New York Journal of Mathematics

New York J. Math. 14 (2008) 643–717.

The relative Riemann–Roch theorem from Hochschild homology

Ajay C. Ramadoss

Abstract. This paper attempts to clarify a preprint of Markarian (2001). Markarian’s preprint proves the relative Riemann–Roch theorem using a result describing how the HKR map fails to respect comultiplication. This paper elaborates on the core computations in Markarian’s preprint. These computations show that the HKR map twisted by the square root of the Todd genus “almost preserves” the Mukai pairing. This settles a part of a conjecture of Caldararu, 2005. The relative Riemann–Roch theorem follows from this and a result of Caldararu, preprint, 2003.

Contents

Introduction 644 1. The main results 646

1.1. The crux of this paper 647 1.2. The Mukai pairing 649 1.3. The relative Riemann–Roch theorem 652

2. Two “connections” on the Hochschild chain complex Ĉ^{•}(X) 653 2.1. The completed bar and Hochschild chain complexes 653 2.2. Two connections on Ĉ^{•}(X) 656 2.3. Remark — the beginning of a dictionary 659 2.4. More on the maps α_{R} and α_{L} 660 2.5. A long remark — enlarging the dictionary 663

3. Some essential linear algebra 667 3.1. Some propositions and a lemma of linear algebra 668 3.2. Applying the linear algebra to S^{•}(Ω[1]) 682

Received April 10, 2008. Mathematics Subject Classification. 19L10, 16E40, 14F40. Key words and phrases. Riemann–Roch theorem, Mukai pairing, Hochschild–Kostant–

Rosenberg map.

643^{ISSN 1076-9803/08}

4. The adjoint of Φ_{R} 688 4.1. Stating the main lemma of this section 688 4.2. Comparing two “sections” of D_{n} ⊗ Ω^{n}[n] 690 4.3. Another long remark — Lemma 4 and our dictionary 696 4.4. Proof of Lemma 3 698

5. Proof of Theorem 1 701 5.1. Unwinding some definitions 701 5.2. The final (long) computation proving Theorem 1 708 5.3. Proving Lemma 5 714

References 717

Introduction

The purpose of this paper is to explain in detail an alternative approach to the relative Riemann–Roch theorem which first appeared in a very interesting but cryptic preprint [5] of Markarian. This approach leads to a proof of the relative Riemann–Roch theorem by a direct computation of the pairing on Hodge cohomology to which the Mukai pairing on Hochschild homology defined by Caldararu [1] descends via the Hochschild–Kostant–Rosenberg map multiplied by the square root of the Todd genus. In the framework of this approach, the fundamental reason for the appearance of the Todd genus in the Riemann–Roch theorem is the failure of the Hochschild–Kostant– Rosenberg map I_{HKR} introduced in Section 2 to respect comultiplication. One of the main theorems in [5] (Theorem 1 of [5]) describes the Duflo like error term that measures by how much I_{HKR} fails to respect comultiplication. The proof supplied in [5] however, has a nontrivial error. A “dual” version of this result equivalent to the original theorem has since been correctly proven by the author in [7]. A correct proof of another version of this result has been outlined by Markarian himself in a revised version [6] of [5]. Theorem 1 of [5] appears as Theorem 2′ in this paper.

Let X be a smooth proper scheme over a field 𝕂 of characteristic 0. We use Theorem 2′ of this paper to prove the main result (Theorem 1) of the current paper. This explicitly interrelates the Hochschild–Kostant–Rosenberg map, the twisted HKR map introduced in Section 1 and a map which we call the duality map between RHom_{X}(O_{X}, Δ^{∗}O_{Δ}) and RHom_{X}(Δ^{∗}O_{Δ}, S_{X}). Here, S_{X} is the shifted line bundle on X tensoring with which is the Serre duality functor on D^{b}(X). Theorem 1 of this paper is equivalent to a corrected version of an erroneous result (Theorem 8 of [5]) that appears in [5].

Theorem 1 enables us to compute the pairing on Hodge cohomology to which the Mukai pairing on Hochschild homology defined by Caldararu [1] descends via the HKR-map twisted by the square root of the Todd genus. Given Theorem 1, doing this computation is fairly easy. This pairing on Hodge cohomology (see Proposition 5 of this paper) is very similar to the generalized Mukai pairing Caldararu defined in [2]. In particular, it satisfies the adjointness property one expects from the Mukai pairing. Moreover, it coincides with the Mukai pairing defined by Caldararu [2] on Mukai vectors (see [2]) of elements of D^{b}(X). However, the pairing obtained in Proposition 5 is not exactly the same as Caldararu’s Mukai pairing on Hodge cohomology. This settles a part of Caldararu’s conjecture in [2] regarding the equivalence between the Hochschild and Hodge structures of a smooth proper complex variety — to be able to say that the HKR map twisted by the square root of the Todd genus preserves the Mukai pairing, one has to replace the Mukai pairing defined by Caldararu [2] with the pairing that shows up in Proposition 5. Since pairing in Proposition 5 does not in general coincide with the Mukai pairing on Hodge cohomology for K3-surfaces, we stay contented by saying that the HKR map twisted by the square root of the Todd genus “almost preserves” the Mukai pairing.

The relative Riemann–Roch theorem follows from Proposition 5 and the adjointness property of the Mukai pairing on Hochschild homology. The adjointness property of the Mukai pairing on Hochschild homology was proven in a paper [1] of Caldararu.

In order to prove Theorem 1, we elaborate upon the core computations in [5]. These computations appeared in [5] in a very cryptic way. Some of these computations do not appear in [6], whose approach differs in some details from [5]. In particular, unlike [5], [6] does not contain a result equivalent to Theorem 1 and does not compute the Mukai pairing on Hochschild homology at the level of Hodge cohomology.

The key steps in this computational approach are covered by Theorem 2′ and Lemmas 2, 3 and 4 of this paper. The most crucial computations, Theorem 2′ and Lemma 4 are related to very familiar computations in elementary Lie theory. Theorem 2′ is related to computing the pullback of a left invariant 1-form on a Lie group G via the exponential map. Similarly, Lemma 4 is related computing the pullback of a left invariant volume form on G via the the map \overline{exp} where \overline{exp}(Z) = exp(−Z) for any element Z of the Lie algebra 𝔤 of G. We aim to make these relations transparent by developing a “dictionary” in this paper in three separate subsections containing remarks meant for this purpose only.

The layout of this paper. Section 1 begins by introducing the notations and conventions that shall be used for the rest of this paper. It then goes on to state Theorem 1 after defining the various maps involved in Theorem 1. The pairing on Hodge cohomology to which the Mukai pairing on Hochschild homology descends (via the HKR map twisted by the square root of the Todd genus) is then computed. Finally, Section 1 uses this computation to prove the relative Riemann–Roch theorem. The remaining sections of this paper are devoted to proving Theorem 1.

Section 2 introduces two “connections” on the complex of completed Hochschild chains of a smooth scheme X. Their properties are proven in various propositions in this section. This section also proves Theorem 2′. Sections 2.3 and 2.5 develop the “dictionary” making the analogy between Theorem 2′ and its counterpart in elementary Lie theory more transparent.

Section 3 consists of a number of definitions, technical propositions and two lemmas (Lemma 1 and Lemma 2) pertaining mainly to linear algebra. These are used in later sections at various points. The definitions of this section are important to understand later sections. Proofs of propositions in later sections time and again refer to propositions in this section.

The key result of Section 4 is Lemma 3. This in turn follows from Lemma 4 and Lemma 2. Besides proving Lemma 3 and Lemma 4, Section 4 has a subsection (Section 4.3) which explains the analogy between Lemma 4 and its counterpart in basic Lie theory. Section 4.3 is the last of the three sections (2.3, 2.5 and 4.3) developing the “dictionary” in this paper.

Section 5 undertakes the final computations leading to the proof of Theorem 1.

Acknowledgements. I thank Prof. Victor Ginzburg for introducing me to the works of Markarian and Caldararu and for useful discussions. I also thank Prof. Madhav Nori for useful discussions and Prof. Andy Magid and Dr. Victor Protsak for useful suggestions regarding the presentation of this paper.

1. The main theorem, the Mukai pairing and the relative Riemann–Roch theorem

We begin this section by clarifying some notation and conventions that shall be followed throughout this paper. Immediately after that, in Section 1.1, we state the main theorem (Theorem 1) of this paper after describing the maps involved. This section then goes on to explain in detail why Theorem 1 implies the relative Riemann–Roch theorem. This is done in Sections 1.2 and 1.3.

Notation and conventions. Let X be a smooth proper scheme over a field 𝕂 of characteristic 0. All schemes and complex varieties that we encounter in this paper are assumed to be proper. Let Δ : X → X × X denote the diagonal embedding. Let p_{1} and p_{2} denote the projections from X × X onto the first and second factors respectively. As usual, O_{X} denotes the structure sheaf of X.

Ch^{b}(O_{X} − mod) (resp. Ch^{−}(O_{X} − mod) and Ch^{+}(O_{X} − mod)) denotes the category of bounded (resp. bounded above and bounded below) chain complexes of O_{X}-modules with coherent cohomology. D^{b}(X) denotes the bounded derived category of complexes of O_{X}-modules with coherent cohomology. Similarly, D^{b}(X × X) denotes the bounded derived category of complexes of O_{X×X}-modules with coherent cohomology.

Whenever f : X → Y is a morphism of schemes, f^{∗} : D^{b}(Y) → D^{b}(X) denotes the left derived functor of the pullback via f. Similarly, f_{∗} : D^{b}(X) → D^{b}(Y) denotes the right derived functor of the push-forward via f. O_{Δ} will denote Δ_{∗}O_{X}. Similarly, when we refer to a tensor product, we will mean the corresponding left-derived functor unless stated otherwise explicitly. Also, if E is an object of D^{b}(X) and φ is a morphism in D^{b}(X), E ⊗ φ shall denote the morphism 1_{E} ⊗ φ. At times, E shall be used to denote the morphism 1_{E}.

If E and F are objects of D^{b}(X), we shall denote RHom_{D^{b}(X)}(E, F) by RHom_{X}(E, F). Also, we denote RHom_{D^{b}(X×X)}(−, −) by RHom_{X×X}(−, −).

Ω_{X} denotes the cotangent bundle of X. S^{•}(Ω_{X}[1]) will denote the symmetric algebra generated over O_{X} by Ω_{X} concentrated in degree −1. Note that S^{•}(Ω_{X}[1]) = ⊕_{i} ∧^{i} Ω_{X}[i]. We shall often denote ∧^{i}Ω_{X} by Ω_{X}^{i}. From Section 2 onwards, Ω_{X} and S^{•}(Ω_{X}[1]) shall be denoted by Ω and S^{•}(Ω[1]) respectively. The tangent bundle of X shall be denoted by T_{X} in this section and T from Section 2 onwards.

Where convenient,we shall denote the Hodge cohomology ⊕_{p,q}H^{q}(X, Ω_{X}^{p}) by H^{∗}(X) and H^{q}(X, Ω_{X}^{p}) by H^{p,q}(X). Note that

H^{∗}(X) = RHom_{X}(O_{X}, S^{•}(Ω_{X}[1])).

The product on H^{∗}(X) induced by the wedge product on S^{•}(Ω_{X}[1]) will be denoted by ∧. However, we shall often suppress the ∧: If a, b ∈ H^{∗}(X), ab should be understood to mean a ∧ b ∈ H^{∗}(X).

Despite our attempts to minimize abuse of notation, it does happen at times. There are many situations in this paper where we encounter maps between tensor products of various objects in D^{b}(X) that rearrange factors. Very often, such maps are denoted by the symbol τ. In each such situation, we specify what τ means unless we feel it is obvious to the reader.

1.1. The crux of this paper. Recall from Yekutieli [9] that the completed complex of Hochschild chains Ĉ^{•}(X) ∈ Ch^{−}(O_{X} − mod) is a complex of flat O_{X}-modules that represents Δ^{∗}O_{Δ} in D^{b}(X). Upper indexing is used here to convert what was originally a chain complex into a cochain complex.

The Hochschild–Kostant–Rosenberg (HKR) map I_{HKR} from Ĉ^{•}(X) to S^{•}(Ω_{X}[1]) is a map of complexes of O_{X} modules. We describe this in greater detail in Section 2. We identify Δ^{∗}O_{Δ} with Ĉ^{•}(X). Thus, the HKR map I_{HKR} can be thought of as a map in D^{b}(X) from Δ^{∗}O_{Δ} to S^{•}(Ω_{X}[1]).

Let S_{X} denote the object Ω^{n}[n] in D^{b}(X). Let π_{n} denote the projection from S^{•}(Ω_{X}[1]) to the direct summmand Ω^{n}[n]. Consider the pairing

⟨−, −⟩ : S^{•}(Ω_{X}[1]) ⊗ S^{•}(Ω_{X}[1]) → S_{X}

given by the composite

S^{•}(Ω_{X}[1]) ⊗ S^{•}(Ω_{X}[1]) ⟶^{(−∧−)} S^{•}(Ω_{X}[1]) ⟶^{π_{n}} S_{X}

of morphisms in D^{b}(X). One also has a twisted HKR map from S^{•}(Ω_{X}[1]) to RHom_{X}(Δ^{∗}O_{Δ}, S_{X}). This arises out of the composite

(1) Δ^{∗}O_{Δ} ⊗ S^{•}(Ω[1]) ⟶^{I_{HKR}⊗1_{S^{•}(Ω_{X} [1])}} S^{•}(Ω_{X}[1]) ⊗ S^{•}(Ω_{X}[1]) ⟶^{⟨,⟩} S_{X}.

of morphisms in D^{b}(X). We denote the twisted HKR map by { \atop I_{HKR}}.

The duality map. The material in this paragraph is recalled from Caldararu [1]. Recall that Δ^{∗} : D^{b}(X × X) → D^{b}(X) is the left adjoint of Δ_{∗} : D^{b}(X) → D^{b}(X × X). Also recall that the functor of tensoring by the shifted line bundle S_{X} is the Serre duality functor on D^{b}(X). Similarly, tensoring by the shifted line bundle S_{X×X} is the Serre duality functor on D^{b}(X × X). We denote the functor given by tensoring by a shifted line bundle L by L itself. The left adjoint Δ_{!} : D^{b}(X) → D^{b}(X × X) of Δ^{∗} is given by S_{X×X}^{−1}Δ_{∗}S_{X}.

Since Δ_{!} is the left adjoint of Δ^{∗} we have an isomorphism

(2)  I : RHom_{X}(O_{X}, Δ^{∗}O_{Δ}) ≃ RHom_{X×X}(Δ_{!}O_{X}, O_{Δ}).

Now, Δ_{!}O_{X} = S_{X×X}^{−1}Δ_{∗}S_{X} ≃ Δ_{∗}S_{X}^{−1}. We also have an isomorphism

(3)  T : RHom_{X×X}(Δ_{!}O_{X}, O_{Δ}) ≃ RHom_{X×X}(O_{Δ}, Δ_{∗}S_{X})

given by tensoring an element of RHom_{X×X}(Δ_{!}O_{X}, O_{Δ}) on the right by the shifted line bundle p_{2}^{∗}S_{X} and making the obvious identifications. Now, since Δ^{∗} is the left adjoint of Δ_{∗} we have an isomorphism

(4)  J : RHom_{X×X}(O_{Δ}, Δ_{∗}S_{X}) ≃ RHom_{X}(Δ^{∗}O_{Δ}, S_{X}).

Let D_{Δ} : RHom_{X}(O_{X}, Δ^{∗}O_{Δ}) → RHom_{X}(Δ^{∗}O_{Δ}, S_{X}) denote the composite J ∘ T ∘ I. We refer to D_{Δ} as the duality map.

The main theorem. The main theorem of this paper relates the HKR, twisted HKR and duality maps. This is a corrected version of Theorem 8 of Markarian’s preprint [5].

Note that I_{HKR} induces a map

I_{HKR} : RHom_{X}(O_{X}, Δ^{∗}O_{Δ}) → RHom_{X}(O_{X}, S^{•}(Ω_{X}[1])) = H^{∗}(X).

Similarly, { \atop I_{HKR}} induces a map

{ \atop I_{HKR}} : H^{∗}(X) = RHom_{X}(O_{X}, S^{•}(Ω_{X}[1])) → RHom_{X}(Δ^{∗}O_{Δ}, S_{X}).

Let J denote the endomorphism on S^{•}(Ω_{X}[1]) that multiplies ∧^{i}Ω_{X}[i] by (−1)^{i}. J induces an endomorphism on H^{q}(X, Ω_{X}^{p}). Let

td(T_{X}) ∈ ⊕_{i}H^{i}(X, Ω_{X}^{i})

denote the Todd genus of the tangent bundle of X. Recall that the wedge product ( - ∧ - ) : S^{•}(Ω_{X}[1])^{⊗2} → S^{•}(Ω_{X}[1]) induces a product on H^{∗}(X). We are now in a position to state the main theorem.

Theorem 1. The following diagram commutes.

RHom_{X}(O_{X}, Δ^{∗}O_{Δ}) ⟶^{D_{Δ}} RHom_{X}(Δ^{∗}O_{Δ}, S_{X})
↓^{I_{HKR}}  { \atop I_{HKR}}↑
H^{∗}(X)  ⟶^{(−∧td(T_{X}))∘J}  H^{∗}(X).

The map in the bottom row of the above diagram takes an element α ∈ H^{∗}(X) to

J(α) ∧ td(T_{X}).

Theorem 1 can be thought of as an explicit computation of the duality map.

1.2. The Mukai pairing. We now try to understand how Theorem 1 leads to the relative Riemann–Roch theorem. It is in this attempt that we see how Theorem 1 helps us calculate what the Mukai pairing on Hochschild homology [1] descends to in Hodge cohomology. This settles a part of a conjecture by Caldararu in [2].

Let HH_{i}(X) denote Hom_{D^{b}(X)}(O_{X}, Δ^{∗}O_{Δ}[i]). HH_{i}(X) is called the ith Hochschild homology of X. Let I, T and J be as in (2), (3), and (4) respectively. Let tr_{X} and tr_{X×X} denote the canonical identifications of Hom_{D^{b}(X)}(O_{X}, S_{X}) and Hom_{D^{b}(X×X)}(Δ_{∗}S_{X}^{−1}, Δ_{∗}S_{X}) with 𝕂 respectively. We recall that Caldararu [1] defined a Mukai pairing on Hochschild homology. This was a pairing

HH_{i}(X) × HH_{−i}(X) → 𝕂 (5)

(v, w) ↦ tr_{X×X}(T (I(v)) ∘ I(w)).

On the other hand we can consider the pairing

HH_{i}(X) × HH_{−i}(X) → 𝕂 (6)

(v, w) ↦ tr_{X}(D_{Δ}(v) ∘ w).

Proposition 1. The pairings on Hochschild homology defined in (5) and (6) are identical.

Proof. By definition, D_{Δ}(v) = J (T (I(v))). The proposition would follow if we can check that

(7)  tr_{X}(J (α) ∘ β) = tr_{X×X}(α ∘ I(β))

for any α ∈ Hom_{D^{b}(X×X)}(O_{Δ}, Δ_{∗}S_{X}[i]) and β ∈ Hom_{D^{b}(X)}(O_{X}[i], Δ^{∗}O_{Δ}). This is just saying that I is the map “dual” to the map J in (4).

We remark here that the assertion (7) is similar to the last part of Proposition 3.1 of Caldararu’s paper [1]. Proposition 3.1 of [1] describes the construction of a right adjoint to a functor from D^{b}(X) to D^{b}(Y) given a left adjoint (via Serre duality). In our situation, Δ_{!} is a left adjoint to Δ^{∗} : D^{b}(X × X) → D^{b}(X). Δ_{!} was constructed in [1] using the right adjoint Δ_{∗} of Δ^{∗} and Serre duality. □

Moreover, let ∫_{X} : H^{∗}(X) → 𝕂 denote the linear functional that is 0 on H^{p,q}(X) if (p, q) ≠ (n, n) and coincides with the identification of H^{n}(X, Ω_{X}^{n}) = Hom_{D^{b}(X)}(O_{X}, S_{X}) with 𝕂 on H^{n,n}(X). Recall the defini-

tion of the twisted HKR map { \atop I_{HKR}}. The following proposition is immediate from the definition of { \atop I_{HKR}}.

Proposition 2. If a ∈ H^{∗}(X) and b ∈ RHom_{X}(O_{X}, Δ^{∗}O_{Δ}), then

tr_{X}({ \atop I_{HKR}}(a) ∘ b) = ∫_{X} I_{HKR}(b) ∧ a.

Let J be as in Theorem 1. Let ⟨, ⟩ denote the Mukai pairing on Hochschild homology in this subsection only. The following proposition is immediate from Proposition 1, Proposition 2 and Theorem 1.

Proposition 3. If a ∈ HH_{i}(X) and b ∈ HH_{−i}(X) then

⟨a, b⟩ = ∫_{X} I_{HKR}(b) ∧ J(I_{HKR}(a)) ∧ td(T_{X}).

Note that the product on H^{∗}(X) is graded commutative. Also note that ∫_{X} is nonvanishing only on H^{2n}(X). Therefore,

∫_{X} v ∧ w = ∫_{X} \overline{w} ∧ v

where \overline{w} is obtained from w by multiplying its component in H^{k}(X) by (−1)^{k}. Note that if w ∈ H^{∗}(X), \overline{J(w)} = K(w) where K is the endomorphism on H^{∗}(X) multiplying H^{q}(X, Ω_{X}^{p}) by (−1)^{q}. Since td(T_{X}) ∈ ⊕_{i}H^{2i}(X), td(T_{X}) commutes with every element of H^{∗}(X). Proposition 3 may therefore, be rewritten as

(8)  ⟨a, b⟩ = ∫_{X} K(I_{HKR}(a)) ∧ I_{HKR}(b) ∧ td(T_{X}).

A Mukai like pairing on Hodge cohomology. Now suppose that X is a smooth complex variety. Recall that a generalized Mukai pairing ⟨, ⟩_{C} has been defined by Caldararu [2] on the Hodge cohomology H^{∗}(X). Let ω_{X} = Ω_{X}^{n} and let τ denote the endomorphism on H^{∗}(X) that is multiplication by \sqrt{(−1)}^{p+q} on H^{q}(X, Ω_{X}^{p}). Let ch : D^{b}(X) → H^{∗}(X) denote the Chern character. Recall from [2] that \sqrt{ch(ω_{X})} is a well-defined element of H^{∗}(X). Then, if v, w ∈ H^{∗}(X),

(9)  ⟨v, w⟩_{C} = ∫_{X} \frac{τ(v)}{\sqrt{ch(ω_{X})}} ∧ w.

Let \overline{τ} denote the endomorphism on H^{∗}(X) given by multiplication by \sqrt{(−1)}^{q−p} on H^{q}(X, Ω_{X}^{p}). Then, K = τ ∘ \overline{τ} = \overline{τ} ∘ τ. Define a pairing ⟨, ⟩_{M} on H^{∗}(X) by setting

(10)  ⟨v, w⟩_{M} = ⟨\overline{τ}(v), w⟩_{C} = ∫_{X} \frac{K(v)}{\sqrt{ch(ω_{X})}} ∧ w.

Proposition 4. If f : X → Y is a proper morphism of smooth complex varieties, then

⟨f^{∗}(v), w⟩_{M} = ⟨v, f_{∗}(w)⟩_{M}

for all v ∈ H^{∗}(Y) and w ∈ H^{∗}(X).

Proof. We recall from Caldararu [2] that ⟨f^{∗}(v), w⟩_{C} = ⟨v, f_{∗}(w)⟩_{C} for all v ∈ H^{∗}(Y) and w ∈ H^{∗}(X). Now, f^{∗}(\overline{τ}v) = \overline{τ}(f^{∗}(v)) for any v ∈ H^{∗}(Y). Thus,

⟨f^{∗}(v), w⟩_{M} = ⟨\overline{τ}(f^{∗}(v)), w⟩_{C} = ⟨f^{∗}(\overline{τ}(v)), w⟩_{C}
= ⟨\overline{τ}(v), f_{∗}(w)⟩_{C} = ⟨v, f_{∗}(w)⟩_{M}

for all v ∈ H^{∗}(Y) and w ∈ H^{∗}(X). □

Proposition 5. If a ∈ HH_{i}(X) and b ∈ HH_{−i}(X) then

⟨a, b⟩ = ⟨I_{HKR}(a) ∧ \sqrt{td(T_{X})}, I_{HKR}(b) ∧ \sqrt{td(T_{X})}⟩_{M} .

Proof. Since \sqrt{td(T_{X})} is a linear combination of elements in H^{i}(X, Ω_{X}^{i}), it commutes with other elements in H^{∗}(X). The RHS of the equation in this proposition is therefore,

∫_{X} \frac{K(I_{HKR}(a) ∧ \sqrt{td(T_{X})})}{\sqrt{ch(ω_{X})}} ∧ \sqrt{td(T_{X})} ∧ I_{HKR}(b).

But K is a ring endomorphism of H^{∗}(X). Thus

K(I_{HKR}(a) ∧ \sqrt{td(T_{X})}) = K(I_{HKR}(a)) ∧ K(\sqrt{td(T_{X})}).

But K(\sqrt{td(T_{X})}) = τ(\sqrt{td(T_{X})}) since both K and τ are multiplication by (−1)^{i} on H^{i}(X, Ω_{X}^{i}). It has also been shown in Caldararu [2] that

\frac{τ(\sqrt{td(T_{X})})}{\sqrt{ch(ω_{X})}} = \sqrt{td(T_{X})}.

It follows that

∫_{X} \frac{K(I_{HKR}(a) ∧ \sqrt{td(T_{X})})}{\sqrt{ch(ω_{X})}} ∧ \sqrt{td(T_{X})} ∧ I_{HKR}(b)

= ∫_{X} K(I_{HKR}(a)) ∧ td(T_{X}) ∧ I_{HKR}(b)

= ∫_{X} K(I_{HKR}(a)) ∧ I_{HKR}(b) ∧ td(T_{X}).

The desired proposition now follows from (8). □

Remark 1. Recall that if E ∈ D^{b}(X), then ch(E).\sqrt{td(T_{X})} is called the Mukai vector of E (Caldararu [2]). The pairing ⟨, ⟩_{M} is slightly different from the generalized Mukai pairing ⟨, ⟩_{C} defined by Caldararu [2]. However, if v and w are Mukai vectors of elements of D^{b}(X), then ⟨v, w⟩_{M} = ⟨v, w⟩_{C}.

Remark 2. Let X and Y be smooth complex varieties. Recall the definition of an integral transform Φ : D^{b}(X) → D^{b}(Y) from Caldararu [2], [1]. An integral transform Φ induces a map Φ_{∗} : H^{∗}(X) → H^{∗}(Y). We remark that the pairing ⟨, ⟩_{M} satisfies the adjointness one expects from a Mukai pairing. More precisely, if Φ : D^{b}(X) → D^{b}(Y) and Ψ : D^{b}(Y) → D^{b}(X) are integral transforms such that Ψ is a left adjoint of Φ, then

⟨Ψ_{∗}v, w⟩_{M} = ⟨v, Φ_{∗}w⟩_{M}

for all v ∈ D^{b}(Y) and w ∈ D^{b}(X). This follows from the analogous property for the pairing ⟨, ⟩_{C} and the fact (see Caldararu [2]) that integral transforms preserve the columns of the Hodge diamond. We are thus justified when we refer to the pairing ⟨, ⟩_{M} as a Mukai like pairing.

Remark 3. A part of the main conjecture of Caldararu [2] was that the HKR map twisted by the square root of the Todd genus of X preserves the Mukai pairing. However, instead of taking the Mukai pairing on Hochschild homology to ⟨, ⟩_{C}, it takes it to ⟨, ⟩_{M} by Proposition 5. The latter pairing is itself a Mukai like pairing and is very similar to the former pairing. However, ⟨, ⟩_{M} does not coincide with the Mukai pairing on the Hodge cohomology of a K3-surface in general. In particular, if v ∈ H^{2,0}(X) and w ∈ H^{0,2}(X) then ⟨v, w⟩_{M} ≠ ⟨v, w⟩_{C}. This is why we do not go so far as to call ⟨, ⟩_{M} a generalized Mukai pairing. We can however, justifiably say that the HKR map twisted by the square root of the Todd genus of X “almost preserves” the Mukai pairing.

