CYCLIC HOMOLOGY

B.TSYGAN

1. Introduction

Many geometric objects associated to a manifold M can be expressed in terms of an appropriate algebra A of functions on M (measurable, continuous, smooth, holomorphic, algebraic, ...). Very often those objects can be defined in a way that is applicable to any algebra A, commutative or not. Study of associative algebras by means of such objects of geometric origin is the subject of non-commutative geometry [C], [M1]. The Hochschild and cyclic (co)homology theory is the part of non-commutative geometry which generalizes the classical differential and integral calculus. The geometric objects being generalized to the non-commutative setting are differential forms, densities, multi-vector fields, etc.

In our exposition, the primary object is the negative cyclic complex CC_{•}^{−}(A). Other complexes, namely the Hochschild chain complex C_{•}(A), the periodic cyclic complex CC_{•}^{per}(A), and the cyclic complex CC_{•}(A), are defined as results of some natural procedure applied to CC_{•}(A). The cyclic homology is the homology of the cyclic complex CC_{•}(A). It was originally defined using another standard complex which we denote by C_{•}^{λ}(A). The study of this latter complex has a distinctly different flavor, mainly coming from the fact that it is related to the Lie algebra homology.

The above complexes are non-commutative versions of the space of differential forms (the ) and of the De Rham complex. One also defines the Hochschild cochain complex C^{•}(A, A) which is a non-commutative analogue of the space of multi-vector fields.

In the first three sections we study rather systematically various algebraic structures on the above complexes. These structures are supposed to generalize the classical algebraic structures arising in calculus. Sections 2 and 4.3 are devoted to the structures on the complex CC_{•}^{−}(A) and related complexes; section 3 deals with the cyclic complex C_{•}^{λ}(A).

These algebraic structures can be studied on three levels of complexity. One can start by defining basic algebaric operations generalizing the product and the bracket of multivector fields and various pairings between multivectors and forms; these are operations at the level of complexes. They acquire the standard prorerties from calculus only if one passes from complexes to their cohomology. When the algebra A is the algebra of functions on a manifold, then the homology of the Hochschild chain complex is the space of forms, the cohomology of the Hochschild cochain complex is the space of multivectors, etc. So, in a more naive sense, even at the first level of non-commutative calculus we already achieve our goal: to generalize the basic definitions and structures of calculus to the non-commutative setting (subsections 2.3, 2.4). But this generalization is not interesting when the algebra becomes non-commutative. For example, for the algebra of differential operators on ℝ^{n} both the spaces of “non-commutative forms” and “non-commutative multivectors” are one-dimensional.

Supported in part by NSF grant.

To make non-commutative calculus useful for applications, one has to pass to the second level of complexity. Namely, one tries to reproduce more of the standard algebraic structure from the classical calculus not at the level of cohomology but rather at the level of complexes. This plan succeeds partially, in the following sense: one can define certain algebraic structures on Hochschild chains and cochains. These structures are only part of what one could expect from the classical calculus. The advantage of these structures is that they are defined by explicit, canonical constructions (2.5, 3.5). They are also adequate for some applications, namely for the index theorems for symplectic deformations.

To construct a richer algebraic structure on Hochschild chains and cochains, one that generalizes a fuller version of the classical calculus, one has to go to a different level of complexity. The work in this direction was started by Tamarkin in [T] and then continued in [TT1], [TT2], [TT3]. We discuss these methods in Section 4.3. They provide a considerable refinement of the results of 2.5, 3.5, but there is no canonical and explicit construction anymore. The “non-comutative differential calculus” can be constructed using some inexplicit formulas; a choice of coefficients in these formulas depends on a choice of a Drinfeld associator [Dr]. In particular, the Grothendieck-Teichmüller group acts on the space of all such calculi. This version of non-commutative calculus allows one to generalize the index theorem form symplectic deformations to arbitrary deformations.

Note that in [C], [CM],[CM1],[CM2] a different, though clearly related, version of non-commutative calculus is used. In particular, there the renormalization group appears as a hidden group of symmetries of the calculus, whereas in our construction the Grothendieck-Teichmüller group acts on the space of universal formulas defining a calculus. This group is closely related to the Galois group Gal(\overline{ℚ}/ℚ). Note that finding a unified symmetry group incorporating both the renormalization group and the Galois group of ℚ is one of the important aims of Connes’ non-commutative geometry program. In light of this, it seems to be an interesting problem to find a unified framework to the two approaches to non-commutative calculus (the author would like to thank Alain Connes for his remarks on this subject).

In section 6, we study various examples of computations of the Hochschild and cyclic homology. The examples that we choose in this presentation revolve mostly around several related classes of algebras: functions on manifolds or on algebraic varieties; operators on functions; deformed algebras of functions; group algebras. Many interesting examples, in particular the ones related to algebraic topology, are not considered here. We compute the Hochschild homology and various versions of the cyclic homology of any formal deformation of the algebra of smooth functions on a C^{∞} manifold. This computation, as well as the classification of all such deformations, is based on the noncommutative differential calculus from Section 4.3. The constructions that we use are neither explicit nor canonical; they depend on a choice of a Drinfeld associator.

In Section 7, index theorems for formal deformations are discussed. An index theorem is an equality that compares two maps from the (periodic) cyclic complex of a deformed algebra of functions on a manifold to the De Rham complex of this manifold. One of these maps is the principal symbol; it is, essentially, reduction modulo deformation parameter, followed by the Connes isomorphism between the cyclic homology and the De Rham cohomology. The other is the trace density map. It is given by an explicit and canonical construction in the case when the quasi-classical limit of the deformation is symplectic; in the general case, its definition depends on a choice of an associator.

In Section 8 we outline an application of the above theorem (in the symplectic case) to a general index theorem. This theorem, conjectured in [SS] and proven in [BNT], contains many classical index theorems as partial cases, such as the Atiyah-Singer theorem for real analytic elliptic operators, the Boutet de Monvel index theorem for boundary value problems, the Kashiwara index theorem for holonomic D-modules, and others.

2. Hochschild and cyclic homology of algebras

Let k denote a commutative algebra over a field of characteristic zero and let A be a flat k-algebra with unit, not necessarily commutative. Let \overline{A} = A/k · 1, and let C_{p}(A) =^{def} A ⊗_{k} \overline{A}^{⊗_{k}p}. Define

b : C_{p}(A) ⟶ C_{p−1}(A)  (2.1)
a_{0} ⊗ ⋯ ⊗ a_{p} ↦ (−1)^{p}a_{p}a_{0} ⊗ ⋯ ⊗ a_{p−1} +

∑_{i=0}^{p−1}(−1)^{i}a_{0} ⊗ ⋯ ⊗ a_{i}a_{i+1} ⊗ ⋯ ⊗ a_{p} .

Then b^{2} = 0 and one gets the complex (C_{•}, b), called the standard Hochschild complex of A. The homology of this complex is denoted by H_{•}(A, A), or by HH_{•}(A).

Proposition 2.0.1. The map

B : C_{p}(A) ⟶ C_{p+1}(A)  (2.2)
a_{0} ⊗ ⋯ ⊗ a_{p} ↦ ∑_{i=0}^{p}(−1)^{pi}1 ⊗ a_{i} ⊗ ⋯ ⊗ a_{p} ⊗ a_{0} ⊗ ⋯ ⊗ a_{i−1}

satisfies B^{2} = 0 and bB + Bb = 0 and therefore defines a map of complexes

B : C_{•}(A) ⟶C_{•}(A)[−1]

Definition 2.0.2. For i, j, p ∈ ℤ let

CC_{p}^{−}(A) = ∏_{i=p mod 2^{i≥p}} C_{i}(A)

CC_{p}^{per}(A) = ∏_{i=p mod 2} C_{i}(A)

CC_{p}(A) = ⨁_{i=p mod 2^{i≤p}} C_{i}(A)

The complex (CC_{•}^{−}(A), B + b) (respectively (CC_{•}^{per}(A), B + b), respectively (CC_{•}(A), B + b)) is called the negative cyclic (respectively periodic cyclic, respectively cyclic) complex of A. The homology of these complexes is denoted by HC_{•}^{−}(A), respectively by HC_{•}^{per}(A), respectively by HC_{•}(A).

There are inclusions of complexes

CC_{•}^{−}(A)[−2] ↪ CC_{•}^{−}(A) ↪ CC_{•}^{per}(A)  (2.3)

and the short exact sequences

0 ⟶CC_{•}^{−}(A)[−2] ⟶CC_{•}^{−}(A) ⟶C_{•}(A) ⟶0  (2.4)
0 ⟶C_{•}(A) ⟶CC_{•}(A) ⟶^{S} CC_{•}(A)[2] ⟶0  (2.5)

The operator S is called the Bott periodicity operator. To the double complex CC_{•}(A) one associates the spectral sequence

E_{pq}^{2} = H_{p−q}(A, A)  (2.6)

converging to HC_{p+q}(A).

In what follows we will use the notation of Getzler and Jones ([GJ]). Let u denote a variable of degree −2. Then the negative and periodic cyclic complexes are described by the following formulas:

CC_{•}^{−}(A) = (C_{•}(A)[[u]], b + uB)  (2.7)
CC_{•}^{per}(A) = (C_{•}(A)[[u, u^{−1}], b + uB)  (2.8)
CC_{•}(A) = (C_{•}(A)[[u, u^{−1}]/uC_{•}(A)[[u]], b + uB)  (2.9)

In this language, the Bott periodicity map S is just multiplication by u.

Remark 2.0.3. For an algebra A without unit, let Ã = A + k · 1 and put

CC_{•}(A) = Ker(CC_{•}(Ã) ⟶CC_{•}(k));

similarly for the negative and periodic cyclic complexes. If A is a unital algebra then these complexes are quasi-isomorphic to the ones defined above.

2.1. Homology of differential graded algebras. One can easily generalize all the above constructions to the case when A is a differential graded algebra (DGA) with the differential δ (i.e A is a graded algebra and δ is a derivation of degree 1 such that δ^{2} = 0).

