Local analog of the Deligne--Riemann--Roch isomorphism for line bundles in relative dimension~1

D. V. Osipov

Izvestiya. Mathematics , Tome 88 (2024) no. 5, pp. 930-976

Résumé

We prove a local analog of the Deligne–Riemann–Roch isomorphism in the case of line bundles and relative dimension $1$. This local analog consists in computation of the class of $12$th power of the determinant central extension of a group ind-scheme $\mathcal G$ by the multiplicative group scheme over $\mathbb Q$ via the product of $2$-cocyles in the second cohomology group. These $2$-cocycles are the compositions of the Contou-Carrère symbol with the $\cup$-product of $1$-cocycles. The group ind-scheme $\mathcal{G}$ represents the functor which assigns to every commutative ring $A$ the group that is the semidirect product of the group $A((t))^*$ of invertible elements of $A((t))$ and the group of continuous $A$-automorphisms of $A$-algebra $A((t))$. The determinant central extension naturally acts on the determinant line bundle on the moduli stack of geometric data (proper quintets). A proper quintet is a collection of a proper family of curves over $\operatorname{Spec} A$, a line bundle on this family, a section of this family, a relative formal parameter at the section, a formal trivialization of the bundle at the section that satisfy further conditions.

Détail

Keywords: determinant central extension, $\cup$-products of $1$-cocycles, determinant linear bundle.
Mots-clés : Deligne–Riemann–Roch isomorphism, Contou-Carrère symbol