Partial Galois cohomology and related homomorphisms (expanded version)
aa r X i v : . [ m a t h . R A ] S e p PARTIAL GALOIS COHOMOLOGY AND RELATEDHOMOMORPHISMS (EXPANDED VERSION)
M. DOKUCHAEV, A. PAQUES, AND H. PINEDO
Abstract.
For a partial Galois extension of commutative rings we give a seven termssequence, which is an analogue of the Chase-Harrison-Rosenberg sequence. Introduction
The concept of a Galois extension of commutative rings was introduced in the samepaper by M. Auslander and O. Goldman [2], in which they laid the foundations forseparable extensions of commutative rings and defined the Brauer group of a commu-tative ring. Later, in [9], S. U. Chase, D.K. Harrison and A. Rosenberg developedGalois theory of commutative rings by giving several equivalent definitions of a Galoisextension, establishing a Galois correspondence, and specifying, to the case of a Galoisextension, the Amitsur cohomology seven terms exact sequence, given by S. U. Chaseand A. Rosenberg in [8]. The Chase-Harrison-Rosenberg sequence can be viewed as acommon generalization of the two most fundamental facts from Galois cohomology offields: the Hilbert’s Theorem 90 and the isomorphism of the relative Brauer group withthe second cohomology group of the Galois group. Since then much attention have beenpayed to the sequence and its parts subject to more constructive proofs, generalizationsand analogues in various contexts.Our point of view is to replace global actions by partial ones. The latter are be-coming an object of intensive research and have their origins in the theory of operatoralgebras, where they, together with the corresponding crossed products and partial rep-resentations, form the essential ingredients of a new and successful method to study C ∗ -algebras generated by partial isometrics, initiated by R. Exel in [25], [26], [27], [28]and [29]. The first algebraic results on these new concepts, established in [29], [18], [44],[45], [34] and [14], and the development of a Galois theory of partial actions in [19], stim-ulated a growing algebraic activity around partial actions (see the surveys [13] and [31]).In particular, partial Galois theoretic results have been obtained in [3], [6], [7], [32], [38],[42], and applications of partial actions were found to graded algebras in [14] and [17],to tiling semigroups in [35], to Hecke algebras in [30], to automata theory in [24], to Date : September 19, 2018.The first author was partially supported by FAPESP and CNPq of Brazil. The second author waspartially supported by FAPESP of Brazil. The third author was supported by FAPESP of Brazil.
Mathematics Subject Classification : Primary 13B05; Secondary 13A50; 16H05; 16S35; 16W22;20M18.
Key words and phrases:
Partial action, partial Galois extension, partial cohomology, crossed product,Azumaya algebra, Brauer group, Picard group, Picard monoids. restriction semigroups in [10] and [37] and to Leavitt path algebras in [33]. In addi-tion, the interpretation of the famous R. Thompson’s groups as partial action groupson finite binary words permitted J.-C. Birget to study algorithmic problems for them[5]. Amongst the recent advances we mention a remarkable application of the theoryof partial actions to paradoxical decompositions and to algebras related to separatedgraphs [1], its efficient use in the study of the Carlsen-Matsumoto C ∗ -algebras associ-ated to arbitrary subshifts [15] and of the Steinberg algebras [4], as well as the proofof an algebraic version of the Effros-Hahn conjecture on the ideals in partial crossedproducts [16].The general notion of a continuous twisted partial action of a locally compact groupon a C ∗ -algebra introduced in [27], and adapted to the abstract ring theoretic contextin [17], contains multipliers which satisfy a sort of 2-cocycle identity, and it was naturalto ask what kind of cohomology theory would suite it. The answer was given in [20],where the partial cohomology of groups were introduced and studied together with theirrelation to cohomology of inverse semigroups, showing also that it fits nicely the theoryof partial projective group representations developed in [21], [22] and [23]. Note thatpartial group cohomology turned out to be useful to study ideals of global reduced C ∗ -crossed products [36].Having at hand partial Galois theory and partial group cohomology we may ask nowwhat would be the analogue of the Chase-Harrison-Rosenberg exact sequence in thecontext of a partial Galois extension of commutative rings. The purpose of the presentpaper is to answer this question. The additional new ingredients include a partial actionof the Galois group G on the disjoint union of the Picard groups of all direct summandsof R (see Section 4), as well as partial representations of G (see Section 6).In Section 2 we recall for reader’s convenience some facts used in the paper, whereasthe homomorphisms of the sequence are given in Sections 4, 5, 6.Our proofs are constructive and the partial case is essentially more laborious than theclassical one, so that in this article we only build up the homomorphisms, and the proofof the exactness of the sequence will be given in a forthcoming paper.Throughout this work the word ring means an associative ring with an identity el-ement. For any ring R by an R -module we mean a left unital R -module. If R iscommutative, we shall consider an R -module M as a central R - R -bimodule, i.e, an R - R -bimodule M with mr = rm for all m ∈ M and r ∈ R. We write that M is a f.g.p. R -module if M is a (left) projective and finitely generated R -module, and by a faithfullyprojective R -module we mean a faithful, f.g.p. R -module. For a monoid (or a ring) T, the group of its units (i.e, invertible elements) is denoted by U ( T ) . In all what follows,unless otherwise stated, R will denote a commutative ring and unadorned ⊗ will mean ⊗ R . Preliminaries
In this section we give some definitions and results which will be used in the paper.All modules over commutative rings are considered as central bimodules.2.1.
The Brauer group of a commutative ring.
Recall that an R -algebra A is called separable if A is a projective module over its enveloping algebra A e = A ⊗ A op , where ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 3 A op denotes the opposite algebra of A. If A is faithful as an R -module we can identify R with R A , and if, in addition, its center C ( A ) is equal to R we say that A is central .Moreover, A is called an Azumaya R -algebra if A is central and separable. Equivalently, A is a faithfully projective R -module and A ⊗ A op ≃ End R ( A ) as R -algebras, (see [2,Theorem 2.1(c)]).In [2] the following equivalence relation was defined on the class of all Azumaya R -algebras: A ∼ B if there exist faithfully projective R -modules P and Q such that A ⊗ End R ( P ) ∼ = B ⊗ End R ( Q ) , as R -algebras.Let [ A ] denote the equivalence class containing A , and B ( R ) the set of all such equi-valence classes. Then B ( R ) has a natural structure of a multiplicative abelian group,whose multiplication is induced by the tensor product of R -algebras, that is,[ A ][ B ] = [ A ⊗ B ] , for all [ A ] , [ B ] ∈ B ( R ) . Its identity element is [ R ] and [ A ] − = [ A op ], for all [ A ] ∈ B ( R ). This group is calledthe Brauer group of R.
According to [2], for any commutative R -algebra S, the map from B ( R ) to B ( S ),given by [ A ] [ A ⊗ S ], is a well defined group homomorphism. Its kernel is denoted by B ( S/R ) and called the relative Brauer group of S over R . If A is an Azumaya R -algebrawhose equivalence class in B ( R ) belongs to B ( S/R ) , we say that A is split by S, or that S is a splitting ring for A. For any nonempty subset X of a ring B and any subring V of B, we denote by C V ( X ) = { y ∈ V | xy = yx for all x ∈ X } the centralizer of X in V. In particular,if X = V, then C V ( V ) is the center C ( V ) of V. It is known that a commutative R -subalgebra B of an R -algebra A is a maximal commutative subalgebra if and only if C A ( B ) = B. Partial cohomology of groups.
Let G be a group. A unital twisted partial action of G on R is a triple α = ( { D g } g ∈ G , { α g } g ∈ G , { ω g,h } ( g,h ) ∈ G × G ) , such that for every g ∈ G, D g is an ideal of R generated by a non-necessarily non-zeroidempotent 1 g , α g : D g − → D g is a ring isomorphism, for each pair ( g, h ) ∈ G × G,ω g,h ∈ U ( D g D gh ) and for all g, h, l ∈ G the following statements are satisfied:(i) D = R and α is the identity map of R, (ii) α g ( D g − D h ) = D g D gh , (iii) α g ◦ α h ( t ) = ω g,h α gh ( t ) ω − g,h for any t ∈ D h − D ( gh ) − , (iv) ω ,g = ω g, = 1 g and(v) α g (1 g − ω h,l ) ω g,hl = ω g,h ω gh,l . Using (v) one obtains(1) α g ( ω g − ,g ) = ω g,g − , for any g ∈ G. M. DOKUCHAEV, A. PAQUES, AND H. PINEDO
In [17] the authors defined twisted partial actions of groups on algebras in a moregeneral setting in which the D g ’s are not necessarily unital rings. In all what follows weshall use only unital twisted partial actions.If ( { D g } g ∈ G , { α g } g ∈ G , { ω g,l } ( g,l ) ∈ G × G ) is a twisted partial action of G on R, the familyof partial isomorphisms ( D g , α g ) g ∈ G forms a partial action which we denote by α. Then,the family ω = { ω g,h } ( g,h ) ∈ G × G is called a twisting of α, and the above twisted partialaction will be denoted by ( α, ω ) . If R , in particular, is a multiplicative monoid, then one obtains from the above defi-nition the concept of a unital twisted partial action of a group on a monoid.We recall from [20] the following. Definition 2.1.
Let T be a commutative ring or monoid, n ∈ N , n > and α =( T g , α g ) g ∈ G a unital partial action of G on T. An n -cochain of G with values in T is afunction f : G n → T, such that f ( g , . . . , g n ) ∈ U ( T g g g . . . g g ...g n ) . A -cochain isan element of U ( T ) . Denote the set of all n -cochains by C n ( G, α, T ). This set is an abelian group via thepoint-wise multiplication. Its identity is the map ( g , . . . , g n ) g g g . . . g g ...g n andthe inverse of f ∈ C n ( G, α, T ) is f − ( g , . . . , g n ) = f ( g , . . . , g n ) − , where f ( g , . . . , g n ) − is the inverse of f ( g , . . . , g n ) in T g g g . . . g g ...g n , for all g , . . . , g n ∈ G . Definition 2.2. ( The coboundary homomorphism ) Given n ∈ N , n > , f ∈ C n ( G, α, T ) and g , . . . , g n +1 ∈ G , set ( δ n f )( g , . . . , g n +1 ) = α g (cid:16) f ( g , . . . , g n +1 )1 g − (cid:17) n Y i =1 f ( g , . . . , g i g i +1 , . . . , g n +1 ) ( − i f ( g , . . . , g n ) ( − n +1 . (2) Here, the inverse elements are taken in the corresponding ideals. If n = 0 and t is aninvertible element of T , we set ( δ t )( g ) = α g ( t g − ) t − , for all g ∈ G . Proposition 2.3. [20, Proposition 1.5] δ n is a group homomorphism from C n ( G, α, T ) to C n +1 ( G, α, T ) such that ( δ n +1 δ n f )( g , g , . . . , g n +2 ) = 1 g g g . . . g g ...g n +2 , for any n ∈ N , f ∈ C n ( G, α, T ) and g , g , . . . , g n +2 ∈ G. Definition 2.4.
For n ∈ N , we define the groups Z n ( G, α, T ) = ker δ n of partial n -cocycles, B n ( G, α, T )= im δ n − of partial n -coboundaries, and H n ( G, α, T ) = ker δ n im δ n − ofpartial n -cohomologies of G with values in T , n ≥ . For n = 0 we define H ( G, α, T ) = Z ( G, α, T ) = ker δ . Example 2.5. H ( G, α, T ) = Z ( G, α, T ) = { t ∈ U ( T ) | α g ( t g − ) = t g , ∀ g ∈ G } ,B ( G, α, T ) = { f ∈ C ( G, α, T ) | f ( g ) = α g ( t g − ) t − , for some t ∈ U ( T ) } . In the sense defined in [14].
ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 5
We have ( δ f )( g, h ) = α g ( f ( h )1 g − ) f ( gh ) − f ( g ) for f ∈ C ( G, α, T ) , so that Z ( G, α, T ) = { f ∈ C ( G, α, T ) | f ( gh )1 g = f ( g ) α g ( f ( h )1 g − ) , ∀ g, h ∈ G } , moreover B ( G, α, T ) is the group (cid:8) w ∈ C ( G, α, T ) | ∃ f ∈ C ( G, α, T ) , with w ( g, h ) = α g ( f ( h )1 g − ) f ( g ) f ( gh ) − (cid:9) . For n = 2 we obtain ( δ w )( g, h, l ) = α g ( w ( h, l )1 g − ) w ( gh, l ) − w ( g, hl ) w ( g, h ) − , with w ∈ C ( G, α, T ) , and Z ( G, α, T ) is { w ∈ C ( G, α, T ) | α g ( w ( h, l )1 g − ) w ( g, hl ) = w ( gh, l ) w ( g, h ) , ∀ g, h, l ∈ G } . Hence, the elements of Z ( G, α, T ) are exactly the twistings for α. (cid:4) Two cocycles f, f ′ ∈ Z n ( G, α, T ) are called cohomologous if they differ by an n -coboundary. Remark 2.6.
Notice that a -cocycle is always normalized, i.e. f (1) = 1 T . Indeed,taking g = h = 1 in the -cocycle equality we immediately see that f (1) = f (1) , so f (1) must be T , as f (1) ∈ U ( T ) . Partial Galois extensions.
Let G be a finite group and α = ( D g , α g ) g ∈ G a unital(non twisted) partial action of G on R . The subring of invariants of R under α wasintroduced in [19] as(3) R α = { r ∈ R | α g ( r g − ) = r g for all g ∈ G } . Notice that R α = H ( G, α, T ) . The ring extension R ⊇ R α is called an α -partial Galois extension if for some m ∈ N there exists a subset { x i , y i | ≤ i ≤ m } of R such that m X i =1 x i α g ( y i g − ) = δ ,g , g ∈ G. As in [19], we call the set { x i , y i | ≤ i ≤ m } a partial Galois coordinate system of R ⊇ R α . The trace map tr R/R α : R → R α is given by x P g ∈ G α g ( x g − ) . By [19, Remark3.4] there exists c ∈ R such that tr R/R α ( c ) = 1 , provided that the extension R ⊇ R α is α -partial Galois (here we write 1 = 1 R = 1 R α ).The partial skew group ring R ⋆ α G is defined as the set of all formal sums P g ∈ G r g δ g , r g ∈ D g , with the usual addition and the multiplication determined by the rule( r g δ g )( r ′ h δ h ) = r g α g ( r ′ h g − ) δ gh . It is shown in [19, Theorem 4.1] that R ⊇ R α is a partial Galois extension if and onlyif R is a f.g.p. R α -module and the map(4) j : R ⋆ α G → End R α ( R ) , j X g ∈ G r g δ g ( r ) = X g ∈ G r g α g ( r g − ) M. DOKUCHAEV, A. PAQUES, AND H. PINEDO is an R α -algebra and an R -module isomorphism.3. On generalizations of the Picard group
To construct our version of the seven term sequence, we need some generalizations ofthe concept of the Picard group of a commutative ring. First, we recall the next.
Definition 3.1.
The abelian group of all R -isomorphism classes of f.g.p. R -modules ofrank 1, with binary operation given by [ P ][ Q ] = [ P ⊗ Q ] is denoted by Pic ( R ) . The identityin
Pic ( R ) is [ R ] and the inverse of [ P ] in Pic ( R ) is [ P ∗ ] , where M ∗ = Hom R ( M, R ) forany R -module M. Recall that if P is a faithfully projective R -module, then [ P ] ∈ Pic ( R ) exactly whenthe map R → End R ( P ) , given by r m r , where m r ( p ) = rp for all p ∈ P , is anisomorphism of rings (see [12, Lemma I.5.1]). We also recall the next. Proposition 3.2. [12, Hom-Tensor Relation I.2.4]
Let A and B be R -algebras. Let M be a f.g.p. A -module and N be a f.g.p. B -module. Then for any A -module M ′ and any B -module N ′ , the map ψ : Hom A ( M, M ′ ) ⊗ Hom B ( N, N ′ ) → Hom ( A ⊗ B ) ( M ⊗ N, M ′ ⊗ N ′ ) , induced by ( f ⊗ g )( m ⊗ n ) = f ( m ) ⊗ g ( n ) , for all m ∈ M, n ∈ N , is an R -module isomor-phism. If M = M ′ and N = N ′ , then ψ is an R -algebra isomorphism. (cid:4) Let Λ be a unital commutative R -algebra. We give the following. Definition 3.3. A Λ - Λ -bimodule P is called R -partially invertible if P is central as an R - R -bimodule, and • P is a left f.g.p. Λ -module, • There is an R -algebra epimorphism Λ op → End Λ ( P ) . Let [ P ] = { M | M is a Λ - Λ -bimodule and M ∼ = P as Λ - Λ -bimodules } . We denote by
PicS R (Λ) the set of the isomorphism classes [ P ] of R -partially invertible Λ - Λ -bimodules. Finally, we set PicS R ( R ) := PicS ( R ) . Proposition 3.4.
The product [ P ][ Q ] = [ P ⊗ Λ Q ] endows PicS R (Λ) with the structureof a semigroup. Proof.
