Timeline for The cohomology of the relative dualizing sheaf of a relative curve
Current License: CC BY-SA 3.0
11 events
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| May 29, 2012 at 7:41 | history | edited | Harry | CC BY-SA 3.0 |
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| May 28, 2012 at 12:00 | comment | added | naf | @Harry: What you say is indeed right. | |
| May 28, 2012 at 9:44 | comment | added | Thomas Riepe | @ulrich & @Harry: Thanks for the excellent link, leading to many good other texts! | |
| May 28, 2012 at 9:31 | comment | added | Harry | @myself. I think I got it. The push-forward f_\ast coincides with taking global sections and then exact functor "tilde" which associates to a $\mathbf{Z}$-module $M$ the coherent sheaf $\widetilde{M}$ on $\mathrm{Spec} \mathbf{Z}$. | |
| May 28, 2012 at 9:23 | comment | added | Harry | @ulrich. Let $n\geq 2$ and let $f:X\to S$ be a semi-stable curve. It's clear that the higher cohomology of $\omega^{\otimes n}$ vanishes on the generic fibre by duality. By loc. cit., this implies that $R^1 f_\ast \omega^{\otimes n} =0$. But why does this imply that $H^1(X,\omega^{\otimes n}) =0 $? I don't see how $f_\ast $ and the global sections functor are related in this case, because the base is $\mathrm{Spec} \mathbf{Z}$. | |
| May 28, 2012 at 8:07 | comment | added | Harry | Here's the link : math.ucdavis.edu/~osserman/math/cohom-base-change.pdf There was just a small typo in your hyperlink. | |
| May 28, 2012 at 8:05 | history | edited | Harry | CC BY-SA 3.0 |
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| May 28, 2012 at 8:05 | comment | added | Harry | Thanks for your answer. The link doesn't work, unfortunately. What do you mean when you answer by yes? $n=3$ works? | |
| May 28, 2012 at 8:01 | comment | added | naf | Yes, this is a simple consequence of "cohomology and base change". See, for example, math.ucdavis.edu/~osserman/math/cohomom-base-change.pdf Theorem 1.2. The point is that one has vanishing for the generic fibre for $n>1$ (by duality) and since the fibres are all $1$-dimensional and the base is affine, the result follows from the Theorem. | |
| May 28, 2012 at 7:41 | history | edited | Harry | CC BY-SA 3.0 |
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| May 27, 2012 at 23:19 | history | asked | Harry | CC BY-SA 3.0 |