# Direct image functor commuting with infinite direct sum of sheaves

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.

• Can you also add what you have tried? – Praphulla Koushik Aug 4 at 6:26
• @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. – Luke Aug 4 at 6:33