VERTEX ALGEBRAS AND THE FORMAL LOOP SPACE

M. Kapranov and E. Vasserot

1. Introduction

One of the salient mathematical features of string theory is the importance of vertex algebras. Their role in the theory can be compared to that of Lie algebras in the “ordinary” physics of point particles.

Mathematically, the approach of string theory can be cast in terms of analysis on the space of free loops, i.e., smooth maps S^{1} → X where X is a given “spacetime” manifold. Accordingly, one has the folklore principle that constructions involving the space of free loops lead to vertex algebras. One class of such constructions is provided by the spaces of highest weight representations of loop groups. Another is Ω_{X}^{ch}, the chiral de Rham complex of an algebraic variety X, introduced by Malikov, Schectman and Vaintrob [MSV]. Heuristically, this complex should be interpreted in terms of LX, the space of free loops and its subvariety L^{0}X consisting of loops extending holomorphically into the unit disk. More precisely, Ω_{X}^{ch} can be thought of as the semiinfinite de Rham complex with coefficients in the space of distributions on LX supported on L^{0}X. This is not, however, the way Ω_{X}^{ch} has been defined mathematically. The definition given in [MSV] is of more computational nature and proceeds by constructing the action of the group of diffeomorphisms on the irreducible module over the Heisenberg algebra. In that approach it seems miraculous that such an action exists at all.

The aim of this paper is twofold. First, to give a precise mathematical theorem underlying the above folklore principle about vertex algebras. For this, we introduce an algebro-geometric version of the free loop space L(X) for any scheme X of finite type over a field. This is an ind-scheme containing L^{0}(X), the scheme of formal germs of curves on X studied in [DL]. We prove that both L(X) and L^{0}(X) themselves possess a non-linear version of the vertex algebra structure (which makes it clear that any natural linear construction applied to them should give a vertex algebra in the usual sense). More precisely, we use the geometric approach

Typeset by AMS-TEX

to vertex algebras developed by Beilinson and Drinfeld [BD1] and based on the concepts of chiral algebras and factorization algebras. The latter concept has a natural nonlinear version, that of a factorization semigroup. What we prove is that natural “global”versions of L(X), L^{0}(X) have natural structures of factorization semigroups. An earlier known example of a factorization semigroup is given by the affine Grassmannian [G], and this explains why the spaces of representations of loop groups are vertex algebras. Our construction is similar in spirit.

To give a good definition of the algebraic analog of the full loop space LX one has to overcome a certain subtlety. Namely, a natural approach would be to try to (ind-)represent a functor which to any commutative ring R associates the set of R((t))-points of X. (This is exactly how one defines the scheme L^{0}(X), with R[[t]] instead of R((t)).) If X is affine, this indeed gives a good ind-scheme which we denote L̃(X). But when X is, say, projective, then (for R a field) there is no difference between R[[t]]-points and R((t))-points of X (valuative criterion of properness), so it may seem that nothing is gained by allowing Laurent series. To state this phenomenon differently, the ind-schemes L̃(U) for affine U ⊂ X do not glue together well. This is in fact understandable on general grounds: the loop space LX is not the union of the LU since a loop need not spend all its time in any given U.

To get around this difficulty we adopt the following strategy. For an affine X we consider L(X), the formal neighborhood of L^{0}(X) in L̃(X). So we are dealing with formal loops which are “infinitesimal in the Laurent direction”.Then, we prove that the L(U), U ⊂ X, do indeed possess the right gluing properties. This is due to the infinitesimal nature of our loops.

Our second goal is to give a direct geometric construction of Ω_{X}^{ch} (for smooth X) in terms of our model for the loop space. By the above, this construction explains also the fact that Ω_{X}^{ch} is a sheaf of vertex algebras. In order to achieve this, we represent L(X) as an ind-pro-object in the category of schemes of finite type and then show that the shifted de Rham complexes of the terms of this ind-pro-system arrange naturally into a double inductive system whose inductive limit is identified with Ω_{X}^{ch}.

As with the study of formal arcs and motivic integration [DL], one can view our considerations as algebro-geometric analogs of the basic constructions of p-adic analysis. The difference between our ind-scheme L(X) and the more familiar scheme L^{0}(X) is similar to the difference between ℚ_{p} and ℤ_{p}: while the latter is a pro-object in the category of schemes of finite type (resp. finite sets), the former is an ind-pro-object. Further, our approach to Ω_{X}^{ch} is similar to the construction of the space of locally constant functions with compact support on ℚ_{p} = lim_{⟶}_{i}lim_{⟵}_{j}p^{−i}ℤ_{p}/p^{j}ℤ_{p} as

the double inductive limit of the spaces of functions on the finite sets p^{−i}ℤ_{p}/p^{j}ℤ_{p}, cf [P]. Notice that the reason that these spaces of functions indeed form a double inductive system (with respect to the maps of inverse image in the j-direction and direct image in the i-direction) is that the commutative squares in the indpro-system p^{−i}ℤ_{p}/p^{j}ℤ_{p} are Cartesian (so that we have base change). This is an algebraic counterpart of the property of the local compactness of ℚ_{p}, see [Kat]. In our situation it is equally important that the ind-scheme L(X) satisfies a certain formal analog of local compactness.

This work has been done over a period of several years during which the first author benefitted from visits to and the financial support of the Université Cergy-Pontoise, the IHES, the Ecole Normale Superieure and the Max-Planck-Institut für Mathematik. His research was also partially supported by grants from NSF and NSERC. The second author is partially supported by EEC grant no. ERB. FMRX-CT97-0100.

2. Construction of the formal loop space

(2.1) Generalities on schemes and ind-schemes. If C is a category, then we denote by Ind(C) and Pro(C) the categories of ind- and pro-objects of C, see [AM] [GV] for background. Thus, objects of Ind(C) (resp. Pro(C)) are symbols “lim_{⟶}_{i}”C_{i} (resp. “lim_{⟵}_{i}”C_{i}) where (C_{i}) is a filtering inductive (resp. projective)

system over C, with morphisms defined as in loc. cit. Recall that Ind(C) can be considered as a full subcategory in Fun^{∘}(C, Sets), the category of contravariant functors.

Throughout the paper we fix a field k of characteristic 0. We denote by Sch ⊂ Lrs the categories of schemes and locally ringed spaces over k. If R is a commutative ring, we will write \underline{Spec} R for the topological space (the set of prime ideals with the Zariski topology) underlying the affine scheme Spec R which is thus the ringed space (\underline{Spec} R, O_{Spec R}).

By an ind-scheme we will mean in this paper an ind-object of Sch represented by an inductive system of closed embeddings. The category of ind-schemes will be denoted by Isch. Let us make the category Sch into a Grothendieck site by using Zariski open coverings and let Shf be the category of sheaves of sets on Sch. For any ind-scheme Y the functor η_{Y} on schemes represented by Y is then a sheaf, so we have the embeddings

(2.1.1)  Sch ⊂ Isch ⊂ Shf ⊂ Fun^{∘}(Sch, Sets).

Since the category Sch has finite inverse limits, so do all the categories in (2.1.1) and the embeddings preserve them. On the contrary, finite direct limits, such as cokernels (when they exist) are preserved by the first two of the embeddings but not by the third one: cokernels in the category of sheaves are not the same as in the category of all functors (presheaves).

We denote by Alg the category of k-algebras and by Aff ⊂ Sch the dual category of affine schemes. Note that Isch can be as well realized as a full subcategory in Fun^{∘}(Aff, Sets) = Fun(Alg, Sets).

Given two contravariant functors ϕ, ϕ′ : Sch → Sets, and a morphism F : ϕ′ → ϕ, we will say that F is formally smooth (resp. formally étale), if for any nilpotent extension of affine schemes S ⊂ S′ the natural map

ϕ′(S′) → ϕ(S′) ×_{ϕ(S)} ϕ′(S)

is surjective (resp. bijective).

We will say that F is an open embedding, if for any scheme S and any u ∈ ϕ(S) (which is the same as a morphism η_{S} → ϕ) the fiber product functor η_{S} ×_{ϕ} ϕ′ is representable by a scheme S′ whose natural morphism to S is an open embedding.

We define formal smoothness and openness for morphisms of ind-schemes by considering their representable functors.

For a scheme Z we denote by Z_{red} ⊂ Z the corresponding reduced subscheme. We extend this notation to ind-schemes by applying it term by term in inductive systems.

Let X be a k-scheme of finite type. We denote by Aff_{X} (resp. Aff_{X}^{ft}) the category of schemes affine over X (resp. affine of finite type over X). For future use let us quote the following fact [EGAIV, Corollary 8.13.2].

(2.1.2) Proposition. The category Pro(Aff_{X}^{ft}) is equivalent to Aff_{X} via the functor “lim_{⟵}_{n}”S_{n} ↦ lim_{⟵}_{n}S_{n}.

(2.2) The scheme of germs of arcs.

Let X be a scheme. We denote by L^{0}(X) the scheme of germs of arcs on X, see [DL] [BLR, Theorem 7.6.4]. It represents the following covariant functor λ_{X}^{0} on the category Alg:

λ_{X}^{0} : R ↦ Hom_{Sch}(Spec R[[t]], X).

Here are some of the well-known properties of L^{0}(X). Note that if R is a local ring with maximal ideal M, then R[[t]] is a local ring with maximal ideal M[[t]]+tR[[t]]. Thus for any scheme S the pair (S, O_{S}[[t]]) is a locally ringed space.

(2.2.1) Proposition. (a) For any scheme S we have

Hom_{Sch}(S, L^{0}(X)) = Hom_{Lrs}((S, O_{S}[[t]]), X).

(b) The scheme L^{0}(X) is the projective limit of the schemes L_{n}^{0}(X), n ∈ ℕ, representing the functors

λ_{n,X}^{0} : R ↦ Hom_{Sch}(Spec R[t]/t^{n+1}, X).

If X is of finite type, than so is each L_{n}^{0}(X).

(c) Denote π_{n} : L_{n}^{0}(X) → X, π : L^{0}(X) → X the natural projections. They are affine morphisms. For an open subset U ⊂ X we have π^{−1}(U) = L^{0}(U) and π_{n}^{−1}(U) = L_{n}^{0}(U).

(d) If X is smooth, then so is L_{n}^{0}(X) and L^{0}(X) is formally smooth.

Let us give the proof of (a), as it will serve as a template for a similar statement about the space of formal loops. It is enough to assume S = Spec R affine. The embedding of constant series and the evaluation at 0 give ring homomorphisms R →^{α} R[[t]] →^{β} R and hence morphisms of topological spaces

(2.2.2)  \underline{Spec} R ⟵^{p} \underline{Spec} R[[t]] ⟵^{i} \underline{Spec} R.

(2.2.3) Lemma. We have

i^{−1}O_{Spec R[[t]]} = O_{Spec R}[[t]] = p_{∗}O_{Spec R[[t]]}.

Proof: Let us prove the first equality. If 𝔭 ∈ \underline{Spec} R is a prime ideal in R, then the stalk of i^{−1}O_{Spec R[[t]]} at 𝔭 is the localization of R[[t]] with respect to the multiplicative subset β^{−1}(R−𝔭) while the stalk of O_{Spec R}[[t]] is R[[t]]⊗_{R} R_{𝔭} where R_{𝔭} denotes, as usual, the localization of R with respect to R − 𝔭. Now, to see that the two rings are the same, it suffices to use the following obvious property of formal power series rings: if A is a commutative ring and f(t) ∈ A[[t]] is such that f(0) is invertible in A, then f(t) is invertible in A[[t]].

The second equality is obvious: the stalk of p_{∗}O_{Spec R[[t]]} at 𝔭 is immediately seen to coincide with R[[t]] ⊗_{R} R_{𝔭}. ⊓ ⊔

Now, composing with i defines a map of sets

ϕ : Hom_{Sch}(Spec R[[t]], X) → Hom_{Lrs}((Spec R, O_{Spec R}[[t]]), X).

A map ψ in the other direction comes from the second equality in (2.3.3). One verifies easily that ϕ and ψ are mutually inverse.

We will also need the following generalization of Proposition 2.2.1(c).

(2.2.4) Proposition. If ϕ : X → Y is an étale morphism, then the natural squares

\begin{aligned}& L_{n}^{0}(X) & ⟶^{π_{n}} & X \\ L_{n}^{0}(ϕ) & ↓ & & ↓ϕ \\ & L_{n}^{0}(Y) & ⟶^{π_{n}} & Y \end{aligned} , n ≥ 0

as well as the analogous square for L^{0}(Y), are Cartesian.

Proof: We can assume that X = Spec (A), Y = Spec (B) are affine. It is enough to prove that for each n ≥ 0 the natural morphism

α : L_{n}^{0}(X) → L_{n}^{0}(Y) ×_{Y} X

is an isomorphism. Let us construct the inverse morphism β. Let R be a k-algebra and f be a morphism S → L_{n}^{0}(Y) ×_{Y} X. Thus, f corresponds to a pair of ring homomorphisms forming the horizontal arrows of the commutative diagram

B ⟶^{u} R[t]/t^{n+1}
ϕ^{∗} ↓  ↓ π
A ⟶^{v}  R.

Here π is natural projection. Since ϕ is étale and π is nilpotent, there is a unique homomorphism w : A → R[t]/t^{n+1} such that both resulting triangles are commutative. Let g be the morphism S → L_{n}^{0}(X) represented by w. Then we set β(f) = g. The verifications are obvious and left to the reader.

(2.3) Nil-Laurent series. Let R be a commutative ring. A nil-Laurent series is, by definition, a Laurent series a(t) = ∑_{i≫−∞} a_{i}t^{i} ∈ R((t)) such that all the a_{i} with i < 0 are nilpotent. The set of such series will be denoted R((t))^{√} .

It is clear that R((t))^{√} is a subring in R((t)), the ring of all Laurent series. Indeed, let \sqrt{R} be the radical of R (the set of all nilpotent elements) and set R_{red} = R/\sqrt{R}. Consider the homomorphism ρ : R((t)) → R_{red}((t)) induced by the projection R → R_{red}. Then R((t))^{√} = ρ^{−1}(R_{red}[[t]]).

(2.3.1) Proposition. For a = ∑ a_{i}t^{i} ∈ R((t))^{√} the following are equivalent : (i) a is invertible in R((t))^{√} . (ii) ρ(a) is invertible in R_{red}[[t]]. (iii) The element a_{0} is invertible in R.

Proof: (i)⇒(ii) is obvious. To see that (ii)⇔(iii), note that the invertibility of ρ(a) implies the invertibility of the image of a_{0} in R_{red} which certainly implies the invertibility of a_{0} and the converse is equally obvious. Let us prove that (ii)⇒(i). Indeed, if ρ(a) is invertible in R_{red}[[t]], we have ab = 1 + c for some b ∈ R((t))^{√} , c ∈ Ker ρ = \sqrt{R}((t)). But every element c ∈ \sqrt{R}((t)) is topologically nilpotent (setting c = c_{−} + c_{+} with c_{−} ∈ t^{−1}\sqrt{R}[t^{−1}] and c_{+} ∈ \sqrt{R}[[t]], we have that c_{−} is nilpotent while c_{+} is topologically nilpotent), thus ab is invertible in R((t))^{√} , and a is invertible, too. ⊓ ⊔ (2.3.2) Corollary. If R is a local ring with maximal ideal M, then R((t))^{√} is a local ring with maximal ideal ρ^{−1}(M_{red}[[t]] + tR_{red}[[t]]).

(2.4) The formal loop space. We now describe our main construction. Let X be a scheme of finite type over k. Define a covariant functor λ_{X} from Alg to sets as follows:

(2.4.1)  λ_{X}(R) = Hom_{Sch}(Spec R((t))^{√} , X).

(2.4.2) Theorem. (a) The functor λ_{X} is represented by an ind-scheme L(X), containing L^{0}(X).

(b) L(X) is an inductive limit of nilpotent extensions of L^{0}(X). In particular, for any open set Y ⊂ L^{0}(X) there is a well-defined ind-scheme L(X)|_{Y}.

(c) If U is an open subset in X, then the ind-scheme L(U) is identified with L(X)|_{p^{−1}(U)}.

(d) If X is smooth, then L(X) is formally smooth.

The proof will be finished in the next subsection.

Let us now define a contravariant functor λ′_{X} on the category Sch by

(2.4.3)  λ′_{X}(S) = Hom_{Lrs}((S, O_{S}((t))^{√} ), X).

(2.4.4) Proposition. For an affine scheme S = Spec(R) we have λ′_{X}(S) = λ_{X}(R).

Proof: Since R((t))^{√} is a nilpotent extension of R[[t]], we have \underline{Spec} R((t))^{√} = \underline{Spec} R[[t]], so Spec R((t))^{√} is the ringed space formed by \underline{Spec} R[[t]] and a certain sheaf of rings O_{Spec R((t))^{√}} on it.

(2.4.5) Lemma. With respect to the maps in (2.2.2) we have

i^{−1}O_{Spec R((t))^{√}} = O_{Spec R}((t))^{√} = p_{∗}O_{Spec R((t))^{√}} .

The proof is analogous to that of Lemma 2.2.3. Instead of the property of A[[t]] quoted there, we use Proposition 2.3.1. ⊓ ⊔

Having Lemma 2.4.5, we prove Proposition 2.4.4 similarly to Proposition 2.2.1(a). In virtue of Proposition 2.4.4, it suffices to show that the functor λ′_{X} on Sch is

ind-representable. We start by estabilishing some of its properties.

(2.4.6) Proposition. (a) For every scheme X the functor λ′_{X} is a sheaf on Sch.

(b) If U ⊂ X is an open subset, then the induced morphism of functors λ′_{U} → λ′_{X} is open.

