Geodesic
Arithmetic surfaces and adelic quotient groups
Izvestiya. Mathematics , Tome 82 (2018) no. 4, pp. 817-836

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We explicitly calculate an arithmetic adelic quotient group for a locally free sheaf on an arithmetic surface when the fibre over the infinite point of the base is taken into account. The result is stated in the form of a short exact sequence. We relate the last term of this sequence to the projective limit of groups which are finite direct products of copies of the one-dimensional real torus and are connected with the first cohomology groups of locally free sheaves on the arithmetic surface.
Keywords: arithmetic surface, Parshin–Beilinson adeles, arithmetic adeles. 
@article{IM2_2018_82_4_a4,
     author = {D. V. Osipov},
     title = {Arithmetic surfaces and adelic quotient groups},
     journal = {Izvestiya. Mathematics },
     pages = {817--836},
     publisher = {mathdoc},
     volume = {82},
     number = {4},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2018_82_4_a4/}
}
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D. V. Osipov. Arithmetic surfaces and adelic quotient groups. Izvestiya. Mathematics , Tome 82 (2018) no. 4, pp. 817-836. http://geodesic.mathdoc.fr/item/IM2_2018_82_4_a4/