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

TL;DR

This paper establishes a local version of the Deligne-Riemann-Roch isomorphism for line bundles over relative dimension 1, involving computations of classes of determinant central extensions via cocycles and symbols.

Contribution

It introduces a local analog of the Deligne-Riemann-Roch isomorphism for line bundles in relative dimension 1, with explicit cocycle computations of determinant extensions.

Findings

Computed the class of the 12th power of the determinant central extension.

Expressed the class via 2-cocycles from the Contou-Carrère symbol and cup product.

Connected the extension to the action on the determinant line bundle over moduli stacks.

Abstract

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-Carr\`{e}re 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 $\mathop{\rm Spec} A$, 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.