On the Hochschild homology of convolution algebras of proper Lie groupoids
aa r X i v : . [ m a t h . K T ] S e p ON THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OFPROPER LIE GROUPOIDS
M.J. PFLAUM, H. POSTHUMA, AND X. TANG
Abstract.
We study the Hochschild homology of the convolution algebra of a proper Lie groupoidby introducing a convolution sheaf over the space of orbits. We develop a localization result forthe associated Hochschild homology sheaf, and prove that the Hochschild homology sheaf at eachstalk is quasi-isomorphic to the stalk at the origin of the Hochschild homology of the convolutionalgebra of its linearization, which is the transformation groupoid of a linear action of a compactisotropy group on a vector space. We then explain Brylinski’s ansatz to compute the Hochschildhomology of the transformation groupoid of a compact group action on a manifold. We verifyBrylinski’s conjecture for the case of smooth circle actions that the Hochschild homology is givenby basic relative forms on the associated inertia space.
Introduction
Let M be a smooth manifold, and C ∞ ( M ) be the algebra of smooth functions on M . Connes’version [Con85] of the seminal Hochschild-Kostant-Rosenberg theorem [HKR62] states that theHochschild homology of C ∞ ( M ) is isomorphic to the graded vector space of differential forms on M .In this paper, we aim to establish tools for a general Hochschild-Kostant-Rosenberg type theoremfor proper Lie groupoids.Recall that a Lie groupoid G ⇒ M is proper if the map G → M × M , g ( s ( g ) , t ( g )) is a propermap, where s ( g ) and t ( g ) are the source and target of g ∈ G . When the source and target maps areboth local diffeomorphisms, the groupoid G ⇒ M is called ´etale. In efforts by many authors, e.g.[BDN17, BN94, Con94, Cra99, FT87, Pon18, Was88], the Hochschild and cyclic homology theory of´etale Lie groupoids has been unvealed. The Hochschild and cyclic homology of a proper ´etale Liegroupoid was explicitly computed by Brylinksi and Nistor [BN94]. Let us explain this result in thecase of a finite group Γ action on a smooth manifold M , the transformation groupoid Γ ⋉ M ⇒ M for a finite group Γ action on M .The convolution groupoid algebra associated to the transformation groupoid Γ ⋉ M ⇒ M is thecrossed product algebra C ∞ ( M ) ⋊ Γ, which consists of C ∞ ( M )-valued functions on Γ equipped withthe convolution product, e.g. for f, g ∈ C ∞ ( M ) ⋊ Γ, f ∗ g ( γ ) = X αβ = γ β ∗ (cid:0) f ( α ) (cid:1) · g ( β ) . The algebra C ∞ ( M ) ⋊ Γ is naturally a Fr´echet algebra. The Hochschild homology of the algebra C ∞ ( M ) ⋊ Γ as a bornological algebra is given by the following formula the proof of which is recalledin Corollary B.6. HH • (cid:0) C ∞ ( M ) ⋊ Γ (cid:1) ∼ = M γ ∈ Γ Ω • ( M γ ) Γ , where M γ is the γ -fixed point submanifold, and Γ acts on the disjoint union ` γ ∈ Γ M γ by γ ′ ( γ, x ) =( γ ′ γ ( γ ′ ) − , γ ′ x ). Recall that the so called loop space Λ (Γ , M ) of the transformation groupoidΓ ⋉ M ⇒ M is defined as Λ (Γ , M ) := a γ ∈ Γ M γ , equipped with the same action of Γ as above. In other words, the Hochschild homology of C ∞ ( M ) ⋊ Γis the space of differential forms on the quotient Λ (Γ , M ) / Γ, which is called the associated inertiaorbifold. We would like to remark that just as the classical Hochschild-Kostant-Rosenberg theorem,the above identification can be realized as an isomorphism of sheaves over the quotient M/ Γ. Thismakes Hochschild and cylic homology of C ∞ ( M ) ⋊ Γ the right object to work with in the study oforbifold index theory, see e.g. [PPT10].Our goal in this project is to extend the study of Hochschild homology of proper ´etale groupoidsto general proper Lie groupoids, which are natural generalizations of transformation groupoids forproper Lie group actions. The key new challenge from the study of (proper) ´etale groupoids isthat orbits of a general proper Lie groupoid have different dimensions. This turns the orbit spaceof a proper Lie groupoid into a stratified space with a significantly more complicated singularitystructure than an orbifold.Our main result is to introduce a sheaf
H H • on the orbit space X := M/ G of a proper Liegroupoid G ⇒ M , whose space of global sections computes the Hochschild homology of the convo-lution algebra of G . To achieve this, we start with introducing a sheaf A of convolution algebras onthe orbit space X in Definition 1.1. Using the localization method from [BP08] we introduce theHochschild homology sheaf H H • ( A ) for A as a sheaf of bornological algebras over X . Moreover, weprove the following sheafification theorem for the Hochschild homology of the convolution algebra A of the groupoid G . Theorem 2.3.
Let A be the convolution sheaf of a proper Lie groupoid G . Then the natural mapin Hochschild homology HH • (cid:0) A ( X ) (cid:1) → H H • ( A )( X ) = Γ (cid:0) X, H H • ( A ) (cid:1) is an isomorphism. To determine the homology sheaf
H H • ( A ), we study its stalk at an orbit O ∈ X . Using thelinearization result of proper Lie groupoid developed by Weinstein and Zung (c.f. [CS13, dHF18,PPT14, Wei02, Zun06]), we obtain a linear model of the stalk H H • , O ( A ) in Proposition 3.5 asa linear compact group action on a vector space. This result leads us to focus on the Hochschildhomology of the convolution algebra C ∞ ( M ) ⋊ G associated to a compact Lie group action on asmooth manifold M in the second part of this article.The Hochschild homology of compact Lie group actions was studied by several authors, e.g.[BG94], [Bry87a, Bry87b]. Brylinski [Bry87a, Bry87b] proposed a geometric model of basic relativeforms along the idea of the Grauert-Grothendieck forms to compute the Hochschild homology. How-ever, a major part of the proof is missing in [Bry87a, Bry87b]. We decided to turn this result intothe main conjecture of this paper in Section 5. Conjecture 5.6.
The Hochschild homology of the crossed product algebra C ∞ ( M ) ⋊ G associatedto a compact Lie group action on a smooth manifold M is isomorphic to the space of basic relativeforms on the loop space Λ ( G ⋉ M ) = { ( g, p ) ∈ G × M | gp = p } . Block and Getzler [BG94] introduced an interesting Cartan model for the cyclic homology ofthe crossed product algebra C ∞ ( M ) ⋊ G . However, the Block-Getzler model is not a sheaf on theorbit space M/G , but a sheaf on the space of conjugacy classes of G . This makes it impossible tolocalize the sheaf to an orbit of the group action in the orbit space. It is worth pointing out thatthe truncation of the Block-Getzler Cartan model at E -page provides a complex to compute theHochschild homology of C ∞ ( M ) ⋊ G . However, the differential ι introduced in [BG94, Section 1] isnontrivial, and makes it challenging to explicitly identify the Hochschild homology of C ∞ ( M ) ⋊ G as the space of basic relative forms. We refer the reader to Remark 4.3 for a more detail discussionabout the Block-Getzler model.In the last part of this paper, we prove Conjecture 5.6 in the case where the group G is S ;see Proposition 6.9. Our proof relies on a careful study of the stratification of the loop space N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 3 Λ ( S ⋉ M ) ⊂ S × M . The crucial property we use in our computation is that at its singular point,Λ ( S ⋉ M ) locally looks like the union of the hyperplane { x = 0 } and the line { x = · · · = x n = 0 } in R n +1 , which are transverse to each other. The loop space Λ ( G ⋉ M ) for a general G -manifold M is much more complicated to describe. This has stopped us from extending our result for S -actions to more general compact group actions. It is foreseeable that some combinatorial structuresdescribing the stratifications of the loop spaces and real algebraic geometry tools characterizing basicrelative forms on the loop spaces are needed to solve Conjecture 5.6 in full generality. We plan tocome back to this problem in the near future.As is mentioned above, the study of Hochschild and cyclic homology of the convolution algebraof a proper Lie groupoid is closely related to the study of the groupoid index theory, e.g. [PPT10],[PPT15]. We expect that the study of the Hochschild homology and the generalized Hochschild-Kostant-Rosenberg theorem will eventually lead to the correct definition of basic relative forms forproper Lie groupoids, where the right index theorem will be established. Acknowledgements : We would like to thank Marius Crainic, Ralf Meyer, Rapha¨el Ponge andMichael Puschnigg for inspiring discussions. Pflaum’s research is partially supported by SimonsFoundation award number 359389 and NSF award OAC 1934725. Tang’s research is partially sup-ported by the NSF awards DMS 1800666, 1952551.1.
The convolution sheaf of a proper Lie groupoid
Throughout this paper, G ⇒ M denotes a Lie groupoid over a base manifold M . Elements of M are called points of the groupoid, those of G its arrows. The symbols s, t : G → M denote the sourceand target map, respectively, and u : M → G the unit map. By definition of a Lie groupoid, s and t are assumed to be smooth submersions. This implies that the space of k -tuples of composablearrows G k := { ( g , . . . , g k ) ∈ G k | s ( g i ) = t ( g i +1 ) for i = 1 , . . . , k − } is a smooth manifold, and multiplication of arrows m : G → G , ( g , g ) g g a smooth map.If g ∈ G is an arrow with s ( g ) = x and t ( g ) = y , we denote such an arrow sometimes by g : y ← x ,and write G ( y, x ) for the space of arrows with source x and target y . The s -fiber over x , i.e. themanifold s − ( x ), will be denoted by G ( − , x ), the t -fiber over y by G ( y, − ). Note that for each object x ∈ M multiplication of arrows induces on G ( x, x ) a group structure. This group is called the isotropy group of x and is denoted by G x . The union of all isotropy groupsΛ G := [ x ∈ M G x = { g ∈ G | s ( g ) = t ( g ) } will be called the loop space of G .Given a Lie groupoid G ⇒ M two points x, y ∈ M are said to lie in the same orbit if there is anarrow g : y ← x . In the following, we will always write O x for the orbit containing x , and M/ G forthe space of orbits of the groupoid G . We assume further that the orbit space always carries thequotient topology with respect to the canonical map π : M → M/ G . Note that M/ G need not beHausdorff unless G is a proper Lie groupoid, which means that the map ( s, t ) : G → M × M is aproper map.Sometimes, we need to specify to which groupoid a particular structure map belongs to. In sucha situation we will write s G , m G , π G and so on.In the following, we will define a sheaf of algebras A on M/ G in such a way that the algebra A c ( M/ G ) of compactly supported global sections of A coincides with the smooth convolution algebraof the groupoid. To this end, we use a smooth left Haar measure on G . M.J. PFLAUM, H. POSTHUMA, AND X. TANG
Recall that by a smooth left Haar measure on G one understands a family of measures ( λ x ) x ∈ M such that the following properties hold true:(H1) For every x ∈ G , λ x is a positive measure on G ( x, − ) with supp λ x = G ( x, − ).(H2) For every g ∈ G , the family ( λ x ) x ∈ M is invariant under left multiplication L g : G ( s ( g ) , − ) → G ( t ( g ) , − ) , h gh or in other words Z G ( s ( g ) , − ) u ( gh ) dλ s ( g ) ( h ) = Z G ( t ( g ) , − ) u ( h ) dλ t ( g ) ( h ) for all u ∈ C ∞ c ( G ) . (H3) The system is smooth in the sense that for every u ∈ C ∞ c ( G ) the map M → C , x Z G ( x, − ) u ( h ) dλ x ( h )is smooth.Let us fix a smooth left Haar measure ( λ x ) x ∈ M on G . Given an open set U ⊂ M/ G we first put(1.1) U := π − ( U ) , U := s − ( U ) ⊂ G and U k +1 := k \ i =1 σ − i ( U k ) ⊂ G k +1 for all k ∈ N ∗ , where σ i : G k +1 → G k , ( g , . . . , g k +1 ) ( g , . . . , g i g i +1 , . . . , g k ). Then we define(1.2) A ( U ) := (cid:8) f ∈ C ∞ (cid:0) U (cid:1) | supp f is longitudinally compact (cid:9) . Hereby, a subset K ⊂ G is called longitudinally compact , if for every compact subset C ⊂ M/ G theintersection K ∩ s − π − ( C ) is compact. Obviously, every A ( U ) is a linear space, and the map whichassigns to an open U ⊂ M/ G the space A ( U ) forms a sheaf on M/ G which in the following will bedenoted by A or by A G if we want to emphasize the underlying groupoid. The section space A ( U )over U ⊂ M/ G open becomes an associative algebra with the convolution product (1.3) f ∗ f ( g ) := Z G ( t ( g ) , − ) f ( h ) f ( h − g ) dλ t ( g ) ( h ) , f , f ∈ A ( U ) , g ∈ G . The convolution product is compatible with the restriction maps, hence A becomes a sheaf of algebrason M/ G .Let us assume from now on that the groupoid G is proper. Recall from [PPT14] that then theorbit space M/ G carries the structure of a differentiable stratified space in a canonical way. Thestructure sheaf C ∞ M/ G coincides with the sheaf of continuous functions ϕ : U → R with U ⊂ M/ G open such that ϕ ◦ π ∈ C ∞ ( U ). Now observe that the action C ∞ M/ G ( U ) × A ( U ) → A ( U ) , ( ϕ, f ) ϕf := (cid:16) U ∋ g ϕ (cid:0) πs ( g ) (cid:1) f ( g ) ∈ R (cid:17) commutes with the convolution product, and turns A into a C ∞ M/ G -module sheaf. Proposition and Definition 1.1.
Given a proper Lie groupoid G ⇒ M , the associated sheaf A is a fine sheaf of algebras over the orbit space M/ G which in addition carries the structure of a C ∞ M/ G -module sheaf. The space A c ( M/ G ) of global sections of A with compact support coincides withthe smooth convolution algebra of G . We call A the convolution sheaf of G . For later purposes, we equip the spaces A ( U ) with a locally convex topology and a convex bornol-ogy. To this end, observe first that for every longitudinally compact subset K ⊂ U the space A ( M/ G ; K ) := (cid:8) f ∈ C ∞ ( G ) | supp f ⊂ K (cid:9) inherits from C ∞ ( G ) the structure of a Fr´echet space. Moreover, since C ∞ ( G ) is nuclear, A ( M/ G ; K )has to be nuclear as well by [Tr`e67, Prop. 50.1]. By separability of U there exists a (countable) N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 5 exhaustion of U by longitudinally compact sets, i.e. a family ( K n ) n ∈ N of longitudinally compactsubset of U such that K n ⊂ K ◦ n +1 for all n ∈ N , and such that S n ∈ N K n = U . The space A ( U ) canthen be identified with the inductive limit of the strict inductive system of nuclear Fr´echet spaces (cid:0) A ( M/ G ; K n ) (cid:1) n ∈ N . It is straightforward to check that the resulting inductive limit topology on A ( U ) does not depend on the particular choice of the exhaustion ( K n ) n ∈ N . Thus, A ( U ) becomes anuclear LF-space, where nuclearity follows from [Tr`e67, Prop. 50.1]. As an LF-space, A ( U ) carriesa natural bornology given by the von Neumann bounded sets, i.e. by the sets S ⊂ A ( U ) which areabsorbed by each neighborhood of 0. In other words, a subset S ⊂ A ( U ) is bounded if all f ∈ S are supported in a fixed longitudinally compact subset K ⊂ U , and if the set of functions D ( S ) isuniformly bounded for every compactly supported differential operator D on U .The bornological point of view is particularly convenient when considering tensor products. Inparticular one has the following fundamental property. Proposition 1.2.
Let G ⇒ M , and H ⇒ N be proper Lie groupoids. Denote by M/ G and N/ H their respective orbit spaces. Then M/ G × N/ H is diffeomorphic as a differentiable stratified space tothe orbit space of the product groupoid G × H ⇒ M × N . Moreover, there is a natural isomorphism (1.4) A G ( U ) ˆ ⊗A H ( V ) ∼ = A G × H ( U × V ) for any two open sets U ⊂ M/ G and V ⊂ N/ H .Proof. The first claim is a consequence of the fact that two elements ( x, y ) , ( x ′ , y ′ ) ∈ M × N lie inthe same ( G × H )-orbit if and only if x and x ′ lie in the same G -orbit and y and y ′ lie in the same H -orbit. Let us prove the second claim. Let ( K n ) n ∈ N be an exhaustion of U := s − G π − G ( U ) bylongitudinally compact subsets and ( L m ) m ∈ N an exhaustion of V := s − H π − H ( V ) by such sets. Since A G ( U ) coincides with the inductive limit colim n ∈ N A G ( M/ G ; K n ) and A H ( V ) with colim m ∈ N A H ( N/ H ; L m ),[Mey99, Cor. 2.30] entails that(1.5) A G ( U ) ˆ ⊗A H ( V ) ∼ = colim n ∈ N A G ( M/ G ; K n ) ˆ ⊗A H ( N/ H ; L n ) . Now observe that A G ( M/ G ; K n ) ˆ ⊗A H ( N/ H ; L m ) ∼ = A G × H ( M/ G × N/ H ; K n × L m ) by [Tr`e67, Prop. 51.6],and that ( K n × L n ) n ∈ N is an exhaustion of U × V by longitudinally compact subsets. Together withEq. (1.5) this proves the claim. (cid:3) Localization of the Hochschild chain complex
In this section, we apply the localization method in Hochschild homology theory, partially follow-ing [BP08], to the Hochschild chain complex of the convolution algebra.2.1.
Sheaves of bornological algebras over a differentiable space.
We start with a (reducedseparated second countable) differentiable space ( X, C ∞ ) and assume that A is a sheaf of R -algebrason X . We will denote by A = A ( X ) its space of global sections. We assume further that A is a C ∞ X -module sheaf and that every section space A ( U ) with U ⊂ X open carries the structure of a nuclearLF-space such that each of the restriction maps A ( U ) → A ( V ) is continuous and multiplication in A ( U ) is separately continuous. Finally, it is assumed that the action C ∞ ( U ) × A ( U ) → A ( U ) iscontinuous.As a consequence of our assumptions, each of the spaces A ( U ) carries a natural bornology, namelythe one consisting of all von Neumann bounded subsets, i.e. of all subsets B ⊂ A ( U ) which are ab-sorbed by every neighborhood of the origin. Moreover, by [Mey07, Lemma 1.30], separate continuityof multiplication in A ( U ) entails that the product map is a jointly bounded map, hence induces abounded map A ( U ) ˆ ⊗A ( U ) → A ( U ) on the complete (projective) bornological tensor product of A ( U ) with itself. Remark 2.1. (1) We refer to Appendix B for basic definitions and to [Mey07] for further detailson bornological vector spaces, their (complete projective) tensor products, and the use of
M.J. PFLAUM, H. POSTHUMA, AND X. TANG these concepts within cyclic homology theory. We always assume the bornologies in thispaper to be convex vector bornologies.(2) In this paper, we will often silently make use of the fact, that for two nuclear LF-spaces V and V their complete bornological tensor product V ˆ ⊗ V naturally concides (up to naturalequivalence) with the complete inductive tensor product V ˆ ⊗ ι V endowed with the bornologyof von Neumann bounded sets. Moreover, V ˆ ⊗ ι V is again a nuclear LF-space. We refer to[Mey99, A.1.4] for a proof of these propositions. Note that for Fr´echet spaces the projectiveand inductive topological tensor product coincide. Definition 2.2.
A sheaf of algebras A defined over a differentiable space ( X, C ∞ X ) such that theabove assumptions are fulfilled will be called a sheaf of bornological algebras over ( X, C ∞ X ). If all A ( U ) are unital and the restriction maps A ( U ) → A ( V ) are unital homomorphisms, we say that A is a sheaf of unital bornological algebras or just that A ( U ) is unital . If every section space A ( U ) isan H-unital algebra, we call A a sheaf of H-unital bornological algebras or briefly H-unital . Finally,we call A an admissible sheaf of bornological algebras if A is H-unital and if for each k ∈ N ∗ thepresheaf assigning to an open U ⊂ X the k -times complete bornological tensor product A ( U ) ˆ ⊗ k iseven a sheaf on X . Example 2.1. (1) The structure sheaf C ∞ X of a differentiable space ( X, C ∞ X ) is an example ofan admissible sheaf of unital bornological algebras over ( X, C ∞ X ).(2) Given a proper Lie groupoid G , the convolution sheaf A is an admissible sheaf of bornologicalalgebras over the orbit space ( X, C ∞ X ) of the groupoid. This follows by construction of A ,Prop. 1.2 and [CM01, Prop. 2], which entails H-unitality of each of the section spaces A ( U ).2.2. The Hochschild homology sheaf.
Assume that A is a sheaf of bornological algebras over thedifferentiable space ( X, C ∞ X ). We will construct the Hochschild homology sheaf H H • ( A ) associatedto A as a generalization of Hochschild homology for algebras; see [Lod98] for the latter and AppendixB for basic definitions and notation used.For each k ∈ N ∗ let C k ( A ) denote the presheaf on X which assigns to an open U ⊂ X the ( k +)-times complete bornological tensor product A ( U ) ˆ ⊗ ( k +1) . Note that in general, C k ( A ) is not a sheaf.We denote by ˆ C k ( A ) the sheafification of C k ( A ). Observe that for V ⊂ U ⊂ X open the Hochschildboundary b : C k ( A )( U ) → C k − ( A )( U )commutes with the restriction maps r UV : C k ( A )( U ) → C k ( A )( V ), hence we obtain a complex ofpresheaves (cid:0) C • ( A ) , b (cid:1) and by the universal property of the sheafification a sheaf complex (cid:0) ˆ C • ( A ) , b (cid:1) .The Hochschild homology sheaf H H • ( A ) is now defined as the homology sheaf of (cid:0) ˆ C • ( A ) , b (cid:1) thatmeans H H k ( A ) := ker (cid:0) b : ˆ C k ( A ) → ˆ C k − ( A ) (cid:1) / im (cid:0) b : ˆ C k +1 ( A ) → ˆ C k ( A ) (cid:1) . By construction, the stalk
H H k ( A ) O , O ∈ X coincides with the k -th Hochschild homology HH k ( A O )of the stalk A O . On the other hand, HH k ( A ( X )) need in general not coincide with the space H H k ( A )( X ) of global sections of the k -th Hochschild homology sheaf. The main goal of thissection is to prove the following result which is crucial for our study of the Hochschild homology ofthe convolution algebra of a proper Lie groupoid, but also might be intersting by its own. Its proofwill cover the remainder of Section 2. Theorem 2.3.
