1.1.1 Ind-schemes: definitions

In this paper by ind-scheme we mean a functor ${\mathcal {F}}:{\mathsf {CAlg}}_{k}\longrightarrow \mathsf {Set}$ that can be realized as a colimit of schemes $X_i$ indexed by a partially ordered filtered set $(I,<)$ and such that for $i<j$ the map $X_i\longrightarrow X_j$ is a closed immersion.

It is important to notice, however, that we do not assume that the colimit is over a countable poset: unfortunately none of our ind-schemes will have this property.

We say that an ind-scheme ${\mathcal {F}}$ is an ind-affine ind-scheme if, furthermore, the $X_i$ ’s can be chosen to be affine schemes.

1.1.2 Jets and loops

Recall that given a functor $X:{\mathsf {CAlg}}_{k}\longrightarrow \mathsf {Set}$ then the loop space $LX:{\mathsf {CAlg}}_{k}\longrightarrow \mathsf {Set}$ of ${\mathcal {F}}$ and the jet space $JX:{\mathsf {CAlg}}_{k}\longrightarrow \mathsf {Set}$ of ${\mathcal {F}}$ are defined as the functors

1.1.3 The affine Grassmannian

If $H:{\mathsf {CAlg}}_{k}\longrightarrow \mathsf {Groups}$ then its affine Grassmannian is defined as the presheaf on ${\mathsf {CAlg}}_{k}$ given by the quotient

$$ \begin{align*}\mathcal G r_H=\frac{LH}{JH}(R)=\frac{LH(R)}{JH(R)}.\end{align*} $$

If H is a smooth algebraic reductive group, by [Reference Česnavičius4, Theorem 3.4] it is known that this presheaf is already a sheaf for the fppf topology. In the case $H=G$ is a solvable group this result is easier and follows from the following Lemma.

Lemma 1.2. If G is solvable group then there exists an ind-affine ind-scheme

$$ \begin{align*}\Sigma_{G} \subset LG \end{align*} $$

such that for all k algebras R, every element in $LG(R)/JG(R)$ has a unique representative in $\Sigma _{G} (R)$ . This implies that $LG/JG$ is an ind-affine ind-scheme and, in particular, a sheaf for the fppf topology.

The proof of Lemma 1.2 will be given in the next sections: in Section 1.2.2 we reduce the proof to the case of the additive and of the multiplicative group, and in Section 1.3.1 we will study the case of $\mathbb G_m$ , the case of $\mathbb G_a$ being trivial.

Lemma 1.2, together with [Reference Česnavičius4, Theorem 3.4], implies the following fact that is maybe worth noticing.

1.2.1 Representability for solvable G

In the rest of this Section we prove the following results.

Theorem 1.7. Assume G is solvable and let $G'$ be equal to $JLG$ , $LJG, JJG$ or $G^{(2)}$ . Then there exists an ind-affine ind-scheme

$$ \begin{align*}\Sigma[G']\subset LLG \end{align*} $$

such that for all k algebras R, every element in $LLG(R)/G'(R)$ has a unique representative in $\Sigma [G'](R)$ .

A similar result holds in the case of the affine Grassmannian of the jet group $LJG/JJG$ : there exists $\Sigma ^{J}\subset LJG$ with analogous properties.

This implies that the Grassmannians $\mathcal G r^L_G$ , $ ^L\mathcal G r_{G}$ , $\mathcal G r^J_{G}$ , $\mathcal G r^{big}_{G}$ and $\mathcal G r^{(2)}_{G}$ from ${\mathsf {CAlg}}_{k}$ to $\mathsf {Set}$ of Definition 1.5 are ind-affine ind-schemes and, in particular, they are sheaves for the fppf topology.

Depending on which affine Grassmannian from Definition 1.5 we are considering, we denote the corresponding ind-schemes by $\Sigma ^L_G$ , ${}^L\Sigma _G, \Sigma _G^{\mathrm {big}}, \Sigma _G^{(2)}$ , $\Sigma _G^J$ .

1.2.2 Reduction to the case of the multiplicative group

The proof of Theorem 1.7 and of Lemma 1.2 reduces to the case of the multiplicative group. In this section we assume the results true for $\mathbb G_m$ and we deduce the results for any solvable group.

We start by remarking that Theorem 1.7, and Lemma 1.2 are easily verified in the case of $G=\mathbb G_a$ . In fact, for Lemma 1.2 it is enough to take $\Sigma _{G}(R)=tR[t^{-1}]$ , while Theorem 1.7 is a consequence of the following lemma.

Lemma 1.9. Let $G=\mathbb G_a$ , and let $\mathcal G r$ be any of the Grassmannians introduced in Definition 1.5. Then there exists an ind-affine ind-scheme

$$ \begin{align*}\Sigma \subset LLG\end{align*} $$

(or $\Sigma \subset LJG$ , in the case of the jet Grassmannian) such that for all k-algebras R, every element in $\mathcal G r(R)$ has a unique representative in $\Sigma (R)$ . Moreover $\Sigma $ is always the colimit of infinite dimensional affine spaces.

Proof. All Grassmannians are treated similarly, and we only give details for the case of $\mathcal G r^L$ . Recall that by definition we have

If we set $\Sigma _G^L(R) = t^{-1}R[t^{-1}](\!(s)\!)$ , it is clear that every element $f \in \mathcal G r^L(R)$ has a unique representative in $\Sigma _G^L(R)$ , as desired. Moreover, $\Sigma _G^L$ can be represented by an ind-affine ind-scheme as follows. Let h be an integer, and let $\alpha $ be a function $\mathbb Z \to \mathbb Z$ . Denote by $ (\Sigma _G^L)_{h,\alpha }(R) $ the set of series of the form

$$\begin{align*}\sum_{i\geq h} f_i(t^{-1})s^i \end{align*}$$

where $f_i(t^{-1}) \in t^{-1}R[t^{-1}]$ has degree at most $-\alpha _{i}$ . The functor $ R \mapsto (\Sigma _G^L)_{h,\alpha }(R)$ is obviously represented by an infinite dimensional affine space whose algebra of functions is given by

$$\begin{align*}k[x_{i,j}: i\geq -h \text{ and } \alpha_i \leq j\leq -1] \end{align*}$$

where the coordinates $x_{i,j}$ corresponds to the coefficients of $f_i$ . On the sets of all couples $(h,\alpha )$ , we define a partial order where $(h,\alpha ) \geq (h', \alpha ')$ if and only if $h \geq h'$ and $\alpha _i \leq \alpha _i'$ for all i. The functor $\Sigma _G^L$ is then the colimit of the $(\Sigma _G^L)_{h,\alpha }$ over this poset.

