Following the idea of Galois-type extensions and entwining structures, we
define the notion of a principal extension of noncommutative algebras. We show
that modules associated to such extensions via finite-dimensional
corepresentations are finitely generated projective, and determine an explicit
formula for the Chern character applied to the thus obtained modules.Comment: 4 pages, LaTe

Brzezinski, Tomasz

Hajac, Piotr M.

English

arXiv

Tomasz Brzeziński

Piotr M. Hajac

Comptes Rendus Mathématique

sion oftations areained.éfinir laensions auicite pourficientlyties. This. To anyFor finite-, and thusd,groupfully flatorphismC. R. Acad. Sci. Paris, Ser. I 338 (2004) 113–116Algebra/TopologyThe Chern–Galois characterTomasz Brzezínskia, Piotr M. Hajacb,ca Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UKb Instytut Matematyczny, Polska Akademia Nauk, ul.Śniadeckich 8, Warszawa, 00-956 Polandc Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski ul. Ho˙za 74, Warszawa, 00-682 PolandReceived 31 July 2003; accepted after revision 12 November 2003Presented by Alain ConnesAbstractFollowing the idea of Galois-type extensions and entwining structures, we define the notion of a principal extennoncommutative algebras. We show that modules associated to such extensions via finite-dimensional corepresenfinitely generated projective, and determine an explicit formula for the Chern character applied to the modules so obtTocite this article: T. Brzeziński, P.M. Hajac, C. R. Acad. Sci. Paris, Ser. I 338 (2004). 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméLe caractère de Chern–Galois.Nous nous inspirons des extensions de type Galois et des structures enlacées pour dnotion d’extension principale d’algèbres non commutatives. Nous montrons que les modules associés à de telles exttravers de coreprésentations de dimension finie sont projectifs et de type fini, et nous déterminons une formule explle caractère de Chern appliqué aux modules ainsi obtenus.Po r citer cet article : T. Brzeziński, P.M. Hajac, C. R. Acad. Sci.Paris, Ser. I 338 (2004). 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.1. IntroductionThe aim of this paper is twofold. First we need to determine a class of Galois-type extensions that are sufgeneral to accommodate interesting examples and sufficiently specific to derive a number of desired properleads to the concept of principal extensions. They play the role of algebraic analogues of principal bundlessuch extension one can associate modules much as vector bundles are associated to principal bundles.dimensional corepresentations these modules are always finitely generated projective (see Theorem 3.1)fit the formalism of the Chern–Connes pairing betweenK-theory and cyclic cohomology [6]. On the other hanjust as the commutative faithfully flat Hopf–Galois extensions with bijective antipodes coincide with affinescheme torsors of algebraic geometry, the principal extensions are precisely the (noncommutative) faithHopf–Galois extensions with bijective antipodes whenever the defining coaction is an algebra homomE-mail address:T.Brzezinski@swansea.ac.uk (T. Brzeziński).URL: http://www.fuw.edu.pl/~pmh (P.M. Hajac).1631-073X/$ – see front matter 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2003.11.009114 T. Brzeziński, P.M. Hajac / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 113–116ding aGaloismologytructiontensionitelya [6] toingrHopf–t itropertiession [3,tyt theasrections(cf. Theorem 2.5 and [12]). A very interesting and non-trivial example of a principal extension encononcommutative version of the instanton fibration SU(2) → S7 → S4 was recently constructed in [1].The second and main outcome of this work is the construction of and an explicit formula for the Chern–character for a given principal extension. This is a homomorphism of Abelian groups that assigns the hoclass of an even cyclic cycle to the isomorphism