The action of δ extends to an action on Hochschild chains by the Leibnitz rule:

δ(a_{0} ⊗ ⋯ ⊗ a_{p}) = ∑_{i=1}^{p} (−1)^{∑^{k<i} (|a_{k}|+1)+1}(a_{0} ⊗ ⋯ ⊗ δa_{i} ⊗ ⋯ ⊗ a_{p})

The maps b, B, τ are modified to include signs:

b(a_{0} ⊗ … ⊗ a_{p}) = ∑_{k=0}^{p−1}(−1)^{∑{k \atop i=0} (|a_{i}|+1)+1}a_{0} … a_{k}a_{k+1} ⊗ … a_{p}(2.10)

+(−1)^{|a_{p}|+(|a_{p}|+1) ∑_{i=0}^{p−1}(|a_{i}|+1)}a_{p}a_{0} ⊗ … ⊗ a_{p−1}

B(a_{0} ⊗ … ⊗ a_{p}) = ∑_{k=0}^{p}(−1)^{∑_{i≤k}(|a_{i}|+1) ∑_{i≥k}(|a_{i}|+1)}1 ⊗ a_{k+1} ⊗ … a_{p} ⊗ (2.11)

a_{0} ⊗ … ⊗ a_{k}

The complex C_{•}(A) now becomes the total complex of the double complex with the differential b+δ. The negative and the periodic cyclic complexes are defined as before in terms of the new definition of C_{•}(A). All the results of this section extend to the differential graded case.

2.2. The Hochschild cochain complex. Let A be a graded algebra with unit over a commutative unital ring k of characteristic zero. A Hochschild d-cochain is a linear map A^{⊗d} → M. Put, for d ≥ 0,

C^{d}(A) = C^{d}(A, A) = Hom_{k}(\overline{A}^{⊗d}, M)

where \overline{A} = A/k · 1. Put

|D| = (degree of the linear map D) + d

Put for cochains D and E from C^{•}(A, A)

(D ⌣ E)(a_{1}, …, a_{d+e}) = (−1)^{|E| ∑_{i≤d}(|a_{i}|+1)}D(a_{1}, …, a_{d})×

×E(a_{d+1}, …, a_{d+e});

(D ∘ E)(a_{1}, …, a_{d+e−1}) = ∑_{j≥0}(−1)^{(|E|+1) ∑_{i=1}^{j}(|a_{i}|+1)}×

×D(a_{1}, …, a_{j}, E(a_{j+1}, …, a_{j+e}), …);
[D, E] = D ∘ E − (−1)^{(|D|+1)(|E|+1)}E ∘ D

These operations define the graded associative algebra (C^{•}(A, A) , ⌣) and the graded Lie algebra (C^{•+1}(A, A), [ , ]) (cf. [CE]; [G]). Let

m(a_{1}, a_{2}) = (−1)^{deg a_{1}} a_{1}a_{2};

this is a 2-cochain of A (not in C^{2}). Put

δD = [m, D];
(δD)(a_{1}, …, a_{d+1}) = (−1)^{|a_{1}||D|+|D|+1}×
×a_{1}D(a_{2}, …, a_{d+1})+

+ ∑_{j=1}^{d}(−1)^{|D|+1+∑_{i=1}^{j}(|a_{i}|+1)}D(a_{1}, …, a_{j}a_{j+1}, …, a_{d+1})

+(−1)^{|D| ∑_{i=1}^{d}(|a_{i}|+1)}D(a_{1}, …, a_{d})a_{d+1}

One has

δ^{2} = 0; δ(D ⌣ E) = δD ⌣ E + (−1)^{|degD|}D ⌣ δE
δ[D, E] = [δD, E] + (−1)^{|D|+1} [D, δE]

(δ^{2} = 0 follows from [m, m] = 0).

Thus C^{•}(A, A) becomes a complex; we will denote it also by C^{•}(A). The cohomology of this complex is H^{•}(A, A) or the Hochschild cohomology. We denote it also by H^{•}(A). The ⌣ product induces the Yoneda product on H^{•}(A, A) = Ext_{A⊗A^{0}}^{•}(A, A). The operation [ , ] is the Gerstenhaber bracket [G].

If (A, ∂) is a differential graded algebra then one can define the differential ∂ acting on C^{•}(A) by:

∂D = [∂, D]

Theorem 2.2.1. [G] The cup product and the Gerstenhaber bracket induce a Gerstenhaber algebra structure on H^{•}(A).

For cochains D and D_{i} define a new Hochschild cochain by the following formula of Gerstenhaber ([G]) and Getzler ([Ge1]):

D_{0}{D_{1}, …, D_{m}}(a_{1}, …, a_{n}) =
= ∑(−1)^{∑_{k≤ip}(|a_{k}|+1)(|D_{p}|+1)}D_{0}(a_{1}, …, a_{i_{1}}, D_{1}(a_{i_{1}+1}, …), …,
D_{m}(a_{i_{m}+1}, …), …)

Proposition 2.2.2. One has

(D{E_{1}, …, E_{k}}){F_{1}, …, F_{l}} = ∑(−1)^{∑_{q≤ip}(|E_{p}|+1)(|F_{q}|+1)}×

×D{F_{1}, …, E_{1}{F_{i_{1}+1}, …, }, …, E_{k}{F_{i_{k}+1}, …, }, …, }

The above proposition can be restated as follows. For a cochain D let D^{(k)} be the following k-cochain of C^{•}(A):

D^{(k)}(D_{1}, …, D_{k}) = D{D_{1}, …, D_{k}}

Proposition 2.2.3. The map

D ↦ ∑_{k≥0} D^{(k)}

is a morphism of differential graded algebras

C^{•}(A) → C^{•}(C^{•}(A))

2.3. Products on Hochschild and cyclic complexes.

2.3.1. Product and coproduct; the Künneth exact sequence. For an algebra A define the shuffle product

sh : C_{p}(A) ⊗ C_{q}(A) ⟶C_{p+q}(A)  (2.12)

as follows.

(a_{0} ⊗ … ⊗ a_{p}) ⊗ (c_{0} ⊗ … ⊗ c_{q}) = a_{0}c_{0} ⊗ sh_{pq}(a_{1}, …, a_{p}, c_{1}, …, c_{q}) (2.13)

where

sh_{pq}(x_{1}, …, x_{p+q}) = ∑_{σ∈Sh(p,q)} sgn(σ)x_{σ^{−1}1} ⊗ … ⊗ x_{σ^{−1}(p+q)} (2.14)

and

Sh(p, q) = {σ ∈ Σ_{p+q} |σ1 < … < σp; σ(p + 1) < … < σ(p + q)}

In the graded case, sgn(σ) gets replaced by the sign computed by the following rule: in all transpositions, the parity of a_{i} is equal to |a_{i}|+1 if i ≠ 0 and |a_{0}| if i = 0, and similarly for c_{i}. A transposition contributes a product of parities.

The shuffle product is not a morphism of complexes unless A is commutative. It defines, however, an exterior product as shown in the following theorem. For two unital algebras A and C, let i_{A}, i_{C} be the embeddings a ↦ a ⊗ 1, resp. c ↦ 1 ⊗ c of A, resp. C, to A ⊗ C. We will use the same notation for the embeddings that i_{A}, i_{C} induce on all the chain complexes considered by us.

Theorem 2.3.1. For two unital algebras A and C the composition

C_{p}(A) ⊗ C_{q}(C) ⟶^{i_{A}⊗i_{C}} C_{p}(A ⊗ C) ⊗ C_{q}(A ⊗ C) ⟶^{sh} C_{p+q}(A ⊗ C)

defines a quasi-isomorphism

sh : C_{•}(A) ⊗ C_{•}(C) ⟶C_{•}(A ⊗ C)

To extend this theorem to cyclic complexes, define

sh′ : C_{p}(A) ⊗ C_{q}(A) ⟶C_{p+q+2}(A)  (2.15)

as follows.

(a_{0} ⊗ … ⊗ a_{p}) ⊗ (c_{0} ⊗ … ⊗ c_{q}) ↦ 1 ⊗ sh′_{p+1, q+1}(a_{0}, …, a_{p}, c_{0}, …, c_{q}) (2.16)

where

sh′_{p+1,q+1}(x_{0}, …, x_{p+q+1}) = ∑_{σ∈Sh′(p+1,q+1)} sgn(σ)x_{σ^{−1}0} ⊗ … ⊗ x_{σ^{−1}(p+q+1)} (2.17)

and Sh′(p + 1, q + 1) is the set of all permutations σ ∈ Σ_{p+q+2} such that σ0 < … < σp, σ(p + 1) < … < σ(p + q + 1), and σ0 < σ(p + 1)

sh′ : C_{•}(A) ⊗ C_{•}(C) ⟶C_{•+2}(A ⊗ C)

Now define (2.15) to be the composition

C_{p}(A) ⊗ C_{q}(C) ⟶^{i_{A}⊗i_{C}} C_{p}(A ⊗ C) ⊗ C_{q}(A ⊗ C) ⟶^{sh′} C_{p+q+2}(A ⊗ C)

In the graded case, the sign rule is as follows: any a_{i} has parity |a_{i}| + 1, and similarly for c_{i}.

Theorem 2.3.2. The map sh+ush′ defines a k[[u]]-linear, (u)-adically continuous quasi-isomorphism

(C_{•}(A) ⊗ C_{•}(C))[[u]] ⟶CC_{•}^{−}(A ⊗ C)

as well as

(C_{•}(A) ⊗ C_{•}(C))[[u^{−1}, u]] ⟶CC_{•}^{per}(A ⊗ C)

(C_{•}(A) ⊗ C_{•}(C))[[u^{−1}, u]]/u(C_{•}(A) ⊗ C_{•}(C))[[u]] ⟶CC_{•}(A ⊗ C) (the differentials on the left hand sides are equal to b⊗1+1⊗b+u(B ⊗ 1 + 1 ⊗ B).

Note that the left hand side of the last formula maps to the tensor product CC_{•}(A) ⊗ CC_{•}(C) : Δ(u^{−p}c ⊗ c′) = (u^{−1} ⊗ 1 + 1 ⊗ u^{−1})^{p}c ⊗ c′. One checks that this map is an embedding whose cokernel is the kernel of the map u⊗1−1⊗u, or S ⊗1−1⊗S where S is the Bott periodicity operator. ¿From this we get

Theorem 2.3.3. There is a long exact sequence

⟶HC_{n}(A ⊗ C) ⟶^{Δ} ⨁_{p+q=n} HC_{p}(A) ⊗ HC_{q}(C) ⟶^{S⊗1−1⊗S}

⨁_{p+q=n−2} HC_{p}(A) ⊗ HC_{q}(C) ⟶^{×} HC_{n−1}(A ⊗ C) ⟶^{Δ}

2.4. Pairings between chains and cochains. For a graded algebra A, for D ∈ C^{d}(A, A), define

i_{D}(a_{0} ⊗ … ⊗ a_{n}) = (−1)^{|D| ∑_{i≤d}(|a_{i}|+1)}a_{0}D(a_{1}, …, a_{d}) ⊗ a_{d+1} ⊗ … ⊗ a_{n} (2.18)

Proposition 2.4.1.

