Timeline for Flasque sheaves on a site
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Apr 21, 2022 at 7:40 | comment | added | Jochen Wengenroth | This is out of my comfort zone, but one should get an exact sequence $0=H^1(U,\mathscr F)\to H^1(U,\mathscr F'')\to H^2(U,\mathscr F')\to \cdots$. Without knowing $H^2(U,\mathscr F')$ this looks completely hopeless -- I would rather look for a counter example than for a proof. | |
| Apr 20, 2022 at 15:07 | comment | added | Zhen Lin | @Jehu314 If you have a sheaf $F$ such that $\textrm{Ext}^1 (E, F) = 0$ for a sufficiently large class of sheaves $E$, then $F$ will have the property that "restrictions along monomorphisms are surjective": given $U \subseteq V$, we can form the short exact sequence of sheaves $0 \to \mathbb{Z} U \to \mathbb{Z} V \to \mathbb{Z} U / \mathbb{Z} V \to 0$ and apply $\textrm{Ext}^*(- , F)$ to get an exact sequence $F (V) \to F (U) \to \textrm{Ext}^1 (\mathbb{Z} V / \mathbb{Z} U, F)$. The point is that in a general site, morphisms in the site are not always monomorphisms. | |
| Apr 20, 2022 at 15:02 | history | edited | Zhen Lin |
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| Apr 19, 2022 at 18:50 | comment | added | R. van Dobben de Bruyn | @JochenWengenroth Note that the same statement for the other definition (restrictions are surjective) is an exercise in Hartshorne (II.1.16(c)), at least for topological spaces. But we already concluded that the notions do not agree. | |
| Apr 19, 2022 at 16:16 | comment | added | Jehu314 | @JochenWengenroth No, the book uses this particular statement in a later proof. | |
| Apr 19, 2022 at 15:48 | comment | added | Jochen Wengenroth | Could this be a typo? The statement If $\mathscr F'$ and $\mathscr F''$ are flasque then so is $\mathscr F$ looks much better. | |
| Apr 19, 2022 at 13:03 | comment | added | Jehu314 | @Johan I see. For any $U$, take the covering $U\sqcup U\rightarrow U$. | |
| Apr 19, 2022 at 12:30 | comment | added | Johan | Exercise: on the \'etale site of a scheme, there are no "flasque" sheaves of abelian groups apart from the zero sheaf, if flasque is defined by "surjective restriction maps". | |
| Apr 19, 2022 at 12:16 | history | edited | Jehu314 | CC BY-SA 4.0 |
added 2 characters in body
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| Apr 19, 2022 at 12:09 | comment | added | Jehu314 | @JasonStarr The book makes the claim only for the étale site, but I never thought the choice of site would matter. | |
| Apr 19, 2022 at 10:56 | comment | added | Jason Starr | I am just clarifying: are you asking whether this is true for every site, or just for the etale site of a scheme (the only site where Lei Fu claims that it is true)? | |
| Apr 19, 2022 at 9:27 | comment | added | R. van Dobben de Bruyn | The Stacks project calls a sheaf $\mathscr F$ totally acyclic if $H^i(\mathbf T/U,\mathscr F) = 0$ for all $i > 0$ and all objects $U$ of the topos $\mathbf T$, and claims without details that (1) it is not enough to check on representable objects $U$ (coming from the site), and (2) this clashes with the 'restrictions are surjective' definition. See Tag 079X. There is no remark on whether $H^1$ suffices. | |
| Apr 19, 2022 at 7:59 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals, added tag
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| Apr 19, 2022 at 4:24 | history | asked | Jehu314 | CC BY-SA 4.0 |