If $X$ is a scheme and $M,N$ are $\mathcal{O}_X$-modules, then we can consider the $\mathcal{O}_X$-module $\underline{\hom}(M,N)$. I would like to have a reference for the following easy and well-known observation:

If $M$ is of finite presentation and $N$ is quasi-coherent, then $\underline{\hom}(M,N)$ is quasi-coherent.

(Proof: We may work locally and therefore assume that there is an exact sequence $\mathcal{O}^p \to \mathcal{O}^q \to M \to 0$. Using $\underline{\hom}(\mathcal{O}^q,N)=N^q$, we get an exact sequence $0 \to \underline{\hom}(M,N) \to N^q \to N^p$. Since quasi-coherent modules are closed under direct sums and kernels, the claim follows)

I couldn't find it in the usual books (EGA, Görtz-Wedhorn, Liu, Bosch, Stacks Project). For locally noetherian schemes this is Exercise 5.1.6.(a) in Liu's book. I ask because I want to use this result in a paper and don't want to waste any time with it.


It is Proposition 9.1.1 in the old version of EGA I (1960). In the new version (1971) I can only find the special case that $M$ and $N$ are coherent. Strange!


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