1.3. The relative Riemann–Roch theorem. Recall that Caldararu [1] defined a Chern character

Ch : D^{b}(X) → HH_{0}(X).

He also showed in [1] that if f : X → Y is a proper morphism of smooth schemes, then

(11)  Ch(f_{∗}E) = f_{∗}Ch(E)

for any E ∈ D^{b}(X). Also, in [2], it was shown that

I_{HKR} ∘ Ch(E) = ch(E).

The relative Riemann–Roch theorem follows from Proposition 5.

Relative Riemann–Roch Theorem. Let f : X → Y be a proper morphism of smooth proper complex varieties. Then, if E is a vector bundle on X,

∫_{X} f^{∗}(l)ch(E)td(T_{X}) = ∫_{Y} lch(f_{∗}E)td(T_{Y})

for any l ∈ H^{∗}(Y).

Proof. Note that I_{HKR} : HH_{∗}(X) → H^{∗}(X) is an isomorphism of complex vector spaces. Let

a = I_{HKR}^{−1}(K(l)) ∈ HH_{∗}(Y).

Then,

(12)  ⟨f^{∗}a, Ch(E)⟩ = ⟨a, f_{∗}Ch(E)⟩ = ⟨a, Ch(f_{∗}E)⟩.

The first equality in (12) is due to the adjointness property of the Mukai pairing (see Caldararu [1]). The second equality in (12) is due to (11). By Proposition 5 and the fact that td(T_{X}) commutes with other elements of H^{∗}(X),

(13)  ⟨f^{∗}a, Ch(E)⟩ = ∫_{X} K(I_{HKR}(f^{∗}a))I_{HKR}(Ch(E))td(T_{X})
⟨a, Ch(f_{∗}E)⟩ = ∫_{Y} K(I_{HKR}(a))I_{HKR}(Ch(f_{∗}E))td(T_{Y}).

Now recall that I_{HKR} ∘ f^{∗} = f^{∗} ∘ I_{HKR} (Theorem 7 of [5]) and note that K ∘ f^{∗} = f^{∗} ∘ K. Also I_{HKR} ∘ Ch = ch. Now, applying these facts to (13) and using (12), the desired theorem follows. □

Remark 4. Note that the Chern character to Hochschild homology actually commutes with push-forwards as shown in Caldararu [1]. The Todd genus in the relative Riemann–Roch theorem thus occurs as a consequence of the fact that the Mukai pairing on Hochschild homology does not correspond to a Mukai like pairing on Hodge cohomology under I_{HKR}. For the Mukai pairing on Hochschild homology to be “preserved” in any sense, one has to twist I_{HKR} by \sqrt{td(T_{X})}.

2. Two “connections” on the Hochschild chain complex Ĉ^{•}(X)

2.1. The completed bar and Hochschild chain complexes. Let U = Spec R be an open affine subscheme of X. The restriction of O_{Δ} to U × U has a free R ⊗ R-module resolution given by the bar resolution:

B^{−n}(R) = R^{⊗(n+2)}
d(r_{0} ⊗ ⋯ ⊗ r_{n+1}) = r_{0}r_{1} ⊗ ⋯ ⊗ r_{n+1} − r_{0} ⊗ r_{1}r_{2} ⊗ ⋯ ⊗ r_{n+1}
+ ⋯ + (−1)^{n}r_{0} ⊗ ⋯ ⊗ r_{n}r_{n+1}

r_{i} ∈ R.

The R ⊗ R-module structure is given by multiplication with the extreme factors. Let I_{n} denote the kernel of the (n+2)-fold multiplication R^{⊗(n+2)} → R. Let

B̂^{−n}(R) = lim_{k} \frac{B^{−n}(R)}{I_{n}^{k}}.

Note that each summand of the differential d takes I_{n} to I_{n−1}. It follows that the differential on B^{•}(R) extends to yield a differential on B̂^{•}(R). Yekutieli [9] shows that completing the bar resolution in this manner yields a complex B̂^{•}(X) of coherent sheaves on X × X. He also shows that B̂^{•}(X) is a resolution of O_{Δ} by flat O_{X×X}-modules.

It follows that Δ^{∗}O_{Δ} is represented by the complex

Ĉ^{•}(X) := O_{X} ⊗_{Δ^{−1}O_{X×X}} Δ^{−1}B̂^{•}(X).

Ĉ^{•}(X) is called the complex of completed Hochschild chains on X. On an open subscheme U = Spec R of X before completion,

C^{−n}(R) = R^{⊗(n+1)}
d(r_{0} ⊗ ⋯ ⊗ r_{n}) = r_{0}r_{1} ⊗ ⋯ ⊗ r_{n} − r_{0} ⊗ r_{1}r_{2} ⊗ ⋯ ⊗ r_{n}
+ ⋯ + (−1)^{n−1}r_{0} ⊗ ⋯ ⊗ r_{n−1}r_{n}
+ (−1)^{n}r_{n}r_{0} ⊗ ⋯ ⊗ r_{n−1}

Ĉ^{−n}(R) = lim_{k} \frac{B^{−n}(R)}{I_{n}^{k}} ⊗_{R^{⊗(n+2)}} C^{−n}(R).

Yekutieli [9] also showed that RD(Ĉ^{•}(X)) is represented in D^{b}(X) by the complex D_{poly}^{•}(X) of poly-differential operators on X equipped with Hochschild coboundary.

Let us describe some operations on Ĉ^{•}(X) that endow it with the structure of a Hopf algebra in Ch^{−}(O_{X} − mod) (and therefore in D^{b}(X)).

Product on Ĉ^{•}(X): The product m : Ĉ^{•}(X)⊗_{O_{X}} Ĉ^{•}(X) → Ĉ^{•}(X) is given by the signed shuffle product. Recall that a (p, q)-shuffle σ is a permutation of {1, …, p + q} such that σ(1) < ⋯ < σ(p) and σ(p + 1) < ⋯ < σ(p + q). Denote the set of (p, q)-shuffles by Sh_{p,q}. On an open subscheme U = Spec R of X before completion, this product is given by

(r_{0} ⊗ r_{1} ⊗ ⋯ ⊗ r_{p}) ⊗_{R} (r′_{0} ⊗ r_{p+1} ⊗ ⋯ ⊗ r_{p+q})

↦ ∑_{σ∈Sh_{p,q}} sgn(σ)r_{0}r′_{0} ⊗ r_{σ^{−1}(1)} ⊗ ⋯ ⊗ r_{σ^{−1}(p+q)}.

This is easily seen to be a (graded) commutative product.

Coproduct on Ĉ^{•}(X). The coproduct Ĉ^{•}(X) → Ĉ^{•}(X) ⊗_{O_{X}} Ĉ^{•}(X) is given by the cut coproduct . Contrary to the usual practise, we denote the coproduct by C to avoid confusion with Δ which denotes the diagonal map X → X × X in this paper. On an open subscheme U = Spec R of X before completion,

C(r_{0} ⊗ ⋯ ⊗ r_{n}) = ∑_{p+q=n} r_{0} ⊗ r_{1} ⊗ ⋯ ⊗ r_{p} ⊗_{R} 1 ⊗ r_{p+1} ⊗ ⋯ ⊗ r_{n}.

Unit for Ĉ^{•}(X). There is a unit map ϵ : O_{X} → Ĉ^{•}(X). On an open subscheme U = Spec R of X before completion, ϵ is given by the composite

R ≃ C^{0}(R) ↪ C^{•}(R).

Counit for Ĉ^{•}(X). There is a counit η : Ĉ^{•}(X) → O_{X}. On an open subscheme U = Spec R of X before completion, this is given by the projection from C^{•}(R) to C^{0}(R).

Proposition 6. Ĉ^{•}(X) is a Hopf algebra in Ch^{−}(O_{X} − mod) via M, C, ϵ and η.

Proof. Yekutieli [9] showed that applying the functor Hom_{O_{X}}^{cont}(−, O_{X}) (O_{X} is given the discrete topology) to Ĉ^{•}(X) ∈ Ch^{−}(O_{X} − mod) yields

D_{poly}^{•}(X) ∈ Ch^{+}(O_{X} − mod).

It is easy to verify that this functor takes the product, coproduct, unit and counit of Ĉ^{•}(X) to the coproduct, product, counit and unit of D_{poly}^{•}(X) respectively. The desired proposition then follows from Proposition 2 of [7]. □

Antipode on Ĉ^{•}(X). Ĉ^{•}(X) also comes equipped with an antipode map. We will denote this map by S. On U = Spec R before completion,

S(r_{0} ⊗ ⋯ ⊗ r_{n}) = (−1)^{\frac{n(n+1)}{2}} r_{0} ⊗ r_{n} ⊗ r_{n−1} ⊗ ⋯ ⊗ r_{1}.

The Hochschild–Kostant–Rosenberg (HKR) map. There is a quasiisomorphism (see Yekutieli [9])

I_{HKR} : Ĉ^{•}(X) → S^{•}(Ω[1])

of complexes of O_{X}-modules. On an open subscheme U = Spec R of X before completion,

I_{HKR}(r_{0} ⊗ ⋯ ⊗ r_{n}) = \frac{1}{n!}r_{0}dr_{1} ∧ ⋯ ∧ dr_{n}.

2.2. Two connections on Ĉ^{•}(X). Let π_{k} : S^{•}(Ω[1]) → Ω^{k}[k] denote the natural projection. Denote by α_{R} the composite

Ĉ^{•}(X) ⟶^{C} Ĉ^{•}(X) ⊗ Ĉ^{•}(X) ⟶^{(1_{Ĉ^{•}(X)}⊗π_{1}∘I_{HKR})} Ĉ^{•}(X) ⊗ Ω[1].

More concretely, on an open subscheme U = Spec R of X before completion,

α_{R}(r_{0} ⊗ ⋯ ⊗ r_{n}) = r_{0} ⊗ ⋯ ⊗ r_{n−1} ⊗ dr_{n}.

Let α_{L} : Ĉ^{•}(X) → Ĉ^{•}(X) ⊗ Ω[1] be the map such that

α_{L}(r_{0} ⊗ ⋯ ⊗ r_{n}) = (−1)^{n−1}r_{0} ⊗ r_{2} ⊗ ⋯ ⊗ r_{n} ⊗ dr_{1}

on any open subscheme U = Spec R of X before completion.

Then, α_{L} = −(S ⊗ Ω[1]) ∘ α_{R} ∘ S. Let α_{R} ⊗ Ĉ^{•}(X) denote the composite

Ĉ^{•}(X) ⊗ Ĉ^{•}(X)
α_{R}⊗Ĉ^{•}(X)↓
Ĉ^{•}(X) ⊗ Ω[1] ⊗ Ĉ^{•}(X) ⟶^{Ĉ^{•}(X)⊗τ} Ĉ^{•}(X) ⊗ Ĉ^{•}(X) ⊗ Ω[1]

where τ : Ω[1] ⊗ Ĉ^{•}(X) → Ĉ^{•}(X) ⊗ Ω[1] is the map that swaps factors. Similarly, let α_{L} ⊗ Ĉ^{•}(X) denote the composite

Ĉ^{•}(X) ⊗ Ĉ^{•}(X)
α_{L}⊗Ĉ^{•}(X)↓
Ĉ^{•}(X) ⊗ Ω[1] ⊗ Ĉ^{•}(X) ⟶^{Ĉ^{•}(X)⊗τ} Ĉ^{•}(X) ⊗ Ĉ^{•}(X) ⊗ Ω[1].

We now have the following proposition.

Proposition 7. The following diagrams commute in Ch^{−}(O_{X} − mod).

\begin{matrix}Ĉ^{•}(X) ⊗ Ĉ^{•}(X) & ⟶^{α_{R}⊗Ĉ^{•}(X)+Ĉ^{•}(X)⊗α_{R}} & Ĉ^{•}(X) ⊗ Ĉ^{•}(X) ⊗ Ω[1] \\ ↓^{m} & & ^{m⊗Ω[1]}↓ \\ Ĉ^{•}(X) & ⟶^{α_{R}} & Ĉ^{•}(X) ⊗ Ω[1] \end{matrix} (14)

\begin{matrix}Ĉ^{•}(X) ⊗ Ĉ^{•}(X) & ⟶^{α_{L}⊗Ĉ^{•}(X)+Ĉ^{•}(X)⊗α_{L}} & Ĉ^{•}(X) ⊗ Ĉ^{•}(X) ⊗ Ω[1] \\ ↓^{m} & & ^{m⊗Ω[1]}↓ \\ Ĉ^{•}(X) & ⟶^{α_{L}} & Ĉ^{•}(X) ⊗ Ω[1]. \end{matrix} (15)

Proof. This proposition is proven by a combinatorial argument. On an open subscheme U = Spec R of X before completion,

(16) (α_{R}⊗Ĉ^{•}(X) + Ĉ^{•}(X) ⊗ α_{R})
((r_{0} ⊗ r_{1} ⊗ ⋯ ⊗ r_{p}) ⊗_{R} (1 ⊗ r_{p+1} ⊗ ⋯ ⊗ r_{p+q}))

= (−1)^{q}(r_{0} ⊗ r_{1} ⊗ ⋯ ⊗ r_{p−1}) ⊗_{R} (1 ⊗ r_{p+1} ⊗ ⋯ ⊗ r_{p+q}) ⊗ dr_{p} + (r_{0} ⊗ r_{1} ⊗ ⋯ ⊗ r_{p}) ⊗_{R} (1 ⊗ r_{p+1} ⊗ ⋯ ⊗ r_{p+q−1}) ⊗ dr_{p+q}.

Note that if σ is a (p, q)-shuffle, then σ^{−1}(p+q) = p or σ^{−1}(p+q) = p+q. Let Sh_{p,q}^{1} denote the set of all (p, q)-shuffles σ such that σ^{−1}(p + q) = p. Let Sh_{p,q}^{2} denote Sh_{p,q} \ Sh_{p,q}^{1}.

Also note that there is a sign preserving bijection from Sh_{p,q−1} to Sh_{p,q}^{2} the inverse of which takes an element σ of Sh_{p,q}^{2} to its restriction to the set {1, …, p + q − 1}. Denote this bijection by I : Sh_{p,q−1} → Sh_{p,q}^{2}.

We also have a bijection from Sh_{p,q}^{1} to Sh_{p−1,q}. Let

ψ : {1, …, p + q − 1} → {1, …, p − 1, p + 1, …, p + q − 1}

be the unique order-preserving map.The permutation in Sh_{p−1,q} corresponding to a permutation σ in Sh_{p,q}^{1} is given by the composite

{1, …, p + q − 1} →^{ψ} {1, …, p − 1, p + 1, …, p + q} →^{σ} {1, …, p + q − 1}.

This bijection from Sh_{p,q}^{1} to Sh_{p−1,q} however, changes the sign by (−1)^{q}. Denote the inverse of this bijection by J : Sh_{p−1,q} → Sh_{p,q}^{1}.

Then

m((r_{0} ⊗ r_{1} ⊗ ⋯ ⊗ r_{p}) ⊗_{R} (1 ⊗ r_{p+1} ⊗ ⋯ ⊗ r_{p+q}))

= ∑_{σ∈Sh_{p,q}} sgn(σ)r_{0} ⊗ r_{σ^{−1}(1)} ⊗ … r_{σ^{−1}(p+q)}

= ∑_{σ∈Sh_{p,q}^{1}} sgn(σ)r_{0} ⊗ r_{σ^{−1}(1)} ⊗ … r_{σ^{−1}(p+q)}

+ ∑_{σ∈Sh_{p,q}^{2}} sgn(σ)r_{0} ⊗ r_{σ^{−1}(1)} ⊗ … r_{σ^{−1}(p+q)}

= ∑_{σ∈Sh_{p−1,q}} (−1)^{q}sgn(σ)r_{0} ⊗ r_{J(σ)^{−1}(1)} ⊗ ⋯ ⊗ r_{J(σ)^{−1}(p+q−1)} ⊗ r_{p}

+ ∑_{σ∈Sh_{p,q−1}} sgn(σ)r_{0} ⊗ r_{I(σ)^{−1}(1)} ⊗ ⋯ ⊗ r_{I(σ)^{−1}(p+q−1)} ⊗ r_{p+q}

= (−1)^{q}m((r_{0} ⊗ r_{1} ⊗ ⋯ ⊗ r_{p−1}) ⊗_{R} (1 ⊗ r_{p+1} ⊗ ⋯ ⊗ r_{p+q})) ⊗ r_{p} + m((r_{0} ⊗ r_{1} ⊗ ⋯ ⊗ r_{p}) ⊗_{R} (1 ⊗ r_{p+1} ⊗ ⋯ ⊗ r_{p+q−1})) ⊗ r_{p+q}.

It follows that

α_{R}(m((r_{0} ⊗ r_{1} ⊗ ⋯ ⊗ r_{p}) ⊗_{R} (1 ⊗ r_{p+1} ⊗ ⋯ ⊗ r_{p+q})))

= (−1)^{q}m((r_{0} ⊗ r_{1} ⊗ ⋯ ⊗ r_{p−1}) ⊗_{R} (1 ⊗ r_{p+1} ⊗ ⋯ ⊗ r_{p+q})) ⊗ dr_{p}

+ m((r_{0} ⊗ r_{1} ⊗ ⋯ ⊗ r_{p}) ⊗_{R} (1 ⊗ r_{p+1} ⊗ ⋯ ⊗ r_{p+q−1})) ⊗ dr_{p+q}.

It follows from (16) that this is precisely

(m ⊗ Ω[1])((α_{R} ⊗ Ĉ^{•}(X) + Ĉ^{•}(X) ⊗ α_{R})) ((r_{0} ⊗ r_{1} ⊗ ⋯ ⊗ r_{p}) ⊗_{R} (1 ⊗ r_{p+1} ⊗ ⋯ ⊗ r_{p+q})).

This proves that the diagram (14) commutes. Proving that the diagram (15) commutes is very similar and left to the reader. □

Let α_{R}^{∘i} denote the composite

Ĉ^{•}(X)
α_{R}↓
Ĉ^{•}(X) ⊗ Ω[1]
α_{R}⊗Ω[1]↓
Ĉ^{•}(X) ⊗ Ω[1]^{⊗2} ⟶ ⋯ ⟶^{α_{R}⊗Ω[1]^{⊗i−1}} Ĉ^{•}(X) ⊗ Ω[1]^{⊗i}.

Let p : Ω[1]^{⊗i} → Ω^{i}[i] be the standard projection. On an open subscheme U = Spec R, of X,

p(r_{0}dr_{1} ⊗ ⋯ ⊗ dr_{i}) = r_{0}dr_{1} ∧ ⋯ ∧ dr_{i}.

Let α_{R}^{i} denote the composite

Ĉ^{•}(X) ⟶^{α_{R}^{∘i}} Ĉ^{•}(X) ⊗ Ω[1]^{⊗i} ⟶^{Ĉ^{•}(X)⊗p} Ĉ^{•}(X) ⊗ Ω^{i}[i].

Let exp(α_{R}) denote the sum

∑_{i} \frac{1}{i!}α_{R}^{i} : Ĉ^{•}(X) → Ĉ^{•}(X) ⊗ S^{•}(Ω[1]).

We then have the following proposition.

Proposition 8.

(Ĉ^{•}(X) ⊗ I_{HKR}) ∘ C = exp(α_{R}).

Proof. On an open subscheme U = Spec R before completion,

exp(α_{R})(r_{0} ⊗ ⋯ ⊗ r_{n})

= \frac{1}{i!} ∑_{i} r_{0} ⊗ ⋯ ⊗ r_{n−i} ⊗_{R} dr_{n−i+1} ∧ ⋯ ∧ dr_{n}

= ∑_{i} r_{0} ⊗ ⋯ ⊗ r_{n−i} ⊗_{R} I_{HKR}(1 ⊗ r_{n−i+1} ⊗ ⋯ ⊗ r_{n})

= (Ĉ^{•}(X) ⊗ I_{HKR})(∑_{i} r_{0} ⊗ ⋯ ⊗ r_{n−i} ⊗_{R} 1 ⊗ r_{n−i+1} ⊗ ⋯ ⊗ r_{n})

= (Ĉ^{•} ⊗ I_{HKR}) ∘ C(r_{0} ⊗ ⋯ ⊗ r_{n}).

This verifies the desired proposition. □

Let τ : Ω[1] ⊗ Ω[1] → Ω[1] ⊗ Ω[1] denote the swap map. The following proposition tells us that α_{L} and α_{R} “commute” with each other.

Proposition 9.

(α_{R} ⊗ Ω[1]) ∘ α_{L} − (Ĉ^{•}(X) ⊗ τ) ∘ (α_{L} ⊗ Ω[1]) ∘ α_{R} = 0.

Proof. On an open subscheme U = Spec R before completion,

(α_{R} ⊗ Ω[1]) ∘ α_{L}(r_{0} ⊗ ⋯ ⊗ r_{n}) = (−1)^{n−1}(r_{0} ⊗ r_{2} ⊗ ⋯ ⊗ r_{n−1}) ⊗_{R} dr_{n} ⊗_{R} dr_{1}

(α_{L} ⊗ Ω[1]) ∘ α_{R}(r_{0} ⊗ ⋯ ⊗ r_{n}) = (−1)^{n−2}(r_{0} ⊗ r_{2} ⊗ ⋯ ⊗ r_{n−1}) ⊗_{R} dr_{1} ⊗_{R} dr_{n}

(Ĉ^{•}(X) ⊗ τ)((r_{0} ⊗ r_{2} ⊗ ⋯ ⊗ r_{n−1}) ⊗_{R} dr_{1} ⊗_{R} dr_{n}) = −(r_{0} ⊗ r_{2} ⊗ ⋯ ⊗ r_{n−1}) ⊗_{R} dr_{n} ⊗_{R} dr_{1}.

The desired proposition is now immediate. □

2.3. Remark — the beginning of a dictionary. For reasons that will become clear later in this section, the reader should think of Ĉ^{•}(X) as analogous to the ring of functions on an open “symmetric” neighborhood U_{G} of the identity of a Lie group G. By “symmetric” we mean that if g ∈ U_{G} then g^{−1} ∈ U_{G}. T [−1] is the analog of the Lie algebra 𝔤 of the Lie group G. Thus, Ω[1] is the analog of 𝔤^{∗}. Proposition 7 says that in this picture, both α_{L} and α_{R} are analogs of “connections” on the ring of functions of G.

In the same picture, S^{•}(Ω[1]) is to be thought of as analogous to the ring of functions on a neighborhood V of 0 in 𝔤 that is diffeomorphic to U_{G} via the exponential map. The Hochschild–Kostant–Rosenberg map is then the analog of the pullback by the exponential map exp^{∗}.

The antipode map S is the analog of the pullback by the map which takes an element of G to its inverse.

2.4. More on the maps α_{R} and α_{L}. The question that arises at this stage is, “Can α_{L} and α_{R} be described by explicit formulae as maps in D^{b}(X) from S^{•}(Ω[1]) to S^{•}(Ω[1])⊗Ω[1] ?”. Markarian sought to answer this in Theorem 1 of [5]. The proof there was however erroneous. A result dual to what we want is available in [7]. A later version [6] of [5] also contains a result equivalent to Theorem 1 of [5].

Recall from Kapranov [3] that T [−1] is a Lie algebra object in D^{b}(X). The Lie bracket of T [−1] is given by the Atiyah class

At_{T} : T [−1] ⊗ T [−1] → T [−1]

of the tangent bundle of X. It is also known that the universal enveloping algebra of T [−1] in D^{b}(X) is represented by the complex D_{poly}^{•}(X). This was proven in [7]. Equivalent results have been proven using methods different from that in [7] by Markarian [6] and Roberts and Willerton [8].

Let μ denote the wedge product on S^{•}(T [−1]). Let

δ : S^{•}(T [−1]) ⊗ T [−1] → S^{•}(T [−1]) ⊗ T [−1] ⊗ T [−1]

be the map

δ(v_{1} ∧ ⋯ ∧ v_{k} ⊗ y) = ∑_{i=1}^{i=k} (−1)^{k−i}{ \atop v_{1} ∧ ⋯ i ⋯ ∧ v_{k}} ⊗ v_{i} ⊗ y

for sections v_{1}, …, v_{k}, y of T over an open subscheme U = Spec R of X. We have a map

\overline{ω} : S^{•}(T [−1]) ⊗ T [−1] → S^{•}(T [−1]) ⊗ T [−1]
\overline{ω} = (1_{S^{•}(T [−1])} ⊗ At_{T} ) ∘ δ.

Note that μ∘\overline{ω} yields the right adjoint action of the Lie algebra object T [−1] on S^{•}(T [−1]).

The Hochschild–Kostant–Rosenberg map I_{HKR} : S^{•}(T [−1]) → D_{poly}^{•}(X) is the “dual” of the HKR map I_{HKR} : Ĉ^{•}(X) → S^{•}(Ω[1]). The following theorem, which figures as Corollary 1 in [7] and Theorem 2 in [6], describes the Duflo-like error term that measures how the map

I_{HKR} : S^{•}(T [−1]) → D_{poly}^{•}(X)

fails to commute with multiplication.

Theorem 2 (Recalled from [7]; [6] has a similar result). The following diagram commutes in D^{b}(X).

D_{poly}^{•}(X) ⊗ D_{poly}^{•}(X) ⟶^{m} D_{poly}^{•}(X)
↑^{I_{HKR}⊗I_{HKR}}  ^{I_{HKR}}↑
S^{•}(T [−1]) ⊗ T [−1] ⟶^{μ∘ \frac{\overline{ω}}{1−e^{−\overline{ω}}}} S^{•}(T [−1]).

Note that applying the functor RD to \overline{ω} gives us a morphism in D^{b}(X)

\overline{ω} : S^{•}(Ω[1]) ⊗ Ω[1] → S^{•}(Ω[1]) ⊗ Ω[1].

Further, denote the comultiplication on S^{•}(Ω[1]) by C_{Ω}. Denote the map

(S^{•}(Ω[1]) ⊗ π_{1}) ∘ C_{Ω} : S^{•}(Ω[1]) → S^{•}(Ω[1]) ⊗ Ω[1]

by \overline{C}.

We denote the map

\frac{\overline{ω}}{e\overline{ω} − 1} ∘ \overline{C} : S^{•}(Ω[1]) → S^{•}(Ω[1]) ⊗ Ω[1]

by Φ_{L}. The map

\frac{\overline{ω}}{1 − e^{−\overline{ω}}} ∘ \overline{C} : S^{•}(Ω[1]) → S^{•}(Ω[1]) ⊗ Ω[1]

will be denoted by Φ_{R}.

We can now state the following theorem. This can be thought of as the starting point for the computations leading to Theorem 1.

Theorem 2′. The following diagrams commute in D^{b}(X).

Ĉ^{•}(X) ⟶^{α_{R}} Ĉ^{•}(X) ⊗ Ω[1]
↓^{I_{HKR}}  ^{I_{HKR}⊗Ω[1]}↓
S^{•}(Ω[1]) ⟶^{Φ_{R}} S^{•}(Ω[1]) ⊗ Ω[1]

(17)

Ĉ^{•}(X) ⟶^{α_{L}} Ĉ^{•}(X) ⊗ Ω[1]
↓^{I_{HKR}}  ^{I_{HKR}⊗Ω[1]}↓
S^{•}(Ω[1]) ⟶^{Φ_{L}} S^{•}(Ω[1]) ⊗ Ω[1].

(18)

Proof. The fact that the diagram (17) commutes in D^{b}(X) is obtained by applying the functor RD to the diagram in Theorem 2.

Note that since α_{L} = −(S ⊗Ω[1])∘α_{R}∘S the following diagram commutes in Ch^{−}(O_{X} − mod) (and therefore in D^{b}(X)).

