CHAPTER 8 Hochschild homology

8.1. Formal Hochschild homology

The Hochschild homology is defined, formally, for any associative algebra. Thus if A is the algebra then the space of formal k-chains, for k ∈ ℕ_{0} is the (k + 1)-fold tensor product

A^{⊗(k+1)} = A ⊗ A ⊗ ⋯ ⊗ A.

The ‘formal’ here refers to the fact that for the ‘large’ topological algebras we shall consider it is wise to replace this tensor product by an appropriate completion, usually the ‘projective’ tensor product. At the formal level the differential defining the cohomolgy is given in terms of the product, ⋆, by

\begin{matrix}b(a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{k}) = b′(a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{k}) + (−1)^{k}(a_{0} ⋆ a_{k}) ⊗ a_{1} ⊗ ⋯ ⊗ a_{k−1}, \\ b′(a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{k}) = ∑_{j=0}^{k−1}(−1)^{j}a_{0} ⊗ ⋯ ⊗ a_{j−1} ⊗ a_{j+1} ⋆ a_{j} ⊗ a_{j+2} ⊗ ⋯ ⊗ a_{k}. \end{matrix} (8.1)

Lemma 8.1. Both the partial map, b′, and the full map, b, are differentials, that is

(b′)^{2} = 0 and b^{2} = 0. (8.2)

Proof. This is just a direct computation. From (8.1) it follows that

(8.3) (b′)^{2}(a_{0} ⊗ a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{m})

= ∑_{j=2}^{m−1} ∑_{p=0}^{j−2}(−1)^{j}(−1)^{p}(⋯ ⊗ a_{p+1} ⋆ a_{p} ⊗ ⋯ ⊗ a_{j−1} ⊗ a_{j+1} ⋆ a_{j} ⊗ a_{j+2} ⊗ ⋯ ⊗ a_{m})

− ∑_{j=1}^{m−1}(⋯ ⊗ a_{j+1} ⋆ a_{j} ⋆ a_{j−1} ⊗ ⋯) − ∑_{j=0}^{m−2}(⋯ ⊗ a_{j+21} ⋆ a_{j+1} ⋆ a_{j} ⋆ ⊗ ⋯)

+ ∑_{j=0}^{m−3} ∑_{p=j+2}^{m−1}(−1)^{j}(−1)^{p−1}(a_{0}⊗⋯⊗a_{j−1}⊗a_{j+1}⋆a_{j}⊗a_{j+2}⊗⋯⊗a_{p+1}⋆a_{p}⊗⋯) = 0.

Similarly, direct computation shows that

\begin{matrix}(b − b′) & b′(a_{0} ⊗ ⋯ ⊗ a_{m}) = (−1)^{m−1}(a_{1} ⋆ a_{0} ⋆ a_{m} ⊗ ⋯ a_{m−1}) \\ + & ∑_{i=1}^{m−2}(−1)^{i+m−1}(a_{0} ⋆ a_{m} ⊗ ⋯ ⊗ a_{i+1} ⋆ a_{i} ⊗ ⋯) + (a_{0} ⋆ a_{m} ⋆ a_{m−1} ⊗ ⋯), \end{matrix}

\begin{matrix}b′(b − b′) & (a_{0} ⊗ ⋯ ⊗ a_{m}) = (−1)^{m}(a_{1} ⋆ a_{0} ⋆ a_{m} ⊗ ⋯ a_{m−1}) \\ + ∑_{i=1}^{m−2}(−1)^{i+m}(a_{0} ⋆ a_{m} ⊗ ⋯ ⊗ a_{i+1} ⋆ a_{i} ⊗ ⋯) and \end{matrix}

(b − b′)^{2}(a_{0} ⊗ ⋯ ⊗ a_{m}) = −(a_{0} ⋆ a_{m} ⋆ a_{m−1} ⊗ ⋯)

so

(b − b′)b′ + b′(b − b′) = −(b − b′)^{2}. (8.4)

The difference between these two differentials is fundamental, roughly speaking b′ is ‘trivial’.

Lemma 8.2. For any algebra with identity the differential b′ is acyclic, since it satifies

b′s + sb′ = Id where (8.5)

s(a_{0} ⊗ ⋯ ⊗ a_{m}) = Id ⊗a_{0} ⊗ ⋯ ⊗ a_{m}. (8.6)

Proof. This follows from the observation that

b′(Id ⊗a_{0} ⊗ ⋯ ⊗ a_{m}) = a_{0} ⊗ ⋯ ⊗ a_{m} + ∑_{i=1}^{m}(−1)^{i}(Id ⊗ ⋯ a_{i} ⋆ a_{i−2} ⊗ ⋯). (8.7)

Definition 8.1. An associative algebra is said to be H-unital if its b′ complex is acyclic.

Thus the preceeding lemma just says that every unital algebra is H-unital.

8.2. Hochschild homology of polynomial algebras

Consider the algebra ℂ[x] of polynomials in n variables^{1}, x ∈ ℝ^{n} (or x ∈ ℂ^{n} it makes little difference). This is not a finite dimensional algebra but it is filtered by the finite dimensional subspaces, P_{m}[x], of polynomials of degree at most m;

ℂ[x] = ⋃_{m=0}^{∞} P_{m}[x], P_{m}[x] ⊂ P_{m+1}[x].

Furthermore, the Hochschild differential does not increase the total degree so it is enough to consider the formal Hochschild homology.

The chain spaces, given by the tensor product, just consist of polynomials in n(k + 1) variables

(ℂ[x])^{ˆ⊗(k+1)} = ℂ[x_{0}, x_{1}, …, x_{k}], x_{j} ∈ ℝ^{n}.

1The method used here to compute the homology of a polynomial algebra is due to Sergiu Moroianu; thanks Sergiu.

Furthermore composition acts on the tensor product by

p(x_{0})q(x_{1}) = p ⊗ q ⟼ p(x_{0})q(x_{0})

which is just restriction to x_{0} = x_{1}. Thus the Hochschild differential can be written

b : ℂ[x_{0}, …, x_{k}] ⟶ ℂ[x_{0}, …, x_{k−1}],

\begin{aligned}(bq)(x_{0}, x_{1}, …, x_{k−1}) & = ∑_{j=0}^{k−1}(−1)^{j}p(x_{0}, …, x_{j−1}, x_{j}, x_{j}, x_{j+1}, …, x_{k−1}) \\ & + (−1)^{k}q(x_{0}, x_{1}, …, x_{k−1}, x_{0}). \end{aligned}

One of the fundamental results on Hochschild homology is

Theorem 8.1. The Hochschild homology of the polynomail algebra in n variables is

HH_{k}(ℂ[x]) = ℂ[x] ⊗ Λ^{k}(ℂ^{n}), (8.8)

with the identification given by the map from the chain spaces

ℂ[x_{0}, …, x_{k}] ∋ q ⟶ ∑_{1≤j_{i}≤n} \frac{∂}{∂x_{1}^{j_{1}}} … \frac{∂}{∂x_{k}^{j_{k}}} p|_{x=x_{0}=x_{1}=···=x_{k}}dx_{1}^{j_{1}} ∧ ⋯ ∧ dx_{k}^{j_{k}}.

Note that the appearance of the original algebra ℂ[x] on the left in (8.8) is not surprising, since the differential commutes with multilplication by polynomails in the first variable, x_{0}

b(r(x_{0})q(x_{0}, …, x_{k})) = r(x_{0})(bq(x_{0}, …, x_{k})).

Thus the Hochschild homology is certainly a module over ℂ[x].

