HOMOLOGY OF COMPLETE SYMBOLS AND NON-COMMUTATIVE GEOMETRY

MOULAY-TAHAR BENAMEUR AND VICTOR NISTOR

Version: 3.0A; Revised: 6/17/2000; Run: June 22, 2000

Abstract. We identify the periodic cyclic homology of the algebra of complete symbols on a differential groupoid G in terms of the cohomology of S^{∗}(G), the cosphere bundle of A(G), where A(G) is the Lie algebroid of G. We also relate the Hochschild homology of this algebra with the homogeneous Poisson homology of the space A^{∗}(G) ∖ 0 ≅ S^{∗}(G) × (0, ∞), the dual of A(G) with the zero section removed. We use then these results to compute the Hochschild and cyclic homologies of the algebras of complete symbols associated with manifolds with corners, when the corresponding Lie algebroid is rationally isomorphic to the tangent bundle.

Contents

Introduction 1 1. Hochschild and cyclic homology of filtered algebras 3 2. A rational Laurent de Rham complex 8 3. Homology of complete symbols 11 4. Hochschild homology for manifolds with corners 18 5. Applications 19 Appendix A. Projective limits 21 References 22

Introduction

Singular cohomology is often used in Algebraic Topology to obtaining invariants of topological spaces. In the same spirit, Hochschild and cyclic homology often provide interesting invariants of algebras. A possible important application of these algebra invariants is to the study of spaces with additional structures; these include, for instance, spaces with singularities or spaces endowed with group actions. This is one of the fundamental ideas of non-commutative geometry; see Connes’ book [9] and the references therein.

Let G be a differentiable groupoid with units M, a manifold with corners, and Lie algebroid A(G) → M, (see [14] in this volume for definitions, notation, and background material). To G one can associate several algebras: the convolution algebras C_{c}^{∞}(G), L^{1}(G), Ψ^{∞}(G), or other variants of these algebras. These algebras have always been a favorite toy model for non-commutative geometry and have several applications, see [5, 8, 9, 10, 22]. The algebra Ψ^{∞}(G) of pseudodifferential operators on G, for example, is expected to play an important role in the analysis on singular spaces [14]. The ideal Ψ^{−∞}(G) of regularizing operators in Ψ^{∞}(G) identifies with C_{c}^{∞}(G) and gives rise to index invariants via the boundary map in algebraic K-theory:

Partially supported by the NSF Young Investigator Award DMS-9457859, NSF Grant DMS-9971951 and ”collaborative research.” http://www.math.psu.edu/nistor/.

∂ : K_{1}^{alg}(ℭ) → K_{0}^{alg}(Ψ^{−∞}(G)),

where ℭ = A(M) := Ψ^{∞}(G)/Ψ^{−∞}(G) or ℭ = A_{0}(M) := Ψ^{0}(G)/Ψ^{−∞}(G), depending on the nature of the problem. Typically, much more attention has been devoted to finding cohomological invariants of Ψ^{−∞}(G), because its (algebraic) K_{0} is so hard to compute. The computation of the K-theory of algebras associated with groupoids or their C^{∗}-closures is far from solved and is part of the larger program involving the Baum-Connes Conjecture(s) [3].

In this paper, we concentrate on the homology of A(M) or variants of this algebra. We are interested in computing the Hochschild, the cyclic, and the periodic cyclic homology groups of A(M), denoted respectively by HH_{∗}(A(M)), by HC_{∗}(A(M)), or by HP_{∗}(A(M)). Of all these, the periodic cyclic homology is the easiest to compute. The result is in terms of A(G), the Lie algebroid of G. Let S^{∗}(G) be the cosphere bundle of A^{∗}(G), that is, the set of unit vectors in the dual of the Lie algebroid of G, and denote H^{[q]} = ⊕_{k∈ℤ} H^{q+2k}.

Theorem 1. Assume that the base M is σ-compact, then

(1) HP_{q}(A(M)) ≅ H_{c}^{[q]}(S^{∗}(G) × S^{1}) and HP_{q}(A_{0}(M)) ≅ H_{c}^{[q]}(S^{∗}(G)).

The assumption that M be σ-compact can be replaced with the assumption that M be paracompact, but then we have to work with more complicated directed sets, and this is usually unnecessary in practice.

Recall that A^{∗}(G) has a natural Poisson structure. We do show that the natural filtration on the complex computing the Hochschild homology of the algebra A(M) gives rise to a spectral sequence with E^{2}-term identified with the homogeneous Poisson homology (Definition 2) of A^{∗}(G) ∖ 0, the dual vector bundle of A(G) with the zero section removed. We expect this spectral sequence to degenerate at E^{2} and to be convergent to the Hochschild homology of A(M). The quantization of A^{∗}(G) with this Poisson structure was studied in [?].

For certain algebras associated to manifolds with corners, we identify the homogeneous Poisson homology of A^{∗}(G) ∖ 0 in terms of a space L(S^{∗}M) functorially associated to the base M. Moreover, the particular form of the resulting spectral sequence guarantees its convergence. This leads to an identification of the Hochschild homology of the Laurent complete symbols algebras A_{L}(M).

Theorem 2. Let O(M) be the ring of functions with only rational (i.e. Laurenttype) singularities at the faces of M. Assume that O(M)Γ(A(G)) ≅ O(M)Γ(T M) via the anchor map, then with n = dim(M)

HH_{q}(A_{L}(M)) ≅ H_{c}^{2n−q}(L(S^{∗}M) × S^{1}). (2)

A similar result holds true in the relative case of symbols vanishing to infinite order at some subset of M, thus extending results of [20] from the case of manifolds with boundary to that of manifolds with corners.

In [19], the norm closure of the algebra of pseudodifferential operators on a manifold with corners was studied from the point of view of K-theory. However, the K-theory is sometimes too rough to identify more subtle invariants – like the ηinvariant of Atiyah, Patodi, and Singer [2] – that are not homotopy invariant. This was partly remedied in [19], where the Hochschild 1-cocycle that gives the index was identified in terms of residues. This cocycle was then split into two parts that are direct analogues of the Atiyah-Singer integrand and, respectively, the η-invariant.

We now describe briefly the contents of each section. In Section 1, we introduce the class of algebras we shall work with, that is, the class of “topologically filtered algebras,” (Definition 1), a class of algebras for which the multiplication is not jointly continuous, but which still has a weak continuity property for multiplication. Because of this, the usual definitions of the Hochschild and cyclic complex of a topologically filtered algebra have to be adapted to our more general framework. Namely, we have to use iterated inductive and projective limits. Then we establish some results on the spectral sequences associated to the natural filtrations of the resulting complexes. In Section 2, we establish some technical results on de Rham complexes with singularities for manifolds with corners, in the spirit of [20]. In Section 3 we identify the periodic cyclic cohomology of the algebra of complete symbols and relate the Hochschild homology of those algebras with the (homogeneous) Poisson cohomology of A^{∗}(G) \ 0. In Section 4, we compute the Hochschild homology of the algebra of complete symbols when A(G) is rationally isomorphic to T M. For manifolds without corners (or boundary), these results are due to Wodzicki and Brylinski-Getzler [7]. Other related results were obtained by Lauter-Moroianu [13] and Moroianu [15]. We then use these results in Section 5 to study residues and to determine the cyclic homology of the algebra of complete symbols, still assuming that A(G) is rationally isomorphic to T M. The appendix contains a short review of projective and inductive limits.

Acknowledgements. We would like to thank Alain Connes, Thierry Fack, Robert Lauter, Sergiu Moroianu, and Serge Parmentier for useful discussions.

1. Hochschild and cyclic homology of filtered algebras

We begin by recalling the definitions of Hochschild and cyclic homology groups of a topological algebra A. A good reference is Connes’ book [9]. 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 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, 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. 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 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}a_{n}a_{0} ⊗ … ⊗ a_{n−1}. \end{matrix} (3)

The Hochschild homology groups of the topological 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 [25].

We now define cyclic homology. We shall use the notation of [8]:

\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}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 = B_{0}(1 − t). \end{matrix} (4)

Then [b, B]_{+} := bB + Bb = B^{2} = b^{2} = 0, and hence, if we define

C(A)_{n} = ⨁_{k≥0} H_{n−2k}(A), (5)

(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 [8] and [24].

