HOMOLOGY OF PSEUDO-DIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBERED BOUNDARIES

ROBERT LAUTER AND SERGIU MOROIANU

Abstract. We compute the Hochschild homology of the algebra of double-edge pseudo-differential operators on a manifold whose boundary is the total space of a fibration of closed manifolds. This algebra was presented in detail in our previous paper [6]. We introduce residue-type traces on this algebra and on various ideals and quotients. As an application, we show that these traces are unique up to a constant. We use these traces to compute the index of Fredholm operators in the double-edge calculus.

1. Introduction

A major motivation of non-commutative geometry is that properties of a space can be detected by studying algebras associated to that space. For instance, the deRham complex resp. cohomology of a smooth closed manifold M are described by a result of Connes [4] as the Hochschild, respectively cyclic homology groups of the algebra of C^{∞} functions on M. Thus, it is reasonable to expect, and it is indeed the case, that we get more information by studying the Hochschild and cyclic homology of other, more complicated algebras. A natural candidate is the algebra Ψ(M) of pseudodifferential operators, because of its intimate relation to the analytic structure of M. Brylinski and Getzler [3] after preliminary work of Wodzicki [23] computed the Hochschild homology of the algebra Ψ(M) for a closed manifold M, explaining in particular the existence of a unique “residue” trace on the algebra Ψ(M) found earlier by Wodzicki [23], Guillemin [5] and (in the one-dimensional case) Adler [1], Manin [9]. Their result states that HH_{k}(Ψ(M)/Ψ^{−∞}(M)) ≅ H^{2n−k}(S^{∗}M × S^{1}), where S^{∗}M is the cosphere bundle of the n-dimensional manifold M. A “noncommutative Euler class” was introduced in [24]. Melrose and Nistor [18] explained the Atiyah-Singer index formula in this homological context.

Let us note that the residue trace has been extended to higher Hochschild homology groups [20] in the case of the adiabatic algebra, defined when the manifold M is the total space of a fibration.

For manifolds with boundary one has to make a choice of an appropriate algebra of pseudodifferential operators. The b- and cusp algebras of Melrose have been extensively studied in [13], [14], [18], [19]. These algebras, in addition to the usual filtration by the order, have another filtration given by the order of vanishing at the boundary. It is natural to consider the residual ideals with respect to these filtrations, their intersection and also the quotients by these ideals. The quotients by the residual ideals are called the symbol algebra, resp. the boundary algebra. Melrose and Nistor [18] have computed the homology of the cusp algebra and of its various ideals and quotients. The result for the cusp symbol algebra is very similar to the case of closed manifolds, namely,

HH_{k}(Ψ_{c}(X)/Ψ_{c}^{−∞,ℤ}(X)) ≅ H_{Laurent}^{2n−k}(^{c}S^{∗}X × S^{1}),

i.e. the deRham cohomology of forms with Laurent expansions at the boundary, where ^{c}S^{∗}X is the sphere bundle in the so-called cusp cotangent bundle. The ideal Ψ_{c}^{−∞,ℤ}(X) of “smoothing” cusp operators does not see (in homology) the boundary ∂X; in fact its homology only lives in dimensions 0 and 1. Melrose and Nistor have found an extension of the Atiyah-Patodi-Singer index formula for fully elliptic cusp pseudodifferential operators, using new traces similar to the residue trace.

Date: May 7, 2000.

If moreover the boundary has some additional structure, one can build a maze of different algebras modeled on this geometry. Let us assume that this extra structure is a fibration of the boundary. Our choice, the double-edge algebra described below, is in some sense the least degenerate such algebra with the property that it distinguishes between fibers and base, and that the product becomes local at the boundary. We have as for the cusp algebra a filtration by the operator order and another one by the order of vanishing at the boundary, both of them compatible with the algebra structure. The homology of the “smoothing” ideal Ψ_{de}^{−∞,ℤ}(X) is independent of the fibration but captures the deRham cohomology of the boundary. However, we can recover the dimension of the fiber and the base of the fibration at the boundary by considering traces on this ideal and the residue trace. Thus, the double-edge smoothing ideal gives more information than its cusp cousin. Let us stress that for the usual algebra Ψ(M) on a closed manifold M, Melrose and Nistor remarked that the only nontrivial boundary map (described already in [24]), δ : HH_{1}(Ψ(M)/Ψ^{−∞}(M)) ⟶ HH_{0}(Ψ^{−∞}(M)), is related to the index of elliptic operators. This leads to a Hochschild-homological version of the Atiyah-Singer index formula. In the double-edge case, the boundary maps in the homology long exact sequence arising from

0 ⟶ Ψ_{de}^{−∞,ℤ}(X) ⟶ Ψ_{de}^{ℤ,ℤ}(X) ⟶ Ψ_{de}^{ℤ,ℤ}(X)/Ψ_{de}^{−∞,ℤ}(X) ⟶ 0

are non-trivial in higher dimensions, and can be described geometrically.

In the present paper we compute the Hochschild homology of the double-edge algebra and of its significant ideals and quotients. We then describe the index of fully elliptic double-edge operators in terms of certain functionals in Hochschild homology. This part follows formally the paper [18]. As expected, our index formula reduces to the Atiyah-Singer formula for an operator which is a bundle isomorphism near the boundary.

Let us now give a brief description of the double-edge algebra and a more detailed plan of the paper. Let X be a compact manifold whose boundary is the total space of a fibration of closed manifolds. Consider those vector fields on X (called double-edge vector fields) which vanish at the boundary, and whose first Taylor coefficients at the boundary are tangent to the fibers of this fibration. They form a Lie algebra, that we denote by V_{de}(X). This is a special case of the Lie algebra of doubly warped vector fields considered in [17]. The space of double-edge differential operators is by definition the enveloping algebra of V_{de}(X). Melrose [16] introduced a method of constructing pseudo-differential calculi associated to certain Lie sub-algebras of V(X). As usual, the corresponding calculus in our case contains the double-edge differential operators. We have studied this calculus and in particular its Fredholm theory in [6]. If the base of the fibration of ∂X degenerates to a point, the double-edge calculus is also known as scattering calculus; this calculus occurs naturally when dealing with scattering problems on asymptotically euclidean spaces [15].

The paper is structured as follows: in Section 2 we recall the definition and the main properties of the double-edge algebra from [6]. We give a brief overview of Hochschild homology of topological algebras. We recall Wodzicki’s notion of H-unitality [23] and show that our algebras have this property. In doing this we rely on a result from [20], which allows us to check only that the associated graded algebras are H-unital

In Section 3 we compute the homology of the algebra of complete double-edge symbols. This is very similar to the computations from [3] and [18], hence we treat it briefly.

Section 4 is devoted to the computation of the homology of the double-edge boundary algebra. This algebra is local, in the sense that the product is given by an asymptotic sum of bi-differential operators. This property makes it similar to the semi-classical limit boundary algebra, a particular case of the adiabatic algebra studied in [20]. Melrose suggested to compute the homology of this algebra by the spectral sequence associated to the filtration by the order of vanishing at the boundary, which would lead to the Leray-Serre spectral sequence of the fibration. We have not been able to overcome the technical difficulties of this approach; instead, using locality, we compute first the homology of a local boundary algebra. We prove then that Hochschild homology for this algebra can be computed by a Čech-Hochschild double complex; finally, we show that the spectral sequence of this double complex degenerates at E^{2}. This allows us to extract the result by using Čech cohomology. The last step of this argument relies on a very fortunate numerical identity. This limits the possible applications of the techniques developed in this section.

In Section 5 we define traces on the double-edge algebras as residue functionals, by analogy with the residue trace of Wodzicki and Guillemin. This part is formally identical to [18, Section 5].

The description of the long exact sequences in Hochschild homology (which exist as a consequence of H-unitality) is done in Section 6. As a corollary, we deduce that the traces on the double-edge algebras are unique up to a multiplicative constant.

Finally, in Section 7 we describe the index of double-edge Fredholm operators in terms of functionals on Hochschild homology. The index formula is formally identical to [7], which slightly improves Melrose and Nistor’s Index Theorem [18]. However, both terms of the formula are local in the double-edge case. It follows from the results of [6] that we can remove the boundary part by a homotopy. The index is then given by the Atiyah-Singer formula.

Acknowledgments: The present work and [6] sprang from a suggestion by Richard Melrose. His comments and help were invaluable throughout our journey in double-edge land.

The second author is grateful to Carmen Young, Victor Nistor, Gustavo Granja and Phil Bradley for their help and for many interesting discussions.

The first author was supported by a scholarship of the German Academic Exchange Service (DAAD) within the Hochschulsonderprogramm III von Bund und Ländern. He wishes to thank the Massachusetts Institute of Technology, where most of the results of this paper were obtained for the invitation and warm hospitality.

2. Review of background material

2.1. The double-edge calculus. Let X be a compact manifold with boundary. Assume that the boundary ∂X is the total space of a fibration π : ∂X → Y^{n} of compact, boundaryless manifolds, with fiber type F^{m}. Let N = dim X = n + m + 1. The algebra Ψ_{de}^{ℤ,ℤ}(X) of double-edge pseudodifferential operators was introduced in [17]. Let us review the definition and some notation from [6], where this calculus is presented in detail.

Let x : X → \overline{ℝ}_{+} be a boundary defining function on X, i.e. ∂X = \{x = 0\} and dx ≠ 0 on ∂X. A vector field v ∈ C^{∞}(X, T X) is called an edge vector field if it is tangent to the fibers of π on the boundary. A vector field is called double-edge if it is of the form xv, where v is an edge vector field. This definition is independent of the choice of the boundary-defining function x. The double-edge vector fields form a Lie algebra V_{de}(X). Moreover, they are naturally all the sections of a vector bundle ^{de}T X → X. Since every section of ^{de}T X is also a section of T X, there exists a natural map of vector bundles j^{de} : ^{de}T X → T X which is an isomorphism over the interior of X. Let ϕ : T^{∗}X → ^{de}T^{∗}X be the dual of this map.