Let A be the convolution sheaf of a proper Lie groupoid G . Then the natural mapin Hochschild homology HH • (cid:0) A ( X ) (cid:1) → H H • ( A )( X ) = Γ (cid:0) X, H H • ( A ) (cid:1) is an isomorphism. Before we can spell out the proof we need several auxiliary tools and results.
N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 7
The localization homotopies.
Throughout this paragraph we assume that A ( X ) is an ad-missible sheaf of bornological algebras over the differentiable space ( X, C ∞ X ).To construct the localization morphisms, observe that the complex C • ( A ) inherits from A = A ( X )the structure of a C ∞ ( X )-module. More precisely, the corresponding action is given by(2.1) C ∞ ( X ) × C k ( A ) → C k ( A ) , ( ϕ, a ⊗ . . . ⊗ a k ) ( ϕa ) ⊗ a ⊗ . . . ⊗ a k . By definition, it is immediate that the C ∞ ( X )-action commutes with the operators b and b ′ henceinduces a chain map C ∞ ( X ) × C • ( A ) → C • ( A ). In a similar fashion we define an action of C ∞ ( X k +1 ) ∼ = (cid:0) C ∞ ( X ) (cid:1) ˆ ⊗ ( k +1) on C k ( A ) by(2.2) ( ϕ ⊗ . . . ⊗ ϕ k , a ⊗ . . . ⊗ a k ) ( ϕ a ) ⊗ . . . ⊗ ( ϕ k a k ) . This allows us to speak of the support of a chain c ∈ C k ( A ). It is defined as the complement of thelargest open subset U in X k +1 such that ϕ · c = 0 for all ϕ ∈ C ∞ ( X ) with supp ϕ ⊂ U .Next choose a metric d : X × X → R such that the function d lies in C ∞ ( X × X ). Such a metricexists by Corollary A.4. Then fix a smooth function ̺ : R → [0 ,
1] which has support in ( −∞ , ]and satisfies ̺ ( r ) = 1 for r ≤ . For ε > ̺ ε the rescaled function r ̺ ( sε ). Nowdefine functions Ψ k,i,ε ∈ C ∞ ( X k +1 ) for k ∈ N and i = 0 , . . . , k by(2.3) Ψ k,i,ε ( x , . . . , x k ) = i − Y j =0 ̺ ε (cid:0) d ( x j , x j +1 ) (cid:1) , where x , . . . , x k ∈ X and x k +1 := x . Moreover, put Ψ k,ε := Ψ k,k +1 ,ε . Using the C ∞ ( X k +1 )-action on C k ( A ) we obtain for each ε > ε : C • ( A ) → C • ( A ) , C k ( A ) ∋ c Ψ k,ε c . One immediately checks that Ψ ε commutes with the face maps b i and the cyclic operator t k . Hence,Ψ ε is a chain map. One even has more. Lemma 2.4.
Let A be an admissible sheaf of bornological algebras over the differentiable space ( X, C ∞ ) , and put A := A ( X ) . Let d be a metric on X such that d is smooth and fix a smooth map ̺ : R → [0 , with support in ( −∞ , ] such that ̺ | ( ∞ , ] = 1 . Then, for each ε > , the chain map Ψ ε : C • ( A ) → C • ( A ) is homotopic to the identity morphism on C • ( A ) .Proof. Let us first consider the case, where A is a sheaf of unital algebras. The Hochschild chaincomplex then is a simplicial module with face maps b i and the degeneracy maps s k,i : C k ( A ) → C k +1 ( A ) , a ⊗ . . . ⊗ a k a ⊗ . . . ⊗ a i ⊗ ⊗ a i +1 ⊗ . . . ⊗ a k , where k ∈ N , i = 0 , . . . , k . Define C ∞ ( X )-module maps η k,i,ε : C k ( A ) → C k +1 ( A ) for k ∈ N , i = 1 , · · · , k + 2 and ε > η k,i,ε ( c ) := ( Ψ k +1 ,i,ε · ( s k,i − c ) for i ≤ k + 1 , i = k + 2 . Moreover, put C − ( A ) := { } and let η − , ,ε : C − ( A ) → C ( A ) be the 0-map. For k ≥ i = 2 , · · · , k one then computes( bη k,i,ε + η k − ,i,ε b ) c = ( − i − Ψ k,i − ,ε c + Ψ k,i − ,ε i − X j =0 ( − j s k − ,i − b k,j c ++ ( − i Ψ k,i,ε c + Ψ k,i,ε i − X j =0 ( − j s k − ,i − b k,j c . For the case i = 1 one obtains( bη k, ,ε + η k − , ,ε b ) c = c − Ψ k, ,ε c + Ψ k, ,ε s k − , b k, c , M.J. PFLAUM, H. POSTHUMA, AND X. TANG and for i = k + 1( bη k,k +1 ,ε + η k − ,k +1 ,ε b ) c = Ψ k,k,ε ( − k c + Ψ k,k,ε k − X j =0 ( − j s k − ,k − b k,j c + ( − k +1 Ψ k,ε c. Finally, one checks for k = 0 and i = 1( bη , ,ε + η − , ,ε b ) c = bη , ,ε c = 0 . These formulas immediately entail that the maps H k,ε = k +1 X i =1 ( − i +1 η k,i,ε : C k ( A ) → C k +1 ( A )form a homotopy between the identity and the localization morphism Ψ ε . More precisely, (cid:0) bH k,ε + H k − ,ε b (cid:1) c = c − Ψ ε c for all k ∈ N and c ∈ C k ( A ) . (2.5)This finishes the proof of the claim in the unital case.Now let us consider the general case, where A is assumed to be a sheaf of H-unital but notnecessarily unital algebras. Consider the direct sum of sheaves A ⊕ C ∞ X , denote it by e A , and put e A := e A ( X ). We turn e A into a sheaf of unital bornological algebras by defining the product of( f , h ) , ( f , h ) ∈ e A ( U ) as(2.6) ( f , h ) · ( f , h ) := ( h f + h f + f f , h h ) . One obtains a split short exact sequence in the category of bornological algebras0 / / A / / e A q / / C ∞ ( X ) i o o ❴ ❴ ❴ / / . This gives rise to a diagram of chain complexes and chain maps(2.7) 0 / / ker • q ∗ (cid:31) (cid:127) / / κ (cid:15) (cid:15) ✤✤✤ C • ( e A ) q ∗ / / C • ( C ∞ ( X )) i ∗ o o ❴ ❴ ❴ / / C • ( A ) , ι O O where the row is split exact, and ι denotes the canonical embedding. Since A is H-unital, ι is aquasi-isomorphsm. Because the chain complexes ker • q ∗ and C • ( A ) are bounded from below, thereexists a chain map κ which is left inverse to ι . Note that the components κ k need not be boundedmaps between bornological spaces. By construction, Ψ ε acts on each of the chain complexes withinthe diagram, and all chain maps (besides possibly κ ) commute with this action. By the first part ofthe proof we have an algebraic homotopy H : C • ( e A ) → C • +1 ( e A ) such thatid − Ψ ε = bH + Hb .
Define F : C • ( A ) → C • +1 ( A ) by F := κ (id − i ∗ q ∗ ) Hι . Note that F is well-defined indeed, since q ∗ (id − i ∗ q ∗ ) = 0. Now compute for c ∈ C k ( A )( bF + F b ) c = κ (id − i ∗ q ∗ )( bH + Hb ) ιc = κ (id − i ∗ q ∗ )( ιc − Ψ ε ιc ) = c − Ψ ε c . Hence F is a homotopy between the identity and Ψ ε and the claim is proved. (cid:3) Lemma 2.5.
Let A be an admissible sheaf of bornological algebras over the differentiable space ( X, C ∞ ) , put A := A ( X ) , and let the metric d and the cut-off function ̺ as in the preceding lemma.Assume that ( ϕ l ) l ∈ N is a smooth locally finite partition of unity and that ( ε l ) l ∈ N a sequence of positivereal numbers. Then (2.8) Ψ : C • ( A ) → C • ( A ) , C k ( A ) ∋ c X l ∈ N ϕ l Ψ ε l c . N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 9 is a chain map and there exists a homotopy between the identity on C • ( A ) and Ψ .Proof. Recall that the action of C ∞ ( X ) commutes with the Hochschild boundary and that each Ψ ε l is a chain map. Since ( ϕ l ) l ∈ N is a locally finite smooth partition of unity, Ψ then has to be a chainmap by construction.Now assume that A is a sheaf of unital algebras. Let H • ,ε l : C • ( A ) → C • +1 be the homotopyfrom the preceding lemma which fulfills Equation (2.5) with ε = ε l . For all k ∈ N let H k be the map H k : C k ( A ) → C k +1 ( A ) , c X l ∈ N H k,ε l ϕ l c . Then (cid:0) bH k + H k − b (cid:1) c = X li ∈ N ( ϕ l c − Ψ ε l ϕ l c ) = c − Ψ c for all k ∈ N and c ∈ C k ( A ) . (2.9)Hence H is a homotopy between the identity and Ψ which proves the claim in the unital case.In the non-unital case define the unitalizations e A and e A as before and let q ∗ , i ∗ , ι , κ denote thechain maps as in Diagram (2.7). Let H : C • ( e A ) → C • +1 ( e A ) be the algebraic homotopy constructedfor the unital case. In particular this means thatid − Ψ = bH + Hb .
Defining F : C • ( A ) → C • +1 ( A ) by F := κ (id − i ∗ q ∗ ) Hι then gives a homotopy between the identityon C • ( A ) and Ψ. (cid:3) Lemma 2.6.
Let A be an admissible sheaf of bornological algebras over the differentiable space ( X, C ∞ ) , put A := A ( X ) and let c ∈ C k ( A ) be a Hochschild cycle. If the support of c does not meetthe diagonal, then c is a Hochschild boundary.Proof. Assume that the support of the Hochschild cycle c does not meet the diagonal and let U = X k +1 \ supp c . Then U is an open neighborhood of the diagonal. By Corollary A.4 there existsa complete metric d : X × X → R such that d ∈ C ∞ ( X × X ). Choose a compact exhaustion( K n ) n ∈ N of X which means that each K n is compact, K n ⊂ K ◦ n +1 for all n ∈ N and S n ∈ N K n = X .For each n ∈ N there then exists an ε n > x , . . . , x k ) ∈ K k +1 n are in U whenever d ( x j , x j +1 ) < ε n for j = 0 , . . . , k and x k +1 := x . Choose a locally finite smooth partition of unity( ϕ l ) l ∈ N subordinate to the open covering ( K ◦ n ) n ∈ N and let Ψ : C • ( A ) → C • ( A ) be the associatedchain map defined by (2.8). According to Lemma 2.5 there then exists a chain homotopy H betweenthe identity on C • ( A ) and Ψ. Since the support of c does not meet U one obtains c = c − Ψ ε c = bH ( c ) , so c is a Hochschild boundary indeed. (cid:3) Proposition 2.7.
Assume to be given a proper Lie groupoid with orbit space X and convolutionsheaf A . Let A = A ( X ) and ˆ C • ( A ) be the sheaf complex of Hochschild chains. Denote for each O ∈ X and each chain c ∈ C • (cid:0) A ( U ) (cid:1) defined on a neighborhood U ⊂ X of O by [ c ] O the germ of c at O that is the image of c in the stalk ˆ C • , O ( A ) = colim V ∈N ( O ) C • ( A ( V )) , where N ( O ) denotes the filter basisof open neighborhoods of O . Then the chain map η : C • ( A ) → Γ (cid:0) X, ˆ C • ( A ) (cid:1) , c (cid:0) [ c ] O (cid:1) O ∈ X is a quasi-isomorphism.Proof. Consider a section s ∈ Γ (cid:0) X, ˆ C k ( A ) (cid:1) . Then there exists a (countable) open covering ( U i ) i ∈ I of the orbit space X and a family ( c i ) i ∈ I of k -chains c i ∈ C k (cid:0) A ( U i ) (cid:1) such that [ c i ] O = s ( O ) forall i ∈ I and O ∈ U i . After possibly passing to a finer (still countable) and locally finite coveringone can assume that there exists a partition of unity ( ϕ i ) i ∈ I by functions ϕ i ∈ C ∞ ( X ) such thatsupp ϕ i ⊂⊂ U i for all i ∈ I . If s is a cycle, then we can achieve after possible passing to an even finer locally finite covering that each c i is a Hochschild cycle as well. Choose a metric d : X × X → R such that d ∈ C ∞ ( X × X ). For each i there then exists ε i > O ∈ X with d ( O , supp ϕ i ) ≤ ( k + 1) ε i is a compact subset of U i . The chain Ψ ε i ( ϕ i c i ) then has compact supportin U k +1 i . Extend it by 0 to a smooth function on X k +1 and denote the thus obtained k -chain alsoby Ψ ε i ( ϕ i c i ). Now put(2.10) c := X i ∈ I Ψ ε i ( ϕ i c i ) . Then c ∈ C k ( A ) is well-defined since the sum in the definition of c is locally finite. For every O ∈ X now choose an open neigborhood W O meeting only finitely many of the elements of the covering( U i ) i ∈ I . Denote by I O the set of indices i ∈ I such that U i ∩ W O = ∅ . Then each I O is finite. Nextlet H i : C • (cid:0) A ( U i ) (cid:1) → C • +1 (cid:0) A ( U i ) (cid:1) be the homotopy operator constructed in the proof of Lemma2.4 such that bH i + H i b = id − Ψ ε i . Let e i = H i ( ϕ i c i ) for i ∈ I O and put e O = P i ∈ I O e i | W k +2 O . Then e O ∈ C k +1 (cid:0) A ( W O ) (cid:1) . Now computefor Q ∈ W O s ( Q ) − [ c ] Q = X i ∈ I O [ ϕ i c i ] ( Q ) − [Ψ ε i ( ϕ i c i )] Q = X i ∈ I O [ be i ] Q + [ H i ( ϕ i bc i )] Q == [ be O ] Q + X i ∈ I O [ H i ( ϕ i bc i )] Q . Hence one obtains, whenever s is a cycle, s ( Q ) − [ c ] Q = [ be O ] Q for all O ∈ X, Q ∈ W O . This means that s and η ( c ) define the same homology class. So the induced morphism betweenhomologies H • η : HH • ( A ) → H • (cid:0) Γ (cid:0) X, ˆ C • ( A ) (cid:1)(cid:1) is surjective. It remains to show that H • η isinjective. To this end assume that e ∈ C k ( A ) is a cycle such that H • η ( e ) = 0. Then η ( e ) = bs for some s ∈ Γ (cid:0) X, ˆ C k +1 ( A ) (cid:1) . As before, associate to s a sufficiently fine locally finite open cover( U i ) i ∈ I together with a subordinate smooth partition of unity ( ϕ i ) i ∈ I and c i ∈ C k +1 ( A ( U i )) suchthat [ c i ] O = s ( O ) for all O ∈ U i . Let W O and I O also be as above. Define c ∈ C k +1 ( A ) by Eq. (2.10).Now compute for Q ∈ W O [ bc − e ] Q = X i ∈ I O [ b Ψ ε i ( ϕ i c i )] Q − [ ϕ i e ] Q = X i ∈ I O [Ψ ε i ( ϕ i bc i )] Q − [ ϕ i e ] Q == X i ∈ I O [ ϕ i bc i ] Q − [ ϕ i e ] Q = X i ∈ I O ( ϕ i bs )( Q ) − ( ϕ i bs )( Q ) = 0 . Therefore, bc − e ∈ C k ( A ) is a k -cycle such that its support does not meet the diagonal. ByLemma 2.6, bc − e is a boundary which means that the homology of e is trivial. Hence H • η is anisomorphism. (cid:3) Now we have all the tools to verify our main localization result.
Proof of Theorem 2.3.
First note that we can regard every chain complex of sheaves D • as a cochaincomplex of sheaves under the duality D n := D − n for all integers n . We therefore have the hyperco-homology H n ( X, D • ) := H − n ( X, D • ); see [Wei96, Appendix], where the case of cochain complexesof sheaves not necessarily bounded below as we have it here is considered. Observe that (cid:0) ˆ C • ( A ) , b (cid:1) and (cid:0) H H • ( A ) , (cid:1) are quasi-isomorphic sheaf complexes, hence their hypercohomologies coincide.Recall that for a cochain complex of fine sheaves D • H n ( X, D • ) = H n (cid:0) Γ( X, D • ) (cid:1) . N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 11
Since both ˆ C • ( A ) and H H • ( A ) are complexes of fine sheaves, these observations together withProposition 2.7 now entails for natural nHH n (cid:0) A ( X ) (cid:1) = H n (cid:0) Γ( X, ˆ C • ( A ) (cid:1) = H n ( X, ˆ C • ( A )) == H n ( X, H H • ( A )) = H n (cid:0) Γ( X, H H • ) (cid:1) = Γ (cid:0) X, H H n ( A ) (cid:1) . This is the claim. (cid:3) Computation at a stalk
Recall that G ⇒ M is a proper Lie groupoid, X is its orbit space, and A G is the convolution sheafof G (Definition 1.1). Given an orbit O ∈ X of G , we introduce in this section a linear model of thegroupoid around the stalk and use it in Proposition 3.5 to construct a quasi-isomorphism betweenthe stalk complex ˆ C • , O ( A G ) and the corresponding of the linear model. We divide the constructioninto two steps.3.1. Reduction to the linear model.
Let us recall the linearization result for the groupoid G around an orbit O . Let N O → O be the normal bundle of the closed submanifold O in M , and G | O ⇒ O be the restriction of the groupoid G to O . G | O acts on N O canonically. And we use G | O ⋉ N O ⇒ N O to denote the associated transformation groupoid. As in Definition 1.1, let A N O be the sheaf of convolution algebras on X N O = N O / G | O , the orbit space associated to the groupoid G | O ⋉ N O . Accordingly, we can consider the presheaf of chain complexes C • ( A N O ) and the associatedsheaf complex ˆ C • ( A N O ) as in Proposition 2.7. In what follows in this subsection, we will identifythe stalk ˆ C • , O ( A G ) with the linearized model ˆ C • , O ( A N O ), which is the stalk of the sheaf ˆ C • ( A N O ) atthe zero section of N O .The main tool to identify the above two stalks is the linearization result of proper Lie groupoidsdeveloped by Weinstein [Wei02] and Zung [Zun06] (See also [CS13, PPT14, dHF18]). The particularapproach we take below is from [PPT14]. Fix a transversely invariant metric g on M . Given afunction δ : O → R > , let T δ O ,N O be the δ -neighborhood of the zero section in N O . According to[PPT14, Theorem 4.1], there exists a continuous map δ : O → R > such that the exponential mapexp | T δ O ,N O : T δ O ,N O → T δ O := exp( T δ O ,N O ) ⊂ M is a diffeomorphism. Furthermore, the exponential mapexp | T δ O ,N O lifts to an isomorphism Θ of the following groupoids(3.1) Θ : (cid:0) G | O ⋉ N O (cid:1) | T δ O ,N O → G | T δ O . Lemma 3.1.
For each orbit O ⊂ M , the pullback map Θ ∗ defines a quasi-isomorphism Θ • , O fromthe stalk complex ˆ C • , O ( A G ) to the stalk complex ˆ C • , O ( A N O ) .Proof. We explain how Θ • , O is defined on ˆ C • , O ( A G ). Let [ f ⊗ · · · ⊗ f k ] ∈ ˆ C k, O ( A G ) be a germ ofa k -chain at O ∈ X . Let U be a neighborhood of O in X such that f ⊗ · · · ⊗ f k is a section of C k ( A ( U )) which is mapped to [ f ⊗ · · · ⊗ f k ] in the stalk complex ˆ C • , O ( A G ) under the canonicalmap η from Proposition 2.7. By (1.2), the support of each of the maps f , · · · , f k is longitudinallycompact. In particular, supp( f i ) ∩ s − ( O ) ( i = 0 , · · · , k ) is compact. Therefore, s (cid:0) supp( f i ) ∩ s − ( O ) (cid:1) = t (cid:0) supp( f i ) ∩ s − ( O ) (cid:1) and the union K f , ··· ,f k := S ki =0 s (cid:0) supp( f i ) ∩ s − ( O ) (cid:1) is also compact in O .Let K be a precompact open subset of O containing K f , ··· ,f k as a proper subset. Observe thatthe closure of K is compact in O . Hence, there is a positive constant ε such that the ε -neighborhood T εK of K is contained inside the δ -neighborhood T δ O , the range of the linearization map Θ in (3.1).Applying the homotopy map Ψ ε defined in Lemma 2.4 to f ⊗ · · · ⊗ f k , we may assume withoutloss of generality that the support of f , · · · , f n is contained inside T εK , and therefore inside the δ -neighborhood T δ O . Accordingly, the pullback function Θ ∗ ( f ⊗ · · · ⊗ f k ) is well defined and supportedin (cid:0) G | O ⋉ N O (cid:1) | Θ − ( T εK ) × · · · × (cid:0) G | O ⋉ N O (cid:1) | Θ − ( T εK ) . Let U ε O be the ε -neighborhood of O in N O / G | O . By the definition of Θ, it is not difficult to checkthat Θ ∗ ( f i ) is supported inside (cid:0) G | O ⋊ N O (cid:1) | Θ − ( T εK ) for i = 0 , · · · , k and therefore Θ ∗ ( f ⊗ · · · ⊗ f k )is a well defined k -chain in C k (cid:0) A N O ( U ε O ) (cid:1) . Define Θ • , O (cid:0) [ f ⊗ · · ·⊗ f k ] (cid:1) ∈ ˆ C • , O ( A N O ) to be the germ ofΘ ∗ ( f ⊗ · · · ⊗ f k ) at the point O in the orbit space X N O = N O / G | O . It is worth pointing out that theconstruction of Θ • , O (cid:0) [ f ⊗ · · · ⊗ f k ] (cid:1) is independent of the choices of the subset K and the constant ε . Analogously, using the inverse map Θ − , we can construct the inverse morphism (Θ − ) • , O fromˆ C • , O ( A N O ) to ˆ C • , O ( A G ), and therefore prove that Θ • , O is a quasi-isomorphism. We leave the detailsto the diligent reader. (cid:3) Computation of the linear model.