We shall show that [ P ⊗ Λ Q ] ∈ PicS R (Λ) , for any [ P ] , [ Q ] ∈ PicS R (Λ) . Noticethat P ⊗ Λ Q is a left f.g.p. Λ-module. Indeed, there are free f.g left Λ-modules F , F and Λ-modules M , M such that P ⊕ M = F , Q ⊕ M = F . Now consider M and F as central Λ-Λ-bimodules, then by tensoring the two previous relations we see that thereexists a left Λ-module M such that ( P ⊗ Λ Q ) ⊕ M = F ⊗ Λ F , and the assertion follows.By assumption there are R -algebra epimorphisms ξ : Λ op → End Λ ( P ) and ξ : Λ op → End Λ ( Q ) . It follows from Proposition 3.2 that ξ ⊗ ξ : Λ op ⊗ Λ Λ op → End Λ ( P ⊗ Λ Q ) isan R -algebra epimorphism. Since Λ op ∋ λ λ ⊗ Λ Λ ∈ Λ op ⊗ Λ Λ op is an R -algebraisomorphism, we conclude that Λ op ∋ λ ξ ( λ ) ⊗ ξ (1 Λ ) ∈ End Λ ( P ⊗ Λ Q ) is an R -algebra epimorphism. (cid:4) ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 7
Throughout the paper, by Spec( R ) we mean, as usual, the set of all prime ideals of R . Definition 3.5.
We say that a f.g.p. central R - R -bimodule P has rank less than orequal to one, if for any p ∈ Spec( R ) one has P p = 0 or P p ∼ = R p as R p -modules. In thiscase we write rk R ( P ) ≤ . The following result characterizes
PicS ( R ) . Proposition 3.6.
We have
PicS ( R ) = { [ E ] | E is a f.g.p. central R - R -bimodule and rk R ( E ) ≤ } . Proof.
Let E be a f.g.p. central R - R -bimodule such that rk R ( E ) ≤ , and considerthe map m R : R ∋ r m r ∈ End R ( E ) , m r ( x ) = rx, r ∈ R, x ∈ E. Via localization it is easy to show that m R is an R -algebra epimorphism.Conversely if [ E ] ∈ PicS ( R ) , then E is a f.g.p. central R - R -bimodule and there is an R -algebra epimorphism R → End R ( E ) . Thus, for any p ∈ Spec( R ) there is an R -algebraepimorphism R p → End R p ( R n p p ) ≃ M n p ( R p ) , where n p = rk R p ( E p ) , which gives n p ≤ (cid:4) Remark 3.7.
Notice that U ( PicS ( R )) = Pic ( R ) . Indeed, the inclusion U ( PicS ( R )) ⊇ Pic ( R ) is trivial. On the other hand, for [ E ] ∈ U ( PicS ( R )) there exists [ P ] ∈ PicS ( R ) with E ⊗ P ∼ = R, so that E p ⊗ P p ∼ = R p for each prime p in R. Then E p = 0 and we seeby Proposition 3.6 that rk R p ( E p ) = 1 for each prime p , and thus [ E ] ∈ Pic ( R ) . Given an inverse semigroup S, we denote the inverse of s ∈ S by s ∗ . We proceed withthe following fact.
Proposition 3.8.
The set
PicS ( R ) with the binary operation induced by the tensorproduct is a commutative inverse monoid with 0. Moreover [ E ∗ ] = [ E ] ∗ , for all [ E ] ∈ PicS ( R ) . Proof.
It follows from Proposition 3.4 and Proposition 3.6 that
PicS ( R ) is a com-mutative monoid with 0 . Take [ M ] ∈ PicS ( R ) . By Proposition 3.2 we obtain ( M ∗ ) p ∼ = ( M p ) ∗ = Hom R p ( M p , R p ) , for all p ∈ Spec( R ) , and hence [ M ∗ ] ∈ PicS ( R ) thanks to Proposition 3.6.Now we prove that [ M ][ M ∗ ][ M ] = [ M ] and [ M ∗ ][ M ][ M ∗ ] = [ M ∗ ] . Recall that M ⊗ M ∗ ∼ = End R ( M ) , since M is a f.g.p. R -module (see [12, Lemma I.3.2 (a)]), and we get[ M ][ M ∗ ][ M ] =[End R ( M )][ M ] . There is an R -module homomorphism κ : End R ( M ) ⊗ M ∋ f ⊗ m f ( m ) ∈ M, and via localization we will prove that κ is an isomorphism. Indeed, take p ∈ Spec( R )then there are two cases to consider. Case 1: M p = 0 . In this case κ p : 0 → R p -module isomorphism. Case 2: M p ∼ = R p . Here we have κ p : R p ⊗ R p R p ∋ r ′ p ⊗ R p r p → r ′ p r p ∈ R p is an R p -module isomorphism. M. DOKUCHAEV, A. PAQUES, AND H. PINEDO
From this we conclude that [ M ][ M ∗ ][ M ] = [ M ] , for all [ M ] ∈ PicS ( R ) . Finally since M is a f.g.p. R -module, there is an R -module isomorphism M ∼ = ( M ∗ ) ∗ (see [40, TheoremV.4.1]), and consequently [ M ∗ ][ M ][ M ∗ ] = [ M ∗ ][( M ∗ ) ∗ ][ M ∗ ] = [ M ∗ ] . (cid:4) By Proposition 3.8 and Clifford’s Theorem (see for instance [11]),
PicS ( R ) is a semi-lattice of abelian groups. In particular, PicS ( R ) = [ ζ ∈ F ( R ) PicS ζ ( R ) , where F ( R ) is a semilattice isomorphic to the semilattice of the idempotents of PicS ( R ) . Therefore, to describe
PicS ( R ) we need to know its idempotents.We recall that given an inverse semigroup S, its idempotents form a commutativesubsemigroup which is a semilattice with respect to the natural order, given by e ≤ f ⇔ ef = e. Let T be a commutative ring. For a T -module M denote by Ann T ( M ) the annihi-lator of M in T . If M is a finitely generated T -module, the sets Supp T ( M ) = { p ∈ Spec( T ) | M p = 0 } and V (Ann T ( M )) = { p ∈ Spec( T ) | p ⊇ Ann T ( M ) } coincide, (see e.g.[41, p. 25-26]).The following lemma characterizes the idempotents of PicS ( R ) . Lemma 3.9.
Let M be a f.g.p. R -module and I M = Ann R ( M ) . Then, the followingstatements are equivalent: (i) M ⊗ M ∼ = M . (ii) M ∼ = R/I M . (iii) M ∼ = Re (and I M = R (1 − e ) ), for some idempotent e of R . Proof. (i) ⇒ (ii) It easily follows from the dual basis lemma that M is a faith-fully projective ( R/I M )-module. Moreover, the R -module isomorphism M ⊗ M ∼ = M, implies M ⊗ R/I M M ∼ = M as R/I M -modules, and hence rk R/I M ( M ) ≤
1. Moreover,Supp
R/I M ( M ) = V (¯0) = Spec( R/I M ) , and since M is a faithfully projective R/I M -module, then [ M ] ∈ Pic ( R/I M ) . Being an idempotent, [ M ] must be the identity ele-ment of Pic ( R/I M ) , so that there is an R/I M -module isomorphism M ∼ = R/I M , whichis clearly an isomorphism of R -modules.(ii) ⇒ (iii) Since M ∼ = R/I M is f.g.p. as an R -module, then the exact sequence0 → I M → R → R/I M → I M is a direct summand of R and the assertion easily follows.(iii) ⇒ (i) It is clear. (cid:4) Let I p ( R ) be the semilattice of the idempotents of R with respect to the product. Ifthe R -modules Re ∼ = Rf are isomorphic where e, f ∈ I p ( R ), then their annihilators in R coincide, i.e. (1 − e ) R = (1 − f ) R. This yields e = f, and it follows by Lemma 3.9 thatthe map e [ eR ] is an isomorphism of I p ( R ) with the semilattice of the idempotents of ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 9
PicS ( R ) . Consequently, the components of
PicS ( R ) can be indexed by the idempotentsof R, and PicS ( R ) = [ e ∈ I p ( R ) PicS e ( R )gives the decomposition of PicS ( R ) as a semilattice of abelian groups. Thus, if for each e ∈ Ip ( R ) we denote by [ M e ] the identity element of PicS e ( R ) , then [ M e ][ M f ] = [ M ef ]and PicS e ( R ) PicS f ( R ) ⊆ PicS ef ( R ) , for all e, f ∈ I p ( R ) (this also can be seen directlyfrom Lemma 3.9).Now we will describe the components of PicS ( R ) . For this, note that for any R -module N we have Ann R ( N ) = Ann R (End R ( N )) , and if N is projective it follows from the dualbasis lemma that Ann R ( N ) = Ann R ( N ∗ ) . Lemma 3.10. PicS e ( R ) = { [ N ] ∈ PicS ( R ) | Ann R ( N ) = R (1 − e ) } ∼ = Pic ( Re ) , for all e ∈ Ip ( R ) . In particular, the identity element of
PicS e ( R ) is [ Re ] . Proof.
For the first equality let [ N ] ∈ PicS e ( R ) . Then, there are R -module isomor-phisms End R ( N ) ∼ = N ⊗ N ∗ ∼ = M e ∼ = Re, where the last isomorphism follows from Lemma3.9, and hence Ann R ( N ) = Ann R (End R ( N )) = R (1 − e ) . Conversely, if Ann R ( N ) = R (1 − e ) we have N ⊗ M e ∼ = N ⊗ Re ∼ = N e = N e ⊕ N (1 − e ) = N, as R -modules, so[ N ] ∈ PicS e ( R ) . Thus for any [ N ] ∈ PicS e ( R ) , its representative is a faithfully projec-tive Re -module and the isomorphism PicS e ( R ) ∼ = Pic ( Re ) is now trivial. Finally, noticethat the image of e in R p is either 0 (if e ∈ p ) or the identity of R p (if e / ∈ p ). Hence( eR ) p is free of rank 0 or 1 , and, consequently, [ eR ] ∈ PicS ( R ) . Summarizing, we have.
Theorem 3.11.
The (disjoint) union (5)
PicS ( R ) = [ e ∈ I p ( R ) PicS e ( R ) ∼ = [ e ∈ I p ( R ) Pic ( Re ) gives the decomposition of PicS ( R ) as a semilattice of abelian groups, whose structuralhomomorphisms are given by ε e,f : Pic ( Re ) → Pic ( Rf ) , [ M ] [ M ⊗ Rf ] , where e, f ∈ I p ( R ) , e ≥ f. (cid:4) We point out the following.
Lemma 3.12.
For any g ∈ G, we have: (i) PicS g ( R ) ∼ = Pic ( D g ) . (ii) Let g ∈ G and [ M ] ∈ PicS ( R ) . If g m = m, for all m ∈ M and M p ∼ = ( D g ) p as R p -modules, for all p ∈ Spec( R ) , then [ M ] ∈ Pic ( D g ) . (iii) PicS ( D g ) = S e ∈ Ip ( R ) e g = e Pic ( Re ) , for any g ∈ G. Proof.
Item (i) is clear from Theorem 3.11.(ii) Notice that M is a f.g.p. D g -module. Let p ∈ Spec( D g ) . Since we have a ringisomorphism D g ∼ = R/ Ann R ( D g ) , we may consider M as an R/ Ann R ( D g )-module andmake the identification p = ¯ p = p / Ann R ( D g ) , where p ∈ Spec( R ) and p contains Ann R ( D g ) . Thus, it follows from the assumption that there exist a ( D g ) ¯ p -module iso-morphisms M p ∼ = M ¯ p ∼ = ( D g ) ¯ p ∼ = ( D g ) p , which imply [ M ] ∈ Pic ( D g ) . (iii) Since (5) holds for any commutative ring, we get PicS ( D g ) = [ e ∈ I p ( D g ) Pic ( D g e ) = [ e ∈ I p ( D g ) Pic ( Re ) . Moreover, e ∈ I p ( D g ) exactly when e is an element of I p ( R ) and e g = e. (cid:4) A partial action on
PicS ( R ) and the sequence H ( G, α, R ) ϕ → Pic ( R α ) ϕ → Pic ( R ) ∩ PicS ( R ) α ∗ ϕ → H ( G, α, R )4.1.
A partial action on PicS( R ). Let α = ( D g , α g ) g ∈ G be a unital partial action ofa group G on R. It is known that α g (1 h g − ) = 1 g gh , for all g, h ∈ G (see [14, p. 1939]).Then for any y ∈ R, we have(6) α g ( α h ( y h − )1 g − ) = α gh ( y ( gh ) − )1 g , for all g, h ∈ G. In all what follows α = ( D g , α g ) g ∈ G will be a fixed unital partial action of the group G on the ring R. The next result will help us in the construction of a partial action on
PicS ( R ) . Lemma 4.1.
Let E and F be central R - R -bimodules and g ∈ G. Suppose that g − x = x and g − y = y for all x ∈ E and y ∈ F. Denote by E g the set E where the (central)action of R is given by r • x g = α g − ( r g ) x = x g • r, r ∈ R, x g ∈ E g . Then (i) E g is an R -module and ( E g ) p = ( E p ) g as R -modules, where the action of R on ( E p ) g is r • xs = α g − ( r g ) xs , for any x ∈ E, p ∈ Spec( R ) , s ∈ R \ p . (ii) Hom R ( E, F ) = Hom R ( E g , F g ) as sets. In particular, we have Iso R ( E, F ) =Iso R ( E g , F g ) and End R ( E ) = End R ( E g ) . (iii) If E is a f.g.p. R -module, so too is E g . (iv) There is an R-module isomorphism ( E ⊗ F ) g ∼ = E g ⊗ F g . (v) If rk( E ) ≤ , then rk( E g ) ≤ . (vi) For any [ M ] ∈ Pic ( D g − ) , [ M g ] ∈ Pic ( D g ) . Proof.
Item (i) is clear.(ii) Obviously Hom R ( E, F ) ⊆ Hom R ( E g , F g ) . Let f ∈ Hom R ( E g , F g ) , r ∈ R and x ∈ E g . Then f ( rx ) = f ( r g − x ) = f ( α g − ( r ′ g ) x ) = f ( r ′ • x ) = r ′ • f ( x ) = rf ( x ) , where r ′ ∈ R is such that α g − ( r ′ g ) = r g − . (iii) For any f ∈ E ∗ , the map α g ◦ f : E g ∋ x α g ◦ f ( x ) = α g ( f ( x )1 g − ) ∈ R ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 11 is an element of ( E g ) ∗ . Indeed, α g ◦ f ( r • x ) = α g ( f ( r • x )1 g − ) = α g ( α g − ( r g ) f ( x )) = r ( α g ◦ f )( x ) . Suppose that E is a f.g.p. R -module. Then, there are f i ∈ E ∗ and x i ∈ E such that x = P i f i ( x ) x i = P i f i ( x )1 g − x i = P i ( α g ◦ f i ( x )) • x i , for any x ∈ E , which implies that E g is a f.g.p. R -module, with dual basis { α g ◦ f i , x i } . (iv) The map E g × F g ∋ ( x, y ) ( x ⊗ y ) g ∈ ( E ⊗ F ) g is R -balanced, therefore it inducesa well defined R -module map(7) ι g : E g ⊗ R F g ∋ x ⊗ y ( x ⊗ y ) g ∈ ( E ⊗ F ) g which is clearly bijective.(v) Take p ∈ Spec( R ) . We have two cases to consider.
Case 1: E p = 0 . In this case we have ( E g ) p (i) = ( E p ) g = 0 . Case 2: E p ∼ = R p as R p -modules. Since 1 g − x = x, for all x ∈ E we have 1 g − r p = r p for all r p ∈ R p , which implies 1 g − / ∈ p , because the image of 1 g − in R p is either 0 orthe identity of R p , and thus E p ∼ = R p = ( D g − ) p as R p -modules.Finally, using (i) we get( E g ) p = ( E p ) g ∼ = (( D g − ) p ) g = (( D g − ) g ) p ∼ = ( D g ) p , where the latter isomorphism is given by α g . This ensures that rk( E g ) ≤ . vi) In the proof of item (iii) we saw that if { f i , m i } is a dual basis for M, then { α g ◦ f i , m i } is a dual basis for the D g -module M g . On the other hand, by the same reasonas in the proof of item (ii), we have that D g ∼ = D g − ∼ = End D g − ( M ) = End D g ( M g ) asrings. Since M is faithful, so too is M g , and the ring isomorphism D g ∼ = End D g ( M g )implies [ M g ] ∈ Pic ( D g ) . (cid:4) Lemma 4.2.
For any g ∈ G set X g = { [1 g E ] | [ E ] ∈ PicS ( R ) } = [ D g ] PicS ( R ) . Then, X g is an ideal of PicS ( R ) and (i) X g = { [ E ] ∈ PicS ( R ) | E = 1 g E } , (ii) For any [ E ] ∈ X g − we have [ E g ] ∈ X g . Proof. (i) It is clear that X g ⊇ { [ E ] ∈ PicS (R) | E = 1 g E } . On the other hand, given[ E ] ∈ X g there exists [ F ] ∈ PicS ( R ) and an R -module isomorphism ϕ g : 1 g F → E. Thisleads to E = ϕ g (1 g F ) = 1 g ϕ g (1 g F ) = 1 g E. (ii) Notice that 1 g • x g = 1 g − x g = x g for any x g ∈ E g , so 1 g • E g = E g . By Lemma4.1 E g is a f.g.p. R -module and rk( E g ) ≤ , hence [ E g ] ∈ PicS ( R ) . Thus, using item (i)we conclude that [ E g ] ∈ X g . (cid:4) Theorem 4.3.
The family α ∗ = ( X g , α ∗ g ) g ∈ G , where α ∗ g : X g − ∋ [ E ] [ E g ] ∈ X g defines a partial action of G on PicS ( R ) . Proof.
