On partial skew groupoids rings
aa r X i v : . [ m a t h . R A ] M a y ON PARTIAL SKEW GROUPOIDS RINGS
DIRCEU BAGIO, ANTONIO PAQUES, AND H´ECTOR PINEDO
Abstract.
Given a partial action α of a connected groupoid G on an associative ring A we investigate under what conditions the partial skew groupoid ring A ⋆ α G can berealized as a partial skew group ring. In such a case applications concerning to theseparability, semisimplicity and Frobenius property of the ring extension A ⊂ A ⋆ α G as well as to the artinianity of A ⋆ α G are given. Introduction
In this work we will consider partial actions of groupoids on rings. We are interestedin studying the structure of the corresponding partial skew groupoids rings.Partial groupoids actions on rings were introduced in [3] and they are a natural gener-alization of partial group actions. It is well known that every groupoid is a direct sum ofits connected component. A partial action of a groupoid on a ring A is completely deter-mined by the partial actions of its connected components on A . Thence, we can reducethe study of groupoid partial action on rings to the context of connected groupoids.The structure of a connected groupoid is also well known. If G is a connected groupoidthen G ≃ G ×G ( x ), where G is the coarse groupoid associated to the set G of the objectsof G and G ( x ) is the isotropy group of an object x of G .For a partial action α of a connected groupoid G on a ring A we can construct thepartial skew groupoid ring A ⋆ α G . If G is finite and α is unital then A ⋆ α G is anassociative and unital ring which is an extension of A .The partial skew groupoid rings have an important role in the partial Galois theory forgroupoids as it is explicit in Theorem 5.3 of [3]. They also are examples of Leavitt pathalgebras, which are important in the theory of C ∗ -algebras (see Theorem 3.11 of [7]).In the last years, algebraic properties to the extension A ⊂ A ⋆ α G have been studied.For example, the separability and semisimplicity properties of the extension A ⊂ A ⋆ α G were studied in [4] whereas in [11] the authors investigate chain conditions between A and A ⋆ α G .Our purpose in this work is to study the following problem. Let G be a connectedgroupoid such that G is finite and α a unital partial action of G on a ring A . Doesthe factorization G ≃ G × G ( x ) induce a factorization of A ⋆ α G ? Theorem 4.1 providessufficient conditions for the answer to this question to be affirmative. Precisely, when α Date : May 30, 2019.
Mathematics Subject Classification : Primary 20L05, 16W22, 16S99. Secondary 18B40, 20N02.
Key words and phrases:
Groupoid, partial groupoid action, partial skew groupoid ring, separabil-ity, semisimplicity, Frobenius property, artinianity. is a group-type partial action, we construct a groupoid action β of G on A and a partialgroup action γ of G ( x ) on A ⋆ β G and we prove that A ⋆ α G ≃ ( A ⋆ β G ) ⋆ γ G ( x ).We organize our work as follows. The background about groupoids is presented inSection 2. The topics of partial groupoid actions that will be used are in Section 3. InSection 4, we construct the actions β and γ which allow us to prove the factorization of A ⋆ α G mentioned in the previous paragraph. Applications of this result concerning tothe separability, semisimplicity and Frobenius property of the extension A ⊂ A ⋆ α G aswell as to the artinianity of A ⋆ α G are given in Section 5. Conventions.
Throughout this work, by ring we mean an associative and not neces-sarily unital ring. The center of a ring A will be denoted by C ( A ). We will denote thecardinality of a finite set X by | X | . 2. Groupoids
We recall that a groupoid is a small category in which every morphism is an isomor-phism. The set of the objects of a groupoid G will be denoted by G . If g : x → y is a morphism of G then s ( g ) = x and t ( g ) = y are called the source and the tar-get of g respectively. We will identify any object x of G with its identity morphism,that is, x = id x . The isotropy group associated to an object x of G is the group G ( x ) = { g ∈ G : s ( g ) = t ( g ) = x } .The composition of morphisms of a groupoid G will be denoted via concatenation.Hence, for g, h ∈ G , there exists gh if and only if t ( h ) = s ( g ). Notice that, if g ∈ G then s ( g ) = g − g and t ( g ) = gg − . Also, s ( gh ) = s ( h ) and t ( gh ) = t ( g ) for all g, h ∈ G with t ( h ) = s ( g ).A groupoid G is said to be connected if for any x, y ∈ G there exists a morphism g ∈ G such that s ( g ) = x and t ( g ) = y , that is, the morphism g connects the objects x and y . It is well-known that any groupoid is a disjoint union of connected subgroupoids. Inorder to justify this fact, we consider the following equivalence relation on G : for any x, y ∈ G , x ∼ y if and only if there exists g ∈ G such that s ( g ) = x and t ( g ) = y . Everyequivalence class X ∈ G / ∼ determines a full connected subgroupoid G X of G . The setof objects of G X is X . The set G X ( x, y ) of morphisms of G X from x to y is equal to G ( x, y ), for all x, y ∈ X . By construction, G is the disjoint union of the subgroupoids G X , i. e.(1) G = ˙ ∪ X ∈G / ∼ G X . For the convenience of the reader, we will prove a well-known result about the structureof connected groupoids. In order to do this, we need to introduce some extra notation.Let X be a nonempty set and X = X × X . Then X is a groupoid. The source andtarget maps of X are, respectively, s ( x, y ) = x and t ( x, y ) = y , for all x, y ∈ X . Therule of composition is given by: ( y, z )( x, y ) = ( x, z ), for all x, y, z ∈ X . The groupoid X is called the coarse groupoid associated to X . Proposition 2.1.
