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Normally I would think this kind of question doesn't belong on overflow, but I haven't been able to find an answer anywhere else, so perhaps it is not so trivial.

Let $f: X \rightarrow Y$ be a morphism of schemes, where $Y = \text{spec}A$ is affine. Let $\mathcal{L}$ be an invertible sheaf on $X$, globally generated by sections. Is it true that the direct image functor $f_{*}$ respects arbitrary direct sums in the sense that, $$ f_{*} \bigoplus_{d \geq 0} \mathcal{L}^{\otimes d} \simeq \bigoplus_{d \geq 0} f_{*} \mathcal{L}^{\otimes d} $$

If it is true, how would one prove it? If not, what extra conditions are needed on $f$ that would make this true?

I ask because it seems to be what's happening in the stacks project here. I have asked the question in a comment there as well. If $f$ was quasicompact and quasiseparated I can see it being true, but that is not assumed there. Initially I thought it was a mistake, but it seems to be used repeatedly in the following results on the stacks project.

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    $\begingroup$ Can you also add what you have tried? $\endgroup$
    – Praphulla Koushik
    Commented Aug 4, 2019 at 6:26
  • $\begingroup$ @PraphullaKoushik At this point I am not even sure what to try. The only thought I have had is to try to find a family of morphisms, $\psi_{d}: f_{*} \mathcal{L}^{\otimes d} \longrightarrow f_{*} \oplus_{d \geq 0} \mathcal{L}^{\otimes d}$ which is equivalent to a family of maps, $\epsilon_{d}: \mathcal{L}^{\otimes d} \longrightarrow f^{*}f_{*} \oplus_{d \geq 0} \mathcal{L}^{\otimes d}$ which I expect would be the counit of adjunction. But the fact that $f$ is not quasicompact or quasiseparated means that the pushforward may not even be quasicoherent. $\endgroup$
    – Luke
    Commented Aug 4, 2019 at 6:33
  • $\begingroup$ Can you say exactly where in the stacks project (exactly which sentence of the proof) this isomorphism is used in your opinion? As you have already guessed, the isomorphism doesn't hold without finiteness assumptions, even for the case $\mathcal{L} = \mathcal{O}_X$. $\endgroup$
    – Martin Brandenburg
    Commented Jan 30, 2020 at 17:47

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