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Local analog of the Deligne–Riemann–Roch isomorphism for line bundles in relative dimension 1 D. V. Osipov Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2024, Volume 88, Issue 5, DOI: https://doi.org/10.4213/im9532 
Bibliographic databases:
  § 1. IntroductionThe goal of this paper is to prove a local analog of the Deligne–Riemann–Roch theorem for linear bundles in relative dimension $1$. The parts for this local analog of the Deligne–Riemann–Roch theorem consist of the central extensions of the group ind-scheme that is the semidirect product of the group of invertible functions on the formal punctured disc and the group of automorphisms of this disc. These central extensions are by the multiplicative group scheme $\mathbb{G}_m$. We prove an equivalence of these central extensions after extensions of scalars of all schemes and ind-schemes from the ring $\mathbb{Z}$ to the field $\mathbb{Q}$. 1.1. Deligne–Riemann–Roch theoremWe recall the Deligne–Riemann–Roch isomorphism (or, in other words, the Deligne–Riemann–Roch theorem). Let $\pi\colon X \to S$ be a smooth proper morphism of relative dimension $1$ of schemes, that is, $\pi$ gives a family of smooth proper curves over a scheme $S$. For any two invertible sheaves $L$ and $M$ on $X$, Deligne constructed (see [8], § 6, [31], Exposé XVIII, § 1.3) an invertible sheaf $\langle L, M \rangle$ on $X$. This Deligne pairing (or, in other words, Deligne bracket) is symmetric and bilinear with respect to the tensor products of invertible sheaves. It is also functorial in $L$ and $M$ (with respect to isomorphisms of sheaves) and compatible with base change. When $S$ is a smooth algebraic variety over a field, then inside the group $\mathrm{Pic}(S)$ the Deligne bracket for invertible sheaves $L$ and $M$ on $X$ is
$$
\begin{equation}
c_1(\langle L, M \rangle) = \pi_* (c_1(L) \cdot c_1(M)),
\end{equation}
\tag{1}
$$
where “$\,{\cdot}\,$” is the product in the Chow ring $\mathrm{CH}^*(X)$, and $\pi_* : \mathrm{CH}^2(X) \to \mathrm{CH}^1(S) =\mathrm{Pic}(S) $ is the direct image between Chow groups. Let $\omega = \Omega^1_{X/S}$ be an invertible sheaf of relative differential $1$-forms on $X$, and $L$ be any invertible sheaf on $X$. When geometric fibres of $\pi$ are connected, Deligne constructed a Riemann–Roch isomorphism of line bundles (see Théorème 9.9 in [8]):
$$
\begin{equation}
(\det R\pi_* L)^{\otimes 12} \simeq \langle L, L \rangle^{\otimes 6} \otimes \langle L, \omega \rangle^{\otimes -6} \otimes \langle \omega, \omega \rangle.
\end{equation}
\tag{2}
$$
Isomorphism (2) is functorial in $L$ and compatible with base change. When $S$ is a smooth algebraic variety over a field, then in $\mathrm{Pic}(S) \otimes_{\mathbb{Z}} \mathbb{Q}$, using (1), isomorphism (2) becomes an equality
$$
\begin{equation}
\begin{aligned} \, c_1(\det R\pi_* L) &= \pi_* \biggl( \frac{1}{2} c_1(L) \cdot c_1(L) - \frac{1}{2} c_1(L) \cdot c_1(\omega) + \frac{1}{12} c_1(\omega) \cdot c_1(\omega) \biggr) \nonumber \\ &=\pi_* \bigl( \bigl( \mathrm{ch}(L) \cdot \operatorname{td}(\omega^{\otimes -1}) \bigr)_2 \bigr), \end{aligned}
\end{equation}
\tag{3}
$$
where
$$
\begin{equation*}
\mathrm{ch}(L) = 1 + c_1(L) + \frac{1}{2} c_1^2(L) + \cdots
\end{equation*}
\notag
$$
is the Chern character,
$$
\begin{equation*}
\operatorname{td}(\omega^{\otimes -1}) = 1 - \frac{1}{2} c_1(\omega) + \frac{1}{12} c_1^2(\omega) + \cdots
\end{equation*}
\notag
$$
is the Todd class, and $(\ )_2$ means the component of degree $2$ in the ring $\mathrm{CH}^*(X) \otimes_{\mathbb{Z}} \mathbb{Q}$, that is, an element from the group $\mathrm{CH}^2(X) \otimes_{\mathbb{Z}} \mathbb{Q}$. Thus we see that equality (3) is the corollary of the Grothendieck–Riemann–Roch theorem. 1.2. Invertible functions and automorphisms of the punctured formal discWe consider the functor $L \mathbb{G}_m $ from the category of commutative rings to the category of Abelan groups where $A((t))^*$ is the group of invertible elements in the $A$-algebra of Laurent series $A((t))= A[[t]][t^{-1}]$. This functor is represented by a group ind-scheme (see § 2.3).The group ind-scheme $L \mathbb{G}_m $ can be considered as the group of invertible functions on the formal punctured disc. On the $A$-algebra $A((t))$, there is the natural $t$-adic topology that makes $A((t))$ a topological $A$-algebra, where $A$ has discrete topology. We consider the functor ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})$ from the category of commutative rings to the category of groups that assigns to every commutative ring $A$ the group of all continuous $A$-automorphisms of the $A$-algebra $A((t))$. This is indeed a functor, and this functor is represented by a group ind-scheme, see §§ 2.2 and 2.3. The group ind-scheme ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}) $ can be considered as the group of automorphisms of the formal punctured disc. There is the natural action of the group ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}}(\mathcal{L})(A)$ on the Abelian group $A((t))^*$, where $A$ is any commutative ring. Therefore, the group ind-scheme
$$
\begin{equation*}
\mathcal{G} = L \mathbb{G}_m \rtimes {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})
\end{equation*}
\notag
$$
makes sense. This group ind-scheme has the following geometric meaning. There is a natural action of $\mathcal{G}$ on the moduli stack of quintets $\mathcal M$ and on the moduli stack of proper quintets ${\mathcal M}_\mathrm{pr}$ (see Theorem 3), and we explain now the notion of a quintet and of a proper quintet. By a quintet over a commutative ring $A$, we mean a collection that consists of a separated family $C$ of curves over $A$, a section of this family such that this family is smooth near this section, a sheaf of $\mathcal{O}_C$-modules that is an invertible sheaf near the section, a relative formal parameter at the section, a formal trivialization of the sheaf at the section (in other words, the last two conditions mean that the topological $A$-algebra of functions on the formal neighbourhood of $C$ at the section is isomorphic to the topological $A$-algebra $A[[t]]$ and the sheaf obtained as the restriction of the sheaf of $\mathcal{O}_C$-modules to this formal neighbourhood is trivial, and we fix an element $t$ and a formal trivialization of the sheaf); see § 6.1. A proper quintet is a quintet such that, in addition to the previous conditions, the morphism $C \to \operatorname{Spec} A$ is a flat proper finitely presented morphism such that all geometric fibers are integral one-dimensional schemes, and the sheaf is an invertible sheaf of $\mathcal{O}_C$-modules, see also § 6.1. Informally speaking, elements of $\mathcal{G}(A)$ reglue the family of curves and the sheaf in a quintet along the section (cf. [11], § 17.3, § 18.1.3, and [22], § 4.1, but we provide detailed proofs of the construction, see Theorem 3). The action of $\mathcal{G}$ on a proper quintet with a smooth curve $C$ over a field $k$ is transitive, that is, the map from the corresponding Lie algebra to the tangent space is surjective. We note that in complex analytic category some analog of functor of proper quintets modulo isomorphisms (when all fibers of $C$ from a quintet are smooth curves of genus $g$) is represented by an infinite-dimensional complex manifold, see [2]. Such quintets are related to the infinite-dimensional Sato Grassmanian and the Krichever map that maps quintets over fields to the points of the Sato Grassmanian, see, for example, [2] and [27]. Through the Krichever map there is also further relationships with the moduli spaces of curves, soliton equations, etc. 1.3. Local analog of the left-hand side of the Deligne–Riemann–Roch isomorphism: determinant central extensionThere is a natural central extension, which we call the determinant central extension, of group ind-schemes
$$
\begin{equation*}
1 \to \mathbb{G}_m \to \widetilde{\mathcal{G}} \xrightarrow{\eta} \mathcal{G} \to 1,
\end{equation*}
\notag
$$
where the morphism $\eta$ admits a section (as ind-schemes, not as group ind-schemes). The construction of the determinant central extension is as follows (see also § 5). Let $A$ be any commutative ring. The group $\mathcal{G}(A)$ naturally acts on $A((t))$. Let $(h, \varphi) \in L \mathbb{G}_m(A) \rtimes {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A) $ and $f \in A((t))$. Then $(h, \varphi) (f)= h \varphi(f)$. The group $\widetilde{\mathcal{G}}(A)$ consists of pairs $(g,s)$, where $g \in \mathcal{G}(A)$ and an element $s$ belongs to the free $A$-module $\det(g (A[[t]]) \mid A[[t]])$ of rank $1$, and also generates this $A$-module. Here, $\det(g(A[[t]]) \mid A[[t]])$ is the relative determinant of $A$-modules $g(A[[t]])$ and $A[[t]]$, which is canonically isomorphic to the $A$-module1
$$
\begin{equation*}
\operatorname{Hom}_A \biggl( \bigwedge^{\mathrm{max}} (g(A[[t]])/ t^l A[[t]]), \, \bigwedge^{\mathrm{max}} (A[[t]]/ t^l A[[t]]) \biggr),
\end{equation*}
\notag
$$
where an integer $l$ satisfies $t^l A[[t]] \subset (g(A[[t]]) \cap A[[t]])$. Now $\eta((g,s)){=}\,g$. We note that this construction originates (when $A=k$ is a field) from [15], where the group $\mathcal{G}(A)$ is changed to the group of continuous automorphisms of topological $k$-vector space $k((t))$. A (non-group) section of morphism $\eta$ can be chosen in a natural way, see Remark 3. On the moduli stack of proper quintets ${\mathcal M}_\mathrm{pr}$, there is a natural determinant linear bundle, see § 6.2. This means that, for any proper quintet with family of curves $\pi \colon C \to \operatorname{Spec} A$ and an invertible sheaf of $\mathcal{O}_C$-modules $\mathcal{F}$, there is a linear bundle on $\operatorname{Spec} A$ that satisfy further compatibilty conditions. This linear bundle on $\operatorname{Spec} A$ is exactly an invertible sheaf $\det R \pi_* \mathcal{F}$ from § 1.1. In Proposition 6, we construct an explicit complex of finitely generated projective $A$-modules, which consists of two terms and whose determinant gives $\det R \pi_* \mathcal{F}$. Using this complex, we construct a natural action of the group ind-scheme $\widetilde{\mathcal{G}}$ on the determinant line bundle on the moduli stack of proper quintets ${\mathcal M}_\mathrm{pr}$ that lifts an action of the group ind-scheme $\mathcal{G}$ on ${\mathcal M}_\mathrm{pr}$, see Theorem 4. We can consider cohomology groups of group ind-schemes with coefficients in commutative group ind-schemes. More generally, instead of ind-schemes we consider functors from the category of commutative rings, see § 3.1. Then we write an analog of the bar complex (or, in other words, an analog of the standard complex) from the group cohomology, where we will use the Abelian groups of morphisms of functors, see complex (19). Thus, for any integer $q \geqslant 0$, we obtain the notions of $q$-cocycles, $q$-coboundaries, $q$th cohomology groups of a group functor with coefficients in a commutative group functor, where functors are from the category of commutative rings. We outline also another approach to compute these cohomology groups. This approach is by resolution by induced modules (certain commutative group functors) that are acyclic, see § 3.2. Similarly to group cohomology, we have the notion of $\cup$-products of cocycles. Besides, any central extension of a group functor by a commutative group functor that admits a section (as functors) defines a $2$-cocycle. The equivalence classes of such extensions are in one-to-one correspondence with elements of the second cohomology group. For any group functor $G$ that acts on a commutative group functor $F$ and any commutative ring $A$, there is a natural homomorphism of Abelian groups We will denote by $D$ the $2$-cocycle on $\mathcal{G}$ with coefficients in $\mathbb{G}_m$ given by the determinant central extension and its natural section. The determinant central extension (and the $2$-cocycle $D$) give the local analog of the left-hand side of the Deligne–Riemann–Roch isomorphism. 1.4. Local analog of the right-hand side of the Deligne–Riemann–Roch isomorphism: $\cup$-products of $1$-cocyclesThere is the Contou-Carrère symbol $\operatorname{CC}$
$$
\begin{equation*}
\operatorname{CC}\colon L \mathbb{G}_m \times L \mathbb{G}_m \to L \mathbb{G}_m \otimes L \mathbb{G}_m \to \mathbb{G}_m,
\end{equation*}
\notag
$$
that is a bimultiplicative and antisymmetric morphism (see § 7.1 on the definition of the Contou-Carrère symbol $\operatorname{CC}$, some its properties, and the references). The Contou-Carrère symbol $\operatorname{CC}$ is invariant under the diagonal action of the group functor ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})$. Through the natural morphism of group ind-schemes $\mathcal{G} \to {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}) $ we find that the group ind-scheme $\mathcal{G}$ naturally acts on the commutative group ind-scheme $L \mathbb{G}_m$. Let $\lambda_1$ and $\lambda_2$ be arbitrary $1$-cocycles on $\mathcal{G}$ with coefficients in $L \mathbb{G}_m$. We construct the $2$-cocycle $\langle \lambda_1, \lambda_2 \rangle$ on $\mathcal{G}$ with coefficients in $\mathbb{G}_m$ (where $\mathcal{G}$ acts trivially on $\mathbb{G}_m$)
$$
\begin{equation*}
\langle \lambda_1, \lambda_2 \rangle = \operatorname{CC} \circ\, (\lambda_1 \cup \lambda_2),
\end{equation*}
\notag
$$
where “$\circ$” means the composition of the morphism $\lambda_1 \cup \lambda_2$ from $\mathcal{G} \times \mathcal{G}$ to $L \mathbb{G}_m \otimes L \mathbb{G}_m $ and the $\mathcal{G}$-equivariant morphism $\operatorname{CC}$ from $ L \mathbb{G}_m \otimes L \mathbb{G}_m $ to $\mathbb{G}_m$. There are distinct $1$-cocycles $\Lambda$ and $\Omega$ on $\mathcal{G}$ with coefficients in $L \mathbb{G}_m$ (see also § 7.2):
$$
\begin{equation*}
\Lambda((h, \varphi)) = h \quad \text{and} \quad \Omega( (h, \varphi) ) = \frac{d \varphi(t)}{dt} = \varphi(t)',
\end{equation*}
\notag
$$
where $(h, \varphi) \in \mathcal{G}(A) = L \mathbb{G}_m (A) \rtimes {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}}(\mathcal{L})(A)$. The $1$-cocycle $\Lambda$ is the universal $1$-cocycle (see Remark 13) that will replace $L$ in the right-hand side of formula (2), and the $1$-cocycle $\Omega$ will replace $\omega$ in the right-hand side of formula (2). Now we obtain distinct $2$-cocycles on $\mathcal{G}$ with coefficients in $\mathbb{G}_m$ (where $\mathcal{G}$ acts trivially on $\mathbb{G}_m$):
$$
\begin{equation*}
\langle \Lambda, \Lambda \rangle, \quad \langle \Lambda, \Omega \rangle, \quad \langle \Omega, \Omega \rangle.
\end{equation*}
\notag
$$
Here, the first $2$-cocycle is a local analog of $\langle L, L \rangle $ from formula (2), the second $2$-cocycle is a local analog of $\langle L, \omega \rangle $ from (2), and the third $2$-cocycle is a local analog of $\langle \omega, \omega \rangle $ from (2). Besides, the Contou-Carrère symbol $\operatorname{CC}$ is an analog of the direct image $\pi_*$, see formula (1). 1.5. Equality in the second cohomology groupWe obtain in Theorem 7 the local analog of the Deligne–Riemann–Roch isomorphism (2)
$$
\begin{equation*}
D^{12} = \langle \Lambda, \Lambda \rangle^6 \cdot \langle \Lambda, \Omega \rangle^{-6} \cdot \langle \Omega, \Omega \rangle,
\end{equation*}
\notag
$$
where this equality is in the group $H^2(\mathcal{G}_{\mathbb{Q}}, {\mathbb{G}_m}_{\mathbb{Q}})$, and ${\mathcal{G}}_{\mathbb{Q}}$ acts trivially on ${\mathbb{G}_m}_{\mathbb{Q}}$. Besides, we extend scalars from $\mathbb{Z}$ to $\mathbb{Q}$ for all the schemes and ind-schemes. We use also the multiplicative notation for the group law in the Abelian group $H^2 (\mathcal{G}_{\mathbb{Q}}, {\mathbb{G}_m}_{\mathbb{Q}})$. We note that we stated Theorem 7 without proof in the short note [20]. The proof of the above equality in $H^2 (\mathcal{G}_{\mathbb{Q}}, {\mathbb{G}_m}_{\mathbb{Q}})$ is based on Corollary 2. This result generalizes Theorem 5.1 from [19] from the case of the group ind-scheme ${{\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})}_{\mathbb{Q}}$ to the group ind-scheme ${\mathcal{G}}_{\mathbb{Q}}$. Besides, for the case of the group of orientation-preserving diffeomorphisms of the circle in the theory of infinite-dimensional Lie groups, see Corollary (7.5) from [26]. Corollary 2 says that an element from the group $H^2 (\mathcal{G}_{\mathbb{Q}}, {\mathbb{G}_m}_{\mathbb{Q}})$, where ${\mathbb{G}_m}_{\mathbb{Q}}$ is a trivial ${\mathcal{G}}_{\mathbb{Q}}$-module, is uniquely defined by its image in $H^2(\operatorname{Lie} \mathcal{I} \mathcal{G}^0_{\mathbb{Q}}(\mathbb{Q}), \mathbb{Q})$ together with its restriction to $H^2( {\mathcal{G}_+}_{\mathbb{Q}}, {\mathbb{G}_m}_{\mathbb{Q}})$. Here, $\mathcal{G}_+$ is the group ind-scheme that represents the functor that assigns to every commutative ring $A$ the group which is the semidirect product of the group $A[[t]]^*$ of invertible elements of the $A$-algebra $A[[t]]$ and the group of continuous $A$-automorphisms of the $A$-algebra $A[[t]]$. The group $\mathcal{G}_+ (A)$ is a subgroup of the group $ \mathcal{G}(A)$. The Lie $\mathbb{Q}$-algebra $\operatorname{Lie} \mathcal{I} \mathcal{G}^0_{\mathbb{Q}}(\mathbb{Q})$ is the $\mathbb{Q}$-subalgebra of Lie $\mathbb{Q}$-algebra $\operatorname{Lie} \mathcal{G}(\mathbb{Q})$ of the group ind-scheme $\mathcal{G}_{\mathbb{Q}}$. The algebra $\operatorname{Lie} \mathcal{G} (\mathbb{Q})$ is the algebra of continuous differential operators of order $\leqslant 1$ acting on $\mathbb{Q}((t))$. The subalgebra $\operatorname{Lie} \mathcal{I} \mathcal{G}^0_{\mathbb{Q}}(\mathbb{Q})$ is
$$
\begin{equation*}
\mathbb{Q}[t,t^{-1}] \rtimes \operatorname{Der}_{\mathbb{Q}} (\mathbb{Q}[t, t^{-1}]),
\end{equation*}
\notag
$$
where $ \operatorname{Der}_{\mathbb{Q}} (\mathbb{Q}[t, t^{-1}]) = \mathbb{Q}[t, t^{-1}] \, \frac{\partial}{\partial t} $ is the Lie $\mathbb{Q}$-algebra of $\mathbb{Q}$-derivations. Now, to obtain the local analog of the Deligne–Riemann–Roch isomorphism, we note that the $2$-cocycles in the left-hand and right-hand sides are trivial after restriction to ${\mathcal{G}_+}_{\mathbb{Q}}$. And for the comparison of the corresponding Lie algebra $2$-cocyles it is enough to check the corresponding equalities for elements of type $t^n$ and $t^m \, \frac{\partial}{\partial t}$ from $\operatorname{Lie} \mathcal{I} \mathcal{G}^0_{\mathbb{Q}}(\mathbb{Q})$. We note that this check is reduced to the finite sum of constant elements in case of $\langle \Lambda, \Lambda \rangle$, to the sum of finite arithmetic progression in case of $\langle \Lambda, \Omega \rangle$, and to the sum of quadratic function over a finite set of consecutive non-negative integers in case of $\langle \Omega, \Omega \rangle$, see the proof of Theorem 5. 1.6. Organization of the paperThe paper is organized as follows. In § 2, we introduce and give some properties for the group functors $L \mathbb{G}_m$, ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}}(\mathcal{L})$, $\mathcal{G}$. In § 3, we introduce and prove properties for cohomology of group functors. In § 4, we investigate the Lie algebra valued functor $\operatorname{Lie} \mathcal{G}$, in particular, for any commutative ring $A$, the Lie $A$-algebra $\operatorname{Lie} \mathcal{G} (A)$, constructed from the ind-scheme $\mathcal{G}$. In § 5, we give various constructions of the determinant central extension of $\mathcal{G}$ and prove its properties. In § 6, we define quintets and proper quintets and describe the action of $\mathcal{G}$ on the moduli stack of quintets $\mathcal M$ and on the moduli stack of proper quintets ${\mathcal M}_\mathrm{pr}$, and describe also the action of $\widetilde{\mathcal{G}}$ on the determinant line bundle on ${\mathcal M}_\mathrm{pr}$. In § 7, we define and investigate $2$-cocycles $\langle \Lambda, \Lambda \rangle $, $\langle \Lambda, \Omega \rangle $ and $\langle \Omega, \Omega \rangle $ on $\mathcal{G}$ with coefficients in $\mathbb{G}_m$. In § 8, we prove a local analog of the Deligne–Riemann–Roch isomorphism by the comparison of two central extensions of ${\mathcal{G}}_{\mathbb{Q}}$ by ${\mathbb{G}_m}_{\mathbb{Q}}$. § 2. Groups related with the ring of Laurent seriesLet $A$ be any commutative ring. Let $A((t))= A[[t]][t^{-1}]$ be the ring of Laurent series over $A$. We set $\mathcal{L} (A) = A((t))$. By a group functor (or commutative group functor), we mean a covariant functor from the category of commutative rings to the category of groups (or Abelian groups). 2.1. Loop functor of multiplicative groupAs usual, let $\mathbb{G}_m$ be a commutative group functor such that $\mathbb{G}_m(A)= A^*$ for any commutative ring $A$. By $L \mathbb{G}_m$ we denote the group loop functor of $\mathbb{G}_m$, that is, There is a canonical isomorphism of commutative group functors (see Lemma 1.3 in [6]) and Lemma 0.8 in [7]
$$
\begin{equation}
\underline{\mathbb{Z}} \times \mathbb{G}_m \times \mathbb{V}_+ \times \mathbb{V}_- \simeq L \mathbb{G}_m,
\end{equation}
\tag{4}
$$
where the group functors in the left-hand side and the corresponding embeddings are described as follows. Let $A$ be any commutative ring. The group $\underline{\mathbb{Z}}(A)$ is the group of locally constant $\mathbb{Z}$-valued functions on $\operatorname{Spec} A$. Any $\underline{n} \in \underline{\mathbb{Z}}(A)$ gives the decomposition $A = A_1 \times \dots \times A_l$ such that the function $\underline{n}$ restricted to any $\operatorname{Spec} A_i$ is the constant function with value $n_i \in \mathbb{Z}$. Then $\underline{n} \mapsto t^{\underline{n}} = t^{n_1} \times \dots \times t^{n_l}$ defines the embedding $\underline{\mathbb{Z}} \to L \mathbb{G}_m$ of the corresponding group functors in (4). The group embedding $\mathbb{G}_m(A) \hookrightarrow L \mathbb{G}_m(A)$ is given by mapping of an invertible element from $A$ to the series consisting of only a constant term. The subgroups $\mathbb{V}_+(A)$ and $\mathbb{V}_-(A)$ of the group $L \mathbb{G}_m(A) $ are defined as
$$
\begin{equation}
\mathbb{V}_+(A) = \biggl\{ 1 + \sum_{l > 0} a_l t^l \biggm| a_l \in A \text{ for any } l > 0 \biggr\},
\end{equation}
\tag{5}
$$
$$
\begin{equation}
\mathbb{V}_-(A) = \biggl\{ 1 + \sum_{l < 0} a_l t^l \biggm| \sum_{l < 0} a_l t^l \in A((t)), \ a_l \in \operatorname{Nil}(A) \text{ for any } l< 0 \biggr\},
\end{equation}
\tag{6}
$$
where $\operatorname{Nil}(A)$ is the nil-radical of $A$, that is, the set of all nilpotent elements of $A$. We denote
$$
\begin{equation}
(L \mathbb{G}_m)^0 = \mathbb{G}_m \times \mathbb{V}_+ \times \mathbb{V}_-.