Let G be a connected groupoid. Then G ≃ G × G ( x ) as groupoids.Proof. Let x ∈ G a fixed object of G . For each y ∈ G , we choose a morphism τ y : x → y of G . We also choose τ x = x . Define ϕ : G → G × G ( x ) by ϕ ( g ) = (( s ( g ) , t ( g )) , g x ), N PARTIAL SKEW GROUPOIDS RINGS 3 where g x = τ − t ( g ) gτ s ( g ) , for all g ∈ G . It is straightforward to prove that ϕ in a groupoidmorphism. Suppose that ϕ ( g ) is an identity of G × G ( x ). Then, ϕ ( g ) = (( y, y ) , x )for some y ∈ G . Hence, s ( g ) = t ( g ) = y and x = g x = τ − y gτ y which implies that g = τ y τ − y = x . This ensures that ϕ is injective. Given an element (( y, z ) , h ) ∈ G ×G ( x ),consider g = τ z hτ − y ∈ G . Notice that g x = h and whence ϕ ( g ) = (( y, z ) , h ), that is, ϕ issurjective, so an isomorphism of groupoids. (cid:3) Partial actions
In this section we recall the notion of partial actions of groupoids. Some propertiesrelated to partial actions, that will be used later, are presented. The definition ofgroup-type partial groupoid actions, which has a central role for our purposes, will beintroduced.3.1.
Partial groupoid action.
We recall from [3] that a partial action of a groupoid G on a ring A is a pair α = ( A g , α g ) g ∈G such that(i) A g is an ideal of A t ( g ) and A t ( g ) is an ideal of A , for all g ∈ G ,(ii) α g : A g − → A g is an isomorphism of rings, for all g ∈ G ,(iii) α x = id A x , for all x ∈ G ,(iv) α g α h ≤ α gh , for all g, h ∈ G such that t ( h ) = s ( g ).The condition (iv) means that α gh is an extension of α g α h . Since the domain of α g α h is α − h ( A g − ∩ A h ), it follows that (iv) is equivalent to(v) α − h ( A g − ∩ A h ) ⊂ A ( gh ) − and α gh ( a ) = α g α h ( a ) , for all a ∈ α − h ( A g − ∩ A h ) . The partial action α is said to be global if α g α h = α gh , for all g, h ∈ G such that t ( h ) = s ( g ). Also, α is called unital if each A g is a unital ring, i. e., there exists a centralelement 1 g of A such that A g = A g , for all g ∈ G .Now we recall Lemma 1.1 of [3] which give us some useful properties of partial actionsthat will be used in what follows. Lemma 3.1.
Let α = ( A g , α g ) g ∈G be a partial action of a groupoid G on a ring A . Then: (i) α is global if and only if A g = A t ( g ) , for all g ∈ G ; (ii) α g − = α − g , for all g ∈ G ; (iii) α g ( A g − ∩ A h ) = A g ∩ A gh , for all g, h ∈ G such that t ( h ) = s ( g ) . Remark 3.2.
Let α = ( A g , α g ) g ∈G be a partial action of a groupoid G on a ring A . Noticethat α induces by restriction a partial action α ( x ) = ( A h , α h ) h ∈G ( x ) of the isotropy group G ( x ) on the ring A x , for each x ∈ G . Remark 3.3.
Let G be a groupoid. Using the decomposition of G given in (1), it isstraightforward to check that partial actions of G on a ring A induce by restriction partialactions of G X on A , for all X ∈ G / ∼ . Conversely, partial actions of G on A are uniquelydetermined by partial actions of G X , X ∈ G / ∼ , on A . Hence, we can reduce the studyof partial groupoid actions to the connected case. D. BAGIO, A. PAQUES, AND H. PINEDO
Group-type partial groupoid action.
Let G be a connected groupoid, x ∈ G and S x = { h ∈ G : s ( h ) = x } . Consider the following equivalence relation on S x : g ∼ x l if and only if t ( g ) = t ( l ) , g, l ∈ S x . A transversal τ ( x ) = { τ y : y ∈ G } for ∼ x such that τ x = x will be called a transversalfor x . Hence, τ y : x → y is a chosen morphism of G , for each y ∈ G and τ x = x .A partial action α = ( A g , α g ) g ∈G of a connected groupoid G on A will be called group-type if there exist x ∈ G and a transversal τ ( x ) = { τ y : y ∈ G } for x such that A τ − y = A x and A τ y = A y , for all y ∈ G . (2) Remark 3.4. (i) Notice that the notion of group-type partial action not depend onthe choice of object x . Indeed, for another object z of G , consider ˜ τ y := τ y τ − z , for all y ∈ G . Clearly, ˜ τ ( z ) = { ˜ τ y : y ∈ G } is a transversal for z . From (2) follows that α τ y α τ − z = α τ y τ − z = α ˜ τ y . Thus, A (˜ τ y ) − = A z and A ˜ τ y = A y , for all y ∈ G .(ii) We use the term “group-type partial actions” since by Theorem 4.4, proved in thenext section for this kind of partial actions, the corresponding partial skew groupoidring is indeed a partial skew group ring.By Lemma 3.1 (i), any global groupoid action is group-type. The converse is not trueas we can see in the next example. Example 3.5.