Proof. Consider first the cases of small k. If k = 0 then b is identically 0. If k = 1 then again

(bq)(x_{0}) = q(x_{0}, x_{0}) − q(x_{0}, x_{0}) = 0

vanishes identically. Thus the homology in dimension 0 is indeed ℂ[x].

Suppose that k > 1 and consider the subspace of ℂ[x_{0}, x_{1}, …, x_{k}] consisting of the elements which are independent of x_{1}. Then the first two terms in the definition of b cancel and

(bq)(x_{0}, x_{1}, …, x_{k−1}) = ∑_{j=2}^{k−1}(−1)^{j}p(x_{0}, …, x_{j−1}, x_{j}, x_{j}, x_{j+1}, …, x_{k−1})

+ (−1)^{k}q(x_{0}, x_{1}, …, x_{k−1}, x_{0}), ∂_{x_{1}}q ≡ 0.

It follows that bq is also independent of x_{1}. Thus there is a well-defined subcomplex on polynomails independend of x_{1} given by

ℂ[x_{0}, x_{2}, …, x_{k}] ∋ q ⟼ (b̃q)(x_{0}, x_{2}, …, x_{k−1})

= ∑_{j=2}^{k−1}(−1)^{j}p(x_{0}, x_{2}, x_{2}, x_{3} …, x_{k−1}) + ∑_{j=3}^{k−1}(−1)^{j}

p(x_{0}, …, x_{j−1}, x_{j}, x_{j}, x_{j+1}, …, x_{k−1}) + (−1)^{k}q(x_{0}, x_{2}, …, x_{k−1}, x_{0})

The reordering of variables (x_{0}, x_{2}, x_{3}, …, x_{k}) ⟶ (x_{2}, x_{3}, …, x_{k}, x_{0}) for each k, transforms b̃ to the reduced Hochschild differential b′ acting in k variables. Thus b̃ is acyclic.

Similarly consider the subspace of ℂ[x_{0}, x_{1}, …, x_{k}] consisting of the polynomials which vanish at x_{1} = x_{0}. Then the first term in the definition of b vanishes and the action of the differential becomes

(8.9) (bq)(x_{0}, x_{1}, …, x_{k−1}) = p(x_{0}, x_{1}, x_{1}, x_{2}, …, x_{k−1}) +

∑_{j=2}^{k−1}(−1)^{j}p(x_{0}, …, x_{j−1}, x_{j}, x_{j}, x_{j+1}, …, x_{k−1})

+ (−1)^{k}q(x_{0}, x_{1}, …, x_{k−1}, x_{0}), if b(x_{0}, x_{0}, x_{2}, …) ≡ 0.

It follows that bq also vanishes at x_{1} = x_{0}.

By Taylor’s theorem any polynomial can be written uniquely as a sum

q(x_{0}, x_{1}, x_{2}, …, x_{k}) = q′_{1}(x_{0}, x_{1}, x_{2}, …, x_{k}) + q′′(x_{0}, x_{2}, …, x_{k})

of a polynomial which vanishes at x_{1} = x_{0} and a polynomial which is independent of x_{1}. From the discussion above, this splits the complex into a sum of two subcomplexes, the second one of which is acyclic. Thus the Hochschild homology is the same as the homology of b, which is then given by (8.9), acting on the spaces

{q ∈ ℂ[x_{0}, x_{1}, …, x_{k}]; q(x_{0}, x_{1}, …) = 0} . (8.10)

This argument can be extended iteratively. Thus, if k > 2 then b maps the subspace of (8.10) of functions independent of x_{2} to functions independent of x_{2} and on these subspaces acts as b′ in k −2 variables; it is therefore acyclic. Similar it acts on the complementary spaces given by the functions which vanish on x_{2} = x_{1}. Repeating this argument shows that the Hochschild homology is the same as the homology of b acting on the smaller subspaces

\begin{matrix}\{q ∈ ℂ[x_{0}, x_{1}, …, x_{k}]; q(…, x_{j−1}, x_{j}, …) = 0, j = 1, …, k\} , \\ (bq)(x_{0}, x_{1}, …, x_{k−1}) = (−1)^{k}q(x_{0}, x_{1}, …, x_{k−1}, x_{0}). \end{matrix} (8.11)

Note that one cannot proceed further directly, in the sense that one cannot reduce to the subspace of functions vanishing on x_{k} = x_{0} as well, since this subspace is not linearly independent of the previous ones^{2}

x_{k} − x_{0} = ∑_{j=0}^{k−1}(x_{j_{1}} − x_{j}).

It is precisely this ‘non-transversality’ of the remaining restriction map in (8.11) which remains to be analysed.

Now, let us we make the following change of variable in each of these reduced chain spaces setting

y_{0} = x_{0}, y_{1} = x_{j} − x_{j−1}, for j = 1, …, k.

2Hence Taylor’s theorem cannot be applied.

Then the differential can be written in terms of the pull-back operation

E_{P} : ℝ^{nk} ↪ ℝ^{n(k+1)}, E_{P} (y_{0}, y_{1}, …, y_{k−1}) = (y_{0}, y_{1}, …, y_{k−1}, − ∑_{j=1}^{k−1} y_{j}),

bq = (−1)^{k}E_{P}^{∗} q,

The variable x_{0} = y_{0} is a pure parameter, so can be dropped from the notation (and restored at the end as the factor ℂ[x] in (8.8)). Also, as already noted, the degree of a polynomial (in all variables) does not increase under any of these pull-back operations, in fact they all preserve the total degree of homogeneity so it suffices to consider the differential b acting on the spaces of homogeneous polynomials which vanish at the origin in each factor

Q_{k}^{m} = {q ∈ ℂ^{m}[y_{1}, …, y_{k}]; q(sy) = s^{m}q(y), q(y_{1}, …, y_{j−1}, 0, y_{j+1}, …, y_{k}) = 0}

b : Q_{k}^{m} ⟶ Q_{k−1}^{m}, bq = (−1)^{∗}E_{P}^{∗} q.

To analyse this non-transversality further, let J_{i} ⊂ ℂ[y_{1}, …, y_{k}] be the ideal generated by the n monomials y_{i}^{l}, l = 1, …, n. Thus, by Taylor’s theorem,

J_{i} = {q ∈ ℂ[y_{1}, …, y_{k}]; q(y_{1}, y_{2}, …, y_{j−1}, 0, y_{j}, y_{k}) = 0.

Similary set

J_{P} = {q ∈ ℂ[y_{1}, …, y_{k}]; q(y_{1}, …, − ∑_{j=1}^{k−1} y_{j}) = 0)

For any two ideals I and J, let I · J be the span of the products. Thus for these particular ideals an element of the product is a sum of terms each of which has a factor vanishing on the corresponding linear subspace. For each k there are k + 1 ideals and, by Taylor’s theorem, the intersection of any k of them is equal to the span of the product of those k ideals. For the k coordinate ideals this is Taylor’s theorem as used in the reduction above. The general case of any k of the ideals can be reduced to this case by linear change of coordinates. The question then, is structure of the intersection of all k+1 ideals. The proof of the theorem is therefore completed by the following result.

Lemma 8.3. The intersection Q_{k}^{m} ∩ J_{P} = Q^{m} · J_{P} for every m ≠ k and

Q_{k}^{k} ∩ J_{P} = Λ^{k}(ℂ^{n}). (8.12)

Proof. When m < k the ideal Q_{k}^{m} vanishes, so the result is trivial.

