Timeline for Comparing locally ringed spaces and presheaves on Aff
Current License: CC BY-SA 3.0
5 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jan 4, 2014 at 0:37 | comment | added | user76758 | The answer to your Zariski-sheaf question is "yes", and this is an exercise in definitions with maps of ringed spaces, so I prefer to say nothing more and let you figure it out for yourself. The rest is just saying that algebraic spaces which aren't schemes probably provide counterexamples to Q2, but I didn't find a rigorous proof and so just put those comments there in case another reader who is also familiar with algebraic spaces might be able to turn that idea into a justified counterexample. | |
| Jan 3, 2014 at 22:41 | comment | added | jmc | @user76758 Do you mean to say that the essential image of the functor $F$ is contained in the subcategory of Zariski sheaves? Do you have a proof/reference for this? I can not follow the rest of your comment (after "Less trivial:"). Would you please explain it a bit more (also the notation...) | |
| Jan 3, 2014 at 22:20 | comment | added | user76758 | Q2 fails if not a Zariski sheaf. Less trivial: consider q-c separated algebraic spaces $X$. Via an etale chart $R\rightrightarrows U$ in Aff, $|X| :=|U|/|R|$ and $O:=\ker(q_{\ast}(O_U)\rightrightarrows p_{\ast}(O_R))$ (using $p:|U|\rightarrow|X|$ and $p=q\circ p_i:|R|\rightarrow|X|$) make $X':=(|X|,O)$ an LRS. The map $\underline{X}\rightarrow\underline{X}'$ on Aff is initial for $\underline{X}\rightarrow\underline{Y}$ with LRS $Y$. Via $\underline{X}\simeq\underline{Y}$, $X'\rightarrow Y$ is bijective on spaces and an isomorphism on residue fields. Maybe this violates Q2 for $X$ not a scheme? | |
| Jan 3, 2014 at 13:32 | history | asked | jmc | CC BY-SA 3.0 |