Let G = { g, h, l, m, l − , m − } be the groupoid with objects G = { x, y } and the following composition rules g = x, h = y, lg = m = hl, g ∈ G ( x ) , h ∈ G ( y ) and l, m : x → y. The diagram bellow illustrates the structure of G : x l / / y h (cid:15) (cid:15) x g O O m / / y Consider A = C e ⊕ C e ⊕ C e ⊕ C e , where C denotes the complex number field, e i e j = δ i,j e i and e + . . . + e = 1. We define the following partial action α = (cid:0) A p , α p (cid:1) p ∈G of G on A : A x = C e ⊕ C e = A l − , A y = C e ⊕ C e = A l ,A g = C e = A g − = A m − , A m = A h = C e = A h − , and α x = id A x , α y = id A y , α g : ae ae , α h : ae ae , α m : ae ae ,α m − : ae ae , α l : ae + be ae + be , α l − : ae + be ae + be , where a denotes the complex conjugate of a , for all a ∈ C . Notice that α is a group-type (not global) partial action. Indeed, to see this it is enough to take the transversal τ ( x ) = { τ x = x, τ y = l } for x . N PARTIAL SKEW GROUPOIDS RINGS 5 The partial skew groupoid ring
In this section, we will assume that G is a connected groupoid such that G is finite, x ∈ G is a fixed object of G and α = ( A g , α g ) g ∈G is a unital partial action of G on aring A with A g = A g , where 1 g is a central idempotent of A , for all g ∈ G . We will alsoassume that α is group-type and τ ( x ) = { τ y : y ∈ G } is a transversal for x such that(2) is satisfied.The partial skew groupoid ring A ⋆ α G associated to α is the set of all formal sums P g ∈G a g δ g , where a g ∈ A g , with the usual addition and multiplication induced by thefollowing rule ( a g δ g )( a h δ h ) = ( a g α g ( a h g − ) δ gh if s ( g ) = t ( h ) , , for all g, h ∈ G , a g ∈ A g and a h ∈ A h . The partial skew groupoid ring A ⋆ α G isan associative ring. Since by assumption G is finite, A ⋆ α G is unital with identity1 A⋆ α G = P y ∈G y δ y (see § A ⋆ α G only depends on the choice of the ideals A y , y ∈ G . Hence, we can choose A to be anyring having the ideals as above described. In this sense, we will assume for the rest ofthis paper that A = ⊕ y ∈G A y . The main theorem.
In this subsection we will prove that the factorization of G ,given by Proposition 2.1, induces a factorization of A ⋆ α G . Particularly, we obtain that A ⋆ α G is a partial skew group ring. In order to prove this result we will use some lemmasthat will be proved in the sequel. Lemma 4.1.
The pair β = ( B u , β u ) u ∈G , where B u = A t ( u ) and β u = α τ t ( u ) α τ − s ( u ) , is aglobal action of G on A .Proof. For any identity e = ( y, y ) of G we have that B e = A y and β e = α τ y α τ − y is theidentity map of A τ y = A y . Also, if u = ( y, z ) and v = ( r, y ) are elements in G then uv = ( r, z ) and β u β v = α τ z α τ − y α τ y α τ − r = α τ z α τ − r = β uv . (cid:3) Thanks to Lemma 4.1 we can consider the skew groupoid ring C := A ⋆ β G . In thesequel we will see that the group G ( x ) acts partially on C . Let z ∈ G . Since α isgroup-type, it follows from (2) that A τ − z = A x . Then, for all h ∈ G ( x ), A h ⊂ A x and C z,h := α τ z ( A h ) , (3)is well-defined. Hence, we can set C h := ⊕ u ∈G C t ( u ) ,h δ u . (4) Lemma 4.2. C h is a unital ideal of C , for all h ∈ G ( x ) . Moreover, C x = C . D. BAGIO, A. PAQUES, AND H. PINEDO
Proof.
Note that C = ⊕ u ∈G B u δ u = ⊕ u ∈G A t ( u ) δ u (2) = ⊕ u ∈G A τ t ( u ) δ u = ⊕ u ∈G α τ t ( u ) ( A τ − t ( u ) ) δ u (2) = ⊕ u ∈G α τ t ( u ) ( A x ) δ u (3) = ⊕ u ∈G C t ( u ) ,x δ u (4) = C x . Observe also that 1 ′ h = P z ∈G α τ z (1 h ) δ ( z,z ) is the identity element of C h , for all h ∈ G ( x ).Indeed, let u = ( y, w ) ∈ G and a ∈ C t ( u ) ,h δ u = C w,h δ u . By (3), there exists a h ∈ A h such that a = α τ w ( a h ) δ ( y,w ) and consequently a ′ h = X z ∈G α τ w ( a h ) δ ( y,w ) α τ z (1 h ) δ ( z,z ) = α τ w ( a h ) δ ( y,w ) α τ y (1 h ) δ ( y,y ) = α τ w ( a h ) β ( y,w ) ( α τ y (1 h )) δ ( y,w ) = α τ w ( a h ) α τ w α τ − y α τ y (1 h ) δ ( y,w ) (see Lemma 4.1))= α τ w ( a h ) α τ w (1 h ) δ ( y,w ) = α τ w ( a h ) δ ( y,w ) = a. Similarly, 1 ′ h a = a . Hence 1 ′ h is a central idempotent of C . A straightforward calculationshows that C h = 1 ′ h C . Thus, C h is an ideal of C . (cid:3) Let ( z, h ) ∈ G × G ( x ). We define γ z,h : C z,h − → C z,h , α τ z ( a ) α τ z ( α h ( a )), for all a ∈ A h − . Clearly θ z,h is a bijection. Moreover, these maps induce the following bijectivemap γ h : C h − → C h , γ h ( α τ t ( u ) ( a ) δ u ) = γ t ( u ) ,h ( a ) δ u , for all a ∈ A h − and u ∈ G . Lemma 4.3.