We compute in this subsection the cohomology of C • ( A N O ). Our method is inspired by the work of Crainic and Moerdijk [CM01].To start with, recall that we prove in [PPT14, Cor. 3.11 ] that for a proper Lie groupoid G ⇒ M ,given x ∈ M , there is a neighborhood U of x in M diffeomorphic to O × V x where O is an open ballin the orbit O through x centered at x , and V x is a G x –the isotropy group of G at x – invariant openball in N x O centered at the origin. Under this diffeomorphism G | U is isomorphic to the productof the pair groupoid O × O ⇒ O and the transformation groupoid G x ⋉ V x ⇒ V x . Applying thisresult to the the transformation groupoid G | O ⋉ N O ⇒ N O , we conclude that given any x ∈ O , thereis an open ball O of x in O such that the restricted normal bundle U x := N O | O is diffeomorphicto N x O × O and (cid:0) G | O ⋉ N O (cid:1) | U x is isomorphic to the product of the pair groupoid O × O and thetransformation groupoid G x ⋉ N x O .Following the above local description of G | O ⋉ N O , we choose a covering ( O x ) x ∈ O of the orbit O ,and therefore also a covering U := ( U x ) x ∈ O , U x := O x × N x O , of N O . We choose a locally finitecountable subcovering ( O i ) i ∈ I of O and the associated covering ( U i ) i ∈ I of N O . Choose ϕ i ∈ C ∞ c ( O )such that ϕ i is a partition of unity subordinate to the open covering ( O i ) i ∈ I of O . Lift ϕ i ∈ C ∞ c ( O )to ˜ ϕ i ∈ C ∞ ( N O ) that is let it be constant along the fiber direction. As ϕ i is compactly supported,˜ ϕ i is longitudinally compactly supported and therefore belongs to A N O . Now consider the groupoid H U over the disjoint union ⊔ U i , such that arrows from U i to U j are arrows in G | O ⋉ N O starting from U i and ending in U j . Composition of arrows in G | O ⋉ N O equips H U with a natural Lie groupoidstructure that is Morita equivalent to G | O ⋉ N O . As a consequence of this the orbit spaces of thegroupoids G | O ⋉ N O and H U are therefore naturally homeorphic, actually even diffeomorphic in thesense of differentiable spaces. We therefore identify them.The following lemma is essentially due to Crainic and Moerdijk [CM01]. Lemma 3.2.
The map
Λ : A ( G | O ⋉ N O ) := Γ (cid:0) A G | O ⋉ N O (cid:1) → A ( H U ) := Γ (cid:0) A H U (cid:1) defined by Λ( f ) := ( ˜ ϕ i f ˜ ϕ j ) i,j is an algebra homomorphism which induces a quasi-isomorphism Λ • from C • (cid:0) A ( G | O ⋉ N O ) (cid:1) to C • (cid:0) A ( H U ) (cid:1) . In addition, Λ induces a quasi-isomorphism of sheaf complexes Λ • : ˆ C • ( A G | O ⋉ ) → ˆ C • ( A H U ) over their joint orbit space N O / G | O ∼ = ( H U ) / H U .Proof. The proof of the claim is a straightforward generalization of the one of [CM01, Lemma 5].The slight difference here is that we work with the algebras A ( G | O ⋉ N O ) and A ( H U ) instead of thealgebra of compactly supported functions. We skip the proof here to avoid repetition. (cid:3) Next, the groupoid H U can be described more explicitly as follows. Firstly, index the open sets inthe covering ( U i ) i ∈ I by natural numbers so in other words assume I ⊂ N ∗ . After possibly reindexingagain, one can assume that if k ∈ I , then l ∈ I for all 1 ≤ l ≤ k . Secondly, given i , write x ∈ U i as( x v , x o ) where x v ∈ N x i O and x o ∈ O i . Choose a diffeomorphism ψ i : O i → R k , where k = dim( O ).Thirdly, for any 1 < i ∈ I , choose an arrow g i ∈ G from x to x i . The arrow g i induces anisomorphism between N x O and N x i O , and conjugation by g i defines an isomorphism from G x i to G x . Accordingly, g i induces a groupoid isomorphism between G x ⋉ N x O and G x i ⋉ N x i O . N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 13
Lemma 3.3.
The groupoid H U is isomorphic to the product groupoid (cid:0) G x ⋉ N x O (cid:1) × ( I × I ) × ( R k × R k ) . Proof.
We define groupoid morphismsΦ : H U → (cid:0) G x ⋉ N x O (cid:1) × ( I × I ) × ( R k × R k )and Ψ : (cid:0) G x ⋉ N x O (cid:1) × ( I × I ) × ( R k × R k ) → H U . Given an arrow h ∈ H U with source in U i and target in U j , we consider ( s ( h ) o , x i ) ∈ O i × O i and( t ( h ) o , x j ) ∈ O j × O j . Define h x i ∈ ( G x i ⋉ N x i O ) × ( O i × O i ) (and h x j ∈ ( G x j ⋉ N x j O ) × ( O i × O i ))by h x i = (cid:0) (id , , ( s ( h ) o , x i ) (cid:1) (and h x j = (cid:0) (id , , ( t ( h ) o , x j ) (cid:1) ). The arrow g − j h − x j hh x i g i belongs to H U | U and its component in O × O is ( x , x ). The arrow Φ( h ) now is defined to beΦ( h ) := (cid:0) g − j h − x j hh x i g i , ( i, j ) , ( ψ ( s ( h ij ) , t ( h ij )) (cid:1) ∈ (cid:0) G x ⋉ N x O (cid:1) × ( I × I ) × ( R k × R k ) . Similarly, given ( k, ( i, j ) , ( y i , y j )) ∈ (cid:0) G x ⋉ N x O (cid:1) × ( I × I ) × ( R k × R k ), define h y i := (cid:0) (id , , ( ψ − i ( y i ) , x i ) (cid:1) ∈ G | U i , h y j := (cid:0) (id , , ( ψ − j ( y j ) , x j ) (cid:1) ∈ G | U j , and h := (cid:0) k, ( x , x ) (cid:1) ∈ G | U . Notice g j h g − i is an arrow in H U starting from x i and ending at x j .We can now define Ψ( k, ( i, j ) , ( y i , y j )) to beΨ( k, ( i, j ) , ( y i , y j )) := h y j g j h g − i h − y i ∈ H U . It is straightforward to check that Φ and Ψ are groupoid morphisms and inverse to each other. (cid:3)
Let A (cid:0) ( G x ⋉ N x O ) × ( I × I ) × ( R k × R k ) (cid:1) be the space of global sections of the convolutionsheaf A ( G x ⋉ N x O ) × ( I × I ) × ( R k × R k ) . With the maps Φ and Ψ introduced in Lemma 3.3, we have thefollowing induced isomorphisms of chain complexes,Φ • : C • (cid:16) A (cid:0)(cid:0) G x ⋉ N x O (cid:1) × ( I × I ) × ( R k × R k ) (cid:1)(cid:17) → C • (cid:0) A ( H U ) (cid:1) , Ψ • : C • (cid:0) A ( H U ) (cid:1) → C • (cid:16) A (cid:0)(cid:0) G x ⋉ N x O (cid:1) × ( I × I ) × ( R k × R k ) (cid:1)(cid:17) . Since they are induced by an ismorphism of groupoids, we also obtain a pair of mutually inverseisomorphisms of complexes of sheaves which are denoted by the same symbols,Φ • : ˆ C • (cid:16) A ( G x ⋉ N x O ) × ( I × I ) × ( R k × R k ) (cid:17) → ˆ C • ( A H U ) , Ψ • : ˆ C • ( A H U ) → ˆ C • (cid:16) A ( G x ⋉ N x O ) × ( I × I ) × ( R k × R k ) (cid:17) . Observe that both groupoids I × I and R k × R k have only one orbit. Therefore, longitudinallycompactly supported functions on them are the same as compactly supported functions. Observethat C ∞ ( G x ⋉ N x O ) is the algebra of longitudinally compactly supported smooth functions on G x ⋉ N x O . By Lemma 3.3, the groupoid algebra A ( H U ) is isomorphic to A (cid:0) ( G x ⋉ N x O ) × ( I × I ) × ( R k × R k ) (cid:1) . The latter can be identified with C ∞ ( G x ⋉ N x O ) ˆ ⊗ R I × I ˆ ⊗ C ∞ c ( R k × R k ), where R I × I is the space of finitely supported functions on I × I . Note that I × I and R k × R k both carrythe structure of a pair groupoid, so the corresponding products on R I × I and C ∞ c ( R k × R k ) are givenin both cases by convolution which we denote as usual by ∗ . Let τ I be the trace on R I × I defined by τ I ( d ) := X i d ii , d = ( d ij ) i,j ∈ I ∈ R I × I and let τ R k be the trace on C ∞ c ( R k × R k ) given by τ R k ( α ) := Z R k α ( x, x ) dx , α ∈ C ∞ c ( R k × R k ) , where dx is the Lebesgue measure on R k . Define a map τ m : C m (cid:0) C ∞ ( G x ⋉ N x O ) ˆ ⊗ R I × I ˆ ⊗ C ∞ c ( R k × R k ) (cid:1) → C m (cid:0) C ∞ ( G x ⋉ N x O ) (cid:1) as follows: τ m (cid:0) ( f ⊗ · · · ⊗ f m ) ⊗ ( d ⊗ · · · ⊗ d m ) ⊗ ( α ⊗ · · · ⊗ α m ) (cid:1) := τ I ( d ∗ · · · ∗ d m ) τ R k ( α ∗ · · · ∗ α m ) f ⊗ · · · ⊗ f m ,f , · · · , f m ∈ C ∞ ( G x ⋉ N x O ) , d , · · · , d m ∈ R I × I , α , · · · , α m ∈ C ∞ c ( R k × R k ) . It is easy to check using the tracial property of τ I and τ R k that τ • is a chain map. Moreover, observethat the whole argument works not only for the global section algebra C ∞ ( G x ⋉ N x O ) but for anyof the section algebras C ∞ ( G x ⋉ V ) with V ⊂ N x O an open G x -invariant subspace. So eventuallywe obtain a morphism of sheaf complexes τ • : ˆ C • (cid:0) A C ∞ ( G x ⋉ N x O ) ˆ ⊗ R I × I ˆ ⊗ C ∞ c ( R k × R k ) (cid:1) → ˆ C • (cid:0) A C ∞ ( G x ⋉ N x O ) (cid:1) . over the orbit space N x O / G x . Lemma 3.4.
The chain map τ • : C • (cid:0) C ∞ ( G x ⋉ N x O ) ˆ ⊗ R I × I ˆ ⊗ C ∞ c ( R k × R k ) (cid:1) → C • (cid:0) C ∞ ( G x ⋉ N x O ) (cid:1) is a quasi-isomorphism. More generally, τ • : ˆ C • (cid:0) A C ∞ ( G x ⋉ N x O ) ˆ ⊗ R I × I ˆ ⊗ C ∞ c ( R k × R k ) (cid:1) → ˆ C • (cid:0) A C ∞ ( G x ⋉ N x O ) (cid:1) is an isomorphism of complexes of sheaves.Proof. Choose a function β ∈ C ∞ c ( R k ) such that Z R k β ( x ) dx = 1 . Let α ∈ C ∞ c ( R k × R k ) be the function β ⊗ β . Define an algebra morphism j α : C ∞ ( G x ⋉ N x O ) → C ∞ ( G x ⋉ N x O ) ˆ ⊗ R I × I ˆ ⊗ C ∞ c ( R k × R k )by j α ( f ) = f ⊗ δ (1 , ⊗ α , where δ (1 , is the function on I × I that is 1 on (1 ,
1) and 0 otherwise. j α, • is the induced map onthe cochain complex. It is easy to check τ • ◦ j α, • = id. Applying j α, • ◦ τ • to( f ⊗ · · · ⊗ f m ) ⊗ ( d ⊗ · · · ⊗ d m ) ⊗ ( α ⊗ · · · ⊗ α m )gives τ I ( d ∗ · · · ∗ g m ) τ R k ( α ∗ · · · ∗ α m ) (cid:0) f ⊗ · · · ⊗ f m (cid:1) ⊗ (cid:0) δ , ⊗ · · · ⊗ δ , (cid:1) ⊗ (cid:0) α ⊗ · · · ⊗ α (cid:1) . Following the proof of Lemma 2.4, we consider the unital algebra e C ∞ ( G x ⋉ N x O ) which is thedirect sum of C ∞ ( G x ⋉ N x O ) with C ∞ ( N x O ) G x and product structure given by Eq. (2.6). Wethen have the following split exact sequence in the category of bornological algebras(3.2) 0 → C ∞ ( G x ⋉ N x O ) → e C ∞ ( G x ⋉ N x O ) → C ∞ ( N x ) G x → . It is not hard to see that the chain maps τ • and j α, • extend to the corresponding versions ofthe algebras e C ∞ ( G x ⋉ N x O ) and C ∞ ( N x ) G x . As both algebras are unital, the homotopy mapsconstructed in the proof of [CM01, Lemma 6] can be applied to conclude that j α, • τ • is a quasi-isomorphism for e C ∞ ( G x ⋉ N x O ) and C ∞ ( N x ) G x . As the algebra C ∞ ( G x ⋉ N x ) is H -unital, weconsider the long exact sequence associated to the short exact sequence (3.2). As j α, • and τ • arequasi-isomorphisms on e C ∞ ( G x ⋉ N x O ) and C ∞ ( N x ) G x , we conclude by the five lemma that τ • and j α, • are also quasi-isomorphisms for C ∞ ( G x ⋉ N x O ). The argument generalizes immediatelyto the sheaf case. (cid:3) N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 15
Summarizing Lemma 3.1 – Lemma 3.4, we thus obtain the following local model for the stalkcomplex ˆ C • , O ( A G ). Proposition 3.5.
For every orbit O ∈ X the composition L • , O := τ • , ◦ Ψ • , ◦ Λ • , ◦ Θ • , O , where τ • , , Ψ • , , and Λ • , denote the respective sheaf morphisms localized at the zero sections, is a quasi-isomorphism, L • , O : ˆ C • , O ( A G ) Θ • , O −→ ˆ C • , O (cid:0) A G | O ⋉ N O (cid:1) Λ • , −→ ˆ C • , (cid:16) A H U (cid:17) Ψ • , −→ ˆ C • , (cid:0) A ( G x ⋉ N x O ) × ( I × I ) × ( R k × R k ) (cid:1) τ • , −→ ˆ C • , (cid:0) A G x ⋉ N x O (cid:1) . Basic relative forms
Let M be a smooth manifold equipped with a left action of a compact Lie group G which wewrite as ( g, x ) gx, for g ∈ G, x ∈ M . Associated to this action is the Lie groupoid G ⋉ M ⇒ M with source map given by the projection ( g, x ) x and target given by the action ( g, x ) gx . The loop space Λ ( G ⋉ M ) ⊂ G × M coincides in this case with the disjoint union of all fixed point sets M g ⊂ M for g ∈ G : Λ ( G ⋉ M ) := (cid:8) ( g, p ) ∈ G × M | gp = p (cid:9) = [ g ∈ G { g } × M g . For fixed g ∈ G , the fixed point subset M g ⊂ M is a closed submanifold but it can wildly vary as g varies over G . Therefore, the loop space Λ ( G ⋉ M ) is a singular subset of G × M . If we let G acton G × M by h · ( g, p ) := ( hgh − , hp ) , h ∈ G, ( g, p ) ∈ G × M , this action preserves Λ ( G ⋉ M ) ⊂ G × M sending M g to M hgh − . In [Bry87a, Bry87b], Brylinskiintroduces the notion of basic relative forms . Intuitively, a basic relative k -form is a smooth family( ω g ) g ∈ G ∈ Q g ∈ G Ω k ( M g ) of differential forms on fixed point subspaces which are(i) horizontal that is i ξ Mg ω g = 0 for all g ∈ G and ξ ∈ Lie( G g ), and(ii) G - invariant which means that h ∗ ω g = ω h − gh for all g, h ∈ G .Here, G g := Z G ( g ) denotes the centralizer of g ∈ G , which acts on M g . Because of the singularnature of Λ , one needs to make sense of what is exactly meant by a smooth family of differentialforms. There are two solutions for this: (A) Sheaf theory. In the sense of Grauert–Grothendieck and following Brylinski [Bry87b], wedefine the sheaf of relative forms on Λ ( G ⋉ M ) as the quotient sheafΩ k rel , Λ := ι − (cid:0) Ω kG ⋉ M → G / (cid:0) J Ω kG ⋉ M → G + d rel J ∧ Ω k − G ⋉ M → G (cid:1)(cid:1) . Here, Ω kG ⋉ M → G denotes the sheaf of k -forms on G × M relative to the projection pr : G × M → G and ι the canonical injection Λ ( G ⋉ M ) ֒ → G ⋉ M . A form ω ∈ Ω kG ⋉ M → G ( e U ) for e U ⊂ G ⋉ M open is given by a smooth global section of the vector bundle s ∗ V k T ∗ M that is by an element ω ∈ Γ ∞ ( e U , s ∗ V k T ∗ M ). The de Rham differential on M defines a differential d rel : Ω kG ⋉ M → G → Ω k +1 G ⋉ M → G . Finally, J denotes the vanishing ideal of smooth functions on G × M that restrict to zeroon Λ ( G ⋉ M ) ⊂ G × M . Note that J Ω • G ⋉ M → G + d rel J ∧ Ω • G ⋉ M → G is a differential graded idealin the sheaf complex (cid:0) Ω kG ⋉ M → G , d rel (cid:1) , so Ω • rel , Λ becomes a sheaf of differential graded algebrason the loop space. For open U ⊂ Λ ( G ⋉ M ), an element of Ω k rel , Λ ( U ) can now be understoodas an equivalence class [ ω ] Λ of forms ω ∈ Ω kG ⋉ M → G ( e U ) defined on some open e U ⊂ G ⋉ M suchthat U = e U ∩ Λ ( G ⋉ M ). This explains the definition of the sheaf complex of relative forms onthe singular space Λ ( G ⋉ M ); confer also [PPT17]. Next observe that the map which associatesto each p ∈ M the conormal space N ∗ p := (cid:0) T p M/T p O p (cid:1) ∗ is a generalized subdistribution of thecotangent bundle T ∗ M in the sense of Stefan-Suessmann, cf. [Ste80, Sus73, JR´S11]. In the language of [DLPR12], N ∗ is a cosmooth generalized distribution. The restriction of N ∗ to each orbit, andeven to each stratum of M of a fixed isotropy type, is a vector bundle, cf. [PPT14]. Henceforth, thepullback distribution s ∗ V k N ∗ is naturally a cosmooth generalized subdistribution of V k T ∗ G ⋉ M .We define the space Ω k hrel , Λ G ( U ) of horizontal relative k -forms on the loop space (over U ) as thesubspace Ω k hrel , Λ G ( U ) := (cid:8) [ ω ] Λ ∈ Ω k rel , Λ G ( U ) | ω ( g,p ) ∈ ^ k N ∗ p for all ( g, p ) ∈ U (cid:9) . This implements the above condition (i). Observe that the action of G on T N leaves the orbitsinvariant, hence induces also an action on the conormal distribution N ∗ in a canonical way [PPT14,Sec. 3]. Call a section [ ω ] Λ ∈ Ω k hrel , Λ ( U ) invariant , if(4.1) ω hgh − ,hp ( hv , . . . , hv k ) = ω ( g,p ) ( v , . . . , v k )for all ( g, p ) ∈ U ⊂ Λ G , h ∈ G such that ( hgh − , hp ) ∈ U and v , . . . , v k ∈ N p . Note that theinvariance of [ ω ] Λ does not depend on the particular choice of the representative ω such that ω p ∈ V k N ∗ p . Condition (ii) is covered by defining the space Ω k brel , Λ ( U ) of basic relative k -forms on theloop space (over U ) now as the space of all invariant horizontal relative k -forms [ ω ] Λ ∈ Ω k hrel , Λ G ( U ).Obviously, one thus obtains sheaves Ω k hrel , Λ and Ω k brel , Λ on the loop space Λ ( G ⋉ M ). We will callthe push forward π ∗ s ∗ Ω k brel , Λ by the source map s and canonical projection π : M → X = M/G sheaf of basic relative functions as well and denote it also by the symbol Ω k brel , Λ . This will not leadto any confusion. The interpretion of basic relative forms as smooth families of forms on the fixedpoint manifolds is still missing, but will become visible in the following approach. (B) Differential Geometry. From a more differential geometric perspective, we consider thefamily of vector bundles F → Λ defined by F ( g,p ) := T ∗ p M g for ( g, p ) ∈ Λ ( G ⋉ M ). Of course, thisdoes not define a (topological) vector bundle over the inertia space Λ ( G ⋉ M ) because in general therank jumps discontinuously but it is again a cosmooth generalized distribution. Using the canonicalprojection s ∗ T ∗ M | Λ → F we say that a local section ω ∈ Γ( U, V k F ) over U ⊂ Λ is smooth iffor each ( g, p ) ∈ U there exist open neighborhoods O ⊂ G of g and V ⊂ M of p together witha locally representing smooth k -form ω O,V ∈ Γ ∞ ( O × V, V k s ∗ T ∗ M ) such that ( O × V ) ∩ Λ ⊂ U and ω ( h,q ) = (cid:2) ω O,V (cid:3) ( h,q ) for all ( h, q ) ∈ ( O × V ) ∩ Λ ( G ⋉ M ). Hence a smooth section ω can beidentified with the smooth family ( ω g ) g ∈ pr G ( U ) of forms ω g ∈ Ω k (cid:16) s (cid:0) U ∩ ( { g } × M g ) (cid:1)(cid:17) wich areuniquely defined by the condition that ω g | V g = ι ∗ V g ω O,V for all g ∈ O and all pairs ( O, V ) withlocally representing forms ω O,V as before. The ι V g : V g ֒ → V hereby are the canonical embeddingsof the fixed point manifolds V g . We denote the space of all smooth sections of V k F over U byΓ ∞ ( U, V k F ) or Γ ∞ V k F ( U ). Obviously, Γ ∞ V k F becomes a sheaf on Λ . Proposition 4.1.