(19)  \begin{matrix}Ĉ^{•}(X) & ⟶^{α_{R}} & Ĉ^{•}(X) ⊗ Ω[1] \\ ↓^{S} & & ^{S⊗Ω[1]}↓ \\ Ĉ^{•}(X) & ⟶^{−α_{L}} & Ĉ^{•}(X) ⊗ Ω[1]. \end{matrix}

Recall that J is the endomorphism of S^{•}(Ω[1]) multiplying ∧^{i}Ω[i] by (−1)^{i}. Then,

(20)  I_{HKR} ∘ S = J ∘ I_{HKR}.

To see (20), note that on an open subscheme U = Spec R of X before completion,

I_{HKR} ∘ S(r_{0} ⊗ ⋯ ⊗ r_{n}) = (−1)^{\frac{n(n+1)}{2}} \frac{1}{n!}r_{0}dr_{n} ∧ ⋯ ∧ dr_{1}

= (−1)^{\frac{n(n+1)}{2}} (−1)^{\frac{n(n−1)}{2}} \frac{1}{n!}r_{0}dr_{1} ∧ ⋯ ∧ dr_{n}

= (−1)^{n}I_{HKR}(r_{0} ⊗ ⋯ ⊗ r_{n}).

Further, note that applying the functor RD to J yields the endomorphism I of S^{•}(T [−1]) that multiplies ∧^{i}T_{X}[−i] by (−1)^{i}. For sections v_{1}, …, v_{k}, y of T over an open subscheme U = Spec R of X,

δ(I(v_{1} ∧ ⋯ ∧ v_{k}) ⊗ y)

= (−1)^{k} ∑_{i=1}^{i=k} (−1)^{k−i}{ \atop v_{1} ∧ ⋯ i ⋯ ∧ v_{k}} ⊗ v_{i} ⊗ y

= (−1)^{k}(−1)^{k−1} ∑_{i=1}^{i=k} (−1)^{k−i}I({ \atop v_{1} ∧ ⋯ i ⋯ ∧ v_{k}}) ⊗ v_{i} ⊗ y.

It follows from the above computation and the fact that

\overline{ω} = (S^{•}(T [−1]) ⊗ At_{T}) ∘ δ

that the following diagram commutes in D^{b}(X).

(21)  \begin{matrix}S^{•}(T [−1]) ⊗ T [−1] & ⟶^{\overline{ω}} & S^{•}(T [−1]) ⊗ T [−1] \\ ↓^{I⊗T [−1]} & & ↓^{I⊗T [−1]} \\ S^{•}(T [−1]) ⊗ T [−1] & ⟶^{−\overline{ω}} & S^{•}(T [−1]) ⊗ T [−1]. \end{matrix}

Applying the functor RD to the diagram (21) we obtain the following diagram.

(22)

S^{•}(Ω[1]) ⊗ Ω[1] ⟶^{\overline{ω}} S^{•}(Ω[1]) ⊗ Ω[1]
↓^{J⊗Ω[1]}  ^{J⊗Ω[1]}↓
S^{•}(Ω[1]) ⊗ Ω[1] ⟶^{−\overline{ω}} S^{•}(Ω[1]) ⊗ Ω[1].

A calculation similar to the one made to verify (21) also shows that the following diagram commutes.

(23)

S^{•}(Ω[1]) ⟶^{\overline{C}} S^{•}(Ω[1]) ⊗ Ω[1]
↓^{J}  ^{J⊗Ω[1]}↓
S^{•}(Ω[1]) ⟶^{−\overline{C}} S^{•}(Ω[1]) ⊗ Ω[1].

Combining (23) and (22) we obtain the following commutative diagram.

(24)

S^{•}(Ω[1]) ⟶^{\frac{\overline{ω}}{1−e^{−\overline{ω}}} ∘\overline{C}} S^{•}(Ω[1]) ⊗ Ω[1]
↓^{J}  ^{J⊗Ω[1]}↓
S^{•}(Ω[1]) ⟶^{− \frac{\overline{ω}}{e\overline{ω}−1} ∘\overline{C}} S^{•}(Ω[1]) ⊗ Ω[1].

It follows from (17) and (24) that all squares in the diagram below commute in D^{b}(X).

(25)

Ĉ^{•}(X) ⟶^{α_{R}}  Ĉ^{•}(X) ⊗ Ω[1]
↓^{I_{HKR}}  ^{I_{HKR}⊗Ω[1]}↓
S^{•}(Ω[1]) ⟶^{\frac{\overline{ω}}{1−e^{−\overline{ω}}} ∘\overline{C}} S^{•}(Ω[1]) ⊗ Ω[1]
↓^{J}  ^{J⊗Ω[1]}↓
S^{•}(Ω[1]) ⟶^{− \frac{\overline{ω}}{e\overline{ω}−1} ∘\overline{C}} S^{•}(Ω[1]) ⊗ Ω[1].

The diagram (25) says that

−Φ_{L} ∘ J ∘ I_{HKR} = [(J ∘ I_{HKR}) ⊗ Ω[1]] ∘ α_{R}.

But by (20)

J ∘ I_{HKR} = I_{HKR} ∘ S ⟹ −Φ_{L} ∘ I_{HKR} ∘ S = (I_{HKR} ⊗ Ω[1]) ∘ (S ⊗ Ω[1]) ∘ α_{R}.

Since α_{L} = −(S ⊗ Ω[1]) ∘ α_{R} ∘ S and S ∘ S = 1Ĉ^{•}(X),

Φ_{L} ∘ I_{HKR} = (I_{HKR} ⊗ Ω[1]) ∘ α_{L}.

This proves that the diagram (18) commutes. □

2.5. A long remark — enlarging the dictionary. This subsection is a continuation of Section 2.3. Recall that Ĉ^{•}(X) should be thought of as analogous to the ring of functions on a “symmetric” neighborhood U_{G} of the identity in a Lie group G. T [−1] is analogous to the Lie algebra 𝔤 of G. S^{•}(Ω[1]) is analogous to the ring of functions on a neighborhood V of 0 in 𝔤 that is diffeomorphic to U_{G} via the exponential map.

I_{HKR} : Ĉ^{•}(X) → S^{•}(Ω[1])

is analogous to the pullback of functions by the map exp : 𝔤 → G.

2.5.1. The classical picture. I. Let us look at the classical picture for now. Choose a basis {X_{1}, …, X_{n}} of 𝔤 and a basis {Y_{1}, …, Y_{n}} of 𝔤^{∗} dual to {X_{1}, …, X_{n}}. Then, ∑_{i=1}^{i=n} X_{i} ⊗ Y_{i} yields an element 𝔤 ⊗ 𝔤^{∗}. Denote the ring of functions on U_{G} by C(G). We identify elements of 𝔤 (resp. elements of 𝔤^{∗}) with left-invariant vector fields (resp. left-invariant 1-forms) on U_{G} whenever necessary. Given any element of 𝔤 ⊗ 𝔤^{∗}, letting 𝔤 act on C(G) as a differential operator yields us a map from C(G) to C(G) ⊗ 𝔤^{∗} satisfying the Leibniz rule. Therefore, any element of 𝔤 ⊗ 𝔤^{∗} yields a global connection on C(G). It is easy to verify that the connection on C(G) yielded by ∑_{i=1}^{n} X_{i} ⊗ Y_{i} is precisely d_{G}, the de Rham differential from C(G) to the sections of the cotangent bundle of U_{G}.

Denote the ring of functions on V by C(𝔤). Note that replacing C(G) by C(𝔤) in the previous paragraph enables us to conclude ∑_{i=1}^{n} X_{i} ⊗ Y_{i} yields the connection d_{𝔤} on C(𝔤) where d_{𝔤} is precisely the de Rham differential from C(𝔤) to the sections of the cotangent bundle of V.

Note that elements of C(G) ⊗ 𝔤^{∗} are smooth functions from U_{G} to 𝔤^{∗}, i.e, sections of the cotangent bundle of U_{G}. Similarly, elements of C(𝔤) ⊗ 𝔤^{∗} are smooth functions from V to 𝔤^{∗}, i.e, sections of the cotangent bundle of V.

Given a smooth function A from V to End(𝔤) and a smooth function h from V to 𝔤 (resp. 𝔤^{∗}), one can obtain a smooth function A(h) from V to 𝔤 (resp. 𝔤^{∗}) by setting

A(h)(Z) = A(Z)h(Z)

for every Z ∈ V. Denote the smooth function Z ↦ ad(Z) from V to End(𝔤) by \overline{ad}.

Consider the connection Φ on C(𝔤) such that the following diagram commutes.

C(G) ⟶^{d_{G}} C(G) ⊗ 𝔤^{∗}
↓^{exp^{∗}}  ^{exp^{∗}⊗𝔤^{∗}}↓
C(𝔤) ⟶^{Φ} C(𝔤) ⊗ 𝔤^{∗}.

We are interested in comparing Φ with d_{𝔤}. This is done as follows:

Let f be any function on U_{G}. Then, d_{G}(f) is a 1-form on U_{G}. The pullback of the 1-form d_{G}(f) via the exponential map is precisely the 1form d_{𝔤}(exp^{∗}(f)). Recall the formula

d(exp)_{Z} = \frac{1 − e^{−ad(Z)}}{ad(Z)} ⟹ (exp^{∗} ⊗ d(exp)^{∗}) = \frac{1 − e^{−\overline{ad}}}{\overline{ad}} ∘ (exp^{∗} ⊗ 𝔤^{∗}).

By definition,

Φ(exp^{∗}(f)) = (exp^{∗} ⊗ 𝔤^{∗})d_{G}(f).

The fact that d_{𝔤}(exp^{∗}(f)) is the pullback of d_{G}(f) via the exponential map implies that

d_{𝔤}(exp^{∗}(f)) = (exp^{∗} ⊗ d(exp)^{∗})(d_{G}(f)) = \frac{1 − e^{−\overline{ad}}}{\overline{ad}} ∘ (exp^{∗} ⊗ 𝔤^{∗})(d_{G}(f))
= \frac{1 − e^{−\overline{ad}}}{\overline{ad}} ∘ Φ(exp^{∗}(f)).

Since any smooth function on V is of the form exp^{∗}(f),

(26)  d_{𝔤} = \frac{1 − e^{−\overline{ad}}}{\overline{ad}} ∘ Φ ⟹ Φ = \frac{\overline{ad}}{1 − e^{−\overline{ad}}} ∘ d_{𝔤}.

Now assume that U_{G} and V are chosen so that V is a sufficiently small open disc in 𝔤 containing 0. Also assume that G is not 1-dimensional. Then, any closed 1-form on U_{G} is also exact. Let Y ∈ 𝔤^{∗}. Since the left-invariant 1-form 1 ⊗ Y ∈ C(G) ⊗ 𝔤^{∗} is closed, it is exact as well. Thus, there is a function f_{Y} on U_{G} such that d_{G}(f_{Y}) = 1 ⊗ Y. Then, Φ(exp^{∗}(f_{Y})) = 1 ⊗ Y. On the other hand, d_{𝔤}(exp^{∗}(f_{Y})) is the pullback of the 1-form 1 ⊗ Y via the exponential map. The formula (26) thus amounts to the formula for the pullback of a left-invariant 1-form on U_{G} via the exponential map.

Note that the bracket ad : 𝔤⊗𝔤 → 𝔤 induces a map ad : 𝔤^{∗} → 𝔤^{∗} ⊗𝔤^{∗}. Let μ : C(𝔤) ⊗ 𝔤^{∗} → C(𝔤) denote the product taken after treating an element of 𝔤^{∗} as a function on V. We now claim that as an endomorphism in the space of smooth sections of V × 𝔤^{∗}, \overline{ad} is given by the following composite map

C(𝔤) ⊗ 𝔤^{∗} ⟶^{C(𝔤)⊗ad} C(𝔤) ⊗ 𝔤^{∗} ⊗ 𝔤^{∗} ⟶^{μ⊗𝔤^{∗}} C(𝔤) ⊗ 𝔤^{∗}.

Denote the above composite map by ω_{𝔤}.

To verify this, choose a basis {X_{i}} of 𝔤 and a basis {Y_{i}} of 𝔤^{∗} dual to {X_{i}}. Suppose that [X_{i}, X_{j}] = ∑_{k} C_{ij}^{k}X_{k}. Then, ad(Y_{k}) = ∑_{i,j} C_{ij}^{k}Y_{i} ⊗ Y_{j}. Therefore, if f ∈ C(𝔤), then

(μ ⊗ 𝔤^{∗}) ∘ (C(𝔤) ⊗ ad)(f ⊗ Y_{k})(∑_{i} a_{i}X_{i}) = f(∑_{i} a_{i}X_{i}) ∑_{i,j} a_{i}C_{ij}^{k}Y_{j}.

On the other hand,

\overline{ad}(f ⊗ Y_{k})(∑_{i} a_{i}X_{i}) = f(∑_{i} a_{i}X_{i})ad(∑_{i} a_{i}X_{i})(Y_{k}).

But ad(X_{i})(Y_{k}) = ∑_{j} C_{ij}^{k}Y_{j}. Therefore,

\overline{ad}(f ⊗ Y_{k})(∑_{i} a_{i}X_{i}) = f(∑_{i} a_{i}X_{i}) ∑_{i,j} a_{i}C_{ij}^{k}Y_{j}.

It follows that

\frac{\overline{ad}}{1 − e^{−\overline{ad}}} = \frac{ω_{𝔤}}{1 − e^{−ω_{𝔤}}}

and that

Φ = \frac{ω_{𝔤}}{1 − e^{−ω_{𝔤}}} ∘ d_{𝔤}.

2.5.2. The classical picture. II. Let \overline{exp} denote the map from V to U_{G} such that Z ↦ exp(Z)^{−1} = exp(−Z). The discussion in Section 2.5.1 with exp replaced by \overline{exp} together with the fact that

d(\overline{exp})_{Z} = −\frac{1 − e^{−ad(−Z)}}{ad(−Z)} = −\frac{e^{ad(Z)} − 1}{ad(Z)}

tells us that if

Ψ = −\frac{ω_{𝔤}}{e^{ω_{𝔤}} − 1} ∘ d_{𝔤}

then the following diagram commutes.

C(G) ⊗ 𝔤^{∗} ⟶^{d_{G}} C(G) ⊗ 𝔤^{∗}
↓^{\overline{exp}^{∗}}  ^{\overline{exp}^{∗}⊗𝔤^{∗}}↓
C(𝔤) ⟶^{Ψ} C(𝔤) ⊗ 𝔤^{∗}.

This is equivalent to the formula,

d_{𝔤} = −\frac{e\overline{ad} − 1}{\overline{ad}} ∘ Ψ.

The above formula is equivalent to giving a formula for the pullback by \overline{exp} of a left-invariant 1-form on U_{G}.

2.5.3. Enlarging our dictionary. The discussion in the previous two subsections helps us understand Theorem 2′ better. The discussion in Section 2.5.1 says that if

Φ = \frac{ω_{𝔤}}{1 − e^{−ω_{𝔤}}} ∘ d_{𝔤} : C(𝔤) ⊗ 𝔤^{∗} → C(𝔤) ⊗ 𝔤^{∗}

then the diagram

C(G) ⟶^{d_{G}} C(G) ⊗ 𝔤^{∗}
↓^{exp^{∗}}  ^{exp^{∗}⊗𝔤^{∗}}↓
C(𝔤) ⟶^{Φ} C(𝔤) ⊗ 𝔤^{∗}

commutes. The analogy between this and the diagram (17) of Theorem 2′ is now fairly explicit. Ĉ^{•}(X) is analogous to C(G). The map α_{R} is analogous to the connection d_{G}. I_{HKR} is analogous to exp^{∗}. T [−1] is analogous to 𝔤. Ω[1] is analogous to 𝔤^{∗}. S^{•}(Ω[1]) is analogous to C(𝔤). The map \overline{C} is analogous to d_{𝔤}. The map \overline{ω} : S^{•}(Ω[1]) ⊗ Ω[1] → S^{•}(Ω[1]) ⊗ Ω[1] is analogous to ω_{𝔤}, and the map Φ_{R} is analogous to Φ. In short, the diagram in (17) is analogous to the computation of the Duflo-like term describing the correction that needs to be applied to d_{𝔤} to yield the map Φ. This in turn is equivalent to the formula for the pullback of a left-invariant 1-form on U_{G} via the exponential map.

S : Ĉ^{•}(X) → Ĉ^{•}(X) is analogous to the map S_{G} : C(G) → C(G) given by pulling back an element of C(G) by the map g ↦ g^{−1}. The map

−α_{L} = (S ⊗ Ω[1]) ∘ α_{R} ∘ S

is therefore analogous to (S_{G} ⊗ 𝔤^{∗}) ∘ d_{G} ∘ S_{G}. Moreover, exp^{∗} ∘ S_{G} = \overline{exp}^{∗}. Therefore I_{HKR} ∘ S is analogous to \overline{exp}^{∗}. It follows that Φ_{L} is analogous to the map −Ψ of the previous subsection. In short, the diagram (18) is analogous to the computation of the term describing the correction that has to be applied to d_{𝔤} to describe the map −Ψ. This in turn is equivalent to the formula for the pullback of a left-invariant 1-form on U_{G} via the map \overline{exp}.

Finally we recall from [7] that the universal enveloping algebra in D^{b}(X) of the Lie algebra object T [−1] was shown to be represented by the complex D_{poly}^{•}(X) of polydifferential operators with Hochschild coboundary. Results equivalent to this statement were obtained using different methods by Roberts and Willerton [8] and Markarian [6] as well. Yekutieli [9] showed that the functor RD applied to D_{poly}^{•}(X) yields Ĉ^{•}(X). It follows that the Hopf algebra object Ĉ^{•}(X) of D^{b}(X) is the “dual” in D^{b}(X) of the universal enveloping algebra in D^{b}(X) of the Lie algebra object T [−1] of D^{b}(X).

We warn the reader that the material in the remaining part of this subsection is rather hazy. It can be verified that Ĉ^{•}(X) satisfies all the formal properties that a ring of functions on a Lie group is required to satisfy. It might therefore be possible to say that Ĉ^{•}(X) corresponds to a “Lie group object” in D^{b}(X).

One has to be very careful here. Since the concept of a geometric object like a manifold in D^{b}(X) does not make sense by itself, the best we can do is to try to define such a notion by attempting to define a ring of functions on a manifold in D^{b}(X). This has to be a commutative algebra object in D^{b}(X). If such a definition is possible, T [−1] thought of as a manifold in D^{b}(X) should correspond to the algebra object S^{•}(Ω[1]) of D^{b}(X).

The Lie algebra of the Lie group object Ĉ^{•}(X) should be T [−1]. The diagrams (17) and (18) in Theorem 2′ could then be thought of as being equivalent to “computing the pullback of a 1-form on on the Lie group object Ĉ^{•}(X) via the maps exp and \overline{exp} respectively”. Of course, exp and \overline{exp} are defined solely by what they are as maps from Ĉ^{•}(X) to S^{•}(Ω[1]). However, for this to make any sense, one should be able to define the notion of a differential form on a manifold in D^{b}(X).

3. Some essential linear algebra

The first subsection of this section proves some propositions and a lemma in linear algebra. The second subsection describes their extensions pertaining to the object S^{•}(Ω[1]) of D^{b}(X). We remind the reader that maps between tensor products of graded 𝕂-vector spaces (or graded O_{X}-modules) that rearrange factors are assumed to take the appropriate signs into account. If x is a homogenous element of a graded 𝕂-vector space W, |x| will denote the degree of x.

3.1. Some propositions and a lemma of linear algebra. In this subsection, we will work with differential graded vector spaces over a field 𝕂 of characteristic 0. Almost every dg-vector space in this section has 0 differential. We shall assume that the differential on a graded vector space is 0 unless we explicitly say otherwise. Let V be a finite-dimensional vector space over 𝕂. Denote the dual of V by V^{∗}. Denote the dimension of V in this subsection by m.

As usual S^{•}(V [1]) denotes the symmetric algebra ⊕_{i} ∧^{i} V [i] generated over 𝕂 by V concentrated in degree −1. Similarly, S^{•}(V^{∗}[−1]) denotes the symmetric algebra ⊕_{i} ∧^{i} V^{∗}[−i] generated over 𝕂 by V^{∗} concentrated in degree 1. Note that the products on S^{•}(V [1]) and S^{•}(V^{∗}[−1]) are wedge products.

There is a map

i_{V} : S^{•}(V [1]) → End(S^{•}(V [1]))
Z ↦ (W ↦ W ∧ Z).

i_{V} takes an element Z of S^{•}(V [1]) to the endomorphism of S^{•}(V [1]) given by multiplication by Z on the right.

We will often denote the related map

S^{•}(V [1]) ⊗ S^{•}(V [1]) → S^{•}(V [1])
W ⊗ Z ↦ W ∧ Z

by ( - ∧ - )_{V}. The subscript V may be dropped at times when it is obvious.

Choose a basis {x_{1}, …, x_{m}} of V. Let {y_{1}, …, y_{m}} be a basis of V^{∗} dual to {x_{1}, …, x_{m}}. Let j_{V^{∗}}(y_{i}) be the endomorphism of S^{•}(V [1]) given by

j_{V^{∗}}(y_{i})(x_{j}) = δ_{ij}
j_{V^{∗}}(y_{i})(1) = 0

(27) j_{V^{∗}}(y_{i})(x_{i_{1}} ∧ ⋯ ∧ x_{i_{k}})

= δ_{ii_{k}}x_{i_{1}} ∧ ⋯ ∧ x_{i_{k−1}} − j_{V^{∗}}(y_{i})(x_{i_{1}} ∧ ⋯ ∧ x_{i_{k−1}}) ∧ x_{i_{k}}.

Note that j_{V^{∗}} extends by linearity to a map from V^{∗}[−1] to End(S^{•}(V [1])). Let

∘ : End(S^{•}(V [1]))^{⊗2} → End(S^{•}(V [1]))

denote the composition map. Extend j_{V^{∗}} to a map

j_{V^{∗}} : S^{•}(V^{∗}[−1]) → End(S^{•}(V [1]))

by setting

(28)  j_{V^{∗}}(Y_{1} ∧ Y_{2}) = j_{V^{∗}}(Y_{2}) ∘ j_{V^{∗}}(Y_{1}) ∀ Y_{1}, Y_{2} ∈ S^{•}(V^{∗}[−1]).

We will often denote the related map

S^{•}(V [1]) ⊗ S^{•}(V^{∗}[−1]) → S^{•}(V [1])
W ⊗ Y ↦ j_{V^{∗}}(Y)(W)

by ( - • - )_{V}. The subscript V may be dropped at times when it is obvious.

Remark 5 (A geometric analogy). One can think of S^{•}(V [1]) as the ring of functions on an odd supermanifold M_{V}. For an element Z of S^{•}(V [1]), i_{V} (Z) is just the operator given by “multiplication on the right by Z”. All operators on S^{•}(V [1]) act on the right in our viewpoint. From this viewpoint, elements of V^{∗}[−1] yield vector fields on M_{V}. The map j_{V^{∗}} takes an element Y of V^{∗}[−1] to the constant vector field on M_{V} associated with Y. The reader can observe that the Leibniz rule is part of the definition of the map j_{V^{∗}} : V^{∗}[−1] → End(S^{•}(V [1])) (see Equation (27)). The equation (28) in the definition of j_{V^{∗}} just says that the map j_{V^{∗}} takes an element Y of S^{•}(V^{∗}[−1]) to the constant differential operator on M_{V} associated with Y.

Note that on End(S^{•}(V [1])) we have a natural product given by the composition ∘. Let ∘^{op} denote the product on End(S^{•}(V [1]))^{op}. If a, b ∈ End(S^{•}(V [1])) then a ∘^{op} b = b ∘ a. We have the following proposition.

Proposition 10. The composite

S^{•}(V^{∗}[−1]) ⊗ S^{•}(V [1]) ⟶^{j_{V^{∗}}⊗i_{V}} End(S^{•}(V [1])) ⊗ End(S^{•}(V [1])

⟶^{∘^{op}} End(S^{•}(V [1]))

is an isomorphism of graded 𝕂 vector spaces with 0 differential.

Proof. Denote the given composite by G_{r}.

Since all vector spaces involved in this proposition have 0 differential, it suffices to check that G_{r} is an isomorphism of graded 𝕂-vector spaces. Further, it is easy to check that i_{V}, j_{V^{∗}} and ∘^{op} are degree preserving. It therefore, suffices to check that G_{r} is an isomorphism of 𝕂-vector spaces. All the 𝕂-vector spaces involved in this proposition are finite-dimensional. Further, S^{•}(V^{∗}[−1])⊗S^{•}(V [1]) and End(S^{•}(V [1])) have the same dimension as 𝕂-vector spaces. It therefore, suffices to check that G_{r} injective.

Choose a basis {x_{1}, …, x_{m}} of V. Let {y_{1}, …, y_{m}} be a basis of V^{∗} dual to {x_{1}, …, x_{m}}. By convention, we list the elements of any subset of {1, …, m} in ascending order. We can then define an ordering ≺ on the set of subsets of {1, …, m} by setting

S ≺ T if |S| < |T |

and {i_{1}, …, i_{k}} ≺ {j_{1}, …, j_{k}} if (i_{1}, …, i_{k}) ≺ (j_{1}, …, j_{k}) in the lexicographic order.

For S = {i_{1}, …, i_{k}}, let x_{S} denote x_{i_{1}} ∧⋯∧x_{i_{k}}. Similarly, y_{S} will denote y_{i_{1}} ∧ ⋯ ∧ y_{i_{k}}. It is easy to verify that

(29)  j_{V^{∗}}(y_{S})(x_{S}) = ±1
j_{V^{∗}}(y_{S})(x_{T}) = 0 if T ≺ S.

An element of S^{•}(V^{∗}[−1])⊗S^{•}(V [1]) is given by an expression of the form

∑_{S⊂{1,...,m}} y_{S} ⊗ a_{S}, a_{S} ∈ S^{•}(V [1]).

Choose S_{0} ⊂ {1, …, m} to be the least subset (under the ordering ≺) of {1, …, m} such that a_{S_{0}} ≠ 0. Then

G_{r}( ∑_{S⊂{1,...,m}} y_{S} ⊗ a_{S})(x_{S_{0}}) = ±a_{S_{0}}

by (29). It follows that G_{r} is injective. □

Remark 6. Let M_{V} be the supermanifold of whose ring of functions is S^{•}(V [1]). Under our convention that all operators on S^{•}(V [1]) act on the right, G_{r} identifies ∧^{i}T [−i] ⊗ S^{•}(V [1]) with the space of principal symbols of differential operators of order i on M_{V}. Proposition 10 says that every endomorphism of S^{•}(V [1]) is given by a differential operator on M_{V}.

Notation. For the rest of this paper, G_{r} shall denote the isomorphism in Proposition 10 and F_{r} shall denote its inverse. In addition, if

τ : S^{•}(V^{∗}[−1]) ⊗ S^{•}(V [1]) → S^{•}(V [1]) ⊗ S^{•}(V^{∗}[−1])

denotes the swap map the composite S^{•}(V^{∗}[−1]) ⊗ S^{•}(V [1])

τ↓
S^{•}(V [1]) ⊗ S^{•}(V^{∗}[−1]) ⟶^{i_{V} ∘^{op}j_{V^{∗}}} End(S^{•}(V [1]))

shall be denoted by G_{l}. The inverse of G_{l} will be denoted by F_{l}.

Let π_{k} : S^{•}(V [1]) → ∧^{k}V [k] denote the natural projection. Note that we have a nondegenerate pairing ⟨, ⟩ on S^{•}(V [1]). This is given by the composite

S^{•}(V [1]) ⊗ S^{•}(V [1]) ⟶^{∧} S^{•}(V [1]) ⟶^{π_{m}} ∧^{m}V [m]

where m is the dimension of V.