The pair γ = ( C h , γ h ) h ∈G ( x ) is a unital partial action of G ( x ) on C .Proof. By definition, γ x is the identity map of C x = C . Note that γ h preserves theoperation of multiplication. In fact, for all a, b ∈ A h − , γ h (( α τ z ( a ) δ ( y,z ) )( α τ y ( b ) δ ( w,y ) )) = γ h ( α τ z ( a ) α ∗ ( y,z ) ( α τ y ( b )) δ ( w,z ) )= γ h ( α τ z ( a ) α τ z ( α τ − y ( α τ y ( b ))) δ ( w,z ) )= γ h ( α τ z ( ab ) δ ( w,z ) )= α τ z ( α h ( ab )) δ ( w,z ) = α τ z ( α h ( a )) α τ z ( α h ( b )) δ ( w,z ) . N PARTIAL SKEW GROUPOIDS RINGS 7
On the other hand, γ h ( α τ z ( a ) δ ( y,z ) ) γ h ( α τ y ( b ) δ ( w,y ) ) = α τ z ( α h ( a )) δ ( y,z ) α τ y ( α h ( b )) δ ( w,y ) = α τ z ( α h ( a )) α τ z ( α τ − y ( α τ y ( α h ( b )))) δ ( w,z ) = α τ z ( α h ( a )) α τ z ( α h ( b )) δ ( w,z ) . Hence, γ h is a ring isomorphism. It remains to show that γ satisfies the condition (v)given in § . γ l − ( C l ∩ C h − ) ⊂ C ( hl ) − , for all h, l ∈ G ( x ). Indeed, by definition, γ l − is additive and whence γ l − ( C l ∩ C h − ) = ⊕ u ∈G γ l − ( α τ t ( u ) ( A l ∩ A h − ) δ u )= ⊕ u ∈G α τ t ( u ) ( α l − ( A l ∩ A h − )) δ u ⊂ ⊕ u ∈G α τ t ( u ) ( A l − ∩ A l − h − ) δ u = C ( hl ) − , where the last inclusion above holds because α ( x ) is a partial action of G ( x ) on A x asdefined in Remark 3.2. Finally, let c = α τ z ( α l − ( a )) δ ( y,z ) ∈ γ l − ( C l ∩ C h − ). Then γ h ( γ l ( c )) = γ h ( γ l ( α τ z ( α l − ( a )) δ ( y,z ) ))= γ h ( α τ z ( a ) δ ( y,z ) )= α τ z ( α h ( a )) δ ( y,z ) . On the other hand, since α hl = α h α l in α l − ( A l ∩ A h − ) we have γ hl ( c ) = α τ z ( α hl ( α l − ( a ))) δ ( y,z ) = α τ z ( α h ( a )) δ ( y,z ) , and consequently γ satisfies (v) of § . (cid:3) By Lemma 4.3 we can consider the partial skew group ring (
A ⋆ β G ) ⋆ γ G ( x ) and thuspresent the main result of this section which give us a factorization of the ring A ⋆ α G . Theorem 4.4.
A ⋆ α G ≃ ( A ⋆ β G ) ⋆ γ G ( x ) . Proof.
