Proof. Part (i) is true for every affine scheme ${\mathcal{G}}$ of finite presentation over ${\mathcal{O}}$: Let ${\mathcal{G}}=\mathbb{A}_{{\mathcal{O}}}^{1}$ first. Denote by $I_{D}$ the invertible ideal defined by $D$ in ${\mathcal{O}}[\![D]\!]$. By the preceding discussion, the ring ${\mathcal{O}}[\![D]\!]/I_{D}$ is isomorphic to the global sections of $D$ and both are finite locally free ${\mathcal{O}}$-modules; cf. [StaPro, Tag 0B9C]. For any $a\in \mathbb{Z}$, we form $I_{D}^{a}$ as an invertible ${\mathcal{O}}[\![D]\!]$-module. For $a\leqslant b$, denote by $E_{[a,b]}$ the ${\mathcal{O}}$-module $I_{D}^{a}/I_{D}^{b}$ which is also finite locally free (hence reflexive) by an induction argument. As $b$ varies, the set of ${\mathcal{O}}$-modules $\{{E_{[a,b]}\}}_{b\geqslant a}$ forms an inverse system, and ${\mathcal{O}}[\![D]\!]=\text{lim}_{b\geqslant 0}E_{[0,b]}$ by definition. It follows that $I_{D}^{a}=\text{lim}_{b\geqslant a}E_{[a,b]}$ for any $a\in \mathbb{Z}$. In particular, we get ${\mathcal{O}}((D))=\text{colim}_{a}\text{lim}_{b\geqslant a}E_{[a,b]}$. As $E_{[a,b]}$ is a reflexive ${\mathcal{O}}$-module, we get for every ${\mathcal{O}}$-algebra $R$ that

(3.6)$$\begin{eqnarray}E_{[a,b]}\otimes _{{\mathcal{O}}}R=\operatorname{Hom}_{{\mathcal{O}}\text{-}\text{Mod}}((E_{[a,b]})^{\ast },R)=\operatorname{Hom}_{{\mathcal{O}}\text{-}\text{Sch}}(\operatorname{Spec}(R),\mathbb{V}_{[a,b]}),\end{eqnarray}$$

where $\mathbb{V}_{[a,b]}=\operatorname{Spec}(\text{Sym}^{\otimes }(E_{[a,b]})^{\ast })$ for every pair $b\geqslant a$. Taking limits shows that

$$\begin{eqnarray}\mathbb{A}_{{\mathcal{O}}}^{1}(R[\![D]\!])=R[\![D]\!]=\text{lim}_{b\geqslant 0}(E_{[0,b]}\otimes _{{\mathcal{O}}}R)\end{eqnarray}$$

is identified with the $R$-points of the affine ${\mathcal{O}}$-scheme $\text{lim}_{b\geqslant 0}\mathbb{V}_{[0,b]}$. The same argument shows that $R\mapsto \mathbb{A}_{{\mathcal{O}}}^{1}(R((D)))$ is representable by the ind-affine ind-scheme $\text{colim}_{a}\text{lim}_{b\geqslant a}\mathbb{V}_{[a,b]}$. This gives part (i) in the case ${\mathcal{G}}=\mathbb{A}_{{\mathcal{O}}}^{1}$. For the general case, one verifies that the $L^{+}$-construction (respectively, $L$-construction) commutes with taking finite products and equalizers, and that finite products and equalizers are constructed termwise in the category of ind-schemes. Hence, the lemma follows for $L^{+}\mathbb{A}_{{\mathcal{O}}}^{n}$ (respectively, $L\mathbb{A}_{{\mathcal{O}}}^{n}$). A finite presentation ${\mathcal{G}}=\operatorname{Spec}({\mathcal{O}}[t_{1},\ldots ,t_{n}]/(f_{1},\ldots ,f_{m}))$ realizes ${\mathcal{G}}$ as the equalizer of the two maps $\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}:\mathbb{A}_{{\mathcal{O}}}^{n}\rightarrow \mathbb{A}_{{\mathcal{O}}}^{m}$ where $\unicode[STIX]{x1D711}$ is given by the functions $f_{1},\ldots ,f_{m}$ and $\unicode[STIX]{x1D713}$ is the composition of the structure map with the zero section. Hence, $L^{+}{\mathcal{G}}$ (respectively, $L{\mathcal{G}}$) is the equalizer of $L^{+}\unicode[STIX]{x1D711}$ and $L^{+}\unicode[STIX]{x1D713}$ (respectively, $L\unicode[STIX]{x1D711}$ and $L\unicode[STIX]{x1D713}$) in the category of schemes (respectively, ind-schemes). As equalizers define closed subschemes and $L^{+}\mathbb{A}_{{\mathcal{O}}}^{n}$ is affine (respectively, $L\mathbb{A}_{{\mathcal{O}}}^{n}$ ind-affine), (i) follows.

