HOMOLOGY OF ALGEBRAS OF FAMILIES OF PSEUDODIFFERENTIAL OPERATORS

MOULAY-TAHAR BENAMEUR AND VICTOR NISTOR

Version: 3.0B; Revised: 6/11/2001; Run: June 18, 2001

Abstract. We compute the Hochschild, cyclic, and periodic cyclic homology groups of algebras of families of Laurent complete symbols on manifolds with corners. We show in particular that the spectral sequence associated with Hochschild homology degenerates at E^{2} and converges to Hochschild homology. As a byproduct, we deduce an identification of the space of residue traces on fibrations by manifolds with corners. In the process, we prove several general results about algebras of complete symbols on manifolds with corners.

Contents

Introduction 1 1. Topological filtered algebras 3 2. Algebras of complete symbols 8 3. Families of manifolds with corners 10 4. Hochschild homology of families 13 5. The relative case 23 6. Examples and applications 25 References 27

Introduction

Some of the main tools in the applications of Non-commutative geometry to index theory and other areas of mathematics are the Hochschild and periodic cyclic homology groups. Hochschild homology, for example, can be used to understand the residue trace introduced by Guillemin and Wodzicki [16, 50]. Other higher residue cocycles appear when studying more complicated singular spaces.

In this paper, we study the Hochschild homology of certain algebras of complete symbols. Recall that an algebra of complete symbols is the quotient of the algebra of all pseudodifferential operators by the ideal of regularizing (or order −∞) operators. Previously, results in this direction were obtained in [4, 10, 22, 32, 27, 37], and by Wodzicki [51] (unpublished). See also [38].

Our algebras of complete symbols can be obtained as algebras of complete symbols on differentiable groupoids [23, 34, 41]. For this large class of examples, it has been shown in [4] that the periodic cyclic homology can be computed, without any further assumption on the groupoid under consideration, in terms of the Laurent cohomology spaces of the cosphere bundle of the associated Lie algebroid. The Hochschild homology groups of algebras of complete symbols on differentiable groupoids, however, cannot be described in general in a simple, uniform way for all differentiable groupoids. Finding the right language in which to express these Hochschild homology groups seems to be a problem in itself–clearly an interesting one.

Partially supported by the NSF Young Investigator Award DMS-9457859, NSF Grant DMS-9971951 and “collaborative CNRS/NSF research grant” DMS-9981251. This manuscript is available from http://www.math.psu.edu/nistor/.

In the present paper, we are concerned with a class of algebras of complete symbols that consist of families of pseudodifferential operators on a fibration by manifolds with corners and with Laurent singularities at the hyperfaces (see Section 3 for precise definitions). Let us consider then a fibration π : M → B of a manifold with corners over a smooth manifold B. Let A_{L}(M|B; E) be the algebra of complete symbols associated to the algebra of smooth families of pseudodifferential operators on the fibers of π and with Laurent singularities at the hyperfaces. (The precise construction of this algebra is done using groupoids, see Section 3.) To our surprise, the resulting algebra turns out not to depend on the groupoid G used to define it, but only on π : M → B, as long as the groupoid G satisfies the assumptions (13) and (14) of Section 3.

Let S_{vert}^{∗}(M) := (T_{vert}^{∗}M ∖ 0)/ℝ_{+}^{∗} denote the sphere bundle of the vertical cotangent bundle to the fibration π : M → B. The periodic cyclic homology of the algebra A_{L}(M|B; E) of Laurent vertical complete symbols with coefficients in a ℤ/2ℤ-graded vector bundle E, is given by Theorem 3:

HP_{j}(A_{L}(M|B; E)) ≃ ⨁_{k∈ℤ} H_{c,L}^{j+2k}(S_{vert}^{∗}(M) × S^{1}), j = 0, 1, (1)

where H_{c,L} denotes compactly supported Laurent cohomology (it is the compactly supported cohomology of a certain space functorially associated to S_{vert}^{∗}(M)).

We also compute the Hochschild homology groups of the algebras A_{L}(M|B; E). For B reduced to a point, this was done in [4]. For families, the computation requires new ingredients, and, in particular, we may obtain infinite dimensional groups. The result is then:

HH_{m}(A_{L}(M|B; E)) ≃ ⨁_{k+h=m} Ω_{c}^{h}(B, F^{2p−k}), (2)

where F^{∗} is the local coefficient system over B given by the Laurent cohomology of the fibers of S_{vert}^{∗}(M) × S^{1} and p is the dimension of the fibers of π : M → B (Theorem 6). This result leads, in particular, to an explicit description of the space of residue traces on the algebras of families of pseudodifferential operators that we consider. For example, when M is smooth (no corners) and π has connected fibers, we obtain that

HH^{0}(A(M|B; E)) ≃ C^{−∞}(B) := C_{c}^{∞}(B)′, (3)

that is, that the space of (super)traces on A(M|B; E) identifies with the space of distributions on B. (Note that in this case A_{L}(M|B; E) = A(M|B; E).) The space of traces in the general case of fibrations π : M → B when M has corners is also explicitly described in terms of the minimal faces of the fibers of π (Theorem 9).

Let us now briefly describe the contents of each section. In Section 1, we briefly recall the basic definitions of ℤ/2ℤ-graded homologies for topologically filtered algebras and give an appropriate criterion for the convergence of the associated spectral sequences. Section 2 is devoted to the description of the algebras of complete symbols that we are interested in. In Section 3, we fix the assumptions on our groupoids and also prove that these algebras depend only on π : M → B, as long as Assumptions (13) and (14) are satisfied. Section 4 is devoted to the computation of the Hochschild homology of our algebras of complete symbols. Along the way, we compute several other homology groups associated to certain Poissson manifolds. In Section 5, we extend the main results of the previous sections to the relative case. The last section, Section 6, treats in detail a few examples. In particular, we obtain a description of the space of traces on our algebras of complete symbols. Note that in this paper almost all results are formulated in the ℤ/2ℤ-graded case, in view of some possible applications.

We hope that the results of this paper will find applications to the index theorem for families [1, 7] or to its generalization to families of fibrations by manifolds with boundary [8].

We would like to thank A. Connes, R. Lauter, R. Melrose, and B. Monthubert for useful discussions. The second named author would like to thank the Max Planck Institute for Mathematics in Bonn for its kind hospitality while parts of this work were completed. He would also like to thank M. Crainic for explaining his results with R. Fernandes on the integration of Lie algebroids [14] (see also Section 6).

1. Topological filtered algebras

Topologically filtered algebras were introduced in [4] to provide a natural framework for the algebras of complete symbols associated to algebras of pseudodifferential operators. In this section we review the definition of topologically filtered algebras and a few other relevant facts. The complexes computing the various homologies of these algebras have to be defined appropriately. In view of the applications that we have in mind, we have found it necessary to extend our setting to include that of ℤ/2ℤ-graded algebras. For basic facts about pseudodifferential operators, see one of the many nice monographs available [36], [43], or [45].

We begin by recalling the definitions of Hochschild and cyclic homology groups of a topological algebra A. A good reference is Connes’ book [12]. See also [20, 26]. See [18] for the homology of ℤ/2ℤ-graded algebras. These definitions have to be (slightly) modified when the multiplication of our algebra is only separately continuous. We thus discuss also the changes necessary to handle the class of algebras that we are interested in, that of “topologically filtered algebras” (Definition 1), and then we prove some results on the homology of these algebras.

First we consider the case of a topological algebra A. Here “topological algebra” has the usual meaning, that is, A is a real or complex algebra, which is at the same time a locally convex space such that the multiplication A × A → A is continuous when A × A is endowed with the product topology. Denote by ˆ⊗ the projective tensor product and H_{n}(A) := A^{ˆ⊗n+1}, the completion of A^{⊗n+1} in the topology of the projective tensor product. Also, we denote as usual by ∂a ∈ ℤ/2ℤ the degree of an element in the ℤ/2ℤ-graded algebra and by b′ and b the Hochschild differentials:

\begin{matrix}b′(a_{0} ⊗ a_{1} ⊗ … ⊗ a_{n}) = ∑_{i=0}^{n−1}(−1)^{i}a_{0} ⊗ … ⊗ a_{i}a_{i+1} ⊗ … ⊗ a_{n}, \\ b(a_{0} ⊗ a_{1} ⊗ … ⊗ a_{n}) = b′(a_{0} ⊗ a_{1} ⊗ … ⊗ a_{n}) + (−1)^{n+μ}a_{n}a_{0} ⊗ … ⊗ a_{n−1}, \end{matrix} (4)

where μ = ∂a_{n}(∂a_{0} + … + ∂a_{n−1}).

The Hochschild homology groups of the algebra A, denoted HH_{∗}(A), are then the homology groups of the complex (H_{n}(A), b). By contrast, the complex (H_{n}(A), b′) is often acyclic, for example when A has a unit. A topological algebra A for which (H_{n}(A), b′) is acyclic is called H-unital (or, better, topological H-unital), following Wodzicki [50].

We now define cyclic homology. Assume first that A is unital. We shall use the notation of [11]. See also [20].

\begin{matrix}s(a_{0} ⊗ a_{1} ⊗ … ⊗ a_{n}) = 1 ⊗ a_{0} ⊗ a_{1} ⊗ … ⊗ a_{n}, \\ t(a_{0} ⊗ a_{1} ⊗ … ⊗ a_{n}) = (−1)^{n+μ}a_{n} ⊗ a_{0} ⊗ … ⊗ a_{n−1}, \\ B_{0}(a_{0} ⊗ a_{1} ⊗ … ⊗ a_{n}) = s ∑_{k=0}^{n} t^{k}(a_{0} ⊗ a_{1} ⊗ … ⊗ a_{n}), and B = (1 − t)B_{0}, \end{matrix} (5)

where μ = ∂a_{n}(∂a_{0} + … + ∂a_{n−1}), as above. Then [b, B]_{+} := bB + Bb = B^{2} = b^{2} = 0, and hence, if we define

C(A)_{n} = ⨁_{k≥0} H_{n−2k}(A), (6)

(C(A), b + B), is a complex, called the cyclic complex of A, whose homology is by definition the cyclic homology of A, as introduced in [11] and [46]. The cyclic homology groups of the algebra A are denoted HC_{n}(A). For algebras without unit A one considers the algebra with adjoint unit A^{+} and then the cyclic homology of A is the kernel of the map HC_{n}(A^{+}) → HC_{n}(ℂ) induced by the augmentation morphism A^{+} → ℂ.

Consideration of the natural periodicity morphism C_{n}(A) → C_{n−2}(A) easily shows that cyclic and Hochschild homology are related by a long exact sequence

(7) … → HH_{n}(A) ⟶^{I} HC_{n}(A) ⟶^{S} HC_{n−2}(A) ⟶^{B} HH_{n−1}(A) ⟶^{I} …,

with the maps I, B, and S explicitly determined. The map S is also called the periodicity operator. See [11, 28]. This exact sequence exists whether or not A is endowed with a topology.

