  For an algebraic variety $X$ we introduce generalized first Chern classes,
which are defined for coherent sheaves on $X$ with support in codimension $p$
and take values in $CH^p(X)$. We use them to provide an explicit formula for
the differentials ${d_2^p: E_2^{p,-p-1} \to E_2^{p+2, -p-2} \cong CH^{p+2}(X)}$
in the Quillen spectral sequence.