(c) Let (U_{α})_{α∈A} be an open covering of X. Then λ′_{X} is equal to the cokernel, in the category Shf, of the pair of morphisms

∐_{α,β} λ_{U_{α}∩U_{β}}⇉ ∐_{α} λ_{U_{α}},

Proof: The proposition follows from simple properties of representable functors on the category Lrs. If T = (T, O_{T}) is a locally ringed space, we will call an open part of T a ringed space of the form (U, O_{T} |_{U}) where U ⊂ T is an open subset in the usual sense. An open embedding is, by definition, a morphism isomorphic to the inclusion of an open part. Accordingly, we have the concept of an open covering of T . This makes Lrs into a Grothendieck site. For X ∈ Lrs let η_{X} be the corresponding representable functor on Lrs. As in (2.1), a morphism F : ϕ′ → ϕ of functors Lrs → Sets will be called open, if for any S ∈ Lrs and any element u ∈ ϕ(S) (i.e., a morphism η_{S} → ϕ) the fiber product functor η_{S} ×_{ϕ} ϕ′ is representable by a locally ringed space S′ whose natural morphism to S is an open embedding. Let us recall the following basic facts.

(2.4.7) Lemma. (a) For any X ∈ Lrs the representable functor η_{X} is a sheaf on Lrs.

(b) If U ⊂ X is an open embedding in Lrs, then η_{U} → η_{X} is an open morphism of functors.

(c) Let X ∈ Lrs and (U_{α})_{α∈A} be an open covering of X. Then in the category of sheaves of sets on Lrs we have the equality

(2.4.8)  η_{X} = Coker{∐_{α,β} η_{U_{α}∩U_{β}}⇉ ∐_{α} η_{U_{α}}}

or, explicitly, for any S ∈ Lrs,

(2.4.9)  η_{X} (S) = lim_{S_{α}^{⟶}} Ker{∏_{α} η_{U_{α}}(S_{α})⇉ ∏_{αβ} η_{U_{α}∩U_{β}}(S_{α} ∩ S_{β})},

where the limit is taken over the set of open coverings (S_{α})_{α∈A} of S (the indexing set A being fixed) ordered by simultaneous inclusion.

Proposition 2.4.6 follows from Lemma 2.4.7 applied to X = X (scheme considered as a ringed space) and the ringed spaces of the form S = (S, O_{S}((t))^{√} ) where S is a scheme.

⊓ ⊔

Proof of (2.4.7) : Part (a) is obvious, and Part (b) is proved in [EGA0, (4.5.2)]. Let us prove Part (c). For any ϕ ∈ η_{X} (S) and any α, consider the ringed space S_{α} = (ϕ^{−1}(U_{α}), O_{S}|_{ϕ^{−1}(U_{α})}). Clearly, (S_{α}) is an open covering of S. Let ϕ_{α} be the restriction of ϕ to S_{α}. Then, (ϕ_{α}) is an element of ∏_{α} η_{U_{α}}(S_{α}) lying in the kernel (2.4.9). Thus we have constructed a map from η_{X} (S) to the right hand side of (2.4.9). On the other hand, assume that (ϕ_{α}) ∈ ∏_{α} η_{U_{α}}(S_{α}) is such that ϕ_{α}(S_{α}∩S_{β}) ⊂ U_{α}∩U_{β} and ϕ_{α}|_{S_{α}∩S_{β}} = ϕ_{β}|_{S_{α}∩S_{β}}. The corresponding maps S_{α,red} → U_{α,red} glue together, giving a continuous map ϕ_{red} : S_{red} → X_{red}. Moreover, there is an obvious sheaf homomorphism ϕ^{∗} : ϕ_{red}^{−1}O_{X} → O_{S} : the restriction of ϕ^{∗} to S_{α} is the composition of the chain of maps

(ϕ_{red}^{−1}O_{X})|_{S_{α}} = (ϕ_{α,red}^{−1}O_{U_{α}})|_{S_{α}} →^{ϕ_{α}^{∗}}O_{S_{α}} = O_{S}|_{S_{α}}.

This establishes (2.4.9).

Let C be the cokernel as in (2.4.8) but taken in the category of presheaves. Then the cokernel in the category of sheaves is just the sheaf \overline{C} associated to the presheaf C. By definition, for S ∈ Lrs the set C(S) consists of pairs (α, ϕ : S → U_{α}) taken modulo the identifications coming from morphisms of S into U_{α} ∩ U_{β}. Now, by the definition of the sheaf associated to the presheaf,

\overline{C} = lim_{⟶}_{S=⋃_{γ∈Γ} S_{γ}} Ker\{∏_{γ∈Γ} C(S_{γ}) → ∏_{γ,γ′∈Γ} C(S_{γ} ∩ S_{γ′})\},

where the limit is over all possible open coverings of S (with arbitrary indexing sets Γ) ordered by refinement. Notice now that specifying a section of C over S_{γ} includes specifying an index α from the set A indexing the covering {U_{α}}, so we get a map p : Γ → A. Denoting S_{α} = ⋃_{p(γ)=α} S_{γ}, we get an element of the right hand side of (2.4.9). This establishes the equivalence of (2.4.8) and (2.4.9). ⊓ ⊔

Note that Proposition 2.4.6(c) implies the following.

(2.4.10) Corollary. We have λ_{X} = lim_{⟶}_{U⊂X affine}λ_{U}, the limit being taken in the category Shf.

(2.5) Proof of Theorem 2.4.2. We first assume that X = Spec A is an affine scheme of finite type. Consider the larger functor λ̃_{X} on Alg defined by

λ̃_{X}(R) = Hom_{Alg}(A, R((t))).

(2.5.1) Proposition. (a) The functor λ̃_{X} is represented by an ind-scheme L̃(X), which is an inductive limit of affine schemes of infinite type.

(b) If X is smooth, then L̃(X) is formally smooth.

Proof: (a) Consider the k-ind-scheme

\underline{k((t))} = lim_{⟶}_{N} Spec k[a_{l}; −N ≤ l].

We can think of the a_{i} as the coefficients of an indeterminate Laurent series ∑ a_{i}t^{i}. It is clear that \underline{k((t))} represents the functor λ̃_{𝔸^{1}}. Since 𝔸^{1} is a k-algebra object in the category of schemes, multiplication of Laurent series makes \underline{k((t))} into a kalgebra object in the category of ind-schemes. In particular, each polynomial f ∈ k[x_{1}, ..., x_{d}] defines a morphism of ind-schemes L̃(f) : \underline{k((t))}^{d} → \underline{k((t))}. Hereafter we will write k[x] instead of k[x_{1}, ..., x_{d}] to simplify.

Suppose now that X is given in 𝔸^{d} by a system of algebraic equations, say

f_{m}(x_{1}, ..., x_{d}) = 0, m = 1, 2, ..., l.

The ind-scheme L̃(X) is realized as the closed sub-ind-scheme in \underline{k((t))}^{d} defined as the intersection of the preimages of 0 under the L̃(f_{m}). More explicitly, replacing x_{i} by x_{i}(t) = ∑_{j≥−N} a_{j}^{(i)}t^{j} in the equations above, we get, for each N, a system

of algebraic equations in k[a_{l}^{(i)}; −N ≤ l] for each N which defines a subscheme in (Spec k[a_{l}; −N ≤ l])^{d}. Our desired ind-scheme L̃(X) is the direct limit of these schemes as N → ∞.

(b) The infinitesimal lifting condition for λ̃_{X} is formulated for affine schemes S = Spec(R). We need to prove that for any surjection of k-algebras R′ → R whose kernel I satisfies I^{n} = 0 for some n, the map of sets λ̃_{X}(R′) → λ̃_{X}(R) is surjective. But the kernel of R′((t)) → R((t)) is I((t)) which is also nilpotent of order n. So the smoothness of A implies that any morphism A → R((t)) can be lifted to a morphism A → R′((t)). ⊓ ⊔

(2.5.2) Corollary. The functor λ_{X} is represented by an ind-scheme L(X) which is the inductive limit of the formal neighborhoods of L^{0}(X) in the schemes forming an inductive system for L̃(X). If X is smooth, then L(X) is formally smooth.

This proves parts (a), (b) and (d) of Theorem 2.4.2 for the case of affine X. Part (c) of the theorem follows from Proposition 2.4.6(c).

To prove Theorem 2.4.2 for general X, it is enough to establish the existence of the limit lim_{⟶}_{U⊂X affine}L(U) in the category of ind-schemes. In fact, it is enough to take the limit over a finite set of U consisting of elements of some finite covering and their intersections. Indeed, given this, all the other properties follow from the affine case and from Proposition 2.4.6. But using Proposition 2.4.6(b) again, we see that for any pair U′ ⊂ U of affine open subsets in X the corresponding morphism of ind-schemes L(U′) → L(U) is an open embedding. Further, the ind-scheme L(U)_{red} is actually a scheme, namely L^{0}(U). So our statement follows from the next general fact.

(2.5.3) Lemma. Let (Z_{i})_{i∈I} be a finite diagram of ind-schemes in which all the arrows are open embeddings. Assume that the inductive limit of Z_{i,red} exists in the category of ind-schemes. Then so does the inductive limit of the Z_{i}.

Theorem 2.4.2 is proved. Let us also note the following fact.

(2.5.4) Proposition. If ϕ : X → Y is an étale morphism of schemes of finite type, then L(ϕ) : L(X) → L(Y) is formally étale.

Proof: By the above we can assume that X = Spec (A), Y = Spec (B) are affine. If S is a scheme and we have two compatible maps

α : S_{red} → L(X), β : S → L(Y),

we must construct the unique map S → L(X). If S = Spec R, then we have a diagram

A →^{α^{∗}} R_{red}((t))^{√} = R_{red}[[t]]
ϕ^{∗} ↑  ↑
B →^{β^{∗}} R((t))^{√} ,

where the right arrow, which is the projection ρ in (2.3), has nilpotent kernel. Therefore there is a unique homomorphism of rings A → R((t))^{√} making the diagram commute. ⊓ ⊔

(2.6) The formal loop space as an ind-object. For future purposes we construct a certain class of local realizations of L(X) as an ind-object in the category of schemes.

Let x_{1}, x_{2}, ..., x_{d} be the coordinates on the affine space 𝔸^{d}. For any k-algebra A and any morphism ϕ : Spec A → 𝔸^{d} let ϕ^{∗} be the corresponding map k[x_{1}, ..., x_{d}] → A.

(2.6.1) Proposition. Assume that X = Spec A is an affine scheme and that ϕ : X → 𝔸^{d} is an étale morphism of schemes.

(a) There is a unique morphism θ_{ϕ} : L(X) → X such that θ_{ϕ}(f)(ϕ^{∗}x_{i}) = f(ϕ^{∗}x_{i})_{0} for any f ∈ λ_{X}(R). The restriction of θ_{ϕ} to L_{0}(X) is equal to the projection p from (2.2.1)(b).

(b) We have L(𝔸^{d}) ×_{𝔸^{d}} X ≃ L(X).

Proof: Consider the morphism of functors λ̃_{𝔸^{d}} → η_{𝔸^{d}} which maps a homomorphism f ∈ Hom_{Alg}(k[x], R((t))) to the unique morphism in Hom_{Alg}(k[x], R) such that x_{i} ↦ f(x_{i})_{0}. This morphism of functors can be seen as a morphism of ind-schemes θ̃_{𝔸^{d}} : L̃(𝔸^{d}) → 𝔸^{d}. It is clear that its restriction to L^{0}(𝔸^{d}) coincides with the projection p from (2.2.1)(b). Let θ_{𝔸^{d}} be the restriction of θ̃_{𝔸^{d}} to the ind-subscheme L(𝔸^{d}) ⊂ L̃(𝔸^{d}). Since L(X) is an inductive limit of nilpotent extensions of the scheme L^{0}(X) and ϕ is formally étale, there is a map θ_{ϕ} : L(X) → X such that the diagram

L^{0}(X) → X
↓  ↓ ϕ
L(X) → 𝔸^{d}

commutes. Here, the lower horizontal arrow is the composition of maps

θ_{𝔸^{d}} ∘ L(ϕ) : L(X) → L(𝔸^{d}) → 𝔸^{d}.

Let ψ be the resulting map L(X) → L(𝔸^{d}) ×_{𝔸^{d}} X. We claim that ψ is an isomorphism. For this, we construct a map χ : L(𝔸^{d}) ×_{𝔸^{d}} X → L(X) inverse to ψ. Let S = Spec R be a scheme. A morphism S → L(𝔸^{d}) ×_{𝔸^{d}} X is a compatible pair

(α : S → L(𝔸^{d}), β : S → X).

We need to construct a map χ(α, β) : S → L(X). First, α(S_{red}) ⊂ L^{0}(𝔸^{d}), and β(S_{red}) ⊂ X. Thus, by Proposition 2.2.2, we have a map γ : S_{red} → L^{0}(X). The composition γ̃ of γ and the embedding L^{0}(X) ⊂ L(X) gives a commutative diagram

(2.6.3)  \begin{matrix}S_{red} & →^{γ̃} & L(X) \\ ↓ & & ↓ L(ϕ) \\ S & →^{α} & L(𝔸^{d}). \end{matrix}

To complete the construction, we notice that L(ϕ) is formally étale by Proposition 2.5.4 and so we have a map S → L(X) splitting (2.6.2) into two commutative triangles. We take this map to be χ(α, β). The verification that χ is inverse to ψ is straightforward. Proposition 2.6.1 is proved. ⊓ ⊔

Let 𝔼 be the set of ε = (ε_{−1}, ε_{−2}, ...), ε_{j} ∈ ℤ_{+} such that ε_{j} = 0 for almost all j. It is equipped with the natural partial order: ε ≤ ε′ if ε_{j} ≤ ε′_{j} for all j. In the remainder of this section we assume that the k-scheme X is smooth. Thus X can be covered by affine open sets U = Spec A possessing étale maps ϕ : U → 𝔸^{d}. For every such U, ϕ we consider the functor

λ_{ϕ}^{ε} : R ↦ {f ∈ λ_{U}(R) | f(ϕ^{∗}x_{i})_{j}^{1+ε_{j}} = 0, ∀i ∈ [1, d], ∀j < 0}.

(2.6.3) Proposition. The functor λ_{ϕ}^{ε} is representable by a closed subscheme L^{ε}(ϕ) ⊂ L(U), such that L(U) is the inductive limit of the schemes L^{ε}(ϕ).

Proof: We first consider the case U = 𝔸^{d}, ϕ = Id. Let N be such that ϵ_{j} = 0 for j < N. Define the scheme L^{ε}(𝔸^{d}) by

(2.6.4)  L^{ε}(𝔸^{d}) = Spec (k[a_{l}^{(i)}; −N ≤ l]/((a_{l}^{(i)})^{1+ϵ_{l}})).

It is immediate that this scheme represents the functor λ_{Id}^{ε}, and that λ_{ϕ}^{ε} = λ_{Id}^{ε} ×_{λ_{𝔸^{d}}} λ_{U}. The proof of Proposition 2.6.1 implies that that the map

λ_{ϕ}^{ε} → λ_{Id}^{ε} ×_{η_{𝔸^{d}}} η_{U}, f ↦ (f ∘ ϕ^{∗}, θ_{ϕ}(f)),

is an isomorphism of functors. Thus, the closed subscheme

(2.6.5)  L^{ε}(𝔸^{d}) ×_{𝔸^{d}} U ⊂ L(𝔸^{d}) ×_{𝔸^{d}} U = L(U),

see Proposition 2.6.1.(b), represents the functor λ_{ϕ}^{ε}. It is clear by the definition that λ_{U} = lim_{⟶}_{ε}λ_{ϕ}^{ε} in the category of functors Alg → Sets and thus we have that

L(U) = lim_{⟶}_{ε}L^{ε}(ϕ) in the category of ind-schemes. ⊓ ⊔

Let us note the following reformulation of this fact, to be used later.

(2.6.6) Corollary. If ϕ, ψ are two étale maps U → 𝔸^{d}, then the ind-objects “lim_{⟶}_{ε}”L^{ε}(ϕ), “lim_{⟶}_{ε}”L^{ε}(ψ) are isomorphic, i.e., for any ε there is ε′ such that

L^{ε}(ϕ) ⊂ L^{ε′}(ψ) and vice versa.

Proof: Given ψ, any map S → L(U) with S an affine scheme, factors through some L^{ε′}(ψ). Now take S = L^{ε}(ϕ) which we know to be an affine scheme by (2.6.4-5). ⊓ ⊔

(2.7) The formal loop space as an ind-pro-object. We keep the notations of (2.6). Thus U = Spec (A) is affine and ϕ : U → 𝔸^{d} is étale. The schemes L^{ε}(ϕ) are of infinite type. In fact, each of these schemes is a projective limit of schemes of finite type, so L(U) can be viewed as an ind-pro-object in the category of schemes of finite type over k. In this subsection we construct explicit ind-pro-systems for L(U).

Consider the functor λ_{nϕ}^{ε} : R ↦ λ_{ϕ}^{ε}(R)/ ∼_{nϕ}, where

f ∼_{nϕ} g ⟺ f(ϕ^{∗}x_{i}) − g(ϕ^{∗}x_{i}) ∈ t^{n+1}R[[t]], ∀i.

(2.7.1) Proposition. (a) The functor λ_{nϕ}^{ε} is representable by a scheme L_{n}^{ε}(ϕ) of finite type which is a nilpotent extension of the scheme L_{n}^{0}(U) = L_{n}^{0}(ϕ). Moreover,

L^{ε}(ϕ) = lim_{⟵}_{n}L_{n}^{ε}(ϕ).

(b) The schemes L_{n}^{ε}(ϕ) form an ind-pro-system with Cartesian squares (n′ ≥ n, ε′ ≤ ε)

L_{n′}^{ε}(ϕ) ↪ L_{n′}^{ε′}(ϕ)
↓  ↓
L_{n}^{ε}(ϕ) ↪ L_{n}^{ε′}(ϕ)

where the vertical arrows are smooth affine morphisms.

(c) The ind-pro-object “lim_{⟶}_{ε}”“lim_{⟵}_{n}”L_{n}^{ε}(ϕ) is independent, up to isomorphism, on ϕ.