\end{equation}
\tag{7}
$$
Thus, we obtain decomposition of group functors
$$
\begin{equation}
\underline{\mathbb{Z}} \times (L \mathbb{G}_m)^0 \, \simeq \, L \mathbb{G}_m.
\end{equation}
\tag{8}
$$
By $\nu \colon L \mathbb{G}_m \to \underline{\mathbb{Z}}$ we denote the morphism of group functors given by the corresponding projection in decomposition (8). 2.2. Automorphism group and semidirect productOn the ring $A((t))$, where $A$ is any commutative ring, there is the natural topology with the base of neighbourhoods of zero consisting of $A$-submodules $U_n = t^n A[[t]]$, $n \in \mathbb{Z}$. This topology makes the ring $A((t))$ into a topological ring. By ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}) (A)$ we denote the group of all $A$-automorphisms of the $A$-algebra $A((t))$ that are homeomorphisms. We have the following facts on elements of this group (see § 2.1 in [19] and references therein). Any continuous $A$-automorphism of the $A$-algebra $A((t))$ is a homeomorphism. There is the following isomorphism of sets:
$$
\begin{equation}
{\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}) (A) \simeq \{ h \in L \mathbb{G}_m(A) \mid \nu(h) =1\},
\end{equation}
\tag{9}
$$
where the map from the left-hand side to the right-hand side is $\varphi \mapsto \widetilde{\varphi}= \varphi(t)$, and the map from the right-hand side to the left-hand side is
$$
\begin{equation*}
\widetilde{\varphi} \mapsto \{ f \mapsto f \circ \widetilde{\varphi}\},
\end{equation*}
\notag
$$
where $f \in A((t))$, and $f \circ \widetilde{\varphi}$ denotes the series from $A((t))$ obtained by substitution of the series $\widetilde{\varphi}$ into the series $f$ instead of variable $t$. For any $\varphi_1$ and $\varphi_2$ from ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}) (A)$, we have
$$
\begin{equation*}
\widetilde{\varphi_1 \varphi_2} = \widetilde{\varphi_2} \circ \widetilde{\varphi_1}.
\end{equation*}
\notag
$$
Isomorphism (9) defines the structure of the functor on the correspondence
$$
\begin{equation*}
A \mapsto {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A),
\end{equation*}
\notag
$$
where $A$ is any commutative ring. This is the group functor, which we denote by ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})$. We will use the following unique decomposition of the group functor ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})$ (see [19], § 3.1):
$$
\begin{equation}
{\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}) = {\mathcal Aut}_+^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}) \cdot {\mathcal Aut}_-^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}) = {\mathcal Aut}_-^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}) \cdot {\mathcal Aut}_+^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}).
\end{equation}
\tag{10}
$$
Here the subgroups ${\mathcal Aut}_+^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$ and ${\mathcal Aut}_-^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$ of the group ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$, where $A$ is any commutative ring, are described via isomorphism (9) as the set of elements
$$
\begin{equation}
\biggl\{ \sum_{l \geqslant 0} a_l t^l \biggm| a_l \in A \text{ for any } l \geqslant 0, \ a_0 \in \operatorname{Nil}(A), \ a_1 \in A^* \biggr\}
\end{equation}
\tag{11}
$$
and
$$
\begin{equation}
\biggl\{ \sum_{n \leqslant l \leqslant -1} a_l t^l + t \biggm| n < 0 \text{ is any}, \ a_l \in \operatorname{Nil}(A) \text{ for any } l < 0 \biggr\},
\end{equation}
\tag{12}
$$
correspondingly. We note that the group ${\mathcal Aut}_+^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$ consists of all continuous $A$-automorphisms of the $A$-algebra $A[[t]]$. Besides, decomposition (10) is the direct product decomposition as functors, but it is not the direct product or semi-direct product decomposition as group functors (the corresponding subgroups are not normal for some rings $A$). We have the natural action of the group functor ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})$ on the commutative group functor $L \mathbb{G}_m$. This leads to the following group functor that will play one of the main roles in this article
$$
\begin{equation}
{\mathcal G} = L \mathbb{G}_m \rtimes {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}),
\end{equation}
\tag{13}
$$
where, for any commutative ring $A$, for any elements $h_1, h_2 \in L \mathbb{G}_m(A)$ and any elements $\varphi_1, \varphi_2 \in {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$, we have
$$
\begin{equation*}
(h_1, \varphi_1)(h_2, \varphi_2)= (h_1 \varphi_1(h_2), \varphi_1 \varphi_2).
\end{equation*}
\notag
$$
Besides, the group functor ${\mathcal G}$ acts on the commutative group functor $\mathcal{L}$ in the natural way: $(h,\varphi) (f) = h \varphi(f)$, where $h \in L\mathbb{G}_m(A)$, $\varphi \in {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$, $f \in \mathcal{L}(A)=A((t))$ for any commutative ring $A$. We note that the group ${\mathcal G}(A)$ acts by $A$-module homeomorphisms on $A((t))$ for any commutative ring $A$. We note that, for any commutative ring $A$, any $\varphi \in {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$, and any $h \in L \mathbb{G}_m(A)$, we have $\nu (\varphi(h)) = \nu(h)$. Therefore, the group functor ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})$ preserves the group subfunctor $(L \mathbb{G}_m)^0 \hookrightarrow L \mathbb{G}_m$ under the natural action. Hence the group subfunctor $\mathcal{G}^0 \hookrightarrow \mathcal{G}$ defined by
$$
\begin{equation}
{\mathcal G}^0 = (L \mathbb{G}_m)^0 \rtimes {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})
\end{equation}
\tag{14}
$$
is well-defined. Lemma 1. We have $\mathcal{G} = \mathcal{G}^0 \rtimes \underline{\mathbb{Z}} $, where $\underline{\mathbb{Z}} \hookrightarrow L \mathbb{G}_m \hookrightarrow \mathcal{G}$ as in § 2.1. Proof. For any commutative ring $A$, any element $h \in (L \mathbb{G}_m)^0(A)$, and any element $\varphi \in {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$ we have inside the group $\mathcal{G}(A)$ the following equality: and $(th\varphi(t)^{-1}, \varphi) \in \mathcal{G}^0(A)$, since $\nu(th\varphi(t)^{-1})=0$. This proves the lemma. 2.3. Ind-schemesBy an ind-affine ind-scheme, we mean an ind-object $M= ``\varinjlim_{i \in I}\!" \operatorname{Spec} C_i$ of the category of affine schemes such that all transition maps in the ind-object are closed embeddings of schemes. We say that ind-affine ind-scheme is ind-flat if any $C_i$ is a flat $\mathbb{Z}$-module. By $\mathcal{O}(M)= \varprojlim_{i \in I} C_i$ we will denote the ring of regular functions of $M$. We note that all the functors described in §§ 2.1 and 2.2 are represented by ind-flat ind-affine ind-schemes. Indeed (see formulas (5), (6), (11), (12)),
$$
\begin{equation}
\mathbb{G}_m = \operatorname{Spec} \mathbb{Z}[x,x^{-1}], \qquad \underline{\mathbb{Z}} = ``\varinjlim_{n \in \mathbb{N}}\!" \coprod_{-n \leqslant l \leqslant n} (\operatorname{Spec} \mathbb{Z})_l,
\end{equation}
\tag{15}
$$
$$
\begin{equation}
\mathbb{V}_+ = \operatorname{Spec} \mathbb{Z}[a_1, a_2, \dots], \qquad \mathbb{V}_- = ``\varinjlim_{\{\epsilon_i\}}\!" \operatorname{Spec} \mathbb{Z}[a_{-1}, a_{-2}, \dots]/ I_{\{\epsilon_i\}},
\end{equation}
\tag{16}
$$
$$
\begin{equation}
{\mathcal Aut}_+^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}) = ``\varinjlim_{n \in \mathbb{N}}\!" \operatorname{Spec} \mathbb{Z}[a_0]/ (a_0^n) \times \operatorname{Spec} \mathbb{Z}[a_1, a_1^{-1}] \times \operatorname{Spec} \mathbb{Z}[a_2, a_3, \dots ],
\end{equation}
\tag{17}
$$
$$
\begin{equation}
{\mathcal Aut}_-^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}) = ``\varinjlim_{\{\epsilon_i\}}\!" \operatorname{Spec} \mathbb{Z}[a_{-1}, a_{-2}, \dots]/ I_{\{\epsilon_i\}},
\end{equation}
\tag{18}
$$
where $(\operatorname{Spec} \mathbb{Z})_l = \operatorname{Spec} \mathbb{Z} $ and the limits in $\mathbb{V}_-$ and ${\mathcal Aut}_-^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})$ are taken over all the sequences $\{\epsilon_i\}$ with integers $i < 0$ and $\epsilon_i$ are non-negative integers such that all but finitely many $\epsilon_i$ equal zero, the ideal $I_{\{\epsilon_i\}}$ is generated by elements $a_i^{\epsilon_i +1}$ for all integers $i < 0$. The other functors $L \mathbb{G}_m$, $(L \mathbb{G}_m)^0$, ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})$, $\mathcal{G}$, $\mathcal{G}^0$ are represented by products of some above ind-schemes. § 3. Cohomology of group functors3.1. General definitions and propertiesWe fix a commutative ring $R$. We say that a functor is over $R$ if this functor is from the category of commutative $R$-algebras. All functors (correspondingly, group or commutative group functors) in this section will be over $R$. We will omit this indication on $R$. For any functor $G$ and any commutative $R$-algebra $R'$, by $F_{R'}$ we denote the restriction of the functor $F$ to the category of commutative $R'$-algebras. For any functor $G$ and any commutative group functor $F$, by $\operatorname{Hom}(G, F)$ we denote the Abelian group of all morphisms from the functor $G$ to the functor $F$. In this definition, we do not take into account any additional structure (if it exists) on the functor $G$. For any integer $n \geqslant 1$ and any functor $G$, by $G^{\times n}$ we denote the functor which is the $n$-fold direct product of the functor $G$. A group functor $G$ acts on a commutative group functor $F$ if for any commutative $R$-algebra $A$ the group $G(A)$ acts on the Abelian group $F(A)$ such that this action is compatible with any $R$-algebra homomorphism of commutative $R$-algebras. We will also say that $F$ is a $G$-module. Now we recall the definition and the main properties (see [19], § 2.3.1) for cohomology of group functors. We will fix a group functor $G$ until the end of this section. Let $F$ be a $G$-module. Then there is the cochain complex of Abelian groups (this complex is similar to the complex obtained from the bar or standard resolution in the group cohomology)
$$
\begin{equation}
C^0 (G,F) \xrightarrow{\delta_0} C^{1}(G, F) \xrightarrow{\delta_1} \cdots \xrightarrow{\delta_{k-1}} C^{k}(G,F) \xrightarrow{\delta_k} \cdots,
\end{equation}
\tag{19}
$$
where $C^0(G, F)= F(R)$, $C^k(G, F)= \operatorname{Hom} (G^{\times k}, F)$ when $k \geqslant 1$, and
$$
\begin{equation*}
\begin{aligned} \, \delta_{q}c(g_1, \dots, g_{q+1}) &*= g_1 c(g_2, \dots, g_{q+1}) \cdot \prod_{i=1}^q c(g_1, \dots, g_i g_{i+1}, \dots, g_{q+1})^{(-1)^i} \\ &\qquad\times c(g_1, \dots, g_q)^{(-1)^{q+1}}, \end{aligned}
\end{equation*}
\notag
$$
where integers $q \geqslant 0$, $c \in C^q(G,F)$, $g_j \in G(A)$ with $1 \leqslant j \leqslant q+1$ for any commutative $R$-algebra $A$. Besides, we use the multiplicative notation for the group laws in Abelian groups $F(A)$ and $C^{q}(G,F)$. Let $q$ be a non-negative integer. A $q$-cocycle on $G$ with coefficients in $F$ is an element of the subgroup $\operatorname{Ker} \delta_q \subset C^{q}(G,F)$. For $q \geqslant 1$, a $q$-coboundary on $G$ with coefficients in $F$ is an element of the subgroup $\operatorname{Im} \delta_{q-1} \subset C^{q}(G,F)$. The $q$th cohomology group of $G$ with coefficients in $F$ is the Abelian group
$$
\begin{equation*}
H^q(G,F)= \operatorname{Ker} \delta_q / \operatorname{Im} \delta_{q-1}.
\end{equation*}
\notag
$$
To give a $q$-cocycle on $G$ with coefficients in $F$ is the same as to give a collection of $q$-cocycles $\{ c_A \}$ on $G(A)$ with coefficients in $F(A)$ for any commutative $R$-algebra $A$ with obvious compatibility condition for $c_{A_1}$ and $c_{A_2}$ for any $R$-algebra homomorphism $A_1 \to A_2$. Besides, for any fixed commutative $R$-algebra $A$, there is a natural morphism from complex (19) to the corresponding complex that calculates the cohomology of $G(A)$ with coefficients in $F(A)$. This gives us the natural homomorphism Any morphism of $G$-modules $F_1 \to F_2$ induces the homomorphism of Abelian groups $H^q(G,F_1) \to H^q(G, F_2)$. We can speak about a short exact sequence (or an extension) of group functors, this is equivalent that this sequence becomes a short exact sequence of groups after taking the values of group functors on any commutative $R$-algebra $A$. By Proposition 2.2 in [19], the elements of the group $H^2(G,F)$ are in one-to-one correspondence with equivalence classes of extensions of the group functor $G$ by the commutative group functor $F$
$$
\begin{equation}
1 \to F \to \widetilde{G} \xrightarrow{\vartheta} G \to 1
\end{equation}
\tag{20}
$$
such that the morphism $\vartheta$ has a section $G \to \widetilde{G}$ as a morphism of functors (in general, not as a morphism of group functors), and the action of $G$ on $F$ comes from inner automorphisms in the group functor $\widetilde{G}$. Besides, as usual, extension (20) is central if and only if $G$ acts trivially on $F$. As in the group cohomology (see [4], Ch. V, § 3), for any $1$-cocycles $\lambda_1$ and $\lambda_2$ on $G$ with coefficients in $G$-modules $F_1$ and $F_2$, correspondingly, we obtain a $2$-cocycle $\lambda_1 \cup \lambda_2$ on $G$ with coefficients in $F_1 \otimes F_2$ such that
$$
\begin{equation*}
(\lambda_1 \cup \lambda_2) (g_1, g_2) = \lambda_1 (g_1) \otimes g_1 ( \lambda_2(g_2) ),
\end{equation*}
\notag
$$
where, for any commutative $R$-algebra $A$, we have $(F_1 \otimes F_2)(A) = F_1(A) \otimes_{\mathbb{Z}} F_2(A)$, and $g_1, g_2 $ are any elements from $G(A)$. This induces the well-defined homomorphism, which is called the $\cup$-product:
$$
\begin{equation*}
\cup \colon H^1(G, F_1) \otimes_{\mathbb{Z}} H^1(G, F_2) \to H^2(G, F_1 \otimes F_2).
\end{equation*}
\notag
$$
3.2. Resolution by induced modulesWe will explain that it is also possible to calculate $H^q(G,F)$ via the resolution of $F$ by induced $G$-modules, which are acyclic. For any $G$-module $F$, we denote the group $F^{G}= H^0(G, F)$ which consists of elements $f \in F(R)$ such that, for any commutative $R$-algebra $A$ and any element $g \in G(A)$, we have $g(f)=f$. Now $F \mapsto F^{G}$ is a left exact functor from the Abelian category of $G$-modules to the category of Abelian groups. Let $L$ be any commutative group functor. By $\underline{\operatorname{Hom}}(G, L)$ we denote a $G$-module defined as
$$
\begin{equation*}
\underline{\operatorname{Hom}}(G, L)(A) = \operatorname{Hom}(G_A, L_A),
\end{equation*}
\notag
$$
where $A$ is any commutative $R$-algebra, and the $G$-module structure is the following: for any $g \in G(A)$, $h \in \underline{\operatorname{Hom}}(G, L)(A)$, $x \in G(A')$, where $A'$ is any commutative $A$-algebra, we define $g(h)(x)= h(x g)$. There is an embedding of $G$-modules
$$
\begin{equation}
\alpha\colon L \hookrightarrow \underline{\operatorname{Hom}}(G, L)
\end{equation}
\tag{21}
$$
given for any commutative $R$-algebra $A$ as $L(A) \ni m \mapsto \alpha(m) \in \underline{\operatorname{Hom}}(G, L)(A)$, where $\alpha(m)(x)= x m$ for any $x \in G(A')$ and any commutative $A$-algebra $A'$. Besides, there is a morphism of group functors (in general, not a morphism of $G$-modules)
$$
\begin{equation}
\beta\colon \underline{\operatorname{Hom}}(G, L) \to L \quad \text{such that} \quad \beta \alpha = \operatorname{Id},
\end{equation}
\tag{22}
$$
where, for any commutative $R$-algebra $A$, any $h \in \underline{\operatorname{Hom}}(G, L)(A)$, we have $\beta(h)= h(e)$, where $e$ is the identity element of the group $G(A)$. We will call also the $G$-module $\underline{\operatorname{Hom}}(G,L)$ an induced $G$-module. Proposition 1. We fix a group functor $G$. 1. Any short exact sequence of $G$-modules such that $\gamma$ admits a section as functors (not necessarily as group functors) induces the long exact sequence of Abelian groups
$$
\begin{equation*}
\cdots \to H^q(G, F_1) \to H^q(G, F_2) \to H^q(G, F_3) \to H^{q+1}(G, F_1) \to \cdots.
\end{equation*}
\notag
$$
2. For any commutative group functor $L$ and any integer $q > 0$, 3. For any $G$-module $F$, taking via (21) the resolution of $F$ by induced modules, then applying to this resolution the functor $K \mapsto K^{G}$ (where $K$ is a $G$-module), and then taking the cohomology groups of this complex, we obtain the groups $H^*(G,F)$. Proof. 1. The proof follows from a long exact sequence of cohomology groups induced by a short exact sequence of cochain complexes (19) for $F_1$, $F_2$, $F_3$ that is obtained from short exact sequences of Abelian groups which follow because of a section $F_3 \to F_2$ as functors.
2. For complex (19) with $F = \underline{\operatorname{Hom}}(G, L)$, we construct a contracting homotopy for any $q \geqslant 1$ (cf. [25], Exposé I, Lemma 5.2.2)
$$
\begin{equation*}
s_q \colon C^{q}(G,\underline{\operatorname{Hom}}(G, L)) \to C^{q-1}(G,\underline{\operatorname{Hom}}(G, L))
\end{equation*}
\notag
$$
by the rule
$$
\begin{equation*}
(s_q h) (g_1, g_2, \dots, g_{q-1})(g)= h(g, g_1, \dots, g_{q-1})(e),
\end{equation*}
\notag
$$
where $h \in C^q(G, \underline{\operatorname{Hom}}(G, L))$, $g_1, \dots, g_{q-1} \in G(A)$ for a commutative $R$-algebra $A$, $g \in G(A')$ for any commutative $R$-algebra $A'$, $e \in G(A')$ is the identity element of the group $G(A')$. The necessary equalities $\delta_{q-1} s_q + s_{q+1} \delta_q = \operatorname{Id}$ follow by direct calculation.