Let ∂X ×_{π} ∂X = {(p, q) ∈ ∂X × ∂X; π(p) = π(q)} ⊂ X^{2} be the fiber diagonal of the fibration π, and let X_{e}^{2} = [X^{2}; ∂X ×_{π} ∂X] be the blow-up of X^{2} along the fiber diagonal. This space was introduced in [10]. By definition, there is a natural blow-down map β_{e} : X_{e}^{2} → X^{2} which is the identity outside the pre-image of ∂X ×_{π} ∂X. Let Δ_{e} be the closure in X_{e}^{2} of the interior of the diagonal (or rather its pre-image in X_{e}^{2}). The boundary of Δ_{e} is a submanifold of X_{e}^{2}. Define the double-edge double space, X_{de}^{2}, to be the blow-up [X_{e}^{2}; ∂Δ_{e}]. Let Δ_{de} be the closure in X_{de}^{2} of the lift through the blow-down map of the interior of the diagonal.

Call the new face introduced by the last blow-up the double-edge front face, or ff^{de}. Denote by ρ_{ff}_{de} a boundary defining function for this front face. Define the double-edge kernel half-density bundle ^{de}KD^{\frac{1}{2}} over X_{de}^{2} by its space of sections:

C^{∞}(X_{de}^{2},^{de}KD^{\frac{1}{2}} ) = ρ_{ff^{de}}^{− \frac{2+2n+m}{2}} C^{∞}(X_{de}^{2}, Ω^{\frac{1}{2}} (T X_{de}^{2})),

where Ω^{\frac{1}{2}} is the half-density functor.

Definition 2.1. Let Ψ_{de}^{m,0} be the space of distributional sections of ^{de}KD^{\frac{1}{2}} over X_{de}^{2} which are (classical) conormal to the lifted diagonal of order m ∈ ℂ, vanish to infinite order at boundary faces other than ff^{de}, and are smoothly extendible across this face.

By a form of the Schwartz kernel theorem, these distributions act on Ċ^{∞}(X, Ω^{\frac{1}{2}} (^{de}T X)), the space of sections of Ω^{\frac{1}{2}} (^{de}T X) over X which are vanishing with all derivatives at ∂X. For p ∈ ℂ, we finally define Ψ_{de}^{m,p}(X) := x^{−p}Ψ_{de}^{m,0}(X) = Ψ_{de}^{m,0}(X)x^{−p} [6, Lemma 3.4].

In this paper we mostly consider double-edge operators of integral order (however, operators of any complex order are needed in order to construct trace functionals in Section 5). In this case,

Ψ_{de}^{ℤ,ℤ}(X) = ⋃_{m∈ℤ} ⋃_{p∈ℤ} Ψ_{de}^{m,p}(X)

becomes a topological algebra which is a direct limit of Fréchet spaces. The two indices correspond to the increasing filtrations by the operator order, respectively by the negative of the order of vanishing at ∂X. Namely, an operator belongs to Ψ_{de}^{m,p}(X) if its order (of conormality to the lifted diagonal) is at most m, and it vanishes at least to order −p to the front face.

The filtrations are compatible with the algebra structure, i.e. they are preserved by multiplication. Let Ψ_{de}^{−∞,ℤ}(X) = ⋂_{k∈ℤ} Ψ_{de}^{k,ℤ}(X), Ψ_{de}^{ℤ,−∞}(X) = ⋂_{k∈ℤ} Ψ_{de}^{ℤ,k}(X), A_{σ} = Ψ_{de}^{ℤ,ℤ}(X)/Ψ_{de}^{−∞,ℤ}(X), I_{σ} = Ψ_{de}^{ℤ,−∞}(X)/Ψ_{de}^{−∞,−∞}(X) and A_{∂,σ} = A_{σ}/I_{σ}. Analogously, let A_{∂} = Ψ_{de}^{ℤ,ℤ}(X)/Ψ_{de}^{ℤ,−∞}(X) and I_{∂} = Ψ_{de}^{−∞,ℤ}(X)/Ψ_{de}^{−∞,−∞}(X). The following diagram summarizes the situation, where the horizontal and vertical sequences are exact:

(1)

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & Ψ_{de}^{−∞,−∞}(X) \ar[r] \ar[d] & Ψ_{de}^{ℤ,−∞}(X) \ar[r] \ar[d] & I_{σ} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Ψ_{de}^{−∞,ℤ}(X) \ar[r] \ar[d] & Ψ_{de}^{ℤ,ℤ}(X) \ar[r] \ar[d] & A_{σ} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & I_{∂} \ar[r] \ar[d] & A_{∂} \ar[r] \ar[d] & A_{∂,σ} \ar[r] \ar[d] & 0. \\  & 0 & 0 & 0}

The product on Ψ_{de}^{−∞,−∞}(X) is the usual composition of kernels. Let us recall from [6] the form of the product on Ψ_{de}^{ℤ,ℤ}(X)/Ψ_{de}^{−∞,−∞}(X).

The front face ff^{de} is canonically diffeomorphic to the radial compactification^{de}\overline{T} X_{|∂X} of^{de}T X_{|∂X}, and Δ_{de} is canonically diffeomorphic to X. A double-edge quantization is an extension ^{de}Θ of these diffeomorphisms from a neighborhood of X ∪ ^{de}\overline{T} X_{|∂X} inside the radial compactification of ^{de}T X to a neighborhood of Δ_{de} ∪ ff^{de} in X_{de}^{2}, that induces in addition an isomorphism of ^{de}T X, i.e. the normal bundle to the zero section, onto the normal bundle NΔ_{de} of the double edge diagonal.

Let us fix the following notation: For any vector bundle V over a closed manifold Y, denote by S^{ℤ}(V), 𝒮(V), resp. S(V) the space of classical symbols, of Schwartz functions, resp. of formal symbols on V. The last space is the set of formal sums of homogeneous functions of integral order on V \ {0}, bounded above but not below. Any such sum can be realized as the asymptotic expansion of an actual classical symbol, hence we have the following short exact sequence:

0 ⟶ S^{ℤ}(V) ⟶ 𝒮(V) ⟶ S(V) ⟶ 0.

A choice of a double-edge quantization ^{de}Θ, Fourier transform in the fibers of ^{de}T X and contraction with the double-edge symplectic volume form fixes an isomorphism

Ψ_{de}^{ℤ,ℤ}(X)/Ψ_{de}^{−∞,−∞}(X) ⟶ S^{ℤ}(^{de}T^{∗}X)[x^{−1}]/𝒮̇(^{de}T^{∗}X)

(the dot denotes rapid vanishing at ∂X). We are going to describe the induced product on this last space. Let X^{∘} denote the interior of X. Recall that j^{de} : T X_{|X^{∘}} → ^{de}T X_{|X^{∘}} is a vector bundle isomorphism. Together with the blow-down map, this conjugates ^{de}Θ to a “local quantization” T X_{|X^{∘}} ⊃ U ⟶^{Θ} V ⊂ (2X)^{2}, where U is a neighborhood of the zero-section, and V is a neighborhood of the diagonal in X^{∘} × X^{∘}. Let Ψ_{cpt}(X^{∘}) denote the algebra of pseudo-differential operators on X whose Schwartz kernels are supported inside X^{∘}×X^{∘} (the distinction between usual and double-edge operators disappears in this case). As above, Θ induces an isomorphism

Ψ_{cpt}^{ℤ}(X^{∘})/Ψ_{cpt}^{−∞}(X^{∘}) ⟶ S_{cpt}^{ℤ}(T^{∗}X)/𝒮_{cpt}(T^{∗}X),

and hence a star-product on compactly supported formal symbols on T^{∗}X_{|X^{∘}}, i.e. a sequence of bi-differential operators with polynomial coefficients in the fibers:

a ∗ b = ∑_{j=0}^{∞} P_{j}(a, b).

Moreover, P_{0}(a, b) = ab and P_{1}(a, b) − P_{1}(b, a) = \frac{1}{i}\{a, b\}. The main point here is that this product is local in T^{∗}X. Then the product on Ψ_{de}^{ℤ,ℤ}(X)/Ψ_{de}^{−∞,−∞}(X) is given over the interior of X by the push-forward of ∗ under ϕ:

a ∗_{de} b = ∑_{j=0}^{∞} ϕ_{∗}(P_{j})(a, b). (2)

In particular, this implies that the differential operators ϕ_{∗}(P_{j}) have a Taylor expansion at ∂X, ϕ_{∗}(P_{0})(a, b) = ab and ϕ_{∗}(P_{1})(a, b) − ϕ_{∗}(P_{1})(b, a) = \frac{1}{i}\{a, b\}_{de}, the double-edge Poisson bracket. We have proved in [6] that the product on the boundary algebra A_{∂} is given by the Taylor expansion at ∂X of ∗_{de}.

2.2. Hochschild homology. A full treatment of this topic can be found in [8]. Let A be a unital algebra over ℂ. Let C_{i}(A) = A^{⊗i+1} be the space of Hochschild chains. For j = 0, …, i, define b_{j} : C_{i}(A) → C_{i−1}(A) by

a_{0} ⊗ … ⊗ a_{j} ⊗ a_{j+1} ⊗ … ⊗ a_{i} ⟼^{b_{j}} a_{0} ⊗ … ⊗ a_{j}a_{j+1} ⊗ … ⊗ a_{i} for j < i,
a_{0} ⊗ … ⊗ a_{i} ⟼^{b_{i}} a_{i}a_{0} ⊗ … ⊗ a_{i−1}.

The Hochschild boundary map b : C_{i}(A) → C_{i−1}(A) is defined by b = ∑_{j=0}^{i}(−1)^{j}b_{j}. By direct computation, b^{2} = 0. The homology of the complex (C_{•}, b) is called the Hochschild homology of the algebra A (relative to A).

If A is not unital, let Ā be the augmented algebra A ⊕ ℂ. We define the chain spaces of A by C_{k}(A) = ker(Ā^{⊗k+1} → ℂ^{⊗k+1}), with the boundary maps given by the same formula. This definition, when applied to a unital algebra, gives different chain spaces from the ones above. Nevertheless, there are canonical chain maps which induce isomorphisms on homology.