Proof: Claim (a) is entirely similar to Proposition 2.6.3. We first consider the case U = 𝔸^{d}, ϕ = Id and define the scheme L_{n}^{ε}(𝔸^{d}) by

(2.7.2)  L_{n}^{ε}(𝔸^{d}) = Spec (k[a_{l}^{(i)}; −N ≤ l ≤ n]/((a_{l}^{(i)})^{1+ε_{l}})), N ≫ 0.

It represents λ_{nId}^{ε}. As before, we see that for an arbitrary étale ϕ : U → 𝔸^{d} the fiber product scheme

(2.7.3)  L_{n}^{ε}(ϕ) := L_{n}^{ε}(𝔸^{d}) ×_{𝔸^{d}} X

represents the functor λ_{nϕ}^{ε}.

Claim (b) is obvious in the case ϕ = Id. The general case follows from (2.7.3) since the since the base change of a smooth affine morphism is still smooth affine, and the base change of a Cartesian square is Cartesian.

Claim (c) follows from Corollary 2.6.6 and Proposition 2.3.2. ⊓ ⊔

Passing to the limit in ϵ we get the ind-scheme L_{n}(ϕ) = lim_{⟶}_{ϵ}L_{n}^{ϵ}(ϕ). It represents

the functor R ↦ λ_{U}(R)/ ∼_{nϕ}. As in (2.7.2) we get

(2.7.4)  L_{n}(𝔸^{d}) = lim_{⟶}_{N} Spf k[a_{l}^{(i)}, 0 ≤ l ≤ n][[a_{l}^{(i)}, −N ≤ l < 0]],

and we see that L_{n}(𝔸^{d}) is formally smooth. Further, as in (2.7.3), we see that

(2.7.5)  L_{n}(ϕ) = L_{n}(𝔸^{d}) ×_{𝔸^{d}} X,

from which we see that L_{n}(ϕ) is also formally smooth.

3. The global loop space

(3.1) Localization on a curve. Consider the Lie algebra Der k[[t]] and the group scheme Aut k[[t]] over k. They form a Harish-Chandra pair [BB] [F, §6.1]. By construction, we have the action of this pair on the scheme L^{0}(X) and on the ind-scheme L(X) constructed in §2.

Let C be a smooth curve over k. The well known procedure of localization, see loc. cit. and [GKF], gives then a scheme L^{0}(X)_{C} and an ind-scheme L(X)_{C} over C defined as follows. Let C^{∧} → C be the scheme of pairs (c, t_{c}) where c ∈ C and t_{c} is a formal coordinate near c. Then C^{∧} has a natural (Der k[[t]], Aut k[[t]])-structure, i.e. a simply transitive Der k[[t]]-action extending the fiberwise Aut k[[t]]-action. We define

(3.1.1)  L^{0}(X)_{C} = C^{∧} ×_{Aut k[[t]]} L^{0}(X), L(X)_{C} = C^{∧} ×_{Aut k[[t]]} L(X).

If X is affine, we define, in a similar way, the ind-scheme L̃(X)_{C} starting from L̃(X). Because of the simple transitivity of the Der k[[t]]-action on C^{∧}, the (ind-)schemes thus constructued possess natural connections along C which are compatible with the embeddings L^{0}(X)_{C} ⊂ L(X)_{C} and, for X affine, L^{0}(X)_{C} ⊂ L̃(X)_{C}.

Note that L^{0}(X)_{C} is nothing but the scheme of infinite jets of morphisms C → X, so it is easy to describe explicitly the functor represented by L^{0}(X)_{C} (and also by L(X)_{C}, L̃(X)_{C}). Let us introduce some notations. For a scheme S and a morphism f : S → C we denote by Γ(f) ⊂ S × C the graph of f. Let f_{red} : S_{red} → C be the restriction of f to S_{red}. We have then the following sheaves of k-algebras on S × C supported on Γ(f):

– O_{f}^{∧}, the completion of O_{S×C} along Γ(f), i.e. the sheaf of functions on the formal neighborhood of Γ(f).

– K_{f}^{∧}, the sheaf of functions on the punctured formal neighborhood of Γ(f). Thus K_{f}^{∧} is obtained from O_{f}^{∧} by inverting the local equations of the divisor Γ(f) ⊂ S × C.

– K_{f}^{√} ⊂ K_{f}^{∧}, the subsheaf of sections whose restriction to S_{red} × C lies in O_{f_{red}}^{∧} ⊂ K_{f_{red}}^{∧}.

The following is straightforward and left to the reader, part (a) being proved similarly to Proposition 2.1.1.

(3.1.2) Proposition. (a) The sheaves O_{f}^{∧} and K_{f}^{√} are sheaves of local rings.

(b) The scheme L^{0}(X)_{C} represents the functor λ_{X,C}^{0} : Sch → Sets which associates to a scheme S the set of pairs (f, ρ) where f : S → C is a morphism of schemes and ρ : (Γ(f), O_{f}^{∧}) → X is a morphism of ringed spaces.

(c) The ind-scheme L(X)_{C} ind-represents the functor λ_{X,C} defined as in (b) but with O_{f}^{∧} replaced by K_{f}^{√} .

(d) Similarly, when X is affine, the ind-scheme L̃(X)_{C} ind-represents the functor λ̃_{X,C} defined as in (b) but with O_{f}^{∧} replaced by K_{f}^{∧}.

(3.2) Factorization semigroups. Let C be a smooth curve, as before. For any surjection J ↠ I of finite sets and i ∈ I we denote by J_{i} the preimage of i. To such a surjection one associates, in a standard way, the “diagonal” embedding Δ^{(J/I)} : C^{I} ↪ C^{J}. Let U^{(J/I)} ⊂ C^{J} be the locus of (c_{j})_{j∈J} such that c_{j} ≠ c_{j′} whenever the images of j and j′ in I are different. We denote by j^{(J/I)} : U^{(J/I)} ↪ C^{J} the embedding.

(3.2.1) Definition. Let Y be an ind-scheme over C. A factorization semigroup structure on Y is a collection of ind-schemes Y_{I} over C^{I} for each nonempty finite set I such that Y_{{1}} = Y, equipped with the following data:

(a) An identification Δ^{(J/I)∗}Y_{J} →^{∼} Y_{I} for every surjection J ↠ I, compatible with compositions of surjections.

(b) An isomorphism of U^{(J/I)}-ind-schemes

κ^{(J/I)} : j^{(J/I)∗}(∏_{i∈I}Y_{J_{i}}) →^{∼} j^{(J/I)∗}Y_{J}

for every J ↠ I, such that for K ↠ J the isomorphism κ^{(K/I)} coincides with the composition κ^{(J/I)} ∘ (∏_{i∈I} κ^{(K_{i}/J_{i})}).

(3.2.2) Remarks. (a) This is a nonlinear counterpart of the concept of a factorization algebra due to Beilinson and Drinfeld [BD1] [G], see also (6.1) below. Factorization semigroups can be used to construct factorization algebras by applying “natural” linear constructions.

(b) More generally, we can speak about a factorization semigroup structure on any functor Y : Sch → Sets (not necessarily representable by an ind-scheme) which is equipped with a morphism to C (i.e. to the representable functor η_{C}).

(3.2.3) Example. Let G be an affine algebraic group over k. Then L^{0}(G) is a group scheme and L̃(G) is a group ind-scheme over C. The quotient ind-scheme 𝔊r_{G} = L̃(G)/L^{0}(G) is known as the affine Grassmannian associated to G. The natural family of such Grassmannians over C, i.e. the ind-scheme 𝔊r_{G}(C) = L̃(G)_{C}/L^{0}(G)_{C} is known to have a structure of a factorization semigroup, see [G, §5.2.2].

Now, the main result of this section is the following.

(3.2.4) Theorem. Let X be a scheme of finite type. Then the C-scheme L^{0}(X)_{C} and the C-ind-scheme L(X)_{C} possess natural structures of factorization semigroups so that the embedding L^{0}(X)_{C} ⊂ L(X)_{C} is a factorization homomorphism. Similarly, if X is affine, then the C-ind-scheme L̃(X)_{C} has a natural structure of a factorization semigroup so that the embedding L^{0}(X)_{C} ⊂ L̃(X)_{C} is a factorization homomorphism.

(3.3) Factorization semigroup structure on the functors represented by L^{0}(X)_{C}, L̃(X)_{C} and L(X)_{C}. Let S be a scheme and f_{I} be I-uple of morphisms f_{i} : S → C, i ∈ I. We denote by Γ(f_{I}) ⊂ S × C the union of the graphs of the f_{i}’s and by f_{I,red} ⊂ S_{red} × C the union of the graphs of the f_{i,red}. Let us introduce, similarly to (3.1), the following sheaves of rings on S × C with support in Γ(f_{I}) :

– O_{f_{I}}^{∧}, the completion of O_{S×C} along Γ(f_{I}), i.e. the sheaf of functions on the formal neighborhood of Γ(f_{I}).

– K_{f_{I}}^{∧}, the sheaf of functions on the punctured formal neighborhood of Γ(f_{I}).

– K_{f_{I}}^{√} ⊂ K_{f_{I}}^{∧}, the subsheaf of sections whose restriction to S_{red} × C lies in O_{f_{I,red}}^{∧} ⊂ K_{f_{I,red}}^{∧}.

(3.3.1) Proposition. The sheaves O_{f_{I}}^{∧} and K_{f_{I}}^{√} are sheaves of local rings.

The proof is analog to the proof of Proposition 2.1.1, and is left to the reader.

(3.3.2) Definition. Let I be a nonempty finite set.

(a) We define the contravariant functors λ_{X,C^{I}}^{0} , λ_{X,C^{I}} from Sch to Sets as follows. For a scheme S the set λ_{X,C^{I}}^{0} (S) consists of pairs (f_{I}, ρ) such that

f_{I} ∈ Hom_{Sch_{k}}(S, C^{I}) and ρ ∈ Hom_{Lrs_{k}}((Γ(f_{I}), O_{f_{I}}^{∧}), X).

The contravariant functor λ_{X,C^{I}} is defined similarly but with O_{f_{I}}^{∧} replaced by K_{f_{I}}^{√} .

(b) If X is affine, the functor λ̃_{X,C^{I}} from Alg to Sets is such that the set λ̃_{X,C^{I}} (R) consists of pairs (f_{I}, ρ) with

f_{I} ∈ Hom_{Sch_{k}}(Spec R, C^{I}) and ρ ∈ Hom_{Alg_{k}}(k[X], H^{0}(Γ(f_{I}), K_{f_{I}}^{∧})).

The embeddings of sheaves of rings O_{f_{I}}^{∧} ↪ K_{f_{I}}^{√} ↪ K_{f_{I}}^{∧} induce embeddings of functors λ_{X,C^{I}}^{0} ↪ λ_{X,C^{I}} ↪ λ̃_{X,C^{I}} .

(3.3.3) Proposition. Let X be a fixed scheme of finite type. The families of functors (λ_{X,C^{I}}^{0} ), (λ̃_{X,C^{I}}), (λ_{X,C^{I}}) with I running over nonempty finite sets, form each a factorization semigroup in the category of functors.

Proof: This is almost obvious by construction. Indeed, let J ↠ I be a surjection of nonempty sets. Then Δ^{(J/I)∗} of the Jth functor in any of the three families takes S into the set of (f_{J}, ρ) where f_{J} is a morphism S → C^{J} which in fact lie in the image of Δ^{(J/I)}. Thus f_{J} comes from a uniquely defined I-tuple f_{I}. Now, Γ(f_{J}) = Γ(f_{I}) and so each of the three sheaves O^{∧}, K^{∧}, K^{√} associated to them coincide. This gives the datum (a) of Definition 3.2.1. Similarly, j^{(J/I)^{∗}} applied to the Jth functor in any of the families, takes S into the set of (f_{J}, ρ) where f_{J} : S → U^{(J/I)}. But this means that Γ(f_{J}) = ∐_{i∈I} Γ(f_{J_{i}}) and hence on the level of set of morphisms of Γ(f_{J}) equipped with any of the three sheaves of rings, we get a direct product. This gives the datum (b), i.e. the isomorphism κ^{(J/I)}. The associativity of these isomorphisms is obvious. ⊓ ⊔

Therefore, to establish Theorem 3.2.4, we need only to prove the representability of the functors λ_{X,C^{I}}^{0} , λ_{X,C^{I}} and, when X is affine, of λ̃_{X,C^{I}}.

(3.4) The global space of germs of arcs.

(3.4.1) Proposition. The functor λ_{X,C^{I}}^{0} is representable by a scheme L^{0}(X)_{C^{I}} of infinite type over C^{I}.

Proof: For n > 0 let C^{(n)} be the nth symmetric product of C. As C is a smooth curve, C^{(n)} is identified with the Hilbert scheme Hilb^{n}(C) parametrizing subschemes in C of finite length n. Explicitly, to a point of C^{(n)}, i.e. an effective divisor D on C of degree n, there corresponds the subscheme Z_{D} = Spec (O_{C}/O_{C}(−D)). The following lemma is well-known.

(3.4.2) Lemma. Let T be a k-scheme, and X, Z be any T -schemes. Assume that the morphism Z → T is finite, and that X → T is of finite type. The contravariant functor

Sch → Sets, S ↦ Hom_{T} (S × Z, X),

is represented by a T -scheme \underline{Hom}_{T} (Z, X).

Let Z be the total space of the family of the schemes Z_{D}, D ∈ C^{(n)}. We set Map^{n}(C, X) = \underline{Hom}_{C^{(n)}}(Z, X × C^{(n)}). Let u_{I}^{n} be the composition of the maps

C^{I} → C^{I×[1,n+1]} → C^{((n+1)|I|)}

where the first map takes the I-uple (c_{i}, i ∈ I) to the I×[1, n+1]-uple (c_{i}, c_{i}, ..., c_{i}, i ∈ I), each c_{i} counted n + 1 times, and the second map is the projection from the Cartesian product to the symmetric product. Let L_{n}^{0}(X)_{C^{I}} be the fiber product

(3.4.3)  L_{n}^{0}(X)_{C^{I}} = C^{I} ×_{C^{((n+1)|I|)}} Map^{(n+1)|I|}(C, X).

For any I-uple c ∈ C^{I} we have inclusions of subschemes of C

(3.4.4)  u_{I}^{0}(c) ⊂ u_{I}^{1}(c) ⊂ u_{I}^{2}(c) ⊂ ⋯

Thus we get a projective system of schemes

L_{1}^{0}(X)_{C^{I}} ← L_{2}^{0}(X)_{C^{I}} ← L_{3}^{0}(X)_{C^{I}} ← ⋯

The morphisms in this projective system are affine because the embeddings u_{I}^{n}(c) ⊂ u_{I}^{n+1}(c) are purely nilpotent. Therefore we have the limit scheme

L^{0}(X)_{C^{I}} = lim_{⟵}_{n}L_{n}^{0}(X)_{C^{I}} .

We claim that the scheme L^{0}(X)_{C^{I}} represents the functor λ_{X,C^{I}}^{0} . Indeed a morphism from a k-scheme S to L_{n}^{0}(X)_{C^{I}} is a pair (f_{I}, ψ) where f_{I} is a morphism S → C^{I} and

ψ : (Γ(f_{I}), O_{u_{I}^{n}(f_{I})}) → (X, O_{X})

is a morphism of ringed spaces. Here u_{I}^{n}(f_{I}) ⊂ S × C is the subscheme of relative length n|I| over S corresponding to the S-point f_{I} of C^{I} via u_{I}^{n}. When we pass to

the limit we get lim_{⟵}_{n}O_{u_{I}^{n}(f_{I})} = O_{f_{I}}^{∧} . ⊓ ⊔

(3.4.5) Remark. When X = Spec A is affine, the scheme L^{0}(X)_{C}, being the scheme of infinite jets of maps C → A, is the spectrum of a commutative O_{C}- algebra with connection along C (in fact, of the universal such algebra with connection generated by O_{C} ⊗_{k} A). According to Beilinson and Drinfeld [BD1], [G], commutative O_{C}-algebras with connection are particular case of chiral algebras which, in their turn, give factorization algebras. So our construction in this case is a particular case of theirs.

(3.4.6) Proposition. If ϕ : X → Y is an étale morphism of schemes of finite type and π : C → D is an étale morphism of smooth curves, then the square

L^{0}(X)_{C^{I}}  ⟶^{L^{0}(ϕ)_{π^{I}}} L^{0}(Y)_{D^{I}}
↓ ↓
L_{0}^{0}(X)_{C^{I}} ⟶ L_{0}^{0}(Y)_{D^{I}}

is Cartesian. Here L_{0}^{0}(X)_{C^{I}} is the same as in (3.4.3).

Proof: Similarly to the proof of Proposition 2.2.2, this follows from the fact that all the embeddings in (3.4.4) are purely nilpotent. ⊓ ⊔

(3.5) The meromorphic loop space of an affine scheme. Here we prove the ind-representability of the functor λ̃_{X,C^{I}} for X affine. We first treat the case when X = 𝔸^{1}. Let I be fixed. To every point (c_{i})_{i∈I} ∈ C^{I} we associate the effective divisor ∑ c_{i} on C. For every m, n ≥ 0 let A_{mn} be the vector bundle on C^{I} whose fiber over (c_{i}) is the vector space of global sections of the coherent 0-dimensional sheaf

O_{C}(m∑c_{i})/O_{C}(−n∑c_{i})

on C. Thus rk(A_{mn}) = |I|(m + n). Let A_{mn} be the total space of the bundle A_{mn}, considered as an algebraic variety over C^{I}. When m, n vary, these varieties form a double inductive-projective system and the following is then obvious.

(3.5.1) Proposition. (a) For every m > 0 the limit lim_{n^{⟵}} A_{mn} exists as a scheme.

The ind-scheme L̃(𝔸^{1})_{C^{I}} = lim_{⟶}_{m}lim_{⟵}_{n}A_{mn} represents the functor λ̃_{𝔸^{1},C^{I}}.

(b) L̃(𝔸^{1})_{C^{I}} has a natural structure of a k-algebra object in the category of indschemes over C^{I}.

