1. Introduction

Given a connected smooth complex projective curve $C$ and ${\bf G}$ a smooth affine $\mathbb{C}$ -group scheme, the corresponding (global) affine Grassmannian is the functor

\begin{align*}\underline {{\rm Gr}}_{C,{\bf G}}: (\mathsf{Aff}_{\mathbb{C}})^{{\rm op}} \to {\bf Sets} \, : \, S \longmapsto \left \{ (x, \mathcal{E}, \varphi ) \, |\, x \in C(S), \mathcal{E}\in {\rm Bun}^{{\bf G}}(C \times S), \varphi \;:\; \mathcal{E}_{| (C\times S)\smallsetminus \Gamma _{x}} \simeq \mathcal{E}_0 \right \},\end{align*}

where ${\rm Bun}^{{\bf G}}(C \times S)$ is the groupoid of ${\bf G}$ -torsors on $C \times S$ , $\Gamma _x \subset C \times S$ denotes the graph of $x \in C(S)$ and $\mathcal{E}_0$ denotes the trivial ${\bf G}$ -bundle (see, e.g., [Reference ZhuZhu17, 1.4]). A local version of $\underline {{\rm Gr}}_{C,{\bf G}}$ can also be defined using ${\bf G}$ -bundles only defined on a formal neighbourhood of $\Gamma _x$ and trivializations defined on the corresponding affine punctured neighbourhood [Reference ZhuZhu17, (1.2.1)], and Beauville–Laszlo theorem [Reference Beauville and LaszloBL95] ensures that such a local version of the affine Grassmannian is in fact equivalent to the global version recalled above (see, e.g., [Reference ZhuZhu17, Theorem 1.4.2]).

A brilliant idea of A. Beilinson and V. Drinfeld was to generalize the construction of the affine Grassmannian by allowing an arbitrary, finite number of points $x$ . This can be done by first introducing the so called Ran space of the curve $C$ as the presheaf

\begin{align*}\mathsf{Ran}_C: (\mathsf{Aff}_{\mathbb{C}})^{{\rm op}} \to {\bf Sets} \, : \, S \longmapsto \left \{ \textrm {finite non-empty subsets of } C(S)\right \},\end{align*}

and then defining

\begin{align*}\underline {{\rm Gr}}^{{\rm BD}}_{C,{\bf G}}: (\mathsf{Aff}_{\mathbb{C}})^{{\rm op}} \to {\bf Sets} : S \longmapsto \left \{ (\underline {x}, \mathcal{E}, \varphi ) |\, \underline {x} \in \mathsf{Ran}_C (S), \mathcal{E}\in {\rm Bun}^{{\bf G}}(C \times S), \varphi : \mathcal{E}_{| (C\times S)\smallsetminus \Gamma _{\underline {x}}} \simeq \mathcal{E}_0 \right \}\!,\end{align*}

where $\Gamma _{\underline {x}} \subset C \times S$ denotes the joint graph of $\underline {x} \in \mathsf{Ran}_C(S)$ . The functor $\underline {{\rm Gr}}^{{\rm BD}}_{C,{\bf G}}$ is called the (Ran version of the) Beilinson–Drinfeld affine Grassmannian of the pair $(C, {\bf G})$ . There is an obvious morphism of presheaves $p: \underline {{\rm Gr}}^{{\rm BD}}_{C,{\bf G}} \to \mathsf{Ran}_C$ , and the trivial ${\bf G}$ -bundle endows it with a canonical section $\mathsf{triv}: \mathsf{Ran}_C \to \underline {{\rm Gr}}^{{\rm BD}}_{C,{\bf G}}$ . Although $\mathsf{Ran}_C$ is not representable,Footnote ¹ not even by an ind-scheme, the map $p$ is itself ind-representable and it is even ind-proper when ${\bf G}$ is reductive [Reference ZhuZhu17, 3.3]. The presheaf $\underline {{\rm Gr}}^{{\rm BD}}_{C,{\bf G}}$ is not represented by an ind-scheme but it is the presheaf (non-filtered) colimit of ind-schemes under proper maps (i.e. a pseudo-ind-scheme as in [Reference GaitsgoryGai13, 1.2]). Although passing from $\underline {{\rm Gr}}_{C,{\bf G}}$ to $\underline {{\rm Gr}}^{{\rm BD}}_{C,{\bf G}}$ might seem an easy and obvious step, it does make it possible for $\underline {{\rm Gr}}^{{\rm BD}}_{C,{\bf G}}$ to exhibit an important property called factorization. First of all notice that $\mathsf{Ran}_C$ carries a (non-unital) semigroup structure given by the union

\begin{align*}\cup \;:\; \mathsf{Ran}_C \times \mathsf{Ran}_C \to \mathsf{Ran}_C: (\{\underline {x}\}, \{\underline {y}\}) \mapsto \{\underline {x}, \underline {y}\}.\end{align*}

Let $(\mathsf{Ran}_C \times \mathsf{Ran}_C)_{{\rm disj}} \subset \mathsf{Ran}_C \times \mathsf{Ran}_C$ be the sub-presheaf defined by those pairs $(\{\underline {x}\}, \{\underline {y}\})$ such that $\{\underline {x}\} \cap \{\underline {y}\}= \emptyset$ . We define similarly $(\mathsf{Ran}_C \times \mathsf{Ran}_C \times \mathsf{Ran}_C )_{{\rm disj}} \subset \mathsf{Ran}_C \times \mathsf{Ran}_C \times \mathsf{Ran}_C$ . Then, the factorization property of $\underline {{\rm Gr}}^{{\rm BD}}_{C,{\bf G}}$ can be stated as the existence of a natural isomorphism (called factorization isomorphism) between the pullbacks of the two diagrams

(1.1)

which is compatible, in an obvious way, with the map $\mathsf{triv}: \mathsf{Ran}_C \to \underline {{\rm Gr}}^{{\rm BD}}_{C,{\bf G}}$ and satisfies a natural cocycle condition on $(\mathsf{Ran}_C \times \mathsf{Ran}_C \times \mathsf{Ran}_C )_{{\rm disj}}$ [Reference ZhuZhu17, Theorem 3.3.3]. We may denote $\mathsf{Gr}_{C,{\bf G}; 2}^{{\rm BD}}$ the common (up to canonical isomorphism) pullback of these two diagrams. The map $p:\mathsf{Gr}_{C,{\bf G}; 2}^{{\rm BD}} \to \mathsf{Ran}_C$ carries a flat connection ([Reference ZhuZhu17, Theorem 3.1.20] and [Reference Beilinson and DrinfeldBD99, 2.8]) and actions of suitable Ran-versions ${\rm L}_{\mathsf{Ran}}{\bf G}$ and ${\rm L}^+_{\mathsf{Ran}}{\bf G}$ of the loop group ${\rm L}{\bf G}$ and of the positive loop group ${\rm L}^+{\bf G}$ , respectively ([Reference GaitsgoryGai22, 1.1.2] and [Reference Mirkovic and VilonenMV07] § 5, (5.5)).