As in [3], define the topological Hochschild chain spaces C_{k}(Ψ_{de}^{ℤ,ℤ}(X)) as the direct limit of projective tensor products of Fréchet spaces:

C_{k}(Ψ_{de}^{ℤ,ℤ}(X)) = lim_{j→∞^{⟶}} Ψ_{de}^{j,j}(X) ˆ⊗_{π} … ˆ⊗_{π}Ψ_{de}^{j,j}(X) (3)

(k + 1 terms). Any filtration F_{k} on Ψ_{de}^{ℤ,ℤ}(X) (e.g. the filtrations by the operator order and by the order of vanishing at the boundary) is inherited by C_{j}(Ψ_{de}^{ℤ,ℤ}(X)) in the following way: a pure tensor a_{0} ⊗ … ⊗ a_{j} is said to belong to F_{k} if a_{0} ∈ F_{k_{0}}, …, a_{j} ∈ F_{k_{j}} with k_{0} + … + k_{j} = k. Define F_{k} to be the closure of the span of pure tensors in F_{k}. The boundary map extends to these topological chain spaces. Moreover, it preserves the induced filtration F_{j}, provided that the filtration on the algebra is compatible with multiplication. Hence any such filtration induces a spectral sequence associated to the Hochschild complex of Ψ_{de}^{ℤ,ℤ}(X) (see [11]). This also holds for the induced filtrations on the other spaces from diagram (1).

2.3. H-unitality. The aim of this section is to prove that the ideals I_{∂}, I_{σ}, Ψ_{de}^{ℤ,−∞}(X), Ψ_{de}^{−∞,ℤ}(X) and Ψ_{de}^{−∞,−∞}(X) are H-unital.

Let A be an algebra over ℂ. The bar resolution of A is the complex (C_{i}(A), b′)_{i≥0}, where b′ : C_{i}(A) → C_{i−1}(A) is given by b′ = ∑_{j=0}^{i−1}(−1)^{j}b_{j} (this is the Hochschild boundary map without the last term). As before, if A is a topological algebra, then C_{•}(A) is defined by using a topological tensor product.

Definition 2.2 (Wodzicki [25]). The algebra A is called H-unital if the bar resolution of A is acyclic.

Every unital algebra is H-unital. Wodzicki [25] proved that H-unital algebras are excisive for Hochschild homology. In other words, let I be a H-unital algebra. Then every extension

0 → I → A → A/I → 0

leads to a long exact sequence in Hochschild homology. In turn, this property implies that I is H-unital. We refer the reader to [20] or [21] for a discussion of this result for topological algebras. Essentially, Wodzicki’s proof holds in this case as well.

Proposition 2.3. The algebras I_{∂}, I_{σ}, Ψ_{de}^{−∞,−∞}(X), Ψ_{de}^{ℤ,−∞}(X) and Ψ_{de}^{−∞,ℤ}(X) are H-unital.

Proof: The graded algebra GI_{∂} associated to I_{∂} with respect to the x-filtration is naturally isomorphic to the algebra 𝒮(^{de}T^{∗}X_{|∂X})[x, x^{−1}] of Schwartz functions on ^{de}T^{∗}X_{|∂X}, tensored with the ring of Laurent polynomials in the boundary-defining function x. Similarly, the graded algebra GI_{σ} of I_{σ} with respect to the order filtration is naturally isomorphic to the algebra of finite sums of homogeneous functions on ^{de}T^{∗}X, with x^{−1} adjoined.

Melrose and Nistor [18] proved that for a manifold with boundary M, the algebra Ċ^{∞}(M) is H-unital. Their homotopy [18, Lemma 1], works also for the algebra Ċ^{∞}(M) ⊗ ℂ[x, x^{−1}]. Hence GI_{∂} and GI_{σ} are H-unital. Since the filtrations are complete on I_{∂} and I_{σ}, we apply [21, Theorem A.2], and we get that I_{∂} and I_{σ} themselves are H-unital.

The following result appears implicitly in [18]; we include the proof for the sake of completeness.

Lemma 2.4. The residual ideal Ψ_{de}^{−∞,−∞}(X) is H-unital.

Proof: Let Ψ̇^{−∞}(X) denote the algebra of smoothing operators on X, whose Schwartz kernels are rapidly vanishing to the boundary of X^{2}. Remark that the algebras Ψ_{de}^{−∞,−∞}(X) and Ψ̇^{−∞}(X) are isomorphic (the isomorphism is given at the level of Schwartz kernels by the blow-down map). Fix a density dx on X, so that smoothing operators become identified with rapidly vanishing functions on X^{2}. Let f ∈ Ċ^{∞}(X) be a function of square integral 1. We construct a homotopy for the bar resolution as follows:

s_{k} : C_{k} → C_{k+1} : (s_{k}a)(x_{0}, y_{0}, …, x_{k+1}, y_{k+1}) = f(y_{0})f(x_{1})a(x_{0}, y_{1}, …, x_{k+1}, y_{k+1}).

Using this lemma and [21, Theorem A.8], it follows that Ψ_{de}^{ℤ,−∞}(X) and Ψ_{de}^{−∞,ℤ}(X) are also H-unital.

3. The homology of the double-edge symbol algebras

Consider the short exact sequence

0 ⟶ I_{σ} ⟶ A_{σ} ⟶ A_{∂,σ} ⟶ 0, (4)

where A_{σ} is the algebra of complete double-edge symbols A_{σ} = Ψ_{de}^{ℤ,ℤ}(X)/Ψ_{de}^{−∞,ℤ}(X), and I_{σ} is the ideal of symbols in A_{σ} which vanish rapidly at ∂X. In this section we compute the Hochschild homology of these algebras. We use the results of [3]. A similar computation was done in [18] for the algebra of cusp pseudodifferential operators on a compact manifold with boundary.

In the following theorem, the absolute and relative cohomology spaces can be thought of in terms of singular cohomology. By the deRham Theorem, these can be computed with smooth differential forms, resp. with forms supported in the interior. The last space can also be computed as the cohomology of rapidly vanishing forms, or alternately with forms vanishing at the boundary.

Theorem 3.1.

(5)  HH_{k}(A_{σ}) ≅ H_{abs}^{2N−k}(^{de}S^{∗}X × S^{1}) ⊕ H^{2N−1−k}(^{de}S^{∗}X|_{∂X} × S^{1});
HH_{k}(A_{∂,σ}) ≅ H^{2N−k}(^{de}S^{∗}X_{|∂X} × S^{1} × S^{1});
HH_{k}(I_{σ}) ≅ H_{rel}^{2N−k}(^{de}S^{∗}X × S^{1}).

Proof: The algebra A_{σ} admits the usual filtration F_{j} by the operator order. We form a spectral sequence abuting to HH_{∗}(A_{σ}) as explained in Subsection 2.2. Let GA_{σ} be the associated graded algebra. In [6] we showed that GA_{σ} ≅ S(^{de}T^{∗}X)[x^{−1}], where S(^{de}T^{∗}X) is the algebra of formal symbols on ^{de}T^{∗}X_{|∂X} with the usual commutative product. From the definitions, and using a modified form of the result of Connes [4], we get

E_{i,k}^{1}(A_{σ}) ≅ (HH_{i+k}(GA_{σ}))_{i} ≅ Ω_{(i)}^{i+k}(^{de}T^{∗}X \ {0})[x^{−1}].

The subscript (i) denotes homogeneous forms of homogeneity i. The second isomorphism is realized by the Hochschild-Kostant-Rosenberg map, denoted HKR:

a_{0} ⊗ a_{1} ⊗ … ⊗ a_{k} ⟼^{HKR} \frac{1}{k!}a_{0}da_{1} ∧ … ∧ da_{k}. (6)

Note that HKR is a local operator. We have seen in Section 2 that the double-edge product on the algebra A_{σ} is also local in the base X. It follows that so is d^{1}, the first differential in the spectral sequence. Let us first localize outside ∂X. Let α ∈ Ω_{(i)}^{i+k}(^{de}T^{∗}X \ {0})[x^{−1}] be a form supported outside ∂X. We can assume that α is represented by a chain ∑ a_{0} ⊗ a_{1} ⊗ … ⊗ a_{i+k} where all the symbols a_{j} are supported outside ∂X. Extending by 0, we can think of them as symbols of (usual) pseudo-differential operators on the closed manifold 2X, the double of X. Let us recall how this is done. The blow-down map β : X_{de}^{2} → X^{2} is a diffeomorphism outside the boundary, hence it preserves kernels supported in the interior. It also induces a map between the normal bundles to the diagonal, which is just the double-edge map ^{de}T X → T X. The correspondence above is given by pull-back via the (dual) structure map ϕ : T^{∗}X → ^{de}T^{∗}X. This map pulls back the double-edge star product to a star product on T^{∗}X^{∘} [6]. Note that ϕ is homogeneous and that HKR commutes with pull-back. It follows that, in the interior of X, d^{1} = (ϕ^{−1})^{∗}d_{2X}^{1} ϕ^{∗}, where d_{2X}^{1} is the first differential in the spectral sequence of pseudo-differential operators on 2X [3].

Let ∗_{ϕ} be the de-symplectic duality operator (not to confuse with the product ∗_{de}!):

∗_{ϕ} = (ϕ^{−1})^{∗} ∗ ϕ^{∗}, (7)

where ∗ is the symplectic duality operator defined by Brylinski [2]. Since ∗^{2} = 1, it follows that ∗_{ϕ}^{2} = 1.

Lemma 3.2. ∗_{ϕ} extends down to x = 0 to an involution of Ω^{•}(^{de}T^{∗}X)[x^{−1}], and interchanges homogeneous forms in the following way:

Ω_{(i)}^{k}(^{de}T^{∗}X \ {0})[x^{−1}] ⟷ Ω_{(i−k+N)}^{2N−k}(^{de}T^{∗}X \ {0})[x^{−1}].

Proof: Since ∗_{ϕ} is C^{∞}(X^{∘})-linear and multiplicative, it is enough to check the claim locally for a basis of covectors in ^{de}T^{∗}X. For these, apply the local coordinate formulas from [6].

By a result from [3], d_{2X}^{1} = (−1)^{•} ∗ d∗ on Ω^{•}. Hence, in the interior of X,

d^{1} = (−1)^{•} ∗_{ϕ} d ∗_{ϕ} .

