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?
Best Answer
Any submodule of a quasicoherent $O_X$-module satisfies (#): this is clear via reduction to principal open sets, and the fact that localization is exact. More generally, as Neil observes, if $F$ satisfies (#) then so does every submodule of $F$.
For instance, if $F$ is quasicoherent on $X$ and $j:U\hookrightarrow X$ is open, then the extension by zero $j_!(F_{\mid U})$ satisfies (#) but is not quasicoherent in general.