Globalization of partial cohomology of groups

arXiv:1706.02546v3 [math.RA] 7 Jul 2020 GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIMÓN Abstract. We study the relations between partial and global group cohomology with values in a commutative unital ring A. In particular, for a unital partial action of a group G on A, such that A is a direct product of commutative indecomposable rings, we show that any partial n-cocycle of G with values in A is globalizable. Introduction Given a partial action it is natural to ask whether there exists a global action which restricts to the partial one. This question was first considered in the PhD Thesis [1] (see also [2]) and independently in [46] and [40] for partial group actions, with subsequent developments in [3, 12, 17, 18, 21, 22, 24, 34, 35, 41, 45]. More generally the problem was investigated for partial semigroup actions in [38, 39, 42, 44], for partial groupoid actions in [10, 11, 37] and in the context of partial (weak) Hopf (co)actions in [5, 6, 7, 8, 14, 15, 16]. Globalization results help one to use known facts on global actions in the studies involving partial ones. Thus the first purely ring theoretic globalization fact [22, Theorem 4.5] stimulated intensive algebraic activity, permitting, in particular, to develop a Galois Theory of commutative rings [25]. The latter, in its turn, inspired the 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 starting point for interesting Hopf theoretic developments. Moreover, globalizable partial actions are more manageable, so that the great majority of ring theoretic studies on the subject deal with the globalizable case. Among the recent applications of globalization facts we mention their remarkable use to paradoxical decompositions in [9] and to restriction semigroups in [42]. The reader is referred to the surveys [19, 20, 36] and to the recent book by R. Exel [33] for more information about partial actions and their applications. In [32] R. Exel introduced the general concept of a continuous twisted partial action of a locally compact group on a C ∗ -algebra and proved that any second countable C ∗ -algebraic bundle, which is regular in a certain sense, is isomorphic to the C ∗ -algebraic bundle constructed from a twisted partial group action. The purely algebraic version of this result was obtained in [23]. The concept involves a twisting which satisfies a kind of 2-cocycle equality needed for associativity purposes. Thus, it was natural to work out a cohomology theory, encompassing such twistings, and this was done in [26]. The partial cohomology from [26] is strongly related to 2010 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), FAPESP of Brazil (Proc. 2012/01554-7, 2015/09162-9), MINECO (MTM2016-77445-P) and Fundación Séneca of Spain. 1 2 MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIMÓN H. Lausch’s cohomology of inverse semigroups [43] and nicely fits the theory of partial projective group representations developed in [29], [30] and [31]. 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 global action β of G on a ring B such that A can P be identified with a two-sided ideal in B, α is the restriction of β to A and B = g∈G βg (A). Moreover, if B has an identity element, then any two globalizations of α are equivalent in a natural sense. If A is commutative, 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 structure on B) and a usual 2-cocycle of G with values in the group of units of the multiplier algebra of B. The above mentioned results from [24] mean in this context that given 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 (see Definition 2.2). Moreover, if B has an identity element, then any two globalizations of w are cohomologous. The purpose of the present article is to extend the results from [24] in the commutative case to arbitrary n-cocycles. The technical difficulties coming from [24] are being overcome by improvements and notation. In Section 1 we recall some notions needed in the sequel. The main result of Section 2 is Theorem 2.5, in which we prove that given a unital partial G-module structure on a commutative ring A, a partial n-cocycle w with values in A is globalizable if and only if w can be extended to an n-cochain w̃ of G with values in the unit group U(A) which satisfies 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 concentrated in Section 3, in which we assume that A is a product of blocks, and this assumption is maintained for the rest of the paper. Our goal is to construct a more manageable partial n-cocycle w′ which is cohomologous to w (see Theorem 3.13). In Section 4 we prove our main existence result Theorem 4.3. The defining formula for w′ perf′ : Gn → U(A) which satisfies our mits us to extend easily w′ to an n-cochain w f′ by a “co-boundary “more global” n-cocycle identity (see Lemma 4.2). Modifying w looking” function we define in (83) a function w̃ : Gn → U(A) and show that w̃ is a desired extension of w fitting Theorem 2.5, and permitting us to conclude that w is globalizable. The uniqueness of a globalization is treated in 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 commutative ring A, which is a product of blocks, and a partial n-cocycle w related to α, any two globalizations of w are cohomologous. More generally, arbitrary globalizations of cohomologous partial n-cocycles are also cohomologous. This results in Corollary 5.4 which establishes an isomorphism between the partial cohomology group H n (G, A) and the global one H n (G, U(M(B))), where U(M(B)) stands for the unit group of the multiplier ring M(B) of B. Section 6 serves as a demonstration of our technique. In Example 6.1 we give an explicit construction of a globalization of an arbitrary partial 2-cocycle associated with a “shift” partial action of a GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 3 group of order 3 on the direct product of 2 copies of a commutative unital ring. In Remark 6.2 we also show (independently from the result of Example 6.1) that the corresponding partial and global 2-cohomology groups are isomorphic. 1. Background on globalization and cohomology of partial actions In all what follows G will stand for an arbitrary group whose identity element will be denoted by 1, and by a ring we shall mean an associative ring, which is not unital in general. Nevertheless, our main attention will be paid to partial actions on commutative and unital rings. In this section we recall a couple of concepts around partial actions. Definition 1.1 (see [22]). Let A be a ring. A partial action α of G on A is a collection of two-sided ideals Dg ⊆ A (g ∈ G) and ring isomorphisms αg : Dg−1 → Dg such that (i) D1 = A and α1 is the identity automorphism of A; (ii) for all g, h ∈ G: αg (Dg−1 ∩ Dh ) = Dg ∩ Dgh ; (iii) for all g, h ∈ G and a ∈ Dh−1 ∩ Dh−1 g−1 : αg ◦ αh (a) = αgh (a). An equivalent form to state (i)–(iii) is as follows: (i) α1 = idA ; (iv) for all g, h ∈ G and a ∈ A: if αh (a) and αg ◦ αh (a) are defined, then αgh (a) is defined and αg ◦ αh (a) = αgh (a). Partial actions can be obtained as restrictions of global ones, i.e. those satisfying Dg = A for all g ∈ G, as follows. Let β be a global action of G on a ring B and A a two-sided ideal in B. Then setting Dg = A ∩ βg (A) and denoting by αg the restriction of βg to Dg−1 for all g ∈ G, we readily see that α = {αg : Dg−1 → Dg | g ∈ G} is a partial action of G on A, called the P restriction of β to A, and α is said to be an admissible restriction of β if B = g∈G βg (A). Clearly, P P if B = 6 g∈G βg (A), the partial action α can be g∈G βg (A), then replacing B by viewed as an admissible restriction. Partial actions isomorphic to restrictions of global ones are called globalizable. The notion of an isomorphism of partial action is defined as follows. Definition 1.2 (see p. 17 from [2] and Definition 4 from [30]). Let A and A′ be rings and α = {αg : Dg−1 → Dg | g ∈ G}, α′ = {α′g : Dg′ −1 → Dg′ | g ∈ G} be partial actions of G on A and A′ , respectively. A morphism (A, α) → (A′ , α′ ) of partial actions is a ring homomorphism ϕ : A → A′ such that for any g ∈ G and a ∈ Dg−1 the next two conditions are satisfied: (i) ϕ(Dg ) ⊆ Dg′ ; (ii) ϕ(αg (a)) = α′g (ϕ(a)). We say that a morphism ϕ : (A, α) → (A′ , α′ ) of partial actions is an isomorphism1 if ϕ : A → A′ is an isomorphism of rings and ϕ(Dg ) = Dg′ for each g ∈ G. By [22, Theorem 4.5] a partial action α on a unital ring A is globalizable exactly when each ideal Dg is a unital ring, i.e. Dg is generated by an idempotent which is central in A, and which will be denoted by 1g . In order to guarantee the uniqueness of a globalization one considers the following. 1This was called equivalence in [22, Definition 4.1]. 