C n ( G, U ( A )) → C n +1 ( G, U ( A )) the coboundary operator whichcorresponds to the trivial G -module, i. e.(˜ δ n e w )( x , . . . , x n +1 ) = e w ( x , . . . , x n +1 ) n Y i =1 e w ( x , . . . , x i x i +1 , . . . , x n +1 ) ( − i e w ( x , . . . , x n ) ( − n +1 . (11)We see from (8) that u ( x , . . . , x n ) | t = e w ( x , . . . , x n )(˜ δ n e w )( t − , x , . . . , x n ) − . Therefore, (10) becomes e w ( x , . . . , x n +1 )(˜ δ n e w )( t − x , x , . . . , x n +1 ) − n Y i =1 e w ( x , . . . , x i x i +1 , . . . , x n +1 ) ( − i n Y i =1 (˜ δ n e w )( t − , x , . . . , x i x i +1 , . . . , x n +1 ) ( − i +1 e w ( x , . . . , x n ) ( − n +1 (˜ δ n e w )( t − , x , . . . , x n ) ( − n . Regrouping the factors and using (11), we obtain(˜ δ n e w )( x , . . . , x n +1 )(˜ δ n e w )( t − x , x , . . . , x n +1 ) − n Y i =1 (˜ δ n e w )( t − , x , . . . , x i x i +1 , . . . , x n +1 ) ( − i +1 (˜ δ n e w )( t − , x , . . . , x n ) ( − n , which is (˜ δ n +1 ˜ δ n e w )( t − , x , . . . , x n ) = 1 A . (cid:3) Let now α be a unital partial action of G on A . Then ϕ ( a ) | t = α t − (1 t a ) , (12) a ∈ A , defines an embedding of A into F , and ( β, B ) is an enveloping action for( α, A ), where B = P g ∈ G β g ( ϕ ( A )) (see the proof of [22, Theorem 4.5]). Since( β, B ) is unique up to an isomorphism, it follows by Theorem 3.1 from [21] that B is left s -unital. Hence there is a canonical embedding of B into the multiplierring M ( B ) and, moreover, B is commutative as so too is A . In addition, B isidempotent because B is left s -unital, which implies that M ( B ) is commutative MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIM ´ON thanks to Remark 1.7. Using the commutativity of M ( B ), the global action β of G on B can be extended by (6) to a global action β ∗ of G on M ( B ) by setting β ∗ g ( u ) = β g uβ − g = ( β − g Rβ g , β g Lβ − g ) , (13)where u = ( R, L ) ∈ M ( B ) and g ∈ G . This permits us to consider the group ofunits U ( M ( B )) as a G -module via β ∗ . Definition 2.2.
Let α = { α g : D g − → D g | g ∈ G } be a unital partial action of G on a commutative ring A and w ∈ Z n ( G, A ). Denote by β the enveloping action of G on B and by ϕ : A → B the embedding which transforms α into an admissiblerestriction of β . A globalization of w is a (classical) n -cocycle u ∈ Z n ( G, U ( M ( B ))),where G acts on U ( M ( B )) via β ∗ , such that ϕ ( w ( x , . . . , x n )) = ϕ (1 ( x ,...,x n ) ) u ( x , . . . , x n ) , (14)for any x , . . . , x n ∈ G . If n = 0, then by 1 ( x ,...,x n ) we mean 1 A in (14).Observe from (5) that (14) implies ϕ ( w ( x , . . . , x n )) = u ( x , . . . , x n ) ϕ (1 ( x ,...,x n ) ) . With the notation in Definition 2.2 we say, following [24], that α is unitallyglobalizable if B has 1 B . In this case β is called a unital enveloping action of α .Note that by the uniqueness, up to an isomorphism, of an enveloping action (see[22, Theorem 4.5]), all enveloping actions of a unitally globalizable partial actionare unital. Notice, furthermore, that unitally globalizable partial actions are alsocalled partial actions of finite type (see [35, Definition 1.1]).It is readily seen that if B contains 1 B , then the isomorphism M ( B ) ∼ = B trans-forms β ∗ into β , and the globalization u is an n -cocycle with values in U ( B ). Remark 2.3.
Observe that any partial 0-cocycle w is globalizable, and its glo-balization is the constant function u ∈ U ( F ) with u | t = w . Moreover, such u isunique.Indeed, (14) reduces to ϕ ( w ) = ϕ (1 A ) u , which is the 0-cocycle identity for w by (12).Moreover, u is an (invertible) multiplier of B , as β g ( ϕ ( a )) | t u | t = ϕ ( a ) | g − t w = α t − g (1 g − t a ) w = α t − g (1 g − t a ) · t − g w = α t − g (1 g − t aw ) = β g ( ϕ ( aw )) | t thanksto (7) and (12) and the 0-cocycle identity for w .Now if u , u ∈ M ( U ( B )) are globalizations of w , then ϕ (1 A ) u = ϕ (1 A ) u by (14). Applying β x to this equality and using the 0-cocycle identity for u i whichmeans that β x u i β − x = u i , i = 1 ,
2, one gets β x ( ϕ (1 A )) u = β x ( ϕ (1 A )) u for all x ∈ G . Consequently, β x ( ϕ ( a )) u = β x ( ϕ ( a )) u for all x ∈ G and a ∈ A . It followsthat u = u , as B = P g ∈ G β g ( A ).Given an arbitrary n >
0, as in the case n = 2 (see [24, Theorem 4.1]), we areable to reduce the globalization problem for partial n -cocycles to an extendibilityproperty. Theorem 2.4.
