Timeline for Is the higher direct image sheaf of a locally free sheaf over $\mathbb{P}^1$ locally free?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Dec 5, 2013 at 6:59 | comment | added | abx | It is of course, but not a separable algebra. | |
| Dec 5, 2013 at 6:41 | comment | added | Jana | The last line of the example says it is not a closed subscheme (see last line on pp. 259). Isnt $H^0(\mathcal{O}_{X_0})$ a finite dimensional $\mathbb{C}$-vector space? | |
| Dec 5, 2013 at 6:21 | comment | added | abx | Projectivity is not a problem, look at Hartshorne, example 9.8.4. And for your EGA corollary, the key hypothesis, namely that $H^0(X_0,\mathcal{O}_{X_0})$ is a separable $\mathbb{C}$-algebra (i.e. $\mathbb{C}\times \ldots \times \mathbb{C}$) is not satisfied. | |
| Dec 5, 2013 at 6:07 | comment | added | Jana | I am a bit confused right now. I noticed in the example that you state $X_0$ is not a closed subscheme of $\mathbb{P}^2$. So I do not totally understand why $f$ is projective. Adding to my confusion is the corollary $7.8.7$ in EGA-III which if I understand correctly means that the dimension of the global sections of the structure sheaf remains contant in this example if it were projective. Could you tell me what is it that I am getting wrong? | |
| Nov 18, 2013 at 8:31 | vote | accept | Jana | ||
| Nov 18, 2013 at 5:37 | history | edited | abx | CC BY-SA 3.0 |
Replaced "rational curve" by "nodal plane cubic curve".
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| Nov 17, 2013 at 20:51 | history | answered | abx | CC BY-SA 3.0 |