Derived functor of global sections of coherent sheaves - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T10:01:45Z http://mathoverflow.net/feeds/question/73217 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73217/derived-functor-of-global-sections-of-coherent-sheaves Derived functor of global sections of coherent sheaves Descartes 2011-08-19T12:27:32Z 2011-08-21T14:08:10Z <p>Hi,</p> <p>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.</p> <p>I consider the left exact functor</p> <p>$\Gamma: Qcoh(X) \rightarrow Vec(k)$</p> <p>Then, as $Qcoh(X)$ has enough Injectives one has a right derived functor</p> <p>$R\Gamma: D^{+}(Qcoh(X)) \rightarrow D^{+}(Vec(k))$.</p> <p>My question: how can I get the right derived functor of</p> <p>$\Gamma: Coh(X) \rightarrow Vec_{f}(k)$?</p> <p>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.</p> <p>Of course one first thinks of just composing</p> <p>$D^{+}(Coh(X)) \rightarrow D^{+}(Qcoh(X)) \rightarrow D^{+}(Vec(k))$,</p> <p>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.)</p> <p>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.</p> <p>Thanks a lot!</p>