Let α = { α g : D g − → D g | g ∈ G } be a unital partial action of G on a commutative ring A and w ∈ Z n ( G, A ) . Then w is globalizable if and only ifthere exists a function e w : G n → U ( A ) which satisfies the equalities α x (cid:16) x − e w ( x , . . . , x n +1 ) (cid:17) n Y i =1 e w ( x , . . . , x i x i +1 , . . . , x n +1 ) ( − i LOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 9 e w ( x , . . . , x n ) ( − n +1 = 1 x , (15) and w ( x , . . . , x n ) = 1 ( x ,...,x n ) e w ( x , . . . , x n ) , (16) for all x , . . . , x n +1 ∈ G .Proof. We shall assume that n >
0, as n = 0 was considered in Remark 2.3.Suppose that w ∈ Z n ( G, A ) is globalizable. Denote by ( β, B ) an envelopingaction of ( α, A ) and let β ∗ be the corresponding action of G on M ( B ) (see (13)).Let u ∈ Z n ( G, U ( M ( B ))) be a globalization of w and define e w ( x , . . . , x n ) ∈ U ( A )by ϕ ( e w ( x , . . . , x n )) = ϕ (1 A ) u ( x , . . . , x n ) = u ( x , . . . , x n ) ϕ (1 A ) . (17)Evidently, e w ( x , . . . , x n ) ∈ U ( A ), as u ( x , . . . , x n ) is an invertible multiplier, and ϕ ( e w ( x , . . . , x n ) − ) = ϕ (1 A ) u − ( x , . . . , x n ) = u − ( x , . . . , x n ) ϕ (1 A ). Then (16)clearly holds by (14), and for (15) notice first that ϕ (1 g ) = β g ( ϕ (1 A )) ϕ (1 A ) , (18)and consequently (and in fact more generally), ϕ ( α g (1 g − a )) = β g ( ϕ ( a )) ϕ (1 A ) , (19)for all g ∈ G and a ∈ A (see [25, p. 79]). The (global) n -cocycle identity for u is ofthe form β ∗ x ( u ( x , . . . , x n +1 )) n Y i =1 u ( x , . . . , x i x i +1 , . . . , x n +1 ) ( − i u ( x , . . . , x n ) ( − n +1 = 1 M ( B ) . (20)Applying the first multiplier in (20) to ϕ (1 x ) and using (13), (17) and (19), weobtain β ∗ x ( u ( x , . . . , x n +1 )) ϕ (1 x ) = ( β x u ( x , . . . , x n +1 ) β − x )( β x ( ϕ (1 A )) ϕ (1 A ))= ( β x ( u ( x , . . . , x n +1 ) ϕ (1 A ))) ϕ (1 A )= ( β x [ ϕ ( e w ( x , . . . , x n +1 ))]) ϕ (1 A )= ϕ ( α x (1 x − e w ( x , . . . , x n +1 ))) . Then applying both sides of (20) to ϕ (1 x ) and using axioms of a multiplier, wereadily see that (15) is a consequence of (20).Suppose now that there exists e w : G n → U ( A ) such that (15) and (16) hold. Let( β, B ) be the globalization of ( α, A ), with β , B ⊆ F = F ( G, A ) and ϕ : A → B asdescribed above. In particular, it follows from (1) that ϕ (1 ( x ,...,x n ) ) | t = 1 ( t − ,x ,...,x n ) . (21)Taking our e w , define u : G n → U ( F ) by formula (8). We are going to showthat u is a globalization of w . By Lemma 2.1 one has u ∈ Z n ( G, U ( F )). We nowcheck (14). By (12) ϕ ( w ( x , . . . , x n )) | t = α t − (1 t w ( x , . . . , x n )) , which by the partial n -cocycle identity for w equals w ( t − x , x , . . . , x n ) n − Y i =1 w ( t − , x , . . . , x i x i +1 , . . . , x n ) ( − i w ( t − , x , . . . , x n − ) ( − n . In view of (8), (16) and (21) the latter is1 ( t − ,x ,...,x n ) u ( x , . . . , x n ) | t = ϕ (1 ( x ,...,x n ) ) | t u ( x , . . . , x n ) | t , for arbitrary t, x , . . . , x n ∈ G , proving (14).It remains to see that u ( x , . . . , x n ) and u ( x , . . . , x n ) − are multipliers of B .Notice first that using (15) for ( t − , x , . . . , x n ) we obtain from (8) that α t − (1 t e w ( x , . . . , x n )) = 1 t − e w ( t − x , x , . . . , x n ) n − Y i =1 e w ( t − , x , . . . , x i x i +1 , . . . , x n ) ( − i e w ( t − , x , . . . , x n − ) ( − n = 1 t − u ( x , . . . , x n ) | t . (22)Then by (12) u ( x , . . . , x n ) | t ϕ ( a ) | t = α t − (1 t e w ( x , . . . , x n )) α t − (1 t a ) , so that u ( x , . . . , x n ) ϕ ( a ) = ϕ ( a e w ( x , . . . , x n )) , (23)for all x , . . . , x n ∈ G and a ∈ A . Equalities (8) and (23) readily imply u ( x , . . . , x n ) − ϕ ( a ) = ϕ ( a e w ( x , . . . , x n ) − ) . (24)Furthermore, applying the n -cocycle identity for u to ( t − , x , . . . x n ) we see that β t − ( u ( x , . . . , x n )) ϕ ( a ) = u ( t − x , x , . . . , x n ) n − Y i =1 u ( t − , x , . . . , x i x i +1 , . . . , x n +1 ) ( − i u ( t − , x , . . . , x n − ) ( − n ϕ ( a ) , which belongs to ϕ ( A ) thanks to (23) and (24). Thus β t − ( u ( x , . . . , x n )) ϕ ( A ) ⊆ ϕ ( A ), which yields u ( x , . . . , x n ) β t ( ϕ ( A )) ⊆ β t ( ϕ ( A )). Since B = P t ∈ G β t ( ϕ ( A )),it follows that u ( x , . . . , x n ) B ⊆ B , and hence u ( x , . . . , x n ) ∈ M ( B ). Similarly, u ( x , . . . , x n ) − ∈ M ( B ), as desired. (cid:3) Note that taking t = 1 in (22) we obtain u ( x , . . . , x n ) | = e w ( x , . . . , x n ).3. From w to w ′ Our next purpose is to show that e w in Theorem 2.4 exists, provided that A is aproduct of blocks, and we need first some technical preparation for this fact, whichwe do in the present section.Suppose that A = Q λ ∈ Λ A λ , where A λ is an indecomposable (commutative)unital ring, called a block . We identify the unity element of A µ , µ ∈ Λ, with theindecomposable idempotent 1 A µ of A which is the function Λ → S λ ∈ Λ A λ whose LOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 11 value at µ is the identity of A µ and the value at any λ = µ is the zero of A λ . Then A µ is identified with the ideal of A generated by the idempotent 1 A µ . Denote by pr µ the projection of A onto A µ , namely, pr µ ( a ) = 1 A µ a . Thus, any a ∈ A is identifiedwith the set of its projections { pr λ ( a ) } λ ∈ Λ , and we write a = Q λ ∈ Λ pr λ ( a ) in thissituation. If there exists Λ ⊆ Λ, such that pr λ ( a ) = 0 A for all λ ∈ Λ \ Λ , then weshall also write a = Q λ ∈ Λ pr λ ( a ), and such elements a form an ideal in A whichwe denote by Q λ ∈ Λ A λ .Since A λ is indecomposable, the only idempotents of A λ are 0 A and 1 A λ . Hence,for any idempotent e of A the projection pr λ ( e ) is either 0 A , or 1 A λ . In particular, e A = Y λ ∈ Λ A λ , (25)where Λ = { λ ∈ Λ | pr λ ( e ) = 1 A λ } . Thus, the unital ideals of A are exactly theproducts of blocks A λ over all Λ ⊆ Λ. Lemma 3.1.
Let I = Q λ ∈ Λ A λ and J = Q λ ∈ Λ A λ be unital ideals of A and ϕ : I → J an isomorphism. Then there exists a bijection σ : Λ → Λ , such that ϕ (pr λ ( a )) = pr σ ( λ ) ( ϕ ( a )) for all a ∈ I and λ ∈ Λ .Proof. Note that { A λ } λ ∈ Λ and { A λ } λ ∈ Λ are the sets of indecomposable idem-potents of I and J , respectively. Since ϕ is an isomorphism, ϕ (1 A λ ) = 1 A σ ( λ ) forsome bijection σ : Λ → Λ . Then ϕ (pr λ ( a )) = ϕ (1 A λ a ) = 1 A σ ( λ ) ϕ ( a ) = pr σ ( λ ) ( ϕ ( a )) . (cid:3) Let α = { α x : D x − → D x | x ∈ G } be a unital partial action of G on A . Bythe observation above each domain D x is a product of blocks, and α x maps a blockof D x − onto some block of D x . As in [24] we call α transitive , when for any pair λ ′ , λ ′′ ∈ Λ there exists x ∈ G , such that A λ ′ ⊆ D x − and α x ( A λ ′ ) = A λ ′′ ⊆ D x .In all what follows, if otherwise is not stated, we assume that α is transitive. Thenwe may fix λ ∈ Λ, so that each A λ is α x ( A λ ) for some x ∈ G with A λ ⊆ D x − .Observe that, whenever A λ ⊆ D x − D x ′− and α x ( A λ ) = α x ′ ( A λ ), it follows that A λ ⊆ D ( x ′ ) − x and α x − x ′ ( A λ ) = A λ . Hence, introducing as in [24] the subgroup H = { x ∈ G | A λ ⊆ D x − and α x ( A λ ) = A λ } and choosing a left transversal Λ ′ of H in G , one may identify Λ with a subset ofΛ ′ , namely, λ ∈ Λ corresponds to (a unique) g ∈ Λ ′ , such that A λ ⊆ D g − and α g ( A λ ) = A λ . Assume, moreover, that Λ ′ contains the identity element 1 of G .Then λ is identified with 1 and thus A g = α g ( A ) for g ∈ Λ ⊆ Λ ′ . (26)Given x ∈ G , we use the notation ¯ x from [24] for the element of Λ ′ with x ∈ ¯ xH .We recall the following useful fact. Lemma 3.2 (Lemma 5.1 from [24]) . Given x ∈ G and g ∈ Λ ′ , one has(i) g ∈ Λ ⇔ A ⊆ D g − ;(ii) if g ∈ Λ , then xg ∈ Λ ⇔ A g ⊆ D x − , and in this situation α x ( A g ) = A xg . Notice that taking g = 1 in (ii), one gets x ∈ Λ ⇔ A ⊆ D x − . Then using (ii)once again, we see that for any g ∈ Λ A g ⊆ D x ⇔ x − g ∈ Λ ⇔ A ⊆ D g − x . (27) In particular, A x ⊆ D x for all x ∈ G , such that x ∈ Λ.For any g ∈ Λ and a ∈ A define θ g ( a ) = α g (pr ( a )) . (28)Note that by (i) of Lemma 3.2 the block A is a subset of D g − , so α g (pr ( a ))makes sense and belongs to A g . Thus, θ g is a correctly defined homomorphism A → A g . Clearly, θ g ( a ) = θ g (1 x a ) (29)for any x ∈ G , such that A ⊆ D x . In particular, this holds for x ∈ H and for x = g − .Observe also that θ g ( a ) = θ g (1 g − a ) = pr g ( α g (1 g − a )) (30)in view of Lemma 3.1. It follows that θ g ( α g − (1 g a )) = pr g (1 g a ) = pr g ( a ), as A g ⊆ D g . Therefore, a = Y g ∈ Λ θ g ( α g − (1 g a )) . (31) Lemma 3.3.