Definition 1. The adjoint Φ^{+} ∈ End(S^{•}(V [1])) of a homogenous element Φ of End(S^{•}(V [1])) is the unique element of End(S^{•}(V [1])) satisfying

⟨Φ(a), b⟩ = (−1)^{|Φ||b|}⟨a, Φ^{+}(b)⟩ ∀ homogenous a, b ∈ S^{•}(V [1]).

The adjoint of an arbitrary element of End(S^{•}(V [1]) is the sum of the adjoints of its homogenous components.

Let

ev_{V} : S^{•}(V [1]) ⊗ End(S^{•}(V [1]))

be the map

W ⊗ Φ ↦ Φ(W).

Denote the map which takes an element of End(S^{•}(V [1])) to its adjoint by A_{V}. Let ⟨ , ev^{+}⟩_{V} denote the composite

S^{•}(V [1]) ⊗ S^{•}(V [1]) ⊗ End(S^{•}(V [1])
↓^{S^{•}(V [1])⊗S^{•}(V [1])⊗A_{V}}
S^{•}(V [1]) ⊗ S^{•}(V [1]) ⊗ End(S^{•}(V [1])
↓^{S^{•}(V [1])⊗ev_{V}}
S^{•}(V [1]) ⊗ S^{•}(V [1])
↓^{⟨ , ⟩}
∧^{m}V [m].

Then the following diagram commutes.

S^{•}(V [1]) ⊗ End(S^{•}(V [1])) ⊗ S^{•}(V [1]) ⟶^{ev_{V} ⊗S^{•}(V [1])} S^{•}(V [1]) ⊗ S^{•}(V [1])
↓^{S^{•}(V [1])⊗τ}  ^{⟨,⟩}↓
S^{•}(V [1]) ⊗ S^{•}(V [1]) ⊗ End(S^{•}(V [1])) ⟶^{⟨ ,ev^{+}⟩_{V}}  ∧^{m}V [m].

The map τ : End(S^{•}(V [1])) ⊗ S^{•}(V [1]) → S^{•}(V [1]) ⊗ End(S^{•}(V [1])) in the above diagram swaps factors.

The following proposition describes the basic properties of the adjoint.

Proposition 11. (1) Let L be an element of End(S^{•}(V [1])). Then

L^{++} = L.

(2) If L_{1} and L_{2} are homogenous elements of End(S^{•}(V [1])), then

(L_{1} ∘ L_{2})^{+} = (−1)^{|L_{1}||L_{2}|}(L_{2})^{+} ∘ (L_{1})^{+}.

(3) If Z ∈ S^{•}(V [1]) is a homogenous element, then

i_{V} (Z)^{+} = i_{V} (Z).

(4) If Y ∈ S^{•}(V^{∗}[−1]) is a homogenous element, then

j_{V^{∗}}(Y)^{+} = (−1)^{|Y |}j_{V^{∗}}(Y).

(5) If Z ∈ S^{•}(V [1]) and Y ∈ S^{•}(V^{∗}[−1]) are homogenous elements, then

G_{l}(Y ⊗ Z)^{+} = (−1)^{|Y |}G_{r}(Y ⊗ Z).

Proof. Observe that if L ∈ End(S^{•}(V [1])) is homogenous, then |L^{+}| = |L|. Also note that if a, b ∈ S^{•}(V [1]) are homogenous, then ⟨a, b⟩ = (−1)^{|a||b|}⟨b, a⟩. Also recall that the pairing ⟨, ⟩ is nondegenerate.

If a, b ∈ S^{•}(V [1]) are homogenous elements and if L ∈ End(S^{•}(V [1])) is homogenous, note that

⟨L^{++}(a), b⟩ = (−1)^{|b||a|+|b||L|}⟨b, L^{++}(a)⟩ = (−1)^{|b||a|+|b||L|+|a||L|}⟨L^{+}(b), a⟩
= (−1)^{|b||a|+|b||L|+|a||L|+|a||b|+|a||L|}⟨a, L^{+}(b)⟩ = ⟨L(a), b⟩.

Part (1) of this proposition now follows immediately from this calculation. For the rest of this proof a and b shall be homogenous elements of S^{•}(V [1]). If L_{1} and L_{2} are homogenous elements of End(S^{•}(V [1])), then

(−1)^{|L_{1}||L_{2}|}⟨a, (L_{2})^{+} ∘ (L_{1})^{+}(b)⟩
= (−1)^{|L_{1}||L_{2}|+|L_{2}|(|b|+|L_{1}|)}⟨L_{2}(a), L_{1}^{+}(b)⟩
= (−1)^{|L_{1}||L_{2}|+|L_{2}|(|b|+|L_{1}|)+|L_{1}||b|}⟨L_{1} ∘ L_{2}(a), b⟩
= (−1)^{|b|(|L_{1}|+|L_{2}|)}⟨L_{1} ∘ L_{2}(a), b⟩.

Part (2) of this proposition now follows from the observation that

|L_{1} ∘ L_{2}| = |L_{1}| + |L_{2}|.

Part (3) of this proposition is immediate from the relevant definitions and the fact that

a ∧ Z ∧ b = (−1)^{|b||Z|}a ∧ b ∧ Z.

To verify part (4), choose a basis {x_{1}, …, x_{m}} of V. Let {y_{1}, …, y_{m}} be a basis of V^{∗} dual to {x_{1}, …, x_{m}}. For an ordered subset S = {i_{1}, …, i_{k}} of {1, …, m} let y_{S} denote y_{i_{1}} ∧ ⋯ ∧ y_{i_{k}} and let x_{S} denote x_{i_{1}} ∧ ⋯ ∧ x_{i_{k}}. If T is disjoint from S, note that

j_{V^{∗}}(y_{S})(x_{T} ∧ x_{S}) = x_{T} ∧ j_{V^{∗}}(y_{S})(x_{S}) = (−1)^{\frac{k(k−1)}{2}} x_{T} .

Let T and T′ be subsets of {1, …, m} disjoint from S. Then,

j_{V^{∗}}(y_{S})(x_{T} ∧ x_{S}) ∧ (x_{T′} ∧ x_{S})
= (−1)^{\frac{k(k−1)}{2}} x_{T} ∧ x_{T′} ∧ x_{S}
= (−1)^{\frac{k(k−1)}{2}} (−1)^{|T′||S|}x_{T} ∧ x_{S} ∧ x_{T′}
= (−1)^{|T′||S|}x_{T} ∧ x_{S} ∧ j_{V^{∗}}(y_{S})(x_{T′} ∧ x_{S})
= (−1)^{(|T′|+|S|)|S|+|S|}x_{T} ∧ x_{S} ∧ j_{V^{∗}}(y_{S})(x_{T′} ∧ x_{S}).

Putting a = x_{T} ∧ x_{S}, b = x_{T′} ∧ x_{S} we see that |b| = |T′| + |S|. Further, |Y | = −|S|. Part (4) now follows from the above computation once we recall that ⟨a, b⟩ = π_{m}(a ∧ b).

Part (5) follows from part (2), part (3) and part (4).

G_{l}(Y ⊗ Z) = (−1)^{|Z||Y |}j_{V^{∗}}(Y) ∘ i_{V} (Z).

Thus,

\begin{aligned}G_{l}(Y ⊗ Z)^{+} & = (−1)^{|Z||Y |}(j_{V^{∗}}(Y) ∘ i_{V} (Z))^{+} \\ & = (−1)^{|Z||Y |}(−1)^{|Z||Y |}(i_{V} (Z))^{+} ∘ (j_{V^{∗}}(Y))^{+} \\ & = (−1)^{|Y |}(i_{V} (Z)) ∘ (j_{V^{∗}}(Y)) = (−1)^{|Y |}G_{r}(Y ⊗ Z). \end{aligned}  □

Recall that π_{j} : S^{•}(V [1]) → ∧^{j}V [j] denotes the natural projection. We will denote the projection

S^{•}(V^{∗}[−1]) ⊗ π_{j} : S^{•}(V^{∗}[−1]) ⊗ S^{•}(V [1]) → S^{•}(V^{∗}[−1]) ⊗ ∧^{j}V [j]

by π_{j} itself. Let I denote the endomorphism of S^{•}(V^{∗}[−1]) that multiplies ∧^{i}V^{∗}[−i] by (−1)^{i}. The following proposition is really a corollary of Proposition 11.

Proposition 12. If L ∈ End(S^{•}(V [1])), then

π_{0}(F_{l}(L)) = I(π_{0}(F_{r}(L^{+}))).

Proof. This is almost immediate from part (5) of Proposition 11. Let Y ∈ S^{•}(V^{∗}[−1]) and Z ∈ S^{•}(V [1]) be homogenous. By part (5) of Proposition 11

G_{l}(Y ⊗ Z)^{+} = (−1)^{|Y |}G_{r}(Y ⊗ Z).

By definition, F_{r}(G_{r}(Y ⊗ Z)) = Y ⊗ Z. On the other hand,

F_{l}(G_{r}(Y ⊗ Z)^{+}) = F_{l}((−1)^{|Y |}G_{l}(Y ⊗ Z) = (−1)^{|Y |}Y ⊗ Z.

Since G_{r} and A_{V} are degree preserving isomorphisms of 𝕂-vector spaces, any homogenous element in End(S^{•}(V [1])) is of the form G_{r}(Y ⊗ Z)^{+}. It follows from the above computation that

F_{l}(M) = (I ⊗ S^{•}(V [1]))F_{r}(M^{+})

for any homogenous M ∈ End(S^{•}(V [1])). Therefore,

π_{j}(F_{l}(L)) = I(π_{j}(F_{r}(L^{+})))

for any j. When j = 0, we get the desired proposition. □

Convention To simplify notation, we follow the following convention: if a ∈ S^{•}(V^{∗}[−1]) ⊗ S^{•}(V [1]), then G_{r}(a) will be denoted by a itself. Keeping this convention in mind:

Proposition 13. If a, b ∈ S^{•}(V^{∗}[−1]) ⊗ S^{•}(V [1]), then

π_{0}(F_{r}(a ∘ b)) = π_{0}(F_{r}(π_{0}(a) ∘ b)).

Proof. It suffices to check that if Z ∈ ∧^{i}V [i]with i > 0, then

π_{0}(Y ⊗ Z ∘ Y′ ⊗ Z′) = 0

for any Z′ ∈ S^{•}(V [1]), Y, Y′ ∈ S^{•}(V^{∗}[−1]). Note that by our convention,

Y ⊗ Z ∘ Y′ ⊗ Z′ = i_{V} (Z) ∘ j_{V^{∗}}(Y) ∘ i_{V} (Z′) ∘ j_{V^{∗}}(Y′).

For subsets S and T of {1, …, m}, let x_{S} and y_{T} be as in the proof of Proposition 11, part (4). By Proposition 10,

j_{V^{∗}}(Y) ∘ i_{V} (Z′) ∘ j_{V^{∗}}(Y′) = ∑_{S,T ⊂{1,...,m}} a_{S,T} y_{T} ⊗ x_{S}

for some a_{S,T} ∈ 𝕂. Then,

i_{V} (Z) ∘ j_{V^{∗}}(Y) ∘ i_{V} (Z′) ∘ j_{V^{∗}}(Y′) = ∑_{S,T ⊂{1,...,m}} a_{S,T} y_{T} ⊗ x_{S} ∧ Z.

Since Z ∈ ∧^{i}V [i], x_{S} ∧ Z ∈ ⊕_{k≥i} ∧^{k} V [k] ⊂ ⊕_{k>0} ∧^{k} V [k]. It follows that

π_{0}( ∑_{S,T ⊂{1,...,m}} a_{S,T} y_{T} ⊗ x_{S} ∧ Z) = 0.

This proves the desired proposition. □

Remark 7. Proposition 10 said that every element of End(S^{•}(V [1])) can be thought of as a differential operator on M_{V}. The isomorphism F_{r} makes this identification. The map π_{0} : S^{•}(V^{∗}[−1]) ⊗ S^{•}(V [1]) → S^{•}(V^{∗}[−1]) should be thought of as the map which “takes the constant term” of a differential operator. Proposition 13 says that

const. term(D_{1} ∘ D_{2}) = const. term(( const. term (D_{1})) ∘ D_{2})

for any two differential operators D_{1} and D_{2} on M_{V}.

Recall that by Proposition 10, F_{r} identifies End(S^{•}(V [1])) with

S^{•}(V^{∗}[−1]) ⊗ S^{•}(V [1]).

We also remarked (in Remark 6) that the direct summand

∧^{i}V^{∗}[−i] ⊗ S^{•}(V [1])

of S^{•}(V^{∗}[−1])⊗S^{•}(V [1]) can be thought of as the space of principal symbols of differential operators of order i on S^{•}(V [1]). Reflecting this understanding, we denote ∧^{i}V^{∗}[−i] ⊗ S^{•}(V [1]) by D_{i}.

Note that the composition ∘ : End(S^{•}(V [1]))^{⊗2} → End(S^{•}(V [1])) equips End(S^{•}(V [1])) with the structure of a graded associative 𝕂-algebra. This algebra structure induces a Lie superalgebra structure on End(S^{•}(V [1])). If a, b ∈ End(S^{•}(V [1])) are homogenous, then

[a, b]_{V} = a ∘ b − (−1)^{|a||b|}b ∘ a.

Also note that the map ∘^{op} : End(S^{•}(V [1]))^{⊗2} → End(S^{•}(V [1])) equips End(S^{•}(V [1])) with the structure of a graded associative 𝕂-algebra. We denote this 𝕂-algebra by End(S^{•}(V [1]))^{op}. Its algebra structure induces the structure of a Lie superalgebra on End(S^{•}(V [1]))^{op}. If a, b ∈ End(S^{•}(V [1])) are homogenous, then

[a, b]_{V}^{op} = a ∘^{op} b − (−1)^{|a||b|}b ∘^{op} a = [b, a]_{V}.

Proposition 14.

[D_{1}, D_{1}]_{V} ⊂ D_{1}.

Proof. Let H ∈ S^{•}(V [1]),let Z ∈ S^{•}(V [1]) and let Y ∈ S^{•}(V^{∗}[−1]). Recall that (H • Y) denotes j_{V^{∗}}(Y)(H). Note that HZ := H ∧ Z = i_{V} (Z)(H). Keep in mind that Y ⊗ Z is identified with G_{r}(Y ⊗ Z). Then

Y ⊗ Z(H) = (H • Y)Z.

Let Z_{1}, Z_{2} ∈ S^{•}(V [1]) be homogenous and let y_{1}, y_{2} ∈ V^{∗}[−1]. Then, if H ∈ S^{•}(V [1]),

(y_{1} ⊗ Z_{1})∘(y_{2} ⊗ Z_{2})(H)
= ((H • y_{2})Z_{2} • y_{1})Z_{1}
= (H • y_{2})(Z_{2} • y_{1})Z_{1} + (−1)^{|Z_{2}|}((H • y_{2}) • y_{1})Z_{2}Z_{1}.

Therefore,

(30) (y_{1} ⊗ Z_{1}) ∘ (y_{2} ⊗ Z_{2}) = y_{2} ⊗ (Z_{2} • y_{1})Z_{1} + (−1)^{|Z_{2}|}y_{2} ∧ y_{1} ⊗ Z_{2}Z_{1}.

Similarly,

(31) (y_{2} ⊗ Z_{2}) ∘ (y_{1} ⊗ Z_{1}) = y_{1} ⊗ (Z_{1} • y_{2})Z_{2} + (−1)^{|Z_{1}|}y_{1} ∧ y_{2} ⊗ Z_{1}Z_{2}.

If D_{1} = y_{1} ⊗Z_{1} and D_{2} = y_{2} ⊗Z_{2} then |D_{1}| = |Z_{1}|−1 and |D_{2}| = |Z_{2}|−1. It then follows from (30) and (31) that

(32) D_{1} ∘ D_{2} − (−1)^{|D_{1}||D_{2}|}D_{2} ∘ D_{1}

= y_{2} ⊗ (Z_{2} • y_{1})Z_{1} − (−1)^{|D_{1}||D_{2}|}y_{1} ⊗ (Z_{1} • y_{2})Z_{2}.

Note that the right-hand side is an element of D_{1}. The desired proposition now follows immediately. □

We will assume that the Lie superalgebra End(S^{•}(V [1])) is equipped with the bracket [,]_{V} unless we explicitly state otherwise.

Proposition 14 tells us that D_{1} is a Lie subalgebra of End(S^{•}(V [1])). It also follows immediately from Proposition 14 that

[D_{1}, D_{1}]_{V}^{op} ⊂ D_{1}.

It follows that D_{1} equipped with the bracket [,]_{V}^{op} is a Lie subalgebra of End(S^{•}(V [1]))^{op}. D_{m} can be identified with the top symmetric power of D_{1} over S^{•}(V [1]). In other words,

D_{m} ≃ S_{S^{•}(V [1])}^{m}D_{1}.

One can then study the right adjoint action of the Lie algebra D_{1} (equipped with the bracket [,]_{V}^{op}) on D_{m}. For an element L of D_{1}, let ad(L) denote the right adjoint action of L on D_{m} with respect to the bracket [,]_{V}^{op}. Then

ad(L)(D_{m}D_{m−1} … D_{1})
= D_{m}D_{m−1} … [D_{1}, L]_{V}^{op} + (−1)^{|L||D_{1}|}D_{m}D_{m−1} … [D_{2}, L]_{V}^{op}D_{1}
+ ⋯ (−1)^{|L|(|D_{1}|+···+|D_{m−1}|)}[D_{m}, L]_{V}^{op} … D_{1}
= D_{m}D_{m−1} … [L, D_{1}] + (−1)^{|L||D_{1}|}D_{m}D_{m−1} … [L, D_{2}]D_{1}
+ ⋯ (−1)^{|L|(|D_{1}|+···+|D_{m−1}|)}[L, D_{m}] … D_{1}

for homogenous elements D_{i} ∈ D_{1}. In defining ad(L), D_{m} is treated as S_{S^{•}(V [1])}^{m}D_{1}.

One also has the right adjoint action \overline{ad}(L) of L on D_{1}^{⊗m} (with respect to the bracket [,]_{V}^{op} on D_{1}).

\overline{ad}(L)(D_{m} ⊗ D_{m−1} ⊗ ⋯ ⊗ D_{1})

= D_{m} ⊗ D_{m−1} ⊗ ⋯ ⊗ [D_{1}, L]_{V}^{op}

+ (−1)^{|L||D_{1}|}D_{m} ⊗ D_{m−1} ⊗ ⋯ ⊗ [D_{2}, L]_{V}^{op} ⊗ D_{1}

+ ⋯ (−1)^{|L|(|D_{1}|+···+|D_{m−1}|)}[D_{m}, L]_{V}^{op} ⊗ ⋯ ⊗ D_{1}

= D_{m} ⊗ D_{m−1} ⊗ ⋯ ⊗ [L, D_{1}] + (−1)^{|L||D_{1}|}D_{m} ⊗ D_{m−1} ⊗ ⋯ ⊗ [L, D_{2}] ⊗ D_{1}

+ ⋯ (−1)^{|L|(|D_{1}|+···+|D_{m−1}|)}[L, D_{m}] ⊗ ⋯ ⊗ D_{1}

for homogenous elements D_{i} ∈ D_{1}.

Let p : D_{1}^{⊗m} → D_{m} denote the map

D_{m} ⊗ D_{m−1} ⊗ ⋯ ⊗ D_{1} ↦ D_{m}D_{m−1} … D_{1}.

The following proposition is clear from the definitions.

Proposition 15. The following diagram commutes.

D_{1}^{⊗m} ⟶^{p} D_{m}
↓^{\overline{ad}(L)}  ↓^{ad(L)}
D_{1}^{⊗m} ⟶^{p} D_{m}.

Choose a basis {x_{1}, …, x_{m}} of V and a basis {y_{1}, …, y_{m}} of V^{∗} dual to {x_{1}, …, x_{m}}. Let 1^{m} : 𝕂 → ∧^{m}V^{∗}[−m] ⊗ ∧^{m}V [m] be the map

1 ↦ y_{m} ∧ ⋯ ∧ y_{1} ⊗ x_{1} ∧ ⋯ ∧ x_{m}.

Let τ : S^{•}(V [1]) ⊗ ∧^{m}V^{∗}[−m] → ∧^{m}V^{∗}[−m] ⊗ S^{•}(V [1]) denote the swap map. Denote the composite map

S^{•}(V [1]) ⟶^{(τ⊗∧^{m}V [m])∘(S^{•}(V [1])⊗1^{m})} ∧^{m}V^{∗}[−m] ⊗ S^{•}(V [1]) ⊗ ∧^{m}V [m]

by 1^{m} itself.

Lemma 1. Let L ∈ D_{1} be a homogenous element. The following diagram commutes.

S^{•}(V [1]) ⟶^{1^{m}} ∧^{m}V^{∗}[−m] ⊗ S^{•}(V [1]) ⊗ ∧^{m}V [m]
↓_{−L^{+}}  ↓_{(−1)^{|L|m}ad(L)⊗∧^{m}V [m]}
S^{•}(V [1]) ⟶^{1^{m}} ∧^{m}V^{∗}[−m] ⊗ S^{•}(V [1]) ⊗ ∧^{m}V [m].

Proof. This lemma is proven by a direct computation. We once more recall that if H, Z ∈ S^{•}(V [1]) and if Y ∈ S^{•}(V^{∗}[−1]) then HZ denotes

H ∧ Z = i_{V} (Z)(H)

and (H • Y) denotes j_{V^{∗}}(Y)(H).

Choose a basis {y_{1}, …, y_{m}} of V^{∗} and a basis {x_{1}, …, x_{m}} of V dual to {y_{1}, …, y_{m}}. We may assume without loss of generality that L = y_{1} ⊗ Z where Z ∈ S^{•}(V [1]) is homogenous. Then, if H ∈ S^{•}(V [1]) is homogenous, by Proposition 11, parts (2), (3), and (4),

L^{+}(H) = (−1)^{−|Z|−1}((HZ) • y_{1}) = (−1)^{|Z|+1}((HZ) • y_{1}).

Thus,

(33) 1^{m}(L^{+}(H))

= (−1)^{(|H|+|Z|−1)m}(−1)^{|Z|+1}y_{m} ∧ ⋯ ∧ y_{1} ⊗ ((HZ) • y_{1}) ⊗ x_{1} ∧ ⋯ ∧ x_{m}.

On the other hand, if we treat H as an element of D_{0}, then

(y_{1} ⊗ Z) ∘ H(P) = (P H • y_{1})Z = P (H • y_{1})Z + (−1)^{|H|}(P • y_{1})HZ

H ∘ (y_{1} ⊗ Z)(P) = (P • y_{1})ZH.

Note that the degree of the operator y_{1} ⊗ Z is |Z| − 1. It follows that

[y_{1} ⊗ Z, H](P) = P (H • y_{1})Z.

Thus,

[y_{1} ⊗ Z, H] = (H • y_{1})Z.

By (32) in the proof of Proposition 14,

[y_{1} ⊗ Z, y_{i}] = (−1)^{|Z|}y_{1} ⊗ (Z • y_{i}).

Therefore,

y_{m} ∧ ⋯ ∧ [y_{1} ⊗ Z, y_{i}] ∧ ⋯ ∧ y_{1} = 0 ∀ i ≠ 1.

Thus,

ad(L) ⊗ ∧^{m}V [m](y_{m} ∧ ⋯ ∧ y_{1} ⊗ H ⊗ x_{1} ∧ ⋯ ∧ x_{m})

= y_{m} ∧ ⋯ ∧ y_{1} ⊗ ((H • y_{1})Z

+ (−1)^{|H|(|Z|−1)}(−1)^{|Z|}(Z • y_{1})H) ⊗ x_{1} ∧ ⋯ ∧ x_{m}

= y_{m} ∧ ⋯ ∧ y_{1} ⊗ ((H • y_{1})Z

+ (−1)^{|H|(|Z|−1)}(−1)^{|Z|}(−1)^{|H|(|Z|−1)}H(Z • y_{1})) ⊗ x_{1} ∧ ⋯ ∧ x_{m}

= y_{m} ∧ ⋯ ∧ y_{1} ⊗ ((H • y_{1})Z + (−1)^{|Z|}H(Z • y_{1})) ⊗ x_{1} ∧ ⋯ ∧ x_{m}.

Note that

(HZ • y_{1}) = H(Z • y_{1}) + (−1)^{|Z|}(H • y_{1})Z.

Further recall that

1^{m}(H) = (−1)^{|H|m}y_{m} ∧ ⋯ ∧ y_{1} ⊗ H ⊗ x_{1} ∧ ⋯ ∧ x_{m}.

Therefore,

(−1)^{|L|m}ad(L) ⊗ ∧^{m}V [m](1^{m}(H))

= (−1)^{(|H|+|Z|−1)m}ad(L) ⊗ ∧^{m}V [m](y_{m} ∧ ⋯ ∧ y_{1} ⊗ H ⊗ x_{1} ∧ ⋯ ∧ x_{m})

= (−1)^{(|H|+|Z|−1)m}y_{m} ∧ ⋯ ∧ y_{1} ⊗ ((H • y_{1})Z + (−1)^{|Z|}H(Z • y_{1})) ⊗ x_{1} ∧ ⋯ ∧ x_{m}

= −(−1)^{(|H|+|Z|−1)m}(−1)^{|Z|+1}y_{m} ∧ ⋯ ∧ y_{1} ⊗ (HZ • y_{1}) ⊗ x_{1} ∧ ⋯ ∧ x_{m}

= −1^{m}(L^{+}(H)).

This proves the desired lemma. □

Remark 8. Lemma 1 seems to be a phenomenon occurring in purely odd supergeometry only. Let M_{V} be the supermanifold whose ring of functions is S^{•}(V [1]). The pairing ⟨, ⟩ on S^{•}(V [1]) is the pairing

⟨f, g⟩ = ∫_{M_{V}} fg ∀ f, g ∈ S^{•}(V [1]).

By ∫_{M_{V}}, we of course mean a Berezinian integral. We can think of the usual geometric analog of S^{•}(V [1]) to be the ring of compactly supported functions on a smooth oriented manifold M. The analog of the pairing ⟨, ⟩ on S^{•}(V [1]) is the pairing

(f, g) ↦ ∫_{M} fgdμ

where dμ is the measure arising out of a volume form on M.On M_{V} there is a constant top-order differential operator ∂ that is unique upto scalar. Lemma 1 then says that if D is a differential operator on M_{V} that is purely of first-order, and if f is a function on M_{V}, the Lie bracket of D with f∂ is ±(D^{+}f)∂ where D^{+} is the adjoint of D. In the usual geometric setting, the analog of ∂ would be a global, nowhere vanishing section of the top wedge power of the tangent bundle of M. Even if such a section exists on M, the analog of Lemma 1 does not hold even in the 1-dimensional case. For example, if M = ℝ, then the adjoint of the operator f \frac{d}{dx} is the operator −(\frac{df}{dx} + f \frac{d}{dx}). This follows from the standard integration by parts. But

[f \frac{d}{dx}, g \frac{d}{dx}] = f \frac{dg}{dx} − g \frac{df}{dx} ≠ ± (−g \frac{df}{dx} − f \frac{dg}{dx}).

We get back to proving more propositions in linear algebra that we require for future use.

Choose a basis {x_{1}, …, x_{m}} of V and a basis {y_{1}, …, y_{m}} of V^{∗} dual to {x_{1}, …, x_{m}}. Define k_{V} : S^{•}(V [1]) → End(S^{•}(V^{∗}[−1])) by the formulae

k_{V} (x_{i})(y_{j}) = δ_{ij}
k_{V} (x_{i})(1) = 0
k_{V} (x_{i})(Y_{1} ∧ Y_{2}) = k_{V} (x_{i})(Y_{1}) + (−1)^{|Y_{1}|}Y_{1} ∧ k_{V} (x_{i})(Y_{2})
k_{V} (X_{1} ∧ X_{2})(Y) = k_{V} (X_{1})(k_{V} (X_{2})(Y)).

To simplify notation, we shall denote k_{V} (Z)(Y) by (Z|Y).

