Abstract: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 $\mathcal{O}_X$-modules we obtain a local (adelic) decomposition for the difference of Euler characteristics of this sheaf and the sheaf $\mathcal{O}_X^n$. Two distinct calculations of this difference lead to the Riemann-Roch theorem on $X$ (wit… Show more
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