4 MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIMÓN Definition 1.3 (Definition 4.2 from [22]). A global action β of G on a ring B is said to be an enveloping action for the partial action α of G on a ring A if α is isomorphic to an admissible restriction of β. By the above mentioned [22, Theorem 4.5], an enveloping action β for a globalizable partial action of G on a unital ring A is unique up to an isomorphism. Denote by F = F (G, A) the ring of functions from G to A, i.e. F is the Cartesian product of copies of A indexed by the elements of G. Note that by the proof of [22, Theorem 4.5], the ring under the global action is a subring B of F , and consequently B is commutative if and only if A is. Every ring is a semigroup with respect to multiplication, and if in Definition 1.1 we assume that A is a (multiplicative) semigroup and the maps αg are isomorphisms of semigroups satisfying (i)–(iii), then we obtain the concept of a partial action of G on a semigroup (see [29]). Furthermore, the concept of a morphism of partial actions on semigroups is obtained from Definition 1.2 by assuming that ϕ : A → A′ is a homomorphism of semigroups satisfying (i) and (ii). Partial cohomology was defined in [26] as follows. Let α = {αg : Dg−1 → Dg | g ∈ G} be a partial action of G on a commutative monoid A. Assume that each ideal Dg is unital, i.e. Dg is generated by an idempotent 1g = 1A g . In this case we shall say that α is a unital partial action. Then Dg ∩ Dh = Dg Dh , for all g, h ∈ G, so the properties (ii) and (iii) from Definition 1.1 can be replaced by (ii’) αg (Dg−1 Dh ) = Dg Dgh ; (iii’) αg ◦ αh = αgh on Dh−1 Dh−1 g−1 . Note also that (iii’) implies a more general equality αx (Dx−1 Dy1 . . . Dyn ) = Dx Dxy1 . . . Dxyn , (1) for any x, y1 , . . . , yn ∈ G, which easily follows by observing that Dx−1 Dy1 . . . Dyn = Dx−1 Dy1 . . . Dx−1 Dyn . 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 that its restriction to each Dg is a homomorphism of monoids Dg → Dg′ , g ∈ G. For simplicity, we shall often omit α from the pair (A, α), if no confusion arises. Definition 1.5 (see [26]). Let A be a partial G-module and n a positive integer. An n-cochain of G with values in A is a function f : Gn → A, such that f (x1 , . . . , xn ) is an invertible element of the ideal D(x1 ,...,xn ) = Dx1 Dx1 x2 . . . Dx1 ...xn for any (x1 , . . . , xn ) ∈ Gn . By a 0-cochain we shall mean an invertible element of A, i.e. a ∈ U(A), where U(A) stands for the group of invertible elements of A. Denote the set of n-cochains by C n (G, A). It is an abelian group under the pointwise multiplication. Indeed, its identity is en which is the n-cochain defined by en (x1 , . . . , xn ) = 1(x1 ,...,xn ) := 1x1 1x1 x2 . . . 1x1 ...xn , and the inverse of f ∈ C n (G, A) is f −1 (x1 , . . . , xn ) = f (x1 , . . . , xn )−1 , where f (x1 , . . . , xn )−1 means the inverse of f (x1 , . . . , xn ) in D(x1 ,...,xn ) . The multiplicative form of the classical coboundary homomorphism now can be adapted to our context by replacing the global action by a partial one, and taking inverse elements in the corresponding ideals, as follows. GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 5 Definition 1.6 (see [26]). Let (A, α) be a partial G-module and n a positive integer. For any f ∈ C n (G, A) and x1 , . . . , xn+1 ∈ G define (δ n f )(x1 , . . . , xn+1 ) = αx1 (1x−1 f (x2 , . . . , xn+1 )) 1 · n Y f (x1 , . . . , xi xi+1 , . . . , xn+1 )(−1) i i=1 · f (x1 , . . . , xn )(−1) n+1 . (2) If n = 0 and a is an invertible element of A, we set (δ 0 a)(x) = αx (1x−1 a)a−1 . (3) According to [26, Proposition 1.5] the coboundary map δ n is a homomorphism C (G, A) → C n+1 (G, A) of abelian groups, such that n δ n+1 δ n f = en+2 (4) for any f ∈ C n (G, A). As in the classical case one defines the abelian groups of partial 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−1 and H n (G, A) = ker δ n / im δ n−1 , n ≥ 1 (H 0 (G, A) = Z 0 (G, A) = ker δ 0 ). Then two partial n-cocycles which represent the same element of H n (G, A) are called cohomologous. Taking n = 0, we see that H 0 (G, A) = Z 0 (G, A) = {a ∈ U(A) | ∀x ∈ G : αx (1x−1 a) = 1x a}, B 1 (G, A) = {f ∈ C 1 (G, A) | ∃a ∈ U(A) ∀x ∈ G : f (x) = αx (1x−1 a)a−1 }. Notice that H 0 (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 twisted partial actions, consider the cases n = 1 and n = 2. In the first case we have (δ 1 f )(x, y) = αx (1x−1 f (y))f (xy)−1 f (x) with f ∈ C 1 (G, A), so that Z 1 (G, A) = {f ∈ C 1 (G, A) | ∀x, y ∈ G : αx (1x−1 f (y))f (x) = 1x f (xy)}, B 2 (G, A) = {f ∈ C 2 (G, A) | ∃g ∈ C 1 (G, A) ∀x, y ∈ G : f (x, y) = αx (1x−1 g(y))g(xy)−1 g(x)}, and for n = 2 (δ 2 f )(x, y, z) = αx (1x−1 f (y, z))f (xy, z)−1 f (x, yz)f (x, y)−1 , with f ∈ C 2 (G, A), and Z 2 (G, A) = {f ∈ C 2 (G, A) | ∀x, y, z ∈ G : αx (1x−1 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 commutative ring A splits into two parts: a unital partial action of G on A, and a twisting which, 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, and for this purpose we remind the reader that the multiplier ring of an associative not necessarily unital ring A is the set M(A) = {(R, L) ∈ End (A A) × End (AA ) : (aR)b = a(Lb) for all a, b ∈ A} 6 MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIMÓN 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 ∈ End (A A), L ∈ End (AA ) and a ∈ A, we write a 7→ aR and a 7→ La. For a 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 = (e2 )u = e(eu) = (eu)e = e(ue) = (ue)e = ue. Any a ∈ A determines a multiplier ua by setting ua b = ab and bua = ba, b ∈ A, so that a 7→ ua gives the canonical homomorphism A → M(A), which is an isomorphism if A has 1A (in this case the inverse isomorphism is given by M(A) ∋ u 7→ u1A = 1A u ∈ A). According to [22] a ring A is said to be nondegenerate if the canonical map A → M(A) is injective. This is guaranteed if A is left (or right) s-unital, i.e. for any a ∈ A one has a ∈ Aa (respectively, a ∈ aA). Furthermore, given a ring isomorphism φ : A → A′ , the map M(A) ∋ u 7→ φuφ−1 ∈ M(A′ ), where φuφ−1 = (φ−1 Rφ, φLφ−1 ), u = (R, L), is an isomorphism of rings. In particular, an automorphism φ of A gives rise to an automorphism u 7→ φuφ−1 (6) of M(A). We shall also use the following. Remark 1.7 (see Remark 5.2 from [27] and Lemma 3.1 from [28]). If A is a commutative idempotent ring, then M(A) is also commutative and for each w ∈ M(A) and a ∈ A one has aw = wa. 2. The notion of a globalization of a partial cocycle and its relation with an extendibility property In this section we introduce the concept of a globalization of a partial n-cocycle with values in a commutative unital ring A and show that a partial n-cocycle w is globalizable, provided that an extendibility property for w holds. We start with a general 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 βx : F → F by βx (f )|t = f (x−1 t), (7) where x, t ∈ G. Then β is a global action of G on F which was used in [22] to deal with the globalization problem for partial actions on unital rings. Let w e : Gn → U(A) be a function, i.e. w e 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 : Gn → U(F ) by n u(x1 , . . . , xn )|t = w(t e −1 , x1 , . . . , xn−1 )(−1) w(t e −1 x1 , x2 , . . . , xn ) · n−1 Y i=1 i w(t e −1 , x1 , . . . , xi xi+1 , . . . , xn )(−1) , n > 0. (8) GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 7 We proceed with a technical fact which will be used in the main result of this section. 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 n Y i n+1 u(x1 , . . . , xi xi+1 , . . . , xn+1 )(−1) u(x1 , . . . , xn )(−1) (9) βx1 (u(x2 , . . . , xn+1 )) i=1 is the identity, i.e. it equals 1F for any x1 , . . . , xn+1 ∈ G. Evaluating (9) at t and using (7), we get n Y i n+1 u(x1 , . . . , xi xi+1 , . . . , xn+1 )(−1) |t u(x1 , . . . , xn )(−1) |t . u(x2 , . . . , xn+1 )|x−1 t 1 i=1 (10) Denote by δ̃ n : C n (G, U(A)) → C n+1 (G, U(A)) the coboundary operator which corresponds to the trivial G-module, i.e. (δ̃ n w)(x e 1 , . . . , xn+1 ) = w(x e 2 , . . . , xn+1 ) · n Y i=1 w(x e 1 , . . . , xi xi+1 , . . . , xn+1 )(−1) · w(x e 1 , . . . , xn )(−1) We see from (8) that n+1 i . (11) u(x1 , . . . , xn )|t = w(x e 1 , . . . , xn )(δ̃ n w)(t e −1 , x1 , . . . , xn )−1 . Therefore, (10) becomes w(x e 2 , . . . , xn+1 )(δ̃ n w)(t e −1 x1 , x2 , . . . , xn+1 )−1 · n Y i=1 (−1)i w(x e 1 , . . . , xi xi+1 , . . . , xn+1 ) · w(x e 1 , . . . , xn )(−1) n+1 n Y i=1 (δ̃ n w)(t e −1 , x1 , . . . , xi xi+1 , . . . , xn+1 )(−1) i+1 n (δ̃ n w)(t e −1 , x1 , . . . , xn )(−1) . Regrouping the factors and using (11), we obtain (δ̃ n w)(x e 1 , . . . , xn+1 )(δ̃ n w)(t e −1 x1 , x2 , . . . , xn+1 )−1 · n Y i=1 (δ̃ n w)(t e −1 , x1 , . . . , xi xi+1 , . . . , xn+1 )(−1) e −1 , x1 , . . . , xn ) = 1A . which is (δ̃ n+1 δ̃ n w)(t i+1 n (δ̃ n w)(t e −1 , x1 , . . . , xn )(−1) ,  Let now α be a unital partial action of G on A. Then ϕ(a)|t = αt−1 (1t a), (12) where t ∈ G and a ∈ A, defines an embedding of A into F , and (β, B) is an P β enveloping action for (α, A), where B = g∈G g (ϕ(A)) (see the proof of [22, Theorem 4.5]). Since (β, B) is unique up to an isomorphism, it follows