Remark 9. The map j_{V^{∗}} identifies an element Y of S^{•}(V^{∗}[−1]) with the operation “differentiation on the right by Y ”. The map k_{V} identifies an element Z of S^{•}(V [1]) with “differentiation on the left by Z”.

We have the following proposition.

Proposition 16. Let Y ∈ S^{•}(V^{∗}[−1]) and let Z ∈ S^{•}(V [1]). Then,

π_{0}(F_{r}(j_{V^{∗}}(Y) ∘ i_{V} (Z))) = (Z|Y).

Proof. This is again proven by a direct computation. We may assume without loss of generality that Y and Z are homogenous.

Choose a basis {x_{1}, …, x_{m}} of V and a basis {y_{1}, …, y_{m}} of V^{∗} dual to {x_{1}, …, x_{m}}. Let H ∈ S^{•}(V [1]). Let Z = x_{S} and let Y = y_{T} for some S, T ⊂ {1, …, m}. If S and T are disjoint and if S is nonempty, j_{V^{∗}}(Y) and i_{V} (Z) commute upto sign. It follows that π_{0}(F_{r}(j_{V^{∗}}(Y) ∘ i_{V} (Z))) = 0. Also, (Z|Y) = 0 if S and T are disjoint and S is nonempty. If S is empty, x_{S} = 1 by convention. Therefore, π_{0}(F_{r}(j_{V^{∗}}(Y) ∘ i_{V} (Z))) = Y and (Z|Y) = Y. We therefore, have to prove this proposition for the case when S and T are not disjoint.

Let S and T be arbitrary subsets of {1, …, m}. Suppose that j ∉ S ∪ T.

j_{V^{∗}}(y_{j} ∧ Y) ∘ i_{V} (Z ∧ x_{j})(H) = (HZ ∧ x_{j} • y_{j} ∧ Y)
= (HZ • Y) − ((HZ • y_{j}) ∧ x_{j} • Y).

Since j is not in T. It follows that

Z′ ∧ x_{j} • Y = ±(Z′ • Y) ∧ x_{j}.

Therefore

j_{V^{∗}}(y_{j} ∧ Y) ∘ i_{V} (Z ∧ x_{j})(H) = (HZ • Y) ± ((HZ • y_{j}) • Y) ∧ x_{j}.

It follows that

π_{0}(F_{r}(j_{V^{∗}}(y_{j} ∧ Y) ∘ i_{V} (Z ∧ x_{j}))) = π_{0}(F_{r}(j_{V^{∗}}(Y) ∘ i_{V} (Z))).

Note that

(Z ∧ x_{j}|y_{j} ∧ Y) = (Z|Y)

since j ∉ S ∪ T. The desired proposition follows for homogenous Z = x_{S} and Y = y_{T} by induction on |S ∩ T |. For general Y and Z the proposition follows from the fact that the maps

S^{•}(V^{∗}[−1]) × S^{•}(V [1]) → S^{•}(V^{∗}[−1])
(Y, Z) ↦ π_{0}(F_{r}(j_{V^{∗}}(Y) ∘ i_{V} (Z)))

and

S^{•}(V^{∗}[−1]) × S^{•}(V [1]) → S^{•}(V^{∗}[−1])
(Y, Z) ↦ (Z|Y)

are both 𝕂-bilinear. □

Let ( - || - )_{V} : S^{•}(V [1]) ⊗ S^{•}(V^{∗}[−1]) → 𝕂 denote map

Z ⊗ Y ↦ p_{0}(Z|Y)

where p_{0} : S^{•}(V^{∗}[−1]) → 𝕂 denotes the projection to the degree 0 direct summand.

Let γ_{V} : S^{•}(V [1]) → S^{•}(V^{∗}[−1]) ⊗ ∧^{m}V [m] be the isomorphism such that

π_{m}( - ∧ - )_{V} = [( - || - )_{V} ⊗ ∧^{m}V [m]] ∘ (S^{•}(V [1]) ⊗ γ_{V}).

Let ζ_{V} denote γ_{V}^{−1}.

Proposition 17. If Z, W ∈ S^{•}(V [1]), then

ζ_{V} ({(Z| - ) ⊗ ∧^{m}V [m]}(γ(W))) = Z ∧ W.

Proof. This proposition is verified by a direct computation. Choose a basis {x_{1}, …, x_{m}} of V and a basis {y_{1}, …, y_{m}} of V^{∗} dual to {x_{1}, …, x_{m}}. Without loss of generality, Z = x_{1} ∧ ⋯ ∧ x_{k} and W = x_{l+1} ∧ ⋯ ∧ x_{m}.

Then

γ_{V} (W) = y_{l} ∧ ⋯ ∧ y_{1} ⊗ x_{1} ∧ ⋯ ∧ x_{m}

((Z| - ) ⊗ ∧^{m}V [m])(γ_{V} (W)) = (x_{1} ∧ ⋯ ∧ x_{k}|y_{l} ∧ ⋯ ∧ y_{1}) ⊗ x_{1} ∧ ⋯ ∧ x_{m}.

If k > l then (x_{1}∧⋯∧x_{k}|y_{1}∧⋯∧y_{l}) = 0 and x_{1}∧⋯∧x_{k}∧x_{l+1}∧⋯∧x_{m} = 0, proving this proposition. We may thus assume that k ≤ l. Then

((Z| - ) ⊗ ∧^{m}V [m])(γ_{V} (W)) = (x_{1} ∧ ⋯ ∧ x_{k}|y_{l} ∧ ⋯ ∧ y_{1}) ⊗ x_{1} ∧ ⋯ ∧ x_{m}

= (−1)^{k(l−k)}y_{l} ∧ ⋯ ∧ y_{k+1} ⊗ x_{1} ∧ ⋯ ∧ x_{m}

= γ_{V} (x_{1} ∧ ⋯ ∧ x_{k} ∧ x_{l+1} ∧ ⋯ ∧ x_{m}).

This proves the desired proposition. □

Let J_{V} denote the endomorphism of S^{•}(V [1]) multiplying ∧^{i}V [i] by (−1)^{i}. We also have the following proposition. Let

τ : ∧^{m}V [m] ⊗ S^{•}(V^{∗}[−1]) ⊗ ∧^{m}V [m] → S^{•}(V^{∗}[−1]) ⊗ ∧^{m}V [m] ⊗ ∧^{m}V [m]

denote the map that interchanges ∧^{m}V [m] and S^{•}(V^{∗}[−1]) ⊗ ∧^{m}V [m]. Let

≃: ∧^{m}V [m] ⊗ ∧^{m}V^{∗}[−m] → 𝕂
x_{1} ∧ ⋯ ∧ x_{m} ⊗ y_{m} ∧ ⋯ ∧ y_{1} ↦ 1.

Proposition 18. The following diagram commutes.

∧^{m}V [m] ⊗ S^{•}(V^{∗}[−1]) ⊗ ∧^{m}V [m] ⊗ ∧^{m}V^{∗}[−m] ⟶^{( - • - )⊗≃} S^{•}(V [1])
↓^{τ⊗∧^{m}V^{∗}[−m]}  ^{J_{V}} ↓
S^{•}(V^{∗}[−1]) ⊗ ∧^{m}V [m] ⊗ ∧^{m}V [m] ⊗ ∧^{m}V^{∗}[m] ⟶^{ζ_{V} ⊗≃} S^{•}(V [1]).

Proof. This is verified by a direct computation as well. Let {x_{i}}, {y_{i}} be as in the proof of the previous proposition.

(x_{1} ∧ ⋯ ∧ x_{m} • y_{m} ∧ ⋯ ∧ y_{k+1}) = x_{1} ∧ ⋯ ∧ x_{k}

implies that

(( - • - )⊗ ≃)(x_{1} ∧⋯∧x_{m} ⊗y_{m} ∧⋯∧y_{k+1} ⊗x_{1} ∧⋯∧x_{m} ⊗y_{m} ∧⋯∧y_{1}) = x_{1} ∧ ⋯ ∧ x_{k}.

Also,

τ(x_{1} ∧ ⋯ ∧ x_{m} ⊗ y_{m} ∧ ⋯ ∧ y_{k+1} ⊗ x_{1} ∧ ⋯ ∧ x_{m})

= (−1)^{mk}(y_{m} ∧ ⋯ ∧ y_{k+1} ⊗ x_{1} ∧ ⋯ ∧ x_{m} ⊗ x_{1} ∧ ⋯ ∧ x_{m})

and

ζ_{V} (y_{m} ∧ ⋯ ∧ y_{k+1} ⊗ x_{1} ∧ ⋯ ∧ x_{m}) = (−1)^{k(m−k)}x_{1} ∧ ⋯ ∧ x_{k}

imply that (ζ_{V} ⊗ ≃) ∘ (τ ⊗ ∧^{m}V^{∗}[−m]) takes

(x_{1} ∧ ⋯ ∧ x_{m} ⊗ y_{m} ∧ ⋯ ∧ y_{k+1} ⊗ x_{1} ∧ ⋯ ∧ x_{m} ⊗ y_{m} ∧ ⋯ ∧ y_{1})

to

(−1)^{−k^{2}}x_{1} ∧ ⋯ ∧ x_{k} = J_{V} (x_{1} ∧ ⋯ ∧ x_{k}).

This proves the desired proposition. □

Let C_{V} denote the comultiplication on S^{•}(V [1]). Think of C_{V} as an element of End(S^{•}(V [1])) ⊗ S^{•}(V [1]).

Proposition 19. If Y ∈ S^{•}(V^{∗}[−1]) then

(End(S^{•}(V [1])) ⊗ ( - ||Y) ∘ C_{V}) = j_{V^{∗}}(Y).

Proof. This is yet another proposition that is verified by a direct computation. Let {x_{i}}, {y_{i}} be as in the proof of the previous proposition. Without loss of generality, Y = y_{1} ∧ ⋯ ∧ y_{k}. Then

C_{V} (x_{l} ∧ ⋯ ∧ x_{1}) = x_{l} ∧ ⋯ ∧ x_{k+1} ⊗ x_{k} ∧ ⋯ ∧ x_{1} + ∑_{S≠\{l,...,k+1\}} ±x_{S} ⊗ x\overline{S}

implies that

(End(S^{•}(V [1])) ⊗ ( - ||Y) ∘ C_{V})(x_{l} ∧ ⋯ ∧ x_{1}) = x_{l} ⊗ ⋯ ⊗ x_{k+1}
= j_{V^{∗}}(Y)(x_{l} ∧ ⋯ ∧ x_{1}).

The desired proposition follows immediately. □

3.2. Applying the linear algebra to S^{•}(Ω[1]). Let M be a locally free coherent O_{X}-module. We begin with the remark that every proposition in the previous subsection holds in Ch^{b}(O_{X} −mod) (and hence in D^{b}(X)) with V replaced by M, V^{∗} replaced by M^{∗} = Hom_{O_{X}} (M, O_{X}). We are interested in the case when M = Ω. All graded O_{X}-modules in this subsection are to be thought of as complexes of O_{X}-modules with 0-differential.

All 𝕂-vector spaces in this section are finite-dimensional. In this subsection and in future sections, n shall denote the dimension of X.

3.2.1. An important point for the reader to note. Each proposition or lemma in this subsection is proven by proving it for an arbitrary open subscheme U of X such that Ω is trivial over U. Then, Ω|_{U} = O_{U} ⊗_{𝕂} V and T |_{U} = O_{U} ⊗_{𝕂} V^{∗} for some finite-dimensional 𝕂-vector space V. After observing that every map involved in the proposition/lemma is O_{U}- linear, proving the proposition/lemma reduces to proving the corresponding proposition/lemma in Section 3.1. The proposition/lemma in Section 3.1 corresponding to each proposition here can be thought of as the “local computation” required to prove the corresponding proposition/lemma in this subsection.

With the above announcement we can just state the propositions and lemma that we wish to state.

Note that there is a morphism

i_{Ω} : S^{•}(Ω[1]) → End(S^{•}(Ω[1]))

such that whenever U is an open subscheme on X such that Ω ≃ O_{U} ⊗_{𝕂} V for some 𝕂-vector space V, then

i_{Ω}|_{U} = i_{V} ⊗_{𝕂} O_{U}.

We denote i_{Ω} by i to simplify notation.

Similarly, we have a morphism

j_{T} : S^{•}(T [−1]) → End(S^{•}(Ω[1]))

such that whenever U is an open subscheme on X such that Ω ≃ O_{U} ⊗_{𝕂} V for some 𝕂-vector space V, then

j_{T} |_{U} = j_{V^{∗}} ⊗_{𝕂} O_{U}.

We denote j_{T} by j to simplify notation.

The following proposition corresponds to Proposition 10.

Proposition 20. The composite

S^{•}(T [−1]) ⊗ S^{•}(Ω[1]) ⟶^{j⊗i} End(S^{•}(Ω[1])) ⊗ End(S^{•}(Ω[1]) ⟶^{∘^{op}} End(S^{•}(Ω[1]))

is an isomorphism in Ch^{b}(O_{X} − mod).

Notation. G_{r} shall denote the isomorphism in Proposition 20 and F_{r} shall denote its inverse. In addition, if

τ : S^{•}(T [−1]) ⊗ S^{•}(Ω[1]) → S^{•}(Ω[1]) ⊗ S^{•}(T [−1])

denotes the swap map the composite

S^{•}(T [−1]) ⊗ S^{•}(Ω[1]) ⟶^{τ} S^{•}(Ω[1]) ⊗ S^{•}(T [−1]) ⟶^{i∘^{op}j} End(S^{•}(Ω[1]))

shall be denoted by G_{l}. The inverse of G_{l} will be denoted by F_{l}.

Let π_{k} : S^{•}(Ω[1]) → Ω^{k}[k] denote the natural projection. Note that we have a nondegenerate pairing ⟨, ⟩ on S^{•}(Ω[1]). This is given by the composite

S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]) ⟶^{( - ∧ - )} S^{•}(Ω[1]) ⟶^{π_{n}} Ω^{n}[n].

Recall the definition of the adjoint of an element of End(S^{•}(V [1])) from Section 3.1. Let A_{V} : End(S^{•}(V [1])) → End(S^{•}(V [1])) denote the map taking an element of End(S^{•}(V [1])) to its adjoint. We have a morphism A_{Ω} : End(S^{•}(Ω[1])) → End(S^{•}(Ω[1])) in Ch^{b}(O_{X} − mod) such that if U is an open subscheme of X such that Ω ≃ O_{U} ⊗_{𝕂} V for some 𝕂-vector space V, then

A_{Ω} = A_{V} ⊗_{𝕂} O_{U}.

Similarly, the map

ev_{V} : S^{•}(V [1]) ⊗ End(S^{•}(V [1]))
Z ⊗ Φ ↦ Φ(Z)

yields a map

ev : S^{•}(Ω[1]) ⊗ End(S^{•}(Ω[1])) → S^{•}(Ω[1])

such that

ev|_{U} = ev_{V} ⊗_{𝕂} O_{U}

on any open subscheme U of X such that Ω|_{U} = V ⊗_{𝕂}O_{U} for some 𝕂-vector space V.

We denote A_{Ω} by A to simplify notation. Let ⟨ , ev^{+}⟩ denote the composite

S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]) ⊗ End(S^{•}(Ω[1])
↓^{S^{•}(Ω[1])⊗S^{•}(Ω[1])⊗A}
S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]) ⊗ End(S^{•}(Ω[1])
↓^{S^{•}(Ω[1])⊗ev}
S^{•}(Ω[1]) ⊗ S^{•}(Ω[1])
↓^{⟨ , ⟩}
Ω^{n}[n].

By the corresponding fact for a finite-dimensional 𝕂-vector space V, the following diagram commutes in Ch^{b}(O_{X} − mod) (and hence in D^{b}(X)).

(34)

S^{•}(Ω[1]) ⊗ End(S^{•}(Ω[1])) ⊗ S^{•}(Ω[1]) ⟶^{ev⊗S^{•}(Ω[1])} S^{•}(Ω[1]) ⊗ S^{•}(Ω[1])
↓^{S^{•}(Ω[1])⊗τ}  ^{⟨,⟩}↓
S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]) ⊗ End(S^{•}(Ω[1])) ⟶^{⟨ ,ev^{+}⟩}  Ω^{n}[n].

The map τ : End(S^{•}(Ω[1])) ⊗ S^{•}(Ω[1]) → S^{•}(Ω[1]) ⊗ End(S^{•}(Ω[1])) in the above diagram swaps factors.

The following proposition corresponds to Proposition 11 in Section 3.1.

Proposition 21. (1) To begin with,

A ∘ A = 1_{End(S^{•}(Ω[1]))}.

(2) If τ : End(S^{•}(Ω[1]))⊗End(S^{•}(Ω[1])) → End(S^{•}(Ω[1]))⊗End(S^{•}(Ω[1])) is the swap map, then the following diagram in Ch^{b}(O_{X} − mod) commutes.

End(S^{•}(Ω[1])) ⊗ End(S^{•}(Ω[1])) ⟶^{(A⊗A)∘τ} End(S^{•}(Ω[1])) ⊗ End(S^{•}(Ω[1]))
↓^{∘}  ^{∘}↓
End(S^{•}(Ω[1]))  ⟶^{A}  End(S^{•}(Ω[1])).

(3) Also,

A ∘ i = i.

(4) If I : S^{•}(T [−1]) → S^{•}(T [−1]) denotes the endomorphism multiplying ∧^{i}T [−i] by (−1)^{i} then

A ∘ j = j ∘ I.

(5) Finally,

A ∘ G_{l} = G_{r} ∘ (I ⊗ S^{•}(Ω[1])).

Recall that π_{j} : S^{•}(Ω[1]) → Ω^{j}[j] denotes the natural projection. We will denote the projection

S^{•}(T [−1]) ⊗ π_{j} : S^{•}(T [−1]) ⊗ S^{•}(Ω[1]) → S^{•}(T [−1]) ⊗ Ω^{j}[j]

by π_{j} itself. Let I be as in part (4), Proposition 21. The following proposition corresponds to Proposition 12 of Section 3.1.

Proposition 22.

π_{0} ∘ F_{l} = I ∘ π_{0} ∘ F_{r} ∘ A.

Denote the composite

(S^{•}(T [−1]) ⊗ S^{•}(Ω[1]))^{⊗2}
G_{r}∘G_{r}↓
End(S^{•}(Ω[1]))  ⟶^{F_{r}} S^{•}(T [−1]) ⊗ S^{•}(Ω[1])

by m to simplify notation. The proposition below corresponds to Proposition 13 of Section 3.1.

Proposition 23. As morphisms in Ch^{b}(O_{X} − mod),

π_{0} ∘ m = π_{0} ∘ m ∘ (π_{0} ⊗ S^{•}(T [−1]) ⊗ S^{•}(Ω[1])).

Denote the direct summand ∧^{i}T [−i] ⊗ S^{•}(Ω[1]) of End(S^{•}(Ω[1])) by D_{i}. Note that the map [,]_{V} : End(S^{•}(V [1]))^{⊗2} → End(S^{•}(V [1])) extends to a morphism [,] : End(S^{•}(Ω[1]))^{⊗2} → End(S^{•}(Ω[1])). If U is an open subscheme on X such that Ω ≃ O_{U} ⊗_{𝕂} V for some 𝕂-vector space V, then

[,]|_{U} = [,]_{V} ⊗_{𝕂} O_{U}.

The following proposition corresponds to Proposition 14 of Section 3.1.

Proposition 24. The composite

D_{1} ⊗ D_{1} ⟶ End(S^{•}(Ω[1]))^{⊗2} ⟶^{[,]} End(S^{•}(Ω[1]))

factors through the inclusion of D_{1} in End(S^{•}(Ω[1])).

Let V be a 𝕂-vector space of dimension n. Denote the direct summands V^{∗}[−1] ⊗ S^{•}(V [1]) and ∧^{n}V^{∗}[−n] ⊗ S^{•}(V [1]) of S^{•}(V^{∗}[−1]) ⊗ S^{•}(V [1]) by \overline{D}_{1} and \overline{D}_{n} respectively, unlike in Section 3.1. Recall that in Section 3.1 we defined maps

ad(L) : ∧^{n}V^{∗}[−n] ⊗ S^{•}(V [1]) → ∧^{n}V^{∗}[−n] ⊗ S^{•}(V [1])
\overline{ad}(L) : \overline{D}_{1}^{⊗n} → \overline{D}_{1}^{⊗n}

for any L ∈ \overline{D}_{1}. These yield maps

ad_{V} : ∧^{n}V^{∗}[−n] ⊗ S^{•}(V [1]) ⊗ \overline{D}_{1} → ∧^{n}V^{∗}[−n] ⊗ S^{•}(V [1])
Y ⊗ L ↦ ad(L)(Y)
\overline{ad}_{V} : \overline{D}_{1}^{⊗n} ⊗ \overline{D}_{1} → \overline{D}_{1}^{⊗n}
W ⊗ L ↦ \overline{ad}(L)(W).

These yield morphisms

ad : ∧^{n}T [−n] ⊗ S^{•}(Ω[1]) ⊗ D_{1} → ∧^{n}T [−n] ⊗ S^{•}(Ω[1])
\overline{ad} : D_{1}^{⊗n} ⊗ D_{1} → D_{1}^{⊗n}

in Ch^{b}(O_{X} −mod). If U is an open subscheme on X such that Ω ≃ O_{U} ⊗_{𝕂}V for some 𝕂-vector space V, then

ad|_{U} = ad_{V} ⊗_{𝕂} O_{U}
\overline{ad}|_{U} = \overline{ad}_{V} ⊗_{𝕂} O_{U}.

The map p_{V} : \overline{D}_{1}^{⊗n} → \overline{D}_{n} also yields a map

p : D_{1}^{⊗n} → ∧^{n}T [−n] ⊗ S^{•}(Ω[1])

such that if Ω ≃ V ⊗_{𝕂} O_{U} for some open subscheme U of X and some 𝕂-vector space V, then,

p = p_{V} ⊗_{𝕂} O_{U}.

Note that the dimension n of X is the rank of Ω as well. Let

1^{n} : O_{X} → ∧^{n}T [−n] ⊗ Ω^{n}[n]

denote the map dual to the evaluation map. There are maps

τ : ∧^{n}T [−n] ⊗ Ω^{n}[n] ⊗ S^{•}(Ω[1]) ⊗ D_{1} → ∧^{n}T [−n] ⊗ S^{•}(Ω[1]) ⊗ D_{1} ⊗ Ω^{n}[n]

and

τ′ : ∧^{n}T [−n] ⊗ Ω^{n}[n] ⊗ S^{•}(Ω[1]) → ∧^{n}T [−n] ⊗ S^{•}(Ω[1]) ⊗ Ω^{n}[n].

τ is obtained by swapping Ω^{n}[n] and S^{•}(Ω[1]) ⊗ D_{1}. τ′ is obtained by swapping Ω^{n}[n] and S^{•}(Ω[1]). Denote the composites

τ ∘ (1^{n} ⊗ S^{•}(Ω[1]) ⊗ D_{1})

and

τ′ ∘ (1^{n} ⊗ S^{•}(Ω[1]))

by 1^{n}. The following proposition corresponds to Proposition 15 of Section 3.1.

Proposition 25. The following diagram commutes in Ch^{b}(O_{X} − mod).

D_{1}^{⊗n} ⊗ D_{1} ⟶^{p⊗D_{1}} D_{n} ⊗ D_{1}
\sqrt{ad}  _{ad}↓
D_{1}^{⊗n}  ⟶^{p} D_{n}.

The following lemma corresponds to Lemma 1 of Section 3.1. In the following lemma, A denotes the composite

D_{1} ⟶ End(S^{•}(Ω[1])) ⟶^{A} End(S^{•}(Ω[1])).

Lemma 2. The following diagram commutes in Ch^{b}(O_{X} − mod).

S^{•}(Ω[1]) ⊗ D_{1} ⟶^{1^{n}} ∧^{n}T [−n] ⊗ S^{•}(Ω[1]) ⊗ D_{1} ⊗ Ω^{n}[n]
↓^{−ev∘(S^{•}(Ω[1])⊗A)}  ^{ad⊗Ω^{n}[n]}↓
S^{•}(Ω[1]) ⟶^{1^{n}} ∧^{n}T [−n] ⊗ S^{•}(Ω[1]) ⊗ Ω^{n}[n].

The map

( - | - )_{V} : S^{•}(V [1]) ⊗ S^{•}(V^{∗}[−1]) → S^{•}(V^{∗}[−1])
Z ⊗ Y ↦ (Z|Y)

yields a map

( - | - ) : S^{•}(Ω[1]) ⊗ S^{•}(T [−1]) → S^{•}(T [−1]).

As usual, if U is an open subscheme on X such that Ω ≃ O_{U} ⊗_{𝕂} V for some 𝕂-vector space V, then

( - | - ) = ( - | - )_{V} ⊗_{𝕂} O_{U}.

Recall that the composition product on End(S^{•}(Ω[1])) was denoted by ∘. Let ∘^{op} denote the product on End(S^{•}(Ω[1]))^{op}. The following proposition corresponds to Proposition 16 of Section 3.1.

Proposition 26.

π_{0} ∘ F_{r} ∘ (i ∘^{op} j) = ( - | - ) : S^{•}(Ω[1]) ⊗ S^{•}(T [−1]) → S^{•}(T [−1]).

Let ( - || - ) : S^{•}(Ω[1]) ⊗ S^{•}(T [−1]) → O_{X} denote the map such that for any open U ⊂ X such that Ω ≃ V ⊗_{𝕂} O_{U} for some 𝕂-vector space V,

( - || - ) = ( - || - )_{V} ⊗_{𝕂} O_{U}.

Let γ : S^{•}(Ω[1]) → S^{•}(T [−1]) ⊗ S_{X} be the isomorphism such that

π_{n} ∘ ( - ∧ - ) = (( - || - ) ⊗ S_{X}) ∘ (S^{•}(Ω[1]) ⊗ γ).

Let ζ denote the inverse of γ.

The following proposition corresponds to Proposition 17 of Section 3.1.

Proposition 27.

ζ([( - | - )⊗S_{X}]∘[S^{•}(Ω[1])⊗γ]) = ( - ∧ - ) : S^{•}(Ω[1])⊗S^{•}(Ω[1]) → S^{•}(Ω[1]).

Let τ : S_{X} ⊗S^{•}(T [−1])⊗S_{X} → S^{•}(T [−1])⊗S_{X} ⊗S_{X} be the map swapping S_{X} with S^{•}(T [−1]) ⊗ S_{X}. Let ≃ be the identification of S_{X} ⊗ S_{X}^{−1} with O_{X}. The following proposition corresponds to Proposition 18 of Section 3.1. Recall that J is the endomorphism of S^{•}(Ω[1]) that multiplies Ω^{j}[j] by (−1)^{j}.

Proposition 28. The following diagram commutes in Ch^{b}(O_{X} − mod).

S_{X} ⊗ S^{•}(T [−1]) ⊗ S_{X} ⊗ S_{X}^{−1} ⟶^{τ⊗S_{X}^{−1}} S^{•}(T [−1]) ⊗ S_{X} ⊗ S_{X} ⊗ S_{X}^{−1}
↓^{( - • - )⊗≃}  ^{ζ⊗≃}↓
S^{•}(Ω[1])  ⟶^{J}  S^{•}(Ω[1]).

Let C_{Ω} denote the coproduct on S^{•}(Ω[1]). Think of C_{Ω} as a morphism in D^{b}(X) from O_{X} to End(S^{•}(Ω[1])) ⊗ S^{•}(Ω[1]). The following proposition corresponds to Proposition 19 of Section 3.1:

Proposition 29.

End(S^{•}(Ω[1])) ⊗ ( - || - ) ∘ C_{Ω} = j : S^{•}(T [−1]) → End(S^{•}(Ω[1])).

4. The adjoint of Φ_{R}

This section is a continuation of Section 2. All maps in this section are in D^{b}(X) unless explicitly stated otherwise. Both in this section and the next, we use results from Section 3.2. All morphisms in Ch^{b}(O_{X} −mod) described in Section 3.2 induce morphisms in D^{b}(X). The diagrams that were shown to commute in Ch^{b}(O_{X} − mod) in Section 3.2 also commute in D^{b}(X).