Consider the map ϕ : A ⋆ α G → ( A ⋆ β G ) ⋆ γ G ( x ) given by aδ g aδ ( s ( g ) ,t ( g )) δ g x ,where g x = τ − t ( g ) gτ s ( g ) . In order to prove that ϕ is a ring isomorphism we proceed by aseries of steps. Step 1: ϕ is well defined. D. BAGIO, A. PAQUES, AND H. PINEDO
By Lemma 4.1, A g ⊆ A t ( g ) = B ( s ( g ) ,t ( g )) , for all g ∈ G . Hence, we only need to show that aδ ( s ( g ) ,t ( g )) ∈ C g x , for all a ∈ A g . Notice that α τ t ( g ) ( A g x ) = α τ t ( g ) ( A τ − t ( g ) gτ s ( g ) )= α τ t ( g ) ( A τ − t ( g ) gτ s ( g ) ∩ A x )= A gτ s ( g ) ∩ A τ t ( g ) x (by Lemma 3.1 (iii))= A gτ s ( g ) ∩ A τ t ( g ) (2) = A gτ s ( g ) ∩ A t ( g ) = A gτ s ( g ) ( A gτ s ( g ) ⊆ A t ( gτ s ( g ) ) = A t ( g ) ) . Since A g − = A g − ∩ A s ( g ) (2) = A g − ∩ A τ s ( g ) we have A g = α g ( A g − ∩ A τ s ( g ) ) = A g ∩ A gτ s ( g ) .Hence a ∈ A g ⊆ A gτ s ( g ) = α τ t ( g ) ( A g x ) which implies aδ ( s ( g ) ,t ( g )) ∈ C g x by (3) and (4). Step 2: ϕ is a ring homomorphism. It is enough to prove that ϕ preserves the operation of multiplication. Let g, h ∈ G suchthat s ( g ) = t ( h ). It is easy to see that ( gh ) x = g x h x . Hence ϕ (( aδ g )( bδ h )) = ϕ ( aα g ( b g − ) δ gh )= aα g ( b g − ) δ ( s ( gh ) ,t ( gh )) δ ( gh ) x = aα g ( b g − ) δ ( s ( h ) ,t ( g )) δ g x h x , for all a ∈ A g and b ∈ A h . On the other hand ϕ ( aδ g ) ϕ ( bδ h ) = ( aδ ( s ( g ) ,t ( g )) δ g x )( bδ ( s ( h ) ,t ( h )) δ h x )= ( aδ ( s ( g ) ,t ( g )) )( γ g x ( bδ ( s ( h ) ,t ( h )) ′ g − x )) δ g x h x . As in Step 1, we have that A h ⊆ A hτ s ( h ) = α τ t ( h ) ( A h x ). Since b ∈ A h , there is b ′ ∈ A h x such that b = α τ t ( h ) ( b ′ ) and bδ ( s ( h ) ,t ( h )) ′ g − x = α τ t ( h ) ( b ′ ) δ ( s ( h ) ,t ( h )) X z ∈G α τ z (1 g − x ) δ ( z,z ) = α τ t ( h ) ( b ′ ) δ ( s ( h ) ,t ( h )) α τ s ( h ) (1 g − x ) δ ( s ( h ) ,s ( h )) = α τ t ( h ) ( b ′ ) α τ t ( h ) (1 g − x ) δ ( s ( h ) ,t ( h )) = α τ t ( h ) ( b ′ g − x ) δ ( s ( h ) ,t ( h )) . N PARTIAL SKEW GROUPOIDS RINGS 9
Hence, ϕ ( aδ g ) ϕ ( bδ h ) = ( aδ ( s ( g ) ,t ( g )) )( γ g x ( bδ ( s ( h ) ,t ( h )) ′ g − x )) δ g x h x = ( aδ ( s ( g ) ,t ( g )) )( γ g x ( α τ t ( h ) ( b ′ g − x ) δ ( s ( h ) ,t ( h )) )) δ g x h x = ( aδ ( s ( g ) ,t ( g )) )( α τ t ( h ) ( α g x ( b ′ g − x )) δ ( s ( h ) ,t ( h )) ) δ g x h x = aα τ t ( g ) α τ − s ( g ) α τ t ( h ) α g x ( b ′ g − x ) δ ( s ( h ) ,t ( g )) δ g x h x = aα τ t ( g ) α g x ( b ′ g − x ) δ ( s ( h ) ,t ( g )) δ g x h x (2) = aα τ t ( g ) ( α g x ( b ′ g − x )1 τ − t ( g ) ) δ ( s ( h ) ,t ( g )) δ g x h x = aα τ t ( g ) g x ( b ′ ( τ t ( g ) g x ) − )1 τ t ( g ) δ ( s ( h ) ,t ( g )) δ g x h x (by (v) of § aα gτ s ( g ) ( b ′ ( gτ s ( g ) ) − )1 τ t ( g ) δ ( s ( h ) ,t ( g )) δ g x h x (2) = aα gτ s ( g ) ( b ′ ( gτ s ( g ) ) − )1 g δ ( s ( h ) ,t ( g )) δ g x h x = aα g ( α τ s ( g ) ( b ′ τ − s ( g ) )1 g − ) δ ( s ( h ) ,t ( g )) δ g x h x (by (v) of § (2) = aα g ( α τ t ( h ) ( b ′ )1 g − ) δ ( s ( h ) ,t ( g )) δ g x h x = aα g ( b g − ) δ ( s ( h ) ,t ( g )) δ g x h x = ϕ (( aδ g )( bδ h )) . Step 3: ϕ is injective. Let v = P g ∈G a g δ g ∈ A ⋆ α G such that ϕ ( v ) = 0. Then0 = X g ∈G a g δ ( s ( g ) ,t ( g )) δ g x = X h ∈G ( x ) X g ∈G g x = h a g δ ( s ( g ) ,t ( g )) δ h Since
C ⋆ θ G ( x ) is a direct sum, it follows that X g ∈G g x = h a g δ ( s ( g ) ,t ( g )) = 0 , for all h ∈ G ( x ) . (5)Consider h ∈ G ( x ) and g, g ′ ∈ G such that g x = g ′ x = h . It is straightforward to checkthat ( s ( g ) , t ( g )) = ( s ( g ′ ) , t ( g ′ )) if and only if g = g ′ . Therefore (5) holds if and only if a g = 0, for all g ∈ G . Thus v = 0. Step 4: ϕ is surjective. It is enough to check that given any element of the type α τ z ( a ) δ ( y,z ) δ h , with h ∈ G ( x )and a ∈ A h , there exists an element w ∈ A ⋆ α G such that ϕ ( w ) = α τ z ( a ) δ ( y,z ) δ h . To do that observe that α τ z ( a ) ∈ α τ z ( A h ) = α τ z ( A h ∩ A x ) (2) = α τ z ( A h ∩ A τ − z )= A τ z h ∩ A τ z (by Lemma 3.1 (iii)) (2) = A τ z h ∩ A z = A τ z h (because A τ z h ⊆ A t ( τ z h ) = A t ( τ z ) = A z ) . Therefore, for g = τ z hτ − y we have t ( g ) = t ( τ z ) = z , s ( g ) = s ( τ − y ) = y and α τ z ( a ) ∈ A τ z h = A τ t ( g ) h ( ∗ ) = α τ t ( g ) ( A h ) ( ∗∗ ) ⊆ α τ t ( g ) ( A hτ − s ( g ) ) ( ∗ ) = A τ t ( g ) hτ − s ( g ) = A τ z hτ − y = A g , where ( ∗ ) is ensured by α τ t ( g ) ( A h ) = α τ t ( g ) ( A h ∩ A x ) (2) = α τ t ( g ) ( A h ∩ A τ − s ( g ) ) = A τ t ( g ) h , and ( ∗∗ ) by A h = α h ( A h − ∩ A x ) (2) = α h ( A h − ∩ A τ − s ( g ) ) = A h ∩ A hτ − s ( g ) ⊆ A hτ − s ( g ) . Now, taking w = α τ z ( a ) δ g we are done. (cid:3) Remark 4.5.