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

(6) … → 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 [8, 16]. 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 exist an increasing multi-filtration F_{n,l}^{m} ⊂ A of A,

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

by closed subspaces 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} ∩ ( ∪_{n} F_{n,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 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, does not make much sense. 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}) , (7)

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)) , (8)

where k_{0} + … + k_{n} ≤ p defines the filtration. The projective and inductive limits are such that first k, l → −∞ and then m → ∞.

For any topologically filtered algebra, we denote

Gr(A) := ⊕_{n} A_{n}/A_{n−1}

the graded algebra associated to A (recall that A_{n} was defined as ∪_{l,m}F_{n,l}^{m}A). 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

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 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 induced 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.

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 of spectral sequences S : EC_{k,h}^{r} → EC_{k,h−2}^{r}, which for r = 1 is the graded map associated to the periodicity operator S : HC_{n}(Gr(A)) → HC_{n−2}(Gr(A)).

Proof. We shall write F_{p} = F_{p}H(A), for simplicity, where the filtration is as defined in Equation (8).

The filtration of the complex computing the Hochschild homology of A then gives rise to a spectral sequence with E_{k,h}^{1} = H_{k+h}(F_{k}/F_{k−1}), by standard homological algebra. By the definition of the Hochschild complex of Gr(A),

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

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

Lemma 2. Let N and M be integers. If A is a topologically filtered algebra with F_{n,l}^{m}A independent of m and EH_{k,h}^{1} = 0 for all pairs (k, h) such that k < N and k + h ≥ M, then

HH_{q}(A) ≅ ⊕_{k=N}^{∞} EH_{k,q−k}^{∞},

if q ≥ M. A similar result is true for the cyclic homology spectral sequence.

The above isomorphism is not natural, in general, but comes from a filtration of HH_{q}(A) whose subquotients identify naturally with EH_{k,q−k}^{∞}, see [17, 18]. What the above lemma says, put differently, is that the spectral sequence EH_{k,h}^{r} converges to HH_{k+h}(A), for k + h ≥ M .

Proof. We shall use

H_{q}(A) = lim_{→}( lim_{←} F_{p}H_{q}(A)/F_{p′}H_{q}(A)).

Denote F_{r} = F_{r}H(A) for simplicity. The projective limits give rise to a lim^{1} exact sequence

0 ⟶ lim_{←}^{1} H_{m+1}(F_{p}/F_{p′}) ⟶ H_{m}(F_{p}) ⟶ lim_{←} H_{m}(F_{p}/F_{p′}) ⟶ 0

for every fixed p (see Lemma 6 from the Appendix). From the assumption that H_{q}(F_{p′}/F_{p′−1}) = 0 for p′ < N and q ≥ M, we know that H_{q}(F_{p}/F_{p′}) becomes stationary for p′ < N and q ≥ M. This shows that the lim^{1} term above vanishes for q ≥ M, and hence

H_{q}(F_{p}) ≃ lim_{←} H_{q}(F_{p}/F_{p′}) = H_{q}(F_{p}/F_{N−1}),

if q ≥ M. It also gives that the natural morphism H(A) → H(A)/F_{N−1} induces an isomorphism of the E_{k,h}^{r}-terms of the corresponding spectral sequences, for r ≥ 1 and k + h ≥ M.

Using then the fact that homology and inductive limits commute we obtain

HH_{q}(A) = H_{q}( lim_{p→∞} lim_{←} F_{p}/F_{p′}) ≅ lim_{p→∞}H_{q}(lim_{←} F_{p}/F_{p′})

≅ lim_{p→∞}H_{q}(F_{p}/F_{N−1}) ≅ H_{q}(H(A)/F_{N−1}) ≅ ⊕_{l=N}^{∞}E_{k,q−k}^{∞},

p′ → −∞, where for the last isomorphism we have used that the spectral sequence associated to H(A)/F_{N−1} is convergent because H(A)/F_{N−1} = ∪_{q}F_{q}/F_{N−1}.

Theorem 3. Fix an integer N and M. If A is a topologically filtered algebra such that each of the algebras A^{m} := ∪_{n,l}F_{n,l}^{m}A satisfies the assumptions of Lemma 2 for the given N and M, then

HH_{q}(A) ≅ ⊕_{k=N}^{∞} EH_{k,q−k}^{∞}, q ≥ M.

A similar result is true for the cyclic homology spectral sequence.

Proof. Each of the algebras A^{m} = ∪_{n,l}A^{m} is a topologically filtered algebra in its own if we let F_{n′,l′}^{m′}A^{m} = F_{n′,l′}^{m}A (so the filtration depends only on n and l).

For the algebras A^{m}, the Hochschild complex is defined as for any topologically filtered algebra, except that there is no need to take an additional direct limit with respect to m.

The Hochschild complex of A is then given by

H_{q}(A) = lim_{→} H_{q}(A^{m}), m → ∞

Because taking homology is compatible with inductive limits, we obtain that

HH_{q}(A) = lim_{→} HH_{q}(A^{m}), m → ∞.

Denote by EH_{k,h}^{r}(A^{m}) the spectral sequence associated by Lemma 1 to the topologically filtered algebra A^{m}. Again because homology is compatible with inductive limits, EH_{k,h}^{r}(A) ≅ lim_{→} EH_{k,h}^{r}(A^{m}). The result then follows from Lemma 2.

Lemma 3. Fix an integer N and a ≥ 1. If A is a topologically filtered algebra with F_{n,l}^{m}A independent of m and EH_{k,h}^{a} = 0, for all k < N, then the spectral sequence EH_{k,h}^{r} converges to HH_{k+h}(A). More precisely,

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

A similar result is true for the cyclic homology spectral sequence.

Proof. The assumption that F_{n,l}^{m}A is independent of m has as a consequence that we need not take inductive limits with respect to m in the definition of the Hochschild complex of A.

Denote F_{p} = F_{p}H(A) for simplicity. The Hochschild complex of A is complete, in the sense that

H(A) = lim_{→}(lim_{←} F_{p}H(A)/F_{p′}H(A)), p′ → −∞ and p → ∞.

The projective limits give rise to a lim^{1} exact sequence

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

as p → −∞, for every fixed q (see Lemma 7).

The spectral sequence E_{k,h}^{r}(p) associated to the complex H(A)/F_{p} is convergent because H(A)/F_{p} = ∪_{j}F_{j}/F_{p}. Consequently, the homology groups of H(A)/F_{p} are endowed with a filtration F_{t} H_{q}(H(A)/F_{p}) such that

F_{t} H_{q}(H(A)/F_{p})/F_{t−1} H_{q}(H(A)/F_{p}) ≅ E_{t,q−t}^{∞}(p) (9)

Assume now that p < N, the assumption that E_{k,h}^{a} = 0 for all k < N gives that E_{k,h}^{r}(p) = E_{k,h}^{r} for all k < p < N or p + a < k. Let A_{n} = H_{q}(H(A)/F_{N−na}), B_{n} = F_{N−na+a} H_{q}(H(A)/F_{N−na}), and C_{n} = A_{n}/B_{n}, n ≥ 2. Then equation (9) gives that the natural map A_{n+1} → A_{n} gives an isomorphism C_{n+1} ≅ C_{n} and induces the zero map B_{n+1} → B_{n}. Using Lemma 6 from the Appendix, we obtain that lim_{←}^{1}A_{n} = 0 and lim_{←} A_{n} = C_{n_{0}} for any fixed n_{0}.

Because C_{n_{0}} = ⊕_{l=N}^{∞}E_{l,q−l}^{∞}, the result follows.

Theorem 4. Fix an integer N and a ≥ 1. If A is a topologically filtered algebra such that each of the algebras A^{m} := ∪_{n,l}F_{n,l}^{m}A satisfies the assumptions of Lemma 3 for the given N and a, then the spectral sequence EH_{k,h}^{r} converges to HH_{k+h}(A). More precisely, we have

HH_{q}(A) ≅ ⊕_{h=N}^{∞} EH_{k,q−k}^{∞} .

A similar result is true for the cyclic homology spectral sequence.

Proof. Each of the algebras A^{m} is a topologically filtered algebra in its own, if we let F_{n′,l′}^{m′}A^{m} = F_{n′,l′}^{m}A (so the filtration really depends only on n and l). We shall write F_{n,l}A^{m} instead of F_{n,l}^{m′}A^{m}, and hence F_{n,l}^{m}A = F_{n,l}A^{m}.