The reduction will also be based on the following two lemmas.

Proof. The proof is the same for all Grassmannians. For example, suppose Theorem 1.7 for $\mathcal G r_G^{L}$ holds for H and K. Let $\varphi $ be the section of $G \to K$ . Notice that $\varphi $ induces maps

$$\begin{align*}LL(\varphi) : LLK \to LLG, \quad \quad LJ(\varphi) : LJK \to LJG \end{align*}$$

which are sections of $LLG \to LLK$ and $LJG \to LJK$ , respectively. In particular, we have exact sequences

$$\begin{align*}1 \to LLH \to LLG \to LLK \to 1 \end{align*}$$

and

$$\begin{align*}1 \to LJH \to LJG \to LJK \to 1. \end{align*}$$

For every k-algebra R, we now set

$$ \begin{align*}\Sigma^L_{G}(R)= LL(\varphi)(\Sigma_{K}^L(R))\times \Sigma_{H}^L(R).\end{align*} $$

This defines a subsheaf of $LLG$ , and it is straightforward to check that it has the desired property.

1.3.1 Proof of Lemma 1.2

For $G=\mathbb G_m$ , Lemma 1.2 is a consequence of the ind-schematic description of $LG$ given in [Reference Osipov and Zhu17, Lemma 2.1]. We provide here an explicit proof that will be used in the proof of Theorem 1.7. For $m\in \mathbb Z$ , consider the sub-presheaf of $LG$ given by

(1.1) $$ \begin{align} \Sigma^{}_{G,m}(R)=\{f(t)\in R(\!(t)\!)^*: f \text{ is of the form } t^m+t^{m-1}\varepsilon(t)\text{ where }\varepsilon(t)\in R[t^{-1}] \text{ is nilpotent}\} \end{align} $$

It is easily seen that $\Sigma _{G,m}$ is an ind-affine ind-scheme. The disjoint union as sheaves (not as presheaves) of these ind-schemes is also an ind-affine ind-scheme that we will denote by $\Sigma _G$ . We notice that the reduced scheme structure on $\Sigma _{G,m}$ is just a point, hence $\Sigma _{G,m}$ is a formal thickening of $\operatorname {\mathrm {Spec}} k$ and, as a topological space, $\Sigma _G$ is just $\mathbb Z$ with the discrete topology.

We now prove that, for all R, every element $LG(R)/JG(R)$ has a unique representative in $\Sigma _G(R)$ .

By Lemma 1.12 given $f(t)\in R(\!(t)\!)^*$ there exists a decomposition $R=R_1\times \cdots \times R_h$ such that $f=f_1+\dots +f_h$ and $f_i(t)\in R_{i}(t)$ is of the form

$$ \begin{align*}f_i(t)=t^{m_i}+t^{m_{i}-1}\varepsilon_i(t)+t^{m_{i}+1}g_i(t) \end{align*} $$

where and $\varepsilon _i(t)\in R_i[t^{-1}]$ is nilpotent. Moreover if the $m_i$ are distinct then this decomposition is uniquely determined up to permutation.

Therefore, it remains to prove that, given $f(t)\in R(\!(t)\!)$

$$ \begin{align*}f(t)=t^{m}+t^{m-1}\varepsilon (t)+t^{m+1}g(t) \end{align*} $$

with and $\varepsilon (t)\in R[t^{-1}]$ nilpotent, there exists a unique such that $f(t)a(t)\in \Sigma _{G,m}(R)$ . In this proof we can assume $m=0$ .

Uniqueness of a(t). We prove that if $f(t)=1+\varepsilon _{-1}t^{-1}+\dots +\varepsilon _{-n}t^{-n}$ and is such that $f(t)a(t)\in \Sigma _{G,m}(R)$ then $a=1$ . Let I be the ideal generated by $\varepsilon _{-1},\dots , \varepsilon _{-n}$ . There exists k such that $I^k=0$ . If $a(t)=\sum a_i t^i$ and $f(t)a(t)\in \Sigma _{G,m}$ we immediately see that $m=0$ and that

$$ \begin{align*} a_1&=-\varepsilon_{-1}a_2-\dots-\varepsilon_{-n}a_{n+1}\\ a_2&=-\varepsilon_{-1}a_3-\dots-\varepsilon_{-n}a_{n+2}\\ \cdots& \end{align*} $$

Hence we obtain $a_i\in I$ for all i and inductively we have $a_i\in I^h$ for all h. Hence $a_i=0$ for all $i>0$ . Finally we have $1=a_0$ .

Existence of a(t). Let

$$ \begin{align*}f(t)=1+\varepsilon_{-1}t^{-1}+\dots+\varepsilon_{-n}t^{-n}+\sum_{i>0}c_it^i \end{align*} $$

and let I, as above, be the ideal generated by the $\varepsilon _i$ so that there exists k be such that $I^k=0$ . In order to argue by induction on k we prove a stronger statement: there exists $a(t)$ such that

$$ \begin{align*}f(t)a(t)=1+\tilde \varepsilon_{-1}t^{-1}+\dots+\tilde \varepsilon_{-N}t^{-N} \end{align*} $$

with $\tilde \varepsilon _i\in I$ .

For $k=1$ the claim is trivial since . Assume now $k>1$ , let $\overline R:=R/I^{k-1}$ and let $\overline f$ be the reduction of f modulo $I^{k-1}$ . By induction there exists such that $\overline f(t)\overline b(t)$ has the required form. Hence

$$ \begin{align*}f(t)b(t)=c_0'+\varepsilon^{\prime\prime}_{-1}t^{-1}+\dots+\varepsilon^{\prime\prime}_{-n}t^{-n}+\sum_{i>0}c^{\prime}_it^i \end{align*} $$

where $\varepsilon ^{\prime \prime }_i\in I$ , $c_i'\in I^{k-1}$ and $c_0'\in R^*$ . Up to multiplying by the inverse of $c_0'$ , we can also assume that $c_0'=1$ . Notice that $1-c(t)=1-\sum _{i>0} c_i'(t)t^i$ is an invertible element of and that

$$ \begin{align*}f(t)b(t)(1-c(t))=c_0"+\varepsilon^{\prime\prime}_{-1}t^{-1}+\dots+\varepsilon^{\prime\prime}_{-n}t^{-n}.\end{align*} $$

with $c_0"$ invertible and $\varepsilon _i"\in I$ . Finally, we may multiply by the inverse of $c_0"$ .