class of a finite-dimensional corepresentation. This consis in analogy with the Chern–Weil formalism for principal bundles and bridges the coalgebra-Galois ex[13,4,3] andK-theoretic formalisms (cf. [8] for the Hopf–Galois version). In particular, with the help of finsummable Fredholm modules, it allows one to apply the analytic tool of the noncommutative index formulcompute theK0-invariants of line bundles over generic Podleś spheres [9], which are among prime examples gobeyond the Hopf–Galois framework.Except for the last formula, we work over a general fieldk. We use the usual notations∆c = c(1)⊗c(2) ∈ C⊗C,((id⊗∆) ◦∆)(c) = c(1) ⊗ c(2) ⊗ c(3) ∈ C ⊗C ⊗C, etc.,∆V (v) = v(0) ⊗ v(1) ∈ V ⊗C (summation understood) fothe coproduct of a coalgebraC, its iterations and a rightC-coaction onV , respectively. We denote the counit ofCby ε. For an algebraA, A Hom(V ,W) stands for the space of leftA-linear maps. Similarly, for a coalgebraC, wewrite HomC(V,W) for the space of rightC-colinear maps.2. Principal extensions and strong connectionsThe concept of a faithfully flat Hopf–Galois extension with a bijective antipode is a cornerstone ofGalois theory. The following notion of aprincipal extensiongeneralizes this key concept in such a way thaencompasses interesting examples escaping Hopf–Galois theory, yet still enjoys a number of crucial pof the aforementioned class of Hopf–Galois extensions. It is an elaboration of the Galois-type extenDefinition 2.3] (see the condition (1) below), which evolved from [13, p. 182], [4] and other papers.Definition 2.1. Let C be a coalgebra andP an algebra and a rightC-comodule via∆P :P → P ⊗ C. PutB = PcoC := {b ∈ P | ∆P (bp) = b∆P (p), ∀p ∈ P }. We say that the inclusionB ⊆ P is a C-extension.A C-extensionB ⊆ P is calledprincipal if and only if(1) canR :P ⊗B P→P ⊗ C, p ⊗B p′ → p∆P (p′) is bijective (Galois or freeness condition);(2) ψ :C ⊗P→P ⊗C, c ⊗ p → can(can−1(1⊗ c)p) is bijective (invertibility of the canonical entwining);(3) there is a group-like elemente ∈ C such that∆P (p) = ψ(e ⊗ p), for all p ∈ P (co-augmentation);(4) P is C-equivariantly projective as a leftB-module (existence of a strong connection).The meaning of the last condition in Definition 2.1 is as follows. LetX be a leftB-module and a righC-comodule such that the coaction isB-linear. We say thatX is a C-equivariantly projectiveB-moduleif andonly if for everyB-linear C-colinear epimorphismπ :M → N that is split as aC-comodule map, and for anB-linear C-colinear homomorphismf :X → N , there exists aB-linear C-colinear mapg :X → M such thatπ ◦ g = f . For the trivialC we recover the usual concept of projectivity. Much as for the trivialC, one can showthat equivariant projectivity is equivalent to the existence of aB-linearC-colinear splitting of the multiplicationmapm :B ⊗ X → X. If we takeA = Hom(C, k)op to be the opposite of the convolution algebra ofC, then sucha splitting is the same as a(B,A)-bimodule splitting ofm. Now one can reverse the argument and prove thaexistence of such a(B,A)-bimodule splitting is equivalent toA-equivariant projectivity defined analogouslyC-equivariant projectivity. IfA is a commutative ring andB is an algebra overA, then we obtain the familiaconcept of relative projectivity [5, p. 197]). On the other hand, as explained in [8, p. 314], a(B,A)-bimodulesplitting ofm can be interpreted as a Cuntz–Quillen type connection [7]. The unitalized version of such conncorresponds to strong connections. More precisely, ifB ⊆ P is a principalC-extension, astrong connectionis aunital leftB-linear rightC-colinear splitting of the multiplication mapB⊗P → P [8, Remark 2.11]. The followinglemma allows us to conclude that principal extensions always admit strong connections.T. Brzeziński, P.M. Hajac / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 113–116 115f of the)–(3) ofe a leftap.thetrongthat, fortcturesup-ewn thatltsings theyresult ofem 