Proof: A morphism of k-schemes Spec R → A_{mn} is a pair of a morphism of kschemes f_{I} : S → C^{I}, and a morphism of sheaves of k-algebras O_{A_{mn}} ⊗_{O_{C^{I}}} R → R (here, the right hand side is identified with the constant sheaf on C^{I}). There is a canonical isomorphism

lim_{⟶}_{m}lim_{⟵}_{n}Hom(O_{A_{mn}}⊗_{O_{C^{I}}} R, R) ≃ H^{0}(Γ(f_{I}), K_{f_{I}} ) ≃ Hom_{Alg}(k[x], H^{0}(Γ(f_{I}), K_{f_{I}})).

Claim (a) is proved. In part (b), the variety A_{mn} is not a ring scheme. However there is an obvious map A_{mn} ×A_{m′n′} → A_{m+m′,n+n′} which induces a ring structure on the limit of the ind-pro system. ⊓ ⊔

(3.5.2) Proposition. For any affine X of finite type the functor λ̃_{X,C^{I}} is representable by an ind-scheme L̃(X)_{C^{I}} over C^{I}.

Proof: By part (b) of Proposition 3.5.1, for every f ∈ k[x_{1}, ..., x_{d}] we have a morphism of ind-schemes

L̃(f)_{C^{I}} : (L̃(𝔸^{1})_{C^{I}} )^{d} → L̃(𝔸^{1})_{C^{I}} .

If now the scheme X is given in some 𝔸^{d} be equations f_{j}(x_{1}, ..., x_{d}) = 0, then λ̃_{X,C^{I}} is represented by the ind-scheme L̃(X)_{C^{I}} which is the intersection, in (L̃(𝔸^{1})_{C^{I}} )^{d}, of the preimages of 0 under the L̃(f_{j})_{C^{I}}, i.e. the inverse limit of an obvious diagram in the category of ind-schemes. ⊓ ⊔

(3.5.3) Corollary. For an affine scheme X the functor λ_{X,C^{I}} is represented by an ind-scheme L(X)_{C^{I}} which is the inductive limit of the formal neighborhoods of L^{0}(X)_{C^{I}} in the schemes of an inductive system for L̃(X)_{C^{I}}.

(3.6) The global loop space of an arbitrary scheme. Let now X be an arbitrary scheme of finite type. The (ind-)representability of the functor λ_{X,C^{I}} follows from Corollary 3.5.3 for the affine case and from the general gluing properties of the functors summarized in the next proposition.

(3.6.1) Proposition. (a) The functor λ_{X,C^{I}} is a sheaf on Sch.

(b) If U ⊂ X is an open subset, then the induced morphism of functors λ_{U,C^{I}} → λ_{X,C^{I}} is open.

(c) Let {U_{α}}_{α∈A} be an open covering of X. Then λ_{X,C^{I}} is equal to the cokernel, in the category Shf, of the pair of morphisms

∏_{α,β} λ_{U_{α}∩U_{β},C^{I}} ⇉ ∏_{α} λ_{U_{α},C^{I}} .

The proof is similar to that of Proposition 2.4.2, using Lemma 2.4.3 and the fact that K_{f_{I}}^{√} is a sheaf of local rings on Γ(f_{I}). An immediate corollary of Proposition 3.6.1 and Corollary 3.5.3 is that the functor λ_{X,C^{I}} is representable by an indscheme L(X)_{C^{I}} over C^{I}. Then, Proposition 3.3.3 implies that the collections of schemes and ind-schemes L^{0}(X)_{C^{I}} and L(X)_{C^{I}} are factorization semigroups in the categories of schemes and ind-schemes. Theorem 3.2.4 is proved.

(3.6.2) Proposition. If ϕ : X → Y, π : C → D are étale morphisms, then the induced morphism

L(ϕ)_{π^{I}} : L(X)_{C^{I}} → L(Y)_{D^{I}}

is formally étale.

The proof is similar to that of Proposition 2.5.4.

(3.7) Example : the cases C = 𝔸^{1} and X = 𝔸^{d}. Let C = Spec k[t]. If S = Spec R, a morphism f_{I} = (f_{i}) : S → C^{I} is the same as a collection of elements b_{i} = f_{i}^{∗}(t) ∈ R. Assume these have been fixed. Then the subscheme u_{I}^{n}(f_{I}) ⊂ S ×C from the proof of Proposition 3.4.1 is described explicitly :

(3.7.1)  u_{I}^{n}(f_{I}) = Spec (R[t]/∏_{i∈I}(t − b_{i})^{n+1}).

This implies the following.

(3.7.2) Proposition. (a) The ring

H^{0}(Γ(f_{I}), O_{f_{I}}^{∧}) = lim_{⟵}_{n}(R[t]/∏_{i}(t − b_{i})^{n+1})

is identified with the set of series ∑_{l=0}^{∞} a_{l}(t) ∏_{i}(t − b_{i})^{l}, where a_{l}(t) ∈ R[t] are polynomials of degree less than |I| (such polynomials form a set of representatives for R[t]/ ∏_{i}(t − b_{i})).

(b) The ring H^{0}(Γ(f_{I}), K_{f_{I}}) is identified with the set of series ∑_{l≫−∞} a_{l}(t) ∏_{i}(t− b_{i})^{l}, where a_{l}(t) are as in (a).

(c) The subring H^{0}(Γ(f_{I}), K_{f_{I}}^{√} ) is identified with the series as in (b) but with the condition that all the coefficients of the polynomials a_{l}(t), l < 0, are nilpotent elements of R.

(3.7.3) Notation. For fixed b_{i} ∈ R and a ∈ H^{0}(Γ(f_{I}), K_{f_{I}}) we denote by a_{l}(t) ∈ R[t] the lth coefficient of the series corresponding to a by Proposition 3.7.2(b) and by a_{lν} ∈ R, ν = 0, ..., |I| − 1, the νth coefficient of the polynomial a_{l}(t).

Assume moreover that X = 𝔸^{d} with coordinates x_{1}, ..., x_{d}. In this case we can give a completely explicit description of the ind-schemes L̃(𝔸^{d})_{𝔸^{I}} and L(𝔸^{d})_{𝔸^{I}}, using Proposition 3.7.2. Indeed,

λ̃_{𝔸^{d},𝔸^{I}} (R) ={(b_{i}, ρ) | (b_{i}) ∈ R^{I}, ρ : k[x] → H^{0}(Γ(f_{I}), K_{f_{I}})}.

The algebra homomorphism ρ is uniquely determined by the choice of

ρ(x_{j}) = ∑_{l≫−∞} ∑_{ν=0}^{|I|−1} a_{lν}^{(j)}t^{ν} ∏_{i∈I}(t − b_{i})^{l}, a_{lν}^{(j)} ∈ R.

A choice of ρ(x_{j}) is the same as a choice of elements a_{lν}^{(j)} ∈ R. Thus the universal case corresponds to b_{i}, a_{lν}^{(j)}, with i ∈ I, j ∈ [1, d], l ∈ ℤ and ν ∈ [0, |I| − 1], being independent variables, i.e.

(3.7.4)  L̃(𝔸^{d})_{𝔸^{I}} = lim_{⟶}_{N}Spec k[b_{i}, a_{lν}^{(j)}; l ≥ −N].

and

(3.7.5)  L(𝔸^{d})_{𝔸^{I}} = lim_{⟶}_{N} Spf k[b_{i}, a_{lν}^{(j)}; l ≥ 0][[a_{lν}^{(j)}; −N ≤ l ≤ −1]].

Notice further that

L_{n}^{0}(𝔸^{d})_{𝔸^{I}} = Spec k[b_{i}, a_{lν}^{(j)}; 0 ≤ l ≤ n].

In particular, we have a natural morphism

(3.7.6)  θ_{𝔸^{d},I} : L(𝔸^{d})_{𝔸^{I}} → L_{0}^{0}(𝔸^{d})_{𝔸^{I}}

taking all a_{lν}^{(j)}, l ≠ 0, to 0.

(3.8) The global loop space as an ind-object. We first consider the case X = 𝔸^{d}, C = 𝔸^{1}, employing the notations of (3.7). Let ε = (ε_{−1}, ε_{−2}, ...) ∈ 𝔼 be as in (2.6). Define the scheme

(3.8.1)  L^{ε}(𝔸^{d})_{𝔸^{I}} = Spec (k[b_{i}, a_{lν}^{(j)}]/(a_{l,ν_{1}}^{(j)} ⋯ a_{l,ν_{1+ε_{l}}}^{(j)} )),

where j ∈ [1, d], l < 0 and ν_{1}, ..., ν_{1+ε_{l}} ∈ [0, |I| − 1] are arbitrary. It is clear that

L(𝔸^{d})_{𝔸^{I}} = lim_{⟶}_{ε}L^{ε}(𝔸^{d})_{𝔸^{I}} .

next, assume that X is an arbitrary smooth scheme of finite type and C is an arbitrary smooth curve. Then X can be covered by open U = Spec A possessing an étale map ϕ : U → 𝔸^{d} and similarly C can be covered by open V possessing an étale π : V → 𝔸^{1}. For each such (U, ϕ), (V, π) we get a formally étale map of ind-schemes

L(ϕ)_{π^{I}} : L(U)_{V^{I}} → L(𝔸^{d})_{𝔸^{I}}

and we define the subscheme L^{ε}(ϕ)_{π^{I}} ⊂ L(U)_{V^{I}} to be the pullback of L^{ε}(𝔸^{d})_{𝔸^{I}}.

(3.8.2) Proposition. The ind-scheme L^{ε}(ϕ)_{π^{I}} is in fact an affine scheme, fitting into a Cartesian diagram

L^{ε}(ϕ)_{π^{I}}  ↪  L(U)_{V^{I}}
θ_{ϕ,πI} ↓  ↓L(ϕ)_{πI}
L_{0}^{0}(U)_{V^{I}}  ⟶^{L_{0}^{0}(ϕ)_{π^{I}}} L_{0}^{0}(𝔸^{d})_{𝔸^{I}} .

The ind-object “lim_{⟶}_{ε}”L^{ε}(ϕ)_{π^{I}} in Sch is isomorphic to L(U)_{V^{I}}.

The proof (including the construction of θ_{ϕ,π^{I}}) is similar to that of Propositions 2.6.1 and 2.6.3 and is left to the reader.

(3.9) The global loop space as an ind-pro-object. We keep the notation of (3.8). For ε ∈ 𝔼, n ≥ 1 consider the scheme

(3.9.1)  L_{n}^{ε}(𝔸^{d})_{𝔸^{I}} = Spec (k[b_{i}, a_{lν}^{(j)}; l ≤ n]/(a_{lν_{1}}^{(j)} ⋯ a_{l,ν_{1+ε_{l}}}^{(j)} )),

which we regard as a quotient scheme of L^{ε}(𝔸^{d})_{𝔸^{I}}, see (3.8.1). Denoting by R_{n} ⇉ L^{ε}(𝔸^{d})_{𝔸^{I}} the corresponding equivalence relation, let R_{nϕπ} ⇉ L^{ε}(ϕ)_{π^{I}} be its pullback under the map L^{ε}(ϕ)_{π^{I}} → L^{ε}(𝔸^{d})_{𝔸^{I}}.

(3.9.2) Proposition. (a) The quotient L_{n}^{ε}(ϕ)_{π^{I}} = L^{ε}(ϕ)_{π^{I}} /R_{nϕπ} exists and is a scheme of finite type fitting into a Cartesian square

L_{n}^{ε}(ϕ)_{π^{I}} ⟶ L_{n}^{ε}(𝔸^{d})_{𝔸^{I}}
↓ ↓
L_{0}^{0}(U)_{V^{I}} ⟶ L_{0}^{0}(𝔸^{d})_{𝔸^{I}} .

In particular, the natural map L_{n}^{ε}(ϕ)_{π^{I}} → L_{n}^{ε}(𝔸^{d})_{𝔸^{I}} is étale.

(b) The scheme L_{n}^{ε}(ϕ)_{π^{I}} is a nilpotent extension of L_{n}^{0}(U)_{V^{I}}.

(c) The schemes L_{n}^{ε}(ϕ)_{π^{I}} form a double ind-pro-system with Cartesian squares (n′ ≥ n, ε ≤ ε′)

L_{n′}^{ε}(ϕ)_{π^{I}} ↪ L_{n′}^{ε′}(ϕ)_{π^{I}}
↓  ↓
L_{n}^{ε}(ϕ)_{π^{I}} ↪ L_{n}^{ε′}(ϕ)_{π^{I}}

where the vertical arrows are smooth affine morphisms.

(d) The ind-pro-object “lim_{⟶}_{ε}”“lim_{⟵}_{n}”L_{n}^{ε}(ϕ)_{π^{I}} is independent on ϕ, π.

Proof: The proof is similar to that of Proposition 2.7.1. The proof of the second part of Claim (a) is as follows. The natural morphism of schemes L_{0}^{0}(U)_{V^{I}} → L_{0}^{0}(𝔸^{d})_{𝔸^{I}} being locally of finite presentation, by base change and [EGAIV, §17] it suffices to check that it is is formally étale. If S is a scheme and we have two compatible pairs

(f_{I}, ρ) ∈ λ_{0,ϕ,π^{I}}^{0} (S_{red}), (f′_{I}, ρ′) ∈ λ_{0,𝔸^{d},𝔸^{I}}^{0} (S),

we must construct the unique pair (f̃_{I}, ρ̃) ∈ λ_{0,ϕ,π^{I}}^{0} (S). By definition, we have a commutative square

S_{red} ⟶^{f_{I}} C^{I}
↓  ↓ π^{I}
S ⟶^{f′_{I}} 𝔸^{I}.

Thus, π being étale, there is a unique lift f̃_{I} ∈ Hom_{Sch}(S, C^{I}). Then, Γ(f̃_{I})_{red} = Γ(f_{I}). We have also a commutative diagram

Γ(f_{I}) ⟶^{ρ} U
↓  ↓ ϕ
Γ(f̃_{I}) ⟶ 𝔸^{d}

where the lower arrow is the composition of the chain of maps

Γ(f̃_{I})⟶^{Id×π}Γ(f′_{I})⟶^{ρ′}𝔸^{d}.

Thus, ϕ being étale, there is a unique lift ρ̃ ∈ Hom_{Sch}(Γ(f̃_{I}), U). We are done. The smoothness of the vertical arrows in Claim (c) is proved as follows. Set

A_{n}^{N} = k[𝔸^{I}][a_{ν}^{(j)}; −N ≤ l ≤ n].

Then, L_{n}^{ε}(𝔸^{d})_{𝔸^{I}} is a closed subset of Spec A_{n}^{N} if N >> 0. We have a Cartesian square

Spec A_{n′}^{N} → Spec A_{n}^{N}
↑  ↑
L_{n′}^{ε}(ϕ)_{π^{I}} → L_{n}^{ε}(ϕ)_{π^{I}}

where the horizontal arrow is the obvious smooth affine projection, and the vertical arrows are the composition of the chain of maps

L_{n′}^{ε}(ϕ)_{π^{I}} →^{∼} L_{n}^{ε}(𝔸^{d})_{𝔸^{I}} ×_{L_{0}^{0}(𝔸^{d})_{𝔸^{I}}} L_{0}^{0}(U)_{V^{I}} → L_{n}^{ε}(𝔸^{d})_{𝔸^{I}} ↪ Spec A_{n}^{N}.

Thus, the projection L_{n′}^{ε}(ϕ)_{π^{I}} → L_{n}^{ε}(ϕ)_{π^{I}} is smooth and affine, by base change. ⊓ ⊔

4. D-modules over ind-schemes

(4.1) Reminder on D-modules. Let Sch^{ft} denote the category of k-schemes of finite type. For any such scheme X let O_{X} be the category of all quasi-coherent O_{X}-modules. For a morphism f : X → Y in Sch^{ft} we denote by f_{∗}, f^{∗} the functors of the direct and inverse images on O_{X}, O_{Y}. If f is a closed embedding and E ∈ O_{Y}, let

(4.1.1)  f^{!}E = f^{−1}Hom_{O_{Y}} (f_{∗}O_{X}, E)

be the inverse image of the subsheaf of E consisting of sections supported schemetheoretically on f(X) ⊂ Y.

For X ∈ Sch^{ft} let D_{X} be the category of right D-modules on X. It is defined as follows, see [BD2, §7.10] or [G, §0.2.2]. If X is smooth, then we have the sheaf of rings D_{X} of differential operators from O_{X} to itself. An object of D_{X} is then a sheaf of right D_{X}-modules quasicoherent over O_{X}. Next, if X admits a closed embedding into a smooth scheme Y, one defines D_{X} as D_{Y,X}, the full subcategory of D_{Y} consisting of modules supported (as sheaves) on X. This definition is independent on the choice of embedding: if X is embedded into two smooth schemes Y_{1} and Y_{2}, then one has an equivalence D_{Y_{1},X} → D_{Y_{2},X} which is unique up to a unique isomorphism of functors. Now, given any X, an embedding into a smooth scheme always exists locally on X. Therefore we have an open covering X = ⋃ U_{α}, the categories D_{U_{α}}, D_{U_{αβ}} etc. and the obvious restriction functors among them. One then defines an object M of D_{X} as a collection of objects M_{α} ∈ D_{U_{α}} together with isomorphisms of their images in D_{U_{αβ}} whose images in Mor(D_{U_{αβγ}}) satisfy the obvious compatibility conditions.

Given X ∈ Sch^{ft} and M ∈ D_{X}, we define a sheaf M^{O} ∈ O_{X} as follows. If X admits a closed embedding i : X → Y with Y smooth and M is represented by a sheaf of right D_{Y} -modules supported on X (which we also denote M) then we set M^{O} = i^{!}M. This definition is easily seen to be independent on the choice of Y. In the case of a general X one defines M^{O} by gluing the sheaves given by above procedures on open parts of X admitting embeddings into scmooth schemes.

Any morphism f : X → Y in Sch^{ft} induces functors f_{•}, f^{•} of the direct and inverse image on D_{X}, D_{Y}. In the particular case when X, Y are smooth, we can view M ∈ D_{X} as a sheaf on X and

f_{•}(M) = f_{∗}(M ⊗_{D_{X}} D_{X→Y}).

