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This article is cited in 1 scientific paper (total in 1 paper) Brief communications Determinant central extension and D. V. Osipov Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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     In this note the class of the determinant central extension of some group functor defined over the commutative $\mathbb{Q}$-algebras is calculated as a product of $2$-cocycles that consist of the Contou-Carrère symbol applied to pairwise $\cup$-products of $1$-cocycles. This is a local Riemann–Roch theorem for invertible sheaves in relative dimension $1$, as written in the second cohomology group of group functors. By a functor (commutative group functor or group functor) $H$ we mean a covariant functor from the category of commutative rings to the category of sets (Abelian groups or groups, respectively). Let $H_{\mathbb{Q}}$ denote the restriction of $H$ to the commutative $\mathbb{Q}$-algebras. Let $H^{\times n}$ denote the functor that is the $n$th direct product of $H$ with itself. In all groups we use the multiplicative notation for the group laws. Let $R$ be a commutative ring. Let $G$ and $F$ be a group functor and a commutative group functor, respectively, defined over the commutative $R$-algebras, and assume that $G$ acts on $F$ (so that $F$ is a $G$-module). We denote by $\operatorname{Hom}(G^{\times n},F)$ the Abelian group of all (without regard to group structures) morphisms of functors $G^{\times n } \to F$. Then $H^n(G, F)$ (see [4], § 2.3.1) is the $n$th cohomology group of the complex
$$
\begin{equation}
C^0 (G,F) \xrightarrow{\delta_0} C^{1}(G, F) \xrightarrow{\delta_1} \cdots \xrightarrow{\delta_{n-1}} C^{n}(G,F) \xrightarrow{\delta_n} \cdots,
\end{equation}
\tag{1}
$$
where $C^0(G,F)=F(R)$, $C^k(G,F)=\operatorname{Hom}(G^{\times k},F)$ for $k\geqslant 1$, and
$$
\begin{equation*}
\begin{aligned} \, \delta_{n}c(g_1,\dots,g_{n+1})&=g_1 c(g_2,\dots,g_{n+1}) \\ &\qquad\times\prod_{i=1}^n c(g_1,\dots,g_i g_{i+1},\dots,g_{n+1})^{(-1)^i}\cdot c(g_1,\dots,g_n)^{(-1)^{n+1}}, \end{aligned}
\end{equation*}
\notag
$$
where $c\in C^n(G,F)$ and $g_j \in G(A)$, $1 \leqslant j \leqslant n+1$, for an arbitrary commutative $R$-algebra $A$. An element of $\operatorname{Ker}\delta_n$ is called an $n$-cocycle on $G$ with coefficients in $F$.
For any $1$-cocycles $\lambda_1$ and $\lambda_2$ on $G$ with coefficients in $G$-modules $F_1$ and $F_2$, respectively, we obtain a $2$-cocycle $\lambda_1 \cup \lambda_2$ on $G$ with coefficients in $F_1 \otimes F_2$ such that $(\lambda_1 \cup \lambda_2) (g_1, g_2) = \lambda_1 (g_1) \otimes g_1 ( \lambda_2(g_2))$, 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 where $g_1$ and $ g_2 $ are arbitrary elements of $G(A)$. This induces a $\cup$-product between the first cohomology groups. Any morphism of $G$-modules induces a homomorphism of the corresponding cohomology groups of $G$. Fixing a commutative $R$-algebra $A$ gives us a map from the complex (1) to the bar-complex for the $G(A)$-module $F(A)$ and a homomorphism $H^n(G,F) \to H^n(G(A), F(A))$ compatible with the $\cup$-products. In what follows $A$ is an arbitrary commutative ring. A central extension of group functors is a short exact sequence of group functors that becomes a central extension of groups after fixing any $A$. Central extensions of $G$ by $F$ that admit a section from $G$ (just as functors) are classified up to isomorphism by elements of $H^2(G,F)$, where $F$ is a trivial $G$-module. Let ${\mathbb G}_m(A)=A^*$. Let $L{\mathbb G}_m $ be a commutative group functor such that $L {\mathbb G}_m (A)=A((t))^*$, where $A((t))=A[[t]][t^{-1}]$. Then $A((t))$ is a topological ring with the following neighbourhood basis of zero: $U_l=t^l A[[t]]$, $l \in \mathbb{Z}$. Let ${{\mathcal Aut}^{\rm c, alg} ({\mathcal L})}$ be a group functor such that ${{\mathcal Aut}^{\rm c,alg}({\mathcal L})}(A)$ is the group of all $A$-automorphisms of the $A$-algebra $A((t))$ that are homeomorphisms; see [4], § 2.1. Each element $\varphi \in {{\mathcal Aut}^{\rm c,alg}({\mathcal L})}(A)$ is uniquely defined by the element $\widetilde{\varphi}=\varphi(t)\in A((t))^*$ (this gives the structure of a functor). Since $L {\mathbb