by [21, Theorem 3.1] that B is left s-unital. Hence there is a canonical embedding of B into the multiplier ring M(B) and, moreover, B is commutative because A is. In 8 MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIMÓN addition, B is idempotent because B is left s-unital, which implies that M(B) is commutative thanks to Remark 1.7. Observe that 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 −1 βg∗ (u) = βg uβ g−1 = (β −1 g Rβg , βg Lβ g ), (13) where u = (R, L) ∈ M(B) and g ∈ G. Moreover, the commutativity of M(B) permits us to consider the group of units U(M(B)) as a G-module via β ∗ . Definition 2.2. Let α = {αg : Dg−1 → Dg | 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 admissible restriction 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(x1 , . . . , xn )) = ϕ(1(x1 ,...,xn ) )u(x1 , . . . , xn ), (14) for any x1 , . . . , xn ∈ G. If n = 0, then by 1(x1 ,...,xn ) we mean 1A in (14). Observe from (5) that (14) implies ϕ(w(x1 , . . . , xn )) = u(x1 , . . . , xn )ϕ(1(x1 ,...,xn ) ). It is readily seen that if B contains 1B , then the isomorphism M(B) ∼ = B transforms β ∗ into β, and the globalization u is an n-cocycle with values in U(B). Proposition 2.3. Observe that any partial 0-cocycle w is globalizable, and its globalization is the constant function u ∈ U(F ) with u|t = w for all t ∈ G. Moreover, such u is unique. Proof. Indeed, (14) reduces to ϕ(w) = ϕ(1A )u, which is the partial 0-cocycle identity for w by (12). Moreover, u is an (invertible) multiplier of B, as βg (ϕ(a))|t u|t = ϕ(a)|g−1 t w = αt−1 g (1g−1 t a)w = αt−1 g (1g−1 t a) · 1t−1 g w = αt−1 g (1g−1 t aw) = βg (ϕ(aw))|t thanks to (7) and (12) and the 0-cocycle identity for w. Applying βx uβ −1 x to an arbitrary βy (ϕ(a)) ∈ B and evaluating the result at any t ∈ G, we obtain by (7) (βx uβ −1 x )(βy (ϕ(a)))|t = βx uβx−1 y (ϕ(a))|t = (uβx−1 y (ϕ(a)))|x−1 t = u|x−1 t βx−1 y (ϕ(a))|x−1 t = wϕ(a)|y−1 t = wβy (ϕ(a))|t = u|t βy (ϕ(a))|t , so that βx uβ −1 x coincides with u as a multiplier on B, i.e. u is a (global) 0-cocycle with respect to the action β ∗ of G on M(B). Now if the restrictions of u1 , u2 ∈ H 0 (G, U(M(B))) to the ideal ϕ(A) coincide, then ϕ(1A )u1 = ϕ(1A )u2 . Applying βx to this equality and using the 0-cocycle identity for ui which means that βx ui β −1 x = ui , i = 1, 2, one gets βx (ϕ(1A ))u1 = βx (ϕ(1A ))u2 for all x ∈ G. Consequently, βP x (ϕ(a))u1 = βx (ϕ(a))u2 for all x ∈ G and a ∈ A. It follows that u1 = u2 , as B = g∈G βg (A). In particular, this holds for any two globalizations of the same w ∈ H 0 (G, A).  Corollary 2.4. We have H 0 (G, A) ∼ = H 0 (G, U(M(B))). GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 9 Proof. By Proposition 2.3 there is an injective map from H 0 (G, A) to H 0 (G, U(M(B))) sending w ∈ H 0 (G, A) to its globalization u ∈ H 0 (G, U(M(B))), which is readily seen to be a group homomorphism. It follows from the uniqueness of the globalization that this map is also surjective, since any u ∈ H 0 (G, U(M(B))) is the globalization of its restriction to ϕ(A).  Given an arbitrary n > 0, as in the case n = 2 (see [24, Theorem 4.1]), we are able to reduce the globalization problem for partial n-cocycles to an extendibility property. Theorem 2.5. Let α = {αg : Dg−1 → Dg | 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 if there exists a function w e : Gn → U(A) which satisfies the equalities n  Y i w(x e 1 , . . . , xi xi+1 , . . . , xn+1 )(−1) e 2 , . . . , xn+1 ) αx1 1x−1 w(x 1 i=1 · w(x e 1 , . . . , xn )(−1) and n+1 = 1x1 , (15) w(x1 , . . . , xn ) = 1(x1 ,...,xn ) w(x e 1 , . . . , xn ), (16) ϕ(w(x e 1 , . . . , xn )) = ϕ(1A )u(x1 , . . . , xn ) = u(x1 , . . . , xn )ϕ(1A ). (17) ϕ(1g ) = βg (ϕ(1A ))ϕ(1A ), (18) for all x1 , . . . , xn+1 ∈ G. Proof. We shall assume that n > 0, as n = 0 was considered in Proposition 2.3. Suppose that w ∈ Z n (G, A) is globalizable. Denote by (β, B) an enveloping action 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 w(x e 1 , . . . , xn ) ∈ U(A) by Evidently, w(x e 1 , . . . , xn ) ∈ U(A), as u(x1 , . . . , xn ) is an invertible multiplier, and ϕ(w(x e 1 , . . . , xn )−1 ) = ϕ(1A )u−1 (x1 , . . . , xn ) = u−1 (x1 , . . . , xn )ϕ(1A ). Then (16) clearly holds by (14), and for (15) notice first that and consequently (and in fact more generally), ϕ(αg (1g−1 a)) = βg (ϕ(a))ϕ(1A ), (19) for all g ∈ G and a ∈ A (see [25, p. 79]). The (global) n-cocycle identity for u is of the form n Y i u(x1 , . . . , xi xi+1 , . . . , xn+1 )(−1) βx∗1 (u(x2 , . . . , xn+1 )) i=1 · u(x1 , . . . , xn )(−1) n+1 = 1M(B) . (20) Applying the first multiplier in (20) to ϕ(1x1 ) and using (13), (17) and (19), we obtain βx∗1 (u(x2 , . . . , xn+1 ))ϕ(1x1 ) = (βx1 u(x2 , . . . , xn+1 )β −1 x1 )(βx1 (ϕ(1A ))ϕ(1A )) = (βx1 (u(x2 , . . . , xn+1 )ϕ(1A )))ϕ(1A ) e 2 , . . . , xn+1 ))])ϕ(1A ) = (βx1 [ϕ(w(x 10 MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIMÓN e 2 , . . . , xn+1 ))). = ϕ(αx1 (1x−1 w(x 1 Then applying both sides of (20) to ϕ(1x1 ) and using axioms of a multiplier, we readily see that (15) is a consequence of (20). Suppose now that there exists w e : Gn → 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 as described above. In particular, it follows from (1) that for arbitrary t, x1 , . . . , xn : ϕ(1(x1 ,...,xn ) )|t = 1(t−1 ,x1 ,...,xn ) . (21) Taking our w, e define u : Gn → U(F ) by formula (8). We are going to show that u is a globalization of w. By Lemma 2.1 one has u ∈ Z n (G, U(F )). We now check (14). By (12) ϕ(w(x1 , . . . , xn ))|t = αt−1 (1t w(x1 , . . . , xn )), which by the partial n-cocycle identity for w equals w(t−1 x1 , x2 , . . . , xn ) n−1 Y w(t−1 , x1 , . . . , xi xi+1 , . . . , xn )(−1) i i=1 n · w(t−1 , x1 , . . . , xn−1 )(−1) . In view of (8), (16) and (21) the latter is 1(t−1 ,x1 ,...,xn ) u(x1 , . . . , xn )|t = ϕ(1(x1 ,...,xn ) )|t u(x1 , . . . , xn )|t , for arbitrary t, x1 , . . . , xn ∈ G, proving (14). We proceed with a proof that u(x1 , . . . , xn ) and u(x1 , . . . , xn )−1 are multipliers of B. Notice first that using (15) for (t−1 , x1 , . . . , xn ) we obtain from (8) that αt−1 (1t w(x e 1 , . . . , xn )) = 1t−1 w(t e −1 x1 , x2 , . . . , xn ) · n−1 Y i=1 w(t e −1 , x1 , . . . , xi xi+1 , . . . , xn )(−1) · w(t e −1 , x1 , . . . , xn−1 )(−1) i n = 1t−1 u(x1 , . . . , xn )|t . Then by (12) so that (22) u(x1 , . . . , xn )|t ϕ(a)|t = αt−1 (1t w(x e 1 , . . . , xn ))αt−1 (1t a), u(x1 , . . . , xn )ϕ(a) = ϕ(aw(x e 1 , . . . , xn )), (23) for all x1 , . . . , xn ∈ G and a ∈ A. Equalities (8) and (23) readily imply u(x1 , . . . , xn )−1 ϕ(a) = ϕ(aw(x e 1 , . . . , xn )−1 ). (24) Furthermore, applying the n-cocycle identity for u to (t−1 , x1 , . . . xn ) we see that βt−1 (u(x1 , . . . , xn ))ϕ(a) = u(t−1 x1 , x2 , . . . , xn ) · n−1 Y u(t−1 , x1 , . . . , xi xi+1 , . . . , xn+1 )(−1) i=1 −1 · u(t n , x1 , . . . , xn−1 )(−1) ϕ(a), i GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 11 which belongs to ϕ(A) thanks to (23) and (24). Thus βt−1 (u(x1 , .P . . , xn ))ϕ(A) ⊆ ϕ(A), which yields u(x1 , . . . , xn )βt (ϕ(A)) ⊆ βt (ϕ(A)). Since B = t∈G βt (ϕ(A)), it follows that u(x1 , . . . , xn )B ⊆ B, and hence u(x1 , . . . , xn ) ∈ M(B). Similarly, u(x1 , . . . , xn )−1 ∈ M(B), as desired. It remains to see that u ∈ Z n (G, U(M(B))). Observe that M(F ) ∼ = F and ∗ β coincides with β up to this isomorphism, so the n-cocycle identity for u as an element of C n (G, U(M(B))) reduces to the n-cocycle identity for u as an element of C n (G, U(F )).  Note that taking t = 1 in (22) we obtain u(x1 , . . . , xn )|1 = w(x e 1 , . . . , xn ). 3. From w to w′ Our next purpose is to show that w e in Theorem 2.5 exists, provided that A is a product of blocks, and we need first some technical preparation for this fact, which we do in the present section. Q Suppose that A = λ∈Λ Aλ , where Aλ is an indecomposable unital ring, called a block. So far, we do not assume that A is commutative. We identify the unity element of Aµ , µ ∈ Λ,S with the centrally primitive idempotent 1Aµ of A which is the function Λ → λ∈Λ Aλ whose value at µ is the identity of Aµ and the value at any λ 6= µ is the zero of Aλ . Then Aµ is identified with the ideal of A generated by the idempotent 1Aµ . Denote by prµ the projection of A onto Aµ , namely, prµ (a) = 1Aµ a. Thus, any aQ∈ A is identified with the set of its projections {prλ (a)}λ∈Λ , and we write a = λ∈Λ prλ (a) in this situation. If there existsQΛ1 ⊆ Λ, such that prλ (a) = 0A for all λ ∈ Λ \ Λ1 , then we shall also write a = λ∈Λ1 prλ (a), and such elements a form an ideal in A which we denote by Q λ∈Λ1 Aλ . Since Aλ is indecomposable, the only central idempotents of Aλ are 0A and 1Aλ . Hence, for any central idempotent e of A the projection prλ (e) is either 0A , or 1Aλ . In particular, Y Aλ , (25) eA = λ∈Λ1 where Λ1 = {λ ∈ Λ | prλ (e) = 1Aλ }. Thus, the unital ideals of A are exactly the products of blocks Aλ over all Λ1 ⊆ Λ. Q Q Lemma 3.1. Let I = λ∈Λ1 Aλ and J = λ∈Λ2 Aλ be unital ideals of A and ϕ : I → J an isomorphism. Then there exists a bijection σ : Λ1 → Λ2 , such that ϕ(prλ (a)) = prσ(λ) (ϕ(a)) for all a ∈ I and λ ∈ Λ1 . Proof. Note that {1Aλ }λ∈Λ1 and {1Aλ }λ∈Λ2 are the sets of centrally primitive idempotents of I and J, respectively. Since ϕ is an isomorphism, ϕ(1Aλ ) = 1Aσ(λ) for some bijection σ : Λ1 → Λ2 . Then ϕ(prλ (a)) = ϕ(1Aλ a) = 1Aσ(λ) ϕ(a) = prσ(λ) (ϕ(a)).  