Documenta Math

Denis Perrot

Description

Documenta Math. 275 Secondary Invariants for Frechet Algebras and Quasihomomorphisms Denis Perrot Received: March 3, 2008 Communicated by Joachim Cuntz Abstract. A Frechet algebra endowed with a multiplicatively con- vex topology has two types of invariants: homotopy invariants (topo- logical K-theory and periodic cyclic homology) and secondary in- variants (multiplicative K-theory and the non-periodic versions of cyclic homology). The aim of this paper is to establish a Riemann- Roch-Grothendieck theorem relating direct images for homotopy and secondary invariants of Frechet m-algebras under finitely summable quasihomomorphisms. 2000 Mathematics Subject Classification: 19D55, 19K56, 46L80, 46L87. Keywords and Phrases: K-theory, bivariant cyclic cohomology, index theory. 1 Introduction For a noncommutative space described by an associative Frechet algebra A over C, we distinguish two types of invariants. The first type are (smooth) homotopy invariants, for example topological K-theory [27] and periodic cyclic homology [5]. The other type are secondary invariants; they are no longer stable under homotopy and carry a finer information about the “geometry” of the space A . Typical examples of secondary invariants are algebraic K-theory [29] (which will not be used here), multiplicative K-theory [17] and the unstable versions of cyclic homology [18].

• chern character


• invariants appear


• secondary invariants


• algebras


• constructions involving


• unstable versions


• algebra situation


• baum-connes construction


• index theorems


• assembly maps

Sujets

• Perrot
• Chern class
• Algebra

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 40
• Langue: English

Extrait

Documenta Math.

Secondary

Invariants for Frechet Algebras

and

Quasihomomorphisms

Denis Perrot

Received: March 3, 2008

Communicated by Joachim Cuntz

Abstract.FAlgebraenr´echetaumahpitlewodtiwdcolyn-caliveti vex topology has two types of invariants: homotopy invariants (topo- logicalK-theory and periodic cyclic homology) and secondary in- variants (multiplicativeK-theory and the non-periodic versions of cyclic homology). The aim of this paper is to establish a Riemann- Roch-Grothendieck theorem relating direct images for homotopy and secondaryinvariantsofFr´echetm-algebras under finitely summable quasihomomorphisms.

2000 Mathematics Subject Classification: 19D55, 19K56, 46L80, 46L87. Keywords and Phrases:K-theory, bivariant cyclic cohomology, index theory.

1 Introduction

275

ForanoncommutativespacedescribedbyanassociativeFre´chetalgebraA overC first type are (smooth) The, we distinguish two types of invariants. homotopy invariants, for example topologicalK-theory [27] and periodic cyclic homology [5]. The other type are secondary invariants; they are no longer stable under homotopy and carry a finer information about the “geometry” of the spaceA. Typical examples of secondary invariants are algebraic K-theory [29] (which will not be used here), multiplicativeK-theory [17] and the unstable versions of cyclic homology [18]. The aim of this paper is to define push-forward maps for homotopy and secondary invariants between two Fr´echetalgebrasAandB, induced by a smooth finitely summable quasiho- momorphism [8]. The compatibility between the different types of invariants is expressed through a noncommutative Riemann-Roch-Grothendieck theorem

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(Theorem 6.3). The present paper is the first part of a series on secondary characteristic classes. In the second part we will show how to obtainlocal formulasfor push-forward maps, following a general principle inspired by renormalization which establishes the link with chiral anomalies in quan- tum field theory [25]; in order to keep a reasonable size to the present paper, these methods will be published in a separate survey with further examples [26].