In this paper we propose and study an analogue of Beilinson-Drinfeld Grassmannian, called the flag Grassmannian, where the curve $C$ is replaced by a smooth complex projective surface $X$ , and points in the Ran space of $C$ are replaced with (families) of flags of subschemes in $X$ . Our investigation on the flag Grassmannian has its origins in some conversations of the third author with Mauro PortaFootnote ² triggered by § 3.4.6 of [Reference Beilinson and DrinfeldBD04] where an equivalent approach to factorization algebras sheaves is based on effective Cartier divisors (possibly considered up to reduced equivalence) on a curve, rather than on the Ran space of the curve. This remark immediately gave us the idea of a possible replacement of the Ran space in the higher dimensional case of a surface, with the further possibility of including arbitrary flags consisting of an effective Cartier divisor together with a codimension 1 subscheme in this divisor.

Let us summarize the results of this paper.

Formal glueing. In proving several properties of our flag Grassmannian, we will need a higher-dimensional version of Beauville–Laszlo gluing, therefore we devote the first three sections to establish a general version of formal gluing. Let $k$ be field. For a $k$ -scheme $Y$ and a closed subscheme $Z$ , with open complement $U$ , roughly speaking, formal gluing (whenever it exists) is a way of describing geometric objects of some kind defined on $Y$ , as geometric objects of the same kind defined on $U$ and on the formal completion $\widehat {Z}$ of $Z$ in $Y$ , that are suitably compatible. The notion of compatibility here is a delicate one. Morally, guided by a purely topological intuition, one would like to say that a geometric object $\mathcal{E}_U$ and a geometric object $\mathcal{E}_{{\widehat {Z}}}$ are compatible if they restrict to the same object (or to an equivalent one) on the intersection $U \cap {\widehat {Z}}= {\widehat {Z}} \smallsetminus Z$ . However, strictly speaking, in algebraic geometry this intersection is empty, so the notion of compatibility cannot be the naive one, at least if we want to remain within the realm of algebraic geometry, i.e. we want to keep viewing $Y$ , $U$ and $\widehat {Z}$ as algebro-geometric objects. Many formal glueing results have been proved, and we give here only a probably non-exhaustive list: [Reference WeilWei82], [Reference ArtinArt70], [Reference Ferrand and RaynaudFR70], [Reference Beauville and LaszloBL95], [Reference Moret-BaillyMB96], [Reference Ben-Bassat and TemkinBBT13], [Reference BhattBha16], [Reference SchäppiSch15], [Reference Hall and RydhHR16] and [Reference MathewMat22]. In the first sections, based on some of these previous works (mainly [Reference BhattBha16] and [Reference MathewMat22]), we address the general formal glueing problem when the geometric objects above are

• • almost perfect modules (also known as pseudo-coherent complexes) on $Y$ , whose derived moduli stack is denoted by ${\bf Coh}^-_Y$ in the paper,

• • perfect modules on $Y$ , whose derived moduli stack is denoted by ${\bf Perf}_Y$ in the paper,

• • ${\bf G}$ -bundles on $Y$ , for ${\bf G}$ an affine $k$ -group scheme, whose derived moduli stack is denoted by ${\bf Bun}^{{\bf G}}_Y$ in the paper,

and $Y$ is a locally Noetherian stack over $k$ . If ${\bf F} = {\bf Coh}^-, {\bf Perf}$ or ${\bf Bun}^{{\bf G}}$ , and $Z\subset Y$ is a closed substack, restriction induces an equivalence of derived stacks over $Y$ (i.e. defined on the category $\mathsf{dAff}_{Y}$ of derived affine Noetherian schemes ${\rm Spec}\, R$ with a map ${\rm Spec}\, R \to Y$ )

(1.2)

This is proved in Proposition 3.3 and Theorem 3.5. In this formula, $\widehat {Z}$ denotes the formal completion of $Z$ inside $Y$ , while ${\widehat {Z}}^{{\rm aff}} \smallsetminus Z$ is defined using the fibre functor Footnote ³ of affinization (see Definitions 2.10 and 2.16). In particular, when $R$ is an underived commutative $k$ -algebra together with a map ${\rm Spec}\, R \to Y$ , the value of ${\widehat {Z}}^{{\rm aff}} \smallsetminus Z$ on $({\rm Spec}\, R \to Y)$ is the expected one

\begin{align*}({\widehat {Z}}^{{\rm aff}} \smallsetminus Z)({\rm Spec}\, R \to Y) = {\rm Spec}(\widehat {R}_{{\rm I}_Z})\smallsetminus {\rm Spec}(R/{\rm I}_Z),\end{align*}

where ${\rm I}_Z\subset R$ denotes the ideal of the pullback of $Z$ to ${\rm Spec}\, R$ , and $\widehat {R}_{{\rm I}_Z}$ is the ring completion of $R$ at the ideal ${\rm I}_Z$ . Note that, by algebraization (Proposition 3.1), the canonical map ${\bf F}_{{\widehat {Z}}} \to {\bf F}_{{\widehat {Z}}^{{\rm aff}}}$ is an equivalence of derived stacks over $Y$ (where ${\widehat {Z}}^{{\rm aff}}({\rm Spec}\, R \to Y)= {\rm Spec}(\widehat {R}_{{\rm I}_Z})$ , when ${\rm Spec}\, R$ is a Noetherian underived affine $k$ -scheme endowed with a map to $Y$ ). In the presence of a longer nonlinear flag $(Z_k\subset Z_{k-1} \subset \cdots \subset Z_1 \subset Y)$ of closed substacks in $Y$ , the formal glueing formula (1.2) can be iterated yielding what we call a flag decomposition of ${\bf F}$ (see § $\,$ 3.4).

