Timeline for When will the pushforward of a structure sheaf still be a structure sheaf?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 24, 2017 at 3:55 | comment | added | R. van Dobben de Bruyn | And for those of us lucky enough to have non-algebraically closed fields in our lives: the assumption should be that $f$ has geometrically connected fibres, rather than merely connected. An easy counterexample is a field extension. | |
| Oct 17, 2016 at 12:41 | comment | added | David Holmes | I think the reference to III.10.3 is incorrect, or at least it points to something irrelevant in the edition I have to hand. Also, the statement `if $Y$ is in addition normal, then $f_∗\mathcal O_X=\mathcal O_S$ holds' needs $X$ reduced even in characteristic zero, see comment above. | |
| Jan 3, 2013 at 1:40 | comment | added | minimax | Is the reference to Hartshorne III.10.3 correct? | |
| Apr 28, 2011 at 18:10 | comment | added | Karl Schwede | Another quick comment, one can do even better than normality. Let's work over an algebraically closed field of characteristic zero. With J.C. Ottem's notation, assume that $f$ has connected fibers and $Y$ is seminormal. Doing Stein factorization as above, one has $g : Z \to Y$ a finite map, which is birational (at least in characteristic zero). Furthermore, one can show that $g$ also has connected fibers because $f'$ is surjective (since it is proper and dominant). Thus $g$ is an isomorphism by the defining property of seminormality (over algebraicaly closed field of char. 0). | |
| Apr 28, 2011 at 18:08 | comment | added | J.C. Ottem | Yes, that's true. In that case $g$ is bijective, but not necessarily an isomorphism. | |
| Apr 28, 2011 at 18:04 | comment | added | Karl Schwede | A minor correciton, if $Y$ is normal and $f$ has connected fibers, then $f_* \mathcal{O}_X = \mathcal{O}_Y$ holds in characteristic zero. In characteristic $p$, this is not the case (for example the Frobenius morphism). | |
| Apr 28, 2011 at 17:10 | history | edited | J.C. Ottem | CC BY-SA 3.0 |
added 20 characters in body
|
| Apr 28, 2011 at 16:48 | history | edited | J.C. Ottem | CC BY-SA 3.0 |
added 329 characters in body; deleted 55 characters in body
|
| Apr 28, 2011 at 16:34 | history | answered | J.C. Ottem | CC BY-SA 3.0 |