  We study canonical central extensions of the general linear group of 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 ${\mathcal O}_X$-modules we obtain the local (adelic)
decomposition for the difference of Euler characteristics of this sheaf and the
sheaf ${\mathcal O}_X^n$. Two various calculations of this difference lead to
the Riemann-Roch theorem on $X$ (without the Noether formula).