3. This is a standard statement, which follows from two previous assertions and splitting of the resolution into short exact sequences such that each of them has splitting as functors in virtue of (22). This proves the proposition. § 4. Lie algebra valued functors from group functorsWe suppose that a group functor $G$ over a commutative ring $R$ is represented by an ind-affine ind-scheme over $R$. Then (see, for example, [19], Appendix A) we have the tangent space functor $\operatorname{Lie} G $ at the unit of the group $G(R)$:
$$
\begin{equation*}
\operatorname{Lie} G(A) = \operatorname{Ker} \bigl( G(A[\varepsilon]/ (\varepsilon^2)) \to G(A) \bigr),
\end{equation*}
\notag
$$
where $A$ is any commutative $R$-algebra. The functor $\operatorname{Lie} G$ is a commutative group functor over $R$, which has a natural structure of ${\mathbb A}^1_R$-module, where ${\mathbb A}^1_R$ is a ring functor over $R$ such that ${\mathbb A}^1_R(A) = A$. And there is a bracket
$$
\begin{equation*}
[\,{\cdot}\,, {\cdot}\,] \colon \operatorname{Lie} G \times \operatorname{Lie} G \to \operatorname{Lie} G
\end{equation*}
\notag
$$
which defines on $\operatorname{Lie} G (A)$ the structure of the Lie $A$-algebra for any commutative $R$-algebra $A$ (that is, $\operatorname{Lie} G$ is a Lie algebra valued functor over $R$). For the definition of $[\,{\cdot}\,, {\cdot}\,]$, we take elements $d_i \in \operatorname{Lie} G (A) \subset G (A[\varepsilon_i]/ (\varepsilon_i^2)) $, where $i=1$ and $i=2$. Hence $d_1, d_2 \in G(A[\varepsilon_1, \varepsilon_2] / (\varepsilon_1^2, \varepsilon_2^2))$ via the natural embedding $G (A[\varepsilon_i]/ (\varepsilon_i^2)) \hookrightarrow G(A[\varepsilon_1, \varepsilon_2] / (\varepsilon_1^2, \varepsilon_2^2))$, where $i=1$ and $i=2$. Now we have
$$
\begin{equation}
[d_1, d_2] = d_1 d_2 d_1^{-1} d_2^{-1} \in G (A[\varepsilon_1, \varepsilon_2] / (\varepsilon_1^2, \varepsilon_2^2))
\end{equation}
\tag{23}
$$
is the image of the element from $\operatorname{Lie} G (A)$ under the embedding induced by the embedding of rings $A [\varepsilon_1 \varepsilon_2] / (\varepsilon_1^2 \varepsilon_2^2) \hookrightarrow A[\varepsilon_1, \varepsilon_2]/ (\varepsilon_1^2, \varepsilon_2^2)$, where we consider
$$
\begin{equation*}
\operatorname{Lie} G(A) = \operatorname{Ker} \bigl( G(A[\varepsilon_1 \varepsilon_2]/ (\varepsilon_1^2 \varepsilon_2^2)) \to G(A) \bigr).
\end{equation*}
\notag
$$
The correspondence $G \mapsto \operatorname{Lie} G$ is a functor from the category of group functors over $R$ represented by group ind-affine ind-schemes over $R$ to the category of Lie algebra valued functors over $R$. We consider the group functor $\mathcal{G}^0$ (see (14)), which is represented by an ind-affine ind-scheme. Proposition 2. Let $A$ be any commutative ring. Then we have a natural isomorphism of Lie $A$-algebras where $A((t))$ is an Abelian Lie $A$-algebra and $\operatorname{Der}^{\mathrm{c}}_A (A((t)))$ is the Lie $A$-algebra of continuous $A$-derivations on the commutative $A$-algebra $A((t))$ with the commutator bracket. This means that $\operatorname{Lie} \mathcal{G}^0(A)$ is $A((t)) + A((t)) \, \frac{\partial}{\partial t}$ as an $A$-module with the bracket where $s_i $ and $r_i$ are from $A((t))$, and $r_i'$ or $s_i'$ means the derivative with respect to $t$ of the corresponding element. In other words, the Lie $A$-algebra $\operatorname{Lie} \mathcal{G}^0 (A)$ is naturally isomorphic to the Lie algebra of continuous differential operators of order $\leqslant 1$ acting on functions on the punctured disc over $A$: Proof. Let $A$ be a commutative ring.
By definition, $\operatorname{Lie} \mathcal{G}^0 (A) $ consists of elements $(1 + s \varepsilon, \varphi) \in \mathcal{G}^0 (A[\varepsilon]/ (\varepsilon^2))$, where $s \in A((t))$ and $\varphi \in {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}) (A[\varepsilon]/ (\varepsilon^2))$ such that $\varphi(t)= t + r \varepsilon $ with $r \in A((t))$. For any $f \in A((t))$ we have the action of $(1 + s \varepsilon, \varphi)$ on $f$ in $(A[\varepsilon]/ (\varepsilon^2))((t))$:
$$
\begin{equation*}
\begin{aligned} \, (1 + s \varepsilon, \varphi)(f) &= (1 + s\varepsilon) \varphi(f)= (1 + s \varepsilon) \cdot (f \circ (t + r \varepsilon)) \\ &= (1 + s \varepsilon)(f + r f' \varepsilon)= f + (sf + rf') \varepsilon. \end{aligned}
\end{equation*}
\notag
$$
Hence the map $(1 + s \varepsilon, \varphi) \mapsto s + r \, \frac{\partial}{\partial t} $ gives isomorphism (24) as $A$-modules and formula (26).
To obtain the Lie bracket (25), we use formula (23). For the bracket between elements $r_1 \, \frac{\partial}{\partial t}$ and $r_2\, \frac{\partial}{\partial t}$, see [19], § 4.1. The bracket between elements $s_1$ and $s_2$ is zero, because they come from the commutative group functor $(L \mathbb{G}_m)^0$. Now we calculate $[r_1 \, \frac{\partial}{\partial t}, s_2]$. The answer follows from the following calculations in $(A[\varepsilon_1, \varepsilon_2] / (\varepsilon_1^2, \varepsilon_2^2))((t))$:
$$
\begin{equation*}
\begin{aligned} \, (1 + s_2 \varepsilon_2) \circ (t + r_1 \varepsilon_1) &= 1+ (s_2 \circ (t + r_1 \varepsilon_1)) \varepsilon_2 = 1 + (s_2 + r_1 s_2' \varepsilon_1) \varepsilon_2 \\ &= (1+ r_1s_2' \varepsilon_1 \varepsilon_2)(1 + s_2 \varepsilon_2), \end{aligned}
\end{equation*}
\notag
$$
since in the group $\mathcal{G}^0(A[\varepsilon_1, \varepsilon_2]/ ( \varepsilon_1^2, \varepsilon_2^2))$ we have
$$
\begin{equation*}
\varphi_1 \cdot (1 + s_2 \varepsilon_2) \cdot \varphi_1^{-1} = (1 + s_2 \varepsilon_2) \circ (t + r_1 \varepsilon_1),
\end{equation*}
\notag
$$
where $\varphi_1 \in {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}) (A[\varepsilon_1, \varepsilon_2]/ (\varepsilon_1^2, \varepsilon_2^2))$ such that $\varphi_1(t) = t + r_1 \varepsilon_1$. Proposition 2 is proved. Remark 1. We consider the group functor $\mathcal{G}$ (see (13)), which is represented by ind-affine ind-scheme. Clearly, by definition, for any commutative ring $A$, We will need the following lemma, which is a punctured disc over $A$ analog of [2], Lemma 4.3, in the holomorphic case. Proof. We note that $[\operatorname{Lie} \mathcal{G}^0(A), \operatorname{Lie} \mathcal{G}^0(A)]$ is an $A((t))$-module, because
$$
\begin{equation*}
f \biggl[ s_1 + r_1 \, \frac{\partial}{\partial t}, \, s_2 + r_2 \, \frac{\partial}{\partial t} \biggr] = \biggl[ f r_1 \, \frac{\partial}{\partial t},\, s_2 + \frac{1}{2} \, r_2 \, \frac{\partial}{\partial t} \biggr] + \biggl[ s_1 + \frac{1}{2} \, r_1 \, \frac{\partial}{\partial t},\, f r_2 \, \frac{\partial}{\partial t} \biggr],
\end{equation*}
\notag
$$
where $f, r_i, s_i \in A((t))$ for $i=1$ and $i=2$. Therefore, it is enough to show that elements $1$ and $\frac{\partial}{\partial t}$ belong to $[ \operatorname{Lie} \mathcal{G}^0(A), \operatorname{Lie} \mathcal{G}^0(A)]$. We have This proves the lemma. § 5. Determinant central extension of $\mathcal{G}$We construct the determinant central extension of the group functor $\mathcal{G}$ (see (13)) by the group functor $\mathbb{G}_m$. 5.1. Block matrixWe consider the group functor $\mathcal{G}^0$ (see (14)). We will use the following decomposition into the direct sum. Proof. We note that if an element $g_1$ belongs to the subgroup $\mathbb{G}_m(A) \times \mathbb{V}_+ (A)$ or to the subgroup ${\mathcal Aut}_+^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$ of the group $\mathcal{G}^0(A)$, then the action of the element $g_1$ restricted to the $A$-submodule $A[[t]]$ is an isomorphism of this submodule.
If an element $g_2$ belongs to the subgroup $\mathbb{G}_m(A) \times \mathbb{V}_-(A)$ or to the subgroup ${\mathcal Aut}_-^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$ of the group $\mathcal{G}^0(A)$, then the action of the element $g_2$ restricted to the $A$-submodule $t^{-1}A[t^{-1}]$ is an isomorphism of this submodule. From decompositions (14), (7), and (10), it follows that, for any $g \in \mathcal{G}^0(A)$, there is a decomposition where $h_- \in \mathbb{V}_-(A)$, $h_+ \in \mathbb{G}_m(A) \times \mathbb{V}_+(A)$, $\varphi_- \in {\mathcal Aut}_-^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$, and $\varphi_+ \in {\mathcal Aut}_+^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$.The element $\varphi_-^{-1} h_+ \varphi_-$ belongs to the subgroup $(L \mathbb{G}_m)^0(A)$. We denote this element by $\widetilde{h}$. According to (7), we have a decomposition $\widetilde{h} = \widetilde{h}_- \widetilde{h}_+ $, where elements $\widetilde{h}_- \in \mathbb{V}_-(A) $ and $\widetilde{h}_+ \in \mathbb{G}_m(A) \times \mathbb{V}_+(A)$. Hence we obtain
$$
\begin{equation*}
g = h_- \varphi_- \varphi_-^{-1} h_+ \varphi_- \varphi_+ = h_- \varphi_- \widetilde{h} \, \varphi_+ = h_- \varphi_- \widetilde{h}_- \widetilde{h}_+ \varphi_+ = p_- p_+,
\end{equation*}
\notag
$$
where $p_- = h_- \varphi_- \widetilde{h}_-$ and $p_+ = \widetilde{h}_+ \varphi_+$. Besides, the action of the element $p_-$ restricted to the $A$-submodule $t^{-1}A[t^{-1}]$ is an isomorphism of this submodule, and the action of the element $p_+$ restricted to the $A$-submodule $A[[t]]$ is an isomorphism of this submodule.
Now we obtain
$$
\begin{equation*}
\begin{aligned} \, A((t)) &= p_- (A((t))) = p_-( t^{-1}A[t^{-1}] \oplus A[[t]]) = p_- ( p_-^{-1} (t^{-1}A[t^{-1}]) \oplus A[[t]]) \\ &= t^{-1}A[t^{-1}] \oplus p_- ( A[[t]]) \,{=}\, t^{-1}A[t^{-1}] \oplus p_- p_+ ( A[[t]]) \,{=}\, t^{-1}A[t^{-1}] \oplus g( A[[t]]). \end{aligned}
\end{equation*}
\notag
$$
This proves the proposition. For any commutative ring $A$ and any element $g \in \mathcal{G}(A)$, we introduce the following block matrix with respect to the decomposition $A((t)) = t^{-1} A[t^{-1}] \oplus A[[t]]$ for the action of $g$ on $A((t))$:
$$
\begin{equation}
\begin{pmatrix} a_g & b_g \\ c_g & d_g \end{pmatrix},
\end{equation}
\tag{28}
$$
where $d_g \colon A[[t]] \to A[[t]]$, $d_g= \operatorname{pr} \cdot (g |_{A[[t]]})$, and $\operatorname{pr}\colon A((t)) \to A[[t]]$ is the projection. (Here, the matrix acts from the left on an element-column from $A((t))$.) We note that the $A$-module map $d_g$ is continuous as the composition of continuous maps. As an immediate corollary of Proposition 3 we have Corollary 1. For any commutative ring $A$ and any $g \in \mathcal{G}^0(A)$, the map $d_g$ is bijective. 5.2. Construction of the determinant central extension of $\mathcal{G}^0$We will construct a central extension of the group functor $\mathcal{G}^0$ by the group functor $\mathbb{G}_m$. We call this central extension the determinant central extension. (This construction is similar to the corresponding constructions for the group ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$ from [19], § 3.3, and to the constructions in the smooth case from [23], § 6.6.) Let $A$ be any commutative ring. Let $\operatorname{GL}_A(A[[t]])$ be the group of all $A$-module automorphisms of $A[[t]] $. We construct a group $\widehat{\mathcal{G}^0}(A)$ that as a set consists of all pairs $(g, r)$, where $g \in \mathcal{G}^0(A)$ and $r \in \operatorname{GL}_A(A[[t]])$ such that there is an integer $n > 0$, which depends on $g$ and $r$, with the property $d_g |_{t^nA[[t]]} = r |_{t^nA[[t]]}$. We note that $r$ is a continuous map, since so are $d_g -r$ and $d_g$. Since $g$, $g^{-1}$ and $r$ are continuous maps, it is easy to see that the set $\widehat{\mathcal{G}^0}(A)$ is a subgroup in $\mathcal{G}^0(A) \times \operatorname{GL}_A(A[[t]])$. This gives a group structure on the set $\widehat{\mathcal{G}^0}(A)$. Now we have the homomorphism $\widehat{\mathcal{G}^0}(A) \xrightarrow{\alpha} \mathcal{G}^0(A)$ given as $(g,r) \mapsto g$. By Corollary 1, this homomorphism is surjective. Therefore, we obtain an exact sequence of groups
$$
\begin{equation}
1 \to \operatorname{GL}_f(A) \to \widehat{\mathcal{G}^0}(A) \xrightarrow{\alpha} \mathcal{G}^0(A) \to 1,
\end{equation}
\tag{29}
$$
where $\operatorname{GL}_f(A)$ is a group that consists of all elements $r \in \operatorname{GL}_A(A[[t]])$ such that there is an integer $n >0$, which depends on $r$, with the property $r |_{t^n A[[t]]} = \mathrm{id}$. We note that there is a well-defined determinant homomorphism where $\det(r)$ for $r \in \operatorname{GL}_f(A)$ is the determinant of the action of $r $ on a free $A$-module $A[[t]]/ t^n A[[t]]$, where $r |_{t^n A[[t]]} = \mathrm{id}$.Lemma 3. For any $r_1 \in \operatorname{GL}_f(A)$ and $(g,r) \in \widehat{\mathcal{G}^0}(A)$, consider where $r_2 \in \operatorname{GL}_f(A)$. Then $\det(r_2) = \det(r_1)$. Proof. We consider an integer $n > 0$ such that $r_1 |_{t^n A[[t]]} = \mathrm{id}$ and an integer $l > 0$ such that $r(t^l A[[t]]) \subset t^n A[[t]]$. Then $r_2 |_{t^l A[[t]]} = \mathrm{id}$ and $\det(r_2)$ comes from the following composition of homomorphisms of $A$-modules of rank $1$:
$$
\begin{equation*}
\begin{aligned} \, \bigwedge^l (A[[t]]/ t^l A[[t]]) &\to \bigwedge^l (A[[t]]/ r( t^l A[[t]])) \to \bigwedge^l (A[[t]]/ r( t^l A[[t]])) \\ &\to \bigwedge^l (A[[t]]/ t^l A[[t]]), \end{aligned}
\end{equation*}
\notag
$$
where the first arrow is induced by the map $r$, the second arrow is induced by the map $r_1$, the third arrow is induced by the map $r^{-1}$. Besides, the second arrow is the multiplication by $\det(r_1)$, since the homomorphism $\det$ does not depend on the choice of a basis of a free $A$-module of finite rank and addition to this $A$-module another projective $A$-module of finite rank, see [17], § 3. This proves the lemma. We denote a subgroup $T(A) = \operatorname{Ker} \det$ of the group $\operatorname{GL}_f(A)$. By Lemma 3, $T(A)$ is a normal subgroup in $\widehat{\mathcal{G}^0}(A)$. We introduce a group $\widetilde{{\mathcal{G}^0}}(A) = \widehat{\mathcal{G}^0}(A)/ T(A)$. From exact sequence (29), we obtain a central extension of groups
$$
\begin{equation}
1 \to A^* \to \widetilde{{\mathcal{G}^0}}(A) \to \mathcal{G}^0(A) \to 1.
\end{equation}
\tag{30}
$$
We call this central extension the determinant central extension. We note that the map $\alpha$ in (29) has a section $g \mapsto (g, d_g)$ as sets. This induces a section $\sigma$ of the determinant central extension (30). Using the section $\sigma$, in the standard way we obtain a $2$-cocycle $D$ for the determinant central extension such that $D(x,y) = \sigma(x) \sigma(y) \sigma(xy)^{-1}$, where $x,y \in \mathcal{G}^0(A)$. From the construction of $\sigma$ we find that the $2$-cocycle $D$ is explicitly written in the following way: where $d_{xy}\,{=}\, c_x \,{\cdot}\, b_y + d_x \,{\cdot}\, d_y$ (recall (28)). The determinant $\det$ in (31) is well-defined, since there is an integer $n \geqslant 0$ such that $(d_x \,{\cdot}\, d_y \,{\cdot}\, d_{xy}^{-1}) |_{t^n A[[t]] = \mathrm{id}}$.Clearly, the constructions of central extension (30) and $2$-cocycle (31) are functorial with respect to the ring $A$. Therefore, we obtain the following proposition. Proposition 4. The central extension (30) gives the determinant central extension of the group functor $\mathcal{G}^0$ by the commutative group functor $\mathbb{G}_m$. This central extension admits a section $\sigma$ which gives a $2$-cocycle $D$ of $\mathcal{G}_0$ by $\mathbb{G}_m$ given by formula (31). 5.3. Construction of the determinant central extension of $\mathcal{G}$We will construct a central extension of the group functor $\mathcal{G}$ by the group functor $\mathbb{G}_m$. We will call this central extension also the determinant central extension. (In § 5.4, we will prove that this central extension restricted to $\mathcal{G}^0$ coincides with the determinant central extension from Proposition 4.) Let $A$ be any commutative ring. By [19], § 3.2, for any element $g$ from the group $\mathcal{G}(A)$ and any integer $l$ such that $t^l A[[t]] \subset g (A[[t]]) $, the $A$-module $g(A[[t]]) / t^l A[[t]]$ is projective and finitely generated. For any $g_1, g_2 \in \mathcal{G}(A)$, the projective $A$-module of rank $1$
$$
\begin{equation*}
\operatorname{Hom}_A \biggl(\,\bigwedge^{\mathrm{max}}(g_1(A[[t]])/ t^l A[[t]]), \, \bigwedge^{\mathrm{max}}(g_2(A[[t]])/ t^lA[[t]]) \biggr)
\end{equation*}
\notag
$$
does not depend on the choice of an integer $l$ up to a unique isomorphism. We identify over all such $l$ all these projective $A$-modules via the following definition of the projective $A$-module of rank $1$:
$$
\begin{equation*}
\begin{aligned} \, &\det(g_1(A[[t]]) \mid g_2(A[[t]])) \\ &\qquad= \varinjlim_l \operatorname{Hom}_A \biggl(\, \bigwedge^{\mathrm{max}}(g_1(A[[t]])/ t^l A[[t]]), \, \bigwedge^{\mathrm{max}}(g_2(A[[t]])/ t^lA[[t]]) \biggr). \end{aligned}
\end{equation*}
\notag
$$
For any elements $g_1, g_2, g_3 \in \mathcal{G}(A)$, we have a canonical isomorphism of $A$-modules
$$
\begin{equation}
\det(g_1(A[[t]]) \mid g_2(A[[t]])) \otimes_A \det(g_2(A[[t]]) \mid g_3(A[[t]])) \xrightarrow{\sim} \det(g_1(A[[t]]) \mid g_3(A[[t]]))
\end{equation}
\tag{32}
$$
that satisfies the associativity diagram for any four elements from $\mathcal{G}(A)$. This gives also the following canonical isomorphisms of $A$-modules:
$$
\begin{equation*}
\begin{gathered} \, \det(g_1(A[[t]]) \mid g_1(A[[t]])) \xrightarrow{\sim} A, \\ \det(g_1(A[[t]]) \mid g_2(A[[t]])) \otimes_A \det(g_2(A[[t]]) \mid g_1(A[[t]])) \xrightarrow{\sim} A. \end{gathered}
\end{equation*}
\notag
$$
Besides, any element $f \in \mathcal{G}(A)$ defines an isomorphism of $A$-modules
$$
\begin{equation}
\det(g_1(A[[t]]) \mid g_2(A[[t]])) \xrightarrow{\sim} \det(fg_1(A[[t]]) \mid fg_2(A[[t]])).
