Central extensions and the Riemann-Roch theorem on algebraic surfaces

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