For the algebras that we are interested in, however, the multiplication is usually only separately continuous, but there will usually exist an increasing multi-filtration F_{n,l}^{m}A ⊂ A of A,

F_{n,l}^{m}A ⊂ F_{n′,l′}^{m′}A, if n ≤ n′, l ≤ l′, and m ≤ m′,

by closed, complemented subspaces, invariant under the ℤ_{2}-grading, and satisfying the following properties:

1. A = ∪_{n,l,m}F_{n,l}^{m}A;

2. The union A_{n} := ∪_{m,l}F_{n,l}^{m}A is a closed subspace such that

F_{n,l}^{m}A = A_{n} ∩ ( ∪_{j} F_{j,l}^{m}A);

3. Multiplication maps F_{n,l}^{m}A ⊗ F_{n′,l′}^{m}A to F_{n+n′,l+l′}^{m}A;

4. The maps

F_{n,l}^{m}A/F_{n−1,l}^{m}A ⊗ F_{n′,l′}^{m}A/F_{n′−1,l′}^{m}A → F_{n+n′,l+l′}^{m}A/F_{n+n′−1,l+l′}^{m}A

induced by multiplication are continuous;

5. The quotient F_{n,l}^{m}A/F_{n−1,l}^{m}A is a nuclear Frechet space in the induced topology;

6. The natural map

F_{n,l}^{m}A → lim_{←} F_{n,l}^{m}/F_{n−j,l}^{m}A, j → ∞

is a homeomorphism; and

7. The topology on A is the strict inductive limit of the subspaces F_{n,n}^{n}A, as n → ∞ (recall that F_{n,n}^{n}A is assumed to be closed in F_{n+1,n+1}^{n+1}A).

Definition 1. An algebra A satisfying the conditions 1–7, above, will be called a topologically filtered algebra.

It follows from the definition that if A^{[m]} := ∪_{n,l}F_{n,l}^{m}A, then A^{[m]} is actually a subalgebra of A which is topologically filtered in its own, but with multi-filtration independent of m.

For topologically filtered algebras, the multiplication is not necessarily continuous, and the definition of the Hochschild and cyclic homologies using the projective tensor product of the algebra A with itself, as above, is not very useful. For this reason, we change the definition of the space H_{m}(A) to be an inductive limit:

H_{q}(A) ≃ lim_{→}(F_{n,n}^{n}A)^{ˆ⊗q+1},

the tensor product being the (complete) projective tensor product. The Hochschild homology of A is then still the homology of the complex (H(A), b). Since the projective tensor product is compatible with the projective limits, we also have

H_{q}(A) = lim_{→} (lim_{←} (F_{n,n}^{n}A/F_{k,n}^{n}A)^{ˆ⊗q+1}) , (8)

with the induced topology, where first k → −∞ (in the projective limit) and then n → ∞ (in the inductive limit). The operator B extends to a well defined map B : H_{q}(A) → H_{q+1}(A), which allows us to define the cyclic complex and the cyclic homology of the algebra A as the homology of the complex (C_{∗}(A), b + B), with C_{q}(A) := ⊕H_{q−2k}(A), as for topological algebras.

We also observe that both the Hochschild and cyclic complexes have natural filtrations given by

F_{p}H_{q}(A) := lim_{→} (lim_{←} ˆ⊗_{j=0}^{q} (F_{k_{j},m}^{m}A/F_{k,m}^{m}A)) , (9)

where k_{0} + … + k_{n} ≤ p defines the filtration. The projective and inductive limits are such that first k, l → −∞ (in the projective limit) and then m → ∞ (in the inductive limit). It follows from the definitions that the operators S, B and I preserve these filtrations.

For any topologically filtered algebra, we denote

Gr(A) := ⊕_{n} A_{n}/A_{n−1}

the graded algebra associated to A, where A_{n} is the union ∪_{l,m}F_{n,l}^{m}A, as before. Its topology is that of an inductive limit of Frechet spaces:

Gr(A) ≃ lim_{N,m,l→∞} ⊕_{n=−N}^{N} F_{n,l}^{m}A/F_{n−1,l}^{m}A.

For the algebras like Gr(A), we need yet a third way of topologizing its iterated tensor products. The correct definition is then

H_{q}(Gr(A)) ≃ lim_{N,m,l→∞}(⊕_{n=−N}^{N}F_{n,l}^{m}A/F_{n−1,l}^{m}A)^{ˆ⊗q+1}.

The Hochschild homology of Gr(A) is the homology of the complex (H_{∗}(Gr(A)),b). The operator B again extends to a map B : H_{q}(Gr(A)) → H_{q+1}(Gr(A)) and we can define the cyclic homology of Gr(A) as above. The operators S, B and I associated to H_{q}(Gr(A)) are the graded operators associated with the corresponding operators (also denoted S, B and I) on H_{q}(A).

The Hochschild and cyclic complexes of the algebra Gr(A) decompose naturally as direct sums of complexes indexed by p ∈ ℤ. For example, H_{q}(Gr(A)) is the direct sum of the subspaces H_{q}(Gr(A))_{p}, where

H_{q}(Gr(A))_{p} = lim_{m,N,l→∞} ⨁_{k_{j}} (ˆ⊗_{j=0}^{n}F_{k_{j},l}^{m}A/F_{k_{j}−1,l}^{m}A) ,

where k_{0} + k_{1} + … + k_{n} = p and − N ≤ k_{j} ≤ N,

with the inductive limit topology. The corresponding subcomplexes of the cyclic complex are defined similarly. We denote by HH_{∗}(Gr(A))_{p} and HC_{∗}(Gr(A))_{p} the homologies of the corresponding complexes (Hochschild and, respectively, cyclic).

The following two results are well known consequences of standard results in homological algebra (for topologically filtered algebras they were proved in [4]).

Lemma 1. Let A be a topologically filtered algebra. Then the natural filtrations on the Hochschild and cyclic complexes of A define spectral sequences EH_{k,h}^{r} and EC_{k,h}^{r} such that

EH_{k,h}^{1} ≃ HH_{k+h}(Gr(A))_{k} and EC_{k,h}^{1} ≃ HC_{k+h}(Gr(A))_{k}.

Moreover, the periodicity morphism S induces a morphism S′ : EC_{k,h}^{r} → EC_{k,h−2}^{r} of spectral sequences. For r = 1, the morphism S′ is the graded map associated to the periodicity operator S : HC_{n}(Gr(A)) → HC_{n−2}(Gr(A)) and the natural filtration of the groups HC_{n}(Gr(A)).

Proof. The filtration F_{p}H_{q}(A) of the complex computing the Hochschild homology of A gives rise to a spectral sequence (E^{r})_{r≥1} with

E_{k,h}^{1} = H_{k+h}(F_{k}H(A)/F_{k−1}H(A)),

by standard homological algebra. By the definition of the Hochschild complex of Gr(A), we have:

H_{k+h}(F_{k}H(A)/F_{k−1}H(A)) ≃ HH_{k+h}(Gr(A))_{k}.

This completes the proof for Hochschild homology. The proof for cyclic homology is similar.

In our considerations below, we shall need the following classical result, which was proved for topologically filtered algebras in [4]. Due to the importance of this result in what follows and for the convenience of the reader, we include a proof of it.

Theorem 1. Fix an integer N and a ≥ 1. Let A be a topologically filtered algebra and let A^{[m]} := ∪_{n,l}F_{n,l}^{m}A be such that EH_{k,h}^{a}(A^{[m]}) = 0, for all k < N. Then the spectral sequence EH_{k,h}^{r} = EH_{k,h}^{r}(A) defined in Lemma 1 converges to HH_{k+h}(A). More precisely, we have

HH_{j}(A) ≃ ⊕_{k=N}^{∞} EH_{k,j−k}^{∞} .

A similar result holds for the cyclic homology spectral sequence.

Proof. Assume first that the filtration of A is independent of m. We have

H(A) = lim_{←} H(A)/F_{p}H(A).

This enables to write, for every fixed q, the well known associated lim^{1} exact sequence (see [4][Lemma 6], for example)

0 → lim_{←}^{1} H_{q+1}(H(A)/F_{p}H(A)) → H_{q}(H(A)) → lim_{←} H_{q}(H(A)/F_{p}H(A)) → 0.

Now the spectral sequence E_{k,h}^{r}(p) associated with the Hochschild homology of the filtered complex H(A)/F_{p}H(A) converges because it is the shift of a first quadrant spectral sequence and we have:

H_{q}(H(A)/F_{p}H(A)) ≅ ⊕_{t+s=q}E_{t,s}^{∞}(p).

Therefore, the homology groups H_{q}(H(A)/F_{p}H(A)) are endowed with a filtration F_{t}(p) so that

F_{t}(p)/F_{t−1}(p) ≅ E_{t,q−t}^{∞}(p). (10)

Furthermore, the spectral sequence E_{k,h}^{r}(p) satisfies:

(11) E_{k,h}^{r}(p) = \begin{cases}0 & if k ≤ p and \\ EH_{k,h}^{r} & if k > p + r \end{cases}

Let us choose a particular projective system by setting

A_{n} := H_{q}(H(A)/F_{N−na}H(A)), B_{n} := F_{N−na+a}(N − na) , and C_{n} := A_{n}/B_{n}.

Then the ker-coker lemma [2] for the short exact sequence

0 → ΠB_{n} ↪ ΠA_{n} → ΠC_{n} → 0

gives rise to the following exact sequence [4][Lemma 7]:

0 → lim_{←} B_{n} → lim_{←} A_{n} → lim_{←} C_{n} → lim_{←}^{1}B_{n} → lim_{←}^{1}A_{n} → lim_{←}^{1}C_{n} → 0.

By conditions (10) and (11), the natural map A_{n+1} → A_{n} induces the zero map B_{n+1} → B_{n} and an isomorphism C_{n+1} → C_{n}, for n ≥ 2. Therefore we get:

lim_{←}^{1}A_{n} = 0 and lim_{←} A_{n} = C_{n_{0}}, ∀n_{0} ≥ 2.

And hence, finally,

HH_{q}(A) ≅ C_{n_{0}} = ⊕_{l≥N}E_{l,q−l}^{∞} ≅ ⊕_{l∈ℤ} EH_{l,q−l}^{∞} .

The proof in the general case when the filtration F_{n,l}^{m}A of A depends on m is deduced using that homology is compatible with taking direct limits.

The above isomorphism is not natural, in general, but comes from a filtration F_{k} HH_{q}(A) of HH_{q}(A) whose subquotients F_{k} HH_{q}(A)/F_{k−1} HH_{q}(A) identify naturally with EH_{k,q−k}^{∞}, see [30, 31].

It is useful to mention here that the composite map

HH_{q}(A) →^{I} HC_{q}(A) →^{B} HH_{q+1}(A)

preserves the filtrations and hence it induces natural maps

(B ∘ I)^{(r)} : EH_{k,h}^{r}(A) → EH_{k,h+1}^{r}(A).

For r = 1, this map is the composition of the corresponding morphisms

HH_{q}(Gr(A)) → HC_{q}(Gr(A)) → HH_{q+1}(Gr(A))

for the graded algebra of A.

2. Algebras of complete symbols

We now introduce the algebras of complete symbols that we study in this paper. We shall follow the standard notation for groupoids and Lie algebroids, using

the conventions of [23]. In particular, if G is a differentiable groupoid with space of units M, then d, r : G → M denote the domain and range maps, respectively, so that the composition gg′ of two elements g, g′ ∈ G is defined if, and only if, d(g) = r(g′).

We shall also follow [4] for some specific constructions involving manifolds with corners, some of which are recalled below. As in that paper, we are interested in certain specific groupoid algebras associated to manifolds with corners. If G is a differentiable groupoid with space of units M and E → M is a ℤ/2ℤ-graded vector bundle, then we shall denote by

Ψ^{∞}(G; E) = ∪_{m∈ℤ}Ψ^{m}(G; E)

the algebra of pseudodifferential operators on G acting on sections of the vector bundle r^{∗}E. We also define

Ψ^{−∞}(G; E) := ∩_{m∈ℤ}Ψ^{m}(G; E).