By continuity, this formula is also valid near the boundary ∂X. It follows that

E_{i,k}^{2}(A_{σ}) ≅ H_{i}(Ω_{(•)}^{•+k}(^{de}T^{∗}X \ {0})[x^{−1}], ∗_{ϕ}d∗_{ϕ}) ≅ H^{i}(Ω_{(N−k)}^{2N−•−k}(^{de}T^{∗}X \ {0})[x^{−1}], d).

The second isomorphism is obtained by conjugation with ∗_{ϕ}. Let R be the radial vector field in the fibers of ^{de}T^{∗}X. On forms of homogeneity k, we have kI = L_{R} = dı_{R} + ı_{R}d. Hence if k ≠ 0, \frac{1}{k} ı_{R} is a homotopy to the identity, and so the cohomology vanishes. Therefore E_{i,k}^{2}(A_{σ}) = 0 unless k = N. This proves the degeneracy of the spectral sequence at E^{2}. A straightforward computation (see [18], [12]) yields

H^{k}(Ω_{(0)}^{•}(^{de}T^{∗}X \ {0})[x^{−1}], d) = H_{abs}^{k}(^{de}S^{∗}X × S^{1}) ⊕ H^{k+1}(^{de}S^{∗}X|_{∂X} × S^{1}).

By Lemma 3.2, this implies the result. The convergence of the spectral sequence follows from the following observation: let b^{j} be the component of b which decreases the homogeneity degree by j. Then the map b^{1} increases the x degree by at most 2, while the maps b^{2}, b^{3}, … increase the x degree by at least 2.

Exactly the same proof applies for the other two isomorphisms.

From the shape of the spectral sequences, it follows that these isomorphisms are realized by edge maps, hence are natural. Since I_{σ} is H-unital, the short exact sequence (4) induces a long exact sequence in Hochschild homology.

Proposition 3.3. The long exact sequence induced from (4) is isomorphic to the long exact sequence in deRham cohomology derived from

0 ⟶ (Ω̇_{(0)}^{2N−•}(^{de}T^{∗}X \ {0})[x^{−1}], d) ⟶ (Ω_{(0)}^{2N−•}(^{de}T^{∗}X \ {0})[x^{−1}], d)

⟶ ((Ω_{(0)}^{2N−•}(^{de}T^{∗}X \ {0}))/Ω̇_{(0)}^{2N−•}(^{de}T^{∗}X \ {0})[x^{−1}], d) ⟶ 0. (8)

Proof: As usual, the dot denotes rapid vanishing to the boundary. From the computation of E^{1}, the sequence (4) induces short exact sequences in the E_{•,k}^{1} terms, hence long exact sequences of the E_{•,k}^{2} terms. The following diagram is then commutative:

(9)

\xymatrix{… HH_{i}(I_{σ}) \ar[r] \ar[d] & HH_{i}(A_{σ}) \ar[r] \ar[d] & HH_{i}(A_{∂,σ}) \ar[r] \ar[d] & HH_{i−1}(I_{σ}) … \ar[d] \\ … E_{i−N,N}^{2}(I_{σ}) \ar[r] & E_{i−N,N}^{2}(A_{σ}) \ar[r] & E_{i−N,N}^{2}(A_{∂,σ}) \ar[r] & E_{i−1−N,N}^{2}(I_{σ}) …}

The vertical arrows are the edge isomorphisms. It is enough to show that the lower row is isomorphic to the long exact sequence induced from (8). Notice that the complexes (E_{•−N,N}^{1}, d^{1}) and (Ω_{(0)}^{2N−•}, d) are isomorphic through the map ∗_{ϕ} ∘ HKR. This map commutes with the maps between the three complexes of E_{1}’s, and the natural maps between the three deRham complexes, hence it induces an isomorphism between the two long exact sequences.

Remark 3.4. In addition to Theorem 3.1, this Proposition describes the maps in the long exact sequence in Hochschild homology.

4. The homology of the boundary algebras

4.1. The E^{1} terms. The algebra A_{∂} has a natural filtration R_{i}(A_{∂})_{{i∈ℤ}} by the order of vanishing to the boundary. Let GI_{∂}, GA_{∂} and GA_{∂,σ} be the associated graded algebras with respect to this filtration. As explained in Subsection 2.2, the filtration induces spectral sequences for the Hochschild complexes. We denote by a subscript S^{ℤ}, 𝒮 and S those forms whose coefficients are classical symbols, Schwartz functions, resp. formal symbols.

Proposition 4.1.

E_{i,j}^{1}(A_{∂}) ≅ E_{i,j}^{1}(GA_{∂}) ≅ x^{−i}Ω_{S^{ℤ}}^{i+j}(^{de}T^{∗}X_{|∂X}) ⊕ x^{−i−1}dx ∧ Ω_{S^{ℤ}}^{i+j−1}(^{de}T^{∗}X_{|∂X})
E_{i,j}^{1}(I_{∂}) ≅ E_{i,j}^{1}(GI_{∂}) ≅ x^{−i}Ω_{𝒮}^{i+j}(^{de}T^{∗}X_{|∂X}) ⊕ x^{−i−1}dx ∧ Ω_{𝒮}^{i+j−1}(^{de}T^{∗}X_{|∂X})

E_{i,j}^{1}(A_{∂,σ}) ≅ E_{i,j}^{1}(GA_{∂,σ}) ≅ x^{−i}Ω_{S}^{i+j}(^{de}T^{∗}X_{|∂X} \ {0}) ⊕ x^{−i−1}dx ∧ Ω_{S}^{i+j−1}(^{de}T^{∗}X_{|∂X} \ {0}).

Proof: Let us prove the statement for I_{∂}. Recall that ff^{de} is canonically diffeomorphic to ^{de}\overline{T} X_{|∂X}. We have GI_{∂} ≅ 𝒮(^{de}T^{∗}X_{|∂X}) ⊗ ℂ[x, x^{−1}], where x is the boundary defining function. From the definitions, E_{i,j}^{1}(I_{∂}) ≅ HH_{i+j}(GI_{∂})_{(i)}, where the subscript i denotes the degree in GI_{∂}. Note that this statement is true when we replace I_{∂} by any filtered algebra. This proves the first isomorphism. The second one is realized by the Hochschild-Kostant-Rosenberg map (6). The other two statements are completely similar, so we leave them as an exercise.

By H-unitality of I_{∂}, the map of complexes

C_{•}(I_{∂}) ⟶ ker(C_{•}(A_{∂}) → C_{•}(A_{∂}/I_{∂}))

induced from the short exact sequence

0 ⟶ I_{∂} ⟶ A_{∂} ⟶ A_{∂,σ} ⟶ 0 (10)

is a quasi-isomorphism. This map induces maps of spectral sequences, which are isomorphisms starting at E^{1} (since GI_{∂} is also H-unital). We can also factor out the acyclic sub-complex spanned by chains which contain 1 on a position other that the first, hence work with the reduced complex, denoted by a subscript red. In conclusion, we can replace in the computation of E(I_{∂}) the complex C_{•}(I_{∂}) with ker(C_{•}(A_{∂}) → C_{•}(A_{∂}/I_{∂}))_{red}. See [8] for algebraic details.

4.2. The local computation. The product on the boundary algebra A_{∂}, and hence on I_{∂}, is local in the base X. When we identify operators with Laurent series of complete symbols on ^{de}T^{∗}X_{|∂X} via Fourier transform and a double-edge quantization, the product becomes local in all variables. This is more like a product structure on ^{de}T^{∗}X_{|∂X} itself. We proved in [6] that this product is the Taylor series at x = 0 of the push-forward via ϕ of a star-product on homogeneous symbols on T^{∗}X_{|{x>0}}. A different choice of double-edge quantizations transforms S^{ℤ}(^{de}T^{∗}X_{|∂X})[x^{−1}] in the following way:

∑_{k≥k_{0}}^{∞} x^{k}a_{k} ⟼ ∑_{k} ∑_{j=0}^{∞} x^{j}H_{j}(x^{k}a_{k}),

where H_{j} are differential operators on ^{de}T^{∗}X of homogeneity 0 in x, and H_{0} is the identity.

Let O ⊂ ∂X be an open set. Define the local boundary algebra A_{∂}(O) as the vector space S^{ℤ}(^{de}T^{∗}X_{|∂X∩O})[x^{−1}], with the product given by the star-product of a fixed double-edge quantization. Any two products are related by conjugation with a transformation as above, hence the local boundary algebra is well-defined up to isomorphism.

In a local coordinates neighborhood O × \overline{ℝ}_{+} around a point on the boundary, X looks like the manifold \overline{ℝ}_{+} × ℝ_{y}^{n} × ℝ_{z}^{m}, where the variable x ∈ \overline{ℝ}_{+} is the boundary defining function. Then the set \{x^{2} \frac{∂}{∂x}, x^{2} \frac{∂}{∂y_{i}} , x\frac{∂}{∂z_{j}} \} forms a basis for ^{de}T X. The bundle T X has the canonical basis \{\frac{∂}{∂x}, \frac{∂}{∂y_{i}} , \frac{∂}{∂z_{j}} \}. The structure map ϕ takes the following form in the coordinates given by these trivializations:

T^{∗}X ⟶^{ϕ} ^{de}T^{∗}X, (x, y, z, ξ, η, ζ) ⟼ (x, y, z, x^{2}ξ, x^{2}η, xζ). (11)

Consider now the local coordinates system (x, y, z, T, V, W) in X_{de}^{2} near the boundary of the lifted diagonal, and the local coordinates (x, y, z, x′, y′, z′) on the product X^{2}, so that the blow down map takes the following form (see [6]):

(x, y, z, T, V, W) ⟶ (x, y, z, x^{2}T + x, x^{2}V + y, xW + z).