Consider the case in (8.12), when m = k. A homogeneous polynomial of degree k in k variables (each in ℝ^{n}) which vanishes at the origin in each variable is necessarily linear in each variable, i.e. is just a k-multilinear function. Given such a multilinear function q(y_{1}, …, y_{k}) the condition that bq = 0 is just that

q(y_{1}, …, y_{k−1}, −y_{1} − y_{2} − ⋯ − y_{k−1}) ≡ 0. (8.13)

Using the linearity in the last variable the left side can be expanded as a sum of k − 1 functions each quadratic in one variables y_{j} and linear in the rest. Thus the vanishing of the sum implies the vanishing of each, so

q(y_{1}, …, y_{k−1}, y_{j}) ≡ 0 ∀ j = 1, …, k − 1.

This is the statement that the multlinear function q is antisymmetric between the jth and kth variables for each j < k. Since these exchange maps generate the permutation group, q is necessarily totally antisymmetric. This proves the isomorphism (8.12) since Λ^{k}(ℂ^{n}) is the space of complex-valued totally antisymmetric k-linear forms.^{3}

Thus it remains to consider the case m ≥ k + 1. Consider a general element q ∈ Q_{k}^{m} ∩ J_{P} . To show that it is in Q_{k}^{m} · J_{P} we manipulate it, working modulo Q_{k}^{m} ∩ J_{P} , and use induction over k. Decompose q as a sum of terms q_{l}, each homogeneous in the first variable, y_{1}, of degree l. Since q vanishes at y_{1} = 0 the first term is q_{1}, linear in y_{1}. The condition bq = 0, i.e. q ∈ J_{P} , is again just (8.13). Expanding in the last variable shows that the only term in bq which is linear in y_{1} is

q_{1}(y_{1}, …, y_{k−1}, −y_{2} − ⋯ − y_{k−1}).

Thus the coefficient of y_{1,i}, the ith component of y_{1} in q_{1}, is an element of Q_{k−1}^{m−1} which is in the ideal J_{P} (ℝ^{k−1}), i.e. for k − 1 variables. This ideal is generated by the components of y_{2} + ⋯ + y_{k}. So we can proceed by induction and suppose that the result is true for less than k variables for all degrees of homogeneity. Writing y_{2} + ⋯ + y_{k} = (y_{1} + y_{2} + ⋯ + y_{k}) − y_{1} It follows that, modulo Q_{k}^{m} · J_{P} , q_{1} can be replaced by a term of one higher homogeneity in y_{1}. Thus we can assume that q_{i} = 0 for i < 2. The same argument now applies to q_{2}; expanded as a polynomial in y_{1} the coefficients must be elements of Q_{k−1}^{m−2} ∩ J_{P} . Thus, unless m − 2 = k − 1, i.e. m = k + 1, they are, by the inductive hypothesis, in Q_{k−1}^{m−2} · J_{P} (ℝ^{k−1}) and hence, modulo Q_{k}^{m} · J_{P} , q_{2} can be absorbed in q_{3}. This argument can be continued to arrange that q_{i} ≡ 0 for i < m − k + 1. In fact q_{i} ≡ 0 for i > m − k + 1 by the assumption that q ∈ Q_{k}^{m}.

Thus we are reduced to the assumption that q = q_{m−k+1} ∈ Q_{k}^{m} ∩J_{P} is homogeneous of degree m − k + 1 in the first variable. It follows that it is multilinear in the last k −1 variables. The vanishing of bq shows that it is indeed totally antisymmetric in these last k − 1 variables. Now for each non-zero monomial consider the map J : {1, 2, …, n} ⟶ ℕ_{0} such that J(i) is the number of times a variable y_{l,i} occurs for some 1 ≤ l ≤ k. The decomposition into the sum of terms for each fixed J is preserved by b. It follows that we can assume that q has only terms corresponding to a fixed map J. If J(i) > 1 for any i then a factor y_{1,i} must be present in q, since it is antisymmetric in the other k − 1 variables. In this case it can be written y_{1,i}q′ where bq′ = 0. Since q′ is necessarily in the product of the indeals J_{2} · . . . J_{k} · J_{P} it follows that q′ ∈ Q^{m} · J_{P} . Thus we may assume that J(i) = 0 or 1 for all i. Since the extra variables now play no rôle we may assume that n = m is the degree of homogeneity and each index i occurs exactly once.

For convenience let us rotate the last k − 1 variables so the last is moved to the first position. Polarizing q in the first variable, it can be represented uniquely as an n-multilinear function on ℝ^{n} which is symmetric in the first n − k + 1 variables, totally antisymmetric in the last k − 1 and has no monomial with repeated index. Let M_{k−1}(n) be the set of such multilinear funtions. The vanishing of bq now corresponds to the vanishing of the symmetrization of q in the first n−k+2 variables. By the antisymmetry in the second group of variables this gives a complex

[figure: M_{n}(n); b_{n}; M_{n−1}(n); b_{n−1}; …; b_{2}; M_{1}(n); b_{1}; M_{0}; b_{0}; 0.]

3Really on the dual but that does not matter at this stage.

8.3. HOCHSCHILD HOMOLOGY OF C^{∞}(X) 133

The remaining step is to show that this is exact.

Observe that dim(M_{k}(n)) = \binom{n}{k} since there is a basis of M_{k}(n) with elements labelled by the subsets I ⊂ {1, …, n} with k elements. Indeed let ω be a nontrivial k-multilinear function of k variables and let ω_{I} be this function on ℝ^{k} ⊂ ℝ^{n} identified as the set of variables indexed by I. Then if a ∈ M_{0}(n − k) is a basis of this 1-dimensional space and a_{I} is this function on the complementary ℝ^{n−k} the tensor products a_{I}ω_{I} give a basis. Thus there is an isomorphism

M_{k} ∋ q = ∑_{I⊂{1,...,n},|I|=k} c_{I}a_{I} ⊗ ω_{I} ⟼ ∑_{I⊂{1,...,n},|I|=k} c_{I} ⊗ ω_{I} ∈ Λ^{k}(ℝ^{n}).

Transfered to the exterior algebra by this isomorphism the differential b is just contraction with the vector e_{1} + e_{2} + ⋯ + e_{n} (in the first slot). A linear transformation reducing this vector to e_{1} shows immediately that this (Koszul) complex is exact, with the null space of b_{k} on Λ^{k}(ℝ^{n}) being spanned by those ω_{I} with 1 ∈ I and the range of b_{k+1} spanned by those with 1 ∉ I. The exactness of this complex completes the proof of the lemma.

8.3. Hochschild homology of C^{∞}(X)

The first example of Hochschild homology that we shall examine is for the commutative algebra C^{∞}(X) where X is any C^{∞} manifold (compact or not). As noted above we need to replace the tensor product by some completion. In the present case observe that for any two manifolds X and Y

C^{∞}(X) ⊗ C^{∞}(Y) ⊂ C^{∞}(X × Y) (8.14)

is dense in the C^{∞} topology. Thus we simply declare the space of k-chains for Hochschild homology to be C^{∞}(X^{k+1}), which can be viewed as a natural completion^{4} of C^{∞}(X)^{⊗(k+1)}. Notice that the product of two functions can be written in terms of the tensor product as

a · b = D^{∗}(a ⊗ b), a, b ∈ C^{∞}(X), D : X ∋ z ⟼ (z, z) ∈ X^{2}.

The variables in X^{k+1} will generally be denoted z_{0}, z_{1}, …, z_{k}. Consider the ‘diagonal’ submanifolds

D_{i,j} = {(z_{0}, z_{1}, …, z_{k}); z_{i} = z_{j}}, i, j = 0, …, m, i ≠ j. (8.15)

We shall use the same notation for the natural embedding of X^{k} as each of these submanifolds, at least for j = i + 1 and i = 0, j = m,

D_{i,i+1}(x_{0}, …, z_{m−1}) = (z_{0}, …, z_{i}, z_{i}, z_{i+1}, …, z_{m−1}) ∈ D_{i,i+1}, i = 0, …, m − 1

D_{m,0}(z_{0}, …, z_{m−1}) = (z_{0}, …, z_{m−1}, z_{0}).

