In this paper we present an intrinsic description of the structure of an Azumaya skew groupoid ring, having its center contained in the respective ground ring, in terms of suitable central Galois algebras and commutative Galois extensions

Flores, D.

Paques, A.

Algebra and Discrete Mathematics

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Algebra and Discrete Mathematics RESEARCH ARTICLEVolume 16 (2013). Number 1. pp. 71 – 80© Journal “Algebra and Discrete Mathematics”On the structure of skew groupoid ringswhich are AzumayaDaiana Flôres, Antonio PaquesCommunicated by V. V. KirichenkoAbstract. In this paper we present an intrinsic descriptionof the structure of an Azumaya skew groupoid ring, having its centercontained in the respective ground ring, in terms of suitable centralGalois algebras and commutative Galois extensions.1. IntroductionGroupoids are usually presented as small categories whose morphismsare invertible.The notion of a groupoid action that we use in this paper arose fromthe notion of a partial groupoid action [3], which is a natural extensionof the notion of a partial group action [5]. First, partial ordered groupoidactions on sets were introduced in the literature, as ordered premorphisms,by N. Gilbert [8]. After, partial ordered groupoid actions on rings wereconsidered by D. Bagio and the authors [2] as a generalization of partialgroup actions, as introduced by M. Dokuchaev and R. Exel in [5]. And in[3] this notion was extended to the general context of groupoids.Accordingly [3], an action of a groupoid G on a K-algebra R is a pairβ = ({Eg}g∈G, {βg}g∈G), where for each g ∈ G, Eg = Er(g) is an ideal ofR and βg : Eg−1 → Eg is an isomorphism of K-algebras satisfying somesuitable conditions of compatibility (see the subsection 2.3).2010 MSC: 16H05, 18B40, 20L99.Key words and phrases: groupoid action, skew groupoid ring, Azumaya ring,Galois algebra.72 On the structure of skew groupoid ringsThe notion of groupoid action given by Caenepeel and De Groot in [4]is equivalent to the above one in the case that the set G0 of all identitiesof G is finite, each ideal Ee (e ∈ G0) is unital and R =⊕e∈G0Ee, [7,Proposition 2.2]. This is the groupoid action we will deal with throughoutall this paper.Given such an action β of G on R, we can consider the skew groupoidring A = R ⋆β G, similarly to the construction given in [2], which is anassociative and unital K-algebra.Our aim is to present an intrinsic structural description of A in thecase that G is finite, A is Azumaya and its center is contained in R (seeTheorem 3.3).This paper is organized as follows. In the next section we give pre-liminaries about groupoids, groupoid actions, skew groupoid rings, andseparability, Hirata-separability, Galois, and Azumaya properties. In thatsection we will be concerned only with the results strictly necessary toconstruct the appropriate conditions to prove our main theorem, whoseproof we set in the section 3.Throughout, unless otherwise specified, rings and algebras are asso-ciative and unital.2. Definitions and basic results2.1. GroupoidsThe axiomatic version of groupoid that we adopt in this paper wastaken from [13]. A groupoid is a non-empty set G equipped with a partiallydefined binary operation, that we will denote by concatenation, for whichthe usual axioms of a group hold whenever they make sense, that is:(i) For every g, h, l ∈ G, g(hl) exists if and only if (gh)l exists and inthis case they are equal.