[b, i_{D}] = i_{δD}
i_{D}i_{E} = (−1)^{|D||E|}i_{E⌣D}

Now, put

L_{D}(a_{0} ⊗ … ⊗ a_{n}) = ∑_{k=1}^{n−d} ϵ_{k}a_{0} ⊗ … ⊗ D(a_{k+1}, …, a_{k+d}) ⊗ … ⊗ a_{n} + (2.19)

∑_{k=n+1−d}^{n} η_{k}D(a_{k+1}, …, a_{n}, a_{0}, …) ⊗ … ⊗ a_{k}

(The second sum in the above formula is taken over all cyclic permutations such that a_{0} is inside D). The signs are given by

ϵ_{k} = (|D| + 1) ∑_{i=0}^{k}(|a_{i}| + 1)

and

η_{k} = |D| + 1 + ∑_{i≤k}(|a_{i}| + 1) ∑_{i≥k}(|a_{i}| + 1)

Proposition 2.4.2.

[L_{D}, L_{E}] = L_{[D,E]}
[b, L_{D}] + L_{δD} = 0
[L_{D}, B] = 0

Now let us extend the above operations to the cyclic complex. Define

S_{D}(a_{0} ⊗ … ⊗ a_{n}) = ∑_{j≥0; k≥j+d} ϵ_{jk}1 ⊗ a_{k+1} ⊗ … a_{0} ⊗ … ⊗ (2.20)

D(a_{j+1}, …, a_{j+d}) ⊗ … ⊗ a_{k}

(The sum is taken over all cyclic permutations for which a_{0} appears to the left of D). The signs are as follows:

ϵ_{jk} = |D|(|a_{0}|+∑_{i=1}^{n}(|a_{i}|+1))+(|D|+1) ∑_{j+1}^{k}(|a_{i}|+1)+∑_{i≤k}(|a_{i}|+1) ∑_{i≥k}(|a_{i}|+1)

As we will see later, all the above operations are partial cases of a unified algebraic structure for chains and cochains; the sign rule for this unified construction will be explained in 2.5.

Proposition 2.4.3. ([R])

[b + uB, i_{D} + uS_{D}] − i_{δD} − uS_{δD} = L_{D}

Proposition 2.4.4. ([DGT]) There exists a linear transformation T (D, E) of the Hochschild chain complex, bilinear in D, E ∈ C^{•}(A, A), such that

[b + uB, T (D, E)] − T (δD, E) − (−1)^{|D|}T (D, δE) =

= [L_{D}, i_{E} + uS_{E}] − (−1)^{|D|+1}(i_{[D,E]} + uS_{[D,E]})

2.5. A_{∞} structure on C_{•}(C^{•}(A)). All the pairings described above are in fact different parts of the same algebraic structure that we are going to describe next.

Recall that an A_{∞} algebra is a graded vector space C together with a Hochschild cochain m of total degree 1,

m = ∑_{n=1}^{∞} m_{n}

where m_{n} ∈ C^{n}(C) and

[m, m] = 0

To define A_{∞} modules over A_{∞} algebras, note that for a graded space M, the Gerstenhaber bracket [ , ] can be extended to the space

Hom(C^{⊗•}, C) ⊕ Hom(M ⊗ C^{⊗•}, M)

For a graded space M, a structure of an A_{∞} module over an A_{∞} algebra C on M is a cochain

μ = ∑_{n=1}^{∞} μ_{n}

μ_{n} ∈ Hom(M ⊗ C^{⊗n−1}, M)

such that

[m + μ, m + μ] = 0

Consider the Hochschild cochain complex of a graded algebra A as a differential graded associative algebra (C^{•}(A), ⌣, δ). Consider the Hochschild chain complex of this differential graded algebra. The total differential in this complex is b + δ; the degree of a chain is given by

|D_{0} ⊗ … ⊗ D_{n}| = |D_{0}| + ∑_{i=1}^{n}(|D_{i}| + 1)

where D_{i} are Hochschild cochains.

The complex C_{•}(C^{•}(A)) contains the Hochschild cochain complex C^{•}(A) as a subcomplex (of zero-chains) and has the Hochschild chain complex C_{•}(A) as a quotient complex:

C^{•}(A) ⟶^{i} C_{•}(C^{•}(A)) ⟶^{π} C_{•}(A)

(this sequence is not by any means exact). The projection on the right splits if A is commutative. If not, C_{•}(A) is naturally a graded subspace but not a subcomplex.

Theorem 2.5.1. There is an A_{∞} structure on C_{•}(C^{•}(A))[[u]] such that:

• All m_{n} are k[[u]]-linear, (u)-adically continuous

• m_{1} = b + δ + uB For x, y ∈ C_{•}(A):

• (−1)^{|x|}m_{2}(x, y) = (sh +u sh′)(x, y)

For D, E ∈ C^{•}(A):

• (−1)^{|D|}m_{2}(D, E) = D ⌣ E

• m_{2}(1 ⊗ D, 1 ⊗ E) + (−1)^{|D||E|}m_{2}(1 ⊗ E, 1 ⊗ D) = (−1)^{|D|}1 ⊗ [D, E]

• m_{2}(D, 1 ⊗ E) + (−1)^{(|D|+1)|E|}m_{2}(1 ⊗ E, D) = (−1)^{|D|+1}[D, E]

Theorem 2.5.2. On C_{•}(A)[[u]], there exists a structure of an A_{∞} module over the A_{∞} algebra C_{•}(C^{•}(A))[[u]] such that:

• All μ_{n} are k[[u]]-linear, (u)-adically continuous

• μ_{1} = b + uB on C_{•}(A)[[u]] For a ∈ C_{•}(A)[[u]]:

• μ_{2}(a, D) = (−1)^{|a||D|+|a|}(i_{D} + uS_{D})a

• μ_{2}(a, 1 ⊗ D) = (−1)^{|a||D|}L_{D}a For a, x ∈ C_{•}(A)[[u]]: (−1)^{|a|}μ_{2}(a, x) = (sh +u sh′)(a, x)

Here is an explicit description of the above A_{∞} structures. We define for n ≥ 2

m_{n} = m_{n}^{(1)} + m_{n}^{(2)} + um_{n}^{(3)}

where, for

a^{(k)} = D_{0}^{(k)} ⊗ … ⊗ D_{N_{k}}^{(k)},
m_{n}^{(1)} = 0

for n ≥ 3;

m_{2}^{(1)}(a^{(1)}, a^{(2)}) = (−1)^{|a^{(1)}|} ∑ ±D_{0}^{(1)} ⌣ D_{0}^{(2)}\underline{\{ \}} ⊗ \underline{…}⊗
⊗D_{1}^{(2)}\underline{\{ \}} ⊗ \underline{…} ⊗ D_{N_{2}}^{(2)}\underline{\{ \}} ⊗ \underline{…}

The space designated by is filled with D_{1}^{(1)}, …, D_{N_{1}}^{(1)} whose order is preserved. The sign rule is as follows: the parity of D_{j}^{(i)} is |D_{j}^{(i)}| for j = 0 and |D_{j}^{(i)}| + 1 otherwise.

m_{n}^{(2)}(a^{(1)}, …, a^{(n)}) =

(−1)^{∑_{i=1}^{n−1} |a_{1}|+n} ∑ ±D_{N_{n}}^{(n)}\{\underline{…}, D_{0}^{(1)}, \underline{…}, D_{0}^{(n−1)}, \underline{…}\} ⌣ D_{0}^{(n)}

⊗\underline{…} ⊗ D_{1}^{(n)}\underline{\{ \}} ⊗ \underline{…} ⊗ D_{N_{n}−1}^{(n)}\underline{\{ \}} ⊗ \underline{…}

The space designated by is filled with D_{i}^{(j)} for j < n in such a way that:

• the cyclic order of each group D_{0}^{(k)}, …, D_{N_{k}}^{(k)} is preserved

• D_{0}^{(1)}, …, D_{0}^{(n−1)} are all inside the braces in D_{N_{n}}^{(n)}{ }

• D_{0}^{(i)} is to the left of D_{0}^{(j)} for i < j

• any cochain D_{j}^{(i)} may contain some of its neighbors on the right inside the braces, provided that all of these neighbors are of the form D_{q}^{(p)} with p < i

The parity of D_{j}^{(i)} is |D_{j}^{(i)}| if i = n and j = 0; it is |D_{j}^{(i)}| + 1 otherwise.

m_{n}^{(3)}(a^{(1)}, …, a^{(n)}) = (−1)^{n+1} ∑ ±1 ⊗ \underline{…} ⊗ D_{0}^{(0)} ⊗ \underline{…} ⊗ D_{0}^{(n)} ⊗ \underline{…}

The space designated by is filled with D_{i}^{(j)} for in such a way that: • the cyclic order of each group D_{0}^{(k)}, …, D_{N_{k}}^{(k)} is preserved

• D_{0}^{(i)} is to the left of D_{0}^{(j)} for i < j

• any cochain D_{j}^{(i)} may contain some of its neighbors on the right inside the braces, provided that all of these neighbors are of the form D_{q}^{(p)} with p < i. The parity of D_{j}^{(i)} is always |D_{j}^{(i)}| + 1.

The above formulas provide an A_{∞} structure from Theorem 2.5.1. To obtain a structure of an A_{∞} module from Theorem 2.5.2, one has to assume that all D_{j}^{(1)} are elements of A and to replace braces { } by the usual parentheses ( ) symbolizing evaluation of a multi-linear map at elements of A. Remark 2.5.3. Let A be a commutative algebra. Then C_{•}(A)[[u]] is not only a subcomplex but an A_{∞} subalgebra of C_{•}(C^{•}(A))[[u]]. This A_{∞} structure on C_{•}(A)[[u]] was introduced in [GJ].

The corresponding binary product was defined by Hood and Jones [HJ]. One can define it for any algebra A, commutative or not. If A is not commutative then this product is not compatible with the differential. Nevertheless, we will use it in 3.5.

3. The cyclic complex C_{•}^{λ}

3.1. Definition. Recall the original definition of the cyclic complex from [C1], [Ts1]. Let τ = τ_{p} denote the endomorphism of A^{⊗_{k}(p+1)} given by the formula

τ(a_{0} ⊗ ⋯ ⊗ a_{p}) = (−1)^{p}a_{p} ⊗ a_{0} ⋯ ⊗ a_{p−1}

Let

C_{p}^{λ}(A) = A^{⊗_{k}(p+1)}/Im(id −τ) .  (3.1)

Define

b′ : C_{p}(A) ⟶ C_{p−1}(A)
a_{0} ⊗ ⋯ ⊗ a_{p} ↦ ∑_{i=0}^{p−1}(−1)^{i}a_{0} ⊗ ⋯ ⊗ a_{i}a_{i+1} ⊗ ⋯ ⊗ a_{p}

and

N : C_{p}(A) ⟶ C_{p−1}(A)
N = id +τ + … τ^{p−1}

One has

b(id −τ) = (id −τ)b′; b′N = Nb.  (3.2)

Therefore the differential b descends to a map

b : C_{p}^{λ}(A) ⟶C_{p−1}^{λ}(A) .

