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?


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.

| cite | improve this answer | |
  • $\begingroup$ 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? $\endgroup$ – Strongart Apr 4 '14 at 12:12
  • 1
    $\begingroup$ 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. $\endgroup$ – aegbert Apr 4 '14 at 14:07
  • $\begingroup$ Thanks, I see. Maybe the Hartshorne II.5.5 is also helpful. $\endgroup$ – Strongart Apr 5 '14 at 11:24
  • $\begingroup$ Yep II.5.5 also applies. $\endgroup$ – aegbert Apr 5 '14 at 12:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.