1.3.2 Proof of Theorem 1.7 in the case of $\mathcal G r^J_G$

We will now prove Theorem 1.7 in the case of $G=\mathbb G_m$ (as already noticed this implies the Theorem for any solvable G). We analyze the possible Grassmannians one by one. We start with the affine Grassmannian of the jet group $JG$ . We give the details of the ind-scheme structure only in this case. All the other cases are analogous.

We start by defining a family of infinite dimensional subschemes of $LJG$ as follows. Fix an integer m, a positive number p, and a function $\alpha : \mathbb N \to \mathbb Z$ .

We let $\Sigma ^J_{G,m,p,\alpha }(R)$ be the set of series of the form $t^m+\sum _{i,j} f_{ij}t^is^j$ , where the sums runs over $\alpha _j \leq i < m$ and $j\geq 0$ , and such that $(f_{i0})^p=0$ for all i. It is immediate that we have an isomorphism

$$ \begin{align*}\Sigma^J_{G,m,p,\alpha}=\operatorname{\mathrm{Spec}}\left( \frac{k[x_{ij} \colon j\geq 0,\quad \alpha_j\leq i<m]}{\left(x_{i0}^p=0\text{ for } \alpha_0\leq i< m \right)} \right). \end{align*} $$

In particular, $\Sigma ^J_{G,m,p,\alpha }$ is an affine scheme; in fact, more precisely, it is an infinitesimal neighborhood of a finite-codimension linear subspace inside an infinite dimensional affine space.

Let us denote by $\mathcal {I}$ the set of couples $(p,\alpha )$ as above. We say that $(p,\alpha )\geq (q,\beta )$ if $p\geq q$ and $\alpha _i \leq \beta _i$ for every i. Notice that, for a fixed m, if $(p,\alpha )\geq (q,\beta )$ then $\Sigma ^J_{G,m,p,\alpha } \subset \Sigma ^J_{G,m,q,\beta }$ as a closed subscheme. The set $\mathcal {I}$ is filtered, and we denote by $\Sigma ^J_{G,m}$ the ind-affine ind-scheme

$$ \begin{align*}\Sigma^J_{G,m}=\operatorname*{\mathrm{colim}}_{\mathcal I} \Sigma^J_{G,m,p,\alpha}. \end{align*} $$

We denote by $\Sigma ^J_{G}$ the disjoint union of the ind-schemes $\Sigma ^J_{G,m}$ . We notice that $\Sigma ^J_{G}$ has both a discrete and continuous structure. The discrete structure is given by the fact that $\Sigma ^J_{G}$ is the disjoint union of its components $\Sigma ^J_{G,m}$ , the continuous structure is given by the fact that the reduced structure of $\Sigma ^J_{G,m,p,\alpha }$ is a infinite dimensional affine space. The space $\Sigma ^J_{G}$ can also be realized as the filtered colimit over $\mathcal I$ of the schemes

$$ \begin{align*}\tilde{\Sigma}^J_{G,p,\alpha}=\bigsqcup_{m=-p}^p \Sigma^J_{G,m,p,\alpha}. \end{align*} $$

hence it is an ind-affine ind-scheme. We now prove that every class in $LJG(R)/JJG(R)$ has a unique representative in $\Sigma ^J_G(R)$ . This will prove Theorem 1.7 for the Grassmannian of the jet group.

The evaluation in $s=0$ define a map $p:LJG/JJG\longrightarrow LG/JG$ . This map has a section induced by the inclusion . Hence, given a class of an invertible series $f=\sum f_i(t)s^i$ in , we may assume that $f(t,0)\in \Sigma _{G}(R)$ . In particular, up to decomposing R as a product, we can use the description (1.1) of $\Sigma _{\mathbb G_m}$ given in the proof of Lemma 1.2, and further assume that $f=f_0(t)+\sum _{i>0}f_i(t)s^i$ with $f_0(t)\in \Sigma _{G,m}$ and $f_i(t)\in R(\!(t)\!)$ . Therefore, we are left to prove that there exists a unique $h=\sum _{i\geq 0} h_i(t)s^i$ with such that $f\cdot h\in \Sigma ^J_{G,m}(R)$ .

We need the following lemma.

Proof. Let $\varepsilon =\varepsilon _1t^{-1}+\dots +\varepsilon _Nt^{-N}$ and let I be the ideal generated by the $\varepsilon _i$ . Then $I^p=0$ for some integer p.

We prove first the existence of $\gamma $ . We define a sequence of series by induction. We set $\beta _0=\beta $ . Given $\beta _n$ we write $\beta _n=a_n+b_n$ where and $b_n\in t^{-1}R[t^{-1}]$ . Then we set $\beta _{n+1}=\beta _n-(1+\varepsilon )a_n=b_n-\varepsilon a_n$ . Notice that by construction the coefficients of $a_n$ are in $I^n$ , hence $a_n=0$ for $n\geq p$ . Finally $\gamma :=\alpha _0+\dots +\alpha _k$ has the required properties.

Now we prove the uniqueness of $\gamma $ . It enough to prove that if $\beta \in t^{-1}R[t^{-1}]$ and $\beta -\alpha \gamma \in t^{-1}R[t^{-1}]$ then $\gamma =0$ . Write $\gamma =\sum _{i\geq h} c_i t^i$ . For all $i\geq 0$

$$ \begin{align*} c_i+\varepsilon_1 c_{i+1}+\dots+\varepsilon_N c_{i+N}=0. \end{align*} $$

Hence $c_i\in I$ for all i, and arguing by induction, $c_i\in I^m$ for all m. This implies $c_i=0$ for all i.

We now argue by induction proving both uniqueness and existence of $h_\ell $ . For $\ell =0$ we must have $h_0=1$ since we are already assuming that $f_0\in \Sigma _{G,m}(R)$ , hence this follows from the proof of Lemma 1.2 in the case of $G=\mathbb G_m$ . Suppose we have determined $h_0,\dots ,h_\ell $ so that the coefficient of $s^i$ in the product $f\cdot h$ for $i=0,\dots ,\ell $ has the required form. Notice that the coefficient of $s^{\ell +1}$ is equal to

$$ \begin{align*}f_0 h_{\ell+1}+\cdots+ f_{\ell}h_1+f_{\ell+1} \end{align*} $$

We then apply the previous Lemma 1.14 with $\alpha =-f_0$ and $\beta =f_1 h_{\ell }+\cdots + f_{\ell }h_1+f_{\ell +1}$ .

