In Blochs "Cycles on arithmetic schemes and Euler characteristics of curves" he defines the bivariant chern class for bundles on a scheme.
Now I wanted to calculate a bivariant chern class with $n=1$ for what I need to calculate the first chern class $c_1(\xi)$, where $$\xi=pr_0^*\xi_0-pr_1^*\xi_1=G\times_{G_0} E_0-G\times_G \xi_1=G\times_Y E_0-\xi_1$$ for the canonical rank-$e_i$-subbundles $\xi_0=E_0$ and $\xi_1$.
Question: How can I find a more explicit (as explicit as possible) representation of $c_1(\xi$)? I want to calculate $deg(c_1(\xi))$. Any hint can help!
I read parts of Fultons "Intersection theory", where Fulton defines the bivariant chern class in a slightly more general setting, but it did not help that much.
(Notation: He considers a scheme Y of finite type over a regular noetherian base, a closed subscheme $X\subset Y$, a two-term-complex $E_1\rightarrow E_0$ of vector bundles and $e_i=rk(E_i)$. $G=Grass_{e_1}(E_1\oplus E_0)$ is the Grassmannian of rank-$e_1$-subbundles of $E_1\oplus E_0$, $pr_0:G\rightarrow G_0=Grass_{e_0}(E_0)=Y$ the structure map and $pr_1=id:G\rightarrow G$.)