The complex C_{•}^{λ}(A) is quasi-isomorphic to CC_{•}(A). (Sketch of the proof: in the cyclic double complex CC_{•}(A) one can replace C_{•}(A) by the cone of id −τ and B by N. One gets a double complex which is quasi-isomorphic to CC_{•} and which is in turn quasi-isomorphic to C_{•}^{λ}. Cf. [L] for more detail).

Lemma 3.1.1. For algebras without unit, the cyclic homology defined in Remark 2.0.3 is isomorphic to the homology of the complex C_{•}^{λ}(A).

3.2. The reduced cyclic complex. Let \overline{A} = A/k and let

\overline{C}_{p}^{λ}(A) = \overline{A}^{⊗p+1}/Im(id −τ) .  (3.3)

It is easy to see that the diferential b descends to \overline{C}_{•}^{λ}(A). We denote the homology of the complex \overline{C}_{•}^{λ}(A) by \overline{HC}_{•}(A).

Proposition 3.2.1. There is an exact triangle

C_{•}^{λ}(k) ⟶C_{•}^{λ}(A) ⟶\overline{C}_{•}^{λ}(A) ⟶C_{•}^{λ}(k)[1] .  (3.4)

Cf. [L] for the proof. (Note that the above proposition could be deduced from the Hochschild-Serre spectral sequence

E_{pq}^{2} = H_{p}(𝔤𝔩(A), 𝔤𝔩(k)) ⊗ H_{q}(𝔤𝔩(k))

converging to H_{p+q}(𝔤𝔩(A)) and from theorem 3.3.1).

3.3. Relation to Lie algebra homology. For an algebra (A, δ) over k let 𝔤𝔩(A) = \underrightarrow{lim}_{n}𝔤𝔩_{n}(A) denote the Lie algebra of finite matrices. Note

that 𝔤𝔩(k) is a DGL-subalgebra of 𝔤𝔩(A). Theorem 3.3.1 below identifies their respective subcomplexes of primitive elements of the DG coalgebras C_{•}(𝔤𝔩(A))_{𝔤𝔩(k)} and C_{•}(𝔤𝔩(A), 𝔤𝔩(k)).

Let E_{pq}^{a} denote the elementary matrix with (E_{pq}^{a})_{pq} = a and other entries equal to zero.

Theorem 3.3.1. The map

A^{⊗p+1} ⟶ ⋀^{p+1}𝔤𝔩(A)
a_{0} ⊗ ⋯ ⊗ a_{p} ↦ E_{01}^{a_{0}} ∧ E_{12}^{a_{1}} ∧ ⋯ ∧ E_{p−1,0}^{a_{p}}

induces isomorphisms of complexes

C_{•}^{λ}(A) ⟶ Prim C_{•}(𝔤𝔩(A))_{𝔤𝔩(k)}[1]
\overline{C}_{•}^{λ}(A) ⟶ Prim C_{•}(𝔤𝔩(A), 𝔤𝔩(k))[1]

The proof is contained in [Ts], [LQ], [L]

3.4. The connecting morphism. Here we give an explicit formula for the connecting morphism ∂ of the exact triangle from Proposition 3.2.1. For a DGA (A, δ) we denote by T (A) the tensor algebra of A and by j a k-linear map A ⟶k such that j(1) = 1.

Let ρ : T (A) ⟶k denote the map defined by

ρ(a) = j(δa) for a ∈ A,
ρ(a_{1} ⊗ a_{2}) = j(a_{1})j(a_{2}) − j(a_{1}a_{2}) for a_{1}, a_{2} ∈ A, and
ρ = 0 on A^{⊗p} for p ≠ 1, 2.

For a_{i} ∈ A, a_{0} ⊗ ⋯ ⊗ a_{p} ∈ A^{⊗p+1} with (∑_{i}(deg a_{i} + 1)) − 1 = 2n + 1 let

br_{2n+1}(a_{0} ⊗ ⋯ ⊗ a_{p}) =

\frac{1}{n!} ∑_{i=0}^{p}(−1)^{∑_{k<i}(deg a_{k}+1) ∑_{k≥i}(deg a_{k}+1)}(ρ⊗⋯⊗ρ)(a_{i}⊗⋯ a_{0}⊗⋯⊗a_{i−1}) . (3.5)

This defines a map br_{2n+1} : \overline{C}_{2n+1}^{λ}(A) ⟶ k (cf. [Br]). Let br_{2n} = 0 on \overline{C}_{2n}^{λ}(A).

Let 1^{(n+1)} = n!(n + 1)! · 1^{⊗2n+1} ∈ C_{2n}^{λ}(k). Let Br_{2n+1}^{A} = br_{2n+1} ·1^{(n)}, Br_{2n}^{A} = 0, Br^{A} = ∑_{p} Br_{p}.

One gets the map of graded spaces

Br^{A} : \overline{C}_{•}^{λ}(A) ⟶C_{•}^{λ}(k)[1]

which is, in fact, a map of complexes as follows from the alternative description given below.

The splitting j induces the splitting ρρρ : 𝔤𝔩(A) ⟶𝔤𝔩(k) of the inclusion

𝔤𝔩(k) ↪ 𝔤𝔩(A). Let P_{n} denote the invariant polynomial X ↦ \frac{1}{n!}tr(X^{n}) on 𝔤𝔩(k). Put

R(ρρρ) = (∂^{Lie} + δ)R(ρρρ) + \frac{1}{2}[R(ρρρ), R(ρρρ)]  (3.6)

Lemma 3.4.1. The restriction of the pairing with P_{n}(R(ρρρ)) to the subspace of all primitive elements (see Theorem 3.3.1) coincides with the map br_{2n+1}.

The proof is straightforward.

Corollary 3.4.2. The map Br^{A} is a morphism of complexes.

Proposition 3.4.3. The morphism Br^{A} : \overline{C}_{•}^{λ}(A) ⟶ C_{•}^{λ}(k)[1] represents the connecting morphism in the triangle (3.4).

3.4.1. Explicit formula for the product HC_{p} ⊗ HC_{q} → HC_{p+q+1}. Here we give an explicit formula for the product × from Theorem 2.3.3 in one of the realizations of the cyclic complex. Note first that, because of (3.2), the map N induces an isomorphism

C^{λ}(A) ≃ (Ker(id −τ), b′)  (3.7)

where in the right hand side id −τ is considered as an operator on A^{⊗(•+1)}.

Proposition 3.4.4. If one identifies C_{•}^{λ} with the right hand side of (3.7), then the shuffle product

(a_{0} ⊗ … ⊗ a_{p}) × (c_{0} ⊗ … ⊗ c_{q}) = sh_{p+1,q+1}((a_{0}, … ⊗ a_{p}, c_{0}, …, c_{q})

is a morphism of complexes

× : C_{•}^{λ}(A) ⊗ C_{•}^{λ}(C) ⟶C_{•+1}^{λ}(A ⊗ C)  (3.8)

which induces on homology the × product from Theorem 2.3.3.

By the same construction one defines the product

× : \overline{C}_{•}^{λ}(A) ⊗ \overline{C}_{•}^{λ}(C) ⟶\overline{C}_{•}^{λ}(A ⊗ C)

on the reduced cyclic homology. Cf. [L] for the explicit formula for the × products at the level of complexes CC_{•}.

3.5. Other operations on the cyclic complexes. Here we recall, in a modified version, some results from [NT2].

Let 𝔤_{A}^{•}[ϵ] = 𝔤_{A}^{•} + 𝔤_{A}^{•}ϵ, ϵ^{2} = 0, degϵ = 1. Let u be a formal parameter of degree 2 and consider the graded algebra U(𝔤_{A}^{•}[ϵ])[u] with the differential u · \frac{∂}{∂ϵ} + δ (recall that δ is the differential in 𝔤_{A}^{•}).

The motivation for the theorem below is the following. If A = C^{∞}(M) then HC_{•}^{−}(A) is isomorphic to the cohomology of the complex Ω^{•}(M)[[u]], ud; the cohomology of 𝔤_{A}^{•} is the graded Lie algebra 𝔤_{M}^{•} of multivector fields on M; one can define an action of 𝔤_{M}^{•}[ϵ][u] on HC_{•}^{−}(A): for two multivector fields X, Y the action of X + ϵY is given by L_{X} + ι_{Y} where ι_{Y} is the contraction operator and L_{X} = [d, ι_{X}].

We would like to have a non-commutative analog of the above action. In fact, because of Theorem 4.4.1, we know that there is an L_{∞} action of 𝔤_{A}^{•}[ϵ][u] on CC_{•}^{−}(A), i.e., a DGLA which is quasi-isomorphic to the former acts on a complex quasi-isomorphic to the latter. However, the proof of Theorem 4.4.1 is inexplicit, and we need explicit operations for index-theoretical applications. Below we provide an explicit formula for an action of the complex U(𝔤_{A}^{•}[ϵ])[u] on CC_{•}^{−}(A).

Denote by E_{A}^{•} the DGA C^{•}(A) with the cup product and the differential δ. Let ⋆ be the Hood - Jones product on CC_{•}^{−}(E_{A}^{•}) (Remark 2.5.3). In other words,

a ⋆ b = (−1)^{|a|}(m_{2}^{(1)}(a, b) + um_{2}^{(3)}(a, b))

where m_{i}^{(j)} are defined in 2.5 and applied here to the subspace CC_{•}^{−}(C^{0}(E)) of CC_{•}^{−}(C^{•}(E))

Theorem 3.5.1. Formula

(ϵD_{1} · · · · · ϵD_{m}) • α = I(D_{1}, …, D_{m})α =

(−1)^{deg α ∑_{i=1}^{m} degD_{i}} \frac{1}{m!}∑_{σ}ϵ_{σ}α • (D_{σ_{1}} ⋆ (D_{σ_{2}} ⋆ (… ⋆ D_{σ_{m}})) …)

defines morphisms of complexes of k[u]-modules

U(𝔤_{A}^{•}[ϵ])[u] ⊗_{U(𝔤_{A}^{•})[u]} CC_{•}^{−}(A) ⟶ CC_{•}^{−}(A)
U(𝔤_{A}^{•}[ϵ])[u, u^{−1}] ⊗_{U(𝔤_{A}^{•})[u,u^{−1}]} CC_{•}^{per}(A) ⟶ CC_{•}^{per}(A)

Let u be a formal parameter of degree two. Consider the differential graded algebra A[η] = A + Aη, degη = −1, η^{2} = 0 with the differential \frac{∂}{∂η}. Consider the complex \overline{C}_{•}^{λ}(A[η])[u] with the differential \frac{∂}{∂η} + u · b.