(see [23] or [41] for definitions). These two algebras are naturally ℤ/2ℤ-graded.

We shall denote by O(M) the space of smooth functions on the interior of M that have only Laurent singularities at the boundary faces. If every hyperface H of M has a defining function x_{H}, then O(M) is the ring generated by C^{∞}(M) and x_{H}^{−1}. Let then

A(G; E) := Ψ^{∞}(G; E)/Ψ^{−∞}(G; E) and A_{L}(G; E) = O(M)A(G; E).

The ℤ/2ℤ-grading on E then provides us with a natural ℤ/2ℤ-grading on the algebras A(G; E) and A_{L}(G; E) too.

Proposition 1. Assume that G and M are as above and that M is σ-compact. Then the quotients A(G; E) and A_{L}(G; E) are topologically filtered algebras.

Proof. The proof is exactly the same as the one given for E = ℂ in [4].

Let A(G) be the Lie algebroid of G (see [23]) and let S^{∗}(G) be the sphere bundle of A^{∗}(G), that is, the set of unit vectors in the dual of the Lie algebroid of G, and denote H_{c}^{[q]} = ⊕_{k∈ℤ} H_{c}^{q+2k}.

Theorem 2. Assume that the base M is σ-compact, then the periodic cyclic homology of the algebra A(G; E) is given by

HP_{m}(A(G; E)) ≃ H_{c}^{[m]}(S^{∗}(G) × S^{1}). (12)

Proof. An argument similar to that of Lemma 5 in Section 4 shows that the Hochschild homology is unchanged by introducing the extra vector bundle E and the ℤ/2ℤ-grading. Standard homological algebra arguments then show that the same is true for cyclic and periodic cyclic homology. The result follows then from the case E = ℂ that was proved in [4].

To state the result for the algebra A_{L}(G; E), we need first to recall a construction from [4] that will be used several times in what follows.

Let P be a manifold with corners. Then L(P) is a space naturally associated to P and defined as follows. Consider for each face F of P the space F × (S^{1})^{k}, where k is the codimension of the face. We establish a one-to-one correspondence between the canonical k copies of the unit circle in (S^{1})^{k} and the faces F′ of P containing F, of dimension one higher than that of F. We then identify the points of the disjoint union ∪F × (S^{1})^{k} as follows. If F ⊂ F′ and F′ corresponds to the variable θ_{i} ∈ S^{1} we identify (x, θ_{1}, …, θ_{i−1}, 1, θ_{i+1}, …, θ_{k}) ∈ F × (S^{1})^{k} to the point (x, θ_{1}, …, θ_{i−1}, θ_{i+1}, …, θ_{k}) ∈ F′×(S^{1})^{k−1} (same x). The resulting quotient space is by definition L(P).

By construction, there exists a continuous map p_{L} : L(P) → P. Let J_{ϵ} = S^{1} ∪ [1, 1 + ϵ) ⊂ ℂ, for some ϵ > 0, with S^{1} identified with a subset of the complex plane. Then the space L(P) is locally modeled by J_{ϵ}^{k} × ℝ^{n−k}, above each point of P belonging to an open face of codimension k.

Suppose now that P → B is a fibration by manifolds with corners with B smooth. In particular, B is a smooth manifold. Let Q be the typical fiber of the fibration P → B. Then we obtain by the above construction, a fibration L(P) → B, with typical fiber the locally compact space L(Q).

In [4], the periodic cyclic homology of several algebras of complete symbols was computed. These results include our algebras A_{L}(G; E), when E is trivial. The result is the same in general.

Theorem 3. For j = 0, 1, we have

HP_{j}(A_{L}(G; E)) ≃ H_{c}^{[j]}(L(S^{∗}(G)) × S^{1}).

Proof. When E is trivial one-dimensional, this result was proved in [4]. The general case is proved in the same way, using the same argument as in the proof of Lemma 5, which shows that the Hochschild homology of the algebras A_{L}(G; E) does not depend on the bundle E, thanks to the Morita invariance of Hochschild homology.

In particular, when M has no corners and G = M ×_{B} M, the algebra Ψ^{∞}(G; E) is the algebra of smooth families of pseudodifferential operators along the fibers of π : M → B that have compactly supported Schwartz kernel. Let us denote by A(M|B; E) = A(G; E) the algebra associated to this groupoid .

Corollary 1. For any fibration π : M → B of smooth manifolds (without boundary), we obtain

HP_{j}(A(M|B; E)) ≃ H_{c}^{[j]}(S_{vert}^{∗}(M) × S^{1}), j = 0, 1.

This leads to a complete determination of the periodic cyclic homology of the algebras A(G; E) := Ψ^{∞}(G; E)/Ψ^{∞}(G; E) and A_{L}(G; E) := O(M)A(G; E). The result is moreover easily expressed in a uniform manner for all differentiable groupoids G. The Hochschild homology of these algebras seems to be more difficult to compute, in general. Finding the groups HH_{∗}(A_{L}(G; E)) in general seems to depend on finding the right language in which to express this result. Needless to say, finding the right language to express the groups HH_{∗}(A_{L}(G; E)) and then determining them in general is a worthy problem.

We shall determine the groups HH_{∗}(A_{L}(G; E)) for a class of groupoids that, roughly speaking, consists of families of groupoids of the kind considered in [4]. We now proceed to describe this class in detail.

3. Families of manifolds with corners

To describe the class of differentiable groupoids G for which we shall determine the groups HH_{∗}(A_{L}(G; E)), we first describe the assumptions on the space of units of G. We shall denote the space of units of G by M, where M is a differentiable manifold, possibly with corners, as before. We shall assume that there exists a smooth manifold (without corners) B and a map π : M → B that makes M a differentiable fiber bundle over B with fiber F. We regard this fiber bundle as a being the fiber bundle associated to a principal bundle with structure group Diffeo(F), the group of diffeomorphisms of F that map faces to faces. From now on and throughout the paper, we shall denote n = dim(M), q = dim(B). Also, we shall denote by p the dimension of the fibers of π : M → B, so, in particular, n = p + q.

Fix M as above. We now describe our three assumptions on the groupoid G. Our first assumption on G is that for any arrow g ∈ G, the domain and range of g are in the same fiber of π : M → B, that is,

π(d(g)) = π(r(g)), ∀g ∈ G. (13)

The intuitive meaning of this condition is that the natural action of Ψ^{∞}(G; E) on C_{c}^{∞}(M) via the vector representation [21, 41] is given by families of operators acting on the fibers of π.

Let T_{vert}M be the vertical tangent bundle to the fibration π : M → B. Denote as above by O(M) the space of smooth functions on the interior M_{0} of M that have only Laurent (or rational) type singularities at the faces of M. Let us denote by ϱ : A(G) → T M the anchor map of the Lie algebroid of G. Our second assumption on G is that the map ϱ_{Γ} : Γ(A(G)) → Γ(T_{vert}M) defined by ϱ induces an isomorphism

O(M) ⊗_{C^{∞}(M)} Γ(A(G)) ≃ O(M) ⊗_{C^{∞}(M)} Γ(T_{vert}M), (14)

of vector spaces. Clearly the above map preserves the Lie bracket, so we get an isomorphism of Lie algebras also.

Our third and last assumption on G is a local triviality condition on the algebra A_{L}(G; E). To state this assumption, we need to fix the notation. For any open set V ⊂ B, we denote by G_{V} the reduction of G to π^{−1}(V). Our previous assumptions on G give that G_{V} = (π ∘ d)^{−1}(V). Similarly, for every point b ∈ B, we denote by G_{b} the reduction of G to π^{−1}(b). Again, our assumptions give us that G_{b} = (π ∘ d)^{−1}(b) = (π ∘ r)^{−1}(b). Let us observe that

O(π^{−1}(V)) ⊗ (Ψ^{∞}(V × G_{b}; E)/Ψ^{−∞}(V × G_{b}; E))

has a natural filtration and a natural completion to a topologically filtered algebra, denoted

O(π^{−1}(V)) ⊗_{tf} (Ψ^{∞}(V × G_{b}; E)/Ψ^{−∞}(V × G_{b}; E)) .

Then, for any b ∈ B, we assume the existence of an open neighborhood V ⊂ B of b and a C^{∞}(B)-linear isomorphism

(15) A_{L}(π^{−1}(V)|V; E_{|V}) := O(π^{−1}(V)) (Ψ^{∞}(G_{V}; E)/Ψ^{−∞}(G_{V}; E))

≃ O(π^{−1}(V)) ⊗_{tf} (Ψ^{∞}(V × G_{b}; E)/Ψ^{−∞}(V × G_{b}; E))

of topologically filtered algebras, where V × G_{b} is the product groupoid, with V consisting of just units and the operations being defined pointwise.

The three assumptions above, Equations (13), (14), and (15) are not independent, as we shall see shortly. See [52] for some basic facts about Poisson manifolds.

Lemma 2. Assume that (14) is satisfied. Then the morphism ϱ above induces an isomorphism

O(M) ⊗_{C^{∞}(M)} C^{∞}(A^{∗}(G)) ≃ O(M) ⊗_{C^{∞}(M)} C^{∞}(T_{vert}^{∗}M)

of Poisson algebras.

Proof. Let X, Y, and Z ∈ Γ(A(G)). Then X, Y, and Z define functions (denoted by the same letter) X, Y, Z : A^{∗}(G) → ℝ. Assume Z = [X, Y]. Then the Poisson bracket on C^{∞}(A^{∗}(G)) is uniquely determined by {X, Y } = Z. The equation ϱ([X, Y]) = [ϱ(X), ϱ(Y)] shows that the natural map

C^{∞}(A^{∗}(G)) → C^{∞}(T_{vert}^{∗}M)

is a Poisson map. The proof is completed by including O(M)-coefficients.

Let M_{0} := M ∖∂M be as above the interior of M and let T_{vert}M_{0} be the vertical tangent bundle to the smooth fibration M_{0} → B. Our second assumption implies, in particular, that the anchor map ϱ restricts to an isomorphism

A(G)|_{M_{0}} ≃ T_{vert}M_{0},

of vector bundles.

We now discuss the relation between our three assumptions on G. It turns out that these assumptions do not play equal roles. In fact, the second assumption implies the third one, and, under some weak assumptions on G (d-connectivity) it also implies the first assumption. The following considerations are however somewhat independent from the rest of the paper, and, for the purpose of Hochschild homology computations, the reader can just ignore some of the results below, but instead consider all three assumptions on G.

First, let us notice that, in the same spirit as the above lemma, we get an isomorphism of the algebras of differential operators corresponding to G and to M. More precisely, let Diff(M, G) be the algebra of differential operators on M generated by C^{∞}(M) and Γ(A(G)). Similarly, let Diff(M) be the algebra of differential operators on M generated by C^{∞}(M) and Γ(T M). The anchor map ϱ then gives rise to an inclusion

(16) ϱ_{Diff} : Diff(M, G) → Diff(M).

Proposition 2. Suppose that the map ϱ_{Γ} : Γ(A(G)) → Γ(T M) defined by ϱ is injective. Then our second assumption on G, Equation (14), is equivalent to the fact that ϱ_{Diff} : Diff(M, G) → Diff(M) induces an isomorphism

O(M) Diff(M, G) → O(M) Diff(M).