Since global actions are group-type actions, the factorization given inTheorem 4.4 holds for all unital global action α of G on A . In such a case, the partialgroup action γ of G ( x ) on A x is indeed a global action and consequently A ⋆ α G is a skewgroup ring. 5. Applications
The aim of this section is to present some applications of Theorem 4.4. In what follows, G is connected and G is finite. The partial action α , the ring A and the transversal τ ( x )are assumed as in the previous section. Also, β is the global action of G on A given inLemma 4.1 and γ is the partial action of G ( x ) on A ⋆ β G given in Lemma 4.3.Note that ϕ : A → A⋆ α G , a P y ∈G ( a y ) δ y , is a monomorphism of rings and whence A⋆ α G is a ring extension of A . By Theorem 4.4, A ⊂ A⋆ β G ⊂ ( A⋆ β G ) ⋆ γ G ( x ) ≃ A⋆ α G .Therefore, we will investigate some properties of the extension A ⊂ A ⋆ α G using theintermediate extensions and the results known for partial group actions.5.1. Separability.
In this subsection we will study the separability property to the ringextension A ⊂ A ⋆ α G . We recall that a unital ring extension R ⊂ S is called separable ifthe multiplication map m : S ⊗ R S → S is a splitting epimorphism of S -bimodules. Thisis equivalent to saying that there exists an element x ∈ S ⊗ R S such that sx = xs , for all s ∈ S , and m ( x ) = 1 S . Such a element x is usually called an idempotent of separability of S over R . N PARTIAL SKEW GROUPOIDS RINGS 11
Throughout this subsection, we will assume that G is finite . As in [4], consider themaps t y,z : A → A and t z : A → A given by t y,z ( a ) = X g ∈G ( y,z ) α g ( a g − ) , t z ( a ) = X y ∈G t y,z ( a ) , y, z ∈ G , a ∈ A. Particularly, if G is a group then G = { x } and t x : A → A is the trace map for partialgroup actions as defined in Section 2 of [6].Now, we recall Theorem 4.1 of [4] which will be useful for our purposes. Theorem 5.1.
The ring extension A ⊂ A ⋆ α G is separable if and only if there is anelement a in the center C ( A ) of A such that t z ( a ) = 1 z , for all z ∈ G . (cid:3) In order to apply Theorem 4.4 to determine when the extension A ⊂ A⋆ α G is separable,we consider the separability problem for the extensions A ⊂ A ⋆ β G and A ⋆ β G ⊂ ( A ⋆ β G ) ⋆ γ G ( x ). Lemma 5.2.
The extension A ⊂ A ⋆ β G is separable if and only if there exists a ∈ C ( A ) such that P z ∈G α τ − z ( a z ) = 1 x .Proof. Let z ∈ G and a ∈ A . Then t z ( a ) = X y ∈G t y,z ( a ) = X y ∈G X u ∈G ( y,z ) β u ( a u − )= X y ∈G β ( y,z ) ( a y ) = X y ∈G α τ z α τ − y ( a y )= α τ z ( X y ∈G α τ − y ( a y )) . Consequently t z ( a ) = 1 z if and only if P y ∈G α τ − y ( a y ) = 1 x and the result follows byTheorem 5.1. (cid:3) Lemma 5.3. C ( A ⋆ β G ) = (cid:8)P z ∈G α τ z ( a x ) δ ( z,z ) : a x ∈ C ( A x ) (cid:9) . Proof.