Using the map $Y \to {\rm Spec}\, k$ , and the induced functor

(1.3) \begin{align} \mathsf{dSt}_Y \to \mathsf{dSt}_k \;:\; {\bf F} \mapsto \underline {\mathsf{F}}, \end{align}

from derived stacks defined over $\mathsf{dAff}_{Y}$ to derived stacks defined over $\mathsf{dAff}_{k}$ , formula (1.2) translates into the corresponding obvious formal glueing equivalence between derived stacks over $k$

(1.4)

(see Construction 3.7).

In our analysis, the case of ${\bf G}$ -bundles follows from the case of perfect modules (since ${\bf Bun}^{{\bf G}}_Y$ is Perf-local in the variable $Y$ , see Lemma 2.24), while the case of almost perfect modules is similar to that of perfect modules. Therefore, we have decided to present these three formal gluing results together, although in the rest of the paper we will mainly concentrate on the study of ${\bf G}$ -bundles. Moreover, for the sake of completeness, and since this does not require an excessive effort, we actually establish formal gluing and flag decomposition results, in the more general case where $Y$ a locally Noetherian derived stack over $k$ , $Z \subset Y$ is a derived closed substack.

Section 3.5 explains how to deduce from the previous results formal gluing for the underived versions of ${\bf F} = {\bf Coh}^-, {\bf Perf}$ or ${\bf Bun}^{{\bf G}}$ .

Nonlinear flags. Starting with § 4, we will let $k=\mathbb{C}$ , and fix a smooth complex projective surface $X$ . In § 4 we introduce the simplicial object $\underline {\mathsf{Fl}}_{X,\bullet }$ of families of nonlinear flags on $X$ . Here is the basic idea of the construction of $\underline {\mathsf{Fl}}_{X,\bullet }$ . We first define $\operatorname {\underline {\mathsf{Fl}}}_{X,1}$ (which we also denote simply by $\operatorname {\underline {\mathsf{Fl}}}_X$ ) as the presheaf of posets on $\mathsf{Aff}_{\mathbb{C}}$ assigning to $S$ the poset whose elements are pairs $(D, Z)$ consisting of a relative effective Cartier divisor $D$ on $X \times S/ S$ , and an element $Z$ in the Hilbert scheme of $0$ -dimensional subschemes of $X \times S /S$ , subject to the nesting condition $Z \subset D$ . The partial order on $\operatorname {\underline {\mathsf{Fl}}}_{X,1}(S)$ is defined as follows: we say that $(D,Z) \leq (D',Z')$ (also denoted as a morphism $(D,Z) \to (D',Z')$ ) if there exists a commutative square of closed subschemes of $X \times S$

such that the canonical induced map $Z \to D\times _{D'} Z'$ yields an isomorphism on the induced reduced scheme structures. Note that by forgetting the partial order structure, $\underline {\mathsf{Fl}}_{X,1}$ is just a variant of the so-called flag Hilbert scheme of $X$ (see, e.g., [Reference SernesiSer06, 4.5]). The main step in the construction of $\underline {\mathsf{Fl}}_{X,\bullet }$ is to define $\underline {\mathsf{Fl}}_{X,2}$ in such a way that it supports a union map $\bigcup \;:\; \underline {\mathsf{Fl}}_{X,2} \to \underline {\mathsf{Fl}}_{X,1}$ of the form $(D_1, Z_1) \cup (D_2, Z_2)= (D_1 +D_2, Z_1 \cup Z_2)$ . In order to ensure that such a map is well defined (e.g. that $Z_1 \cup Z_2$ is flat over the test scheme $S$ ), we define $\underline {\mathsf{Fl}}_{X,2}(S)$ as the sub-poset of pairs $((D_1, Z_1), (D_2, Z_2))$ in $\underline {\mathsf{Fl}}_{X,1}(S) \times \underline {\mathsf{Fl}}_{X,1}(S)$ satisfying the following good position (or transversality) condition: $D_1 \cap D_2$ is a relative effective Cartier divisor inside both $D_1 /S$ and $D_2 / S$ , and $Z_1 \cap Z_2 = D_1 \cap D_2$ (Definition 4.9). Note that, if $X$ is a curve, then $\underline {\mathsf{Fl}}_{X,2}$ classifies pairs of disjoint Cartier divisors. Finally, we build our full simplicial object $\underline {\mathsf{Fl}}_{X,\bullet }$ as the classifying simplicial object associated with the partially defined unital monoid structure $\cup$ on $\underline {\mathsf{Fl}}_{X,1}$ (see § 4.4). Note that $\operatorname {\underline {\mathsf{Fl}}}_{X, 0}= \left \{ * \right \}$ , $\partial _0, \partial _2: \operatorname {\underline {\mathsf{Fl}}}_{X,2} \to \operatorname {\underline {\mathsf{Fl}}}_{X,1}$ are, respectively, the projection on the second and first flag, while $\partial _1: \operatorname {\underline {\mathsf{Fl}}}_{X,2} \to \operatorname {\underline {\mathsf{Fl}}}_{X,1}$ is the union map. The main result of § 4 (Theorem 4.21) is that the simplicial object $\underline {\mathsf{Fl}}_{X,\bullet }$ has the so-called $2$ -Segal property [Reference Dyckerhoff and KapranovDK19], i.e. the diagram

defines an associative algebra structure in correspondences. This will induce, later in the paper, the same property on the simplicial flag Grassmannian $\mathcal{G}r_{X, \bullet }$ .

Flag Grassmannian. Fix, once and for all, a smooth affine group scheme ${\bf G}$ over $\mathbb{C}$ , that will be most often omitted from the notation. In §§ 5 and 6 we build the simplicial flag Grassmannian $\mathcal{G}r_{X,\bullet }$ for ${\bf G}$ -bundles, together with a simplicial map $\mathcal{G}r_{X,\bullet } \to \underline {\mathsf{Fl}}_{X,\bullet }$ . We sketch here, in a slightly simplified way, the basic ideas of the constructions, where furthermore the schemes parametrizing families of flags will be underived. The actual rigorous constructions in the main body of the paper follow a slightly different and more technical path (mainly in order to take care of higher coherences and derived structures), and giving here all such details would obscure the intuition. Moreover, in this Introduction, we will focus on the flag Grassmannian for ${\bf G}$ -bundles; variations involving perfect complexes are considered in the main text.

