  We give a construction of the second Chern number of a vector bundle over a
smooth projective surface by means of adelic transition matrices for the vector
bundle. The construction does not use an algebraic $K$-theory and depends on
the canonical $\mathbb{Z}$-torsor of a locally linearly compact $k$-vector
space. Analogs of certain auxiliary results for the case of an arithmetic
surface are also discussed.