Let n > and w ∈ Z n ( G, A ) . Then w ( x , . . . , x n ) = Y g ∈ Λ θ g [ w ( g − x , x , . . . , x n ) n − Y k =1 w ( g − , . . . , x k x k +1 , . . . , x n ) ( − k w ( g − , x , . . . , x n − ) ( − n ] . (32) Proof.
By (31) w ( x , . . . , x n ) = Y g ∈ Λ θ g ( α g − (1 g w ( x , . . . , x n ))) . As w ∈ Z n ( G, A ), one has1 ( g − ,x ,...,x n ) = ( δ n w )( g − , x . . . , x n )= α g − (1 g w ( x , . . . , x n )) w ( g − x , x , . . . , x n ) − n − Y k =1 w ( g − , . . . , x k x k +1 , . . . , x n ) ( − k − w ( g − , x , . . . , x n − ) ( − n − . Hence, α g − (1 g w ( x , . . . , x n )) = w ( g − x , x , . . . , x n ) n − Y k =1 w ( g − , . . . , x k x k +1 , . . . , x n ) ( − k w ( g − , x , . . . , x n − ) ( − n . Observe that this θ differs from the one introduced in [24]. More precisely, denoting θ from [24]by θ ′ , we may write θ ′ g ( a ) = θ g − ( a ) + 1 A − A g − for g − ∈ Λ. LOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 13 (cid:3)
Given x ∈ G , denote by η ( x ) the element x − ¯ x ∈ H . Let n > g ∈ Λ ′ .Define η gn : G n → H by η gn ( x , . . . , x n ) = η ( x − n x − n − . . . x − g ) (33)and τ gn : G n → H n by τ gn ( x , . . . , x n ) = ( η g ( x ) , η g ( x , x ) , . . . , η gn ( x , . . . , x n )) . (34)Observe that η g ( x ) η g ( x , x ) . . . η gn ( x , . . . , x n ) = η ( x − n . . . x − g ) = η g ( x . . . x n ) . (35)We shall also need the functions σ gn,i : G n → G n +1 , 0 ≤ i ≤ n , defined by σ gn, ( x , . . . , x n ) = ( g − , x , . . . , x n ) , (36) σ gn,i ( x , . . . , x n ) = ( τ gi ( x , . . . , x i ) , ( x − i . . . x − g ) − , x i +1 , . . . , x n ) , < i < n, (37) σ gn,n ( x , . . . , x n ) = ( τ gn ( x , . . . , x n ) , ( x − n . . . x − g ) − ) . (38)In the formulas above we may allow n to be equal to zero, meaning that σ g , = g − ∈ G .With any n > w ∈ C n ( G, A ) we shall associate w ′ ( x , . . . , x n ) = 1 ( x ,...,x n ) Y g ∈ Λ θ g ◦ w ◦ τ gn ( x , . . . , x n ) , (39) ε ( x , . . . , x n − ) = 1 ( x ,...,x n − ) Y g ∈ Λ θ g n − Y i =0 w ◦ σ gn − ,i ( x , . . . , x n − ) ( − i ! . (40) Lemma 3.4.
One has w ′ ∈ C n ( G, A ) and ε ∈ C n − ( G, A ) .Proof. Notice by (34) and (35) that w ◦ τ gn ( x , . . . , x n ) ∈ U ( D η ( x − g ) D η ( x − x − g ) . . . D η ( x − n ...x − g ) ) . Since η ( x − k . . . x − g ) ∈ H , then A ⊆ D η ( x − k ...x − g ) , 1 ≤ k ≤ n , sopr ◦ w ◦ τ gn ( x , . . . , x n ) ∈ U ( A )and hence by (28) θ g ◦ w ◦ τ gn ( x , . . . , x n ) = α g ◦ pr ◦ w ◦ τ gn ( x , . . . , x n ) ∈ U ( A g ) . Therefore, the product of the values of θ g on the right-hand side of (39) belongs to U ( A ) and thus w ′ ∈ C n ( G, A ).To prove that ε ∈ C n − ( G, A ) for n > , observe first that the right-hand sideof (40) depends only on θ g with g ∈ Λ satisfying A g ⊆ D ( x ,...,x n − ) (41)(if there is no such g , then D ( x ,...,x n − ) is zero and thus ε ( x , . . . , x n − ) is auto-matically invertible in this ideal). Now n − Y i =0 w ◦ σ gn − ,i ( x , . . . , x n − ) ∈ U ( D ( g − ,x ,...,x n − ) D η ( x − g ) . . . D η ( x − n − ...x − g ) ) . As above, A ⊆ D η ( x − k ...x − g ) , 1 ≤ k ≤ n −
1, because η ( x − k . . . x − g ) ∈ H .Moreover, by (27) condition (41) is equivalent to A ⊆ D ( g − ,x ,...,x n − ) . The restof the proof now follows as for w ′ . If n = 1, then ε = Y g ∈ Λ θ g ( w ( g − )) ∈ U ( A ) , (42)as D g − ⊇ A by (i) of Lemma 3.2. (cid:3) The following notation will be used in the results below.Π( l, m ) = n − Y k = l,i = m w ◦ σ gn − ,i ( x , . . . , x k x k +1 , . . . , x n ) ( − k + i n − Y i = m w ◦ σ gn − ,i ( x , . . . , x n − ) ( − n + i , (43)where 1 ≤ l ≤ n − ≤ m ≤ n − Lemma 3.5.
For all w ∈ Z ( G, A ) and x ∈ G we have: ( δ ε )( x ) α x (1 x − ε ) − w ( x ) − = Y g ∈ Λ θ g ( w ( g − x ) − ) . (44) Moreover, for n > , w ∈ Z n ( G, A ) and x , . . . , x n ∈ G : ( δ n − ε )( x , . . . , x n ) α x (1 x − ε ( x , . . . , x n )) − w ( x , . . . , x n ) − = Y g ∈ Λ θ g ( w ( g − x , x , . . . , x n ) − Π(1 , . (45) Proof.