For any affine (underived) Noetherian test scheme $S$ , and any flag $(D,Z)\in \underline {\mathsf{Fl}}_{X,1}(S)$ , so that $(D,Z)$ is a flag inside $X \times S$ , we consider (Definition 5.7) the derived stack over $X\times S$ (notation being the one introduced in the previous paragraph)

(1.5) \begin{align} {\bf Gr}_X(S)(D,Z) := {\bf Bun}_{X \times S} \times _{{\bf Bun}_{(X \times S) \smallsetminus D} \times _{{\bf Bun}_{(X \times S)\smallsetminus D \times _{X\times S} {\widehat {Z}}^{{\rm aff}}}} {\bf Bun}_{{\widehat {Z}}}} \left \{ * \right \} \,\,\, \in \, \mathsf{dSt}_{X \times S}, \end{align}

where the map from $\left \{ * \right \}$ represents the trivial ${\bf G}$ -bundle $\mathcal{E}_0$ . This derived stack over $X \times S$ is called the flag Grassmannian of $X\times S \,/ S$ at the fixed flag $(D,Z)$ , and it classifies triples $(\mathcal{E}, \varphi , \psi )$ where $\mathcal{E}$ is a family of ${\bf G}$ -bundles on $X \times S$ , $\varphi$ a family of trivializations of $\mathcal{E}$ on $(X\times S)\smallsetminus D$ and $\psi$ a family of trivializations of $\mathcal{E}$ on $\widehat {Z}$ , such that $\varphi$ and $\psi$ are compatible on the ‘intersection’ $(X \times S)\smallsetminus D \times _{X\times S} {\widehat {Z}}^{{\rm aff}}$ . Like in the classical case where $X$ is a curve, there is also a corresponding local version (Definition 5.8)

(1.6) \begin{align} {\bf Gr}^{{\rm loc}}_X(S)(D,Z) := {\bf Bun}_{\widehat {D}} \times _{{\bf Bun}_{\widehat {D} \smallsetminus D} \times _{{\bf Bun}_{\widehat {D}\smallsetminus D \times _{\widehat {D}} {\widehat {Z}}^{{\rm aff}}}} {\bf Bun}_{{\widehat {Z}}}} \left \{ * \right \} \,\,\, \in \, \mathsf{dSt}_{X \times S}, \end{align}

and formal gluing implies that the obvious restriction

\begin{align*}{\bf Gr}_X(S)(D,Z) \longrightarrow {\bf Gr}^{{\rm loc}}_X(S)(D,Z)\end{align*}

is an equivalence of derived stacks over $X \times S$ (Lemma 5.9). Translating these constructions into absolute derived stacks (i.e. stacks over $k=\mathbb{C}$ ) by using the functor (1.3) defined above, we get derived stacks $\underline {\mathsf{Gr}}_X(S)(D,Z)$ and $\underline {\mathsf{Gr}}^{{\rm loc}}_X(S)(D,Z)$ over $k$ , together with a restriction equivalence $\underline {\mathsf{Gr}}_X(S)(D,Z) \simeq \underline {\mathsf{Gr}}^{{\rm loc}}_X(S)(D,Z)$ . Since $D$ is a relative effective Cartier divisor in $X \times S \, / S$ , one can prove that the truncation ${\rm t}_0(\underline {\mathsf{Gr}}_X(S)(D,Z))$ , hence ${\rm t}_0(\underline {\mathsf{Gr}}^{{\rm loc}}_X(S)(D,Z))$ , is actually a sheaf of sets (Lemma 5.12). In particular the global sections ${\rm Gr}_X(S)(D,Z)$ of both $\underline {\mathsf{Gr}}_X(S)(D,Z)$ and $\underline {\mathsf{Gr}}^{{\rm loc}}_X(S)(D,Z)$ , form a set. Since $(X\times S)\smallsetminus D$ , $\widehat {D}$ , $\widehat {D}^{{\rm aff}}$ , $\widehat {Z}$ , ${\widehat {Z}}^{{\rm aff}}$ and $\widehat {D}\smallsetminus D$ are all insensitive to the scheme structure on $D$ and $Z$ , ${\bf Gr}_X(S)(D,Z)$ , $ {\bf Gr}^{{\rm loc}}_X(S)(D,Z)$ , $\underline {\mathsf{Gr}}_X(S)(D,Z)$ and $\underline {\mathsf{Gr}}^{{\rm loc}}_X(S)(D,Z)$ are topologically invariant (§ $\,$ 5.4). With the test scheme $S$ fixed, by using again formal gluing, one can prove that the assignment $(D,Z) \mapsto {\rm Gr}_X(S)(D,Z)$ defines, for each test scheme $S$ , a functor ${\rm Gr}_X(S): \operatorname {\underline {\mathsf{Fl}}}_{X}(S) \to {\bf Sets},$ and that the assignment $S\mapsto {\rm Gr}_X(S)$ is suitably functorial in the argument $S$ (see § $\,$ 5.3).

The Grothendieck construction applied to ${\rm Gr}_X(S): \operatorname {\underline {\mathsf{Fl}}}_{X}(S) \to {\bf Sets}$ defines $p_1(S): \mathcal{G}r_X(S) \to \operatorname {\underline {\mathsf{Fl}}}_X(S)$ , and the fuctoriality in $S$ of ${\rm Gr}_X(S)$ (established in § $\,$ 5.3), builds for us $\mathcal{G}r_X$ as a functor $\mathsf{dAff}^{{\rm op}} \to {\rm Cat}_\infty$ , together with a morphism of functorsFootnote ⁴

(1.7) \begin{align} p_1: \mathcal{G}r_X \longrightarrow \operatorname {\underline {\mathsf{Fl}}}_X. \end{align}

In other words, when $S$ is an underived test affine scheme, the objects of $\mathcal{G}r_X(S)$ consist of pairs $(F, \underline {\mathcal{E}})$ where $F$ is a flag in $X \times S$ , and $\underline {\mathcal{E}} \in {\rm Gr}_X(S)$ . Note that the trivial bundle induces an obvious section $\mathsf{triv}: \operatorname {\underline {\mathsf{Fl}}}_X \to \mathcal{G}r_X$ of $p_1$ .

