Central extensions and Riemann-Roch theorem on algebraic surfaces

We study canonical central extensions of the general linear group over the ring of adeles on a smooth projective algebraic surface X by means of the group of integers. By these central extensions and adelic transition matrices of a rank n locally free sheaf of O X -modules we obtain the local (adelic) decomposition for the difference of Euler characteristics of this sheaf and the sheaf O nX . Two various calculations of this difference lead to the Riemann-Roch theorem on X (without the Noether formula).