WedealwithFr´echetalgebrasendowedwithamultiplicativelyconvextopology, orFr´echetm These-algebras for short. algebras can be presented as inverse lim- its of sequences of Banach algebras, and as a consequence many constructions validforBanachalgebrascarryoverFr´echetm-algebras. In particular Phillips [27] defines topologicalK-theory groupsK n top (A) for any such algebraAand n∈Z fundamental properties of interest for us are (smooth) homotopy. The invariance and Bott periodicity, i.e.K n p +2 to (A) = ∼ K n top (A). Hence there are es- sentially two topologicalKt-Fy´rceehtheorygroupsforanm-algebra,K 0top (A) whose elements are roughly represented by idempotents in the stabilization of Aby the algebraKof ”smooth compact operators”, andK top 1 (A) whose elementsarerepresentedbyinvertibles.Fre´chetm-algebras naturally arise in many situations related to differential geometry, commutative or not, and the formulation of index problems. In the latter situation one usually encounters an algebraI´rceehtof”fibammpoeletinusylor,faFusaters”orm-algebra provided with a continuous trace on itsp-th power for somep≥1. A typ- ical example is the Schatten classI=L p (H) ofp-summable operators on an infinite-dimensional separable Hilbert spaceH.Acan be stabilized by the completed projective tensor produI ˆ ⊗Aand its ory ct topologicalK-the K n top (I⊗ ˆ A Other important topolog-) is the natural receptacle for indices. ical invariants ofA(as a locally convex algebra) are provided by the peri- odic cyclic homology groupsH P n (A), which is the correct version sharing the properties of smooth homotopy invariance and periodicity mod 2 with topolog- icalK For any finitely summable algebra-theory [5].Ithe Chern character K n top (I ˆ ⊗A)→H P n (A) allows to obtain cohomological formulations of index theorems. If one wants to go beyond differential topology and detectsecondaryinvari- ants as well, which are no longer stable under homotopy, one has to deal with algebraicK In-theory [29] and the unstable versions of cyclic homology [18]. principle the algebraicK-theory groupsK a n lg (A) defined for anyn∈Zpro- vide interesting secondary invariants for any ringA, but are very hard to calculate. It is also unclear if algebraicK-theory can be linked to index theory in a way consistent with topologicalK-theory, and in particular if it is possible to construct direct images of algebraicK-theory classes in a reasonable con- text. Instead, we will generalize slightly an idea of Karoubi [16, 17] and define foranyFr´echetm-algebraAthe multiplicativeK-theory groupsM K n I (A), n∈Zteedni,sulyabmmFrlech´ebdexigayfinevetinm-algebraI. Depending on the parity of the degreen, multiplicativeK-theory classes are represented by idempotents or invertibles in certain extensions ofI ˆ ⊗A, together with

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a transgression of their Chern character in certain quotient complexes. Mul- tiplicativeK-theory is by definition a mixture of the topologicalK-theory K n top (I ˆ ⊗A) and the non-periodic cyclic homologyH C n (A provides a). It “good” approximation of algebraicK In-theory but is much more tractable. addition, the Jones-Goodwillie Chern character innegative cyclic homology K a n lg (A)→H N n (A) factors through multiplicativeK-theory. The precise relations between topological, multiplicativeK-theory and the various versions of cyclic homology are encoded in a commutative diagram whose rows are long exact sequences of abelian groups

K t n1 o + p (I⊗ ˆ A)

H C n−1 (A) δ M K n I (A)

K n top (I⊗ ˆ A) (1)

e H P n+1 (A) S H C n−1 (A) B H N n (A) I H P n (A) The particular caseI=Cwas already considered by Karoubi [16, 17] after the construction by Connes and Karoubi of regulator maps on algebraicK-theory [6]. The incorporation of a finitely summable algebraIis rather straightfor- ward. This diagram describes the primary and secondary invariants associated to the noncommutative “manifold”A mention that the restriction to. We Fr´echetm-algebras is mainly for convenience. these constructions In principle could be extended to all locally convex algebras overC, however the subsequent results, in particular the proof of the Riemann-Roch-Grothendieck theorem would become much more involved.

If nowAandBatwret´rFoehcem-algebras, it is natural to consider the ad- equate “morphisms” mapping the primary and secondary invariants fromA toB. LetIbe apecheeFr´tamlbs-mum-algebra. By analogy with Cuntz’ description of bivariantK-theory forC ∗ -algebras [8], ifE⊲I ˆ ⊗Bdenotes a Fr´echetm-algebra containingI⊗ ˆ Bas a (not necessarily closed) two-sided ideal, we define ap-summable quasihomomorphismfromAtoBas a contin- uous homomorphism ρ:A→E s ⊲I s ⊗ ˆ B, whereE s andI s are certainZ 2 -graded algebras obtained fromEandIby a standard procedure. Quasihomomorphisms come equipped with a parity (even or odd) depending on the construction ofE s andI s . In general, we may suppose that the parity ispmod 2. We say thatIis multiplicative if it is provided with a homomorphism⊠:I ˆ ⊗I→I, possibly defined up to adjoint action of multipliers onI, and compatible with the trace. A basic example of multiplicativep-summable algebra is, once again, the Schatten classL p (H). Then it is easy to show that such a quasihomomorphism induces a pushforward map in topologicalK-theoryρ ! :K n top (I⊗ ˆ A)→K t n o − p p (I ˆ ⊗B), whose degree coincides with the parity of the quasihomomorphism. This is what one expects from bivariantK Our goal is to extend this-theory and is not really new.