One can show that the map $p_1: \mathcal{G}r_X \to \operatorname {\underline {\mathsf{Fl}}}_X$ carries a canonical flat connection (§ $\,$ 5.3). This is conveniently and compactly rephrased by asserting that the map $p_1: \mathcal{G}r_X \to \operatorname {\underline {\mathsf{Fl}}}_X$ is actually the pullback along the canonical map $\operatorname {\underline {\mathsf{Fl}}}_X \to \operatorname {\underline {\mathsf{Fl}}}^{DR}_X$ of a suitable derived stack $\mathcal{G}r'_X$ mapping to the de Rham stack $\operatorname {\underline {\mathsf{Fl}}}^{DR}_X$ of $\operatorname {\underline {\mathsf{Fl}}}_X$ . This should be seen as the analogue for surfaces of the existence of a canonical flat connection on Beilinson–Drinfeld Grassmannian (see [Reference ZhuZhu17, Proposition 3.1.20]). A further important fact (Theorem 6.20) is that $\mathcal{G}r_X \to \operatorname {\underline {\mathsf{Fl}}}^{DR}_X$ (and also $p_1: \mathcal{G}r_X \to \operatorname {\underline {\mathsf{Fl}}}_X$ ) is, in an appropriate sense, represented by a derived ind-scheme (not of finite typeFootnote ⁵ ).

We are now able to construct the simplicial flag Grassmannian $\mathcal{G}r_{X, \bullet }$ together with a simplicial map $p_{\bullet }: \mathcal{G}r_{X, \bullet } \to \operatorname {\underline {\mathsf{Fl}}}_{X, \bullet }$ (see § $\,$ 6.4). First of all we put $\mathcal{G}r_{X, 0}= \left \{ * \right \}$ , $\mathcal{G}r_{X, 1}=\mathcal{G}r_X$ , and define $\mathcal{G}r_{X, 2}$ via the following pullback diagram.

(1.8)

A non-trivial result (see § 6.1), obtained through formal gluing inside punctured formal neighbourhoods, allows us to defineFootnote ⁶ projection maps

(1.9) \begin{align} q_i (S; F_1, F_2): {\rm Gr}_X(S)(F_1 \cup F_2) \longrightarrow {\rm Gr}_X(S)(F_i), \quad i=1,2, \end{align}

for arbitrary flags $(F_1, F_2) \in \operatorname {\underline {\mathsf{Fl}}}_{X,2}(S)$ that are suitably functorial both with respect to maps $(F_1, F_2) \to (F'_1, F'_2)$ in $\operatorname {\underline {\mathsf{Fl}}}_{X,2}(S)$ , and with respect to maps $S' \to S$ between test schemes. Thus, the projection maps of (1.9) induce morphisms $\partial _0, \partial _2: \mathcal{G}r_{X,2} \to \mathcal{G}r_{X,1}$ such that the diagram

commutes, for $j=0, 2$ . Using the empty flag to build degeneracy maps, we get a 2-truncated simplicial object $\mathcal{G}r_{X,\leq 2}$ . Now, via a general extension result (§ Appendix A), we complete $\mathcal{G}r_{X,\leq 2}$ to a full simplicial object $\mathcal{G}r_{X,\bullet }$ , called the simplicial flag Grassmannian of the pair $(X, {\bf G})$ , equipped with a canonical simplicial map $p_{\bullet }: \mathcal{G}r_{X, \bullet } \to \operatorname {\underline {\mathsf{Fl}}}_{X, \bullet }$ , extending $p_1$ and $p_2$ . The $2$ -Segal property of $\operatorname {\underline {\mathsf{Fl}}}_{X, \bullet }$ , and the properties of $p_{\bullet }$ , imply that the simplicial flag Grassmannian $\mathcal{G}r_{X, \bullet }$ is again $2$ -Segal. One can furthermore prove that the simplicial map $p_{\bullet }$ is endowed with a canonical flat connection (extending the one already established on $p_1$ ), or equivalently, that the previous construction yields in fact a map of simplicial objects $\mathcal{G}r'_{X, \bullet } \to \operatorname {\underline {\mathsf{Fl}}}^{DR}_{X, \bullet }$ inducing $p_{\bullet }: \mathcal{G}r_{X, \bullet } \to \operatorname {\underline {\mathsf{Fl}}}_{X, \bullet }$ via pullback along the canonical map $\operatorname {\underline {\mathsf{Fl}}}_{X, \bullet } \to \operatorname {\underline {\mathsf{Fl}}}^{DR}_{X, \bullet }$ .

Flag-factorization. In § 6.2 we prove the flag-factorization property for the flag Grassmannian $\mathcal{G}r_X$ . More precisely, and in complete analogy with (1.1), we prove (see Remark 6.17) that the map $(\partial _0, \partial _2): \mathcal{G}r_{X,2} \to \mathcal{G}r_{X, 1} \times \mathcal{G}r_{X,1}$ establishes an isomorphismFootnote ⁷ (called the factorization isomorphism) between the pullbacks of the following diagrams.

(1.10)

The proof boils down to showing that the projection maps (1.9) induce an isomorphism

(1.11)

for any pair $(F_1,F_2) \in \operatorname {\underline {\mathsf{Fl}}}_{X, 2}(S)$ , and for any $S$ (Theorem 6.2). One important and non-trivial ingredient of the proof (Lemma 6.7) consists in the equivalence

for arbitrary (derived) closed subschemes $T_1, T_2$ of an arbitrary affine (derived) scheme.

Flag-dependent exotic structures on the stack of ${\bf G}$ -bundles. For a (underived) Noetherian scheme $S$ and a smooth morphism $Y \to S$ of relative dimension $\geq 2$ , we say that a flag $(D,Z)$ in $Y/S$ is large if $Z_s$ meets any irreducible component of $D_s$ , for any $s \in S$ . As A. Beilinson observed (private communication), the set of global sections

\begin{align*}{\rm Gr}(Y; D,Z)= {\rm Bun}^{{\bf G}}(Y) \times _{{\rm Bun}^{{\rm G}}(Y\setminus D) \times _{{\rm Bun}^{{\bf G}}(\widehat {Z} \setminus D)} {\rm Bun}^{{\bf G}}(\widehat {Z}) } \left \{ \overline {\mathcal{E}} \right \}\end{align*}

is a singleton,Footnote ⁸ for any fixed ${\bf G}$ -bundle $\overline {\mathcal{E}}$ on $Y$ (Proposition 7.2). This observation allows us, in § 7, to associate with any large flag on a smooth projective surface $X$ a derived structure on the classical stack of ${\bf G}$ -bundles on $X$ (Proposition 7.6). These derived structures (given by the corresponding derived Hecke stacks, see § 7.2) are, in general, different from the usual one, i.e. $\underline {\mathsf{Bun}}_{X}$ (see Examples 7.7 and 7.8), and therefore we call them exotic derived structures on the truncation ${\rm t}_0(\underline {\mathsf{Bun}}_{X})$ , i.e. on the classical stack of ${\bf G}$ -bundles on $X$ . We remark that different choices of large flags in $X$ yield different exotic structures.