The Hochschild complex of A is then given by

H_{q}(A) = lim_{→} H_{q}(A^{m}), m → ∞

Because taking homology is compatible with inductive limits, we obtain that

HH_{q}(A) = lim_{→} HH_{q}(A^{m}), m → ∞.

Denote by EH_{k,h}^{r}(A^{m}) the spectral sequence associated to the natural filtration of the Hochschild complex the algebra A^{m}. Because homology is compatible with inductive limits, EH_{k,h}^{r}(A) ≅ lim_{→} EH_{k,h}^{r}(A^{m}). The result then follows from Lemma 3.

We conclude this section with a result that is in the same spirit with the above results and was independently observed by Sergiu Moroianu in a different setting.

Proposition 1. If the graded algebra Gr(A) of the topologically filtered algebra A is H-unital, then A is H-unital, in the sense that the complex (H_{n}(A), b′) is acyclic.

Proof. This is completely analogous to the previous results, so we will be sketchy. The natural filtration on the complex (H_{n}(A), b′) induces a spectral sequence whose E^{1} term is the b′-homology of Gr(A). This spectral sequence is proved to be convergent as in any of the above two theorems.

2. A rational Laurent de Rham complex

We now recall some notation and obtain some preliminary results on de Rham type complexes used in our computations. Examples of topologically filtered algebras will be discussed in the following section.

Consider a σ–compact manifold with corners M. Let O(M) be the ring of functions on M such that f ∈ O(M) if, and only if, on every open subset of M diffeomorphic to [0, 1)^{k} × ℝ^{n−k}, the function f is of the form x_{1}^{−p_{1}} … x_{k}^{−p_{k}}g, with g a smooth function on M. (So O(M) is a quotient ring.) By abuse of terminology, we shall call O(M) the ring of Laurent rational functions on M. Also, we shall call x_{1}, …, x_{k} the local defining functions of the hyperfaces of M at a point x belonging to a corner of codimension k.

If there exists a boundary defining function ρ_{H} ∈ C^{∞}(M) for each hyperface H (that is, H = {ρ_{H} = 0}, ρ_{H} ≥ 0 on M, and dρ_{H} ≠ 0 on H), then

O(M) = ρ^{−ℤ}C^{∞}(M) = ⋃_{j∈ℤ} x^{j}C^{∞}(M),

where ρ^{−ℤ} stands for the multiplicative set consisting of all products ρ_{H_{1}}^{n_{1}}ρ_{H_{2}}^{n_{2}} ⋯ ρ_{H_{m}}^{n_{m}}, n_{k} ∈ ℤ. We then say that M has embedded faces. A nice discussion of how to avoid the assumption of M having embedded faces is due to Monthubert (see [?, ?] and the references within). It is essential then to use groupoids.

For each k, let Ω_{c,L}^{k}(M) := O(M) ⊗_{C^{∞}(M)} Ω_{c}^{k}(M) be the space of k-forms with compact support on M with only Laurent singularities at the faces. Recall then that the Laurent–de Rham cohomology of M is the cohomology of the Laurent–de Rham complex.

⋯ ⟶ Ω_{c,L}^{k}(M) ⟶ Ω_{c,L}^{k+1}(M) ⟶ ⋯ (10)

with respect to the de Rham differential. We denote by H_{L}^{∗}(M) these cohomology groups. (These groups were denoted H_{b}^{∗}(M) in [20], if M is a manifold with boundary.)

We shall use also the complex

⋯ ⟶ Ω_{c,L}^{k}(M)_{x} ⟶ Ω_{c,L}^{k+1}(M)_{x} ⟶ · · · . (11)

of germs of the Laurent-de Rham complex at a point x ∈ M.

Lemma 4. The cohomology of the complex of stalks of the Laurent-de Rham complex at the point x ∈ M belonging to a corner of codimension k is the exterior algebra generated by d log x_{i}, where x_{i}, i = 1, …, k, are locally defining functions of the hyperfaces containing x.

Proof. This statement is local because we are dealing with germs. So we can assume that x = (0, 0, …, 0) ∈ ℝ^{n} and M = [0, ∞)^{k} × ℝ^{n−k} ⊂ ℝ^{n}. The complex whose cohomology we have to compute is then the projective tensor product of the corresponding complexes for [0, ∞) and ℝ an appropriate number of times. By the Poincaré Lemma, the complex corresponding to ℝ has cohomology only in dimension 0. The cohomology of germs at 0 of Laurent forms on [0, ∞) is seen to be generated by 1 in dimension 0 and by d log t in dimension 1 (t > 0). The cohomology of a tensor product of these complexes is isomorphic to the tensor product of their cohomologies as graded vector spaces (by the topological Künneth theorem). This proves the theorem.

We shall denote by M_{j}(M) the set of codimension j faces of M.

Theorem 5. Let M be a manifold with corners all of whose hyperfaces H are embedded submanifolds with defining function ρ_{H}. Then the Laurent-de Rham cohomology spaces of M can be naturally decomposed in terms of the cohomology of its faces as

H_{L}^{k}(M) = ⨁_{j=0}^{k} ⨁_{F ∈M_{j}(M)} H^{k−j}(F). (12)

Proof. Consider for each face F of M the usual de Rham complex

⋯ ⟶ Ω^{k}(F) ⟶^{d} Ω^{k+1}(F) ⟶ ⋯ (13)

whose homology is, by (a variant of) de Rham’s theorem, the absolute cohomology of F. Denote by H_{1}, H_{2}, …, H_{m} the hyperfaces containing F and by ρ_{i} their defining functions. Fix a local product structure in a neighborhood of the face F and choose a smooth cutoff function ϕ with support in that neighborhood and equal to 1 in a smaller neighborhood of F. The map

Φ_{F} : C^{∞}(F; Λ^{k}) ∋ α ⟶ α ∧ d(ϕ log ρ_{1}) ∧ ⋯ ∧ d(ϕ log ρ_{m}) ∈ ρ^{−ℤ}C^{∞}(M; Λ^{k+m}) (14)

where the local product decomposition near F is used to lift α to a smooth form on M, is a chain map.

The de Rham complex is a resolution of the constant sheaf on a manifold with corners (no factors of x_{i}^{−1} are allowed). From this and the previous lemma we obtain that the cochain map ⊕Φ_{F}, where the sum is take over all faces of M, gives an isomorphism in cohomology. This proves the proposition.

The cohomology of the above complex can be described as the cohomology of a space L(M) naturally associated to M and defined as follows. Consider for each face F of M the space F × (S^{1})^{k}, where k is the codimension of the face. Moreover we establish a one-to-one correspondence between the k copies of the unit circle and the faces F′ of M 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(M). By construction there exists a continuous map p_{L} : L(M) → M. Let J = S^{1} ∪ [1, 1 + ϵ), for some ϵ > 0, with S^{1} identified with a subset of the complex plane. Then the space L(M) is locally modelled by J_{ϵ}^{k} × ℝ^{n−k}, above each point of M belonging to an open face of codimension k. Using the space L(M), one can describe also the homology of other complexes associated to M.

Let X ⊂ M be a closed subset consisting of a closed union of faces of M. Denote by Ω_{c}^{k}(M, X) the space of compactly supported smooth forms on M that vanish to infinite order on X. We then obtain complexes

⋯ ⟶ Ω_{c}^{k}(M, X) ⟶^{d} Ω_{c}^{k+1}(M, X) ⟶ ⋯ (15)

and

⋯ ⟶ Ω_{c,L}^{k}(M, X) ⟶^{d} Ω_{c,L}^{k+1}(M, X) ⟶ ⋯, (16)

where Ω_{c,L}^{k}(M, X) := O(M)⊗_{C^{∞}(M)}Ω_{c}^{k}(M, X), using the same convention as above. We denote by H_{L}^{∗}(M, X) the cohomology of this complex. When X = ∅, this recovers the old definitions.

Proposition 2. The homology of the complex (15) is the relative (de Rham) cohomology group H_{c}^{∗}(M, X) and the cohomology of the complex (16) is

H_{L}^{∗}(M, X) ≅ H_{c}^{∗}(L(M), p_{L}^{−1}(X)) = H_{c}^{∗}(L(M) ∖ p_{L}^{−1}(X)).