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map to the entire diagram (1). Direct images for the unstable versions of cyclic homology are necessarily induced by a bivariantnon-periodiccyclic cohomology class ch p (ρ)∈H C p (A,B bivariant Chern character exists only under). This certain admissibility properties about the algebraE(note that it is sufficient for Ito be (p instead of+ 1)-summablep-summable). In particular, the bivariant Chern caracter constructed by Cuntz for any quasihomomorphism in [9, 10] cannot be used here because it provides a bivariant periodic cyclic cohomology class, which does not detect the secondary invariants ofAandB give. We the precise definition of an admissible quasihomomorphism and construct the bivariant Chern character ch p (ρ) in section 3, on the basis of previous works [23]. An analogous construction was obtained by Nistor [20, 21] or by Cuntz and Quillen [12]. However the bivariant Chern character of [23] is related to other constructions involving the heat operator and can be used concretely for establishing local index theorems, see for example [24]. The pushforward map in topologicalK-theory combined with the bivariant Chern character leads to a pushforward map in multiplicativeK-theoryρ ! :M K n I (A)→M K n I −p (B). Our first main result is the following non-commutative version of the Riemann- Roch-Grothendieck theorem (see Theorem 6.3 for a precise statement):

Theorem 1.1Letρ:A→E s ⊲I s ⊗ ˆ Bbe an admissible quasihomomorphism of paritypmod 2. that SupposeIis(p+ 1)-summable in the even case and p Then one has a graded-commutative diagram-summable in the odd case.

K n p +1 to (I ˆ ⊗A) H C n−1 (A)

ρ !

K t n op +1−p (I ˆ ⊗B)

ch p (ρ)

H C n−1−p (B)

M K n I (A)K t n op (I⊗ ˆ A) ρ ! ρ ! M K n I −p (B) K n to − p p (I⊗ ˆ B)

compatible with the cyclic homologySBIexact sequences after taking the Chern charactersK ∗ top (I ˆ ⊗)→H P ∗ andM K ∗ I →H N ∗ . At this point it is interesting to note that the pushforward mapsρ ! and the bivariant Chern character ch p (ρ) enjoy some invariance properties with respect to equivalence relations among quasihomomorphisms. Two types of equivalence relations are defined: smooth homotopy and conjugation by invertibles. The second relation corresponds to “compact perturbation” in KasparovKK-theory forC ∗  the latter situation, the In-algebras [2].M 2 - stable version of conjugation essentially coincide with homotopy, at least for separableAandσ-unitalB.ecr´rFFoeblgtahevewohsar,reM 2 -stable conjugation isstrictly strongerthan homotopy as an equivalence relation. ThisisindeedinthecontextofFre´chetalgebrasthatsecondaryinvariants appear. The pushforward maps in topologicalK-theory and periodic cyclic homology are invariant under homotopy of quasihomomorphisms. The maps in multiplicativeK-theory and the non-periodic versions of cyclic homology H C ∗ andH N ∗ are only invariant under conjugation and not homotopy. Also

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note that in contrast with theC ∗ -algebra situation, the Kasparov product of two quasihomomorphismsρ:A→E s ⊲I s ˆ ⊗Bandρ ′ :B→F s ⊲I s ˆ ⊗C is not defined as a quasihomomorphism fromAtoC. The various bivariant K-theories constructed form-algebras [9, 10] or even for general bornological algebras [11] cannot be used here, again because they are homotopy invariant by construction. We leave the construction of a bivariantK-theory compatible with secondary invariants as an open problem.