Part (ii) is true for every smooth affine ${\mathcal{O}}$-scheme ${\mathcal{G}}$, necessarily of finite presentation. For $n\geqslant 0$, let $D_{n}=\operatorname{Spec}({\mathcal{O}}[\![D]\!]/I_{D}^{n+1})$ be the $n$th infinitesimal neighborhood of $D$ in $X$. The Weil restriction of scalars ${\mathcal{G}}_{n}:=\operatorname{Res}_{D_{n}/{\mathcal{O}}}({\mathcal{G}}\times _{X}D_{n})$ is a smooth affine ${\mathcal{O}}$-group scheme; cf. [Reference Bosch, Lütkebohmert and RaynaudBLR90, § 7.6, Theorem 4, Proposition 5]. For varying $n$, these groups fit into an inverse system ${\mathcal{G}}_{m}\rightarrow {\mathcal{G}}_{n}$ for $m\geqslant n$, and the natural map of functors

(3.7)$$\begin{eqnarray}L^{+}{\mathcal{G}}\,\overset{\simeq }{\longrightarrow }\,\operatorname{lim}_{n\geqslant 0}{\mathcal{G}}_{n}\end{eqnarray}$$

is an isomorphism. This proves (ii), and the lemma follows. ◻

Proof. Part (i) is deduced from the Beauville–Laszlo theorem [Reference Beauville and LaszloBL95]; cf. [Reference Beilinson and DrinfeldBD99, § 2.12] for a further discussion (cf. also [Reference Pappas and ZhuPZ13, Lemma 6.1]). For (ii), it is enough to prove that the projection $L{\mathcal{G}}\rightarrow \operatorname{Gr}_{{\mathcal{G}}}$ admits sections étale locally.

Let $R$ be an ${\mathcal{O}}$-algebra, and let ${\mathcal{F}}\rightarrow \hat{D}_{R}$ be a ${\mathcal{G}}$-torsor. We have to show that ${\mathcal{F}}$ is trivial étale locally on $R$, that is, admits a $\hat{D}_{R}$-section étale locally on $R$. By applying the lifting criterion for smoothness and an algebraization result for sections (algebraization is easy because ${\mathcal{F}}$ is affine), it is enough to show that the restriction ${\mathcal{F}}|_{D_{R}}\rightarrow D_{R}$ admits a section étale locally on $R$. Since the functor ${\mathcal{F}}|_{D_{R}}:R\text{-Alg}\rightarrow \text{Sets},B\mapsto {\mathcal{F}}(D_{B})$ commutes with filtered colimits (because ${\mathcal{F}}$ is a scheme locally of finite presentation [StaPro, 01ZC]), we may assume without loss of generality that $R$ is a strictly Henselian local ${\mathcal{O}}$-algebra. Now by assumption on $D$ the $R$-algebra $R^{\prime }:=\unicode[STIX]{x1D6E4}(D_{R},{\mathcal{O}}_{D_{R}})$ is finite, and hence a direct product of Henselian local rings $R^{\prime }=R_{1}\times \cdots \times R_{n}$; cf. [StaPro, 04GH]. As $R$ is strictly Henselian, each $R_{i}$ is strictly Henselian as well (because a finite extension of a separably closed field is separably closed). But each non-empty smooth scheme over a finite product of strictly Henselian local rings admits a section by Hensel’s lemma. This finishes the proof.◻