\end{equation}
\tag{33}
$$
Lemma 4. For all $g_1, g_2 \,{\in}\, \mathcal{G}(A)$, the $A$-module $\det(g_1(A[[t]]) \mid g_2(A[[t]]))$ is free of rank $1$. Proof. By (33) it is enough to suppose that, for example, $g_1 =1$ and $g_2=g$. If $g \in \mathcal{G}^0(A)$, then this statement follows from Corollary 1 and [19], Proposition 3.3 (see also the inverse isomorphism to the isomorphism given below by (37) for $r = d_g$). For any $g \in \mathcal{G}(A)$, we have $g = t^{\underline{n}} f$, where $\underline{n} \in \underline{\mathbb{Z}}(A)$ and $f \in \mathcal{G}^0(A)$ (see Lemma 1). Therefore, the required result now follows from isomorphisms of $A$-modules
$$
\begin{equation*}
\begin{aligned} \, \det(A[[t]] \mid g(A[[t]])) &\simeq \det(A[[t]] \mid t^{\underline{n}}(A[[t]])) \otimes_A \det( t^{\underline{n}}(A[[t]]) \mid t^{\underline{n}} f (A[[t]])) \\ &\simeq A \otimes_A \det( t^{\underline{n}}(A[[t]]) \mid t^{\underline{n}} f (A[[t]])) \simeq \det( A[[t]] \mid f (A[[t]])). \end{aligned}
\end{equation*}
\notag
$$
This proves the lemma. Using Lemma 4, we define a determinant central extension
$$
\begin{equation}
1 \to A^* \to \widetilde{\mathcal{G}}(A) \to \mathcal{G}(A) \to 1,
\end{equation}
\tag{34}
$$
where the group $\widetilde{\mathcal{G}}(A)$ consists of pairs $(g, s)$, where $g \in \mathcal{G}(A)$ and $s$ is an element of free $A$-module $\det( g(A[[t]]) \mid A[[t]])$ of rank $1$ such that $s$ is a generator of the $A$-module $\det( g(A[[t]]) \mid A[[t]])$. The group law is as follows:
$$
\begin{equation*}
(g_1, s_1) (g_2, s_2)= (g_1 g_2, g_1(s_2) \otimes s_1),
\end{equation*}
\notag
$$
and we map $(g,s) \in \widetilde{\mathcal{G}}(A)$ to $g \in \mathcal{G}(A)$ to obtain the central extension. The correspondence $A \mapsto \widetilde{\mathcal{G}}(A)$ is functorial with respect to $A$ (compare with the analogous fact for $\widetilde{{\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)}$ in [19], § 3.3). This follows from the natural isomorphism
$$
\begin{equation*}
(g (A_1[[t]]) / h (A_1[[t]])) \otimes_{A_1} A_2 \xrightarrow{\sim} u(g) (A_2[[t]]) / u(h) (A_2[[t]]),
\end{equation*}
\notag
$$
where $u\colon A_1 \to A_2$ is any homomorphism of commutative rings, which induces the homomorphism $\mathcal{G}(A_1) \to \mathcal{G}(A_2)$ denoted by the same letter $u$, and elements $g, h$ are from $ \mathcal{G}(A_1)$ such that $g (A_1[[t]]) \supset h (A_1[[t]])$. This isomorphism is satisfied, since by [19], Proposition 3.2, the projective $ A_1$-module $g (A_1[[t]]) / h (A_1[[t]])$ is a direct summand of an $A_1$-module $t^lA_1[[t]]/ t^mA_1[[t]]$ for appropriate $l, m \in \mathbb{Z}$, and, for the $A_1$-module $t^lA_1[[t]]/ t^mA_1[[t]]$, this isomorphism is evident. Thus we obtain the group functor $\widetilde{\mathcal{G}}$ and the determinant central extension of $\mathcal{G}$ by $\mathbb{G}_m$ from (34). 5.4. Uniqueness of an extension of the determinant central extension from $\mathcal{G}^0$ to $\mathcal{G}$We will prove that the determinant central extension of $\mathcal{G}$ by $\mathbb{G}_m$ after restriction to $\mathcal{G}^0$ coincides with the determinant central extension from Proposition 4 and an extension from $\mathcal{G}^0$ to $\mathcal{G}$ is unique. The last statement will follow from the fact that any central extension of $\mathcal{G}$ by $\mathbb{G}_m$ is uniquely defined by its restriction to $\mathcal{G}^0$. Proposition 5. Any morphism of group functors $\mathcal{G}^0_{\mathbb{Q}} \to {\mathbb{G}_m}_{\mathbb{Q}}$ (that is, which preserves the group structures) is trivial. Analogously, any morphism of group functors ${\mathcal{G}^0} \to {\mathbb{G}_m}$ is trivial. Proof. Let $\beta$ be a morphism of group functors $\mathcal{G}^0 \to \mathbb{G}_m$. The functor $\mathbb{G}_m$ is represented by the affine scheme, and the functor $\mathcal{G}^0$ is represented by an ind-affine ind-scheme that is ind-flat. Therefore, to prove that $\beta$ is trivial, it is enough to prove that the morphism $\beta_{\mathbb{Q}} \colon \mathcal{G}^0_{\mathbb{Q}} \to {\mathbb{G}_m}_{\mathbb{Q}}$ (that is, after restrictions of functors to the category of commutative $\mathbb{Q}$-algebras) is trivial, since these morphisms are determined by the corresponding homomorphisms of rings of regular functions of schemes and ind-schemes.
So, we will prove the first statement of the proposition and suppose that we have a morphism of group functors $\beta \colon \mathcal{G}^0_{\mathbb{Q}} \to {\mathbb{G}_m}_{\mathbb{Q}}$. Let $A$ be any commutative $\mathbb{Q}$-algebra. We claim that, for any $z \in \mathcal{G}^0(A)$, the differential $(d \beta)_z$ of $\beta$ is zero as the map from the tangent $A$-module $T_z \mathcal{G}^0 (A)$ of $\mathcal{G}^0$ at $z$ to the tangent $A$-module $T_{\beta(z)} {\mathbb{G}_m}(A)$, where $T_z \mathcal{G}^0 (A) = \rho^{-1}(z)$ and $\rho \colon \mathcal{G}^0(A[\varepsilon]/ (\varepsilon^2)) \to G^0(A)$ is the natural map, and the tangent space
$$
\begin{equation*}
T_{\beta(z)} {\mathbb{G}_m}(A)= \beta(z) + A \varepsilon \subset (A[\varepsilon] / (\varepsilon^2))^*
\end{equation*}
\notag
$$
is naturally identified with $A$.
Indeed, if $z$ is the identity element $e$ of the group $\mathcal{G}^0(A)$, then $T_e \mathcal{G}^0 (A) = \operatorname{Lie} \mathcal{G}^0 (A)$ and $(d \beta)_e$ is a homomorphism from the Lie $A$-algebra $\operatorname{Lie} \mathcal{G}^0 (A)$ to the Lie algebra $\operatorname{Lie} \mathbb{G}_m(A)$. But by Lemma 2, $[\operatorname{Lie} \mathcal{G}^0 (A), \operatorname{Lie} \mathcal{G}^0 (A)] = \operatorname{Lie} \mathcal{G}^0 (A)$, and $\operatorname{Lie} \mathbb{G}_m(A) \simeq A$ is an Abelian Lie algebra. Therefore, $(d \beta)_e = 0$. For any $z \in \mathcal{G}^0(A)$, we have $(d \beta)_z = 0$, since $\beta$ is the morphism of group functors and therefore, the following diagram is commutative: where $r_z$ or $r_{\beta(z)}$ is the multiplication on the right by the element $z$ or $\beta(z)$, and the horizontal arrows are isomorphisms.According to § 2.3, the functor $\mathcal{G}^0_{\mathbb{Q}}$ is represented by the ind-scheme $``\varinjlim_{i \in I}\!" \operatorname{Spec} C_i$ for concrete $\mathbb{Q}$-algebras $C_i$. Therefore, the morphism $\beta$ is uniquely defined by the homomorphism of $\mathbb{Q}$-algebras
$$
\begin{equation*}
\beta^* \colon \mathbb{Q}[x,x^{-1}] \to \mathcal{O}(\mathcal{G}^0_{\mathbb{Q}}) = \varprojlim_{i \in I} C_i,
\end{equation*}
\notag
$$
which is uniquely defined by the image $F = \beta^* (x)$ in $\mathcal{O}(\mathcal{G}^0_{\mathbb{Q}})$.
By (14), (7), (10), this ind-scheme is the product of another ind-schemes written explicitly as some ind-schemes from (15)–(18) after extension of scalars to $\mathbb{Q}$. Thus, the $\mathbb{Q}$-algebra $\mathcal{O}(\mathcal{G}^0_{\mathbb{Q}})$ is the algebra of formal power series in infinite (countable) number of variables with coefficients in the $\mathbb{Q}$-algebra of polynomials in infinite (countable) number of variables with coefficients in the $\mathbb{Q}$-algebra of Laurent polynomials in two variables. A formal power series from this algebra is an infinite sum of monomials, where every monomial is the product of a finite number of variables in powers, and the convergence condition for this infinite sum comes from the description of ideals $I_{\{ \epsilon_i \}}$ in § 2.3. We recall that, for any commutative $\mathbb{Q}$-algebra $A$ and any $z \in \mathcal{G}^0(A)$, we have $(d \beta)_z = 0$. Therefore, from explicit description of the tangent $A$-module $T_z \mathcal{G}^0(A)$ as an inductive limit of $A$-modules $A^{\mathbb{N}}$ and by the Taylor formula (which is enough to see for any monomial)
$$
\begin{equation*}
F (z + \varepsilon_c) = F(z) + \biggl(\frac{\partial F}{ \partial c} \biggr) (z) \varepsilon_c, \qquad \varepsilon_c^2 =0, \quad \varepsilon_c \text{ is placed in }c\text{th coordinate},
\end{equation*}
\notag
$$
we find that the series $\frac{\partial F}{ \partial c}$ obtained as the partial derivative of the series $F$ with respect to any variable $c$ (including variables from the algebra of polynomials and the algebra of Laurent polynomials) applied to the element $z$ equals zero.
However, if a series applied to any element from $\mathcal{G}^0(A)$ for any commutative $\mathbb{Q}$-algebra $A$ equals zero, then this series is itself zero. Indeed, it is enough to take algebras of the form
$$
\begin{equation*}
A= \mathbb{Q}[x_1, x_2, x_1^{-1}, x_2^{-1}] \otimes_{\mathbb{Q}} \mathbb{Q}[y_1, y_2, \dots ] \otimes_{\mathbb{Q}} (\mathbb{Q}[z_1, \dots, z_l]/ (z_1^{n_1}, \dots, z_l^{n_l}))
\end{equation*}
\notag
$$
and put $x_i$, $y_j$, $z_k$ and zero instead the variables in the series $F$ (including variables from the algebra of polynomials and the algebra of Laurent polynomials).
Thus, $\frac{\partial F}{\partial c} =0$ with respect to any variable $c$. Hence $F = \mathrm{const} \in \mathbb{Q} $. Since the morphism $\beta$ preserves the group structure, this constant equals $1$, that is, $F =1$, and $\beta$ is trivial. Proposition 5 is proved. Theorem 1. Any central extension of the group functor $\mathcal{G}$ by the group functor $\mathbb{G}_m$ is uniquely (up to isomorphism) defined by its restriction to the group subfunctor $\mathcal{G}^0$. The same is also true after restrictions of all these functors to the category of commutative $\mathbb{Q}$-algebras. Proof. By Lemma 1, we have $\mathcal{G} = \mathcal{G}^0 \rtimes \underline{\mathbb{Z}}$.
Similarly to Construction 1.7 in [5], a central extension $\widetilde{C}$ of a group functor $C = G \rtimes H$ by a group functor $E$ is equivalent to the following data: 1) a central extension of $H$ by $E$; 2) a central extension $\widetilde{G}$ of $G$ by $E$; 3) an action of $H$ on the group functor $\widetilde{G}$, lifting the action of $H$ on $G$, and trivial on $E$. In our case, any central extension $\widetilde{\underline{\mathbb{Z}}}$ of $\underline{\mathbb{Z}}$ is trivial, since there is a group functor splitting $\underline{\mathbb{Z}} \to \widetilde{\underline{\mathbb{Z}}}$. This splitting is given by any map
$$
\begin{equation}
\underline{\mathbb{Z}} (\mathbb{Z}) = \mathbb{Z} \ni 1 \mapsto a\in\widetilde{\underline{\mathbb{Z}}}(\mathbb{Z})
\end{equation}
\tag{35}
$$
that maps $1$ to a preimage $a$ of $1$ in $\widetilde{\underline{\mathbb{Z}}}(\mathbb{Z})$. Now, for any commutative ring $A$ and any function $\underline{n} \in \underline{\mathbb{Z}}(A)$, we have $A = A_1 \times \dots \times A_l$ such that the function $\underline{n}$ restricted to any $\operatorname{Spec} A_i$ is the constant function with value $n_i \in \mathbb{Z}$. Besides, we have $\widetilde{\underline{\mathbb{Z}}}(A) = \widetilde{\underline{\mathbb{Z}}}(A_1) \times \dots \times \widetilde{\underline{\mathbb{Z}}}(A_l)$. Now $n_i \in \underline{\mathbb{Z}}(A_i)$ is the image of $n_i \in \underline{\mathbb{Z}} (\mathbb{Z}) = \mathbb{Z}$. Therefore, the splitting
$$
\begin{equation*}
\underline{\mathbb{Z}}(A_i) \ni n_i \mapsto a^{n_i} \in \widetilde{\underline{\mathbb{Z}}}(A_i)
\end{equation*}
\notag
$$
follows from formula (35).
We claim uniqueness of the action of $1 \in \underline{\mathbb{Z}} (\mathbb{Z})$ on the group functor that gives a central extension of $\mathcal{G}^0$ by $\mathbb{G}_m$ such that this action lifts the action of $1 \in \underline{\mathbb{Z}} (\mathbb{Z})$ on $\mathcal{G}^0$ and is trivial on $\mathbb{G}_m$. Indeed, if there are at least two such actions, then they differ by the automorphism of the central extension (this automorphism induces identically action on $\mathbb{G}_m$ and $\mathcal{G}^0$). But such an automorphism can be identified with the morphism of the group functor $\mathcal{G}^0 \to \mathbb{G}_m$. By Proposition 5, this morphism is trivial. Now, for any commutative ring $A$, by a similar reasoning as above, the action of $1 \in \underline{\mathbb{Z}} (\mathbb{Z})$ induces that of $\underline{n} \in \underline{\mathbb{Z}}(A)$ on the group which defines the central extension after specifying the ring $A$. Therefore, the action of the group functor $\underline{\mathbb{Z}}$ is unique. The statement about the restricted to commutative $\mathbb{Q}$-algebras functors is proved by the same reasoning with the help of Proposition 5. Theorem 1 is proved. Theorem 2. The determinant central extension from § 5.3 Proof. By Theorem 1, it is enough to prove that the determinant central extension from § 5.3 restricted to $\mathcal{G}^0$ is isomorphic to the central extension from Proposition 4.
Let $A$ be a commutative ring. We construct a homomorphism from exact sequence of groups (29) to the central extension (34) restricted to $\mathcal{G}^0(A)$. Let $(g, r) \in \widehat{\mathcal{G}^0}(A)$. Then there is an $A$-submodule $L =t^m A[[t]] $ with $m \geqslant 0$ such that $g(L) \subset A[[t]]$ and $r |_L = d_g |_L$ (we recall the notation from (28)). Since $g(L) \subset A[[t]]$, we have $d_g |_L = g |_L$. Therefore, $r(L)= g(L)$. We have the following isomorphism:
$$
\begin{equation}
\frac{A[[t]]}{g(L)}= \frac{A[[t]]}{r(L)} \xleftarrow{r} \frac{A[[t]]}{L} \xleftarrow{g^{-1}} \frac{g(A[[t]])}{g(L)}.
\end{equation}
\tag{37}
$$
By [19], § 3.2, all the above $A$-modules are projective. Therefore, taking some $N =t^k A[[t]] \subset g(L)$, we obtain the isomorphism of $A$-modules $g(A[[t]])/ N \to A[[t]]/N$, which gives the isomorphism of the corresponding top exterior powers of these projective $A$-modules. This gives us an element $s$ from the $A$-module $\det(g (A[[t]]) \mid A[[t]])$, and this element generates this $A$-module. Thus we have mapped the element $(g, r) \in \widehat{\mathcal{G}^0}(A)$ to the element $(g,s) \in \widetilde{\mathcal{G}}(A)$. This map is a group homomorphism. Besides, this map restricted to the subgroup $\operatorname{GL}_f(A)$ is the determinant homomorphism $\det$. Theorem 2 is proved. Remark 2. From the proof of Theorem 1 it follows that, to extend the determinant central extension from $\mathcal{G}^0$ to $\mathcal{G}$, it is enough to lift the action of $1 \in \underline{\mathbb{Z}} (\mathbb{Z})$ on $\mathcal{G}^0$ to $\widetilde{{\mathcal{G}^0}}$ such that it will be trivial on $\mathbb{G}_m$. It is possible also to introduce this extension through the groups $\operatorname{GL}_f(A)$ and $\widehat{\mathcal{G}^0}(A)$ introduced in § 5.2 (this is similar to the smooth case from [23], § 6.6). Indeed, let $A$ be any commutative ring. The action of $1 \in \underline{\mathbb{Z}} (\mathbb{Z})$ on $A((t))$ is the multiplication by the element $t$, and the action on $\mathcal{G}^0 (A)$ is by means of the conjugation by $t$. Now we consider the following map on $\widehat{\mathcal{G}^0}(A)\colon (g,r) \mapsto (t g t^{-1}, r_t )$, where $r_t \colon A[[t]] \to A[[t]]$ equals to $\mathrm{id} \oplus t r t^{-1} $ for the decomposition of $A$-module $A[[t]]= A \oplus t A[[t]]$. This map is an injective endomorphism of the group $\widehat{G^0}(A)$, but this endomorphism is not surjective. But it is easy to see that the induced endomorphism of the group $ \widetilde{{\mathcal{G}^0}}(A)$ is the automorphism. This automorphism is the lift of the action of $1 \in \underline{\mathbb{Z}} (\mathbb{Z})$ to the group $\widetilde{{\mathcal{G}^0}}(A)$. Remark 3. We note that the determinant central extension (36) has a natural section $\varsigma \colon \mathcal{G} \to \widetilde{\mathcal{G}}$ as functors (not as group functors), since $\mathcal{G} = \mathcal{G}^0 \rtimes \underline{\mathbb{Z}}$, and the central extension has a canonical section $\sigma$ over $\mathcal{G}^0$ by Proposition 4 and a group section $\varrho$ over $\underline{\mathbb{Z}}$ as Now, for all $g \in \mathcal{G}^0(A)$, $\underline{n} \in \underline{\mathbb{Z}}(A)$, we put The corresponding $2$-cocycle on the group functor $\mathcal{G}$ with coefficients in $\mathbb{G}_m$ coincides with the $2$-cocycle $D$ after restriction to $\mathcal{G}^0$. We denote the $2$-cocycle on $\mathcal{G}$ by the same letter $D$. By Remark 3 we have a non-group section of the determinant central extension. Therefore, we have a decomposition of functors (not as group functors)
$$
\begin{equation*}
\widetilde{\mathcal{G}} \simeq \mathbb{G}_m \times \mathcal{G}.
\end{equation*}
\notag
$$
Since the functors $\mathbb{G}_m$ and $\mathcal{G}$ are represented by ind-affine ind-schemes, the functor $\widetilde{\mathcal{G}}$ is also represented by an ind-affine ind-scheme. § 6. Geometric action of $\mathcal{G}$ and of $\widetilde{\mathcal{G}}$We will construct a natural action of $\mathcal{G}$ on the moduli stack of some geometric data (including proper families of curves with some conditions and invertible sheaves on these families) and a natural action of $\widetilde{\mathcal{G}}$ on the determinant line bundle on this moduli stack. In this section, $A$ is any commutative ring (unless a concrete ring is specified). 6.1. Quintets and proper quintetsLet $C$ be a separated family of curves over $A$, that is, we have a separated morphism $C \to \operatorname{Spec} A$ whose fibres are one-dimensional schemes. We consider an $A$-point ${p \in C(A)}$ such that $C$ is smooth near $p$, that is, there is an open $V \supset p$ such that $V$ is smooth over $\operatorname{Spec} A$ (we will denote by the same letter an $A$-point and its image in $C$). Since the morphism $C \to \operatorname{Spec} A$ is separated, $p$ is a closed subscheme of $C$. Besides, by Theorem 17.12.1 in [14], $p$ is an effective Cartier divisor on $C$. Remark 4. There is also a statement in the opposite direction. According to Theorem 17.12.1 in [14], if an $A$-point $q \in C(A)$ defines the Cartier divisor on $C$ and the morphism $C \to \operatorname{Spec} A$ is locally of finite presentation near $q$, then the morphism $C \to \operatorname{Spec} A$ is smooth near $q$. Consider the $A$-algebra
$$
\begin{equation*}
\widehat{\mathcal{O}}_{C, p} = \varprojlim_{n > 0} H^0(C, \mathcal{O}_C / \mathcal{O}_C( - np)),
\end{equation*}
\notag
$$
which is the $A$-algebra of functions on the formal neighbourhood of $C$ at $p$. It is clear that $\widehat{\mathcal{O}}_{C, p}$ has a natural topology that makes $\widehat{\mathcal{O}}_{C, p}$ into a topological $A$-algebra with discrete topology on $A$. We also consider the topological $A$-algebra
$$
\begin{equation*}
\mathcal{K}_{C,p} = \varinjlim_{m > 0} \varprojlim_{n > 0} H^0(C, \mathcal{O}_C (mp) / \mathcal{O}_C(-np)),
\end{equation*}
\notag
$$
which is an $A$-algebra of functions on the punctured formal neighbourhood of $C$ at $p$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_C$-modules such that $\mathcal{F}$ is an invertible sheaf near $p$. Consider the $\widehat{\mathcal{O}}_{C,p}$-module
$$
\begin{equation*}
\widehat{\mathcal{F}}_{C, p} = \varprojlim_{n > 0} H^0(C, \mathcal{F} / \mathcal{F}( - np)),
\end{equation*}
\notag
$$
which is the module of sections of $\mathcal{F}$ over the formal neighbourhood of $C$ at $p$. Definition 1. By a quintet over $A$ we mean a collection $(C, p, \mathcal{F}, t, e)$, where $\bullet$ $C$ is a separated family of curves over $A$, $\bullet$ $p \in C(A)$ is an $A$-point of $C$ such that $C$ is smooth near $p$, $\bullet$ $\mathcal{F}$ a sheaf of $\mathcal{O}_C$-modules such that $\mathcal{F}$ is an invertible sheaf near $p$, $\bullet$ $t$ is a relative formal parameter at $p$, that is, $t$ is an element of ideal of $p$ of $\widehat{\mathcal{O}}_{C, p}$ that induces an isomorphism of topological $A$-algebras
$$
\begin{equation}
A[[t]] \xrightarrow{\sim} \widehat{\mathcal{O}}_{C, p},
\end{equation}
\tag{38}
$$
$\bullet$ $e$ is a formal trivialization of $\mathcal{F}$ at $p$, that is, $\widehat{\mathcal{F}}_{C, p}$ is a free $\widehat{\mathcal{O}}_{C, p}$-module of rank $1$, and $e$ is its basis. A quintet $(C, p, \mathcal{F}, t, e)$ is called a proper quintet if, in addition, the natural morphism $C \to \operatorname{Spec} A$ is a flat proper finitely presented morphism such that all geometric fibers are integral one-dimensional schemes, and $\mathcal{F}$ is an invertible sheaf of $\mathcal{O}_C$-modules. Remark 5. Since $p$ is a Cartier divisor on $C$, isomorphism (38) exists locally on $\operatorname{Spec} A$ (see Corollary 16.9.9 in [14], where there is also a statement in the opposite direction). In the definition of a quintet (and of a proper quintet) we demand that this isomorphism exists globally on $\operatorname{Spec} A$ and fix a relative formal parameter $t$. Analogously, a formal trivialization $e$ exists always locally on $\operatorname{Spec} A$, but we demand that it exists globally on $\operatorname{Spec} A$ and fix it. We note that, for a quintet $(C, p, \mathcal{F}, t, e)$, the relative formal parameter $t$ defines an isomorphism Clearly, when we vary $A$ we obtain the moduli stack (as an abstract stack in Zariski topology) of quintets $\mathcal{M}$ and, similarly, the moduli stack of proper quintets $\mathcal{M}_\mathrm{pr}$. Using descent data and morphisms of descent data we can consider $\mathcal{M}$ and $\mathcal{M}_\mathrm{ pr}$ as stacks over the category of all schemes (these are stacks associated with the prestacks that are given as stacks over affine schemes and empty over other schemes). We also consider the corresponding functors $\mathcal{Q}$ and $\mathcal{Q}_\mathrm{pr}$ of isomorphism classes of objects of these stacks as the covariant functors from the category of commutative rings to the category of sets:
$$
\begin{equation*}
\begin{aligned} \, \mathcal{Q} (A) &= \{ \text{quintets over }A \} / \text{ isomorphisms}, \\ \mathcal{Q}_\mathrm{pr}(A) &= \{ \text{proper quintets over }A \} / \text{ isomorphisms}. \end{aligned}
\end{equation*}
\notag
$$
There is a natural notion of an action of a group functor from the category of schemes (which will be in our case the sheaf of groups that is represented by an ind-scheme) on a category fibered in groupoids, see, for example, [24]. In particular, we can consider a stack as an example of such category. Clearly, this action induces the corresponding action of the group functor on the functor of isomorphism classes of objects of this stack. Theorem 3. 1. There is a natural action of the group ind-scheme $\mathcal{G}$ on the moduli stack $\mathcal{M}$ and on the moduli stack $\mathcal{M}_\mathrm{pr}$. This action applied to a quintet $(C, p, \mathcal{F}, t, e)$ does not change the geometric fibers of $C \to \operatorname{Spec} A$ and does not change the topological space of $C$. 2. Fixing a field $k$, the corresponding action of the Lie algebra $\operatorname{Lie} \mathcal{G}(k)$ is transitive after restriction to a proper quintet $(C, p, \mathcal{F}, t, e)$ with a smooth curve $C$ over $k$. In other words, $\operatorname{Lie} \mathcal{G}(k)$ is mapped surjectively to the tangent space of $\mathcal{Q}_\mathrm{pr}$ at $(C, p, \mathcal{F}, t, e)$ under the action on the quintet obtained from $(C, p, \mathcal{F}, t, e)$ as the base change from $k$ to $k[\varepsilon]/ (\varepsilon^2)$. (The tangent space consists of all elements from $\mathcal{Q}_\mathrm{pr}( k[\varepsilon]/(\varepsilon^2))$ giving $(C, p, \mathcal{F}, t, e)$ after reduction to $k$.) Remark 6. An action of $\mathcal{G}$ on $\mathcal{M}$, which will constructed in the proof below, depends on the choice of an affine open covering of a neighbourhood of $p$ in $C$ for every quintet $(C, p, \mathcal{F}, t, e)$ over $A$. The result of the action of $\mathcal{G}(A)$ on $(C, p, \mathcal{F}, t, e)$ depends on this covering up to a canonical isomorphism. Therefore, the corresponding action of $\mathcal{G}$ on $\mathcal Q$ does not depend on these choices. Proof of Theorem 3.
