Let $X$ be a scheme (quasi-projective over a field if it helps). Let $\mathcal F$ and $\mathcal G$ be $\mathcal O_X$-modules and $U$ an open subset of $X$ such that $\mathcal F(U)$ and $\mathcal G(U)$ are $\mathcal O_X(U)$-modules of finite type. Let $\mathcal F \otimes_{\mathcal O_X}\mathcal G$ be the tensor product sheaf. Is $(\mathcal F \otimes_{\mathcal O_X}\mathcal G)(U)$ an $\mathcal O_X(U)$-module of finite type?

  • $\begingroup$ You certainly like this kind of question! Anyway, I think you can do something dumb like take $X = U = \mathbb P^1_k$, $\mathcal{F} = \bigoplus_{n\in\mathbb Z} \mathcal{O}(-1)$, $\mathcal{G} = \mathcal{O}(1)$, so coherence seems helpful. $\endgroup$ – Hoot Jun 1 '16 at 22:25
  • $\begingroup$ Even though $F$ is not coherent, that answers completely what I had in mind. Thank you (I will edit the question removing the coherence so that it is completely answered by your comment and accept it if you want to write it as an answer). $\endgroup$ – A.G Jun 1 '16 at 22:35

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