In the last part of the paper we illustrate the Riemann-Roch-Grothendieck theorem by constructing assembly maps for certain crossed product algebras. If ΓisadiscretegroupactingonaFr´echetm-algebraA, under certain conditions the crossed productA⋊Γisaganiae´rFtehcm-algebra and one would like to obtain multiplicativeK-theory classes out of a geometric model inspired by the Baum-Connes construction [1]. Thus letP→ Γ Mbe a principal Γ- bundle over a compact manifoldM, and denote byA P the algebra of smooth sections of the associatedA-bundle. IfDis aK-cycle forMrepresented by a pseudodifferential operator, we obtain a quasihomomorphism fromA P to A⋊Γ and hence a map

M K n I (A P )→M K I −p (A⋊Γ) n

for suitablepand Schatten idealI general this map cannot exhaust. In the entire multiplicativeK-theory of the crossed product but nevertheless interesting secondary invariants arise in this way. In the case whereAis the algebra of smooth functions on a compact manifold,A P is commutative and its secondary invaiants are closely related to (smooth) Deligne cohomology. From this point of view the pushforward map in multiplicativeK-theory should be considered as a non-commutative version of “integrating Deligne classes along the fibers” of a submersion. We perform the computations for the simple example provided by the noncommutative torus.

The paper is organized as follows. In section 2 we review the Cuntz-Quillen formulation of (bivariant) cyclic cohomology [12] in terms of quasi-free exten- sions formis new but we take the opportunity to fix the -algebras. Nothing notations and recall a proof of generalized Goodwillie theorem. In section 3 we define quasihomomorphisms and construct the bivariant Chern character. The formulas are identical to those found in [23] but in addition we carefully es- tablish their adic properties and conjugation invariance. In section 4 we recall Phillips’ topologicalKyforheorchetFr´et-m-algebras, and introduce the peri- odic Chern characterK t n op (I ˆ ⊗A)→H P n (A) whenIis a finitely summable algebra. The essential point here is to give explicit and simple formulas for subsequent use. Section 5 is devoted to the definition of the multiplicative K-theory groupsM K n I (A) and the proof of the long exact sequence relating them with topologicalK also construct the-theory and cyclic homology. We negative Chern characterM K n I (A)→H N n (A) and show the compatibility

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with theSBI images of topological and multiplica- Directexact sequence. tiveK-theory under quasihomomorphisms are constructed in section 6 and the Riemann-Roch-Grothendieck theorem is proved. The example of assembly maps and crossed products is treated in section 7.

2 Cyclic homology

Cyclic homology can be defined for various classes of associative algebras over C, in particular complete locally convex algebras. us, a locally convex For algebraAtopology induced by a family of continuous seminormshas a p: A→R + , for which the multiplicationA×A→Ais jointly continuous. Hence for any seminormpthere exists a seminormqsuch thatp(a 1 a 2 )≤ q(a 1 )q(a 2 ),∀a i ∈A. For technical reasons however we shall restrict ourselves tomultiplicatively convexwhose topology is generated by a familyalgebras [5], of submultiplicative seminorms

p(a 1 a 2 )≤p(a 1 )p(a 2 )∀a i ∈A. A complete multiplicatively convex algebra is calledm-algebra, and may equiv- alently be described as a projective limit of Banach algebras. The unitalization A + =C⊕Aof anm-algebraAis again anm-algebra, for the seminorms p˜(λ1 +a) =|λ|+p(a),∀λ∈C, a∈A. In the same way, ifBis another m-algebra, the direct sumA⊕Bis anm-algebra for the topology generated by the seminorms (p⊕q)(a, b) =p(a) +q(b), wherepis a seminorm onAandq a seminorm onB the algebraic tensor product. Also,A⊗Bmay be endowed with the projective topology induced by the seminorms n n (p⊗q)(c) = inf n X p(a i )q(b i ) such thatc= X a i ⊗b i ∈A⊗B o .(2) i=1i=1 ThecompletionA ˆ ⊗B=A ˆ ⊗ π Bof the algebraic tensor product under this family of seminorms is the projective tensor product of Grothendieck [14], and is again anm-algebra. Cyclic homology, cohomology and bivariant cyclic cohomology form-algebras can be defined either within the cyclic bicomplex formalism of Connes [5], or theX-complex of Cuntz and Quillen [12]. We will make an extensive use of both formalisms throughout this paper. In general, we suppose that all linear maps or homomorphims betweenm-algebras are continuous, tensor products are completed projective tensor products, and extensions ofm-algebras 0→ I→R→A→0 always admit a continuous linear splittingσ:A→R.