Proof. For (i), we may by (3.2) assume ${\mathcal{O}}=F$. It is immediate from Remark 3.3 that for any ${\mathcal{O}}$-algebra $R$, we have $R[\![D_{\operatorname{red}}]\!]\simeq R[\![D]\!]$ (respectively, $R((D_{\operatorname{red}}))\simeq R((D))$). Further, there is a canonical isomorphism

$$\begin{eqnarray}R[\![D_{\operatorname{red}}]\!]\,\overset{\simeq }{\longrightarrow }\,\mathop{\prod }_{i=1}^{n}R[\![D_{i}]\!]\quad \biggl(\text{respectively, }R((D_{\operatorname{red}}))\,\overset{\simeq }{\longrightarrow }\,\mathop{\prod }_{i=1}^{n}R((D_{i}))\biggr)\end{eqnarray}$$

because $X$ is of dimension $1$, and hence $D_{i}\cap D_{j}=\varnothing$ for $i\not =j$. Part (i) follows from Lemma 3.4(ii).

For (ii), first note that if we consider $D$ as the divisor on $X_{K}$ defined by the $K$-point $x$, then $\operatorname{Gr}_{(X_{K},{\mathcal{G}}_{X_{K}},D)}$ is the twisted affine Grassmannian over $K$; cf. Example 3.1(i). Let $\tilde{K}/F$ be the splitting field of $K$ which is a finite Galois extension with Galois group $\tilde{\unicode[STIX]{x1D6E4}}$. There is a canonical isomorphism of $\tilde{K}$-algebras

$$\begin{eqnarray}K\otimes _{F}\tilde{K}\,\overset{\simeq }{\longrightarrow }\,\mathop{\prod }_{\unicode[STIX]{x1D713}:K{\hookrightarrow}\tilde{K}}\tilde{K},\quad a\otimes b\,\longmapsto \,(\unicode[STIX]{x1D713}(a)\cdot b)_{\unicode[STIX]{x1D713}},\end{eqnarray}$$

which is $\tilde{\unicode[STIX]{x1D6E4}}$-equivariant for the action $\unicode[STIX]{x1D6FE}\ast (c_{\unicode[STIX]{x1D713}})_{\unicode[STIX]{x1D713}}\mapsto (\unicode[STIX]{x1D6FE}(c_{\unicode[STIX]{x1D713}}))_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D713}}$ on the target. Applying this isomorphism to $D\otimes _{F}\tilde{K}$, we obtain by (i) a $\tilde{\unicode[STIX]{x1D6E4}}$-equivariant isomorphism

(3.9)$$\begin{eqnarray}\operatorname{Gr}_{(X,{\mathcal{G}},D)}\otimes _{F}\tilde{K}\,\overset{\simeq }{\longrightarrow }\,\mathop{\prod }_{\unicode[STIX]{x1D713}}\operatorname{Gr}_{(X_{K},{\mathcal{G}}_{X_{K}},D)}\otimes _{K,\unicode[STIX]{x1D713}}\tilde{K}\end{eqnarray}$$

compatible with the actions of the loop groups. The canonical descent datum on the source in (3.9) induces a descent datum on the target of (3.9) which implies (ii). ◻

Proof. Since $X^{\prime }\rightarrow X$ is finite flat surjective, the closed subscheme $D^{\prime }\subset X^{\prime }$ is a relative effective Cartier divisor which is finite flat over ${\mathcal{O}}$. Hence, the functor $\operatorname{Gr}_{(X^{\prime },{\mathcal{G}}^{\prime },D^{\prime })}$ is well defined. Using Lemma 3.4(ii), the map (3.10) is induced for any ${\mathcal{O}}$-algebra $R$ by the canonical map of $R$-algebras

$$\begin{eqnarray}\operatorname{Spec}(R[\![D^{\prime }]\!])\,\rightarrow \,\operatorname{Spec}(R[\![D]\!])\times _{X}X^{\prime }\quad (\text{respectively, }\operatorname{Spec}(R((D^{\prime })))\,\rightarrow \,\operatorname{Spec}(R((D)))\times _{X}X^{\prime }).\end{eqnarray}$$

If $R$ is Noetherian, then the first map (hence the second map) is an isomorphism by [StaPro, 00MA] because $X^{\prime }\rightarrow X$ is finite. In particular, (3.10) is an isomorphism for any Noetherian ${\mathcal{O}}$-algebra $R$. As both functors in (3.10) commute with filtered colimits of ${\mathcal{O}}$-algebras, the corollary follows.◻

