Abstract
Let X be a smooth algebraic variety over k. We prove that any flat quasicoherent sheaf on \({\text {Ran}}(X)\) canonically acquires a \(\mathscr {D}\)-module structure. In addition, we prove that, if the geometric fiber \(X_{\overline{k}}\) is connected and admits a smooth compactification, then any line bundle on \(S \times {\text {Ran}}(X)\) is pulled back from S, for any locally Noetherian k-scheme S. Both theorems rely on a family of results which state that the (partial) limit of an n-excisive functor defined on the category of pointed finite sets is trivial.
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Notes
In particular, X is smooth and geometrically integral over k.
Tangential point: Beilinson and Drinfeld also impose a mild flatness hypothesis, which could be summarized as ‘flatness along the diagonals,’ see [1, Lem. 3.4.3(i)] and also Remark 3.2.5.
Note that G lands in the category of strictly commutative Picard groupoids, which identifies with the extension-closed subcategory of \({\text {D}}^b({\text {AbGrp}})\) consisting of objects concentrated in cohomological degrees \([-1, 0]\). Thus, this example can be analyzed in the context of functors which land in the stable \((\infty , 1)\)-category \({\text {D}}^b({\text {AbGrp}})\). However, note that the composition of G with the full embedding into \({\text {D}}^b({\text {AbGrp}})\) need not be 2-excisive.
The functor \(\iota \) was defined in 2.1 and it appears here to indicate that the limit is over \({\text {fSet}}^{{\text {surj}}}_{{\text {n.e.}}}\), not \({\text {fSet}}_*\).
This is the diagram obtained from \(\mathscr {D}_0\) by freely adjoining an initial object.
The value of \(F : {\text {Sch}}_{k}^{{\text {aff}}, {{\text {op}}}} \rightarrow \mathscr {C}\) on a prestack is defined by right Kan extension along the fully faithful embedding \({\text {Sch}}_{k}^{{\text {aff}}} \hookrightarrow {\text {Fun}}({\text {Sch}}_{k}^{{\text {aff, op}}}, {\text {Set}}) = {\text {PreStk}}_{k}\), so we have
$$\begin{aligned}F({\text {Ran}}(X)^S) \simeq \lim _{I \in {\text {fSet}}^{{\text {surj}}}_{{\text {n.e.}}}} F(X^{I \times S}) \simeq \lim _{I \in {\text {fSet}}^{{\text {surj}}}_{{\text {n.e.}}}} G(\{*\} \sqcup (I \times S)) =: M(\{*\} \sqcup S).\end{aligned}$$For the first isomorphism, see 3.2.2.
If it were not a constant diagram, the proof would break at this point because, although it is a limit diagram in \(\mathscr {C}^{\le d-1}\), the inclusion \(\mathscr {C}^{\le d-1} \hookrightarrow \mathscr {C}^{\le d}\) is a left adjoint and does not preserve limits.
The notation \(\le N/3\) is shorthand for \(\le \lfloor N/3 \rfloor \).
Remark: \(\mathscr {C}\) is the fiber over Y of the category defined in [12, Def. 2.4.9] and denoted ‘\({\text {Ran}}(X)\)’ therein.
Remark: These lax colimits admit a less ad hoc interpretation. Namely, \(\lim _{I \in {\text {fSet}}^{{\text {surj}}}_{{\text {n.e.}}}}^{{\text {lax}}} {\text {QCoh}}(X^I)\) is equivalent to the category of natural transformations (i.e. pseudonatural transformations) \({\text {Ran}}^{{\text {(iv, lax)}}}(X) \Rightarrow {\text {QCoh}}(-)\) of functors \({\text {Sch}}^{{\text {aff, ft, op}}}_k \rightarrow (\infty , 1){\text {-Cat}}\). One can replace \({\text {QCoh}}(-)\) by another functor to \((\infty , 1){\text {-Cat}}\).