Here D_{X→Y} is the sheaf of differential operators from f^{−1}O_{Y} to O_{X}, see [BB] [G].

If f is a closed embedding (X and Y are enot assumed smooth), the functor f_{•} is exact, and we have a canonical embedding

(4.1.2)  M^{O} ≃ f^{!}f_{•}(M) ↪ f^{∗}f_{•}(M).

For example, if X is non-reduced and i : X_{red} ↪ X is the reduced part, then i_{•} identifies D_{X_{red}} with D_{X}.

If f is a smooth morphism and ω_{X/Y} is its relative canonical bundle, then

f^{•}(M) = f^{∗}(M^{O}) ⊗_{O_{X}} ω_{X/Y} .

We have then a canonical embedding

(4.1.3)  M^{O} ↪ f_{∗}((f^{•}M)^{O} ⊗ ω_{X/Y}^{−1} ).

We have the following base change property.

(4.1.4) Lemma. Suppose that in a Cartesian diagram of schemes of finite type

\begin{matrix}X & ↪^{i} & X′ \\ f↓ & & ↓f′ \\ Y & ↪^{j} & Y′ \end{matrix}

the morphisms f, f′ are smooth, i, j are closed embeddings. Then for any M ∈ D_{Y}, we have

i_{•}f^{•}M ≃ f′^{•}j_{•}M.

The projection formula holds for right D-modules. More precisely, if f is an embedding of smooth k-schemes and M ∈ D_{X}, N ∈ D_{Y}, then there is a canonical isomorphism

f_{•}(f^{•}(N) ⊗_{O_{X}} M ⊗_{O_{X}} ω_{X}^{−1}) ≃ N ⊗_{O_{Y}} f_{•}(M) ⊗_{O_{Y}} ω_{Y}^{−1}.

(4.2) D-modules over pro-schemes.

(4.2.1) Definition. (cf. [Kap].) Let A be a filtering poset and (C_{α})_{α∈A} be an inductive system of categories labelled by A. In other words, for each α ≤ β we have a functor i_{αβ} : C_{α} → C_{β}, for each α ≤ β ≤ γ a natural isomorphism i_{βγ} ∘i_{αβ} ⇒ i_{αγ} and these isomorphisms satisfy the obvious coherence conditions for any α ≤ β ≤ γ ≤ δ.

The inductive limit 2lim_{⟶}_{α}C_{α} is the category whose objects are pairs (α, x_{α}),

α ∈ A, x_{α} ∈ Ob(C_{α}) and

Hom(x_{α}, y_{β}) = lim_{⟶}_{γ≥α,β} Hom_{C_{α}}(i_{αγ}(x_{α}), i_{βγ}(y_{β})).

(4.2.2) Definition. A k-scheme X_{∞} (possibly of infinite type) is called compact if it can be represented as lim_{⟵}_{α} X_{α} where (X_{α})_{α∈A} is a filtering projective system

over Sch_{k}^{ft} such that all the maps p_{αβ} : X_{β} → X_{α}, α ≤ β, are affine morphisms.

(4.2.3) Proposition. The category of compact k-schemes can be identified with a full subcategory in Pro(Sch_{k}^{ft}), via X_{∞} = lim_{⟵}_{α} X_{α} ↦ “lim_{⟵}_{α}”X_{α}.

(4.2.4) Definition. A compact scheme X_{∞} is called smooth if it can be represented as lim_{⟵}_{α} X_{α} for some (X_{α}) as in (4.2.2) with the extra property that each X_{α} is a smooth scheme and each p_{αβ} is a smooth affine morphism.

(4.2.5) Example. For any smooth X ∈ Sch_{k}^{ft}, the scheme L^{0}(X) is smooth and compact.

Let X_{∞} be a compact scheme and (X_{α}) be as in Definition 4.2.2. We have two inductive systems of categories (O_{X_{α}}, p_{αβ}^{∗}), (D_{X_{α}}, p_{αβ}^{•}). We set D_{X_{∞}} = 2lim_{⟶}_{α}D_{X_{α}},

see also [KT, §1.9]. Informally, an object of D_{X_{∞}} is a “D-module pulled back from some X_{α}”. The category D_{X_{∞}} is independent, up to a canonical equivalence of categories, on the choice of (X_{α}) as in (4.2.2). Let p_{α} : X_{∞} → X_{α} be the projection. Let p_{α}^{•} : D_{X_{α}} → D_{X_{∞}}, M^{α} ↦ (α, M^{α}) be the canonical functor. If the compact scheme X_{∞} is smooth, there is the functor

D_{X_{∞}} → 2lim_{⟶}_{α}O_{X_{α}} ⊂ O_{X_{∞}}, p_{α}^{•}M_{α} ↦ (p_{α}^{•}M_{α})^{O} := (p_{αβ}^{•}M_{α} ⊗ ω_{X_{β}}^{−1})_{β≥α}.

In particular, we can associate to M its space of global sections

(4.2.6)  Γ(X_{∞}, M) = lim_{⟶}_{β≥α}Γ(X_{β}, p_{αβ}^{•}M_{α} ⊗ ω_{X_{β}}^{−1}).

(4.3) D-modules over ind-schemes.

(4.3.1) Definition. Let A be a filtering poset and (C_{α})_{α∈A} be a projective system of categories labelled by A. In other words, for each α ≤ β we have a funtctor j_{αβ} : C_{β} → C_{α} and for any α ≤ β ≤ γ a natural isomorphism j_{αβ} ∘ j_{βγ} ⇒ j_{αγ} satisfying the obvious compatibility conditions.

The projective limit 2lim_{⟵}C_{α} is the category whose objects are systems consisting of objects x_{α} ∈ C_{α} given for all α ∈ A and isomorphisms j_{αβ}(x_{β}) → x_{α} given for each α ≤ β and satisfying the compatibility condition for each α ≤ β ≤ γ. Morphisms are defined in the obvious way.

(4.3.2) Definition. Let X^{∞} be an ind-scheme. We say that X_{∞} is discrete if it can be represented as X^{∞} = “lim_{⟶}_{α}”X^{α} where (X^{α})_{α∈A} is a filtering inductive sys-

tem over Sch_{k}^{ft} such that each map i_{αβ} : X^{α} → X^{β}, α ≤ β, is a closed embedding.

(4.3.3) Example. For any ϕ, U as in (2.7) the ind-scheme L_{n}(ϕ) is discrete.

Let X^{∞} be a discrete ind-scheme and (X^{α}) be as in Definition 4.3.2. We have then the projective system of categories (O_{X^{α}}, i_{αβ}^{!}). We define O_{X^{∞}} = 2lim_{⟵}O_{X^{α}}.

If (E^{α}, γ_{αβ} : E^{α} → i_{αβ}^{!}E^{β}) is an object of O_{X^{∞}}, then the spaces of global sections Γ(X^{α}, E^{α}) form an inductive system and we define

(4.3.4)  Γ(X^{∞}, E) = lim_{⟶}_{α} Γ(X^{α}, E^{α}).

We will also use the category Ô_{X^{∞}} which is the limit of the projective system of categories (O_{X^{α}}, i_{αβ}^{∗}). There is a functor

O_{X^{∞}} × Ô_{X^{∞}} → O_{X^{∞}}, (E, F̂) ↦ E ⊗ F̂ = (E^{α} ⊗_{O_{X^{α}}} F̂^{α}).

See [BD2, §7.11.4] for more details on O_{X^{∞}}, Ô_{X^{∞}}.

We set also D_{X^{∞}} = 2lim_{⟶}_{α}(D_{X^{α}}, i_{αβ•}). Let i_{α} be the embedding X^{α} ↪ X^{∞}.

Let i_{α•} : D_{X^{α}} → D_{X∞}, M^{α} ↦ (α, M^{α}) be the canonical functor. It is exact. There is also the functor

D_{X^{∞}} → O_{X^{∞}}, i_{α•}M^{α} ↦ (i_{α•}M^{α})^{O} := (i_{αβ•}M^{α})_{β≥α},

see (4.1.2). In particular, to any M ∈ D_{X^{∞}} we can associate its space of global sections: If M is represented by M^{α} ∈ D_{X^{α}}, then

(4.3.4)  Γ(X^{∞}, M) = lim_{⟶}_{β≥α}Γ(X^{β}, (i_{αβ•}M^{α})^{O}).

(4.3.5) Remark. (a) We have an exact functor

2lim_{⟶}_{α}O_{X^{α}} → O_{X^{∞}}, (α, E^{α}) ↦ (i_{αβ∗}E^{α})_{β≥α}.

The two categories are not equivalent in general.

(b) The category O_{X^{∞}} is closed by inductive limits.

(c) If the ind-scheme X^{∞} is not discrete anymore, the O_{X^{α}}-module i_{αβ}^{!}E^{β} may not be quasi-coherent. However the category O_{X^{∞}} is still well-defined.

Let X^{∞} = lim_{⟶}_{α}X^{α} be a formally smooth discrete ind-scheme. Following [BD2,

Prop. 7.11.8 and 7.12.13] we define the tangent sheaf of X to be the object Θ̂_{X^{∞}} ∈ Ô_{X^{∞}} such that

(4.3.6)  E ⊗ Θ̂_{X^{∞}} = lim_{⟶}_{α}Hom(Ω_{X^{α}}^{1}, E^{α}), ∀E ∈ O_{X^{∞}}.

Denoting i^{α} : X^{α} → X^{∞} the canonical embedding, we have that i^{α∗}Θ̂_{X^{∞}} is, locally, a projective O_{X^{α}}-module (possibly of infinite rank).

(4.4) D-modules over ind-pro-schemes. The following definition is inspired by the paper of K. Kato [Kat].

(4.4.1) Definition. An ind-scheme X_{∞}^{∞} is called locally compact if it can be represented as

X_{∞}^{∞} = lim_{⟶}_{α∈A}lim_{⟵}_{β∈B}X_{β}^{α}

where (X_{β}^{α}) is a bi-filtering ind-pro-system over Sch^{ft} with the following properties:

(1) For each β ∈ B and α ≤ α′ ∈ A the structure map i_{b}^{αα′} : X_{β}^{α} → X_{b}^{a′} is a closed embedding.

(2) For each α ∈ A and β ≤ β′ ∈ B the structure map p_{ββ′}^{α} : X_{b′}^{α} → X_{β}^{α} is an affine morphism.

(3) For each α ≤ α′ ∈ A and β ≤ β′ ∈ B the commutative square

X_{β′}^{α} ↪ X_{β′}^{α′}
↓  ↓
X_{β}^{α} ↪ X_{β}^{α′}

is Cartesian.

We denote by Lc ⊂ Isch the full subcategory formed by locally compact indschemes.

Let (X_{β}^{α}) be an ind-pro-system as above. The maps p_{ββ′}^{α} being affine, the projective limit lim_{⟵}_{β}X_{β}^{α} is represented by a compact scheme (possibly of infinite type)

denoted by X_{∞}^{α}. Similarly, we have the discrete ind-schemes X_{β}^{∞} = lim_{⟶}_{α}X_{β}^{α}. By definition X_{∞}^{∞} = lim_{⟶}_{α}X_{∞}^{α}.

(4.4.2) Proposition. For an ind-pro-system satisfying the conditions of (4.4.1) we also have X_{∞}^{∞} = lim_{⟵}_{β}X_{β}^{∞} = lim_{⟵}_{β}lim_{⟶}_{α}X_{β}^{α}.

Proof: By passing to the functors represented by our (ind-)schemes, we reduce the statement (a) to the following lemma whose proof we leave to the reader.

(4.4.3) Lemma. Let (S_{β}^{α})_{α∈A,β∈B} be a bi-filtering ind-pro-system of sets. Then there is a canonical map

c : lim_{⟶}_{α}lim_{⟵}_{β}S_{β}^{α} → lim_{⟵}_{β}lim_{⟶}_{α}S_{β}^{α}.

If, moreover, all the squares in (S_{β}^{α}) are Cartesian, then c is an isomorphism.

(4.4.4) Definition. We say that a locally compact ind-scheme X_{∞}^{∞} is smooth if it admits a presentation as in (4.1.1) where:

(1) All the p_{ββ′}^{α} are smooth morphisms of relative dimension d_{ββ′} independent on α and at least one X_{β}^{α} is smooth.

(2) All the ind-schemes X_{β}^{∞} = lim_{⟶}_{α}X_{β}^{α} are formally smooth.

(4.4.5) Example. If X is a smooth affine variety admitting an etale map to 𝔸^{d}, then the ind-scheme L(X) is locally compact and smooth.

Let X_{∞}^{∞} be a locally compact smooth ind-scheme and (X_{β}^{α}) be an ind-pro-system as in (4.4.1), (4.4.4). By Lemma 4.1.4, we have then a double inductive system of categories (D_{X_{β}^{α}}, i_{β•}^{αα′}, p_{ββ′}^{α•}) and we define the category of (right) D-modules on X_{∞}^{∞} to be D_{X_{∞}^{∞}} = 2lim_{⟶}_{α,β}D_{X_{β}^{α}}.

(4.4.6) Remark. Although we have defined D_{X_{∞}^{∞}} as an abstract category, it is impossible, in general, to view its objects as sheaves in a more conventional sense. For example, it is impossible to associate to an object of D_{X_{∞}^{∞}} a vector space of global sections. Indeed, if such an object M is represented by some M_{β}^{α} ∈ D_{X_{β}^{α}}, then the spaces of global sections of the coherent sheaves

(p_{ββ′}^{α′•}i_{b•}^{αα′}M_{β}^{α})^{O} = p_{ββ′}^{α′∗}(i_{β•}^{αα′}M_{β}^{α})^{O} ⊗ ω_{X_{β′}^{α′} /X_{β}^{α′}}

do not form an inductive system because of the twist by the relative canonical class. For a compact smooth pro-scheme it is possible to get around this problem by untwisting by the absolute canonical classes of the terms of the projective system (4.2.6). To achieve the same effect in the ind-pro-case one would need to make sense of the canonical class of the ind-scheme X_{∞}^{α}, i.e., of the determinant of the (possibly infinite-dimensional) vector bundle Θ̂_{X_{∞}^{α}}. The impossibility of doing this (“the determinantal anomaly”) is precisely the reason why there is no natural space Γ(X_{∞}^{∞}, M).

5. De Rham complexes on ind-schemes

(5.1) Reminder on the De Rham complexes. Let X be a smooth algebraic variety over k and M ∈ D_{X} be a right D_{X}-module. Its de Rham complex DR(M) is given by

(5.1.1)  DR^{i}(M) = Hom_{O_{X}} (Ω_{X}^{−i}, M) = M ⊗_{O_{X}} ⋀^{−i} Θ_{X}, i ≤ 0.

If (x_{j}) is an etale coordinate system on an open part of X, then the differential is given by the formula d = ∑ ∂_{x_{j}} ⊗ dx_{j} where dx_{j} ∈ Ω_{X}^{1} is considered as the contraction operator ⋀^{−i} Θ_{X} → ⋀^{−i−1} Θ_{X}. We denote by DR(M) = Γ(X, DR(M)) the complex of global sections.

Let i : X → Y be a closed embedding of smooth varieties and M ∈ D_{X}. The embedding (4.1.2) induces an embedding of the de Rham complexes DR(M) ↪ i^{∗}DR(i_{•}M) and therefore an embedding of the complexes of global sections:

(5.1.2)  DR(M) ↪ DR(i_{•}(M)).

Let p : X → Y be a smooth morphism of smooth varieties of relative dimension d and M ∈ D_{Y}. The embedding (4.1.3) induces an embedding of de Rham complexes which now involves a shift in the degrees:

p^{∗}DR(M) ↪ DR(p^{•}(M)[d]).

It is induced by the map

p^{∗} ⋀^{i} Θ_{Y} → ω_{X/Y} ⊗ ⋀^{i+d} Θ_{X}.

In particular, we get an embedding of the complexes of global sections

(5.1.3)  DR(M) ↪ DR(p^{•}M)[d].

Note that without passing to the de Rham complexes there is no embedding of the space of sections of M into that of p^{•}M. The map (5.1.3) can be seen as a D-module manifestation of the fact that “fermions cancel the determinantal anomaly”.

(5.1.4) Example. Let Y = 𝔸^{d} with coordinates a_{1}, ..., a_{d} and i : X ↪ Y be the embedding of the affine subspace {a_{1} = ... = a_{l} = 0}, l ≤ d. The algebra Γ(Y, D_{Y}) is just the Heisenberg (Weyl) algebra D_{Y} generated by a_{1}, ..., a_{d} and a_{1}^{∗}, ..., a_{d}^{∗} subject to the relations

[a_{m}, a_{n}] = [a_{m}^{∗}, a_{n}] = 0, [a_{m}^{∗}, a_{n}] = δ_{mn}.

The space of global sections Γ(Y, i_{•}ω_{X}) is the right D_{Y} -module

ω_{XY} = D_{Y} /(a_{m}^{∗}, a_{n}, n ≤ l ≤ m)D_{Y} .

Let also C_{Y} be the Clifford algebra generated by odd elements b_{1}, ..., b_{d}, b_{1}^{∗}, ..., b_{d}^{∗} subject to the relations

[b_{m}, b_{n}]_{+} = [b_{m}^{∗}, b_{n}^{∗}]_{+} = 0, [b_{m}^{∗}, b_{n}]_{+} = δ_{mn}.

Denote CD_{Y} = C_{Y} ⊗_{k} D_{Y} the tensor product algebra. Then the global de Rham complex of i_{•}ω_{X} is identified with the right CD_{Y} -module

DR_{XY} = CD_{Y} /(a_{m}^{∗}, a_{n}, b_{p}, n ≤ l ≤ m, p = 1, ..., d)CD_{Y} .