By Lemmas 4.1 and 4.2 the map α ∗ g is a well defined semigroup homomor-phism, for all g ∈ G. Clearly X = PicS ( R ) and α ∗ = id PicS ( R ) . We need to show that α ∗ gh is an extension of α ∗ g ◦ α ∗ h . If [ E ] ∈ X h − is such that [ E h ] ∈ X g − , then E = 1 h − E and E = E h = 1 g − • E h = α h − (1 g − h ) E h = 1 ( gh ) − h − E = 1 ( gh ) − E. Thus E = 1 ( gh ) − E, which shows that dom( α ∗ g ◦ α ∗ h ) ⊆ dom α ∗ gh , thanks to item (i)of Lemma 4.2. Furthermore, we have α ∗ g ◦ α ∗ h ([ E ]) = [( E h ) g ] , α ∗ gh ([ E ]) = [ E gh ] , and( E h ) g = E gh as sets. Now, for any r ∈ R, x = ( x h ) g ∈ ( E h ) g we get r • ( x h ) g = α h − ( α g − ( r g )1 h ) x (6) = α ( gh ) − ( r gh )1 h − x = α ( gh ) − ( r gh ) x = r • gh x, and ( E h ) g ∼ = E gh as R -modules. In particular, α ∗ g has an inverse α ∗ g − , so that each α ∗ g is an isomorphism. (cid:4) Remark 4.4.
It follows from Theorem 4.3 and (6) that there is an R -module isomor-phism ( D ( gh ) − ⊗ P ) gh ⊗ D g ∼ = ( D g − ⊗ ( D h − ⊗ P ) h ) g , for any R -module P and g, h ∈ G. The subset of invariants of
PicS ( R ) (see equation (3)) is given by(8) PicS ( R ) α ∗ = { [ E ] ∈ PicS ( R ) | ( D g − ⊗ E ) g ∼ = D g ⊗ E, for all g ∈ G } . Proposition 4.5. PicS ( R ) α ∗ has an element 0 and is a commutative inverse submonoidof PicS ( R ) . Proof.
Evidently 0 ∈ PicS ( R ) α ∗ . Moreover, for any [ E ] , [ N ] ∈ PicS ( R ) α ∗ , we have α ∗ g ([ D g − ⊗ ( E ⊗ N )]) = α ∗ g ([ D g − ⊗ E ]) α ∗ g ([ D g − ⊗ N )]) = [ D g ⊗ E ][ D g ⊗ N ] = [ D g ⊗ ( E ⊗ N )]and [ E ][ N ] ∈ PicS ( R ) α ∗ . Given any element [ E ] of PicS ( R ) α ∗ , we need to show that [ E ∗ ] is also in PicS ( R ) α ∗ . If [ E ] ∈ X g − , for some g ∈ G, then [ E ∗ ] = [ E ∗ ][ E ][ E ∗ ] ∈ X g − . Since [ E ∗ ][ E ][ E ∗ ] = [ E ∗ ]and [ E ][ E ∗ ][ E ] = [ E ] , then [( E ∗ ) g ][ E g ][( E ∗ ) g ] = [( E ∗ ) g ] and [ E g ][ E ∗ g ][ E g ] = [ E g ] , thanksto (iv) of Lemma 4.1 . Thus,(9) [( E ∗ ) g ] = [ E g ] ∗ = [( E g ) ∗ ] , where the last equality follows from Proposition 3.8. Therefore for any [ E ] ∈ PicS ( R ) α ∗ we get { α ∗ g ([ D g − ⊗ E ∗ ]) } ∗ = [( D g − ⊗ E ∗ ) g ] ∗ = [(( D g − ⊗ E ) ∗ ) g ] ∗ (9) = [( D g − ⊗ E ) g ]= [ D g ⊗ E ] = [ D g ⊗ E ] ∗∗ = [ D g ⊗ E ∗ ] ∗ , hence α ∗ g ([ D g − ⊗ E ∗ ]) = [ D g ⊗ E ∗ ] , and [ E ∗ ] ∈ PicS ( R ) α ∗ . Finally, since α ∗ g is a ringisomorphism, we have α ∗ g ([ D g − ]) = [ D g ] , g ∈ G, or equivalently [ R ] ∈ PicS ( R ) α ∗ . (cid:4) ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 13
Remark 4.6.
Recall that by definition X g = PicS ( R )[ D g ] , and thus by Theorem 3.11we have that X g = S e ∈ I p ( R ) Pic ( Re )[ D g ] = S e ∈ I p ( R ) Pic ( D g e ) , where the last equality holdsbecause the map Pic ( Re ) ∋ [ M ] [ M ⊗ D g e ] ∈ Pic ( D g e ) is a group epimorphism.Consequetly, X g = [ e ∈ I p ( R ) Pic ( D g e ) = [ e ∈ I p ( R ) Pic ( Re g ) = [ e ∈ Ip ( Dg ) e g = e Pic ( Re ) = PicS ( D g ) , thanks to (iii) of Lemma 3.12. Finally, Remark 3.7 implies that U ( X g ) = Pic ( D g ) . In all what follows G will stand for a finite group and R ⊇ R α will be an α -partialGalois extension. In particular, it follows from [19, (ii) Theorem 4.1] that R and any D g , g ∈ G, are f.g.p R α -modules. We also recall two well known results that will be usedseveral times in the sequel. The proof of the first can be easily obtained by localization.For the second one we refer to the literature. Lemma 4.7.
Let
M, N be R-modules such that M and M ⊗ N are f.g.p. R-modules, and M p = 0 for all p ∈ Spec( R ) . Then, N is also a f.g.p. R-module.
Lemma 4.8. [12, Chapter I, Lemma 3.2 (b)]
Let M be a faithfully projective R-module.Then, there is a R - R -bimodule isomorphism M ∗ ⊗ End R ( M ) M ∼ = R. Consequently if
N, N ′ are R -modules such that M ⊗ N ∼ = M ⊗ N ′ as R -modules, then N ∼ = N ′ as R -modules. (cid:4) The sequence H ( G, α, R ) ϕ → Pic ( R α ) ϕ → PicS ( R ) α ∗ ∩ Pic ( R ) ϕ → H ( G, α, R ) . For any
R ⋆ α G -module M, as in [19, page 82] we denote M G = { m ∈ M | (1 g δ g ) m = 1 g m, for all g ∈ G } . It can be seen that R is an R ⋆ α G -module via ( r g δ g ) ✄ r = r g α g ( r g − ) , for each g ∈ G and r ∈ R, r g ∈ D g , and this action induces an R ⋆ α G -module structure on R ⊗ R α M G . We have the following.
Theorem 4.9.
There is a group homomorphism ϕ : H ( G, α, R ) → Pic ( R α ) . Proof.
Take f ∈ Z ( G, α, R ) and define θ f ∈ End R α ( R ⋆ α G ) by θ f ( r g δ g ) = r g f ( g ) δ g for all r g ∈ D g , g ∈ G. Then, θ f is an R α -algebra homomorphism because θ f ( r g δ g ) θ f ( r h δ h ) = ( r g f ( g ) δ g )( r h f ( h ) δ h ) = r g f ( g ) α g ( r h f ( h )1 g − ) δ gh = r g α g ( r h g − ) f ( g ) α g ( f ( h )1 g − ) δ gh = r g α g ( r h g − ) f ( gh ) δ gh = θ f (( r g δ g )( r h δ h )) , for all r g ∈ D g , r h ∈ D h , g, h ∈ G. Hence, we may define an R⋆ α G -module R f by R f = R as sets and ( r g δ g ) · r = θ f ( r g δ g ) ✄ r, for any r ∈ R, r g ∈ D g , g ∈ G. In particular R f = R as R -modules, as f is normalized in view of Remark 2.6. By (iii)of [19, Theorem 4.1] there is an R -module isomorphism R ⊗ R α R Gf ∼ = R f . Since R is af.g.p. R α -module we conclude that R Gf is a f.g.p. R α -module by Lemma 4.7. Finally, via localization we see from the last isomorphism that rk R α ( R Gf ) = 1 , so [ R Gf ] ∈ Pic( R α ) . Define ϕ : H ( G, α, R ) ∋ cls( f ) → [ R Gf ] ∈ Pic( R α ) . We will check that ϕ is a well defined group isomorphism. If f ∈ B ( G, α, R ) , thereexists a ∈ U ( R ) such that f ( g ) = α g ( a g − ) a − , for all g ∈ G. In this case one has r ∈ R Gf if and only if, ar ∈ R α . Thus, multiplication by a gives an R α -module isomorphism R Gf ∼ = R α , which yields [ R Gf ] = [ R α ] ∈ Pic( R α ) . Therefore, to prove that ϕ is welldefined, it is enough to show that ϕ preserves products. For any f, g ∈ Z ( G, α, R )there is a chain of R α -module isomorphisms R ⊗ R α ( R Gf ⊗ R α R Gg ) ∼ = ( R ⊗ R α R Gf ) ⊗ ( R ⊗ R α R gG ) ∼ = R f ⊗ R g ∼ = R ⊗ R ∼ = R fg ∼ = R ⊗ R α R Gfg , and recalling that R is a f.g.p. R α -module we have R Gf ⊗ R α R Gg ∼ = R Gfg as R α -modules,by Lemma 4.8. (cid:4) Proposition 4.10.
There is a group homomorphism
Pic ( R α ) ϕ → PicS ( R ) α ∗ ∩ Pic ( R ) . Proof.
For any [ E ] ∈ Pic ( R α ) set ϕ ([ E ]) = [ R ⊗ R α E ] . Clearly ϕ is a well definedgroup homomorphism from Pic ( R α ) to Pic ( R ). We shall check that im ϕ ⊆ PicS ( R ) α ∗ . There are R -module isomorphisms(10) D g ⊗ ( R ⊗ R α E ) ∼ = D g ⊗ R α E and ( D g − ⊗ ( R ⊗ R α E )) g ∼ = ( D g − ⊗ R α E ) g . Furthermore, the map determined by(11) D g ⊗ R α E ∋ d ⊗ R α x α g − ( d ) ⊗ R α x ∈ ( D g − ⊗ R α E ) g , is also an R -module isomorphism. Then, combining (10) and (11) we obtain an R -module isomorphism D g ⊗ ( R ⊗ R α E ) ∼ = ( D g − ⊗ ( R ⊗ R α E )) g , for all g ∈ G, and hence ϕ ([ E ]) ∈ PicS ( R ) α ∗ . (cid:4) Now, we proceed with the construction of ϕ . First, for any R -module M we identify M ⊗ D g with M D g , via the R -module isomorphism M ⊗ D g ∋ x ⊗ d xd ∈ M D g , for any g ∈ G. Now, let [ E ] ∈ PicS ( R ) α ∗ ∩ Pic ( R ) . Then, by (8) there is a family of R -moduleisomorphisms(12) { ψ g : ED g → ( ED g − ) g } g ∈ G with ψ g ( rx ) = α g − ( r g ) ψ g ( x ) , where r ∈ R, x ∈ ED g , g ∈ G. Thus the maps ψ − g : ( ED g − ) g → ED g , g ∈ G, satisfy(13) ψ − g ( rx ) = ψ − g ( α g ( r g − ) • x ) = α g ( r g − ) ψ − g ( x ) , for all r ∈ R, x ∈ ED g − . We shall prove that ψ ( gh ) − ψ − h − ψ − g − : ED g D gh → ED g D gh is well defined and is anelement of U (End D g D gh ( ED g D gh )) . From (13) we have ψ − g − ( ED g D gh ) ⊆ ED g − D h ⊆ dom ψ − h − ∩ dom ψ g − , and also ψ − h − ( ED g − D h ) ⊆ ED h − g − D h − ⊆ dom ψ h − g − ∩ dom ψ h − . This yields that the map ψ ( gh ) − ψ − h − ψ − g − is well defined and ψ g − ψ h − ψ − gh ) − is its inverse. ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 15
Now we check that ψ ( gh ) − ψ − h − ψ − g − is D g D gh -linear. Take d ∈ D g D gh and x ∈ ED g D gh . Using (13) and (12) we get the following ψ ( gh ) − ψ − h − ψ − g − ( dx ) d ∈ D g D gh = ψ ( gh ) − ( α h − ( α g − ( d ))) ψ − h − ψ − g − ( x ) = ψ ( gh ) − ( α h − g − ( d )) ψ − h − ψ − g − ( x ) = dψ ( gh ) − ψ − h − ψ − g − ( x ) . Since [ E ] ∈ Pic ( R ) then [ ED g D gh ] ∈ Pic ( D g D gh ), and thus End D g D gh ( ED g D gh ) ∼ = D g D gh . Moreover, ψ ( gh ) − ψ − h − ψ − g − is an invertible element of End D g D gh ( ED g D gh ) , andhence there exists ω g,h ∈ U ( D g D gh ) such that ψ ( gh ) − ψ − h − ψ − g − ( x ) = ω g,h x, for all x ∈ ED g D gh . Summarizing, for an element [ E ] ∈ PicS ( R ) α ∗ ∩ Pic ( R ) we have found a map ω [ E ] = ω : G × G ∋ ( g, h ) ω g,h ∈ U ( D g D gh ) ⊆ R. We shall see that ω ∈ Z ( G, α, R ) . Take g, h, l ∈ G and x ∈ ED g D gh D ghl . Then ω g,hl α g ( ω h,l g − ) x (12) = ω g,hl ψ g − ( ω h,l ψ − g − ( x ))= ψ ( ghl ) − ψ − hl ) − ψ − g − ψ g − ( ω h,l ψ − g − ( x ))= ψ ( ghl ) − ψ − hl ) − ( ω h,l ψ − g − ( x ))= ψ ( ghl ) − ψ − hl ) − ( ψ ( hl ) − ψ − l − ψ − h − ψ − g − ( x ))= ψ ( ghl ) − ψ − l − ψ − h − ψ − g − ( x )= ψ ( ghl ) − ψ − l − ψ − gh ) − ψ ( gh ) − ψ − h − ψ − g − ( x )= ω gh,l ω g,h x. Notice that [ ED g D gh D ghl ] ∈ Pic ( D g D gh D ghl ) and, in particular, ED g D gh D ghl is afaithful D g D gh D ghl -module. Since ω g,hl α g ( ω h,l g − ) , ω gh,l ω g,h ∈ D g D gh D ghl , we obtainthat ω g,hl α g ( ω h,l g − ) = ω gh,l ω g,h as desired. Claim 4.11. cls ( ω ) does not depend on the choice of the isomorphisms. Let { λ g | g ∈ G } be another choice of R -isomorphisms ED g → ( ED g − ) g . Then λ g ( rx ) = α g − ( r g ) λ g ( x ) , λ − g ( rx ) = α g ( r g − ) λ − g ( x ), for all g ∈ G, r ∈ R and x belonging to the correspondent domain. Let ˜ ω : G × G → R also be defined by˜ ω ( g, h ) x = λ ( gh ) − λ − h − λ − g − ( x ) , for any x ∈ ED g D gh . We shall prove that cls( ω ) = cls(˜ ω ) in H ( G, α, R ) . Since λ g ψ − g : ( ED g − ) g → ( ED g − ) g is D g -linear and [( ED g − ) g ] ∈ Pic ( D g ) , there exists u g ∈ U ( D g ) such that λ g ψ − g is themultiplication by u g . Then, the map u : G ∋ g → u g ∈ R belongs to C ( G, α, R ), and for any x ∈ ED g D gh D ghl we have ω − g,h ˜ ω g,h x = ψ g − ψ h − ψ − gh ) − (˜ ω g,h x )= ( ψ g − λ − g − ) λ g − ( ψ h − λ − h − ) λ − g − ( λ g − λ h − ψ − gh ) − )(˜ ω g,h x )= u − g − ( λ g − u − h − λ − g − )( λ g − λ h − ψ − gh ) − )( ˜ ω g,h |{z} ∈ D g D gh x )= u − g − ( λ g − u − h − λ − g − )˜ ω g,h ( λ g − λ h − ψ − gh ) − )( x )= u − g − ( λ g − u − h − λ − g − )( λ ( gh ) − λ − h − λ − g − λ g − λ h − ψ − gh ) − )( x )= u − g − ( λ g − u − h − λ − g − )( λ ( gh ) − ψ − gh ) − )( x )= u − g − ( λ g − u − h − λ − g − )( u ( gh ) − x )= u − g − λ g − [ u − h − α g − ( u ( gh ) − g ) λ − g − ( x )]= u − g − α g ( u − h − g − ) u ( gh ) − x = v g α g ( v h g − ) v − gh ) x, where v g = u − g − . Since the map v : G ∋ g → v g ∈ R belongs to C ( G, α, R ) , this showsthat cls( ω ) = cls(˜ ω ) in H ( G, α, R ) as desired.Let ϕ : PicS ( R ) α ∗ ∩ Pic ( R ) ∋ [ E ] cls( ω ) ∈ H ( G, α, R ) . Claim 4.12. ϕ does not depend on the choice of the representative of [ E ] , for any [ E ] ∈ PicS ( R ) α ∗ ∩ Pic ( R ) . Let [ E ] = [ F ] ∈ PicS ( R ) α ∗ ∩ Pic ( R ) , and { ψ g | g ∈ G } , { λ g | g ∈ G } be familiesof R -isomorphisms ED g → ( ED g − ) g , F D g → ( F D g − ) g inducing the (2 , α )-cocycles ω, ˜ ω respectively. Let also Ω : E → F be an R -module isomorphism. Then Ω( ED g ) =Ω( E ) D g = F D g and one obtains the R -module isomorphisms Ω | ED g : ED g → F D g andΩ | ED g − : ( ED g − ) g → ( F D g − ) g (thanks to (ii) of Lemma 4.1), for any g ∈ G. Thus,the family { Ω ψ g Ω − | F D g : F D g → ( F D g − ) g | g ∈ G } induces a (2 , α )-cocycle which iscohomologous to ˜ ω, in view of Claim 4.11, and we may suppose that Ω ψ g Ω − = λ g on F D g , g ∈ G. Hence, given x ∈ F D g D gh we have˜ ω g,h x = λ ( gh ) − λ − h − λ − g − ( x ) ≡ Ω ψ ( gh ) − ψ h − ψ g − Ω − ( x ) = Ω( ω g,h Ω − ( x )) = ω g,h x, which implies ˜ ω g,h = ω g,h , because F D g D gh , being an element of Pic ( D g D gh ), is afaithful D g D gh -module. We conclude that in general cls( ω ) = cls(˜ ω ) and ϕ is welldefined. Claim 4.13. ϕ is a group homomorphism. Let [ E ] , [ F ] ∈ PicS ( R ) α ∗ ∩ Pic ( R ) with ϕ ([ E ]) = cls( ω ) and ϕ ([ F ]) = cls( ω ′ ) . Con-sider families of R -module isomorphisms { φ g : ED g → ( ED g − ) g } g ∈ G and { λ g : F D g → ( F D g − ) g } g ∈ G defining cls( ω ) and cls( ω ′ ) respectively. ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 17
Notice that ED g ⊗ F D g = ( E ⊗ F ) D g , for all g ∈ G, and by (ii) and (iv) of Lemma 4.1,( ED g − ) g ⊗ ( F D g − ) g ∼ = (( E ⊗ F ) D g − ) g , g ∈ G, via the map ι g defined in (7). Then, ι g ◦ ( φ g ⊗ λ g ) : ( E ⊗ F ) D g → (( E ⊗ F ) D g − ) g , g ∈ G is an R -module isomorphism which induces an element u ∈ Z ( G, α, R ) , where u g,h x ⊗ y = [ ι ( gh ) − ◦ ( φ ( gh ) − ⊗ λ ( gh ) − )][ ι h − ◦ ( φ h − ⊗ λ h − )] − [ ι g − ◦ ( φ g − ⊗ λ g − )] − ( x ⊗ y )= ( φ ( gh ) − φ − h − φ − g − ⊗ λ ( gh ) − λ − h − λ − g − )( x ⊗ y )= ( ω g,h ⊗ ω ′ g,h )( x ⊗ y ) = ( ω g,h ω ′ g,h )( x ⊗ y ) , for all x ∈ ED g D gh , y ∈ F D g D gh , g, h ∈ G. Then, u = ωω ′ and ϕ is a group homomor-phism. 5. The sequence H ( G, α, R ) ϕ → B ( R/R α ) ϕ → H ( G, α ∗ , PicS ( R ))We start this section by giving some preliminary results that help us to construct thehomomorphism ϕ .First of all we recall from [17] that the partial crossed product R ⋆ α,ω G for the unitaltwisted partial action ( α, ω ) of G on R is the direct sum L g ∈ G D g δ g , in which the δ ′ g sare symbols, with the multiplication defined by the rule:( r g δ g )( r ′ h δ h ) = r g α g ( r ′ h g − ) ω g,h δ gh , for all g, h ∈ G , r g ∈ D g and r ′ h ∈ D h . If, in particular, the twisting ω is trivial, then werecover the partial skew group ring R ⋆ α G as given in Subsection 2.3. Proposition 5.1. If ω, ˜ ω ∈ Z ( G, α, R ) are cohomologous, there is an isomorphism of R α -algebras and R -modules R ⋆ α,ω G ∼ = R ⋆ α, ˜ ω G. Proof.