Proof. If $T\rightarrow \operatorname{Spec}({\mathcal{O}}^{\prime })$ is a test scheme, then $X\times _{\operatorname{Spec}({\mathcal{O}})}(\operatorname{Spec}({\mathcal{O}})\times _{\operatorname{Spec}({\mathcal{O}}^{\prime })}T)=X\times _{\operatorname{Spec}({\mathcal{O}}^{\prime })}T$. The lemma follows immediately from the definitions.◻

Proof. Let $R$ be an ${\mathcal{O}}$-algebra. If ${\mathcal{G}}=\operatorname{Gl}_{n,X}$, then $\operatorname{Gr}_{{\mathcal{G}}}(R)$ classifies rank-$n$ vector bundles ${\mathcal{E}}$ on $X_{R}$ together with an isomorphism ${\mathcal{E}}|_{U_{R}}\simeq {\mathcal{O}}_{U_{R}}^{n}$ where $U_{R}:=(X\backslash D)_{R}$. Let ${\mathcal{I}}_{D_{R}}\subset {\mathcal{O}}_{X_{R}}$ be the invertible ideal sheaf defined by $D_{R}\subset X_{R}$. For $N\geqslant 1$, let $\operatorname{Gr}_{{\mathcal{G}},N}$ be the ${\mathcal{O}}$-space whose $R$-valued points are rank-$n$ vector bundles ${\mathcal{E}}$ on $X_{R}$ such that, as ${\mathcal{O}}_{X_{R}}$-modules,

$$\begin{eqnarray}({\mathcal{I}}_{D_{R}}^{N})^{n}\,\subset \,{\mathcal{E}}\,\subset \,({\mathcal{I}}_{D_{R}}^{-N})^{n}.\end{eqnarray}$$

Every vector bundle is locally free and by bounding the poles (respectively, zeros) of basis elements, one gets, as ${\mathcal{O}}$-spaces,

$$\begin{eqnarray}\operatorname{colim}_{N\geqslant 1}\operatorname{Gr}_{{\mathcal{G}},N}\,\overset{\simeq }{\longrightarrow }\,\operatorname{Gr}_{{\mathcal{G}}}.\end{eqnarray}$$

We claim that each $\operatorname{Gr}_{{\mathcal{G}},N}$ is representable by a $\operatorname{Quot}$-scheme as follows. The ${\mathcal{O}}_{X_{R}}$-module ${\mathcal{E}}_{N,R}:=({\mathcal{I}}_{D}^{-N}/{\mathcal{I}}_{D}^{N})^{n}\otimes _{{\mathcal{O}}}R$ is coherent and locally free over $R$. Let $\operatorname{Quot}_{N}$ be the ${\mathcal{O}}$-space whose $R$-points are coherent ${\mathcal{O}}_{X_{R}}$-module quotients ${\mathcal{E}}_{N,R}{\twoheadrightarrow}{\mathcal{Q}}$ which are locally free $R$-modules. The functor $\operatorname{Quot}_{N}$ is representable by a projective ${\mathcal{O}}$-scheme by the theory of $\operatorname{Quot}$-schemes applied to the finite flat ${\mathcal{O}}$-scheme $2ND$, and the coherent ${\mathcal{O}}_{2ND}={\mathcal{O}}_{X}/{\mathcal{I}}_{D}^{2N}$-module ${\mathcal{E}}_{N,{\mathcal{O}}}$. More precisely, in the notation of [Reference FantechiFan05, § 5.1.4], one has a finite disjoint union

$$\begin{eqnarray}\operatorname{Quot}_{N}=\coprod _{r\in \mathbb{Z}_{{\geqslant}0}}\operatorname{Quot}_{{\mathcal{E}}_{N,{\mathcal{O}}}/2ND/\operatorname{Spec}({\mathcal{O}})}^{r,{\mathcal{O}}_{2ND}},\end{eqnarray}$$

and the representability result is then a theorem of Grothendieck [Reference FantechiFan05, § 5.5.2, Theorem 5.14]. Note that the structure sheaf ${\mathcal{O}}_{2ND}$ is relatively ample for $2ND\rightarrow \operatorname{Spec}({\mathcal{O}})$ because the map is finite (cf. [StaPro, Tag 01VG, 28.35.6]). Concretely, $\operatorname{Quot}_{N}$ is the closed subscheme of the Grassmannian

