Timeline for Direct image functor commuting with infinite direct sum of sheaves
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Jan 30, 2020 at 17:47 | comment | added | Martin Brandenburg | 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 $\mathcal{L} = \mathcal{O}_X$. | |
| Aug 4, 2019 at 6:33 | comment | added | Luke | @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. | |
| Aug 4, 2019 at 6:26 | comment | added | Praphulla Koushik | Can you also add what you have tried? | |
| Aug 4, 2019 at 5:23 | history | asked | Luke | CC BY-SA 4.0 |