Proof. The space of vector fields on a manifold coincides with the space of first order differential operators without constant term (i.e. that send the function constant equal to 1 to 0). Thus, the isomorphism O(M) Diff(M, G) ≃ O(M) Diff(M) is equivalent to the fact that O(M)Γ(A(G)) maps surjectively onto O(M)Γ(T M). Since this map is injective by assumption, the result follows.

The algebras A_{L}(G; E) turn out to depend only on π : M → B.

Theorem 4. The algebras A_{L}(G; E) are independent of G, as long as assumptions (13) and (14) are satisfied.

Proof. Assume E is trivial, for simplicity. Let Ψ_{prop}^{∞}(M|B) be the algebra of smooth, properly supported families of operators acting on the fibers of

π_{0} : M_{0} := M ∖ ∂M → B.

Recall [21, 41] that the vector representation π_{v} : Ψ^{∞}(G) → End(C_{c}^{∞}(M_{0})) is defined uniquely by

(π_{v}(P)f) ∘ r = P (f ∘ r),

see [41] for details. Assumption (13) shows that π_{v} factors through a morphism Ψ^{∞}(G) → Ψ_{prop}^{∞}(M|B). Assumption (14) implies that d^{−1}(x) ≃ π_{0}^{−1}(π_{0}(x)) for all x ∈ M_{0}, so the action of some P ∈ Ψ^{∞}(G) on G is completely determined by the action of P on the fibers of π_{0}. This means that the vector representation π_{v} of Ψ^{∞}(G) is injective, so Ψ^{∞}(G) identifies with a subalgebra of Ψ_{prop}^{∞}(M|B). Consequently, A_{L}(G) identifies with a subalgebra of B := Ψ_{prop}^{∞}(M|B)/Ψ_{prop}^{−∞}(M|B).

We now argue that Proposition 2 and asymptotic completeness implies that the image of Ψ^{∞}(G) in B is independent of G. Indeed, it is enough to check that the image of Ψ^{m}(G) → B is independent of G, for any m. Let D ∈ O(M) Diff(M; G) be an elliptic differential operator in O(M)Ψ^{k}(G), for some fixed k ≥ 1. Let Q be a parametrix of D. Then Proposition 2 implies that

O(M) Diff(M; G)[Q] = O(M) Diff(M)[Q]. (17)

Let G_{1} be another differentiable groupoid satisfying Assumptions (13) and (14), then

O(M) Diff(M; G_{1})[Q] = O(M) Diff(M; G)[Q],

also. Because O(M)(Ψ^{m}(G)/Ψ^{−∞}(G)) → B is continuous and injective and the image of the space of operators of order at most m of O(M) Diff(M; G)[Q] is dense in O(M)Ψ^{m}(G)/Ψ^{−∞}(G)), we obtain that the closure of the range of Ψ^{m}(G) in B does not depend on G. By looking at the complete symbols of the images of O(M)(Ψ^{m}(G)) and Ψ^{m}(G_{1}) in B and using asymptotic completeness of the algebras of pseduodifferential operators O(M)(Ψ^{∞}(G)) and Ψ^{∞}(G_{1}), we obtain that the actual range of Ψ^{m}(G) in B is independent of G, as desired.

Let us recall that G is d-connected if, and only if, all the sets G_{x} := d^{−1}(x) are connected.

Corollary 2. Suppose G is a differentiable groupoid with units M. Then assumption (14) implies assumption (15). If G is also d-connected, then (14) implies also assumption (13).

Proof. By Theorem 4, it is enough to check (15) for any fixed groupoid G satisfying (13). In particular, we can choose G to be locally a product, in which case (15) is trivially satisfied. (For example, we could take G = G_{M,b}, the b-groupoid defined in Section 6.)

Let X_{1}, …, X_{m} be sections of A(G). We shall write ϱ(X_{j}) for ϱ_{Γ}(X_{j}). Then

π(exp(ϱ(X_{1})) … exp(ϱ(X_{m}))x) = π(x),

for all x ∈ M. The assumption that G be d-connected is equivalent to the assumption that, for any g ∈ G, there exist X_{1}, …, X_{m} as above such that

r(g) = exp(ϱ(X_{1})) … exp(ϱ(X_{m}))d(g),

see [29].

It also follows from the above discussion that it is enough for our computations to consider a “typical” algebra for each fibration π : M → B. There are several choices of “typical” algebras, in general. One possible choice, the b-calculus, as well as the result of our computations for these algebras, will be described in Section 6.

4. Hochschild homology of families

In this section, we compute the Hochschild homology of the algebras A_{L}(M|B; E) introduced in the previous section. Recall that these algebras are algebras of complete symbols associated with a groupoid G with units M and a fibration π : M → B by manifolds with corners satisfying the assumptions of Equations (13) and (14). The results of this section are already interesting when the manifold M has no boundary. Recall that n = dim(M), q = dim(B), and n = p + q.

In addition to helping us eliminate our third assumption on G, Equation (15), the introduction of the Laurent-type factors also simplifies the calculations, as in [32] and [4]. When B is reduced to a point ∗, this also ensures that the Hochschild homology of A_{L}(M) := A_{L}(M|∗) is finite dimensional. For example, the dimension of the space of traces on A_{L}(M|∗) is the number of minimal faces of M [4]. Moreover, the “cone algebras” described for example in [42] are more closely related to the algebras A_{L}(M|B; E) than to the algebras A(M|B; E). See also [24, 37].

Our computations will use the Poisson structure of A^{∗}(G) and, more precisely, the “homogeneous Laurent-Poisson homology” of A^{∗}(G)∖0, where A^{∗}(G)∖0 is the dual of the Lie algebroid of G, with the zero section removed. The homogeneous Laurent-Poisson homology of A^{∗}(G) ∖ 0 is defined below and will be identified in terms of the “homogeneous, vertical Laurent-de Rham cohomology” of the fibration A^{∗}(G)∖0 → B (this cohomology is also defined below). The homogeneous Laurent-Poisson homology and the homogeneous vertical Laurent-de Rham cohomology are natural analogues of the Poisson and, respectively, de Rham cohomology, which are obtained, roughly speaking, by introducing Laurent type singularities at the corners of M and by considering homogeneous forms (on A^{∗}(G) ∖ 0, for example). See [35, 53, 54] for more on Poisson cohomology.

We begin with the definition of the groups H_{c,L}^{i,j}(A^{∗}(G)∖0|B)_{l}, the homogeneous, vertical Laurent-de Rham cohomology of the fibration A^{∗}(G) ∖ 0 → B. Then we shall discuss Poisson homology and its variant, the homogeneous Laurent-Poisson homology.

Let X be a manifold with corners and let π_{0} : X → B be a fiber bundle with B smooth. Let us call the sections of T_{vert}X vertical vector fields, as it is customary. Then the sections of the dual T_{vert}^{∗}X are called vertical differential forms. There exists a natural (i.e. independent of any choices) differential operator

d_{vert} : Γ(Λ^{k}T_{vert}^{∗}X) → Γ(Λ^{k+1}T_{vert}^{∗}X), (18)

the vertical de Rham differential.

Every vertical vector field on X is also a vector field on X in the usual sense. On the other hand, a form on X restricts to a vertical form on X. Moreover, every vertical form on X is the restriction of a form on X, but we cannot choose that form in a canonical way. A convenient way to choose extensions of vertical forms is to consider a splitting of T X into vertical and horizontal parts. We shall hence fix from now an isomorphism (or splitting)

Θ : T X ≃ T_{vert}X ⊕ π_{0}^{∗}T B. (19)

The splitting Θ of Equation (19) gives rise to an embedding Θ_{k} : Γ(Λ^{k}T_{vert}^{∗}X) → Ω^{k}(X). More generally, we get isomorphisms

Λ^{k}T^{∗}X ≃ ⨁_{i+j=k} Λ^{i}T_{vert}^{∗}X ⊗ π_{0}^{∗}Λ^{j}T^{∗}B.

Let Ω^{i,j}(X) := Γ(X, Λ^{i}T_{vert}^{∗}X ⊗ π_{0}^{∗}Λ^{j}T^{∗}B). Then Ω^{k}(X) ≃ ⨁_{i+j=k} Ω^{i,j}(X), and we also have isomorphisms

Ω^{i,0}(X) ⊗_{C^{∞}(B)} Ω^{j}(B) ∋ ω ⊗ η → ω ∧ π_{0}^{∗}η ∈ Ω^{i,j}(X). (20)

The embedding Θ_{k} can then be used to define a map d_{vert} : Ω^{i,0}(X) → Ω^{i+1,0}(X) (using the same notation is unlikely to cause any confusion in our case). We extend then d_{vert} to a map

d_{vert} : Ω^{i,j}(X) → Ω^{i+1,j}(X),

by using the isomorphisms of the Equation (20) above and setting

d_{vert}(ω ∧ π_{0}^{∗}η) = d_{vert}(ω) ∧ π_{0}^{∗}η

if η ∈ Ω^{i,0}(X) and ω ∈ Ω^{j}(B). Clearly d_{vert}^{2} = 0. The extension d_{vert} that we obtain depends on the splitting Θ of Equation (19). The isomorphism class of the resulting complex, however, does not depend on Θ.

Let us denote by Ω_{L}^{k}(X) the space of k-differential forms on the interior of X that have only rational (or Laurent) singularities near the corners. We shall sometimes call forms with these properties Laurent-differential forms. The above definitions and properties extend to Ω_{L}^{k}(X) as follows. Let Ω_{L}^{i,j}(X) := O(X)Γ(X, Λ^{i}T_{vert}^{∗}X ⊗ π_{0}^{∗}Λ^{j}T^{∗}B). Then

Ω_{L}^{k}(X) ≃ ⨁_{i+j=k} Ω_{L}^{i,j}(X),

and, as before, we obtain a differential

d_{vert} : Ω_{L}^{i,j}(X) → Ω_{L}^{i+1,j}(X).

We shall denote by H_{L}^{i,j}(X) = ker(d_{vert})/d_{vert}Ω_{L}^{i−1,j}(X) the homology of the above complex. Similarly, if compactly supported forms are considered, we obtain a complex whose homology we denote by H_{c,L}^{i,j}(X).

Define the horizontal differential d_{hor} : Ω_{L}^{i,j}(X) → Ω_{L}^{i,j+1}(X) as the component of bidegree (0, 1) of d. Then ∂ := d − d_{vert} − d_{hor} is known to be a differential and to have bidegree (−1, 2). See for instance [47].

The equality d^{2} = 0 is equivalent to the following relations:

d_{vert}d_{hor} + d_{hor}d_{vert} = 0, d_{hor}^{2} + ∂d_{vert} + d_{vert}∂ = 0

d_{vert}^{2} = 0, and ∂d_{hor} + d_{hor}∂ = 0.

The vertical Laurent-de Rham cohomology can be computed in a fairly explicit manner. Indeed, let F^{k} be the local coefficient system determined by the Laurent cohomology groups of the fibers of X → B. Thus F^{k} is a canonically flat vector bundle over B whose fiber at b ∈ B is

F^{k}(b) = H_{c,L}^{k}(π_{0}^{−1}(b)) := H_{c,L}^{k}(π_{0}^{−1}(b)|b). (21)

Let Ω_{c}^{k}(B) be the space of compactly supported k-forms on B.