There exists u ∈ C ( G, α, R ) , u : G ∋ g u g ∈ U ( D g ) ⊆ R such that ω g,h = ˜ ω g,h u g α g ( u h g − ) u − gh , for all g, h ∈ G. Take a g ∈ D g , set ϕ ( a g δ g ) = a g u g δ g andextend ϕ to ϕ : R ⋆ α,ω G → R ⋆ α, ˜ ω G by R -linearity. Clearly ϕ is bijective with inverse a g δ g a g u − g δ g , then we only need to prove that ϕ preserves products. ϕ (( a g δ g )( b h δ h )) = ϕ ( a g α g ( b h g − ) ω g,h δ gh ) = a g α g ( b h g − ) ω g,h u gh δ gh = a g α g ( b h g − )˜ ω g,h u g α g ( u h g − ) δ gh = ( a g u g δ g )( b h u h δ h ) = ϕ ( a g δ g ) ϕ ( b h δ h ) . (cid:4) We also give the next.
Proposition 5.2. If ϕ : R ⋆ α,ω G → R ⋆ α G is an isomorphism of R α -algebras and R -modules, then ω is cohomologous to α = { g gh } g,h ∈ G . Proof.
To avoid confusion, we write
R ⋆ α,ω G = L g ∈ G D g δ g , R ⋆ α G = L g ∈ G D g δ ′ g andidentify R = Rδ = Rδ ′ . For r ∈ R we have ϕ ( rδ ) = rδ ′ ∈ R and(1 g δ g ) r ( ω − g − ,g δ g − ) = α g ( r g − ) α g ( ω − g − ,g ) ω g,g − δ = α g ( r g − ) δ . Hence, ϕ (1 g δ g ) rϕ ( ω − g − ,g δ g − ) = ϕ ( α g ( r g − ) δ ) = α g ( r g − ) δ ′ = (1 g δ ′ g ) r (1 g − δ ′ g − ) . From this we obtain(1 g − δ ′ g − ) ϕ (1 g δ g ) rϕ ( ω − g − ,g δ g − ) = (1 g − δ ′ g − )(1 g δ ′ g ) r (1 g − δ ′ g − ) = r (1 g − δ ′ g − ) , and, multiplying by the right both sides of the equality by ϕ (1 g δ g ), we get r [(1 g − δ ′ g − ) ϕ (1 g δ g )] = [(1 g − δ ′ g − ) ϕ (1 g δ g ) rϕ ( ω − g − ,g δ g − )] ϕ (1 g δ g )= (1 g − δ ′ g − ) ϕ (1 g δ g ) rϕ [( ω − g − ,g δ g − )(1 g δ g )]= (1 g − δ ′ g − ) ϕ (1 g δ g ) rϕ (1 g − δ )= (1 g − δ ′ g − ) ϕ (1 g δ g ) ϕ (1 g − δ ) r = (1 g − δ ′ g − ) ϕ ((1 g δ g )(1 g − δ )) r = (1 g − δ ′ g − ) ϕ (1 g δ g ) r, so (1 g − δ ′ g − ) ϕ (1 g δ g ) ∈ C R⋆ α G ( R ) . Since R is commutative and R ⊇ R α is an α -partial Galois extension, [43, Lemma2.1(vi) and Proposition 3.2] imply that R ⋆ α, ˜ ω G is R α -Azumaya and C R⋆ α, ˜ ω G ( R ) = R for arbitrary ˜ ω, in particular this is true for R ⋆ α G. Thus,(1 g − δ ′ g − ) ϕ (1 g δ g ) = r g , for some r g ∈ R, and, multiplying from the left both the sides of the last equality by 1 g δ ′ g we obtain ϕ (1 g δ g ) = u g δ ′ g , where u g = α g ( r g g − ) ∈ D g . Therefore, ϕ (1 g δ g ) = u g δ ′ g , g ∈ G. On the other hand there exists W = P h ∈ G a h δ h such that 1 g δ ′ g = ϕ ( W ) = P h ∈ G a h u h δ ′ h , then 1 g δ ′ g = a g u g δ ′ g and u g ∈ U ( D g ) . Set u : G ∋ g u g ∈ U ( D g ) ⊆ R, then u ∈ C ( G, α, R ) and ω g,h u gh δ ′ gh = ϕ ( ω g,h δ gh ) = ϕ (1 g δ g ) ϕ (1 h δ h ) = ( u g δ ′ g )( u h δ ′ h ) = u g α g ( u h g − ) δ ′ gh . From this we conclude that ω g,h u gh = u g α g ( u h g − ), hence ω is cohomologous to 1 α . (cid:4) Proposition 5.3.
Let ω ∈ Z ( G, α, R ) . Then
R ⋆ α,ω G is an Azumaya R α -algebra and [ R ⋆ α,ω G ] ∈ B ( R/R α ) . Proof.
As mentioned above, the facts that R is commutative and R ⊇ R α is an α -partial Galois extension imply that R ⋆ α,ω G is R α -Azumaya and C R⋆ α,ω G ( R ) = R. Thelatter implies that R is a maximal commutative R α -subalgebra of R ⋆ α,ω G. On the otherhand, by [19, Theorem 4.2] the extension R ⊇ R α is separable, and finally [2, Theorem5.6] tells us that R ⋆ α,ω G is split by R, which means that [ R ⋆ α,ω G ] ∈ B ( R/R α ) . (cid:4) It follows from Propositions 5.1 and 5.3 that there is a well defined function(14) ϕ : H ( G, α, R ) ∋ cls( ω ) [ R ⋆ α,ω G ] ∈ B ( R/R α ) . ϕ is a homomorphism. In this subsection we follow the ideas of [12, chapter IV]to prove that the map ϕ defined in (14) is a group homomorphism. For this we need aseries of lemmas. ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 19
Lemma 5.4.
Let ω ∈ Z ( G, α, R ) . Then, there exists an R α -algebra isomorphism ( R ⋆ α,ω G ) op ∼ = R ⋆ α,ω − G. Proof.
Define φ : ( R ⋆ α,ω G ) op → R ⋆ α,ω − G by φ ( r g δ g ) = α g − ( r g ω g,g − ) δ g − , for all g ∈ G, r g ∈ D g . Note that φ is an R α -module isomorphism with inverse R ⋆ α,ω − G ∋ r g − δ g − α g ( r g − ω − g − ,g ) δ g ∈ ( R ⋆ α,ω G ) op . For g, h ∈ G, r g ∈ D g and t h ∈ D h we have φ [( r g δ g ) ◦ ( t h δ h )] = φ ( t h α h ( r g h − ) ω h,g δ hg )= α g − h − ( t h α h ( r g h − ) ω h,g ω hg,g − h − ) δ g − h − = α g − [ α h − ( t h α h ( r g h − ) ω h,g ω hg,g − h − )] δ g − h − (v) = α g − [ α h − ( t h α h ( r g h − ) α h ( ω g,g − h − ) ω h,h − )] δ g − h − (1) = α g − ( α h − ( t h ) r g ω g,g − h − ω h − ,h ) δ g − h − . On the other hand φ ( r g δ g ) φ ( t h δ h ) = ( α g − ( r g ω g,g − ) δ g − )( α h − ( t h ω h,h − ) δ h − )= ( α g − ( r g ) ω g − ,g )( α g − ( α h − ( t h ) ω h − ,h ) ω − g − ,h − δ g − h − )= α g − ( r g α h − ( t h ) ω h − ,h ) ω g − ,g ω − g − ,h − δ g − h − (v) = α g − ( r g α h − ( t h ) ω h − ,h ) α g − ( ω g,g − h − ) δ g − h − = α g − ( r g α h − ( t h ) ω h − ,h ω g,g − h − ) δ g − h − , and we conclude that φ is multiplicative. (cid:4) From now on, in order to simplify notation we will denote R e = R ⊗ R α R . Lemma 5.5.
There is a family of orthogonal idempotents e g ∈ R e with g ∈ G, satisfyingthe following properties: (15) (1 ⊗ R α α g ( x g − )) e g = ( x ⊗ R α e g , for all x ∈ R. (16) X g ∈ G e g = 1 R e . Proof.
By (iv) of [19, Theorem 4.1] the map ψ : R e → Q g ∈ G D g , given by ψ ( x ⊗ R α y ) =( xα g ( y g − )) g ∈ G , is an isomorphism of R -algebras. Take v g = ( x h ) h ∈ G ∈ Q h ∈ G D h , g ∈ G, where x h = δ h,g g . Then, the set { e g = ψ − ( v g − ) | g ∈ G } is a family of orthogonalidempotents in R e . Using the isomorphism ψ we check (15) and (16). (cid:4) Lemma 5.6.
For any ω ∈ Z ( G, α, R ) there is an R -module isomorphism R⋆ α,ω G ∼ = R e . Proof.
Let { e g | g ∈ G } be the family of pairwise orthogonal idempotents constructedin Lemma 5.5, and set η : R⋆ α,ω G → R e defined by η ( P g ∈ G r g δ g ) = P g ∈ G ( r g ⊗ R α e g − . Clearly η is R -linear and we only need to check that η is an isomorphism. If P g ∈ G ( r g ⊗ R α e g − = 0 , then ( r h ⊗ R α e h − = 0 , for any h ∈ G. Applying theisomorphism ψ from the proof of Proposition 5.5 we conclude that r h = 0 and η isinjective.Now we prove the surjectivity. By applying ψ we get (1 ⊗ R α r g − ) e g − = (1 ⊗ R α r ) e g − ,r ∈ R, g ∈ G. Then for any r, s ∈ R, we obtain r ⊗ R α s (16) = X g ∈ G ( r ⊗ R α s ) e g − = X g ∈ G ( r ⊗ R α s g − ) e g − (15) = X g ∈ G ( rα g ( s g − ) ⊗ R α e g − = η X g ∈ G rα g ( s g − ) δ g . (cid:4) By Lemma 5.6 the map ¯ η : End R α ( R ⋆ α,ω G ) ∋ f ηf η − ∈ End R α ( R e ) , is an R α -algebra isomorphism.Now we prove that the tensor product of partial Galois extensions is also a partialGalois extension. More precisely we have the following. Proposition 5.7.
Let G and H be finite groups and α = ( D g , α g ) g ∈ G , θ = ( I h , θ h ) h ∈ H (unital) partial actions of G and H on commutative rings R and R , respectively. As-sume that R α = R θ = k and suppose that the ring extensions R ⊇ k, R ⊇ k are α -and θ -partial Galois, respectively. Then R ⊗ k R ⊇ k ⊗ k k = k is an ( α ⊗ k θ ) -partialGalois extension. Proof.
In this proof unadorned ⊗ means ⊗ k . Note that α ⊗ θ = ( D g ⊗ I h , α g ⊗ θ h ) ( g,h ) ∈ G × H , is partial action of G × H on the ring R ⊗ R . Consider x = P i u i ⊗ v i ∈ k ⊗ k = R α ⊗ R θ , then for all ( g, h ) ∈ G × H we have α g ⊗ θ h ( x (1 g − ⊗ h − )) = P i u i g ⊗ v i h = x (1 g ⊗ h ) , and ( R ⊗ R ) α ⊗ θ ⊇ R α ⊗ R θ . Conversely, take c ∈ R and c ∈ R such that tr R /R α ( c ) = 1 k = tr R /R θ ( c ) . Nowlet x ∈ ( R ⊗ R ) α ⊗ θ and write x ( c ⊗ c ) = m P i =1 t i ⊗ s i . Then, x = x (1 R ⊗ R ) = P ( g,h ) ∈ G × H x ( α g (1 g − c ) ⊗ θ h (1 h − c ))= P ( g,h ) ∈ G × H x (1 g ⊗ h )( α g ⊗ θ h )(1 g − c ⊗ h − c )= P ( g,h ) ∈ G × H ( α g ⊗ θ h )( x ( c ⊗ c )(1 g − ⊗ h − ))= P mi =1 P ( g,h ) ∈ G × H ( α g ⊗ θ h )( t i g − ⊗ s i h − )= P mi =1 P ( g,h ) ∈ G × H α g ( t i g − ) ⊗ θ h ( s i h − )= P mi =1 tr R /R α ( t i ) ⊗ tr R /R θ ( s i ) ∈ R α ⊗ R θ . Thus ( R ⊗ R ) α ⊗ θ = R α ⊗ R θ = k. Finally, let m, n ∈ N and { x i , y i | ≤ i ≤ m } , { u i , z i | ≤ i ≤ n } be the α - and θ -partial Galois systems for the extensions R ⊇ k and R ⊇ k, respectively. Then for ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 21 all ( g, h ) ∈ G × H we see that X i,j ( x i ⊗ u j )( α g ⊗ θ h )(( y i ⊗ z j )(1 g − ⊗ h − )) = X i,j ( x i ⊗ u j )( α g ( y i g − ) ⊗ θ h ( z j h − ))= X i x i α g ( y i g − ) ⊗ X j u j θ h ( z j h − ) = δ ,g ⊗ δ ,h = δ (1 , , ( g,h ) . We conclude that the set { x i ⊗ u j , y i ⊗ z j | ≤ i ≤ m, ≤ j ≤ n } is an α ⊗ θ -partialGalois coordinate system for the extension R ⊗ R ⊇ k ⊗ k. (cid:4) With the same hypothesis and notations given in Proposition 5.7, we have.
Proposition 5.8.
There is an isomorphism of k -algebras ( R ⋆ α,ω G ) ⊗ k ( R ⋆ θ, ˜ ω H ) ∼ = ( R ⊗ k R ) ⋆ α ⊗ θ,ω ⊗ ˜ ω ( G × H ) . Proof.
Here again unadorned ⊗ means ⊗ k . We denote S = ( R ⊗ R ) ⋆ α ⊗ θ,ω ⊗ ˜ ω ( G × H ) = M ( g,h ) ∈ G × H ( D g ⊗ I h ) ǫ ( g,h ) ,S = R ⋆ α,ω G = L g ∈ G D g δ g and S = R ⋆ θ, ˜ ω H = L h ∈ H I h δ ′ h . For ( g, h ) ∈ G × H, the map D g δ g × I h δ ′ h ∋ ( a g δ g , b h δ ′ h ) ( a g ⊗ b h ) ǫ ( g,h ) ∈ S extended by k linearity to S × S is clearly a bilinear k -balanced map. Hence, it inducesa bijective k -linear map ξ : S ⊗ S → S such that a g δ g ⊗ b h δ ′ h ( a g ⊗ b h ) ǫ ( g,h ) . The factthat ξ preserves products is straightforward. (cid:4) Proposition 5.9.
