Quasi-coherent sheaves $\supset$ locally free sheaves?

Question

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?

Answer 1

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.