Proposition 3. Using the above notation, we have that

H_{c,L}^{k,h}(X|B) ≃ Ω_{c}^{h}(B) ⊗_{C^{∞}(B)} Γ(F^{k}) =: Ω_{c}^{h}(B; F^{k}), (22)

the space of compactly supported h-forms on B with values in F^{k}. In particular, the vertical Laurent-de Rham cohomology groups H_{c,L}^{k,h}(X|B) are independent of the splitting T X ≃ T_{vert}X ⊕ π_{0}^{∗}T B used to define them, see Equation (19).

In addition, the action induced by the horizontal de Rham differential d_{hor} on H_{c,L}^{k,h}(X|B) is isomorphic under (22) to the de Rham differential on B with coefficients in the locally constant sheaf F^{k}.

Proof. The above formula is checked right away when π : X → B is a trivial fiber bundle (i.e. X = B × F) by using the Künneth formula for the tensor product of complexes of nuclear vector spaces, [19]. Moreover, an automorphism of the trivial fiber bundle X = B × F does not affect the isomorphism of the proposition. A partition of unity argument then completes the proof.

Let A^{∗}(G) ∖ 0 be obtained from the vector bundle A^{∗}(G), as before, by removing the zero section. We are interested in the above constructions when X = A^{∗}(G) ∖ 0 and π_{0} : A^{∗}(G) ∖ 0 → B is obtained from the composition of the maps A^{∗}(G) → M and π : M → B. More precisely, for us, the relevant cohomology groups are the cohomology groups obtained by considering homogeneous forms. Let then Ω_{rc,L}^{i,j}(A^{∗}(G) ∖ 0)_{l} be the space of l-homogeneous forms in

Ω_{L}^{i,j}(A^{∗}(G) ∖ 0) = O(M)Γ(A^{∗}(G) ∖ 0, Λ^{i}T_{vert}(A^{∗}(G) ∖ 0)^{∗} ⊗ π_{0}^{∗}Λ^{j}T^{∗}B)

whose support projects onto a compact subset of M. Here the homogeneity is considered with respect to the natural action of ℝ_{+}^{∗} on A^{∗}(G) ∖ 0 by dilations. We denote then by

H_{c,L}^{i,j}(A^{∗}(G) ∖ 0|B)_{l}

the homology of the complex Ω_{rc,L}^{i,j}(A^{∗}(G)∖0)_{l} with respect to the vertical de Rham differential d_{vert}. We shall call these groups the homogeneous, vertical Laurent-de Rham cohomology groups of A^{∗}(G).

Similar constructions and definitions are obtained with T_{vert}^{∗}M in place of A^{∗}(G). Our second assumption on the groupoid G, Equation (14), gives that the two cohomologies are isomorphic.

Lemma 3. The anchor map ϱ : A(G) → T_{vert}M induces a natural isomorphism

H_{c,L}^{i,j}(A^{∗}(G) ∖ 0|B)_{l} ≃ H_{c,L}^{i,j}(T_{vert}^{∗}M ∖ 0|B)_{l}.

These groups vanish if l ≠ 0 and, for l = 0, we have

H_{c,L}^{i,j}(A^{∗}(G) ∖ 0|B)_{0} ≃ H_{c,L}^{i,j}(T_{vert}^{∗}M ∖ 0|B)_{0}

≅ H_{c,L}^{i,j}(S_{vert}^{∗}(M) × S^{1}|B) ≅ H_{c,L}^{i,j}(S^{∗}(G) × S^{1}|B).

Proof. The map ϱ induces an isomorphism of the corresponding complexes, by Equation (14) and the definition of the spaces Ω_{rc,L}^{i,j}(A^{∗}(G) ∖ 0)_{l}. The vanishing of the groups H_{c,L}^{i,j}(T_{vert}^{∗} ∖ 0|B)_{l} for l ≠ 0 follows from the homotopy invariance of de Rham cohomology. The computation of the 0-homogeneous cohomology spaces is classical, see for instance [10, 32].

We can thus replace A^{∗}(G) with T_{vert}^{∗}M for the rest of our computations of the homogeneous, vertical Laurent-de Rham cohomology groups of A^{∗}(G).

The homogeneous, vertical Laurent-de Rham cohomology can be computed using a method similar to the one we used to determine the non-homogeneous homology. Indeed, let F^{k} be the local coefficient system determined by the Laurent cohomology groups of the fibers of π_{0} : S_{vert}^{∗}(M)×S^{1} → B. Thus F^{k} is a canonically flat vector bundle over B whose fiber at b ∈ B is

F^{k}(b) = H_{c,L}^{k}(π_{0}^{−1}(b)) = H_{c}^{k}(L(π_{0}^{−1}(b))). (23)

Proposition 4. Using the above notation, we have that

H_{c,L}^{k,h}(A^{∗}(G) ∖ 0|B)_{0} ≃ Ω_{c}^{h}(B) ⊗_{C^{∞}(B)} Γ(F^{k}) =: Ω_{c}^{h}(B; F^{k}). (24)

For l ≠ 0, the groups H_{c,L}^{k,h}(A^{∗}(G) ∖ 0|B)_{l} vanish. The horizontal de Rham differential d_{hor} induces a differential on H_{c,L}^{k,h}(X|B) which is isomorphic under (24) to the de Rham differential on B with coefficients in the locally constant sheaf F^{k}.

Proof. The proof is completely similar to that of Proposition 3. The vanishing of the groups H_{c,L}^{k,h}(A^{∗}(G)∖0|B)_{l} for l ≠ 0 follows again from the homotopy invariance of Laurent-de Rham cohomology.

Let us now introduce the Poisson homology groups that we are interested in. The following considerations apply to any regular Poisson structure. Recall that the Poisson structure on A^{∗}(G) is defined by a two tensor

G ∈ C^{∞}(A^{∗}(G), Λ^{2}T A^{∗}(G))

so that {f, g} = i_{G}(df ∧ dg). Clearly, the tensor G must satisfy some non-trivial conditions for the map { , } to satisfy the Jacobi identity. These conditions turn out to be equivalent to [G, G]_{SN} = 0, where [ , ]_{SN} is the Schouten-Nijenhuis bracket [48]. The formula for the Poisson bracket is determined in terms of the Lie algebra structure on the space of sections of A(G). (This was recalled in Lemma 2.)

Let i_{G} : Ω^{k}(A^{∗}(G)) → Ω^{k−2}(A^{∗}(G)) be the contraction with the tensor G. Then we obtain as in [9] a differential

δ := i_{G} ∘ d − d ∘ i_{G} : Ω^{k}(A^{∗}(G)) → Ω^{k−1}(A^{∗}(G)). (25)

Explicitly, for any (f_{0}, …, f_{k}) ∈ C^{∞}(A^{∗}(G))^{k+1}, the differential δ is given by the formula

(26) δ(f_{0}df_{1}df_{2} … df_{k}) = ∑_{1≤j≤k}(−1)^{j+1}{f_{0}, f_{j}}df_{1} … d̂f_{j} … df_{k}

+ ∑_{1≤i<j≤k}(−1)^{i+j}f_{0}d{f_{i}, f_{j}}df_{1} … d̂f_{i} … d̂f_{j} … df_{k}.

When G is our groupoid and under the Assumptions (13) and (14), we can use the splitting Θ of Equation (19). One then shows that δ decomposes into the sum of two bihomogeneous differentials [48]

δ = δ_{vert} + α,

where δ_{vert} := i_{G} ∘ d_{vert} − d_{vert} ∘ i_{G} is the vertical Poisson differential. The vertical Poisson differential has bidegree (−1, 0). The extra term α has bidegree (−2, +1) and is in fact given by [5]:

α = i_{G} ∘ d_{hor} − d_{hor} ∘ i_{G}.

See [5]. In particular, the commutator [i_{G}, ∂] is trivial. It is straightforward to see that the restriction of δ_{vert} to vertical differential forms is given by the following local expression

(27) δ_{vert}(f_{0}d_{vert}f_{1}d_{vert}f_{2} … d_{vert}f_{k})

= ∑_{1≤j≤k}(−1)^{j+1}\{f_{0}, f_{j}\}d_{vert}f_{1} … { \atop d_{vert}f_{j}} … d_{vert}f_{k}

+ ∑_{1≤i<j≤k}(−1)^{i+j}f_{0}d_{vert}\{f_{i}, f_{j}\}d_{vert}f_{1} … { \atop d_{vert}f_{i}} … { \atop d_{vert}f_{j}} … d_{vert}f_{k}.

The above formula determines δ_{vert} on Ω^{j,0}(A^{∗}(G)). To determine δ_{vert} in general, we can use the following lemma.

Lemma 4. Let α ∈ Ω^{i}(A^{∗}(G)), let β ∈ Ω^{j}(B), and let π_{0} : A^{∗}(G) → B be the composite projection. Then

δ(α ∧ π_{0}^{∗}(β)) = δ(α) ∧ π_{0}^{∗}(β) and δ_{vert}(α ∧ π_{0}^{∗}(β)) = δ_{vert}(α) ∧ π_{0}^{∗}(β).

Proof. It is enough to check the first equation when β = g or β = dg, for some smooth function g on B. Our claim then follows from the fact that {f, g ∘ π_{0}} = 0, for any smooth function f on M and from the explicit formula for δ, Equation (26).

The equation for δ_{vert} := i_{G} ∘ d_{vert} − d_{vert} ∘ i_{G} follows from the equation for δ by checking bidegrees.

The formula of Equation (26) is valid also when M has corners and it is easy to check that the differential δ is homogeneous of degree −1 with respect to the action of ℝ_{+}^{∗} on A^{∗}(G) ∖ 0. Let Ω^{k}(A^{∗}(G) ∖ 0)_{l} be the space of k-forms on A^{∗}(G) ∖ 0 that are homogeneous of order l. We hence obtain a differential

δ : Ω^{k}(A^{∗}(G) ∖ 0)_{l} → Ω^{k−1}(A^{∗}(G) ∖ 0)_{l−1}.

Let Ω_{rc}^{k}(A^{∗}(G) ∖ 0)_{l} be the subspace of Ω^{k}(A^{∗}(G) ∖ 0)_{l} consisting of forms whose support projects onto a compact subset of M, as before. Because δ preserves the support, it maps the space Ω_{rc}^{k}(A^{∗}(G) ∖ 0)_{l} to Ω_{rc}^{k−1}(A^{∗}(G) ∖ 0)_{l−1}.

The same result holds with δ_{vert} and we have:

δ_{vert} : Ω_{rc,L}^{k}(A^{∗}(G) ∖ 0)_{l} → Ω_{rc,L}^{k−1}(A^{∗}(G) ∖ 0)_{l−1}.

We obtain in this way a direct sum of complexes (P^{k})_{k∈ℤ}

P^{k} : 0 → P_{2p+q}^{k} →^{δ} P_{2p+q−1}^{k} →^{δ} … →^{δ} P_{−k}^{k} → 0, (28)

where P_{l}^{k} = Ω_{rc,L}^{k+l}(A^{∗}(G) ∖ 0)_{l}.

We shall denote the homology groups of the above complex by

H_{L,k+l}^{δ}(A^{∗}(G) ∖ 0|B)_{l} := \frac{ker(δ : P_{l}^{k} → P_{l−1}^{k})}{δ(P_{l+1}^{k})}.

In the same way we define the vertical homogeneous Laurent-Poisson homology groups using δ_{vert} instead of δ and denote them by

H_{L,k+l}^{δ_{vert}}(A^{∗}(G) ∖ 0|B)_{l} := \frac{ker(δ_{vert} : P_{l}^{k} → P_{l−1}^{k})}{δ_{vert}(P_{l+1}^{k})}.