Flag-chiral structures. In § 8 we define flag analogues of chiral algebras and chiral categories. In the classical case, i.e. when $C$ is a smooth projective curve, one can define the so called chiral tensor product on appropriate categories $\mathsf{Shv}(\mathsf{Ran}_C)$ of sheaves on the Ran space of $C$ , essentially by performing the pull–push construction (of the exterior tensor product of such sheaves) along the diagram

(1.12)

where $q$ is the obvious inclusion (see [Reference Francis and GaitsgoryFG12] for details). When $X$ is a smooth projective surface, in order to get a flag analogue of this chiral product, one might naively perform the same construction by replacing (1.12) by the diagram

(1.13)

where recall that $\partial _1$ is the union map of good flags. This can certainly be done if we forget maps between flags (i.e. considering $\operatorname {\underline {\mathsf{Fl}}}_{X}$ and $\operatorname {\underline {\mathsf{Fl}}}_{X,2}$ as sets-valued rather than posets-valued), see § 8.1. In order to deal with the general case, we use the tools developed by S. RaskinFootnote ⁹ [Reference RaskinRas19], together with the Beilinson–Drinfeld idea in [Reference Beilinson and DrinfeldBD04, 3.4.6], to get a definition of flag-chiral category and of quasi-coherent flag-factorization algebra, together with their crystal and $\mathcal{D}$ -Modules counterparts (Definition 8.2). Some examples of these structures are given.

Why this flag Grassmannian? There are various reasons for considering our flag Grassmannian $\mathcal{G}r_X \to \underline {\mathsf{Fl}}_X$ , for $X$ a smooth projective algebraic surface, as a reasonable generalization of the Beilinson–Drinfeld Ran Grassmannian $\underline {{\rm Gr}}^{{\rm BD}}_{C,{\bf G}} \to \mathsf{Ran}_C$ , defined for a smooth projective algebraic curve $C$ . First of all, if we take $X$ to be a curve, and we only keep the divisorial part $D$ of the flag (the lowest-dimensional part $Z$ being then pushed into dimension $-1$ , so reasonably considered to be empty), then ${\bf Gr}_X(S)(D,Z=\emptyset )$ , out of which $\mathcal{G}r_X$ is constructed by varying the flags, reduces to the usual affine Grassmannian of the curve with respect to the divisor $D$ . Secondly, and most importantly, as we have tried to show above in this Introduction, our flag Grassmannian $\mathcal{G}r_X \to \underline {\mathsf{Fl}}_X$ enjoys most of the nice properties ofthe Beilinson–Drinfeld Ran Grassmannian. Namely, it has (a flags’ version of) the factorization property, it carries a natural flat connection and it naturally accommodates (flags’ versions of) chiral algebra structures. It is also worth mentioning that our general construction can be performed by replacing the stack ${\bf Bun}^{{\bf G}}$ by the stack ${\bf Perf}$ (see Remark 5.6 and 5.11).

On the other hand, as already observed, for ‘large’ flags on the surface (Definition 7.1), the corresponding flag Grassmannian is a purely derived stackFootnote ¹⁰ (Proposition 7.2). This is a consequence of imposing compatibility between the two trivializations in formula (1.5), expressed by the lowest term ${\bf Bun}_{(X \times S)\smallsetminus D \times _{X\times S} {\widehat {Z}}^{{\rm aff}}}$ in the fibre product. Although this property has the effect of producing interesting exotic derived structures on the stack of ${\bf G}$ -bundles on the surface (as explained earlier in this Introduction), it may well be considered as a weakness of our construction. We believe that moving from a curve to a higher-dimensional variety, and assuming that in such higher-dimensional cases flags of subvarieties should enter in the construction, naturally forces the derived geometrical part of any corresponding flag version of the affine Grassmannian to be predominant in some sense. This is basically due to the existence of Hartogs-like phenomena, like the example of ${\bf G}$ -bundles on $\mathbb{A}^2$ and on $\mathbb{A}^2 \setminus 0$ shows (the corresponding derived stacks are different while their truncations agree, see Remark 7.4). Nonetheless, there are variations on our definition of the flag Grassmannian that behave better from this specific point of view. Some of them are discussed in the next paragraph, and they form the object of a forthcoming paper [Reference Hennion, Melani and VezzosiHMV].

Leftovers and future directions. We list here some possible further lines of research naturally suggested by this paper.

(i) Actions. In Appendix B we sketch the construction of flag versions of the loop and positive loop group (both fibred over $\operatorname {\underline {\mathsf{Fl}}}_X$ ), well known in the curve case, together with their actions on the flag Grassmannian. This construction should be rigorously enhanced to the derived setting, and one should then be able to make sense, at least for ${\bf Shv}$ denoting $\ell$ -adic sheaves or $\mathcal{D}$ -Modules, of the equivariant $\infty$ -category ${\bf Shv}^{\mathcal{L}^{+}G_X}(\mathcal{G}r_{X})$ of $\mathcal{L}^{+}G_X$ -equivariant sheaves on $\mathcal{G}r_{X}$ , endowed with an appropriate convolution monoidal structure. This should also be extended to the other flag Grassmannians introduced in (ii) and (iii) below.

