I would like to prove the "if" part in the following equivalence:
Given a scheme $X$ and an ideal sheaf $\mathcal{I}$ of $O_X$, then $\mathcal{I}$ is quasi-coherent if and only if $\mathcal{I}$ is locally generated by sections.
By locally generated by sections I mean that $X$ has a cover by open subsets $U_i,\ i\in I$ such that $\mathcal{I}_{|U_i}$ is generated by global sections (i.e. there exists a family of sections $(s_{\alpha})_{\alpha\in A}$ such that for all $x\in U_i,\ \mathcal{I}_x$ is generated as an $O_{X,x}-$module by $s_{\alpha_x},\ \alpha\in A$) and that for all $i\in I$.
Let $U_i,\ i\in I$ be an open cover of $X$ such that $\mathcal{I}_{|U_i}$ is generated by global sections for all $i\in I.$ We can suppose that each one of the open subsets $U_i$ is affine, say $U_i\cong \operatorname{Spec}A_i$. We want to prove that $\mathcal{I}_{|U_i}\cong \widetilde{I_i}$ where $I_i=\mathcal{I}(U_i).$ Let $D(f),\ f\in A_i$ be the basis of distinguished open subsets of $\operatorname{Spec}A_i$. The restriction morphism $I_i=\mathcal{I}(U_i)\rightarrow \mathcal{I}(D(f))$ induces a unique morphism $\varphi(D(f)):\widetilde{I_i}(D(f))=I_{i_f}\rightarrow \mathcal{I}(D(f))$ defined by $\frac{s}{f^r}\mapsto \frac{s_{|D(f)}}{f^r}\ \forall s\in I_i.$ These morphisms $\varphi(D(f))$ define a morphism $\varphi:\widetilde{I_i}\rightarrow \mathcal{I}_{|U_i}$. The corresponding morphisms on stalks are $\varphi_p:I_{i_p}\rightarrow \mathcal{I}_p,\ \frac{t}{s}\mapsto \frac{t_p}{s}\ \forall (t,s)\in I_i\times(A_i\setminus p)$ and they are surjective since $\mathcal{I}_{|U_i}$ is generated by global sections. I'm not sure about the injectivity though, if $\varphi(D(f))(\frac{s}{f^r})=0$ then $f^ms_{|D(f)}=0$ for some $m\in\mathbb{N}$. $s_{D(f)}=s$ in $A_{i_f}$ as $\mathcal{I}$ is an ideal of $O_X$ so $s=0$ in $A_{i_f}.$ Is that right ?
Thank you for your help!
Best Answer
I think your proof is correct, since you can use that $\mathcal{I}$ is a subsheaf of $\mathcal{O}_X$ to show injectivity.
Alternativey, you can use exactness properties. Since quasi-coherence is local, assume that $X = U_i$ so that $\mathcal{I}$ is generated by global sections(indexed by $A$). Then, there is a surjective morphism of $\mathcal{O}_X$-modules $\mathcal{O}_X^{\oplus A} \to \mathcal{I}$. Taking the composite with the inclusion $\mathcal{I} \hookrightarrow \mathcal{O}_X$ yields $f: \mathcal{O}_X^{\oplus A} \to \mathcal{O}_X$, which is a morphism of quasi-coherent sheaves with image $\mathcal{I}.$
Such sheaves are always quasi-coherent since $\operatorname{Im}f \cong \operatorname{ker}(\operatorname{coker}(f))$, and quasi-coherence is closed under kernels and cokernels.