The canonical sheaf morphism θ k : ι − Γ ∞ V k s ∗ T ∗ M → Γ ∞ V k F factors through aunique epimorphism of sheaves Θ k : Ω • rel , Λ → Γ ∞ V k F making the following diagram commutative: ι − Γ ∞ V k s ∗ T ∗ M θ k / / (cid:15) (cid:15) Γ ∞ V k F Ω • rel , Λ Θ k ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ Proof.
The claim follows by showing that for open e U ⊂ G × M and U := e U ∩ Λ ( G ⋉ M ) thecanonical map θ k e U : Γ ∞ ( e U , V k s ∗ T ∗ M ) → Γ ∞ ( U, V k F ), ω [ ω ] is surjective and has K ( e U ) := J ( e U ) Γ ∞ ( e U , ^ k s ∗ T ∗ M ) + d rel J ( e U ) ∧ Γ ∞ ( e U , ^ k − s ∗ T ∗ M ) N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 17 contained in its kernel.The sheaf Γ ∞ V k F is a C ∞ Λ -module sheaf, hence a soft sheaf. This entails surjectivity of θ k e U . Assumethat ω ∈ Γ ∞ ( e U , V k s ∗ T ∗ M ) is of the form ω = f ̺ for some f ∈ J ( e U ) and ̺ ∈ Γ ∞ ( e U , V k s ∗ T ∗ M ).Then θ k e U ( ω ) ( g,p ) = θ kU ( f ̺ ) ( g,p ) = f ( q, p ) ̺ ( q,p ) = 0 for all ( g, p ) ∈ U .
Now assume ω = d rel f ∧ ̺ with f as before and ̺ ∈ Γ ∞ ( e U , V k − s ∗ T ∗ M ). To prove that θ kU ( ω ) = 0it suffices to show that ι ∗ U gg ω = 0 for all g ∈ pr G ( U ). Fix some g ∈ pr G ( U ) and p ∈ U gg and choose anopen coordinate neighborhood V ⊂ M with coordinates ( x , . . . , x d ) : V ֒ → R d such that V ⊂ U g ,( x | V g , . . . , x k | V g ) : V g ֒ → R k is a local coordinate system of M g over V g and such that V g is thezero locus of the coordinate functions ( x k +1 , . . . , x d ) : V ֒ → R d − k . After possibly shrinking V thereexists an open neighborhood O of g in G such that O × V ⊂ e U . Extend the coordinate functions( x , . . . , x d ) to smooth functions on O × V constant along the fibers of the source map. Then wehave d rel f = P dl =1 ∂f∂x l dx l . Since ∂f∂x l ( g, p ) = 0 for p ∈ V g and 1 ≤ l ≤ k and since ι ∗ V g dx l = 0 for k < l ≤ d one gets ι ∗ V g ι ∗ U gg ω = ι ∗ V g (cid:0) d rel f ∧ ̺ (cid:1) = d X l =1 (cid:16) ι ∗ V g ∂f∂x l (cid:17) (cid:0) ι ∗ V g dx l (cid:1) ∧ (cid:0) ι ∗ V g ̺ (cid:1) = 0 , where, by slight abuse of notation, we have also used the symbol ι V g for the embedding V g ֒ → U , p ( g, p ). So ι ∗ U gg ω = 0 and K ( e U ) is in the kernel of θ k e U . Hence θ k e U factors through some linear mapΘ kU : Ω k rel , Λ ( U ) → Γ ∞ ( U, ^ k F ) . This proves the claim. (cid:3)
Remark 4.2.
Conjecturally, the morphism Θ k is an isomorphism, showing that the sheaf theoreticapproach (A) and the differential geometric approach (B) above leads to the same definition of basicrelative forms. Below, in Section 6, we prove this conjecture for the case of an S -action. In thegeneral case this conjecture remains open.Note that the image of the sheaf of horizontal relative k -forms under Θ k coincides exactly withthose families of forms ( ω g ) g ∈ pr G ( U ) fulfilling condition (i) above. Since G naturally acts on thegeneralized distribution F and Θ k is obviously equivariant by construction, the original conditionsby Brylinski are recovered now also in the differential geometric picture of relative forms. Remark 4.3.
In [BG94], Block and Getzler define a sheaf on G whose stalk at g ∈ G is given bythe space of G g -equivariant differential forms on M g . There are two differentials on this sheaf, d and ι , together constituting the equivariant differential D := d + ι , which, under an HKR-type mapcorrespond to the Hochschild and cyclic differential on the crossed product algebra G ⋉ C ∞ ( M ).Taking cohomology with respect to ι only leads to a very similar definition of basic relative formsas above, however notice that the basic relative forms defined above form a sheaf over the quotient M/G , not the group G . 5. The group action case
In this section we consider the action of a compact Lie group G on a complete bornological algebra A and then specialize to the case where A is the algebra of smooth functions on a smooth G -manifold M . The general assumption hereby is always that the action α : G × A , ( g, a ) g · a is smoothin the sense of [KM97] that is if each smooth curve in G × A is mapped by α to a smooth curvein A . This is automatically guaranteed when G acts by diffeomorphisms on the manifold M and A = C ∞ ( M ). Under the assumptions made the associated smooth crossed product G ⋉ A is given by C ∞ ( G, A ) equipped with the product(5.1) ( f ∗ f )( g ) := Z G f ( h ) ( h · f ( h − g )) dh , f , f ∈ C ∞ ( G, A ) , g ∈ G .
The equivariant Hochschild complex.
To compute the Hochschild homology of the smoothcrossed product G ⋉ A , consider the bigraded vector space C = M p,q ≥ C p,q , with C p,q := C ∞ ( G ( p +1) , A ⊗ ( q +1) ) . There exists a bi-simplicial structure on C given by face maps δ vi : C p,q → C p,q − , 0 ≤ i ≤ q and δ hj : C p,q → C p − ,q , 0 ≤ j ≤ p defined as follows. The vertical maps are given by δ vi ( F )( g , . . . , g p ) := ( b i ( F ( g , . . . , g p )) for 0 ≤ i ≤ q − ,b ( g ··· g p ) − q ( F ( g , . . . , g p )) for i ≤ q, where the b i for 0 ≤ i ≤ q − q − i ’th and i + 1’thentry in A ⊗ ( q +1) underlying the Hochschild chain complex of A , and b gq is the g -twisted version ofthe last one: b gq ( a ⊗ . . . ⊗ a q ) := ( g · a q ) a ⊗ a ⊗ . . . ⊗ a q − , a , . . . , a q ∈ A, g ∈ G .
The horizontal maps are defined by δ hj ( F )( g , . . . , g p − ) := (R G F ( g , . . . , h, h − g j , . . . g p − ) dh for 0 ≤ j ≤ p − , R G h · F ( h − g , g , . . . , g p − , h ) dh for j = p, where, in the second line, h acts diagonally on A ⊗ ( q +1) . The following observations now hold true.( i ). The diagonal complex diag( C • , • ) := L k ≥ C k,k equipped with the differential d diag := X i ( − i δ hi δ vi is isomorphic to the Hochschild complex C k ( G ⋉ A ) = C ∞ (cid:0) G ( k +1) , A ⊗ ( k +1) (cid:1) of the smooth crossedproduct algebra G ⋉ A via the isomorphism : diag( C • , • ) → C • ( G ⋉ A )), F F defined by(5.2) F ( g , . . . , g k ) := ( g − k · · · g − ⊗ g − k · · · g − ⊗ . . . ⊗ g − k ) · F ( g , . . . , g k ) , F ∈ C k,k , where the pre-factor on the right hand side acts componentwise via the action of G on A .( ii ). The vertical differential δ v in the total complex is given by a twisted version of the standardHochschild complex of the algebra A . The horizontal differential δ h in the q -th row can be interpretedas the Hochschild differential of the convolution algebra C ∞ ( G ) with values in the G -bimodule C ∞ ( G, A ⊗ ( q +1) ) with bimodule structure( g · f )( h ) := g ( f ( g − h )) , ( f · g )( h ) := f ( hg ) , f ∈ C ∞ ( G, A ⊗ ( k +1) ) , g, h ∈ G .
The homology of this complex is isomorphic to the group homology of G with values in the adjointmodule C ∞ ( G, A ⊗ ( k +1) ) ad given by C ∞ ( G, A ⊗ ( k +1) ) equipped with the diagonal action: H • (cid:0) C ∞ ( G ) , C ∞ ( G, A ⊗ ( q +1) ) (cid:1) ∼ = H diff • (cid:0) G, C ∞ ( G, A ⊗ ( q +1) ) ad (cid:1) . Because G is a compact Lie group, its group homology vanishes except for the zeroth degree: H diff k (cid:0) G, C ∞ ( G, A ⊗ ( k +1) ) ad (cid:1) = ( C ∞ ( G, A ⊗ ( k +1) ) invad for k = 0 , k > . N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 19 ( iii ). Filtering the total complex by rows, we obtain a spectral sequence with E -terms E ,q ∼ = C ∞ ( G, A ⊗ ( q +1) ) inv , E p,q = 0 for p ≥ . The spectral sequence therefore collapses and the cohomology of the total complex is computed bythe complex C G • ( A ) := C ∞ ( G, A ⊗ ( • +1) ) inv equipped with the twisted Hochschild differential( b tw f )( g ) := q X i =0 ( − i b i ( f ( g )) + ( − q +1 b g − q +1 ( f ( g )) , f ∈ C ∞ ( G, A ⊗ ( q +1) ) , g ∈ G .
This complex is called the equivariant Hochschild complex in [BG94].( iv ). By the Eilenberg–Zilber theorem, the diagonal complex is quasi-isomorphic to the total complexTot( C • , • ) with δ Tot := δ h + δ v where the horizontal and vertical differentials are given by theusual formulas δ h,v := P i ( − i δ h,vi . There is an explicit formula for the map EZ : diag( C • , • ) → Tot( C • , • ) implementing this quasi-isomorphism. Combining items ( i ) − ( iv ) above we conclude thatthe following holds. Proposition 5.1.
Given a complete bornological algebra A with a smooth left G -action, the compo-sition e : C • ( G ⋉ A ) −→ diag( C ) • EZ −→ Tot( C • , • ) −→ C G • ( A ) is a quasi-isomorphism of complexes. The explicit formula is given by mapping a chain F ∈ C k ( C ∞ ( G, A )) to the equivariant Hochschild chain e F ∈ C Gk ( A ) defined by e F ( g ) := Z G k ( g − h · · · h k ⊗ ⊗ h ⊗ . . . ⊗ h · · · h k − ) F ( h − k · · · h − g, h , . . . , h k ) dh · · · dh k . Remark 5.2.
This result has originally been proved by Brylinski in [Bry87a, Bry87b]. Observethat a right G -action β on an algebra A can be changed to a left G -action α on an algebra A by α ( g )( a ) := β ( g − )( a ). Let A op be the opposite algebra of A and assume that β defines a right G action on A op . Use A op ⋊ β G to denote the (right) crossed product algebra defined by the right G action on A op . Define a map Φ : G α ⋉ A → A op ⋊ β G by Φ( f )( g ) := f ( g − ). One directly checks thefollowing identity, Φ( f ∗ G α ⋉ A f ) = Φ( f ) ∗ A op ⋊ β G Φ( f ) , and concludes that the map Φ induces an isomorphism of algebras G α ⋉ A ∼ = (cid:0) A op ⋊ β G (cid:1) op . Furthermore notice that for a general algebra A , the algebra A ⊗ A op is naturally isomorphic to A op ⊗ A and therefore HH • ( A ) ∼ = HH • ( A op ) since the corresponding Bar resolutions coincide.Applying this observation to (cid:0) A op ⋊ β G (cid:1) op , one concludes that HH • ( G α ⋉ A ) ∼ = HH • (cid:0) A op ⋊ β G (cid:1) , and that Proposition 5.1 holds also true for a smooth right G -action on an algebra A meaning thatthere is a quasi-isomorphism of chain complexes b : C • ( A ⋊ G ) −→ C G • ( A op ) . Note that for a right G -action the convolution product on C ∞ ( G, A ) is given by(5.3) ( f ∗ f )( g ) := Z G ( f ( h ) · ( h − g )) f ( h − g ) dh , f , f ∈ C ∞ ( G, A ) , g ∈ G .
Throughout this paper, as it is more natural to have a left G -action on a manifold M , we will workwith a right G -action on C ∞ ( M ). The G -manifold case. Let M be a manifold endowed with a smooth left G -action. Denoteby X = M/G the space of G -orbits in M and by π : M → X the canonical projection. We considerthe action groupoid G = G ⋉ M ⇒ M and the corresponding convolution sheaf A = A G ⋉ M over X . It is straightforward to check that in the case of A = C ∞ ( M ) the product defined by Eq. (5.3)coincides with the convolution product on A ( M/G ) ∼ = C ∞ ( G ⋉ M ) ∼ = C ∞ ( G, A ) given by Eq. (1.3).Hence A ( M/G ) coincides with A ⋊ G . According to Proposition 5.1 and Remark 5.2, we then havefor each G -invariant open V ⊂ M a quasi-isomorphism between Hochschild chain complexes b | V/G : C • (cid:0) A ( V /G ) (cid:1) → C G • (cid:0) C ∞ ( V ) (cid:1) ∼ = C • (cid:0) C ∞ ( V ) , A ( V /G ) (cid:1) . To compute the Hochschild homology HH • (cid:0) A ( V /G ) (cid:1) it therefore suffices to determine the homologyof the complex C • (cid:0) C ∞ ( V ) , A ( V /G ) (cid:1) which we will consider in the following. Recall that A ( V /G ) isisomorphic as a bornological vector space to the completed tensor product C ∞ ( G ) ˆ ⊗C ∞ ( V ) and that A ( V /G ) carries the (twisted) C ∞ ( V )-bimodule structure C ∞ ( V ) ˆ ⊗A ( V /G ) ˆ ⊗C ∞ ( V ) → A ( V /G ) , f ⊗ a ⊗ f ′ (cid:16) G × V ∋ ( g, v ) f ( gv ) a ( g, v ) f ′ ( v ) ∈ R (cid:17) . Since the bimodule structure is compatible with restrictions r UV for G -invariant open subsets V ⊂ U ⊂ M one obtains a complex of presheaves which assigns to every open V /G with V ⊂ M open and G -invariant the complex C • ( C ∞ ( V ) , A ( V /G )). Sheafification gives rise to a sheaf complex which wedenote by ˆ C • (cid:0) C ∞ M , A (cid:1) . Since C • (cid:0) C ∞ ( V ) , A ( V /G ) (cid:1) ∼ = A ( V /G ) ˆ ⊗ C • ( C ∞ ( V )) for all G -invariant open V ⊂ M , this sheaf complex can be written asˆ C • (cid:0) C ∞ M , A (cid:1) = A ˆ ⊗ π ∗ ˆ C • (cid:0) C ∞ M (cid:1) , where, as before, ˆ C • (cid:0) C ∞ M (cid:1) denotes the Hochschild sheaf complex of C ∞ M . We now have the followingresult. Proposition 5.3.
Assume to be given a G -manifold M , let A be the convolution sheaf of theassociated action groupoid G ⋉ M ⇒ M on the orbit space X = M/G , and put A = A ( X ) . Thenthe chain map ̺ : C • (cid:0) C ∞ ( M ) , A (cid:1) → Γ (cid:0) X, ˆ C • ( C ∞ M , A ) (cid:1) , c ([ c ] O ) O ∈ X which associates to every k -chain c ∈ C k (cid:0) C ∞ ( M ) , A (cid:1) the section ([ c ] O ) O ∈ X , where [ c ] O denotes thegerm of c in the stalk ˆ C • , O ( C ∞ M , A ) , is a quasi-isomorphism.Proof. Observe that the sheaves ˆ C k ( C ∞ M , A ) are fine and that ̺ : C ( C ∞ ( M ) , A ) → Γ (cid:0) X, ˆ C ( C ∞ M , A ) (cid:1) is the identity morphism. Using again the homotopies from Section 2.3, the proof is completelyanalogous to the one of Proposition 2.7, hence we skip the details. (cid:3) Next, we compare the sheaf complex ˆ C • ( C ∞ M , A ) (cid:1) with the complex of relative forms by construct-ing a morphism of sheaf complexes between them. Proposition 5.4.
Under the assumptions of the preceding proposition define for each open G -invariant subset V ⊂ M and k ∈ N a C ∞ ( V /G ) -module map by Φ k,V/G : C k (cid:0) C ∞ ( V ) , A ( V /G ) (cid:1) ∼ = A ( V /G ) ˆ ⊗ C k (cid:0) C ∞ ( V ) (cid:1) → Ω k rel , Λ (cid:0) Λ ( G ⋉ V ) (cid:1) ,f ⊗ f ⊗ . . . ⊗ f k (cid:2) f d ( s ∗ G ⋉ V f ) ∧ . . . ∧ d ( s ∗ G ⋉ V f k ) (cid:3) Λ . Then the Φ k,V/G are the components of a morphism of sheaf complexes Φ • : ˆ C • ( C ∞ M , A ) (cid:1) → π ∗ ( s | Λ ) ∗ Ω • rel , Λ , where the differential on Ω • rel , Λ is given by the zero differential. The image of a cycle under Φ • liesin the sheaf complex of horizontal relative forms Ω • hrel , Λ . N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 21
Proof.
Let f ∈ A ( V /G ) and f , . . . , f k ∈ C ∞ ( V ). Observe first that Φ k,V/G ( f ⊗ f ⊗ . . . ⊗ f k ) isa relative form indeed since d ( s ∗ G ⋉ V f ) ∈ Ω G ⋉ V → G ( G ⋉ M ) for all f ∈ C ∞ ( V ). Now let ( g, p ) ∈ Λ ( G ⋉ V ) and compute:Φ k − ,V/G b ( f ⊗ f ⊗ . . . ⊗ f k )( g, p ) = f ( g, p ) f ( p ) (cid:2) d ( s ∗ G ⋉ V f ) ∧ . . . ∧ d ( s ∗ G ⋉ V f k ) (cid:3) ( g,p ) ++ k − X i =1 ( − i f ( g, p ) f i ( p ) (cid:2) d ( s ∗ G ⋉ V f ) ∧ . . . ∧ d ( s ∗ G ⋉ V f i − ) ∧ d ( s ∗ G ⋉ V f i +1 ) ∧ . . . ∧ d ( s ∗ G ⋉ V f k ) (cid:3) ( g,p ) ++ k − X i =1 ( − i f ( g, p ) f i +1 ( p ) (cid:2) d ( s ∗ G ⋉ V f ) ∧ . . . ∧ d ( s ∗ G ⋉ V f i ) ∧ d ( s ∗ G ⋉ V f i +2 ) ∧ . . . ∧ d ( s ∗ G ⋉ V f k ) (cid:3) ( g,p ) ++ ( − k f k ( gp ) f ( g, p ) (cid:2) d ( s ∗ G ⋉ V f ) ∧ . . . ∧ d ( s ∗ G ⋉ V f k − ) (cid:3) ( g,p ) = 0 . Hence Φ • ,V/G is a chain map in the sense that it intertwines the Hochschild boundary with the zerodifferential.It remains to show that the image of Φ • ,V/G is in the space of horizontal relative forms. To thisend assume for a moment that V is a G -invariant open ball around the origin in some euclideanspace R n which is assumed to carry an orthogonal G -action. Consider the Connes–Koszul resolutionof C ∞ ( V ) provided in (B.2). A chain map between the Connes–Koszul resolution and the Barresolution of C ∞ ( V ) over the identity map id C ∞ ( V ) in degree 0 is given by the family of mapsΨ k,V : Γ ∞ ( V × V, E k ) → B k (cid:0) C ∞ ( V ) (cid:1) = C ∞ ( V × V ) ˆ ⊗C ∞ ( V k ) ,ω (cid:16) ( v, w, x , . . . , x k ) ω ( v,w ) (cid:0) Y ( x , w ) , . . . , Y ( x k , w ) (cid:1)(cid:17) . Tensoring the Connes–Koszul resolution of C ∞ ( V ) with A ∞ ( V /G ) results in the following complex:(5.4) Ω dG ⋉ V → G ( V ) i YG ⋉ V −→ . . . i YG ⋉ V −→ Ω G ⋉ V → G ( V ) i YG ⋉ V −→ C ∞ ( G ⋉ V ) −→ , where Y G ⋉ V : G ⋉ V → s ∗ T V is defined by Y G ⋉ V ( g, v ) = v − gv . The composition of id A ∞ ( V/G ) ˆ ⊗ Ψ k,V with Φ k,V/G then is the map which associates to each relative form ω ∈ Ω kG ⋉ V → G ( V ) its restriction[ ω ] Λ to the loop space. It therefore suffices to show that for ω ∈ Ω kG ⋉ V → G ( V ) with i Y G ⋉ V ω = 0 therestriction to the loop space is a horizontal relative form. To verify this let ξ be an element of theLie algebra g of G and again ( g, v ) ∈ Λ ( G ⋉ V ). Then0 = ddt (cid:0) i Y G ⋉ V ω (cid:1) ( e − tξ g,v ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = (cid:0) − i Y G ⋉ V i ξ G d G ω + i ξ V ω (cid:1) ( g,v ) = (cid:0) i ξ V ω (cid:1) ( g,v ) , where d G denotes the exterior differential with respect to G and ξ G and ξ V are the fundamentalvector fields of ξ on G and V , respectively. So i ξ V ω ∈ J ( V )Ω k − G ⋉ V → G ( V ), which means that[ ω ] Λ ∈ Ω k hrel , Λ ( G ⋉ V ). (cid:3) Proposition 5.5.
