Timeline for Picard group of a singular projective curve
Current License: CC BY-SA 3.0
11 events
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| S May 23, 2017 at 7:20 | history | suggested | user347489 | CC BY-SA 3.0 |
backslash * doesn't seem to work
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| May 23, 2017 at 7:09 | review | Suggested edits | |||
| S May 23, 2017 at 7:20 | |||||
| Jul 18, 2012 at 21:59 | history | edited | Karl Schwede | CC BY-SA 3.0 |
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| Jul 17, 2012 at 12:05 | comment | added | Jason Starr | Bhargav is correct, I was wrong: pushforward by a finite morphism is not necessarily acyclic (for the Zariski topology). You can see this quite clearly for a non-local, semilocal domain if you use Exercise 1.19 of Hartshorne with the locally constant sheaf $\mathbb{Z}$. As Bhargav says, you can use the \'etale topology to compute Pic, and pushforward by a finite morphism is acyclic for the \'etale topology. | |
| Jul 17, 2012 at 3:33 | comment | added | Bhargav | @Jason: Are finite morphisms really acyclic for the Zariski topology? Perhaps (likely!) I'm mistaken, but there are semi-local schemes (obtained via finite covers of local ones) with non-trivial Zariski cohomology, which I thought would contradict acyclicity (see Example 1.10 of Morel-Voevodsky). Of course, for Justin's question, one may work with the etale/Nisnevich topology, so the acyclicity is fine. | |
| Jul 17, 2012 at 3:25 | comment | added | Justin Campbell | @Jason Starr: I had hoped something like this was true. Would you mind elaborating a bit, perhaps in an answer? What property of $\pi$ is responsible for this fact? I know there is a theorem about higher direct images for proper maps that would give me what I want e.g. in the complex topology, but I'm wary of applying this in the Zariski topology. | |
| Jul 17, 2012 at 2:11 | comment | added | Karl Schwede | I guess that might be a better way to go about it, to prove directly that $\pi_*$ is exact on arbitrary sheaves of Abelian groups. | |
| Jul 17, 2012 at 2:05 | history | edited | Karl Schwede | CC BY-SA 3.0 |
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| Jul 17, 2012 at 2:01 | comment | added | Karl Schwede | Hi Jason, I was trying to avoid $\pi_*$ exactness (since that seemed to be what Just Campbell was worried about)? Instead I was claiming that all this work can be done on a different ringed space on $X$, and we don't need to consider the scheme $(X, O_X)$ at all. | |
| Jul 17, 2012 at 1:59 | history | edited | Karl Schwede | CC BY-SA 3.0 |
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| Jul 17, 2012 at 1:39 | history | answered | Karl Schwede | CC BY-SA 3.0 |