1. We prove assertion 1 of Theorem 3.
Given a quintet $(C, p, \mathcal{F}, t, e)$ over $A$ and an element $(h, \varphi ) \in \mathcal{G}(A)$, we will define a new quintet $(C'', p'', \mathcal{F}'', t'', e'')$ over $A$. First, we define a quintet $(C', p', \mathcal{F}', t', e')$ over $A$ from the quintet $(C, p, \mathcal{F}, t, e)$ and the element $\varphi \in {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$ (see [11], § 17.3, [22], § 4.1, but we provide detailed proofs below). Then we will define a quintet $(C'', p'', \mathcal{F}'', t'', e'')$ from the quintet $(C', p', \mathcal{F}', t', e')$ and the element $h \in L \mathbb{G}_m(A)$ (cf. [11], § 18.1.3). The procedure is as follows. We can suppose that there is an open affine neighbourhood $U$ of $p$ in $C$ such that $U$ is smooth over $\operatorname{Spec} A$, the ideal of effective Cartier divisor $p$ restricted to $\operatorname{Spec} \mathcal{O}(U)$ is generated by nonzerodivisor of $\mathcal{O}(U)$, and $\mathcal{F} |_U \simeq \mathcal{O}_U$. Indeed, we can do it by taking smaller affine open subset in $ \operatorname{Spec} A$, and then to obtain new quintets over this new base. Then we can glue the resulting quintets to obtain a new quintet over $\operatorname{Spec} A$. From the properties of $U$ it follows that the natural map $\mathcal{O}(U) \to {\mathcal{O}(U \setminus p)}$ is an embedding. Besides, since $U$ is smooth over $\operatorname{Spec} A$, we see that the homomorphism $A \to \mathcal{O}(U)$ is smooth, and hence this homomorphism is of finite presentation and flat, see, for example, [29], Tag 01V4, Tag 01TO. Construction of $(C', p', \mathcal{F}', t', e')$. We have the natural homomorphisms of rings
$$
\begin{equation*}
\mathcal{O}(U) \to \widehat{\mathcal{O}}_{C, p}, \qquad \gamma \colon \mathcal{O}(U \setminus p) \to \mathcal{K}_{C, p},
\end{equation*}
\notag
$$
which lead to the following Cartesian diagram: Using the fixed relative formal parameter $t$, we can replace the bottom row of this diagram by the embedding $A[[t]] \hookrightarrow A((t))$.
Now we define $C'$ by gluing the new affine curve $U'$ with $C \setminus p$ along $U \setminus p$, where the ring $\mathcal{O}(U')$ is defined as the fibered product (more on existence and properties of such affine curve $U'$ we will write a little bit later) Here, $\varphi \, \gamma$ is the composition of the map $\gamma$ and the map $\varphi \colon A((t)) \to A((t))$. The composition of the homomorphisms $\mathcal{O}(U') \to A[[t]] \to A$ defines an $A$-point $p' \in U'(A) \subset C'(A)$. We will prove a little bit later that $p'$ defines an effective Cartier divisor on $U'$ such that $U' \setminus p' \simeq U \setminus p$, and hence $C' \setminus p' \simeq C \setminus p$.From the properties of the construction, which we describe below, it will follow that the homomoprhism $\mathcal{O}(U') \to A[[t]]$ (written as the left vertical arrow in (40)) gives an isomorphism of the completion of $\mathcal{O}(U')$ with respect to the ideal of $p'$ with $A[[t]]$. Via this isomorphism we define $t'$ as the element that is mapped to $t$. We define $\mathcal{F}'$ and $e'$ such that $\mathcal{F}' |_{C' \setminus p'} = \mathcal{F} |_{C \setminus p}$ and using that $\mathcal{F} |_U \simeq \mathcal{O}_U$, and so that $\mathcal{F}'$ preserves this property, that is, $\mathcal{F}' |_{U' \setminus p'} \simeq \mathcal{O}_{U' \setminus p'}$ and we extend this sheaf trivially to $U'$ by means of this isomorphism. We put $e'=e$. First properties of the construction. Now we say more on the ring $\mathcal{O}(U')$. We note that if $\psi \in {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$ and $\psi (t A[[t]]) = t A[[t]]$, then the Cartesian diagram (40) can be changed to the following Cartesian diagram: and we can consider the ring $A((\psi(t))) \simeq A((t))$ with the changed relative formal parameter from $t$ to $\psi(t)$. Therefore, by this procedure, to investigate $\mathcal{O}(U')$ and $C'$ we can and will suppose further that $t \in \mathcal{O}(U)$ such that $\mathcal{O}(U)/t \mathcal{O}(U) \simeq A$ (see above our assumptions on $U$). From (38) we also haveBy (10), we have $\varphi \,{=}\, \varphi_+ \varphi_-$, where $\varphi_+ {\in}\, {\mathcal Aut}_+^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$ and $\varphi_- {\in}\, {\mathcal Aut}_-^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$. Since $\varphi_+ (A[[t]])= A[[t]]$, the ring $\mathcal{O}(U')$ depends only on the element $\varphi_-$, which is characterized by the property (12), which implies
$$
\begin{equation*}
\varphi_-(t) \in tA[t^{-1}] \subset A[t,t^{-1}] \subset \mathcal{O}(U \setminus p).
\end{equation*}
\notag
$$
Therefore, using (41), we can and will suppose further that $\varphi \in {\mathcal Aut}_-^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$, and therefore, $\varphi(t) \in \mathcal{O}(U \setminus p)$. It also implies that $\varphi^{-1} \in {\mathcal Aut}_-^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})(A)$, and therefore, by (12) we have
$$
\begin{equation*}
\varphi^{-1}(t) \in tA[t^{-1}] \subset A[t,t^{-1}] \subset \mathcal{O}(U \setminus p).
\end{equation*}
\notag
$$
The definition of $\mathcal{O}(U')$ from the diagram (40) can be written as the following:
$$
\begin{equation}
\mathcal{O}(U') = \{ w \mid w \in \mathcal{O}(U \setminus p) \text{ such that } \varphi \gamma (w) \in A[[t]]\}.
\end{equation}
\tag{43}
$$
Since $\varphi$ is a continuous automorphism of $A((t))$, there is an integer $l > 0$ such that $\varphi (t^l A[[t]]) \subset A[[t]]$. We have $\gamma(t^l \mathcal{O}(U)) \subset t^l A[[t]]$. Therefore, by (43),
$$
\begin{equation}
\varphi \gamma(t^l \mathcal{O}(U)) \subset A[[t]] \quad \text{and} \quad t^l \mathcal{O}(U) \subset \mathcal{O}(U').
\end{equation}
\tag{44}
$$
From our choice of $t$, we have a canonical isomorphism of $A$-modules
$$
\begin{equation*}
A[t, t^{-1}]/ t^l A[t] \simeq \mathcal{O}(U \setminus p) / t^l \mathcal{O}(U).
\end{equation*}
\notag
$$
Hence and from (42) we see that the $A$-module $\mathcal{O}(U \setminus p)$ is generated by the following two $A$-submodules:
$$
\begin{equation}
\mathcal{O}(U \setminus p) = A[t, t^{-1}] + t^l \mathcal{O}(U).
\end{equation}
\tag{45}
$$
We set
$$
\begin{equation}
b = \varphi^{-1}(t) \in tA[t^{-1}] \subset A[t,t^{-1}] \subset \mathcal{O}(U \setminus p).
\end{equation}
\tag{46}
$$
From (43) we have $b \in \mathcal{O}(U')$.
Since $\varphi$ is an automorphism of $A[t,t^{-1}]$, from (43), (45), and (44) it is easy to see that the $A$-submodule $\mathcal{O}(U')$ of $\mathcal{O}(U \setminus p)$ is generated by the following two $A$-submodules: Hence we immediately find that the subring $\mathcal{O}(U') $ of ring $\mathcal{O}(U \setminus p)$ is generated by the subrings $A[b]$ and $A + t^l \mathcal{O}(U)$. Flattness property and base change. We claim that the $A$-module $\mathcal{O}(U')$ given by (47) is decomposed into the following direct sum of $A$-submodules:
$$
\begin{equation}
\mathcal{O}(U')= \bigoplus_{i =0}^{l-1} A b^i \oplus t^l \mathcal{O}(U).
\end{equation}
\tag{48}
$$
Indeed, we have $\varphi\bigl(\bigoplus_{i =0}^{l-1} A b^i\bigr) = \bigoplus_{i =0}^{l-1} A t^i $. Besides, $\bigl(\bigoplus_{i =0}^{l-1} A b^i\bigr) \cap t^l \mathcal{O}(U) = 0$, since $b = \varphi^{-1}(t) \in tA[t^{-1}] $ and therefore, $\bigl(\bigoplus_{i =0}^{l-1} A b^i\bigr) \subset t^{l-1} A[t^{-1}]$, and we use that $t^{l-1} A[t^{-1}] \cap t^l \mathcal{O}(U) =0$. Now (47) implies that (48) will follow now if we will prove that any element $b^m$, where $m \geqslant l$, belongs to the $A$-module in the right-hand side of (48). We will prove it by induction on $m$, where the base of induction is the evident case $m=l-1$. Suppose we proved it for some $m \geqslant l-1$ and all natural numbers less than $m$. We have $t^{m+1} \in t^l \mathcal{O}(U) \subset \mathcal{O}(U')$. Therefore, using (43) and since $\varphi(t) $ is of type (12), the element $s=\varphi(t^{m+1}) $ is a polynomial from $A[t]$ whose degree is $m+1$ and the coefficient at $t^{m+1}$ is $1$. We apply $\varphi^{-1}$ to the polynomial $s$, and using that $b= \varphi^{-1}(t)$, we find that $t^{m+1} = \varphi^{-1}(\varphi(t^{m+1})) = \varphi^{-1}(s)$ can be written as the element from $A[b]$, where this polynomial in variable $b$ has degree $m+1$ and the coefficient at $b^{m+1}$ is $1$. Using that $t^{m+1} \in t^l \mathcal{O}(U)$ and the induction hypothesis, we verify that $b^{m+1}$ belongs to the $A$-module in the right-hand side of (48).
Since $\mathcal{O}(U)$ is a flat $A$-module, we obtain from (48) that $\mathcal{O}(U')$ is also a flat $A$-module. From the description of the construction $\mathcal{O}(U') \subset \mathcal{O}(U \setminus p)$ by formula (48) it follows that this construction commutes with base change from $A$ to another commutative ring. Cartier divisor $p'$, completion at $p'$, and complement to $p'$. The $A$-algebra $\varphi(\mathcal{O}(U'))$ is an $A$-subalgebra of $A[[t]]$. The point $p' \in U'(A)$ is defined through the following homomorphism:
$$
\begin{equation*}
\mathcal{O}(U') \xrightarrow{\varphi} \varphi(\mathcal{O}(U')) \xrightarrow{\sigma} A,
\end{equation*}
\notag
$$
where $\sigma$ is induced by the natural homomorphism $A[[t]] \to A$. We claim that the ideal $\operatorname{Ker} \sigma $ that is the kernel of $\sigma$ is generated by the element $t = \varphi(b) \in \varphi(\mathcal{O}(U')) $. Indeed, suppose that $v \in \mathcal{O}(U')$ such that $\varphi(v) \in tA[[t]]$. We have $b^{-1} = (\varphi^{-1}(t))^{-1} \in t^{-1}A[t^{-1}]$, since $b \in tA[t^{-1}] $ is of type (12). Therefore, $b^{-1} \in \mathcal{O}(U \setminus p)$. Hence $\varphi(v) = (\varphi(v) t^{-1}) t$, where $\varphi(v)t^{-1} = \varphi(vb^{-1}) \in \varphi(\mathcal{O}(U'))$, because $vb^{-1} \in \mathcal{O}(U \setminus p)$ and $\varphi(vb^{-1}) \in A[[t]]$. Thus we proved that $ \operatorname{Ker} \sigma = t \varphi(\mathcal{O}(U')) $, and hence $p'$ is an effective Cartier divisor on $U'$.
Besides, from these reasonings it follows that the completion of $\varphi(\mathcal{O}(U'))$ with respect to the ideal $t \varphi(\mathcal{O}(U'))$ is $A[[t]]$. We have $\mathcal{O}(U') \subset \mathcal{O}(U \setminus p)$. We claim that $\mathcal{O}(U' \setminus p') = \mathcal{O}(U \setminus p)$. Indeed, we have proved that $\mathcal{O}(U' \setminus p') = \mathcal{O}(U')[b^{-1}]$. Since, for any integer $m$, we have $b^m \in t^mA[t^{-1}]$, we obtain $\mathcal{O}(U' \setminus p') \subset \mathcal{O}(U \setminus p)$. To obtain the reverse inclusion, we note that $t^{-1} \in A[b, b^{-1}]$, since $\varphi$ is the automorphism of $A[t, t^{-1}]$ and hence $A[t,t^{-1}] = A[b, b^{-1}]$. Finite presentability and smoothness property. Now we prove that the ring homomorphism $A \to \mathcal{O}(U')$ is of finite presentation. We know that $A \to \mathcal{O}(U)$ is of finite presentation. Hence there is a surjective $A$-algebra homomorphism $\kappa \colon A[y_1, \dots, y_r] \to \mathcal{O}(U) $ such that $\kappa(y_i) \in t\mathcal{O}(U)$ for any $1 \leqslant i \leqslant r$. Let $\omega_1, \dots, \omega_m$ be all monomials of elements $y_1, \dots, y_r$ of total degrees from $l$ to $2l-1$. Since $A[t]/ t^l A[t] \simeq \mathcal{O}(U)/ t^l \mathcal{O}(U)$, we obtain a surjective $A$-algebra homomorphism
$$
\begin{equation*}
\theta' \colon A[x_0, \dots, x_m] \to \mathcal{O}(U),
\end{equation*}
\notag
$$
where $\theta'(x_0) =t $, $\theta'(x_i) = \kappa(\omega_i)$ for $1 \leqslant i \leqslant m$. We note that $\theta'(T) =t^l \mathcal{O}(U)$, where $T$ is the $A$-submodule generated by all monomials of elements $x_1, \dots, x_m$ of total positive degrees.
Let $ \theta \colon A[x_0, x_0^{-1}, x_1, \dots, x_m] \to \mathcal{O}(U \setminus p)$ be the natural surjective $A$-algebra homomorphism that extends $\theta'$. Since $A \to \mathcal{O}(U)$ is of finite presentation, the ideal $\operatorname{Ker} \theta'$ is finitely generated by polynomials $f_1, \dots, f_s$, see [29], Tag 00F2. Let $a_j$, where $1 \leqslant j \leqslant n$, be elements of $A$ which are all the coefficients of the polynomials $f_i$, where $1 \leqslant i \leqslant s$, and all the coefficients of the Laurent polynomial $b$, see (46). Let $B \subset A$ be a subring generated over $\mathbb{Z}$ by elements $a_j$, where $1 \leqslant j \leqslant n$. Then $B$ is a Noetherian subring. Since elements $f_1, \dots, f_s$ are from $B[x_0, \dots, x_m]$, it is easy to see that $\theta(B[x_0, \dots, x_m]) \otimes_B A \simeq \mathcal{O}(U) $. Since $\theta(B[x_0, \dots, x_m]) / \theta(T) \simeq \bigoplus_{i=0}^{l-1} Bt^i$ and $\mathrm{Tor}^B_1 \bigl(\bigoplus_{i=0}^{l-1} Bt^i, A\bigr) =0$, we have that $\theta(T) {\otimes_B} A \hookrightarrow \theta(B[x_0, \dots, x_m]) {\otimes_B} A$. Hence $\theta(T) {\otimes_B} A \simeq t^l\mathcal{O}(U) $. We consider the subring $B[{b_0}, x_1, \dots, x_m] \subset B [x_0, x_0^{-1}, x_1, \dots, x_m]$, where the element $b_0$ is from $B[x_0, x_0^{-1}] $ such that $\theta({b_0}) =b$. From (48) and arguing as above we obtain
$$
\begin{equation}
\theta (B[{b_0}, x_1, \dots, x_m]) \otimes_B A \simeq \mathcal{O}(U').
\end{equation}
\tag{49}
$$
The polynomial ring $B[z_0, \dots, z_m]$ is a Noetherian ring. Therefore, the kernel of the homomorphism which is the composition
$$
\begin{equation*}
B[z_0, \dots, z_m] \xrightarrow{\tau} B[b_0, x_1, \dots, x_m] \xrightarrow{\theta} \mathcal{O}(U \setminus p),
\end{equation*}
\notag
$$
where $\tau(z_0) = b_0 $, $\tau(z_i) = x_i$ for $1 \leqslant i \leqslant m$, is a finitely generated ideal. Hence and from (49) we see that the ring homomorphism $A \to \mathcal{O}(U')$ is of finite presentation.
Since $A \to \mathcal{O}(U')$ is of finite presentation and $p'$ is an effective Cartier divisor on $U'$, the morphism $U' \to \operatorname{Spec} A$ is smooth near $p'$, see Remark 4. Since $\mathcal{O}(U)$ is a smooth $A$-algebra and $\mathcal{O}(U' \setminus p') = \mathcal{O}(U \setminus p)$, the $A$-algebra $\mathcal{O}(U')$ is also smooth (see, for example, [29], Tag 01V4). Independence on the choice of $U$. Now we prove that the construction of $C'$ does not depend on the choice of $U$ (up to a canonical isomorphism). By means of subdividing of $\operatorname{Spec} A$ and taking a smaller open affine subset of $\operatorname{Spec} A$ if necessary, we can suppose that we change the open set $U$ to the affine open subset
$$
\begin{equation*}
D_{U, f} = \{ x \in U \mid f(x) \ne 0 \} = \operatorname{Spec} \mathcal{O}(U)_f
\end{equation*}
\notag
$$
of $U$, where $f \in \mathcal{O}(U)$ such that $f \mod t \mathcal{O}(U) \in A^*$. By (7), we have the decomposition $\varphi(\gamma (f) ) = r_- r_+$, where $r_- \in \mathbb{V}_-(A) $ and $r_+ \in \mathbb{G}_m(A) \times \mathbb{V}_+(A)$. As above, we find that the elements $r_-$, $r_-^{-1}$, and $\varphi^{-1} (r_-^{-1})$ belong to the ring $A[t, t^{-1}] \subset \mathcal{O}(U \setminus p)$. Let $g = \varphi^{-1} (r_-^{-1}) f$ be an element from $\mathcal{O}(U \setminus p)$. We have $D_{ U, f} \setminus p = D_{U \setminus p, g} $ and $\varphi(\gamma(g)) =r_+$. Besides, $r_+ A[[t]]= A[[t]]$. Therefore, using (40) or (43) written also for $D_{U, f}$, by changing of $U$ to $D_{U, f}$ we obtain the changing of $U'$ to $D_{U', g}$, which is an affine open subset. (It is important that from the explicit kind of $D_{U', g}$ we know that it will be an open subset.) Therefore, the construction of $C'$ does not depend on the choice of open $U$ (up to a canonical isomorphism).
Further properties. We have already checked that the construction of $\mathcal{O}(U')$ commutes with base change from $A$ to another commutative ring. Hence it is easy to see that the construction of $(C', p', \mathcal{F}', t', e')$ commutes with base change. From (48), (46), (10) (all the coefficients of $b$ except for the coefficient at $t$ are nilpotent elements from $A$) it follows that $C' \times_A A_\mathrm{red} = C \times_A A_\mathrm{red}$, where $A_\mathrm{red} = A/ \mathrm{Nil} A$. Hence this construction does not change the reduced structures of the schemes, that is, $C'_\mathrm{red} = C_\mathrm{red}$. Hence, for any scheme $Z$ over $A$,
$$
\begin{equation*}
(C \times_{\operatorname{Spec} A} Z)_\mathrm{red} = (C' \times_{\operatorname{Spec} A} Z)_\mathrm{ red}.
\end{equation*}
\notag
$$
It implies that the topological spaces of schemes $C \times_{\operatorname{Spec} A} Z$ and $C' \times_{\operatorname{Spec} A} Z$ coincide. In particular, $C$ and $C'$ has the same topological spaces. From these reasonings it also follows that the geometric fibres of $C \to \operatorname{Spec} A$ do not change under this construction.
