Let $R$ be a commutative Noetherian ring, and $X=$Spec$(R)$ the associated affine scheme. Let $F$ be a sheaf of $O_X$-modules. Consider the following condition

- (#) For all containments $V \subseteq U$ of affine open subschemes of $X$, the natural map $O(V) \otimes_{O(U)} F(U) \rightarrow F(V)$ of $O(V)$-modules is injective.

One can reduce to the case where $V = D(f)$ where $f\in \Gamma(U,O_X)$.

One of the equivalent conditions for quasi-coherence is that the maps in (#) are *isomorphisms*. Curiously, though, the examples I know of sheaves that are not quasi-coherent also fail the condition (#).

My question is: Are there any (natural) examples of $O_X$-module sheaves that satisfy (#) but fail to be quasi-coherent? And if this is impossible, would the answer be different if the requirement that $X$ be affine were relaxed?

Also, does anyone know a name for this condition?

submodulesof quasicoherent $O_X$-modules? $\endgroup$