2.1 Cyclic bicomplex

Non-commutative differential forms.LetAbe anm-algebra. The space of non-commutative differential forms overAis the algebraic direct sum ˆ ΩA= L n≥0 Ω n Aof then-forms subspaces Ω n A=A + ⊗ ˆ A ⊗n forn≥1

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and Ω 0 A=A, whereA + is the unitalization ofA of the subspaces. Each Ω n A is customary Itis complete but we do not complete the direct sum. to use the differential notationa 0 da 1 . . . da n (resp.da 1 . . . da n ) for the string a 0 ⊗a 1 . . .⊗a n (resp. 1⊗a 1 . . .⊗a n ). A continuous differentiald: Ω n A→ Ω n+1 Ais uniquely specified byd(a 0 da 1 . . . da n ) =da 0 da 1 . . . da n andd 2 = 0. A continuous and associative product Ω n A×Ω m A→Ω n+m Ais defined as usual and fulfills the Leibniz ruled(ω 1 ω 2 ) =dω 1 ω 2 + (−) |ω 1 | ω 1 dω 2 , where|ω 1 | is the degree ofω 1 . This turns ΩAinto a differential graded (DG) algebra. On ΩA of all, the Hochschild boundary Firstare defined various operators. mapb: Ω n+1 A→Ω n Areadsb(ωda) = (−) n [ω, a] forω∈Ω n A, andb= 0 on Ω 0 A=A easily shows that. Onebis continuous andb 2 = 0, hence ΩAis a complex graded overN. The Hochschild homology ofA(with coefficients in the bimoduleA) is the homology of this complex:

H H n (A) =H n (ΩA, b),∀n∈N.

(3)

Then the Karoubi operatorκ: Ω n A→Ω n Ais defined by 1−κ=db+bd. Thereforeκis continuous and commutes withbandd. One hasκ(ω da) = (−) n da ωfor anyω∈Ω n Aanda∈A. The last operator is Connes’B: Ω n A→Ω n+1 A +, equal to (1κ+. . .+κ n )don Ω n A is also continuous. It and verifiesB 2 = 0 =Bb+bBandBκ=κB=B. Thus ΩAendowed with the two anticommuting differentials (b, B splits as a direct It) becomes a bicomplex. sum ΩA= ΩA + ⊕ΩA − odd degree differential forms, hence is aof even and Z 2 -graded complex for the total boundary mapb+B. However its homology is trivial [18]. The various versions of cyclic homology are defined using the natural filtrations on ΩA Cuntz and Quillen [12], we define the. Following Hodge filtrationon ΩAas the decreasing family ofZ 2 -graded subcomplexes for the total boundaryb+B

F n ΩA=bΩ n+1 A⊕ M Ω k A,∀n∈Z, k>n with the convention thatF n ΩA= ΩAforn < completion of Ω0. TheAis defined as the projective limit ofZ 2 -graded complexes

Ω b A= li ← m − ΩAF n ΩA= Y Ω n A n n≥0

.

(4)

b b b Hence ΩA= Ω + A⊕Ω − Ais aZ 2 -graded complex endowed with the total boundary mapb+ b B is its. It b elf filtered by the decreasing family ofZ 2 -graded subcomplexesF n ΩA= Ker(ΩA→ΩAF n ΩA), which may be written

F n b bΩ n+1 A⊕ Y Ω k A,∀n∈Z. ΩA= k>n

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b b In particular the quotient ΩAF n ΩAis aZ 2 -graded complex isomorphic to ΩAF n ΩA, explicitly n−1 b F n Ω b A= M Ω k A⊕Ω n Ab( +1 6) ΩAΩ n A),( k=0

b b and it vanishes forn < a topological vector space, Ω0. AsAF n ΩAmay fail to be separated because the imageb(Ω n+1 A) is not closed in general.

