Central extensions and the 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. Via these central extensions and the adelic transition matrices of a rank n locally free sheaf of O-X-modules we obtain a local (adelic) decomposition for the difference of Euler characteristics of this sheaf and the sheaf O-X(n). Two distinct calculations of this difference lead to the Riemann-Roch theorem on X (without Noether's formula). Bibliography: 21 titles.