Formal Bott-Thurston cocycle and part of a formal Riemann-Roch theorem

TL;DR

This paper introduces a formal analog of the Bott-Thurston cocycle, relates it to determinantal extensions, and proves a partial formal Riemann-Roch theorem with applications to formal neighborhoods in algebraic geometry.

Contribution

It defines a formal Bott-Thurston cocycle, establishes its equivalence to a multiple of the determinantal extension, and derives a partial formal Riemann-Roch theorem for schemes over al.

Findings

Formal Bott-Thurston cocycle is equivalent to 12-fold Baer sum of determinantal extension.

Proves a partial formal Riemann-Roch theorem.

Applications to ringed spaces and formal neighborhoods in algebraic geometry.

Abstract

The Bott-Thurston cocycle is a $2$-cocycle on the group of orientation-preserving diffeomorphisms of the circle. We introduce and study a formal analog of Bott-Thurston cocycle. The formal Bott-Thurston cocycle is a $2$-cocycle on the group of continuous $A$-automorphisms of the algebra $A((t))$ of Laurent series over a commutative ring $A$ with values in the group $A^*$ of invertible elements of $A$. We prove that the central extension given by the formal Bott-Thurston cocycle is equivalent to the $12$-fold Baer sum of the determinantal central extension when $A$ is a $\mathbb Q$-algebra. As a consequence of this result we prove a part of new formal Riemann-Roch theorem. This Riemann-Roch theorem is applied to a ringed space on a separated scheme $S$ over $\mathbb Q$, where the structure sheaf of the ringed space is locally on $S$ isomorphic to the sheaf ${\mathcal O}_S((t))$ and the transition automorphisms are continuous. Locally on $S$ this ringed space corresponds to the punctured formal neighbourhood of a section of a smooth morphism to $U$ of relative dimension $1$, where an open subset $U \subset S$.