Let a x ∈ C ( A x ) and Λ = P z ∈G α τ z ( a x ) δ ( z,z ) . Given ( y, w ) ∈ G and a w ∈ A w wehave that a w δ ( y,w ) · Λ = a w δ ( y,w ) · α τ y ( a x ) δ ( y,y ) = a w α τ w ( a x ) δ ( y,w ) and Λ · a w δ ( y,w ) = α τ w ( a x ) δ ( w,w ) · a w δ ( y,w ) = α τ w ( a x ) a w δ ( y,w ) . Since a x ∈ C ( A x ) and α τ w is an isomorphism it is clear that α τ w ( a x ) ∈ C ( A w ). Thus,Λ ∈ C ( A ⋆ β G ). Conversely, consider Λ = P y,z ∈G a ( y,z ) δ ( y,z ) ∈ C ( A ⋆ β G ), with a ( y,z ) ∈ A ( y,z ) = A z , for all y ∈ G . From Λ · w δ ( w,w ) = 1 w δ ( w,w ) · Λ, it follows that X z ∈G a ( w,z ) δ ( w,z ) = X y ∈G a ( y,w ) δ ( y,w ) , for all w ∈ G . Hence a ( y,z ) = 0 if y = z and whence Λ = P z ∈G a ( z,z ) δ ( z,z ) . Moreover, for all y, w ∈ G and a ∈ A w ,Λ · aδ ( y,w ) = a ( w,w ) aδ ( y,w ) and aδ ( y,w ) · Λ = aα τ w ( α τ − y ( a ( y,y ) )) δ ( y,w ) . Therefore a ( w,w ) a = aα τ w ( α τ − y ( a ( y,y ) )) , for all y, w ∈ G , a ∈ A w . (6)When a = 1 w we obtain that a ( w,w ) = α τ w ( α τ − y ( a ( y,y ) )), for all y, w ∈ G . Particularly, a ( w,w ) = α τ w ( a ( x,x ) ) for all w ∈ G . Given b ∈ A x , consider a := α τ w ( b ) ∈ A w . By (6), α τ w ( a ( x,x ) b ) = α τ w ( ba ( x,x ) ). Thus a ( x,x ) b = ba ( x,x ) and consequently a ( x,x ) ∈ C ( A x ). (cid:3) Lemma 5.4.
The extension
A ⋆ β G ⊂ ( A ⋆ β G ) ⋆ γ G ( x ) is separable if and only if theextension A x ⊂ A x ⋆ α ( x ) G ( x ) is separable.Proof. Let Λ ∈ C ( A ⋆ β G ). By Lemma 5.3, Λ = P z ∈G α τ z ( a x ) δ ( z,z ) , with a x ∈ C ( A x ).Notice that t x (Λ) = X h ∈G ( x ) γ h ( X z ∈G α τ z ( a x ) δ ( z,z ) · X w ∈G α τ w (1 h − ) δ ( w,w ) )= X h ∈G ( x ) X z ∈G γ h ( α τ z ( a x ) δ ( z,z ) · α τ z (1 h − ) δ ( z,z ) )= X h ∈G ( x ) X z ∈G γ h ( α τ z ( a x h − ) δ ( z,z ) )= X h ∈G ( x ) X z ∈G α τ z ( α h ( a x h − )) δ ( z,z ) = X z ∈G α τ z ( X h ∈G ( x ) α h ( a x h − )) δ ( z,z ) = X z ∈G α τ z ( t x ( a x )) δ ( z,z ) . Hence, there is Λ ∈ C ( A ⋆ β G ) such that t x (Λ) = 1 = P z ∈G z δ ( z,z ) if and only if thereis a x ∈ C ( A x ) such that t x ( a x ) = 1 x . it Then the result follows from Theorem 5.1 andTheorem 3.1 of [2]. (cid:3) Theorem 5.5.
The following statements hold: (i) if A x ⊂ A x ⋆ α ( x ) G ( x ) is a separable extension and there exists a ∈ C ( A ) suchthat P z ∈G α τ − z ( a z ) = 1 x then A ⊂ A ⋆ α G is a separable extension; (ii) if A ⊂ A ⋆ α G is a separable extension then A x ⊂ A x ⋆ α ( x ) G ( x ) also is.Proof. (i) By Lemmas 5.2 and 5.4, A ⊂ A ⋆ β G and A ⋆ β G ⊂ ( A ⋆ β G ) ⋆ γ G ( x ) areseparable extensions. By the transitivity of the separability property (see Proposition2.5 of [8]), A ⊂ ( A ⋆ β G ) ⋆ γ G ( x ) is separable. Thus the result follows from Theorem4.4. N PARTIAL SKEW GROUPOIDS RINGS 13 (ii) From Theorem 4.4 and Proposition 2.5 of [8] it follows that
A ⋆ β G ⊂ ( A ⋆ β G ) ⋆ γ G ( x ) is a separable extension. Hence, Lemma 5.4 implies that A x ⊂ A x ⋆ α ( x ) G ( x ) isseparable. (cid:3) Corollary 5.6.
Suppose that |G | A is invertible in A and ( |G | A ) − = n A , with n ∈ N .If A x ⊂ A x ⋆ α ( x ) G ( x ) is separable then A ⊂ A ⋆ α G is separable.Proof. Let a = n A ∈ C ( A ). Then X z ∈G α τ − z ( a z ) = X z ∈G α τ − z ( n z ) = n X z ∈G α τ − z (1 z )= n |G | x = 1 x , and whence the result follows by Theorem 5.5 (1). (cid:3) Example 5.7.