1.3.3 Proof of Theorem 1.7 in the case of $ ^L\mathcal G r_G$

This case follows by what we have already recalled in Section 1.1.2: if X is an ind-affine ind-scheme then $LX$ is also an ind-affine ind-scheme. More precisely, we define

$$ \begin{align*}{}^L\Sigma_G =L(\Sigma_G)\subset LLG. \end{align*} $$

It follows immediately, from the properties of $\Sigma _G$ , that ${}^L\Sigma _{G}$ has the required properties.

Recall from the discussion in 1.3.1 that the reduced structure of $\Sigma _G$ is given by a countable union of isolated points. Since the reduced scheme structure of the loop of a fat point is $\operatorname {\mathrm {Spec}} \, k$ , this implies that the reduced structure of $L(\Sigma _G)$ is the same as the one of $\Sigma _G$ . In particular ${}^L\Sigma _G\simeq \coprod _{n\in \mathbb {Z}} \xi _n$ where each $\xi _n$ is a formal thickening of a point $\operatorname {\mathrm {Spec}} k$ .

1.3.4 Proof of Theorem 1.7 in the case of $\mathcal G r^{\mathrm {big}}_G$

The presheaf $LLG/JJG$ fibers over $LLG/LJG=L(\mathcal G r_G)$ , the loop space of the affine Grassmannian, and the fiber is $LJG/JJG$ the affine Grassmannian of the jet group. As we have seen in the previous sections, for both spaces we have canonical representatives. In the first case we have the space $ ^L\Sigma _{G}=L(\Sigma _G)\subset LLG$ and in the second case $\Sigma ^J_{G}\subset LJG$ . Hence we set

$$ \begin{align*}\Sigma^{big}_{G}= \Sigma^J_{G}\times L(\Sigma_G)\subset LLG \end{align*} $$

We have a map from this presheaf to $LLG$ , given by the inclusion of $LJG\subset LLG$ and product in $LLG$ :

$$ \begin{align*}(g,h)\longmapsto g \cdot h \end{align*} $$

It follows easily, from the properties of $L(\Sigma _G)$ and of $\Sigma _{G}^J$ that this map is injective and that $\Sigma _{G}^{big}$ has the required properties. Moreover by the description of these ind-schemes, given in 1.3.2 and 1.3.3, we notice that $\Sigma _G^{big}$ will be a countable disjoint union of formal thickenings of infinite dimensional affine spaces.

1.3.5 Proof of Theorem 1.7 in the case of $\mathcal G r^{L}_G$

We now deal with the case of the affine Grassmannian of the loop group. As already observed, we are reduced to study the case of $G=\mathbb G_m$ .

We start by noticing that in the case of the multiplicative group $\Sigma _G\subset LG$ is a subgroup. Hence $^L\Sigma _G=L(\Sigma _G)\subset LLG$ and $J(\Sigma _G)\subset JLG\subset LLG$ are also subgroups, and moreover $J(\Sigma _G)\subset L(\Sigma _G)$ . Firstly, we describe representatives of the quotient $L(\Sigma _G)/ J(\Sigma _G)$ . Define

$$ \begin{align*}\Sigma^{L,-}_G(R)=\big\{ 1+\varepsilon_1(t)s^{-1}+\cdots+\varepsilon_n(t)s^{-n} : n\in \mathbb N\text{ and } \varepsilon_i\in t^{-1}R[t^{-1}] \text{ are nilpotent}\big\}. \end{align*} $$

which is easily seen to be an ind-affine ind-scheme that is a formal thickening of $\operatorname {\mathrm {Spec}} k$ .

To prove this Proposition we need the following Lemma. Given a series $a(t)=\sum a_it^i \in R(\!(t)\!)$ we denote by $a_+(t)$ the series $\sum _{i\geq 0}a_it^i$ .

Lemma 1.16. Let $\varepsilon _1,\dots ,\varepsilon _N$ be nilpotent series in $R(\!(t)\!)$ . Then there exists M such that for all choices of indexes $i_1,\dots ,i_M$ in $\{1,\dots ,N\}$ we have

(1.2) $$ \begin{align} \bigg(\varepsilon_{i_1}\cdots\Big(\varepsilon_{i_{M-2}}\cdot \big(\varepsilon_{i_{M-1}}\cdot (\varepsilon_{i_M})_+\big)_+\Big)_+ \! \! \cdots \!\!\bigg)_+ =0. \end{align} $$

Proof. Let $\varepsilon _{ij} \in R$ be the coefficient of $t^j$ in the series $\varepsilon _i$ . Note that the $\varepsilon _{ij}$ ’s are all nilpotent. Fix an integer d such that $\varepsilon _{ij} = 0$ whenever $j<d$ . Let I be the ideal in $R(\!(t)\!)$ generated by $\varepsilon _1,\dots ,\varepsilon _N$ and by $\varepsilon _{ij}$ for $i=1,\dots ,N$ and $-d\leq j\leq d$ . This is an ideal generated by a finite number of nilpotent elements, hence it is nilpotent. Let Q be such that $I^Q=0$ . We claim that we can choose $M=3Q$ .

For $\ell \geq 0$ the coefficient of $t^\ell $ in the left hand term of formula (1.2) is equal to

$$ \begin{align*}a_\ell:=\sum_ {\substack{j_M\geq 0\\ j_{M-1}+j_M\geq 0\\ \cdots \\j_{2}+\cdots +j_M\geq 0 \\j_{1}+\cdots +j_M=\ell }} \varepsilon_{i_1,j_1}\cdots \varepsilon_{i_M,j_M} \end{align*} $$

In order to prove that $a_\ell = 0$ , we compare it with the coefficient of $t^\ell $ in $\varepsilon _{i_1}\cdots \varepsilon _{i_M}$ (which vanishes as $M \geq Q$ ). If we denote by $c_h(p)$ is the coefficient of $t^p$ in $\varepsilon _{i_1}\cdots \varepsilon _{i_h}$ , then

$$ \begin{align*} a_\ell =\: c_M(\ell)&-\sum_{j_M<0}\,c_{M-1}(\ell-j_M)\,\varepsilon_{i_M,j_M} \\ &-\sum_{\substack{j_M\geq 0\\j_{M-1}+j_M<0}} \,c_{M-2}(\ell-j_M-j_{M-1}) \,\varepsilon_{i_{M-1},j_{M-1}}\,\varepsilon_{i_M,j_M} \\ &-\sum_{\substack{j_M\geq 0 \\j_{M-1}+j_M\geq 0\\j_{M-2}+j_{M-1}+j_M<0}} \,c_{M-3}(\ell-j_M-j_{M-1}-j_{M-2})\,\varepsilon_{i_{M-2},j_{M-2}}\,\varepsilon_{i_{M-1},j_{M-1}}\,\varepsilon_{i_M,j_M} \\ &\cdots \\ & -\sum_{\substack{j_M\geq 0\\\cdots \\ j_2+\cdots+j_M<0}} \,c_{1}(\ell-j_M-\cdots -j_{2})\,\varepsilon_{i_{2},j_{2}}\cdots \varepsilon_{i_M,j_M} \end{align*} $$