(ii) For every g, h, l ∈ G, g(hl) exists if and only if gh and hl exist.(iii) For each g ∈ G there exist (unique) elements d(g), r(g) ∈ G suchthat gd(g) and r(g)g exist and gd(g) = g = r(g)g.(iv) For each g ∈ G there exists an element g−1 ∈ G such that d(g) =g−1g and r(g) = gg−1.We will denote by G2 the subset of the pairs (g, h) ∈ G×G such thatthe element gh exists.D. Flôres, A. Paques 73An element e ∈ G is called an identity of G if e = d(g) = r(g−1), forsome g ∈ G. In this case e is called the domain identity of g and the rangeidentity of g−1. We will denote by G0 the set of all identities of G and wewill denote by Ge the set of all g ∈ G such that d(g) = r(g) = e. Clearly,Ge is a group, called the isotropy (or principal) group associated to e.The assertions listed in the following lemma are straightforward fromthe above definition. Such assertions will be freely used along this paper.Lemma 2.1. Let G be a groupoid. Then,(i) for every g ∈ G, the element g−1 is unique satisfying g−1g = d(g)and gg−1 = r(g),(ii) for every g ∈ G, d(g−1) = r(g) and r(g−1) = d(g),(iii) for every g ∈ G, (g−1)−1 = g,(iv) for every g, h ∈ G, (g, h) ∈ G2 if and only if d(g) = r(h),(v) for every g, h ∈ G, (h−1, g−1) ∈ G2 if and only if (g, h) ∈ G2 and,in this case, (gh)−1 = h−1g−1,(vi) for every (g, h) ∈ G2, d(gh) = d(h) and r(gh) = r(g),(vii) for every e ∈ G0, d(e) = r(e) = e and e−1 = e,(viii) for every (g, h) ∈ G2, gh ∈ G0 if and only if g = h−1,(ix) for every g, h ∈ G, there exists l ∈ G such that g = hl if and onlyif r(g) = r(h),(x) for every g, h ∈ G, there exists l ∈ G such that g = lh if and onlyif d(g) = d(h).2.2. Groupoid actions and skew groupoid ringsLet G be a groupoid and R an algebra over a commutative ring K.Following [3], an action of G on R is a pairβ = ({Eg}g∈G, {βg}g∈G),where for each g ∈ G, Eg = Er(g) is an ideal of R, βg : Eg−1 → Eg isan isomorphism of K-algebras (not necessarily unital), and the followingconditions hold:(i) βe is the identity map IEe of Ee, for all e ∈ G0,(ii) βgβh(r) = βgh(r), for all (g, h) ∈ G2 and r ∈ Eh−1 = E(gh)−1 .74 On the structure of skew groupoid ringsIn particular, β induces an action of the group Ge on Ee, for every e ∈ G0.Accordingly [2, Section 3], the skew groupoid ring R⋆βG correspondingto β is defined as the direct sumR ⋆β G =⊕g∈GEgδgin which the δg’s are symbols, with the usual addition, and multiplicationdetermined by the rule(xδg)(yδh) ={xβg(y)δgh if (g, h) ∈ G20 otherwise,for all g, h ∈ G, x ∈ Eg and y ∈ Eh.This multiplication is well defined. Indeed, if (g, h) ∈ G2 then d(g) =r(h) (see Lemma 2.1(iv)). So, Eg−1 = Er(g−1) = Ed(g) = Er(h) = Eh,βg(y) makes sense, and xβg(y) ∈ Eg = Er(g)(⋆)= Er(gh) = Egh, where theequality (⋆) is ensured by Lemma 2.1(vi).By a routine calculation one easily