Furthermore, we define for any (i, j)

P_{l}^{i,j} := Ω_{rc,L}^{i+l,j}(T_{vert}^{∗}M ∖ 0)_{l}.

From the results of Section 3, we deduce that

P_{l}^{k} ≃ ⨁_{i+j=k} P_{l}^{i,j}.

Note that with respect to the splitting (19), we have:

δ_{vert} : P_{l}^{i,j} → P_{l−1}^{i,j},

and the vertical Laurent Poisson homology can be computed by fixing (i, j) and restricting to each P^{i,j} := ⊕_{l}P_{l}^{i,j}. However, the extra term α does not preserve P^{i,j}, and sends P^{i,j} to P^{i−1,j+1}.

The relevance for us of Poisson homology, in general, and of homogeneous Laurent-Poisson homology, in particular, is that they are related to the Hochschild homology groups of the algebras A_{L}(M|B; E) := O(M)(Ψ^{∞}(G; E)/Ψ^{−∞}(G; E)) introduced in the previous section, where E is a ℤ/2ℤ-graded vector bundle. The following lemma makes this connection precise.

Lemma 5. The algebra A_{L}(M|B; E) is topologically H-unital. The EH^{2} term of the spectral sequence associated to A_{L}(M|B; E) by Lemma 1 is given by

EH_{k,h}^{2} ≃ H_{L,k+h}^{δ}(A^{∗}(G) ∖ 0|B)_{k}.

Proof. When E is a trivial one-dimensional vector bundle and G is an arbitrary groupoid, the above proposition was proved in [4, Proposition 7].

The extension to a non-trivial vector bundle and the ℤ/2ℤ-graded case is obtained as a consequence of Künneth formula as follows. Let us assume first that E is trivially graded. The graded algebra of A_{L}(M|B; E) is the algebra generated by the homogeneous sections of the lift of E to A^{∗}(G) ∖ 0. All these algebras are Morita equivalent, so their Hochschild homology is the same as that of the algebra corresponding to a trivial line bundle. Since the Hochschild homology of a commutative algebra can be represented as the homology of the space of global sections of a complex of sheaves, the statement about Morita equivalence is a local statement, which can then be proved locally using the Künneth formula. The spectral sequence of Lemma 1 then tells us that the Hochschild homology of the algebra A_{L}(M|B; E) is independent of E.

In general, let E = E_{+} ⊕ E_{−} be the decomposition of E into the direct sum of the +1 and, respectively, −1 eigenvalue of the grading automorphism. As above, we observe that the statement of the lemma is local, so we can assume that E = E_{+} ⊕ E_{−} is such that both E_{+} and E_{−} are trivial bundles. Denote by N the rank of E. Then A_{L}(M|B; E) ≃ M_{N}(A_{L}(M|B)).

Let A be a (topologically filtered) algebra A and N an integer. Assume that the grading automorphism of the algebra M_{N}(A) is given by conjugation with a matrix in M_{N}(ℂ). Then HH_{∗}(M_{N}(A)) ≃ HH_{∗}(A). This follows for example from the Künneth formula in (ℤ/2ℤ-graded) Hochschild homology (see [18]).

As with the homogeneous, vertical de Rham homology, we can replace A^{∗}(G) in H_{L,k}^{δ}(A^{∗}(G) ∖ 0|B)_{l} with T_{vert}^{∗}M, the vertical cotangent bundle to π : M → B.

Lemma 6. The anchor map ϱ : A(G) → T_{vert}M induces an isomorphism

H_{L,k}^{δ}(A^{∗}(G) ∖ 0|B)_{l} ≃ H_{L,k}^{δ}(T_{vert}^{∗}M ∖ 0|B)_{l}.

Proof. This follows from Lemma 2 and the explicit formula for the Poisson bracket, Equation (26).

We now proceed as usual and construct a chain map ∗_{vert} from the complex that defines vertical Poisson homology to the complex that defines de Rham cohomology. The chain map ∗_{vert} is, in a certain sense, a vertical symplectic ∗-operator. It corresponds to the canonical symplectic forms on the cotangent spaces of the fibers of π : M → B. Denote by M_{b} := π^{−1}(b), b ∈ B, and by ω_{b} the symplectic form on T^{∗}M_{b}. There exists then a 2-form ω on T_{vert}^{∗}M := ∪_{b∈B}T^{∗}M_{b} that restricts on each fiber T^{∗}M_{b} to the form ω_{b}. This form is certainly not unique. There will be, however, a unique form ω ∈ Ω_{L}^{2,0}(T_{vert}^{∗}M) with this property. We shall call this form ω the vertical symplectic form of T_{vert}^{∗}M. It depends on the splitting of Equation (19).

The vertical symplectic volume form on T_{vert}^{∗}M is defined by analogy to be vol_{vert}(M) := ω^{p}/p!. Next, we define ∗_{vert} : Ω_{L}^{k,0}(T_{vert}^{∗}M) → Ω_{L}^{2p−k,0}(T_{vert}^{∗}M) by the equation

β ∧ (∗_{vert}α) = (β, α)_{ω} · vol_{vert}(M), ∀ α, β ∈ Ω_{L}^{k,0}(T_{vert}^{∗}M),

where ( , )_{ω} is the bilinear form induced by the symplectic form. Then we obtain that ∗_{vert}(Ω_{L}^{k,0}(T_{vert}^{∗}M)_{l}) = Ω_{L}^{2p−k,0}(T_{vert}^{∗}M)_{l+p−k}. Finally, to define

∗_{vert} : Ω_{L}^{i,j}(T_{vert}^{∗}M) → Ω_{L}^{2p−i,j}(T_{vert}^{∗}M)

in general, it is enough to define ∗_{vert}α when α = η ∧ π_{0}^{∗}β, with η ∈ Ω_{L}^{i,0}(M) and β ∈ Ω^{j}(B), where π_{0} : T_{vert}^{∗}M → B is the induced projection. We set then

∗_{vert}(α) = ∗_{vert}(η ∧ π_{0}^{∗}β) := ∗_{vert}(η) ∧ π_{0}^{∗}β. (29)

Similarly, we obtain again that ∗_{vert}(Ω_{L}^{i,j}(T_{vert}^{∗}M)_{l}) = Ω_{L}^{2p−i,j}(T_{vert}^{∗}M)_{l+p−i}.

The usual properties of the symplectic ∗-operator in relation to de Rham and Poisson homology extend to ∗_{vert}.

Proposition 5. Let ∗_{vert} be the operator defined above. Then (1) ∗_{vert}^{2} = id;

(2) (−1)^{i+1}d_{vert} ∘ ∗_{vert} = ∗_{vert} ∘ δ_{vert} on Ω^{i,j}(T_{vert}^{∗}M). We can extend the range of both formulas to include homogeneous forms or forms with Laurent type singularities.

Proof. Both formulas are well known when B is reduced to a point [9]. Using a version with parameters of this particular case, we obtain that the two formulas are correct on Ω^{i,0}(T_{vert}^{∗}M).

For the general case, let α ∈ Ω^{i,0}(T_{vert}^{∗}M) be of the form α = η ∧ π_{0}^{∗}β, where η ∈ Ω_{L}^{i,0}(M) and β ∈ Ω_{c}^{j}(B), and π_{0} : T_{vert}^{∗}M → B is the induced projection. Then, using Equation (29), we obtain

∗_{vert}^{2}(α) = ∗_{vert}^{2}(η) ∧ π_{0}^{∗}β = α. (30)

Similarly, using the definition of d_{vert}, Lemma 4, and Equation (29), we obtain

(−1)^{i+1}d_{vert} ∘ ∗_{vert}(α) = (−1)^{i+1}d_{vert} ∘ ∗_{vert}(η) ∧ π_{0}^{∗}β

= ∗_{vert} ∘ δ_{vert}(η) ∧ π_{0}^{∗}β = ∗_{vert} ∘ δ_{vert}(α).

This is enough to complete the proof.

We are ready now to determine the homogeneous, Laurent-Poisson homology groups of A^{∗}(G). Recall that p denotes the dimension of the fibers of M → B and q is the dimension of the manifold B. We set for any fixed k ∈ ℤ,

K^{j,l} := P_{l−j}^{k−j,j},

so that

δ_{vert} : K^{j,l} → K^{j,l−1} and α : K^{j,l} → K^{j+1,l}.

To compute the homogeneous Laurent δ-homology of A^{∗}(G) ∖ 0, we use that the complex splits into subcomplexes (P^{k}, δ). Thus we can fix the integer k ∈ ℤ and define a filtration of the above bicomplex K^{j,l} by

F_{h} := ⨁_{l∈ℤ,j≤h} K^{j,l}.

Proposition 6. The spaces H_{L,k+l}^{δ}(A^{∗}(G) ∖ 0|B)_{l} and H_{L,k+l}^{δ_{vert}}(A^{∗}(G) ∖ 0|B)_{l} are isomorphic, that is

H_{L,k+l}^{δ}(A^{∗}(G) ∖ 0|B)_{l} ≃ H_{L,k+l}^{δ_{vert}}(A^{∗}(G) ∖ 0|B)_{l}.

Proof. Recall that we have

δ = δ_{vert} + α and δ_{vert} ∘ α + α ∘ δ_{vert} = 0.

We then use for any fixed k the decomposition P^{k} ≅ ⨁_{i+j=k} P^{i,j} into a finite double complex and the decreasing filtration F_{h} defined above. This yields a spectral sequence (E^{r})_{r≥1} which converges to the δ-homology by classical homological arguments. The E^{1} term of this spectral sequence is given by

E_{u,v}^{1} = H_{L}^{2p−v−k+u,u}(T_{vert}^{∗}M ∖ 0|B)_{p−k+u}.

But again by a homotopy argument, the homogeneous vertical de Rham cohomology space H_{L}^{2p−v−k+u,u}(T_{vert}^{∗}M ∖ 0|B)_{p−k+u} is trivial unless u = k − p. Therefore, we get:

E_{u,v}^{1} = 0 if v ≠ −k − p.

Hence for any r ≥ 1, we see that d^{r} = 0 and the spectral sequence collapses at E^{1}. The proof is thus complete.

Theorem 5. The homogeneous, Laurent-Poisson homology groups of A^{∗}(G) are given by

H_{L,k}^{δ}(A^{∗}(G) ∖ 0|B)_{l} ≃ H_{c,L}^{p−l,k−l−p}(T_{vert}^{∗}M ∖ 0|B)_{0}.

Proof. The vertical symplectic Hodge operator ∗_{vert} yields isomorphisms

∗_{vert} : ⨁_{i+j=k} Ω_{L}^{i,j}(T_{vert}^{∗}M ∖ 0)_{l} → ⨁_{i+j=k} Ω_{L}^{2p−i,j}(T_{vert}^{∗}M ∖ 0)_{l+p−i}

which intertwine the δ_{vert} and d_{vert} differentials (up to a sign). Proposition 6 shows then that

H_{L,k}^{δ}(A^{∗}(G) ∖ 0|B)_{l} ≃ ⨁_{i+j=k} H_{c,L}^{2p−i,j}(T_{vert}^{∗}M ∖ 0|B)_{l+p−i}.

But for l + p − i ≠ 0, the cohomology spaces H^{2p−i,j}(T_{vert}^{∗}M ∖ 0|B)_{l+p−i} are trivial by the homotopy invariance of de Rham cohomology. Hence the only non trivial term is:

H_{c,L}^{p−l,k−l−p}(T_{vert}^{∗}M ∖ 0|B)_{0},

and this completes the proof.