By definition $c_h(p)=0$ for $h\geq Q$ and for all p, hence the first $M-Q$ rows of the sum above vanish. For the remaining $Q-1$ rows, notice that each summand contains a product of the form

$$ \begin{align*}\varepsilon_{i_{k},j_{k}}\cdots \varepsilon_{i_M,j_M} \end{align*} $$

with $k\leq Q$ and such that $j_k+\dots +j_M<0$ . If at least Q among the indexes $j_k,\dots ,j_M$ are less or equal than d, then by the definition of the ideal I, the product vanishes. If this is not the case then at least $M-k-Q+2$ of the indexes $j_k,\dots ,j_M$ are bigger or equal to $d+1$ . Then

$$ \begin{align*}j_{k}+\cdots+j_M\geq (d+1)(M-k-Q+2)-d(Q-1)\geq (d+1)(Q+2)-d(Q-1)\geq 0.\end{align*} $$

Hence the condition $j_k+\dots +j_M<0$ cannot be satisfied in this case. It follows that the summands always involve products of at least Q elements of I, and therefore vanish.

We can now prove Proposition 1.15.

Proof of Proposition 1.15.

Let

$$ \begin{align*}f=s^m-\varepsilon_1(t)s^{m-1}\cdots-\varepsilon_N(t)s^{m-N}\in L(\Sigma_{G,m})(R)=\Sigma_{G,m}\big(R(\!(t)\!)\big), \end{align*} $$

in particular $\varepsilon _1,\dots ,\varepsilon _N$ are nilpotent series. We need to prove that there exists a unique such that $f\cdot h\in \Sigma _{G}^{L,-}(R)$ . Since $s^m$ is invertible in , we can assume $m=0$ . Let $h=1+\sum _{i<0} h_{-i}(t)s^i \in J(\Sigma _G)$ . The coefficient of $s^\ell $ in $f\cdot h$ is $ h_{-\ell }- \varepsilon _1h_{-\ell -1}\cdots -\varepsilon _{N}h_{-\ell -N}$ where we set for convenience $h_i=0$ for $i< 0$ and $h_0=1$ . If $fh \in \Sigma _G^{L,-}(R)$ , then the coefficient of $s^\ell $ lives in $t^{-1}R[t^{-1}]$ for every $\ell \leq 1$ , and therefore

$$ \begin{align*}h_\ell=\big(\varepsilon_1h_{\ell-1}\cdots+\varepsilon_{N}h_{\ell-N} \big)_+ \end{align*} $$

for all $\ell \geq 1$ . By induction, this uniquely determines the coefficients $h_i$ , and hence proves the unicity of h. To prove the existence of such an h, we must show that $h_{i}=0$ for i big enough. Notice that, by induction, we get that $h_\ell $ is a sum of expressions of the form

$$ \begin{align*}\bigg(\varepsilon_{i_1}\cdots\Big(\varepsilon_{i_{M-2}}\cdot \big(\varepsilon_{i_{M-1}}\cdot (\varepsilon_{i_M})_+\big)_+\Big)_+ \! \! \cdots \!\!\bigg)_+ \end{align*} $$

where $i_1+i_2+\dots +i_M=\ell $ . Hence by Lemma 1.16 for $\ell $ big enough we have $h_\ell =0$ .

We now describe a representative for the quotient $LLG/JLG$ . Define the ind-affine ind-scheme

$$ \begin{align*}\Sigma_G^L=\Sigma_G^J \times \Sigma_G^{L,-}. \end{align*} $$

First notice that, by definition, both $\Sigma _G^{L,-}(R)\subset LLG(R)$ and $\Sigma _G^J(R)\subset LJG(R)\subset LLG(R)$ are subgroups, and that their intersection is the neutral element $1$ . Therefore the multiplication from $\Sigma _G^{L,-}(R)\times \Sigma _G^J(R)$ to $LLG(R)$ is injective. In particular, we view $\Sigma ^L_G(R)$ as a subgroup of $LLG(R)$ .

Note that $\Sigma _G^{big}= \Sigma ^J_G\times L(\Sigma _G)$ is a subgroup of $LLG$ and in particular, by what proved in the case of the quotient $LLG/JJG$ , we have $LLG=\Sigma ^J_G\times L(\Sigma _G)\times JJG$ . Moreover, since $LG=\Sigma _G\times JG$ , we have $JLG=J(\Sigma _G)\times JJG\subset L(\Sigma _G)\times JJG$ . Hence

$$ \begin{align*}\frac{LLG}{JLG}=\frac{\Sigma^J_G\times L(\Sigma_G)\times JJG}{J(\Sigma_G)\times JJG}= \Sigma^J_G\times\frac{L(\Sigma_G)}{J(\Sigma_G)}=\Sigma^J_G\times\Sigma^{L,-}_G. \end{align*} $$

proving the claim.

From the description of $\Sigma _G^J$ and $\Sigma _G^{L,-}$ , it follows that $\Sigma _G^L$ is the countable disjoint union of ind-schemes which are colimit of schemes whose reduced structure is an infinite dimensional affine space.

1.3.6 Proof of Theorem 1.7 in the case of $\mathcal G r^{(2)}_G$

Since $G^{(2)}\subset LJG$ , this Grassmannian fibers over $LLG/LJG$ , the loop space of the affine Grassmannian of G, and the fiber is isomorphic to the presheaf given by

Given an element in , we have that $g(t,0) \in R(\!(t)\!)^*$ . In particular, given , the element $\bar g(t,0)$ is a well defined element of $\mathcal G r_G (R)$ . Viceversa $\mathcal G r_G\subset LJG/G^{(2)}$ . This easily implies that every element of the fiber has a unique representative in

$$ \begin{align*}\Sigma_G\subset LG\subset LLG\end{align*} $$

where the last inclusion is given by $g(t)\mapsto h(t,s)$ where $h(t,s):=g(t)$ . From this, and the properties of ${}^L\Sigma _{G}=L(\Sigma _G)$ , we easily deduce that

$$ \begin{align*}\Sigma^{(2)}_{G}=\Sigma_G\times L(\Sigma_{G}) \end{align*} $$

has the required properties. Recall that the reduced structure of both $\Sigma _G$ and $L(\Sigma _G)$ is a countable union of isolated points. It follows that $\Sigma ^{(2)}_{G}\simeq \coprod _{(n, m) \in \mathbb {Z}\times \mathbb {Z}} \xi _{n,m}$ where $\xi _{m,n}$ are formal thickenings of $\operatorname {\mathrm {Spec}} \, k$ .

