Russian Mathematical Surveys, 2023, Volume 78, Issue 4, Pages 791–793 DOI: https://doi.org/10.4213/rm10138e (Mi rm10138)

DOI: https://doi.org/10.4213/rm10138e

Funding agency	Grant number
Russian Science Foundation	23-11-00033
This work was supported by the Russian Science Foundation under grant no. 23-11-00033, https://rscf.ru/en/project/23-11-00033/.

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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

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):

Let $g_1, g_2, g_3, g \in {\mathcal G}(A)$. Then there are canonical isomorphisms of $A$-modules:

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 $(h,\varphi) \in {\mathcal G}(A)$.

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 ${\mathcal O}_{C_A}$.

Bibliography
1.	C. Contou-Carrère, C. R. Acad. Sci. Paris Sér. I Math., 318:8 (1994), 743–746
2.	P. Deligne, Current trends in arithmetical algebraic geometry (Arcata, CA 1985), Contemp. Math., 67, Amer. Math. Soc., Providence, RI, 1987, 93–177
3.	S. O. Gorchinskiy and D. V. Osipov, Funct. Anal. Appl., 50:4 (2016), 268–280
4.	D. V. Osipov, Proc. Steklov Inst. Math., 320 (2023), 226–257
5.	D. Osipov and Xinwen Zhu, J. Algebraic Geom., 25:4 (2016), 703–774

Citation in format AMSBIB

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\by D.~V.~Osipov
\paper Determinant central extension and $\cup$-products of 1-cocycles
\jour Russian Math. Surveys
\yr 2023
\vol 78
\issue 4
\pages 791--793
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\crossref{https://doi.org/10.4213/rm10138e}
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