The morphism $C \to \operatorname{Spec} A$ is separated. This is equivalent to the property that the diagonal is a closed subset in the topological space of $C \times_{\operatorname{Spec} A} C$, see, for example, [29], Tag 01KH. By the above reasoning, we have the same condition for the diagonal subset in $C' \times_{\operatorname{Spec} A} C'$. Therefore, the morphism $C' \to \operatorname{Spec} A$ is also separated. Suppose that the morphism $C \to \operatorname{Spec} A$ is proper. Then this morphism is universally closed. From the properties of the construction described above it follows that, to prove that $C' \to \operatorname{Spec} A$ is proper, it is enough to prove that this morphism is universally closed. But this follows, because the topological spaces of $C \times_{\operatorname{Spec} A} Z$ and $C' \times_{\operatorname{Spec} A} Z$ coincide. Construction and properties of $(C'', p'', \mathcal{F}'', t'', e'')$. Note that $\mathcal{F}' |_{U'} \simeq \mathcal{O}_{U'}$. We have the natural homomorphisms of $\mathcal{O}(U')$-modules
$$
\begin{equation*}
\mathcal{F}'(U') \to \widehat{\mathcal{F}}_{C', p'} \xrightarrow{\sim} \widehat{\mathcal{O}}_{C',p'}, \qquad \beta \colon \mathcal{F}'(U' \setminus p') \to {{\mathcal K }_{C', p'}}
\end{equation*}
\notag
$$
that are constructed by means of the formal trivialization $e'$. These homomorphisms lead to the following Cartesian diagram of $\mathcal{O}(U')$-modules: Now we define $C''=C'$, $p''=p'$, $t''=t'$, and the sheaf $\mathcal{F}''$ by gluing the sheaf $\mathcal{F}''|_{U'} \simeq \mathcal{O}_{U'}$ with the sheaf $\mathcal{F}'|_{C' \setminus p'}$ over $U' \setminus p'$, where the gluing data is uniquely defined by the fibered product of $\mathcal{O}(U')$-modules Here, $h\beta$ is the composition of the map $\beta$ and the map $\mathcal{K}_{C', p'} \to \mathcal{K}_{C', p'}$ given by the multiplication on $h \in L \mathbb{G}_m(A) \simeq A((t'))^*$. We define $e''$ as the preimage of $1$ under the isomorphism which is the completion of the map $\mathcal{F}''(U') \to \widehat{\mathcal{O}}_{C',p'}$ (that was constructed by $e'$).
To check that this construction is well-defined we choose a relative formal parameter $t'$ at $p'$ from $\mathcal{O}(U')$ (as we did it in the previous arguments) and use the decompositions (7) and (8) for $h$. According to these decompositions, we have $h = h_+ h'$, where $h_+ \in \mathbb{G}_m \times \mathbb{V}_+(A)$, and $h' \in \mathbb{V}_-(A) \times \underline{\mathbb{Z}}(A)$. We have $h' \in A[t',t'^{-1}] \subset \mathcal{O}(U' \setminus p')$. Besides, by (51)
$$
\begin{equation*}
\mathcal{F}''(U') = \{ z \mid z \in \mathcal{F}'(U' \setminus p') \text{ such that } h' \beta (z) \in A[[t']]\}.
\end{equation*}
\notag
$$
Hence $\mathcal{F}''(U') = h'^{-1} \mathcal{F}'(U') \subset \mathcal{F}'(U' \setminus p')$, since if $h' \beta (z) \in A[[t']]$ for $z \in \mathcal{F}'(U' \setminus p')$, then $h' z \in \mathcal{F}'(U' \setminus p')$ and $\beta(h' z) \in A[[t']]$, and from (50) we have $h' z \in \mathcal{F}'(U')$.
Now to check the independence of the construction on the choice of $U'$ it is enough to check it when we change $U'$ to the affine open subset $D_{U', f}$, where $f \in \mathcal{O}(U')$ such that $f \ \operatorname{mod} t' \mathcal{O}(U') \in A^*$ (after subdividing of $\operatorname{Spec} A$ and taking a smaller open affine subset of $\operatorname{Spec} A$ if necessary). In this case, $\mathcal{F}'(U')$ is changed to $\mathcal{F}'(D_{U', f}) = \mathcal{F}'(U')_f $, and $\mathcal{F}''(U')$ is changed to $\mathcal{F}''(D_{U', f}) = \mathcal{F}''(U')_f $. The construction maps $\mathcal{F}'(U')_f $ to $\mathcal{F}''(U')_f $. So, the construction maps quintets to quintets and proper quintets to proper quintets. It is also easy to see that we have an action of the group functor $\mathcal{G}$ (which is represented by the group ind-scheme) on the moduli stack $\mathcal{M}$ and the moduli stack $\mathcal{M}_\mathrm{pr}$. 2. Let us prove assertion 2 of Theorem 3. Let $(C, p, \mathcal{F}, t, e)$ be a proper quintet with smooth curve $C$ over the field $k$. Let $(\widetilde{C}, \widetilde{p}, \widetilde{\mathcal{F}}, \widetilde{t}, \widetilde{e})$ be a proper quintet over $A$, where $A = k[\varepsilon]/ (\varepsilon^2)$, such that its reduction to $k$ be the quintet $(C, p, \mathcal{F}, t, e)$. The affine scheme $(C \setminus p) \times_k (k[\varepsilon]/(\varepsilon^2))$ is smooth and, consequently, formally smooth over $k[\varepsilon]/(\varepsilon^2)$, and the scheme $\widetilde{C} \setminus \widetilde{p}$ is also affine. Therefore, by the definition of formally smooth morphism, there is a morphism $\kappa$ such that the following diagram is commutative From the flatness property of $\widetilde{C}$ it is easy to see that $\kappa$ is an isomorphism. Using $\kappa$ and two above quintets, we have two embeddings of the ring $\mathcal{O}(\widetilde{C} \setminus \widetilde{p}\,)$ into the ring $(k[\varepsilon]/ (\varepsilon^2)) ((t))$, which differ by an element from ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}) (A)$ (the constructed isomorphism is continuous, since it depends only on finite number of elements from the ring $\mathcal{O}(\widetilde{C} \setminus \widetilde{p}\,)$), see [11], § 17.3.4. Analogously, we obtain an element from $(\mathcal{K}_{C, p} \otimes_k (k[\varepsilon]/ (\varepsilon^2)))^*$ when compare two invertible sheaves from quintets over $k[\varepsilon]/ (\varepsilon^2)$ which have the same curves, points, and relative formal parameters and coincide after reduction to $k$. The constructed elements are from the corresponding Lie algebras. This gives us an element from $\operatorname{Lie} \mathcal{G}(k)$, which maps the quintet obtained as the base change from $k$ to $k[\varepsilon]/(\varepsilon^2)$ of the quintet $(C, p, \mathcal{F}, t, e)$ to the quintet $(\widetilde{C}, \widetilde{p}, \widetilde{\mathcal{F}}, \widetilde{t}, \widetilde{e})$. Theorem 3 is proved. Remark 7. Note that $\mathbb{G}_m \subset L\mathbb{G}_m$ (see (4)). It is easy to see that via this embedding we have $\mathbb{G}_m $ is a normal group subfunctor of $\mathcal{G}$, and, for any commutative ring $A$, any element from $\mathbb{G}_m(A)$ maps any quintet over $A$ to an isomorphic quintet over $A$. Remark 8. An analog of assertion 2 of Theorem 3 in complex analytic category was given in Proposition 3.19 of [2]. Remark 9. It was proved in [2], § 3, that in complex analytic category, some analog of the functor $\mathcal{Q}_\mathrm{pr}$ (when all fibers of $C$ from a quintet are smooth projective curves of genus $g$) is represented by an infinite-dimensional complex manifold. 6.2. Determinant line bundle on the moduli stack of proper quintets $\mathcal{M}_\mathrm{pr}$There is a natural notion of a quasicoherent sheaf on a category fibered in groupoids over the category of schemes (see, for example, Definition 2.1 from Chapter XIII, § 2 of [1]). In particular, the moduli stack of proper quintets $\mathcal{M}_\mathrm{pr}$ is a category fibered in groupoids. To define a linear bundle on $\mathcal{M}_\mathrm{pr}$ it is enough to define a linear bundle on $\operatorname{Spec} A$ for every proper quintet over any commutative ring $A$ such that the corresponding compatibilty conditions between these linear bundles are satisfied. Proposition 6. Let $C \to \operatorname{Spec} A$ be a family of curves and $p \in C(A)$ such that $C$, $p$ satisfy the corresponding conditions from the definition of proper quintet (see Definition 1). Let $\mathcal{F}$ be a locally free sheaf of $\mathcal{O}_C$-modules. Then there is an integer $n >0$ such that the following properties are satisfied. 1. For any $m \geqslant n $, the complex
$$
\begin{equation}
H^0(C, \mathcal{F}(mp)) \to H^0(C, \mathcal{F}(mp)/\mathcal{F})
\end{equation}
\tag{52}
$$
is a complex of finitely generated projective $A$-modules whose cohomology groups coincide with the cohomology groups $H^*(C, \mathcal{F})$. 2. For any $m \geqslant n $, for any Cartier divisor $\widetilde{p} $ on $C$ such that $\mathcal{O}(mp) \subset \mathcal{O}(\widetilde{p}\,) $, we have an isomorphism
$$
\begin{equation*}
H^0(C, \mathcal{F}(\widetilde{p}\,)) / H^0(C, \mathcal{F}(mp)) \xrightarrow{\sim} H^0(C, \mathcal{F}(\widetilde{p}\,)/\mathcal{F}(mp)).
\end{equation*}
\notag
$$
3. Suppose that $(C, p, \mathcal{F})$ is a part of a proper quintet $(C, p, \mathcal{F}, t, e)$. Then, for any $m \geqslant n $, complex (52) is quasi-isomorphic to the complex where the map in this complex is induced by the formal trivialization $e$, isomorphism (39), and the fact that Besides, for any $k_1 \leqslant k_2$, we have canonical isomorphisms of $A$-modules Proof. 1. We note that the restriction of $p$ to any fiber of $ C \to \operatorname{Spec} A$ is an ample divisor on a curve. Therefore, by [13], Corollaire 9.6.4, $\mathcal{O}(p)$ is an ample sheaf on $C$. Therefore, by [30], III, Corollaire 2.3, [29], Tag 0695 (see also Theorem A6 from [3] and the explanations before this theorem), there is an integer $n > 0$ such that $H^q(C, \mathcal{F}(mp)) =0 $ for any $q > 0$ and any $m \geqslant n $.
By [16], Theorem 2.9, and [29], Tag 0B91 (see also Theorem A7 from [3]) it follows that $H^0(C, \mathcal{F}(mp))$ is a pseudo-coherent, and even perfect, $A$-module. Hence $H^0(C, \mathcal{F}(mp))$ is a finitely presented $A$-module. The scheme $C$ is quasicompact and separated. Therefore, we can write the Čech complex for the sheaf $\mathcal{F}(mp)$ and a finite affine covering of $C$, where the all intersections will be also affine open subsets. This complex is of finite length and all the terms of this complex are flat $A$-modules. Since $H^q(C, \mathcal{F}(mp)) =0 $ for any $q > 0$, this complex is acyclic in positive degrees, see [29], Tag 01XD. Therefore, $H^0(C, \mathcal{F}(mp))$ is a flat $A$-module. Since this module is also finitely presented, it is a finitely generated projective $A$-module. The $A$-module $H^0(C, \mathcal{F}(mp)/\mathcal{F})= H^0(p, \mathcal{F}(mp)/\mathcal{F})$ is a finitely-generated projective $A$-module, since it is an extension of $A$-modules $H^0(p, \mathcal{F}((k+1)p)/ \mathcal{F}(kp))$ of the same type. Now the statement on the cohomology groups $H^*(C, \mathcal{F})$ follows from the long exact cohomological sequence associated with the exact sequence of sheaves:
$$
\begin{equation*}
0 \to \mathcal{F} \to \mathcal{F}(mp) \to \mathcal{F}(mp)/\mathcal{F} \to 0.
\end{equation*}
\notag
$$
2. Assertion 2 follows from $H^1(C, \mathcal{F}(mp)) =0$ and the long exact cohomological sequence associated with exact sequence of sheaves
$$
\begin{equation*}
0 \to \mathcal{F}(mp) \to \mathcal{F}(\widetilde{p}\,) \to \mathcal{F}(\widetilde{p}\,)/\mathcal{F}(mp) \to 0.
\end{equation*}
\notag
$$
3. Complex (53) is an inductive limit of complexes (52) over all integers $m \geqslant n$. Isomorphisms (54) follow from the existence of $t$ and $e$ in the definition of proper quintet. Proposition 6 is proved. Remark 10. Complex (52) is the complex (4.15) from Chapter XIII, § 4 of [1]. We added the proofs and the corresponding references when the ring $A$ is non-Noetherian. Remark 11. Complex (53) is quasi-isomorphic to the complex Remark 12. From the proof of Proposition 6 it follows that the statement in assertion 3 about complex (53) will remain true if we change the $A$-module $A((t))/ A[[t]]$ to the $A$-module $\varinjlim_{m > 0} H^0(C, \mathcal{F} (mp)/ \mathcal{F})$ and take $C$ and $p$ as in the proposition, and $\mathcal{F}$ any locally free sheaf of $\mathcal{O}_C$-modules. Besides, we do not demand an existence of $t$ and $e$. By assertion 1 of Proposition 6 for any proper quintet $(C, p, \mathcal{F}, t, e)$ over $A$, for any integer $m $ (as in the proposition) we have the projective $A$-module of rank $1$
$$
\begin{equation}
\operatorname{Hom}_A \biggl( \, \bigwedge^{\mathrm{max}} H^0(C, \mathcal{F}(mp)/\mathcal{F}),\, \bigwedge^{\mathrm{max}} H^0(C, \mathcal{F}(mp)) \biggr).
\end{equation}
\tag{55}
$$
By assertion 2 of Proposition 6б this $A$-module does not depend on the choice of $m$ up to a canonical isomorphism (or we can choose the minimal $m$ that we need in Proposition 6), and this $A$-module is compatible with base change such that the obvious diagram of compatibility is satisfied. This $A$-module is the determinant of the cohomology of $\mathcal{F}$, which was denoted by $\det R \pi_* \mathcal{F}$ in § 1.1, where $\pi \colon C \to \operatorname{Spec} A$ is the morphism from the definition of the quintet. Thus, by a proper quintet $(C, p, \mathcal{F}, t, e)$ over $A$, we have constructed a projective $A$-module of rank $1$, which is the determinant of the cohomology of $\mathcal{F}$. This defines a linear bundle on $\mathcal{M}_\mathrm{pr}$ which we call the determinant line bundle on $\mathcal{M}_\mathrm{pr}$. A linear bundle on a stack (or on a category fibered in groupoids over the category of schemes) can be considered also as a category fibered in groupoids over the category of schemes with the natural functor between two categories fibered in groupoids. Therefore, when a group functor from the category of schemes (which will be in our case the sheaf of groups that is represented by an ind-scheme) acts on a stack, we can speak about an action of this functor on a line bundle on the stack which lifts the action on the stack (see, for example, [24], § 1). Theorem 4. There is a natural action of the group ind-scheme $\widetilde{\mathcal{G}}$ on the determinant line bundle on the moduli stack $\mathcal{M}_\mathrm{pr}$ that lifts an action of the group ind-scheme $\mathcal{G}$ on $\mathcal{M}_\mathrm{pr}$ constructed in Theorem 3. Proof. Let $(C,p,\mathcal{F}, t, e)$ be a quintet over $A$ with $\pi \colon C \to \operatorname{Spec} A$. The determinant line bundle on $\operatorname{Spec} A$ that corresponds to this quintet is determined by a projective $A$-module $\det R \pi_* \mathcal{F}$ (see formula (55)). Let $\upsilon \colon \mathcal{F}(C \setminus p) \to A((t))$ be the natural map (see assertion 3 of Proposition 6).
Let an element $g$ be from $\mathcal{G}(A)$. We denote the new quintet over $A$ after the action by $g$ as $(g(C), g(p), g (\mathcal{F}), g(t), g(e)) $ with $g(\pi) \colon g(C) \to \operatorname{Spec} A$. The determinant line bundle on $\operatorname{Spec} A$ that corresponds to the new quintet is determined by a projective $A$-module $\det R g(\pi)_* g(\mathcal{F})$. Since $g(A((t))) = A((t))$, after the action by $g$, complex (53) for the quintet $(C, p, \mathcal{F}, g, e)$ is isomorphic to the following complex:
$$
\begin{equation}
\mathcal{F}(C \setminus p) \xrightarrow{\Theta_1} A((t))/ g(A[[t]]),
\end{equation}
\tag{56}
$$
where the map $\Theta_1$ is the composition of the map $g \upsilon$ with the quotient map from $A((t))$ to $A((t))/ g (A[[t]])$ (here and in what follows, $g \upsilon$ is the composition of $\upsilon$ and the action by $g$ on $A((t))$).
From the construction of the quintet $(g(C), g(p), g (\mathcal{F}), g(t), g(e))$, since
$$
\begin{equation*}
g(\mathcal{F})(g(C) \setminus g(p)) \simeq \mathcal{F}(C \setminus p),
\end{equation*}
\notag
$$
complex (53) written for the quintet $(g(C), g(p), g (\mathcal{F}), g(t), g(e))$, is isomorphic to the complex
$$
\begin{equation}
\mathcal{F}(C \setminus p) \xrightarrow{\Theta_2} A((t))/ A[[t]],
\end{equation}
\tag{57}
$$
where the map $\Theta_2$ is the composition of the map $g \upsilon$ with the quotient map from $A((t))$ to $A((t))/ A[[t]]$.
From Proposition 3.2 in [19] and Proposition 6 it is easy to see that there is an integer $r > 0$ such that $g(t^{-r}A[[t]]) \supset A[[t]]$ and complex (56) contains a quasi-isomorphic subcomplex of finitely generated projective $A$-modules
$$
\begin{equation}
H^0(C, \mathcal{F}(rp)) \xrightarrow{\Theta_1} g(t^{-r}A[[t]])/ g(A[[t]])
\end{equation}
\tag{58}
$$
that is isomorphic to a complex (52), and complex (57) contains a quasi-isomorphic subcomplex of finitely generated projective $A$-modules
$$
\begin{equation}
H^0(C, \mathcal{F}(rp)) \xrightarrow{\Theta_2} g(t^{-r}A[[t]])/ A[[t]]
\end{equation}
\tag{59}
$$
that contains a quasi-isomorphic subcomplex (52) written for the quintet $(g(C), g(p), g (\mathcal{F}), g(t), g(e))$.
From complexes (58), (59), we have the canonical isomorphisms
$$
\begin{equation*}
\begin{aligned} \, \det R \pi_* \mathcal{F} &\simeq \operatorname{Hom}_A \biggl( \,\bigwedge^{\mathrm{max}} (g(t^{-r}A[[t]])/ g( A[[t]])),\, \bigwedge^{\mathrm{max}} H^0(C, \mathcal{F}(rp)) \biggr) \\ &=\operatorname{Hom}_A \biggl(\det(g(A[[t]]) \mid g(t^{-r}A[[t]])), \, \bigwedge^{\mathrm{max}} H^0(C, \mathcal{F}(rp)) \biggr), \\ \det R g(\pi)_* g(\mathcal{F}) &\simeq \operatorname{Hom}_A \biggl( \, \bigwedge^{\mathrm{max}} (g(t^{-r}A[[t]])/ A[[t]]), \, \bigwedge^{\mathrm{max}} H^0(C, \mathcal{F}(rp)) \biggr) \\ &= \operatorname{Hom}_A \biggl(\det(A[[t]] \mid g(t^{-r}A[[t]])), \, \bigwedge^{\mathrm{max}} H^0(C, \mathcal{F}(rp)) \biggr). \end{aligned}
\end{equation*}
\notag
$$
From (32) we have
$$
\begin{equation*}
\det(g(A[[t]]) \mid g(t^{-r}A[[t]])) \simeq \det(g(A[[t]]) \mid A[[t]]) \otimes_A \det ( A[[t]] \mid g(t^{-r}A[[t]])).
\end{equation*}
\notag
$$
Hence we obtain a canonical isomorphism
$$
\begin{equation}
\det R \pi_* \mathcal{F} \otimes_A \det(g(A[[t]]) \mid A[[t]]) \xrightarrow{\sim} \det R g(\pi)_* g(\mathcal{F})
\end{equation}
\tag{60}
$$
such that the following diagram of isomorphisms is commutative for all $g_1$ and $g_2$ from $\mathcal{G}(A)$ where
$$
\begin{equation*}
P = \det R \pi_* \mathcal{F} \otimes_A \det(g_1 g_2 ( A[[t]]) \mid g_1(A[[t]])) \otimes_A \det(g_1 ( A[[t]]) \mid A[[t]]).
\end{equation*}
\notag
$$
Now the proof of Theorem 4 concludes by using the definition of $\widetilde{\mathcal{G}}$ from § 5.3 and recalling the compatibility of isomorphism (60) with base change. § 7. $2$-cocycle on $\mathcal{G}$ obtained from $\cup$-products of $1$-cocylesWe will construct a $2$-cocycle on $\mathcal{G}$ by means of products of $\cup$-products of $1$-cocyles on $\mathcal{G}$ and the Contou-Carrère symbol $\operatorname{CC}$. We also calculate the corresponding Lie algebra $2$-cocycle on $\operatorname{Lie} \mathcal{G}$. 7.1. Contou-Carrère symbolWe recall the definition of the Contou-Carrère symbol (see [6], [9], § 2.9, [21], § 2):
$$
\begin{equation*}
\operatorname{CC} \colon L\mathbb{G}_m \times L \mathbb{G}_m \to L \mathbb{G}_m \otimes L \mathbb{G}_m \to \mathbb{G}_m.