As the notation suggests, we have \(\mathbb {D}^n \simeq (\mathbb {D})^{\times n}\).
More precisely, for any test scheme \(S \in {\text {Sch}}^{{\text {aff}}}_k\), we have
$$\begin{aligned} {\text {Hom}}_{{\text {Sch}}^{{\text {aff}}}_k} (S, (\mathbb {A}^n)^I) \simeq {\text {Hom}}_{{\text {Sch}}^{{\text {aff}}}_k} (S, \mathbb {A}^n)^I \simeq {\text {Hom}}_{{\text {Set}}_*}(\{*\} \sqcup I, {\text {Hom}}_{{\text {Sch}}^{{\text {aff}}}_k}(S, \mathbb {A}^n)), \end{aligned}$$where \({\text {Set}}_*\) is the category of pointed sets, and the distinguished point of \({\text {Hom}}_{{\text {Sch}}^{{\text {aff}}}_k}(S, \mathbb {A}^n)\) is the constant map to \(0 \in \mathbb {A}^n\). This shows how to make the map \(\{*\} \sqcup I \mapsto (\mathbb {A}^n)^I\) functorial.
Namely, \(\mathscr {F}\) is a sheaf on \({\text {Ran}}_{\langle d \rangle }^n\) and \(\overline{\tau }\) is an isomorphism with \(\underline{V}\) on \({\text {Ran}}_{\langle d-1 \rangle }^n\).
This statement is true because, in the definition of the infinitesimal Ran space from 4.1, we used the formal neighborhood \(\mathbb {D}^n\) rather than \({\text {Spec}}k[[ x_1, \ldots , x_n ]]\).
We emphasize that the simplicial category \({\varvec{\Delta }}\) is written in bold font, while the small diagonal \(\Delta : Y \hookrightarrow Y^I\) is not. In this proof, the latter meaning is intended only when \(\Delta \) appears as a subscript \((-)_{\widehat{\Delta }}\) indicating completion.
For details regarding this point, see [7, Sect. 1.4].
The e-superscript notation for rigidified line bundles is inspired by [21, Sect. 3.4].
Equivalently, no element of T vanishes on an associated point of \(A_f\). In fact, we will only use this construction when f vanishes on the embedded points of A, so \(A_f\) has no embedded points, in which case the requirement on T is simply that its elements do not vanish on any generic point of A.
In geometric terms, f does not vanish on any associated point of \({\text {Spec}}A'\), and the hypothesis on T then implies that none of its elements vanish on any associated point of \({\text {Spec}}A'\).
Proof: the map \(\widehat{S} \times V(\mathfrak {m}^{N-2}) \xrightarrow {{\text {pr}}_1} \widehat{S}\) is flat and induces a bijection on the underlying topological spaces, so Exercise 24.2.J in [19] yields the result.
The scheme \((S \times X^n)^\wedge \) is defined as in 6.2.6, completing with respect to \(f_n\) in place of f.
A comment about notation: since we are in the special case of 6.2.6 when \(f_n = f \in T\), we can replace the subscripts fT by T, and we have done so above.
The additional superscript indicates completion with respect to \((f_n)\).
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Acknowledgements
The author would like to thank Dennis Gaitsgory for suggesting the problem of showing that \({\text {Ran}}(X)\) is Pic-contractible, Yifei Zhao for providing a wealth of helpful comments on an earlier draft, and Charles Fu for providing comments and corrections on a later draft. Much of the paper in its current form was inspired by Yifei Zhao’s suggestion that the proof of triviality of \(\mathbf {Pic}({\text {Ran}}(X))\) could be interpreted as a vanishing theorem for limits of polynomial functors. This work was supported by the National Science Foundation Graduate Research Fellowship.
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Tao, J. n-Excisive functors, canonical connections, and line bundles on the Ran space. Sel. Math. New Ser. 27, 2 (2021). https://doi.org/10.1007/s00029-020-00611-4
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DOI: https://doi.org/10.1007/s00029-020-00611-4