Central extensions and Riemann-Roch theorem on algebraic surfaces

Abstract

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).

25 pages; minor chnages; to appear in Sbornik: Mathematics

References in corpus (4)

• Representations of Higher Adelic Groups and Arithmetic
• Harmonic analysis on local fields and adelic spaces I
• Unramified two-dimensional Langlands correspondence
• Harmonic analysis and the Riemann-Roch theorem