Moreover, the morphism on cohomology induced by the inclusion of the first complex into the second complex identifies, up to isomorphism, with the morphism p_{L}^{∗}.

Proof. That the cohomology of the first complex is isomorphic to H_{c}^{∗}(M, X) := H_{c}^{∗}(M ∖ X) is of course well known. The second isomorphism follows from the Serre-Leray spectral sequence applied to the map p_{L} : L(M) → M and the above proposition.

In particular, the above theorem gives formula H_{L}^{∗}(M) ≅ H_{c}^{∗}(L(M)) for the Laurent–de Rham cohomology of the manifold with corners M (assumed to be σ-compact).

3. Homology of complete symbols

Let G be a differentiable groupoid with space of units denoted M, a manifold with corners, in general. We obtain in this section some results on the homologies of algebras of complete symbols on G. In particular, we get a complete determination of the periodic cyclic cohomology of the algebras of complete symbols A(M, X), A_{0}(M, X), and A_{L}(M, X) defined below.

Let X be a locally finite union of closed faces of the manifold with corners M. Also, let I_{X} ⊂ C^{∞}(M) be the subset of functions that vanish to infinite order on X. Then we define A(M, X) := I_{X}(Ψ^{∞}(G)/Ψ^{−∞}(G)) the algebra of complete symbols on G, supported above a compact set of M and vanishing to infinite order at X. Similarly, we let

\begin{matrix}A_{0}(M, X) := I_{X}(Ψ^{0}(G)/Ψ^{−∞}(G)), \\ A_{L}(M, X) := O(M)I_{X}(Ψ^{∞}(G)/Ψ^{−∞}(G)). \end{matrix}

The algebra A_{L}(M, X) will be sometimes referred to as the algebra of “Laurent complete symbols vanishing rapidly at X.” Note that we have A_{L}(M, X) = A(M)O(M)I_{X}, too. If X = ∅, we do not include it in the notation.

Let us recall some notation from [14], in this volume. We denote by T_{vert}(G) = ∪_{x∈M}T (G_{x}) the vertical tangent bundle of G to the fibers of the domain map d : G → M. Then A(G) can be identified with the restriction of T_{vert}(G) to M, that is, A(G) = ∪_{x∈M}T_{x}(G_{x}). In particular, the smooth sections of A^{∗}(G) are canonically identified with the right G-invariant smooth sections of T_{vert}^{∗}(G).

Proposition 3. Suppose M, the space of units of G is σ-compact. Then the algebras A(M, X), A_{0}(M, X), and A_{L}(M, X) are topologically filtered algebras.

Proof. We shall prove this lemma for A_{L}(M, X), the other case being completely similar, and actually even simpler. The subspaces F_{n,l}^{m}A are defined in the following way. Fix an increasing, countable exhaustion M = ∪K_{m} by compact submanifolds. For any vector bundle E → M, we denote by S_{rc}^{n}(E) the space of classical symbols supported above some compact set in the base M. We then choose a quantization map q : S_{rc}^{n}(A^{∗}(G)) → Ψ^{n}(G) as in [23], whose main property is that it induces a bijection

S_{rc}^{n}(A^{∗}(G))/S_{rc}^{n′}(A^{∗}(G)) ≅ Ψ^{n}(G)/Ψ^{n′}(G),

for all n′ < n (including n = ∞ or n′ = −∞). Denote by π : A^{∗}(G) → M the natural projection and by S_{K_{m}}^{n}(A^{∗}(G)) the set of symbols with support in π^{−1}(K_{m}), then an increasing triple filtration for A is defined, for l = 0 first, by

F_{n,0}^{m}A = (q(S_{K_{m}}^{n}(A^{∗}(G))) + Ψ_{K_{m}}^{−∞}(G))/Ψ_{K_{m}}^{−∞}(G).

Choose now a radial completion of A^{∗}(G), which is then a diffeomorphism of A^{∗}(G) onto the interior of the ball bundle B^{∗}(G) := {ξ ∈ A^{∗}(G), ∥ξ∥ ≤ 1}. This identifies S_{K}^{0}(A^{∗}(G)) with the subset C_{K}^{∞}(B^{∗}(G)) ⊂ C^{∞}(B^{∗}(G)) of those smooth functions on B^{∗}(G) with support above the compact set K ⊂ M. We use this identification to define the topology on S_{K_{m}}^{0}(A^{∗}(G)), which in turn gives F_{0,0}^{m}A the induced topology.

To define F_{n,l}^{m}A in general, let S_{K_{m}}^{n,l}(A^{∗}(G)) be the space of Laurent symbols of order ≤ n, with support in π^{−1}(K_{m}), and only rational singularities with total order ≤ l in each defining function of a hyperface of M outside X (we do not count negative orders). Then we define F_{n,l}^{m}A as the image of q(S_{K_{m}}^{n,l}(A^{∗}(G)), with the induced topology.

This definition is such that F_{n,l}^{m}A = F_{n,0}^{m}A, if l < 0. The topology on this space is defined similarly. In this way, F_{n,l}^{m}A becomes a closed subset of F_{n′,l′}^{m′}A whenever n ≤ n′, l ≤ l′, and m ≤ m′. Then we endow A with the strict inductive limit topology. Conditions 1–7 of a topologically filtered algebra are then satisfied.

Remark. When M is compact, there is no need to consider the additional filtration with respect to m. Also, for the algebras A_{0}(M, X), there is no need to consider the variable l, as the filtration is independent of l

We now apply some of the results of the previous sections to the algebras A(M, X) and A_{L}(M, X).

In this section, we first concentrate on the cyclic homology, because this will lead to a complete determination of the periodic cyclic homology of these algebras. We concentrate first on the algebra A(M) = Ψ^{∞}(G)/Ψ^{−∞}(G) of complete symbols on G, which we shall denote by A throughout the rest of this section, for simplicity. The groupoid G is arbitrary but fixed in what follows.

Fix a metric on A(G) and let P be a pseudodifferential operator of order one such that σ_{1}(P) ≡ r (modulo lower order symbols), where r ∈ C^{∞}(A^{∗}(G)) is the distance function to the origin. We know that the graded algebra Gr(A) associated to A is commutative. Denote by S^{∗}(G) = S^{∗}(A(G)) the set of vectors of length one in A^{∗}(G), the dual of the Lie algebroid of G. Then Gr(A) ≃ C_{c}^{∞}(S^{∗}(G)) ⊗ ℂ[r, r^{−1}], with grading given by the powers of r.

As noted in Section 1, the tensor products appearing in the Hochschild complex are such that F_{k}H_{n}(A)/F_{k−1}H_{n}(A) is a direct sum of spaces, each of which is isomorphic to C_{c}^{∞}(S^{∗}(G) × S^{∗}(G) × ⋯ × S^{∗}(G)), such that the natural map

F_{k}H_{n}(A) → lim_{←} F_{k}H_{n}(A)/F_{k−j}H_{n}(A), j → ∞,

is an isomorphism. The same comments are valid for the cyclic complex.

The Hochschild, cyclic, and periodic cyclic homologies of Gr(A) are identified using a combination of the Hochschild-Kostand-Rosenberg (HKR) isomorphism and a result of Connes, which is the analog of the HKR-isomorphism for algebras of C^{∞}-functions. We denote by Ω_{rc}^{l}(A^{∗}(G) ∖ 0)_{d} the set of l−differential forms on the manifold A^{∗}(G) ∖ 0 that are positively d-homogeneous in the radial direction and whose support projects onto a compact subset of M. (Here A^{∗}(G) ∖ 0 stands for the dual of A(G) with the zero section removed.) Then, using the grading of Gr(A), we have

HH_{l}(Gr(A))_{d} ≅ Ω_{rc}^{l}(A^{∗}(G) ∖ 0)_{d} ≅ Ω_{c}^{l}(S^{∗}(G))r^{d} ⊕ Ω_{c}^{l−1}(S^{∗}(G))r^{d−1}dr,

the isomorphism being obtained via the Hochschild-Kostant-Rosenberg-Connes map

χ(a_{0}, …, a_{l}) = (1/l!) a_{0}da_{1} ∧ da_{2} ∧ … ∧ da_{l}.

(We shall often omit the “wedge” ∧ in what follows.)

