If $X$ is a scheme, is it always possible to find a basis $\mathcal{B}$ for the topology of $X$ (for example, the affine open subsets) with the following property?

For every quasi-coherent sheaf $M$ on $X$, every basic-open subset $U \in \mathcal{B}$ and every local section $s \in \Gamma(U,M)$ there is a homomorphism $$\alpha : M \to M'$$ into a quasi-coherent sheaf such that

- $\alpha|_{U}$ is injective,
- the image $\alpha(s) \in \Gamma(U,M')$ lifts to a global section in $\Gamma(X,M')$.

This is correct when $X$ is quasi-separated: Take $\mathcal{B}$ to be the quasi-compact open subsets of $X$ and let $M' = j_* j^* M$, where $j$ denotes the inclusion $U \hookrightarrow X$. We need quasi-separatedness in this approach because this implies that the direct image $j_*$ preserves quasi-coherence. In general, there is still a right adjoint of $j^*$, namely the quasi-coherator of $j_*$, but it lacks the necessary properties to carry out the argument. But this does not exclude other approaches which could work for general schemes.

A standard example of a scheme which is not quasi-separated is $X=X_1 \sqcup_U X_2$, where $X_1=X_2$ is a scheme which has an open subset $U$ which is not quasi-compact; for example, $X_1=\mathrm{Spec}(k[x_1,x_2,\dotsc])$ and $U=D(x_1) \cup D(x_2) \cup \dotsc$. But here the property is trivially satisfied, since we may take $\alpha=\mathrm{id}_M$.