sees that A = R⋆βG is associative,and by [2, Proposition 3.3] it is unital if G0 is finite and Ee is unital, forall e ∈ G0. In this case, the identity element of A is 1A =∑e∈G01eδe,where 1e denotes the identity element of Ee, for all e ∈ G0.2.3. Invariants, Galois extensions and Galois algebrasLet R, G and β as in the preceding subsection.For our purposes we will assume hereafter that G is finite, R =⊕e∈G0Ee and every ideal Eg is unital, with its identity element denotedby 1g (in particular, each 1g is a central idempotent of R).We will denote by Rβ the subalgebra of the invariant elements of Runder the action β, that is,Rβ = {r ∈ R | βg(r1g−1) = r1g, for all g ∈ G}.For any non-empty subset X of R, any subring Y of R containing Xand any (Y, Y )-bimodule V we will denote by CV (X) the centralizer ofX in V , that is, the set of all v ∈ V such that xv = vx for all x ∈ X. If,in particular, X = Y = V = R, then CV (X) is the center of R and wewill denote it simply C(R).A non-empty subset X of R is called β-invariant if βg(Eg−1 ∩X) =Eg ∩X, for every g ∈ G.D. Flôres, A. Paques 75In particular, C(R) and CR(Rβ) are β-invariant. Indeed, if r ∈ Eg−1 ∩C(R), thenβg(r)x = βg(r)1gx = βg(rβg−1(x1g)) == βg(βg−1(x1g)r) = x1gβg(r) = xβg(r),for all x ∈ R, and so βg(r) ∈ Eg ∩ C(R), for all g ∈ G. Hence, βg(Eg−1 ∩C(R)) = Eg ∩ C(R). By similar arguments one also gets βg(Eg−1 ∩CR(Rβ)) = Eg ∩ CR(Rβ).Moreover, since 1g ∈ C(R) ⊆ CR(Rβ), β induces by restriction anaction on X ′ = CR(X), with X = R or Rβ, given by the pair β|X′ =({E′g = X′1g}g∈G, {βg|E′g−1}g∈G). Notice that E′g = E′r(g), for all g ∈ G,and X ′ =⊕e∈G0E′e.We say that R is a β-Galois extension (resp., G-Galois extension, if Gis a group) of Rβ (resp., RG) if there exist elements xi, yi ∈ R, 1 ≤ i ≤ m,such that ∑1≤i≤mxiβg(yi1g−1) = δe,g1e,for all e ∈ G0 and g ∈ G. In particular, in this case, Ee is a Ge-Galoisextension of EGee for every e ∈ G0. The set {xi, yi | 1 ≤ i ≤ n} is called aGalois coordinate system of R over Rβ (resp., RG).We also say thatR is a β-Galois algebra (resp., β-central Galois algebra)if R is a β-Galois extension of Rβ and Rβ ⊆ C(R) (resp., Rβ = C(R)).In the particular case that G is a group, we replace “β-" by “G-".Remark 2.2. Observe that Rβ ⊆⊕EGee . Indeed, any element x ∈ Ris of the form x =∑e∈G0xe, with xe ∈ Ee, and x ∈ Rβ if and onlyif βg(x1g−1) = x1g if and only if βg(xd(g)) = xr(g), for all g ∈ G. Inparticular, βg(xe) = xe, for all e ∈ G0 and g ∈ Ge. Hence, each xe belongsto EGee and x ∈⊕e∈G0EGee .In general the inclusion Rβ ⊆⊕EGee is strict, as it is shown inthe following example: take G = {g, g−1, r(g), d(g)}, R = Se1 ⊕ Se2 ⊕Se3⊕Se4, where S is a ring and e1, e2, e3 and e4 are pairwise orthogonalidempotents with sum 1R, Eg = Er(g) = Se1 ⊕ Se2, Eg−1 = Ed(g) =Se3 ⊕ Se4, βd(g) = IEd(g), βr(g) = IEr(g), βg(xe3 + ye4) = xe1 + ye2and βg−1(xe1 + ye2) = xe3 + ye4, para todo x, y ∈ S. It is immediateto check that β = ({El}l∈G, {βl}l∈G) is an action of G on R, Rβ =S(e1 + e3)⊕ S(e2 + e4), EGd(g)d(g) = Se3 ⊕ Se4 and EGr(g)r(g) = Se1 ⊕ Se2. Inthis example we also notice that Rβ1d(g) = EGd(g)d(g) and Rβ1r(g) = EGr(g)r(g) .76 