We can identify Ω_{c}^{l}(S^{∗}(G))r^{d}⊕Ω^{l−1}(S^{∗}(G))r^{d−1}dr with the subspace Ω_{rc}^{l}(A^{∗}(G)∖ 0)_{d} ⊂ Ω_{rc}^{l}(A^{∗}(G) ∖ 0) consisting of d-homogeneous forms, as above. Also, we shall sometimes complexify r and restrict it to the unit circle, so that r becomes e^{iθ} and dr becomes ie^{iθ}dθ. Let Ω_{fin}^{∗}(S^{1}) be the set of polynomial forms on S^{1}, that is, of finite linear combinations of r^{l} and r^{m}dr. The complex Ω_{fin}^{∗}(S^{1}) has the same cohomology as the de Rham complex of all forms on S^{1}, which also gives the cohomology of S^{1} (with complex coefficients).

Then, we see that the complex (⊕_{m}Ω_{rc}^{∗}(A^{∗}(G)∖0)_{m}, d), can be identified with the complex Ω_{c}^{∗}(S^{∗}(G))⊗Ω_{fin}^{∗}(S^{1}), and hence it has the same cohomology as S^{∗}(G)×S^{1}. Next we use that

Ω_{rc}^{k+1}(A^{∗}(G) ∖ 0)_{m}/dΩ_{rc}^{k}(A^{∗}(G) ∖ 0)_{m} ≅ Ω_{c}^{k+1}(S^{∗}(G)), for m ≠ 0, and

Ω_{rc}^{k+1}(A^{∗}(G) ∖ 0)_{0}/dΩ_{rc}^{k}(A^{∗}(G) ∖ 0)_{0} ≅
Ω_{c}^{k+1}(S^{∗}(G))/dΩ_{c}^{k}(S^{∗}(G)) ⊕ Ω_{c}^{k}(S^{∗}(G))/dΩ_{c}^{k−1}(S^{∗}(G)).

The first isomorphism above is given by

(α, β) ∈ Ω_{c}^{k+1}(S^{∗}G)r^{m} ⊕ Ω_{c}^{k}(S^{∗}G)r^{m−1}dr ↦ m^{−1}α − dβ.

Lemma 5. The E^{1}-term of the spectral sequence associated by Lemma 1 to the topologically filtered algebra A is given by EC_{k,h}^{1} ≅ Ω_{c}^{k+h}(S^{∗}(G)) if k ≠ 0 and

EC_{0,h}^{1} ≅ Ω_{c}^{h}(S^{∗}(G))/dΩ_{c}^{h−1}(S^{∗}(G)) ⊕ Ω_{c}^{h−1}(S^{∗}(G))/dΩ_{c}^{h−2}(S^{∗}(G)) ⊕

⊕_{j>0} H_{c}^{h−2j}(S^{∗}(G) × S^{1}).

Moreover, the periodicity morphism S : EC_{k,h}^{1} → EC_{k,h−2}^{1} vanishes if k ≠ 0 and is the natural projection if k = 0.

Proof. We know from Lemma 1 that EC_{k,h}^{1} ≃ HC_{k+h}(Gr(A))_{k}. Using again the HKR-isomorphism, we obtain that HC_{m}(Gr(A))_{d} is isomorphic to

Ω_{rc}^{m}(A^{∗}(G) ∖ 0)_{d}/dΩ_{rc}^{m−1}(A^{∗}(G) ∖ 0)_{d} ⊕ ⊕_{j>0} H^{m−2j}(A^{∗}(G) ∖ 0)_{d},

and the operator S : EC_{k,h}^{1} → EC_{k,h−2}^{1} is the projection which sends Ω_{rc}^{k+h}(A^{∗}(G)∖ 0)_{k}/dΩ_{rc}^{k+h−1}(A^{∗}(G) ∖ 0)_{k} to 0, is the inclusion

H^{k+h−2}(A^{∗}(G) ∖ 0)_{k} ↪ Ω_{rc}^{k+h−2}(A^{∗}(G) ∖ 0)_{k}/dΩ_{rc}^{k+h−3}(A^{∗}(G) ∖ 0)_{k}

and the identity on the other factors. The above computation of Ω_{rc}^{m}(A^{∗}(G) ∖ 0)/dΩ_{rc}^{m−1}(A^{∗}(G) ∖ 0) then gives the result.

From the above lemma we obtain the following immediate consequence.

Corollary 1. We have EC_{k,h}^{1} = 0 if k < 0 and k + h > dim S^{∗}(G).

The following proposition is essential in determining the periodic cyclic homology of A.

Proposition 4. If q > dim S^{∗}(G), then HC_{q}(A) ≅ ⊕_{k∈ℤ} H_{c}^{q−2k}(S^{∗}(G) × S^{1}) with S : HC_{q+2}(A) → HC_{q}(A) also an isomorphism.

Proof. We shall use Theorem 3 and Lemma 5. For q > dim(S^{∗}G) + 1, Lemma 5 gives

EC_{k,q−k}^{1} = 0 for all k ≠ 0, and EC_{0,q}^{1} = ⊕_{j∈ℤ} H_{c}^{q−2j}(S^{∗}G × S^{1}).

Moreover, the assumptions of Theorem 3 are satisfied, by Corollary 1, and this gives

HC_{q}(A) ≃ ⊕_{k} EC_{k,q−k}^{1} ≅ ⊕_{j>0} H_{c}^{q−2j}(S^{∗}(G) × S^{1}).

This gives the result.

From this we obtain

Theorem 6. The periodic cyclic homology groups of A = Ψ^{∞}(G)/Ψ^{−∞}(G), the algebra of complete symbols on G, are given by HP_{q}(A) ≅ H_{c}^{[q]}(S^{∗}(G) × S^{1}).

Proof. Whenever the periodicity operator S of the exact sequence (6) is surjective, we have HP_{q}(A) ≅ lim_{←} HC_{q+2j}, the projective limit being taken with respect to the operator S. The conclusion then follows from Proposition 4.

Similarly, we have the following determination of the periodic cyclic homology groups of the algebra A_{0} = Ψ^{0}(G)/Ψ^{−∞}(G).

Theorem 7. HP_{q}(A_{0}) ≅ H_{c}^{[q]}(S^{∗}(G)).

Proof. The proof is essentially the same as for the corresponding result for the algebra A, so we will be brief. The filtration on A_{0} is induced from the filtration on A. This and the specific form of the EC^{1}-terms then give

EC_{k,h}^{1}(A_{0}) ≃ EC_{k,h}^{1}(A) if k < 0, and EC_{k,h}^{1}(A_{0}) ≃ {0} if k > 0,
and EC_{0,h}^{1}(A_{0}) ≅ ⊕_{j∈ℤ} H_{c}^{h−2j}(S^{∗}(G)) if h ≥ dim S^{∗}(G).

Moreover, the differential d^{1} is the same as that for A if k ≤ 0, but is trivial for k > 0. Thus, if q > dim(S^{∗}G), we get as above that

∀k ∈ ℤ ∖ {0}, EC_{k,q−k}^{1}(A_{0}) ≃ 0.

Consequently,

EC_{0,q}^{1}(A_{0}) ≃ ⊕_{j∈ℤ} H_{c}^{q−2j}(S^{∗}(G)) =: H^{[q]}(S^{∗}(G)).

A similar approach can be used to treat variants of the algebra A when the Schwarz symbols of our operators vanish to infinite order at certain hyperfaces of M.

Let X be a closed union of faces of the manifold with corners M. Let I_{X} ⊂ C^{∞}(M) be the ideal of functions that vanish to infinite order on X. We now study the algebras A_{L}(M, X) of “Laurent complete symbols vanishing rapidly at X,” supported above a compact subset of M, which, we recall, are given by

A_{L}(M, X) := O(M)I_{X}A(G) = O(M)I_{X}(Ψ^{∞}(G)/Ψ^{−∞}(G)),

and A_{L}(G) = A_{L}(G, ∅).