Indeed, by (3), (32) and (42) we see that( δ ε )( x ) α x (1 x − ε ) − w ( x ) − = ε − w ( x ) − = Y g ∈ Λ θ g ( w ( g − ) − w ( g − x ) − w ( g − ))= Y g ∈ Λ θ g ( w ( g − x ) − ) . For (45) observe from (2), (40) and (43) that( δ n − ε )( x , . . . , x n ) α x (1 x − ε ( x , . . . , x n )) − = n − Y k =1 ε ( x , . . . , x k x k +1 , . . . , x n ) ( − k ! ε ( x , . . . , x n − ) ( − n = Y g ∈ Λ θ g (Π(1 , . Now in (32) one has w ( g − , x , . . . , x k x k +1 , . . . , x n ) ( − k = w ◦ σ gn − , ( x , . . . , x k x k +1 , . . . , x n ) ( − k +0 ,w ( g − , x , . . . , x n − ) ( − n = w ◦ σ gn − , ( x , . . . , x n − ) ( − n +0 , which are the factors of Π(1 ,
0) corresponding to i = 0 and 1 ≤ k ≤ n −
1. Hence, Y g ∈ Λ θ g (Π(1 , w ( x , . . . , x n ) Y g ∈ Λ θ g ( w ( g − x , x , . . . , x n ) − Π(1 , . LOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 15 (cid:3)
Lemma 3.6.
For all n > , w ∈ Z n ( G, A ) , g ∈ Λ and x , . . . , x n ∈ G : w ( g − x , x , . . . , x n ) − Π(1 ,
1) = α η g ( x ) (1 η g ( x ) − w ◦ σ x − gn − , ( x , . . . , x n )) − w ( τ g ( x ) , ( x − g ) − x , x , . . . , x n ) − Π(2 , n − Y i =1 w ◦ σ gn − ,i ( x x , x , . . . , x n ) ( − i +1 . (46) Proof.
Since w is a partial n -cocycle, one has that (see (33), (34) and (37))( δ n w ) ◦ σ gn, ( x , . . . , x n ) = ( δ n w )( τ g ( x ) , ( x − g ) − , x , . . . , x n )= ( δ n w )( η g ( x ) , ( x − g ) − , x , . . . , x n )= ( δ n w )( g − x · x − g, ( x − g ) − , x , . . . , x n ) (47)= 1 g − x · x − g ( g − x ,x ,...,x n ) . Applying (2), we expand (47) as follows:1 g − x · x − g ( g − x ,x ,...,x n ) = α g − x · x − g (1 ( x − g ) − x − g w (( x − g ) − , x , . . . , x n )) w ( g − x , x , . . . , x n ) − w ( g − x · x − g, ( x − g ) − x , x , . . . , x n ) n − Y k =2 w ( g − x · x − g, ( x − g ) − , x , . . . , x k x k +1 , . . . x n ) ( − k +1 w ( g − x · x − g, ( x − g ) − , x , . . . , x n − ) ( − n +1 . Using our notations (33)–(34), (37) and (38), we conclude that1 η g ( x ) w ( g − x , . . . , x n ) − = α η g ( x ) (1 η g ( x ) − w ◦ σ x − gn − , ( x , . . . , x n )) − (48) w ( τ g ( x ) , ( x − g ) − x , x , . . . , x n ) − (49) n − Y k =2 w ◦ σ gn − , ( x , . . . , x k x k +1 , . . . x n ) ( − k (50) w ◦ σ gn − , ( x , x , . . . , x n − ) ( − n , (51)the lines (50) and (51) being the inverses of the factors of Π(1 , i = 1 and 2 ≤ k ≤ n −
1. Thus, after the multiplication of the right-hand side ofequality (48)–(51) by Π(1 , ,
1) which correspond to k = 1, and the factors of Π(2 ,
2) (i. e.those of Π(1 ,
1) with indexes 2 ≤ i, k ≤ n − η g ( x ) Π(1 ,
1) = Π(1 ,
1) and the idempotents which appear inthe cancellations are absorbed by the element (49), except 1 x − g which is absorbedby the element in the right hand side of (48). (cid:3) Lemma 3.7.
For all < j < n , w ∈ Z n ( G, A ) , g ∈ Λ and x , . . . , x n ∈ G : w ( τ gj − ( x , . . . , x j − ) , ( x − j − . . . x − g ) − x j , x j +1 , . . . , x n ) − Π( j, j ) = α η g ( x ) (1 η g ( x ) − w ◦ σ x − gn − ,j − ( x , . . . , x n )) ( − j w ( τ gj ( x , . . . , x j ) , ( x − j . . . x − g ) − x j +1 , x j +2 , . . . , x n ) − Π( j + 1 , j + 1) n − Y i = j w ◦ σ gn − ,i ( x , . . . , x j x j +1 , . . . , x n ) ( − i + j j − Y s =1 w ◦ σ gn − ,j − ( x , . . . , x s x s +1 , . . . , x n ) ( − s + j (52) (here by Π( n, n ) we mean the identity element A ).Proof. We use the same idea as in the proof of Lemma 3.6:( δ n w ) ◦ σ gn,j ( x , . . . , x n )= ( δ n w )( τ gj ( x , . . . , x j ) , ( x − j . . . x − g ) − , x j +1 , . . . , x n )= ( δ n w )( η g ( x ) , η g ( x , x ) , . . . , η gj ( x , . . . , x j ) , ( x − j . . . x − g ) − , x j +1 , . . . , x n )= ( δ n w )( g − x x − g, ( x − g ) − x x − x − g, . . . , ( x − j − . . . x − g ) − x j x − j . . . x − g, ( x − j . . . x − g ) − , x j +1 , . . . , x n ) (53)= 1 g − x x − g g − x x x − x − g . . . g − x ...x j x − j ...x − g ( g − x ...x j ,x j +1 ,...,x n ) . Expanding (53), we obtain by (2)1 g − x x − g g − x x x − x − g . . . g − x ...x j x − j ...x − g ( g − x ...x j ,x j +1 ,...,x n ) = α g − x x − g (1 ( x − g ) − x − g w (( x − g ) − x x − x − g, . . . , ( x − j − . . . x − g ) − x j x − j . . . x − g, ( x − j . . . x − g ) − , x j +1 , . . . , x n )) w ( g − x x x − x − g, . . . , ( x − j − . . . x − g ) − x j x − j . . . x − g, ( x − j . . . x − g ) − , x j +1 , . . . , x n ) − j − Y s =2 w ( g − x x − g, . . . , ( x − s − . . . x − g ) − x s x s +1 x − s +1 . . . x − g, . . . , ( x − j − . . . x − g ) − x j x − j . . . x − g, ( x − j . . . x − g ) − , x j +1 , . . . , x n ) ( − s w ( g − x x − g, . . . , ( x − j − . . . x − g ) − x j − x j x − j . . . x − g, ( x − j . . . x − g ) − , x j +1 , . . . , x n ) ( − j − w ( g − x x − g, . . . , ( x − j − . . . x − g ) − x j − x − j − . . . x − g, ( x − j − . . . x − g ) − x j , x j +1 , . . . , x n ) ( − j w ( g − x x − g, . . . , ( x − j − . . . x − g ) − x j x − j . . . x − g, ( x − j . . . x − g ) − x j +1 , x j +2 , . . . , x n ) ( − j +1 n − Y t = j +1 w ( g − x x − g, . . . , ( x − j − . . . x − g ) − x j x − j . . . x − g, LOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 17 ( x − j . . . x − g ) − , x j +1 , . . . , x t x t +1 , . . . , x n ) ( − t +1 w ( g − x x − g, . . . , ( x − j − . . . x − g ) − x j x − j . . . x − g, ( x − j . . . x − g ) − , x j +1 , . . . , x n − ) ( − n +1 . We rewrite this in our shorter notations (33)–(34), (37) and (38):1 η g ( x ) η g ( x x ) . . . η g ( x ...x j ) ( g − x ...x j ,x j +1 ,...,x n ) = α η g ( x ) (1 η g ( x ) − w ◦ σ x − gn − ,j − ( x , . . . , x n )) w ◦ σ gn − ,j − ( x x , x , . . . , x n ) − (54) j − Y s =2 w ◦ σ gn − ,j − ( x , . . . , x s x s +1 , . . . , x n ) ( − s (55) w ◦ σ gn − ,j − ( x , . . . , x j − x j , . . . , x n ) ( − j − (56) w ( τ gj − ( x , . . . , x j − ) , ( x − j − . . . x − g ) − x j , x j +1 , . . . , x n ) ( − j w ( τ gj ( x , . . . , x j ) , ( x − j . . . x − g ) − x j +1 , x j +2 , . . . , x n ) ( − j +1 n − Y t = j +1 w ◦ σ gn − ,j ( x , . . . , x t x t +1 , . . . , x n ) ( − t +1 w ◦ σ gn − ,j ( x , . . . , x n − ) ( − n +1 . Note that the factors (54) and (56) may be included into the product (55), permit-ting thus s run from 1 to j − η g ( x ...x j ) w ( τ gj − ( x , . . . , x j − ) , ( x − j − . . . x − g ) − x j , x j +1 , . . . , x n ) − (57)= α η g ( x ) (1 η g ( x ) − w ◦ σ x − gn − ,j − ( x , . . . , x n )) ( − j j − Y s =1 w ◦ σ gn − ,j − ( x , . . . , x s x s +1 , . . . , x n ) ( − s + j w ( τ gj ( x , . . . , x j ) , ( x − j . . . x − g ) − x j +1 , x j +2 , . . . , x n ) − n − Y t = j +1 w ◦ σ gn − ,j ( x , . . . , x t x t +1 , . . . , x n ) ( − t + j +1 (58) w ◦ σ gn − ,j ( x , . . . , x n − ) ( − n + j +1 . (59)The lines (58) and (59) are the inverses of the factors of Π( j, j ) corresponding to i = j and j + 1 ≤ k ≤ n −
1. Therefore, multiplication by Π( j, j ) replaces these twolines by the factors of Π( j, j ) with j ≤ i ≤ n − k = j , and, whenever j < n − j + 1 , j + 1), giving the right-hand side ofequality (52). Finally, the left-hand side of (52) coincides with (57) multiplied byΠ( j, j ), as 1 η g ( x ...x j ) Π( j, j ) = Π( j, j ). (cid:3) Lemma 3.8.
For all w ∈ Z ( G, A ) , g ∈ Λ and x ∈ G : η g ( x ) w ( g − x ) − = α η g ( x ) (1 η g ( x ) − w (( x − g ) − ) − )( w ◦ τ g )( x ) − . (60) Moreover, for all n > , w ∈ Z n ( G, A ) , g ∈ Λ and x , . . . , x n ∈ G : w ( τ gn − ( x , . . . , x n − ) , ( x − n − . . . x − g ) − x n ) − = α η g ( x ) (1 η g ( x ) − w ◦ σ x − gn − ,n − ( x , . . . , x n )) ( − n n − Y s =1 w ◦ σ gn − ,n − ( x , . . . , x s x s +1 , . . . , x n ) ( − s + n w ◦ τ gn ( x , . . . , x n ) − . (61) Proof.
For (60) write1 η g ( x ) g − x = ( δ w ) ◦ σ g , ( x )= ( δ w )( τ g ( x ) , ( x − g ) − )= ( δ w )( η g ( x ) , ( x − g ) − )= α η g ( x ) (1 η g ( x ) − w (( x − g ) − )) w ( g − x ) − ( w ◦ τ g )( x ) . To get (61), analyze the proof of Lemma 3.7 (we skip the details):1 η g ( x ) η g ( x x ) . . . η g ( x ...x n ) g − x · ... · x n = ( δ n w ) ◦ σ gn,n ( x , . . . , x n )= α η g ( x ) (1 η g ( x ) − w ◦ σ x − gn − ,n − ( x , . . . , x n )) n − Y s =1 w ◦ σ gn − ,n − ( x , . . . , x s x s +1 , . . . , x n ) ( − s w ( τ gn − ( x , . . . , x n − ) , ( x − n − . . . x − g ) − x n ) ( − n w ◦ τ gn ( x , . . . , x n ) ( − n +1 . (cid:3) Lemma 3.9.
For all n > , w ∈ Z n ( G, A ) and x , . . . , x n ∈ G : ( δ n − ε )( x , . . . , x n ) α x (1 x − ε ( x , . . . , x n )) − w ( x , . . . , x n ) − = Y g ∈ Λ θ g ◦ α η g ( x ) η g ( x ) − n − Y j =0 w ◦ σ x − gn − ,j ( x , . . . , x n ) ( − j +1 w ′ ( x . . . , x n ) − . (62) Proof. If n = 1, then the result follows from (29), (39), (44) and (60) and the factthat η g ( x ) ∈ H .Let n >
1. Using the recursion whose base is (46), an intermediate step is (52)and the final step is (61), we have w ( g − x , . . . , x n ) − Π(1 , α η g ( x ) η g ( x ) − n − Y j =0 w ◦ σ x − gn − ,j ( x , . . . , x n ) ( − j +1 w ◦ τ gn ( x , . . . , x n ) − n − Y j =1 n − Y i = j w ◦ σ gn − ,i ( x , . . . , x j x j +1 , . . . , x n ) ( − i + j (63) n Y j =2 j − Y s =1 w ◦ σ gn − ,j − ( x , . . . , x s x s +1 , . . . , x n ) ( − s + j . (64)After the change of indexes j ′ = j − n − Y j ′ =1 j ′ Y s =1 w ◦ σ gn − ,j ′ ( x , . . . , x s x s +1 , . . . , x n ) ( − s + j ′ +1 . Now switching the order in this double product, we come to n − Y s =1 n − Y j ′ = s w ◦ σ gn − ,j ′ ( x , . . . , x s x s +1 , . . . , x n ) ( − s + j ′ +1 . The latter is exactly the inverse of (63). Hence, w ( g − x , x , . . . , x n ) − Π(1 , α η g ( x ) η g ( x ) − n − Y j =0 w ◦ σ x − gn − ,j ( x , . . . , x n ) ( − j +1 w ◦ τ gn ( x , . . . , x n ) − . It remains to substitute this into (45) and to apply (39). (cid:3)
Lemma 3.10.
For all x ∈ G and a : Λ ′ → A one has α x x − Y g ∈ Λ θ g ( a ( g )) = 1 x Y g ∈ Λ θ g ◦ α η g ( x ) (cid:16) η g ( x ) − a (cid:16) x − g (cid:17)(cid:17) . (65) Proof.