\end{equation*}
\notag
$$
Let $A$ be any commutative ring and $\underline{n} \in \underline{\mathbb{Z}}(A)$. The element $\underline{n}$ defines a decomposition $A = A_1 \times \dots \times A_k$ into the finite direct product of rings such that $\underline{n}$ restricted to every $\operatorname{Spec} A_l$ equals a constant function with value $n_l \in \mathbb{Z}$. Now, for any $a\in A^*$, we define the element $a^{\underline{n}} = a^{n_1} \times \dots \times a^{n_k}$ from $ A^*$. We define a free $A((t))$-module of rank $1$:
$$
\begin{equation}
\widetilde{\Omega}^1_{A((t))} = \Omega^1_{A((t))} / N,
\end{equation}
\tag{61}
$$
where $\Omega^1_{A((t))}$ is the $A((t))$-module of absolute Kähler differentials, the $A((t))$-submodule $N$ is generated by all elements $df - f'\, dt$, where $f \in A((t))$ and $f'= \frac{\partial{f}}{\partial{t}}$. It is clear that $N$ contains elements $da$, where $a \in A$, and that $dt$ is a basis of the $A((t))$-module $\widetilde{\Omega}^1_{A((t))}$. We define the residue
$$
\begin{equation*}
\operatorname{res} \colon \Omega^1_{A((t))} \to \widetilde{\Omega}^1_{A((t))} \to A,
\end{equation*}
\notag
$$
where the first map is the natural map, and the second map is $\sum_{i \in \mathbb{Z}} a_it^i \, dt \mapsto a_{-1}$. The Contou-Carrère symbol $\operatorname{CC}$ is a bimultiplicative antisymmetric morphism. It has the following additional properties. Let $f,g \in A((t))^*$. 1. If $a \in A^*$, then $\operatorname{CC}(a, g) = a^{\nu(g)}$. 2. $\operatorname{CC}(t,t)=-1$. 3. If $\mathbb{Q} \subset A$ and $ f \in \mathbb{V}_+(A) \times \mathbb{V}_-(A) $, then
$$
\begin{equation}
\operatorname{CC}(f,g) = \exp \operatorname{res} \biggl(\log f \cdot \frac{dg}{g} \biggr),
\end{equation}
\tag{62}
$$
where $\exp (x)$ and $\log(1+y)$ are the usual formal series, the series $\log$ in above formula converges in the topology of $A((t))$, and an application of series $\exp$ in above formula makes sense, because $\operatorname{res} \bigl(\log f \cdot \frac{dg}{g} \bigr) \in \mathrm{Nil }(A)$. After restriction to commutative $\mathbb{Q}$-algebras, the Contou-Carrère symbol
$$
\begin{equation*}
\operatorname{CC}_{\mathbb{Q}} \colon {L\mathbb{G}_m}_{\mathbb{Q}} \times {L \mathbb{G}_m}_{\mathbb{Q}} \to {\mathbb{G}_m}_{\mathbb{Q}}
\end{equation*}
\notag
$$
is uniquely defined by the above properties. For an arbitrary $A$, and all $f, g \in A((t))^*$, there are unique decompositions
$$
\begin{equation*}
f = \prod_{i < 0} (1 - a_it^i) \cdot a_0 \cdot t^{\nu(f)} \cdot \prod_{i > 0} (1 - a_it^i), \quad g = \prod_{j < 0} (1 - b_jt^j) \cdot b_0 \cdot t^{\nu(g)} \cdot \prod_{j > 0} (1 - b_jt^j),
\end{equation*}
\notag
$$
where $a_0, b_0 \in A^*$, $a_i, b_j \in \mathrm{Nil}(A) $ when $i, j <0$, and the products over negative $i$ and over negative $j$ are finite products. Now the Contou-Carrère symbol is defined by the formula
$$
\begin{equation}
\operatorname{CC}(f,g)= (-1)^{\nu(f) \nu(g)} \frac{a_0^{\nu(g)} \prod_{i > 0} \prod_{j>0 } \bigl(1 - a_i^{j /(i,j)} b_{-j}^{i/(i,j)} \bigr)^{(i,j)} }{b_0^{\nu(f)} \prod_{i > 0} \prod_{j>0 } \bigl(1 - a_{-i}^{j /(i,j)} b_{j}^{i/(i,j)} \bigr)^{(i,j)}},
\end{equation}
\tag{63}
$$
where the products in the numerator and denominator actually consist of a finite number of factors, and, therefore, the formula makes sense. (Formula (63) can be derived from the above properties and by an application of formula (62) to the elements $1 - a_it^i$ and $1 - b_jt^j$.) We recall (see (4)) that we have the embeddings $\mathbb{G}_m \times \mathbb{V}_+ \hookrightarrow L\mathbb{G}_m$ and $\mathbb{G}_m \times \mathbb{V}_- \hookrightarrow L\mathbb{G}_m$. From the above formulas for the Contou-Carrère symbol $\operatorname{CC}$ we have
$$
\begin{equation}
\operatorname{CC} |_{(\mathbb{G}_m \times \mathbb{V}_+) \times (\mathbb{G}_m \times \mathbb{V}_+)} =1 \quad \text{and} \quad \operatorname{CC} |_{(\mathbb{G}_m \times \mathbb{V}_-) \times (\mathbb{G}_m \times \mathbb{V}_-)} =1.
\end{equation}
\tag{64}
$$
It is important for application of the Contou-Carrère symbol $\operatorname{CC}$ in this article that $\operatorname{CC}$ is invariant under diagonal action of the group functor ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})$ on the commutative group functor $L\mathbb{G}_m \times L \mathbb{G}_m$, see [21], § 2, and [12]. 7.2. $\cup$-products of $1$-cocyles on $\mathcal{G}$ and the corresponding Lie algebra $2$-cocyclesSince $L\mathbb{G}_m$ is an ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})$-module, from the natural morphism of group functors $\mathcal{G} \to {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})$, we find that $L\mathbb{G}_m$ is also a $\mathcal{G}$-module. Besides, the Contou-Carrère symbol $\operatorname{CC}$ is a morphism of $\mathcal{G}$-modules $L \mathbb{G}_m \otimes L \mathbb{G}_m \to \mathbb{G}_m$, where we consider the trivial action of $\mathcal{G}$ on $\mathbb{G}_m$. Hence, the following definition is well-defined. Definition 2. For any two $1$-cocycles $\lambda_1$ and $\lambda_2$ on the group functor $\mathcal{G}$ with coefficients in the $\mathcal{G}$-module $L \mathbb{G}_m$ we define the $2$-cocycle on the group functor $\mathcal{G}$ with coefficients in the trivial $\mathcal{G}$-module $\mathbb{G}_m$, where $\circ$ means the composition of morphisms of functors. Definition 3. We define $1$-cocycles $\Lambda$ and $\Omega$ on the group functor $\mathcal{G}$ with coefficients in the $\mathcal{G}$-module $ L\mathbb{G}_m$: where $ (h, \varphi) \in \mathcal{G}(A) $ for any commutative ring $A$. It is easy to see that Definition 3 is well-defined, that is, $\Lambda$ and $\Omega$ are $1$-cocycles. Remark 13. The $1$-cocycle $\Lambda$ is universal in the following sense. Let $\lambda$ be any $1$-cocycle on $\mathcal{G}$ with values in the $\mathcal{G}$-module $L\mathbb{G}_m$. We define the new morphism of group functors where $(h, \varphi) \in \mathcal{G}(A) $ for any commutative ring $A$. Now, under the inverse image, we have $1$-cocycles $\Phi_{\lambda}^*(\Lambda) = \lambda$, $\Phi_{\lambda}^* (\Omega) = \Omega$, and correspondingly we have $2$-cocycle $\Phi_{\lambda}^* (\langle \lambda_1, \lambda_2 \rangle) = \langle \Phi_{\lambda}^* (\lambda_1), \Phi_{\lambda}^* (\lambda_2) \rangle$. Any $2$-cocycle $\Upsilon$ on $\mathcal{G}$ with coefficients in the trivial $\mathcal{G}$-module $\mathbb{G}_m$ induces the Lie algebra $2$-cocycle (see Appendix A.3 in [19])
$$
\begin{equation*}
\operatorname{Lie} \Upsilon \colon \operatorname{Lie} \mathcal{G} \times \operatorname{Lie} \mathcal{G} \to \mathbb{G}_a,
\end{equation*}
\notag
$$
where $\mathbb{G}_a {=}\operatorname{Lie} \mathbb{G}_m $, $\mathbb{G}_a (A) \,{=}\,A$ for any commutative ring $A$. We take elements $d_i $ from $ \operatorname{Lie} \mathcal{G} (A) \,{\subset}\, \mathcal{G} (A[\varepsilon_i]/ (\varepsilon_i^2)) $ and consider them as $d_i {\in}\, \mathcal{G}(A[\varepsilon_1, \varepsilon_2] / (\varepsilon_1^2, \varepsilon_2^2))$ through the natural embeddings $\mathcal{G} (A[\varepsilon_i]/ (\varepsilon_i^2)) \hookrightarrow \mathcal{G}(A[\varepsilon_1, \varepsilon_2] / (\varepsilon_1^2, \varepsilon_2^2))$, where $i=1$ and $i=2$. Now the element
$$
\begin{equation}
\operatorname{Lie} \Upsilon (d_1, d_2) = \Upsilon(d_1,d_2) \Upsilon(d_2, d_1)^{-1} \in (A[\varepsilon_1, \varepsilon_2] / (\varepsilon_1^2, \varepsilon_2^2))^*
\end{equation}
\tag{65}
$$
is the image of the element from $A$ as $a \mapsto 1 + a\varepsilon_1 \varepsilon_2$. We recall Remark 1 and Proposition 2 for the description of $\operatorname{Lie} \mathcal{G}(A)$. The following result holds. Proposition 7. Let $A$ be any commutative ring. For any elements $s_i, r_i$ from $A((t))$, where $i=1$ and $i=2$, we have explicit expressions for the Lie algebra $2$-cocycles: Proof. The $2$-cocycle $\langle \Omega, \Omega \rangle $ is the formal Bott–Thurston cocycle from Proposition 2.3 of [19]. Now (68) is the result of Proposition 4.1 from [19].
Formulas (66), (67) follow by direct calculations with formulas (65) and (62), (63), by the similar reasonings as in the proof of Proposition 4.1 from [19]. Indeed, we use that, for any $f_1$, $f_2$ from $A((t))$ and $i=1$, $i=2$, we have, for the Contou-Carrère symbol $\operatorname{CC}$ in the ring $E((t))$, where $E = A[\varepsilon_1, \varepsilon_2] / (\varepsilon_1^2, \varepsilon_2^2)$,
$$
\begin{equation*}
\operatorname{CC}( 1 + f_1 \varepsilon_i, 1 + f_2 \varepsilon_1 \varepsilon_2) =1.
\end{equation*}
\notag
$$
Now the calculations in case of (66) lead to the following equalities using the ring $E((t))$:
$$
\begin{equation*}
\begin{aligned} \, &\operatorname{CC}(1 + s_1 \varepsilon_1, 1+ s_2 \varepsilon_2) \operatorname{CC}(1 + s_2 \varepsilon_2, 1+ s_1 \varepsilon_1)^{-1} \\ &\qquad= \bigl(1 + \operatorname{res} (s_1 \, ds_2) \varepsilon_1 \varepsilon_2 \bigr) \bigl(1 + \operatorname{res} (s_2 \, ds_1) \varepsilon_1 \varepsilon_2 \bigr)^{-1}= 1 + 2 \operatorname{res} (s_1 \, ds_2) \varepsilon_1 \varepsilon_2. \end{aligned}
\end{equation*}
\notag
$$
And in case of formula (66) we obtain
$$
\begin{equation*}
\begin{aligned} \, &\operatorname{CC}(1 + s_1 \varepsilon_1, 1+ r_2' \varepsilon_2) \operatorname{CC}(1 + s_2 \varepsilon_2, 1+ r_1' \varepsilon_1)^{-1} \\ &\qquad= \bigl(1 + \operatorname{res} (s_1 \, dr_2') \varepsilon_1 \varepsilon_2 \bigr) \bigl(1 + \operatorname{res} (s_2 \, dr_1') \varepsilon_1 \varepsilon_2 \bigr)^{-1}= 1 + \operatorname{res} (s_1 \, dr_2' - s_2 \, dr_1') \varepsilon_1 \varepsilon_2. \end{aligned}
\end{equation*}
\notag
$$
This proves the proposition. Remark 14. Let $ A=k$ be a field of characteristic zero. Proceeding as in the proof of Proposition (2.1) from [2], we can verify that the continuous Lie algebra cohomology $H^2_\mathrm{c}(\operatorname{Lie} \mathcal{G}(k), k)$ is a three-dimensional vector space over $k$ with the basis given by the $2$-cocycles from formulas (66)–(68). § 8. Comparison of central extensions of $\mathcal{G}$We will prove a local analog of the Deligne–Riemann–Roch isomorphism (2). This is the comparison of two central extensions of $\mathcal{G}_{\mathbb{Q}}$ by ${\mathbb{G}_m}_{\mathbb{Q}}$. The first central extension is the determinant central extension of $\mathcal{G}$ from § 5, and the second central extension is given by some products of $2$-cocycles constructed in § 7.2. At first, we prove the corresponding result for the Lie algebra $2$-cocycles. We will also use the theory of infinitesimal formal groups over $\mathbb{Q}$. 8.1. Comparison of Lie algebra $2$-cocyclesWe recall that by Proposition 4 the determinant central extension of $\mathcal{G}^0$ admits a natural section $\mathcal{G}^0 \to \widetilde{\mathcal{G}^0}$ (as functors) with the corresponding explicit $2$-cocycle $D$ given by formula (31). Let $A$ be any commutative ring. By Proposition 2 any element $z $ from $\operatorname{Lie} \mathcal{G}^0(A)$ acts naturally on the $A$-module $A((t))$ by continuous $A$-endomorphisms. We write this action as a block matrix (cf. formula (28)) with respect to the decomposition $A((t)) = t^{-1} A[t^{-1}] \oplus A[[t]]$, and where the matrix acts on elements-columns from $A((t))$ on the left.The following proposition is an analog of Proposition 4.2 from [19] for ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})$ and also a formal analog of Proposition 6.6.5 from [23] from the theory of smooth loop groups. Proposition 8. For any commutative ring $A$ and any elements $z,w$ from $\operatorname{Lie} \mathcal{G}^0(A)$, Here the trace of the $A$-linear map $A[[t]] \to A[[t]]$ is well-defined (because there is an integer $n \geqslant 0$ such that $b_z |_{t^n A[[t]]} = b_w |_{t^nA[[t]]} = 0$). Proof. By (31) and (65), we have to calculate over the ring $A[\varepsilon_1, \varepsilon_2]/ (\varepsilon_1^2, \varepsilon_2^2)$:
$$
\begin{equation*}
\begin{aligned} \, &D (\mathrm{id} + z \varepsilon_1, \mathrm{id} + w \varepsilon_2) D(\mathrm{id} + w \varepsilon_2, \mathrm{id} + z \varepsilon_1)^{-1} \\ &\qquad= \det (\mathrm{id} - c_z b_w \varepsilon_1 \varepsilon_2) \det(\mathrm{id} - c_w b_z \varepsilon_1 \varepsilon_2)^{-1} \\ &\qquad= \bigl(1 - \operatorname{tr}(c_z b_w) \varepsilon_1 \varepsilon_2 \bigr) \bigl(1 - \operatorname{tr}(c_w b_z) \varepsilon_1 \varepsilon_2 \bigr)^{-1} = 1 + \operatorname{tr} (c_w b_z - c_z b_w) \varepsilon_1 \varepsilon_2. \end{aligned}
\end{equation*}
\notag
$$
Here, $\mathrm{id}$ is the identity map. Proposition 8 is proved. Theorem 5. We have an equality between Lie algebra $2$-cocycles on $\operatorname{Lie} \mathcal{G}^0$ with coefficients in $\mathbb{G}_a$ Proof. We will use Propositions 7 and 8.
We fix a commutative ring $A$. There is a natural topology on the $A$-module $\operatorname{Lie} \mathcal{G}^0(A) = A((t)) + A((t)) \, \frac{\partial}{\partial t}$. It is easy to see that $2$-cocycles given by formulas (66)–(68) are continuous in each argument when we consider the discrete topology on $A$. Besides, $\operatorname{Lie} D$ is also continuous, since $\operatorname{tr} (c_w b_z)$ and $\operatorname{tr} (c_z b_w)$ are continuous in each argument (see formula (70)). Therefore, it is enough to check equality (71) on elements of type $t^i$ and $t^{j+1} \frac{\partial}{\partial t}$ from $\operatorname{Lie} \mathcal{G}^0(A)$. We will use the Kronecker delta $\delta_{r,s}$ that is equal to $1$ when $r =s$ and is equal to $0$ otherwise. We have the following cases (using also that Lie algebra cocycles are antisymmetric). The first case. We calculate
$$
\begin{equation*}
12 \operatorname{Lie} D (t^n, t^m ) = 12 \operatorname{tr} (c_{t^m} b_{t^n} - c_{t^n} b_{t^m}) = 12 \biggl( \sum_{l=0}^{m-1} 1 \biggr) \delta_{m, -n} = 12 m\delta_{m, -n}.
\end{equation*}
\notag
$$
Here, the result is equal to the first summand $12 \operatorname{tr}( c_{t^m} b_{t^n})$ when $m > 0$ and it is equal to the second summand $12(- c_{t^n} b_{t^m})$ when $m \leqslant 0$ (the other summands are equal to zero).
The right-hand side of (71) applied to the elements $t^n$ and $t^m$ is equal to
$$
\begin{equation*}
6 \operatorname{Lie} \langle \Lambda, \Lambda \rangle (t^n, t^m) = 6 \cdot 2 \operatorname{res} (t^n \, d t^m)= 12 m\delta_{m, -n}.
\end{equation*}
\notag
$$
The second case. Now we calculate
$$
\begin{equation*}
\begin{aligned} \, &12 \operatorname{Lie} D \biggl(t^n, \, t^{m+1} \, \frac{\partial}{\partial t} \biggr) = 12 \operatorname{tr} \bigl( c_{t^{m+1} \, \frac{\partial}{\partial t}} b_{t^n} - c_{t^n} b_{t^{m+1} \, \frac{\partial}{\partial t} } \bigr) \\ &\qquad= 12 \biggl( \sum_{l=0}^{m-1} (l-m) \biggr)\delta_{m, -n} = 6 (-m- m^2) \delta_{m, -n}. \end{aligned}
\end{equation*}
\notag
$$
Here, the result is equal to the first summand $12 \operatorname{tr} \bigl( c_{t^{m+1} \frac{\partial}{\partial t}} b_{t^n} \bigr)$ when $m > 0$ and it is equal to the second summand $12 \bigl(- c_{t^n} b_{t^{m+1} \frac{\partial}{\partial t}} \bigr)$ when $m \leqslant 0$ (the other summands are equal to zero).
The right-hand side of (71) applied to the elements $t^n$ and $t^{m+1} \frac{\partial}{\partial t}$ is equal to
$$
\begin{equation*}
-6 \operatorname{Lie} \langle \Lambda, \Omega \rangle \biggl( t^n,\, t^{m+1} \, \frac{\partial}{\partial t} \biggr) = -6 \operatorname{res} \bigl(t^n \, d(t^{m+1})' \bigr)= 6(-m -m^2)\delta_{m, -n}.
\end{equation*}
\notag
$$
The third case. At the end, we calculate
$$
\begin{equation*}
\begin{aligned} \, &12 \operatorname{Lie} D \biggl( t^{n+1} \, \frac{\partial}{\partial t},\, t^{m+1} \, \frac{\partial}{\partial t} \biggr) = 12 \operatorname{tr} \bigl( c_{t^{m+1} \, \frac{\partial}{\partial t}} b_{t^{n+1} \, \frac{\partial}{\partial t}} - c_{t^{n+1} \, \frac{\partial}{\partial t}} b_{t^{m+1} \, \frac{\partial}{\partial t}} \bigr) \\ &\qquad= 12 \biggl( \sum_{l=0}^{m-1} l(l-m) \biggr)\delta_{m, -n} = 2 (m- m^3) \delta_{m, -n}. \end{aligned}
\end{equation*}
\notag
$$
Here, the result is equal to the first summand $12 \operatorname{tr} \bigl( c_{t^{m+1} \frac{\partial}{\partial t}} b_{t^{n+1} \frac{\partial}{\partial t}} \bigr)$ when $m > 0$ and it is equal to the second summand $12 \bigl( - c_{t^{n+1} \frac{\partial}{\partial t}} b_{t^{m+1} \frac{\partial}{\partial t}} \bigr)$ when $m \leqslant 0$ (the other summands are equal to zero).
The right-hand side of (71) applied to the elements $t^{n+1} \frac{\partial}{\partial t}$ and $t^{m+1} \frac{\partial}{\partial t}$ is equal to
$$
\begin{equation*}
\operatorname{Lie} \langle \Omega, \Omega \rangle \biggl( t^{n+1} \, \frac{\partial}{\partial t}, \, t^{m+1} \, \frac{\partial}{\partial t} \biggr) = 2 \operatorname{res} \bigl((t^{n+1})' \, d(t^{m+1})' \bigr)= 2 (m -m^3)\delta_{m, -n}.
\end{equation*}
\notag
$$
Theorem 5 is proved. 8.2. Correspondence between infinitesimal formal groups and Lie algebrasLet $k$ be a field of zero characteristic. It is well-known that the category of finite-dimensional formal groups over $k$ is equivalent to the category of finite-dimensional Lie algebras over $k$, see, for example, Theorem 3 in Part II, Ch. V, § 6 of [28]. We will need the generalization of this statement to the infinite-dimensional case. An infinitesimal formal group over $k$ is a group ind-scheme $G = ``\varinjlim_{i \in I}\!" \operatorname{Spec} A_i$ such that every $A_i$ is a finite-dimensional $\mathbb{Q}$-algebra and the corresponding profinite algebra of regular functions $\mathcal{O}(G)= \varprojlim_{i \in I} A_i$ is a local $k$-algebra with the residue field $k$. By [10], Ch. I, § 1.7, § 2.14; Ch. II, § 2.2, § 2.5, and [25], Exposé $\textrm{VII}_B$, § 2.7, the functor $G \mapsto \mathcal{O}(G)$ is an equivalence between the category of infinitesimal formal groups over $k$ and the opposite of the category of commutative linearly compact local $k$-algebras $A= k \oplus U$ (decomposition as $k$-vector spaces), where $U$ is an open ideal, such that the topological dual vector space consisting of continuous linear functionals $\operatorname{Hom}^\mathrm{c}_k (A, k)$ is a (discrete) bialgebra over $k$. Now, similarly to the finite-dimensional case, by [25], Exposé $\textrm{VII}_B$, § 3, there is an isomorphism of bialgebras
$$
\begin{equation*}
U(\operatorname{Lie} G (k)) \xrightarrow{\sim} \operatorname{Hom}^\mathrm{c}_k (\mathcal{O}(G), k),
\end{equation*}
\notag
$$
where $U(\operatorname{Lie} G (k) )$ is the universal enveloping algebra of the Lie algebra $\operatorname{Lie} G(k)$ over $k$ with the standard bialgebra structure. Moreover, this isomorphism is induced by the natural linear embedding of $\operatorname{Lie} G (k)$ to $\operatorname{Hom}^\mathrm{c}_k (\mathcal{O}(G), k)$, and $\operatorname{Lie} G(k) \subset U(\operatorname{Lie} G (k))$ is the set of primitive elements of the bialgebra. These constructions lead to the theorem (see [25], Exposé $\textrm{VII}_B$, § 3.3.2) that the functor $G \mapsto \operatorname{Lie} G (k)$ gives an equivalence of the category of infinitesimal formal groups over $k$ and the category of Lie algebras over $k$. 8.3. The class of the determinant central extension of $\mathcal{G}_{\mathbb{Q}}$ in the group $H^2(\mathcal{G}_{\mathbb{Q}}, {\mathbb{G}_m}_{\mathbb{Q}})$We recall, see § 2, that we have
$$
\begin{equation*}
\mathbb{G}_m \times \mathbb{V}_+ \hookrightarrow (L\mathbb{G}_m)^0 \quad \text{and} \quad {\mathcal Aut}_+^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}) \hookrightarrow {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}).
