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I notice that some books say that for arbitrary quasi-coherent sheaves $F$, $G$ over a scheme $X$, the $\mathcal O_X$-module $\mathrm{Hom}_{\mathcal O_X}(F,G)$ maybe not quasi-coherent, who can give me a counterexample?

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Consider an affine scheme the spectrum of a DVR $M$, then $N=\oplus_{\mathbb{N}}M$ and $M$ are q.c. and via 4.21, Algebraic_Geometry:_Sheaves_and_cohomology $Hom_{O_{spec M}} (N,M)$ is q.c. iff isomorphic to its associated sheaf. Localizing at a non-unit should preserve this ismorphism via Hartshorne II.5.1.c, but it doesn't.

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I do not understand why "Localizing at a non-unit should preserve this ismorphism", is it possible that Hom(N,M) is q.c. but not induced by the Hom functor? –  Strongart Apr 4 at 12:12
    
I'll use the notation of Hartshorne II.5.1. Think of $Hom_{O_X}(N,M)$ as $\tilde{A}$ for some associated sheafand $M$-module $A$. Then by (d), $\Gamma(\tilde{A}) = A$ and by (d) $\Gamma(\tilde{A})_f = A_f$ on the other hand, by (c) this should be equal to $Hom_{O_X}(N,M)(D(f)) = Hom_{O_X}(N(D(f)), M(D(f)))=Hom_{O_X}(N_f, M_f)$ Thus in total, we should have $Hom_{O_X}(N,M)_f=Hom_{O_X}(N_f, M_f)$, but there might be maps in the latter which don't exist in the former. –  aegbert Apr 4 at 14:07
    
Thanks, I see. Maybe the Hartshorne II.5.5 is also helpful. –  Strongart Apr 5 at 11:24
    
Yep II.5.5 also applies. –  aegbert Apr 5 at 12:15

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