Given ω ∈ Z ( G, α, R ) , we have ω ⊗ R α ω − ∈ B ( G × G, α ⊗ R α α, R e ) . Thus if ˜ ω ∈ Z ( G, α, R ) , then ω ⊗ R α ˜ ω is cohomologous to ω ˜ ω ⊗ R α α . Proof.
By Proposition 5.7, R e is an α ⊗ R α α -partial Galois extension of R α withthe partial action of G × G. Then using the isomorphisms appeared in Proposition 5.8,Lemmas 5.4, 5.6, [2, Theorem 2.1 (c)] and iv) of [19, Theorem 4.1] we obtain a chain of R α -algebra isomorphisms( R e ) ⋆ α ⊗ Rα α,ω ⊗ Rα ω − ( G × G ) → ( R ⋆ α,ω G ) ⊗ R α ( R ⋆ α,ω − G ) id ⊗ Rα φ − → ( R ⋆ α,ω G ) ⊗ R α ( R ⋆ α,ω G ) op Γ → End R α ( R ⋆ α,ω G ) ¯ η → End R α ( R e ) j − → ( R e ) ⋆ α ⊗ α ( G × G ) , where Γ is given by Γ( x ⊗ y ) : z xzy , for all x, y, z ∈ R ⋆ α,ω G , and j is given by (4).By Proposition 5.2 one only needs to show that the above composition restricted to R e is the identity. We have( r ⊗ R α t ) δ (1 , r δ ⊗ R α t δ y = r δ ⊗ R α t δ Γ( y ) j − ( η Γ( y ) η − ) . For a fixed a ∈ G we compute the image of η Γ( y ) η − on ( r a ⊗ R α e a − ∈ R e , r a ∈ D a . We see that η Γ( y ) η − (( r a ⊗ R α e a − )= η Γ( y )( r a δ a ) = η (( r δ )( r a δ a )( t δ )) = η ( r r a α a ( t a − ) δ a )=( r r a α a ( t a − ) ⊗ R α e a − = ( r r a ⊗ R α α a ( t a − ) ⊗ R α e a − (15) = ( r r a ⊗ R α ⊗ R α t a − ) e a − = ( r r a ⊗ R α t )(1 ⊗ R α α a − (1 a )) e a − (15) = ( r ⊗ R α t )( r a ⊗ R α e a − = j (( r ⊗ R α t ) δ (1 , )[( r a ⊗ R α e a − ] . Then, η Γ( y ) η − ( x ) = j (( r ⊗ R α t ) δ (1 , ) x, for all x ∈ R e , and we conclude that j − ( η Γ( y ) η − ) = ( r ⊗ R α t ) δ (1 , . Hence, the composition is R e -linear and ω ⊗ R α ω − ∈ B ( G × G, α ⊗ R α α, R e ) . Finally, for any ˜ ω ∈ Z ( G, α, R ) , we have ( ω ⊗ R α ˜ ω )(˜ ω ⊗ R α ˜ ω − ) = ω ˜ ω ⊗ R α α , and thisyields that ω ⊗ R α ˜ ω is cohomologous to ω ˜ ω ⊗ R α α . (cid:4) Theorem 5.10.
Let R be an α -partial Galois extension of R α . Then the map ϕ : H ( G, α, R ) ∋ cls( ω ) [ R ⋆ α,ω G ] ∈ B ( R/R α ) is a group homomorphism. Proof.
In this proof unadorned ⊗ will mean ⊗ R α . Let cls( ω ) , cls(˜ ω ) ∈ H ( G, α, R ) . By Propositions 5.8, 5.1 and 5.9 we have[
R ⋆ α,ω G ][ R ⋆ α, ˜ ω G ] = [( R ⊗ R ) ⋆ α ⊗ α,ω ⊗ ˜ ω ( G × G )] = [( R ⊗ R ) ⋆ α ⊗ α,ω ˜ ω ⊗ α ( G × G )]= [ R ⋆ α,ω ˜ ω G ][ R ⋆ α, α G ] = [ R ⋆ α,ω ˜ ω G ][End R α ( R )] = [ R ⋆ α,ω ˜ ω G ] . which gives [ R ⋆ α,ω G ][ R ⋆ α, ˜ ω G ] = [ R ⋆ α,ω ˜ ω G ] and the assertion follows. (cid:4) The construction of B ( R/R α ) ϕ → H ( G, α ∗ , PicS ( R )) . We remind that un-adorned ⊗ stands for ⊗ R . Let α ∗ = ( α ∗ g , X g ) g ∈ G be the partial action of G on PicS ( R ) constructed in Theorem4.3. Since U ( PicS ( R )) = Pic ( R ) we have that B ( G, α ∗ , PicS ( R )) is the group { f ∈ C ( G, α ∗ , PicS ( R )) | f ( g ) = α ∗ g ([ P ][ D g − ])[ P ∗ ] , for some [ P ] ∈ Pic ( R ) } and Z ( G, α ∗ , PicS ( R )) is given by { f ∈ C ( G, α ∗ , PicS ( R )) | f ( gh )[ D g ] = f ( g ) α ∗ g ( f ( h )[ D g − ]) , ∀ g, h ∈ G } . Remark 5.11.
Let f ∈ C ( G, α,
PicS ( R )) , g ∈ G and p ∈ Spec( R ) . We shall make alittle abuse of notation by writing f ( g ) p for a representative of the class f ( g ) localized at p . We proceed with the construction of ϕ . Take [ A ] ∈ B ( R/R α ) . Then by [2, Theo-rem 5.7] there is an Azumaya R α -algebra equivalent to A containing R as a maximalcommutative subalgebra. Hence, we assume that A contains R as a maximal commuta-tive subalgebra. By [19, Theorem 4.2] R ⊇ R α is separable, moreover [2, Theorem 5.6]tells us that A is a faithfully projective R -module and there is a R -algebra isomorphism R ⊗ R α A op ∼ = End R ( A ) . ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 23
On the other hand, D g ⊗ A is a faithfully projective D g -module, thus by Proposition3.2 there is an R -algebra isomorphism End D g ( D g ⊗ R A ) ∼ = D g ⊗ End R ( A ) , for any g ∈ G. Consequently, we have an R -algebra isomorphism End D g ( D g ⊗ A ) ∼ = D g ⊗ R α A op . Therefore, by [12, Proposition I.3.3] the functor(17) ⊗ D g ( D g ⊗ A ) : D g Mod → ( D g ⊗ Rα A op ) Mod , determines a category equivalence.It is clear that D g ⊗ A is a left R ⊗ R α A op -module via ( r ⊗ R α a )( d ′ ⊗ a ′ ) = rd ′ ⊗ a ′ a. Moreover, let ( D g − ⊗ A ) g = D g − ⊗ A (as sets) endowed with a left R ⊗ R α A op -modulestructure via(18) ( r ⊗ R α a ) • ( d ′ ⊗ a ′ ) = α g − ( r g ) d ′ ⊗ a ′ a, for any g ∈ G, r ∈ R, a ∈ A, d ′ ∈ D g − . Restricting, we obtain left D g ⊗ R α A op -modulestructures on D g ⊗ A and ( D g − ⊗ A ) g , respectively. Moreover D g − ⊗ A is also a right R ⊗ R α A op -module via(19) ( d ⊗ a )( r ⊗ R α a ′ ) = dr ⊗ a ′ a. Furthermore, we denote by h ( D g − ⊗ A ) I , the R - R -bimodule D g − ⊗ A, where the ac-tions of R are induced by (18) and (19), and h ∈ { g, G } . It follows from (18) that ( D g − ⊗ A ) g is an object in D g ⊗ Rα A op Mod and by (17) thereis a D g -module M ( g ) such that(20) ( D g − ⊗ A ) g ∼ = M ( g ) ⊗ D g ( D g ⊗ A ) ∼ = M ( g ) ⊗ A, as ( D g ⊗ R α A op )-modules , where M ( g ) is considered as an R -module via the map r r g , r ∈ R. Our aim is toshow that [ M ( g ) ] ∈ PicS ( R ) , for all g ∈ G. From (20) we see that(21) ( D g − ⊗ A ) g ∼ = M ( g ) ⊗ A as R ⊗ R α A op -modules.As R -modules we have ( D g − ⊗ A ) g = ( D g − ⊗ A ) g . Since D g − is a f.g.p. R -module,we have that D g − ⊗ A is a f.g.p. R -module too, and by (iii) of Lemma 4.1 we concludethat ( D g − ⊗ A ) g is also a f.g.p. R -module. Since A p = 0 , for all p ∈ Spec( R ) , Lemma4.7 and (21) imply that M ( g ) is a f.g.p. R -module.Now we prove that rk p ( M ( g ) p ) ≤ , for all p ∈ Spec( R ) . Since A p ∼ = R n p p , for some n p ≥
1, then ( D g − ⊗ A ) p ∼ = ( D g − ) n p p . By (i) of Lemma 4.1 we get (( D g − ⊗ A ) g ) p ∼ = ( D g ) n p p , which using (21) implies( M ( g ) p ) ⊗ R p A p ∼ = ( D g ) n p p ∼ = ( D g ) p ⊗ R p R p n p ∼ = ( D g ) p ⊗ R p A p . We conclude that(22) rk p ( M ( g ) p ) = rk p (( D g ) p ) ≤ , for any p ∈ Spec( R ) , and by Proposition 3.6 we have [ M ( g ) ] ∈ PicS ( R ) . In addition, since M ( g ) is a (unital) D g -module, we also have that [ M ( g ) ] ∈ X g = [ D g ] PicS ( R ) . Set f A : G ∋ g [ M ( g ) ] ∈ PicS ( R ) . Notice that f = f A is well defined by Lemma 4.7and M (1) = R satisfies (20). We shall check that f ∈ Z ( G, α ∗ , PicS ( R )) . Using (21),Remark 4.4 and (iv) of Lemma 4.1 we obtain R -module isomorphisms( D g ⊗ M ( gh ) ) ⊗ A ∼ = D g ⊗ ( M ( gh ) ⊗ A ) ∼ = D g ⊗ ( D ( gh ) − ⊗ A ) gh ∼ = [ D g − ⊗ ( D h − ⊗ A ) h ] g ∼ = [ D g − ⊗ ( M ( h ) ⊗ A )] g ∼ = ( D g − ⊗ M ( h ) ) g ⊗ ( D g − ⊗ A ) g ∼ = ( D g − ⊗ M ( h ) ) g ⊗ M ( g ) ⊗ A. Finally, by Lemma 4.8 we get(23) M ( gh ) ⊗ D g ∼ = ( M ( h ) ⊗ D g − ) g ⊗ M ( g ) , which gives f ( gh )[ D g ] = f ( g ) α ∗ g ( f ( h )[ D g − ]) . Taking h = g − in (23) we obtain that D g ∼ = ( M ( g − ) ⊗ D g − ) g ⊗ M ( g ) , as R -modules. Thus, [ M ( g ) ] ∈ U ( X g ) and we concludethat f ∈ Z ( G, α ∗ , PicS ( R )) . We define ϕ : B ( R/R α ) ∋ [ A ] cls( f A ) ∈ H ( G, α ∗ , PicS ( R )) . Claim 5.12. ϕ is well defined. Suppose [ A ] = [ B ] ∈ B ( R/R α ) , where A and B contain R as a maximal commutativesubalgebra (see [2, Theorem 5.7]). There are faithfully projective R α -modules P, Q suchthat A ⊗ R α End R α ( P ) ∼ = B ⊗ R α End R α ( Q ) , as R α -algebras. It is proved in [12, page 127]that this leads to the existence of a f.g.p. R -module N with rk( N ) = 1 satisfying(24) ( A ⊗ R α P ∗ ) ⊗ N ∼ = B ⊗ R α Q ∗ as R -modules.We know that there are D g -modules M ( g ) , W ( g ) such that(25) ( D g − ⊗ A ) g ∼ = M ( g ) ⊗ R A as R ⊗ R α A op -modulesand(26) ( D g − ⊗ B ) g ∼ = W ( g ) ⊗ R B as R ⊗ R α B op -modules , for each g ∈ G. Let f A , f B : G → PicS ( R ) be defined by f A ( g ) = [ M ( g ) ] , f B ( g ) = [ W ( g ) ] , g ∈ G. We must show that cls( f A ) = cls( f B ) in H ( G, α ∗ , PicS ( R )) . Since for any R α -module P one has ( D g − ⊗ A ⊗ R α P ∗ ) g ∼ = ( D g − ⊗ A ) g ⊗ R α P ∗ as R -modules, there are R -module isomorphisms[ D g − ⊗ ( B ⊗ R α Q ∗ )] g (24) ∼ = [ D g − ⊗ ( A ⊗ R α P ∗ ) ⊗ N ] g ∼ = [ D g − ⊗ N ⊗ ( A ⊗ R α P ∗ )] g ∼ = ( N ⊗ D g − ) g ⊗ ( D g − ⊗ A ) g ⊗ R α P ∗ (25) ∼ = ( N ⊗ D g − ) g ⊗ M g ⊗ A ⊗ R α P ∗ . On the other hand,(27) [ D g − ⊗ ( B ⊗ R α Q ∗ )] g ∼ = ( D g − ⊗ B ) g ⊗ R α Q ∗ (26) ∼ = W ( g ) ⊗ B ⊗ R α Q ∗ . ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 25
But [ N ] ∈ Pic ( R ) , and, as R -modules, one has[( D g − ⊗ N ) g ⊗ N ∗ ⊗ M ( g ) ] ⊗ ( N ⊗ A ⊗ R α P ∗ ) ∼ = [( D g − ⊗ N ) g ⊗ M ( g ) ⊗ A )] ⊗ R α P ∗ ∼ = [ D g − ⊗ ( B ⊗ R α Q ∗ )] g (27) ∼ = W ( g ) ⊗ B ⊗ R α Q ∗ (24) ∼ = W ( g ) ⊗ ( A ⊗ R α P ∗ ) ⊗ N ∼ = W ( g ) ⊗ ( N ⊗ A ⊗ R α P ∗ ) . Since A is a faithfully projective R -module and P ∗ is a faithfully projective R α -module,then A ⊗ R α P ∗ is a faithfully projective R = R ⊗ R α R α -module. Therefore, N ⊗ A ⊗ R α P ∗ is a also a faithfully projective R -module, and by Lemma 4.8( D g − ⊗ N ) g ⊗ N ∗ ⊗ M ( g ) ∼ = W ( g ) as R -modules, which is equivalent to say that f B ( g ) = α ∗ g ([ N ][ D g − ])[ N ] − f A ( g ) . Thisshows that ϕ is well defined. Theorem 5.13. ϕ is a group homomorphism. Proof.
Let [ A ] , [ A ] ∈ B ( R/R α ) and suppose that R is a maximal commutativesubalgebra of A i , i = 1 , . Let B = A ⊗ R α A . By [19, Theorem 4.2] the extension R ⊇ R α is separable, so let e ∈ R e be a separability idempotent for R. Then End R α ( Be ) isan Azumaya R α -algebra with C End Rα ( Be ) ( B ) = End B ( Be ) ∼ = ( eBe ) op , via the R α -algebramap f ef ( e ) . It follows from [12, Theorem II.4.3] that ( eBe ) op is an Azumaya R α -algebra and B ⊗ R α ( eBe ) op ∼ = End R α ( Be ) as R α -algebras. We conclude that [ B ] = [ eBe ]in B ( R/R α ) . Following the procedure given in [12, page 128] we also check that R isa maximal commutative R α -subalgebra of eBe. Thus, [ A ][ A ] = [ eBe ] and there are D g -modules M ( g ) , W ( g )1 , W ( g )2 such that(28) ( D g − ⊗ eBe ) g ∼ = M ( g ) ⊗ eBe, ( D g − ⊗ A ) g ∼ = W ( g )1 ⊗ A , ( D g − ⊗ A ) g ∼ = W ( g )2 ⊗ A , for any g ∈ G, as R ⊗ R α ( eBe ) op -, R ⊗ R α ( A ) op - and R ⊗ R α ( A ) op -modules, respectively.Let ˜ α = ( ˜ D g , ˜ α g ) g ∈ G denote the partial action of G on R ⊗ R α R, where˜ D g = D g ⊗ R α D g and ˜ α g : ˜ D g − → ˜ D g is induced by x ⊗ R α y α g ( x ) ⊗ R α α g ( y ) . Since e g = (1 g ⊗ R α g ) e satisfies e g ( d ⊗ R α g − g ⊗ R α d ) = e ( d ⊗ R α − ⊗ R α d )(1 g ⊗ R α g ) = 0 , for all d ∈ D g , g ∈ G, then e g is a separability idempotent for the commutative R α -algebra D g . The fact that ˜ α g is a ring isomorphism implies that ˜ α g ( e g − ) ∈ D g ⊗ R α D g isanother separability idempotent for D g . Since separability idempotents for commutativealgebras are unique,(29) ˜ α g ( e g − ) = e g , for all g ∈ G. On the other hand,( ˜ D g − ⊗ R e Be ) g = (( D g − ⊗ R α D g − ) ⊗ R e Be ) g , is an R e ⊗ R α ( eBe ) op -module via the action induced by(30) [( r ⊗ R α r ) ⊗ R α b ] • [( d ⊗ R α d ) ⊗ R e b ′ )] = ˜ α g − [( r ⊗ R α r )1 ˜ g ]( d ⊗ R α d ) ⊗ R e b ′ b, for all r , r ∈ R, d , d ∈ D g − , b ∈ eBe, b ′ ∈ Be and 1 ˜ g = 1 g ⊗ R α g , g ∈ G. Then, there is an R e ⊗ R α ( eBe ) op -module isomorphism( ˜ D g − ⊗ R e Be ) g ∼ = ( ˜ D g − ⊗ R e B ) g e. Notice that for any R e ⊗ R α ( eBe ) op -module M, the abelian group ( e ⊗ R α eBe ) M is an R ⊗ R α ( eBe ) op -module via( r ⊗ R α ebe ) · (( e ⊗ R α eBe ) m ) = (( r ⊗ R α R ) e ⊗ R α ebe ) m, for all m ∈ M, r ∈ R and b ∈ B. In particular, for M = ( ˜ D g − ⊗ R e B ) g e we see that( e ⊗ R α eBe ) • ( ˜ D g − ⊗ R e B ) g e = ˜ α g − ( e g )( ˜ D g − ⊗ R e B ) g e = e g − ( ˜ D g − ⊗ R e B ) g e = e ( ˜ D g − ⊗ R e B ) g e is an R ⊗ R α ( eBe ) op -module via( r ⊗ R α ebe ) · ( e ( d ⊗ R α d ) ⊗ R e b ′ e ) =(( r ⊗ R α R ) e ⊗ R α ebe ) • ( e ( d ⊗ R α d ) ⊗ R e b ′ e ) (30) , (29) = [ ˜ α g − (( r ⊗ R α e g )]( d ⊗ R α d ) ⊗ R e b ′ ebe =( α g − ( r g ) ⊗ R α R ) e ( d ⊗ R α d ) ⊗ R e b ′ ebe = α g − ( r g ) e ( d ⊗ R α d ) ⊗ R e b ′ ebe. Also [ ˜ D g − ⊗ R e eBe ] g = [ e ˜ D g − ⊗ R e Be ] g is an R ⊗ R α ( eBe ) op -module via(31) ( r ⊗ R α ebe ) ◮ ( e ( d ⊗ R α d ) ⊗ R e b ′ e ) = α g − ( r g ) e ( d ⊗ R α d ) ⊗ R e b ′ ebe. We conclude that(32) e ( ˜ D g − ⊗ R e B ) g e ∼ = [ ˜ D g − ⊗ R e eBe ] g , as R ⊗ R α ( eBe ) op -modules.Moreover we have the next. Claim 5.14.