Theorem 3.5.2. There exist natural pairings of k[[u]]-modules

• : \overline{C}_{•−1}^{λ}(A[η])[[u]] ⊗ CC_{−•}^{−}(A) ⟶CC_{−•}^{−}(A)

• : \overline{C}_{•−1}^{λ}(A[η])[[u, u^{−1}] ⊗ CC_{−•}^{per}(A) ⟶CC_{−•}^{per}(A)

such that:

(1) η^{⊗m}•? = \frac{1}{(m−1)!}Id for m > 0.

(2) For x_{i} ∈ A the operation (x_{1} ⊗ ⋯ ⊗ x_{p})•? sends C_{N}(A) to ∑_{i,j≥0} C_{N−p+i}(A)u^{j}.

(3) The component of (x_{1} ⊗ ⋯ ⊗ x_{p}) • (a_{0} ⊗ ⋯ ⊗ a_{N}) in C_{N−p}[[u]] is equal to

\frac{1}{p!} ∑_{i=1}^{p}(−1)^{i(p−1)}a_{0}[x_{i+1}, a_{1}][x_{i+2}, a_{2}] ⋯ [x_{i}, a_{p}] ⊗ a_{p+1} ⊗ ⋯ ⊗ a_{N}

(4) (x_{1} ⊗ ⋯ ⊗ x_{p}) • 1 = ∑_{i=1}^{p}(−1)^{i(p−1)}1 ⊗ x_{i+1} ⊗ x_{i+2} ⊗ ⋯ ⊗ x_{i}.

Note that the formula (4) above defines the map from \overline{C}_{•}^{λ}(A) to Ker(B : C_{•}(A) → C_{•+1}(A)) (one can show that this map is a quasiisomorphism). Clearly, the kernel above embeds into CC_{•}^{−}(A). The above theorem shows that this embedding extends to a pairing •.

The complex \overline{C}_{•−1}^{λ}(A[η])[[u]] is very simple at the level of homology; it is quasi-isomorphic to k[[u]]. Therefore the pairing • does not define any new homological operations. It is, however, very important at the level of chains, as one sees in [BNT].

3.6. Rigidity of periodic cyclic homology. Let A be an associative algebra over a ring k of characteristic zero. Let I be a nilpotent twosided ideal of A. Denote A_{0} = A/I.

Theorem 3.6.1. (Goodwillie) The natural map CC_{•}^{per}(A) → CC_{•}^{per}(A/I) is a quasi-isomorphism.

Proof. Fix a k-linear isomorphism A ≃ A_{0}⊕I. Denote by * the image of the direct sum multiplication under this isomorphism. Put λ(a_{1}, a_{2}) = a_{1}a_{2} − a_{1} ∗ a_{2}; then λ is an element of 𝔤_{A}^{1} = C^{2}(A) such that

δλ + \frac{1}{2}[λ, λ] = 0.

Since λ takes values in I, the operation

χ = ∑_{m=0}^{∞} \frac{1}{m!}I(λ, …, λ)  (3.9)

is well-defined on CC_{•}^{per}(A) (where I(λ, …, λ) are the operators from theorem 3.5.1). One checks that χ is indeed a morphism of complexes ([NT2]). It is easy to check that this morphism is invertible. □

4. Non-commutative differential calculus

4.1. Gerstenhaber algebras. Let k be the ground ring of characteristic zero. A Gerstenhaber algebra is a graded space V^{•} together with

• A graded commutative associative algebra structure on V^{•};

• a graded Lie algebra structure on V^{•+1} such that

[a, bc] = [a, b]c + (−1)^{deg(a) deg(b)}b[a, c]

Example 4.1.1. Let M be a smooth manifold. Then

V_{M}^{•} = ∧^{•}T_{M}

is a sheaf of Gerstenhaber algebras.

The product is the exterior product, and the bracket is the Schouten bracket. We denote by V^{•}(M) the Gerstenhaber algebra of global sections of this sheaf.

Example 4.1.2. Let 𝔤 be a Lie algebra. Then

C_{•}(𝔤) = ∧^{•}𝔤

is a Gerstenhaber algebra.

The product is the exterior product, and the bracket is the unique bracket which turns C_{•}(𝔤) into a Gerstenhaber algebra and which is the Lie bracket on 𝔤 = ∧^{1}(𝔤).

4.2. The Gerstenhaber algebra V^{•}(A). Below is the theorem from [T].

Theorem 4.2.1. For every associative algebra A there exists a dg Gerstenhaber algebra V^{•}(A) such that:

• There is a quasi-isomorphism of dg Lie algebras

V^{•}(A)[1] → C^{•}(A)[1]

• The above quasi-isomorphism induces an isomorphism of Gerstenhaber algebras

H^{•}(V^{•}(A)) → H^{•}(A)

where the Gerstenhaber structure on the right hand side is the standard one from 2.2.

• For A = C^{∞}(M) there is a quasi-isomorphism of dg Gersten-

haber algebras

V^{•}(A) → V^{•}(M)

4.3. Calculi.

Definition 4.3.1. A precalculus is a pair of a Gerstenhaber algebra V^{•} and a graded space Ω^{•} together with

• a structure of a graded module over the graded commutative algebra V^{•} on Ω^{−•} (corresponding action is denoted by i_{a}, a ∈ V^{•});

• a structure of a graded module over the graded Lie algebra V^{•+1} on Ω^{−•} (corresponding action is denoted by L_{a}, a ∈ V^{•}) such that

[L_{a}, i_{b}] = i_{[a,b]}

and

L_{ab} = L_{a}i_{b} + (−1)^{|a|}i_{a}L_{b}

Definition 4.3.2. A calculus is a precalculus together with an operator d of degree 1 on Ω^{•} such that d^{2} = 0 and

[d, i_{a}] = L_{a}.

Example 4.3.3. For any manifold one defines a calculus Calc(M) with V^{•} being the algebra of multivector fields, Ω^{•} the space of differential forms, and d the De Rham differential.

Example 4.3.4. For any associative algebra A one defines a calculus Calc_{0}(A) by putting V^{•} = H^{•}(A, A) and Ω^{•} = H_{•}(A, A). The five operations from Definition 4.3.2 are the cup product, the Gerstenhaber bracket, the pairings i_{D} and L_{D}, and the differential B, as in 2.4. The fact that it is indeed a calculus follows from Theorem 2.4.4.

A differential graded (dg) calculus is a calculus with extra differentials δ of degree 1 on V^{•} and b of degree −1 on Ω^{•} which are derivations with respect to all the structures.

4.4. The calculus Calc(A).

Theorem 4.4.1. [TT1] For any associative algebra A one can define a dg calculus Calc(A) such that:

1). As dg Lie algebras, V^{•+1}(A) is quasi-isomorphic to the Hochschild cochain complex C^{•+1}(A). As dg modules over the dg Lie algebra V^{•+1}(A), Ω^{•}(A)[[u]] with the differential δ + uB is quasi-isomorphic to the negative cyclic complex C_{•}(A)[[u]] with the differential b + uB.

2). If A = C^{∞}(M) then there is a quasi-isomorphism of calculi

Calc(A) → Calc(M)

3). For any A, the calculus H^{•}(Calc(A)) is isomorphic to the calculus Calc_{0}(A) from Example 4.3.4.

Combining 2) and 3), one gets for A = C^{∞}(M) quasi-isomorphisms

C_{•}(A)[[u]] ← Ω^{•}(A)[[u]] → Ω^{•}(M)[[u]].

Let 1 be the canonical homology class of the left hand side. Then

4). The image of 1 in the cohomology of the right hand side is the square root of the Todd class of the tangent bundle T M:

1 ↦ ∑ u^{p}(\sqrt{Td(M)})^{2p}

4.5. Enveloping algebra of a Gerstenhaber algebra. The following construction is motivated by Example 4.1.1. For a Gerstenhaber algebra V^{•}, let Y (V^{•}) be the associative algebra generated by two sets of generators i_{a}, L_{a}, a ∈ V^{•}, both i and L linear in a,

|i_{a}| = |a|; |L_{a}| = |a| − 1

subject to relations

i_{a}i_{b} = i_{ab}; [L_{a}, L_{b}] = L_{[a,b]};
[L_{a}, i_{b}] = i_{[a,b]}; L_{ab} = L_{a}i_{b} + (−1)^{|a|}i_{a}L_{b}

The algebra Y (V^{•}) is equipped with the differential d of degree one which is defined as a derivation sending i_{a} to L_{a} and L_{a} to zero.

For a smooth manifold M one has a homomorphism

Y (V^{•}(M)) → D(Ω^{•}(M))

The right hand side is the algebra of differential operators on differential forms on M. It is easy to see that the above map is in fact an isomorphism.

4.6. Relation to the A_{∞} structure on chains.

Theorem 4.6.1. There is a A_{∞} quasi-isomorphism

Y (V^{•}(A)) → C_{•}(C^{•}(A))

which extends to a k[[u]]-linear, (u)-adically continuous A_{∞} quasi-isomorphism

(Y (V^{•}(A))[[u]], δ + ud) → C_{•}(C^{•}(A))[[u]]

5. Cyclic objects

5.1. Simplicial objects. Denote by Δ the category whose objects are sets [n] = {0, 1, …, n}, n ≥ 0, and morphisms are nondecreasing maps. For a category C define a simplicial object of C an a functor from Δ^{0} to C where Δ^{0} is the category opposite to Δ. Any morphism in Δ can be written as a composition of the following:

1. Embeddings d_{i} : [n] ⟶[n + 1], 0 ≤ i ≤ n + 1 (i is not in the image of d_{i});

2. Surjections s_{i} : [n + 1] ⟶ [n], 0 ≤ i ≤ n (i is the image of two points under s_{i}).

A simplicial object of C can be equivalently described as a collection of objects X_{n} of C, n ≥ 0, together with morphisms

d_{i} : X_{n+1} ⟶X_{n}, 0 ≤ i ≤ n + 1  (5.1)
s_{i} : X_{n} ⟶X_{n+1}, 0 ≤ i ≤ n  (5.2)

subject to

d_{i}d_{j} = d_{j−1}d_{i}, i < j  (5.3)
s_{i}s_{j} = s_{j+1}s_{i}, i ≤ j  (5.4)

d_{i}s_{j} = s_{j−1}d_{j}, i < j; d_{i}s_{i} = d_{i+1}s_{i} = id; d_{i}s_{j} = s_{j}d_{i−1}, i > j + 1 (5.5)

Example 5.1.1. For a topological space X define Sing_{n}(X) to be the set of singular simplices of X. Then Sing(X) has a natural structure of a simplicial set.

