Tensor product of sheaves separated?

Question

Let $(X, O_X)$ be an arbitrary scheme and $\mathcal{F}$ a presheaf of $O_X$-modules. The presheaf $\mathcal{F}$ is said to be separated if the natural map to it's sheafification $\mathcal{F}^+$ is an injection. That is, for each open set $U$, $\mathcal{F}(U) \hookrightarrow \mathcal{F}^+(U)$.

Let $\mathcal{F}, \mathcal{G}$ be two sheaves of $O_X$ modules. Is the presheaf tensor product $\mathcal{F} \otimes_{O_X} \mathcal{G}$ a separated presheaf? Is this true more generally if $(X,O_X)$ is just a ringed space?

This tells you for example, that if $\mathcal{F}, \mathcal{G}$ have some global sections then so does their sheaf tensor product (assuming the presheaf tensor product has global sections).

I wasn't able to find this in the stacks project. Is this proved in books on sheaf theory?

Answer 1

If I understand correctly your question, then the answer is "no, the presheaf tensor product need not be separated." Let $(X,\mathcal{O}_X)$ be the projective line $(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1})$ over a base ring $R$. Let $\mathcal{F}$ and $\mathcal{G}$ both be the invertible sheaf $\mathcal{O}_{\mathbb{P}^1}(1)$. Since these are sheaves, they are separated. Consider the presheaf tensor product $\mathcal{E}$. In particular, the space of global sections of this presheaf is $$ H^0(\mathbb{P}^1,\mathcal{E}) = H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(1))\otimes_R H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(1)).$$ This is a free $R$-module of rank $4$. On the other hand, the sheafification of the presheaf tensor product is $\mathcal{O}_{\mathbb{P}^1}(2)$, whose space of global sections is a free $R$-module of rank $3$.