G}_m$ is an ${{\mathcal Aut}^{\rm c,alg}({\mathcal L})}$-module, we can define the group functor $\mathcal{G}=L{\mathbb G}_m \rtimes{\mathcal Aut}^{\rm c,alg}({\mathcal L})$. Note that $L{\mathbb G}_m$ is a $\mathcal{G}$-module because of the natural morphism ${\mathcal{G} \to {{\mathcal Aut}^{\rm c,alg}({\mathcal L})}}$. We also have a natural continuous action of ${\mathcal G}(A)$ on $A((t))$ such that $(h,\varphi)(f)=h \cdot \varphi(f)$, where $f \in A((t))$, $h \in L{\mathbb G}_m(A)$, and $\varphi \in {\mathcal Aut}^{\rm c,alg}({\mathcal L})(A)$. For any $g_1,g_2\in {\mathcal G}(A)$ there exists $l\in {\mathbb Z}$ such that ${t^{l} A[[t]] \subset g_i(A[[t]])}$ and $g_i(A[[t]]) / t^l A[[t]]$ are projective $A$-modules of finite rank for $i=1$ and $i=2$ (see [4], § 3.2). We obtain the definition of the relative determinant (independent of the choice of $l$ up to canonical isomorphism):
$$
\begin{equation*}
\begin{aligned} \, &\det(g_1(A[[t]]) \mid g_2 (A[[t]])) \\ &\qquad=\operatorname{Hom}_A(\wedge^{\max}_A (g_1(A[[t]])/ t^l A[[t]]), \wedge^{\max}_A (g_2(A[[t]])/ t^l A[[t]])). \end{aligned}
\end{equation*}
\notag
$$
Let $g_1, g_2, g_3, g \in {\mathcal G}(A)$. Then there are canonical isomorphisms of $A$-modules:
$$
\begin{equation*}
\begin{gathered} \, \det(g_1(A[[t]])\,{\mid}\,g_2(A[[t]]))\,{\otimes_A}\,\det(g_2(A[[t]])\,{\mid}\,g_3(A[[t]]))\,{\xrightarrow{\sim}}\, \det(g_1(A[[t]])\,{\mid}\,g_3(A[[t]])), \\ g\colon \det(g_1(A[[t]])\,{\mid}\,g_2(A[[t]])) \xrightarrow{\sim} \det(gg_1(A[[t]])\,{\mid}\,gg_2(A[[t]])). \end{gathered}
\end{equation*}
\notag
$$
Let the group $\widetilde{{\mathcal G}}(A)$ be the set of pairs $(g,s)$ such that $g \in {\mathcal G}(A)$ and $s$ is an element of the $A$-module ${\det(g (A[[t]]) \mid A[[t]])}$ that generates this $A$-module. The group law is as follows: $(g_1,s_1)(g_2,s_2)=(g_1 g_2,g_1(s_2) \otimes s_1)$. The correspondence $A \mapsto \widetilde{{\mathcal G}}(A)$ is a group functor that defines the determinant central extension of ${\mathcal G}$ by ${\mathbb G}_m$, where the homomorphism $\widetilde{{\mathcal G}}(A) \twoheadrightarrow {\mathcal G}(A)$ is defined by $(g,s) \mapsto g$. The Contou-Carrère symbol $\operatorname{CC}$ is a morphism of group functors $ L {\mathbb G}_m \otimes L {\mathbb G}_m \to {\mathbb G}_m$ which has a lot of interesting properties; see [1] and [5], § 2. In particular, $\operatorname{CC}$ is a morphism of $ {\mathcal G}$-modules, where $ {\mathcal G}$ acts diagonally on $ L {\mathbb G}_m \otimes L {\mathbb G}_m $ and trivially on ${\mathbb G}_m$; see [3]. For any two $1$-cocycles $\lambda_1$ and $\lambda_2$ on ${\mathcal G}$ with coefficients in $L{\mathbb G}_m$ we define the $2$-cocycle $\langle \lambda_1,\lambda_2 \rangle= \operatorname{CC} \mathrel{\circ} (\lambda_1 \cup \lambda_2)$ on ${\mathcal G}$ with coefficients in ${\mathbb G }_m$, where $\circ$ means the composition of morphisms of functors. We introduce $1$-cocycles $\Lambda$ and $\Omega$ on ${\mathcal G}$ with coefficients in $L {\mathbb G }_m$, where $\Lambda ((h, \varphi)) =h$ and $\Omega((h,\varphi))={\widetilde{\varphi}}^{\,\prime}=d\varphi(t)/dt$ for $(h,\varphi) \in {\mathcal G}(A)$. Theorem. The determinant central extension of the group functor ${\mathcal G} $ by the group functor ${\mathbb G}_m$ admits a natural section ${\mathcal G} \to \widetilde{\mathcal G}$ as functors, and therefore this central extension yields an element $\mathcal D$ of $H^2({\mathcal G} , {\mathbb G}_m )$. The following equality holds in the group $H^2({\mathcal G}_{\mathbb{Q}},{{\mathbb G}_m}_{\mathbb{Q}})$: Formally, (2) looks like the Deligne–Riemann–Roch isomorphism from [2]. Consider a projective curve $C$ over a field $k$ and a commutative $k$-algebra $A$. Elements of ${\mathcal G}(A)$ reglue the scheme $C_A = C \times_k A$ and the sheaf ${\mathcal O}_{C_A}$ along the punctured formal neighbourhood of a constant section (to a smooth point). Fibres of the determinant central extension over elements of ${\mathcal G}(A)$ are canonically isomorphic to the difference between the determinants of the higher direct images of the reglued sheaves and the sheaf ${\mathcal O}_{C_A}$. 
Citation in format AMSBIB
\Bibitem{Osi23} |
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