Timeline for Sheaf associated to presheaf Aut
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 20, 2017 at 14:46 | vote | accept | Cristian D. Gonzalez-Aviles | ||
| S Feb 20, 2017 at 14:46 | history | bounty ended | Cristian D. Gonzalez-Aviles | ||
| S Feb 20, 2017 at 14:46 | history | notice removed | Cristian D. Gonzalez-Aviles | ||
| Feb 20, 2017 at 13:52 | answer | added | Simon Henry | timeline score: 5 | |
| Feb 19, 2017 at 18:29 | comment | added | Cristian D. Gonzalez-Aviles | Dear Simon, many thanks again for your help. Our definitions of the normalizer presheaf seem to be the same, namely [SGA 3, Exp. I, Definition 2.3.3]. I just hadn't noted that this is a sheaf for any topology. I'm satisfied that you have answered my new question completely. How can I credit you for this since this wasn't my original question (as I did not pose that one well)? | |
| Feb 18, 2017 at 16:55 | comment | added | Simon Henry | So If I understand your question properly, the answer is yes and it has nothing to do with algebraic geometry (it would be true for any site). But I am not completely sure I understand all your notations. For example, you need to clarify what do you mean you consider the presheaf '$N_G(H)$'. for me it is defined by $N_G(H)(X)$ is the set of maps $f$ from $X$ to $G$ such that the maps $X \times H$ to $G$ which send $(x,h)$ to $f(x) h f(x)^{-1}$ factor into $H$ and it is actually a sheaf. | |
| Feb 18, 2017 at 16:45 | comment | added | Simon Henry | Yes, this new question has a lot more chance to be answerable. If $G$ is a group object in a topos, $H$ a subgroup of $G$ and $X$ the quotient object $G/H$ then the internal object of $G$ automorphism of $X$ is always isomorphic to the internal $N_G(H)/H$, by internalization of the usual set theoretic proof of this fact. Now in a topos of sheaves, $N_G(H)/H$ is indeed the sheafification of the pre-sheaf $N_G(H)/H$ | |
| Feb 15, 2017 at 17:05 | comment | added | Cristian D. Gonzalez-Aviles | Dear Simon, Thanks for your insight. Maybe I'm asking the wrong question. Here is my actual problem. I have a trivial homogeneous space $X\simeq a(G/H)$, where $H$ is a closed subgroup scheme of $G$. Let $\underline{N}_{G}(H)$ be the normalizer presheaf of $H$ in $G$. I can show that there exists a canonical isomorphism of presheaves of groups $\underline{N}_{G}(H)/H\simeq\underline{Aut}_{G}(G/H)$ (=presheaf of $G$-equivariant automorphisms of the presheaf $G/H$) and I want to show this induces isom. of sheaves $a(\underline{N}_{G}(H)/H)\simeq\underline{Aut}_{G}(X)$ for any reasonable top. | |
| Feb 15, 2017 at 13:58 | comment | added | Simon Henry | "Aut" is not a functor, so your defintion of Aut(P) does not produces a presheaf... What you want are the aumorphisms of $P \times h_T$ compatible to the projection to $h_T$ which would be functorial under pullback on the $h_T$ component... For your question, it is very likely to be false. Sheafication in general does not preserve formation of internal Hom objects or "Aut" objects, so unless something very special happen in your situation this is not going to work... | |
| Feb 15, 2017 at 13:18 | history | edited | Mikhail Borovoi | CC BY-SA 3.0 |
Formatting, tags added.
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| S Feb 14, 2017 at 17:42 | history | bounty started | Cristian D. Gonzalez-Aviles | ||
| S Feb 14, 2017 at 17:42 | history | notice added | Cristian D. Gonzalez-Aviles | Canonical answer required | |
| Feb 9, 2017 at 13:54 | comment | added | Matthieu Romagny | Ah -- yes, now I see. | |
| Feb 9, 2017 at 11:40 | comment | added | Cristian D. Gonzalez-Aviles | Dear Matthieu, thanks for your efforts. Note that, since a=++, it would be sufficient to show the indicated formula with a replaced by +. I tried to do this using the definition of the + functor (denoted by L in SGA 3, IV), but didn't get very far. | |
| Feb 9, 2017 at 8:12 | comment | added | nfdc23 | @MatthieuRomagny: It isn't clear when an automorphism of $aP$ over some object locally arises from an automorphism of $P$; that is an obstacle to arguing by just stalk calculations (in sites with enough points). | |
| Feb 9, 2017 at 7:58 | comment | added | Matthieu Romagny | There is a canonical map of sheaves $a(Aut(P))\to Aut(aP)$. If your site has enough points, this map is an isomorphism on stalks hence an isomorphism. Am I missing something? | |
| Feb 8, 2017 at 22:45 | history | asked | Cristian D. Gonzalez-Aviles | CC BY-SA 3.0 |