On the structure of skew groupoid ringsBut, neither this fact occurs in general, as we see in the next example:take R as above and G = {g1, g2, g}, with G0 = {g1, g2}, g−1 = g andgg = g2. Setting Eg1 = Se1 ⊕ Se4, Eg2 = Se2 ⊕ Se3 = Ed(g) = Er(g) andβg(ae2+be3) = be2+ae3, one easily sees that Rβ = Se1⊕S(e2+e3)⊕Se4and Rβ1g2 = S(e2 + e3) ⊂ Eg2 = EGg2g2 .2.4. Azumaya, Hirata-separable and separable algebrasLet S be a subalgebra of R. We say that R is separable over S [11]if R is a direct summand of R ⊗S R as an R-bimodule. In this case, ifS ⊆ C(R) (resp., S = C(R)) we also say that R is a separable S-algebra(resp., Azumaya over S or Azumaya algebra or simply Azumaya). Wesay that R is Hirata-separable over S [9] if R⊗S R is isomorphic, as anR-bimodule, to a direct summand of a finite direct sum of copies of R. Itis well known that every Azumaya algebra is Hirata-separable over itscenter [14] as well as every Hirata-separable extension is separable [10].3. The main resultsAlso in this section, R, G and β are as in the later one.We start with the following theorem, which is a generalization of[1, Theorems 1 and 2] to the setting of groupoid actions, and shows anice and closed relation among the notions of Azumaya algebra, Galoisextension and Hirata-separability. The arguments used in the proof ofthe mentioned theorems in [1] are practically the same we will use here.Theorem 3.1. The following statements are equivalent:(i) A is Azumaya and C(A) ⊆ R.(ii) A is Hirata-separable over R and R is separable over C(R)β.(iii) R is a β-Galois extension of Rβ and Rβ is Azumaya over C(A).(iv) CR(Rβ) is a β-Galois extension of C(A) and Rβ is Azumaya overC(A).Moreover, in this case, C(A) = C(R)β = C(Rβ).For the proof of this theorem we need the following lema.Lemma 3.2. C(A) ⊆ R if and only if C(A) = C(R)β.D. Flôres, A. Paques 77Proof. If C(A) ⊆ R, then C(A) ⊆ C(R) and (∑e∈G0x1eδe)1gδg =1gδg(∑e∈G0x1eδe), for every x ∈ C(A) and g ∈ G.Since (∑e∈G0x1eδe)1gδg = (x1r(g)δr(g))(1gδg) = x1gδg and1gδg(∑e∈G0x1eδe) = (1gδg)(x1d(g)δd(g)) = βg(x1g−1)δg, it followsthat βg(x1g−1) = x1g, for every g ∈ G. Therefore x ∈ C(R)β. Theconverse is straightforward.Proof of Theorem 3.1.(i)⇒(ii) By Lemma 3.2 we have C(A) = C(R)β. It follows from[12, Theorem 1] that A is Hirata-separable over R, since A is a finitelygenerated projective R-module. In addition, R is a direct summand ofA as an R-bimodule. Hence, it follows from [14, Proposition 1.3] thatCA(R) is separable over C(A) and CA(CA(R)) = R. Consequently, R isseparable over C(A) by [6, Theorem III.4.3].(ii)⇒(iii) We will proceed by steps.step 1: A is Azumaya and C(A) ⊆ R.Since A is Hirata-separable over R and R is a direct summand ofA as an R-bimodule, it follows that CA(R) is separable over C(A) andCA(CA(R)) = R, by [14, Proposition 1.3]. Furthermore, C(A) = C(R)β byLemma 3.2. On the other hand, by the assumptions and [10, Theorem 2.2]we have that A is separable over R, so over C(R)β too.step 2: E = EndC(R)β (R) is Azumaya over C(R)β .Indeed, A is Azumaya over C(R)β (by step 1) and so a projectivefinitely generated C(R)β-module, by [6, Theorem