(ii) Mapping down and other flag Grassmannians. Fix a smooth divisor $D_0$ in a smooth projective surface $X$ . For each test scheme $S$ , and any flag on $X\times S /S$ of the form $(D_0 \times S, Z)$ , we may consider

\begin{align*}\underline {\mathsf{Gr}}_{D_0}(S;Z) := \underline {\mathsf{Bun}}_{D_0 \times S} \times _{\underline {\mathsf{Bun}}_{(D_0 \times S)\setminus Z}} \{ {\rm triv} \}\end{align*}

i.e. the affine Grassmannian of the curve $D_0$ at the ‘points’ $Z$ . We may try to construct a map $\underline {\mathsf{Gr}}(X\times S)(D_0 \times S, Z) \to \underline {\mathsf{Gr}}_{D_0}(S;Z)$ , and varying both $S$ and $Z \subset D_0 \times S$ , a map

(1.14) \begin{align} \mathcal{G}r_{X;D_0} \longrightarrow \mathcal{G}r_{D_0} ,\end{align}

where $\mathcal{G}r_{D_0}$ is (a Cartier-divisors version of) the Beilinson–Drinfeld Ran Grassmannian of the curve $D_0 \subset X$ . A drawback of the definition of flag Grassmannian in this paper is that this map is necessarily trivial (i.e. factors through the trivial bundle). This is essentially due to the compatibility required between the two trivializations in (1.5). Thus, in order to get a non-trivial map, one may modify our definition of the flag Grassmannian by replacing (1.5) with either of the following:Footnote ¹¹

\begin{align*}\underline {\mathsf{Gr}}_X^{{\rm glob}}(S)(D,Z):= \underline {\mathsf{Bun}}_{X \times S} \times _{\underline {\mathsf{Bun}}_{\widehat {D}\setminus Z} \times \underline {\mathsf{Bun}}_{\widehat {Z}}} \{ {\rm triv} \},\quad \underline {\mathsf{Gr}}^{\rm glob,\ res}_{X}(S)(D,Z):= \underline {\mathsf{Bun}}_{X \times S} \times _{\underline {\mathsf{Bun}}_{\widehat {D}\setminus Z}} \{ {\rm triv} \},\end{align*}

\begin{align*}\underline {\mathsf{Gr}}_X^{{\rm fat}}(S)(D,Z):= \underline {\mathsf{Bun}}_{\widehat {D}} \times _{\underline {\mathsf{Bun}}_{\widehat {D}\setminus Z} \times \underline {\mathsf{Bun}}_{\widehat {Z}}} \{ {\rm triv} \},\qquad \underline {\mathsf{Gr}}^{\rm fat,\ res}_{X}(S)(D,Z):= \underline {\mathsf{Bun}}_{\widehat {D}} \times _{\underline {\mathsf{Bun}}_{\widehat {D}\setminus Z}} \{ {\rm triv} \}.\end{align*}

As done with the flag Grassmannian of this paper, one can show [Reference Hennion, Melani and VezzosiHMV] that there are corresponding $2$ -Segal simplicial objects $\mathcal{G}r^{{\rm glob}}_{X, \bullet }$ , $\mathcal{G}r^{\rm glob,\ res}_{X, \bullet }$ in $\mathsf{dSt}^{{\rm Cat}_{\infty }}/\underline {\mathsf{Bun}}_{X}$ (with $\mathcal{G}r^{{\rm glob}}_{X, 0}= \mathcal{G}r^{\rm glob,\ res}_{X, 0} := \underline {\mathsf{Bun}}_{X}$ ),Footnote ¹² and $2$ -Segal objects $\mathcal{G}r^{{\rm fat}}_{X, \bullet }$ , $\mathcal{G}r^{\rm fat,\ res}_{X, \bullet }$ in $\mathsf{dSt}^{{\rm Cat}_{\infty }}$ , together with simplicial maps $\mathcal{G}r^{{\rm glob}}_{X, \bullet } \to \operatorname {\underline {\mathsf{Fl}}}^{DR, {\rm op}}_{X, \bullet } \times \underline {\mathsf{Bun}}_{X}$ , $\mathcal{G}r^{\rm glob,\ res}_{X, \bullet } \to \operatorname {\underline {\mathsf{Fl}}}^{DR, {\rm op}}_{X, \bullet } \times \underline {\mathsf{Bun}}_{X}$ , $\mathcal{G}r^{{\rm fat}}_{X, \bullet } \to \operatorname {\underline {\mathsf{Fl}}}^{DR, {\rm op}}_{X, \bullet }$ and $\mathcal{G}r^{\rm fat,\ res}_{X, \bullet } \to \operatorname {\underline {\mathsf{Fl}}}^{DR, {\rm op}}_{X, \bullet }$ . Moreover, one can show that $\mathcal{G}r^{{\rm glob}}_{X, \bullet }$ , $\mathcal{G}r^{\rm glob,\ res}_{X, \bullet }$ , $\mathcal{G}r^{{\rm fat}}_{X, \bullet }$ and $\mathcal{G}r^{\rm fat,\ res}_{X, \bullet }$ all satisfy an appropriate flag-factorization property.Footnote ¹³ For this modified flag Grassmannians, the corresponding maps (1.14) are non-trivial. This also suggests the possibility that suitably defined Hecke operators on $X$ induce Hecke operators on the curve $D_0$ , perhaps generating interesting subalgebras of Hecke operators on $D_0$ . Investigations in this directions are ongoing [Reference Hennion, Melani and VezzosiHMV].

(iii) More local flag Grassmannians and the double affine Grassmannian. By considering even more local Grassmannians with respect to those considered above in (ii), one can try to give an algebro-geometrical interpretation of the (usual) affine Grassmannian for the loop group ${\rm L}{\bf G}$ , as discussedFootnote ¹⁴ in [Reference Braverman and FinkelbergBF10, § 3]. Indeed one can run the flag Grassmannian construction of this paper by replacing (1.5) with

\begin{align*}\underline {\mathsf{Gr}}_X^{{\rm double}}(S)(D,Z)=\underline {\mathsf{Bun}}_{(\widehat {Z}^{\widehat {D}})^{{\rm aff}} \setminus D} \times _{\underline {\mathsf{Bun}}_{(\widehat {Z}^{\rm aff,\ \widehat{D}}\setminus Z)^{{\rm aff}} \setminus D}} \{ {\rm triv} \},\end{align*}