V ⊗ W = hom(W′, V)

4One way to justify this is to use results on smoothing operators. For finite dimensional linear spaces V and W the tensor product can be realized as the space of linear maps from the dual of W to V. Identifying the topological dual of C^{∞}(X)

with C_{c}^{−∞}(X; Ω), the space of distributions of compact support, with the weak topology, we can identify the projective tensor product C^{∞}(X) ˆ⊗C^{∞}(X) as the space of continuous linear maps from C_{c}^{−∞}(X; Ω) to C^{∞}(X). These are precisely the smoothing operators, corresponding to kernels in C^{∞}(X × X).

Then the action of b′ and b on the tensor products, and hence on all chains, can be written

b′α = ∑_{i=0}^{m−1}(−1)^{i}D_{i,i+1}^{∗}α, bα = b′α + (−1)^{m}D_{m,0}^{∗}α. (8.16)

Theorem 8.2. The differential b′ is acyclic and the homology^{5} of the complex

… ⟶^{b} C^{∞}(X^{k+1}) ⟶^{b} C^{∞}(X^{k}) ⟶^{b} … (8.17)

is naturally isomorphic to C^{∞}(X; Λ^{∗}).

Before proceeding to the proof proper we note two simple lemmas.

Lemma 8.4. ^{6}For any j = 0, …, m − 1, each function α ∈ C^{∞}(X^{k+1}) which vanishes on D_{i,i+1} for each i ≤ j can be written uniquely in the form

α = α′ + α′′, α′, α′′ ∈ C^{∞}(X^{k+1})

where α′′ vanishes on D_{i,i+1} for all i ≤ j + 1 and α′ is independent of z_{j+1}.

Proof. Set α′ = π_{j+1}^{∗}(D_{j,j+1}^{∗}α) where π_{j} : X^{k+1} ⟶ X^{k} is projection off the jth factor. Thus, essentially by definition, α′ is independent of z_{j+1}. Moreover, π_{j+1}D_{j,j+1} = Id so D_{j,j+1}^{∗}α′ = D_{j,j+1}^{∗}α and hence D_{j,j+1}^{∗}α′′ = 0. The decomposition is clearly unique, and for i < j,

D_{j,j+1} ∘ π_{j+1} ∘ D_{i,i+1} = D_{i,i+1} ∘ F_{i,j} (8.18)

for a smooth map F_{i,j}, so α′ vanishes on D_{i,i+1} if α vanishes there.

Lemma 8.5. For any finite dimensional vector space, V, the k-fold exterior power of the dual, Λ^{k}V^{∗}, can be naturally identified with the space of functions

(8.19) {u ∈ C^{∞}(V^{k}); u(sv) = s^{k}v, s ≥ 0, u ↾ (V^{i}×{0}×V^{k−i−1}) = 0 for i = 0, …, k−1

and u ↾ G = 0, G = {(v_{1}, …, v_{k}) ∈ V^{k}; v_{1} + ⋯ + v_{k} = 0}}.

Proof. The homogeneity of the smooth function, u, on V^{k} implies that it is a homogeneous polynomial of degree k. The fact that it vanishes at 0 in each variable then implies that it is multlinear, i.e. is linear in each variable. The vanishing on G implies that for any j and any v_{i} ∈ V, i ≠ j,

∑_{i≠j} u(v_{1}, …, v_{j−1}, v_{i}, v_{j+1}, …, v_{k}) = 0. (8.20)

Since each of these terms is quadratic (and homogeneous) in the corresponding variable v_{i}, they must each vanish identically. Thus, u vanishes on v_{i} = v_{j} for each i ≠ j; it is therefore totally antisymmetric as a multlinear form, i.e. is an element of Λ^{k}V^{∗}. The converse is immediate, so the lemma is proved.

5This homology is properly referred to as the continuous Hochschild homology of the topological algebra C^{∞}(X).

6As pointed out to me by Maciej Zworski, this is a form of Hadamard’s lemma.

Proof of Theorem 8.2. The H-unitality^{7} of C^{∞}(X) follows from the proof of Lemma 8.59 which carries over verbatim to the larger chain spaces.

By definition the Hochschild homology in degree k is the quotient

HH_{k}(C^{∞}(X)) = {u ∈ C^{∞}(X^{k+1}); bu = 0}/bC^{∞}(X^{k+2}). (8.21)

The first stage in identifying this quotient is to apply Lemma 8.4 repeatedly. Let us carry through the first step separately, and then do the general case.

For j = 0, consider the decomposition of u ∈ C^{∞}(X^{k+1}) given by Lemma 8.4, thus

u = u_{0} + u_{(1)}, u_{0} ∈ π_{1}^{∗}C^{∞}(X^{k}), u_{(1)} ∈ J_{1}^{(k)} = {u ∈ C^{∞}(X^{k+1}); u ↾ D_{0,1} = 0}. (8.22)

Now each of these subspaces of C^{∞}(X^{k+1}) is mapped into the corresponding subspace of C^{∞}(X^{k}) by b; i.e. they define subcomplexes. Indeed,

u ∈ π_{1}^{∗}C^{∞}(X^{k}) ⟹ D_{0,1}^{∗}u = D_{1,2}^{∗}u so

u = π_{1}^{∗}v ⟹ bu = π_{1}^{∗}Bv, B^{∗}v = − ∑_{i=1}^{k−1}(−1)^{i}D_{i,i+1}^{∗}u + (−1)^{k}D_{k−1,0}^{∗}v.

For the other term

bu_{(1)} = ∑_{i=1}^{k−1}(−1)^{i}D_{i,i+1}^{∗}u_{(1)} + (−1)^{k}D_{k,0}^{∗}u_{(1)} ⟹ bu_{(1)} ∈ J_{1}^{(k−1)}. (8.23)

Thus, bu = 0 is equivalent to bu_{0} = 0 and bu_{(1)} = 0. From (8.3), defining an isomorphism by

E_{(k−1)} : C^{∞}(X^{k}) ⟶ C^{∞}(X^{k}), E_{(k−1)}v(z_{1}, …, z_{k}) = v(z_{2}, …, z_{k}, z_{1}), (8.24)

it follows that

B = −E_{(k−1)}^{−1}b′E_{(k−1)} (8.25)

is conjugate to b′. Thus B is acyclic so in terms of (8.22)

bu = 0 ⟹ u − u_{(1)} = bw, w = π_{1}^{∗}v′. (8.26)

As already noted this is the first step in an inductive procedure, the induction being over 1 ≤ j ≤ k in Lemma 8.4. Thus we show inductively that

(8.27) bu = 0 ⟹ u − u_{(j)} = bw,

u_{(j)} ∈ J_{j}^{(k)} = {u ∈ C^{∞}(X^{k+1}); u ↾ D_{i,i+1} = 0, 0 ≤ i ≤ j − 1}.

For j = 1 this is (8.26). Proceeding inductively we may suppose that u = u_{(j)} and take the decomposition of Lemma 8.4, so

u_{(j)} = u′ + u_{(j+1)}, u_{(j+1)} ∈ J_{j+1}^{(k)}, u′ = π_{j+1}^{∗}v ∈ J_{j}^{(k)}. (8.28)

Then, as before, bu_{(j)} = 0 implies that bu′ = 0. Furthermore, acting on the space π_{j+1}^{∗}C^{∞}(X^{k}) ∩ J_{(j)}^{k}, b is conjugate to b′ acting in k + 1 − j variables. Thus, it is again acyclic, so u_{(j)} and u_{(j+1)} are homologous as Hochschild k-cycles.