First of all observe using (ii) of Lemma 3.2 that1 x Y g ∈ Λ c g = Y g ∈ Λ , A g ⊆D x c g = Y g, x − g ∈ Λ c g , (66)where c g is an arbitrary element of A g . Thus, in the right-hand side of (65) we mayreplace the condition g ∈ Λ by a stronger one g, x − g ∈ Λ. Notice also from (27)and (29) that we may put 1 g − x inside of θ g in the right-hand side of (65).Now1 g − x α η g ( x ) (cid:16) η g ( x ) − a (cid:16) x − g (cid:17)(cid:17) = α g − x ◦ α x − g (cid:16) ( x − g ) − η g ( x ) − a (cid:16) x − g (cid:17)(cid:17) , (67)and denoting the argument of α x − g in (67) by b = b ( g, x ), we deduce from (30)that θ g ◦ α g − x ◦ α x − g ( b ) = pr g ◦ α g (cid:16) g − α g − x ◦ α x − g ( b ) (cid:17) = pr g ◦ α g ◦ α g − ◦ α x (cid:16) x − x − g α x − g ( b ) (cid:17) = pr g ◦ α x (cid:16) x − x − g α x − g ( b ) (cid:17) . As xx − g = g ∈ Λ, by (ii) of Lemma 3.2 we have A x − g ⊆ D x − and α x (cid:16) A x − g (cid:17) = A g . Moreover, A x − g ⊆ D x − g by (27). Hence, in view of Lemma 3.1 and (30)pr g ◦ α x (cid:16) x − x − g α x − g ( b ) (cid:17) = α x ◦ pr x − g (cid:16) x − x − g α x − g ( b ) (cid:17) = α x ◦ pr x − g ◦ α x − g ( b )= α x ◦ θ x − g ( b ) , and consequently θ g ◦ α η g ( x ) (cid:16) η g ( x ) − a (cid:16) x − g (cid:17)(cid:17) = α x ◦ θ x − g ( b ) = α x ◦ θ x − g (cid:16) a (cid:16) x − g (cid:17)(cid:17) . Here we used (29) to remove 1 η g ( x ) − and 1 ( x − g ) − from b . It follows that theright-hand side of (65) is Y g,x − g ∈ Λ α x ◦ θ x − g (cid:16) a (cid:16) x − g (cid:17)(cid:17) = α x Y g,x − g ∈ Λ θ x − g (cid:16) a (cid:16) x − g (cid:17)(cid:17) , (68)which is verified by checking the projection of each side of the latter equality ontoan arbitrary block A t , t ∈ Λ. Let g ′ = x − g ∈ Λ. Then g = xx − g = xg ′ ∈ Λ,so (68) becomes, in view of (66), α x Y g ′ ,xg ′ ∈ Λ θ g ′ ( a ( g ′ )) = α x x − Y g ′ ∈ Λ θ g ′ ( a ( g ′ )) , proving (65). (cid:3) Lemma 3.11.
For all n > , w ∈ Z n ( G, A ) and x , . . . , x n ∈ G : ( x ...,x n ) Y g ∈ Λ θ g ◦ α η g ( x ) η g ( x ) − n − Y j =0 w ◦ σ x − gn − ,j ( x , . . . , x n ) ( − j = α x (cid:16) x − ε ( x , . . . , x n ) (cid:17) . (69) Proof.
Let us fix n , w and x , . . . , x n . For arbitrary g ∈ Λ ′ define a ( g ) = n − Y j =0 w ◦ σ gn − ,j ( x , . . . , x n ) ( − j . Then the left-hand side of (69) equals1 ( x ...,x n ) Y g ∈ Λ θ g ◦ α η g ( x ) (cid:16) η g ( x ) − a ( x − g ) (cid:17) . Since 1 ( x ...,x n ) = 1 ( x ...,x n ) x , then applying Lemma 3.10, we transform this into1 ( x ...,x n ) α x x − Y g ∈ Λ θ g n − Y j =0 w ◦ σ gn − ,j ( x , . . . , x n ) ( − j . Rewriting 1 ( x ,...,x n ) as α x (cid:16) x − ( x ,...,x n ) (cid:17) and using (40), we come to the right-hand side of (69). (cid:3) LOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 21
Theorem 3.12.
Let n > and w ∈ Z n ( G, A ) . Then w = δ n − ε · w ′ . In particular, w ′ ∈ Z n ( G, A ) .Proof. This is an immediate consequence of Lemmas 3.9 and 3.11. (cid:3) Existence of a globalization
In this section we construct the cocycle e w whose existence was announced above.Keeping the notations of Section 2, we begin with some auxiliary formulas whoseproof will be left to the reader. Lemma 4.1.
Let g ∈ Λ ′ . Then η gn ( x , . . . , x n ) = η x − gn − ( x , . . . , x n ) , n ≥ , (70) η gn ( x , . . . , x i , x i +1 , . . . , x n ) = η gn − ( x , . . . , x i x i +1 , . . . , x n ) , ≤ i ≤ n − , (71) η gn ( x , . . . , x n − , x n x n +1 ) = η gn ( x , . . . , x n ) η gn +1 ( x , . . . , x n +1 ) , n ≥ . (72)We now define a function f w ′ : G n → A by removing the idempotent 1 ( x ,...,x n ) from the right-hand side of (39), that is f w ′ ( x , . . . , x n ) = Y g ∈ Λ θ g ◦ w ◦ τ gn ( x , . . . , x n ) . (73)As it was observed in the proof of Lemma 3.4, f w ′ ( x , . . . , x n ) ∈ U ( A ), so f w ′ is aclassical n -cochain from C n ( G, U ( A )). It turns out that f w ′ satisfies the “quasi” n -cocycle identity (15). Lemma 4.2.
Let n > , w ∈ Z n ( G, A ) and x , . . . , x n ∈ G . Then α x (cid:16) x − f w ′ ( x , . . . , x n +1 ) (cid:17) n Y i =1 f w ′ ( x , . . . , x i x i +1 , . . . , x n +1 ) ( − i f w ′ ( x , . . . , x n ) ( − n +1 = 1 x . (74) Proof.
According to (73), the left-hand side of (74) is α x x − Y g ∈ Λ θ g ◦ w ◦ τ gn ( x , . . . , x n +1 ) (75) n Y i =1 Y g ∈ Λ θ g ◦ w ◦ τ gn ( x , . . . , x i x i +1 , . . . , x n +1 ) ( − i (76) Y g ∈ Λ θ g ◦ w ◦ τ gn ( x , . . . , x n ) ( − n +1 . (77)Using Lemma 3.10, we rewrite (75) as1 x Y g ∈ Λ θ g ◦ α η g ( x ) (cid:18) η g ( x ) − w ◦ τ x − gn ( x , . . . , x n +1 ) (cid:19) . Moreover, since θ g is a homomorphism, (76) coincides with Y g ∈ Λ θ g n Y i =1 w ◦ τ gn ( x , . . . , x i x i +1 , . . . , x n +1 ) ( − i ! . Therefore, in order to prove (74), one suffices to check the equality1 η ( x − g ) . . . η ( x − n +1 ...x − g ) = α η g ( x ) (cid:18) η g ( x ) − w ◦ τ x − gn ( x , . . . , x n +1 ) (cid:19) (78) n Y i =1 w ◦ τ gn ( x , . . . , x i x i +1 , . . . , x n +1 ) ( − i (79) w ◦ τ gn ( x , . . . , x n ) ( − n +1 . (80)Indeed, each η ( x − i . . . x − g ) belongs to H , so by (28) θ g (cid:16) η ( x − i ...x − g ) (cid:17) = α g ◦ pr (cid:16) η ( x − i ...x − g ) (cid:17) = α g (1 A ) = 1 A g , and consequently, Y g ∈ Λ θ g (cid:16) η ( x − g ) . . . η ( x − n +1 ...x − g ) (cid:17) = Y g ∈ Λ A g = 1 A . We show that (78)–(80) is exactly the partial n -cocycle identity( δ n w ) ◦ τ gn +1 ( x , . . . , x n +1 ) = 1 η ( x − g ) . . . η ( x − n +1 ...x − g ) . (81)By (34) and (70) one has τ x − gn ( x , . . . , x n +1 ) = ( η g ( x , x ) , . . . , η gn +1 ( x , . . . , x n +1 )) , so the right-hand side of (78) is the first factor of the left-hand side of (81) expandedin accordance with (2). Now, the i -th factor of the product (79) is of the form w ( τ gi − ( x , . . . , x i − ) , η gi ( x , . . . , x i − , x i x i +1 ) , . . . , η gn ( x , . . . , x i x i +1 , . . . , x n +1 )) ( − i , which coincides with the i -th factor of the analogous product of the expansion ofthe left-hand side of (81) thanks to (71) and (72). Finally, (80) is literally the lastfactor of the above mentioned expansion. (cid:3) We proceed now with the construction of e w needed in Theorem 2.4. Given n > x , . . . , x n ∈ G , we define˜ ε ( x , . . . , x n − ) = ε ( x , . . . , x n − ) + 1 A − ( x ,...,x n − ) ∈ U ( A ) , (82)understanding that ˜ ε = ε ∈ U ( A ) if n = 1. Define also e w ( x , . . . , x n ) = (˜ δ n − ˜ ε )( x , . . . , x n ) f w ′ ( x , . . . , x n ) ∈ U ( A ) , (83)where (˜ δ n − ˜ ε )( x , . . . , x n ) = ˜ α x (˜ ε ( x , . . . , x n )) n − Y i =1 ˜ ε ( x , . . . , x i x i +1 , . . . , x n ) ( − i ˜ ε ( x , . . . , x n − ) ( − n , (84)and ˜ α x ( a ) = α x (1 x − a ) + 1 A − x , (85)with x ∈ G and a ∈ A .Our main result is as follows. LOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 23
Theorem 4.3.