Let α = {αx : Dx−1 → Dx | x ∈ G} be a unital partial action of G on A. By the observation above each ideal Dx is a product of blocks, and αx maps a block of Dx−1 onto some block of Dx . As in [24] we call α transitive, when for any pair λ′ , λ′′ ∈ Λ there exists x ∈ G, such that Aλ′ ⊆ Dx−1 and αx (Aλ′ ) = Aλ′′ ⊆ Dx . 12 MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIMÓN In all what follows, if otherwise is not stated, we assume that α is transitive. Then we may fix λ0 ∈ Λ, so that each Aλ is αx (Aλ0 ) for some x ∈ G with Aλ0 ⊆ Dx−1 . Observe that, whenever Aλ0 ⊆ Dx−1 Dx′ −1 and αx (Aλ0 ) = αx′ (Aλ0 ), it follows that Aλ0 ⊆ D(x′ )−1 x and αx−1 x′ (Aλ0 ) = Aλ0 . Hence, introducing as in [24] the subgroup H = {x ∈ G | Aλ0 ⊆ Dx−1 and αx (Aλ0 ) = Aλ0 } and choosing a left transversal Λ′ of H in G, one may identify Λ with a subset of Λ′ , namely, λ ∈ Λ corresponds to (a unique) g ∈ Λ′ , such that Aλ0 ⊆ Dg−1 and αg (Aλ0 ) = Aλ . Assume, moreover, that Λ′ contains the identity element 1 of G. Then λ0 is identified with 1 and thus Ag = αg (A1 ) for g ∈ Λ ⊆ Λ′ . (26) Given x ∈ G, we use the notation x̄ from [24] for the element of Λ′ such that x ∈ x̄H. 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 ⇔ A1 ⊆ Dg−1 ; (ii) if g ∈ Λ, then xg ∈ Λ ⇔ Ag ⊆ Dx−1 , and in this situation αx (Ag ) = Axg . Notice that taking g = 1 in (ii), one gets x ∈ Λ ⇔ A1 ⊆ Dx−1 . Then using (ii) once again, we see that for any g ∈ Λ Ag ⊆ Dx ⇔ x−1 g ∈ Λ ⇔ A1 ⊆ Dg−1 x . (27) In particular, Ax ⊆ Dx for all x ∈ G, such that x ∈ Λ. For any g ∈ Λ and a ∈ A define θg (a) = αg (pr1 (a)). (28) Note that by (i) of Lemma 3.2 the block A1 is a subset of Dg−1 , so αg (pr1 (a)) makes sense and belongs to Ag . Thus, θg is a correctly defined homomorphism2 A → Ag . Clearly, θg (a) = θg (1x a) (29) for any x ∈ G, such that A1 ⊆ Dx . In particular, this holds for x ∈ H and for x = g −1 . Observe also that θg (a) = θg (1g−1 a) = prg (αg (1g−1 a)) (30) in view of Lemma 3.1. It follows that θg (αg−1 (1g a)) = prg (1g a) = prg (a), as Ag ⊆ Dg . Therefore, Y a= θg (αg−1 (1g a)). (31) g∈Λ In what follows in this section, we assume A to be commutative, so that (A, α) is a partial G-module. 2Observe that this θ differs from the one introduced in [24]. More precisely, denoting θ from [24] by θ ′ , we may write θg′ (a) = θg−1 (a) + 1A − 1A g−1 for g −1 ∈ Λ. GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 13 Lemma 3.3. Let n > 0 and w ∈ Z n (G, A). Then Y w(x1 , . . . , xn ) = θg [w(g −1 x1 , x2 , . . . , xn ) g∈Λ · n−1 Y w(g −1 , x1 , . . . , xk xk+1 , . . . , xn )(−1) k k=1 n · w(g −1 , x1 , . . . , xn−1 )(−1) ]. Proof. By (31) w(x1 , . . . , xn ) = Y (32) θg (αg−1 (1g w(x1 , . . . , xn ))). g∈Λ As w ∈ Z n (G, A), one has 1(g−1 ,x1 ,...,xn ) = (δ n w)(g −1 , x1 . . . , xn ) = αg−1 (1g w(x1 , . . . , xn ))w(g −1 x1 , x2 , . . . , xn )−1 · n−1 Y w(g −1 , x1 , . . . , xk xk+1 , . . . , xn )(−1) k−1 k=1 · w(g −1 , x1 , . . . , xn−1 )(−1) n−1 . Hence, αg−1 (1g w(x1 , . . . , xn )) = w(g −1 x1 , x2 , . . . , xn ) · n−1 Y w(g −1 , x1 , . . . , xk xk+1 , . . . , xn )(−1) k k=1 n · w(g −1 , x1 , . . . , xn−1 )(−1) .  Given x ∈ G, denote by η(x) the element x−1 x̄ ∈ H. Let n > 0 and g ∈ Λ′ . Define ηng : Gn → H by and τng n −1 −1 ηng (x1 , . . . , xn ) = η(x−1 n xn−1 . . . x1 g) (33) τng (x1 , . . . , xn ) = (η1g (x1 ), η2g (x1 , x2 ), . . . , ηng (x1 , . . . , xn )). (34) n : G → H by Observe that g −1 η1g (x1 )η2g (x1 , x2 ) . . . ηng (x1 , . . . , xn ) = η(x−1 n . . . x1 g) = η1 (x1 . . . xn ). We shall also need the functions g σn,i n n+1 :G →G , 0 ≤ i ≤ n, defined by g σn,0 (x1 , . . . , xn ) = (g −1 , x1 , . . . , xn ), g σn,i (x1 , . . . , xn ) = (τig (x1 , . . . , xi ), (x−1 i (35) (36) −1 . . . x−1 , xi+1 , . . . , xn ), 1 g) 0 < i < n, (37) −1 −1 g σn,n (x1 , . . . , xn ) = (τng (x1 , . . . , xn ), (x−1 ). n . . . x1 g) In the formulas above we may allow n to be equal to zero, meaning that g −1 ∈ G. (38) g σ0,0 = 14 MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIMÓN Definition 3.4. With any n > 0 and w ∈ C n (G, A) we shall associate Y θg ◦ w ◦ τng (x1 , . . . , xn ), w′ (x1 , . . . , xn ) = 1(x1 ,...,xn ) g∈Λ ε(x1 , . . . , xn−1 ) = 1(x1 ,...,xn−1 ) Y θg n−1 Y i g σn−1,i (x1 , . . . , xn−1 )(−1) w◦ i=0 g∈Λ (39) ! . (40) Lemma 3.5. Let n > 0, w ∈ C n (G, A) and w′ , ε be as in Definition 3.4. Then w′ ∈ C n (G, A) and ε ∈ C n−1 (G, A). Proof. Notice by (34) and (35) that −1 w ◦ τng (x1 , . . . , xn ) ∈ U(Dη(x−1 g) Dη(x−1 x−1 g) . . . Dη(x−1 g) ). n ...x 1 Since η(x−1 k . . . x−1 1 g) 2 1 1 ∈ H, then A1 ⊆ Dη(x−1 ...x−1 g) , 1 ≤ k ≤ n, so k pr1 ◦ w ◦ 1 τng (x1 , . . . , xn ) ∈ U(A1 ) and hence by (28) θg ◦ w ◦ τng (x1 , . . . , xn ) = αg ◦ pr1 ◦ w ◦ τng (x1 , . . . , xn ) ∈ U(Ag ). Therefore, the product of the values of θg on the right-hand side of (39) belongs to U(A) and thus w′ ∈ C n (G, A). To prove that ε ∈ C n−1 (G, A) for n > 1, observe first that the right-hand side of (40) depends only on θg with g ∈ Λ satisfying Ag ⊆ D(x1 ,...,xn−1 ) (41) (if there is no such g, then D(x1 ,...,xn−1 ) is zero and thus ε(x1 , . . . , xn−1 ) is automatically invertible in this ideal). Now n−1 Y g w ◦ σn−1,i (x1 , . . . , xn−1 ) ∈ U(D(g−1 ,x1 ,...,xn−1 ) Dη(x−1 g) . . . Dη(x−1 −1 n−1 ...x1 g) 1 ). i=0 −1 As above, A1 ⊆ Dη(x−1 ...x−1 g) , 1 ≤ k ≤ n − 1, because η(x−1 k . . . x1 g) ∈ H. 1 k Moreover, by (27) condition (41) is equivalent to A1 ⊆ D(g−1 ,x1 ,...,xn−1 ) . The rest of the proof now follows as for w′ . If n = 1, then Y ε= θg (w(g −1 )) ∈ U(A), (42) g∈Λ as Dg−1 ⊇ A1 by (i) of Lemma 3.2.  The following notation will be used in the results below. n−1 Y Π(l, m) = g w ◦ σn−1,i (x1 , . . . , xk xk+1 , . . . , xn )(−1) k+i k=l,i=m · n−1 Y g w ◦ σn−1,i (x1 , . . . , xn−1 )(−1) i=m where 1 ≤ l ≤ n − 1 and 0 ≤ m ≤ n − 1. n+i , (43) GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 15 Lemma 3.6. For all w ∈ Z 1 (G, A) and x ∈ G we have: Y (δ 0 ε)(x)αx (1x−1 ε)−1 w(x)−1 = θg (w(g −1 x)−1 ). (44) g∈Λ Moreover, for n > 1, w ∈ Z n (G, A) and x1 , . . . , xn ∈ G: (δ n−1 ε)(x1 , . . . , xn )αx1 (1x−1 ε(x2 , . . . , xn ))−1 w(x1 , . . . , xn )−1 1 Y −1 = 1(x1 ,...,xn ) θg (w(g x1 , x2 , . . . , xn )−1 Π(1, 1)). (45) g∈Λ Proof. Indeed, by (3), (32) and (42) we see that (δ 0 ε)(x)αx (1x−1 ε)−1 w(x)−1 = ε−1 w(x)−1 Y = θg (w(g −1 )−1 w(g −1 x)−1 w(g −1 )) g∈Λ = Y θg (w(g −1 x)−1 ). g∈Λ For (45) observe from (2), (40) and (43) that (δ n−1 ε)(x1 , . . . , xn )αx1 (1x−1 ε(x2 , . . . , xn ))−1 1 ! n−1 Y n (−1)k = ε(x1 , . . . , xn−1 )(−1) ε(x1 , . . . , xk xk+1 , . . . , xn ) k=1 = Y θg (Π(1, 0)). g∈Λ Now in (32) one has k g (x1 , . . . , xk xk+1 , . . . , xn )(−1) w(g −1 , x1 , . . . , xk xk+1 , . . . , xn )(−1) = w ◦ σn−1,0 n g w(g −1 , x1 , . . . , xn−1 )(−1) = w ◦ σn−1,0 (x1 , . . . , xn−1 )(−1) n+0 k+0 , , which are the factors of Π(1, 0) corresponding to i = 0 and 1 ≤ k ≤ n − 1. Hence, Y Y θg (Π(1, 0)) = w(x1 , . . . , xn ) θg (w(g −1 x1 , x2 , . . . , xn )−1 Π(1, 1)). g∈Λ g∈Λ  n Lemma 3.7. For all n > 1, w ∈ Z (G, A), g ∈ Λ and x1 , . . . , xn ∈ G: x−1 g 1 (x2 , . . . , xn ))−1 w(g −1 x1 , x2 , . . . , xn )−1 Π(1, 1) = αη1g (x1 ) (1η1g (x1 )−1 w ◦ σn−1,0 −1 · w(τ1g (x1 ), (x−1 x2 , x3 , . . . , xn )−1 Π(2, 2) 1 g) · n−1 Y g w ◦ σn−1,i (x1 x2 , x3 , . . . , xn )(−1) i+1 . (46) i=1 Proof. Since w is a partial n-cocycle, one has that (see (33), (34) and (37)) g −1 (δ n w) ◦ σn,1 (x1 , . . . , xn ) = (δ n w)(τ1g (x1 ), (x−1 , x2 , . . . , xn ) 1 g) −1 = (δ n w)(η1g (x1 ), (x−1 , x2 , . . . , xn ) 1 g) −1 −1 , x2 , . . . , xn ) = (δ n w)(g −1 x1 · x−1 1 g, (x1 g) (47) 16 MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIMÓN = 1g−1 x −1 1 ·x1 g 1(g−1 x1 ,x2 ,...,xn ) . Applying (2), we expand (47) as follows: 1g−1 x −1 1 ·x1 g 1(g−1 x1 ,x2 ,...,xn ) = αg−1 x −1 1 ·x1 g −1 (1(x−1 g)−1 x−1 g w((x−1 , x2 , . . . , xn )) 1 g) 1 1 · w(g −1 x1 , x2 , . . . , xn )−1 −1 −1 x2 , x3 , . . . , xn ) · w(g −1 x1 · x−1 1 g, (x1 g) · n−1 Y −1 −1 w(g −1 x1 · x−1 , x2 , . . . , xk xk+1 , . . . xn )(−1) 1 g, (x1 g) k=2 −1 −1 · w(g −1 x1 · x−1 , x2 , . . . , xn−1 )(−1) 1 g, (x1 g) n+1 . Using our notation (33)–(34), (37) and (38), we conclude that x−1 g 1 (x2 , . . . , xn ))−1 1η1g (x1 ) w(g −1 x1 , . . . , xn )−1 = αη1g (x1 ) (1η1g (x1 )−1 w ◦ σn−1,0 (48) −1 x2 , x3 , . . . , xn )−1 · w(τ1g (x1 ), (x−1 1 g) · n−1 Y g w ◦ σn−1,1 (x1 , . . . , xk xk+1 , . . . xn )(−1) (49) k (50) k=2 n g · w ◦ σn−1,1 (x1 , x2 , . . . , xn−1 )(−1) , (51) the lines (50) and (51) being the inverses of the factors of Π(1, 1), which correspond to i = 1 and 2 ≤ k ≤ n − 1. Thus, after the multiplication of the right-hand side of equality (48)–(51) by Π(1, 1), they will be reduced, and at their place we shall have the factors of 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 − 1), giving the right-hand side of (46). It remains to note that 1η1g (x1 ) Π(1, 1) = Π(1, 1) and the idempotents which appear in the cancellations are absorbed by the element (49), except 1x−1 g which is absorbed 1 by the element in the right-hand side of (48).  