Hi,

I want to consider a scheme $X$ which is proper over a field $k$. With $Qcoh(X)$ resp. $Coh(X)$ I mean the abelian category of quasicoherent resp. coherent sheaves on $X$. With $Vec(k)$ resp. $Vec_{f}(k)$ I mean the category of k-vector spaces resp. finitedimensional k-vector-spaces.

I consider the left exact functor

$\Gamma: Qcoh(X) \rightarrow Vec(k)$

Then, as $Qcoh(X)$ has enough Injectives one has a right derived functor

$R\Gamma: D^{+}(Qcoh(X)) \rightarrow D^{+}(Vec(k))$.

My question: how can I get the right derived functor of

$\Gamma: Coh(X) \rightarrow Vec_{f}(k)$?

I know that it exists, it is constantly used e.g. in Huybrechts book about Fourier-Mukai. But I don't see how you get it.

Of course one first thinks of just composing

$D^{+}(Coh(X)) \rightarrow D^{+}(Qcoh(X)) \rightarrow D^{+}(Vec(k))$,

but who tells me that this will satisfy the universal property for derived functors? (That you land in the finitedimensional vecs is not that important for me.)

As a stimulus see discussion after Theorem 3.21 in Huybrechts' book about Fourier-Mukai. As I say below in a comment I can't cope with his treatment at that point.

Thanks a lot!