2.5,Lemma 2.2.Let B ⊆ P be aC-extension satisfying conditions(1) and (3) in Definition 2.1. ThenP is C-equi-variantly projective as a leftB-module if and only if there exists a strong connection.The right-to-left part of the assertion is immediate from the discussion preceding the lemma. It is the prooexistence of a strong connection (unital splitting) that requires some work. Next, note that the conditions (2Definition 2.1 allow us to give a symmetric formulation of a strong connection. To begin with, one can defincoactionP∆ :P → C ⊗P , P∆(p) = ψ−1(p⊗ e), and prove thatcanL : P ⊗B P → C ⊗P , p⊗B p′ → P∆(p)p′,is bijective. One can also show thatcan−1L ◦ (id⊗1) = can−1R ◦ (1⊗ id), so that the concept of the translation mτ := can−1R ◦ (1⊗ id), τ (c) =: c[1] ⊗B c[2] (summation suppressed), is left-right symmetric. This leads to:Lemma 2.3.Let B ⊆ P be a principalC-extension, and letπB :P ⊗ P → P ⊗B P be the canonical surjectionThen the formulaes → (" : c → c[1]s(c[2])), " → (s :p → p(0)"(p(1))) define mutually inverse maps betweenspace of strong connections and linear maps" :C → P ⊗ P such thatπB ◦ " = τ , (id ⊗ ∆P ) ◦ " = (" ⊗ id) ◦ ∆,(P∆ ⊗ id) ◦ " = (id ⊗ ") ◦ ∆, "(e) = 1⊗ 1.To avoid multiplying terminology, such unital bicolinear liftings of the translation map are also called sconnections. Among other consequences of the principality of an extension is its coflatness. Recall firstany rightC-comoduleV with a coaction∆V and a leftC-comoduleW with a coactionW∆, thecotensor producis defined asV✷CW := Ker(id ⊗W ∆−∆V ⊗ id)⊆ V ⊗W . A right (resp. left)C-comoduleM is said to becoflatif the functorM✷C− (resp.−✷CM) is exact. Next, recall that there is a general concept of an entwining stru(A,C,ψ), whereA is an algebra,C a coalgebra, andψ : C ⊗A → A⊗C is a linear map satisfying certain axiom[4, Definition 2.1]. With these definitions, we obtain:Lemma 2.4.Let(A,C,ψ) be an entwining structure such thatψ is bijective. Assume also that there exists a grolike e ∈ C such thatA is a rightC-comodule viaψ ◦ (e ⊗ id) and a leftC-comodule viaψ−1 ◦ (id ⊗ e). ThenAis coflat as a right(resp. left) C-comodule if and only if there existsjR ∈ HomC(C,A) (resp.jL ∈ C Hom(C,A))such thatjR(e) = 1 (resp.jL(e) = 1). (HereC is aC-comodule via the coproduct.)The axioms (1)–(3) of a principal extension guarantee that(P,C,ψ) is an entwining structure satisfying thassumptions of the above Lemma [3, Theorem 2.7]. Moreover, with the help of Lemma 2.3, it can be shomapsjL andjR as in Lemma 2.4 can be constructed for any principalC-extension. Combining together the resudescribed in this section, one can prove the following:Theorem 2.5.Let B ⊆ P be a principalC-extension. Then: (1) There exists a strong connection.(2) P is aprojective left and rightB-module.(3) B is a direct summand ofP as a left and rightB-module.(4) P is afaithfully flat left and rightB-module.(5) P is a coflat left and rightC-comodule.3. Associated projective modules and the Chern–Galois characterIf B ⊆ P is a principalC-extension andϕ :Vϕ → Vϕ ⊗ C is a finite-dimensional corepresentation, then, usthe technology from the previous section, one can produce a short proof that the leftB-module HomC(Vϕ,P ) ofall colinear maps fromVϕ to P is finitely generated projective. We call such modules associated modules, aplay the role of sections of vector bundles associated to principal bundles (cf. [2, Theorem 5.4]). The mainthis paper is an explicit formula for an idempotent representing an associated module. By virtue of Theorwe already know that there exists a strong connection" a d a unital leftB-linear mapσL :P → B. Thus we canstate (cf. [11, Corollary 2.4]):116 T. Brzeziński, P.M. Hajac / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 113–116dns ofi-sease Chernn–Galois.ropean. Boths on themh.eom.7.Phys. 220aris,7–190.Theorem 3.1. Let " be a strong connection on a principalC-extensionB ⊆ P and ϕ :Vϕ → Vϕ ⊗ C be