Lemma 3.8. For all 1 < j < n, w ∈ Z n (G, A), g ∈ Λ and x1 , . . . , xn ∈ G: g −1 −1 xj , xj+1 , . . . , xn )−1 Π(j, j) w(τj−1 (x1 , . . . , xj−1 ), (x−1 j−1 . . . x1 g) x−1 g 1 = αη1g (x1 ) (1η1g (x1 )−1 w ◦ σn−1,j−1 (x2 , . . . , xn ))(−1) j −1 −1 xj+1 , xj+2 , . . . , xn )−1 Π(j + 1, j + 1) · w(τjg (x1 , . . . , xj ), (x−1 j . . . x1 g) · n−1 Y g w ◦ σn−1,i (x1 , . . . , xj xj+1 , . . . , xn )(−1) i+j i=j · j−1 Y g w ◦ σn−1,j−1 (x1 , . . . , xs xs+1 , . . . , xn )(−1) s+j s=1 (here by Π(n, n) we mean the identity element 1A ). Proof. We use the same idea as in the proof of Lemma 3.7: g (δ n w) ◦ σn,j (x1 , . . . , xn ) −1 −1 , xj+1 , . . . , xn ) = (δ n w)(τjg (x1 , . . . , xj ), (x−1 j . . . x1 g) (52) k+1 GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 17 −1 −1 = (δ n w)(η1g (x1 ), η2g (x1 , x2 ), . . . , ηjg (x1 , . . . , xj ), (x−1 , xj+1 , . . . , xn ) j . . . x1 g) −1 −1 −1 −1 −1 −1 −1 x2 x−1 xj x−1 = (δ n w)(g −1 x1 x−1 1 g, (x1 g) 2 x1 g, . . . , (xj−1 . . . x1 g) j . . . x1 g, −1 −1 (x−1 , xj+1 , . . . , xn ) j . . . x1 g) = 1g−1 x 1 1 x−1 g g−1 x 1 1 x2 . . . 1g−1 x x−1 x−1 g 2 1 (53) 1 ...xj 1 −1 . x−1 ...x−1 g (g x1 ...xj ,xj+1 ,...,xn ) 1 j Expanding (53), we obtain by (2) 1g−1 x −1 1 x1 g = αg−1 x 1g−1 x −1 1 x1 g −1 −1 1 x2 x2 x1 g . . . 1g−1 x −1 −1 1 ...xj xj ...x1 g 1(g−1 x1 ...xj ,xj+1 ,...,xn ) −1 −1 (1(x−1 g)−1 x−1 g w((x−1 x2 x−1 1 g) 2 x1 g, . . . , 1 1 −1 −1 −1 −1 −1 −1 (x−1 xj x−1 , xj+1 , . . . , xn )) j−1 . . . x1 g) j . . . x1 g, (xj . . . x1 g) −1 −1 −1 −1 −1 · w(g −1 x1 x2 x−1 xj x−1 2 x1 g, . . . , (xj−1 . . . x1 g) j . . . x1 g, −1 −1 , xj+1 , . . . , xn )−1 (x−1 j . . . x1 g) · j−2 Y −1 −1 −1 −1 w(g −1 x1 x−1 xs xs+1 x−1 1 g, . . . , (xs−1 . . . x1 g) s+1 . . . x1 g, . . . , s=2 −1 −1 −1 −1 −1 −1 xj x−1 , xj+1 , . . . , xn )(−1) (x−1 j−1 . . . x1 g) j . . . x1 g, (xj . . . x1 g) s −1 −1 −1 −1 · w(g −1 x1 x−1 xj−1 xj x−1 1 g, . . . , (xj−1 . . . x1 g) j . . . x1 g, −1 −1 , xj+1 , . . . , xn )(−1) (x−1 j . . . x1 g) j−1 −1 −1 −1 −1 · w(g −1 x1 x−1 xj−1 x−1 1 g, . . . , (xj−2 . . . x1 g) j−1 . . . x1 g, −1 −1 xj , xj+1 , . . . , xn )(−1) (x−1 j−1 . . . x1 g) j −1 −1 −1 −1 · w(g −1 x1 x−1 xj x−1 1 g, . . . , (xj−1 . . . x1 g) j . . . x1 g, −1 −1 xj+1 , xj+2 , . . . , xn )(−1) (x−1 j . . . x1 g) · n−1 Y j+1 −1 −1 −1 −1 w(g −1 x1 x−1 xj x−1 1 g, . . . , (xj−1 . . . x1 g) j . . . x1 g, t=j+1 −1 −1 (x−1 , xj+1 , . . . , xt xt+1 , . . . , xn )(−1) j . . . x1 g) t+1 −1 −1 −1 −1 xj x−1 · w(g −1 x1 x−1 1 g, . . . , (xj−1 . . . x1 g) j . . . x1 g, −1 −1 (x−1 , xj+1 , . . . , xn−1 )(−1) j . . . x1 g) n+1 . We rewrite this in our shorter notation (33)–(34), (37) and (38): 1η1g (x1 ) 1η1g (x1 x2 ) . . . 1η1g (x1 ...xj ) 1(g−1 x1 ...xj ,xj+1 ,...,xn ) x−1 g 1 (x2 , . . . , xn )) = αη1g (x1 ) (1η1g (x1 )−1 w ◦ σn−1,j−1 g · w ◦ σn−1,j−1 (x1 x2 , x3 , . . . , xn )−1 · j−2 Y (54) g w ◦ σn−1,j−1 (x1 , . . . , xs xs+1 , . . . , xn )(−1) s (55) s=2 g · w ◦ σn−1,j−1 (x1 , . . . , xj−1 xj , . . . , xn )(−1) j−1 (56) 18 MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIMÓN g −1 −1 · w(τj−1 (x1 , . . . , xj−1 ), (x−1 xj , xj+1 , . . . , xn )(−1) j−1 . . . x1 g) −1 −1 xj+1 , xj+2 , . . . , xn )(−1) · w(τjg (x1 , . . . , xj ), (x−1 j . . . x1 g) n−1 Y · g w ◦ σn−1,j (x1 , . . . , xt xt+1 , . . . , xn )(−1) j j+1 t+1 t=j+1 g · w ◦ σn−1,j (x1 , . . . , xn−1 )(−1) n+1 . Note that the factors (54) and (56) may be included into the product (55), permitting thus s to run from 1 to j − 1 in (55). It follows that g −1 −1 (x1 , . . . , xj−1 ), (x−1 1η1g (x1 ...xj ) w(τj−1 xj , xj+1 , . . . , xn )−1 j−1 . . . x1 g) x−1 1 g = αη1g (x1 ) (1η1g (x1 )−1 w ◦ σn−1,j−1 (x2 , . . . , xn ))(−1) · j−1 Y g w ◦ σn−1,j−1 (x1 , . . . , xs xs+1 , . . . , xn )(−1) (57) j s+j s=1 −1 −1 xj+1 , xj+2 , . . . , xn )−1 · w(τjg (x1 , . . . , xj ), (x−1 j . . . x1 g) · n−1 Y g w ◦ σn−1,j (x1 , . . . , xt xt+1 , . . . , xn )(−1) t+j+1 (58) t=j+1 g · w ◦ σn−1,j (x1 , . . . , xn−1 )(−1) 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 two lines by the factors of Π(j, j) with j ≤ i ≤ n−1 and k = j, and, whenever j < n−1, there will also appear all the factors of Π(j + 1, j + 1), giving the right-hand side of equality (52). Finally, the left-hand side of (52) coincides with (57) multiplied by  Π(j, j), as 1η1g (x1 ...xj ) Π(j, j) = Π(j, j). Lemma 3.9. For all w ∈ Z 1 (G, A), g ∈ Λ and x ∈ G: 1η1g (x) w(g −1 x)−1 = αη1g (x) (1η1g (x)−1 w((x−1 g)−1 )−1 )(w ◦ τ1g )(x)−1 . (60) Moreover, for all n > 1, w ∈ Z n (G, A), g ∈ Λ and x1 , . . . , xn ∈ G: g −1 −1 (x1 , . . . , xn−1 ), (x−1 w(τn−1 xn )−1 n−1 . . . x1 g) x−1 g 1 = αη1g (x1 ) (1η1g (x1 )−1 w ◦ σn−1,n−1 (x2 , . . . , xn ))(−1) · n−1 Y g w ◦ σn−1,n−1 (x1 , . . . , xs xs+1 , . . . , xn )(−1) n s+n s=1 · w ◦ τng (x1 , . . . , xn )−1 . Proof. For (60) write g (x) 1η1g (x) 1g−1 x = (δ 1 w) ◦ σ1,1 = (δ 1 w)(τ1g (x), (x−1 g)−1 ) = (δ 1 w)(η1g (x), (x−1 g)−1 ) = αη1g (x) (1η1g (x)−1 w((x−1 g)−1 ))w(g −1 x)−1 (w ◦ τ1g )(x). (61) GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 19 To get (61), analyze the proof of Lemma 3.8 (we skip the details): 1η1g (x1 ) 1η1g (x1 x2 ) . . . 1η1g (x1 ...xn ) 1g−1 x1 ·...·xn g = (δ n w) ◦ σn,n (x1 , . . . , xn ) x−1 g 1 = αη1g (x1 ) (1η1g (x1 )−1 w ◦ σn−1,n−1 (x2 , . . . , xn )) · n−1 Y g w ◦ σn−1,n−1 (x1 , . . . , xs xs+1 , . . . , xn )(−1) s s=1 g −1 −1 xn )(−1) · w(τn−1 (x1 , . . . , xn−1 ), (x−1 n−1 . . . x1 g) · w ◦ τng (x1 , . . . , xn )(−1) n+1 n .  Lemma 3.10. For all n > 0, w ∈ Z n (G, A) and x1 , . . . , xn ∈ G: (δ n−1 ε)(x1 , . . . , xn )αx1 (1x−1 ε(x2 , . . . , xn ))−1 w(x1 , . . . , xn )−1 1   n−1 Y Y −1 j+1 x1 g = w ◦ σn−1,j (x2 , . . . , xn )(−1)  θg ◦ αη1g (x1 ) 1η1g (x1 )−1 j=0 g∈Λ ′ −1 · w (x1 . . . , xn ) . (62) Proof. If n = 1, then the result follows from (29), (39), (44) and (60) and the fact that η1g (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 −1 x1 , . . . , xn )−1 Π(1, 1)   n−1 Y −1 j+1 x1 g (x2 , . . . , xn )(−1)  w ◦ σn−1,j = αη1g (x1 ) 1η1g (x1 )−1 j=0 ·w◦ · τng (x1 , . . . , xn )−1 n−1 Y n−1 Y g w ◦ σn−1,i (x1 , . . . , xj xj+1 , . . . , xn )(−1) i+j (63) j=1 i=j · n j−1 Y Y g w ◦ σn−1,j−1 (x1 , . . . , xs xs+1 , . . . , xn )(−1) s+j . j=2 s=1 After the change of indexes j ′ = j − 1 the product (64) becomes j′ n−1 Y Y g (−1) w ◦ σn−1,j ′ (x1 , . . . , xs xs+1 , . . . , xn ) s+j ′ +1 . j ′ =1 s=1 Now switching the order in this double product, we come to n−1 Y n−1 Y g (−1) w ◦ σn−1,j ′ (x1 , . . . , xs xs+1 , . . . , xn ) s=1 j ′ =s The latter is exactly the inverse of (63). Hence, w(g −1 x1 , x2 , . . . , xn )−1 Π(1, 1) s+j ′ +1 . (64) 20 MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIMÓN  = αη1g (x1 ) 1η1g (x1 )−1 n−1 Y w◦ j=0 · w ◦ τng (x1 , . . . , xn )−1 .  j+1 x−1 1 g σn−1,j (x2 , . . . , xn )(−1)  It remains to substitute this into (45) and to apply (39).  Lemma 3.11. For all x ∈ G and a : Λ′ → A one has      Y Y θg (a(g)) = 1x αx 1x−1 θg ◦ αη1g (x) 1η1g (x)−1 a x−1 g . g∈Λ (65) g∈Λ Proof. First of all observe using (ii) of Lemma 3.2 that Y Y Y 1x cg = cg = g∈Λ g∈Λ,Ag ⊆Dx g, cg , (66) x−1 g∈Λ where cg is an arbitrary element of Ag . Thus, in the right-hand side of (65) we may replace the condition g ∈ Λ by a stronger one g, x−1 g ∈ Λ. Notice also from (27) and (29) that we may put 1g−1 x inside of θg in the right-hand side of (65). Now       1g−1 x αη1g (x) 1η1g (x)−1 a x−1 g = αg−1 x1 ◦ αx−1 g 1(x−1 g)−1 1η1g (x)−1 a x−1 g , 1 (67) and denoting the argument of αx−1 g in (67) by b = b(g, x), we deduce from (30) 1 that   θg ◦ αg−1 x ◦ αx−1 g (b) = prg ◦ αg 1g−1 αg−1 x ◦ αx−1 g (b)   = prg ◦ αg ◦ αg−1 ◦ αx 1x−1 1x−1 g αx−1 g (b)   = prg ◦ αx 1x−1 1x−1 g αx−1 g (b) .   As xx−1 g = g ∈ Λ, by (ii) of Lemma 3.2 we have Ax−1 g ⊆ Dx−1 and αx Ax−1 g = Ag . Moreover, Ax−1 g ⊆ Dx−1 g by (27). Hence, in view of Lemma 3.1 and (30)     prg ◦ αx 1x−1 1x−1 g αx−1 g (b) = αx ◦ prx−1 g 1x−1 1x−1 g αx−1 g (b) = αx ◦ prx−1 g ◦ αx−1 g (b) = αx ◦ θx−1 g (b), and consequently       = αx ◦ θx−1 g (b) = αx ◦ θx−1 g a x−1 g . θg ◦ αη1g (x) 1η1g (x)−1 a x−1 g Here we used (29) to remove 1η1g (x)−1 and 1(x−1 g)−1 from b. It follows that the right-hand side of (65) is         Y Y αx ◦ θx−1 g a x−1 g θx−1 g a x−1 g  , = αx  (68) g,x−1 g∈Λ g,x−1 g∈Λ GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 21 which is verified by checking the projection of each side of the latter equality onto an arbitrary block At , t ∈ Λ. Let g ′ = x−1 g ∈ Λ. Then g = xx−1 g = xg ′ ∈ Λ, so (68) becomes, in view of (66),     Y Y θg′ (a (g ′ )) = αx 1x−1 αx  θg′ (a (g ′ )) , g′ ∈Λ g′ ,xg′ ∈Λ proving (65).  