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Let $(X, \mathcal{O}_{X})$ be a ringed-space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_{X}$-modules. For any section $s \in \Gamma(X, \mathcal{F})$, we define a morphism, $$ \mathcal{O}_{X} \longrightarrow \mathcal{F} $$ by $f \mapsto f \cdot s|_{U}$ for every open set $U \subseteq X$ with $f \in \mathcal{O}_{X}(U)$. Conversely, any morphism of sheaves $\varphi: \mathcal{O}_{X} \rightarrow X$ as above can be realized in this way by choosing the section $\varphi(1)$. Now if $I$ is any indexing set, any morphism $$ \phi: \mathcal{O}_{X}^{(I)} \longrightarrow \mathcal{F} $$ is determined by a component family, $$ \{ \varphi_{i}: \mathcal{O}_{X} \longrightarrow \mathcal{F} \}_{i \in I} $$ In turn, as described above, we then get sections $s_{i} = \varphi_{i}(1)$ which determine each of these morphisms. But since $\mathcal{O}_{X}$ is a generator in the category of sheaves of $\mathcal{O}_{X}$-modules, we can always find an index set $I$ so that there is a surjection, $$ \phi: \mathcal{O}_{X}^{(I)} \longrightarrow \mathcal{F} $$ as above. But then we could choose out sections like we just did and suddenly we have that $\mathcal{F}$ is generated by those global sections.

Where is the flaw in this reasoning? I know it is obviously wrong because we could find $\mathcal{F}$ having no non-trivial global sections at all, despite being a non-zero sheaf.

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    $\begingroup$ "But since $\mathcal{O}_X$ ia a generator in the category of sheaves of $\mathcal{O}_X$-modules"? $\endgroup$ – Lord Shark the Unknown Feb 10 '18 at 17:04
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    $\begingroup$ @LordSharktheUnknown are you saying this is untrue, or just not sure of what I mean by this statement? $\endgroup$ – Luke Feb 10 '18 at 17:08
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As suggested in the comment by Lord Shark the Unknown, $ \mathcal{O}_X $ is not a generator of the category of sheaves of $ \mathcal{O}_X $-modules though it has a generator, namely $ \bigoplus_{U: \text{ open}} \mathcal{O}_U $.

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  • $\begingroup$ So I've had another look at how generators are defined in Tohoku and you are right, that was a stupid mistake on my part. But even putting that aside, given a sheaf of $\mathcal{O}_{X}$-modules, can you not always find a surjective morphism from some direct sum of copies $\mathcal{O}_{X}$ onto $\mathcal{F}$? $\endgroup$ – Luke Feb 10 '18 at 17:28
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    $\begingroup$ Actually, such a property would \emph{make} it a generator. I guess I have had some major misconceptions for a while. Thanks to you and @Lord Shark the Unknown for the help! $\endgroup$ – Luke Feb 10 '18 at 17:31
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    $\begingroup$ @Luke, you cannot: take O(-1) on the projective line and find its global sections. $\endgroup$ – Mariano Suárez-Álvarez Feb 10 '18 at 17:31
  • $\begingroup$ @Luke Hint: You can think a map O -> F as a global section of F. $\endgroup$ – B. W. Feb 10 '18 at 17:37

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