Example 5.1.2. Let A be a graded algebra. Put A_{n}^{#} = A^{⊗(n+1)}. Define

d_{i}(a_{0} ⊗ … ⊗ a_{n+1}) = (−1)^{∑_{p≤i} |a_{p}|}a_{0} ⊗ … a_{i}a_{i+1} ⊗ … ⊗ a_{n+1}(5.6)

for i ≤ n

d_{n+1}(a_{0} ⊗ … ⊗ a_{n+1}) = (−1)^{|a_{n+1}| ∑_{p≤n} |a_{p}|}a_{n+1}a_{0} ⊗ a_{1} ⊗ … ⊗ a_{n} (5.7)

s_{i}(a_{0} ⊗ … ⊗ a_{n}) = (−1)^{∑_{p≤i} |a_{p}|}a_{0} ⊗ … a_{i} ⊗ 1 ⊗ … ⊗ a_{n} (5.8)

for i ≤ n

Proposition 5.1.3. The above formulas make A^{#} a simplicial graded space.

5.2. Cyclic objects. A cyclic object of a category C is a simplicial object X together with morphisms t : X_{n} ⟶X_{n} for all n ≥ 0 such that

d_{0}t = td_{1}; d_{1}t = td_{2}; …; d_{n−1}t = td_{n}; d_{n}t = d_{0}  (5.9)

as morphisms X_{n} ⟶X_{n−1};

s_{0}t = ts_{1}; s_{1}t = ts_{2}; …; s_{n−1}t = ts_{n}; s_{n}t = t^{2}s_{0}  (5.10)

as morphisms X_{n} ⟶X_{n+1};

t^{n+1} = id  (5.11)

on X_{n}. For a graded algebra A put

t(a_{0} ⊗ a_{1} ⊗ … ⊗ a_{n}) = (−1)^{|a_{0}| ∑_{i≥1} a_{i}}(a_{1} ⊗ a_{2} ⊗ … ⊗ a_{0})

Proposition 5.2.1. The above formula for t, together with (5.6), (5.7), (5.8), makes A^{#} a cyclic graded vector space.

For a cyclic vector space X define

b : X_{n} ⟶X_{n−1}; b = ∑_{i=0}^{n}(−1)^{i}d_{i}; τ = (−1)^{n+1}t : X_{n} ⟶X_{n}; (5.12)

N = ∑_{i=0}^{n} τ^{i}; B : X_{n} ⟶X_{n+1}; B = (1 − τ)sN  (5.13)

where

s = t^{−1}s_{n} : X_{n} ⟶X_{n+1}

Proposition 5.2.2.

b^{2} = bB + Bb = B^{2} = 0

Definition 5.2.3. For a cyclic vector space put

C_{•}(X) = (X_{•}, b)
CC_{•}^{−}(X) = (X_{•}[[u]], b + uB)
CC_{•}(X) = (X_{•}[u^{−1}, u]]/uX_{•}[[u]], b + uB)
CC_{•}^{per}(X) = (X_{•}[u^{−1}, u]], b + uB)

where u is a formal parameter of degree −2.

Let C be a small category and X : C^{op} → k-mod a functor. Let k^{#} : C → k-mod be the constant functor whose value on all objects is k and on all morphisms id_{k}. One defines

X ⊗_{C} k^{#} = ⊕_{i∈Ob(C)}X(i)/ < a − X(f)a|a ∈ X(i) ; f ∈ hom_{C}(j, i) >

One can show that X ↦ X ⊗_{C} k^{#} is a right exact functor. Denote its left derived functors by Tor_{•}^{C}(X, k^{#}).

Theorem 5.2.4. (Connes) One has, for an associative algebra A,

HH_{•}(A) = Tor_{•}^{Δ}(A^{#}, k^{#})
HC_{•}(A) = Tor_{•}^{Λ}(A^{#}, k^{#})

6. Examples

6.1. Smooth functions. For a smooth manifold M one can compute the Hochschild and cyclic homology of the algebra C^{∞}(M) where the tensor product in the definition if the Hochschild complex is one of the following three:

C^{∞}(M)^{⊗n} = C^{∞}(M^{n});  (6.1)
C^{∞}(M)^{⊗n} = germs_{Δ} C^{∞}(M^{n});  (6.2)
C^{∞}(M)^{⊗n} = jets_{Δ} C^{∞}(M^{n})  (6.3)

where Δ is the diagonal.

Theorem 6.1.1. The map

μ : f_{0} ⊗ f_{1} ⊗ … ⊗ f_{n} ↦ \frac{1}{n!}f_{0}df_{1} … df_{n}

defines a quasi-isomorphism of complexes

C_{•}(C^{∞}(M)) → (Ω^{•}(M), 0)

and a ℂ[[u]]-linear, (u)-adically continuous quasi-isomorphism

CC_{•}^{−}(C^{∞}(M)) → (Ω^{•}(M)[[u]], ud)

Localizing with respect to u, we also get quasi-isomorphisms

CC_{•}(C^{∞}(M)) → (Ω^{•}(M)[u^{−1}, u]]/uΩ^{•}(M)[[u]], ud)
CC_{•}^{per}(C^{∞}(M)) → (Ω^{•}(M)[u^{−1}, u]], ud)

This theorem, for the tensor products (6.2, 6.3), is due essentially to Hochschild, Kostant and Rosenberg (the Hochschild case) and to Connes (the cyclic cases). For the tensor product (6.1), see [Te].

6.1.1. Holomorphic functions. Let M be a complex manifold with the structure sheaf O_{M} and the sheaf of holomorphic forms Ω_{M}^{•}. If one uses one of the following definitions of the tensor product, then C_{•}(O_{M}), etc. are complexes of sheaves:

O_{M}^{⊗n} = germs_{Δ} O_{M^{n}};  (6.4)
O_{M}^{⊗n} = jets_{Δ} O_{M^{n}}  (6.5)

where Δ is the diagonal.

Theorem 6.1.2. The map

μ : f_{0} ⊗ f_{1} ⊗ … ⊗ f_{n} ↦ \frac{1}{n!}f_{0}df_{1} … df_{n}

defines a quasi-isomorphism of complexes of sheaves

C_{•}(O_{M}) → (Ω_{M}^{•}, 0)

and a ℂ[[u]]-linear, (u)-adically quasi-isomorphism of complexes of sheaves

CC_{•}^{−}(O_{M}) → (Ω_{M}^{•}[[u]], ud)

Similarly for the complexes CC_{•} and CC^{per}.

6.2. Commutative algebras.

6.2.1. Smooth Noetherian algebras. By definition, a commutative Noetherian k-algebra A is smooth if, for any k-algebra C and any ideal I of C such that I^{2} = 0, the map Hom(A, C) → Hom(A, C/I) is surjective. The class of smooth algebras includes the class of coordinate rings of nonsingular affine varieties over k. Let Ω_{A/k}^{1} be the module of Kähler differentials of A and Ω_{A/k}^{n} = ∧_{A}^{n}Ω_{A/k}^{1}

Theorem 6.2.1. The map

μ : a_{0} ⊗ … a_{n} ↦ \frac{1}{n!}a_{0}da_{1} … da_{n}

defines quasi-isomorphisms of complexes

C_{•}(A) → (Ω_{A/k}^{•}, 0)
CC_{•}^{−}(A) → (Ω_{A/k}^{•}[[u]], ud)
CC_{•}(A) → (Ω_{A/k}^{•}[u^{−1}, u]]/uΩ_{A/k}^{•}[[u]], ud)
CC_{•}^{per}(A) → (Ω_{A/k}^{•}[u^{−1}, u]], ud)

Cf. [L], [LQ]

6.2.2. Finitely generated commutative algebras. For a finitely generated commutative algebra A over k, choose a surjective homomorphism f : P → A where P is a ring of polynomials. Let I be the kernel of f. Consider the complexes

0 ⟶P/I^{n+1} ⟶^{d}Ω_{P/k}^{1}/I^{n} ⟶^{d}… Ω_{P/k}^{n}/I ⟶^{d}0  (6.6)

for all n ≥ 0. Denote their cohomologies by H_{cris}^{∗}(A/k; n) Denote by H_{cris}^{•}(P/k) the cohomology of the complex (Ω̂_{P/k}^{•}, d), the I-adic completion of the De Rham complex of P. It is well known that the cohomologies above are independent on a choice of P and f.

Theorem 6.2.2. 1. There exists the canonical morphism

μ : HC_{n}(A) → ⨁_{i≥0} H_{cris}^{n−2i}(A/k; n − i);

if A is smooth then μ is induced by the map from Theorem 6.2.1. 2. If A is a locally complete intersection then μ is an isomorphism.

3. There exists the canonical isomorphism

μ : HC_{•}^{per}(A) → H_{cris}^{•}(A/k)

Cf. [FT1]

6.3. Rings of differential operators. For a C^{∞} manifold X of dimension n let D(X) be the ring of differential operators on X. We use the tensor products defined analogously to (6.2), (6.3).

Theorem 6.3.1. ([BG], [Wo1]). There is a quasi-isomorphism

C_{•}(D(X)) → (Ω^{2n−•}(T^{∗}X), d)

which extends to a ℂ[[u]]-linear, (u)-adically continuous quasi-isomorphism

CC_{•}^{−}(D(X)) → (Ω^{2n−•}(T^{∗}X)[[u]], d)

As in 6.1, one also has analogous statements for the cyclic and periodic cyclic complexes.

6.3.1. Holomorphic differential operators. Let X be a complex manifold of complex dimension n. For the sheaf D_{X} of holomorphic differential operators, define the Hochschild, cyclic, etc. complexes of sheaves using tensor products analogous to those in 6.1.1. Let π : T^{∗}X → X be the projection.

Theorem 6.3.2. [Bry] There exists an isomorphism

π^{−1}C_{•}(D_{X}) → (Ω_{T^{∗}X}^{•}[2n], d)

in the derived category of the category of sheaves on T^{∗}X, which extends to a ℂ[[u]]-linear, (u)-adically continuous isomorphism in the derived category

π^{−1}CC_{•}^{−}(D_{X}) → (Ω_{T^{∗}X}^{•}[2n][[u]], d)

As in 6.1.1, similar isomorphisms exist for the cyclic and periodic cyclic complexes.