Lemma 3.12. For all n > 0, w ∈ Z n (G, A) and x1 , . . . , xn ∈ G:   n−1 Y Y −1 j x g 1 w ◦ σn−1,j θg ◦ αη1g (x1 ) 1η1g (x1 )−1 (x2 , . . . , xn )(−1)  1(x1 ...,xn ) = αx1  j=0 g∈Λ  1x−1 ε(x2 , . . . , xn ) . 1 (69) Proof. Let us fix n, w and x2 , . . . , xn . For arbitrary g ∈ Λ′ define a(g) = n−1 Y j g w ◦ σn−1,j (x2 , . . . , xn )(−1) . j=0 Then the left-hand side of (69) equals   Y 1(x1 ...,xn ) θg ◦ αη1g (x1 ) 1η1g (x1 )−1 a(x−1 1 g) . g∈Λ Since 1(x1 ...,xn ) = 1(x1 ...,xn ) 1x1 , then applying Lemma 3.11, we transform this into    n−1 Y Y j g w ◦ σn−1,j (x2 , . . . , xn )(−1)  . θg  1(x1 ...,xn ) αx1 1x−1 1 j=0 g∈Λ   Rewriting 1(x1 ,...,xn ) as αx1 1x−1 1(x2 ,...,xn ) and using (40), we come to the right1 hand side of (69).  Theorem 3.13. Let A be a direct product of commutative unital indecomposable rings with a structure of a (unital) partial G-module, n > 0, w ∈ Z n (G, A) and w′ , ε be as in Definition 3.4. Then w = δ n−1 ε · w′ . In particular, w′ ∈ Z n (G, A). Proof. This is an immediate consequence of Lemmas 3.10 and 3.12.  4. Existence of a globalization In this section we construct the cocycle w e whose existence was announced above. Keeping the notation of Section 2, we begin with some auxiliary formulas whose proof will be left to the reader. Lemma 4.1. Let g ∈ Λ′ . Then x−1 g 1 ηng (x1 , . . . , xn ) = ηn−1 (x2 , . . . , xn ), n ≥ 2, ηng (x1 , . . . , xi , xi+1 , . . . , xn ) ηng (x1 , . . . , xn−1 , xn xn+1 ) = = g ηn−1 (x1 , . . . , xi xi+1 , . . . , xn ), 1 ≤ g g ηn (x1 , . . . , xn )ηn+1 (x1 , . . . , xn+1 ), (70) i ≤ n − 2, n ≥ 1. (71) (72) 22 MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIMÓN f′ : Gn → A by removing the idempotent 1(x ,...,x ) We now define a function w 1 n from the right-hand side of (39), that is Y f′ (x1 , . . . , xn ) = θg ◦ w ◦ τng (x1 , . . . , xn ). (73) w g∈Λ f′ (x1 , . . . , xn ) ∈ U(A), so w f′ is a As it was observed in the proof of Lemma 3.5, w n ′ f classical n-cochain from C (G, U(A)). It turns out that w satisfies the “quasi” n-cocycle identity (15). Lemma 4.2. Let n > 0, w ∈ Z n (G, A) and x1 , . . . , xn ∈ G. Then n  Y ′ f′ (x1 , . . . , xi xi+1 , . . . , xn+1 )(−1)i f w αx1 1x−1 w (x2 , . . . , xn+1 ) 1 i=1 f′ (x1 , . . . , xn )(−1) ·w n+1 = 1x1 . Proof. According to (73), the left-hand side of (74) is   Y αx1 1x−1 θg ◦ w ◦ τng (x2 , . . . , xn+1 ) (74) (75) 1 g∈Λ · n Y Y θg ◦ w ◦ τng (x1 , . . . , xi xi+1 , . . . , xn+1 )(−1) i (76) i=1 g∈Λ · Y θg ◦ w ◦ τng (x1 , . . . , xn )(−1) n+1 . (77) g∈Λ Using Lemma 3.11, we rewrite (75) as   Y x−1 g θg ◦ αη1g (x1 ) 1η1g (x1 )−1 w ◦ τn 1 (x2 , . . . , xn+1 ) . 1x1 g∈Λ Moreover, since θg is a homomorphism, (76) coincides with Y n Y θg w◦ i τng (x1 , . . . , xi xi+1 , . . . , xn+1 )(−1) i=1 g∈Λ ! . Therefore, in order to prove (74), it suffices to check the equality   x−1 1 g g g 1η(x−1 g) . . . 1η(x−1 ...x−1 g) = αη1 (x1 ) 1η1 (x1 )−1 w ◦ τn (x2 , . . . , xn+1 ) 1 1 n+1 · n Y w ◦ τng (x1 , . . . , xi xi+1 , . . . , xn+1 )(−1) i (78) (79) i=1 · w ◦ τng (x1 , . . . , xn )(−1) η(x−1 i n+1 . . . . x−1 1 g) Indeed, each belongs to H, so by (28)     θg 1η(x−1 ...x−1 g) = αg ◦ pr1 1η(x−1 ...x−1 g) = αg (1A1 ) = 1Ag , i and consequently, Y g∈Λ 1 i  θg 1η(x−1 g) . . . 1η(x−1 1 −1 n+1 ...x1 g) 1  = Y g∈Λ 1Ag = 1A . (80) GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 23 We show that (78)–(80) is exactly the partial n-cocycle identity g (δ n w) ◦ τn+1 (x1 , . . . , xn+1 ) = 1η(x−1 g) . . . 1η(x−1 −1 n+1 ...x1 g) 1 . (81) By (34) and (70) one has x−1 g τn 1 g (x2 , . . . , xn+1 ) = (η2g (x1 , x2 ), . . . , ηn+1 (x1 , . . . , xn+1 )), so the right-hand side of (78) is the first factor of the left-hand side of (81) expanded in accordance with (2). Now, the ith factor of the product (79) is of the form i g w(τi−1 (x1 , . . . , xi−1 ), ηig (x1 , . . . , xi−1 , xi xi+1 ), . . . , ηng (x1 , . . . , xi xi+1 , . . . , xn+1 ))(−1) , which coincides with the ith factor of the analogous product of the expansion of the left-hand side of (81) thanks to (71) and (72). Finally, (80) is literally the last factor of the above mentioned expansion.  We proceed now with the construction of w e needed in Theorem 2.5. Given n > 0 and x1 , . . . , xn ∈ G, we define ε̃(x1 , . . . , xn−1 ) = ε(x1 , . . . , xn−1 ) + 1A − 1(x1 ,...,xn−1 ) ∈ U(A), (82) understanding that ε̃ = ε ∈ U(A) if n = 1. Define also where f′ (x1 , . . . , xn ) ∈ U(A), w(x e 1 , . . . , xn ) = (δ̃ n−1 ε̃)(x1 , . . . , xn )w (83) (δ̃ n−1 ε̃)(x1 , . . . , xn ) = α̃x1 (ε̃(x2 , . . . , xn )) · n−1 Y ε̃(x1 , . . . , xi xi+1 , . . . , xn )(−1) i i=1 n · ε̃(x1 , . . . , xn−1 )(−1) , (84) and α̃x (a) = αx (1x−1 a) + 1A − 1x , (85) with x ∈ G and a ∈ A. Our main result is as follows. Theorem 4.3. Let A be a commutative unital ring which is a (possibly infinite) direct product of indecomposable rings, and let α = {αg : Dg−1 → Dg | g ∈ G} be a (not necessarily transitive) unital partial action of G on A. Then for any n ≥ 0 each partial cocycle w ∈ Z n (G, A) is globalizable. Proof. Since the case n = 0 has been explained in Proposition 2.3, we assume n > 0. Consider first the transitive case. We will show that our w e defined in (83) satisfies (15) and (16). It directly follows from (39), (73), (82), (84) and (85) that 1(x1 ,...,xn ) w(x e 1 , . . . , xn ) = (δ n−1 ε)(x1 , . . . , xn ) · w′ (x1 , . . . , xn ) for all x1 , . . . , xn ∈ G. By Theorem 3.13 this yields that w e satisfies (16). As for (15), we see that n  Y i w(x e 1 , . . . , xi xi+1 , . . . , xn+1 )(−1) e 2 , . . . , xn+1 ) αx1 1x−1 w(x 1 i=1 · w(x e 1 , . . . , xn )(−1) n+1 (86) 24 MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIMÓN can be written as product of the following two factors n  Y f′ (x1 , . . . , xi xi+1 , . . . , xn+1 )(−1)i f′ (x2 , . . . , xn+1 ) w αx1 1x−1 w 1 i=1 f′ (x1 , . . . , xn )(−1) ·w and n+1 n  Y i (δ̃ n−1 ε)(x1 , . . . , xi xi+1 , . . . , xn+1 )(−1) αx1 1x−1 (δ̃ n−1 ε)(x2 , . . . , xn+1 ) (87) 1 i=1 · (δ̃ n−1 ε)(x1 , . . . , xn )(−1) n+1 . (88) Thanks to Lemma 4.2 the factor (87) is 1x1 , whereas the expansion of (88) has the same form as the usual δ n ◦ δ n−1 in homological algebra, with the difference that instead of a global action we have a mixture of α with α̃. Consequently, all factors in (88) to which neither α, nor α̃ is applied, cancel amongst themselves resulting in 1A . The remaining factors of the expansion are those of   (89) αx1 1x−1 (δ̃ n−1 ε)(x2 , . . . , xn+1 ) 1 and the first factors in each i (δ̃ n−1 ε)(x1 , . . . , xi xi+1 , . . . , xn+1 )(−1) , 1 ≤ i ≤ n, (90) and in (δ̃ n−1 ε)(x1 , . . . , xn )(−1) n+1 . The factors in the expansion of (89) are exactly   αx1 1x−1 α̃x2 (ε̃(x3 , . . . , xn+1 )) , 1   i−1 , 2 ≤ i ≤ n, αx1 1x−1 ε̃(x2 , . . . , xi xi+1 , . . . , xn+1 )(−1) 1 and (91) (92) (93)   n αx1 1x−1 ε̃(x2 , . . . , xn )(−1) , (94) α̃x1 x2 (ε̃(x3 , . . . , xn+1 ))−1 , (95) 1 whereas the first factors in (90) and (91) are which comes from the case i = 1 in (90), i α̃x1 (ε̃(x2 , . . . , xi xi+1 , . . . , xn+1 ))(−1) , 2 ≤ i ≤ n, (96) α̃x1 (ε̃(x2 , . . . , xn )). (97) and Multiplying the elements in (96) and (97) by 1x1 we see that they are canceled with those in (93) and (94), respectively. Now (92) equals 1x1 α̃x1 x2 (ε̃(x3 , . . . , xn+1 )) due to the commutative version of (19) from [24], so that it cancels with (95). It follows e satisfies (15). It remains to apply that (88) also equals 1x1 , and we conclude that w Theorem 2.5. If α is not transitive, then we represent A as a product of ideals, on each of which α acts transitively, so that the construction of w̃ reduces to the transitive case by means of the projection on such an ideal (see [24, Proposition 8.4]).  GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 25 5. Uniqueness of a globalization Our aim is to show that the globalization of w constructed in Section 4 is unique up to cohomological equivalence. We would like to use item (iii) of [24, Lemma 8.3], whose proof was not sufficiently well explained. To clarify it, we need some new terminology. Let R be a ring and Rµ ⊆ R, µ ∈ M , a collection of its unital ideals. Observe from theQdefinition of a direct product that there is a unique homomorphism φ : R → µ∈M Rµ , Q such that φ followed by the natural projection µ∈M Rµ → Rµ′ coincides with the multiplication by 1Rµ′ in R for any µ′ ∈ M . In this situation we say that the homomorphism φ respects projections. Lemma 5.1. Let C be a not necessarily unital ring and {Cµ | µ ∈ M } a family of pairwise distinct unital ideals in C. Suppose that I and J are unital ideals in C such that Y Y I∼ Cµ and J ∼ Cµ , (98) = = µ∈M1 µ∈M2 where M1 , M2 ⊆ M , Cµ ⊆ I for all µ ∈ M1 and Cµ′ ⊆ J for all µ′ ∈ M2 . If the isomorphisms (98) respect projections, then there is a (unique) isomorphism Y I +J ∼ Cµ , (99) = µ∈M1 ∪M2 which also respects projections. Proof. It is readily seen that I + J is a unital ring with unity element 1I + 1J − 1I 1J and I + J = I ⊕ J ′ , where J ′ = J(1J − 1I 1J ). Therefore, the isomorphism J ∼ = Q C restricts to µ∈M2 µ Y Y ∼ Cµ ⊆ J′ = Cµ µ∈M2 \M1 (see (25)). Then  I + J = I ⊕ J′ ∼ = Y µ∈M1   Cµ  ⊕  Q Y µ∈M2 \M1 µ∈M2   Cµ  ∼ = Y µ∈M1   Cµ  ×  Y µ∈M2 \M1  Cµ  , the latter being isomorphic to µ∈M1 ⊔(M2 \M1 ) Cµ , which proves (99). Moreover, the isomorphism can be chosen in such a way that it respects projections, provided that the isomorphisms (98) have this property.  