Remark 1.17. As we noticed during the proof of Theorem 1.7, we have

$$ \begin{align*}^L\mathcal G r_{\mathbb{G}_m} \simeq \coprod_{n\in \mathbb{Z}} \xi_n \qquad \mathcal G r^{(2)}_{\mathbb{G}_m} \simeq \coprod_{(n, m) \in \mathbb{Z}\times \mathbb{Z}} \xi_{n,m}\end{align*} $$

where the $\xi _n$ ’s and the $\xi _{n,m}$ ’s are formal thickenings of $\operatorname {\mathrm {Spec}} \, k$ (whose nonreduced ind-scheme structures depend on n, and on $(n, m)$ , respectively).

For all the other two-dimensional Grassmannians there is always both a discrete set (like above for $\mathbb {Z}$ and $\mathbb {Z}\times \mathbb {Z}$ ) labeling the connected components of the Grassmannian, and a continuous part given by the fact that these connected components are described as ind-schemes of formal neighborhoods of infinite dimensional affine spaces.

For the usual affine Grassmannian, the parametrization of the double coset space $JG\backslash LG /JG$ is at the origin of the Tannakian description of the Langlands dual appearing in the Geometric Satake theorem ([Reference Mirkovic and Vilonen13]). In the case under consideration, since the group $G=\mathbb {G}_m$ is commutative, the double coset space coincides with the affine Grassmannian itself. Therefore, what we have just observed about the reduced structure of our two-dimensional Grassmannians seems to suggest that both $^L\mathcal G r_{\mathbb {G}_m}$ and $\mathcal G r^{(2)}_{\mathbb {G}_m}$ could be better suited for a possible representation theoretic interpretation.

The geometric loop Grassmannian

Take an affine scheme $S=\operatorname {\mathrm {Spec}} R$ over k. By definition,

$$\begin{align*}\underline{\mathbf{Bun}}_{{\widehat Z}^{\widehat D} \smallsetminus Z} (S) \simeq \mathsf{Bun}\left({\mathcal{F}}^{\mathrm{aff}}_{ {\widehat Z}^{\widehat{D}} \smallsetminus Z}(S \times X / X )\right) \end{align*}$$

where $S\times X \simeq \operatorname {\mathrm {Spec}} R[x,y] $ . In particular, ${\mathcal {F}}^{\mathrm {aff}}_{ {\widehat Z}^{D} \smallsetminus Z }( S \times X/X) \simeq \mathsf {Bun} \left ( \operatorname {\mathrm {Spec}}\left ( \lim _n R(\!(x)\!)[y]/y^n \right ) \right ) $ and therefore

Similarly, we have

$$\begin{align*}\underline{\mathbf{Bun}}_{({\widehat Z}^{\widehat D} \smallsetminus Z)\smallsetminus D} (S) \simeq \mathsf{Bun} ( R(\!(x)\!)(\!(y)\!)). \end{align*}$$

In particular, S-points of ${}^L{\underline {\mathbf {Gr}}}_{{\mathbb {A}}^1,0}$ are exactly $R(\!(x)\!)$ -points of the usual affine Grassmannian. By Corollary 1.3, the affine Grassmannian can be identified with the presheaf quotient $\mathcal G r_G$ , and we get

$$\begin{align*}{}^L{\underline{\mathbf{Gr}}}_{{\mathbb{A}}^1,0}(S) \simeq \mathcal G r_{G}(R(\!(x)\!)) \simeq {}^L\mathcal G r_G(R) \end{align*}$$

where the last equivalence follows from the definition of ${}^L\mathcal G r_G$ . As the two presheaves ${}^L{\underline {\mathbf {Gr}}}_{{\mathbb {A}}^1,0}$ and ${}^L\mathcal G r_G$ are equivalent, so are their sheafifications, as claimed.

The geometric 2-dimensional local field Grassmannian

First of all, we observe that since $\mathbb {A}_k^2$ is affine, Definition 2.10 together with Definition 2.5, immediately tell us that

Now, for any $R \in \mathbf {CAlg}_{k}$ , we have a pullback diagram in $\mathbf {CAlg}_{k}$

that, after applying the $\mathrm {Spec}$ functor, yields (by [Reference Alper, Halpern-Leistner, Hall and Rydh1, Lemma 4.5]) a pushout diagram in the category of quasi-compact and quasi-separated algebraic stacks. By applying the internal Hom-stack functor $\mathrm {HOM}_{\mathsf {St}_k}(-, \mathrm {B}G)$ to this pushout, we thus get that

so we do have a canonical map $\Phi : \mathcal {G}r^{(2)}_{G} \to {\underline {\mathbf {Gr}}}_{{\mathbb {A}}^1,0}^{\mathrm {(2)}}$ of prestacks over k, given on R-points by

where $\mathrm {triv}$ stands for the trivial G-bundle. We now prove that ${\underline {\mathbf {Gr}}}_{{\mathbb {A}}^1,0}^{\mathrm {(2)}}$ is set-valued, that is, it is a functor with values in groupoids that are equivalent to sets. To achieve this, we show that for any $(\mathcal {E}, \varphi ) \in {\underline {\mathbf {Gr}}}_{{\mathbb {A}}^1,0}^{\mathrm {(2)}}(R)$ , we have

$$ \begin{align*}\mathrm{Aut}_{{\underline{\mathbf{Gr}}}_{{\mathbb{A}}^1,0}^{\mathrm{(2)}}(R)}(\mathcal{E}, \varphi) = \{ \mathrm{id}_{\mathcal{E}}\}.\end{align*} $$

Let $\rho \in \mathrm {Aut}_{{\underline {\mathbf {Gr}}}_{{\mathbb {A}}^1,0}^{\mathrm {(2)}}(R)}(\mathcal {E}, \varphi )$ , so that $\rho : \mathcal {E} \to \mathcal {E}$ is an isomorphism of G-bundles on that equals the identity when restricted to $U:= \mathrm {Spec}(R(\!(x)\!)(\!(y)\!))$ . Now observe that $U=Y\setminus V(xy)$ where $xy$ is a non zero divisor in . In other words, U is the complement in Y of an effective Cartier divisor, thus U is schematically dense quasi-compact in Y. Moreover, $\mathcal {E} \to Y$ is a separated morphism (since being separated is étale local on the target, G is affine and $\mathcal {E}$ is a G-bundle on Y, hence étale locally trivial on Y). Therefore, $\rho $ and $\mathrm {id}_{\mathcal {E}}$ are Y-morphisms $\mathcal {E} \to \mathcal {E}$ agreeing on U, so they agree.