$$\begin{eqnarray}\operatorname{Quot}_{N}\,{\hookrightarrow}\,\operatorname{Grass}({\mathcal{E}}_{N,{\mathcal{O}}}),\end{eqnarray}$$

which is cut out by the condition that the quotients are stable under the finitely many nilpotent operators $u_{1},\ldots ,u_{n}$ on ${\mathcal{E}}_{N,{\mathcal{O}}}$ induced by some presentation $2ND=\operatorname{Spec}({\mathcal{O}}[u_{1},\ldots ,u_{n}]/J)$. Hence, to prove the lemma it is enough to show that, as functors,

(3.11)$$\begin{eqnarray}\operatorname{Gr}_{{\mathcal{G}},N}\,\overset{\simeq }{\longrightarrow }\,\operatorname{Quot}_{N},\quad {\mathcal{E}}\longmapsto ({\mathcal{I}}_{D_{R}}^{-N})^{n}/{\mathcal{E}}.\end{eqnarray}$$

We need to check that ${\mathcal{Q}}:=({\mathcal{I}}_{D_{R}}^{-N})^{n}/{\mathcal{E}}$ is a locally free $R$-module. This follows from the isomorphism as $R$-modules ${\mathcal{O}}_{U_{R}}^{n}/{\mathcal{E}}\simeq \bigoplus _{k\geqslant 0}{\mathcal{I}}_{D_{R}}^{-k-1}{\mathcal{E}}/{\mathcal{I}}_{D_{R}}^{-k}{\mathcal{E}}$, and the short exact sequence

$$\begin{eqnarray}0\,\rightarrow \,({\mathcal{I}}_{D_{R}}^{-N})^{n}/{\mathcal{E}}\,\rightarrow \,{\mathcal{O}}_{U_{R}}^{n}/{\mathcal{E}}\,\rightarrow \,{\mathcal{O}}_{U_{R}}^{n}/({\mathcal{I}}_{D_{R}}^{-N})^{n}\,\rightarrow \,0;\end{eqnarray}$$

cf. also the argument in [Reference ZhuZhu17, Lemma 1.1.5]. Conversely, let ${\mathcal{Q}}\in \operatorname{Quot}_{N}(R)$, and define the coherent ${\mathcal{O}}_{X_{R}}$-module

$$\begin{eqnarray}{\mathcal{E}}\stackrel{\text{def}}{=}\ker (({\mathcal{I}}_{D_{R}}^{-N})^{n}\rightarrow {\mathcal{E}}_{N,R}\rightarrow {\mathcal{Q}}).\end{eqnarray}$$

We need to show that ${\mathcal{E}}$ is a rank-$n$ vector bundle on $X_{R}$. Covering $X_{R}$ with affine schemes, we may assume $X_{R}=\operatorname{Spec}(S)$ is affine. Let $\mathfrak{p}\subset S$ be a prime ideal lying over a prime ideal $\mathfrak{m}:=\mathfrak{p}\cap R\subset R$. By [StaPro, Tag 00M] applied to the map of local rings $R_{\mathfrak{m}}\rightarrow S_{\mathfrak{p}}$ and the module ${\mathcal{E}}_{\mathfrak{p}}$ (note that ${\mathcal{E}}_{\mathfrak{p}}$ is still $R_{\mathfrak{m}}$-flat), to prove ${\mathcal{E}}_{\mathfrak{p}}$ is free over $S_{\mathfrak{p}}$ we are reduced to the case where $R$ is a field. In the case where $R$ is a field, ${\mathcal{E}}\subset ({\mathcal{I}}_{D_{R}}^{-N})^{n}$ is a torsion-free rank-$n$ submodule, and since $X_{R}\rightarrow \operatorname{Spec}(R)$ is a smooth curve, ${\mathcal{E}}$ is a vector bundle.◻