Definition 2.1In any degreen∈Z, theperiodic, non-periodicandnegative cyclic homologies are respectively the(b+B)-homologies

b H P n (A) =H n+2Z (ΩA), b b H C n (A) =H n+2Z (ΩAF n ΩA),(7) H N n (A) =H nZ ( −1 b +2 F n ΩA). HenceH P n (A) ∼ = H P n+2 (A) is 2-periodic,H C n (A) = 0 forn <0 and H N n (A) = ∼ H P n (A) forn≤ construction these cyclic homology groups0. By fit into a long exact sequence

. . .−→H P n+1 (A)− S →H C n−1 (A)− B →H N n (A)− I →H P n (A)−→. . .(8) whereSis induced by projection,Iby inclusion, and the connecting map cor- responds to the operatorB link between cyclic and Hochshild homology. The may be obtained throughnon-commutative de Rham homology[16], defined as b H D n (A) :=H n+2Z (ΩAF n+1 Ω b A),∀n∈Z.(9) This yields for anyn∈Za short exact sequence ofZ 2 -graded complexes 0−→G n (A)−→Ω b AF n b ΩA−→Ω b AF n−1 Ω b A−→0, whereG n is Ω n AbΩ n+1 Ain degreenmod 2, andbΩ n Ain degreen−1 mod 2. One hasH n+2Z (G n ) =H H n (A) andH n−1+2Z (G n ) = 0, so that the associated six-term cyclic exact sequence in homology reduces to

0→H D n−1 (A)→H C n−1 (A)→H H n (A)→H C n (A)→H D n−2 (A)→0, and Connes’sSBIexact sequence [4] for cyclic homology is actually obtained by splicing together the above sequences for alln∈Z: . . .−→H C n+1 (A)− S →H C n−1 (A)− B →H H n (A)− I →H C n (A)−→. . .(10) Hence the non-commutative de Rham homology groupH D n (A) may be identified with the image of the periodicity shiftS:H C n+2 (A)→H C n (A). Clearly the exact sequence (8) can be transformed to (10) by taking the

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natural mapsH P n (A)→H C n (A) andH N n (A)→H H n (A). b Passing to the dual theory, let Hom(ΩA,C) be theZ 2 -graded complex of linear b b maps ΩA→Cwhich are continuous for the adic topology on ΩAinduced by the Hodge filtration. It is concretely described as the direct sum Hom(ΩA,C) = M Hom(Ω n A,C), b n≥0

where Hom(Ω n A,C) is the space of continuous linear maps Ω n A→C. The b space Hom(ΩA,C) is endowed with the transposed of the boundary operator b b+Bon ΩA the periodic cyclic cohomology of. ThenAis the cohomology of this complex: H P n (A) =H n+2Z (Hom(Ω b A,C)).(11)

One defines analogously the non-periodic and negative cyclic cohomologies which fit into anIBSlong exact sequence.

2.2X-complex and quasi-free algebras

We now turn to the description of theX-complex. It first appeared in the coalgebra context in Quillen’s work [28], and subsequently was used by Cuntz and Quillen in their formulation of cyclic homology [12]. Here we recall the X-complex construction form-algebras. LetRbe anm space of non-commutative one-forms Ω-algebra. The 1 Ris aR- bimodule, hence we can take its quotient Ω 1 R ♮ by the subspace of commutators [R,Ω 1 R] =bΩ 2 R. Ω 1 R ♮  However,may fail to be separated in general. it is automatically separated whenRisquasi-free order to avoid In, see below. confusions in the subsequent notations, we always write a one-formx 0 dx 1 ∈ Ω 1 Rwith a bolddwhen dealing with theX-complex ofR latter is the. The Z 2 -graded complex [12]

X(R) ♮ ⇄ d Ω 1 :R R ♮ , b

(12)

whereR=X + (R) is located in even degree and Ω 1 R ♮ =X − (R) in odd degree. The class of the generic element (x 0 dx 1 mod [,])∈Ω 1 R ♮ is usually denoted by♮x 0 dx 1  map. The♮d:R→Ω 1 R ♮ thus sendsx∈Rto♮dx. Also, the Hochschild boundaryb: Ω 1 R→Rvanishes on the commutator subspace [R,Ω 1 R], hence passes to a well-defined mapb: Ω 1 R ♮ →R the im-. Explicitly age of♮x 0 dx 1 bybis the commutator [x 0 , x 1  maps are continuous and]. These satisfy♮db= 0 andb◦♮d= 0, so that (X(R), ♮db) indeed defines aZ 2 -graded complex. We mention that everything can be formulated whenRitself is a Z 2 -graded algebra: we just have to replace everywhere the ordinary commuta- tors by graded commutators, and the differentials anticommute with elements of odd degree. In particular one getsb♮xdy= (−) |x| [x, y], where|x|is the

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