Let $k$ be an algebraically closed field and $X$ be a normal variety over $k$.
I am trying to show that there is a surjective group homomorphism $G\_0(X)\rightarrow \mathbb{Z}\oplus \mathrm{Cl}(X)$, by using Quillen’s spectral sequence for the $G$-theory of a Noetherian scheme of finite Krull dimension.
But I am not sure about the reasoning.



Relation between $G_0(X)$ and $\mathrm{Cl}(X)$ for a normal variety $X$
You can construct the homomorphism directly by
$$
F \mapsto (\mathrm{rank}(F), c\_1(F)).
$$
Here $c\_1(F)$ is defined by
$$
c\_1(F) = 
c\_1(F\vert\_{X\_0}) \in 
\mathrm{Pic}(X\_0) = 
\mathrm{Cl}(X),
$$
where $X\_0 \subset X$ is the smooth locus of $X$.


