Geodesic
An adelic construction of Chern classes
Sbornik. Mathematics, Tome 202 (2011) no. 11, pp. 1637-1659

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We give a formula expressing the second Chern class $c_2(E)$ in terms of trivializations of a rank two vector bundle $E$ at scheme points of a surface $X$ over a field. To do this, starting with these trivializations, we construct a cocycle in the adelic complex associated with the sheaf $\operatorname{K}_2(\mathscr O_X)$. Furthermore we prove that the Severi formula for the second Chern class is obtained as a special case of the formula constructed in this work. Bibliography: 10 titles.
Keywords: Chern class, adelic complex. 
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     author = {R. Ya. Budylin},
     title = {An adelic construction of {Chern} classes},
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R. Ya. Budylin. An adelic construction of Chern classes. Sbornik. Mathematics, Tome 202 (2011) no. 11, pp. 1637-1659. http://geodesic.mathdoc.fr/item/SM_2011_202_11_a3/