  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.