Proof. Following the proof of [Reference ZhuZhu17, Proposition 1.2.6], it is enough to establish the analogue of [Reference ZhuZhu17, Lemma 1.2.7]. Let $R$ an ${\mathcal{O}}$-algebra, and let $p:V\rightarrow \hat{D}_{R}$ be an affine scheme of finite presentation. Let $s$ be a section of $p$ over $\hat{D}_{R}^{o}$. We need to prove that the presheaf assigning, to any $R$-algebra $R^{\prime }$, the set of sections $s^{\prime }$ of $p$ over $\hat{D}_{R^{\prime }}$ such that $s^{\prime }|_{\hat{D}_{R^{\prime }}^{o}}=s|_{\hat{D}_{R^{\prime }}^{o}}\!$, is representable by a closed subscheme of $\operatorname{Spec}(R)$. Indeed, choosing a closed embedding $V\subset \mathbb{A}_{\hat{D}_{R}}^{n}$ for some $n\gg 0$ and using that $R[\![D]\!]\subset R((D))$ is injective, we reduce to the case $V=\mathbb{A}_{\hat{D}_{R}}^{n}$. The presheaf in question is representable by the locus on $\operatorname{Spec}(R)$ where the class $\bar{s}$ of the section $s\in V(\hat{D}_{R}^{o})=R((D))^{n}$ in $(R((D))/R[\![D]\!])^{n}$ vanishes. With the notation of Lemma 3.2, we have $\bar{s}\in E_{[-N,0]}\otimes _{{\mathcal{O}}}R$ for some $N\gg 0$. As $E_{[-N,0]}$ is a reflexive ${\mathcal{O}}$-module, we see that giving an element of $E_{[-N,0]}\otimes _{{\mathcal{O}}}R$ is equivalent to giving a map of $R$-schemes $\operatorname{Spec}(R)\rightarrow \mathbb{V}(E_{[-N,0]}\otimes _{{\mathcal{O}}}R)$. Then the presheaf in question is representable by the equalizer of the two maps corresponding to the elements $\bar{s},0\in E_{[-N,0]}\otimes _{{\mathcal{O}}}R$ which is a closed subscheme of $\operatorname{Spec}(R)$.◻

Proof. Part (i) is immediate from Lemma 3.8 and Proposition 3.10.

For (ii), we use part (i) together with Lemma 3.12 below. Note that the diagonal of $\operatorname{Gr}_{{\mathcal{G}}}$ being representable by a closed immersion follows from the same property of $\operatorname{Gr}_{\operatorname{Gl}_{n,{\mathcal{O}}}}$ and the effectivity of descent for closed immersions. Further, if ${\mathcal{O}}\rightarrow {\mathcal{O}}^{\prime }$ is étale, then the method of Lemma 3.12 shows that an $L^{+}{\mathcal{G}}\otimes _{{\mathcal{O}}}{\mathcal{O}}^{\prime }$-stable presentation of $\operatorname{Gr}_{{\mathcal{G}}}\otimes _{{\mathcal{O}}}{\mathcal{O}}^{\prime }$ induces an $L^{+}{\mathcal{G}}$-stable presentation of $\operatorname{Gr}_{{\mathcal{G}}}$ (because $L^{+}{\mathcal{G}}$ is affine and flat, and taking the scheme-theoretic image commutes with flat base change).

For (iii), note that after an étale cover ${\mathcal{O}}\rightarrow {\mathcal{O}}^{\prime }$, the group scheme $G_{{\mathcal{O}}^{\prime }}:=G\otimes _{{\mathcal{O}}}{\mathcal{O}}^{\prime }$ is split reductive and, in particular, linearly reductive. If we choose a closed immersion $G_{{\mathcal{O}}^{\prime }}{\hookrightarrow}\operatorname{Gl}_{n,{\mathcal{O}}^{\prime }}$, then the quotient $\operatorname{Gl}_{n,{\mathcal{O}}^{\prime }}/G_{{\mathcal{O}}^{\prime }}$ is representable by an affine scheme by [Reference AlperAlp14, Corollary 9.7.7], and (iii) follows from (ii).◻

Proof. Given a presentation $U=\operatorname{colim}_{i\in I}U_{i}$ with $U_{i}$ being schemes, we need to construct a presentation $X=\operatorname{colim}_{i\in I}X_{i}$ with $X_{i}$ being algebraic spaces. For each $i$, consider $U_{i}\subset U\rightarrow X$. We define $X_{i}^{\prime }$ to be the scheme-theoretic image of the map