We can now apply the results of Section 1 together with Theorem 5.

Proposition 7. The EH^{2}-term of the spectral sequence associated in Lemma 1 to the Hochschild homology of the algebra A_{L}(M|B; E) is given by:

EH_{k,h}^{2} ≃ H_{c,L}^{p−k,h−p}(S_{vert}^{∗}(M) × S^{1}|B),

where S_{vert}^{∗}(M) is the sphere bundle of the vertical cotangent bundle T_{vert}^{∗}M.

Proof. Denote by δ the Poisson differential on the vertically symplectic fibration T_{vert}^{∗}M → B. Lemma 5, Proposition 6 and Theorem 5 give by straightforward computation:

EH_{k,h}^{2} ≃ H_{c,L}^{p−k,h−p}(T_{vert}^{∗}M ∖ 0|B)_{0}.

Now a classical argument shows that [32]:

H_{c,L}^{p−k,h−p}(T_{vert}^{∗}M ∖ 0|B)_{0} ≃ H_{c,L}^{p−k,h−p}(S_{vert}^{∗}(M) × S^{1}|B),

which completes the proof.

In the following lemma, we shall denote by ⊗_{tf} the completion of the tensor product of two algebras in the unique natural way that makes the completed tensor product a topologically filtered algebra. See also Equation (15) where ⊗_{tf} was used before.

Lemma 7. Assume that the fibration π : M → B and the bundle of algebras A_{L}(M|B; E) are trivial; that is, assume that M = B × F and A_{L}(M|B; E) ≃ A_{L}(F; E) ⊗_{tf} C_{c}^{∞}(B) as topologically filtered algebras. Then the spectral sequence associated in Lemma 1 to the Hochschild homology of the algebra A_{L}(M|B; E) collapses at EH^{2}, converges, and we have

HH_{k}(A_{L}(M|B; E)) ≃ ⊕_{i+j=k} H_{c,L}^{2p−i}(S^{∗}(F) × S^{1}) ⊗ Ω_{c}^{j}(B).

Proof. We know from [4] that HH_{j}(A_{L}(F; E)) ≃ H_{c}^{2p−j}(L(S^{∗}(F))×S^{1}). The usual Künneth formula for the Hochschild homology of algebras then gives that

(31) HH_{k}(C_{c}^{∞}(B) ⊗ A_{L}(F; E)) ≃ ⊕_{i+j=k} HH_{i}(C_{c}^{∞}(B)) ⊗ HH_{j}(A_{L}(F; E))

≃ ⊕_{i+j=k}Ω_{c}^{i}(B) ⊗ H_{c,L}^{2p−j}(S^{∗}(F) × S^{1}).

The last equality is a consequence of the computation of HH_{j}(A_{L}(F; E)) carried out in [4]. The inclusion C_{c}^{∞}(B) ⊗ A_{L}(F; E) → C_{c}^{∞}(B) ⊗_{tf} A_{L}(F; E) = A_{L}(M|B; E) preserves the natural filtrations, and hence it induces a morphism of the corresponding spectral sequences. By Proposition 7, Equation (31), and naturality, this morphism is an isomorphism for the E^{2}-terms. We also obtain that all classes in the E^{2} term of the spectral sequence associated to C_{c}^{∞}(B)⊗A_{L}(F; E) survive in the E^{∞} term (they give rise to Hochschild homology classes), and hence the spectral sequence associated to C_{c}^{∞}(B) ⊗ A_{L}(F; E) degenerates at E^{2}. Consequently, the spectral sequence associated to A_{L}(M|B; E) also degenerates at E^{2}.

Then Theorem 1 can be used to complete the proof.

We now extend the above lemma to more general groupoids.

Lemma 8. The spectral sequence EH^{r} associated to the Hochschild homology of A_{L}(M|B; E) by Lemma 1 degenerates at EH^{2} and converges to its Hochschild homology.

Proof. Denote A = A_{L}(M|B; E) in this proof, for simplicity. The differential b of the Hochschild complex of A is C^{∞}(B)-linear, if f ∈ C^{∞}(B) acts on a_{0} ⊗…⊗a_{n} by f(a_{0}⊗…⊗a_{n}) := (fa_{0})⊗…⊗a_{n}. The filtrations of the Hochschild complex are also preserved by the multiplication operators with functions f ∈ C^{∞}(B). This shows that the spectral sequence associated to the Hochschild homology of A_{L}(M|B; E) by Lemma 1 consists of C^{∞}(B)-modules.

Let A^{K} consist of the complete symbols with support in a fixed compact set K ⊂ M. Let K_{m} ⊂ M be an exhausting sequence of M by compact subsets (that is, M = ∪K_{m} and each K_{m} is contained in the interior of K_{m+1}). We can assume that A^{[m]} = A^{K_{m}}. (This sequence can be reduced to one set if M is compact.) To prove that E_{k,h}^{2}(A^{[m]}) = 0 or that d^{r} = 0 for this spectral sequence, it is enough to check that fE_{k,h}^{2}(A^{[m]}) = 0 (or fd^{r} = 0) for any f with support in K_{m}. But fE_{k,h}^{2}(A^{[m]}) = fE_{k,h}^{2}(A) if the support of f is inside K_{m}. This shows that E_{k,h}^{2}(A^{[m]}) = 0 if E_{k,h}^{2}(A) = 0.

We now reverse the argument. To prove that E_{k,h}^{2}(A) = 0 or that d^{r} = 0 for this spectral sequence, it is enough to check that fE_{k,h}^{2}(A) = 0 (or fd^{r} = 0) for any f in a trivializing open subset V of M (that is, a subset satisfying the conditions of our third assumption on the algebra A = A_{L}(M|B; E), Equation (15)), more precisely, satisfying that

(32) A_{V} := O(π^{−1}(V)) (Ψ^{∞}(G_{V}; E)/Ψ^{−∞}(G_{V}; E))

≃ O(π^{−1}(V)) ⊗_{tf} (Ψ^{∞}(V × G_{b}; E)/Ψ^{−∞}(V × G_{b}; E))

is a C^{∞}(B)-linear isomorphism. Then fE_{k,h}^{2}(A) = fE_{k,h}^{2}(A_{V}). By Lemma 7, E_{k,h}^{r} = 0 if k < −p. Similarly, fd^{r} = 0 if r ≥ 2.

This proves that the spectral sequence in Hochschild homology associated to the algebra A_{L}(M|B; E) by Lemma 1 degenerates at EH^{2}. It also proves that the assumptions of Theorem 1 are satisfied, so the spectral sequence EH_{k,h}^{r} converges to Hochschild homology.

Now we can state and prove the main theorem of this section. Let G be a differentiable groupoid whose space of units is a manifold with corners M which is the total space of a fibration π : M → B, as before. Denote by F^{j} the local coefficient system defined by

F^{j}(b) := HH_{2p−j}(A_{L}(π^{−1}(b))) ≃ H_{c,L}^{j}(π_{0}^{−1}(b)) ≃ H_{c}^{j}(L(π_{0}^{−1}(b)))

with π_{0} : S_{vert}^{∗}(M) × S^{1} → B the natural projection.

Theorem 6. Assume that B is smooth (without corners) and that G satisfies the assumptions (13) and (14). Let A_{L}(M|B; E) be the algebra of Laurent type complete symbols on G with coefficients in the ℤ/2ℤ-graded vector bundle E. Then

HH_{m}(A_{L}(M|B; E)) ≃ ⨁_{k+h=m} Ω_{c}^{h}(B, F^{2p−k}). (33)

Proof. We apply Lemma 8 and deduce that:

HH_{m}(A_{L}(M|B; E)) ≃ ⨁_{i+j=m} EH_{i,j}^{2} .

The computation of EH^{2} was carried out in Proposition 7 and the result is:

EH_{i,j}^{2} ≃ H_{c,L}^{p−i,j−p}(S_{vert}^{∗}(M) × S^{1}|B).

On the other hand, Proposition 3 applied to X = S_{vert}^{∗}(M) × S^{1} gives:

H_{c,L}^{p−i,j−p}(S_{vert}^{∗}(M) × S^{1}|B) ≃ Ω_{c}^{j−p}(B, F^{p−i}).

Therefore we get

HH_{m}(A_{L}(M|B; E)) ≃ ⨁_{i+j=m} Ω_{c}^{j−p}(B, F^{p−i}).

The conclusion follows by setting i = k and j = h.

Remark 1. The above proposition has to be modified only slightly if B is also a manifold with corners. For example, when B is compact, the result remains true if we replace Ω_{c}^{j}(B) with O(B)Ω^{j}(B).

5. The relative case

Let again G be a groupoid with corners satisfying the assumptions (13) and (14) with respect to the fibration π : M → B of the manifold with corners M over the smooth manifold B. Let X be a union of faces of M. We shall denote by I_{X} the ideal of smooth functions on M that vanish to infinite order on X.

We shall consider in this section the algebra of Laurent complete symbols on G which vanish to infinite order over X and which represent pseudodifferential operators acting on sections of the ℤ/2ℤ-graded vector bundle E. This algebra is denoted by A_{L}(M|B, X|B; E). Thus we have:

A_{L}(M|B, X|B; E) := O(M)I_{X}A(M|B; E).

Note that if X = ∅, then we recover the algebras A_{L}(M|B; E) studied in the previous sections. The proof of Proposition 3 in [4] extends to show that the algebras A_{L}(M|B, X|B; E) are topologically filtered algebras.

For any fibrations Y → M → B, we denote by p_{Y} the projection Y → M and by π_{Y} the composite projection Y → B. When Y is a manifold with corners, we shall denote by p_{L} the projection L(Y) → M which is the composite map of L(Y) → Y and Y → M. This last notation is intended to simplify the statements of this section.

Let again S_{vert}^{∗}(M) be the quotient bundle of T_{vert}^{∗}M ∖ 0 by the radial action of ℝ_{+}^{∗}. In [4], the computations of periodic cyclic homology recover the case of our algebra A_{L}(M|B, X|B; E). The result can be stated as follows:

Theorem 7. [4] For q = 0, 1, we have:

HP_{q}(A_{L}(M|B, X|B; E)) ≃ H_{c}^{[q]}(L(S_{vert}^{∗}M) × S^{1} ∖ p_{L}^{−1}(X)).

Proof. Again by a Morita equivalence argument we can forget the bundle E. We apply Proposition 5 in [4] and obtain for our groupoid G:

HP_{q}(A_{L}(M|B, X|B)) ≃ H_{c}^{[q]}(L(S^{∗}(G)) × S^{1} ∖ p_{L}^{−1}(X)),

But

H_{c}^{[q]}(L(S^{∗}(G)) × S^{1} ∖ p_{L}^{−1}(X)) ≃ H_{c,L}^{[q]}((A^{∗}(G) ∖ 0) ∖ p_{A^{∗}(G)∖0}^{−1}(X))_{0}.

From Assumption (14), we thus deduce as in Lemma 6 that:

H_{c,L}^{[q]}((A^{∗}(G) ∖ 0) ∖ p_{A^{∗}G∖0}^{−1}(X))_{0} ≃ H_{c,L}^{[q]}((T_{vert}^{∗}M ∖ 0) ∖ p_{T_{vert}^{∗}M∖0}^{−1}(X))_{0}.

