Timeline for restriction and pullback of representable etale sheaf along closed immersion
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Dec 10, 2012 at 16:18 | comment | added | Heer | Now I understand why I got confused. For the small etale site, the restriction and pullback don't coincide; but for big etale site, they do. There is a remark in Milne's book 'etale cohomology', chapter II, remark 3.1 (a), which explains the reason in more genearl setting. Thanks @Kestutis Cesnavicius and @Damian Rössler for the comments | |
| Dec 9, 2012 at 10:42 | comment | added | Heer | I just found that I asked a stupid question. It is trivial that these two coincide by just checking the definition of $i^*G$ | |
| Dec 5, 2012 at 21:56 | comment | added | Damian Rössler | I see now that I answered a different question...sorry for the confusion. | |
| Dec 5, 2012 at 14:04 | comment | added | Heer | I mean over the category $(Sch/S)$, how can you have small etale topology? | |
| Dec 5, 2012 at 14:01 | comment | added | Heer | @ Kestutis Cesnavicius: I think my notation $(Sch/S)_{et}$ is standard for big etale site. | |
| Dec 5, 2012 at 11:54 | comment | added | Damian Rössler | (I am working in the small site). | |
| Dec 5, 2012 at 11:53 | comment | added | Damian Rössler | I think one reason for the confusion is the following. The pull-back of a representable étale sheaf generally only coincides with the pull-back (base-change) of the corresponding scheme if the scheme is étale, so in your situation if $G$ is étale over $S$. To see this, remember that the sheaf pull-back is left adjoint to sheaf push-forward; but the push-forward of a representable étale sheaf is not representable in general - unless the representing scheme is étale, in which case the scheme representing the push-forward is a kind of Weil restriction. See the book by Freitag-Kiehl, I,§3,p29. | |
| Dec 5, 2012 at 11:49 | comment | added | Kestutis Cesnavicius | Are you working with small or big etale sites? Your 1. (1) only makes sense to me in the big etale site, in which case the agreement that you want results from the definition of $i^*G$. | |
| Dec 5, 2012 at 10:40 | history | asked | Heer | CC BY-SA 3.0 |