7Meaning here the continuous H-unitality, that is the acyclicity of b′ on the chain spaces C^{∞}(X^{k+1}).

The end point of this inductive procedure is that each k-cycle is homologous to an element of

J^{(k)} = J_{k}^{(k)} = {u ∈ C^{∞}(X^{k+1}); D_{i,i+1}^{∗}u = 0, i ≤ i ≤ k − 1}. (8.29)

Acting on this space bu = (−1)^{k}D_{k,0}^{∗}u, so we have shown that

HH_{k}(C^{∞}(X)) = M^{(k)}/(M^{(k)} ∩ bC^{∞}(X^{k+1})), M^{(k)} = {u ∈ J^{(k)}; D_{k,o}^{∗}u = 0}. (8.30)

Now consider the subspace

(8.31) M̃^{(k)} = {u ∈ C^{∞}(X^{k+1});

u = ∑_{finite, 0≤j≤k−1} (f(z_{j}) − f(z_{j+1})) u_{f,j}, u_{f,j} ∈ M^{(k)}, f ∈ C^{∞}(X).}.

If u = (f(z_{j}) − f(z_{j+1}))v, with v ∈ M^{(k)} set

(8.32) w(z_{0}, z_{1}, …, z_{j}, z_{j+1}, z_{j+2}, …, z_{k+1})

= (−1)^{j}(f(z_{j}) − f(z_{j+1}))v(z_{0}, …, z_{j}, z_{j+2}, z_{j+3}, …, z_{k}).

Then, using the assumed vanishing of v, bw = u.^{8} Thus all the elements of M̃^{(k)} are exact.

Let us next compute the quotient M^{(k)}/M̃^{(k)}. Linearizing in each factor of X around the submanifold z_{0} = z_{1} = ⋯ = z_{k} in V^{k} defines a map

μ : M^{(k)} ∋ u ⟶ u′ ∈ C^{∞}(X; T X ⊗ ⋯ ⊗ T^{∗}X). (8.33)

The map is defined by taking the term of homogeneity k in a normal expansion around the submanifold. The range space is therefore precisely the space of sections of the k-fold tensor product bundle which vanish on the subbundle defined in each fibre by v_{1} + ⋯ + v_{k} = 0. Thus, by Lemma 8.5, μ actually defines a sequence

0 ⟶ M̃^{(k)} ↪ M^{(k)} ⟶^{μ} C^{∞}(X; Λ^{k}X) ⟶ 0. (8.34)

Lemma 8.6. For any k, (8.34) is a short exact sequence.

Proof. So far I have a rather nasty proof by induction of this result, there should be a reasonably elementary argument. Any offers?

From this the desired identification, induced by μ,

HH_{k}(C^{∞}(X)) = C^{∞}(X; Λ^{k}X) (8.35)

follows, once is is shown that no element u ∈ M^{(k)} with μ(u) ≠ 0 can be exact. This follows by a similar argument. Namely if u ∈ M^{(k)} is exact then write u = bv, v ∈ C^{∞}(X^{k}) and apply the decomposition of Lemma 8.4 to get v = v_{0} + v_{(1)}. Since u = 0 on D_{1,0} it follows that bv_{0} = 0 and hence u = bv_{1)}. Proceeding inductively we conculde that u = bv with v ∈ M^{(k+1)}. Now, μ(bv) = 0 by inspection.

8Notice that v(z_{0}, …, z_{j}, z_{j+2}, …, z_{k+1}) vanishes on z_{i+1} = z_{i} for i < j and i > j + 1 and also on z_{0} + z_{1} + ⋯ + z_{k+1} = 0 (since it is independent of z_{j+1} and bv = 0.

8.5. HOCHSCHILD CHAINS FOR Ψ_{phg}^{ℤ}(X)/Ψ_{phg}^{−∞}(X) 137

8.4. Commutative formal symbol algebra

As a first step towards the computation of the Hochschild homology of the algebra A = Ψ^{ℤ}(X)/Ψ^{−∞}(X) we consider the formal algebra of symbols with commutative product. Thus,

A ={(a_{j})_{j=−∞}^{∞}; a_{j} ∈ C^{∞}(S^{∗}X; P^{(j)}), a_{j} = 0 for j >> 0}. (8.36)

Here P^{(k)} is the line bundle over S^{∗}X with sections consisting of the homogeneous functions of degree k on T^{∗}X \ 0. The multiplication is as functions on T^{∗}X \ 0, so

(a_{j}) · (b_{j}) = (c_{j}), c_{j} = ∑_{k=−∞}^{∞} a_{j−k}b_{k}

using the fact that P^{(l)} ⊗ P^{(k)} ≡ P^{(l+k)}. We take the completion of the tensor product to be

(8.37) B^{(k)} = {u ∈ C^{∞}((T^{∗}X \ 0)^{k+1}); u = ∑_{finite} u_{I},

u_{I} ∈ C^{∞}(S^{∗}X; P^{(I_{0})} ⊗ P^{(I_{1})} ⊗ ⋯ ⊗ P^{(I_{k})}), |I| = k}.

That is, an element of B^{(k)} is a finite sum of functions on the (k + 1)-fold product of T^{∗}X \ 0 which are homogeneous of degree I_{j} on the jth factor, with the sum of the homogeneities being k. Then the Hochschild homology is the cohomology of the subcomplex of the complex for C^{∞}(T^{∗}X)

⋯ ⟶^{b} B^{(k)} ⟶^{b} B^{(k−1)} ⟶^{b} ⋯ (8.38)

Theorem 8.3. The cohomology of the complex (8.38) for the commutative product on A is

(8.39) HH_{k}(A) ≡ {α ∈ C^{∞}(T^{∗}X \ 0; Λ^{k}(T^{∗}X); α is homogeneous of degree k}.

8.5. Hochschild chains for Ψ_{phg}^{ℤ}(X)/Ψ_{phg}^{−∞}(X)

The completion of the tensor product that we take to define the Hochschild homology of the ‘full symbol algebra’ is the same space as in (8.37) but with the non-commutative product derived from the quantization map for some Riemann metric on X. Since the product is given as a formal sum of bilinear differential operators it can be take to act on an pair of factors.

… ⟶^{b^{(⋆)}} B^{(k)} ⟶^{b^{(⋆)}} B^{(k−1)} ⟶^{b^{(⋆)}} … (8.40)

The next, and major, task of this chapter is to describe the cohomology of this complex.

Theorem 8.4. The Hochschild homolgy of the algebra, Ψ_{phg}^{ℤ}(X)/Ψ_{phg}^{−∞}(X), of formal symbols of pseudodifferential operators of integral order, identified as the cohomology of the complex (8.40), is naturally identified with two copies of the cohomology of S^{∗}X^{9}

HH_{k}(A; ∘) ≡ H^{2n−k}(S^{∗}X) ⊕ H^{2n−1−k}(S^{∗}X). (8.41)

9In particular the Hochschild homology vanishes for k > 2 dim X. For a precise form of the identification in (8.41) see (9.1).

8.6. Semi-classical limit and spectral sequence

The ‘classical limit’ in physics, especially quantuum mechanics, is the limit in which physical variables become commutative, i.e. the non-commutative coupling between position and momentum variables vanishes in the limit. Formally this typically involves the replacement of Planck’s constant by a parameter h → 0. A phenomenon is ‘semi-classical’ if it can be understood at least in Taylor series in this parameter. In this sense the Hochschild homology of the full symbol algebra is semi-classical and (following [1]) this is how we shall compute it.