Choose a quantization ^{de}Θ which in the chosen local coordinates looks like the identity. Let Θ be a quantization map T X → X^{2} related to ^{de}Θ by the blow-down map and the induced map on the normal bundles to the diagonal. Over O, Θ has the form (x, y, z, ξ̃, η̃, ζ̃) ⟼ (x, y, z, ξ̃+x, η̃+y, ζ̃+z), therefore the induced product is the usual left-quantization product. Without loss of generality, we can assume that the product on A_{∂}(O) is the push-forward of this product:

∗ = ∑_{j≥0} P_{j}; f ∗_{de} g = ∑_{j≥0} ϕ_{∗}(P_{j})(f, g), (12)

where P_{j} is a bi-differential operator of order (j, j) which contains j partial derivatives in the cotangent directions. Also, P_{0} is the usual multiplication, and P_{1}(f, g) − P_{1}(g, f) = 1/i{f, g}.

Define analogously the local boundary ideal and symbol algebra, I_{∂}(O) and A_{∂,σ}(O).

Proposition 4.2. Every element in E_{•,k−•}^{1}(I_{∂}(O)) (see Proposition 4.1) can be uniquely represented by a chain in ker(C_{k}(A_{∂}(O)) → C_{k}(A_{∂}(O)/I_{∂}(O)))_{red} of the form

a = ∑_{I} f_{I} ⊗ ∑_{σ∈Σ_{k}}(−1)^{|σ|}t_{σ(i_{1})} ⊗ … ⊗ t_{σ(i_{k})}, (13)

where I = (i_{1} < … < i_{k}), the functions {t_{j}}_{j=1,... ,k} ∈ {x, y_{i}, z_{l}, x^{−2}ξ, x^{−2}η_{i}, x^{−1}ζ_{l}} pull back by ϕ to the coordinate functions on T^{∗}X, and f_{I} are Laurent series in x of Schwartz functions on ^{de}T^{∗}X_{|∂X}.

Proof: The structure map ϕ is a diffeomorphism for x > 0. It induces an isomorphism between the spaces of forms with polynomial coefficients after adjoining x^{−1}. Recall that HKR is invariant by pull-back. From (11), we see that ϕ^{∗}t_{j} are the coordinate functions on T^{∗}X_{|O}. The proposition follows.

For a fixed integer p, consider the vector space

L^{k}(p) = x^{p}Ω_{𝒮}^{k}(ℝ_{ξ} × ℝ_{z,ζ}^{2m} × ℝ_{y,η}^{2n}) ⊕ x^{p−1}dx ∧ Ω_{𝒮}^{k−1}(ℝ_{ξ} × ℝ_{z,ζ}^{2m} × ℝ_{y,η}^{2n}) (14)

of Laurent series of (local) Schwartz forms on ^{de}T^{∗}X_{|∂X}, of degree p in x. Here the subscript 𝒮 means that the Schwartz behavior is in the variables ξ, η, ζ. Every form μ ∈ L^{k}(p) can be uniquely written as

μ = ∑_{I} f_{I}dt_{i_{1}} ∧ … ∧ dt_{i_{k}},

in the notation of Proposition 4.2. We endow L^{•} = ⨁_{p} L^{•}(p) with the differential δ = (−1)^{•} ∗_{ϕ} d∗_{ϕ}. Consider the following map:

Φ : (L^{k}, δ) ⟶ (ker(C_{k}(A_{∂}(O)) → C_{k}(A_{∂}/I_{∂}))_{red}, b)
(15)  fdt_{i_{1}} ∧ … ∧ dt_{i_{k}} ⟼ f ⊗ ∑_{σ∈Σ_{k}}(−1)^{|σ|}t_{σ(i_{1})} ⊗ … ⊗ t_{σ(i_{k})}.

Proposition 4.3. The map (15) is a map of complexes.

Proof: The multiplication rule in A_{∂}(O) is given by the star product (2). Let a = fdt_{i_{1}} ∧ … ∧ dt_{i_{k}}. Then

b(Φ(a)) = f ⊗ ∑_{σ∈Σ_{k}} ∑_{j=1}^{k−1}(−1)^{|σ|+j}t_{σ(i_{1})} ⊗ … ⊗ t_{i_{σ(j)}} ∗_{de} t_{i_{σ(j+1)}} ⊗ … ⊗ t_{σ(i_{k})}

+ ∑_{σ∈Σ_{k}}(−1)^{|σ|}f ∗_{de} t_{σ(i_{1})} ⊗ … ⊗ t_{σ(i_{k})} + ∑_{σ∈Σ_{k}}(−1)^{|σ|+k}t_{σ(i_{k})} ∗_{de} f ⊗ t_{σ(i_{1})} ⊗ … ⊗ t_{σ(i_{k−1})}.

The right hand side is anti-symmetric in j, j + 1, so we can replace in the above formula t_{σ(j)} ∗_{de} t_{σ(j+1)} by the commutator (1/2)[t_{σ(j)}, t_{σ(j+1)}]_{de}. We see that (1/2)[t_{σ(j)}, t_{σ(j+1)}]_{∗_{de}} = (1/2)ϕ^{−1}^{∗}[s_{σ(j)}, s_{σ(j+1)}]_{∗} is a constant, where s_{i} are the coordinate functions on T^{∗}X. Hence the terms contracting t_{σ(j)}, t_{σ(j+1)} for j ≠ 0, k vanish in the reduced complex. We are left with the terms contracting f, the first component of a. Again, the expression is anti-symmetric in f, t_{i}, and [f, t_{i}]_{de} = {f, t_{i}}_{de}. It follows that HKR(b(Φ(a))) = ı_{H_{f}} dt_{i_{1}} ∧ … ∧ dt_{i_{k}}. Here H_{f} stands for the Hamiltonian vector field associated to the function f with respect to the symplectic form ω_{de}.

Observe that the map HKR is a left inverse for Φ. Note also that both b(Φ(a)) and Φ(δ(a)) are chains of the form (13). By Proposition 4.2, the map HKR is an isomorphism from the space of chains of the form (13) to L^{•}. Hence it is enough to check that δ(a) = ı_{H_{f}} dt_{i_{1}} ∧ … ∧ dt_{i_{k}}. This follows like in [2], from the identity

∗_{ϕ}ı_{H_{f}} μ = (−1)^{k}df ∧ ∗_{ϕ}μ, (16)

valid for forms μ of degree k.

Note that the map Φ is not invariant under changes of coordinates, hence the argument does not hold for the global boundary ideal I_{∂}.

Theorem 4.4.

HH_{k}(I_{∂}(O)) ≅ \{ \begin{matrix}ℂ & if k = N, N − 1 \\ 0, & otherwise \end{matrix}

Proof: The complex L^{•} is graded as a vector space. The differential δ increases the x degree by at least 1. This follows for instance from (16), by observing that Hamiltonian vector fields gain at least one power of x (see ([6]). The map Φ becomes a map of filtered complexes. Consider the two spectral sequences, E(L) and E(I_{∂}(O)).

Lemma 4.5. E^{1}(L) = E^{0}(L).

Proof: The first differential, d_{0}, in the spectral sequence of L, is induced from the part of degree zero of the differential δ. We have seen above that δ has no component of degree 0, hence d_{0} = 0.

By Proposition 4.1, Φ is an isomorphism at E_{1}. Note that the two spectral sequences are convergent. This follows from the fact that they are constructed on underlying L-vector spaces, with respect to the L-filtration, where L is the field of Laurent series. Recall now the following fact [11]: if a map between differential filtered complexes induces an isomorphism at E_{∞}, and the spectral sequences are convergent, then the map is a quasi-isomorphism. It follows that Φ is an isomorphism on homology. Hence HH_{k}(I_{∂}(O)) ≅ H_{k}(L, δ) ≅ H^{2N−k}(L, d). The last isomorphism is given by the conjugation ∗_{ϕ}.

Clearly, H^{•}(L, d) is concentrated in L(0) (see the end of the proof of Theorem 3.1), and

H^{s}(L(0), d) ≅ \{ \begin{matrix}ℂ & if s = N, N + 1 \\ 0, & otherwise. \end{matrix}

The theorem follows.

Still in the local setting, we can now easily prove the following

Theorem 4.6.

HH_{k}(A_{∂}(O)) ≅ \{ \begin{matrix}ℂ & if k ∈ \{2N, 2N − 2, N, N − 1\} \\ ℂ ⊕ ℂ & if k = 2N − 1 \\ 0, & otherwise \end{matrix}
HH_{k}(A_{∂,σ}(O)) ≅ H^{2N−k}(S^{N−1} × S^{1} × S^{1}).

Proof: By analogy with (14), consider the spaces

(17)  L′^{k}(p) = x^{p}Ω_{S^{ℤ}}^{k}(ℝ_{ξ} × ℝ_{z,ζ}^{2m} × ℝ_{y,η}^{2n}) ⊕ x^{p−1}dx ∧ Ω_{S^{ℤ}}^{k−1}(ℝ_{ξ} × ℝ_{z,ζ}^{2m} × ℝ_{y,η}^{2n})
(18)  L′′^{k}(p) = x^{p}Ω_{S}^{k}(ℝ_{z,y}^{m+n} × (ℝ^{m+n+1} \ {0})_{ζ,η,ξ})
⊕x^{p−1}dx ∧ Ω_{S}^{k−1}(ℝ_{z,y}^{m+n} × (ℝ^{m+n+1} \ {0})_{ζ,η,ξ})

where the symbolic behavior is in the variables ζ, η, ξ. As before, endow L′^{•} and L′′^{•} with the differential δ = (−1)^{•} ∗_{ϕ} d∗_{ϕ}. The map Φ from (15) is a quasi-isomorphism between L′, L′′ and the Hochschild complexes of A_{∂}(O), A_{∂,σ}(O).

It is easy to see that the homology of (L′′, d) is concentrated in homogeneity 0 both in the variable x and in the radial directions of (ℝ^{m+n+1} \ 0)_{ζ,η,ξ}. The second part of the theorem follows immediately. See [3] for a similar computation. For the first part, look at the short exact sequence of complexes

0 → (L, d) → (L′, d) → (L′′, d) → 0.

This induces a long exact sequence in cohomology. From the construction of the long exact sequence, it is straightforward that the boundary map ∂ maps H^{N−1}(S^{N−1}) ⊕ \frac{dx}{x} ∧ H^{N−1}(S^{N−1}) surjectively onto H^{•}(L, d), and the rest of H^{•}(L′′, d) to 0. By conjugation with ∗_{ϕ}, this proves the first part of the theorem as well.

