Timeline for Representability of morphism of stacks
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 31, 2015 at 1:09 | comment | added | user74230 | @Niels: In the above comment I should have said "constant sheaf associated to the strict henselization of $Y$ at $y$ (initially I had planned to take $X = Y \times \mathbf{G}_m$, but then changed my mind). | |
| Mar 30, 2015 at 14:59 | comment | added | user74230 | @Niels: The difference is that base change on geometric objects along a map not "in the site" isn't generally computed by sheaf pullback for the representing functor. For example, if $X = Y \times \mathbf{A}^1$ over a scheme $Y$ then the scheme-theoretic pullback along geometric point $y:{\rm{Spec}}(k) \rightarrow Y$ is the scheme $\mathbf{A}^1_{k}$ whereas the fiber at $y$ of the functor represented by $X$ on the etale site is the constant sheaf over ${\rm{Spec}}(k)$ associated to the group of units in the strict henselization of $Y$ at $y$. So the SP reference seems not quite enough to me. | |
| Mar 30, 2015 at 12:36 | comment | added | Niels | @user74230 : "see this spaces as sheaves" means to a space associate its functor of points ; this is sufficient since this Yoneda-like operation is fully faithful. Are you sure of "that is very different from stalks as in the SP reference" ? | |
| Mar 30, 2015 at 1:43 | comment | added | user74230 | @Niels: Forming fibers of Aut-schemes at geometric points is that the OP wants to do, and that is very different from stalks as in the SP reference concerning "conservative family of points", so your suggestion for how to pass to considerations at geometric points seems a bit unclear. Are you sure that "see these spaces as sheaves" is sufficient for the purposes of the question posed? | |
| Mar 29, 2015 at 16:25 | comment | added | Niels | Yes, it is enough to verify the property on geometric points : you need to check that the kernel say $K/U$ a is trivial, that is that the unit section $U\to K$ is an isomorphism. You can see these spaces as sheaves, and then use the fact that the geometric points form "a conservative family of points" see stacks.math.columbia.edu/tag/00YJ . | |
| Mar 29, 2015 at 14:31 | comment | added | user234 | Thank you for your answer. Just to be sure I'd like to ask one question. In my situation, $S = Spec \mathbb Q$ and I only know that condition 1 holds for $U= Spec \mathbb C$ (or $U=$ spectrum of an alg closed field of char zero). Do I understand correctly that this is enough? I guess to prove this we just replace the target by a geometric point $y$ of $Y$, right? | |
| Mar 29, 2015 at 14:04 | history | answered | Niels | CC BY-SA 3.0 |