Let M be a G -manifold with only one isotropy type and assume that the orbitspace M/G is connected. Then the following holds true. (1)
The quotient space
M/G carries a unique structure of a smooth manifold such that π : M → M/G is a submersion. (2)
The loop space Λ ( G ⋉ M ) is a smooth submanifold of G × M . (3) Let p ∈ M be a point and V p ⊂ M a slice to the orbit through p that is (SL1) V p is a G p -invariant submanifold which is transverse to the orbit O p := Gp at p , (SL2) V := GV p is an open neighborhood of the orbit O p and V p is closed in V , (SL3) there exists a G -equivariant diffeomorphism η : N O p → V mapping the normal space N p = T p M/T p O p onto V p . Then for every k the map Ψ k,V p /G p : Ω k brel , Λ (cid:0) Λ ( G ⋉ GV p ) (cid:1) → Ω k brel , Λ (cid:0) Λ ( G p ⋉ V p ) (cid:1) ω ω | Λ ( G p ⋉ V p ) is an isomorphism and the space of basic relative k -forms Ω k brel , Λ (cid:0) Λ ( G p ⋉ V p ) (cid:1) coincides nat-urally with C ∞ ( G p ) G p ˆ ⊗ Ω k ( V p ) . (4) The chain map Φ • ,M/G : C • (cid:0) C ∞ ( M ) , A ( M/G ) (cid:1) → Ω • hrel , Λ (cid:0) Λ ( G ⋉ M ) (cid:1) is a quasi-isomorphism when the graded module Ω • hrel , Λ (cid:0) Λ ( G ⋉ M ) (cid:1) is endowed with the zerodifferential.Proof. ad (1). It is a well known result about group actions on manifolds that under the assumptionsmade the quotient space M/G carries a unique manifold structure such that π : M → M/G is asubmersion; see e.g. [Bre72, Sec. IV.3] or [Pfl01, Thm. 4.3.10]. ad (2). This has been proved in [FPS15, Prop. 4.4]. Let us outline the argument since we need itfor the following claims, too. By the assumptions made there exists a compact subgroup K ⊂ G such that every point of M has isotropy type ( K ). Let p ∈ M be a point and G p its isotropy group.Without loss of generality we can assume that G p = K . Let V p ⊂ M be a slice to the orbit O through p . The isotropy group of an element q ∈ V p then has to coincide with K , so V Kp = V p .Therefore the map τ : G/K × V p → M, ( gK, q ) gq is a G -equivariant diffeomorphism onto a neighborhood of O . Now choose a small enough openneigborhood of eK in G/K and a smooth section σ : U → G of the fiber bundle G → G/K . Themap e τ : G × U × V p → G × τ ( U × V p ) , ( h, gK, q ) (cid:0) σ ( gK ) hσ ( gK ) − , σ ( gK ) q (cid:1) then is a diffeomorphism onto the open set G × τ ( U × V p ) of G × M . One observes that e τ ( K × U × V p ) = ( G × τ ( U × V p )) ∩ Λ ( G ⋉ M ) , which shows that Λ ( G ⋉ M ) is a submanifold of G × M , indeed. ad (3). Put K = G p as before, let N = GV p , and denote by g and k the Lie algebras of G and K ,respectively. Choose an Ad-invariant inner product on g and let m be the orthogonal complementof k in g . Next choose for each q ∈ N an element h q ∈ G such that h q q ∈ V p . Then π N : N → O p , q h − q p is an equivariant fiber bundle. Let T N → N be the tangent bundle of the total space and V N → N the vertical bundle. Note that T N and
V N inherit from N the equivariant bundle structures. Nowput for q ∈ N H q N := span n(cid:16) Ad h − q ( ξ ) (cid:17) N ( q ) ∈ T q N | ξ ∈ m o , where ξ N denotes the fundamental vector field of ξ on N . Then HN → N becomes an equivariantvector bundle complementary to V N → N . Let P v : T N → V N be the corresponding fiberwiseprojection along HN . By construction, P v is G -equivariant. After these preliminary considerationslet ω ∈ Ω k brel , Λ (cid:0) Λ ( G ⋉ GV p ) (cid:1) . The restriction ω | Λ ( K ⋉ V p ) then is a basic relative form again, soΨ k,V p /K is well defined. Let us show that it is surjective. Assume that ̺ ∈ Ω k brel , Λ (cid:0) Λ ( K ⋉ V p ) (cid:1) .We then put for ( g, q ) ∈ Λ ( G ⋉ N ) and X , . . . , X k ∈ T q N (5.5) ω ( g,q ) ( X , . . . , X k ) := ̺ ( h q gh − q ,h q q ) (cid:0) T h q ( P v ( X )) , . . . , T h q ( P v ( X k )) (cid:1) , where T h : T N → T N for h ∈ G denotes the derivative of the action of h on N . Since T k for k ∈ K acts as identity on T V p ⊂ V N , the value ω ( g,q ) ( X , . . . , X k ) does not depend on theparticular choice of a group element h q such that h q q ∈ V p . Moreover, since for fixed q ∈ N one canfind a small enough neighborhood U and choose h q to depend smoothly on q ∈ U , ω is actually asmooth differential form on N . By construction, it is a relative form. If X l ∈ H q N for some l , then N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 23 ω ( g,q ) ( X , . . . , X k ) = 0 by definition. If X l = (cid:0) Ad h − q ( ξ ) (cid:1) N ( q ) for some ξ ∈ k , then P v X l = X l and T h q X l ( q ) = ξ N ( h q q ) which entails by (5.5) that ω ( g,q ) ( X , . . . , X k ) = 0 again since ̺ is a horizontalform. So ω is a horizontal form. It remains to show that it is G -invariant. Let h ∈ G and ( g, q ) and X , . . . , X k as before. Then ω ( hgh − ,hq ) ( T hX , . . . , T hX k ) = ̺ ( h q gh − q ,h q q ) (cid:0) T h q T h − ( P v ( T hX )) , . . . , T h q T h − ( P v ( T hX k )) (cid:1) = ω ( g,q ) ( X , . . . , X k ) , so ω is G -invariant and therefore a basic relative form. Hence Ψ k,V p /K is surjective. To proveinjectivity of Ψ k,V p /K observe that if ω ∈ Ω k brel , Λ (cid:0) Λ ( G ⋉ GV p ) (cid:1) and ̺ is the restriction ω | Λ ( K ⋉ V p ) ,then Eqn. (5.5) holds true since ω is G -invariant and horizontal. But this implies that if ω | Λ ( K ⋉ V p ) =0, then ω must be 0 as well, so Ψ k,V p /K is injective. It remains to showΩ k brel , Λ (cid:0) Λ ( K ⋉ V p ) (cid:1) ∼ = C ∞ ( K ) K ˆ ⊗ Ω k ( V p ) . To this end observe that Λ ( K ⋉ V p ) = K × V p since V Kp = V p which in other words means thatvery K -orbit in V p is a singleton. The claim now follows immediately. ad (4). By Theorem 2.3 it suffices to verify the claim for the case where M = GV p , where p is a pointand V p a slice to the orbit O through p . As before let K be the isotropy G p . By the slice theoremthere exists a K -equivariant diffeomorphism ϕ : V p → e V p ⊂ N p O onto an open zero neighborhood ofthe normal space N p O . Choose a K -invariant inner product on N p O and a G -invariant inner producton the Lie algebra g . Again as before let m be the orthogonal complement of the Lie algebra k in g . The inner product on g induces a G -invariant riemannian metric on G which then induces a G -invariant riemannian metric on the homogeneous space G/K by the requirement that G → G/K isa riemannian submersion. Now observe that the map
G/K × V p → M , ( gK, v ) gv is a G -invariantdiffeomorphism, so we can identify M with G/K × V p . The chosen riemannian metrics on G/K and V p then induce a G -invariant metric on M . Since C is faithfully flat over R we can assume withoutloss of generality now that smooth functions and forms on M and G ⋉ M are all complex valued,including elements of the convolution algebra. Let e ∈ N p O ∼ = T p V p be a vector of unit length, andlet Z be the vector field on M which maps every point to e (along the canonical parallel transport).Next choose a symmetric open neighborhood U of the diagonal of G/K × G/K such that for eachpair ( gK, hK ) ∈ U there is a unique ξ ∈ Ad h ( m ) such that gK = exp( ξ ) hK . Denote that ξ byexp − hK ( gK ). Let χ : G/K × G/K → [0 ,
1] be a function with support contained in U and such that χ = 1 on a neighborhood of the diagonal. Now define the vector field Y : M × M → pr ∗ ( T M ) by Y (cid:0) ( gK, v ) , ( hK, w ) (cid:1) = χ ( gK, hK ) (cid:0) exp − hK ( gK ) , v − w (cid:1) + √− χ ′ ( gK, hK ) Z (cid:0) ( gK, v ) , ( hK, w ) (cid:1) , where pr : M × M → M is projection onto the second coordinate and where the smooth cut-offfunction χ ′ : G/K × G/K → [0 ,
1] vanishes on a neighborhood of the diagonal and is identical 1 onthe locus where χ = 1. Finally put E k := pr ∗ ( V k T ∗ M ). Then, by [Con85, Lemma 44], the complexΓ ∞ ( M × M, E dim M ) i Y −→ . . . i Y −→ Γ ∞ ( M × M, E ) i Y −→ C ∞ ( M × M ) −→ C ∞ ( M )is a (topologically) projective resolution of C ∞ ( M ) as a C ∞ ( M )-bimodule. Tensoring this resolutionwith the convolution algebra A ( G ⋉ M ) gives the following complex of relative forms:(5.6) Ω dim MG ⋉ M → G ( G ⋉ M ) i YG −→ . . . i YG −→ Ω G ⋉ M → G ( G ⋉ M ) −→ C ∞ ( G ⋉ M ) , where Y G : G × M → pr ∗ T M is the vector field( g, ( hK, v )) χ ( ghK, hK ) (cid:0) exp − hK ( ghK ) , (cid:1) + √− χ ′ ( ghK, hK ) Z (cid:0) ( ghK, v ) , ( hK, v ) (cid:1) . The vector field Y G vanishes on ( g, ( hK, v )) if and only if g ∈ hKh − that is if and only if( g, ( hK, v )) ∈ Λ ( G ⋉ M ). We will use the parametric Koszul resolution Proposition B.8 to showthat the complex (5.6) is quasi-isomorphic to the complex of horizontal relative forms(5.7) Ω dim M hrel , Λ (Λ ( G ⋉ M )) −→ . . . −→ Ω , Λ (Λ ( G ⋉ M )) −→ C ∞ (Λ ( G ⋉ M )) . This will then entail the claim. So it remains to show that (5.6) and (5.7) are quasi-isomorphic. Wefirst consider the case where V p consist just of a point. Then M coincides with the homogeneousspace G/K and Y G is an Euler-like vector field on its set of zeros S = { ( g, hK ) ∈ G × G/K | g ∈ hKh − } ⊂ M .
Note that S is a submanifold on M . That Y G is Euler-like on S indeed follows from the equality ddt exp − hK (cid:0) exp( tξ ) ghK (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) t =0 = ddt exp − hK (cid:0) exp( tξ ) hK (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) t =0 = ξ for all ( g, hK ) ∈ S , ξ ∈ Ad gh ( m ) = Ad h ( m ). Hence, by Proposition B.8, the complexΩ dim G/KG ⋉ G/K → G ( G ⋉ G/K ) i YG −→ . . . i YG −→ Ω G ⋉ G/K → G ( G ⋉ G/K ) −→ C ∞ ( G ⋉ G/K )is quasi-isomorphic to 0 −→ . . . −→ −→ C ∞ ( S ) . Since Ω k hrel , Λ (Λ ( G ⋉ G/K )) = 0 for k ≥
1, the claim follows in the case V p = { p } . Now considerthe case M = G/K × V p with V p an arbitrary manifold on which K acts trivially. Observe that inthis situation Ω kG ⋉ M → G ( G ⋉ M ) ∼ = M ≤ l ≤ k Ω lG ⋉ G/K → G ( G ⋉ G/K ) ˆ ⊗ Ω k − l ( V p )and that Y G acts, near its zero set S = Λ ( G ⋉ M ), only on the first componentsΩ lG ⋉ G/K → G ( G ⋉ G/K ) . Hence the chain complex (5.6) is then quasi-isomorphic to the chain complex C ∞ (Λ ( G ⋉ G/K )) ˆ ⊗ Ω • ( V p )with zero differntial. But sinceΩ k hrel , Λ (Λ ( G ⋉ M )) ∼ = C ∞ (Λ ( G ⋉ G/K )) ˆ ⊗ Ω k ( V p )the claim is now proved. (cid:3) Conjecture 5.6 (Brylinski [Bry87a, Prop. 3.4]& [Bry87b, p. 24, Prop.]) . Let M be G -manifold andregard Ω • hrel , Λ (cid:0) Λ ( G ⋉ M ) (cid:1) as a chain complex endowed with the zero differential. Then the chainmap Φ • ,M/G : C • (cid:0) C ∞ ( M ) , A ( M/G ) (cid:1) → Ω • hrel , Λ (cid:0) Λ ( G ⋉ M ) (cid:1) is a quasi-isomorphism. Remark 5.7.
Proposition 5.5 shows that Brylinski’s conjecture holds true for G -manifolds havingonly one isotropy type. Corollary B.6 tells that Brylinski’s conjecture is true for finite group actions.In the following section we will verify it for circle actions.6. The circle action case
Rotation in a plane.
Let us consider the case of the natural S -action on R by rotation.First we describe the ideal sheaf J ⊂ C ∞ S ⋉R which consists of smooth functions on open sets of S × R vanishing on Λ ( S ⋉ R ). To this end denote by x j : S × R → R , j = 1 ,
2, the projectiononto the first respectively second coordinate of R and by τ : S \{− }× R → ( − π, π ) the coordinatemap ( g, v ) Arg( g ). By r = p x + x we denote the radial coordinate and by B ̺ ( v ) the open discof radius ̺ > v ∈ R . Note that the loop space Λ ( S ⋉ R ) is the disjoint union ofthe strata { (1 , } , { } × ( R \ { } ), and ( S \ { } ) × { } and that the loop space is smooth outsidethe singular point (1 , N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 25
Proposition 6.1.
Around the point (1 , , the vanishing ideal J (cid:0) ( S \ {− } ) × B ̺ (0) (cid:1) consists ofall smooth f : ( S \ {− } ) × B ̺ (0) → R which can be written in the form (6.1) f = f τ x + f τ x , where f , f ∈ C ∞ (cid:0) ( S \ {− } ) × B ̺ (0) (cid:1) . Around the stratum { } × ( R \ { } ) , a function f ∈ C ∞ (cid:0) ( S \ {− } ) × ( R \ { } ) (cid:1) lies in the ideal J (cid:0) ( S \{− } ) × ( R \{ } ) (cid:1) if and only if f is of the form hτ for some h ∈ C ∞ (cid:0) ( S \{− } ) × ( R \{ } ) (cid:1) .Finally, around the stratum ( S \ { } ) × { } , a function f ∈ C ∞ (cid:0) ( S \ { } ) × R (cid:1) vanishes on Λ ( S ⋉ R ) if and only if it is of the form f x + f x with f , f ∈ C ∞ (cid:0) ( S \ { } ) × R (cid:1) .Proof. Since the loop space is smooth at points of the strata { } × ( R \ { } ) and ( S \ { } ) × { } ,only the case where f is defined on a neighborhood of the singular point (1 ,
0) is non-trivial. Solet us assume that f ∈ C ∞ (cid:0) ( S \ {− } ) × B ̺ (0) (cid:1) vanishes on Λ ( S ⋉ R ). Using the coordinatefunctions we can consider f as a function of t ∈ ( − π, π ) and x ∈ R . By the Malgrange preparationtheorem one then has an expansion f ( t, x ) + t = c ( t, x )( t + a ( x )) , where c and a are smooth and a (0) = 0. Since t = c ( t, t for all t ∈ ( − π, π ), one has c ( t,
0) = 1.Putting t = 0 gives 0 = c (0 , x ) a ( x ) for all x ∈ B ̺ (0). Since c (0 ,
0) = 1, one obtains a ( x ) = 0 forall x in a neighborhood of the origin. After possibly shrinking B ̺ (0) we can assume that a = 0.Hence(6.2) f ( t, x ) = ( c ( t, x ) − t . Parametric Taylor expansion of c ( t, x ) − c ( t, x ) − x r ( t, x ) + x r ( t, x ) , where r j ( t, x ) = Z (1 − s ) ∂ j c ( t, sx ) ds, j = 1 , . Since the functions r j are smooth, this expansion together with (6.2) entails (6.1). (cid:3) Lemma 6.2.
The vector fields Y = Y S ⋉R : S × R → R , ( g, x ) x − gx and Z = Z S ⋉R : S × R → R , ( g, x ) x + gx have coordinate representations Y = Y ∂∂x + Y ∂∂x and Z = Z ∂∂x + Z ∂∂x with coefficients givenby Y = x (1 − cos τ ) − x sin τ and Y = x (1 − cos τ ) + x sin τ (6.3) respectively by Z = x (1 + cos τ ) + x sin τ and Z = x (1 + cos τ ) − x sin τ . (6.4) Moreover, the vector fields Y and Z have square norms (6.5) k Y k = 2 r (1 − cos τ ) = r τ ( ξ ◦ τ ) and k Z k = 2 r (1 + cos τ ) , where ξ is holomorphic with positive values over ( − π, π ) and value at the origin.Proof. The representations Y | ( S \{− } ) × R = ( x (1 − cos τ ) − x sin τ ) ∂∂x + ( x (1 − cos τ ) + x sin τ ) ∂∂x and Z | ( S \{− } ) × R = ( x (1 + cos τ ) + x sin τ ) ∂∂x + ( x (1 + cos τ ) − x sin τ ) ∂∂x are immediate by definition of Y and Z and since S acts by rotation. Note that these formulas stillhold true when extending τ to the whole circle by putting τ ( −
1) = π . At g = − τ is not continuous then, but compositions with the trigonometric functions cos and sin are smooth on S . For the norms of Y and Z one now obtains k Y k = x (1 − cos τ ) + x sin τ + x (1 − cos τ ) + x sin τ = 2 r (1 − cos τ )and k Z k = x (1 + cos τ ) + x sin τ + x (1 + cos τ ) + x sin τ = 2 r (1 + cos τ ) . By power series expansion of 1 − cos t one obtains the statement about ξ . (cid:3) Lemma 6.3.
For all open subsets U of the loop space Λ = Λ ( S ⋉ R ) and all k ∈ N the map Θ kU : Ω k rel , Λ ( U ) → Γ ∞ ( U, ∧ k F ) from Prop. 4.1 is injective.Proof. Since Ω , Λ ( U ) = C ∞ ( U ) = Γ ∞ ( U, ∧ F ) and Θ U = id, we only need to prove the claimfor k ≥
1. To this end we have to show that for ω ∈ Γ ∞ ( e U , ∧ k s ∗ T ∗ M ) with [ ω ] F = 0 the relation[ ω ] Λ = 0 holds true. Here, as before, e U ⊂ S × R is an open subset such that U = e U ∩ Λ ( S ⋉ R ).In other words we have to show that each such ω has the form ω = X l ∈ L f l ω l + X j ∈ J d rel h j ∧ η j , where L, J are finite index sets, f l , h j ∈ J ( e U ), ω l ∈ Γ ∞ ( e U , ∧ k s ∗ T ∗ M ), and η j ∈ Γ ∞ ( e U , ∧ k − s ∗ T ∗ M ).Since the involved sheaves are fine, we need to show the claim only locally. So let ( g, v ) ∈ Λ ( S ⋉R ).Choose ̺ > ε > ε < π such that 0 / ∈ B ̺ ( v ) if v = 0 and such that e √− t g = 1 for all t with | t | < ε if g = 1. Let e U = n ( e √− t g, w ) ∈ S × R | | t | < ε & k v − w k < ̺ o . Using the coordinate maps τ, x , x we now consider three cases.
1. Case: g = 1 and v = 0. Then F (1 ,w ) = T ∗ w R , hence ω (1 ,w ) = 0 for all w such that (1 , w ) ∈ e U ∩ Λ . Hence ω = τ X ≤ i <...
2. Case: g = 1 and v = 0. Then F ( h, = 0 for all h ∈ S with ( h, ∈ e U ∩ Λ . Hence ω can beany k -form on e U . But over e U one has x , x ∈ J ( e U ) which entails that ω = X ≤ i <...
3. Case: g = 1 and v = 0. Then F (1 ,w ) = T ∗ R for all w such that (1 , w ) ∈ e U ∩ Λ . Hence ω = τ X ≤ i <...
Lemma 6.4.