Let G , A , α and τ ( x ) be as in Example 3.5. Consider a x = e + e ∈ A x .Observe that t x ( a x ) = α x ( a x ) + α g ( a x g − ) = a x + e = e + e = 1 x . It follows fromTheorem 3.1 of [2] that A x ⊂ A x ⋆ α ( x ) G ( x ) is a separable extension. Moreover, if a = ( e + e + e + e ) ∈ A then α τ − x ( a x ) + α τ − y ( a y ) = α x ( a x ) + α l − ( a l ) = e + e = 1 x . By Theorem 5.5 (1), the extension A ⊂ A ⋆ α G is separable.5.2. Semisimple extension.
In this subsection we investigate the semisimplicity forthe ring extension A ⊂ A ⋆ α G . Recall from [8] that a ring extension R ⊂ S is calledleft (right) semisimple if any left (right) S -submodule N of a left (right) S -module M having an R -complement in M , has an S -complement in M .For the convenience of the reader, we recall Proposition 2.6 of [8]. Proposition 5.8. If R ⊂ S is a separable ring extension then R ⊂ S is a left (right)semisimple ring extension. (cid:3) Theorem 5.9. If G is finite and (i) there exists a x ∈ C ( A x ) such that t x ( a x ) = 1 x and (ii) there exists a ∈ C ( A ) such that P z ∈G α τ − z ( a z ) = 1 x ,then A ⊂ A ⋆ α G is a left (right) semisimple extension.Proof. Using (i), we obtain from Theorem 3.1 of [2] that A x ⊂ A x ⋆ α ( x ) G ( x ) is separable.Then, by Theorem 5.5 (i) the extension A ⊂ A ⋆ α G is separable and so the result followsby Proposition 5.8. (cid:3) Frobenius extension.
In this subsection we will prove that A ⊂ A ⋆ α G is aFrobenius extension. We recall that a ring extension R ⊂ S is called Frobenius if thereexist an element ∆ = P ni =1 s i ⊗ s i ∈ S ⊗ R S and an R -bimodule map ε : S → R suchthat ∆ s = s ∆, for all s ∈ S , and P ni =1 ε ( s i ) s i = P ni =1 s i ε ( s i ) = 1. More details onFrobenius extensions can be seen, for example, in [5] or [9]. Firstly, we note that the natural inclusion A ⊂ A⋆ β G , given by a P z ∈G ( a z ) δ ( z,z ) ,induces the following ( A, A )-bimodule structure on
A ⋆ β G : a · ( a z δ ( y,z ) ) = aa z δ ( y,z ) , ( a z δ ( y,z ) ) · a = a z β ( y,z ) ( a y ) δ ( y,z ) , for all ( y, z ) ∈ G , a z ∈ A z and a ∈ A . Theorem 5.10. If G is finite then A ⊂ A ⋆ α G is a Frobenius extension.Proof. Let ∆ = P z ∈G z δ ( z,z ) ⊗ z δ ( z,z ) ∈ ( A ⋆ β G ) ⊗ A ( A ⋆ β G ) and ε : A ⋆ β G → A given by ε ( a z δ ( y,z ) ) = (cid:26) a z , if y = z , otherwise.It is straightforward to check that ε is an ( A, A )-bimodule map. Also, note that( a w δ ( y,w ) )∆ = a w δ ( y,w ) = ∆( a w δ ( y,w ) ) , for all ( y, w ) ∈ G and a w ∈ A w . Since X z ∈G ε (1 z δ ( z,z ) )(1 z δ ( z,z ) ) = X z ∈G z δ ( z,z ) ε (1 z δ ( z,z ) ) = X z ∈G z δ ( z,z ) = 1 A⋆ β G , it follows that A ⊂ A ⋆ β G is a Frobenius extension. Notice that G ( x ) is finite because G is finite. Then, by Theorem 3.6 of [2], A ⋆ β G ⊂ ( A ⋆ β G ) ⋆ γ G ( x ) is a Frobeniusextension. As Frobenius extension is a transitive notion (see e. g. [9, pg. 6]), we obtainfrom Theorem 4.4 that A ⊂ A ⋆ α G is Frobenius. (cid:3) Artinianity.
The artinian property for partial skew groupoids rings was studied in[11]. In our context, using Theorem 4.4, we obtain the following refinement of Theorem1.3 of [11].
Theorem 5.11.
The partial skew groupoid ring
A ⋆ α G is artinian if and only if A isartinian and A h = { } , for all but finitely many h ∈ G ( x ) .Proof. Assume that A⋆ α G is artinian. By Theorem 1.3 of [11], A is artinian and A g = { } for all but finitely many g ∈ G . Particularly, A h = { } for all but finitely many h ∈ G ( x ).For the converse, consider h ∈ G ( x ). By (3) and (4), C h = { } if and only if thereis z ∈ G such that α τ z ( A h ) = { } . Consequently, C h = { } , for all but finitely many h ∈ G ( x ). Also, since G is finite and A is artinian it follows from Theorem 1.3 of[11] that A ⋆ β G is artinian. Using again the Theorem 1.3 of [11], we conclude that( A ⋆ β G ) ⋆ γ G ( x ) is artinian and the result follows by Theorem 4.4. (cid:3) References [1] D. Bagio, D. Flˆores and A. Paques,
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Departamento de Matem´atica, Universidade Federal de Santa Maria, 97105-900, SantaMaria-RS, Brasil
E-mail address : [email protected] Instituto de Matem´atica e Estat´ıstica, Universidade federal de Porto Alegre, 91509-900, Porto Alegre-RS, Brazil
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