We finally prove that $\Phi ^{\sharp }: \mathcal {G}r^{\mathrm {(2)}, \sharp }_{G} \to {\underline {\mathbf {Gr}}}^{\mathrm {(2)}, \sharp }_{{\mathbb {A}}^1,0} $ is an isomorphism of fppf sheaves of sets on $\mathsf {Aff}_k$ . Clearly $\Phi $ is a monomorphism of presheaves, hence $\Phi ^\sharp $ is a monomorphism of sheaves. It will then be enough to show that $\Phi $ is locally surjective. We will show that for any G-bundle $\mathcal {E}$ on , there exists an fppf covering $R\to R'$ such that the pullback of $\mathcal {E}$ to is trivialisable. By our earlier observation in this proof, $\mathcal {E}$ can be identified with a triple $({\mathcal {E}}_1, {\mathcal {E}}_2, \phi )$ , where ${\mathcal {E}}_1$ is a G-bundle on , ${\mathcal {E}}_2$ a G-bundle on , and $\phi $ is an isomorphism between the restrictions of ${\mathcal {E}}_1$ and ${\mathcal {E}}_2$ to $R(\!(x)\!)$ . Now, we do know that there exists an fppf covering $R\to R'$ such that the pullback on ${\mathcal {E}}_2$ to is trivialisable. Let’s choose one such trivialisation $\alpha ^{\prime }_2$ , and denote by $({\mathcal {E}}^{\prime }_1, {\mathcal {E}}^{\prime }_2, \phi ')$ the pullback of the triple $({\mathcal {E}}_1, {\mathcal {E}}_2, \phi )$ along the maps induced by $R\to R'$ . By composing the restriction of $\alpha ^{\prime }_2$ to $R'(\!(x)\!)$ with the pullback of $\phi $ to $R'(\!(x)\!)$ , we define a trivialisation $\beta ^{\prime }_1$ of ${\mathcal {E}}_1$ restricted to $R'(\!(x)\!)$ . By formal smoothness, we deduce (if needed, by passing to a further fppf covering of $R'$ ) a trivialisation $\alpha ^{\prime }_1$ of all of ${\mathcal {E}}_1$ , extending $\beta ^{\prime }_1$ . By definition of $\beta ^{\prime }_1$ , it is easy to check that the pair $(\alpha ^{\prime }_1 , \alpha ^{\prime }_2)$ defines an isomorphism between $({\mathcal {E}}^{\prime }_1, {\mathcal {E}}^{\prime }_2, \varphi ')$ and the trivial triple $({\mathrm {triv}}^{\prime }_1, {\mathrm {triv}}^{\prime }_2, \mathrm {id})$ . Summing up, we have shown that, for any $R \in {\mathsf {CAlg}}_k$ , and any G-bundle $\mathcal {E}$ on , there exists an fppf cover $R\to R'$ such that the pullback on $\mathcal {E}$ to is trivialisable, as claimed.

The geometric big and the geometric jet Grassmanian

We treat the last two cases together. We start by proving that the prestacks ${\underline {\mathbf {Gr}}}_{{\mathbb {A}}^1,0}^{\mathrm {big}}$ and ${\underline {\mathbf {Gr}}}_{{\mathbb {A}}^1,0}^J$ are set-valued. We only deal with ${\underline {\mathbf {Gr}}}_{{\mathbb {A}}^1,0}^J$ , as the case of the geometric big Grassmannian is similar. Let $S= \operatorname {\mathrm {Spec}} R$ be an affine scheme over k. An object of ${\underline {\mathbf {Gr}}}_{{\mathbb {A}}^1,0}^{J}(R)$ is a couple $(\mathcal E, \varphi )$ , where $\mathcal E$ is a G-bundle on , and $\varphi $ is a trivialisation of the restriction of $\mathcal E$ to . To prove the claim, we need to show that every such object $(\mathcal E, \varphi )$ has trivial automorphisms group in the category ${\underline {\mathbf {Gr}}}_{D,Z}^{J}(R)$ .

First, suppose $\mathcal E = \mathrm {triv}$ is the trivial bundle. Let $\alpha , \beta $ be two automorphisms of $(\mathrm {triv}, \varphi )$ . In particular, $\alpha , \beta $ are elements of . Their compatibility with $\varphi $ implies that their images along the map

coincide. But this map is injective, and therefore $\alpha =\beta $ , as desired.

Let now $(\mathcal E, \varphi )$ be a general object in ${\underline {\mathbf {Gr}}}_{D,Z}^{J}(R)$ , and let $\eta $ be an automorphism of it. We claim that $\eta $ is in fact the identity map. By [Reference Hennion, Melani and Vezzosi11, Section 3.1], the prestack $\mathbf {Bun}_{{\widehat Z}^{\mathrm {aff}}}$ can be identified $\mathbf {Bun}_{{\widehat Z}}$ , which is a stack for the fppf topology on $\mathsf {Aff} /X$ , since ${\mathcal {F}}_{\widehat Z}$ is a representable fiber functor (see [Reference Hennion, Melani and Vezzosi11, Lemma 3.1.2]). It follows in particular that $\underline {\mathbf {Bun}}_{{\widehat Z}^{\mathrm {aff}}}$ is a stack for the fppf topology on $\mathsf {Aff}_k$ . Hence it is enough to find a cover $\{R \to R_i\}_{i \in I}$ for which $\eta _{|R_i}$ is the identity, for every i. In virtue of what we showed earlier in the proof, it would suffice to find a covering family $\{ R \to R_i \}_{i \in I}$ for which $(\mathcal E_i, \varphi _i) := (\mathcal E_{|R_i}, \varphi _{|R_i})$ is isomorphic to an object of the form $(\mathrm {triv}, \psi _i)$ . In fact, $(\mathrm {triv}, \psi _i)$ has trivial automorphisms, hence so does $(\mathcal E_i, \varphi _i)$ , and it would follow that $\eta _{|R_i}$ is the identity, as desired.

