Geodesic
Central extensions and the Riemann-Roch theorem on algebraic surfaces
Sbornik. Mathematics, Tome 213 (2022) no. 5, pp. 671-693 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $X$ (without Noether's formula). Bibliography: 21 titles.
Keywords: central extensions, ring of adeles on an algebraic surface, locally free sheaves, Riemann-Roch theorem. 
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D. V. Osipov. Central extensions and the Riemann-Roch theorem on algebraic surfaces. Sbornik. Mathematics, Tome 213 (2022) no. 5, pp. 671-693. http://geodesic.mathdoc.fr/item/SM_2022_213_5_a5/

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