4.1. Stating the main lemma of this section. Let Φ_{L} and Φ_{R} be as in Section 2. By Theorem 2′, the following diagrams commute in D^{b}(X).

Ĉ^{•}(X) ⟶^{α_{L}} Ĉ^{•}(X) ⊗ Ω[1]
↓^{I_{HKR}}  ^{I_{HKR}⊗Ω[1]}↓
S^{•}(Ω[1]) ⟶^{Φ_{L}} S^{•}(Ω[1]) ⊗ Ω[1]
Ĉ^{•}(X) ⟶^{α_{R}} Ĉ^{•}(X) ⊗ Ω[1]
↓^{I_{HKR}}  ^{I_{HKR}⊗Ω[1]}↓
S^{•}(Ω[1]) ⟶^{Φ_{R}} S^{•}(Ω[1]) ⊗ Ω[1].

Φ_{R} can be thought of as lying in Hom_{D^{b}(X)}(O_{X}, End(S^{•}(Ω[1])) ⊗ Ω[1]). Recall the endomorphism A of End(S^{•}(Ω[1])) (defined in Section 3.2) which takes a section of End(S^{•}(Ω[1])) to its adjoint.

Definition 2. The adjoint Φ_{R}^{+} of Φ_{R} is the element (A ⊗ Ω[1]) ∘ Φ_{R} of Hom_{D^{b}(X)}(O_{X}, End(S^{•}(Ω[1])) ⊗ Ω[1]).

This section is devoted to finding an explicit formula for Φ_{R}^{+}.

Note that the Atiyah class of the tangent bundle T of X yields a morphism At_{T} : Ω[1] → Ω[1] ⊗ Ω[1]. Let p : Ω[1]^{⊗i} → Ω^{i}[i] denote the standard projection. Let

At_{T}^{i} : Ω[1] → S^{•}(Ω[1]) ⊗ Ω[1]

denote the composite

(p ⊗ Ω[1]) ∘ (Ω[1]^{⊗(i−1)} ⊗ At_{T}) ∘ ⋯ ∘ At_{T}.

Then if \frac{z}{e^{z}−1} = 1 + ∑_{i} c_{i}Z^{i} the map

\frac{At_{T}}{exp(At_{T}) − 1} := 1 + ∑_{i} c_{i}At_{T}^{i} : Ω[1] → S^{•}(Ω[1]) ⊗ Ω[1]

makes sense. Denote the element

det (\frac{At_{T}}{exp(At_{T}) − 1}) ∈ Hom_{D^{b}(X)}(O_{X}, S^{•}(Ω[1]))

by f.

Note that i(f) and i(f^{−1}) are elements of Hom_{D^{b}(X)}(O_{X}, End(S^{•}(Ω[1]))). Let i(f) ∘ Φ_{R} ∘ i(f^{−1}) denote the composite

O_{X}
Φ_{R}↓
End(S^{•}(Ω[1])) ⊗ Ω[1] ≃ O_{X} ⊗ End(S^{•}(Ω[1])) ⊗ O_{X} ⊗ Ω[1]
i(f)⊗End(S^{•}(Ω[1]))⊗i(f^{−1})⊗Ω[1]↓
End(S^{•}(Ω[1])) ⊗ End(S^{•}(Ω[1])) ⊗ End(S^{•}(Ω[1])) ⊗ Ω[1]
∘⊗Ω[1]↓
End(S^{•}(Ω[1])) ⊗ Ω[1].

The following lemma is the main lemma of this section.

Lemma 3.

Φ_{R}^{+} = i(f) ∘ Φ_{R} ∘ i(f^{−1}).

The proof of this lemma requires further preparation. The following subsection is devoted to a key lemma (Lemma 4) used to prove Lemma 3. The proof of Lemma 3 itself is at the end of this section (in Section 4.4).

4.2. Comparing two “sections” of D_{n} ⊗ Ω^{n}[n]. Let

p_{k} : S^{•}(T [−1]) → ∧^{k}T [−k]

be the standard projection. View Φ_{L} as an element of

Hom_{D^{b}(X)}(O_{X}, End(S^{•}(Ω[1])) ⊗ Ω[1]).

We have the following proposition.

Proposition 30.

(p_{k} ⊗ S^{•}(Ω[1]) ⊗ Ω[1]) ∘ (F_{r} ⊗ Ω[1])(Φ_{L}) = 0 ∀ k ≠ 1.

Remark 10. The above proposition just states that Φ_{L} can be thought of as an element of Hom_{D^{b}(X)}(O_{X}, D_{1} ⊗ Ω[1]). Recall that D_{1} is like the space of “purely first-order differential operators on S^{•}(Ω[1])”. In principle, this proposition should follow from Proposition 7 and Theorem 2′. I however, give a concrete proof below that uses the definition of Φ_{L} from Section 2, as I cant see how the above proposition follows immediately from Proposition 7.

Proof of Proposition 30. Let

C̃ : S^{•}(T [−1]) → S^{•}(T [−1]) ⊗ S^{•}(T [−1])

denote coproduct on S^{•}(T [−1]. Denote the composite (S^{•}(T [−1]) ⊗ p_{1}) ∘ C̃ by Ĉ. Denote the wedge product on S^{•}(Ω[1]) by μ in this proof only. Note that the dual of the map

\overline{ω} : S^{•}(Ω[1]) ⊗ Ω[1] → S^{•}(Ω[1]) ⊗ Ω[1]

defined in Section 2 is the composite

(S^{•}(T [−1]) ⊗ At_{T}) ∘ (Ĉ ⊗ T [−1]).

It follows that

\overline{ω} = (μ ⊗ Ω[1]) ∘ (S^{•}(Ω[1]) ⊗ At_{T}).

In the latter composite, At_{T} is thought of as a morphism in D^{b}(X) from Ω[1] to Ω[1] ⊗ Ω[1]. Therefore,

(35)  \overline{ω}^{i} = (μ ⊗ Ω[1]) ∘ (S^{•}(Ω[1]) ⊗ At_{T}^{i})

where At_{T}^{i} is as in the previous subsection.

It follows from (35) that (p_{k} ⊗ S^{•}(Ω[1]) ⊗ Ω[1]) ∘ (F_{r} ⊗ Ω[1])(\overline{ω}^{i} ∘ \overline{C}) is given by the composite

(O_{X}) / (\overline{C}↓)

End(S^{•}(Ω[1])) ⊗ Ω[1]
((p_{k}⊗S^{•}(Ω[1]))∘F_{r})⊗At_{T}^{i} ↓
∧^{k}T [−k] ⊗ S^{•}(Ω[1]) ⊗ Ω^{i}[i] ⊗ Ω[1]
∧^{k}T [−k]⊗μ⊗Ω[1]↓
∧^{k}T [−k] ⊗ S^{•}(Ω[1]) ⊗ Ω[1].

Note that from the above description, proving that

(((p_{k} ⊗ S^{•}(Ω[1])) ∘ F_{r}) ⊗ Ω[1]) ∘ \overline{C} = 0

will imply that

(p_{k} ⊗ S^{•}(Ω[1]) ⊗ Ω[1]) ∘ (F_{r} ⊗ Ω[1])(\overline{ω}^{i} ∘ \overline{C}) = 0.

The desired proposition will then follow the fact that

Φ_{L} = ∑_{i} c_{i}\overline{ω}^{i} ∘ \overline{C}

where ∑_{i} c_{i}z^{i} = \frac{z}{e^{z}−1}.

Proving this proposition has therefore been reduced to proving that

(36)  (((p_{k} ⊗ S^{•}(Ω[1])) ∘ F_{r}) ⊗ Ω[1]) ∘ \overline{C} = 0 ∀k ≠ 1.

Let U be an affine open subscheme of X such that Ω ≃ V ⊗_{K} O_{U} for some K-vector space V. Proving (36) reduces to proving that

(37)  (((p_{k} ⊗ S^{•}(V [1])) ∘ F_{r}) ⊗ V [1]) ∘ C_{V} = 0 ∀k ≠ 1

where C_{V} ∈ End(S^{•}(V [1]))⊗V [1] ≃ Hom_{𝕂}(S^{•}(V [1]), S^{•}(V [1])⊗V [1]) is the map

v_{1} ∧ ⋯ ∧ v_{j} ↦ ∑_{i} (−1)^{j−i}{ \atop v_{1} ∧ ⋯ i ⋯ ∧ v_{j}} ⊗ v_{i},

F_{r} is as in Section 3.1 and p_{k} : S^{•}(V^{∗}[−1]) → ∧^{k}V^{∗}[−k] denotes the standard projection. (37) however, follows from the fact that

C_{V} = ∑_{j=1}^{j=n} j_{V} (y_{j}) ⊗ x_{j} ⟹ (F_{r} ⊗ V [1]) ∘ C_{V} = ∑_{j=1}^{j=n} y_{j} ⊗ x_{j}

for any bases {x_{1}, …, x_{n}} of V and {y_{1}, …, y_{n}} of V^{∗} dual to each other. □

The proof of Proposition 30 also helps us understand the map Φ_{L} better: let

id : O_{X} → T [−1] ⊗ Ω[1]

denote the dual of the evaluation map from Ω[1] ⊗ T [−1] to O_{X}.

Proposition 31. As an element of Hom_{D^{b}(X)}(O_{X}, D_{1} ⊗ Ω[1]), Φ_{L} is given by the composite

O_{X} ⟶^{id} T [−1] ⊗ Ω[1] ⟶^{T [−1]⊗ \frac{At_{T}}{exp(At_{T})−1}} T [−1] ⊗ S^{•}(Ω[1]) ⊗ Ω[1].

Proof. By the proof of Proposition 30, \overline{C} : S^{•}(Ω[1]) → S^{•}(Ω[1]) ⊗ Ω[1] is given by the composite

S^{•}(Ω[1]) ⊗ O_{X}
S^{•}(Ω[1])⊗id↓
S^{•}(Ω[1]) ⊗ T [−1] ⊗ Ω[1] ⟶^{( - • - )⊗Ω[1]} S^{•}(Ω[1]) ⊗ Ω[1].

Note that by definition, Φ_{L} = \frac{\overline{ω}}{e\overline{ω}−1} ∘ \overline{C}. Also, Equation (35) in the proof of Proposition 30 says that \overline{ω}^{i} is given by the composite

S^{•}(Ω[1]) ⊗ Ω[1]
S^{•}(Ω[1])⊗At_{T}^{i} ↓
S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]) ⊗ Ω[1] ⟶^{( - ∧ - )⊗Ω[1]} S^{•}(Ω[1]) ⊗ Ω[1].

Therefore \frac{\overline{ω}}{e\overline{ω}−1} is given by the composite

S^{•}(Ω[1]) ⊗ Ω[1]
S^{•}(Ω[1])⊗ \frac{At_{T}}{exp(At_{T})−1} ↓
S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]) ⊗ Ω[1] ⟶^{( - ∧ - )⊗Ω[1]} S^{•}(Ω[1]) ⊗ Ω[1].

It follows that as a morphism in D^{b}(X) from S^{•}(Ω[1]) to S^{•}(Ω[1]) ⊗ Ω[1], Φ_{L} is given by the composite

S^{•}(Ω[1]) ⊗ O_{X}
↓^{S^{•}(Ω[1])⊗id}
S^{•}(Ω[1]) ⊗ T [−1] ⊗ Ω[1]
↓^{S^{•}(Ω[1])⊗T [−1]⊗ \frac{At_{T}}{exp(At_{T})−1}}
S^{•}(Ω[1]) ⊗ T [−1] ⊗ S^{•}(Ω[1]) ⊗ Ω[1] ⟶^{(( - • - )∧ - )⊗Ω[1]} S^{•}(Ω[1]) ⊗ Ω[1].

This proves the desired proposition. □

By Proposition 30, Φ_{L} can be thought of as an element of

Hom_{D^{b}(X)}(O_{X}, D_{1} ⊗ Ω[1]) ≃ Hom_{D^{b}(X)}(O_{X}, T [−1] ⊗ S^{•}(Ω[1]) ⊗ Ω[1]).

Let Φ_{L}^{n} denote the composite

O_{X}
Φ_{L}^{⊗n}↓

(T [−1] ⊗ S^{•}(Ω[1]) ⊗ Ω[1])^{⊗n}

τ↓
T^{⊗n}[−n] ⊗ S^{•}(Ω[1])^{⊗n} ⊗ Ω^{⊗n}[n] ⟶^{p′⊗m⊗p} ∧^{n}T [−n] ⊗ S^{•}(Ω[1]) ⊗ Ω^{n}[n].

The map τ in the above diagram is a rearrangement of factors. The map m : S^{•}(Ω[1])^{⊗n} → S^{•}(Ω[1]) is n-fold multiplication. p′ is the standard projection from T^{⊗n}[−n] to ∧^{n}T [−n] and p is the projection from Ω^{⊗n}[n] to Ω^{n}[n]. Note that Φ_{L}^{n} is an element of Hom_{D^{b}(X)}(O_{X}, D_{n} ⊗ Ω^{n}[n]). Let 1^{n} : S^{•}(Ω[1]) → D_{n} ⊗ Ω^{n}[n] be as in Section 3.2. Then:

Lemma 4. The following diagram commutes in D^{b}(X).

O_{X}  ⟶^{1}  O_{X}
↓_{f}  ^{Φ_{L}^{n}}↓
S^{•}(Ω[1]) ⟶^{1^{n}} D_{n} ⊗ Ω^{n}[n].

Proof. Step 1: The inverse of 1^{n}. Let e : O_{X} → Ω^{n}[n] ⊗ ∧^{n}T [−n] denote the natural isomorphism dual to the evaluation map from ∧^{n}T [−n] ⊗ Ω^{n}[n] to O_{X}. Denote the evaluation map from Ω^{n}[n] ⊗ ∧^{n}T [−n] to O_{X} by b. We claim that the inverse to 1^{n} is given by the following composite.

(38)

O_{X} ⊗ D_{n} ⊗ Ω^{n}[n]
e⊗D_{n}⊗Ω^{n}[n]↓
Ω^{n}[n] ⊗ ∧^{n}T [−n] ⊗ D_{n} ⊗ Ω^{n}[n]
τ↓
Ω^{n}[n] ⊗ D_{n} ⊗ Ω^{n}[n] ⊗ ∧^{n}T [−n]
≃↓
Ω^{n}[n] ⊗ ∧^{n}T [−n] ⊗ S^{•}(Ω[1]) ⊗ Ω^{n}[n] ⊗ ∧^{n}T [−n]
b⊗S^{•}(Ω[1])⊗b↓
S^{•}(Ω[1]).

The map τ in (38) swaps ∧^{n}T [−n] and D_{n} ⊗ Ω^{n}[n].

To see this, note that if U is an open subscheme of X such that

Ω|_{U} ≃ V ⊗_{𝕂} O_{U}

for some n-dimensional 𝕂-vector space V, the maps involved in (38) can be described explicitly. Choose a basis {x_{1}, …, x_{n}} of V and let {y_{1}, …, y_{n}} be the dual basis of V^{∗}. Let

e_{V} : 𝕂 → ∧^{n}V [n] ⊗ ∧^{n}V^{∗}[−n]
1 ↦ x_{1} ∧ ⋯ ∧ x_{n} ⊗ y_{n} ∧ ⋯ ∧ y_{1}.

Let

b_{V} : ∧^{n}V [n] ⊗ ∧^{n}V^{∗}[−n] → 𝕂
x_{1} ∧ ⋯ ∧ x_{n} ⊗ y_{n} ∧ ⋯ ∧ y_{1} ↦ 1.

Let 1_{V}^{n} be as in Section 3.1. The map denoted by 1_{V}^{n} was denoted by 1^{m} in Section 3.1. Then, if H ∈ S^{•}(V [1]) is a homogenous element,

1^{n}(H) = (−1)^{n|H|}y_{n} ∧ ⋯ ∧ y_{1} ⊗ H ⊗ x_{1} ∧ ⋯ ∧ x_{n}.

(e_{V} ⊗ ∧^{n}V^{∗}[−n] ⊗ S^{•}(V [1]) ⊗ ∧^{n}V [n]) ∘ 1^{n}(H)

= (−1)^{n|H|}x_{1} ∧ ⋯ ∧ x_{n} ⊗ y_{n} ∧ ⋯ ∧ y_{1} ⊗ y_{n} ∧ ⋯ ∧ y_{1} ⊗ H ⊗ x_{1} ∧ ⋯ ∧ x_{n}.

τ((−1)^{n|H|}x_{1} ∧ ⋯ ∧ x_{n} ⊗ y_{n} ∧ ⋯ ∧ y_{1} ⊗ y_{n} ∧ ⋯ ∧ y_{1} ⊗ H ⊗ x_{1} ∧ ⋯ ∧ x_{n})

= (−1)^{n|H|}(−1)^{n|H|}x_{1} ∧ ⋯ ∧ x_{n} ⊗ y_{n} ∧ ⋯ ∧ y_{1} ⊗ H ⊗ x_{1} ∧ ⋯ ∧ x_{n} ⊗ y_{n} ∧ ⋯ ∧ y_{1}.

(b_{V} ⊗ S^{•}(V [1]) ⊗ b_{V})((−1)^{n|H|}(−1)^{n|H|}x_{1} ∧ ⋯ ∧ x_{n} ⊗ y_{n} ∧ ⋯ ∧ y_{1}

⊗ H ⊗ x_{1} ∧ ⋯ ∧ x_{n} ⊗ y_{n} ∧ ⋯ ∧ y_{1}) = H.

The fact that the composite given in (38) is the inverse to 1^{n} follows from the facts that b|_{U} = b_{V} ⊗_{𝕂} O_{U}, e|_{U} = e_{V} ⊗_{𝕂} O_{U} and 1^{n}|_{U} = 1_{V}^{n} ⊗_{𝕂} O_{U}.

Step 2. Let us look at the composition of the composite map in (38) with Φ_{L}^{n}. Φ_{L}^{n} can also be identified with the composite

(b ⊗ S^{•}(Ω[1]) ⊗ Ω^{n}[n]) ∘ (Ω^{n}[n] ⊗ Φ_{L}^{n}) : Ω^{n}[n] → S^{•}(Ω[1]) ⊗ Ω^{n}[n].

Thinking of Φ_{L}^{n} as the above composite, it follows from (38) that the inverse of 1^{n} composed with Φ_{L}^{n} : O_{X} → D_{n}⊗Ω^{n}[n] is given by the composite (39)

O_{X}
e↓
Ω^{n}[n] ⊗ ∧^{n}T [−n]
Φ_{L}^{n}⊗∧^{n}T [−n]↓
S^{•}(Ω[1]) ⊗ Ω^{n}[n] ⊗ ∧^{n}T [−n]
S^{•}(Ω[1])⊗b↓
S^{•}(Ω[1]).

Let b_{1} : Ω[1] ⊗ T [−1] → O_{X} denote the evaluation map. Let

m : S^{•}(Ω[1])^{⊗n} → S^{•}(Ω[1])

denote the n-fold product. Let p : Ω[1]^{⊗n} → Ω^{n}[n] denote the natural projection. Note that Φ_{L}^{n} : Ω^{n}[n] → S^{•}(Ω[1]) ⊗ Ω^{n}[n] is also given by the following composite.

(40)

Ω^{n}[n]
↓
(Ω[1] ⊗ O_{X})^{⊗n}
↓^{(Ω[1]⊗Φ_{L})}^{⊗n}
(Ω[1] ⊗ T [−1] ⊗ S^{•}(Ω[1]) ⊗ Ω[1])^{⊗n}
↓^{(b_{1}⊗S^{•}(Ω[1])⊗Ω[1])}^{⊗n}
(S^{•}(Ω[1]) ⊗ Ω[1])^{⊗n}
↓^{τ}
S^{•}(Ω[1])^{⊗n} ⊗ Ω[1]^{⊗n}
↓^{m⊗p}
S^{•}(Ω[1]) ⊗ Ω^{n}[n].

The topmost vertical arrow in the above diagram is induced by the natural inclusion from Ω^{n} to Ω^{⊗n}. The map τ in the above diagram is a rearrangement of factors.

It follows from Proposition 31 and that the composite

(b_{1} ⊗ S^{•}(Ω[1]) ⊗ Ω[1]) ∘ (Ω[1] ⊗ Φ_{L}) : Ω[1] → S^{•}(Ω[1]) ⊗ Ω[1]

is precisely \frac{At_{T}}{exp(At_{T})−1}.

Therefore the composite in (40) is equal to the composite Ω^{n}[n]

↓
Ω[1]^{⊗n}
↓( \frac{At_{T}}{exp(At_{T})−1} )^{⊗n}
(S^{•}(Ω[1]) ⊗ Ω[1])^{⊗n} ⟶^{τ} S^{•}(Ω[1])^{⊗n} ⊗ Ω[1]^{⊗n} ⟶^{m⊗p} S^{•}(Ω[1]) ⊗ Ω^{n}[n].

It follows from Equation (39) that the inverse of 1^{n} composed with Φ_{L}^{n} is given by the composite

O_{X}
↓^{e}
Ω^{n}[n] ⊗ ∧^{n}T [−n]
↓
Ω[1]^{⊗n} ⊗ ∧^{n}T [−n]
↓^{( \frac{At_{T}}{exp(At_{T})−1} )^{⊗n}⊗∧^{n}T [−n]}
(S^{•}(Ω[1]) ⊗ Ω[1])^{⊗n} ⊗ ∧^{n}T [−n]
↓^{τ}
S^{•}(Ω[1])^{⊗n} ⊗ Ω[1]^{⊗n} ⊗ ∧^{n}T [−n]
↓^{m⊗p⊗∧^{n}T [−n]}
S^{•}(Ω[1]) ⊗ Ω^{n}[n] ⊗ ∧^{n}T [−n] ⟶^{S^{•}(Ω[1])⊗b} S^{•}(Ω[1]).

The above composite is det (\frac{At_{T}}{exp(At_{T})−1}) by the definition of det. □

4.3. Another long remark — Lemma 4 and our dictionary. Lemma 4 is the root reason for the Todd genus, an expression having a form similar to the Jacobian of the differential d(exp^{−1}), showing up in the Riemann–Roch theorem. Markarian [6] remarks that a lemma in [6] similar to Lemma 3 in this paper is like pulling back the canonical volume form on a Lie group via the exponential map. He makes a remark in [5] that a formula analogous to that describing the pullback of the canonical (left-invariant) volume form on a Lie group via the exponential map is responsible for the Todd genus showing up in the Riemann–Roch theorem. Lemma 4 is precisely where something like this happens. We will attempt to make the parallel between Lemma 4 and “pulling back the canonical left-invariant volume form by the map \overline{exp}” more transparent. In Lemma 4 of this paper, it is \overline{exp} rather than the exponential map itself that is involved.

We warn the reader that ad and \overline{ad} have the same meaning in this subsection as in Section 2.5. Their meaning in this section is therefore, different from their meaning in Section 3, the rest of Section 4 and Section 5.

4.3.1. The classical situation. Keep the dictionary developed up to Section 2.5 in mind. Let G be a Lie group and 𝔤 its Lie algebra. Choose a basis {X_{i}} of 𝔤 and a basis {Y_{i}} of 𝔤^{∗} dual to {X_{i}}. Let n be the dimension of 𝔤. Let 1_{𝔤} denote the element ∑_{i=1}^{n} X_{i} ⊗ Y_{i} of 𝔤 ⊗ 𝔤^{∗}. Let C(G) and C(𝔤) be as in Section 2.5. Letting an element of 𝔤 act as a differential operator on C(G) (as in Section 2.5) yields a connection on C(G) for each element of 𝔤 ⊗ 𝔤^{∗}. 1_{𝔤} yields the canonical connection d_{G}.

The element 1^{n} := Y_{n} ∧ ⋯ ∧ Y_{1} ⊗ X_{1} ∧ ⋯ ∧ X_{n} of ∧^{n}𝔤^{∗} ⊗ 𝔤^{n} yields a section of the trivial line bundle V × ∧^{n}𝔤^{∗} ⊗ 𝔤^{n} over the neighborhood V of 0 in 𝔤. This section yields a map

1_{𝔤}^{n} : C(𝔤) → C(𝔤) ⊗ ∧^{n}𝔤^{∗} ⊗ 𝔤^{n}
f ↦ f ⊗ Y_{n} ∧ ⋯ ∧ Y_{1} ⊗ X_{1} ∧ ⋯ ∧ X_{n}.

The map 1^{n} in Lemma 4 is analogous to 1_{𝔤}^{n}.

On the other hand, if we think of an element of 𝔤^{∗} as a function on V, we have an (i + 1)-fold multiplication μ_{i} : C(𝔤) ⊗ 𝔤^{∗}^{⊗i} → C(𝔤). Let ad^{∘i} denote the composite

(𝔤^{∗}^{⊗i−1} ⊗ ad) ∘ ⋯ ∘ ad : 𝔤^{∗} → 𝔤^{∗}^{⊗i} ⊗ 𝔤^{∗}.

Note that (μ_{i} ⊗𝔤^{∗})∘(C(𝔤)⊗ad^{∘i}) = \overline{ad}^{i} where \overline{ad} is as in Section 2.5. Given any (convergent) power series f(z) = ∑_{i} c_{i}z^{i} let f(\overline{ad}) denote the map

∑_{i} c_{i}\overline{ad}^{i} : C(𝔤) ⊗ 𝔤^{∗} → C(𝔤) ⊗ 𝔤^{∗}.

Let Ψ be as in Section 2.5. Then

Ψ = \frac{−\overline{ad}}{e\overline{ad} − 1} ∘ d_{𝔤}

as a map from C(𝔤) to C(𝔤) ⊗ 𝔤^{∗}. Therefore, as an element of C(𝔤) ⊗ 𝔤 ⊗ 𝔤^{∗},

Ψ = (𝔤 ⊗ \frac{−\overline{ad}}{e\overline{ad} − 1}) ∘ 1_{𝔤}.

We can therefore think of the element ⋀_{C(𝔤)}^{n}(−Ψ) of C(𝔤) ⊗ ∧^{n}𝔤^{∗} ⊗ 𝔤^{n}. Denote this by (−Ψ)^{n}. Note that (−Ψ)^{n} is a section of the trivial line bundle V × ∧^{n}𝔤^{∗} ⊗ 𝔤^{n} over V. Moreover 1_{𝔤}^{n} is an isomorphism of ℝ-vector spaces. We can therefore ask for the function

f_{𝔤} := (1_{𝔤}^{n})^{−1}((−Ψ)^{n}) ∈ C(𝔤).

One can check from the formula for −Ψ that

(41)  f_{𝔤} = det (\frac{\overline{ad}}{e\overline{ad} − 1}).

At this stage, we remark that the map Φ_{L}^{n} in Lemma 4 is analogous to (−Ψ)^{n}. det (\frac{\overline{ad}}{e\overline{ad}−1}) is analogous to f in Lemma 4. Lemma 4 itself is analogous to Equation (41).

4.3.2. Pulling back the canonical left invariant volume form on G via \overline{exp}. Finally, we observe that an element of C(𝔤) ⊗ ∧^{n}𝔤^{∗} ⊗ 𝔤^{n} and a volume form on V together yield another volume form on V by letting ∧^{n}𝔤^{∗} contract with 𝔤^{n}. In this manner, the canonical volume form Y_{n} ∧ ⋯ ∧ Y_{1} on V and 1_{𝔤}^{n} yield the canonical volume form Y_{n} ∧ ⋯ ∧ Y_{1} on V.

Consider the left invariant volume form ω_{G} on U_{G} arising out of the element Y_{n} ∧ ⋯ ∧ Y_{1} of ∧^{n}𝔤^{∗}. In the same manner the volume form ω yielded by \overline{exp}^{∗}(ω_{G}) and Ψ^{n} equals Y_{n} ∧ ⋯ ∧ Y_{1}. But on the other hand, ω is also equal to det(d(\overline{exp}^{∗}))(−1)^{n}f_{𝔤}. It follows that (41) is equivalent to the formula

det(d(\overline{exp}^{∗})) = det (\frac{−\overline{ad}}{e\overline{ad} − 1}).