\end{equation*}
\notag
$$
It is clear that $\mathbb{G}_m \times \mathbb{V}_+$ is an ${\mathcal Aut}_+^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})$-module. We consider the group subfunctor
$$
\begin{equation*}
\mathcal{G}_+ = (\mathbb{G}_m \times \mathbb{V}_+) \rtimes {\mathcal Aut}_+^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}) \hookrightarrow \mathcal{G}^0.
\end{equation*}
\notag
$$
It is clear that, for any commutative ring $A$, elements of $\mathcal{G}_+(A)$ acts on the $A$-module $A[[t]]$ by continuous automorphisms. We will construct the infinitesimal formal group $\mathcal{I}\mathcal{G}^0_{\mathbb{Q}}$ over $\mathbb{Q}$ from the group ind-affine ind-scheme $\mathcal{G}^0_{\mathbb{Q}}$ (see § 8.2). By definition, for any commutative $\mathbb{Q}$-algebra $A$, the group $\mathcal{I}\mathcal{G}^0_{\mathbb{Q}}(A)$ consists of elements $(h, \varphi) \in (L\mathbb{G}_m)^0(A) \rtimes {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L}) (A)$ such that in the decompositions
$$
\begin{equation}
h = 1 + \sum_i b_i t^i \quad \text{and} \quad \widetilde{\varphi}= \varphi(t) = t + \sum_i c_i t^j
\end{equation}
\tag{72}
$$
all elements $b_i$ and $c_i$ belong to $\operatorname{Nil}(A)$ and they are equal to zero except for a finite number of elements, cf. the definition of the infinitesimal formal group $\mathcal{F} {\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})_{\mathbb{Q}}$ constructed from ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})_{\mathbb{Q}}$ in [19], § 5.1. (The statement that $\mathcal{I} \mathcal{G}^0_{\mathbb{Q}}(A)$ is a group follows easily from decompositions (7) and (10).) By (72), the group functor $\mathcal{I} \mathcal{G}^0_{\mathbb{Q}}(A)$ is represented by the ind-scheme
$$
\begin{equation}
``\varinjlim_{\{\epsilon_i\}}\!" \operatorname{Spec} \mathbb{Q} [b_i; i \in \mathbb{Z}]/ I_{\{\epsilon_i\}} \times ``\varinjlim_{\{\epsilon_i\}}\!" \operatorname{Spec} \mathbb{Q} [c_i; i \in \mathbb{Z}]/ I_{\{\epsilon_i\}},
\end{equation}
\tag{73}
$$
where $\mathbb{Q} [b_i; i \in \mathbb{Z}]$ is the polynomial ring over $\mathbb{Q}$ on a set of variables $b_i$ with $i \in \mathbb{Z}$, and the limit is taken over all the sequences $\{\epsilon_i\}$ with $i \in \mathbb{Z}$ and $\epsilon_i$ are non-negative integers such that all but finitely many $\epsilon_i$ equal zero, the ideal $I_{\{\epsilon_i\}}$ is generated by elements $b_i^{\epsilon_i +1}$ for all $i \in \mathbb{Z}$. The ring $\mathbb{Q} [b_i; i \in \mathbb{Z}]$ and its ideals $I_{\{\epsilon_i\}}$ has the same description. Thus, $\mathcal{I} \mathcal{G}^0_{\mathbb{Q}}$ is an infinitesimal formal group over $\mathbb{Q}$. And we have the natural embedding
$$
\begin{equation*}
\mathcal{I} \mathcal{G}^0_{\mathbb{Q}} \hookrightarrow \mathcal{G}^0_{\mathbb{Q}}
\end{equation*}
\notag
$$
(which is embedding on $A$-points for any commutative $\mathbb{Q}$-algebra $A$). It is easy to see that the Lie $\mathbb{Q}$-algebra $ \operatorname{Lie} \mathcal{I} \mathcal{G}^0_{\mathbb{Q}}(\mathbb{Q})$ has a basis consisting of all elements $d_n = t^{n+1} \frac{\partial}{\partial t}$ and $e_n = t^n$, where $n \in \mathbb{Z}$, by
$$
\begin{equation}
[d_n, d_m]= (m-n) d_{n+m}, \qquad [d_n, e_m]= m e_{n+m}, \qquad [e_n, e_m]= 0.
\end{equation}
\tag{74}
$$
We define the infinitesimal formal group $\mathcal{I}{\mathcal{G}_+}_{\mathbb{Q}}$ over $\mathbb{Q}$ in the following way:
$$
\begin{equation*}
\mathcal{I}{\mathcal{G}_+}_{\mathbb{Q}}(A) = \mathcal{I} \mathcal{G}^0_{\mathbb{Q}}(A) \cap \mathcal{G}_+(A),
\end{equation*}
\notag
$$
where $A$ is any commutative $\mathbb{Q}$-algebra, and the intersection is taken in the group $\mathcal{G}^0(A)$. It is clear that the group functor $\mathcal{I}{\mathcal{G}_+}_{\mathbb{Q}}$ is represented by infinitesimal formal group over $\mathbb{Q}$ (we have to take in formula (73) only indices with $i \geqslant 0$). We define the infinitesimal formal group $\mathcal{I} {\mathbb{G}_m}_{\mathbb{Q}}$ over $\mathbb{Q}$ as
$$
\begin{equation*}
\mathcal{I} {\mathbb{G}_m}_{\mathbb{Q}} (A) = 1 + \operatorname{Nil}(A) \subset A^*,
\end{equation*}
\notag
$$
where $A$ is any commutative $\mathbb{Q}$-algebra. (This formal group is the formal group constructed from the completion of the local ring at the identity element of the algebraic group ${\mathbb{G}_m}_{\mathbb{Q}}$.) Lemma 5. 1. Any morphism of group functors $\mathcal{I}{\mathcal{G}_+}_{\mathbb{Q}} \to {\mathbb{G}_m}_{\mathbb{Q}}$ is trivial. 2. The natural homomorphism of $\mathbb{Q}$-algebras of regular functions $\mathcal{O}(\mathcal{G}^0_{\mathbb{Q}} \times \mathcal{G}^0_{\mathbb{Q}}) \to \mathcal{O}(\mathcal{I} \mathcal{G}^0_{\mathbb{Q}} \times \mathcal{I} \mathcal{G}^0_{\mathbb{Q}})$, that is, induced by the natural morphism of ind-affine ind-schemes $\mathcal{I} \mathcal{G}^0_{\mathbb{Q}} \times \mathcal{I} \mathcal{G}^0_{\mathbb{Q}} \to \mathcal{G}^0_{\mathbb{Q}} \times \mathcal{G}^0_{\mathbb{Q}} $ is an embedding. Proof. 1. The Lie $\mathbb{Q}$-algebra $\operatorname{Lie} \mathcal{I}{\mathcal{G}_+}_{\mathbb{Q}}(\mathbb{Q})$ is a subalgebra of the Lie $\mathbb{Q}$-algebra $ \operatorname{Lie} \mathcal{I}\mathcal{G}^0_{\mathbb{Q}}(\mathbb{Q})$. The basis of this subalgebra consists of all elements $d_n$ and $e_m$, where $n \geqslant -1$ and $m \geqslant 0$. Now from (74) it is easy to see that
$$
\begin{equation*}
\bigl[ \operatorname{Lie} \mathcal{I}{\mathcal{G}_+}_{\mathbb{Q}}(\mathbb{Q}), \operatorname{Lie} \mathcal{I}{\mathcal{G}_+}_{\mathbb{Q}}(\mathbb{Q}) \bigr] = \operatorname{Lie} \mathcal{I}{\mathcal{G}_+}_{\mathbb{Q}}(\mathbb{Q})
\end{equation*}
\notag
$$
(or we can use the similar formulas as in the proof of Lemma 2).
From this fact and since the Lie algebra $\mathbb{G}_a(\mathbb{Q}) = \operatorname{Lie} \mathcal{I} {\mathbb{G}_m}_{\mathbb{Q}} (\mathbb{Q})$ is Abelian, we find that any Lie algebra homomorphism from $\operatorname{Lie} \mathcal{I}{\mathcal{G}_+}_{\mathbb{Q}}(\mathbb{Q})$ to $\operatorname{Lie} \mathcal{I} {\mathbb{G}_m}_{\mathbb{Q}} (\mathbb{Q})$ is equal to zero. Now, using for any commutative $\mathbb{Q}$-algebra $A$ the homomorphism $A\,{\to}\, A/ \operatorname{Nil}(A)$, it is easy to see that any morphism of group functors $\tau \colon \mathcal{I}{\mathcal{G}_+}_{\mathbb{Q}} \to {\mathbb{G}_m}_{\mathbb{Q}}$ comes form the morphism of infinitesimal formal groups $\gamma \colon \mathcal{I}{\mathcal{G}_+}_{\mathbb{Q}} \to \mathcal{I} {\mathbb{G}_m}_{\mathbb{Q}}$ via the composition with the morphism of group functors $\mathcal{I} {\mathbb{G}_m}_{\mathbb{Q}} \to {\mathbb{G}_m}_{\mathbb{Q}}$. By § 8.2, the morphism $\gamma$ is trivial. Therefore, $\tau$ is also trivial. 2. This statement follows easily from the description of the corresponding ind-schemes, since it is embedding of the $\mathbb{Q}$-algebra of mixture polynomials, Laurent polynomials, and formal power series in infinite number of variables to the $\mathbb{Q}$-algebra of formal power series in infinite number of variables, see formula (73) and § 2.3 (compare also with the proof of Proposition 5). Lemma 5 is proved. Any element from the group $H^2({\mathcal{G}}_{\mathbb{Q}}, {\mathbb{G}_m}_{\mathbb{Q}})$, where ${\mathbb{G}_m}_{\mathbb{Q}}$ is a trivial ${\mathcal{G}}_{\mathbb{Q}}$-module, defines a central extension of group functors that allow a section (as functors), see § 3.1. Therefore, it defines the central extension of the corresponding Lie algebras, and a section of central extensions of group functors gives the section of central extension of Lie algebras. Hence, the group of $2$-cocycles of group functor is mapped to the group of Lie algebra $2$-cocycles, see [19], Appendix A.3. Moreover we have the natural homomorphisms
$$
\begin{equation}
H^2({\mathcal{G}}_{\mathbb{Q}}, {\mathbb{G}_m}_{\mathbb{Q}}) \to H^2(\mathcal{G}^0_{\mathbb{Q}}, {\mathbb{G}_m}_{\mathbb{Q}}) \to H^2(\operatorname{Lie} \mathcal{G}^0_{\mathbb{Q}}(\mathbb{Q}), \mathbb{Q}) \to H^2(\operatorname{Lie} \mathcal{I} \mathcal{G}^0_{\mathbb{Q}}(\mathbb{Q}), \mathbb{Q}).
\end{equation}
\tag{75}
$$
Remark 15. From Proposition 7, Remark 14 and [2], Proposition 2.1 (especially its proof) it follows that Theorem 6. Any element from $H^2( \mathcal{G}^0_{\mathbb{Q}}, {\mathbb{G}_m}_{\mathbb{Q}})$, where ${\mathbb{G}_m}_{\mathbb{Q}}$ is a trivial $\mathcal{G}^0_{\mathbb{Q}}$-module, is uniquely defined by its image in $H^2(\operatorname{Lie} \mathcal{I} \mathcal{G}^0_{\mathbb{Q}}(\mathbb{Q}), \mathbb{Q})$ together with its restriction to $H^2( {\mathcal{G}_+}_{\mathbb{Q}}, {\mathbb{G}_m}_{\mathbb{Q}})$. Proof. It is enough to prove that if a central extension of group functors
$$
\begin{equation}
1 \to {\mathbb{G}_m}_{\mathbb{Q}} \to H \xrightarrow{\mu} \mathcal{G}^0_{\mathbb{Q}} \to 1
\end{equation}
\tag{76}
$$
that admits a section $p$ of $\mu$ (as functors) is isomorphic to the trivial central extension after restriction ${\mathcal{G}_+}_{\mathbb{Q}}$ and the corresponding Lie algebra central extension of $\operatorname{Lie} \mathcal{I}\mathcal{G}^0_{\mathbb{Q}}(\mathbb{Q})$ by $\mathbb{Q}$ is isomorphic to the Lie algebra trivial central extension, then central extension (76) is also isomorphic to the trivial central extension.
By changing the section $p$ to $p \cdot p(e)^{-1}$, where $e$ is the identity element of the group ind-scheme $\mathcal{G}^0_{\mathbb{Q}}$, we can suppose that $p(e)$ is the identity element of the group ind-scheme $H$. Hence, the corresponding $2$-cocycle $K$ for (76), constructed by $p$, satisfies $K(e,e)=1$, where $1 \in {\mathbb{G}_m}_{\mathbb{Q}}$ is the identity element. Therefore, by considering, for any commutative $\mathbb{Q}$-algebra $A$, the homomorphism $A \to A/ \operatorname{Nil}(A)$, it is easy to see that $K$ maps $\mathcal{I} \mathcal{G}^0_{\mathbb{Q}} \times \mathcal{I} \mathcal{G}^0_{\mathbb{Q}}$ to the subfunctor $\mathcal{I} {\mathbb{G}_m}_{\mathbb{Q}}$ of the functor $ {\mathbb{G}_m}_{\mathbb{Q}}$. Hence the central extension (76) restricted to $\mathcal{I} \mathcal{G}^0_{\mathbb{Q}} $ comes via $\mathcal{I} {\mathbb{G}_m}_{\mathbb{Q}} \to {\mathbb{G}_m}_{\mathbb{Q}}$ from the central extension
$$
\begin{equation}
1 \to \mathcal{I} {\mathbb{G}_m}_{\mathbb{Q}} \to T \xrightarrow{\kappa} \mathcal{I} \mathcal{G}^0_{\mathbb{Q}} \to 1
\end{equation}
\tag{77}
$$
that admits a section of $\kappa$ (section as functors). Because of this section, $T \simeq \mathcal{I} {\mathbb{G}_m}_{\mathbb{Q}} \times \mathcal{I} \mathcal{G}^0_{\mathbb{Q}}$ (an isomorphism as ind-schemes). Therefore, $T$ is an infinitesimal formal group over $\mathbb{Q}$. By our condition, the corresponding to (77) Lie $\mathbb{Q}$-algebra central extension is trivial, and hence it admits a section-homomorphism from $\operatorname{Lie} \mathcal{I} \mathcal{G}^0_{\mathbb{Q}}(\mathbb{Q})$ to $\operatorname{Lie} T (\mathbb{Q})$. Hence, by § 8.2, there is the corresponding section $s$ from $\mathcal{I} \mathcal{G}^0_{\mathbb{Q}}$ to $T$ that is a morphism of group ind-schemes.
Thus, we have constructed the morphism of group ind-schemes
$$
\begin{equation*}
s \colon \mathcal{I} \mathcal{G}^0_{\mathbb{Q}} \to H
\end{equation*}
\notag
$$
such that $\mu s = \mathrm{id}$ (we denoted it by the same letter $s$ as the above morphism from $\mathcal{I} \mathcal{G}^0_{\mathbb{Q}}$ to $T$).
By our condition, we have also a morphism of group ind-schemes such that $\mu r = \mathrm{id}$.We claim that $r |_{\mathcal{I} {\mathcal{G}_+}_{\mathbb{Q}}} = s |_{\mathcal{I} {\mathcal{G}_+}_{\mathbb{Q}}} $. Indeed, $r/s$ is a morphism of group functors from $\mathcal{I} {\mathcal{G}_+}_{\mathbb{Q}}$ to ${\mathbb{G}_m}_{\mathbb{Q}}$ that is trivial by assertion 1 of Lemma 5. For any commutative $\mathbb{Q}$-algebra $A$ and any element $(h, \varphi) \in \mathcal{G}^0(A)$, by (7) and (10) we have unique decompositions where $ h_- \in \mathbb{V}_-(A)$, $h_+ \in \mathbb{G}_m(A) \times \mathbb{V}_+(A)$, $ \varphi_+ \in {\mathcal Aut}_+^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})$, $ \varphi_- \in {\mathcal Aut}_-^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})$. These decompositions are functorial with respect to $A$. We define the morphism of functors $q \colon \mathcal{G}^0_{\mathbb{Q}} \to H$ such that $\mu q =\mathrm{id}$ in the following way:
$$
\begin{equation*}
q ((h, \varphi)) = s(h_-) r(h_+) r(\varphi_+) s (\varphi_-) = s(h_-) r(h_+ \varphi_+) s (\varphi_-).
\end{equation*}
\notag
$$
To prove the theorem it is enough to show that $q$ is a morphism of group functors, that is, that it preserves the group structure.
We claim that $q |_{\mathcal{I} \mathcal{G}^0_{\mathbb{Q}}} = s$. Indeed, if $(h, \varphi) \in \mathcal{I} \mathcal{G}^0_{\mathbb{Q}}(A)$, then
$$
\begin{equation*}
h_+ \varphi_+ = h_-^{-1} (h, \varphi) \varphi_-^{-1} \in \mathcal{I} \mathcal{G}^0_{\mathbb{Q}}(A),
\end{equation*}
\notag
$$
since $h_-^{-1} $ and $ \varphi_-^{-1} $ are from $ \mathcal{I} \mathcal{G}^0_{\mathbb{Q}}(A)$. Hence, the element $h_+ \varphi_+ $ is from the group $ \mathcal{I} {\mathcal{G}_+}_{\mathbb{Q}} (A)$. Therefore, $r(h_+ \varphi_+) = s (h_+ \varphi_+)$. Since $s$ is a morphism of group functors, we obtain
$$
\begin{equation*}
q((h, \varphi)) = s(h_-) r(h_+ \varphi_+) s (\varphi_-) = s(h_-) s(h_+ \varphi_+) s (\varphi_-) = s((h, \varphi)).
\end{equation*}
\notag
$$
Now we consider a morphism of ind-schemes
$$
\begin{equation*}
\beta \colon \mathcal{G}^0_{\mathbb{Q}} \times \mathcal{G}^0_{\mathbb{Q}} \to {\mathbb{G}_m}_{\mathbb{Q}}, \qquad \beta(g_1 \times g_2)= q(g_1) q(g_2) q(g_1 g_2)^{-1}, \qquad g_1, g_2 \in \mathcal{G}^0_{\mathbb{Q}}(A).
\end{equation*}
\notag
$$
We have to prove that $\beta = \mathbf{1}$, where $\mathbf{1}$ is the constant morphism that is equal to $1$. Note that $\beta |_{\mathcal{I} \mathcal{G}^0_{\mathbb{Q}} \times \mathcal{I} \mathcal{G}^0_{\mathbb{Q}}} =\mathbf{1}$, since $s$ preserves the group structure. The morphism $\beta$ is determined by the homomorphism of $\mathbb{Q}$-algebras of regular functions
$$
\begin{equation*}
\beta^* \colon \mathcal{O}({\mathbb{G}_m}_{\mathbb{Q}}) \to \mathcal{O}(\mathcal{G}^0_{\mathbb{Q}} \times \mathcal{G}^0_{\mathbb{Q}}).
\end{equation*}
\notag
$$
Using the composition of the morphism $\mathcal{I} \mathcal{G}^0_{\mathbb{Q}} \times \mathcal{I} \mathcal{G}^0_{\mathbb{Q}} \to \mathcal{G}^0_{\mathbb{Q}} \times \mathcal{G}^0_{\mathbb{Q}}$ with the morphism $\beta$ and using assertion 2 of Lemma 5, we obtain $\beta^* = \mathbf{1}^*$. Therefore, $\beta = \mathbf{1}$. Theorem 6 is proved. The following corollary is immediate from Theorem 6 and Theorem 1. Corollary 2. Any element from $H^2(\mathcal{G}_{\mathbb{Q}}, {\mathbb{G}_m}_{\mathbb{Q}})$, where ${\mathbb{G}_m}_{\mathbb{Q}}$ is a trivial $\mathcal{G}_{\mathbb{Q}}$-module, is uniquely defined by its image in $H^2(\operatorname{Lie} \mathcal{I} \mathcal{G}^0_{\mathbb{Q}}(\mathbb{Q}), \mathbb{Q})$ together with its restriction to $H^2({\mathcal{G}_+}_{\mathbb{Q}}, {\mathbb{G}_m}_{\mathbb{Q}})$. Remark 16. Theorem 6 is a generalization of the similar Theorem 5.1 from [19] when $\mathcal{G}^0$ is changed to ${\mathcal Aut}^{\mathrm{c}, \mathrm{alg}} (\mathcal{L})$ (for the case of the group of orientation-preserving diffeomorphisms of the circle in the theory of infinite-dimensional Lie groups see Corollary (7.5) from [26]). We recall that by Remark 3 the determinant central extension of $\mathcal{G}$ by $\mathbb{G}_m$ has the natural section that gives the $2$-cocycle $D$. Now we obtain a local analog of Deligne–Riemann–Roch isomorphism (2). Theorem 7. In the group $H^2 (\mathcal{G}_{\mathbb{Q}}, {\mathbb{G}_m}_{\mathbb{Q}})$, where ${\mathbb{G}_m}_{\mathbb{Q}}$ is a trivial ${\mathcal{G}}_{\mathbb{Q}}$-module, we have Proof. We will use Corollary 2. By Theorem 5, we have the corresponding equality of the Lie $\mathbb{Q}$-algebra $2$-cocycles. Besides, by (64) the $2$-cocycle given by the right-hand side of (78) and restricted to $\mathcal{G}_+$ is the constant $2$-cocycle that is equal to $1$. And by Remark 3 and formula (31), the $2$-cocycle $D $ restricted to $\mathcal{G}_+$ is also the constant $2$-cocycle that is equal to $1$. Theorem 7 is proved.
Citation in format AMSBIB
\Bibitem{Osi24} |
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