afinite-dimensional corepresentation. Let{pµ}µ be a basis ofP, {pµ}µ its dual, rµ := (pµ ⊗ id) ◦ ", and{ei}i be a basis ofVϕ , ϕ(ej ) =: ∑dimVϕi=1 ei ⊗ eij . Take anyσL ∈ B Hom(P,B) such thatσL(1) = 1, and setE(µ,i)(ν,j) := σL(rµ(eij )pν), E := (E(µ,i)(ν,j)) ∈ MN(B). ThenE is an idempotent matrix andBNE is isomorphicto HomC(Vϕ,P ) as a leftB-module.It is an immediate corollary of this theorem that a leftB-module HomC(Vϕ,P ) is always finitely generateprojective. On the other hand, take all the isomorphism classes of finite-dimensional corepresentatioCand view them as a semi-group via the direct sum. Denote byRf (C) the Grothendieck group of this semgroup. It is now straightforward to verify that the assignment[ϕ] → [HomC(Vϕ,P )] defines a homomorphismof Abelian groupsRf (C) → K0(B). Combining this homomorphism with the Chern characterh2n :K0(B) →HC2n(B), n ∈ N, (see [10, p. 264]) yields a homomorphismchg2n :Rf (C) → HC2n(B). We call the collectionof homomorphismschg2n, n ∈ N, the Chern–Galois character. The main point of this work is that we can uTheorem 3.1 to determine an explicit formula for the Chern–Galois character.Corollary 3.2. With the assumptions and notation as in Theorem3.1, define the character ofϕ ascϕ := ∑dimVϕi=1 eiiand, for anyc ∈ C, put"(c) =: c〈1〉 ⊗ c〈2〉 (summation understood). If Q⊆ k, then the Chern–Galois character hthe following explicit form:∀ n ∈ N: chg2n([ϕ]) = (−1)n[cϕ (2n+1)〈2〉cϕ (1)〈1〉 ⊗ cϕ (1)〈2〉cϕ (2)〈1〉 ⊗ · · · ⊗ cϕ (2n)〈2〉cϕ (2n+1)〈1〉].Note that, since any strong connection yields an idempotent representing the same module and thcharacter does not dependent on the choice of a representing idempotent, the formula for the Chercharacter is manifestly independent of the choice of a strong connection appearing on the right-hand sideAcknowledgementsT. Brzezínski thanks the EPSRC for an Advanced Research Fellowship. P.M. Hajac thanks the EuCommission for the Marie Curie fellowship HPMF-CT-2000-00523and the KBN for the grant 2 P03A 013 24authors are very grateful to R. Taillefer for the French translation and to R. Matthes for his helpful commentmanuscript. A preliminary version of the full account of this work can be found at http://www.fuw.edu.pl/~pReferences[1] F. Bonechi, L. Da̧browski, N. Ciccoli, M. Tarlini, Bijectivity of the canonical map for the noncommutative instanton bundle, J. GPhys., in press.[2] T. Brzezínski, On modules associated to coalgebra Galois extensions, J. Algebra 215 (1999) 290–317.[3] T. Brzezínski, P.M. Hajac, Coalgebra extensions and algebra coextensions of Galois type, Comm. Algebra 27 (1999) 1347–136[4] T. Brzezínski, S. Majid, Coalgebra bundles, Comm. Math. Phys. 191 (1998) 467–492.[5] H. Cartan, S. Eilenberg, Homological Algebra, Princeton University Press, Princeton, NJ, 1956.[6] A. Connes, Non-commutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62 (1985) 257–360.[7] J. Cuntz, D. Quillen, Algebra extensions and nonsingularity, J. Amer. Math. Soc. 8 (1995) 251–289.[8] L. Da̧browski, H. Grosse, P.M. Hajac, Strong connections and Chern–Connes pairing in the Hopf–Galois theory, Comm. Math.(2001) 301–331.[9] P.M. Hajac, R. Matthes, W. Szymański, Chern numbers for two families of noncommutative Hopf fibrations, C. R. Acad. Sci. PSer. I 336 (2003) 925–930.[10] J.-L. Loday, Cyclic Homology, Second edition, Springer-Verlag, Berlin, 1998.[11] E.F. Müller, H.-J. Schneider, Quantum homogeneous spaces with faithfully flat module structures, Israel J. Math. 111 (1999) 15[12] P. Schauenburg, H.-J. Schneider, Galois-type extensions and Hopf algebras, in preparation.[13] H.-J. Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math. 72 (1990) 167–195.

The Chern–Galois character

Brzeziński, Tomasz

Numérisation de Documents Anciens Mathématiques

The Chern-Galois character

The Chern-Galois character

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Numérisation de Documents Anciens Mathématiques