For every S -invariant open V ⊂ R the restriction morphism [ − ] Λ : Ω • S ⋉ V → S ( S ⋉ V ) → Ω • rel , Λ (cid:0) Λ ( S ⋉ V ) (cid:1) maps the space of cycles Z k (cid:0) Ω • S ⋉ V → S ( S ⋉ V ) , Y y (cid:1) onto the space Ω k hrel , Λ (cid:0) Λ ( S ⋉ V ) (cid:1) of hori-zontal relative forms.Proof. Since the sheaf Ω • hrel , Λ is fine it suffices to verify this claim for V ⊂ R of the form V = B ̺ (0)or V = B ̺ (0) \ B σ (0) where 0 < σ < ̺ . So assume that 1 ≤ k ≤ ω ] Λ ∈ Ω k hrel , Λ (cid:0) Λ ( S ⋉ V ) (cid:1) for some relative form ω ∈ Ω kS ⋉ V → S ( S ⋉ V ). Now observe that N ∗ v = R dr for all v ∈ R \{ } where dr = r ( dx + dx ). Hence, ω | { }× V = 0 if k = 2 and ω | { }× ( V \{ } ) = ϕ dr with ϕ ∈ C ∞ ( V \ { } ) if k = 1. Since the claim for k = 2 therefore has been proved, we assume from now on that k = 1. Incartesian coordinates, ω = ω dx + ω dx with ω j ∈ C ∞ (cid:0) S × ( V \ { } ) (cid:1) , j = 1 ,
2. Comparing withthe expansion in polar coordinates gives the following equality over V \ { } (6.6) ω j (1 , − ) = ϕr x i for j = 1 , . Note that if the origin is an element of V , then ω (1 , = 0, hence ( ω j ) (1 , = 0, j = 1 ,
2. Choosea smooth cut-off function χ : S → [0 ,
1] such that χ is identical 1 on a neighborhood of 1 andidentical 0 on a neighborhood of −
1. Now define the k -form b ω ∈ Ω k ( S × V ) by b ω ( g,x ) = ( χ ( g ) ϕ ( x ) k Z ( g,x ) k h Z ( g, x ) , −i : R → R for g ∈ supp χ and x ∈ V \ { } , g ∈ S \ supp χ or x ∈ V ∩ { } . where h− , −i is the euclidean inner product on R . It needs to be verified that b ω is smooth on aneighborhood of S × { } in case the origin is in V . To simplify notation we denote the compositionof a function f : V → R with the projection S × V → again by f and likewise for a function e f : S → R . With this notational agreement the formula for Z in (6.4) entails by (6.6) over( S \ {− } ) × ( V \ { } ) b ω | ( S \{− } ) × ( V \{ } ) == χ ϕr p τ ) (cid:16) ((1 + cos τ ) x + sin τ x ) dx + ((1 + cos τ ) x − sin τ x ) dx (cid:17) == χ p τ ) (cid:16) ((1 + cos τ ) ω + sin τ ω ) dx + ((1 + cos τ ) ω − sin τ ω ) dx (cid:17) . The right hand side can be extended by 0 to a smooth form on S × V , hence b ω is smooth. Moreover,the restriction of b ω to { } × V coincides with the restriction ω | { }× V . Finally check that for x = 0and g ∈ S \ {− } Y ( g, x ) y b ω ( g,x ) = χ ( g ) ϕ ( x ) k x + gx k h x + gx, x − gx i = 0 . Hence b ω ∈ Z k (cid:0) Ω • S ⋉ V → S ( S ⋉ V ) , Y y (cid:1) and [ b ω ] Λ = [ ω ] Λ . (cid:3) Proposition 6.5.
For each S -invariant open V ⊂ R the chain map (cid:0) Ω • S ⋉ V → S ( S ⋉ V ) , Y y (cid:1) → (cid:0) Ω • hrel , Λ (Λ ( S ⋉ V )) , (cid:1) is a quasi-isomorphism.Proof. It remains to prove that every ω ∈ Z k (cid:0) Ω • S ⋉ V → S ( S ⋉ V ) , Y y (cid:1) which satisfies the condition[ ω ] Λ = 0 is of the form ω = Y y η for some η ∈ Ω k +1 S ⋉ V → S ( S ⋉ V ). Let us show this. We considerthe three non-trivial cases k = 0 , ,
1. Case: k = 0. Then ω is a smooth function on S ⋉ V vanishing on Λ . By Prop. 6.1, thefunction ω can be expanded over S \ {− } × V in the form ω | S \{− }× V = ω τ x + ω τ x , where ω , ω ∈ C ∞ ( S \ {− } × V ) . Moreover, the interior product of a form η = η dx + η dx ∈ Ω S ⋉ V → S ( S ⋉ V ) with the vectorfield Y has the form Y y η = Y η + Y η = ( x (1 − cos τ ) − x sin τ ) η + ( x (1 − cos τ ) − x sin τ ) η . This means that it suffices to find η , η ∈ C ∞ ( S ⋉ V ) which solve the system of equations ω τ = (1 − cos τ ) η + (sin τ ) η ,ω τ = − (sin τ ) η + (1 − cos τ ) η . (6.7)The 1-form η = η dx + η dx will then satisfy Y y η = ω which will prove the first case. Thefunctions η = τ (1 − cos τ )(1 − cos τ ) + sin τ ω − τ sin τ (1 − cos τ ) + sin τ ω = τ ω − τ sin τ − cos τ ) ω η = τ sin τ (1 − cos τ ) + sin τ ω + τ (1 − cos τ )(1 − cos τ ) + sin τ ω = τ sin τ − cos τ ) ω + τ ω now are well-defined and smooth over ( S × V ) \ ( { } × R ). They also solve (6.7). We are donewhen we can show that they can be extended smoothly to the whole domain S × V . But this isclear since the function ( − π, π ) \ { } → R , t t sin t − cos t ) has a holomorphic extension near the originas one verifies by power series expansion.
2. Case: k = 2. Let ω ∈ Ω S ⋉ V → S ( S ⋉ V ) and Y y ω = 0. Then ω = ϕdx ∧ dx for somesmooth function ϕ ∈ S ⋉ V → S . Now compute using (6.3)0 = Y y ω = ϕ · ( Y − Y ) = ϕ · (cid:0) x (1 − cos τ ) − x sin τ − x (1 − cos τ ) − x sin τ (cid:1) == ϕ · ( x − x ) · (1 − cos τ − sin τ ) . Hence ϕ = 0 and ω = 0.
3. Case: k = 1. Observe that in this case ω can be written in the form ω = ω dx + ω dx with ω , ω ∈ J ( S × V ) ⊂ C ∞ ( S × V ). By Lemma 6.1, ω j | ( S \{− } ) × V = τ Ω j for j = 1 , j ∈ C ∞ (( S \ {− } ) × V ). The condition Y y ω = 0 implies(6.8) Y Ω + Y Ω = Y ω + Y ω = 0 . Now define the function ϕ : ( S × V ) \ Λ → R by ϕ = k Y k ( − Y ω + Y ω ) (cid:12)(cid:12)(cid:12) ( S × V ) \ Λ . Since k Y k = 2 r (1 − cos τ ), the vector field Y vanishes nowhere on ( S × V ) \ Λ , so ϕ is well-definedand smooth. By (6.8) one computes ϕ ( g, x ) = ( ω Y ( g, x ) if g = 1 , x = 0 and Y ( g, x ) = 0 , − ω Y ( g, x ) if g = 1 , x = 0 and Y ( g, x ) = 0 . Assume that ϕ can be extended smoothly to S × V . Then η = ϕdx ∧ dx is a smooth form on S × V which satisfies Y y η = ϕ ( Y dx − Y dx ) = ω . So it remains to verify that ϕ can be smoothly extended to S × V . To this end we use the complexcoordinate z = x + √− x of V and introduce the complex valued function Ω = Ω + √− .Moreover, we define y : S × V → C , ( g, z ) z − gz . Then(6.9) y = (1 − e √− τ ) z = Y + √− Y and, by Eq. 6.8,(6.10) 12 ( y Ω + y Ω) = Y Ω + Y Ω = 0 . N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 29
Next observe that 1 − e √− τ = −√− τ (cid:0) − √− τ ( ζ ◦ τ ) (cid:1) for some holomorphic ζ : C → C whichfulfills ζ (0) = . Then Eq. 6.10 entails (cid:0) − √− τ ( ζ ◦ τ ) (cid:1) z Ω = (cid:0) √− τ ( ζ ◦ τ ) (cid:1) z Ω . By power series expansion it follows that ∂ Ω ∂z (cid:12)(cid:12) z =0 = 0 for all k ∈ N . Hence, by Taylor’s Theorem Ω = z Φ for some smooth Φ : S × V → C . Since by Lemma 6.2 k Y k = r τ ( ξ ◦ τ ) for some holomorphicfunction ξ not vanishing on ( − π, π ) the following equality holds over ( S \ {± } ) × ( V \ { } ) ϕ = 1 τ r ( ξ ◦ τ ) ( − Y Ω + Y Ω ) = √− τ r ( ξ ◦ τ ) (cid:0) y Ω − y Ω (cid:1) == 12 r ( ξ ◦ τ ) (cid:16)(cid:0) − √− τ ( ζ ◦ τ ) (cid:1) zz Φ + (cid:0) √− τ ( ζ ◦ τ ) (cid:1) zz Φ (cid:17) == 1( ξ ◦ τ ) (cid:0) − √− τ ( ζ ◦ τ ) (cid:1) Φ (cid:12)(cid:12)(cid:12)(cid:12) ( S \{± } ) × ( V \{ } ) . Since the right hand side has a smooth extension to S \ {− } × V , the function ϕ can be smoothlyextended to S × V and the claim is proved. (cid:3) S rotation in R m . In this subsection, we work with complex-valued functions, and differ-ential forms over complex numbers. Since tensoring an R -vector space with C is a faithfully flatfunctor, our results in this section still hold true for the algebra of real-valued functions.We consider a linear representation of S on R m . We identify R m with C m , and decompose C m into the following two subspaces, i.e.(6.11) C m = V ⊕ V , where V is the subspace of C m on which S acts trivially, and V is the S -invariant subspace of C m orthogonal to V with respect to an S -invariant hermitian metric on C m . Furthermore, V isdecomposed into irreducible unitary representations of S , i.e. V = t M j =1 C w j , where C w j is an irreducible representation ρ w j of S with the weight 0 = w j ∈ Z , i.e. ρ w j (exp(2 π √− t ) (cid:0) z (cid:1) := exp(2 w j π √− t ) z. We observe that C ∞ ( C m ) ⋊ S is isomorphic to (cid:0) C ∞ ( V ) ⊗ C ∞ ( V ) (cid:1) ⋊ S . As S acts on V trivially,we have C ∞ ( C m ) ⋊ S ∼ = C ∞ ( V ) ⊗ (cid:0) C ∞ ( V ) ⋊ S (cid:1) . The K¨unneth formula for Hochschild homology [Lod98, Theorem 4.2.5] gives HH • (cid:16) C ∞ ( C m ) ⋊ S (cid:17) = HH • (cid:0) C ∞ ( V ) (cid:1) ⊗ HH • (cid:0) C ∞ ( V ) ⋊ S (cid:1) . The Hochschild-Kostant-Rosenberg theorem shows HH • (cid:0) C ∞ ( V ) (cid:1) = Ω • ( V ). Hence, we have re-duced the computation of HH • (cid:0) C ∞ ( C m ) ⋊ S (cid:1) to HH • (cid:0) C ∞ ( V ) ⋊ S (cid:1) . Without loss of generality,we assume in the left of this subsection that C m = V , i.e. C m = m M j =1 C w j , = w j ∈ Z . Let w be the lowest common multiplier of w , ..., w m . We observe that for t ∈ [0 , t = jw , j =0 , ..., w −
1, the fixed point subspace of t is { } ; if t = jw , the fixed point subspace of t is C w k ⊕ · · · ⊕ C w kl , for w k , ..., w k l that w divides jw k , · · · , jw k l . Hence the loop space Λ ( S ⋉ C m ) has the followingform,Λ ( S ⋉ C m ) = n(cid:0) exp(2 π √− t ) , (0 , · · · , z w k , · · · , z w kl , , · · · ) (cid:1)(cid:12)(cid:12)(cid:12) (0 , · · · , z w k , · · · , z w kl , , · · · ) ∈ C m , tw k , · · · , tw k l ∈ Z w o . Let σ : Λ ( S ⋉ C m ) → S be the forgetful map mapping (exp(2 π √− t ) , z ) ∈ Λ ( S ⋉ C m ) toexp(2 π √− t ).Following Proposition 5.4 and Eq. (5.4), we can compute the Hochschild homology of C ∞ ( C m ) ⋊ S is computed by the S -invariant part of the cohomology of the following Koszul type complex,(6.12)Ω mS ⋉C m → S ( S ⋉ C m ) i YS ⋉C m −→ . . . i YS ⋉C m −→ Ω S ⋉C m → S ( S ⋉ C m ) i YS ⋉C m −→ C ∞ ( S ⋉ C m ) −→ , where Y S ⋉C m : S ⋉ C m → s ∗ T C m is defined by Y S ⋉C m ( g, v ) = v − gv . Below, we sometimes ab-breviate the symbol Y S ⋉C m by Y by abusing the notations. Fix a choice of coordinates ( z , · · · , z m )for z j ∈ C w j . The vector field Y := Y S ⋉C m (exp(2 π √− t ) , z ) is written as Y := Y S ⋉C m (exp(2 π √− t ) , z ) = m X k =1 (cid:0) exp(2 π √− w k t ) − (cid:1) z k ∂∂z k + (cid:0) exp( − π √− w k t ) − (cid:1) ¯ z k ∂∂ ¯ z k . Define an analytic function a ( z ) on C by a ( z ) := exp(2 π √− z ) − z . Then we have exp(2 π √− w k t ) − w k ta ( w k t ) , exp( − π √− w k t ) − w k t ¯ a ( w k t ) . Observe that for t ∈ R , a ( t ) = ¯ a ( t ), and a ( t ) = 0 for all t sufficiently close to 0. For a sufficientlysmall ǫ , the vector field Y on ( − ǫ, ǫ ) × C m is of the following form Y = t m X k =1 w k (cid:18) a ( w k t ) z k ∂∂z k + a ( w k t )¯ z k ∂∂ ¯ z k (cid:19) . This leads to the following property of the vector field Y . Lemma 6.6.
The vector field Y : S × C m → C m , ( g, z ) z − gz has a coordinate representation Y = P mk =1 Y k z k ∂∂z k + Y k ¯ z k ∂∂ ¯ z k with coefficients given by Y k (cid:0) exp(2 π √− t ) (cid:1) = exp(2 π √− w k t ) − . Denote w = l.c.m. ( w , · · · , w m ) . When t = jw , for ≤ j < w , there is a sufficiently small ǫ > such that on ( jw − ǫ, jw + ǫ ) , Y k is of the following form, Y k (cid:0) exp(2 π √− t ) (cid:1) = w k ( t − jw ) a (cid:0) w k ( t − jw ) (cid:1) , for w k j ∈ Z w, where a (cid:0) w k ( t − jw ) (cid:1) = 0 for all t ∈ ( jw − ǫ, jw + ǫ ) . And for k with w k j / ∈ Z w , Y k (cid:0) exp(2 π √− t ) (cid:1) = 0 for all t ∈ ( jw − ǫ, jw + ǫ ) .When t = jw , there is a sufficiently small ǫ > such that on ( t − ǫ, t + ǫ ) , Y k (cid:0) exp(2 π √− t ) (cid:1) = 0 for all t ∈ ( t − ǫ, t + ǫ ) . Analogous to the expression of the vector field Y , we study in the following lemma the localexpression of the vanishing ideal J of the loop space Λ ( S ⋉ C m ) for the S action on C m definedby Equation (6.11). N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 31
Lemma 6.7.
The vanishing ideal J of Λ ( S ⋉ C m ) has the following local form. • Near (cid:0) exp(2 π √− jw ) , (cid:1) ∈ S × C m , the vanishing ideal J (cid:0) ( jw − ǫ, jw + ǫ ) × B ̺ (0) (cid:1) for asufficiently small ǫ > and a ball B ̺ (0) ⊂ C m centered at with a sufficiently small radius ̺ > consists of all smooth functions f ∈ C ∞ (cid:0) ( jw − ǫ, jw + ǫ ) × B ̺ (0) (cid:1) which can be writtenin the form f = ( t − jw ) X k,w k j ∈ w Z ( z k f k + ¯ z k g k ) + X k,w k j / ∈ w Z ( z k f k + ¯ z k g k ) , for f k , g k ∈ C ∞ (cid:0) ( jw − ǫ, jw + ǫ ) × B ̺ (0) (cid:1) . • Near (cid:0) exp(2 π √− jw ) , Z (cid:1) ∈ S × C m with Z = 0 and exp(2 π √− jw ) Z = Z , the vanishingideal J (cid:0) ( jw − ǫ, jw + ǫ ) × B ̺ ( Z ) (cid:1) for a sufficiently small ǫ > and a ball B ̺ ( Z ) ⊂ C m centered at Z with a sufficiently small radius ̺ > consists of all smooth functions f ∈C ∞ (cid:0) ( jw − ǫ, jw + ǫ ) × B ̺ ( Z ) (cid:1) which can be written in the form f = ( t − jw ) f + X k,w k j / ∈ w Z ( z k f k + ¯ z k g k ) , for f, f k , g k ∈ C ∞ (cid:0) ( jw − ǫ, jw + ǫ ) × B ̺ ( Z ) (cid:1) . • Near (cid:0) exp(2 π √− t ) , (cid:1) ∈ S × C m such that t = jw for all j and ∈ C m , the vanishingideal J (cid:0) ( t − ǫ, t + ǫ ) × B ̺ (0) (cid:1) for a sufficiently small ǫ > and a ball B ̺ (0) ⊂ C m centered at with a sufficiently small radius ̺ > consists of all smooth functions f ∈C ∞ (cid:0) ( t − ǫ, t + ǫ ) × B ̺ (0) (cid:1) which can be written in the form f = m X k =1 ( z k f k + ¯ z k g k ) , for f k , g k ∈ C ∞ (cid:0) ( t − ǫ, t + ǫ ) × B ̺ (0) (cid:1) .Proof. We will prove the case around the most singular point (1 , ∈ S × C m . A similar proofworks for the other points. We leave the details to the reader.For (1 , ∈ S × C m , choose a sufficiently small ǫ > − ǫ, ǫ ) of the form jw for an integer 0 < j < w . We identify ( − ǫ, ǫ ) with a neighborhood of1 in S via the exponential map. For a positive ̺ , the loop space Λ ( S ⋉ C m ) in ( − ǫ, ǫ ) × B ̺ (0) isof the form Λ ( S × C m ) (0 , = { (0 , z ) | z ∈ B ̺ (0) } ∪ { ( t, } . A smooth function f on ( − ǫ, ǫ ) × B ̺ (0) belongs to J (cid:0) ( − ǫ, ǫ ) × B ̺ (0) (cid:1) if and only if f (0 , z ) = f ( t,
0) = 0 . We consider f as a function of t ∈ ( − ǫ, ǫ ). By the Malgrange preparation theorem, we have theexpansion f ( t, z ) + t = c ( t, z )( t + a ( z )) , where c ( t, z ) and a ( z ) are smooth and a (0) = 0. Since t = c ( t, t for all t ∈ ( − ǫ, ǫ ), c ( t,
0) = 1.Putting t = 0 gives 0 = c (0 , z ) a ( z ) for all z ∈ B ̺ (0). Recall that c (0 ,
0) = 1. Therefore, a ( z ) = 0for all z in a neighborhood of 0. After possibly shrinking ̺ , we can assume that a ( z ) = 0 on B ρ (0).Hence, we conclude that f ( t, z ) = t ( c ( t, z ) − . Taking the parametric Taylor expansion of c ( t, z ) − c ( t, z ) − m X j =1 z j f j ( t, z ) + ¯ z j g j ( t, z ) , where f j and g j are smooth functions on ( − ǫ, ǫ ) × B ̺ (0). (cid:3) In the following, we compute the cohomology of the complex (6.12). We observe that the complex(Ω • S ⋉C m → S ( S ⋉ C m ) , i Y ) for Y := Y S ⋉C m forms a sheaf of complexes over S via the map σ : Λ ( S ⋉ C m ) → S . Accordingly, we compute the cohomology (cid:0) Ω • S ⋉C m → S ( S ⋉ C m ) , i Y (cid:1) as asheaf over S . Proposition 6.8.
For all open subsets U of the loop space Λ = Λ ( S ⋉ C m ) and all k ∈ N themap Θ kU : Ω k rel , Λ ( U ) → Γ ∞ ( U, k ^ F ) from Prop. 4.1 is injective.Proof. We will prove the case around the most singular point (1 , ∈ S × C m . A similar proofworks for the other points. We leave the detail to the reader.Recall that we show in Lemma 6.7 that near (1 , J (cid:0) ( − ǫ, ǫ ) × B ̺ (0) (cid:1) for asufficiently small ǫ > B ̺ (0) ⊂ C m centered at 0 with a sufficiently small radius ̺ > f ∈ C ∞ (cid:0) ( − ǫ, ǫ ) × B ̺ (0) (cid:1) which can be written in the form f = t m X k =1 ( z k f k + ¯ z k g k ) , for f k , g k ∈ C ∞ (cid:0) ( − ǫ, ǫ ) × B ̺ (0) (cid:1) . Recall that by definition, Ω p rel , Λ (cid:0) ( − ǫ, ǫ ) × B ̺ (0) (cid:1) is the quotientΩ pS ⋉C m → S (cid:0) ( − ǫ, ǫ ) × B ̺ (0) (cid:1) / J Ω pS ⋉C m → S + d J ∧ Ω pS ⋉C m → S (cid:0) ( − ǫ, ǫ ) × B ̺ (0) (cid:1) . In the following, we will discribe Ω p rel , Λ (cid:0) ( − ǫ, ǫ ) × B ̺ (0) (cid:1) in more detail and, for ease of nota-tion, will use the symbols Ω pS ⋉C m → S and Ω p rel , Λ to stand for Ω pS ⋉C m → S (cid:0) ( − ǫ, ǫ ) × B ̺ (0) (cid:1) andΩ p rel , Λ (cid:0) ( − ǫ, ǫ ) × B ̺ (0) (cid:1) , respectively, and J for the vanishing ideal J (cid:0) ( − ǫ, ǫ ) × B ̺ (0) (cid:1) .In degree p = 0, Ω , Λ coincides with the quotient of C ∞ (cid:0) ( − ǫ, ǫ ) × B ̺ (0) (cid:1) by J (cid:0) ( − ǫ, ǫ ) × B ̺ (0) (cid:1) .In degree p = 1, we know by Lemma 6.7 that d J consists of 1-forms which can be expressed asfollows: t m X k =1 ( f k dz k + g k d ¯ z k ) , f k , g k ∈ C ∞ (cid:0) ( − ǫ, ǫ ) × B ̺ (0) (cid:1) . Hence, d J is of the form t Ω S ⋉C m → S , which contains J Ω S ⋉C m → S . Notice that for (0 , z ) ∈ S × C m , F (0 ,z ) coincides with T ∗ z C m . For ω = P mk =1 f k dz k + g k d ¯ z k ∈ Ω , Λ , if Θ( ω ) = 0, then f k (0 , z ) = g k (0 , z ) = 0 for 1 ≤ k ≤ m . Therefore, taking the parametric Talyor expansion of f k , g k at (0 , z ), we have that there are ˜ f k and ˜ g k in C ∞ (cid:0) ( − ǫ, ǫ ) × B ̺ (0) (cid:1) such that f k = t ˜ f k and g k = t ˜ g k .Hence, ω = t P mk =1 ˜ f k dz k + ˜ g k d ¯ z k ∈ d J and [ ω ] = 0 in Ω , Λ .In degree p >
1, the above description of Ω , Λ generalizes with the above expression for d J .As Ω kS ⋉C m → S is of the form X j dz j ∧ Ω k − S ⋉C m → S + d ¯ z j ∧ Ω k − S ⋉C m → S , we conclude that d J ∧ Ω k − S ⋉C m → S can be identified as t Ω kS ⋉C m → S , which contains J Ω k − S ⋉C m → S as a subspace.We notice that at (0 , z ) ∈ S × C m , V k F (0 ,z ) is V k T ∗ (0 ,z ) C m . For ω = P I,J f I,J dz I ∧ · · · ∧ dz I s ∧ d ¯ z J s +1 ∧ · · · ∧ d ¯ z J k , with 1 ≤ I < · · · < I s ≤ m and 1 ≤ J s +1 < · · · < J k ≤ m , if Θ( ω ) = 0, we thenget f I,J (0 , z ) = 0 for all I, J . And we can conclude from the Taylor expansion that there exists ˜ f I,J such that f I,J = t ˜ f I,J , and ω = t P I,J ˜ f I,J dz I ∧ · · · ∧ dz I s ∧ d ¯ z J s +1 ∧ · · · ∧ d ¯ z J k which is an elementin d J ∧ Ω k − S ⋉C m → S . Therefore, [ ω ] = 0 in Ω k rel , Λ . (cid:3) N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 33
Proposition 6.9.