III.3.4]. Hence, R alsois a projective finitely generated C(R)β-module, as a direct summand ofA. Now, the assertion follows by [6, Theorem III.4.1].step 3: CE(A) = EndA(R) ≃ (Rβ)op as C(R)β-algebras.First of all, we observe that R is a left A-module via the action givenby rgδg · r = rgβg(r1g−1), for all g ∈ G, rg ∈ Eg and r ∈ R. Consequently,E is an A-bimodule in a natural way, that is, af(r) = a · f(r) andfa(r) = f(a · r), for all a ∈ A, f ∈ E and r ∈ R. Now, the equality ofthe assertion is immediate. It is also immediate that R is a ((Rβ)op, A)-bimodule, with (Rβ)op acting on R by the left via the right multiplication,that is, x · r = rx, for all r ∈ R and x ∈ (Rβ)op.The claimed isomorphism of the assertion is given by the map θ :(Rβ)op → EndA(R) defined by θ(x)(r) = x · r, for all x ∈ (Rβ)op andr ∈ R, whose inverse is given by f 7→ f(1R), for all f ∈ EndA(R).Now, we are in condition to prove the statement (iii).78 On the structure of skew groupoid ringsIt follows from the assumptions, step 3 and [12, Lemma 1] that R is agenerator in the category of the left A-modules. Hence, R is a β-Galoisextension of Rβ, as well as A ≃ End(R)Rβ as C(R)β-algebras, by [3,Theorem 3]. Then, A can be seen as a C(R)β-subalgebra of E and, bysteps 1 and 2 and [6, Theorem III.4.3], we have that CE(A) is Azumayaover C(R)β = C(A). So, (Rβ)op (and, therefore, also Rβ) is Azumaya overC(R)β by step 3, which also ensure that C(Rβ) = C(R)β .(iii)⇒(i) By [3, Theorem 3] we have that R is a projective finitelygenerated right Rβ-module, A ≃ End(R)Rβ as C(R)β-algebras. By [6,Theorem III.3.4] we have that Rβ is a projective finitely generatedC(A)(= C(Rβ))-module. By Lemma 3.2, C(A) = C(R)β. Therefore,R is a projective finitely generated C(R)β-module. Moreover, R is a gen-erator in the category of the C(R)β-modules. Hence, E = EndC(R)β (R)is Azumaya over C(R)β by [6, Theorem III.4.1], and CE(Rβ) is Azumayaover C(R)β by [6, Theorem III.4.3].Finally, End(R)Rβ = CE(Rβ). Indeed, it is enough to notice thatEnd(R)Rβ is a subalgebra of E and a Rβ-bimodule via the actionsxfy(r) = f(rx)y, for all f ∈ End(R)Rβ , x, y ∈ Rβ and r ∈ R.(iii)⇒(iv) By assumption Rβ is Azumaya over C(A) and it followsfrom the equivalence (iii)⇔(i) that A is Azumaya. By [6, Theorem III.4.3],CA(Rβ) is Azumaya over C(A).Therefore, A′ = CR(Rβ) ⋆β G is Azumaya and C(A′) = C(A) =C(Rβ) ⊆ CR(Rβ). Then, it follows again from the equivalence (i)⇔(iii)that CR(Rβ) is a β-Galois extension of CR(Rβ)β = C(Rβ) = C(A).(iv)⇒(iii) It is enough to observe that, by assumption, there exist ele-ments xi, yi ∈ CR(Rβ) ⊆ R, 1 ≤ i ≤ m, such that∑1≤i≤m xiβg(yi1g−1) =δe,g1e, for all g ∈ G and e ∈ G0.Theorem 3.3. Assume that A is Azumaya, C(A) ⊆ R, and R′ = CR(Rβ)satisfies R′β1e = (E′e)Ge, where E′e = R′1e, for all e ∈ G0. Then,(i) A ≃ Rβ ⊗C(A) (R′ ⋆β G) as C(A)-algebras.(ii) R′ ⋆β G ≃ End(R′)R′β as C(A)-algebras.(iii) R′ =⊕e∈G0E′e.