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

The formal Bott–Thurston cocycle is a 2-cocycle on the group of continuous automorphisms of the ring of Laurent series over a ring with values in the group of invertible elements of this ring, where we consider the natural topology on the ring of Laurent series. This cocycle is a formal analog of the Bott–Thurston 2-cocycle on the group of orientation-preserving diffeomorphisms of the circle. 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 the basic ring contains the field of rational numbers. As a consequence of this result we prove the part of new formal Riemann–Roch theorem for a ringed space over a scheme, where this ringed space is locally isomorphic to the sheaf of rings of Laurent series over the structure sheaf of this scheme.