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This question has been completely reformulated following the guidelines in the comments below, to make it clearer where I was able to get and where I can't get out of. Such comments helped me a lot.

Let $(X,\mathscr{O}_{X})$ be a ringed space. Let $\mathscr{F}$ be a sheaf of $\mathscr{O}_{X}$-modules locally free.

Show that $\mathscr{F}$ is a quasi-coherent sheaf of $\mathscr{O}_{X}$-modules.

I've been thinking about the following: $\mathscr{F}$ locally free $\Rightarrow$ $X$ can be covered by open sets $U$ for which $\mathscr{F}|_{U}\cong \bigoplus_{i \in I}\mathscr{O}_{X}|_{U}$, for some set $I$.

As with any $ X $ scheme, the structure sheaf $\mathscr{O}_{X}$ is quasi-coherent, then $X$ can be covered by open affine subsets $U_j=$Spec $A_j$, such that for each $j$ there is an $A_j$-module $M_j$ with $\mathscr{O}_{X}|_{U_j} \cong \widetilde{M_j}$.

So, we have to: $\mathscr{F}|_{U \cap U_j}\cong \bigoplus_{i \in I}\mathscr{O}_{X}|_{U\cap U_j}\cong \bigoplus_{j \in I} \widetilde{M_j}\cong \widetilde{(\bigoplus_{j \in I} M_j)}$. Thus, $\mathscr{F}$is quasi-coherent.

But now I have a question. A quasi-coherent sheaf $\mathscr{F}$ is primarily a sheaf of $\mathscr{O}_X$-modules. So so that I can see that $\mathscr{O}_X$ is quasi-coherent, I have to look at $ \mathscr{O}_X(U)$ as $ (\mathscr{O}_X(U), +)$ a group and therefore a $\mathscr{O}_X(U)$ - module, for each open $U$ in $X$. To make sense of the definition of quasi-coherent in sheaf of rings $\mathscr{O}_{X} $.

Is that correct?

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  • $\begingroup$ Assuming I'm reading this right and you're asking why a locally free sheaf is quasi-coherent, this should follow directly from the definitions. What have you tried and where are you stuck? $\endgroup$ – KReiser Nov 24 '20 at 2:05
  • $\begingroup$ I don't ask the questions here without first thinking about what to do, be sure. But I was nowhere near able to see how to move from open coverage that defines a sheaf locally free to coverage that meets the definition of quasi-coherent sheaf. Is it that obvious? $\endgroup$ – Manoel Nov 24 '20 at 2:14
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    $\begingroup$ If you don't write down what you've tried and how that hasn't worked, it's impossible to distinguish your question from someone who hasn't done that work. It's also harder for you to get help that directly addresses the place where you are stuck. For the specifics of this problem, it should be direct from the definitions. Why don't you add your definitions in an edit and explain what's not making sense to you - I think this would greatly increase your chances of getting a helpful answer. $\endgroup$ – KReiser Nov 24 '20 at 2:24
  • $\begingroup$ I'll do it. Thanks :) $\endgroup$ – Manoel Nov 24 '20 at 2:29
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    $\begingroup$ @Manoel while I certainly agree with KReiser that you should expand upon your question, I also agree with him that this should follow from definitions. If you keep the following slogans in mind, this may become apparent: "On an affine cover, a locally free sheaf restricts to sheafifications of free modules" and "On an affine cover, a quasicoherent sheaf restricts to sheafifications of arbitrary modules." $\endgroup$ – Tabes Bridges Nov 24 '20 at 5:43
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You say at first that $(X,\mathcal{O}_X)$ is a locally ringed space, but then talk about schemes afterwards. Note that there are many locally ringed spaces which are far from being schemes.

The most likely definition of quasicoherent sheaf on a locally ringed space is a sheaf $\mathcal{F}$ so that there is an open cover $\{U_i\}_{i\in I}$ of $X$ so that over each $U_i$, we have a (possibly infinite) presentation $$ \mathcal{O}_{U_i}^{\oplus I}\to\mathcal{O}_{U_i}^{\oplus J}\to \mathcal{F}\to 0.$$ Notice that a locally free sheaf $\mathcal{E}$ is of course of this form as there is an open cover $\{U_i\}_{i\in I}$ of the space so that on each $U_i$ $$ 0\to \mathcal{O}_{U_i}^{\oplus J}\xrightarrow{\sim} \mathcal{F}\to 0.$$ In particular, locally free implies quasicoherent.

In the context of schemes, we can define being quasicoherent as being locally of the form $\widetilde{M}_i$ for $M_i$ an $A_i$-module with respect to an open cover $\{\operatorname{spec} A_i\}_{i\in I}$ of the scheme $X$. Then a locally free sheaf can be locally written as $\widetilde{A^{\oplus I}}=\widetilde{A}^{\oplus I}\cong \mathcal{O}_{\operatorname{spec}A_i}^{\oplus I}$ over $\operatorname{spec}A_i$. In particular, a locally free sheaf is a fortiori a quasicoherent sheaf.

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