4.3. The conjugation by log x. .

Lemma 4.7. The commutator

[·, log x]_{de} : Ψ_{de}^{m,k}(X) ⟶ Ψ_{de}^{m−1,k−1}(X)

is well-defined.

Proof: Let x be a boundary defining function and a ∈ Ψ_{de}^{m,k}(X) be arbitrary. Since a as well as multiplication with log x act as bounded operators on Ċ^{∞}(X, Ω^{1/2}(^{de}T X)), composition, hence the commutator [a, log x] are well-defined. It remains to check b := [a, log x] ∈ Ψ_{de}^{m−1,k−1}(X). Since multiplication with log x is a local operator, we can restrict ourselves to operators a with Schwartz kernel κ_{a} supported in a coordinate patch of X_{de}^{2}. Let us first assume that this patch is close to the intersection of the lifted diagonal Δ_{de} with the double-edge front face ff^{de}. Local coordinates are for instance [6, Equation (7)] given by (S, U, Z, x′, y′, z′) with

S = \frac{x − x′}{x′^{2}} , U = \frac{y − y′}{x′^{2}} , and Z = \frac{z − z′}{x′} .

A straightforward computation shows that the kernel κ_{b} of b is given by κ_{b} = − log(1 + x′S)κ_{a}. Since log(1 + x′S) vanishes to first order at Δ_{de} = {S = 0, U = 0, Z = 0} and ff^{de} = {x′ = 0}, we obtain b ∈ Ψ_{de}^{m−1,k−1}(X). The remaining cases are very similar, only simpler.

The effect of the commutation on the coordinate functions t_{i} is, by [6, (3)], [x^{−2}ξ, log x]_{de} = x^{−1}, and all the other commutators are 0. This map is a derivation, hence it induces a map of complexes

(19) e_{log x} : (C_{k}(Ψ_{de}^{ℤ,ℤ}(X)), b) → (C_{k−1}(Ψ_{de}^{ℤ,ℤ}(X)), b) : a_{0} ⊗ … ⊗ a_{k} ↦ [a_{k}, log x]a_{0} ⊗ … ⊗ a_{k−1}.

Proposition 4.8. The map e_{log x} : HH_{N}(I_{∂}(O)) ⟶ HH_{N−1}(I_{∂}(O)) is an isomorphism.

Proof: Consider the following map on the complex L:

fdt_{i_{1}} ∧ … ∧ dt_{i_{k}} ⟼^{e_{log x}} x^{−1}ı_{x^{2}∂_{ξ}}fdt_{i_{1}} ∧ … ∧ dt_{i_{k}}.

It is easy to see that Φ commutes with e_{log x}. Since Φ is a quasi-isomorphism, it is enough to study the effect of e_{log x} on H_{N}(L, δ). We notice that x^{2}∂_{ξ} is the Hamiltonian vector field of x with respect to { , }_{de}. This implies that ∗_{ϕ}x^{−1}ı_{x^{2}∂_{ξ}}(a) = (−1)^{k}x^{−1}dx ∧ ∗_{ϕ}(a). Let τ be a Thom form on the bundle ^{de}T^{∗}X_{|∂X}, i.e. such that the wedge product with τ induces an isomorphism H^{•}(X) → H_{𝒮}^{•+N}(^{de}T^{∗}X_{|∂X}). The previous identity with a replaced by ∗_{ϕ}τ gives e_{log x} ∗_{ϕ} τ = (−1)^{k} ∗_{ϕ} (x^{−1}dx ∧ τ). But ∗_{ϕ}τ and ∗_{ϕ}(x^{−1}dx ∧ τ) generate H_{•}(L(0), δ).

4.4. A Čech complex adapted to Hochschild homology. This subsection is reproduced from [20]. Recall that L denotes the field of Laurent series. Let M be a C^{∞} manifold and A a sheaf of Fréchet algebras over M. Denote by A ˆ⊗L the completion of A ⊗ L in the L-adic topology. Let A be a L-filtered algebra such that F_{0}A/F_{1}A ≅ A as algebras, e.g. a star algebra on a Poisson manifold. Let U be a good cover of M, i.e. such that any intersection of sets in U is contractible. The Hochschild-Čech complex CȞ(A, U) is the following second quadrant homology double complex:

Definition 4.9. CȞ ; i,k(A, U) = Č^{−i}(C_{k}(A), U), with the differentials b and ď.

The augmented complex C̄Ȟ(A, U) is obtained from CȞ(A, U) by adding the bar resolution of A as the 1-column. We will denote by T oT (C̄Ȟ(A, U)) the total complex of this double complex, with the diferential b + (−1)^{k}ď on C̄Ȟ ; •,k. The underlying space of the total complex is defined as follows:

T oT_{i} = ⋃_{k∈ℤ} ∏_{j≥−1} Č^{−j}((F_{k}A)^{ˆ⊗i+1−j}, U).

Note that this definition agrees with the definition (3) of the bar complex of A.

Theorem 4.10. Assume that the following conditions hold:

1. There exists an acyclic sheaf F over M and a sheaf isomorphism HH(A) ≅ F;

2. A and A ˆ⊗L are H-unital.

3. ⋂ F_{i}A = {0}, ⋃ F_{i}A = A and

lim_{i→−∞^{⟶}} lim_{j→∞^{⟵}} F_{i}A/F_{j}A ≅ A;

Then the total homology H_{•}(T oT (C̄Ȟ(A, U))) vanishes.

Proof: Filter the complex C̄Ȟ(A, U) by the total L-degree. This filtration is compatible with the Čech and Hochschild boundary maps, so we can form a spectral sequence, that we call ^{T}E(A).

Proposition 4.11. ^{T}E^{1}(A) = 0.

Proof: By the assumption on the associated graded algebra of A, we have an isomorphism of complexes

(^{T}E_{k,•}^{0}(A), d_{0}) ≅ (T oT_{•−k}(C̄Ȟ(A ˆ⊗L, U)), b + (−1)^{i}ď)_{k}.

We will show that the complex T oT (C̄Ȟ(A ˆ⊗L, , U)_{k} is acyclic for each fixed total degree k. Since the boundary map preserves the total degree, and since the degree in each component is bounded from the definition of T oT, we can replace ˆ⊗ with the usual tensor product

Consider the filtration by columns of the corresponding Čech-Hochschild double complex. This induces a spectral sequence abutting at the homology of T oT (C̄Ȟ(A ⊗ L, U)).

Lemma 4.12. The E_{2} term of this new spectral sequence vanishes.

Proof: Use the Künneth formula for Hochschild homology ([8]). Here we use hypothesis 2 of the Theorem. By assumption,

E_{1}((CȞ(A ⊗ L, U), b + (−1)^{i}ď) ≅ Č^{aug}(HH(A ⊗ L))
≅ Č^{aug}(F) ⊗ HH(L).

We see that d_{1} is just the Čech differential, because all the above isomorphisms are compatible with restrictions. By assumption, F is acyclic, hence E_{2} vanishes.

Lemma 4.13. The spectral sequence from Lemma 4.12 converges.

Proof: Let a be an element in ^{T}E_{1}(A). There is a standard way of constructing a formal sum c such that a = d_{0}(c). We want this c to define an element in T oT, hence to be of degree uniformly bounded below. This is indeed possible: if γ is 0 at E_{2}, then there exists μ such that d_{1}[μ] = [γ] and deg(μ) ≥ min(deg(γ), 0). If γ is 0 at E_{1}, then there exists μ such that d_{0}[μ] = [γ] and deg(μ) ≥ deg(γ). Now the same diagram chasing will produce, for every a ∈ ^{T}E_{1}(A), some formal sum c such that deg(c) ≥ min(deg(a), 0) and (b + (−1)^{i}ď)(c) = a. Using the fact that T oT is defined as a direct product and condition (c) of the theorem, we see that c is summable, hence [a] = 0.

Exactly in the same way we prove that ^{T}E(A) converges. By Proposition 4.11, the theorem follows.

Corollary 4.14. The boundary map in the augmentation complex,

C_{•}(A) ⟶ T oT_{•−1}(CȞ(A, U)) : a ↦ b(a) + (−1)^{•}ď(a)

is a quasi-isomorphism.

Proof: Consider the short exact sequence of complexes coming from the augmentation:

0 → CȞ(A, U) → C̄Ȟ(A, U) → C_{•}(A) → 0.

The homology of the middle term is zero by Theorem 4.10. It follows that the boundary map is an isomorphism.

4.5. The global homology of the boundary algebras. Let OX be the orientation bundle of X. Since the normal bundle to ∂X is trivial, OX is isomorphic to O∂X, the orientation bundle of ∂X.

Theorem 4.15.

HH_{k}(I_{∂}(X)) ≅ H^{N−1−k}(∂X, OX) ⊕ H^{N−k}(∂X, OX);
HH_{k}(A_{∂}(X)) ≅ ⨁_{s=0,1,1,2,N,N+1} H^{2N−k−s}(∂X, OX)
HH_{k}(A_{∂,σ}(X)) ≅ H^{2N−k}(∂X × S^{N−1} × S^{1} × S^{1}, OX)

Proof: Consider a good cover U of ∂X by coordinate patches. Form the Hochschild-Čech double complex of I_{∂} with respect to this cover. The E_{1} term consists of two rows, at height N and N − 1. Each of them is a copy of Č(∂X, U). The differential d_{1} is induced from the restriction maps, hence it is exactly the Čech differential. It follows that the E_{2} term consists of two rows, each of them isomorphic to H(∂X, OX).

This complex has an operation which commutes with the boundaries, namely e_{log x}. This operation is an isomorphism between the two rows. This forces the spectral sequence to degenerate at E_{2}. The first part of the theorem follows.

Remark 4.16. The above decomposition is actually natural: The two components are the image of e_{log x}, respectively the image of the edge homomorphism.

The third part of the theorem was proved in Theorem 3.1. Indeed, even though the fibration deS^{∗}X_{|∂X} → ∂X is not trivial, the cohomology of its total space is isomorphic to the cohomology of ∂X × S^{N−1}. One can see this from the spectral sequence of the fibration, since the dimension of the (sphere) fibers is equal to the dimension of the base.