The parameter h is introduced directly as an isomorphism of the space A

L_{h} : A ⟶ A, L_{h}(a_{j})_{j=−∞}^{∗} = (h^{j}a_{j})_{j=−∞}^{∗}, h > 0.

Clearly L_{h} ∘ L_{h′} = L_{hh′}. For h ≠ 1, L_{h} is not an algebra morphism, so induces a 1-parameter family of products

α ⋆_{h} β = (L_{h}^{−1})(L_{h}α ⋆ L_{h}β). (8.42)

In terms of the differential operators, associated to quantization by a particular choice of Riemann metric on X this product can be written

α ⋆_{h} β = (c_{j})_{j=−∞}^{∗}, c_{j} = ∑_{k=0}^{∗} ∑_{l=−∗}^{∗} h^{k}P_{k}(a_{j−l−k}, b_{l}). (8.43)

It is important to note here that the P_{k}, as differential operators on functions on T^{∗}X, do only depend on k, which is the difference of homogeneity between the product a_{j−l+k}b_{l}, which has degree j + k and c_{j}, which has degree j.

Since A with product ⋆_{h} is a 1-parameter family of algebras, i.e. a deformation of the algebra A with product ⋆ = ⋆_{1}, the Hochschild homology is ‘constant’ in h. More precisely the map L_{h} induces a canonical isomorphism

L_{h}^{∗} : HH_{k}(A; ⋆_{h}) ≡ HH_{k}(A; ⋆).

The dependence of the product on h is smooth, so it is reasonable to expect the cycles to have smooth representatives as h → 0. To investigate the consider Taylor series in h and define

\begin{matrix}F_{p,k} = \{α ∈ B^{(k)}; ∃ α(h) ∈ C^{∞}([0, 1)_{h}; B^{(k)}) with α(0) = α and \\ b_{h}α ∈ h^{p}C^{∞}([0, 1)_{h}; B^{k−1)}\}, \end{matrix} (8.44)

\begin{matrix}G_{p,k} = \{α ∈ B^{(k)}; ∃ β(h) ∈ C^{∞}([0, 1)_{h}; B^{(k+1)}) with \\ b_{h}β(h) ∈ h^{p−1}C^{∞}([0, 1)_{h}; B^{(k)} and (t^{−p+1}b_{h}β)(0) = α\}. \end{matrix} (8.45)

Here b_{h} is the differential defined by the product ⋆_{h}.

Notice that the F_{p,k} decrease with increasing p, since the condition becomes stronger, while G_{p,k} increases with p, the condition becoming weaker.^{10} We define the ‘spectral sequence’ corresponding to this filtration by

E_{p,k} = F_{p,k}/G_{p,k}.

10If α ∈ G_{p,k} and β(h) is the 1-parameter family of chains whose existence is required for the definition then β′(h) = hβ(h) satisfies the same condition with p increased to show that α ∈ G_{p+1,k}.

8.7. THE E_{2} TERM 139

These can also be defined successively, in the sense that if

F′_{p,k} = {u ∈ E_{p−1,k}; u = [u′], u′ ∈ F_{p,k}}
G′_{p,k} = {e ∈ E_{p−1,k}; u = [u′],]u′ ∈ G_{p,k}}
then E_{p,k} ≡ F′_{p,k}/G′_{p,k}.

The basic idea^{11} of a spectral sequence is that each E_{p} = ⨁_{k} E_{p,k}, has defined on it a differential such that the next spaces, forming E_{p+1}, are the cohomology space for the complex. This is easily seen from the definitions of F_{p,k} as follows. If α ∈ F_{p,k} let β(t) be a 1-parameter family of chains as in the defintion. Then consider

γ(t^{−p}b_{h}β)(0) ∈ B^{(k−1)}. (8.46)

This depends on the choice of β, but only up to a term in G_{p,k−1}. Indeed, let β′(t) is another choice of extension of α satisfying the condition that b_{h}β′ ∈ h^{p}C^{∞}([0, 1); B^{(k−1)} and let γ′ be defined by (8.46) with β replaced by β′. Then δ(t) = t^{−1}(β(h) − β′(h)) satisfies the requirements in the definition of G_{p,k−1}, i.e. the difference γ′ − γ ∈ G_{p,k−1}. Similarly, if α ∈ G_{p,k} then γ ∈ G_{p,k}.^{12} The map so defined is a differential

b_{(p)} : E_{p,k} ⟶ E_{p,k−1}, b_{(p)}^{2} = 0.

This follows from the fact that if μ = b_{(p)}α then, by definition, μ = (t^{−p}b_{h}β)(0), where α = β(0). Taking λ(t) = t^{−p}b_{h}β(t) as the extension of μ it follows that b_{h}λ = 0, so b_{(p)}μ = 0.

Now, it follows directly from the definition that F_{0,k} = E_{0,k} = B^{(k)} since G_{0,k} = {0}. Furthermore, the differential b_{(0)} induced on E_{0} is just the Hochschild differential for the limiting product, ⋆_{0}, which is the commutative product on the algebra. Thus, Theorem 8.3 just states that

E_{1,k} = ⨁_{k=−∞}^{∗}{u ∈ C^{∞}(T^{∗}X \ 0; Λ^{k}); u is homogeneous of degree k}.

To complete the proof of Theorem 8.4 it therefore suffices to show that

E_{2,k} ≡ H^{2n−k}(S^{∗}X) ⊕ H^{2n−1−k}(S^{∗}X), (8.47)

E_{p,k} = E_{2,k}, ∀ p ≥ 2, and (8.48)

(8.49) HH_{k}(Ψ_{phg}^{ℤ}(X)/Ψ_{phg}^{−∞}(X)) = lim_{p→∞} E_{p,k}.

The second and third of these results are usually described, respectively, as the ‘degeneration’ of the spectral sequence (in this case at the ‘E_{2} term’) and the ‘convergence’ of the spectral sequence to the desired cohomology space.

8.7. The E_{2} term

As already noted, the E_{1,k} term in the spectral sequence consists of the formal sums of k-forms, on T^{∗}X \ 0, which are homogeneous under the ℝ^{+} action. The E_{2} term is the cohomology of the complex formed by these spaces with the differential b_{(1)}, which we proceed to compute. For simplicity of notation, consider the formal tensor prodoct rather than its completion. As already noted, for any α ∈ B^{(k)} the function b_{h}α is smooth in h and from the definition of b,

11Of Leray I suppose, but I am not really sure.

12Indeed, α is then the value at h = 0 of β(t) = t^{−p+1}b_{h}ϕ(t) which is by hypothesis smooth; clearly b_{h}β ≡ 0.

(8.50) \frac{d}{dh}b_{h}α(0) = ∑_{i=0}^{k−1}(−1)^{i}a_{0} ⊗ ⋯ ⊗ a_{i−1} ⊗ P_{1}(a_{i+1}, a_{i}) ⊗ a_{i+2} ⊗ ⋯ ⊗ a_{k}

+ (−1)^{k}P_{1}(a_{0}, a_{k}) ⊗ a_{1} ⊗ ⋯ ⊗ a_{k−1}, α = a_{0} ⊗ ⋯ ⊗ a_{k}.

The general case is only more difficult to write, not different.^{13} This certainly determines b_{1}α if α is a superposition of such terms with b_{0}α = 0. Although (8.50) is explicit, it is not given directly in terms of the representation of α, assumed to satisfy b_{0}α = 0 as a form on T^{∗}X \ 0.

