Формальный коцикл Ботта–Тёрстона и часть формальной теоремы Римана–Роха
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 the 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 Q-algebra. As a consequence of this result we prove a part of a new formal Riemann–Roch theorem. This Riemann–Roch theorem is applied to a ringed space on a separated scheme S over Q, where the structure sheaf of the ringed space is locally on S isomorphic to the sheaf OS((t)) and the transition automorphisms are continuous. Locally on S this ringed space corresponds to the punctured formal neighborhood of a section of a smooth morphism to U of relative dimension 1, where U⊂S is an open subset.