We shall usually denote by π_{Y} : Y → M the projection associated with a typical fibration Y over M. As before, we denote by Ω_{rc,L}^{t}(A^{∗}(G) ∖ 0, π_{A^{∗}(G)∖0}^{−1}(X)) the space of t-differential forms on A^{∗}(G) ∖ 0 that vanish to infinite order in the base variable on π_{S^{∗}(G)\0}^{−1}(X) and are supported above a compact set in M. The above results on the Hochschild and cyclic homology of the algebras A and A_{0} extend then almost right away to the algebras A_{L}(M, X). Recall that p_{L} : L(Y) → Y is the projection defined in section 2 for a manifold with corners Y. We summarize results in the following two propositions. To simplify notation, we shall denote also by p_{L} : L(S^{∗}(G)) × S^{1} → M the induced projection.

Proposition 5. For q > dim(S^{∗}(G)), we have

HC_{q}(A_{L}) ≅ ⊕_{k≥0} H_{c}^{q−2k}(L(S^{∗}(G)) × S^{1} ∖ p_{L}^{−1}(X)).

Thus, HP_{q}(A_{L}) ≅ H_{c}^{[q]}(L(S^{∗}(G)) × S^{1} ∖ p_{L}^{−1}(X)).

Proof. We first prove the analogue of Lemma 5 in our new settings: namely, the E^{1}- term of the cyclic spectral sequence associated to the topologically filtered algebra A_{L}(M, X) by Lemma 1 is given by

EC_{k,h}^{1} ≅ Ω_{c,L}^{k+h}(S^{∗}(G), π_{S^{∗}(G)}^{−1}(X)),

if k ≠ 0, and otherwise by

EC_{0,h}^{1} ≅ Ω_{c,L}^{h}(S^{∗}(G), π_{S^{∗}(G)}^{−1}(X))/dΩ_{c,L}^{h−1}(S^{∗}(G), π_{S^{∗}(G)}^{−1}(X))

⊕ Ω_{c,L}^{h−1}(S^{∗}(G), π_{S^{∗}(G)}^{−1}(X))/dΩ_{c,L}^{h−2}(S^{∗}(G), π_{S^{∗}(G)}^{−1}(X)) ⊕

⊕_{j>0} H_{c}^{h−2j}(L(S^{∗}(G)) × S^{1}, p_{L}^{−1}(X)).

Moreover, the periodicity morphism S : EC_{k,h}^{1} → EC_{k,h−2}^{1} vanishes if k ≠ 0 and is the natural projection if k = 0.

Indeed, Lemma 1(i) together with a relative version of the HKR isomorphism give that

EC_{k,h}^{1} ≅ \frac{Ω_{rc,L}^{k+h}(A^{∗}(G) ∖ 0, π_{A^{∗}(G)∖0}^{−1}(X))_{k}}{dΩ_{rc,L}^{k+h−1}(A^{∗}(G) ∖ 0, π_{A^{∗}(G)∖0}^{−1}(X))_{k}} ⊕

⊕_{j>0} H_{rc}^{k+h−2j}(A^{∗}(G) ∖ 0, π_{A^{∗}(G)∖0}^{−1}(X))_{k}.

Moreover, it follows that the operator S is the projection just above. Now we can compute each term and use the homotopy invariance of the relative de Rham cohomology to see that the k-homogeneous relative cohomology groups H_{L}^{h−2j}(A^{∗}(G) ∖ 0, π_{A^{∗}(G)∖0}^{−1}(X))_{k} vanish for k ≠ 0 and that for k = 0 we have

H_{rc,L}^{h−2j}(A^{∗}(G) ∖ 0, π_{A^{∗}(G)∖0}^{−1}(X))_{0} ≅ H_{c,L}^{h−2j}(S^{∗}(G) × S^{1}, π_{S^{∗}(G)×S^{1}}^{−1}(X)).

On the other hand, we have

\frac{Ω_{rc,L}^{k+h}(A^{∗}(G) ∖ 0, π_{A^{∗}(G)∖0}^{−1}(X))_{k}}{dΩ_{rc,L}^{k+h−1}(A^{∗}(G) ∖ 0, π_{A^{∗}(G)∖0}^{−1}(X))_{k}} ≅ Ω_{c,L}^{k+h}(S^{∗}(G), π_{S^{∗}(G)}^{−1}(X)),

if k ≠ 0, and, for k = 0, we have

\frac{Ω_{rc,L}^{k+h}(A^{∗}(G) ∖ 0, π_{A^{∗}(G)∖0}^{−1}(X))_{0}}{dΩ_{rc,L}^{k+h−1}(A^{∗}(G) ∖ 0, π_{A^{∗}(G)∖0}^{−1}(X))_{0}}

≅ \frac{Ω_{c,L}^{h}(S^{∗}(G), π_{S^{∗}(G)}^{−1}(X))}{dΩ_{c,L}^{h−1}(S^{∗}(G), π_{S^{∗}(G)}^{−1}(X))} ⊕ \frac{Ω_{c,L}^{h−1}(S^{∗}(G), π_{S^{∗}(G)}^{−1}(X))}{dΩ_{c,L}^{h−2}(S^{∗}(G), π_{S^{∗}(G)}^{−1}(X))}.

The conclusion follows.

The first assertion is a direct consequence of the above discussion and Theorem 4. The computation of HP_{q} then follows as in the proof of Theorem 6.

Proposition 5 is formulated in such a way that it remains true if we drop the subscript L (and if we remove p_{L}). If we proceed then as indicated, we obtain the periodic cyclic homology of A(M, X).

We now take a closer look at the Hochschild homology of A = Ψ^{∞}(G)/Ψ^{−∞}(G) and of the other related algebras.

We shall use below the Poisson structure of A^{∗}(G), which we now recall for the benefit of the reader. The natural regular Poisson structure of T_{vert}^{∗}(G) induces a Poisson structure on A^{∗}(G). (This is recalled in [23], Lemma 7, for example). If r : G → M is the range map, then the image of the differential of r restricted to A(G) determines a possibly singular foliation S on M. On the other hand, the kernel of r_{∗} is a family of Lie algebras whose fiber at x ∈ M is the Lie algebra of the Lie group G_{x}^{x}. When the groups G_{x}^{x} are 0-dimensional, this foliation has no singularities, and A(G) becomes the tangent bundle to the leaves of S. In this case, the Poisson structure of A^{∗}(G) ≅ T^{∗}S is induced from the symplectic structures of the leaves. In general, 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).

Let i_{G} be the contraction by G. Then we obtain as in [4] a differential

δ := i_{G} ∘ d − d ∘ i_{G} : Ω^{k}(A^{∗}(G)) → Ω^{k−1}(A^{∗}(G)), (17)

Explicitly, for any open subset V ⊂ A^{∗}(G) of the form V ≅ [0, 1)^{l} × ℝ^{n−l} and any (f_{0}, ..., f_{k}) ∈ C^{∞}(V)^{k+1}, the differential δ is given locally by the formula

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}.

This formula is valid also when M has corners. 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 and hence maps

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

Moreover, δ preserves the support, so it maps the space Ω_{rc}^{p}(A^{∗}(G) ∖ 0)_{d} of dhomogeneous forms supported above a compact set of M to Ω_{rc}^{p−1}(A^{∗}(G) ∖ 0)_{d−1}.

Definition 2. The d-homogeneous Poisson p-homology space H_{p}^{δ}(A^{∗}(G) ∖ 0)_{d} of the conic Poisson manifold with corners A^{∗}(G) ∖ 0 is defined by

H_{p}^{δ}(A^{∗}(G) ∖ 0)_{d} := \frac{ker(δ : Ω_{rc}^{p}(A^{∗}(G) ∖ 0)_{d} → Ω_{rc}^{p−1}(A^{∗}(G) ∖ 0)_{d−1})}{δ(Ω_{rc}^{p+1}(A^{∗}(G) ∖ 0)_{d+1}}.

Let Ω_{rc,L}^{p}(A^{∗}(G) ∖ 0)_{d} be the set of Laurent differential p-forms which are dhomogeneous under the radial action of ℝ_{+}^{∗} and such that their support projects onto a compact subset of M. Then it can be checked that δ is well defined on Ω_{rc,L}^{∗}(A^{∗}(G) ∖ 0)_{d} and is given by the same formula. In fact the covector G is here tangent to the faces because the groupoid G is differentiable. We thus have the following Laurent complexes indexed by p ∈ ℤ

0 → P^{p,n−p} →^{δ} P^{p,n−p−1} →^{δ} ... →^{δ} P^{p,−p} → 0,

where P^{p,d} = Ω_{rc,L}^{p+d}(A^{∗}(G) ∖ 0)_{d}. The homology of the resulting complex is, by definition, the Laurent-Poisson homogeneous homology of A^{∗}(G)∖0. This homology will be denoted by

H_{L,p+d}^{δ}(A^{∗}(G) ∖ 0)_{d} := \frac{ker(δ : P^{p,d} → P^{p,d−1})}{δ(P^{p,d+1})}.