There is an R ⊗ R α ( eBe ) op -module isomorphism ( D g − ⊗ eBe ) g ∼ = ( ˜ D g − ⊗ R e eBe ) g , where R ⊗ R α ( eBe ) op acts on ( ˜ D g − ⊗ R e eBe ) g as in (31).Indeed, the map ( D g − ⊗ eBe ) g ς → ( ˜ D g − ⊗ R e eBe ) g , determined by d ⊗ ebe (1 g − ⊗ R α d ) ⊗ R e ebe is a well defined R ⊗ R α ( eBe ) op -module isomorphism whose inverse( ˜ D g − ⊗ R e eBe ) g ς ∗ → ( D g − ⊗ eBe ) g , is induced by ( d ⊗ R α d ) ⊗ R e ebe d d ⊗ ebe. ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 27
In fact, since e ∈ R e is the separability idempotent for R, we have ( d ⊗ R α d ) e =(1 g − ⊗ R α d d ) e. Hence, ςς ∗ (( d ⊗ R α d ) ⊗ R e ebe ) = ς ( d d ⊗ ebe ) = (1 g − ⊗ R α d d ) ⊗ R e ebe = (1 g − ⊗ R α d d ) e ⊗ R e ebe = ( d ⊗ R α d ) e ⊗ R e ebe = ( d ⊗ R α d ) ⊗ R e ebe, and ς ∗ ς ( d ⊗ ebe ) = ς ∗ ((1 g − ⊗ R α d ) ⊗ R e ebe ) = d ⊗ ebe. Now we prove that ς is R ⊗ R α ( eBe ) op -linear. For ς (( r ⊗ R α ebe ) • ( d ⊗ eb ′ e )) = ς ( α g − ( r g ) d ⊗ eb ′ ebe ) = (1 g − ⊗ R α α g − ( r g ) d ) ⊗ R e eb ′ ebe, and ( r ⊗ R α ebe ) ◮ ς ( d ⊗ eb ′ e ) = ( r ⊗ R α ebe ) ◮ ((1 g − ⊗ R α d ) ⊗ R e eb ′ e )= α g − ( r g ) e (1 g − ⊗ R α d ) ⊗ R e eb ′ ebe = (1 g − ⊗ R α α g − ( r g )) e (1 g − ⊗ R α d ) ⊗ R e eb ′ ebe = (1 g − ⊗ R α α g − ( r g ))(1 g − ⊗ R α d ) ⊗ R e eb ′ ebe = (1 g − ⊗ R α α g − ( r g ) d ) ⊗ R e eb ′ ebe, which ends the proof of the claim.We still need the following. Claim 5.15.
There is an ( R e ) ⊗ R α ( eBe ) op -module isomorphism [ ˜ D g − ⊗ R e ( A ⊗ R α A )] g ∼ = ( D g − ⊗ A ) g ⊗ R α ( D g − ⊗ A ) g , where the action of ( R e ) ⊗ R α ( eBe ) op on ( D g − ⊗ A ) g ⊗ R α ( D g − ⊗ A ) g is induced by [( r ⊗ R α r ) ⊗ R α ( x ⊗ R α y )] • [( d ⊗ a ) ⊗ R α ( d ⊗ a )]=( α g − ( r g ) d ⊗ a x ) ⊗ R α ( α g − ( r g ) d ⊗ a y ) , for r , r ∈ R, d , d ∈ D g − , x ⊗ R α y ∈ eBe, a ∈ A and a ∈ A . Indeed, we have a well defined (additive) group homomorphism χ : ( ˜ D g − ⊗ R e ( A ⊗ R α A )) g → ( D g − ⊗ A ) g ⊗ R α ( D g − ⊗ A ) g , determined by ( d i ⊗ R α d ′ i ) ⊗ R e ( a j ⊗ R α a ′ j ) ( d i ⊗ a j ) ⊗ R α ( d ′ i ⊗ a ′ j ) , which has ( d i ⊗ a i ) ⊗ R α ( d j ⊗ R α a j ) ( d i ⊗ R α d j ) ⊗ R e ( a i ⊗ a j ) , as an inverse. The fact that χ is ( R e ) ⊗ R α ( eBe ) op -linear is straightforward. By (28), Claim 5.14, (32) and Claim 5.15 we have the R ⊗ R α ( eBe ) op -module isomor-phisms M ( g ) ⊗ eBe ∼ = ( D g − ⊗ eBe ) g ∼ = [ ˜ D g − ⊗ R e eBe ] g ∼ = e [ ˜ D g − ⊗ R e B ] g e ∼ = e [( D g − ⊗ A ) g ⊗ R α ( D g − ⊗ A ) g ] e ∼ = e [( W ( g )1 ⊗ A ) ⊗ R α ( W ( g )2 ⊗ A )] e ∼ = e { W ( g )1 ⊗ [( A ⊗ R α A ) ⊗ W ( g )2 ] } e ∼ = e [( W ( g )1 ⊗ W ( g )2 ) ⊗ ( A ⊗ R α A )] e ∼ = e [( W ( g )1 ⊗ W ( g )2 ) ⊗ ( A ⊗ R α A ) e ] ∼ = e [( A ⊗ R α A ) e ⊗ ( W ( g )1 ⊗ W ( g )2 )] ∼ = e ( A ⊗ R α A ) e ⊗ ( W ( g )1 ⊗ W ( g )2 ) ∼ = ( W ( g )1 ⊗ W ( g )2 ) ⊗ eBe. Finally Lemma 4.8 implies M ( g ) ∼ = W ( g )1 ⊗ W ( g )2 , and we conclude that ϕ is a grouphomomorphism. (cid:4) Two partial representations G → PicS R α ( R ) and the homomorphism H ( G, α ∗ , PicS ( R )) ϕ → H ( G, α, R )For reader’s convenience we recall from [18] the concept of a partial representation.
Definition 6.1.
A (unital) partial representation of G into an algebra (or, more gene-rally, a monoid) S is a map Φ : G → S which satisfies the following properties, for all g, h ∈ G, (i) Φ( g − )Φ( g )Φ( h ) = Φ( g − )Φ( gh ) , (ii) Φ( g )Φ( h )Φ( h − ) = Φ( gh )Φ( h − ) , (iii) Φ(1 G ) = 1 S . For any g ∈ G denote by g ( D g − ) I the R - R -bimodule D g − regarded as an R - R -bimodule with new action ∗ given by r ∗ d = α g − ( r g ) d, and d ∗ r = dr, for any r ∈ R, d ∈ D g − , (analogously we define I ( D g ) g − ; notice that D g = I ( D g ) I as R - R -bimodules). Thenusing (iii) of Lemma 4.1, we get that [ g ( D g − ) I ] ∈ PicS R α ( R ) . We set(33) Φ : G ∋ g [ g ( D g − ) I ] ∈ PicS R α ( R ) . Some useful properties of Φ are given in the next. Proposition 6.2.
Let Φ be as in (33) . Then, • Φ is a partial representation of G in PicS R α ( R ) with (34) Φ ( g )Φ ( g − ) = [ D g ] , for all g ∈ G, • Φ ( g )[ D h ] = [ D gh ]Φ ( g ) , for any g, h ∈ G. In particular, (35) Φ ( g )[ D g − ] = Φ ( g ) = [ D g ]Φ ( g ) , ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 29 • for any g ∈ G and [ P ] ∈ X g − there is an R − R -bimodule isomorphism (36) P g ∼ = Φ ( g ) ⊗ P ⊗ Φ ( g − ) . Proof.
It is clear that Φ (1) = [ R ] . Now let g, h ∈ G and consider the map φ g,h : g ( D g − ) I ⊗ h ( D h − ) I ⊗ h − ( D h ) I → gh ( D ( gh ) − ) I ⊗ h − ( D h ) I , defined by φ g,h ( a g − ⊗ b h − ⊗ c h ) = α h − ( a g − h ) b h − ⊗ c h . Then, φ g,h is R - R -linear because for r , r ∈ R we have φ g,h ( r ∗ a g − ⊗ b h − ⊗ c h ∗ r ) = φ g,h ( α g − ( r g ) a g − ⊗ b h − ⊗ c h r )= α h − ( α g − ( r g ) a g − h ) b h − ⊗ c h r = α ( gh ) − ( r gh ) α h − ( a g − h ) b h − ⊗ c h r = r ∗ φ g,h ( a g − ⊗ b h − ⊗ c h ) ∗ r . Moreover, the map gh ( D ( gh ) − ) I ⊗ h − ( D h ) I → g ( D g − ) I ⊗ h ( D h − ) I ⊗ h − ( D h ) I inducedby x ( gh ) − ⊗ x h → α h ( x ( gh ) − h − ) ⊗ h − ⊗ x h , for all x ( gh ) − ∈ gh ( D ( gh ) − ) I , x h ∈ h − ( D h ) I is the inverse of φ g,h . This yields thatΦ ( g )Φ ( h )Φ ( h − ) = Φ ( gh )Φ ( h − ) . In a similar way, the map g − ( D g ) I ⊗ gh ( D ( gh ) − ) I → g − ( D g ) I ⊗ g ( D g − ) I ⊗ h ( D h − ) I , such that a g ⊗ b ( gh ) − a g ⊗ α h ( b ( gh ) − h − ) ⊗ h − , is an R - R -bimodule isomorphism with inverse x g ⊗ y g − ⊗ z h − x g ⊗ α h − ( y g − h ) z h − , and we obtain Φ ( g − )Φ ( gh ) = Φ ( g − )Φ ( g )Φ ( h ) . To prove (34) one can check that the map determined by g ( D g − ) I ⊗ g − ( D g ) I ∋ a g − ⊗ b g α g ( a g − ) b g ∈ D g is a well defined R - R -bimodule isomorphism whose inverse is D g ∋ d → g − ⊗ d ∈ g ( D g − ) I ⊗ g − ( D g ) I . The second item follows from the first and (2), (3) of [18]. To check the last itemconsider the map P g ∋ p ν → g − ⊗ p ⊗ g ∈ g ( D g − ) I ⊗ P ⊗ g − ( D g ) I . Then, for r , r ∈ R we have ν ( r • p • r ) = 1 g − ⊗ r • p • r ⊗ g = 1 g − ⊗ α g − ( r g ) pα g − ( r g ) ⊗ g = α g − ( r g ) ⊗ p ⊗ r g = r ∗ g − ⊗ p ⊗ g ∗ r = r ∗ ν ( p ) ∗ r . Hence, ν is an R - R -bimodule isomorphism with inverse induced by a g − ⊗ p ⊗ b g = a g − ⊗ p ⊗ α g − ( b g ) ∗ g = 1 g − ⊗ a g − pα g − ( b g ) ⊗ g a g − pα g − ( b g ) , for all a g − ⊗ p ⊗ b g ∈ g ( D g − ) I ⊗ P ⊗ g − ( D g ) I . (cid:4) Now we construct another partial representation of G in PicS R α ( R ) . Lemma 6.3.
For any f ∈ Z ( G, α ∗ , PicS ( R )) set Φ f = f Φ : G → PicS R α ( R ) , that is Φ f ( g ) = f ( g )Φ ( g ) , for any g ∈ G. Then, • Φ f is a partial representation, • Φ f ( g )Φ f ( g − ) = [ D g ] , for all g ∈ G. Moreover, writing Φ f ( g ) = [ J g ] , we have that D g ∼ = End D g ( J g ) , as R - and D g -algebras,for any g ∈ G. Proof.
First of all we have Φ f (1) = [ R ] . Now, let g, h ∈ G. Then,Φ f ( g − )Φ f ( gh ) = f ( g − )Φ ( g − ) f ( gh )Φ ( gh ) (35) = f ( g − )Φ ( g − )[ D g ] f ( gh )Φ ( gh )= f ( g − )Φ ( g − ) f ( g ) α ∗ g ( f ( h )[ D g − ])Φ ( gh ) (36) = f ( g − )Φ ( g − ) f ( g )Φ ( g ) f ( h )[ D g − ]Φ ( g − )Φ ( gh ) (35) = f ( g − )Φ ( g − ) f ( g )Φ ( g ) f ( h )Φ ( g − )Φ ( gh )= f ( g − )Φ ( g − ) f ( g )Φ ( g ) f ( h )Φ ( g − )Φ ( g )Φ ( h ) (34) = f ( g − )Φ ( g − ) f ( g )Φ ( g ) f ( h )[ D g − ]Φ ( h ) (35) = f ( g − )Φ ( g − ) f ( g )Φ ( g ) f ( h )Φ ( h )= Φ f ( g − )Φ f ( g )Φ f ( h ) . Analogously, it can be shown that Φ f ( gh )Φ f ( h − ) = Φ f ( g )Φ f ( h )Φ f ( h − ) . Indeed,since f ∈ C ( G, α ∗ , PicS ( R )) we have [ D h − ] f ( h − ) = f ( h − ) , and by the second itemof Proposition 6.2 we obtain Φ ( gh )[ D h − ] = [ D g ]Φ ( gh ) . Thus,Φ f ( gh )Φ f ( h − ) = f ( gh )Φ ( gh ) f ( h − )Φ ( h − )= f ( gh )[ D g ]Φ ( gh ) f ( h − )Φ ( h − ) (36) = f ( g )Φ ( g ) f ( h )[ D g − ]Φ ( g − )Φ ( gh ) f ( h − )Φ ( h − ) (35) = f ( g )Φ ( g ) f ( h )Φ ( g − )Φ ( gh ) f ( h − )Φ ( h − ) (34) = f ( g )Φ ( g ) f ( h )[ D g − ]Φ ( h ) f ( h − )Φ ( h − )= f ( g )Φ ( g ) f ( h )Φ ( h ) f ( h − )Φ ( h − )= Φ f ( g )Φ f ( h )Φ f ( h − ) . With respect to the second item we have Φ f ( g )Φ f ( g − ) = f ( g )Φ ( g ) f ( g − )Φ ( g − ) (36) = f ( g ) α ∗ g ( f ( g − )) = f ( gg − )[ D g ] = [ R ][ D g ] = [ D g ] . Finally, by definition [ J g ] = f ( g )Φ ( g ) , and J g ∼ = D g ⊗ J g as left R - and D g -modules,for all g ∈ G . Therefore, after identifying D g = End D g ( D g ), the R -algebra epimorphism ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 31 R → End R ( J g ) induces an R - and D g -algebra epimorphism ξ : D g → End D g ( J g ) , thanksto Proposition 3.2. Via localization we will check that ξ is injective.If for any g ∈ G the ring D g is semi-local, then Pic ( D g ) is trivial (see, for example,[39, Ex. 2.22 (D)]), and by Remark 4.6 we have that f ( g ) ∼ = D g , as D g -modules as wellas R -modules. Then(37) J g ∼ = g ( D g − ) I as R - R -bimodules.Moreover, after localizing by a prime ideal of R α , we obtain R α -module isomorphisms D g − ∼ = ( R α ) m ∼ = D g for some m ∈ N , thanks to the facts that the maps α g areisomorphisms of R α -modules and localization is an exact functor. Since in this caseany D g is semi-local, (37) implies that the map ξ : ( R α ) m → End ( R α ) m ( g ( R α ) m ) is anepimorphism of R α -algebras. On the other hand, the left R α -modules ( R α ) m and g ( R α ) m are isomorphic via r α g − ( r ) , and we have End ( R α ) m ( g ( R α ) m ) ∼ = ( R α ) m , and ξ mustbe an isomorphism of R α -modules. Finally, since ξ is R -linear, then it is an isomorphismof R - and D g -algebras, for any g ∈ G. (cid:4) In what follows we shall write Φ f ( g ) = [ J g ] . Remark 6.4.