To get such an explicit formula we shall use the symplectic analogue of the Hodge isomorphism. Recall that in any local coordinates on X, x_{i}, i = 1, …, n, induce local coordinates x_{i}, ξ_{i} in the part of T^{∗}X lying above the coordinate patch. In these canonical coordinates the symplectic form (which determines the Poisson bracket) is given by

ω = ∑_{k=1}^{n} dξ_{k} ∧ dx_{k}. (8.51)

This 2-form is non-degenerate, i.e. the n-fold wedge product ω^{n} ≠ 0. In fact this volume form fixes an orientation on T^{∗}X. The symplectic form can be viewed as a non-degenerate antisymmetric bilinear form on T_{q}(T^{∗}X) at each point q ∈ T^{∗}X, and hence by duality as a bilineear form on T_{q}^{∗}(T^{∗}X). We denote this form in the same way as the Poisson bracket, since with the convention

{a, b}(q) = {da, db}_{q}

they are indeed the same. As a non-degenerate bilinear form on T^{∗}Y, Y = T^{∗}X this also induces a bilinear form on the tensor algebra, by setting

{e_{1} ⊗ ⋯ ⊗ e_{k}, f_{1} ⊗ ⋯ ⊗ f_{k}, } = ∏_{j}{e_{j}, f_{j}}.

These bilinear forms are all antisymmetric and non-degenerate and restrict to be non-degnerate on the antisymmetric part, Λ^{k}Y, of the tensor algebra. Thus each of the form bundles has a bilinear form defined on it, so there is a natural isomorhism

W_{ω} : Λ_{q}^{k}Y ⟶ Λ_{q}^{2n−k}Y, α ∧ W_{ω}β = {α, β}ω^{n}, α, β ∈ C^{∞}(Y, Λ^{k}Y), (8.52)

for each k.

Lemma 8.7. In canonical coordinates, as in (8.51), consider the basis of kforms given by all increasing subsequences of length k, I : {1, 2, …, k} ⟶ {1, 2, …, 2n}, and setting

(8.53) α_{I} = dz_{I(1)} ∧ dz_{I(2)} ∧ ⋯ ∧ dz_{I(k)},

(z_{1}, z_{2}, …, z_{2n}) = (x_{1}, ξ_{1}, x_{2}, ξ_{2}, …, x_{n}, ξ_{n}).

In terms of this ordering of the coordinates

W_{ω}(α_{I}) = (−1)^{N(I)}α_{W (I)} (8.54) where W (I) is obtained from I by considering each pair (2p−1, 2p) for p = 1, …, n, erasing it if it occurs in the image of I, inserting it into I if neither 2p − 1 nor 2p occurs in the range of I and if exactly one of 2p − 1 and 2p occurs then leaving it unchanged; N(I) is the number of times 2p appears in the range of I without 2p−1.

13If you feel it necessary to do so, resort to an argument by continuity towards the end of this discussion.

Proof. The Poisson bracket pairing gives, on 1-forms,

−{dx_{j}, dξ_{j}} = 1 = {dξ_{j}, dx_{j}}

with all other pairings zero. Extending this to k-forms gives

{α_{I}, α_{J}} = 0 unless (I(j), J(j)) = (2p − 1, 2p) or (2p, 2p − 1) ∀ j and
{α_{I}, α_{J}} = (−1)^{N}, if (I(j), J(j)) = (2p − 1, 2p) for N values of j
and (I(j), J(j)) = (2p − 1, 2p) for N − k values of j.

From this, and (8.52), (8.54) follows.

From this proof if also follows that N(W (I)) = N(I), so W_{ω}^{2} = Id . We shall let

δ_{ω} = W_{ω} ∘ d ∘ W_{ω} (8.55)

denote the differential operator obtained from d by conjugation,

δ_{ω} : C^{∞}(T^{∗}X \ 0; Λ^{k}) ⟶ C^{∞}(T^{∗}X \ 0, Λ^{k−1}).

By construction δ_{ω}^{2} = 0. The exterior algebra of a symplectic manifold with this differential is called the Koszul complex.^{14} All the α_{I} are closed so

\begin{matrix}δ_{ω}(aα_{I}) = W_{ω}(∑_{j} \frac{∂a}{∂z_{j}} dz_{j}) ∧ (−1)^{N(I)}α_{W (I)} \\ = ∑_{j} \frac{∂a}{∂z_{j}} (−1)^{N(I)}W_{ω}(dz_{j} ∧ α_{W (I)}), \end{matrix} (8.56)

Observe that^{15}

W_{ω}(dz_{2p−1} ∧ α_{W (I)}) = ι_{∂/∂z_{2p}}α_{I}
W_{ω}(dz_{2p} ∧ α_{W (I)}) = ι_{∂/∂z_{2p−1}}α_{I},

where, ι_{v} denotes contraction with the vector field v. We therefore deduce the following formula for the action of the Koszul differential

δ_{ω}(aα_{I}) = ∑_{i=1}^{2n}(H_{z_{i}}a)ι_{∂/∂z_{i}}α_{I}. (8.57)

Lemma 8.8. With E_{1} identified with the formal sums of homogeneous forms on T^{∗}X \ 0, the induced differential is

b_{(1)} = \frac{1}{i}δ_{ω}. (8.58)

14Up to various sign conventions of course!

15Check this case by case, as the range of I meets the pair {2p − 1, 2p} in {2p − 1, 2p}, {2p − 1}, {2p} or ∅. Both sides of the first equation are zero in the second and fourth case as are both sides of the second equation in the third and fourth cases. In the remaining four individual cases it is a matter of checking signs.

Proof. We know that the bilinear differential operator 2iP_{1} is the Poisson bracket of functions on T^{∗}X. Thus (8.50) can be written

(8.59) 2ib_{1}α = ∑_{i=0}^{k−1}(−1)^{i}a_{0} ⊗ ⋯ ⊗ a_{i−1} ⊗ {a_{i+1}, a_{i}} ⊗ a_{i+2} ⊗ ⋯ ⊗ a_{k}

+ (−1)^{k}{a_{0}, a_{k}} ⊗ a_{1} ⊗ ⋯ ⊗ a_{k−1}, α = a_{0} ⊗ ⋯ ⊗ a_{k}.

The form to which this maps under the identification of E_{2} is just

(8.60) 21b_{1}α = ∑_{i=0}^{k−1}(−1)^{i}a_{0} ∧ da_{i−1} ∧ ⋯ ∧ d{a_{i+1}, a_{i}} ∧ da_{i+2} ∧ a_{k}

+ (−1)^{k}{a_{0}, a_{k}} ∧ da_{1} ∧ ⋯ ∧ da_{k−1}

Consider the basis elements α_{I} for k-forms. These arise as the images of the corresponding functions in local coordinates on X^{k+1}

α̃_{I}(z_{0}, z_{1}, …, z_{k}) = ∑_{σ}(−1)^{sgn σ}(z_{1,σI(1)} − z_{0,σI(1)})

× z_{2,σI(1)} − z_{1,σI(1)}) … (z_{1,σI(m)} − z_{0,σI(m−1)}).