For each S -invariant open V ⊂ C m the chain map R : (cid:0) Ω • S ⋉ V → S ( S ⋉ V ) , Y y (cid:1) → (cid:0) Ω • hrel , Λ (Λ ( S ⋉ V )) , (cid:1) is a quasi-isomorphism.Proof. We consider both sides as sheaves over S , and prove that R is a quasi-isomorphism of sheavesover S . It is sufficient to prove the quasi-isomorphism R at each stalk. We split our proof into twoparts according to the point t in S ,(1) at exp(2 π √− t ) with t = jw for 0 ≤ j < w and t ∈ [0 , π √− jw ) for 0 ≤ j < w . Case (1) . We prove that R exp(2 π √− t ) : (cid:0) Ω • S ⋉ V → S , exp(2 π √− t ) ( S ⋉ V ) , Y y (cid:1) → Ω • hrel , Λ , exp(2 π √− t ) (Λ ( S ⋉ V ))is a quasi-isomorphism for t = jw for 0 ≤ j < w and t ∈ [0 , ǫ >
0, on ( t − ǫ, t + ǫ ) × C m , the vector field Y is of the form Y = m X j =1 (cid:0) exp(2 π √− w j t ) − (cid:1) z j ∂∂z j + (cid:0) exp( − π √− w j t ) − (cid:1) z j ∂∂ ¯ z j . Observe that the vector field Y vanishes exactly at ( t, (cid:0) Ω • S ⋉ V → S , exp(2 π √− t ) (cid:0) ( t − ǫ, t + ǫ ) × C m (cid:1) , Y y (cid:1) is a smooth family of generalized Koszul complexes over t ∈ ( t − ǫ, t + ǫ ). Its cohomology can becomputed using Proposition B.7 as H • (cid:0) Ω • S ⋉ V → S , exp(2 π √− t ) (cid:0) ( t − ǫ, t + ǫ ) × C m (cid:1) , Y y (cid:1) = (cid:26) C ∞ (cid:0) t − ǫ, t + ǫ (cid:1) , • = 0 , , otherwise . At the same time, for every t in ( t − ǫ, t + ǫ ), the fixed point of exp(2 π √− t ) is 0 in C m . Andtherefore, the complex Ω • hrel , Λ (cid:0) ( t − ǫ, t + ǫ ) × C m (cid:1) identified with Γ ∞ (cid:0) ( t − ǫ, t + ǫ ) × { } , V • F (cid:1) is computed as follows,Γ ∞ (cid:0) ( t − ǫ, t + ǫ ) × { } , ^ • F (cid:1) = (cid:26) C ∞ (cid:0) t − ǫ, t + ǫ (cid:1) , • = 0 , , otherwise . From the above computation, it is straight forward to conclude that R exp(2 π √− t ) is a quasi-isomorphism. Case (2) . We prove that at exp(2 π √− jw ), the morphism R exp(2 π √− jw ) is a quasi-isomorphism.Following Lemma 6.6, we write the vector field Y as a sum of two components Y = Y + Y Y = X k,kj / ∈ w Z Y k z k ∂∂z k + Y k ¯ z k ∂∂ ¯ z k Y = ( t − jw ) X k,kj ∈ w Z w k ( a k z k ∂∂z k + ¯ a k ¯ z k ∂∂z k ) , where a k = a (cid:0) w k ( t − jw ) (cid:1) . Define e Y to be P k,kj ∈ w Z w k ( a k z k ∂∂z k + ¯ a k ¯ z k ∂∂z k ). Then we have thefollowing expression for Y , Y = Y + ( t − jw ) e Y . Accordingly, we can decompose C m as a direct sum of two subspaces, that is write C m = S × S with S := M k,kj / ∈ w Z C w k ,S := M k,kj ∈ w Z C w k . Both S and S are equipped with S -actions such that the above decomposition of C m is S -equivariant. As our argument is local, we can assume to work with an open set V , which is of theproduct form V = V × V such that V (and V ) is an S -invariant neighborhood of 0 in S (and S ).We consider (cid:16) Ω • S ⋉ V l → S (cid:0)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V l (cid:1) , i Y l (cid:17) for l = 1 ,
2. Observe that each complexΩ • S ⋉ V l → S (cid:0)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V l (cid:1) is a C ∞ (cid:0) jw − ǫ, jw + ǫ (cid:1) -module, and their tensor product over thealgebra C ∞ (cid:0) jw − ǫ, jw + ǫ (cid:1) defines a bicomplexΩ pS ⋉ V → S (cid:18)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:19) ⊗ C ∞ (cid:0) jw − ǫ, jw + ǫ (cid:1) Ω qS ⋉ V → S (cid:18)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:19) with i Y ⊗ ⊗ i Y being the vertical one. The total complexof this double complex is exactlyΩ • S ⋉ V → S (cid:18)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:19) with the differential i Y = i Y ⊗ ⊗ i Y . The E -page of the spectral sequence associated to thebicomplexΩ • S ⋉ V → S (cid:18)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:19) ⊗ C ∞ (cid:0) jw − ǫ, jw + ǫ (cid:1) Ω • S ⋉ V → S (cid:18)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:19) is H • (cid:18) Ω • S ⋉ V → S (cid:0)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:1) , i Y (cid:19) ⊗ C ∞ (cid:0) jw − ǫ, jw + ǫ (cid:1) Ω qS ⋉ V → S (cid:0)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:1) , with the differential 1 ⊗ i Y . We observe that Y vanishes only at 0 for every fixed t . Therefore, (cid:16) Ω • S ⋉ V → S (cid:0)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:1) , i Y (cid:17) is a smooth family of generalized Koszul complexes. Itscohomology is computed by Proposition B.7 as follows, H • (cid:18) Ω • S ⋉ V → S (cid:0)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:1) , i Y (cid:19) = (cid:26) C ∞ ( jw − ǫ, jw + ǫ ) • = 00 • 6 = 0 . Therefore, we get the following expression of E p,q , E p,q = (cid:26) Ω qS ⋉ V → S (cid:0)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:1) , p = 00 , p = 0 . Next we compute the cohomology of ( E ,q , i Y ). Recall by Lemma 6.6 that Y has the form Y = ( t − jw ) e Y , where e Y vanishes exactly at 0 for every fixed t ∈ ( jw − ǫ, jw + ǫ ). At degree q , wenotice that if an element ω ∈ Ω qS ⋉ V → S (( jw − ǫ, jw + ǫ ) × V ) belongs to ker( i Y ), ( t − jw ) i e Y ω = 0.Hence ω belongs to ker( i e Y ). Hence, we have reached the following equationker( i Y ) = ker( i e Y ) . It is also easy to check that i Y Ω q +1 S ⋉ V → S (cid:18)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:19) = (cid:0) t − jw (cid:1) i e Y Ω q +1 S ⋉ V → S (cid:18)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:19) N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 35
We conclude that the quotient ker( i Y ) /i Y Ω q +1 S ⋉ V → S (cid:0)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:1) is isomorphic toker( i e Y ) / ( t − jw ) i e Y Ω q +1 S ⋉ V → S (cid:18)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:19) Recall that the cohomology of (cid:16) Ω • S ⋉ V → S (cid:0)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:1) , i e Y (cid:17) is computed as follows, H q (cid:18) Ω • S ⋉ V → S (cid:0)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:1) , i e Y (cid:19) = (cid:26) C ∞ ( jw − ǫ, jw + ǫ ) , q = 00 , q = 0 . Therefore, for all q , we conclude that i e Y Ω q +1 S ⋉ V → S (cid:18)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:19) = ker( i e Y ) , and the quotient ker( i Y ) /i Y Ω q +1 S ⋉ V → S (cid:0)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:1) is isomorphic toker( i e Y ) / (cid:0) t − j (cid:1) ker( i e Y ) . As the E page has only nonzero component when p = 0, the spectral sequence collapses at the E page, and we conclude that the cohomology of the total complex, which is the cohomology ofΩ • S ⋉ V → S (cid:0)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:1) with the differential i Y ⊗ ⊗ i Y , is equal to the quotientker( i e Y ) / (cid:0) t − j (cid:1) ker( i e Y )for the contraction i e Y on Ω • S ⋉ V → S (cid:0)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:1) .We now prove that the morphism R : (cid:16) Ω qS ⋉ V → S ( S ⋉ V ) , Y y (cid:17) → (cid:0) Ω • hrel , Λ (Λ ( S ⋉ V )) , (cid:1) is a quasi-isomorphism. The above discussion and description of Λ (( jw − ǫ, jw + ǫ ) × V ) reduces usto prove that the morphism R : (cid:18) Ω • S ⋉ V → S (cid:0)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:1) , Y y (cid:19) → (cid:18) Ω • hrel , Λ (cid:18) Λ (cid:0)(cid:0) jw − ǫ, jw + ǫ (cid:1) × V (cid:1)(cid:19) , (cid:19) is a quasi-isomorphism. We prove this by examination of R in degree q . Hereby, we will work with V • F as it is isomorphic to Ω • rel , Λ by Proposition 6.8. • q ≥
1. Recall that Γ ∞ (cid:0) ( jw − ǫ, jw + ǫ ) × V , V q F (cid:1) is V q F ( jw ,z ) . We observe that the vectorfield e Y at t = jw coincides with the fundamental vector field of the S action on V . Hence, if φ ∈ V q F ( jw ,z ) is horizontal, φ satisfies the equation i e Y ( jw ,z ) φ = 0. As the cohomology of the(Ω • ( V ) , i e Y (0 ,z ) ) at degree q vanishes, there is a degree q +1 form ψ ∈ Ω • ( V ) such that i e Y ( jw ,z ) ψ = φ .Define ω ∈ Ω • S ⋉ V → S (( jw − ǫ, jw + ǫ ) × V ) by ω := i ˜ Y ψ , where ψ is viewed as an element inΩ • S ⋉ V → S (( jw − ǫ, jw + ǫ ) × V ) constant along the t direction. Then we can easily check that ω belongs to the kernel of i e Y and R ( ψ ) = φ . We conclude that R is surjective.For the injectivity of R , we suppose that ω ∈ ker( i e Y ). Hence, R ( ω )( jw , z ) = ω ( jw , z ) = 0. Thenby the parametrized Taylor expansion, we can find a form ˜ ω ∈ Ω • S ⋉ V → S (( jw − ǫ, jw + ǫ ) × V )such that ω = ( t − jw )˜ ω . As 0 = i e Y ω = ( t − jw ) i e Y ˜ ω , i e Y ˜ ω = 0. Hence ω = ( t − jw )˜ ω belongs to( t − jw ) ker i e Y , and [ ω ] is zero in the cohomology of i Y . • q = 0. Recall that e Y is of the form P k w k (cid:0) a ( w k ( t − jw )) z k ∂∂z k + ¯ a ( w k ( t − jw )) (cid:1) ¯ z k ∂∂ ¯ z k , where a ( w k ( t − jw )) = 0 for all t ∈ ( jw − ǫ, jw + ǫ ). Therefore, the space ( t − jw ) i e Y is of the form (cid:0) t − jw (cid:1) X k z k f k + ¯ z k g k , which is exactly the vanishing ideal J (cid:0) ( jw − ǫ, jw + ǫ ) × V (cid:1) . This shows that the cohomology of (cid:0) Ω • S ⋉ V → S (( jw − ǫ, jw + ǫ ) × V ) , Y y (cid:1) at degree 0 coincides with C ∞ (cid:0) Λ ( S ⋉ V ) (cid:1) | ( jw − ǫ, jw + ǫ ) × V .One concludes that R is an isomorphism in degree 0. (cid:3) Stitching it all together.
We are now in a position to prove the Conjecture 5.6 in the caseof circle actions:
Theorem 6.10.
Let M be an S -manifold and regard Ω • hrel , Λ (cid:0) Λ ( S ⋉ M ) (cid:1) as a chain complexendowed with the zero differential. Then the chain map Φ • ,M/S : C • (cid:0) C ∞ ( M ) , A ( M/S ) (cid:1) → Ω • hrel , Λ (cid:0) Λ ( S ⋉ M ) (cid:1) is a quasi-isomorphism.Proof. Since Φ • ,M/S is the global sections of a morphism of fine sheaves on M/S , it suffices toprove that Φ • : ˆ C • ( C ∞ M , A ) (cid:1) → π ∗ ( s | Λ ) ∗ Ω • rel , Λ , is a quasi-isomorphism, i.e., that the induced map on the stalks Φ • , O is. Now there are two cases,depending on the isotropies of the orbit O : when the isotropy subgroup Γ x ⊂ S of a point x ∈ S isa finite group, this follows from the (proof of) Corollary B.6. When the isotropy group is S itself,it follows from Proposition 6.9. (cid:3) Appendix A. Tools from singularity theory
A.1.
Differentiable stratified spaces.
Recall that for every locally closed subset X ⊂ R n ofeuclidean space the sheaf C ∞ X of smooth functions on X is defined as the quotient sheaf C ∞ U / J X,U ,where U ⊂ R n is an open subset such that X ⊂ U is relatively closed, C ∞ U is the sheaf of smoothfunctions on U , and J X,U the ideal sheaf of smooth functions on open subsets of U vanishing on X .Note that C ∞ X does not depend on the particular choice of the ambient open subset U ⊂ R n . Definition A.1.
A commutative locally ringed space ( A, O ) is called an affine differentiable space if there is a closed subset X ⊂ R n and an isomorphism of ringed spaces ( f, F ) : ( A, O ) → ( X, C ∞ X ).By a differentiable stratified space we understand a commutative locally ringed space ( X, C ∞ )consisting of a separable locally compact topological Hausdorff space X equipped with a stratification S on X in the sense of Mather [Mat73] (cf. also [Pfl01, Sec. 1.2]) and a sheaf C ∞ of commutativelocal C -rings on X such that for every point x ∈ X there is an open neighborhood U together with ϕ , . . . , ϕ n ∈ C ∞ ( U ) having the following properties:(DS1) The map ϕ : U → R n , y ( ϕ ( y ) , . . . , ϕ n ( y )) is a homeomorphism onto a locally closedsubset e U := ϕ ( U ) ⊂ R n . and induces an isomorphism of ringed spaces ϕ : ( U, C ∞| U ) → ( e U , C ∞ e U ).(DS2) The map ϕ endows ( U, C ∞| U ) with the structure of an affine differentiable space which meansthat ( ϕ, ϕ ∗ ) : ( U, C ∞| U ) → ( e U , C ∞ e U ) is an isomorphism of ringed spaces, where C ∞ e U denotes thesheaf of smooth functions on e U as defined above.(DS3) For each stratum S ⊂ U , ϕ | S ∩ U is a diffeomorphism of S ∩ U onto a submanifold ϕ ( S ∩ U ) ⊂ R n .A map ϕ : U → R n fulfilling the axioms (DS1) to (DS3) will often be called a singular chart of X (cf. [Pfl01, Sec. 1.3]).A differentiable stratified space is in particular a reduced differentiable space in the sense ofSpallek [Spa69] or Gonz´ales–de Salas [NGSdS03]. Moreover, differentiable stratified spaces definedas above coincide with the stratified spaces with smooth structure as in [Pfl01]. Proposition A.2 (cf. [Pfl01, Thm. 1.3.13]) . The structure sheaf of a differentiable stratified spaceis fine.
N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 37
To formulate the next result, we introduce the commutative ringed space ( R ∞ , C ∞ R ∞ ). It is definedas the limit of the direct system of ringed spaces (cid:0) ( R n , C ∞ R n ) , ι nm (cid:1) n,m ∈ N , n ≤ m , where ι nm : R n ֒ → R m is the embedding given by ι nm ( v , · · · , v n ) = ( v , · · · v n , , · · · , . Note that for each open set U ⊂ R ∞ the section space C ∞ R ∞ ( U ) coincides with the inverse limit ofthe projective system of nuclear Fr´echet algebras (cid:0) C ∞ R n ( U ∩ R n ) , ι ∗ nm (cid:1) n,m ∈ N , n ≤ m . Hence the C ∞ R ∞ ( U )and in particular C ∞ R ∞ ( R ∞ ) are nuclear Fr´echet algebras by [Tr`e67, Prop. 50.1]. Proposition A.3.
For every differentiable stratified space ( X, C ∞ ) there exists a proper embedding ϕ : ( X, C ∞ ) ֒ → ( R ∞ , C ∞ R ∞ ) .Proof. Since X is separable and locally compact there exists a compact exhaustion that is a family( K k ) k ∈ N of compact subsets K k ⊂ X such that K k ⊂ K ◦ k +1 for all k ∈ N and such that S k ∈ N K k = X .By [Pfl01, Lem. 1.3.17] there then exists an inductively embedding atlas that is a family ( ϕ k ) k ∈ N of singular charts ϕ k : K ◦ k +1 → R n k together with a family ( U k ) k ∈ N of relatively compact opensubsets U k ⊂⊂ K ◦ k +1 such that K k ⊂ U k and ϕ k +1 | U k = ι n k n k +1 ◦ ϕ k | U k for all k ∈ N . Now define ϕ : X → R ∞ by ϕ ( x ) = ϕ k ( x ) whenever x ∈ U k . Then ϕ is well defined and an embedding byconstruction. By a straightforward partition of unity argument one constructs a smooth function ψ : X → R such that ψ ( x ) ≥ k for all x ∈ K k +1 \ K ◦ k . The embedding ( ϕ, ψ ) : X → R ∞ × R ∼ = R ∞ then is proper. (cid:3) Corollary A.4. ( X, C ∞ ) be a differential stratified space. Then there exists a complete metric d : X × X → R such that d ∈ C ∞ ( X × X ) .Proof. The euclidean inner product h− , −i R n extends in a unique way to an inner product h− , −i R ∞ on R ∞ such that h j n ( x ) , j n ( y ) i R ∞ = h x, y i R n for all n ∈ N and x, y ∈ R n , where j n : R n ֒ → R ∞ is the canonical embedding ( x , . . . , x n ) ( x , . . . , x n , , . . . , , . . . ). The associated metric d R ∞ : R ∞ × R ∞ → R , ( x, y ) p h x − y, x − y i R ∞ then is related to the euclidean metric d R n by d R ∞ (cid:0) j n ( x ) , j n ( y ) (cid:1) = d R n ( x, y ) for x, y ∈ R n . Now choose a proper embedding X ֒ → R ∞ and denotethe restriction of d R ∞ to X by d . By construction, d then is smooth. Moreover, d is a completemetric since the embedding is proper and each of the metrics d R n is complete. (cid:3) Appendix B. The cyclic homology of bornological algebras
B.1.
Bornological vector spaces and tensor products.
We recall some basic notions from thetheory of bornological vector spaces and their tensor products. For details we refer to [HN77] and[Mey07, Chap. 1].
Definition B.1 (cf. [HN77, Chap. I, 1:1 Def.]) . By a bornology on a set X one understands a set B of subset of X such that the following conditions hold true:(BS) B is a covering of X , B is hereditary under inclusions, and B is stable under finite unions.A map f : X → Y from a set X with bornology B to a set Y carrying a bornology D is called bounded , if the following is satisfied:(BM) The map f preserves the bornologies, i.e. f ( B ) ∈ D for all B ∈ B .If V is a vector space over k = R or k = C , a bornology B is called a convex vector bornology on V , if the following additional properties hold true:(BV) The bornology B is stable under addition, under scalar multiplication, under forming balancedhulls, and finally under forming convex hulls.A set together with a bornology is called a bornological set , a vector space with a convex vectorbornology a bornological vector space . For clarity, we sometimes denote a bornological vector spaceas a pair ( V, B ), where V is the underlying vector space, and B the corresponding convex vectorbornology. A bornological vector space ( V, B ) is called separated , if the condition (S) below is satisfied. If inaddition condition (C) holds true as well, ( V, B ) is called complete .(S) The subspace { } is the only bounded subvector space of V .(C) Every bounded set is contained in a completant bounded disk, where a disk D ⊂ V is called completant , if the space V D spanned by D and semi-normed by the gauge of D is a Banachspace.As for the category of topological vector spaces there exist functors of separation and completionwithin the category of bornological vector spaces. Example B.1.
Let V be a locally convex topological vector space. The von Neumann bornology on V consists of all (von Neumann) bounded subsets of V , ie. of all B ⊂ V which are absorbed byevery 0-neighborhood. One immediately checks that the von Neumann bornology is a convex vectorbornology on V . We sometimes denote this bornology by B vN .Similarly to the topological case, the bornological tensor product is defined by a universal property. Definition B.2.