Localizing by ideals in
Spec( R α ) and using (37) we see that J g is a faithful D g -module. Then if we ignore the right R -module structure of J g and use the last item ofLemma 6.3 we obtain that [ J g ] ∈ Pic ( D g ) , for any f ∈ Z ( G, α ∗ , PicS ( R )) and g ∈ G. In particular, the map m D g : D g → End D g ( J g ) given by left multiplication is a D g -algebraisomorphism. Remark 6.5.
Let f be an element of Z ( G, α ∗ , PicS ( R )) and write f ( g ) = [ M g ] . Then J g = M g ⊗ g ( D g − ) I and (38) x g r = α g ( r g − ) x g , for any x g ∈ J g , r ∈ R, and the map (39) J g ∋ x → x ⊗ g − ∈ J g ⊗ D g − is an R - R -bimodule isomorphism. Furthermore, if D g is a semi-local for any g ∈ G, then by (37) there is an R - R -bimodule isomorphism γ g : g ( D g − ) I ∋ d → γ g ( d ) ∈ J g . Therefore, setting u g = γ g (1 g − ) , we have, for any x ∈ J g , that x = γ g ( d ) = γ g ( α g ( d ) ∗ g − ) = α g ( d ) u g , for some d ∈ D g − . We conclude that J g = D g u g , u g is a free generator of J g over D g and u g r = α g ( r g − ) u g , for all r ∈ R, g ∈ G, in view of (38). We know from Proposition 6.2 and Lemma 6.3 that there are R - R -bimodule iso-morphisms h − ( D h ) I ⊗ g − ( D g ) I ∼ = D h − ⊗ h − g − ( D gh ) I and J g ⊗ D h ∼ = D gh ⊗ J g , for all g, h ∈ G. For further reference, we shall construct these isomorphisms explicitly.
Lemma 6.6.
The map ̺ : h − ( D h ) I ⊗ g − ( D g ) I → D h − ⊗ h − g − ( D gh ) I , induced by x h ⊗ y g h − ⊗ α g ( x h g − ) y g , is an R - R -bimodule isomorphism, and ̺ − : D h − ⊗ h − g − ( D gh ) I → h − ( D h ) I ⊗ g − ( D g ) I is determined by x h − ⊗ y gh α g − ( y gh g ) α h ( x h − ) ⊗ g , for g, h ∈ G. Proof.
Indeed, ̺ is well defined, and using (6) one can show that ̺ is an R - R -bimodule homomorphism. Moreover, x h ⊗ y g ̺ h − ⊗ α g ( x h g − ) y g ̺ − α g − ( α g ( x h g − ) y g )1 h ⊗ g = x h α g − ( y g ) ⊗ g = x h ⊗ y g . On the other hand, x h − ⊗ y gh ̺ − α g − ( y gh g ) α h ( x h − ) ⊗ g ̺ h − ⊗ α g ( α g − ( y gh g ) α h ( x h − ))= 1 h − ⊗ y gh α g ( α h ( x h − )1 g − ) (6) = 1 h − ⊗ y gh α gh ( x h − ( gh ) − )1 g = 1 h − ⊗ y gh α gh ( x h − ( gh ) − ) α gh (1 h − ( gh ) − ) = 1 h − ⊗ y gh α gh ( x h − ( gh ) − )= 1 h − ⊗ ( x h − ∗ y gh ) = x h − ⊗ y gh , as desired. (cid:4) Lemma 6.7.
The map J g ⊗ D h κ g,h → D gh ⊗ J g induced by (40) a g ⊗ b h α g ( b h g − ) ⊗ a g , for any g, h ∈ G, is an R - R -bimodule isomorphism. Proof.
First, κ g,h is well defined by (38). Notice that κ g,h is bijective with inverse ι g,h : D gh ⊗ J g → J g ⊗ D h , a gh ⊗ b g b g ⊗ α g − ( a gh g ) , for all a gh ∈ D gh and b g ∈ J g . Indeed, ι g,h ◦ κ g,h ( a g ⊗ b h ) = ι g,h ( α g ( b h g − ) ⊗ a g ) = a g ⊗ b h g − (38) = 1 g a g ⊗ b h = a g ⊗ b h . In addition, κ g,h ◦ ι g,h ( a gh ⊗ b g ) = κ g,h ( b g ⊗ α g − ( a gh g )) = a gh g ⊗ b g = a gh ⊗ g b g = a gh ⊗ b g . Finally, to prove that κ g,h is R - R -linear take r , r ∈ R. Then, κ g,h ( r · a g ⊗ b h r ) = α g ( b h r g − ) ⊗ r · a g = α g ( b h g − ) α g ( r g − ) ⊗ r · a g = α g ( b h g − ) r ⊗ α g ( r g − ) · a g = r α g ( b h g − ) ⊗ a g r . This completes the proof. (cid:4)
The map H ( G, α ∗ , PicS ( R )) ϕ → H ( G, α, R ) . Let f ∈ Z ( G, α ∗ , PicS ( R )) andΦ f = f Φ . Write Φ f ( g ) = [ J g ] . By Lemma 6.3 the map Φ f is a partial homomorphismsuch that Φ f ( g )Φ f ( g − ) = [ D g ] . Then, there is a family of R - R -bimodule isomorphisms { χ g,h : J g ⊗ J h → D g ⊗ J gh } g,h ∈ G . ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 33
Consider the following diagram(41) J g ⊗ J h ⊗ J lχ g,h ⊗ id l (cid:15) (cid:15) id g ⊗ χ h,l / / J g ⊗ D h ⊗ J hlκ g,h ⊗ id hl / / D gh ⊗ J g ⊗ J hl id Dgh ⊗ χ g,hl / / D gh ⊗ D g ⊗ J ghlτ gh,g ⊗ id ghl (cid:15) (cid:15) D g ⊗ J gh ⊗ J l id Dg ⊗ χ gh,l / / D g ⊗ D gh ⊗ J ghl , for any g, h, l ∈ G, where κ g,h is from Lemma 6.7 and τ gh,g is the twisting u ⊗ v v ⊗ u. We use diagram (41) to construct a cocycle in Z ( G, α, R ) . Let ˜ ω ( g, h, l ) be the mapobtained making a counterclockwise loop in (41):(id D g ⊗ χ gh,l ) ◦ ( χ g,h ⊗ id l ) ◦ (id g ⊗ χ h,l ) − ◦ ( κ g,h ⊗ id hl ) − ◦ (id D gh ⊗ χ g,hl ) − ◦ ( τ gh,g ⊗ id ghl ) − , for all g, h, l ∈ G. Evidently, ˜ ω ( g, h, l ) is a left R -linear automorphism of D g ⊗ D gh ⊗ J ghl . Moreover, from the fact that(42) ( a g ⊗ b gh ⊗ c ghl ) · ( t g ⊗ u gh ⊗ v ghl ) = a g t g ⊗ b gh u gh ⊗ c ghl v ghl = a g b gh c ghl t g ⊗ u gh ⊗ v ghl , for all a g , t g ∈ D g , b gh , u gh ∈ D gh , c ghl ∈ D ghl , v ghl ∈ J ghl , and g, h, l ∈ G, we concludethat ˜ ω ( g, h, l ) is an invertible element ofEnd D g ⊗ D gh ⊗ D ghl ( D g ⊗ D gh ⊗ J ghl ) ∼ = D g ⊗ D gh ⊗ End D ghl ( J ghl ) ∼ = D g ⊗ D gh ⊗ D ghl , where the last ring isomorphism follows from Remark 6.4. Thus there is a unique in-vertible element ω ( g, h, l ) ∈ U ( D g ⊗ D gh ⊗ D ghl ) such that ˜ ω ( g, h, l )( z ) = ω ( g, h, l ) z, for all z ∈ D g ⊗ D gh ⊗ J ghl , and it follows from (42) that there is a unique ω ( g, h, l ) ∈U ( D g D gh D ghl ) satisfying˜ ω ( g, h, l ) z = ω ( g, h, l ) z, g, h, l ∈ G, z ∈ D g ⊗ D gh ⊗ J ghl . We shall check that ω ∈ Z ( G, α, R ) , or equivalently(43) ( δ ω )( g, h, l, t ) = 1 g gh ghl ghlt , for all g, h, l, t ∈ G, where δ is the coboundary operator given by (2).Since ( δ ω )( g, h, l, t ) and 1 g gh ghl ghlt belong to the R α -module R, equality (43) holdsif and only if for every p ∈ Spec( R α ) the image of ( δ ω )( g, h, l, t ) in ( D g D gh D ghl D ghlt ) p is (1 g gh ghl ghlt ) p . But if D g is semi-local for any g ∈ G , Remark 6.5 implies J g = D g u g , and it follows that χ g,h ( a g u g ⊗ b h u h ) = a g α g ( b h g − ) χ g,h ( u g ⊗ u h ) = a g α g ( b h g − )˜ ρ ( g, h )(1 g ⊗ u gh ) , with ˜ ρ ( g, h ) ∈ U ( D g ⊗ D gh ) . One can write ˜ ρ ( g, h ) = 1 g ⊗ ρ ( g, h ) , where ρ ( g, h ) belongsto U ( D g D gh ) . In particular, we have a map ρ ∈ C ( G, α, R ) and χ g,h ( a g u g ⊗ b h u h ) = a g α g ( b h g − ) ρ ( g, h )1 g ⊗ u gh . We conclude that χ − g,h (1 g ⊗ u gh ) = ρ ( g, h ) − u g ⊗ u h , where ρ ( g, h ) − is the inverse of ρ ( g, h ) in D g D gh . Now we apply ω ( g, h, l ) to 1 g ⊗ gh ⊗ u ghl . g ⊗ gh ⊗ u ghl gh ⊗ g ⊗ u ghl gh ⊗ ρ − ( g, hl ) u g ⊗ u hl ρ ( g, hl ) − u g ⊗ h g − ⊗ u hl (38) = ρ ( g, hl ) − u g ⊗ h ⊗ u hl ρ ( g, hl ) − α g ( ρ ( h, l ) − g − ) u g ⊗ u h ⊗ u l ρ ( g, hl ) − α g ( ρ ( h, l ) − g − ) ρ ( g, h )1 g ⊗ u gh ⊗ u l ρ ( g, hl ) − α g ( ρ ( h, l ) − g − ) ρ ( g, h ) ρ ( gh, l )1 g ⊗ gh ⊗ u ghl . Thus, ω ( g, h, l )(1 g ⊗ gh ⊗ u ghl ) = ( δ ρ − )( g, h, l )(1 g ⊗ gh ⊗ u ghl ) , g, h, l ∈ G. Hence, ω ( g, h, l ) = ( δ ρ − )( g, h, l ) and it follows from Proposition 2.3 that( δ ω )( g, h, l, t ) = 1 g gh ghl ghlt . This yields ω ∈ Z ( G, α, R ) . Claim 6.8.
The map ϕ : H ( G, α ∗ , PicS ( R )) ∋ cls( f ) → cls( ω ) ∈ H ( G, α, R ) is welldefined.Proof. If one takes another family { χ ′ g,h : J g ⊗ J h → D g ⊗ J gh } g,h ∈ G of R - R -bimodule iso-morphisms, the map χ ′ g,h ◦ χ − g,h is an invertible element of End D g ⊗ D gh ( D g ⊗ J gh ) , and thereexists σ ( g, h ) ∈ U ( D g D gh ) such that χ ′ g,h ◦ χ − g,h ( z ) = σ ( g, h ) z, for all z ∈ D g ⊗ J gh . Thus, σ ∈ C ( G, α, R ) , χ g,h = σ ( g, h ) χ ′ g,h , and setting ˜ ω ′ ( g, h, l ) = (id D g ⊗ χ ′ gh,l ) ◦ ( χ ′ g,h ⊗ id l ) ◦ (id g ⊗ χ ′ h,l ) − ◦ ( κ g,h ⊗ id hl ) − ◦ (id D gh ⊗ χ ′ g,hl ) − ◦ ( τ gh,g ⊗ id ghl ) − , we see that after local-izaing by ideals in Spec( R α ) that ω ′ ( g, h, l ) = ( δ σ − )( g, h, l ) ω ( g, h, l ) . This implies that cls( ω ) = cls( ω ′ ) in H ( G, α, R ) . On the other hand taking another representative J ′ g ∈ [ J g ] , for any g ∈ G, we havefamilies of R - R -bimodule isomorphisms { χ ′ g,h : J ′ g ⊗ J ′ h → D g ⊗ J ′ gh } g,h ∈ G and { ζ g : J g → J ′ g } g ∈ G . Let χ ′′ g,h = (id D g ⊗ ζ gh ) ◦ χ g,h ◦ ( ζ − g ⊗ ζ − h ) , g, h ∈ G. Thus, if ω ′ and ω ′′ are the corre-sponding cocycles in Z ( G, α, R ) induced by the families { χ ′ g,h : J ′ g ⊗ J ′ h → D g ⊗ J ′ gh } g,h ∈ G and { χ ′′ g,h : J ′ g ⊗ J ′ h → D g ⊗ J ′ gh } g,h ∈ G respectively, by the above we have cls( ω ′ ) = cls( ω ′′ )in H ( G, α, R ) . We shall prove that ω = ω ′′ . By localization we may assume that each D g , g ∈ G, is a semi-local ring. Then, byRemark 6.5 there is u g ∈ J g such that J g = D g u g and J ′ g = D g u ′ g , where ζ g ( u g ) = u ′ g . Hence, the equality χ g,h ( au g ⊗ bu h ) = aα g ( b g − ) ρ ( g, h )1 g ⊗ u gh , for all g, h ∈ G, and thedefinition of χ ′′ g,h imply χ ′′ g,h ( au ′ g ⊗ bu ′ h ) = aα g ( b g − ) ρ ( g, h )1 g ⊗ u ′ gh , and using the construction of ω and ω ′′ we get ω = ω ′′ . ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 35
Finally, if cls( f ) = cls( f ′ ) ∈ H ( G, α ∗ , PicS ( R )) , there exists f ∈ B ( G, α ∗ , PicS ( R ))such that f ′ = f f, and [ P ] ∈ Pic ( R ) with f ( g ) = [ P ] α ∗ g ([ P ∗ ][ D g − ]) , for all g ∈ G. Hence, Φ f ′ ( g ) = f ′ ( g )Φ ( g ) = f ( g ) f ( g )Φ ( g )= [ P ] α ∗ g ([ P ∗ ][ D g − ]) f ( g )Φ ( g ) (35) , (36) = [ P ]Φ ( g )[ P ∗ ] Φ ( g − ) f ( g )Φ ( g ) | {z } ∈ PicS( R ) = [ P ]Φ ( g )Φ ( g − ) f ( g )Φ ( g )[ P ∗ ] (34) = [ P ] f ( g )Φ ( g )[ P ∗ ]= [ P ]Φ f ( g )[ P ∗ ] . Set Φ f ( g ) = [ J g ] . Let { χ g,h : J g ⊗ J h → D g ⊗ J gh } g,h ∈ G be a family of R - R -bimodule iso-morphisms which come from Φ f , and ω ∈ Z ( G, α, R ) determined by the χ g,h . Identify-ing ( P ⊗ J g ⊗ P ∗ ) ⊗ ( P ⊗ J h ⊗ P ∗ ) ∼ = P ⊗ J g ⊗ J h ⊗ P ∗ , we choose the family of R - R -bimoduleisomorphisms { χ ′ g,h : P ⊗ J g ⊗ J h ⊗ P ∗ → P ⊗ D g ⊗ J gh ⊗ P ∗ } g,h ∈ G , where χ ′ g,h = id P ⊗ χ g,h ⊗ id P ∗ : P ⊗ J g ⊗ J h ⊗ P ∗ → P ⊗ D g ⊗ J gh ⊗ P ∗ , for all g, h ∈ G. Theisomorphisms { χ ′ g,h } correspond to Φ f ′ ( g ) . Therefore, if ω ′ ∈ Z ( G, α, R ) is induced bythe family { χ ′ g,h } g,h ∈ G , we get ω ′ ( g, h, l ) = id P ⊗ ω ( g, h, l ) ⊗ id P ∗ ∈ End R ( P ) ⊗ End D g ⊗ D gh ⊗ D ghl ( D g ⊗ D gh ⊗ J ghl ) ⊗ End R ( P ∗ ) . Finally, since the R -algebra isomorphisms End R ( P ) ∼ = R ∼ = End R ( P ∗ ) , send id P andid P ∗ to 1 R , we obtain that ω ′ coincides with ω. This shows that ϕ is well defined. (cid:4) Theorem 6.9. ϕ : H ( G, α ∗ , PicS ( R )) → H ( G, α, R ) is a group homomorphism. Proof.