This in turn is equivalent to the formula for the pullback of a left invariant volume form on G via \overline{exp}.

4.4. Proof of Lemma 3. We are now equipped to prove Lemma 3. Let

ad(Φ_{R}) : D_{n} → D_{n} ⊗ Ω[1]

denote the composite

D_{n} ⊗ O_{X} ⟶^{D_{n}⊗Φ_{R}} D_{n} ⊗ D_{1} ⊗ Ω[1] ⟶^{ad⊗Ω[1]} D_{n} ⊗ Ω[1]

where ad : D_{n} ⊗ D_{1} → D_{n} is as in Section 3.2. We begin with the following proposition.

Proposition 32.

Φ_{R}^{+}(f) = 0.

Proof. The upper square in the commutative diagram below commutes by Lemma 4. The lower square in the diagram below commutes by Lemma 2.

(42)

O_{X}  ⟶^{1}  O_{X}
↓_{f}  ^{Φ_{L}^{n}}↓
S^{•}(Ω[1])  ⟶^{1^{n}}  D_{n} ⊗ Ω^{n}[n]
↓^{−Φ_{R}^{+}}  ^{ad(Φ_{R})⊗Ω^{n}[n]}↓
S^{•}(Ω[1]) ⊗ Ω[1] ⟶^{1^{n}⊗Ω[1]} D_{n} ⊗ Ω[1] ⊗ Ω^{n}[n].

The proof of Proposition 30 with Φ_{R} instead of Φ_{L} would show that

Φ_{R} ∈ Hom_{D^{b}(X)}(O_{X}, D_{1} ⊗ Ω[1]).

Let \overline{ad} be as in Section 3.2. Let \overline{ad}(Φ_{R}) denote the composite

(D_{1} ⊗ Ω[1])^{⊗n} ⊗ O_{X}
↓^{(D_{1}⊗Ω[1])^{⊗n}⊗Φ_{R}}
(D_{1} ⊗ Ω[1])^{⊗n} ⊗ D_{1} ⊗ Ω[1]
↓
D_{1}^{⊗n} ⊗ D_{1} ⊗ Ω[1]^{⊗n} ⊗ Ω[1]
↓^{\overline{ad}⊗Ω[1]^{⊗n}⊗Ω[1]}
D_{1}^{⊗n} ⊗ Ω[1]^{⊗n} ⊗ Ω[1] ⟶ (D_{1} ⊗ Ω[1])^{⊗n} ⊗ Ω[1].

The unlabeled arrows in the above diagram are rearrangements of factors. Note that by the Proposition 25, the following diagram commutes.

(43)

O_{X}  ⟶^{1}  O_{X}
↓^{Φ_{L}^{⊗n}}  ^{Φ_{L}^{n}}↓
(D_{1} ⊗ Ω[1])^{⊗n}  ⟶  D_{n} ⊗ Ω^{n}[n]
↓^{\overline{ad}(Φ_{R})}  ^{ad(Φ_{R})⊗Ω^{n}[n]}↓
(D_{1} ⊗ Ω[1])^{⊗n} ⊗ Ω[1] ⟶ D_{n} ⊗ Ω[1] ⊗ Ω^{n}[n].

By Theorem 2′ and by Proposition 9, Φ_{R} and Φ_{L} are commuting operators on S^{•}(Ω[1]). It follows that \overline{ad}(Φ_{R})(Φ_{L}^{⊗n}) is 0. Thus, ad(Φ_{R}) ⊗ Ω^{n}[n](Φ_{L}^{n}) is 0. The desired proposition now follows from (42) and the fact that 1^{n} ⊗Ω[1] is invertible in D^{b}(X). □

Proof of Lemma 3. Note that the pairing ⟨, ⟩ : S^{•}(Ω[1])^{⊗2} → Ω^{n}[n] induces a nondegenerate pairing

⟨, ⟩ : RHom_{X}(O_{X}, S^{•}(Ω[1]))^{⊗2} → RHom_{X}(O_{X}, Ω^{n}[n]) ≃ 𝕂.

⟨Φ_{R}( - ), - ⟩ : S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]) → Ω^{n}[n] ⊗ Ω[1] denotes the composite

S^{•}(Ω[1]) ⊗ S^{•}(Ω[1])
Φ_{R}⊗S^{•}(Ω[1])↓
S^{•}(Ω[1]) ⊗ Ω[1] ⊗ S^{•}(Ω[1])
S^{•}(Ω[1])⊗τ↓
S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]) ⊗ Ω[1]
⟨,⟩⊗Ω[1]↓
Ω^{n}[n] ⊗ Ω[1]

map in D^{b}(X). The map τ in the above composition of maps in D^{b}(X) swaps Ω[1] and S^{•}(Ω[1]).

Similarly, ⟨ - , Φ_{R}^{+}( - )⟩ : S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]) → Ω^{n}[n] ⊗ Ω[1] denotes the composite

S^{•}(Ω[1]) ⊗ S^{•}(Ω[1])
S^{•}(Ω[1])⊗Φ_{R}^{+}↓
S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]) ⊗ Ω[1] ⟶^{⟨,⟩⊗Ω[1]} Ω^{n}[n] ⊗ Ω[1].

Let a, b ∈ RHom_{X}(O_{X}, S^{•}(Ω[1])). Then,

(44)  ⟨ab.f^{−1}, Φ_{R}^{+}(f)⟩ = 0

by Proposition 32. But,

(45)  ⟨ab.f^{−1}, Φ_{R}^{+}(f)⟩ = ⟨Φ_{R}(ab.f^{−1}), f⟩

by the commutative diagram (34).

By Proposition 7, Theorem 2′ and by the fact that

I_{HKR} : Ĉ^{•}(X) → S^{•}(Ω[1])

is a homomorphism of algebra objects in D^{b}(X),

(46)  Φ_{R}(ab.f^{−1}) = Φ_{R}(a)bf^{−1} + aΦ_{R}(bf^{−1}).

By (44), (45) and (46)

(47)  0 = ⟨Φ_{R}(ab.f^{−1}), f⟩ = ⟨Φ_{R}(a)bf^{−1}, f⟩ + ⟨aΦ_{R}(bf^{−1}), f⟩.

But for any elements u, v, w ∈ RHom_{X}(O_{X}, S^{•}(Ω[1])),

⟨uv, w⟩ = ⟨u, vw⟩

by the definition of ⟨, ⟩. It then follows from (47) that

(48) 0 = ⟨Φ_{R}(a), b.f^{−1}f⟩ + ⟨a, Φ_{R}(b.f^{−1})f⟩ = ⟨Φ_{R}(a), b⟩ + ⟨a, Φ_{R}(b.f^{−1})f⟩.

It follows from the commutative diagram (34) that

Φ_{R}^{+}(b) = −Φ_{R}(bf^{−1})f

for any b ∈ RHom_{X}(O_{X}, S^{•}(Ω[1])). This proves Lemma 3. □

Remark 11. The computation proving Lemma 3 that has been written here can very easily be rewritten in a “canonical” manner without choosing elements of RHom_{X}(O_{X}, S^{•}(Ω[1])). Though that would be the ideal thing to do from the point of view of rigor, we feel that the computation we have depicted conveys the key idea behind the computation more concretely. Computations of a similar nature that show up in Section 5, have however, been written down in a “canonical” manner.

5. Proof of Theorem 1

5.1. Unwinding some definitions. In this subsection, we shall confine ourselves to unwinding the definition of the duality map D_{Δ}. This will help us focus more on what exactly we need to compute to prove Theorem 1. Let

κ : O_{Δ} → Δ_{∗}Δ^{∗}O_{Δ}

denote the unit of the adjunction Δ^{∗} ⊣ Δ_{∗} applied to O_{Δ} ∈ D^{b}(X × X). Also let

β : O_{X} → Δ^{∗}Δ_{!}O_{X}

denote the unit of the adjunction Δ_{!} ⊣ Δ^{∗} applied to O_{X} ∈ D^{b}(X).

Let { \atop D^{b}(X)} denote the category whose objects are those of D^{b}(X) such that

Hom{ \atop D^{b}(X)}(F, G) = RHom_{D^{b}(X)}(F, G)

for any pair of objects F and G of D^{b}(X). Note that any diagram that

commutes in D^{b}(X) also does so in { \atop D^{b}(X)}. We perform a particular cal-

culation in { \atop D^{b}(X)} instead of D^{b}(X) only when absolutely necessary. This enables us to take care of the shifts in grading that occur when an element of RHom_{X}(F, G) shows up instead of an element of Hom_{D^{b}(X)}(F, G). In [8],

{ \atop D^{b}(X)} is called the “extended derived category” of X.

Note that Δ_{!}O_{X} ≃ Δ_{∗}S_{X}^{−1}. It follows that Δ^{∗}Δ_{!}O_{X} ≃ Δ^{∗}O_{Δ} ⊗ S_{X}^{−1}. Let ≃: S_{X} ⊗ S_{X}^{−1} → O_{X} be as in Section 3.2. We now state the following proposition.

Proposition 33. Let ϕ ∈ RHom_{X}(Δ^{∗}O_{Δ}, S_{X}). Then, as a morphism in

{ \atop D^{b}(X)}, D_{Δ}^{−1}(ϕ) is given by the composite O_{X}

β↓
Δ^{∗}O_{Δ} ⊗ S_{X}^{−1}
Δ^{∗}(κ)⊗S_{X}^{−1}↓
Δ^{∗}O_{Δ} ⊗ Δ^{∗}O_{Δ} ⊗ S_{X}^{−1}
Δ^{∗}O_{Δ}⊗ϕ⊗S_{X}^{−1}↓
Δ^{∗}O_{Δ} ⊗ S_{X} ⊗ S_{X}^{−1}
Δ^{∗}O_{Δ}⊗≃↓
Δ^{∗}O_{Δ}.

Proof. By the definition of D_{Δ}, D_{Δ}^{−1} = I^{−1} ∘ T^{−1} ∘ J^{−1} where I, T and J are as in (2), (3) and (4) respectively.

Now,

(49)  J^{−1}(ϕ) = Δ_{∗}ϕ ∘ κ.

Further,

(50)  T^{−1}(α) = α ⊗ p_{2}^{∗}S_{X}^{−1}

and

(51)  I^{−1}(γ) = Δ^{∗}γ ∘ β.

Now, Δ^{∗}(α⊗p_{2}^{∗}S_{X}^{−1}) = Δ^{∗}α⊗S_{X}^{−1} since p_{2} ∘Δ = id. The desired proposition now follows from (49), (50) and (51) and the fact that Δ^{∗}Δ_{∗}ψ = Δ^{∗}O_{Δ} ⊗ ψ for any morphism ψ in D^{b}(X). □

The following propositions help us understand Δ^{∗}(κ) and β explicitly.

Proposition 34. The following diagram commutes in D^{b}(X).

Δ^{∗}O_{Δ} ⟶^{Δ^{∗}(κ)} Δ^{∗}O_{Δ} ⊗ Δ^{∗}O_{Δ}
Δ^{∗}O_{Δ}↓  ↓^{Δ^{∗}O_{Δ}⊗Δ^{∗}O_{Δ}}
Δ^{∗}O_{Δ} ⟶^{C} Δ^{∗}O_{Δ} ⊗ Δ^{∗}O_{Δ}.

Proof. Step 1. Note that if F ∈ D^{b}(X × X), then

Δ_{∗}Δ^{∗}F ≃ O_{Δ} ⊗ F.

Denote the canonical quotient map O_{X×X} → O_{Δ} by h.

Also observe that if G ∈ D^{b}(X), then

Δ^{∗}Δ_{∗}G ≃ Δ^{∗}O_{Δ} ⊗ G.

Recall that Δ^{∗}O_{Δ} is represented by the complex Ĉ^{•}(X). Note that the projection from the graded O_{X}-module Ĉ^{•}(X) to Ĉ^{0}(X) = O_{X} is a map of complexes of O_{X}-modules. This was denoted by η in Section 2. In this proof, we will denote this projection by p.

We claim that tensoring with h constitutes the unit of the adjunction Δ^{∗} ⊣ Δ_{∗} and that tensoring with p constitutes the counit of the adjunction Δ^{∗} ⊣ Δ_{∗}.

To see this, note that Δ^{∗}(h) is just the map

ϵ : O_{X} → Ĉ^{•}(X)

defined in Section 2. This was the unit of the Hopf algebra object Ĉ^{•}(X) of Ch^{−}(O_{X} − mod). It follows that p ∘ Δ^{∗}(h) = O_{X}. Also,

Δ_{∗}(p) ∘ (h ⊗ O_{Δ}) = O_{Δ}

since h ⊗ O_{Δ} can be identified with the map Δ_{∗}(ϵ). It follows that

κ = h ⊗ O_{Δ}.

Step 2. We now show that Δ^{∗}(κ) and C yield the same morphism in D^{b}(X) from Δ^{∗}O_{Δ} to Δ^{∗}O_{Δ} ⊗ Δ^{∗}O_{Δ}.

Note that O_{Δ} is represented by the complex B̂^{•}(X) in D^{b}(X × X). It follows that both B̂^{•}(X)⊗O_{Δ} and O_{Δ}⊗B̂^{•}(X) represent the object O_{Δ}⊗O_{Δ} of D^{b}(X × X). Let

ν : B̂^{•}(X) ⊗ O_{Δ} → B̂^{•}(X) ⊗ B̂^{•}(X) ⊗ O_{Δ}

denote the map such that on an open subscheme U = Spec R × Spec R of X × X before completion,

ν(r_{0} ⊗ ⋯ ⊗ r_{k+1} ⊗_{R⊗R} r′)

= ∑_{p+q=k;p,q≥0} r_{0} ⊗ ⋯ ⊗ r_{p} ⊗ 1 ⊗_{R⊗R} 1 ⊗ r_{p+1} ⊗ ⋯ ⊗ r_{k+1} ⊗_{R⊗R} r′.

ν is easily seen to be a map of complexes of O_{X×X}-modules. Similarly, let

\overline{ν} : O_{Δ} ⊗ B̂^{•}(X) → O_{Δ} ⊗ B̂^{•}(X) ⊗ B̂^{•}(X)

denote the map such that on an open subscheme U = Spec R × Spec R of X × X before completion,

\overline{ν}(r′ ⊗_{R⊗R} r_{0} ⊗ ⋯ ⊗ r_{k+1})

= r′ ⊗_{R⊗R} ∑_{p+q=k;p,q≥0} r_{0} ⊗ ⋯ ⊗ r_{p} ⊗ 1 ⊗_{R⊗R} 1 ⊗ r_{p+1} ⊗ ⋯ ⊗ r_{k+1}.

\overline{ν} is easily seen to be a map of complexes of O_{X×X}-modules.

Let τ : O_{Δ} ⊗ B̂^{•}(X) → B̂^{•}(X) ⊗ O_{Δ} denote the map swapping factors. Let τ′ : O_{Δ} ⊗ B̂^{•}(X) ⊗ B̂^{•}(X) → B̂^{•}(X) ⊗ B̂^{•}(X) ⊗ O_{Δ} denote the map swapping O_{Δ} and B̂^{•}(X) ⊗ B̂^{•}(X). The following diagram then commutes.

O_{Δ} ⊗ B̂^{•}(X) ⟶^{\overline{ν}} O_{Δ} ⊗ B̂^{•}(X) ⊗ B̂^{•}(X)
↓^{τ}  _{τ′}↓
B̂^{•}(X) ⊗ O_{Δ} ⟶^{ν} B̂^{•}(X) ⊗ B̂^{•}(X) ⊗ O_{Δ}.

Note that Δ^{∗}(\overline{ν}) = Δ^{∗}O_{Δ} ⊗ C. It follows from this and from the above commutative diagram that

(52)  Δ^{∗}(ν) = C ⊗ Δ^{∗}O_{Δ}.

Step 3. We use (52) to compare the morphisms ν and κ⊗O_{Δ} in D^{b}(X×X). Recall that

Hom_{D^{b}(X×X)}(O_{Δ} ⊗ O_{Δ}, O_{Δ} ⊗ O_{Δ} ⊗ O_{Δ})

≅ Hom_{D^{b}(X)}(Δ^{∗}O_{Δ} ⊗ Δ^{∗}O_{Δ}, Δ^{∗}O_{Δ} ⊗ Δ^{∗}O_{Δ})

via

α ↦ (p ⊗ Δ^{∗}O_{Δ} ⊗ Δ^{∗}O_{Δ}) ∘ Δ^{∗}(α).

Since p is induced by the counit of the Hopf algebra object Ĉ^{•}(X) of Ch^{−}(O_{X} − mod), and C is induced by the comultiplication of Ĉ^{•}(X),

(p ⊗ Δ^{∗}O_{Δ}) ∘ C = 1_{Δ^{∗}O_{Δ}}.

Also, since Δ^{∗}(κ) = ϵ ⊗ Δ^{∗}O_{Δ},

(p ⊗ Δ^{∗}O_{Δ}) ∘ Δ^{∗}(κ) = 1_{Δ^{∗}O_{Δ}}.

It follows that

(p ⊗ Δ^{∗}O_{Δ} ⊗ Δ^{∗}O_{Δ}) ∘ Δ^{∗}(ν) = (p ⊗ Δ^{∗}O_{Δ} ⊗ Δ^{∗}O_{Δ}) ∘ Δ^{∗}(κ ⊗ O_{Δ})
= 1_{Δ^{∗}O_{Δ}⊗Δ^{∗}O_{Δ}}.

This proves that ν and κ⊗O_{Δ} represent the same morphism in D^{b}(X×X). Therefore, C ⊗ Δ^{∗}O_{Δ} and Δ^{∗}(κ) ⊗ Δ^{∗}O_{Δ} represent the same morphism in D^{b}(X). The desired proposition now follows from the fact that if λ : G → H is a morphism in D^{b}(X), λ is equal to the composite

G ⟶^{G⊗ϵ} G ⊗ Δ^{∗}O_{Δ} ⟶^{λ⊗Δ^{∗}O_{Δ}} H ⊗ Δ^{∗}O_{Δ} ⟶^{H⊗p} H.  □

Recall that p : Ω[1]^{⊗i} → Ω^{i}[i] denotes the standard projection. Let Φ_{R}^{∘i} denote the composite

Φ_{R} ⊗ Ω[1]^{⊗i−1} ∘ ⋯ ∘ Φ_{R} : S^{•}(Ω[1]) → S^{•}(Ω[1]) ⊗ Ω[1]^{⊗i}.

Let Φ_{R}^{i} denote (S^{•}(Ω[1])⊗p)∘Φ_{R}^{∘i}. Then exp(Φ_{R}) := ∑_{i} \frac{1}{i!}Φ_{R}^{i} is a morphism in D^{b}(X) from S^{•}(Ω[1]) to S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]).

The following proposition follows immediately from Proposition 8 and Theorem 2′.

Proposition 35. The following diagram commutes in D^{b}(X).

Δ^{∗}O_{Δ} ⟶^{C} Δ^{∗}O_{Δ} ⊗ Δ^{∗}O_{Δ}
I_{HKR}↓  ↓^{I_{HKR}⊗I_{HKR}}
S^{•}(Ω[1]) ⟶^{exp(Φ_{R})} S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]).

Let ι : S_{X} → S^{•}(Ω[1]) denote the inclusion of S_{X} ≃ Ω^{n}[n] into S^{•}(Ω[1]) as a direct summand. Let \overline{β} denote the composite

O_{X} ⟶ S_{X} ⊗ S_{X}^{−1} ⟶^{ι⊗S_{X}^{−1}} S^{•}(Ω[1]) ⊗ S_{X}^{−1}.

Proposition 36. The following diagram commutes in D^{b}(X).

O_{X} ⟶^{β} Δ^{∗}O_{Δ} ⊗ S_{X}^{−1}
↓^{1}  ^{I_{HKR}⊗S_{X}^{−1}}↓
O_{X} ⟶^{\overline{β}} S^{•}(Ω[1]) ⊗ S_{X}^{−1}.

Proof. Step 1. Note that if F ∈ D^{b}(X × X), then

Δ_{!}Δ^{∗}F ≃ Δ_{∗}S_{X}^{−1} ⊗ F

Also, if G ∈ D^{b}(X), then

Δ^{∗}Δ_{!}G ≃ Δ^{∗}O_{Δ} ⊗ S_{X}^{−1} ⊗ G.

Now, Δ_{∗}S_{X}^{−1} is isomorphic in D^{b}(X ×X) to O_{Δ} ⊗p_{2}^{∗}S_{X}^{−1}. We now refer to the statement of the Serre duality theorem in Markarian [6]. By the Serre duality theorem there is a canonical map in D^{b}(X × X) from O_{Δ} to p_{2}^{∗}S_{X}. We denote this map by q. Tensoring q with p_{2}^{∗}S_{X}^{−1} on the right and making the obvious identifications gives us a morphism from O_{Δ} ⊗ p_{2}^{∗}S_{X}^{−1} to O_{X×X}. We denote this morphism by p in this proof.

Let β : O_{X} → Δ^{∗}O_{Δ} ⊗ S_{X}^{−1} be the morphism in D^{b}(X) such that the diagram in this proposition commutes. β is well-defined in D^{b}(X) since I_{HKR} is a quasi-isomorphism.

We claim that tensoring by β and tensoring by p constitute the unit and counit of the adjunction Δ_{!} ⊣ Δ^{∗} respectively. In order to verify this claim, it suffices to verify that

(53)  Δ^{∗}(p) ∘ β = O_{X}

as morphisms in D^{b}(X) and that

(54)  (p ⊗ Δ_{!}O_{X}) ∘ Δ_{!}β = Δ_{!}O_{X}

as morphisms in D^{b}(X × X).

Step 2. We verify (53) on “good” open subschemes of X. We begin by verifying (53) on an open subscheme U = Spec R of X with local coordinates y_{1}, …, y_{n}. The elements {y_{i} ⊗ 1 − 1 ⊗ y_{i} i = 1, …, m} form a regular sequence generating the ideal I of R ⊗ R defining the diagonal on an open affine neighborhood V = Spec S of the diagonal in U × U. Let

z_{i} = y_{i} ⊗ 1 − 1 ⊗ y_{i}.

This regular sequence gives rise to a Koszul complex K^{•}(z_{1}, …, z_{n}). This Koszul complex is a free S-module resolution of Δ_{∗}R.

The third part of the Serre duality theorem as stated in [6] says that the map q (restricted to V) is equal to the map of complexes

K^{•}(z_{1}, …, z_{n}) → p_{2}^{∗}Ω_{R/𝕂}^{n}[n]
z_{1} ∧ ⋯ ∧ z_{n} ↦ dy_{1} ∧ ⋯ ∧ dy_{n}

as morphisms in D^{b}(V).

Further, let [z_{i}] denote the class of z_{i} in H^{∗}(R ⊗_{S} K^{•}(z_{1}, …, z_{n})). The R-linear map dy_{i} ↦ [z_{i}] induces an isomorphism of graded algebras between S^{•}(Ω_{R/𝕂}[1]) and Tor_{∗}^{S}(R, R) by Proposition 3.4.7 of Loday [4]. Moreover, by the proof of the Hochschild–Kostant–Rosenberg theorem in Section 3.4 of Loday [4], this isomorphism coincides with the isomorphism induced on cohomology by the anti-symmetrization map φ : S^{•}(Ω_{R/𝕂}[1]) → Ĉ^{•}(R). Before completion,

φ(r_{0}dr_{1} ∧ ⋯ ∧ dr_{k}) = ∑_{σ∈S_{k}} sgn(σ)r_{0} ⊗ r_{σ(1)} ⊗ ⋯ ⊗ r_{σ(k)}.

This is immediately seen to be a right inverse of I_{HKR}. It follows from the facts recalled in this paragraph and the description of q in the previous paragraph that Δ^{∗}(q) = π_{n} ∘ I_{HKR} as morphisms in D^{b}(U).

Therefore, in D^{b}(U), Δ^{∗}(p) is given by the composite

Δ^{∗}O_{Δ} ⊗ S_{X}^{−1}|_{U} ⟶^{π_{n}∘I_{HKR}⊗S_{X}^{−1}|_{U}} S_{X}|_{U} ⊗ S_{X}^{−1}|_{U} ⟶ O_{U} = R.

The unlabeled arrow in the above composite is just the identification of S_{X}|_{U} ⊗ S_{X}^{−1}|_{U} with O_{U} = R. It follows that Δ^{∗}(p) ∘ β = O_{U} as morphisms in D^{b}(U).

Step 3. We verify (54) on “good” open subschemes of X × X. Let U and V be as in Step 2 above. Note that

Δ_{∗}I_{HKR} ∘ (Δ_{!}β ⊗ S_{X×X}) = 1_{Δ_{∗}S_{X}} .

Further,

p ⊗ Δ_{!}O_{X} ⊗ S_{X×X} = q ⊗ O_{Δ}.

By the discussion in Step 2, as morphisms in D^{b}(V),

q ⊗ O_{Δ} = Δ_{∗}Δ^{∗}q = Δ_{∗}(π_{n} ∘ I_{HKR}).

Therefore, as morphisms in D^{b}(V),

(q ⊗ O_{Δ}) ∘ (Δ_{!}β ⊗ S_{X×X}) = Δ_{∗}(π_{n} ∘ I_{HKR}) ∘ (Δ_{!}β ⊗ S_{X×X}) = 1_{Δ_{∗}S_{X}} .

Tensoring the morphisms involved in the above equation with S_{X×X}^{−1}, we see that

(p ⊗ Δ_{!}O_{X}) ∘ Δ_{!}β = Δ_{!}O_{X}

as morphisms in D^{b}(V). This is what we set out to verify.

Step 4. Now observe that

Hom_{D^{b}(X)}(O_{X}, Δ^{∗}O_{Δ} ⊗ S_{X}^{−1}) ≃ ⊕_{i}Hom_{D^{b}(X)}(O_{X}, Ω^{i} ⊗ ∧^{n}T [i − n]).

For i < n,

Hom_{D^{b}(X)}(O_{X}, Ω^{i} ⊗ ∧^{n}T [i − n]) = Ext^{i−n}(O_{X}, Ω^{i} ⊗ ∧^{n}T [i − n]) = 0.

For i = n,

Hom_{D^{b}(X)}(O_{X}, Ω^{i} ⊗ ∧^{n}T [i − n]) = Hom_{D^{b}(X)}(O_{X}, Ω^{n} ⊗ ∧^{n}T)
≃ Hom_{D^{b}(X)}(O_{X}, O_{X}) ≃ 𝕂.

It follows that Hom_{D^{b}(X)}(O_{X}, Δ^{∗}O_{Δ} ⊗ S_{X}^{−1}) is a 1-dimensional 𝕂-vector space. Tensoring with β and tensoring with p therefore, do indeed form a valid choice of unit and counit of the adjunction Δ_{!} ⊣ Δ^{∗} upto some scalar factors. In other words, (53) and (54) are satisfied in D^{b}(X) and in D^{b}(X × X) respectively upto some scalar factors. That the scalar factors are indeed 1 follows from the local verifications in Step 2 and Step 3 of this proof.

Step 5. Let J be as in (4) in Section 1. The last detail to be checked is that this particular choice of unit and counit for the adjunction Δ_{!} ⊣ Δ^{∗} satisfies

(55)  tr_{X}(J (ϕ) ∘ β) = tr_{X×X}(ϕ)

for any element ϕ of Hom_{D^{b}(X×X)}(Δ_{∗}S_{X}^{−1}, Δ_{∗}S_{X}). By the arguments in Step 4 of this proof, this “compatibility with traces” is satisfied upto a scalar factor independent of ϕ. We must verify that this scalar factor is 1.