The space H_{c,L}^{[q]}((T_{vert}^{∗}M ∖ 0) ∖ p_{T_{vert}^{∗}M∖0}^{−1}(X))_{0} is again isomorphic to the space H_{c}^{[q]}(L(S_{vert}^{∗}M) × S^{1} ∖ p_{L}^{−1}(X)), and this completes the proof.

Let us state now the corresponding results for Hochschild homology. Let F_{X}^{l} be the coefficient system over B given for any b ∈ B by the relative cohomology space

F_{X}^{l}(b) := H_{L}^{l}(S^{∗}(π^{−1}(b)) × S^{1}, p_{S_{vert}^{∗}M×S^{1}}^{−1}(X)).

Theorem 8. The Hochschild homology spaces of the algebra A_{L}(M|B, X|B; E) are given by:

HH_{m}(A_{L}(M|B, X|B; E)) ≃ ⊕_{k+h=m}Ω_{c}^{h}(B, F_{X}^{2p−k}).

Proof. We can again assume that the graded vector bundle E is trivial, onedimensional. The proof of this theorem is similar to the proof of Theorem 6, replacing cohomology by relative cohomology and using excision. More precisely, the natural filtration of the topologically filtered algebra A_{L}(M|B, X|B) by the order of the symbols gives rise to a spectral sequence for Hochschild homology given by Lemma 1. This spectral sequence was studied in [4] where the E^{2}-term was identified with the homogeneous Laurent Poisson relative homology of the Poisson manifold A^{∗}(G) ∖ 0. Indeed, we have [4][Proposition 7]:

EH_{k,h}^{2} ≃ H_{L,k+h}^{δ}((A^{∗}(G) ∖ 0) ∖ p_{A^{∗}(G)∖0}^{−1}(X))_{k},

Now using the immediate extension of Lemma 6 to the relative case, we obtain:

H_{L,k+h}^{δ}((A^{∗}(G) ∖ 0) ∖ p_{A^{∗}(G)∖0}^{−1}(X))_{k} ≃ H_{L,k+h}^{δ}((T_{vert}^{∗}M ∖ 0) ∖ p_{T_{vert}^{∗}M∖0}^{−1}(X))_{k}.

The vertical symplectic Hodge operator ∗_{vert} preserves the forms vanishing above X, therefore we deduce using the proof of Proposition 7 that:

EH_{k,h}^{2} ≃ H_{c,L}^{p−k,h−p}((S_{vert}^{∗}(M) × S^{1})|B, p_{L}^{−1}(X)|B).

The arguments ensuring the degeneracy of the spectral sequence at the second level in the proof of Lemma 8 again obviously extend to the relative case. In addition, we can apply Theorem 1 to deduce the convergence of the spectral sequence to Hochschild homology. Hence we finally obtain:

HH_{m}(A_{L}(M|B, X|B; E)) ≃ ⊕_{k+h=m} EH_{k,h}^{2}

≃ ⊕_{k+h=m} H_{c,L}^{p−k,h−p}((S_{vert}^{∗}(M) × S^{1})|B, p_{L}^{−1}(X)|B).

To end the proof we simply observe that Proposition 3 extends straightforward to the relative case.

6. Examples and applications

We begin by providing a construction of a groupoid G satisfying the assumptions (13) and (14), for any fibration π : M → B, where B is a smooth manifold (no corners) and M is a manifold, possibly with corners. For each fibration π as above we shall construct a canonical groupoid G_{M,b}, the “B-groupoid,” satisfying the assumptions (13) and (14).

Our examples are obtained by first defining the Lie algebroid of G, and then by integrating it. Let V_{b}(M|B) be the space of vertical vector fields on M that are tangent to all faces of M. Then V_{b}(M|B) is a Lie algebra with respect to the Lie bracket of vector fields and is also a projective C^{∞}(M)-module. By the Serre-Swan theorem [20] there exists a vector bundle A_{b} → M such that V_{b}(M|B) ≅ Γ(A), naturally. (See also Melrose and Piazza [33].)

The procedure in [39] then provides us with groupoids G_{M,b} whose Lie algebroid are isomorphic to A_{b}. The minimal groupoid with this property is obtained as follows. For each face F of M, consider the interior of that face, F_{0} := F ∖ ∂F. Assume F has codimension k. Define then G′_{F} to be the groupoid with units F_{0} and with at most one arrow between any two units. (Thus G′_{F} is the groupoid associated to an equivalence relation: two units are connected by an arrow if, and only if, they are equivalent.) The equivalence relation that we consider is that x, y ∈ F_{0} are equivalent if, and only if, they belong to the same connected component of a set of the form π^{−1}(b) ∩ F_{0}. Let

G_{M,b} = ∪G′_{F} × ℝ^{k}, (34)

the union being a disjoint union, and with the induced groupoid structure. Then it can be checked directly that the charts provided in [39] define a smooth structure on G such that A(G_{M,b}) ≅ A_{b}. This structure must then be unique [39]. See also [14, 34].

Let F be the locally constant sheaf (or coefficient system) that associates to b ∈ B the complex vector space with basis the minimal faces of π^{−1}(b). (All faces of a manifold with corners are connected, by definition.)

Theorem 9. Let π : M → B be as above. Then

HH_{0}(A_{L}(M|B; E)) ≅ C_{c}^{∞}(B, F),

the space of compactly supported sections of the sheaf F. The space of traces of A_{L}(M|B; E) identifies with the dual of this space:

HH^{0}(A_{L}(M|B; E)) ≅ C^{−∞}(B, F) =: C_{c}^{∞}(B, F)′.

Proof. This follows from Theorem 6.

In the particular case when M is smooth, our construction simplifies and we obtain G_{M,b} = M ×_{B} M. Then the algebra Ψ^{∞}(G) consists of differentiable families of pseudodifferential operators along the fibers of M → B. Similarly, let us consider Ψ^{∞}(M|B; E), the algebra of smooth families of pseudodifferential operators along the fibers with coefficients in the ℤ/2ℤ-graded vector bundle E, introduced in [1]. Let A(M|B; E) := Ψ^{∞}(M|B; E)/Ψ^{−∞}(M|B; E) be the algebra of vertical complete symbols. Then A(M|B; E)A_{L}(M|B; E), and we obtain

Corollary 3. Let F^{j} be the locally constant sheaf given by the cohomology of the fibers of S_{vert}^{∗}(M) × S^{1} → B. Then we have:

HH_{m}(A(M|B; E)) ≃ ⊕_{k+h=m}Ω_{c}^{h}(B, F^{2p−k}).

In particular, for m = 0 and provided that the fibers of M → B are connected, this isomorphism becomes

HH_{0}(A(M|B; E)) ≃ C_{c}^{∞}(B).

Therefore, the space of traces is given by

HH^{0}(A(M|B; E)) ≃ C^{−∞}(B) := C_{c}^{∞}(B)′, (35)

the space of distributions on the base manifold B.

Let ω be as before the vertical symplectic form on T_{vert}^{∗}M → B. The isomorphism of Equation (35) can be made more explicit as follows. Let R be the radial vector field on the fibers of T_{vert}^{∗}M → B and α = i_{R}(ω^{p}/p!) the corresponding Liouville form on the fibers of S_{vert}^{∗}(M) → B. Let STr be the graded trace on the endomorphisms of the fibers of E, that is STr(A) = Tr(γ ∘ A), where γ is the involution defining the grading. Also, let π_{∗} be the fiberwise integration on the fibers of T_{vert}^{∗}M. Then, for any a = ∑_{j≤m} a_{j}, we set

τ_{μ}(a) := ⟨μ, π_{∗} STr(a_{−p})α⟩. (36)

This formula defines a super (or graded) trace on A(M|B; E) such that μ → τ_{μ} is the isomorphism described in Corollary 3, for m = 0.

Fix a quantization function

q : ∪S^{s}(T_{vert}^{∗}M; End(E)) → ∪Ψ^{s}(M|B; E), (37)

where S^{s}(T_{vert}^{∗}M; End(E)) denotes classical vertical symbols of order s. The function q is thus assumed to be continuous and to satisfy σ_{s}(q(a)) = a if a is a symbol of order s. Let ρ be a positive symbol on T_{vert}^{∗}M, such that ρ(ξ) = |ξ|, for |ξ| ≥ 1. We consider a family D(z) such that

D(z) = q(ρ^{z})B_{1}(z) + R(z), (38)

where B_{1} is a holomorphic function on ℂ with values in Ψ^{0}(M|B; E), B_{1}(0) = 1, R(z) is a holomorphic function on ℂ with values Ψ^{−∞}(M|B; E), and R(0) = 0. Then D(0) = 1.

Proposition 8. Let μ be a distribution on B and A ∈ Ψ^{m}(M|B; E) and D(z) be as above. Then the function

z → F_{A}(z) := ⟨μ, STr_{b}(AD(z))⟩

is well defined for Re(z) < −m − p, where p is the dimension of the fibers of π : M → B. The function F_{A} extends to a meromorphic function on ℂ, with at most simple poles at the integers. For z = 0, the residue of this function is up to constant τ_{μ}(A).

Proof. This is proved as in the classical case when D(z) is given by the complex powers of a positive elliptic operator (see [44, 17]). One can follow the approach from [40], for example.

We remark that we used only a weak result from [39] (namely Theorem 2). Let us use this opportunity however to mention that there is a missing assumption in the gluing theorem of [39], Theorem 3. Here is the corrected version.

Theorem 10. Let A be a Lie algebroid on a manifold with corners M. Suppose that M has an A-invariant stratification M = ∪S such that, for each stratum S, the restriction A_{S} is integrabl and let G_{S} be d-simply connected differential groupoids such that A(G_{S}) ≃ A_{S}. Then A is integrable if, and only if, the exponential map Exp : A → G = ∪G_{S} is injective on an open neighborhood of the zero section of A for some (equivalently, for any) connection on A. Moreover, if these conditions are satisfied, then the disjoint union G = ∪G_{S} is naturally a differentiable groupoid such that A(G) ≃ A.

The above condition on the map Exp is seen to be necessary from the work of Crainic and Fernandes [14], and also from some earlier results of Weinstein. The map Exp introduced in [39] seems to be essential for both the results of that paper and for the results of [14]. The second named author would like to thank M. Crainic for pointing out a possible problem with the original statement of the above theorem.

Let us mention for completeness the result for the computation of the cyclic homology of the algebras A_{L}(M|B; E). Note first that

HC_{j}(A_{L}(M|B; E)) ≃ HP_{j}(A_{L}(M|B; E)) for j ≥ 2p + 2,

from the SBI-exact sequence, because the Hochschild homology of A_{L}(M|B; E) vanishes above this rank.

Proposition 9. The spectral sequence EC_{k,h}^{r} associated to the cyclic homology complex of A_{L}(M|B; E) has E^{1}-term given by EC_{k,h}^{1} = Ω_{rc,L}^{k+h}(S_{vert}^{∗}M), if k ≠ 0, and by

EC_{0,h}^{1} = Ω^{h}(S_{vert}^{∗}M × S^{1})/dΩ^{h−1}(S_{vert}^{∗}M × S^{1}) ⊕ ⊕_{j>0} H^{h−2j}(S_{vert}^{∗}M × S^{1}).

The d_{1} differential is induced by the Poisson différential and the spectral sequence converges to the cyclic homology of the algebra A_{L}(M|B; E).

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Inst. Desargues, Lyon, France

   E-mail address: benameur@desargues.univ-lyon1.fr

Inst. Desargues and Pennsylvania State University, University Park, PA 16802

   E-mail address: nistor@math.psu.edu