Let $\varepsilon: 0\to A\to B \to C\to 0$ be an exact sequence of ${\cal O}_X$-modules with $X$ a quasi-compact space. $\varepsilon$ is called pure if the induced sequence
$0\rightarrow Hom(F,A)\rightarrow Hom (F,B)\rightarrow Hom(F,C)\rightarrow 0$ (with $Hom$ a functor) is exact for each finitely presented ${\cal O}_X$-module $F$. Do we get the same exact sequence with Hom replace with shef hom?
I have also another question: Is there any relation between a finitely presented sheaf of ${\cal O}_X$-modules and finitely presented quasi-coherent sheaves? It seems that a finitely presented sheaf is quasi-coherent, so what is a finitely presented quasi-coherent sheaf?
Best Answer
If $F$ is finitely presented, then localization commutes with $Hom(F,-)$. Since a sequence is exact iff it is exact stalkwise, the result follows.