Therefore, we are left with proving that we can choose a covering family $\{ R \to R_i \}_{i \in I}$ such that $(\mathcal E_i, \varphi _i)$ is isomorphic to an object $(\mathrm {triv}, \psi _i)$ . To this purpose, consider the restriction $\mathcal E_{|x=y=0}$ of $\mathcal E$ to $\operatorname {\mathrm {Spec}}(R)$ . It is a principal G-bundle on $\operatorname {\mathrm {Spec}}(R)$ , and as such we can find a covering family $\{ R \to R_i \}_{i \in I}$ such that its restriction to $\operatorname {\mathrm {Spec}}(R_i)$ is trivial for every index i. We claim that the restriction $\mathcal E_i$ of $\mathcal E$ to is also trivial. In fact, to construct a trivialization of $\mathcal E_i$ it is enough to construct compatible trivializations of its restrictions $(\mathcal E_i)_{|(x,y)^n=0}$ over . This may be done inductively: we start from the trivialization of $(\mathcal E_i)_{|(x,y)=0}$ , which is given by a section $\sigma _1$ of the structural map $(\mathcal E_i)_{|(x,y)=0} \to \operatorname {\mathrm {Spec}}(R_i)$ . Then we can construct a trivialization $\sigma _2$ as a section in the following diagram

where we used that the right vertical arrow is smooth, and that the bottom map is a first order thickening of affine schemes. Similar constructions yield a sequence of compatible trivializations of the G-bundles $ (\mathcal E_i)_{|(x,y)^n=0}$ , which is what we wanted to construct.

Next, consider the two maps of presheaves

$$\begin{align*}LLG \to {\underline{\mathbf{Gr}}}_{{\mathbb{A}}^1,0}^{\mathrm{big}}, \ \ \ LJG \to {\underline{\mathbf{Gr}}}_{{\mathbb{A}}^1,0}^{J} \end{align*}$$

defined as follows. Given an affine scheme $\operatorname {\mathrm {Spec}}(R)$ and a class $\overline {\varphi } \in LLG(R)$ (or in $LJG(R)$ ), we send $\overline {\varphi }$ to the element $(\mathrm {triv}, \varphi )$ , where $\mathrm {triv}$ denotes the trivial bundle on and $\varphi $ is an automorphism of the trivial bundle on $\operatorname {\mathrm {Spec}}(R(\!(x)\!)(\!(y)\!))$ (or in ). Therefore, we get two morphisms of presheaves

(2.3) $$ \begin{align} \mathcal G r_G^{\mathrm{big}} \to {\underline{\mathbf{Gr}}}_{{\mathbb{A}}^1,0}^{\mathrm{big}}, \ \ \ \mathcal G r^J_G \to {\underline{\mathbf{Gr}}}_{{\mathbb{A}}^1,0}^J\end{align} $$

which in turn induce maps between sheafifications

(2.4) $$ \begin{align} \mathcal G r_G^{\mathrm{big}, \sharp} \to {\underline{\mathbf{Gr}}}_{{\mathbb{A}}^1,0}^{\mathrm{big}, \sharp}, \ \ \ \mathcal G r^{J, \sharp}_G \to {\underline{\mathbf{Gr}}}_{{\mathbb{A}}^1,0}^{J, \sharp}. \end{align} $$

Notice that the maps in (2.3) are monomorphisms of presheaves. Since the sheafification functor is left exact, the maps in (2.4) are monomorphisms of sheaves. To prove that they are also epimorphisms, it is enough to show that the maps (2.3) are locally surjective. But we already observed that every object $(\mathcal E, \varphi ) \in {\underline {\mathbf {Gr}}}_{{\mathbb {A}}^1,0}^{\mathrm {big}}$ (or in ${\underline {\mathbf {Gr}}}_{{\mathbb {A}}^1,0}^J$ ) is locally of the form $(\mathrm {triv}, \psi )$ , where again $\mathrm {triv}$ denotes the trivial bundle. Hence both maps in (2.4) are equivalences, as claimed.

Theorem 2.14 does not include a comparison between the geometric Grassmannian of the loop group (Definition 2.10) for $X= {\mathbb {A}}^2_k$ , at the fixed flag $(D=\{y=0\}, Z= \{x=y=0\})$ , and $\mathcal {G}r_{G}^L$ (1.5). Indeed, in this case we are only able to prove the following weaker analog of Theorem 2.14.

In a first version of this paper, the following Proposition was proved for G solvable, o G special. We thank P. Gille, for explaining to us how to extend this proof to the case of G simply connected.

Proof. We start by proving that, for G as in the statement, any G-bundle on $S=\mathrm {Spec}\, A$ where , is trivial. First of all we notice that A is a PID. Indeed, A is the localization at y of that is a local ring of dimension $2$ . Therefore A is factorial, Noetherian, and of dimension $1$ . Hence A is a PID. This immediately implies the claim for $G=\mathrm {GL}_n$ , and in particular for $G=\mathbb G_m$ , and for $G=\mathrm {SL}_n$ . The group $G=\mathrm {Sp}_{2n}$ is special (i.e., étale torsors and Zariski torsors coincide) and simply connected, thus every (Zariski) G-torsor on S is trivial by [Reference Harder10, Satz 3.3].

The statement in the case G is solvable and connected, reduced by induction on dimension to the case of G a torus (that we have just considered), and $G=\mathbb {G}_a$ which follows from (since S is affine and $\mathbb G_a$ a coherent sheaf).

If k has characteristic zero, consider K the fraction field of A (note that this is different from $k(\!(x))(\!(y)\!)$ ). Then for this field

is trivial by a result of Colliot-Thélène, Gille and Parimala (see [Reference Colliot-Thélène, Gille and Parimala5], Theorem 1.4). Moreover by a result of Harder [Reference Harder10] (see also [Reference Gille9] Proposition 5.2), if T is a maximal torus of G we have the following exact sequence

and the first and last term are trivial, hence every G torsor over $\operatorname {\mathrm {Spec}} A$ is trivial also in this case.

Finally, since the automorphism group of any object of the form $(\mathrm {triv}, \varphi )$ in ${\underline {\mathbf {Gr}}}_{{\mathbb {A}}^1,0}^{L}(k)$ is trivial (because is injective), we deduce from the previous triviality result both that ${\underline {\mathbf {Gr}}}_{{\mathbb {A}}^1,0}^{L}(k)$ is (a groupoid equivalent to) a set, and that the canonical map

$$ \begin{align*}\mathcal{G}r_{G}^L(k) \longrightarrow {\underline{\mathbf{Gr}}}_{{\mathbb{A}}^1,0}^{L}(k), \, [\varphi] \longmapsto (\mathrm{triv}, \varphi)\end{align*} $$

is surjective. Injectivity is obvious, and we conclude.