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

Author:

D. V. Osipov

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Representation Theory (math.RT)

journal:

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date:

2023-08-10 16:00:00

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 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.