(5.2) De Rham complexes for ind-schemes. Let X^{∞} = lim_{⟶}_{α∈A}X^{α} be a

formally smooth discrete ind-scheme. Denote by i^{α} : X^{α} ↪ X the canonical embedding. The considerations of (5.1) generalize easily to give the global de Rham complex of any M ∈ D_{X^{∞}}. Explicitly, let M have the form i_{α•}M^{α}. Then the ith term of its de Rham complex is

(5.2.1)  DR^{i}(M) = lim_{⟶}_{β≥α}Γ(X^{β}, (i_{•}^{αβ}M^{α})^{O} ⊗_{O_{X^{α}}} i^{α∗} ⋀^{−i} Θ̂_{X^{∞}}).

(5.2.2) Proposition. Let f : X^{∞} → Y^{∞} be a morphism of formally smooth discrete ind-schemes which can be included into a Cartesian square

X^{∞} ⟶^{p} Y^{∞}
↓  ↓
X^{0} ⟶^{p^{0}} Y^{0}

where p^{0} : X^{0} → Y^{0} is a smooth morphism of schemes of finite type, of relative dimension d. Then for any M ∈ D_{Y} we have an embedding of the shifted de Rham complexes:

DR(M) ↪ DR(p^{•}M)[d].

Proof: The conditions on p in the proposition are equivalent to the following: we can represent X^{∞} = lim_{⟶}_{α∈A}X^{α}, Y^{∞} = lim_{⟶}_{α∈A}Y^{α} with the same filtering poset A

and represent p by a morphism of inductive systems (p^{α} : X^{α} → Y^{α}) such that for each α ≤ β the arising commutative square is Cartesian and each p^{α} is smooth of relative dimension d.

But in this situation the possibility of the embedding follows from (5.1.3) and from the base change for Cartesian squares (Lemma 2.1.4). ⊓ ⊔

(5.3) De Rham complexes for ind-pro-schemes. Let X_{∞}^{∞} be a locally compact smooth ind-scheme and M be an object of D_{X_{∞}^{∞}}. We fix an ind-pro-system (X_{β}^{α}) for X as in (4.4.1), (4.4.4). We have then the formally smooth discrete ind-schemes X_{β}^{∞} and projections p_{β} : X_{∞}^{∞} → X_{β}^{∞}. The category D_{X_{∞}^{∞}} being the double direct limit of D_{X_{β}^{α}}, we can think of M as being of the form p_{β}^{•}M_{b} for some M_{b} ∈ D_{X_{β}^{∞}}. Recall that d_{ββ′} denotes the relative dimension of the smooth morphism p_{ββ′}^{α} : X_{β′}^{α} → X_{β}^{α}, β ≤ β′. We choose numbers d_{β}, β ∈ B such that d_{ββ′} = d_{β} − d_{β′} (this can be done uniquely up to an overall constant). Proposition 5.2.2 implies then that the shifted global de Rham complexes DR(p_{ββ′}^{•}M_{β})[d_{β′}] form an inductive system of complexes of vector spaces and we define the de Rham complex of M to be

(5.3.1)  DR(M) = lim_{⟶}_{β′≥β}DR(p_{ββ′}^{•}M_{β})[d_{β′}].

(5.4) The de Rham complexes on L(X). We now specialize to the particular case X_{∞}^{∞} = L(X) where X is a smooth affine algebraic variety over k admitting an etale map ϕ to 𝔸^{d}. In this case A = 𝔼, B = ℕ with the terms of the ind-pro-systems being L_{n}^{ε}(ϕ), see (2.6.7). Given an object M ∈ D_{L(X)} we associate to it its de Rham complex DR(M) as in (5.3). Note that it is independent on the choice of an etale map to 𝔸^{d} because two such maps lead to isomorphic ind-pro-objects in the category Sch^{ft} (Proposition 2.7.1c).

Let now X be an arbitrary smooth algebraic variety over k. and M be an object of D_{L(X)}. By covering X with affine open U admitting etale maps to 𝔸^{d}, we get a complex of sheaves U ↦ DR(M|_{L(U)}) which we denote DR(M). Recall that we have the diagram

(5.4.1)  X ⟵^{p} L^{0}(X) ↪^{i} L(X).

Thus, every right D_{X}-module N gives an object i_{•}p^{•}N of D_{L(X)}. We write CDR(N) for DR(i_{•}p^{•}N) and call it the chiral de Rham complex of N. In particular, we write CDR_{X} for DR(i_{•}p^{•}ω_{X}). More generally, denoting p_{n} : L^{0}(X) → L_{n}^{0}(X) the projection, we can start with any right D-module N on the algebraic variety L_{n}^{0}(X): then i_{•}p_{n}^{•}N is an object of D_{L(X)} and we can form its de Rham complex.

(5.4.2) Example. Let X = 𝔸^{1}. Then the complex of global sections of CDR_{X}, i.e., the complex DR(i_{•}p^{•}ω_{𝔸^{1}}) can be found explicitly as follows.

Let V be the topological k-vector space k((t)) and V^{⋆} be its topological dual (over k). Denote by = ⟨l, v⟩ the canonical pairing of l ∈ V^{⋆} and v ∈ V.

Then V^{⋆} can be identified with k((t))dt, the space of 1-forms, the pairing between V and V^{⋆} being (f, ω) → res(f · ω). Let D be the Heisenberg algebra generated by V^{⋆} and V with [l, v] = ⟨lv⟩ and C be the Clifford algebra generated by V^{⋆}, V with [l, v]_{+} = ⟨l, v⟩. Denote CD = C ⊗_{k} D. Explicitly, CD = lim_{⟶}_{M≥0}lim_{⟵}_{N≥0}CD_{M}^{N},

where CD_{M}^{N} is the algebra generated by symbols a_{n}, b_{n} for −N ≤ n ≤ M and a_{n}^{∗}, b_{n}^{∗} for −M ≤ n ≤ N, subject to the relations

[a_{m}, a_{n}] = [a_{m}^{∗}, a_{n}^{∗}] = 0, [a_{m}^{∗}, a_{n}] = δ_{m,−n}
[b_{m}, b_{n}]_{+} = [b_{m}^{∗}, b_{n}^{∗}]_{+} = 0, [b_{m}^{∗}, b_{n}]_{+} = δ_{m,−n},
[a_{m}, b_{n}] = [a_{m}^{∗}, b_{n}] = [a_{m}, b_{n}^{∗}] = [a_{m}^{∗}, b_{n}^{∗}] = 0.

As M, N vary, these algebras obviously form an ind-pro-system. Let I ⊂ CD be the right ideal generated by (possibly infinite) linear combinations of a_{n}, b_{n}, a_{n+1}^{∗}, b_{n+1}^{∗}, n < 0.

(5.4.3) Proposition. The de Rham complex DR(i_{•}p^{•}ω_{𝔸^{1}}) is identified with the (irreducible) CD-module V ac := CD/I.

Remark. Although this description is similar to (5.1.4), there is a difference: here the ideal is generated by b_{n+1}^{∗}, b_{n} for n ≥ 0 while in (5.1.4) all the b_{n} are in the ideal. This is because in our present situation we are dealing with a semiinfinite de Rham complex obtained as an inductive limit of usual de Rham complexes with respect to maps shifting the degrees.

Proof: To simplify, we write L = L(𝔸^{1}), etc. We have

L_{M}^{ε} = Spec (k[a_{l}; −N_{ε} ≤ l ≤ M]/(a_{l}^{1+ε_{l}})),

where, for each ε ∈ 𝔼, we set N_{ε} = max {l; ε_{−l} ≠ 0}. Hence

L_{M} = lim_{⟶}_{N}Spf (k[a_{l}; 0 ≤ l ≤ M][[a_{l}; −N ≤ l < 0]]).

For any M ∈ ℕ, N ∈ ℕ ∪ {∞}, we put

Y_{M}^{N} = Spec (k[a_{l}; −N ≤ l ≤ M]).

Then L_{M} is just the limit, over N > 0, of the formal completions of Y_{M}^{N} along Y_{M}^{0}. Since the de Rham complex with coefficients in the D-module of distributions along a subvariety depends only on the completion along this subvariety, we can write

(5.4.4)  DR(i_{•}p^{•}ω_{𝔸^{1}}) = lim_{⟶}_{M,N}DR(i_{M,N,•}ω_{Y_{M}^{0}} )[M],

where i_{M,N} : Y_{M}^{0} ↪ Y_{M}^{N} is the embedding. Note that i_{M,N} is just the embedding of an affine subspace, so we are in the situation of Example 5.1.4.

Let D_{M}^{N} be the subalgebra in CD_{M}^{N} generated by the allowable a_{n}, a_{n}^{∗} only. It is identified with the algebra of polynomial differential operators on functions of a_{−N}, ..., a_{M}, with a_{−i}^{∗} corresponding to ∂/∂a_{i}. We see therefore that the (M, N)th term of the inductive system in (5.4.4) is identified with

(5.4.5)  V ac_{M}^{N} := CD_{M}^{N}/(a_{i}, i < 0, a_{i}^{∗}, i ≤ 0, b_{i}^{∗})CD_{M}^{N}.

Denote by 1_{M}^{N} the generator of this module. Then for N ≤ N′ and any M the embedding takes 1_{M}^{N} to 1_{M}^{N′} while for M ≤ M′ and any N it takes 1_{M}^{N} into 1_{M′}^{N}b_{M+1}...b_{M′}. This means that the inductive limit of algebras CD = lim_{⟶}_{M} lim_{⟵}_{N}CD_{M}^{N}

will act on the inductive limit of vector spaces (5.4.5) and the image of 1_{0}^{0} will be annihilated by a_{n+1}^{∗}, b_{n+1}^{∗}, a_{n}, b_{n}, n < 0. It is equally clear (from comparison of the bases) that the limit is in fact an irreducible CD-module generated by the above vector. ⊓ ⊔

6. Identification of the chiral de Rham complex

In this section we construct, in a geometric way, the structure of a vertex algebra on the chiral de Rham complex CDR_{X} and compare it with the construction of [MSV].

(6.1) Factorization algebras, and De Rham complexes on L(X)_{C^{I}}. Let C be a smooth curve, as before. For any non-empty finite set I we set U^{(I)} = U^{(I/I)}, Δ^{(I)} = Δ^{(I/{1})} and j^{(I)} = j^{(I/I)} (see §3.2). Hereafter we write U, Δ, j instead of U^{(I)}, Δ^{(I)}, j^{(I)} if I has cardinal 2. We will need the following notation : for any (possibly empty) I let Ī = I ⊔ ♡ be the corresponding pointed set. Any surjection Ī ↠ J̄ maps ♡ to ♡. Let us recall the definition of a factorization algebra, see [BD1, §3.4].

(6.1.1) Definition. (a) Let E be a quasi-coherent sheaf on C. A structure of a factorization algebra on E is a collection of quasi-coherent O_{C^{I}} -modules E_{I} for each non-empty finite set I, such that E_{I} is flat along the diagonal strata, E_{{1}} = E, and

– an isomorphism of O_{C^{I}} -modules Δ^{(J/I)∗}E_{J} →^{∼} E_{I} for every J ↠ I, compatible with the compositions of J ↠ I,

– an isomorphism of O_{U^{(J/I)}}-modules

κ^{(J/I)} : j^{(J/I)∗}(⊠_{I}E_{J_{i}}) →^{∼} j^{(J/I)∗}E_{J}

for every J ↠ I, compatible with the compositions of J ↠ I,

– a global section 1_{E} ∈ H^{0}(C, E) such that for every f ∈ E one has 1_{E} ⊠ f ∈ E_{{1,2}} ⊂ j_{∗}j^{∗}(E ⊠ E) and Δ^{∗}(1_{E} ⊠ f) = f.

(b) A module over E is a quasi-coherent sheaf M on C with a collection of quasicoherent O_{CĪ}-modules MĪ for each non-empty finite set I, such that MĪ is flat along the diagonal strata, M_{{♡}} = M, and

– an isomorphism of O_{CĪ}-modules Δ^{(J̄/Ī)∗}MJ̄ →^{∼} MĪ for every J̄ ↠ Ī, compatible with the compositions of J̄ ↠ Ī,

– an isomorphism of O_{U^{(J̄/Ī)}}-modules

κ^{(J̄/Ī)} : j^{(J̄/Ī)∗}((⊠_{I}E_{J_{i}}) ⊠ MJ̄_{♡}) →^{∼} j^{(J̄/Ī)∗}MJ̄

for every J̄ ↠ Ī, compatible with the compositions of J̄ ↠ Ī, and such that – for any f ∈ M one has 1_{E} ⊠ f ∈ M_{{1,♡}} ⊂ j_{∗}j^{∗}(E ⊠ M) and Δ^{∗}(1_{E} ⊠ f) = f.

We have the following immediate global counterpart of (5.4).

(6.1.2) Proposition. For any right D_{X}-module M there is a unique complex of X × C^{I}-sheaves DR(M)_{C^{I}} such that:

(a) the fiber of DR(M)_{C} at a point of the curve C is isomorphic to the complex DR(M),

(b) the collection (DR(ω_{X})_{C^{I}} ) is a factorization algebra on the curve C. The collection (DR(M)_{C^{I}} ) is a (DR(ω_{X})_{C^{I}} )-module.

Proof: A choice of an element i_{0} ∈ I defines a morphism of schemes p_{i_{0}} : L^{0}(X)_{C^{I}} → X as follows. Recall that Lc^{0}(X)_{C^{I}} represents the functor λ_{X,C^{I}}^{0} which takes a scheme S into the set of pairs (f_{I}, ρ) where f_{I} : S → C^{I} is a morphism of schemes and ρ is a morphism of { \atop Γ(f_{I})}, the formal neighborhood of Γ(f_{I}) ⊂ S × C, into X. Now, restricting ρ onto the graph of f_{i_{0}}, which is a subscheme in { \atop Γ(f_{I})}, isomorphic to S, we get a natural transformation from λ_{X,C^{I}}^{0} into the functor represented by X, so a morphism p_{i_{0}}.

Denote by i : L^{0}(X)_{C^{I}} → L(X)_{C^{I}} the embedding. For any M ∈ D_{X}, we form the object i_{•}p_{♡}^{•}M ∈ D_{L(X)_{CĪ}}. The general construction of (5.3), applied to the restrictions of i_{•}p_{♡}^{•}M onto open subsets in X ×CĪ, gives then a complex of sheaves on X × CĪ which we denote DR(M)_{CĪ}.

Notice that in the particular case where M = ω_{X} the object p_{i_{0}}^{•}M of the category D_{L^{0}(X)_{C^{I}}} is independent (up to a unique isomorphism) of the choice of i_{0} ∈ I. Indeed, objects of the latter category are, by definition, pairs (n, N) where N is a right D-module on L_{n}^{0}(X)_{C^{I}} and two such pairs (n, N) and (n′, N′) are isomorphic, if the pullbacks of N and N′ to L_{m}^{0}(X)_{C^{I}}, m ≥ n, n′ are isomorphic as right Dmodules. Since the pullback for right D-modules is just the O-module pullback tensored with the relative canonical class, p_{i_{0}}^{•}ω_{X} is represented by (n, ω_{L_{n}^{0}(X)_{C^{I}}}) for any n, and thus is clearly independent on i_{0}.

Then, the general construction in (5.3) gives a complex of sheaves on X × C^{I}, denoted by DR(ω_{X})_{C^{I}}.

To prove Claim (a) it is sufficient to observe that the fiber of L(X)_{C} at a point 0 ∈ C is isomorphic to L(X). Recall that L(X)_{C} represents the contravariant functor λ_{X,C} : Sch → Sets such that λ_{X,C}(S) is the set of pairs (f, ρ) such that

f ∈ Hom_{Sch}(S, C) and ρ ∈ Hom_{Lrs}((Γ(f), K_{f}^{√} ), X).

Thus the fiber at 0 represents the subfunctor

S ↦ {(f, ρ) ∈ λ_{X,C}(S) | f(S)_{red} = {0}}.

Let t be a local coordinate on C centered at 0. For any f as above we have (Γ(f), K_{f}^{√} ) = (S, O_{S}((t))^{√} ). Note that the isomorphism of L(X) and the fiber of L(X)_{C} at 0 is compatible with the ind-pro-systems in (2.7), (3.9).

Claim (b) follows from the fact that L^{0}(X)_{C} and L(X)_{C} form factorization algebras, that the inclusion of the former into the latter is a morphism of factorization algebras and that passing to the de Rham complex takes Cartesian products of (ind-)schemes to tensor products of vector spaces. ⊓ ⊔

(6.2) Reminder on chiral and vertex algebras. Let us recall the basic facts on chiral and vertex algebras. See [BD1, §3], [K] for more details. Let C be a smooth curve, as before. For any right D_{C}-module M let

ε_{M} : j_{•}j^{•}(ω_{C} ⊠ M) → Δ_{•}M

be the composition of the projection j_{•}j^{•}(ω_{C} ⊠ M) → j_{•}j^{•}(ω_{C} ⊠ M)/(ω_{C} ⊠ M) and of the isomorphism j_{•}j^{•}(ω_{C} ⊠ M)/(ω_{C} ⊠ M) →^{∼} Δ_{•}Δ^{•}(ω_{C} ⊠ M) →^{∼} Δ_{•}M.

(6.2.1) Definition. (a) A chiral algebra over C is a right D_{C}-module A with two maps μ_{A} ∈ Hom_{D_{C^{2}}} (j_{•}j^{•}(A ⊠ A), Δ_{•}A) and 1_{A} ∈ Hom_{D_{C}} (ω_{C}, A) such that

– the map μ_{A}(1_{A}, id_{A}) coincides with ε_{A},

– the map μ_{A} is antisymmetric, and it satisfies the Jacobi identity.

(b) A module over a chiral algebra A over C is a right D_{C}-module M_{C} with a map μ_{M} ∈ Hom_{D_{C^{2}}} (j_{•}j^{•}(A ⊠ M), Δ_{•}M) such that

– the map μ_{M}(1_{A}, id_{M}) coincides with ε_{M},

– the map μ_{M} is compatible with μ_{A}.