The right hand side of the second isomorphism in the theorem is just the E^{2} term in the Hochschild-Čech spectral sequence of A_{∂}. The degeneracy of the spectral sequence is a consequence of the following facts:

1. dim_{ℂ} HH(A_{∂}) ≤ dim_{ℂ} E^{2}(A_{∂}) = 6 dim_{ℂ} H(∂X, OX);

2. dim_{ℂ} HH(A_{∂,σ}) = 8 dim_{ℂ} H(∂X, OX);

3. dim_{ℂ} HH(I_{∂}) ≤ dim_{ℂ} E^{2}(I_{∂}) = 2 dim_{ℂ} H(∂X, OX) (in fact, we have proved above that this is an equality); 4. By H-unitality of I_{∂}, there is a long exact sequence in Hochschild homology coming from (10);

5. 2+6=8. Hence all the above inequalities must be equalities. This finishes the proof of the theorem.

An easy consequence of this proof is that the long exact homology sequence coming from (10) splits in short exact sequences

0 ⟶ HH_{•}(A_{∂}) ⟶ HH_{•}(A_{∂,σ}) ⟶^{δ} HH_{•−1}(I_{∂}) ⟶ 0. (20)

Since the map A_{∂} → A_{∂,σ} is surjective, the 6 copies of H(∂X, OX) that make up HH_{•}(A_{∂}) map onto the corresponding copies of H(∂X, OX) inside HH_{•}(A_{∂,σ}). This proves the following proposition:

Proposition 4.17. The sequence (20) is the same as the sequence induced in homology from the short exact sequence of complexes of forms

0 ⟶ (Ω_{𝒮}^{2N−•}(∂X)[x]], d) ⟶ (Ω_{S^{ℤ}}^{2N−•}(∂X)[x]], d) ⟶ (Ω_{S}^{2N−•}(∂X)[x]], d) ⟶ 0.

5. Residue traces for the double-edge calculus

The computation of the previous sections show dim_{ℂ} HH_{0}(A) = 1 for A = I_{σ}, I_{∂}, A_{σ}, A_{∂}, A_{∂,σ}; thus, there exists up to a multiplicative constant a unique trace on these algebras. We are going to identify these traces as residues of the analytic continuation of a “double-zeta” functions as well as to give, additionally, explicit formulas. The corresponding results about traces for the cusp calculus were obtained in [18, Section 5]. Since the proofs mainly depend on formal manipulations with zeta-functions, we will only state the results and leave the necessary changes of the proofs to the reader.

Fix a boundary defining function x : X ⟶ \overline{ℝ}_{+}, and let Q ∈ Ψ_{de}^{1,0}(X) be a positive operator. Then the holomorphic family Q^{z} ∈ Ψ_{de}^{z,0}(X), z ∈ ℂ, of complex powers is constructed as in [5].

Proposition 5.1. Let Ω ⊆ ℂ^{2} be open and connected with [R, ∞) × [R, ∞) ⊆ Ω for some R ∈ ℝ, and let a : Ω ⟶ Ψ_{de}^{m_{0},k_{0}}(X) be holomorphic. Then the functions

T_{k} : Ω ∩ {Re(z) > k_{0} + 2n + m + 1} ∩ {Re(λ) > m_{0} + N} ⟶ ℂ

with T_{1} : (λ, z) ↦ Tr(a(λ, z)x^{z}Q^{−λ}) and T_{2} : (λ, z) ↦ Tr(a(λ, z)Q^{−λ}x^{z}) are holomorphic. Moreover, there are meromorphic functions T̃_{k} : Ω ⟶ ℂ with at most simple poles at λ = m_{0} + N − ℓ, ℓ ∈ ℕ_{0}, and z = k_{0} + 2n + m + 1 − j, j ∈ ℕ_{0}, that coincide with T_{k} on the component of Ω containing [R′, ∞) × [R′, ∞) for some R′ ∈ ℝ.

For a ∈ Ψ_{de}^{m_{0},k_{0}}(X), we let Z_{x,Q}(a) be the meromorphic extension of

{Re(z) > k_{0} + 2n + m + 1} ∩ {Re(λ) > m_{0} + N} ∋ (λ, z) ⟼ Tr(ax^{z}Q^{−λ})

by Proposition 5.1; then, (λ, z) ⟼ zλZ_{x,Q}(a)(λ, z) is holomorphic in a neighborhood of 0 ∈ ℂ^{2}, and we can define

zλZ_{x,Q}(a)(λ, z) =: Tr_{∂,σ}(a) + λT̂r_{∂}(a) + zT̂r_{σ}(a) + λ^{2}W (λ, z) + λzW′(λ, z) + z^{2}W′′(λ, z) ,

where W, W′, W′′ are holomorphic near 0 ∈ ℂ^{2}. Furthermore, let ω_{de} be the canonical singular symplectic form on ^{de}T^{∗}X, and let us denote by i_{R} resp. i_{x∂_{x}} the contraction with the radial vector field R in the fibers of ^{de}T^{∗}X resp. with the canonical vector field x∂_{x}, normal to the boundary. Recall that ϱ_{σ} : ^{de}\overline{T}^{∗}X ⟶ ℂ denotes a defining function for the double-edge cosphere bundle deS^{∗}X.

In the rest of this section, we summarize the properties of these functionals.

Proposition 5.2. The double-residue Tr_{∂,σ} : Ψ_{de}^{ℤ,ℤ}(X) ⟶ ℂ is a trace functional that is independent of the choice of Q and x. Moreover, we have Tr_{∂,σ}(Ψ_{de}^{−∞,ℤ}(X) + Ψ_{de}^{ℤ,−∞}(X)) = 0, thus, it defines a trace functional on A_{∂,σ}. If a ∈ A_{∂,σ} is represented by ∑_{k≤k_{0},m≤m_{0}} a_{m,k} with a_{m,k} ∈ x^{−k}ϱ_{σ}^{−m}C^{∞}(^{de}S^{∗}X_{|∂X}), then

Tr_{∂,σ}(a) = (2π)^{−N} ∫_{^{de}S^{∗}X_{|∂X}} a_{−N,−(1+2n+m)}i_{x∂_{x}}i_{R}ω_{de}^{N} .

The “unique” traces on A_{σ} resp. A_{∂} are then given by the composition of Tr_{∂,σ} with the natural projections A_{σ} ⟶ A_{∂,σ} resp. A_{∂} ⟶ A_{∂,σ}.

Since Z_{Q,x}(a) is entire for a ∈ Ψ_{de}^{−∞,−∞}(X), T̂r_{σ} and T̂r_{∂} descend to linear functionals on Ψ_{de}^{ℤ,ℤ}(X)/Ψ_{de}^{−∞,−∞}(X).

Proposition 5.3. The restriction Tr_{σ} : I_{σ} ⟶ ℂ of T̂r_{σ} to I_{σ} is a trace functional; it is independent of the choice of x and Q, and extends the Wodzicki-residue for the double 2X of X by continuity from operators whose kernels are supported in the interior. If a ∈ I_{σ} is represented by ∑_{m≤m_{0}} a_{m} with a_{m} ∈ x^{∞}ϱ_{σ}^{−m}C^{∞}(^{de}S^{∗}X), then

Tr_{σ}(a) = (2π)^{−N} ∫_{^{de}S^{∗}X} a_{−N}i_{R}ω_{de}^{N} .

Proposition 5.4. The restriction Tr_{∂} : I_{∂} ⟶ ℂ of T̂r_{∂} to I_{∂} is a trace functional; it does not depend on the choice of x and Q. If a ∈ I_{∂} is represented by ∑_{k≤k_{0}} a_{k} with a_{k} ∈ x^{−k}Ċ^{∞}(^{de}T^{∗}X_{|∂X}), then

Tr_{∂}(a) = (2π)^{−N} ∫_{^{de}\overline{T}^{∗}X|_{∂X}} a_{−(1+2n+m)}i_{x∂_{x}}ω_{de}^{N} .

6. Long exact sequences

We describe now the long exact sequences arising from the six short exact sequences from diagram (1), as well as from

0 ⟶ Ψ_{de}^{−∞,−∞}(X) ⟶ Ψ_{de}^{ℤ,ℤ}(X) ⟶ Ψ_{de}^{ℤ,ℤ}(X)/Ψ_{de}^{−∞,−∞}(X) ⟶ 0. (21)

Lemma 6.1.

HH_{k}(Ψ_{de}^{−∞,−∞}(X)) = \{ \begin{matrix}ℂ & if k = 0, \\ 0 & otherwise. \end{matrix}

The isomorphism is given by the operator trace map T r on Ψ_{de}^{−∞,−∞}(X).

Proof: Recall the homotopy s_{k} from Lemma 2.4 for the operator b′. We note that bs_{k} + s_{k−1}b = 0 if k > 0, and bs_{0} = I − T r · f ⊗ f, where f is the function used in the definition of s_{k}.

We fix the following notation: a pair (I, A), where I is an ideal in the algebra A, denotes the short exact sequence 0 → I → A → A/I → 0. We also abbreviate “long exact sequence” to “l.e.s.”.

Recall that on I_{∂}, Tr_{∂} is defined by analogy with the residue trace of Wodzicki and Guillemin:

Tr_{∂} : I_{∂} → ℂ : Tr_{∂}(A) = Res_{τ=0}(T r(x^{τ} A)).

Proposition 6.2. The boundary map in the l.e.s. arising from (Ψ_{de}^{−∞,−∞}(X), Ψ_{de}^{−∞,ℤ}(X)) is surjective. Moreover, δ = δ_{1} : HH_{1}(I_{∂}) → HH_{0}(Ψ_{de}^{−∞,−∞}(X)) ≅ ℂ is given by

T r ∘ δ = Tr_{∂} ∘ e_{log x}.

Proof: Let a be a cycle in C_{1}(I_{∂}) and A a lift of a to C_{1}(Ψ_{de}^{−∞,ℤ}(X)). By definition,

δ(a) = [b(A)]_{HH_{0}(Ψ_{de}^{−∞,−∞}(X))}

and hence T r(δ(a)) = T r(b(A)). For each τ ∈ ℂ, define an operation D(τ) on Ψ_{de}^{ℤ,ℤ}(X) by

P ⟼^{D(τ)} \frac{x^{−τ} P x^{τ} − P}{τ} .

