Formal Bott–Thurston Cocycle and Part of a Formal Riemann–Roch Theorem

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