An adelic construction of Chern classes

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.