The ( projective ) bornological tensor product of two bornological spaces vectorspaces (cid:0) V , B (cid:1) and (cid:0) V , B (cid:1) is defined as the up to isomorphism uniquely defined bornologicalvector space (cid:0) V ⊗ V , B ⊗ B (cid:1) together with a bounded map V × V → V ⊗ V such that foreach bornological vector space ( W, D ) and bounded bilinear map λ : V × V → W there is a uniquebounded map λ : V ⊗ V → W making the diagram V ⊗ V λ / / (cid:15) (cid:15) WV ⊗ V λ ; ; ✇✇✇✇✇✇✇✇✇ commute. The completion of the bornological tensor product will be denoted by V ˆ ⊗ V . Remark B.3. (1) Note that the underlying vector space of the bornological tensor productcoincides with the algebraic tensor product of the vector spaces V ⊗ V .(2) Since tensor products of topological vector spaces are also needed in this paper, let usbriefly recall that the complete projective (resp. inductive) topological tensor product ˆ ⊗ π (resp. ˆ ⊗ ι ) can be defined as the (up to isomorphism) unique bifunctor on the category ofcomplete locally convex topological vector spaces which is universal with respect to jointly(resp. separately) continuous bilinear maps with values in complete locally convex topo-logical vector spaces. For Fr´echet spaces, the complete projective and complete inductivetensor products coincide, since separately continuous bilinear maps on Fr´echet spaces areautomatically jointly continuous. See [Gro55] and [Mey07] for details.B.2. The Hochschild chain complex.
In this section we recall the construction of the cyclicbicomplex associated to a complete bornological algebra A which not necessarily is assumed to beunital. To this end observe first that the space of Hochschild k -chains C k ( A ) := A ˆ ⊗ ( k +1) is definedusing the complete projective bornological tensor product ˆ ⊗ . Together with the face maps b k,i : C k ( A ) → C k − ( A ) , a ⊗ . . . ⊗ a k ( a ⊗ . . . ⊗ a i a i +1 ⊗ . . . ⊗ a k , if 0 ≤ i < k,a k a ⊗ . . . ⊗ a k − , if i = k, and the cyclic operators t k : C k ( A ) → C k ( A ) , a ⊗ . . . ⊗ a k ( − k a k ⊗ a ⊗ . . . ⊗ a k − the graded linear space of Hochschild chains C • ( A ) := (cid:0) C k ( A ) (cid:1) k ∈ N then becomes a pre-cyclic object(see for example [Lod98] for the precise commutation relations of the face and cyclic operators).From the pre-cyclic structure one obtains two boundary maps, namely the one of the Bar complex b ′ : C k ( A ) → C k − ( A ), b ′ :=:= P k − i =0 ( − i b i and the Hochschild boundary b : C k ( A ) → C k − ( A ), N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 39 b := b ′ + ( − k b k . The commutation relations for the b i immediately entail b = ( b ′ ) = 0. Thisgives rise to the following two-column bicomplex.... (cid:15) (cid:15) ... (cid:15) (cid:15) C ( A ) b (cid:15) (cid:15) C ( A ) − t o o − b ′ (cid:15) (cid:15) C ( A ) b (cid:15) (cid:15) C ( A ) − t o o − b ′ (cid:15) (cid:15) C ( A ) C ( A ) − t o o We will denote this two-column bicomplex by C • , • ( A ) { } . By definition, the homology of its totalcomplex is the Hochschild homology(B.1) HH • ( A ) := H • (cid:0) Tot • (cid:0) C • , • ( A ) { } (cid:1)(cid:1) . B.3.
A twisted version of the theorem by Hochschild–Kostant–Rosenberg and Connes.
The classical theorem by Hochschild–Kostant–Rosenberg identifies the Hochschild homology of thealgebra of regular functions on a smooth affine variety with the graded module of K¨ahler formsof that algebra [HKR62]. In his seminal paper [Con85], Connes proved that for compact smoothmanifolds an analogous result holds true that is the (continuous) Hochschild homology of the algebraof smooth functions on a manifold coincides naturally with the complex of differential forms overthe manifold (see [Pfl98] for the non-compact case of that result). Here we show a twisted versionof this theorem. That result appears to be folklore, cf. also [BDN17].Assume that h is an orthogonal transformation acting on some euclidean space R d . Let V be anopen ball around the origin of R d . Then we denote by h C ∞ ( V ) the space C ∞ ( V ) with the h -twisted C ∞ ( V )-bimodule structure C ∞ ( V ) ˆ ⊗ h C ∞ ( V ) ˆ ⊗C ∞ ( V ) → h C ∞ ( V ) , f ⊗ a ⊗ f ′ (cid:16) V ∋ v f ( hv ) a ( v ) f ′ ( v ) ∈ R (cid:17) . In the following we compute the twisted
Hochschild homology H • (cid:0) C ∞ ( V ) , h C ∞ ( V ) (cid:1) . Denote by h− , −i the euclidean inner product on R d . By the orthogonality assumption h− , −i is G -invariant,hence V is so, too. Recall that for every topological projective resolution R • → C ∞ ( V ) of C ∞ ( V ) asa C ∞ ( V )-bimodule the Hochschild homology groups H k ( C ∞ ( V ) , h C ∞ ( V ) are naturally isomorphicto the homology groups H k (cid:0) R • , h C ∞ ( V ) (cid:1) , see [Hel89]. Recall further that a topological projectiveresolution of the C ∞ ( V )-bimodule C ∞ ( V ) is given by the Connes-Koszul resolution [Con85, p. 127ff](B.2) Γ ∞ ( V × V, E d ) i Y −→ . . . i Y −→ Γ ∞ ( V × V, E ) i Y −→ C ∞ ( V × V ) −→ C ∞ ( V ) −→ , where E k is the pull-back bundle pr ∗ (cid:0) Λ k T ∗ R d (cid:1) along the projection pr : R d × R d → R d , ( v, w ) w ,and i Y denotes contraction with the vector field Y : V × V → pr ∗ ( T R d ), ( v, w ) w − v . By tensoringthe Connes-Koszul resolution with h C ∞ ( V ) one obtains the chain complex(B.3) Ω d ( V ) i Yh −→ . . . i Yh −→ Ω ( V ) i Yh −→ C ∞ ( V ) −→ , where the vector field Y h : V → T R d is given by Y h ( v ) = v − hv . Denote by V h the fixed pointset of h in V , let ι h : V h ֒ → V be the canonical embedding, and π h : V → V h the restriction ofthe orthogonal projection onto the fixed point space ( R d ) h . One obtains the following commutative diagram.(B.4) Ω d ( V ) i Yh / / ι ∗ h (cid:15) (cid:15) . . . i Yh / / Ω ( V ) i Yh / / ι ∗ h (cid:15) (cid:15) C ∞ ( V ) ι ∗ h (cid:15) (cid:15) Ω d ( V h ) / / π ∗ h (cid:15) (cid:15) . . . / / Ω ( V h ) / / π ∗ h (cid:15) (cid:15) C ∞ ( V h ) π ∗ h (cid:15) (cid:15) Ω d ( V ) i Yh / / . . . i Yh / / Ω ( V ) i Yh / / C ∞ ( V ) Proposition B.4.
The chain maps ι ∗ h and π ∗ h are quasi-isomorphisms.Proof. Since the restriction of the vector field Y h to V h vanishes, the diagram (B.4) commutes,and the ι ∗ h and π ∗ h are chain maps indeed. Let W be the orthogonal complement of ( R d ) h in R d , m = dim W , and π W := id V − π h the orthogonal projection onto W . Since the h -action on W isorthogonal and has as only fixed point the origin, there exists an orthonormal basis w , . . . , w m of W , a natural l ≤ m , and θ , . . . , θ l ∈ ( − π, π ) \ { } such that the following holds: hw k = cos θ i w i − + sin θ i w i if k = 2 i − i ≤ l, − sin θ i w i − + cos θ i w i if k = 2 i with i ≤ l, − w k if 2 l < k ≤ m. Denote by ϕ t : R d → R d , t ∈ R the flow of the complete vector field Y h or in other words the solutionof the initial value problem ddt ϕ t = (id V − h ) ϕ t , ϕ = id V . Then ϕ t v = v for all v ∈ ( R d ) h , and(B.5) ϕ t ( w k ) = e (1 − cos θ i ) t (cid:0) cos( t sin θ i ) w i − + sin( t sin θ i ) w i (cid:1) , if k = 2 i − i ≤ l,e (1 − cos θ i ) t (cid:0) − sin( t sin θ i ) w i − + cos( t sin θ i ) w i (cid:1) , if k = 2 i with i ≤ l,e t w k , if 2 l < k ≤ m. Now let v , . . . , v n be a basis of V h , and denote by v , . . . , v n , w , . . . , w m the basis of V ′ dual to v , . . . , v n , w , . . . , w m . Then every k -form ω on V is the sum of monomials dv i ∧ . . . ∧ dv i l ∧ ω i ,...,i l ,where 1 ≤ i < . . . < i l ≤ n and ω i i ,...,i l = i v i ∧ ... ∧ v il ω ∈ Γ ∞ (cid:0) π ∗ W Λ k − l T ∗ W (cid:1) . Let d W be therestriction of the exterior differential to Γ ∞ (cid:0) π ∗ W Λ • T ∗ W (cid:1) and define S : Ω k ( V ) → Ω k +1 ( V ) by itsaction on the monomials: Sω = k X l =0 X ≤ i <...
Hence one concludes by Cartan’s magic formula( Si Y h + i Y h S ) ω = k X l =0 X ≤ i <...
The proposition immediately entails the following twisted version of the theorem by Hochschild–Kostant–Rosenberg and Connes.
Theorem B.5.
Let h : R d → R d be an orthogonal linear transformation and V ⊂ R d an openball around the origin. Then the Hochschild homology H • (cid:0) C ∞ ( V ) , h C ∞ ( V ) (cid:1) is naturally isomorphicto Ω • ( V h ) , where V h is the fixed point manifold of h in V . A quasi-isomorphism inducing thisidentification is given by h C ∞ ( V ) ˆ ⊗ C k ( C ∞ ( V )) → Ω k ( V h ) , f ⊗ f ⊗ . . . ⊗ f k f | V h df | V h ∧ . . . ∧ df k | V h . We consider a finite subgroup Γ of the orthogonal linear transformation group of R d . Let V ⊂ R d be an open ball around the origin that is invariant with respect to the Γ action on R d . We canapply the quasi-isomorphism from Section 5.2 to compute HH • (cid:0) C ∞ ( V ) ⋊ Γ) by the homology of thecomplex C Γ • ( C ∞ ( V )). Since Γ is a finite group, the homology of C Γ • ( C ∞ ( V )) is computed by (cid:16) M γ ∈ Γ H • (cid:0) C ∞ ( V ) , γ C ∞ ( V ) (cid:1)(cid:17) Γ . As a corollary to Theorem B.5, we thus obtain the following computation of the Hochschild homologyof C ∞ ( V ) ⋊ Γ. Corollary B.6.
The Hochschild homology HH • (cid:0) C ∞ ( V ) ⋊ Γ) is naturally isomorphic to M γ ∈ Γ Ω • ( V γ ) Γ , where Γ acts on the disjoint union ` γ ∈ Γ V γ by γ ′ ( γ, x ) = ( γ ′ γ ( γ ′ ) − , γ ′ x ) . In the case of a smooth affine algebraic variety, Corollary B.6 is proved by [BDN17, Thm. 2.19].We refer the reader to [BN94, FT87, Was88, Pon18] for related developments.
We end with a generalisation of Proposition B.4 which is a useful tool in our computations.Observe that in the complex (B.3)Ω d ( V ) i Yh −→ . . . i Yh −→ Ω ( V ) i Yh −→ C ∞ ( V ) −→ , the vector field Y h can be extended to be a more general linear vector field Y H : R n → T R d of theform Y H ( v ) = H ( v ) ∈ T v R d where H : R d → R d is a diagonalizable linear map. A constructionsimilar to the homotopy operator S in the proof of Proposition B.4 (see also [Was88]) computes thehomology of (Ω • ( V ) , i Y H ) to be (Ω • ( V H ) ,
0) where V H = ker( H ). Furthermore, if H : S → End( R d )is a smooth family of diagonalizable linear operators parametrized by a smooth manifold S , H iscalled regular if H satisfies the following properties:(1) the kernel ker( H ) := { ker( H ( s )) } s ∈ S ⊂ S × R d is a smooth subbundle of the trivial vectorbundle S × R d ;(2) near every s ∈ S , there is a local frame of S × R d on a neighborhood U s of s in S consistingof ξ , · · · , ξ d such that • the collection { ξ , · · · , ξ k } is a local frame of the subbundle ker( H ) on U s , • for every j = k +1 , · · · , d , there is a smooth eigenfunction λ j ( s ) defined on U s satisfying H ( s ) ξ j ( s ) = λ j ( s ) ξ j ( s ) and λ j ( s ) = 0 , ∀ s ∈ U s .The proof of Proposition B.4 generalizes to the following result. Proposition B.7.
Let H : S → End( R d ) be a smooth family of diagonalizable linear operatorsparametrized by a smooth manifold S . Assume that H is regular. Let i ker( H ) : ker( H ) → S × R d bethe canonical embedding, and Ω • (cid:0) ker( H ) (cid:1) the restriction of C ∞ ( S, Ω • ( V )) to ker( H ) along i ker( H ) .Then the restriction map R ker( H ) : (cid:0) C ∞ ( S, Ω • ( V )) , i Y H (cid:1) → (cid:16) Ω • (cid:0) ker( H ) (cid:1) , (cid:17) is a quasi-isomorphism. In a certain sense, the final result is variant of the latter. To formulate it recall that by anEuler-like vector field for an embedded smooth manifold
S ֒ → M one understands a vector field Y : M → T M such that S is the zero set of Y and such that for each f ∈ C ∞ ( M ) vanishing on S the function Y f − f vanishes to second order on S ; cf. [SH18, Def. 1.1]. Proposition B.8.
Let M be a smooth manifold of dimension d , S ֒ → M an embedded submanifoldand Y : M → T M a smooth vector field which is Euler like with respect to S . Then the complex (B.7) Ω d ( M ) i Y −→ . . . i Y −→ Ω ( M ) i Y −→ C ∞ ( M ) −→ C ∞ ( S ) −→ , is exact and will be called the parametrized Koszul resolution of C ∞ ( S ) .Proof. The claim is an immediate consequence of the Koszul resolution as for example stated in[Was88, ]. (cid:3)
References [BDN17] Jacek Brodzki, Shantanu Dave, and Victor Nistor,
The periodic cyclic homology of crossed products offinite type algebras , Adv. Math. (2017), 494–523. MR 3581309[BG94] Jonathan Block and Ezra Getzler,
Equivariant cyclic homology and equivariant differential forms , Ann.Sci. ´Ecole Norm. Sup. (4) (1994), no. 4, 493–527.[BN94] J.-L. Brylinski and Victor Nistor, Cyclic cohomology of ´etale groupoids , K -Theory (1994), no. 4, 341–365. MR 1300545[BP08] Jean-Paul Brasselet and Markus J. Pflaum, On the homology of algebras of Whitney functions oversubanalytic sets , Ann. of Math. (2) (2008), no. 1, 1–52, http://dx.doi.org/10.4007/annals.2008.167.1.[Bre72] G.E. Bredon,
Introduction to Compact Transformation Groups , Academic Press, New York, 1972.[Bry87a] J.-L. Brylinski,
Algebras associated with group actions and their homology , unpublished, Brown Universitypreprint, 1987.[Bry87b] ,
Cyclic homology and equivariant theories , Ann. Inst. Fourier (Grenoble) (1987), no. 4, 15–28.[CM01] Marius Crainic and Ieke Moerdijk, Foliation groupoids and their cyclic homology , Adv. Math. (2001),no. 2, 177–197.[Con85] Alain Connes,
Noncommutative differential geometry , Inst. Hautes ´Etudes Sci. Publ. Math. (1985), no. 62,257–360.
N THE HOCHSCHILD HOMOLOGY OF CONVOLUTION ALGEBRAS OF PROPER LIE GROUPOIDS 43 [Con94] ,
Noncommutative geometry , Academic Press Inc., San Diego, CA, 1994.[Cra99] Marius Crainic,
Cyclic cohomology of ´etale groupoids: the general case , K -Theory (1999), no. 4,319–362. MR 1706117[CS13] Marius Crainic and Ivan Struchiner, On the linearization theorem for proper Lie groupoids , Ann. Sci. ´Ec.Norm. Sup´er. (4) (2013), no. 5, 723–746.[dHF18] Matias del Hoyo and Rui Loja Fernandes, Riemannian metrics on Lie groupoids , J. Reine Angew. Math. (2018), 143–173.[DLPR12] L.D. Drager, J.M. Lee, E. Park, and K. Richardson,
Smooth distributions are finitely generated ,Ann. Glob. Anal. Geom. (2012), 357369.[FPS15] C. Farsi, M. Pflaum, and Ch. Seaton, Stratifications of inertia spaces of compact Lie group actions , J. ofSingularities (2015), 107–140, .[FT87] B. L. Fe˘ıgin and B. L. Tsygan, Additive K -theory , K -theory, arithmetic and geometry (Moscow, 1984–1986), Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 67–209. MR 923136[Gro55] Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucl´eaires , Mem. Amer. Math. Soc. (1955), no. 16, 140.[Hel89] A. Ya. Helemskii,
The homology of Banach and topological algebras , Mathematics and its Applications(Soviet Series), vol. 41, Kluwer Academic Publishers Group, Dordrecht, 1989, Translated from the Russianby Alan West. MR 1093462[HKR62] G. Hochschild, Bertram Kostant, and Alex Rosenberg,
Differential forms on regular affine algebras , Trans.Amer. Math. Soc. (1962), 383–408.[HN77] Henri Hogbe-Nlend,
Bornologies and functional analysis , North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977, Introductory course on the theory of duality topology-bornology and its use infunctional analysis, Translated from the French by V. B. Moscatelli, North-Holland Mathematics Studies,Vol. 26, Notas de Matem´atica, No. 62. [Notes on Mathematics, No. 62].[JR´S11] M. Jotz, T. S. Ratiu, and J. ´Sniatycki,
Singular reduction of Dirac structures , Trans. Amer. Math. Soc. (2011), no. 6, 2967–3013, http://dx.doi.org/10.1090/S0002-9947-2011-05220-7.[KM97] Andreas Kriegl and Peter W. Michor,
The convenient setting of global analysis , Mathematical Surveysand Monographs, vol. 53, American Math. Society, 1997.[Lod98] Jean-Louis Loday,
Cyclic homology , second ed., Grundlehren der Mathematischen Wissenschaften, vol.301, Springer-Verlag, Berlin, 1998.[Mat73] John N. Mather,
Stratifications and mappings , Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador,1971), Academic Press, New York, 1973, pp. 195–232.[Mey99] Ralf Meyer,
Analytic cyclic homology , Ph.D. thesis, Universit¨at M¨unster, 1999, arXiv:9906205.[Mey07] ,
Local and analytic cyclic homology , EMS Tracts in Mathematics, vol. 3, European MathematicalSociety (EMS), Z¨urich, 2007.[NGSdS03] Juan A. Navarro Gonz´alez and Juan B. Sancho de Salas, C ∞ -differentiable spaces , Lecture Notes inMathematics, vol. 1824, Springer-Verlag, Berlin, 2003, http://dx.doi.org/10.1007/b13465 .[Pfl98] Markus J. Pflaum, On continuous Hochschild homology and cohomology groups , Lett. Math. Phys. (1998), no. 1, 43–51.[Pfl01] , Analytic and geometric study of stratified spaces , Lecture Notes in Mathematics, vol. 1768,Springer-Verlag, Berlin, 2001.[Pon18] Rapha¨el Ponge,
Cyclic homology and group actions , J. Geom. Phys. (2018), 30–52. MR 3724774[PPT10] M.J. Pflaum, H.B. Posthuma, and X. Tang,
Cyclic cocycles on deformation quantizations and higherindex theorems , Adv. Math. (2010), no. 6, 1958–2021. MR 2601006[PPT14] ,
Geometry of orbit spaces of proper lie groupoids , Journal f¨ur die Reine und Angewandte Mathe-matik (2014), 49–84, http://dx.doi.org/10.1515/crelle-2012-0092.[PPT15] ,
The localized longitudinal index theorem for Lie groupoids and the van Est map , Adv. Math. (2015), 223–262. MR 3286536[PPT17] ,
The Grauert–Grothendieck complex on differentiable spaces and a sheaf complex of Brylinski ,Methods and Applications of Analysis (2017), no. 2, 321–332.[SH18] Ahmad Reza Haj Saeedi Sadegh and Nigel Higson, Euler-like vector fields, deformation spaces and man-ifolds with filtered structure , Documenta Mathematica (2018), 293325.[Spa69] Karlheinz Spallek, Differenzierbare R¨aume , Math. Ann. (1969), 269–296.[Ste80] P. Stefan,
Integrability of systems of vector fields , J. London Math. Soc. (2) (1980), no. 3, 544–556.[Sus73] H´ector J. Sussmann, Orbits of families of vector fields and integrability of distributions , Trans. Amer.Math. Soc. (1973), 171–188.[Tr`e67] Tr`eves,
Topological Vector Spaces, Distributions and Kernels , Academic Press Inc., New York, 1967.[Was88] Antony Wassermann,
Cyclic cohomology of algebras of smooth functions on orbifolds , Operator alge-bras and applications, Vol. 1, London Math. Soc. Lecture Note Ser., vol. 135, Cambridge Univ. Press,Cambridge, 1988, pp. 229–244. MR 996447[Wei96] Charles Weibel,
Cyclic homology for schemes , Proc. Amer. Math. Soc. (1996), no. 6, 1655–1662. [Wei02] Alan Weinstein,
Linearization of regular proper groupoids , J. Inst. Math. Jussieu (2002), no. 3, 493–511.[Zun06] Nguyen Tien Zung, Proper groupoids and momentum maps: linearization, affinity, and convexity , Ann.Sci. ´Ecole Norm. Sup. (4) (2006), no. 5, 841–869. Markus J. Pflaum, [email protected]
Department of Mathematics, University of Colorado, Boulder, USA
Hessel Posthuma,
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands
Xiang Tang, [email protected]@math.wustl.edu