Define the action e_{D(τ)} of D(τ) on the Hochschild chain spaces by the formula

a_{0} ⊗ … ⊗ a_{k} ↦ D(τ)(a_{k})a_{0} ⊗ … ⊗ a_{k−1}.

By inserting x^{τ} and evaluating at τ = 0, we get

T r(δ(a)) = T r(x^{τ} b(A))_{τ=0} = T r(b(x^{τ} A) + τx^{τ} e_{D(τ)}A)_{τ=0} = Res_{τ=0}T r(x^{τ} e_{log x}A) = Tr_{∂}(e_{log x}a).

A simple computation shows that when we identify HH_{k}(I_{∂}) with H^{N−1−k}(∂X, OX) ⊕

H^{N−k}(∂X, OX), the operation e_{log x} takes the form ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ). Also, under the identification

HH_{0}(I_{∂}) → H^{N−1}(∂X, OX), Tr_{∂} becomes the integral over ∂X, i.e. the canonical isomorphism to ℂ. It follows that δ_{1} is just the integral over ∂X.

From Proposition 5.3, the functional T̂r_{σ} is a trace functional when restricted to I_{σ}:

Tr_{σ} : I_{σ} → ℂ : Tr_{σ}(a) = Res_{z=0}T r(Q^{−z}A),

where A is a double-edge operator in Ψ_{de}^{ℤ,−∞}(X) with total symbol a. Define the derivation [·, log Q] by [A, log Q] = \frac{d}{dz} (Q^{−z}AQ^{z})_{z=0}. As in (19), let e_{log Q} denote the corresponding action on Hochschild homology.

Proposition 6.3. The boundary map δ_{1} in the l.e.s. induced from (Ψ_{de}^{−∞,−∞}(X), Ψ_{de}^{ℤ,−∞}(X)) is surjective. Moreover,

T r ∘ δ_{1} = −Tr_{σ} ∘ e_{log Q}.

Proof: Identical to the proof of Proposition 6.2, after replacing Tr_{∂} with Tr_{σ}, and e_{log x} with e_{log Q}.

We have identified above HH_{k}(I_{σ}) with H_{rel}^{2N−k}(^{de}S^{∗}X × S^{1}). Under this identification, e_{log Q} becomes the exterior product with the generator of S^{1}, and T r_{R} becomes the integral over^{de}S^{∗}X×S^{1}. It follows that δ becomes the integral over ^{de}S^{∗}X.

6.1. The middle sequences. So far we have described in Propositions 3.3, 4.17, 6.2 and 6.3, the long exact sequences arising from the four exterior sequences from diagram (1). The other two are easily deduced from here.

Let us first examine the sequence (Ψ_{de}^{ℤ,−∞}(X), Ψ_{de}^{ℤ,ℤ}(X)). We denote by p_{∗} the maps induced by the horizontal surjections in diagram (1). Since the map p_{∗} : HH_{k}(Ψ_{de}^{ℤ,−∞}(X)) → HH_{k}(I_{σ}) is injective (Proposition 6.3), it is enough to compute p_{∗} ∘ δ. This is the same as δ ∘ p_{∗}, where the last δ map was computed in Proposition 4.17, and the last p_{∗} is the injection from (20).

Proposition 6.4. The map p_{∗} ∘ δ = δ ∘ p_{∗} maps the components with s = 1, 2, N + 1 of HH_{k}(A_{∂}) to H_{rel}^{2N−k}(^{de}S^{∗}X), H_{rel}^{2N−k−1}(^{de}S^{∗}X)\frac{dr}{r} and H_{rel}^{N−k}(^{de}S^{∗}X) via the boundary map in the cohomology l.e.s. of the pair of spaces (X, ∂X) with coefficients in the sheaf OX, tensored with the identity in H^{∗}(S^{N−1} × S^{1}).

Having computed the map δ and the homologies of A_{∂} and Ψ_{de}^{ℤ,−∞}(x), the homology of Ψ_{de}^{ℤ,ℤ}(X) is completely determined. Finally, the sequence (Ψ_{de}^{−∞,ℤ}(X), Ψ_{de}^{ℤ,ℤ}(X)) is determined by the boundary map δ : HH_{k}(A_{σ}) → HH_{k−1}(Ψ_{de}^{−∞,ℤ}(X)). Let p′ be the vertical surjections in diagram 1. From Proposition 6.3, the map p′_{∗} : HH_{k}(Ψ_{de}^{−∞,ℤ}(X)) → HH_{k}(I_{∂}) is injective. Hence it is enough to compute p′_{∗} ∘ δ, which equals δ ∘ p′_{∗}. These last two maps were computed in Propositions 4.17 and 3.3.

Theorem 6.5. The operator trace, resp. the traces defined via residues in Section 5 on the nine algebras from Diagram (1) are unique up to a constant.

Proof: Follows from our computation of Hochschild homologies and boundary maps. Indeed, together with the description of the long exact sequences, our results show that the first Hochschild homology group is 1-dimensional for all the algebras from (1).

It is worth pointing out that the knowledge of these traces gives some basic information about the fibration π : ∂X ⟶ Y of the boundary. Indeed, the dimensions m resp. n of the fiber resp. the base of the fibration can be recovered using for instance the double-residue trace Tr_{∂,σ} as given in Proposition 5.2.

7. The index of fully elliptic double-edge operators

A double-edge operator is Fredholm if and only if its joint symbol is invertible ([6, Theorem 5.15]). By the joint symbol we mean both the leading homogeneous symbol in the usual sense, and the Fourier transform of the leading term in the Laurent expansion at the boundary. Let A be such an operator. Then A is invertible modulo Ψ_{de}^{−∞,−∞}(X), and by [6, Corollary 6.5] we can choose a pseudo-inverse B such that

(22) BA = I − P_{ker A}, AB = I − P_{cokerA}.

Note that A ⊗ B defines a Hochschild cycle in HH_{1}(Ψ_{de}^{ℤ,ℤ}(X)/Ψ_{de}^{−∞,−∞}(X)). This simple but crucial observation was made by Melrose and Nistor [18], in the context of usual or cusp scalar pseudodifferential operators. An analogous statement holds for any other algebra where elliptic operators admit pseudo-inverses. Melrose and Nistor extend the definition of this 1-cycle to operators acting on sections of a bundle E. By Morita invariance, the homology of such an algebra is isomorphic to the homology of the scalar algebra. Then the boundary map of the l.e.s. of the pair (Ψ_{de}^{−∞,−∞}(X), Ψ_{de}^{ℤ,ℤ}(X)) applied to this cycle is a class of trace equal to the index of A.

A slight extension of this construction is carried out in [22] for a totally elliptic operator between sections of arbitrary vector bundles E, F. Replace the cycle A ⊗ B with tr(i_{1}AP_{2} ⊗ i_{2}BP_{1}), where P_{1}, P_{2} are projections from a trivial bundle ℂ^{q} to E, F, i_{1}, i_{2} are embeddings in ℂ^{q}, and tr is the generalized trace functional which induces Morita invariance, tr((a_{ij}) ⊗ (b_{ij})) = ∑ a_{ij} ⊗ b_{ji}. Of course, Morita invariance does not apply in this general case since operators from E to F do not form an algebra.

We follow now formally the computation from [18] modified in [7], and get the following result – the proof is included for the benefit of the reader.

Proposition 7.1. The boundary map δ in the l.e.s. (Ψ_{de}^{−∞,−∞}(X), Ψ_{de}^{ℤ,ℤ}(X)) applied to a cycle c ∈ C_{1}(Ψ_{de}^{ℤ,ℤ}(X)/Ψ_{de}^{−∞,−∞}(X)) is given by

ℂ ∋ T r(δ(c)) = T̂r_{∂}(f_{log x}c) − T̂r_{σ}(e_{log Q}c).

Here f_{log x} is another version of the action of the derivation [·, log x] on 1-chains, namely f_{log x}(a_{0}⊗ a_{1}) = a_{0}[a_{1}, log x]. Proof: Let C be a lift of c to C_{1}(Ψ_{de}^{ℤ,ℤ}(X)). Then, from the definitions, we have δ(c) = [b(C)]_{HH_{0}(Ψ_{de}^{−∞,−∞})}, hence

T r(b(C)) = T r(b(C)x^{z}Q^{−λ})_{z=0,λ=0}.

To avoid introducing heavy algebraic definitions and symbols, write C = U ⊗ V. Of course this is just notation! Then

δ(c) = T r((UV − V U)x^{z}Q^{−λ})_{z=0,λ=0}
= T r [(U(V − x^{z}V x^{−z})x^{z}Q^{−λ}) + (Q^{λ}V Q^{−λ} − V)Ux^{z}Q^{−λ})]_{z=0,λ=0}
= T r [zU([V, log x] + zF_{1}(z))x^{z}Q^{−λ} + λ([log Q, V] + λF_{2}(λ))Ux^{z}Q^{−λ}]_{z=0,λ=0} ,

where F_{1}(z), F_{2}(λ) are entire, of fixed order. By Proposition 5.1, the result follows.

Let us make the (historical) notation η(A) = −2T̂r_{∂}([B, log x]A) and \overline{AS}(A) = −T̂r_{σ}(A[B, log Q]). Note that η(A) and \overline{AS}(A) can be defined using any B which inverts A up to trace class remainders.

Unlike the case of the b- or cusp algebras, our “eta” is local in the boundary ∂X. Proposition 7.1 implies the following formula for the index of A:

Theorem 7.2.

Ind(A) = \overline{AS}(A) − \frac{1}{2}η(A).

None of these two terms is invariant under homotopy. In [6] we showed that A can be deformed inside fully elliptic operators into an operator which is a bundle isomorphism near ∂X. For such an operator, we can choose the pseudo-inverse to be the inverse bundle isomorphism near the boundary. This commutes with log x, hence the η part vanishes and we recover a form of the Atiyah-Singer index formula as in [6] – see also [18].

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