Q Proposition 5.2. Let A be a product g∈Λ Ag of not necessarily commutative indecomposable unital rings, α a transitive unital partial action of G on A and (β, B)Qan enveloping action of (α, A) with A ⊆ B. Then B embeds as an ideal into g∈Λ′ Ag , where Λ′ was defined before formula (26) and Ag denotes the ideal3 Q βg (A1 ) in B. Moreover, M(B) ∼ = g∈Λ′ Ag , and β ∗ is transitive, when seen as an Q action of G on g∈Λ′ Ag . Proof. AsQit was explained before Lemma 5.1, there is a unique homomorphism φ : B → g∈Λ′ Ag , which respects projections. We shall prove that φ is injective. P Since B = g∈G βg (A), each element of B belongs to an ideal I of B of the form 3This does not conflict with (26), because α (A ) = A ⊆ A for g ∈ Λ, so β (A ) = α (A ). g g g g 1 1 1 26 MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIMÓN Pk i=1 βxi (A), x1 , . . . , xk ∈ 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 for any i = 1, . . . , k an isomorphism   Y Y Y βxi (A) = βxi  Ag  ∼ βxi (Ag ) = Axi g , = g∈Λ g∈Λ g∈Λ which respects projections. Notice that it follows from the definition of Λ′ that the ideals Ag , g ∈ Λ′ , are pairwise distinct. Hence by Lemma 5.1 there is an isomorphism Y ψ:I → Ag , (100) g∈Λ′′ where Λ′′ = {xi g | g ∈ Λ, i = 1, . . . , k} ⊆ Λ′ , and it also respects projections. We claim that the restriction of φ to I coincides with Q ψ, if one understands the product in the right-hand side of (100) as an ideal in g∈Λ′ Ag (see (25)). Indeed, for all g ∈ Λ′′ and b ∈ I one has prg ◦ ψ(b) = 1Ag b = prg ◦ φ(b), because φ and ψ respect projections. Now if g ∈ Λ′ \ Λ′′ , then x−1 i g 6∈ Λ for all Pk −1 ′′ i = 1, . . . , k, since otherwise g = xi xi g ∈ Λ . Hence, for all b = i=1 βxi (ai ) ∈ I (ai ∈ A) in view of (ii) of [24, Lemma 8.3]  X  k k X βxi (0) = 0. βxi 1A −1 ai = prg ◦ φ(b) = 1Ag b = i=1 x i g i=1 Q This proves the claim, and thus injectivity of φ. Moreover, since φ(I) = g∈Λ′′ Ag Q Q is an ideal in g∈Λ′ Ag , it follows that φ(B) is also an ideal in g∈Λ′ Ag . Q that each element of Q Regarding the second statement of the proposition, notice let A acts as a multiplier of B, as φ(B) is an ideal in ′ g g∈Λ′ Ag . Conversely, g∈Λ Q w ∈ M(B). Then w1Ag = w1Ag · 1Ag ∈ Ag for all g ∈ Λ′ . Define a ∈ g∈Λ′ Ag by prg (a) = w1Ag . We need to show that φ(wb) = aφ(b) and φ(bw) = φ(b)a. Indeed, using the fact that φ respects projections, we get prg (φ(wb)) = 1Ag · wb = w1Ag · 1Ag b = w1Ag · prg (φ(b)) = prg (aφ(b)) for all g ∈ Λ′ . Similarly prg (φ(bw)) = prg (aφ(b)) for arbitrary g ∈ Λ′ . The transitivity of β ∗ easily follows from the definition of Ag for g ∈ Λ′ .  Q Theorem 5.3. Let A be a product g∈Λ Ag of commutative indecomposable unital rings, α a unital partial action of G on A and wi ∈ Z n (G, A), i = 1, 2 (n > 0). Suppose that (β, B) is an enveloping action of (α, A) and ui ∈ Z n (G, U(M(B))) is a globalization of wi , i = 1, 2. If w1 is cohomologous to w2 , then u1 is cohomologous to u2 . In particular any two globalizations of the same partial n-cocycle are cohomologous. Proof. Let α be transitive. Q Thanks to Proposition 5.2 we may assume, up to an isomorphism, that M(B) = g∈Λ′ Ag ⊇ A. Define Y u′i (x1 , . . . , xn ) = ϑg ◦ ui ◦ τng (x1 , . . . , xn ), i = 1, 2, (101) g∈Λ′ GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 27 where ϑg is a homomorphism M(B) → M(B) given by ϑg = βg ◦ pr1 . (102) Since u′i has the same construction as w′ from Section 3 (see (39)), one has by Theorem 3.13 that u′i ∈ Z n (G, U(M(B))) and ui is cohomologous to u′i , i = 1, 2. It suffices to prove that u′1 is cohomologous to u′2 , provided that w1 is cohomologous to w2 . Observe, in view of (14), that for arbitrary h1 , . . . , hn ∈ H
pr1 ◦ ui (h1 , . . . , hn ) = pr1 ui (h1 , . . . , hn )1(h1 ,...,hn ) = pr1 ◦ w(h1 , . . . , hn ). Together with (101) and (102) this implies that Y u′i (x1 , . . . , xn ) = ϑg ◦ wi ◦ τng (x1 , . . . , xn ), i = 1, 2. (103) g∈Λ′ Let w2 = w1 · δ n−1 ξ for some ξ ∈ C n−1 (G, A). Since ϑg is a homomorphism, one immediately sees from (103) that u′2 = u′1 (δ n−1 ξ)′ , where Y (δ n−1 ξ)′ (x1 , . . . , xn ) = ϑg ◦ (δ n−1 ξ) ◦ τng (x1 , . . . , xn ). g∈Λ′ We shall show that (δ n−1 ξ)′ = δ n−1 ξ ′ with ξ ′ (x1 , . . . , xn−1 ) = Y (104) g ϑg ◦ ξ ◦ τn−1 (x1 , . . . , xn−1 ). g∈Λ′ Taking into account the fact that ϑg is a homomorphism once again and interchanging the left-hand side and the right-hand side of (104), we may reduce (104) to   Y g βx 1  ϑg ◦ ξ ◦ τn−1 (x2 , . . . , xn ) g∈Λ′ = Y ϑg ◦ βη1g (x1 ) ◦ ξ(η2g (x1 , x2 ), . . . , ηng (x1 , . . . , xn )), (105) g∈Λ′ whose right-hand side is Y x−1 g 1 (x2 , . . . , xn ) ϑg ◦ βη1g (x1 ) ◦ ξ ◦ τn−1 g∈Λ′ by (70). Now it is readily seen that (105) follows from the global case of Lemma 3.11 (with α and θ replaced by β ∗ and ϑ, respectively). The non-transitive case reduces to the transitive one, using the same argument as in Theorem 4.3.  Q Corollary 5.4. Let A be a product g∈Λ Ag of commutative indecomposable unital rings, α a unital partial action of G on A and (β, B) an enveloping action of (α, A). Then the partial cohomology group H n (G, A) is isomorphic to the classical (global) cohomology group H n (G, U(M(B))). 28 MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIMÓN Proof. Indeed, when n > 0, it follows from Theorems 4.3 and 5.3 that there is a well-defined map Φ : H n (G, A) → H n (G, U(M(B))) which sends the class of w ∈ Z n (G, A) to the class of a globalization of w. The map Φ is injective as the “restriction” (14) commutes with the coboundary operator, so any two n-cocycles u1 , u2 ∈ Z n (G, U(M(B))) which differ by an n-coboundary v ∈ B n (G, U(M(B))) restrict to two partial n-cocycles from Z n (G, A) which differ by the restriction of f′ v, the latter being a partial n-coboundary from B n (G, A). The constructions of w and u clearly respect products (see (8) and (73)), so Φ is a monomorphism of groups. It is evidently surjective, as any u ∈ Z n (G, U(M(B))) restricts to w ∈ Z n (G, A) by means 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 Corollary 2.4.  6. Example In this section we apply our technique from Sections 2–4 in a concrete example. Q3 Let G = hg | g 3 = ei and B = i=1 Ai , where each Ai is a copy of some commutative indecomposable unital ring R. We write an element of B as a triple a = (a1 , a2 , a3 ), where ai ∈ Ai , i = 1, 2, 3. Consider the “right shift” action β of G on B given by βg (a1 , a2 , a3 ) = (a3 , a1 , a2 ). Denote by A the ideal A1 × A2 × {0A3 } in B, which henceforth will be identified with A1 × A2 for notational purposes, and by α the (admissible) restriction of β to A. Then α = {αg : Dg−1 → Dg | g ∈ G} is a partial action of G on A, where De = A, αe = idA , Dg−1 = A1 × {0A2 }, Dg = {0A1 } × A2 and αg (a, 0A2 ) = (0A1 , a) (B, β) is an enveloping action of (A, α), for all (a, 0A2 ) ∈ Dg−1 . By construction P the isomorphism between B and g∈G βg (ϕ(A)) ⊆ F (G, A) is given by ψ(a1 , a2 , a3 ) = ϕ(a1 , 0A2 ) + βg (ϕ(a2 , 0A2 )) + βg−1 (ϕ(a3 , 0A2 )), where ϕ and β are as in (7) and (12). Observe that ψ(a1 , a2 , a3 ) is the function on G with the following values: ψ(a1 , a2 , a3 )|e = (a1 , a2 ), (106) ψ(a1 , a2 , a3 )|g = (a2 , a3 ), (107) ψ(a1 , a2 , a3 )|g−1 = (a3 , a1 ). (108) Example 6.1. Let w ∈ Z 2 (G, A). Then u ∈ Z 2 (G, B) given by u(e, e) = (w(e, e)1 , w(e, e)2 , w(e, e)1 ), u(e, g) = (w(e, e)1 , w(e, e)2 , w(e, e)1 ), u(e, g −1 ) = (w(e, e)1 , w(e, e)2 , w(e, e)1 ), u(g, e) = (w(e, e)1 , w(e, e)1 , w(e, e)2 ), u(g −1 , e) = (w(e, e)2 , w(e, e)1 , w(e, e)1 ), u(g, g −1 ) = (w(e, e)1 , w(g, g −1 )2 , w(e, e)1 ), 2 u(g −1 , g) = (w(g −1 , g)1 , w(e, e)−1 2 w(e, e)1 , w(e, e)1 ), u(g, g) = (w(e, e)1 , w(g −1 , g)1 , w(g −1 , g)1 ), 2 u(g −1 , g −1 ) = (w(e, e)1 , w(g −1 , g)−1 1 w(e, e)1 , w(e, e)1 ), GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 29 is a globalization of w. Proof. Clearly, α is transitive, H = {e}, Λ′ = {e, g, g −1}, Λ = {e, g} ⊆ Λ′ , x̄ = x for all x ∈ G and θe (a1 , a2 ) = (a1 , 0A2 ), θg (a1 , a2 ) = (0A1 , a1 ). Then by (39) and (40) for all x, y ∈ G Y w′ (x, y) = 1x 1xy θg ◦ w(g −1 x · x−1 g, (x−1 g)−1 y · y −1 x−1 g) g∈Λ = 1x 1xy Y θg ◦ w(e, e) g∈Λ = 1x 1xy (w(e, e)1 , w(e, e)1 ) and Y ε(x) = 1x θg (w(g −1 , x)w(g −1 x · x−1 g, (x−1 g)−1 )−1 ) g∈Λ Y = 1x θg (w(g −1 , x)w(e, g −1 x)−1 ) g∈Λ = 1x (θe (w(e, x)w(e, x)−1 )1 , θg (w(g −1 , x)w(e, g −1 x)−1 )2 ) = 1x (1A1 , w(g −1 , x)1 w(e, g −1 x)−1 1 ). It follows from the partial 2-cocycle identity for w that w(e, g) = 1g w(e, e) = (0A1 , w(e, e)2 ), w(e, g −1 ) = 1g−1 w(e, e) = (w(e, e)1 , 0A2 ), w(g, e) = αg (1g−1 w(e, e)) = (0A1 , w(e, e)1 ), w(g −1 , e) = αg−1 (1g w(e, e)) = (w(e, e)2 , 0A2 ), w(g, g −1 ) = (0A1 , w(g −1 , g)1 w(e, e)−1 2 w(e, e)1 ). (109) So, we have −1 ε(e) = (1A1 , w(g −1 , e)1 w(e, g −1 )−1 1 ) = (1A1 , w(e, e)2 w(e, e)1 ), ε(g) = (0A1 , w(g ε(g −1 −1 , g)1 w(e, e)−1 1 ), ) = 1g−1 , and thus (δ 1 ε)(e, e) = ε(e) = (1A1 , w(e, e)2 w(e, e)−1 1 ), (δ 1 ε)(e, g) = 1g ε(e) = (0A1 , w(e, e)2 w(e, e)−1 1 ), (δ 1 ε)(e, g −1 ) = 1g−1 ε(e) = (1A1 , 0A2 ) = 1g−1 , (δ 1 ε)(g, e) = αg (1g−1 ε(e)) = 1g , (δ 1 ε)(g −1 , e) = αg−1 (1g ε(e)) = (w(e, e)2 w(e, e)−1 1 , 0A2 ), −1 , g)1 ), (δ 1 ε)(g, g −1 ) = αg (1g−1 ε(g −1 ))ε(e)−1 ε(g) = (0A1 , w(e, e)−1 2 w(g (δ 1 ε)(g −1 , g) = αg−1 (1g ε(g))ε(e)−1 ε(g −1 ) = (w(g −1 , g)1 w(e, e)−1 1 , 0A2 ). (110) (111) (112) 30 MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIMÓN Since Dg Dg−1 = {0A }, both w and δ 1 ε are zero at (g, g) and (g −1 , g −1 ). Hence, we explicitly see that w = δ 1 ε · w′ . Removing 1x 1xy from w′ as in (73), we have for all x, y ∈ G f′ (x, y) = (w(e, e)1 , w(e, e)1 ). w Furthermore, by (82) and (110)–(112) ε̃(e) = ε(e) = (1A1 , w(e, e)2 w(e, e)1−1 ), ε̃(g) = ε(g) + 1A − 1g = (1A1 , w(g −1 , g)1 w(e, e)−1 1 ), ε̃(g −1 ) = ε(g −1 ) + 1A − 1g−1 = 1A . Therefore, by (84) and (85) (δ̃ 1 ε̃)(e, e) = ε̃(e) = (1A1 , w(e, e)2 w(e, e)1−1 ), (δ̃ 1 ε̃)(e, g) = ε̃(e) = (1A1 , w(e, e)2 w(e, e)−1 1 ), (δ̃ 1 ε̃)(e, g −1 ) = ε̃(e) = (1A1 , w(e, e)2 w(e, e)−1 1 ), (δ̃ 1 ε̃)(g, e) = αg (1g−1 ε̃(e)) + 1g−1 = 1A , (δ̃ 1 ε̃)(g −1 , e) = αg−1 (1g ε̃(e)) + 1g = (w(e, e)2 w(e, e)−1 1 , 1A2 ), −1 (δ̃ 1 ε̃)(g, g −1 ) = ε̃(e)−1 ε̃(g) = (1A1 , w(e, e)−1 , g)1 ), 2 w(g −1 (δ̃ 1 ε̃)(g −1 , g) = (αg−1 (1g ε̃(g)) + 1g )ε̃(e)−1 = (w(g −1 , g)1 w(e, e)−1 1 , w(e, e)2 w(e, e)1 ), (δ̃ 1 ε̃)(g, g) = ε̃(g) = (1A1 , w(g −1 , g)1 w(e, e)−1 1 ), (δ̃ 1 ε̃)(g −1 , g −1 ) = ε̃(g)−1 = (1A1 , w(g −1 , g)−1 1 w(e, e)1 ). Thus, by (83) w(e, e e) = (w(e, e)1 , w(e, e)2 ) = w(e, e), w(e, e g) = (w(e, e)1 , w(e, e)2 ) = w(e, e), w(e, e g −1 ) = (w(e, e)1 , w(e, e)2 ) = w(e, e), w(g, e e) = (w(e, e)1 , w(e, e)1 ), w(g e −1 , e) = (w(e, e)2 , w(e, e)1 ), −1 w(g, e g −1 ) = (w(e, e)1 , w(e, e)−1 , g)1 w(e, e)1 ) = (w(e, e)1 , w(g, g −1 )2 ), 2 w(g w(g e w(g e −1 , g) = w(g, e g) = −1 (by (109)) 2 (w(g , g)1 , w(e, e)−1 2 w(e, e)1 ), (w(e, e)1 , w(g −1 , g)1 ), −1 2 , g −1 ) = (w(e, e)1 , w(g −1 , g)−1 1 w(e, e)1 ). Finally, to calculate u, we shall use (8) and (106)–(108): ψ(u(e, e))|e = w(e, e e)w(e, e e)w(e, e e)−1 = w(e, e e) = w(e, e) = (w(e, e)1 , w(e, e)2 ), ψ(u(e, e))|g = w(g e −1 , e)w(g e −1 , e)w(g e −1 , e)−1 = w(g e −1 , e) = (w(e, e)2 , w(e, e)1 ), ψ(u(e, e))|g−1 = w(g, e e)w(g, e e)w(g, e e)−1 = w(g, e e) = (w(e, e)1 , w(e, e)1 ), whence u(e, e) = (w(e, e)1 , w(e, e)2 , w(e, e)1 ). ψ(u(e, g))|e = w(e, e e)w(e, e g)w(e, e g)−1 = w(e, e e) = w(e, e) = (w(e, e)1 , w(e, e)2 ), GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 31 ψ(u(e, g))|g = w(g e −1 , e)w(g e −1 , g)w(g e −1 , g)−1 = w(g e −1 , e) = (w(e, e)2 , w(e, e)1 ), ψ(u(e, g))|g−1 = w(g, e e)w(g, e g)w(g, e g)−1 = w(g, e e) = (w(e, e)1 , w(e, e)1 ), whence u(e, g) = (w(e, e)1 , w(e, e)2 , w(e, e)1 ). ψ(u(e, g −1 ))|e = w(e, e e)w(e, e g −1 )w(e, e g −1 )−1 = w(e, e e) = w(e, e) = (w(e, e)1 , w(e, e)2 ), ψ(u(e, g −1 ))|g = w(g e −1 , e)w(g e −1 , g −1 )w(g e −1 , g −1 )−1 = w(g e −1 , e) = (w(e, e)2 , w(e, e)1 ), ψ(u(e, g −1 ))|g−1 = w(g, e e)w(g, e g −1 )w(g, e g −1 )−1 = w(g, e e) = (w(e, e)1 , w(e, e)1 ), whence u(e, g −1 ) = (w(e, e)1 , w(e, e)2 , w(e, e)1 ). ψ(u(g, e))|e = w(e, e g)w(g, e e)w(e, e g)−1 = w(g, e e) = (w(e, e)1 , w(e, e)1 ), ψ(u(g, e))|g = w(g e −1 , g)w(e, e e)w(g e −1 , g)−1 = w(e, e e) = (w(e, e)1 , w(e, e)2 ), ψ(u(g, e))|g−1 = w(g, e g)w(g e −1 , e)w(g, e g)−1 = w(g e −1 , e) = (w(e, e)2 , w(e, e)1 ), whence u(g, e) = (w(e, e)1 , w(e, e)1 , w(e, e)2 ). ψ(u(g −1 , e))|e = w(e, e g −1 )w(g e −1 , e)w(e, e g −1 )−1 = w(g e −1 , e) = (w(e, e)2 , w(e, e)1 ), ψ(u(g −1 , e))|g = w(g e −1 , g −1 )w(g, e e)w(g e −1 , g −1 )−1 = w(g, e e) = (w(e, e)1 , w(e, e)1 ), ψ(u(g −1 , e))|g−1 = w(g, e g −1 )w(e, e e)w(g, e g −1 )−1 = w(e, e e) = (w(e, e)1 , w(e, e)2 ), whence u(g −1 , e) = (w(e, e)2 , w(e, e)1 , w(e, e)1 ). ψ(u(g, g −1 ))|e = w(e, e g)w(g, e g −1 )w(e, e e)−1 = (w(e, e)1 , w(g, g −1 )2 ) ψ(u(g, g −1 ))|g = w(g e −1 , g)w(e, e g −1 )w(g e −1 , e)−1 2 = (w(g −1 , g)1 , w(e, e)−1 2 w(e, e)1 ) · (w(e, e)1 , w(e, e)2 ) −1 · (w(e, e)−1 2 , w(e, e)1 ) = (w(g, g −1 )2 , w(e, e)1 ), ψ(u(g, g −1 ))|g−1 = w(g, e g)w(g e −1 ,g = (w(e, e)1 , w(g −1 −1 · (w(e, e)1 , w(g (by (109)) −1 )w(g, e e) , g)1 ) −1 , g)1−1 w(e, e)21 ) −1 · (w(e, e)−1 1 , w(e, e)1 ) = (w(e, e)1 , w(e, e)1 ), whence u(g, g −1 ) = (w(e, e)1 , w(g, g −1 )2 , w(e, e)1 ). ψ(u(g −1 , g))|e = w(e, e g −1 )w(g e −1 , g)w(e, e e)−1 32 MIKHAILO DOKUCHAEV, MYKOLA KHRYPCHENKO, AND JUAN JACOBO SIMÓN 2 = (w(g −1 , g)1 , w(e, e)−1 2 w(e, e)1 ), ψ(u(g −1 , g))|g = w(g e −1 , g −1 )w(g, e g)w(g e −1 , e)−1 2 = (w(e, e)1 , w(g −1 , g)−1 1 w(e, e)1 ) · (w(e, e)1 , w(g −1 , g)1 ) −1 · (w(e, e)−1 2 , w(e, e)1 ) 2 = (w(e, e)−1 2 w(e, e)1 , w(e, e)1 ), ψ(u(g −1 , g))|g−1 = w(g, e g −1 )w(e, e g)w(g, e e)−1 = (w(e, e)1 , w(g, g −1 )2 ) · (w(e, e)1 , w(e, e)2 ) −1 · (w(e, e)−1 1 , w(e, e)1 ) = (w(e, e)1 , w(g −1 , g)1 ), (by (109)) whence 2 u(g −1 , g) = (w(g −1 , g)1 , w(e, e)−1 2 w(e, e)1 , w(e, e)1 ). ψ(u(g, g))|e = w(e, e g)w(g, e g)w(e, e g −1 )−1 = (w(e, e)1 , w(g −1 , g)1 ), ψ(u(g, g))|g = w(g e −1 , g)w(e, e g)w(g e −1 , g −1 )−1 2 = (w(g −1 , g)1 , w(e, e)−1 2 w(e, e)1 ) · (w(e, e)1 , w(e, e)2 ) · (w(e, e)1−1 , w(g −1 , g)1 w(e, e)−2 1 ) = (w(g −1 , g)1 , w(g −1 , g)1 ), ψ(u(g, g))|g−1 = w(g, e g)w(g e −1 , g)w(g, e g −1 )−1 = (w(e, e)1 , w(g −1 , g)1 ) 2 · (w(g −1 , g)1 , w(e, e)−1 2 w(e, e)1 ) · (w(e, e)1−1 , w(g, g −1 )−1 2 ) = (w(g −1 , g)1 , w(e, e)1 ), (by (109)) whence u(g, g) = (w(e, e)1 , w(g −1 , g)1 , w(g −1 , g)1 ). ψ(u(g −1 , g −1 ))|e = w(e, e g −1 )w(g e −1 , g −1 )w(e, e g)−1 2 = (w(e, e)1 , w(g −1 , g)−1 1 w(e, e)1 ), ψ(u(g −1 , g −1 ))|g = w(g e −1 , g −1 )w(g, e g −1 )w(g e −1 , g)−1 2 = (w(e, e)1 , w(g −1 , g)−1 1 w(e, e)1 ) · (w(e, e)1 , w(g, g −1 )2 ) −2 · (w(g −1 , g)−1 1 , w(e, e)2 w(e, e)1 ) 2 = (w(g −1 , g)−1 1 w(e, e)1 , w(e, e)1 ), (by (109)) GLOBALIZATION OF PARTIAL COHOMOLOGY OF GROUPS 33 ψ(u(g −1 , g −1 ))|g−1 = w(g, e g −1 )w(e, e g −1 )w(g, e g)−1 = (w(e, e)1 , w(g, g −1 )2 ) · (w(e, e)1 , w(e, e)2 ) −1 · (w(e, e)−1 , g)−1 1 , w(g 1 ) = (w(e, e)1 , w(e, e)1 ), (by (109)) whence 2 u(g −1 , g −1 ) = (w(e, e)1 , w(g −1 , g)−1 1 w(e, e)1 , w(e, e)1 ).  2 2 Remark 6.2. The groups H (G, A) and H (G, B) are trivial. Proof. Let w ∈ Z 2 (G, A). Without loss of generality, we may assume w to be normalized (see [26, Remark 2.6]), i.e. w(e, e) = 1A , w(e, g) = w(g, e) = 1g and w(e, g −1 ) = w(g −1 , e) = 1g−1 . Take ε(e) = 1A , ε(g) = 1g and ε(g −1 ) = w(g −1 , g)−1 . Then v := w · δ 1 ε is also normalized and satisfies additionally v(g −1 , g) = 1g−1 . Writing the partial 2-cocycle identity for v with the triple (g, g −1 , g), we obtain v(g, g −1 ) = αg (v(g −1 , g)) = 1g . Since also v(g, g) and v(g −1 , g −1 ) belong to Dg Dg−1 = {0A }, we conclude that v is trivial. Let u ∈ Z 2 (G, B). As in the partial case, multiplying u by a suitable coboundary, we may make u(e, e) = u(e, g) = u(g, e) = u(e, g −1 ) = u(g −1 , e) = u(g −1 , g) = u(g, g −1 ) = 1B . (113) Now, the 2-cocycle identity for u written with the triple (g, g, g) gives βg (u(g, g)) = u(g, g), so that u(g, g)1 = u(g, g)2 = u(g, g)3 . Furthermore, the same identity with (g −1 , g, g) implies u(g −1 , g −1 ) = βg−1 (u(g, g))−1 = u(g, g)−1 . Thus, it suffices to make u(g, g) = 1B maintaining the conditions (113). Take ε(e) = 1B , −1 ) = (1A1 , u(g, g)1 , 1A3 ). Then v := u · δ 1 ε is ε(g) = (1A1 , 1A2 , u(g, g)−1 1 ) and ε(g normalized, v(g −1 , g) = u(g −1 , g)βg−1 (ε(g))ε(e)−1 ε(g −1 ) = βg−1 (ε(g))ε(g −1 ) = 1B , so that v(g, g −1 ) = βg (v(g −1 , g)) = 1B . Finally, v(g, g) = u(g, g)βg (ε(g))ε(g −1 )−1 ε(g) −1 (1A1 , 1A2 , u(g, g)−1 = u(g, g)βg (1A1 , 1A2 , u(g, g)−1 1 ) 1 )(1A1 , u(g, g)1 , 1A3 ) −1 −1 = u(g, g)(u(g, g)−1 1 , u(g, g)1 , u(g, g)1 ) = 1B , so that v(g −1 , g −1 ) = v(g, g)−1 = 1B too.  Acknowledgments The first two authors would like to express their sincere gratitude to the Department of Mathematics of the University of Murcia for its warm hospitality during their visits. 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Insituto de Matemática e Estatı́stica, Universidade de São Paulo, Rua do Matão, 1010, São Paulo, SP, CEP: 05508–090, Brazil E-mail address: [email protected] Departamento de Matemática, Universidade Federal de Santa Catarina, Campus Reitor João David Ferreira Lima, Florianópolis, SC, CEP: 88040–900, Brazil E-mail address: [email protected] Departamento de Matemáticas, Universidad de Murcia, 30071 Murcia, España E-mail address: [email protected]