$\newcommand{\ch}{\mathrm{ch}}$Suppose I have a two-term complex $E:= (E^{-1} \xrightarrow{d} E^0)$ in $D^b(X)$ the derived category of coherent sheaves on some variety $X$. Suppose I know the Chern characters $\ch(E)$, $\ch(\ker d)$, and $\ch ( \mathrm{coker} \, d)$. Does this tell my anything about the Chern characters $\ch(E^i)$ where $i=-1,0$?
Having thought about it for a while, it doesn't seem to me that this would tell me anything about the Chern characters I'm interested in, but perhaps I've missed something simple. The only thing I can think to write down is that $\ch(E) = \ch(E^0)-\ch(E^{-1})$, but I'm unsure how to relate back to the map $d$.
Thanks.