Let f, f ′ ∈ Z ( G, α ∗ , PicS ( R )) . Write Φ f ( g ) = [ J g ] and Φ f ′ ( g ) = [ J ′ g ] . Notice that f ( g ) = f ( g )[ D g ] = f ( g )Φ ( g )Φ ( g − ) = [ J g ⊗ g − ( D g ) I ] . Then Φ ff ′ ( g ) =[ J g ⊗ g − ( D g ) I ⊗ J ′ g ] , for all g ∈ G, and there are R - R -bimodule isomorphisms { F g,h : T g ⊗ T h → D g ⊗ T gh } g,h ∈ G , where T g = J g ⊗ g − ( D g ) I ⊗ J ′ g , g ∈ G. We shall make a specific choice of the F g,h . Noticefirst that T g ⊗ T h (39) ∼ = ( J g ⊗ g − ( D g ) I ⊗ J ′ g ⊗ D g − ) ⊗ [( J h ⊗ h − ( D h ) I ) | {z } ∈ PicS ( R ) ⊗ J ′ h ] ∼ = ( J g ⊗ g − ( D g ) I ⊗ J ′ g ⊗ D g − ) ⊗ [( J h ⊗ h − ( D h ) I ) ⊗ D g − ⊗ J ′ h ] . Moreover, g − ( D g ) I ⊗ g ( D g − ) I ∼ = D g − by (34), and we see that T g ⊗ T h is isomorphic to[( J g ⊗ g − ( D g ) I ⊗ ( J ′ g ⊗ g − ( D g ) I | {z } ∈ PicS ( R ) )] ⊗ [ g ( D g − ) I ⊗ ( J h ⊗ h − ( D h ) I ) ⊗ g − ( D g ) I | {z } ∈ PicS ( R ) ] ⊗ g ( D g − ) I ⊗ J ′ h as R - R -bimodules. Furthermore, since the elements in PicS ( R ) commute, there are R - R -bimodule isomorphisms T g ⊗ T h ∼ = ( J g ⊗ g − ( D g ) I ) ⊗ [ g ( D g − ) I ⊗ ( J h ⊗ h − ( D h ) I ) ⊗ g − ( D g ) I ] ⊗ ( J ′ g ⊗ g − ( D g ) I ) ⊗ g ( D g − ) I ⊗ J ′ h ∼ = J g ⊗ [ g − ( D g ) I ⊗ g ( D g − ) I ] ⊗ [ J h ⊗ h − ( D h ) I ⊗ g − ( D g ) I ] ⊗ J ′ g ⊗ [ g − ( D g ) I ⊗ g ( D g − ) I ] ⊗ J ′ h (34) ∼ = ( J g ⊗ D g − ) ⊗ J h ⊗ h − ( D h ) I ⊗ g − ( D g ) I ⊗ ( J ′ g ⊗ D g − ) ⊗ J ′ h (39) ∼ = ( J g ⊗ J h ) ⊗ [ h − ( D h ) I ⊗ g − ( D g ) I ] ⊗ ( J ′ g ⊗ J ′ h ) . Now, applying χ g,h , χ ′ g,h and Lemma 6.6, we get T g ⊗ T h ∼ = D g ⊗ J gh ⊗ h − ( D h ) I ⊗ ( g − ( D g ) I ⊗ D g ) | {z } ⊗ J ′ gh ∼ = D g ⊗ J gh ⊗ [ h − ( D h ) I ⊗ g − ( D g ) I ] ⊗ J ′ gh ∼ = D g ⊗ J gh ⊗ ( D h − ⊗ h − g − ( D gh ) I ) ⊗ J ′ gh ∼ = D g ⊗ ( J gh ⊗ D h − ) ⊗ h − g − ( D gh ) I ⊗ J ′ gh (40) ∼ = D g ⊗ D g ⊗ J gh ⊗ h − g − ( D gh ) I ⊗ J ′ gh ∼ = D g ⊗ J gh ⊗ h − g − ( D gh ) I ⊗ J ′ gh = D g ⊗ T gh , and we pick the family { F g,h } g,h ∈ G as the composition of the isomorphisms constructedabove. By direct verification we obtain the following: Claim 6.10.
The values of F g,h are given by (44) F g,h (( x g ⊗ d g ⊗ x ′ g ) ⊗ ( x h ⊗ d h ⊗ x ′ h )) = χ g,h ( x g ⊗ x h ) ⊗ d g α g ( d h g − ) χ ′ g,h ( x ′ g ⊗ x ′ h ) , for any ( x g ⊗ d g ⊗ x ′ g ) ⊗ ( x h ⊗ d h ⊗ x ′ h ) ∈ T g ⊗ T h , g, h ∈ G, where d g α g ( d h g − ) χ ′ g,h ( x ′ g ⊗ x ′ h ) is considered in h − g − ( D gh ) I ⊗ J ′ gh , and is given by P i d g α g ( d h g − ) e ′ g,i ⊗ v ′ gh,i , in which χ ′ g,h ( x ′ g ⊗ x ′ h ) = P i e ′ g,i ⊗ v ′ gh,i . We shall also need the next.
Claim 6.11.
The inverse of F g,h is given by (45) d g ⊗ x gh ⊗ d gh ⊗ x ′ gh X i,j ( y g,i ⊗ g ⊗ y ′ g,j ) ⊗ ( z h,i ⊗ α g − ( d gh g ) ⊗ z ′ h,j ) , g, h ∈ G, where χ g,h ( P i y g,i ⊗ z h,i ) = d g ⊗ x gh and χ ′ g,h ( P j y ′ g,j ⊗ z ′ h,j ) = 1 g ⊗ x ′ gh , g, h ∈ G. ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 37
Let V g,h be the map defined by (45). Then, d g ⊗ x gh ⊗ d gh ⊗ x ′ gh V g,h X i,j ( y g,i ⊗ g ⊗ y ′ g,j ) ⊗ ( z h,i ⊗ α g − ( d gh g ) ⊗ z ′ h,j ) F g,h X i,j [ χ g,h ( y g,i ⊗ z h,i ) ⊗ d gh g χ ′ g,h ( y ′ g,j ⊗ z ′ h,j )]= ( X i χ g,h ( y g,i ⊗ z h,i )) ⊗ d gh g X j χ ′ g,h ( y ′ g,j ⊗ z ′ h,j )= d g ⊗ x gh ⊗ d gh g ⊗ x ′ gh = d g ⊗ x gh ⊗ h − • d gh ⊗ x ′ gh = d g ⊗ g x gh ⊗ d gh ⊗ x ′ gh = d g ⊗ x gh ⊗ d gh ⊗ x ′ gh , and since F g,h is invertible, we conclude that V g,h = F − g,h for all g, h ∈ G. The automorphism ˜ ω F induced by the family { F g,h } g,h ∈ G is ˜ ω F ( g, h, l ) = (id D g ⊗ F gh,l ) ◦ ( F g,h ⊗ id l ) ◦ (id g ⊗ F h,l ) − ◦ ( κ g,h ⊗ id hl ) − ◦ (id D gh ⊗ F g,hl ) − ◦ ( τ gh,g ⊗ id ghl ) − . Hence, forany u ghl = 1 g ⊗ gh ⊗ x ghl ⊗ d ghl ⊗ x ′ ghl ∈ dom ˜ ω F ( g, h, l ) , there is a unique ω F ( g, h, l ) ∈U ( D g D gh D ghl ) , g, h, l ∈ G , such that ˜ ω F ( g, h, l )( u ghl ) = ω F ( g, h, l ) u ghl . Claim 6.12. ω F = ωω ′ , or equivalently ˜ ω F ( g, h, l )( u ghl ) = ωω ′ ( g, h, l ) u ghl , for all g, h, l ∈ G. First, we calculate the value ˜ ω ( g, h, l )(1 g ⊗ gh ⊗ x ghl ) , where x ghl ∈ J ghl . For this wedenote χ − g,hl (1 g ⊗ x ghl ) = X i y g,i ⊗ z hl,i , χ − h,l (1 h ⊗ z hl,i ) = X m u ( i ) h,m ⊗ v ( i ) l,m and χ g,h ( y g,i ⊗ u ( i ) h,m ) = X k s ( i,m ) g,k ⊗ t ( i,m ) gh,k , χ gh,l ( t ( i,m ) gh,k ⊗ v ( i ) l,m ) = X p c ( i,m,k ) gh,p ⊗ e ( i,m,k ) ghl,p . Then, 1 g ⊗ gh ⊗ x ghl gh ⊗ g ⊗ x ghl gh ⊗ χ − g,hl (1 g ⊗ x ghl )= X i (1 gh ⊗ y g,i ⊗ z hl,i ) X i ( y g,i ⊗ h ⊗ z hl,i ) X i [ y g,i ⊗ χ − h,l (1 h ⊗ z hl,i )] = X i,m [ y g,i ⊗ ( u ( i ) h,m ⊗ v ( i ) l,m )] X i,m [ χ g,h ( y g,i ⊗ u ( i ) h,m ) ⊗ v ( i ) l,m )] = X i,m,k [( s ( i,m ) g,k ⊗ t ( i,m ) gh,k ) ⊗ v ( i ) l,m )] X i,m,k [ s ( i,m ) g,k ⊗ χ gh,l ( t ( i,m ) gh,k ⊗ v ( i ) l,m )]= X i,m,k,p [ s ( i,m ) g,k ⊗ ( c ( i,m,k ) gh,p ⊗ e ( i,m,k ) ghl,p )]= 1 g ⊗ gh ⊗ X i,m,k,p ( s ( i,m ) g,k c ( i,m,k ) gh,p e ( i,m,k ) ghl,p ) . Since ˜ ω ( g, h, l )(1 g ⊗ gh ⊗ x ghl ) = ω ( g, h, l )(1 g ⊗ gh ⊗ x ghl ) , the uniqueness of ω ( g, h, l )implies(46) 1 g ⊗ gh ⊗ X i,m,k,p ( s ( i,m ) g,k c ( i,m,k ) gh,p e ( i,m,k ) ghl,p ) = 1 g ⊗ gh ⊗ ω ( g, h, l ) x ghl , for all g, h, l ∈ G. Analogously, denoting χ ′− g,hl (1 g ⊗ x ′ ghl ) = X j y ′ g,j ⊗ z ′ hl,j , χ ′− h,l (1 h ⊗ z ′ hl,j ) = X n u ′ ( j ) h,n ⊗ v ′ ( j ) l,n and χ ′ g,h ( y ′ g,j ⊗ u ′ ( j ) h,n ) = X k ′ s ′ ( j,n ) g,k ′ ⊗ t ′ ( j,n ) gh,k ′ , χ ′ gh,l ( t ′ ( j,n ) gh,k ′ ⊗ v ′ ( j ) l,n ) = X p ′ c ′ ( j,n,k ′ ) gh,p ′ ⊗ e ′ ( j,n,k ′ ) ghl,p ′ , we obtain 1 g ⊗ gh ⊗ P j,n,k ′ ,p ′ s ′ ( j,n ) g,k ′ c ′ ( j,n,k ′ ) gh,p ′ e ′ ( j,n,k ′ ) ghl,p ′ = 1 g ⊗ gh ⊗ ω ′ ( g, h, l ) x ′ ghl , which implies(47) X j,n,k ′ ,p ′ s ′ ( j,n ) g,k ′ c ′ ( j,n,k ′ ) gh,p ′ e ′ ( j,n,k ′ ) ghl,p ′ = ω ′ ( g, h, l ) x ′ ghl . Now we use (44), (45) and diagram (41) to calculate ˜ ω F ( g, h, l ) u ghl . We have that u ghl gh ⊗ g ⊗ x ghl ⊗ d ghl ⊗ x ′ ghl gh ⊗ F − g,hl (1 g ⊗ x ghl ⊗ d ghl ⊗ x ′ ghl )= X i,j gh ⊗ ( y g,i ⊗ g ⊗ y ′ g,j ) ⊗ ( z hl,i ⊗ α g − ( d ghl g ) ⊗ z ′ hl,j ) X i,j κ − g,h (1 gh ⊗ ( y g,i ⊗ g ⊗ y ′ g,j )) ⊗ ( z hl,i ⊗ α g − ( d ghl g ) ⊗ z ′ hl,j )= X i,j y g,i ⊗ g ⊗ y ′ g,j | {z } ∈ T g ⊗ h g − ⊗ ( z hl,i ⊗ α g − ( d ghl g ) ⊗ z ′ hl,j )= X i,j y g,i ⊗ g ⊗ y ′ g,j ⊗ h ⊗ ( z hl,i ⊗ α g − ( d ghl g ) ⊗ z ′ hl,j ) X i,j ( y g,i ⊗ g ⊗ y ′ g,j ) ⊗ F − h,l [1 h ⊗ ( z hl,i ⊗ α g − ( d ghl g ) ⊗ z ′ hl,j )]= X i,j,m,n ( y g,i ⊗ g ⊗ y ′ g,j ) ⊗ [( u ( i ) h,m ⊗ h ⊗ u ′ ( j ) h,n ) ⊗ ( v ( i ) l,m ⊗ α h − ( α g − ( d ghl g )1 h ) ⊗ v ′ ( j ) l,n )] ARTIAL GALOIS COHOMOLOGY AND RELATED HOMOMORPHISMS 39 X i,jm,n F g,h [( y g,i ⊗ g ⊗ y ′ g,j ) ⊗ ( u ( i ) h,m ⊗ h ⊗ u ′ ( j ) h,n )] ⊗ ( v ( i ) l,m ⊗ α h − ( α g − ( d ghl g )1 h ) ⊗ v ′ ( j ) l,n )]= X i,jk,n χ g,h ( y g,i ⊗ u ( i ) h,m ) ⊗ g |{z} gh χ ′ g,h ( y ′ g,j ⊗ u ′ ( j ) h,n ) ⊗ ( v ( i ) l,m ⊗ α h − ( α g − ( d ghl g )1 h ) ⊗ v ′ ( j ) l,n )= X i,jk,n χ g,h ( y g,i ⊗ u ( i ) h,m ) ⊗ gh χ ′ g,h ( y ′ g,j ⊗ u ′ ( j ) h,n ) ⊗ ( v ( i ) l,m ⊗ α h − ( α g − ( d ghl g )1 h ) ⊗ v ′ ( j ) l,n )= X i,jm,n χ g,h ( y g,i ⊗ u ( i ) h,m ) ⊗ gh χ ′ g,h ( y ′ g,j ⊗ u ′ ( j ) h,n ) ⊗ ( v ( i ) l,m ⊗ α ( gh ) − ( d ghl gh ) | {z } ∈ l − ( D l ) I h − ⊗ v ′ ( j ) l,n )= X i,jm,n χ g,h ( y g,i ⊗ u ( i ) h,m ) ⊗ gh χ ′ g,h ( y ′ g,j ⊗ u ′ ( j ) h,n ) ⊗ ( v ( i ) l,m |{z} ∈ J l l − h − ⊗ α ( gh ) − ( d ghl gh ) ⊗ v ′ ( j ) l,n )= X i,jm,n χ g,h ( y g,i ⊗ u ( i ) h,m ) ⊗ gh χ ′ g,h ( y ′ g,j ⊗ u ′ ( j ) h,n ) ⊗ ( v ( i ) l,m ⊗ α ( gh ) − ( d ghl gh ) ⊗ v ′ ( j ) l,n )= X i,j,kn,m,k ′ [( s ( i,m ) g,k ⊗ t ( i,m ) gh,k ) ⊗ (1 gh s ′ ( j,n ) g,k ′ ⊗ t ′ ( j,n ) gh,k ′ ) ⊗ ( v ( i ) l,m ⊗ α ( gh ) − ( d ghl gh ) ⊗ v ′ ( j ) l,n )] X i,j,kn,m,k ′ s ( i,m ) g,k ⊗ F gh,l [( t ( i,m ) gh,k ⊗ gh s ′ ( j,n ) g,k ′ ⊗ t ′ ( j,n ) gh,k ′ ) ⊗ ( v ( i ) l,m ⊗ α ( gh ) − ( d ghl gh ) ⊗ v ′ ( j ) l,n )]= X i,j,kn,m,k ′ s ( i,m ) g,k ⊗ χ gh,l ( t ( i,m ) gh,k ⊗ v ( i ) l,m ) ⊗ s ′ ( j,n ) g,k ′ α gh ( α ( gh ) − ( d ghl gh )) χ ′ gh,l ( t ′ ( j,n ) gh,k ′ ⊗ v ′ ( j ) l,n )= X i,j,kn,m,k ′ s ( i,m ) g,k ⊗ χ gh,l ( t ( i,m ) gh,k ⊗ v ( i ) l,m ) ⊗ s ′ ( j,n ) g,k ′ d ghl χ ′ gh,l ( t ′ ( j,n ) gh,k ′ ⊗ v ′ ( j ) l,n )= X i,m,k s ( i,m ) g,k ⊗ χ gh,l ( t ( i,m ) gh,k ⊗ v ( i ) l,m ) ⊗ d ghl X j,n,k ′ s ′ ( j,n ) g,k ′ χ ′ gh,l ( t ′ ( j,n ) gh,k ′ ⊗ v ′ ( j ) l,n )= 1 g ⊗ gh ⊗ X i,m,k,p ( s ( i,m ) g,k c ( i,m,k ) gh,p e ( i,m,k ) ghl,p ) ⊗ d ghl ⊗ X j,n,k ′ ,p ′ ( s ′ ( j,n ) g,k ′ c ′ ( j,n,k ′ ) gh,p ′ e ′ ( j,n,k ′ ) ghl,p ′ ) (46 , = 1 g ⊗ gh ⊗ ω ( g, h, l ) x ghl ⊗ d ghl ⊗ ω ′ ( g, h, l ) x ′ ghl = ω ( g, h, l ) ω ′ ( g, h, l )(1 g ⊗ gh ⊗ x ghl ⊗ d ghl ⊗ x ′ ghl )= ω ( g, h, l ) ω ′ ( g, h, l ) u ghl . 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