  In this paper, we calculate the differential $d^1$ of the rank spectral
sequence. We generalize Quillen's spectral sequence from Dedekind domain to
general integral Noetherian ring $A$ by considering the Q-construction
$Q^{tf}(A)$ of the category of finitely generated torsion-free modules. In
particular, by resolution theorem, if $A$ is regular, then $Q^{tf}(A)$ is
homotopy equivalent to $QP(A)$ where $P(A)$ is the category of finitely
generated projective $A$-modules. We deduce the differential $d^1$ by using
Tits buildings, Steinberg modules, and modular symbols in the sense of
Ash-Rudolph. The spirit of Quillen's categorical homotopy theory will be used
intensively throughout this paper.
