Timeline for Algebraic spaces which are automatically schemes
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 9, 2015 at 21:54 | comment | added | grghxy | @user235: It exists in the same generality as for schemes (e.g., finitely presented and flat with CM fibers) by etale descent from the scheme case since in the scheme case its formation is canonically compatible with etale localization on $X$ as well as with any base change on $S$. | |
| Jun 9, 2015 at 16:57 | vote | accept | user235 | ||
| Jun 9, 2015 at 16:57 | comment | added | user235 | Dear @grghxy , thank you again for this answer. I was merely curious if $\omega_{X/S}$ exists in your more general setting. But you are right, in the smooth proper case this settles it. | |
| Jun 9, 2015 at 16:37 | comment | added | grghxy | In the smooth proper case (what you asked about) the canonical bundle $\Omega^d_{X/S}$ is such an $L$! (Construction same as for schemes.) | |
| Jun 9, 2015 at 16:35 | comment | added | user235 | Thank you for your quick answer. If the fibres of $f:X\to S$ have ample canonical bundle, then do I understand correctly that the mere existence of a line bundle $L$ which restricts to the canonical bundle on each fibre implies that $X$ is a scheme? Is it clear that there exists a relative dualizing sheaf $\omega_{X/S}$ in the setting you consider (i.e., $f$ proper finitely presented)? | |
| S Jun 9, 2015 at 16:24 | history | answered | grghxy | CC BY-SA 3.0 | |
| S Jun 9, 2015 at 16:24 | history | made wiki | Post Made Community Wiki by grghxy |