where notation need some explanation. First of all, $(\widehat {Z}^{\widehat {D}})^{{\rm aff}}$ is the affinization of the ind-scheme $\widehat {Z}^{\widehat {D}}= \{ \widehat {Z}^{D_n}\}_{n\geq 0}$ , where $\widehat {Z}^{D_n}$ is the formal completion of $Z$ inside the $n$ th formal neighbourhood $D_n$ of $D$ inside the ambient scheme $X \times S$ . Secondly, $\widehat {Z}^{\rm aff,\ \widehat{D}}$ is the ind-scheme $\{ \widehat {Z}^{{\rm aff}, D_n}\}_{n\geq 0}$ (where $\widehat {Z}^{{\rm aff}, D_n}$ is the affinization of the completion of $Z$ inside $D_n$ ); then we subtract $Z$ from this ind-scheme, and we affinize it before finally subtracting $D$ : this is the meaningFootnote ¹⁵ of $(\widehat {Z}^{\rm aff,\ \widehat{D}}\setminus Z)^{{\rm aff}} \setminus D$ . Note that, with this definition, for $S= {\rm Spec}\, k$ , $X= \mathbb{A}_k^2$ , $D=\{ y=0 \}$ and $Z=\{ x=y=0 \}$ , we get $(\widehat {Z}^{\widehat {D}})^{{\rm aff}} \setminus D = {\rm Spec} (k[[x]]((y)))$ , and $(\widehat {Z}^{\rm aff,\ \widehat{D}}\setminus Z)^{{\rm aff}} \setminus D = {\rm Spec} (k((x))((y)))$ , and this explains the relation with [Reference Braverman and FinkelbergBF10, § 3]. Further relations between [Reference Braverman and FinkelbergBF10, Reference Braverman and FinkelbergBF12, Reference Braverman and KazhdanBK11] and this very local version of the flag Grassmannian are being investigated [Reference Maffei, Melani and VezzosiMMV25].

(iv) Replacing flags. One could run the construction of this paper (also for the flag Grassmannians listed in (ii) and (iii) above) by replacing $\operatorname {\underline {\mathsf{Fl}}}_{X,\bullet }$ with either the Ran space of nested Cartier divisors in $X$ , the Ran space of $\operatorname {\underline {\mathsf{Fl}}}_X$ , or even the Ran space of families of nested closed subsets of $X$ . This approach might also indicate how to extend our constructions to smooth projective varieties $X$ of dimension $\gt 2$ . This is currently being investigated in [Reference Hennion, Melani and VezzosiHMV].

(v) Open questions. Finally, here are a few questions (inspired by [Reference GaitsgoryGai13] and [Reference Mirkovic and VilonenMV07]) we are considering but have, so far, few clues about.

Let $\mathcal{G}r_X^{{\rm any}}$ be either $\mathcal{G}r_X$ or $\mathcal{G}r^{{\rm glob}}$ or $\mathcal{Gr}^{\rm glob,\ res}$ , and consider the obvious map $\mathcal{G}r^{{\rm any}}_X \to \underline {\mathsf{Bun}}_X$ . Are the fibres of this map de Rham-contractible? If not, does this map induce at least an interesting relation between the de Rham cohomology of $\mathcal{G}r^{{\rm any}}_X$ and the de Rham cohomology of $\underline {\mathsf{Bun}}_X\, ?$

Using (i) above, given any flag Grassmannian $\mathcal{G}r_X$ (i.e. the one in the paper or the ones listed in (ii) and (iii) above), can we interpret in a Tannakian-like way the categoryFootnote ¹⁶ of $\mathcal{L}^{+}G_X$ -equivariant perverse sheaves on $\mathcal{G}r_X$ ? To start with, one might fix a flag, and pose the same question for the flag Grassmannian at this fixed flag.

Related works. It is very much unclear what a topic named ‘Geometric Langlands program for surfaces’ should precisely mean or whether it exists at all, at least in the broad structured sense it has in the case of curves. With no claim of exhaustivity, we will merely list, among the ones we are aware of, those sources that actually inspired and motivated our research project. The paper [Reference Ginzburg, Kapranov and VasserotGKV95] was an inspiring and pioneering work in this direction but, at least to our knowledge, its lines of research have not been pursued since. The article [Reference Kapranov, Gelfand, Lepowsky and WilsonKap95] also contains very interesting suggestions about the possible existence of a Langlands program for higher-dimensional varieties, although here the emphasis is more on higher local fields and formal analogies with topological quantum field theories. As we already mentioned, A. Beilinson and V. Drinfeld’s [Reference Beilinson and DrinfeldBD04] (especially § 3.4.6), although dealing with the curve case, was the actual starting point of our project.

Another question about a possible Geometric Langlands program for surfaces, passed on to the third author by M. Kapranov, is whether it should concern ${\bf G}$ -bundles or ${\bf G}$ -gerbes. In any case, the set-up of formal gluing, nonlinear flags and the flag Grassmannian investigated in our paper for ${\bf G}$ -bundles makes sense, mutatis mutandis, for ${\bf G}$ -gerbes, too. However, algebraization is likely to be false for ${\bf G}$ -gerbes,Footnote ¹⁷ and this is a problem in developing this direction of research.

While finishing the paper, S. Raskin informed us that, in a 2010 conversation with him, I. Mirkovic proposed to use flags of the sort we consider here, in order to extend Belinson-Drinfeld’s Grassmannian to higher-dimensional varieties. We are not aware of the specific definition of the flag Grassmannian he proposed nor of subsequent publicly available work along these lines.

On the other hand, all of the interesting literature on the double affine Grassmannian (e.g. [Reference Braverman and FinkelbergBF10, Reference Braverman and FinkelbergBF12, Reference Braverman and KazhdanBK11] to list a few papers) seems to go in a different but related direction with respect to our paper. An ongoing collaboration between the second and third authors and Andrea Maffei (Pisa) is aiming at establishing a link between the two approaches [Reference Maffei, Melani and VezzosiMMV25].

Notation. Our base field will be $k=\mathbb{C}$ . All our schemes will, unless otherwise stated, be $k$ -schemes. For a scheme $T$ , ${\bf Sh}_T$ will denote the category of flat sheaves of sets on $T$ . Analogously, $\mathsf{dSt}_T$ will denote the $(\infty ,1)$ -category of flat derived stacks in $\infty$ -groupoids on $T$ . We write ${\bf Sh}$ and $\mathsf{dSt}$ when $T={\rm Spec} \, k$ . We will denote by ${\bf PoSets}$ the $1$ -category of (small) posetal categories, i.e. (small) categories whose $\textrm {Hom}$ -sets are either empty or a singleton. In a 1-category $C$ , if $: u: Z\to X$ , and $u': Z'\to X$ are monos, and there exists an isomorphism $u \simeq u'$ in $C/X$ , then this isomorphism is unique, and we will sometimes simply write $u=u'$ . The other relevant notation will be introduced and explained in the main text.