Let A be a commutative unital ring which is a (possibly infinite)direct product of indecomposable rings, and let α = { α g : D g − → D g | g ∈ G } bea (non-necessarily transitive) unital partial action of G on A . Then for any n ≥ each partial cocycle w ∈ Z n ( G, A ) is globalizable.Proof. Since the case n = 0 has been explained in Remark 2.3, we assume n >
0. Consider first the transitive case. We will show that our e w defined in (83)satisfies (15) and (16). It directly follows from (39), (73), (82), (84) and (85) that1 ( x ,...,x n ) e w ( x , . . . , x n ) = ( δ n − ε )( x , . . . , x n ) · w ′ ( x , . . . , x n )for all x , . . . , x n ∈ G . By Theorem 3.12 this yields that e w satisfies (16). As to (15),we see that α x (cid:16) x − e w ( x , . . . , x n +1 ) (cid:17) n Y i =1 e w ( x , . . . , x i x i +1 , . . . , x n +1 ) ( − i e w ( x , . . . , x n ) ( − n +1 (86)can be written as product of the following two factors α x (cid:16) x − f w ′ ( x , . . . , x n +1 ) (cid:17) n Y i =1 f w ′ ( x , . . . , x i x i +1 , . . . , x n +1 ) ( − i f w ′ ( x , . . . , x n ) ( − n +1 (87)and α x (cid:16) x − (˜ δ n − ε )( x , . . . , x n +1 ) (cid:17) n Y i =1 (˜ δ n − ε )( x , . . . , x i x i +1 , . . . , x n +1 ) ( − i (˜ δ n − ε )( x , . . . , x n ) ( − n +1 . (88)Thanks to Lemma 4.2 the factor (87) is 1 x , whereas the expansion of (88) has thesame form as the usual δ n ◦ δ n − in homological algebra, with the difference thatinstead of a global action we have a mixture of α with ˜ α . Consequently, all factorsin (88) to which neither α , nor ˜ α is applied, cancel amongst themselves resultingin 1 A . The remaining factors of the expansion are those of α x (cid:16) x − (˜ δ n − ε )( x , . . . , x n +1 ) (cid:17) (89)and the first factors in each(˜ δ n − ε )( x , . . . , x i x i +1 , . . . , x n +1 ) ( − i , ≤ i ≤ n, (90)and in (˜ δ n − ε )( x , . . . , x n ) ( − n +1 . (91)The factors in the expansion of (89) are exactly α x (cid:16) x − ˜ α x (˜ ε ( x , . . . , x n +1 )) (cid:17) , (92) α x (cid:16) x − ˜ ε ( x , . . . , x i x i +1 , . . . , x n +1 ) ( − i − (cid:17) , ≤ i ≤ n, (93)and α x (cid:16) x − ˜ ε ( x , . . . , x n ) ( − n (cid:17) , (94)whereas the first factors in (90) and (91) are˜ α x x (˜ ε ( x , . . . , x n +1 )) − , (95) which comes from the case i = 1 in (90),˜ α x (˜ ε ( x , . . . , x i x i +1 , . . . , x n +1 )) ( − i , ≤ i ≤ n, (96)and ˜ α x (˜ ε ( x , . . . , x n )) . (97)Multiplying the elements in (96) and (97) by 1 x we see that they are canceled withthose in (93) and (94), respectively. Now (92) equals 1 x ˜ α x x (˜ ε ( x , . . . , x n +1 )) dueto the commutative version of (19) from [24], so that it cancels with (95). It followsthat (88) also equals 1 x , and we conclude that e w satisfies (15). It remains to applyTheorem 2.4.If α is not transitive, then we represent A as a product of ideals, on each ofwhich α acts transitively, so that the construction of ˜ w reduces to the transitivecase by means of the projection on such an ideal (see [24, Proposition 8.4]). (cid:3) Uniqueness of a globalization
Our aim is to show that the globalization of w constructed in Section 4 is uniqueup to cohomological equivalence.We would like to use item (iii) of [24, Lemma 8.3], whose proof was not sufficientlywell explained. To clarify it, we need some new terminology. Let R be a ring and R µ ⊆ R , µ ∈ M , a collection of its unital ideals. Observe from the definitionof a direct product that there is a unique homomorphism φ : R → Q µ ∈ M R µ ,such that φ followed by the natural projection Q µ ∈ M R µ → R µ ′ coincides withthe multiplication by 1 R µ ′ in R for any µ ′ ∈ M . In this situation we say that thehomomorphism φ respects projections . Lemma 5.1.
Let C be a non-necessarily unital ring and {C µ | µ ∈ M } a familyof pairwise distinct unital ideals in C . Suppose that I and J are unital ideals in C such that I ∼ = Y µ ∈ M C µ and J ∼ = Y µ ∈ M C µ , (98) where M , M ⊆ M , C µ ⊆ I for all µ ∈ M and C µ ′ ⊆ J for all µ ′ ∈ M . If theisomorphisms (98) respect projections, then there is a (unique) isomorphism I + J ∼ = Y µ ∈ M ∪ M C µ , (99) which also respects projections.Proof. It is readily seen that I + J is a unital ring with unity element 1 I + 1 J − I J and I + J = I ⊕ J ′ , where J ′ = J (1 J − I J ). Therefore, the isomorphism J ∼ = Q µ ∈ M C µ restricts to J ′ ∼ = Y µ ∈ M \ M C µ ⊆ Y µ ∈ M C µ (see (25)). Then I + J = I ⊕ J ′ ∼ = Y µ ∈ M C µ ⊕ Y µ ∈ M \ M C µ ∼ = Y µ ∈ M C µ × Y µ ∈ M \ M C µ , LOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 25 the latter being isomorphic to Q µ ∈ M ⊔ ( M \ M ) C µ , which proves (99). Moreover,the isomorphism can be chosen in such a way that it respects projections, providedthat the isomorphisms (98) have this property. (cid:3) Proposition 5.2.
