Timeline for Direct images commute with homotopy colimits
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| May 25, 2023 at 9:41 | comment | added | Alexey Do | @crystalline, thanks, noetherian schemes of finite dimension are enough for my purpose. By the way, I think proposition C.12 part (3) of Hoyois has a mistake, if $f^*$ preserves compact generators, then its right adjoint $f_*$ only preserves direct sums. Combining with the fact that $\mathbf{SH}$ is stable, $f_*$ preserves finite limits, hence finite colimits, we deduce that $f_*$ preserves all colimits. | |
| May 25, 2023 at 5:04 | comment | added | crystalline | It does preserve colimits for SH because its left adjoint f^* preserves compact objects, which boils down to the fact that the Nisnevich sheaf condition can be expressed with finite limits instead of Cech nerves (at least in the qcqs case). See eg Hoyois’s Lefschetz formula paper, Appendix C. Note that the statement is false for the Betti sheaf theory for non-proper morphisms though. | |
| May 24, 2023 at 20:33 | history | asked | Alexey Do | CC BY-SA 4.0 |