We begin again with a result on the algebra A = Ψ^{∞}(G)/Ψ^{−∞}(G).

Proposition 6. The algebra A is H-unital. Let

χ : HH_{l}(Gr(A))_{d} → Ω_{rc}^{l}(A^{∗}(G) ∖ 0)_{d}

be the HKR-isomorphism, and let d_{1} : EH_{k,h}^{1} → EH_{k−1,h}^{1} be the first differential of the spectral sequence associated to A by Lemma 1. Then χ ∘ d_{1} ∘ χ^{−1} = −\sqrt{−1}δ, and hence EH_{k,h}^{2} ≃ H_{k+h}^{δ}(A^{∗}(G) ∖ 0)_{k}.

Proof. The H-unitality follows from Propositions 1 and 3. For the second part, we proceed essentially as in [7], Theorem 1. Choose an anti-symmetric tensor in the last m-variables

η = ∑ sign(σ)f_{0} ⊗ f_{σ(1)} ⊗ … ⊗ f_{σ(m)},

with f_{j} ∈ S_{c}^{∞}(A^{∗}(G)). We denote by

q(η) = ∑ sign(σ)q(f_{0}) ⊗ q(f_{σ(1)}) ⊗ … ⊗ q(f_{σ(m)})

the quantization of η. Let k = deg f_{0} + … + deg f_{m} be the total degree. Because

[q(a), q(b)] = −\sqrt{−1}q(\{a, b\}) + …,

where the dots represent terms of order at most deg a + deg b − 2, the quantity b ∘ q(η) is of total order at most k − 1 and hence, modulo terms of order k − 2, χ ∘ b ∘ q(η)) is easily checked to be exactly δ(η).

We think that the above spectral sequence is always convergent and that it actually degenerates at E^{2}. We prove this, for instance, for differentiable groupoids associated with manifolds with corners, see in Section 4. This spectral sequence is also studied in [6] for foliations. In [11] it is proved that the groupoid algebras Ψ^{−∞}(G) are H-unital.

In the same way we can generalize Proposition 6 and state:

Proposition 7. The algebra A_{L}(M, X) is H-unital. The EH^{2} term of the spectral sequence associated by Lemma 1 to A_{L}(M, X) is given by

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

Proof. The H-unitality follows from Propositions 1 and 3, as above.

To prove the rest of this proposition, we can either use the same method as the one used to prove Proposition 6, or we can argue that this proposition actually follows from Proposition 6.

4. Hochschild homology for manifolds with corners

We now restrict our study to groupoids G in a particular class. Thus, we shall assume throughout the rest of this paper that the anchor map ϱ : A(G) → T M is such that the induced map ϱ_{∗} : Γ(A(G)) → Γ(T M) of C^{∞}(M)-modules becomes an isomorphism after tensoring with O(M). That is, we assume that

ϱ_{∗} : O(M) ⊗_{C^{∞}(M)} Γ(A(G)) ≅ O(M) ⊗_{C^{∞}(M)} Γ(T M). (18)

We then say that A(G) is rationally isomorphic to T M. If that is the case, it also follows that ϱ induces an isomorphism of A(G) to T M on the interior M_{0} of M:

A(G)|_{M_{0}} ≅ T M_{0}.

Let O(M) be the ring of smooth functions on the interior M with only Laurent singularities at the corners, as in Section 2. Also, we assume that A(G) is such that where T M is the tangent bundle to M. We proceed in this section to the computation of the Hochschild homology groups of A := Ψ^{∞}(G)/Ψ^{−∞}(G), the algebra of complete symbols on G and of various other related algebras.

Throughout this and the following sections, n denotes the dimension of M, the space of units of G.

Let X ⊂ M be a closed union of faces of M and denote by I_{X} ⊂ C^{∞}(M) the ideal of functions that vanish to infinite order on X. Recall that we have defined the algebra A_{L}(M, X) of “Laurent complete symbols vanishing rapidly at X” by

A_{L}(M, X) := O(M)I_{X}A = O(M)I_{X}(Ψ^{∞}(G)/Ψ^{−∞}(G)),

(A_{L}(M) = A_{L}(M, ∅)).

We keep the notations of Section 3 so that π_{S^{∗}M×S^{1}} : S^{∗}M × S^{1} → M is the natural projection. Also, note that S^{∗}(G) ≅ S^{∗}M, but not canonically. The homotopy class of this homeomorphism is canonical though. Also, p_{L} denotes the structural map L(S^{∗}(G)) × S^{1} → M.

Theorem 8. Let X ⊂ M be a union of hyperfaces of M and assume that A(G) is rationally isomorphic to T M (that is, it satisfies the condition (18) above). Then we have

HH_{∗}(A_{L}(M, X)) ≃ H_{c}^{2n−∗}(L(S^{∗}M) × S^{1} ∖ p_{L}^{−1}(X))

Proof. The computation of HH_{∗}(A_{L}(M, X)) proceeds exactly as in the case of manifolds with boundary. From Proposition 7 we know that the E^{2} term of the spectral sequence associated with the order filtration on the Hochschild complex is given by

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

In our case,

Ω_{rc,L}^{k+h}(A^{∗}(G) ∖ 0, π_{A^{∗}(G)∖0}^{−1}(X))_{k} ≅ Ω_{rc,L}^{k+h}(T^{∗}M ∖ 0, π_{T^{∗}M∖0}^{−1}(X))_{k},

which is the main reason why this result is true, and explains the thinking behind the condition (18). Thus the Poisson structure on A^{∗}(G) ∖ 0 can be related to the natural Poisson structure on the cotangent bundle T^{∗}M ∖ 0. In particular, the restriction of the given Poisson structure to the interior of A^{∗}(G)∖0 is a symplectic structure.

We now recall the definition of the symplectic Hodge operator ∗_{G}. Let ω be the two form on the interior of M that defines the symplectic form. The form ω has only Laurent singularities at the hyperfaces due to our assumptions on A^{∗}(G). This then defines the ∗_{G}-operator by the formula [4]

∗_{G} : Ω_{rc,L}^{t}(T^{∗}M) → Ω_{rc,L}^{2n−t}(T^{∗}M), α ∧ ∗_{G}(α′) = ∧^{n}(G)(α, α′)ω^{n}/n!.

Then ∗_{G}^{2} = 1 and ∗_{G} ∘ δ ∘ ∗_{G} = (−1)^{k}d on the interior of M, by [4], and hence also on Ω_{rc,L}^{k}(T^{∗}M ∖ 0) because the forms in the latter space are determined by their restriction to the interior of M.

Now, keeping in mind that by construction the operator ∗_{G} maps k-homogeneous q-forms into k + n − q-homogeneous (2n − q)-forms, we get

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

≅ H_{L}^{2n−k−h}(T^{∗}M ∖ 0, π_{T^{∗}M∖0}^{−1}(X))_{n−h}.

Using Proposition 2 and the relative homotopy invariance of the relative De Rham cohomology, we see that the only non-zero homogeneous component corresponds to h = n, and then that it is given by

EH_{k,n}^{2} ≅ H_{L}^{n−k}(S^{∗}M × S^{1}, π_{S^{∗}M×S^{1}}^{−1}(X)).

From the fact that EH_{k,h}^{2} = 0, if h ≠ n, we see that d_{r} = 0, for all r ≥ 2, and hence we deduce the result. The convergence of the spectral sequence to HH_{∗}(A_{L}(M, X)) follows from Theorem 4.

By replacing k with k −n we obtain the stated formula for HH_{k}(A_{L}(M, X)).

Let us look now in more detail at one of the constructions used in the above proof in a particular case. Suppose then that the sections of the vector bundle A(G) at the hyperface H = {x = 0} are generated by ∂_{y} and x^{c_{H}} ∂_{x}. Here c_{H} are integers ≥ 1 associated to each face and the y’s are coordinates on H, using a tubular neighborhood of H. (This definition depends on a choice of defining functions for the hyperfaces of M.) Let us call this calculus the c_{n}-calculus.