(iv) for each e ∈ G0 there exist pairwise orthogonal idempotents ve,i ∈C(E′e), 1 ≤ i ≤ me, and subgroups Hi of Ge such that– each E′eve,i is an Hi-central Galois algebra,D. Flôres, A. Paques 79– E′e =⊕1≤i≤meE′eve,i or E′e = (⊕1≤i≤meE′eve,i)⊕C(E′e)ve, withve = 1e −∑1≤i≤meve,i,– Ge|C(E′e)ve ≃ Ge and– C(E′e)ve = E′eve is a commutative Galois extension of (E′eve)Ge .Proof. (i) It follows from Theorem 3.1 and [6, Theorem III.4.3] thatA ≃ Rβ ⊗C(A) CR(Rβ) as C(A)-algebras. It is routine to check thatCA(Rβ) = R′ ⋆β G.(ii) It follows from Theorem 3.1(vi) that R′ is a β-Galois extensionof C(A) = C(Rβ) = R′β. So, R′ ⋆β G ≃ End(R′)R′β as C(A)-algebras.(iii) It is immediate, since R =⊕e∈G0Ee.(iv) It follows from Theorem 3.1 and [6, Theorem III.4.3] thatR′⋆βG =CA(Rβ) is Azumaya over C(A), which implies that C(R′ ⋆β G) = C(A) =C(Rβ) = R′β and, consequently, R′β ⊆ C(R′). Thus, (E′e)Ge = R′β1e ⊆C(R′)1e = C(R′1e) = C(E′e), that is, E′e is a Ge-Galois algebra for alle ∈ G0. The result follows now from [15, Theorem 3.8].References[1] R. Alfaro and G. Szeto, Skew group rings which are Azumaya, Comm. Algebra 23(1995), 2255-2261.[2] D. Bagio, D. Flôres and A. Paques, Partial actions of ordered groupoids on rings,J. Algebra Appl. 9 (2010), 501-517.[3] D. Bagio and A. Paques, Partial groupoid actions: globalization, Morita theoryand Galois theory, Comm. Algebra 40 (2012), 3658-3678.[4] S. Caenepeel and E. De Groot,Galois theory for weak Hopf algebras, Rev. RoumaineMath. Pures Appl. 52 (2007), 151-176.[5] M. Dokuchaev and R. Exel, Associativity of crossed products by partial actions,enveloping actions and partial representations, Trans. AMS 357 (2005), 1931-1952.[6] F. DeMeyer and E. Ingraham, Separable Algebras over Commutative Rings, LNM181, Springer Verlag, Berlin, 1971.[7] D. Flôres and A. Paques, Duality for groupoid (co)actions, to appear in Comm.Algebra (2013).[8] N.D. Gilbert, Actions and expansions of ordered groupoids, J. Pure Appl. Algebra198 (2005), 175-195.[9] M. Harada, Suplementary results on Galois extensions, Osaka J. Math. 2 (1965),287 - 295.[10] K. Hirata, Some types of separable extensions of rings, Nagoya Math. J. 33 (1968),107-115.[11] K. Hirata and K. Sugano, On semisimple extensions and separable extensions overnoncommutative rings, J. Math. Soc. Japan 18 (1966), 360 - 373.80 On the structure of skew groupoid rings[12] S. Ikehata, Note on Azumaya algebras and H-separable extensions, Math. J.Okayama Univ. 23 (1981), 17-18.[13] M.V. Lawson, Inverse Semigroups. The Theory of Partial Symmetries, WorldScientific Pub. Co, London, 1998.[14] K. Sugano, Note on semisimple extensions and separable extensions, Osaka J.Math. 4 (1967), 265-270.[15] G. Szeto and L. Xue. The structure of Galois Algebras, J. Algebra 237 (2001),238-246.Contact informationDaiana Flôres Departamento de Matemática, UniversidadeFederal de Santa Maria, 97105-900, Santa Maria,RS, BrazilE-Mail: flores@ufsm.brAntonio Paques Instituto de Matemática, Universidade Federaldo Rio Grande do Sul, 91509-900, Porto Alegre,RS, BrazilE-Mail: paques@mat.ufrgs.brReceived by the editors: 11.04.2013and in final form 11.04.2013.

On the structure of skew groupoid rings which are Azumaya

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On the structure of skew groupoid rings which are Azumaya

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