I'm trying to show that the functor $\Gamma(X,-)$ from the category of quasi-coherent sheaves maps a quasi-coherent sheaf to an R-Module, and also that for coherent sheaves the same functor takes the sheaf to a finitely generated R-Module (where the sheaf is defined over a Noetherian affine scheme.)

However I'm struggling to see why this is true.

I have that for the structure sheaf we can set $X=Spec(R)=U_1=Spec(R)$ \ (V(1)). Then $O_X(X)=R_1=R.$ Where $R_1$ is the localisation at 1.

Hence $\Gamma(X, O_X)=R$ for $X=Spec(R)$

But am stuck as to where to go from here to do the case for quasi-coherent/coherent sheaves over a Noetherian affine scheme. Can anybody help?

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    $\begingroup$ There must be something funky about your definitions. If I have a module $\mathcal{F}$ on a ringed space $X$ then part of the definition is that $\mathcal{F}(X)$ is an $\mathcal{O}_X(X)$-module, and in your case $\mathcal{O}_X(X) = R$. $\endgroup$ – Hoot Dec 31 '15 at 13:38
  • $\begingroup$ For any module or quasi-coherent/coherent ? Which book are you getting definitions from ? That sounds helpful $\endgroup$ – Joe Jan 3 '16 at 12:48

There are several equivalent definitions of coherent sheaves and as the comment pointed out, some of those trivially addresses aspects of your question.

If you've defined (quasi)-coherent sheaves on affine schemes to be the $\mathcal{O}_X$ module $\tilde{M}$ satisfying $\tilde{M} (D(f)) = M_f$, as in Liu's book, then this is immediately verified.

If you wish to proceed as in Hartshorne, where $\tilde{M}$ is locally defined to be the set of functions from your open set to all the stalks of points in that open set which is locally a fraction, then you need to do more work. Hartshorne covers this in Proposition II.5.4 which immediately gives this result and more, which is in Corollary II.5.5. The arguments, including that of Lemma II.5.3 is an algebraic geometry version of the partitions of unity argument commonly seen in differential topology. As such, you may wish to read at least one of the proofs.

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  • $\begingroup$ Thankyou, that is helpful. So if i were to follow Hartshorne, and not wanting to establish a complete equivalence of categories, but rather just show that for a Noetherian affine scheme and a quasi-coherent sheaf on it that $\Gamma(X,-)$ was indeed an R-module, what would such a proof look like ? Sorry to be specific, but i'm a bit out of my comfort zone with it all at the moment $\endgroup$ – Joe Jan 3 '16 at 12:25
  • $\begingroup$ I think all I have left is the last equivalence of $\Gamma(X,F) = \Gamma(X,\tilde M) = M$ $\endgroup$ – Joe Jan 3 '16 at 15:12
  • $\begingroup$ I explained the idea in my answer, namely Proposition II.5.4. $\endgroup$ – Future Jan 3 '16 at 15:57

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