(3.12)$$\begin{eqnarray}U_{i}\times _{X}U\,\subset \,U\times _{X}U\,\overset{p_{2}\;}{\longrightarrow }\,U.\end{eqnarray}$$

This is well defined for the following reason. Since $U_{i}\times _{X}U$ is a quasi-compact scheme, the map (3.12) factors through $U_{j}\subset U$ for some $j\gg i$. In either case, $U\rightarrow X$ quasi-compact or $U$ quasi-separated, the map (3.12) is quasi-compact. By [StaPro, 01R8], the scheme-theoretic image behaves well for quasi-compact maps, and $X_{i}^{\prime }\subset U_{j}$ is a quasi-compact closed subscheme. As scheme-theoretic images of quasi-compact maps commute with flat base change [StaPro, Tag 081I], the scheme $X_{i}^{\prime }$ is equipped with a descent datum relative to $U\rightarrow X$, and defines a closed ${\mathcal{O}}$-subspace $X_{i}\subset X$ together with an étale surjective map $X_{i}^{\prime }\rightarrow X_{i}$. As $X_{i}\subset X$ is closed, the diagonal of $X_{i}$ is schematic, and $X_{i}$ is a quasi-compact algebraic space. By construction the $X_{i}$ form a filtered direct system indexed by the poset $I$, and the canonical map $\operatorname{colim}_{i\in I}X_{i}\rightarrow X$ is an isomorphism (because $U\rightarrow X$ is a sheaf surjection, and $\operatorname{colim}_{i}X_{i}^{\prime }=U$ by construction).◻

Proof. As in the proof of Lemma 3.16, we reduce to the case where $\unicode[STIX]{x1D712}$ is a standard dominant cocharacter. Then $\unicode[STIX]{x1D712}$ corresponds to a $\mathbb{Z}$-grading on $V:={\mathcal{O}}^{n}$, say $V=\bigoplus _{i\in \mathbb{Z}}V_{i}$, compatible with the standard ${\mathcal{O}}$-basis of $V$. The group $M$ (respectively, $P^{+}$/$P^{-}$) is a standard Levi (respectively, standard parabolic) of automorphisms of $V$ preserving the grading (respectively, the ascending/descending filtration induced from the grading). In the description of Lemma 3.8, the subfunctor $\operatorname{Gr}_{M}$ (respectively, $\operatorname{Gr}_{P^{\pm }}$) consists of those vector bundles ${\mathcal{E}}\in \operatorname{Gr}_{G}(R)$ compatible with the grading (respectively, filtration induced by the grading) on $V\otimes _{{\mathcal{O}}}{\mathcal{O}}_{U_{R}}$. Likewise, the grading on $V$ induces in the notation of Lemma 3.8 gradings on ${\mathcal{E}}_{N,{\mathcal{O}}}=V\otimes _{{\mathcal{O}}}({\mathcal{I}}_{D}^{-N}/{\mathcal{I}}_{D}^{N})$ for each $N\geqslant 1$. As in Lemma 3.16, we have a closed $\mathbb{G}_{m}$-equivariant immersion, and hence the diagram of ${\mathcal{O}}$-schemes

is cartesian, and likewise on attractor (respectively, repeller) schemes. The equality $\operatorname{Grass}({\mathcal{E}}_{N,{\mathcal{O}}})^{0}=\prod _{i\in \mathbb{Z}}\operatorname{Grass}(V_{i}\,\otimes _{{\mathcal{O}}}\,({\mathcal{I}}_{D}^{-N}/{\mathcal{I}}_{D}^{N}))$ is immediate, and one checks that $\operatorname{Grass}(V\otimes _{{\mathcal{O}}}({\mathcal{I}}_{D}^{-N}/{\mathcal{I}}_{D}^{N}))^{\pm }$ is the subfunctor of those subspaces in ${\mathcal{E}}_{N,{\mathcal{O}}}$ compatible with the filtration. The lemma follows.◻

Proof. Replacing ${\mathcal{O}}$ by an étale cover, we may assume that there exists a closed embedding $G{\hookrightarrow}\operatorname{Gl}_{n,{\mathcal{O}}}$. By the proof of Lemma 3.14 (respectively, Lemma 3.2(i)), the induced map $L^{-}G\rightarrow L^{-}\operatorname{Gl}_{n,{\mathcal{O}}}$ is a closed immersion.

