I am a beginner at derived categories and I'm looking for a proof of the following fact:
If $X$ and $Y$ are smooth projective curves such that $D^b(Coh\,X)$ is equivalent to $D^b(Coh\,Y)$ then $X$ and $Y$ are isomorphic.
Can anyone give me the explanation of this fact or provide a good reference for this?
As I understand, the crucial point here is that any object in $D^b(Coh \, X)$ splits as a sum of its cohomology sheaves but I don't know how to finish the proof.
Thank you.