Then the 2-covector is a covector on the manifold with corners A^{∗}(G) ∖ 0 which is given in any open subset of M diffeomorphic to one of the form [0, 1)^{k} × ℝ^{n−k} (with defining functions (x_{1}, …, x_{k})) by

G = ∑_{k+1≤j≤n} ∂_{ξ_{j}} ∧ ∂_{y_{j}} + ∑_{1≤j≤k} x_{j}^{c_{j}}∂_{ξ_{j}} ∧ ∂_{x_{j}}.

Here the convention of sign is that of [4]. The symplectic form is then given by

ω = ∑_{k+1≤j≤n} dy_{j} ∧ dξ_{j} + ∑_{1≤j≤k} x_{j}^{−c_{j}}dx_{j} ∧ dξ_{j}.

These two formulæ can be checked in the case [0, 1) × ℝ and globalized just like in the case of the usual cusp calculus. We have for instance in the case of a manifold with boundary given by {x = 0} that ∗_{G}(f) = ±(f/x^{c})dx ∧ dξ while ∗_{G}(fdx ∧ dξ) = ±fx^{c}.

5. Applications

We keep the assumptions of the previous section. The same methods as the ones in the previous section can be used also to determine the homology of some quotient algebras. Let X be a closed union of faces of M. First we prove a result that allows us to reduce the computation of the homologies of the relative algebras A_{L}(M, X) to the computation of those of an “absolute algebra.”

More precisely, denote by A_{L}(M ∖ X) the algebra of Laurent complete symbols on G|_{M∖X}, the groupoid obtained by restricting G to the invariant subset M ∖ X. Also, let L(S^{∗}M) be the space introduced at the end of Section 2 (with M in place of S^{∗}M), and denote by p_{L} : L(S^{∗}M) × S^{1} → M the resulting natural map. Then we have the following excision result.

Proposition 8. The inclusion A_{L}(M ∖ X) → A_{L}(M, X) induces isomorphisms in Hochschild, cyclic, and periodic cyclic homologies.

Proof. By a standard argument using the ’SBI’-exact sequence, it is enough to prove this statement for Hochschild homology.

Consider then the spectral sequences associated to the two algebras and the order filtration on their Hochschild complexes by Lemma 1. The induced map is then an isomorphism at the E^{2}-term, by Theorem 8, because

L(S^{∗}M) × S^{1} ∖ p_{L}^{−1}(X) = L(S^{∗}(M ∖ X)) × S^{1}.

Proposition 9. The quotient A := A_{L}(M)/A_{L}(M, X) is a topologically filtered algebra with

HH_{q}(A) ≅ H_{c}^{2n−q}(p_{L}^{−1}(X)).

Moreover, the exact sequence in Hochschild homology associated to the exact sequence 0 → A_{L}(M, X) → A_{L}(M) → A → 0 is naturally isomorphic to the long exact sequence in cohomology associated to the pair (L(S^{∗}M) × S^{1}, p_{L}^{−1}(X)).

Proof. Only the last statement is not similar to some other proofs in the previous sections. Choose an open neighborhood U of X in M such that X is a deformation retract of U. Also, let ϕ be a smooth function supported in U which is one in some smaller neighborhood of X. This data gives rise to a lifting of any smooth function on π^{−1}(X) to a function on S^{∗}M with support in U, and hence also to a linear lift ψ : H_{∗}(A) → H_{∗}(A_{L}(M)). The computation of [b, ψ] identifies the boundary map in the Hochschild cohomology spectral sequence and gives the result.

It is interesting to notice that as a consequence of our computations we obtain that the traces live only on the minimal faces. Obviously, the faces of M that are manifolds without corners are exactly the ones that are minimal with respect to inclusion in the set of faces of M. Let then Y be the union of all faces of M that have no corners (and no boundary).

Proposition 10. Let τ ∈ HH^{0}(A_{L}(M)) be a trace. Then it is induced from a trace on A_{L}(M)/A_{L}(M, Y).

Proof. Let n be the dimension of M, as before. The dual of Proposition 9 holds true for Hochschild cohomology. This implies that the inclusion

HH_{0}(A_{L}(M)/A_{L}(M, Y)) → HH_{0}(A_{L}(M))

is isomorphic (in the sense of the above Proposition) to the dual of the map H_{c}^{2n}(L(M)) → H_{c}^{2n}(p_{L}^{−1}(Y)). But this last map is checked to be an isomorphism.

The above proof gives as a corollary then the “number” of residue traces on M.

Corollary 2. The dimension of the space of traces of A_{L}(M) is the number of minimal faces of M.

The above two results can be proved directly, without using singular cohomology groups.

We continue to assume that A(G) is rationally isomorphic to T M (that is, it satisfies Equation (18)), as in the previous section. For the algebras that we have considered, the cyclic homology can be computed directly in terms of the Hochschild homology as we shall see below. Let X ⊂ M be a union of closed faces of M.

Consider Connes’ ‘SBI’-exact sequence associated to the algebra A_{L}(M, X) (see Equation (6)).

Theorem 9. We have B = 0 in the ‘SBI’-exact sequence of the algebra A_{L}(M, X), and hence

HC_{m}(A_{L}(M, X)) = ⊕_{k≥0} HH_{m−2k}(A_{L}(M, X)). (19)

Proof. We will proceed by induction on m to show that the morphism

B : HC_{m−1}(A_{L}(M, X)) → HH_{m}(A_{L}(M, X))

is zero for any m.

Let n denote the dimension of the manifold M, as above. The statement of the theorem is obviously true for m = 0. The proof of Theorem 8, show that the groups HH_{q}(A_{L}(M, X)) are generated by elements of order −n + q with respect to the degree filtration, and that all cycles of order less that −n + q are boundaries. This is a direct consequence of the computation of the E^{1} terms of the spectral sequences associated to the degree filtration.

Assuming now the statement to be true for all values less than m, we obtain from the ‘SBI’ exact sequence that the groups HC_{m−1}(A_{L}(M, X)) are isomorphic to HC_{m−1}(A_{L}(M, X)) ≃ ⊕_{k≥0} HH_{m−2k−1}(A_{L}(M, X)). This shows that the groups HC_{m−1}(A_{L}(M, X)) are generated by elements of order at most −n + m − 1 in the degree filtration. It follows that they map to elements of order less than −n + m in HH_{m}(A_{L}(M, X)), and hence they vanish by the above remark.

The last statement follows directly from the vanishing of B in the Connes’ exact sequence.

Appendix A. Projective limits

In this appendix we recall some well known facts about projective (or inverse) limits and the homology of projective limits, using this also as an opportunity to fix notation. An important role is played by lim^{1} = lim_{←}^{1}, the first (and only) derived functor of the projective limit functor lim_{←}.

Let ϕ_{n} : V_{n+1} → V_{n}, n ∈ ℕ, be an inverse system of vector spaces. Define F : ∏ V_{n} → ∏ V_{n} by F (v_{k}) = (v_{k} − ϕ_{k}(v_{k+1}). Then lim_{←} V_{n} is the kernel of F and

lim_{←}^{1} is the cokernel of F, by definition.

Suppose now that V_{n} are complexes of vector spaces and the maps ϕ_{n} are surjective. By writing the homology long exact sequence associated to the short exact sequence of complexes

0 → lim_{←} V_{n} → ∏ V_{n} ⟶^{F} ∏ V_{n} → 0,

we obtain the following well known exact sequence:

Lemma 6. If the maps ϕ_{n} : V_{n+1} → V_{n} are surjective, then the homology of the inverse limit satisfies

0 → lim_{←}^{1} H_{q+1}(V_{n}) → H_{q}(lim_{←} V_{n}) → lim_{←} H_{q}(V_{n}) → 0.

If B_{n} ⊂ A_{n} is a subspace preserved by ϕ_{n} : A_{n+1} → A_{n} and C_{n} = A_{n}/B_{n}, then we have an exact sequence

0 → ∏ B_{n} → ∏ A_{n} → ∏ C_{n} → 0

and F is an endomorphism of this exact sequence. The ker-coker lemma (see [1], for example) then gives the following exact sequence:

Lemma 7. If A_{n}, B_{n}, and C_{n} are as above, then we have the exact sequence

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

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Pennsylvania State University and 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