Let $\unicode[STIX]{x1D712}^{\prime }:\mathbb{G}_{m,{\mathcal{O}}}\overset{\unicode[STIX]{x1D712}\,}{\rightarrow }G\rightarrow \operatorname{Gl}_{n,{\mathcal{O}}}$, and denote the fixed point group (respectively, attractor/repeller group) by $L$ (respectively, $Q^{\pm }$). It is straightforward to check $L^{-}M=L^{-}G\cap L^{-}L$ (respectively, $L^{-}P^{\pm }=L^{-}G\cap L^{-}Q^{\pm }$) and $(L^{-}G)^{0}=L^{-}G\cap (L^{-}\operatorname{Gl}_{n,{\mathcal{O}}})^{0}$ (respectively, $(L^{-}G)^{\pm }=L^{-}G\cap (L^{-}\operatorname{Gl}_{n,{\mathcal{O}}})^{\pm }$). Hence, we may assume $G=\operatorname{Gl}_{n,{\mathcal{O}}}$.

After passing to a Zariski cover of ${\mathcal{O}}$, we may assume that $\unicode[STIX]{x1D712}$ is a standard dominant cocharacter; cf. proof of Lemma 3.16. We have for every ${\mathcal{O}}$-algebra $R$,

$$\begin{eqnarray}(L^{-}\operatorname{Gl}_{n,{\mathcal{O}}})^{0}(R)=\{g\in G(\mathbb{P}_{R}^{1}\backslash D_{R})\mid \forall S\in \operatorname{(R - Alg)},\unicode[STIX]{x1D706}\in \mathbb{G}_{m}(S):\,\unicode[STIX]{x1D712}(\unicode[STIX]{x1D706})\cdot g\cdot \unicode[STIX]{x1D712}(\unicode[STIX]{x1D706})^{-1}=g\}.\end{eqnarray}$$

Let $g\in (L^{-}\operatorname{Gl}_{n,{\mathcal{O}}})^{0}(R)$. To show $g\in (L^{-}M)(R)$, we can take $S=R[t,t^{-1}]$ to see that the desired entries in the matrix $g$ vanish. The case of $(L^{-}G)^{\pm }$ is similar, and the lemma follows.◻

Proof. By [StaPro, Tag 01QV], the map $\unicode[STIX]{x1D704}$ factors as an open immersion followed by a closed immersion: $Y\rightarrow \bar{Y}\rightarrow Z$. As $\unicode[STIX]{x1D704}$ is bijective, we have $Y=\bar{Y}$ and $\unicode[STIX]{x1D704}$ is a bijective closed immersion. Being an isomorphism is local on the target, and we may assume that $Z=\operatorname{Spec}(A)$ and hence $Y=\operatorname{Spec}(B)$ are affine. The map of ${\mathcal{O}}$-algebras $\unicode[STIX]{x1D704}^{\#}:A\rightarrow B$ is surjective (because closed immersion), and each element in $I:=\ker (\unicode[STIX]{x1D704}^{\#})$ is nilpotent (because $\unicode[STIX]{x1D704}^{\#}$ is bijective on spectra). It is enough to show that for the localization $I_{\mathfrak{m}}=0$ for all maximal ideals $\mathfrak{m}\subset {\mathcal{O}}$. Without loss of generality, we may assume that ${\mathcal{O}}$ is local with maximal ideal $\mathfrak{m}$. If $\mathfrak{m}A=A$, that is, the fiber of $Z$ over $\mathfrak{m}$ is empty, there is nothing to prove, and we may assume that $\mathfrak{m}A\subset A$ is a proper ideal. As $A/\mathfrak{m}^{n}A\rightarrow B/\mathfrak{m}^{n}B$ is an isomorphism for all $n\geqslant 1$, we have $I\subset \bigcap _{n\geqslant 1}\mathfrak{m}^{n}A$. But since $\mathfrak{m}^{n}A=(\mathfrak{m}A)^{n}$ and $\mathfrak{m}A\subset A$ is a proper ideal in a Noetherian ring, we have $\bigcap _{n\geqslant 1}\mathfrak{m}^{n}A=0$ by Krull’s intersection theorem. The lemma follows.◻