Let m : O_{X×X} → O_{Δ} denote the obvious morphism in this proof. Let E and F be objects in D^{b}(X). Let E^{∗} denote the dual RD(E) of E. We recall the second part of the Serre Duality theorem as stated in [6]. It states that if f ∈ Hom_{D^{b}(X)}(E, F), the composite

O_{X×X} ⟶^{(Δ_{∗}f)∘m} p_{1}^{∗}F ⊗ p_{2}^{∗}E^{∗} ⊗ O_{Δ} ⟶^{p_{1}^{∗}F⊗p_{2}^{∗}E^{∗}⊗q} p_{1}^{∗}F ⊗ p_{2}^{∗}(E^{∗} ⊗ S_{X})

is exactly the image of the element f_{∗} of

Hom_{𝕂}(Hom_{D^{b}(X)}(O_{X}, E), Hom_{D^{b}(X)}(O_{X}, F))

under the identification Hom_{𝕂}(Hom_{D^{b}(X)}(O_{X}, E), Hom_{D^{b}(X)}(O_{X}, F))

↓
Hom_{D^{b}(X)}(O_{X}, F) ⊗ Hom_{D^{b}(X)}(O_{X}, E)^{∗}
↓
Hom_{D^{b}(X)}(O_{X}, F) ⊗ Hom_{D^{b}(X)}(O_{X}, E^{∗} ⊗ S_{X})
p_{1}^{∗}⊗p_{2}^{∗}↓
Hom_{D^{b}(X×X)}(O_{X×X}, p_{1}^{∗}F ⊗ p_{2}^{∗}(E ⊗ S_{X})).

Let tr ∈ Hom_{D^{b}(X)}(O_{X}, S_{X}) be the element satisfying tr_{X}(tr) = 1. Applying the above statement with E = F = O_{X} and f = O_{X}, we see that the composite

O_{X×X} ⟶^{m} O_{Δ} ⟶^{q} p_{2}^{∗}S_{X}

is precisely p_{2}^{∗}(tr). It follows that the composite t given by

O_{X×X} ⟶^{m} O_{X×X} ⊗ O_{Δ} ⟶^{p_{1}^{∗}(tr)⊗q} p_{1}^{∗}S_{X} ⊗ p_{2}^{∗}S_{X} = S_{X×X}

satisfies

tr_{X×X}(t) = 1.

Let tr_{Δ} denote the following composite:

O_{X×X} ⊗ O_{Δ} ⟶^{p_{1}^{∗}(tr)⊗q} S_{X×X} ⊗ O_{X×X} ⟶^{S_{X×X}⊗m} S_{X×X} ⊗ O_{Δ}.

By Lemma 2.2 of Caldararu’s paper [1],

tr_{X×X}(tr_{Δ}) = tr_{X×X}(t) = 1.

Therefore,

tr_{X×X}(tr_{Δ} ⊗ p_{2}^{∗}S_{X}^{−1}) = 1.

To verify (55), it suffices to check that

tr_{X}(J (tr_{Δ} ⊗ p_{2}^{∗}S_{X}^{−1}) ∘ β) = 1

where J is as in Section 1.

For this, it is enough to check that

(56)  tr_{X}(J (tr_{Δ}) ∘ (β ⊗ S_{X})) = 1.

Note that Δ^{∗}(tr_{Δ}) is given by the composite

O_{X} ⊗ Δ^{∗}O_{Δ} ⟶^{tr⊗Δ^{∗}(q)} S_{X} ⊗ S_{X} ⟶^{S_{X}⊗S_{X}⊗ϵ} S_{X} ⊗ S_{X} ⊗ Δ^{∗}O_{Δ}

where ϵ is the map induced in D^{b}(X) by the unit O_{X} → Ĉ^{•}(X) of the Hopf algebra object Ĉ^{•}(X) of Ch^{−}(O_{X} − mod). It follows that J (tr_{Δ}) is given by the composite

O_{X} ⊗ Δ^{∗}O_{Δ} ⟶^{tr⊗Δ^{∗}(q)} S_{X} ⊗ S_{X}.

To verify (56) it suffices to check that Δ^{∗}(q) ∘ (β ⊗ S_{X}) = S_{X}. This is an immediate consequence of (53). □

5.2. The final (long) computation proving Theorem 1. We first recall some notation. We remind the reader that as in Section 3.2, the product on S^{•}(Ω[1]) is denoted by ( - ∧ - ) : S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]) → S^{•}(Ω[1]). Let ev be as in Section 3.2. Denote the composite

S^{•}(Ω[1]) ⊗ S^{•}(T [−1]) ⟶^{S^{•}(Ω[1])⊗j} S^{•}(Ω[1]) ⊗ End(S^{•}(Ω[1])) ⟶^{ev} S^{•}(Ω[1])

by ( - • - ) as in Section 3.2. Note that the composite

S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]) ⟶^{S^{•}(Ω[1])⊗i} S^{•}(Ω[1]) ⊗ End(S^{•}(Ω[1])) ⟶^{ev} S^{•}(Ω[1])

is ( - ∧ - ). Let ( - || - ) be as in Section 3.2. We will also denote the isomorphisms S_{X} ⊗ S_{X}^{−1} → O_{X} and S_{X}^{−1} ⊗ S_{X} → O_{X} by ≃.

Also recall from Section 1 that

⟨ - , - ⟩ = π_{n} ∘ ( - ∧ - ) : S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]) → S^{•}(Ω[1]).

Let x ∈ RHom_{X}(O_{X}, S^{•}(Ω[1])).

Outline of the final computation proving Theorem 1. The final computation proving Theorem 1 begins by summarizing the results of the previous subsection to express I_{HKR}(D_{Δ}^{−1}({ \atop I_{HKR}}(x))) as a composite of morphisms

in { \atop D^{b}(X)}. This is done in Proposition 37. After Proposition 37, in (59), (60),

(61) and (62), the composite yielding I_{HKR}(D_{Δ}^{−1}({ \atop I_{HKR}}(x))) is rewritten to express it in a form that is possible to simplify. Lemma 5 is then used to simplify this composite further, yielding the composite (63). The simplification using Lemma 5 is a crucial step. The composite (63) is further simplified to the composite (64). The fact that the composite (64) of mor-

phisms in { \atop D^{b}(X)} yields I_{HKR}(D_{Δ}^{−1}({ \atop I_{HKR}}(x))) together with Proposition 28 and Proposition 27 of Section 3.2 yield Theorem 1. Lemma 5 is then proven in Section 5.3.

With the above notation and outline in mind, view Φ_{R} as an element of

Hom_{D^{b}(X)}(O_{X}, End(S^{•}(Ω[1])) ⊗ Ω[1]).

The following proposition begins the final set of steps towards Theorem 1.

Proposition 37. As a morphism in { \atop D^{b}(X)}, I_{HKR}(D_{Δ}^{−1}({ \atop I_{HKR}}(x))) is given by the composite

(O_{X}) / (\overline{β}↓)

S^{•}(Ω[1]) ⊗ S_{X}^{−1} ≃ S^{•}(Ω[1]) ⊗ O_{X} ⊗ S_{X}^{−1}
S^{•}(Ω[1])⊗exp(Φ_{R})⊗S_{X}^{−1}↓
S^{•}(Ω[1]) ⊗ End(S^{•}(Ω[1])) ⊗ S^{•}(Ω[1]) ⊗ S_{X}^{−1}
≃↓
S^{•}(Ω[1]) ⊗ End(S^{•}(Ω[1])) ⊗ S^{•}(Ω[1]) ⊗ O_{X} ⊗ S_{X}^{−1}
S^{•}(Ω[1])⊗End(S^{•}(Ω[1]))⊗S^{•}(Ω[1])⊗x⊗S_{X}^{−1}↓
S^{•}(Ω[1]) ⊗ End(S^{•}(Ω[1])) ⊗ S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]) ⊗ S_{X}^{−1}
S^{•}(Ω[1])⊗End(S^{•}(Ω[1]))⊗⟨ - , - ⟩⊗S_{X}^{−1}↓
S^{•}(Ω[1]) ⊗ End(S^{•}(Ω[1])) ⊗ S_{X} ⊗ S_{X}^{−1}
ev⊗≃↓

S^{•}(Ω[1]).

Proof. This proposition just amounts to putting together Propositions 33, 34, 35 and 36 and the definition of { \atop I_{HKR}}. □

As in Section 3.2, let γ : S^{•}(Ω[1]) ≃ S^{•}(T [−1]) ⊗ S_{X} be the isomorphism such that

(57)  ⟨ - , - ⟩ = π_{n} ∘ ( - ∧ - ) = (( - || - ) ⊗ S_{X}) ∘ (S^{•}(Ω[1]) ⊗ γ).

Let ζ denote the inverse of γ.

Also note that by the definitions of F_{r} and F_{l}, the following diagrams commute.

S^{•}(Ω[1]) ⊗ End(S^{•}(Ω[1])) ⟶^{S^{•}(Ω[1])⊗τ(F_{l})} S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]) ⊗ S^{•}(T [−1])
ev↓  ↓^{(( - ∧ - )• - )}
S^{•}(Ω[1])  ⟶^{S^{•}(Ω[1])}  S^{•}(Ω[1])

(58)

S^{•}(Ω[1]) ⊗ End(S^{•}(Ω[1])) ⟶^{S^{•}(Ω[1])⊗F_{r}} S^{•}(Ω[1]) ⊗ S^{•}(T [−1]) ⊗ S^{•}(Ω[1])
ev↓  ↓^{(( - • - )∧ - )}
S^{•}(Ω[1])  ⟶^{S^{•}(Ω[1])}  S^{•}(Ω[1]).

In the first diagram in (58),

τ : S^{•}(T [−1]) ⊗ S^{•}(Ω[1]) → S^{•}(Ω[1]) ⊗ S^{•}(T [−1])

interchanges factors. By Proposition 37, I_{HKR}(D_{Δ}^{−1}({ \atop I_{HKR}}(x))) is given by the composite

(59)

(O_{X}) / (\overline{β}↓)

S^{•}(Ω[1]) ⊗ S_{X}^{−1} ≃ S^{•}(Ω[1]) ⊗ O_{X} ⊗ O_{X} ⊗ S_{X}^{−1}
S^{•}(Ω[1])⊗exp(Φ_{R})⊗x⊗S_{X}^{−1}↓
S^{•}(Ω[1]) ⊗ End(S^{•}(Ω[1])) ⊗ S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]) ⊗ S_{X}^{−1}
ev⊗⟨ - , - ⟩⊗S_{X}^{−1}↓
S^{•}(Ω[1]) ⊗ S_{X} ⊗ S_{X}^{−1}
S^{•}(Ω[1])⊗≃↓

S^{•}(Ω[1])

of morphisms in { \atop D^{b}(X)}. Denote the composite O_{X}

exp(Φ_{R})↓
End(S^{•}(Ω[1])) ⊗ S^{•}(Ω[1]) ⟶^{τ(F_{l})⊗S^{•}(Ω[1])} S^{•}(Ω[1]) ⊗ S^{•}(T [−1]) ⊗ S^{•}(Ω[1])

by P.

By (57) and (58) the composite (59) yielding I_{HKR}(D_{Δ}^{−1}({ \atop I_{HKR}}(x))) is the same as the composite

(60)

(O_{X}) / (\overline{β}↓)

S^{•}(Ω[1]) ⊗ S_{X}^{−1} ≃ S^{•}(Ω[1]) ⊗ O_{X} ⊗ O_{X} ⊗ S_{X}^{−1}
S^{•}(Ω[1])⊗P⊗γ(x)⊗S_{X}^{−1}↓

S^{•}(Ω[1]) ⊗ S^{•}(Ω[1]) ⊗ S^{•}(T [−1]) ⊗ S^{•}(Ω[1]) ⊗ S^{•}(T [−1]) ⊗ S_{X} ⊗ S_{X}^{−1}

(( - ∧ - )• - )⊗( - || - )⊗≃↓

S^{•}(Ω[1])

of morphisms in { \atop D^{b}(X)}. Note that the composite

Ω^{j}[j] ⟶^{“\overline{β}⊗Ω^{j}[j]”} S^{•}(Ω[1]) ⊗ Ω^{j}[j] ⊗ S_{X}^{−1} ⟶^{( - ∧ - )⊗S_{X}^{−1}} S^{•}(Ω[1]) ⊗ S_{X}^{−1}

is 0. (“\overline{β} ⊗ Ω^{j}[j]” is a rearrangment of factors composed with \overline{β} ⊗ Ω^{j}[j]). It follows that the composite (60) yielding I_{HKR}(D_{Δ}^{−1}({ \atop I_{HKR}}(x))) is equal to the composite

(61)

(O_{X}) / (\overline{β}↓)

S^{•}(Ω[1]) ⊗ S_{X}^{−1} ≃ S^{•}(Ω[1]) ⊗ O_{X} ⊗ O_{X} ⊗ S_{X}^{−1}
S^{•}(Ω[1])⊗Q⊗γ(x)⊗S_{X}^{−1}↓
S^{•}(Ω[1]) ⊗ S^{•}(T [−1]) ⊗ S^{•}(Ω[1]) ⊗ S^{•}(T [−1]) ⊗ S_{X} ⊗ S_{X}^{−1}
( - • - )⊗( - || - )⊗≃↓

S^{•}(Ω[1])

of morphisms in { \atop D^{b}(X)}. In (61), Q in turn denotes the composite

O_{X}
exp(Φ_{R})↓
End(S^{•}(Ω[1])) ⊗ S^{•}(Ω[1]) ⟶^{π_{0}∘F_{l}⊗S^{•}(Ω[1])} S^{•}(T [−1]) ⊗ S^{•}(Ω[1]).

Let (exp(Φ_{R})|| - ) : S^{•}(T [−1]) → End(S^{•}(Ω[1])) denote the composite

O_{X} ⊗ S^{•}(T [−1])
exp(Φ_{R})⊗S^{•}(T [−1])↓
End(S^{•}(Ω[1])) ⊗ S^{•}(Ω[1]) ⊗ S^{•}(T [−1]) ⟶^{End(S^{•}(Ω[1]))⊗( - || - )} End(S^{•}(Ω[1])).

The composite in the diagram (61) yielding I_{HKR}(D_{Δ}^{−1}({ \atop I_{HKR}}(x))) can be rewritten as

(62)

(O_{X}) / (\overline{β}↓)

S^{•}(Ω[1]) ⊗ S_{X}^{−1} ≃ S^{•}(Ω[1]) ⊗ O_{X} ⊗ S_{X}^{−1}
S^{•}(Ω[1])⊗R⊗S_{X}^{−1}↓
S^{•}(Ω[1]) ⊗ S^{•}(T [−1]) ⊗ S_{X} ⊗ S_{X}^{−1}
( - • - )⊗≃↓

S^{•}(Ω[1])

where

R = [π_{0}(F_{l}(exp(Φ_{R})|| - )) ⊗ S_{X}] ∘ γ(x) : O_{X} → S^{•}(T [−1]) ⊗ S_{X}

in { \atop D^{b}(X)}.

We remind the reader that since (exp(Φ_{R})|| - ) is a morphism in D^{b}(X) from S^{•}(T [−1]) to End(S^{•}(Ω[1])), π_{0}(F_{l}(exp(Φ_{R})|| - )) is a morphism in D^{b}(X) from S^{•}(T [−1]) to S^{•}(T [−1]). It follows from this and the fact that γ(x) ∈ Hom{ \atop D^{b}(X)}(O_{X}, S^{•}(T [−1]) ⊗ S_{X}) that R is indeed an element of

Hom{ \atop D^{b}(X)}(O_{X}, S^{•}(T [−1]) ⊗ S_{X}) as mentioned above. The following lemma simplifies the computation of R. Let

( - | - ) : S^{•}(Ω[1]) ⊗ S^{•}(T [−1]) → S^{•}(T [−1])

be as in Section 3.2. Let (td_{X}^{−1}| - ) denote the composite

( - | - ) ∘ (td_{X}^{−1} ⊗ S^{•}(T [−1])).

This is a morphism in D^{b}(X) from S^{•}(T [−1]) to itself.

Lemma 5.

π_{0}(F_{l}(exp(Φ_{R})|| - )) = (td_{X}^{−1}| - ) : S^{•}(T [−1]) → S^{•}(T [−1]).

We postpone the proof of this lemma to Section 5.3.

It follows from (62) and Lemma 5 that I_{HKR}(D_{Δ}^{−1}({ \atop I_{HKR}}(x))) is given by the composite

(63)

O_{X}
\sqrt{β}
S^{•}(Ω[1]) ⊗ O_{X} ⊗ S_{X}^{−1}
↓^{S^{•}(Ω[1])⊗γ(x)⊗S_{X}^{−1}}
S^{•}(Ω[1]) ⊗ S^{•}(T [−1]) ⊗ S_{X} ⊗ S_{X}^{−1} ⟶^{( - •(td_{X}^{−1}| - ))⊗≃} S^{•}(Ω[1])

of morphisms in { \atop D^{b}(X)}.

Now note that the following diagram commutes in { \atop D^{b}(X)}.

O_{X}  ⟶^{1_{O_{X}}}  O_{X}
↓  \sqrt{β}
S_{X} ⊗ O_{X} ⊗ S_{X}^{−1}  ⟶^{ι⊗O_{X}⊗S_{X}^{−1}}  S^{•}(Ω[1]) ⊗ O_{X} ⊗ S_{X}^{−1}
↓^{S_{X}⊗γ(x)⊗S_{X}^{−1}}  ↓^{S^{•}(Ω[1])⊗γ(x)⊗S_{X}^{−1}}
S_{X} ⊗ S^{•}(T [−1]) ⊗ S_{X} ⊗ S_{X}^{−1} ⟶^{f} S^{•}(Ω[1]) ⊗ S^{•}(T [−1]) ⊗ S_{X} ⊗ S_{X}^{−1}
↓^{( - •(td_{X}^{−1}| - ))⊗≃}  ↓^{( - •(td_{X}^{−1}| - ))⊗≃}
S^{•}(Ω[1])  ⟶^{1_{S^{•}(Ω[1])}}  S^{•}(Ω[1]),

where f = ι ⊗ S^{•}(T [−1]) ⊗ S_{X} ⊗ S_{X}^{−1}.

The topmost square in the above diagram commutes by the definition of \overline{β}. That the remaining squares in the above diagram commute is obvious. The reader should recall that S_{X} = Ω^{n}[n] to make sense out of the map ( - • (td_{X}^{−1}| - )) : S_{X} ⊗ S^{•}(T [−1]) → S^{•}(Ω[1]) in the above diagram.

It follows from the above diagram and (63) that I_{HKR}(D_{Δ}^{−1}({ \atop I_{HKR}}(x))) is given by the composite

(64)  O_{X} ; ↓ ; S_{X} ⊗ O_{X} ⊗ S_{X}^{−1} ; ↓^{S_{X}⊗γ(x)⊗S_{X}^{−1}} ; S_{X} ⊗ S^{•}(T [−1]) ⊗ S_{X} ⊗ S_{X}^{−1} ; ↓^{( - •(td_{X}^{−1}| - ))⊗≃} ; S^{•}(Ω[1])

of morphisms in { \atop D^{b}(X)}.

Let τ : S_{X} ⊗ S^{•}(T [−1]) ⊗ S_{X} → S^{•}(T [−1]) ⊗ S_{X} ⊗ S_{X} denote the map swapping S_{X} and S^{•}(T [−1]) ⊗ S_{X}. Let τ′ : S_{X} ⊗ O_{X} → O_{X} ⊗ S_{X} swap factors. We now have the following proposition.

Proposition 38. The following diagram commutes in { \atop D^{b}(X)}.

O_{X}  ⟶^{1}  O_{X}
↓  ↓
S_{X} ⊗ O_{X} ⊗ S_{X}^{−1}  ⟶^{τ′⊗S_{X}^{−1}}  O_{X} ⊗ S_{X} ⊗ S_{X}^{−1}
S_{X}⊗γ(x)⊗S_{X}^{−1}↓  ↓^{γ(x)⊗S_{X}⊗S_{X}^{−1}}
S_{X} ⊗ S^{•}(T [−1]) ⊗ S_{X} ⊗ S_{X}^{−1} ⟶^{τ⊗S_{X}^{−1}} S^{•}(T [−1]) ⊗ S_{X} ⊗ S_{X} ⊗ S_{X}^{−1}
( - •(td_{X}^{−1}| - ))⊗≃↓  ↓^{ζ((td_{X}^{−1}| - )⊗S_{X})⊗≃}
S^{•}(Ω[1])  ⟶^{J}  S^{•}(Ω[1]).

Proof. The fact that the first two squares in the above diagram commute is clear. The third square commutes by Proposition 28 of Section 3.2. □

Recall that γ : S^{•}(Ω[1]) ≃ S^{•}(T [−1]) ⊗ S_{X} and ζ denotes the inverse of γ. By Proposition 27 of Section 3.2,

ζ(((td_{X}^{−1}| - ) ⊗ S_{X})(γ(x)) = td_{X}^{−1} ∧ x.

Therefore, by Proposition 38, the fact that the composite (64) yields I_{HKR}(D_{Δ}^{−1}({ \atop I_{HKR}}(x))), and the fact that J^{2} = 1_{S^{•}(Ω[1])},

I_{HKR}(D_{Δ}^{−1}({ \atop I_{HKR}}(x))) = J(td_{X}^{−1} ∧ x).

It follows that

D_{Δ}(I_{HKR}^{−1}(y)) = { \atop I_{HKR}}(td_{X} ∧ Jy).

This proves Theorem 1. □

5.3. Proving Lemma 5. Lemma 5 is the only thing left to be proven. The proof of Lemma 5 uses Lemma 3. We first state and prove the following proposition. All statements in this subsection hold in D^{b}(X) and hence in

{ \atop D^{b}(X)}.

Proposition 39.

π_{0}(F_{r}(exp(Φ_{R})|| - )) = 1 : S^{•}(T [−1]) → S^{•}(T [−1]).

Proof. The notation used here is that used while proving Proposition 30. While proving Proposition 30, we noted that \overline{ω}^{i}∘\overline{C} is given by the composite

S^{•}(Ω[1]) ⟶^{\overline{C}} S^{•}(Ω[1]) ⊗ Ω[1] ⟶^{S^{•}(Ω[1])⊗At_{T}^{i}} S^{•}(Ω[1]) ⊗ Ω^{i}[i] ⊗ Ω[1]
( - ∧ - )⊗Ω[1]↓
S^{•}(Ω[1]) ⊗ Ω[1].

It follows by the definition of π_{0} ∘ F_{r} that

(π_{0} ∘ F_{r} ⊗ Ω[1])(\overline{ω}^{i} ∘ \overline{C}) = 0

if i > 0. It follows from the fact that \frac{z}{1−e^{−z}} = 1 + ∑_{i≥1} c_{i}z^{i} that

(π_{0} ∘ F_{r} ⊗ Ω[1])(Φ_{R}) = (π_{0} ∘ F_{r} ⊗ Ω[1])(\overline{C}).

Therefore, by Proposition 23,

(π_{0} ∘ F_{r} ⊗ S^{•}(Ω[1]))(exp(Φ_{R})) = (π_{0} ∘ F_{r} ⊗ S^{•}(Ω[1]))(exp(\overline{C})).

Note that

exp(\overline{C}) = C_{Ω}

where C_{Ω} is the comultiplication on S^{•}(Ω[1]) treated as an element of

Hom_{D^{b}(X)}(O_{X}, End(S^{•}(Ω[1])) ⊗ S^{•}(Ω[1])).

For the rest of this proof let (C_{Ω}|| - ) denote

[End(S^{•}(Ω[1])) ⊗ ( - || - )] ∘ C_{Ω}.

Therefore,

π_{0}(F_{r}(exp(Φ_{R})|| - )) = π_{0}(F_{r}(C_{Ω}|| - )) : S^{•}(T [−1]) → S^{•}(T [−1]).

But (C_{Ω}|| - ) = j : S^{•}(T [−1]) → End(S^{•}(T [−1]) by Proposition 29. This proves the desired proposition. □

Proof of Lemma 5. Let (exp(Φ_{R})|| - )^{+} : S^{•}(T [−1]) → End(S^{•}(Ω[1])) denote the map A ∘ (exp(Φ_{R})|| - ) : S^{•}(T [−1]) → End(S^{•}(Ω[1])) where A is as in Section 3.2.

By Proposition 22,

π_{0}(F_{l}(exp(Φ_{R})|| - )) = I(π_{0}(F_{r}(exp(Φ_{R})|| - )^{+})) : S^{•}(T [−1]) → S^{•}(T [−1]).

By part (2) of Proposition 21,

(exp(Φ_{R})|| - )^{+} = (exp(Φ_{R}^{+})|| - ) : S^{•}(T [−1]) → End(S^{•}(Ω[1])).

By Lemma 3,

(exp(Φ_{R}^{+})|| - )
= (exp(−i(f) ∘ Φ_{R} ∘ i(f)^{−1})|| - )
= i(f) ∘ (exp(−Φ_{R})|| - ) ∘ i(f^{−1}) : S^{•}(T [−1]) → End(S^{•}(Ω[1])).

We remark that i(f) ∘ (exp(−Φ_{R})|| - ) ∘ i(f^{−1}) is precisely the composite

O_{X} ⊗ S^{•}(T [−1]) ⊗ O_{X}
↓^{i(f)⊗(exp(−Φ_{R})|| - )⊗i(f^{−1})}
End(S^{•}(Ω[1])) ⊗ End(S^{•}(Ω[1])) ⊗ End(S^{•}(Ω[1]) ⟶^{∘} End(S^{•}(Ω[1])).

Thus,

(65) π_{0}(F_{l}(exp(Φ_{R})|| - ))

= I(π_{0}(F_{r}(i(f) ∘ (exp(−Φ_{R})|| - ) ∘ i(f^{−1})))) : S^{•}(T [−1]) → S^{•}(T [−1]).

Note that π_{0}(F_{r}(i(f))) = 1 since f = det(1 + ∑_{i>0} c_{i}At_{T}^{i}). It follows from Proposition 23 that

(66) π_{0}(F_{l}(exp(Φ_{R})|| - ))

= I(π_{0}(F_{r}((exp(−Φ_{R})|| - ) ∘ i(f^{−1})))) : S^{•}(T [−1]) → S^{•}(T [−1]).

Another point to note is that

(exp(−Φ_{R})|| - ) = (exp(Φ_{R})|| - ) ∘ I : S^{•}(T [−1]) → End(S^{•}(Ω[1])).

It follows from this observation and (66) that

π_{0}(F_{l}(exp(Φ_{R})|| - ))
= I(π_{0}(F_{r}((exp(Φ_{R})|| - ) ∘ i(f^{−1})))) ∘ I
= I(π_{0}(F_{r}(j(π_{0}(F_{r}(exp(Φ_{R})|| - ))) ∘ i(f^{−1})))) ∘ I by Proposition 23
= I(π_{0}(F_{r}(j(I( - )) ∘ i(f^{−1})))) by Proposition 39
= I(f^{−1}|I( - )) : S^{•}(T [−1]) → S^{•}(T [−1]) by Proposition 26

where ( - | - ) : S^{•}(Ω[1]) ⊗ S^{•}(T [−1]) → S^{•}(T [−1]) is as in Section 3.2.

We remind the reader that (f^{−1}|I( - )) : S^{•}(T [−1]) → S^{•}(T [−1]) is the composite

S^{•}(T [−1])
↓^{I}
O_{X} ⊗ S^{•}(T [−1])
↓^{f^{−1}⊗S^{•}(T [−1])}
S^{•}(Ω[1]) ⊗ S^{•}(T [−1]) ⟶^{( - | - )} S^{•}(T [−1])

of morphisms in D^{b}(X).

Note that

I(f^{−1}|I( - )) = (J(f^{−1})| - ) : S^{•}(T [−1]) → S^{•}(T [−1]).

Also observe that J(f^{−1}) = td_{X}^{−1}. This proves Lemma 5. □

Relative Riemann–Roch theorem 717

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Department of Mathematics, University of Oklahoma, 621 Elm Avenue, Norman, OK-73019 aramadoss@math.ou.edu

This paper is available via http://nyjm.albany.edu/j/2008/14-29.html.