For any factorization algebra on E, each sheaf E_{I} has a canonical left D_{C^{I}} -module structure, compatible with the factorization structure, such that the section 1_{E} is a horizontal (see [BD1, Proposition 3.4.8]). It is proved in [BD1, §3.4.9] that the right D_{C}-module A = E ⊗_{O_{C}} ω_{C} is a chiral algebra over C. The map μ_{A} is the composition of the chain of maps

j_{•}j^{•}(A ⊠ A) = E_{{1,2}} ⊗_{O_{C^{2}}} j_{•}j^{•}ω_{C^{2}} → E_{{1,2}} ⊗_{O_{C^{2}}} Δ_{•}ω_{C} = Δ_{•}A.

Here the 1-st equality is the 2-nd isomorphism in 6.1.1.(a), the second arrow is ε_{ω_{C}}, and the last equality results from the 1-st isomorphism in 6.1.1.(a) and the projection formula for Δ.

(6.2.2) Definition. (a) A vertex algebra is a k-supervector space V with an even vector 1 ∈ V, an even endomorphism ∂ ∈ End (V), and an even linear map V → End (V)[[z, z^{−1}]], a ↦ a(z) = ∑_{n} a_{n}z^{−n−1}. These data satisfy the following axioms:

– ∂(1_{V}) = 0, 1_{V} (z) = id_{V}, a_{n}(1_{V}) = 0 if n ≥ 0, a_{−1}(1_{V}) = a,

– [∂, a(z)] = ∂_{z}a(z),

– we have (z − w)^{N}[a(z), b(w)] = 0 for N >> 0.

(b) A module over a vertex algebra V is a k-supervector space W with an even endomorphism ∂ ∈ End (W), and an event linear map V → End (W)[[z, z^{−1}]], a ↦ a^{W} (z) = ∑_{n} a_{n}^{W} z^{−n−1}. These data satisfy the following axioms :

– 1_{V}^{W} (z) = id_{W}, L_{−1}^{W} = ∂, a_{n}^{W} (1_{W}) = 0 if n ≥ 0, – [∂, a^{W} (z)] = ∂_{z}a^{W} (z),

– we have (z − w)^{N}[a^{W} (z), b^{W} (w)] = 0 for N >> 0.

Let t be a formal coordinate on C centered at the point 0. Let t_{1} = t ⊗ 1, t_{2} = 1 ⊗ t be the corresponding coordinates on C^{2}. Let A_{0} be the fiber at the point 0 ∈ C of a chiral algebra A. Let μ_{0} be the fiber at 0 of the map Δ^{•}(μ_{A}) : Δ^{•}j_{•}j^{•}(A⊠A) → Δ^{•}Δ_{•}(A) → A. We set

(6.2.3)  a(z)(b) = ∑_{n∈ℤ} μ_{0}((a ⊗ b)(t_{1} − t_{2})^{−n})z^{−n−1},  ∀a, b ∈ A_{0}.

Let A^{∧} be the space of section of A on the punctured formal neighborhood of C at the point 0. There is a canonical identification

A_{0} ≃ A^{∧}/A^{∧} · ∂_{t}k((t)).

The operator ∂ on A_{0} is

(6.2.4)  ∂(a) = −a · ∂_{t},  ∀a ∈ A_{0}.

The following basic fact is due to Beilinson-Drinfeld (see [HL], [B] for a proof).

(6.2.5) Lemma. The triple formed by A_{0} and the maps (6.2.3), (6.2.4) is a vertex algebra.

Thus, Proposition 6.1.2 gives the following.

(6.2.6) Theorem. The De Rham complex CDR(ω_{X}) is a sheaf of vertex algebras on X. For any right D_{X}-module M the De Rham complex CDR(M) is a sheaf of DR(ω_{X})-modules.

(6.3) The vacuum module and the chiral de Rham complex. Here we recall the original construction of the chiral de Rham complex of X as given in [MSV]. One first considers the case X = 𝔸^{d}. Similarly to Example 5.4.2, let CD_{M}^{N} be the ℤ/2ℤ-graded k-algebra generated by even elements a_{in}, a_{in}^{∗} and odd elements b_{in}, b_{in}^{∗}, with i = 1, ..., d and −N ≤ n ≤ M for a_{in}, b_{in} and −M ≤ n ≤ N for a_{in}^{∗}, b_{in}^{∗}, modulo the relations

\begin{matrix}a_{im}^{∗}a_{jn} − a_{jn}a_{im}^{∗} = δ_{ij}δ_{m,−n}, & a_{im}a_{jn} − a_{jn}a_{im} = a_{im}^{∗}a_{jn}^{∗} − a_{jn}^{∗}a_{im}^{∗} = 0, \\ b_{im}^{∗}b_{jn} + b_{jn}b_{im}^{∗} = δ_{ij}δ_{m,−n}, & b_{im}b_{jn} + b_{jn}b_{im} = b_{im}^{∗}b_{jn}^{∗} + b_{jn}^{∗}b_{im}^{∗} = 0. \end{matrix}

We further require that the letters a and b commute in all cases. Let CD = lim_{⟶}_{M} lim_{⟵}_{N}CD_{M}^{N}. Let CD^{+} ⊂ CD be the right ideal generated by (possible infi-

nite) linear combinations of a_{i,n+1}^{∗}, b_{i,n+1}^{∗}, a_{in}, b_{in} with n < 0. The vacuum module V ac = CD/CD^{+} has a structure of a vertex algebra such that the generating series associated to the 0- and (-1)-modes of a_{i}, b_{i} are given by:

a_{i,−1}^{∗}(z) = ∑_{n∈ℤ} a_{in}^{∗}z^{−n−1}, a_{i0}(z) = ∑_{n∈ℤ} a_{in}z^{−n},

and similarly with b_{i,−1}^{∗}(z), b_{i0}(z). The generating series associated to other modes are obtained by differentiation, using the action of ∂ given by

∂(a_{in}^{∗}) = −na_{i,n−1}^{∗}, ∂(a_{in}) = −(n − 1)a_{i,n−1},

and idem for b_{in}^{∗}, b_{in}. The map

δ = ∑_{i,n}a_{in}^{∗}b_{i,−n} : V ac → V ac

is a derivation of vertex algebras with zero square.

Setting x_{i} = a_{i0} makes V ac into a module over k[x_{1}, ..., x_{n}], the coordinate ring of 𝔸^{d}. We denote this ring shortly by k[x]. In [MSV] the authors consider the quasicoherent sheaf Ω_{𝔸^{d}}^{ch} := V ac ⊗_{k[x]} O_{𝔸^{d}} corresponding to k[x]-module V ac and extend the vertex algebra structure to it. One also has a vertex algebra structure on

V ac^{∧} := V ac ⊗_{k[x_{i}]} k[[x_{i}]].

Now let X be a smooth algebraic variety, U ⊂ X be an open subset and ϕ : U → 𝔸^{d} be an etale map. Let 0 ∈ X be a point such that ϕ(0) = 0 and let X^{∧} be the formal neighborhood of 0 ion X. Then x′_{i} = ϕ^{∗}x_{i} are the coordinates on X^{∧}.

In [MSV] the authors construct a sheaf Ω_{X}^{ch} of differential vertex algebras on X as the unique such sheaf satisfying the following condition. For any ϕ as above, there is an isomorphism of vertex algebras

ϕ_{ch} : Ω_{X}^{ch} ⊗_{O_{X}} O_{X^{∧}} → ϕ^{∗}Ω_{𝔸^{d}}^{ch} ⊗_{O_{𝔸^{d}}} O_{𝔸^{d∧}} = V ac^{∧}

which coincides, for U ⊂ 𝔸^{d}, with the automorphism of V ac^{∧} introduced in [MSV, Theorem 3.7]. Our aim in the rest of this paper is to prove the following fact.

(6.3.1) Theorem. There is an isomorphism of sheaves of differential vertex algebras Ω_{X}^{ch} ≃ DR(ω_{X}).

(6.4) The factorization algebra associated to the vacuum module. As the first step in proving Theorem 6.3.1, let us describe the factorization algebra corresponding to V ac = Γ(𝔸^{d}, Ω_{𝔸^{d}}^{ch}). In this, we follow [BD1] [G]: the constructions below are a particular instance of the general concept of the chiral enveloping algebra of a Lie*-algebra. For the convenience of the reader we give a self-contained presentation.

Fix a smooth projective curve C. Recall that ω_{C} is the sheaf of 1-forms on C (in the Zariski topology). Let I be a finite set. Consider the product C^{I} × C and its projections p, q to C^{I} and C. Let us specialize the notation of (3.3) to the case when S = C^{I} and f_{I} = Id : C^{I} → C^{I}. We denote the subvariety Γ(f_{I}) simply by

Γ_{I} = {((c_{i}), x) ∈ C^{I} × C|x ∈ {c_{i}}}.

Similarly we write O_{I}^{∧}, K_{I}^{∧} for O_{f_{I}}^{∧}, K_{f_{I}}^{∧}. Set

[[O]]_{I} = p_{∗}O_{I}^{∧}, ((O))_{I} = p_{∗}K_{I}^{∧},
[[ω]]_{I} = p_{∗}(O_{I}^{∧} ⊗ q^{∗}ω_{C}), ((ω))_{I} = p_{∗}(K_{I}^{∧} ⊗ q^{∗}ω_{C}).

These are (non-quasicoherent) sheaves of O-modules on C^{I}. Informally, the “fiber” of, say, ((ω))_{I} at a point (c_{i}) ∈ C^{I} is the space of sections of ω_{C} on the punctured formal neighborhood of the set {c_{i}}, and similarly in the other cases. Note that the sum of residues defines a morphism

res : ((ω))_{I} → O_{C^{I}} ,

trivial on [[ω]]_{I}.

Consider the super-vector space k^{d|d} with basis consisting of even vectors v_{1}, ..., v_{d} and odd vectors v_{d+1}, ..., v_{2d}. The sheaf

𝔥_{I}^{∧} = ((((O))_{I} ⊕ ((ω))_{I}) ⊗ k^{d|d}) ⊕ O_{C^{I}} · γ

is then a Lie superalgebra in the category of left D_{C^{I}} -modules, with respect to the bracket

\begin{matrix}[O_{C}^{d|d}, O_{C}^{d|d}] = [ω_{C}^{d|d}, ω_{C}^{d|d}] = 0, \\ [v_{i} ⊗ f, v_{j} ⊗ ω] = δ_{ij}res(fω) · γ, ∀f ∈ O_{C}, ∀ω ∈ ω_{C}, \end{matrix}

and with γ being a central element. Similarly, let 𝔥_{I}^{∧,+} ⊂ 𝔥_{I}^{∧} be the super-Lie subalgebra ([[O]]_{I} ⊕ [[ω]]_{I}) ⊗ k^{d|d}. For any surjective map J ↠ I there are obvious isomorphisms

Δ^{(J/I)∗}𝔥_{J}^{∧} →^{∼} 𝔥_{I}^{∧}, j^{(J/I)∗}(⊠_{I}𝔥_{J_{i}}^{∧}) →^{∼} j^{(J/I)∗}𝔥_{J}^{∧}.

Let U_{C^{I}}^{∧} be the quotient of the associative enveloping algebra of 𝔥_{I}^{∧}, in the category of left D_{C^{I}} -modules, by the right ideal generated by γ − 1. Consider the sheaf

Vac_{C^{I}} = U_{C^{I}}^{∧} /U_{C^{I}}^{∧,+},

where U_{C^{I}}^{∧,+} ⊂ U_{C^{I}}^{∧} is the right ideal generated by 𝔥_{I}^{∧,+}. The collection (Vac_{C^{I}}) is clearly a factorization algebra. We denote Vac = Vac_{C} ⊗ ω_{C} the chiral algebra associated to the factorization algebra (Vac_{C^{I}}).

(6.4.1) Lemma. V ac is isomorphic, as a vertex algebra, to the fiber of the chiral algebra Vac at any point of C.

Proof: Fix a point 0 ∈ C. Let t be a local coordinate on C centered at 0. Let Vac_{0} be the fiber of Vac at 0. We first identify Vac_{0} with V ac as a vector space and then compare the vertex algebra structures.

The space (k((t)) ⊕ k((t))dt) ⊗ k^{d|d} is identified with the linear span of a_{im}, a_{im}^{∗}, b_{im}, b_{im}^{∗}, where m ∈ ℤ, in such a way that

v_{i}t^{m} ↦ a_{im}^{∗}, v_{i+d}t^{m} ↦ b_{im}^{∗}, v_{i}t^{m−1}dt ↦ a_{im}, v_{i+d}t^{m−1}dt ↦ b_{im}.

This isomorphism gives an isomorphism of k-vector spaces Vac_{0} ≃ V ac.

Next, the operator ∂ on Vac_{0} is the image of ∂^{U} modulo U_{0}^{∧,+}. The vertex algebras Vac_{0} and V ac are completely determined by the operator ∂ and the formal series a_{i,−1}^{∗}(z), b_{i,−1}^{∗}(z), a_{i0}(z), b_{i0}(z). The operators ∂ coincide obviously.

The vertex algebra structure on Vac_{0} is the fiber at the point (0, 0) of the composition of the chain of maps

μ : j_{•}j^{•}(Vac ⊠ Vac) →^{∼} Vac_{C^{2}} ⊗_{O_{C^{2}}} j_{•}j^{•}ω_{C^{2}} ⟶^{ε_{ω_{C}}}Vac_{C^{2}} ⊗_{O_{C^{2}}} Δ_{•}ω_{C} →^{∼} Δ_{•}Vac.

Let μ_{0} : Vac_{0} ⊗ Vac_{0} ⊗ k((t_{1} − t_{2})) → Vac_{0} be this fiber. There is a pair of compatible linear isomorphisms

𝔥_{{1},0}^{∧} ≃ (k((t)) ⊕ k((t))dt)^{d|d} ⊕ k · γ, 𝔥_{{1},0}^{∧,+} ≃ (k[[t]] ⊕ k[[t]]dt)^{d|d}.

Fix elements a(t), b(t) ∈ k[t, t^{−1}]^{d|d}. Recall that t_{1}, t_{2} are the coordinates t⊗1, 1⊗t on C^{2}. Let a(t − t_{1}) ⊗ b(t − t_{2}) be the corresponding element in the stalk of the sheaf Vac_{C} ⊠Vac_{C} at the point (0, 0) ∈ C^{2}. For any f(t) ∈ k((t)), the factorization

morphism j_{•}j^{•}(Vac_{C} ⊠ Vac_{C}) →^{∼} j_{•}j^{•}(Vac_{C^{2}}) maps

a(t − t_{1}) ⊗ b(t − t_{2}) ⊗ f(t_{1} − t_{2}) to a(t − t_{1}) · b(t − t_{2}) ⊗ f(t_{1} − t_{2}),

where · is the product in the algebra U_{C^{2}}^{∧}. Composing it with the map ε_{ω_{C}} in (6.2) we get

μ(a(t − t_{1})dt_{1} ⊗ b(t − t_{2})dt_{2} ⊗ f(t_{1} − t_{2})) =

= ∑_{n} a_{n}(t − t_{2}) · b(t − t_{2}) ⊗ res_{t_{1}=t_{2}}((t_{1} − t_{2})^{n}f(t_{1} − t_{2})dt_{1}dt_{2}),

where a(t_{2} − t_{1}) = ∑_{n} a_{n}(t_{2})t_{1}^{n} is the expansion in k[t_{2}, t_{2}^{−1}][[t_{1}]]^{d|d}. In the particular case where a(t) = at^{−1} with a ∈ k^{d|d}, and f(t) = t^{−n} with n ≥ 0, we have

μ_{0}(at^{−1}dt_{1} ⊗ b(t)dt_{2} ⊗ (t_{1} − t_{2})^{−n}) = (at^{−n}) · b(t) ⊗ dt.

In particular, this formula gives

μ_{0}((a_{i,−1}^{∗} ⊗ b)(t_{1} − t_{2})^{−n}) = a_{in}^{∗}(b), μ_{0}((b_{i,−1}^{∗} ⊗ b)(t_{1} − t_{2})^{−n}) = b_{in}^{∗}(b).

A similar computation gives

μ_{0}((a_{i0} ⊗ b)(t_{1} − t_{2})^{−n}) = a_{i,n+1}(b), μ_{0}((b_{i0} ⊗ b)(t_{1} − t_{2})^{−n}) = b_{i,n+1}(b).

We are done. ⊓ ⊔

(6.4.2) Corollary. Theorem 6.3.1 is true for X = 𝔸^{d}.

Proof: The same argument as in Proposition 5.4.3 identifies the space of global sections of CDR_{𝔸^{d}} with V ac, and the latter is identified with Γ(𝔸^{d}, Ω_{𝔸^{d}}^{ch}).

(6.5) The action of étale morphisms I. To prove Theorem 6.3.1 in full generality, it suffices to establish the following lemma.

(6.5.1) Lemma. (a) For any affine open set U ⊂ X and an étale map ϕ : U → 𝔸^{d}, there is an isomorphism of differential vertex algebras F_{ϕ} : ϕ^{∗}Ω_{𝔸^{d,∧}}^{ch} →^{∼} k[X^{∧}]⊗_{O_{X}} DR(ω_{X}).

(b) If U ⊂ 𝔸^{d}, then F_{ϕ}^{−1} ∘ F_{Id} = ϕ_{ch} is the isomorphism constructed in [MSV].

The plan of the proof is as follows. We will construct an isomorpism (F_{I}) of factorization algebras and obtain F_{ϕ} as the fiber of F_{{1}} at a point of C. It is enough to assume that C = 𝔸^{1}. Set

CD_{I}^{∧} = Γ(𝔸^{I}, U_{C^{I}}^{∧}), CD_{I}^{∧,+} = Γ(𝔸^{I}, U_{C^{I}}^{∧,+}),
DR(ω_{U})_{I} = Γ(L(U)_{𝔸^{I}} , DR(ω_{U})_{𝔸^{I}} ), V ac_{I} = Γ(𝔸^{I}, Vac_{C^{I}}).