6.4. Rings of complete symbols. For a compact smooth manifold X, let CL(X) be the algebra of classical pseudo-differential operators. By L_{∞}(X) denote the algebra of smoothing operators (i.e. integral operators with smooth kernel), and put

CS(X) = CL(X)/L_{∞}(X).

We use the projective tensor products.

For any manifold M, denote by Ω̂^{∗}(M ×S^{1}) the space of power series

∑_{ϵ=0, 1; n=−∞}^{N} α_{i, ϵ}z^{i}dz^{ϵ}

where α_{i} are forms on M. Denote by S^{∗}X the cosphere bundle of X.

Theorem 6.4.1. [Wo2] There exists a quasi-isomorphism

C_{•}(CS(X)) ⟶(Ω̂^{2n−•}(S^{∗}X × S^{1}), d)

which extends to a ℂ[[u]]-linear, (u)-adically continuous quasi-isomorphism

CC_{•}^{−}(CS(X)) ⟶(Ω̂^{2n−•}(S^{∗}X × S^{1})[[u]], d)

Similarly for CC_{•}, CC_{•}^{per}. In particular:

Corollary 6.4.2.

HH_{p}(CS(X)) = H^{2n−p}(S^{∗}X × S^{1})

HC_{p}(CS(X)) = ⨁_{i≥0} H^{2n−p+2i}(S^{∗}X × S^{1})

Combined with the pairing with the fundamental class of S^{∗}X × S^{1}, the first of the above isomorphisms gives an isomorphism

HH_{0}(CS(X)) = CS(X)/[CS(X), CS(X)] → ℂ  (6.7)

This isomorphism is given by the Wodzicki noncommutative residue [Wo]. The above theorem has also a holomorphic version where the ring of complete symbols is replaced by the sheaf of micro-differential operators [BNT1].

6.5. Rings of pseudo-differential operators.

Theorem 6.5.1.

HH_{0}(L_{∞}(X)) ≃ ℂ; HH_{p}(L_{∞}(X)) = 0, p > 0;
HC_{2p}(L_{∞}(X)) ≃ ℂ; HC_{2p+1}(L_{∞}(X)) ⟶0

Theorem 6.5.2. [Wo2]

HH_{p}(CL(X)) ≃ HH_{p}(CS(X)) for p ≠ 1;

there is an exact sequence

0 ⟶HH_{1}(CL(X)) ⟶HH_{1}(CS(X)) ⟶ℂ ⟶0

Theorem 6.5.3. For all p ≥ 0

HC_{2p}(CL(X)) ≃ HC_{2p}(CS(X))

and there is an exact sequence

0 ⟶HC_{2p+1}(CL(X)) ⟶HC_{2p+1}(CS(X)) ⟶HC_{2p}(ℂ) ⟶0

Theorems 6.5.2 and 6.5.3 follow from Theorem 6.5.1 and from Wodzicki excision theorems [Wo3].

The results of the two previous subsections were extended to more general rings of symbols and of pseudo-differential operators by Melrose-Nistor and Benameur-Nistor ([MN], [BN]). A survey of these results can be found in [BBN].

6.6. Non-commutative tori. Let q be a complex number, |q| = 1. By A_{q} denote the algebra over ℂ with two generators u and v subject to one relation

vu = quv  (6.8)

Theorem 6.6.1. [C]

H_{•}(A_{q}) ≃ H_{•}(𝕋^{2})
HC_{•}(A_{q}) ≃ H_{•}(𝕋)^{2}[[u, u^{−1}]/uH_{•}(𝕋)^{2}[[u, u^{−1}]
HC_{•}^{−}(A_{q}) ≃ H_{•}(𝕋^{2})[[u]]
HC_{•}^{per}(A_{q}) ≃ H_{•}(𝕋^{2})[[u, u^{−1}]

where u is a formal variable of degree −2.

6.7. Deformation quantization. Let M be a smooth manifold. By a deformation quantization of M we mean a formal product

f ∗ g = fg + ∑_{k=1}^{∞} t^{k}P_{k}(f, g)  (6.9)

where P_{k} are bidifferential expressions, ∗ is associative, and 1 ∗ f = f ∗ 1 = f for all f. Given such a product (which is called a star product), we define

𝔸^{t}(M) = (C^{∞}(M)[[t]], ∗)  (6.10)

This is an associative algebra over ℂ[[t]]. By 𝔸_{c}^{t}(M) we denote the ideal C_{c}^{∞}(M)[[t]] of this algebra. An isomorphism of two deformations is by definition a power series T (f) = f + ∑_{k=1}^{∞} T_{k}(f) where all T_{k} are differential operators and which is an isomorphism of algebras.

Given a star product on M, for f, g ∈ C^{∞}(M) let

\{f, g\} = P_{1}(f, g) − P_{1}(g, f) = \frac{1}{t}[f, g]|t = 0  (6.11)

This is a Poisson bracket corresponding to some Poisson structure on M. If this Poisson structure is defined by a symplectic form ω, we say that 𝔸^{t}(M) is a deformation of the symplectic manifold (M, ω). In defining the Hochschild and cyclic complexes, we use k = ℂ[[t]] as the ring of scalars, and put

𝔸^{t}(M)^{⊗n} = jets_{Δ} C^{∞}(M^{n})[[t]]  (6.12)

Let 𝔸^{t}(M) be a deformation of a symplectic manifold (M, ω).

Theorem 6.7.1. There exists a quasi-isomorphism

C_{•}(𝔸^{t}(M), 𝔸^{t}(M))[t^{−1}] ⟶(Ω^{2n−•}(M)[[t, t^{−1}], td)

which extends to a ℂ[[t, u]]-linear, (t, u)-adically continuous quasi-isomorphism

CC_{•}^{−}(𝔸^{t}(M))[t^{−1}] ⟶(Ω^{2n−•}(M)[[u, t]][t^{−1}], td)

An analogous theorem holds for 𝔸_{c}^{t}(M) if we replace Ω^{•} by Ω_{c}^{•}.

6.7.1. The canonical trace. Combining the first map from Theorem 6.7.1 in the compactly supported case with integrating over M and dividing by \frac{1}{n!}, one gets the canonical trace of Fedosov

T r : 𝔸_{c}^{t}(M) ⟶ℂ[[t, t^{−1}]

6.7.2. Deformation quantization of complex manifolds. For a complex manifold M with an open cover 𝔘, define a deformation of the structure sheaf of algebras O_{M} to be the following:

a) a collection of holomorphic star products ∗_{U} on every U ∈ 𝔘

b) a collection of holomorphic isomorphisms G_{UV} between ∗_{U} and ∗_{V} for every U, V ∈ 𝔘, such that

G_{UV} G_{V W} = G_{UW}  (6.13)

for all U, V, W.

Such a data defines a sheaf of algebras 𝔸_{M}^{t}.

(The above definition is enough for the purposes of this article; there is, however, a better definition in which (6.13) is replaced by G_{UV} G_{V W} = Ad(c_{UV W})G_{UW} for some invertible elements c_{UV W} subject to a natural cocycle condition on intersections of any four open subsets). Using the definition of tensor products analogous to 6.1, we define complexes of sheafs C_{•}(𝔸_{M}^{t}), etc.

Theorem 6.7.2. Let 𝔸_{M}^{t} be a deformation of a complex manifold M with a holomorphic symplectic structure ω. There exists an isomorphism in the derived category of sheaves

ℂ_{•}(𝔸_{M}^{t})[t^{−1}] → (Ω_{M}^{2n−•}[[t, t^{−1}], td)  (6.14)

which extends to a ℂ[[t, u]]-linear, (t, u)-adically continuous isomorphism in the derived category

CC_{•}^{−}(𝔸_{M}^{t})[t^{−1}] → (Ω_{M}^{2n−•}[[t, u]][t^{−1}], td)  (6.15)

Similarly for CC_{•}, CC_{•}^{per}.

If one wants to avoid the language of derived categories, one can say that there may be no morphism of sheaves (6.15) but there is, canonical up to homotopy, morphism of Cech complexes of corresponding sheaves. One can also use sheaf resolutions other than the Cech resolution, for example the complex of Dolbeault forms with values in Hochschild, cyclic, De Rham, etc. complexes of the algebra of jets [BNT].

6.8. The Brylinski spectral sequence. Consider a deformation quantization 𝔸^{t}(M) of a smooth manifold M. As we saw above, such a deformation gives rise to a Poisson bracket on C^{∞}(M). Denote by π_{0} the corresponding Poisson bivector: π_{0} ∈ Γ(M, ∧^{2}T M) such that {f, g} =< π_{0}, df ∧ dg >. Denote by i_{π_{0}} the operator of contracting a form by π_{0}, and put

L_{π_{0}} = di_{π_{0}} − i_{π_{0}}d  (6.16)

The Hochschild and cyclic complexes of the deformed algebra 𝔸^{t}(M) are complete with respect to the filtration F_{p}C_{•} = t^{p}C_{•}. Computing the spectral sequences corresponding to this filtration, one obtains the following

Theorem 6.8.1. 1. There exists a spectral sequence converging to

HH_{•}(𝔸^{t}(M))

with the E^{1} term equal to the homology of the complex

(Ω^{•}(M)[[t]], tL_{π_{0}})  (6.17)

2. There exists a spectral sequence converging to

HC_{•}^{−}(𝔸^{t}(M))

with the E^{1} term equal to the homology of the complex

(Ω^{•}(M)[[t, u]], tL_{π_{0}} + ud)  (6.18)

and similarly for the cyclic and periodic cyclic homologies.

If π_{0} is symplectic then the complex 6.17, resp. 6.18, is quasiisomorphic to the complex (Ω^{2n−•}(M)[[t]], td), resp. (Ω^{2n−•}(M)[[t, u]], td).

6.9. Deformation quantization: general case. Recall first Kontsevich’s classification of deformation quantizations. We call a series

π = ∑_{i≥0} t^{i}π_{i},

π_{i} ∈ Γ(M, ∧^{2}T_{M}), a formal Poisson structure if [π, π] = 0 where [ , ] is the Schouten bracket; in other words, π is a formal Poisson structure if

{f, g}_{π} =< π, df ∧ dg >

defines a Lie algebra structure on C^{∞}(M)[[t]]. Two formal Poisson structures are called equivalent if there exists a series

X = ∑_{i≥0} t^{i}X_{i},

X_{i} ∈ Γ(M, T_{M}), such that

π′ = exp ad L_{X}(P)

Theorem 6.9.1. (Kontsevich) There is a bijection between the set of isomorphism classes of deformation quantizations of C^{∞}(M) and the set of equivalence classes of formal Poisson structures on M.