Let A be a product Q g ∈ Λ A g of indecomposable unital rings, α atransitive unital partial action of G on A and ( β, B ) an enveloping action of ( α, A ) with A ⊆ B . Then B embeds as an ideal into Q g ∈ Λ ′ A g , where Λ ′ was defined beforeformula (26) and A g denotes the ideal β g ( A ) in B . Moreover, M ( B ) ∼ = Q g ∈ Λ ′ A g ,and β ∗ is transitive, when seen as a partial action of G on Q g ∈ Λ ′ A g .Proof. As it was explained before Lemma 5.1, there is a unique homomorphism φ : B → Q g ∈ Λ ′ A g , which respects projections. We shall prove that φ is injective.Since B = P g ∈ G β g ( A ), each element of B belongs to an ideal I of B of the form P ki =1 β x i ( A ), x , . . . , x k ∈ G . Therefore, it suffices to show that the restriction of φ to any such I is injective. Using (ii) of [24, Lemma 8.3], we may construct anisomorphism β x i ( A ) = β x i Y g ∈ Λ A g ∼ = Y g ∈ Λ β x i ( A g ) = Y g ∈ Λ A x i g , which respects projections. Notice that it follows from the definition of Λ ′ thatthe ideals A g , g ∈ Λ ′ , are pairwise distinct. Hence by Lemma 5.1 there is anisomorphism ψ : I → Y g ∈ Λ ′′ A g , (100)where Λ ′′ = { x i g | g ∈ Λ , i = 1 , . . . , k } ⊆ Λ ′ , and it also respects projections. Weclaim that the restriction of φ to I coincides with ψ , if one understands the productin the right-hand side of (100) as an ideal in Q g ∈ Λ ′ A g (see (25)). Indeed, for all g ∈ Λ ′′ and b ∈ I one has pr g ◦ ψ ( b ) = 1 A g b = pr g ◦ φ ( b ) , because φ and ψ respect projections. Now if g ∈ Λ ′ \ Λ ′′ , then x − i g Λ for all i = 1 , . . . , k , since otherwise g = x i x − i g ∈ Λ ′′ . Hence, for all b = P ki =1 β x i ( a i ) ∈ I ( a i ∈ A ) in view of (ii) of [24, Lemma 8.3]pr g ◦ φ ( b ) = 1 A g b = k X i =1 β x i (cid:18) A x − i g a i (cid:19) = k X i =1 β x i (0) = 0 . This proves the claim, and thus injectivity of φ . Moreover, since φ ( I ) = Q g ∈ Λ ′′ A g is an ideal in Q g ∈ Λ ′ A g , it follows that φ ( B ) is also an ideal in Q g ∈ Λ ′ A g .Regarding the second statement of the proposition, notice that each element of Q g ∈ Λ ′ A g acts as a multiplier of B , as φ ( B ) is an ideal in Q g ∈ Λ ′ A g . Conversely,let w ∈ M ( B ). Then w A g = w A g · A g ∈ A g for all g ∈ Λ ′ . Define a ∈ Q g ∈ Λ ′ A g by pr g ( a ) = w A g . We show that φ ( wb ) = aφ ( b ). Indeed, using the fact that φ respects projections, we getpr g ( φ ( wb )) = 1 A g · wb = w A g · A g b = w A g · pr g ( φ ( b )) = pr g ( aφ ( b )) This does not confuse with (26), because α g ( A ) = A g ⊆ A for g ∈ Λ, so β g ( A ) = α g ( A ). for all g ∈ Λ ′ . The transitivity of β ∗ easily follows from the definition of A g for g ∈ Λ ′ . (cid:3) Theorem 5.3.
Let A be a product Q g ∈ Λ A g of indecomposable unital rings, α apartial action of G on A and w i ∈ Z n ( G, A ) , i = 1 , ( n > ). Suppose that ( β, B ) is an enveloping action of ( α, A ) and u i ∈ Z n ( G, U ( M ( B ))) is a globalization of w i , i = 1 , . If w is cohomologous to w , then u is cohomologous to u . In particularany two globalizations of the same partial n -cocycle are cohomologous.Proof. Let α be transitive. Thanks to Proposition 5.2 we may assume, up to anisomorphism, that M ( B ) = Q g ∈ Λ ′ A g ⊇ A . Define u ′ i ( x , . . . , x n ) = Y g ∈ Λ ′ ϑ g ◦ u i ◦ τ gn ( x , . . . , x n ) , i = 1 , , (101)where ϑ g is a homomorphism M ( B ) → M ( B ) given by ϑ g = β g ◦ pr . (102)Since u ′ i has the same construction as w ′ from Section 3 (see (39)), one has byTheorem 3.12 that u ′ i ∈ Z n ( G, U ( M ( B ))) and u i is cohomologous to u ′ i , i = 1 , u ′ is cohomologous to u ′ , provided that w is cohomol-ogous to w . Observe, in view of (14), that for arbitrary h , . . . , h n ∈ H pr ◦ u i ( h , . . . , h n ) = pr (cid:0) u i ( h , . . . , h n )1 ( h ,...,h n ) (cid:1) = pr ◦ w ( h , . . . , h n ) . Together with (101) and (102) this implies that u ′ i ( x , . . . , x n ) = Y g ∈ Λ ′ ϑ g ◦ w i ◦ τ gn ( x , . . . , x n ) , i = 1 , . (103)Let w = w · δ n − ξ for some ξ ∈ C n − ( G, A ). Since ϑ g is a homomorphism, oneimmediately sees from (103) that u ′ = u ′ ( δ n − ξ ) ′ , where( δ n − ξ ) ′ ( x , . . . , x n ) = Y g ∈ Λ ′ ϑ g ◦ ( δ n − ξ ) ◦ τ gn ( x , . . . , x n ) . We shall show that ( δ n − ξ ) ′ = δ n − ξ ′ (104)with ξ ′ ( x , . . . , x n − ) = Y g ∈ Λ ′ ϑ g ◦ ξ ◦ τ gn − ( x , . . . , x n − ) . Taking into account the fact that ϑ g is a homomorphism once again and inter-changing the left-hand side and the right-hand side of (104), we may reduce (104)to β x Y g ∈ Λ ′ ϑ g ◦ ξ ◦ τ gn − ( x , . . . , x n ) = Y g ∈ Λ ′ ϑ g ◦ β η g ( x ) ◦ ξ ( η g ( x , x ) , . . . , η gn ( x , . . . , x n )) , (105)whose right-hand side is Y g ∈ Λ ′ ϑ g ◦ β η g ( x ) ◦ ξ ◦ τ x − gn − ( x , . . . , x n ) LOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 27 by (70). Now it is readily seen that (105) follows from the global case of Lemma 3.10(with α and θ replaced by β ∗ and ϑ , respectively).The non-transitive case reduces to the transitive one, using the same argumentas in Theorem 4.3. (cid:3) Corollary 5.4.
Let A be a product Q g ∈ Λ A g of indecomposable unital rings, α apartial action of G on A and ( β, B ) an enveloping action of ( α, A ) . Then the partialcohomology group H n ( G, A ) is isomorphic to the classical (global) cohomology group H n ( G, U ( M ( B ))) . Indeed, when n >
0, it follows from Theorems 4.3 and 5.3 that there is a well-definedinjective map H n ( G, A ) → H n ( G, U ( M ( B ))). The constructions of f w ′ and u clearlyrespect products (see (8) and (73)), so the map is a monomorphism of groups. Itis evidently surjective, as any u ∈ Z n ( G, U ( M ( B ))) restricts to w ∈ Z n ( G, A ) bymeans of (14), and a globalization of w is cohomologous to u thanks to Theorem 5.3.For the case n = 0 (which holds in a more general situation) see Remark 2.3. Acknowledgments
The first two authors would like to express their sincere gratitude to the Depart-ment of Mathematics of the University of Murcia for its warm hospitality duringtheir visits.
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Insituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Rua do Mat˜ao,1010, S˜ao Paulo, SP, CEP: 05508–090, Brazil
E-mail address : [email protected] Departamento de Matem´atica, Universidade Federal de Santa Catarina, Campus Re-itor Jo˜ao David Ferreira Lima, Florian´opolis, SC, CEP: 88040–900, Brazil
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