# Does every sheaf embed into a quasicoherent sheaf?

Question. Let $$X$$ be a scheme. Let $$\mathcal{E}$$ be a sheaf of $$\mathcal{O}_X$$-modules. Is there always a quasicoherent sheaf $$\mathcal{E}'$$ together with a monomorphism $$\mathcal{E} \to \mathcal{E}'$$?

Remark. The coherator yields a way to find a quasicoherent sheaf together with a morphism to $$\mathcal{E}$$. But I'm interested in finding a quasicoherent sheaf together with a monomorphism from $$\mathcal{E}$$.

Motivation. There is a way to set up the theory of sheaf cohomology for quasicoherent sheaves without injective or flabby resolutions. If any sheaf of modules would embed into a quasicoherent one, we might be able to extend this development to arbitrary (not necessarily quasicoherent) sheaves of modules.

That already fails for $$X$$ equal to $$\text{Spec}\ R$$, where $$R$$ is a DVR with generic point $$\eta = \text{Spec}\ K$$. Since there are only two nonempty open subsets of $$X$$, namely all of $$X$$ and $$\{\eta\}$$, there is a straightforward equivalence between the category of $$\mathcal{O}_X$$-modules and the category of triples $$(M,V,\phi)$$ of an $$R$$-module $$M$$, a $$K$$-module $$V$$, and an $$R$$-module homomorphism $$\phi:M\to V.$$ This is quasi-coherent if and only if $$\phi$$ induces an isomorphism $$M\otimes_R K \xrightarrow{\cong} V,$$ i.e., the $$\mathcal{O}_X$$-module is equivalent to $$(M,M\otimes_R K,\iota_M).$$ In particular, consider the $$\mathcal{O}_X$$-module $$(R,\{0\},0).$$ For every $$\mathcal{O}_X$$-module homomorphism of this $$\mathcal{O}_X$$-module to a quasi-coherent $$\mathcal{O}_X$$-module, $$(\psi_R,\psi_\eta):(R,\{0\},0) \to (M,M\otimes_R K,\iota_M),$$ the composite $$\iota_M\circ \psi_R$$ equals $$0$$. Thus, the image $$\psi_R(R)$$ is contained in the torsion submodule of $$M$$. Every torsion quotient of $$R$$ has nonzero kernel. Thus, $$(\psi_R,\psi_\eta)$$ is not a monomorphism.