For a formal Poisson structure π the above theorem defines up to isomorphism a deformed associative algebra 𝔸_{π}^{t}(M).

Theorem 6.9.2. There are quasi-isomorphisms of complexes

C_{•}(𝔸_{π}^{t}(M)) → (Ω^{•}(M)[[t]], tL_{π})
CC_{•}^{−}(𝔸_{π}^{t}(M)) → (Ω^{•}(M)[[t, u]], tL_{π} + ud)

In particular, if π is an ordinary Poisson structure, then the Brylinski spectral sequence degenerates.

6.10. Group rings. Let G be a discrete group.

Theorem 6.10.1.

HH_{•}(k[G]) ≃ ⨁_{<x>} H_{•}(G_{x}, k)

where the sum is taken over all conjugacy classes of G and G_{x} is the centralizer of an element x of G.

The Hochschild homology can also be decomposed into components corresponding to conjugacy classes; the construction of these components depends on whether a class is elliptic (i.e. of the form < x > where x is of finite order) or not.

Theorem 6.10.2. [Bu]

HC_{n}(k[G]) ≃ ⨁_{<x>| ord x<∞} ⨁_{i} H_{n−2i}(G_{x}, k)⊕ ⨁_{<x>| ord x=∞} H_{n−2i}(G_{x}/(x), k)

Remark 6.10.3. The exact sequence of the triangle (2.5) splits into a direct sum over the set of conjugacy classes of G. For the summands corresponding to elliptic classes, the component of the differential HH_{•} ⟶ HC_{•} is injective. In the non-elliptic case, the corresponding component of the exact sequence is isomorphic to the exact sequence of the fibration

Bℤ ⟶BG_{x} ⟶B(G_{x}/(x))

Simirarly, the components of the spectral sequence (2.6) which correspond to elliptic classes degenerate at E^{1}. The component corresponding to a non-elliptic class < x > is isomorphic to the spectral sequence of the fibration

BG_{x} ⟶B(G_{x}/(x)) ⟶K(ℤ, 2)

7. Index theorems

7.1. Index theorem for deformations of symplectic structures. Let M be a smooth symplectic manifold. Let 𝔸_{M}^{t} be a deformation of the sheaf of functions on M. Recall that there exists canonical up to homotopy equivalence quasi-isomorphism (6.15)

μ^{t} : CC_{•}^{−}(𝔸^{t}(M))[t^{−1}] → (Ω^{2n−•}(M)[[t, u]][t^{−1}], td)  (7.1)

Localizing in u, we obtain a quasi-isomorphism

μ^{t} : CC_{•}^{per}(𝔸^{t}(M))[t^{−1}] → (Ω^{2n−•}(M)[[t, u, t^{−1}, u^{−1}], td) (7.2)

Definition 7.1.1. The above morphisms are called the trace density morphisms.

The index theorem compares the trace density morphism to the principal symbol morphism. To define the latter, consider the cyclic complex of the deformed algebra where the scalar ring is ℂ instead of ℂ[[t]]. Consider the composition

CC_{•}^{−}(𝔸^{t}(M)) → CC_{•}^{−}(C^{∞}(M)) →
→ (Ω^{•}(M)[[u]], ud) → (Ω^{•}(M)[[u]][[t]], ud)

where the first morphism is reduction modulo t, the second one is μ from Theorem 6.1.1, and the third one is induced by the embedding ℂ → ℂ[[t]]. We will denote this composition, followed by localization in t, by

μ : CC_{•}^{per}(𝔸^{t}(M)) → (Ω^{•}(M)[[u, t, u^{−1}, t^{−1}], ud)  (7.3)

To compare μ and μ^{t}, let us identify the right hand sides by the isomorphism

(Ω^{2n−•}(M)[[u, t, u^{−1}, t^{−1}], td) → (Ω^{•}(M)[[u, t, u^{−1}, t^{−1}], ud)

which is equal to (\frac{t}{u})^{n−k} on Ω^{k}(M)[[u, t, u^{−1}, t^{−1}]. After this identification, we obtain two morphisms

μ, μ^{t} : CC_{•}^{per}(𝔸^{t}(M)) → (Ω^{•}(M)[[u, t, u^{−1}, t^{−1}], ud)

where the right hand side is defined as the periodic cyclic complex with respect to the ground ring ℂ.

Theorem 7.1.2. At the level of cohomology,

μ^{t} = ∑_{p=0}^{∞} u^{p}(Â(M)e^{θ})_{2p} · μ

where Â(M) is the Â class of the tangent bundle of M viewed as a complex bundle (with an almost complex structure compatible to the symplectic form), and θ ∈ \frac{1}{t}[ω] + H^{2}(M, ℂ[[t]]) is the characteristic class of the deformation.

Note that the canonical trace Tr_{can} is the composition of μ^{t} with the integration Ω^{2n}[[t, t^{−1}] → ℂ[[t, t^{−1}]. Let P and Q be N × N matrices over 𝔸^{t}(M) such that P^{2} = P, Q^{2} = Q, and P − Q is compactly supported. Let P_{0}, Q_{0} be reductions of P, Q modulo t. They are idempotent matrix-valued functions; their images P_{0}ℂ^{N}, Q_{0}ℂ^{N} are vector bundles on M. Applying μ^{t} to the the difference of Chern characters of P and Q, we obtain the following index theorem of Fedosov (cf. also [NT4]).

Theorem 7.1.3.

Tr_{can}(P − Q) = ∫_{M}(ch(P_{0}ℂ^{N}) − ch(Q_{0}ℂ^{N}))Â(M)e^{θ}

7.2. Index theorem for holomorphic symplectic deformations. Let M be a complex manifold with a holomorphic symplectic form ω. Let 𝔸_{M}^{t} be a deformation of the sheaf of holomorphic functions on M. One defines two morphisms in the derived category of sheaves

μ, μ^{t} : CC_{•}^{per}(𝔸_{M}^{t}) → (Ω_{M}^{•}[[u, t, u^{−1}, t^{−1}], ud)

exactly as in the smooth case.

Theorem 7.2.1.

μ^{t} = ∑_{p=0}^{∞} u^{p}(\sqrt{Td(M)}e^{θ})_{2p} · μ

in the derived category of sheaves, where Td(M) is the Todd class of the holomorphic tangent bundle of M.

7.3. General index theorem for deformations. Let M be a C^{∞} manifold. Let π be a formal Poisson structure on M and 𝔸_{π}^{t}(M) the deformation corresponding to π under the Kontsevich isomorphism from Theorem 6.9.1. We construct the quasi-isomorphisms

μ, μ^{t} : CC_{•}^{per}(𝔸_{π}^{t}(M)) → (Ω^{•}(M)[[u, t, u^{−1}, t^{−1}], ud)

as follows. The principal symbol map μ is constructed exactly as in (7.3). The trace density morphism μ^{t} is defined as follows. Start with the quasi-isomorphism

CC_{•}^{−}(𝔸_{π}^{t}(M)) → (Ω^{•}(M)[[t, u]], tL_{π} + ud)

from Theorem 6.9.2. Then localize it with respect to t and u, and then follow it by the isomorphism

exp(−\frac{t}{u}i_{π}) : (Ω^{•}(M)[[t, u, t^{−1}, u^{−1}], tL_{π}+ud) → (Ω^{•}(M)[[t, u, t^{−1}, u^{−1}], ud

.

Theorem 7.3.1. At the level of cohomology,

μ^{t} = ∑_{p=0}^{∞} u^{p}(\sqrt{Td(M)})_{2p} · μ

where Td(M) is the Todd class of the tangent bundle of M.

Theorem 7.1.2 can be deduced from the theorem above. Note the absence of the factor e^{θ}, explained by the presence of exp(−\frac{t}{u}i_{π}) in the definition of the trace density map μ^{t}.

Let us stress again that all the ingredients of the theorem above (the construction of the deformation 𝔸_{π}^{t}(M)) from π and the construction of the trace density map) are not canonical but depend on a choice of a Drinfeld associator.

8. Riemann-Roch theorem for D-modules

Let us recall the definition and examples from [SS]. Let X be a complex manifold. An elliptic pair (M, F) on X is a coherent D_{X}- module M and an ℝ-constructible sheaf F on X such that

char(M) ⋂ SS(F) ⊂ T_{X}^{∗}X  (8.1)

More generally, (M, F), can be a pair of complexes, or rather of objects of corresponding derived categories.

Examples

(1) Let M be a coherent D_{X}-module. Then the pair (M, ℂ_{X}) is elliptic.

(2) Let U be an open subset of X. Then the pair (M, ℂ_{U}) is elliptic if and only if ∂U is noncharacteristic for M.

(3) Let X be a complexification of a real analytic manifold X_{0} and M a complexification of a real analytic D_{X_{0}}-module M_{0} on X_{0}. Then (M, ℂ_{X_{0}}) is an elliptic pair if and only if M_{0} is elliptic in the classical sense.

(4) If F is ℝ-constructible on X then (O_{X}, F) is an elliptic pair. We assume that M is endowed with a good filtration. Recall that the

order of differential operators defines a filtration F_{p} on D_{X}; a filtration G_{q} on a module is good if F_{p}D_{X} · G_{q} ⊂ G_{p+q}.

Consider the derived tensor product M ⊗_{D_{X}}^{𝕃} O_{X}. Schapira and Schneiders showed that (M⊗_{D_{X}}^{𝕃} O_{X})⊗_{ℂ}F has finite-dimensional cohomology. Let μ eu(F) be Kashiwara’s microlocal Euler class of F. Let Λ be the characteristic variety of M. The good filtration on M defines a coherent gr_{F} D_{X}-module gr M; since gr_{F} D_{X} is the sheaf of functions on T^{∗}X which are polynomial along the fibres, by extension of scalars we obtain a coherent O_{T^{∗}X}-module σ_{Λ}(M) supported on Λ. One can define Chern character ch_{Λ} of such a module in H_{Λ}^{•}(T^{∗}X).

The following was conjectured in [SS] and proven in [BNT] by reducing it to Theorem 7.2.1.

Theorem 8.0.2.

χ(RΓ((M ⊗_{D_{X}}^{𝕃} O_{X}) ⊗_{ℂ} F)) = ∫ μ eu(F) ⌣ ch_{Λ}(σ_{Λ}(M)) ⌣ π^{∗} Td(X)

where π is the projection from T^{∗}X to X.

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Department of Mathematics, Northwestern University, Evanston, IL, 60208, USA

  E-mail address: tsygan@math.nwu.edu