Since these functions are defined in local coordinates they are not globally defined on (T^{∗}X\0)^{k+1}. Nevertheless they can be localized away from z_{0} = ⋯ = z_{m} and then, with a coefficient (a_{j}(z_{0}))_{j=−∞}^{∗}, a_{j} ∈ C^{∞}(T^{∗}X \ 0) homogeneous of degree j with support in the coordinate patch, unambiguously define elements of E_{1} which we can simply denote as a(z_{0})α̃_{I} ∈ E_{1}. These elements, superimposed over a coordinate cover, span E_{1}. Consider b_{(1)}α̃ given by (8.60). In the sum, the terms with P_{1} contracting between indices other than 0, 1 or m, 0 must give zero because the Poisson bracket is constant in the ‘middle’ variable. Futhermore, by the antisymmetry of α̃, the two remaining terms are equal so

ib_{(1)}(aα̃_{I}) = ∑_{σ∈P_{k}}(H_{z_{σI(1)}}a)(−1)^{sgn(σ)}dz_{σI(2)} ∧ ⋯ ∧ dz_{σI(k)}

= ∑_{i}(H_{z_{i}}a)ι_{∂/∂_{i}}α_{I}.

Since this is just (8.57) the lemma follows.

With this lemma we have identified the differential on the E_{1} term in the spectral sequence with the exterior differential operator. To complete the identification (8.47) we need to compute the corresponding deRham groups.

Proposition 8.1. The cohomology of the complex

… ⟶^{d} ∑_{j=−∞}^{∗} C_{hom(j)}^{∞}(T^{∗}X \ 0; Λ^{k}) ⟶^{d} ∑_{j=−∞}^{∗} C_{hom(j)}^{∞}(T^{∗}X \ 0; Λ^{k+1}) ⟶^{d} …

in dimension k is naturally isomorphic to H^{k}(S^{∗}X) ⊕ H^{k−1}(S^{∗}X).

Proof. Choose a metric on X and let R = |ξ| denote the corresponding length function on T^{∗}X \ 0. Thus, identifying the quotient S^{∗}X = (T^{∗}X \ 0)/ℝ^{+} with {R = 1} gives an isomorphism T^{∗}X \ 0 ≡ S^{∗}X × (0, ∞). Under this map the smooth forms on T^{∗}X \ 0 which are homogeneous of degree j are identified as sums

\begin{matrix}C_{hom(j)}^{∞}(T^{∗}X \ 0, Λ^{k}) ∋ α_{j} \\ = R^{j}(α′_{j} + α′′_{j} ∧ \frac{dR}{R}), α′_{j} ∈ C^{∞}(S^{∗}X; Λ^{k}), α′′_{j} ∈ C^{∞}(S^{∗}X; Λ^{k−1}). \end{matrix} (8.61)

The action of the exterior derivative is then easily computed

dα_{j} = β_{j}, β_{j} = R^{j}(β′_{j} + β_{j} −′′ ∧\frac{dR}{R}),

β′_{j} = dα′_{j}, β′′_{j} = dα′′_{j} + j(−1)^{k−1}α′_{j}.

Thus a k-form (α_{j})_{j=−∞}^{∗} is closed precisely if it satisfies

jα′_{j} = (−1)^{k}dα′′_{j}, dα′_{j} = 0∀j. (8.62)

It is exact if there exists a (k − 1)-form (γ_{j})_{j=−∞}^{∗} such that

α′_{j} = dγ′_{j}, α′′_{j} = dγ′′_{j} + j(−1)^{k}γ′_{j}. (8.63)

Since the differential preserves homogeneity it is only necessary to analyze these equations for each integral j. For j ≠ 0, the second equation in (8.62) follows from the first and (8.63) then holds with γ′_{j} = \frac{1}{j}(−1)^{k}α′′_{j} and γ′′_{j} = 0. Thus the cohomology lies only in the subcomplex of homogeneous forms of degree 0. Then (8.62) and (8.63) become

dα′_{0} = 0, dα′′_{0} = 0 and α′_{0} = dγ′_{0}, α′′_{0} = dγ′′_{0}

respectively. This gives exactly the direct sum of H^{k}(S^{∗}X) and H^{k−1}(S^{∗}X) as the cohomology in degree k. The resulting isomorphism is independent of the choice of the radial function R, since another choice replaces R by Ra, where a is a smooth positive function on S^{∗}X. In the decomposition (8.61), for j = 0, α′′_{0} is unchanged whereas α′_{0} is replaced by α′_{0} + α′′_{0} ∧ d log a. Since the extra term is exact whenever α′′_{0} is closed it has no effect on the identification of the cohomology.

Combining Proposition 8.1 and Lemma 8.8 completes the proof of (8.47). We make the identification a little more precise by locating the terms in E_{2}.

Proposition 8.2. Under the identification of E_{1} with the sums of homogeneous forms on T^{∗}X \ 0, E_{2}, identified as the cohomology of δ_{ω}, has a basis of homogeneous forms with the homogeneity degree j and the form degree k confined to

(8.64) k − j = dim X, − dim X ≤ j ≤ dim X, dim X ≥ 2.

Proof. Provided dim X ≥ 2, the cohomology of S^{∗}X is isomorphic to two copies of the cohomology of X, one in the same degree and one shifted by dim X − 1.^{16} The classes in the first copy can be taken to be the lifts of deRham classes from X, while the second is spanned by the wedge of these same classes with the Todd class of S^{∗}X. This latter, n − 1, class restricts to each fibre to be non-vanishing. Thus in local representations the first forms involve only the base variable and in the second each terms has the maximum number, n − 1, of fibre forms. The cohomology of the complex in Proposition 8.1 therefore consists of four copies of H^{∗}(X) consisting of these forms and the same forms wedged with dR/R.

16That is, just as though S^{∗}X = 𝕊^{n−1} × X, where n = dim X.

With this decomoposition of the cohomology consider the effect on it of the map W_{ω}. In each case the image forms are again homogeneous. A deRham class on X in degree l therefore has four images in E_{2}. One is a form of degree k_{1} = 2n − l which is homogeneous of degree j_{1} = n − l. The second is a form of degree k_{2} = 2n − l − 1 which is homogeneous of degree j_{2} = n − l − 1. The third image is of form degree k_{3} = n−l +1 and homogeneous of degree j_{3} = −l +1 and the final image is of form degree k_{4} = n − l and is homogeneous of degree j_{4} = −l. This gives the relations (8.64).

8.8. Degeneration and convergence

Now that the E_{2} term in the spectral sequence has been explicitly computed, consider the induced differential, b_{(2)} on it. Any homogeneous form representing a class in E_{2} can be represented by a Hochshild chain α of the same homogeneity. Thus an element of E_{2} in degree k corresponds to a function on C^{∞}((T^{∗}X)\)^{k+1}) which is separately homogeneous in each variable and of total homogeneity k − n. Furthermore it has an extension β(t) as a function of the parameter h, of the same homogeneity, such that b_{t}β(t) = t^{2}γ(t). Then b_{(2)}α = [γ(0)], the class of γ(0) in E_{2}. Noting that the differential operator, P_{j}, which is the jth term in the Taylor series of the product ⋆_{h} reduces homogeneity by j and that b_{h} depends multilinearly on ⋆_{h} it follows tha b_{(r)} must decrease homogeneity by r. Thus if the class [γ(0)] must vanish in E_{2} by (8.64). We have therefore shown that b_{(2)} ≡ 0, so E_{3} = E_{2}. The same argument applies to the higher differentials, defining the E_{r} ≡ E_{2} for r ≥ 2, proving the ‘degeneration’ of the spectral sequence, (8.48).

The ‘convergence’ of the spectral sequence, (8.49), follows from the same analysis of homogneities. Thus, we shall define a map from E_{2} to the Hochschild homology and show that it is an isomorphism.

8.9. Explicit cohomology maps

8.10. Hochschild holomology of Ψ^{−∞}(X)

8.11. Hochschild holomology of Ψ^{ℤ}(X)

8.12. Morita equivalence