Thus, V ac_{I} = CD_{I}^{∧}/CD_{I}^{∧,+}. We first construct a right action

(6.5.2)  DR(ω_{U})_{I} ⊗ CD_{I}^{∧} → DR(ω_{U})_{I}

commuting with the factorization maps. To prove that the factorization algebras (Vac_{𝔸^{I}}), (DR(ω_{U})_{𝔸^{I}}) are isomorphic, it is then sufficient to check that the right CD_{I}^{∧}-module DR(ω_{U})_{I} has a cyclic vector whose annihilator is CD_{I}^{∧,+}. Observe that, since the map (6.5.2) depends on ϕ, the resulting isomorphism of sheaves of vertex algebras ϕ^{∗}Ω_{𝔸^{d,∧}}^{ch} →^{∼} k[X^{∧}] ⊗_{O_{X}} DR(ω_{X}) will also depend on ϕ.

(6.5.3) The case U = 𝔸^{1}. First we consider the particular case where U = 𝔸^{d} and ϕ = Id. What we do in this case is to provide an explicit identification of the right CD_{I}^{∧}-module DR(ω_{𝔸^{d}})_{I}. To simplify we set L = L(𝔸^{d})_{𝔸^{I}}, etc. We proceed as in Example 5.4.2. For any n, N ∈ ℕ ∪ {∞} we set

(6.5.4)  B = k[𝔸^{I}], A_{n}^{N} = B[a_{lν}^{(j)}; −N ≤ l ≤ n], Y_{n}^{N} = Spec A_{n}^{N},

where a_{lν}^{(j)} are independent variables such that j ∈ [1, d], l ∈ ℤ, ν ∈ [0, |I| − 1]. Then

L_{n} = lim_{⟶}_{N}Spf (A_{n}^{0}[[a_{lν}^{(j)}; −N ≤ l < 0]]), L_{n}^{ε} = Spec (A_{n}^{N_{ε}}/(a_{lν_{1}}^{(j)} ⋯ a_{lν_{1+ε_{l}}}^{(j)} ; l < 0)),

see (3.9.1), where we set N_{ε} = max {l; ε_{−l} ≠ 0} for each ε ∈ 𝔼. Note that

CD_{I}^{∧} = lim_{⟶}_{n}lim_{⟵}_{N}CD_{Y_{n}^{N}} .

To simplify again we set Y_{n}^{ε} = Y_{n}^{N_{ε}}. Thus Y_{N}^{ε} is an affine space of finite dimension. There are closed embeddings L_{n}^{0} ⊂ L_{n}^{ε} ⊂ Y_{n}^{ε}. Let i_{nε} : L_{n}^{0} ↪ Y_{n}^{ε} is the composite embedding. We write ω_{L_{n}^{0},Y_{n}^{ε}} for the right D-module i_{nε•}ω_{L_{n}^{0}} on Y_{n}^{ε}.

Consider the right D_{𝔸^{d}}-module M = ω_{𝔸^{d}}. Let M_{n}^{ε} ∈ D_{L_{n}^{ε}} be the subsheaf of ω_{L_{n}^{0},Y_{n}^{ε}} consisting of the sections supported (scheme-theoretically) on L_{n}^{ε}. Similarly, let DR(M_{n}^{ε}) ∈ O_{L_{n}^{ε}} be the subsheaf of DR(ω_{L_{n}^{0},Y_{n}^{ε}}) consisting of the sections supported on L_{n}^{ε}. Let DR(ω_{n})_{I} be the space of global sections of DR(M_{n}). By definition,

lim_{⟶}_{ε}Γ(L_{n}^{ε}, DR(M_{n}^{ε})) = DR_{L_{n}^{0},Y_{n}^{∞}}.

Let us denote this space by DR(ω_{n})_{I}.

We get a right lim_{⟵}_{N}CD_{Y_{n}^{N}} -action on DR(ω_{n})_{I} such that DR(ω_{n})_{I} is the quotient

of lim_{⟵}_{N}CD_{Y_{n}^{N}} by lim_{⟵}_{N}CD_{Y_{n}^{N}}^{+} . Since DR(ω_{𝔸^{d}})_{I} = lim_{⟶}_{n}DR(ω_{n})_{I}[nd], we get a right

action of CD_{I}^{∧} = lim_{⟶}_{n}lim_{⟵}_{N}CD_{Y_{n}^{N}} on DR(ω_{𝔸^{d}})_{I} such that

DR(ω_{𝔸^{d}})_{I} ≃ CD_{I}^{∧}/CD_{I}^{∧,+} ≃ V ac_{I},

thus achieving our goal in the case X = 𝔸^{d}.

(6.6) Etale change of coordinates in Clifford algebras. In order to prove Lemma 6.5.1 for general ϕ : U → 𝔸^{d} we need some elementary observations about Clifford algebras.

If Y is a smooth algebraic variety, we denote by CD_{Y} the sheaf of differential operators in Ω_{Y}^{•}, the commutative superalgebra of differential forms.

In particular, if Y = 𝔸^{l} with coordinates y_{1}, ..., y_{l}, then y_{i}, dy_{i} are free generators of (the algebra of global sections of) Ω_{Y}^{•} and we denote ∂_{i} = ∂/∂y_{i} and ξ_{i} = “∂/∂ dy_{i}” the corresponding derivations, which are thus global sections of CD_{𝔸^{l}}.

Let now V be an affine open subset of a smooth variety Y, and let ψ : V → 𝔸^{l} be an étale map. Fix a point 0 ∈ U such that ϕ(0) = 0. Let y′_{i} = ϕ^{∗}y_{i} be the coordinate on the formal neighborhood Y^{∧} of 0 in Y. There are then uniquely determined derivations ∂′_{i}, ξ′_{i} of Ω_{Y^{∧}} such that

[∂′_{i}, y′_{j}] = [ξ′_{i}, dy′_{j}] = δ_{ij}, [∂′_{i}, dy′_{j}] = [ξ′_{i}, y′_{j}] = 0.

Set k[[y]] = k[[y_{1}, y_{2}, ..., y_{l}]], and CD_{Y^{∧}} = k[Y^{∧}] ⊗_{O_{Y}} CD_{Y}. The étale map ψ gives an isomorphism of formal schemes Y^{∧} →^{∼} 𝔸^{l,∧}. Set ψ′ = ψ^{−1}.

(6.6.1) Lemma. For any ψ as above, there is a unique k[Y^{∧}]-algebra isomorphism

ψ_{♯} : CD_{Y^{∧}} →^{∼} ψ^{∗}CD_{𝔸^{l,∧}}

such that

ψ_{♯}(dy′_{i}) = ∑_{j} ∂_{j}ϕ_{i}(y)dy_{j}, ϕ_{♯}(ξ′_{i}) = ∑_{j} ∂′_{i}ϕ′_{j}(ϕ(y))ξ_{j},

ϕ_{♯}(∂′_{i}) = ∑_{j} ∂′_{i}ϕ′_{j}(ϕ(y))∂_{j} + ∑_{j,k,l} ∂′_{i}∂′_{k}ϕ′_{j}(ϕ(y))∂_{l}ϕ_{k}(y)dy_{k}ξ_{j},

where ϕ = (ϕ_{1}, ..., ϕ_{l}).

Proof: It suffices to observe that, since the map ψ is étale, CD_{Y^{∧}} is a free k[Y^{∧}]- module with basis

(∂′_{1})^{r_{1}}(∂′_{2})^{r_{2}} ⋯ (∂′_{l})^{r_{l}} ⊗ (ξ′_{1})^{m_{1}} ⋯ (ξ′_{d})^{m_{l}}(dy′_{1})^{n_{1}} ⋯ (dy′_{d})^{n_{l}},

where r_{i}, m_{i}, n_{i} ∈ ℕ. Then use the coordinates change formulas, see [L, chap. II] for instance. ⊓ ⊔

(6.7) The action of étale morphisms II. Let now ϕ : U → 𝔸^{d} be a general étale morphism. By Proposition 3.9.2 we have an isomorphism of schemes

L_{n}^{ε}(ϕ)_{𝔸^{I}} ≃ L_{n}^{ε} ×_{L_{0}^{0}} L_{0}^{0}(U)_{𝔸^{I}} .

There is an obvious map Y_{n}^{N} → L_{0}^{0}, since L_{0}^{0} = Y_{0}^{0} = Spec A_{0}^{0}, which restricts to the map L_{n}^{ε} → L_{0}^{0} when N = N_{ε}. For any N, consider the fiber product

Y_{nϕ}^{N} = Y_{n}^{N} ×_{L_{0}^{0}} L_{0}^{0}(U)_{𝔸^{I}} .

Denote by ϕ_{n}^{N} : Y_{nϕ}^{N} → Y_{n}^{N} the projection to the first factor. Being a base change of an étale morphism L_{0}^{0}(ϕ)_{Id} : L_{0}^{0}(U)_{𝔸^{I}} → L_{0}^{0}, see Proposition 3.9.2.(a), it is itself étale. Set 0 = {a_{0ν}^{(j)} = 0} ∈ Y_{0}^{0}(B). Fix 0 ∈ Y_{0ϕ}^{0}(B) mapping to 0 by ϕ_{n}^{N}. Let Y_{0}^{0,∧}, Y_{0ϕ}^{0,∧} be the formal neighborhoods of 0, and set

Y_{n}^{N,∧} = Y_{0}^{0,∧} ×_{Y_{0}^{0}} Y_{n}^{N}, Y_{ϕ}^{N,∧} = Y_{0ϕ}^{0,∧} ×_{Y_{0}^{0}} Y_{n}^{N}.

By Lemma 6.6.1 applied to ϕ_{n}^{N}, we have a ring isomorphism

ϕ_{n♯}^{N} : CD_{Y_{nϕ}^{N,∧}} →^{∼} ϕ_{n}^{N∗}CD_{Y_{n}^{N,∧}}.

Thus, by passing to the limits, we get a ring isomorphism

(6.7.1)  ϕ_{∞♯}^{∞} : CD_{I}^{∧} = lim_{⟶}_{n}lim_{⟵}_{N}CD_{Y_{n}^{N∧}} → lim_{⟶}_{n}lim_{⟵}_{N}CD_{Y_{nϕ}^{N∧}}.

Denote the algebra on the right by CD_{I,ϕ}^{∧}. Note that

DR(ω_{U})_{I} = lim_{⟶}_{n,N}DR(ω_{L_{n}^{0}(U)_{𝔸^{I}} ,Y_{nϕ}^{N}})[nd]

is a module over CD_{I,ϕ}^{∧}. We obtain, therefore, a structure of a CD_{I}^{∧}-module on DR(ω_{U})_{I}.

(6.7.2) Lemma. With respect to the above structure of a CD_{I}^{∧}-module, DR(ω_{U})_{I} is isomorphic to V ac_{I}.

Proof: Follows from the fact that each DR(ω_{L_{n}^{0}(U),Y_{nϕ}^{N}}) is isomorphic to the vacuum module over CD_{Y_{nϕ}^{N}}^{∧}.

Note that both DR(ω_{U})_{I} and V ac_{I} have distinguished generators. Namely 1′_{I} ∈ DR(ω_{U})_{I} is the image of the following tensor on L_{0}^{0}(U)_{𝔸^{I}} = U × 𝔸^{I}:

dx_{1}...dx_{d}dt_{1}...dt_{|I|} ⊗ \frac{∂}{∂t_{1}} ∧ ... ∧ \frac{∂}{∂t_{|I|}} ∈ ω_{U×𝔸^{I}} ⊗ ⋀^{|I|} Θ_{U×A^{I}} .

The generator 1′′_{I} ∈ V ac_{I} is the image of 1 ∈ CD_{I}^{∧}. We denote F_{I} : V ac_{I} → DR(ω_{U})_{I} the unique module isomorphism taking 1′′_{I} to 1′_{I}. Let F_{I} : Vac_{I} → DR(ω_{U})_{I} be the corresponding moprhism of quasicoherent sheaves on U × 𝔸^{I}.

(6.7.3) Lemma. The (F_{I}) commute with factorization maps and thus form an isomorphism of factorization algebras.

Proof: Follows from the fact that (1′_{I}) and (1′′_{I}) are compatible with factorization structures: in the notation of (6.1.1) we have κ^{J/I}(⊠1′_{J_{i}}) = 1′_{J} and similarly for 1′′_{I}.

Let F_{ϕ} be the fiber of F_{{1}} at the point 0 ∈ 𝔸^{1}. Lemma 6.7.3 implies that F_{ϕ} is a morphism of vertex algebras. This establishes part (a) of Lemma 6.5.1

We now prove Lemma 6.5.1(b). So we assume U ⊂ 𝔸^{d} and need to compare two automorphisms of the vertex algebra Γ(U, Vac), namely F_{ϕ}^{−1} ∘ F_{Id} and ϕ_{ch}. Notice that the vertex algebra V ac is strongly generated by the fields a_{i,−1}^{∗}(z), a_{i0}(z), b_{i,−1}^{∗}(z), b_{i0}(z), see [K], and Γ(U, Vac) is obtained by localization. Thus it is enough to compare the two automorpisms on the elements

(6.7.4)  a_{i,−1}^{∗}, a_{i0}, b_{i,−1}^{∗}, b_{i0} ∈ V ac.

To simplify we set Y = Y_{∞}^{∞} = lim_{⟶}_{N} lim_{⟵}_{n}Y_{n}^{N}, Y_{ϕ} = Y_{∞ϕ}^{∞} = lim_{⟶}_{N} lim_{⟵}_{n}Y_{nϕ}^{N}, ϕ = ϕ_{∞}^{∞}.

Note that the fiber at 0 ∈ 𝔸^{1} of the B-algebras CD_{Y}, CD_{Y^{∧}} are respectively

CD, CD^{∧} := CD ⊗_{k[a_{i0}]} k[[a_{i0}]].

Following our simplified notation, we denote the isomorphism (6.7.1) simply by ϕ_{♯}. Let ϕ_{♯0} be the fiber of ϕ_{♯} at 0 ∈ 𝔸^{1}. We have ϕ_{♯0}(CD^{+,∧}) ⊂ CD^{+,∧}. In particular, ϕ_{♯0} gives an automorphism of the CD^{∧}-module V ac^{∧} which is compatible with the vertex algebra structure. To prove Claim (b) it suffices therefore to check that

VERTEX ALGEBRAS AND FORMAL LOOPS 47

ϕ_{♯0} = ϕ_{ch} on elements (6.7.4). Recall that L_{0}^{0}(X)_{𝔸^{1}} = X × 𝔸^{1} for any X. We get a commutative diagram

Y_{nϕ}^{N} ↠ Y_{0ϕ}^{0} = U × 𝔸^{1}
ϕ_{n}^{N} ↓ ↓ ↓ ϕ
Y_{n}^{N} ↠ Y_{0}^{0} = 𝔸^{d} × 𝔸^{1}.

This diagram induces a diagram of B-algebra homomorphisms

CD_{Y_{ϕ}^{∧}} ↩ CD_{Y_{0ϕ}^{0∧}}
ϕ_{♯} ↓  ↓ ϕ_{0♯}^{0}
ϕ^{∗}CD_{Y^{∧}} ↩ ϕ_{0}^{0∗}CD_{Y_{0}^{0∧}}.

We see that the images of the elements (6.7.4) by ϕ_{♯} and ϕ_{0♯}^{0} coincide, modulo the identification

a_{i0} ↦ x_{i}, b_{i0} ↦ dx_{i}, b_{i,−1}^{∗} ↦ ξ_{i}, a_{i,−1}^{∗} ↦ ∂_{i}.

It suffices to observe that ϕ_{ch} and ϕ_{0♯}^{0} act similarly on the elements (5.4.9), see the formulas [MSV, (3.17)] for ϕ_{ch} and Lemma 6.6.1 for ϕ_{0♯}^{0}. We are done.

References

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[BB] Beilinson, A., Bernstein, I., A proof of the Jantzen conjectures, I.M. Gelfand Seminar (S. Gelfand, S. Gindikin, ed.), vol. 1, pp. 1-50.

[BD1] Beilinson, A., Drinfeld, V., Chiral algebras, preprint.

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[BLR] Bosch, S., Lütkebohmert, W., Raynaud, M., Néron models, Springer-Verlag, 1990.

[DL] Denef, J., Loeser, F., Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), 201-232.

[EGA0] Grothendieck, A., Dieudonné, J., Eléments de géométrie algébrique, Grund. Math. Wiss., vol. 166, Springer, Boston-Basel-Berlin, 1971.

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[GV] Grothendieck, A., Verdier, J.-L., Prefaisceaux, SGA IV, Exp. I.

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[Kap] Kapranov, M., Double affine Hecke algebras and 2-dimensional local fields, Jour. AMS 14 (2001), 239-262.

[GKF] Gelfand, I.M. Kazhdan, D.A., Fuks, D.B., The actions of infinite-dimensional Lie algebras, Funct. Anal. Appl. 6 (1972), 9-13.

[HL] Huang,Y.-Z., Lepowsky, J., On the D-module and the formal variable approachs to vertex algebras, in Topics in geometry, Birkhäuser, 1996, pp. 175-202.

[Kat] Kato, K., Existence theorem for higher local class field theory, Math.AG/0012150.

[KT] Kashiwara, M., Tanisaki, T., Kazhdan-Lusztig Conjecture for Symmetrizable Kac-Moody Lie Algebra. II Intersection Cohomologies of Schubert varieties.

[L] Leites, D. A., Introduction to the theory of supermanifolds. (Russian), Uspekhi Mat. Nauk 35 (1980), 3-57.

[MSV] Malikov, A., Schechtman, V., Vaintrob, A., Chiral De Rham complex, Comm. Math. Phys. 204 (1999), 439-473.

[P] Parshin, A.N., Higher-dimensional local fields and L-functions, Math.AG/0012151.

Mikhail Kapranov  Eric Vasserot
Department of Mathematics  Département de Mathématiques
University of Toronto  Université de Cergy-Pontoise
100 St. George St.  2 Av. A. Chauvin
Toronto, Ontario M5S 3G3  95302 Cergy-Pontoise Cedex
Canada  France
email: kapranov@math.toronto.edu  email: eric.vasserot@math.u-cergy.fr