Timeline for About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$
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17 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Sep 4, 2015 at 2:45 | comment | added | user40276 | @MarcHoyois Thanks for your patience until now. You're right, I've got confused by my own notation. I was thinking that the realization $|-|: \text{ME} \rightarrow \text{Esg}$, was the inclusion of schemes in locally ringed spaces when restricted to schemes. However, actually, this is the identity. | |
| Sep 4, 2015 at 1:49 | comment | added | Marc Hoyois | @user40276 I don't see how you can conclude from this that Sch ⊂ LRS preserves all limits. What fails for LRS is that the category of sheaves of rings with local stalks over a fixed space is not cocomplete. | |
| Sep 3, 2015 at 21:52 | comment | added | user40276 | @DylanWilson Thanks for your edit. However I still not too convinced that the (co)limits of ringed spaces are preserved by the forgetful functor. What exactly fails if I try the same thing with locally ringed spaces? I mean isn't locally ringed spaces fibered over topological spaces (because there's a pullback)? | |
| Sep 3, 2015 at 21:47 | comment | added | user40276 | (this is somewhere in Demazure and Gabriel's book, I can't find now). Therefore this inclusion preserves $\mathbf{ all}$ limits, since the Yoneda embedding preserves limits. | |
| Sep 3, 2015 at 21:46 | comment | added | user40276 | @MarcHoyois Thanks for your response. Now I could find what you call section 5.1 (it's chapter 1, paragraph 1, number 5). About my justificative, I didn't mean that any morphism from an affine scheme factors through an affine scheme (maybe I didn't expressed myself correctly). Actually, I meant that given a chart for a scheme, if some affine space maps to this chart, then it factors through the inclusion of the chart itself. Anyway, my statement that the realization of $\mathscr{F}$ is an scheme iff $\mathscr{F}$ was already representable by an scheme ... | |
| Sep 2, 2015 at 1:36 | history | edited | Marc Hoyois | CC BY-SA 3.0 |
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| Sep 1, 2015 at 15:31 | comment | added | Dylan Wilson | @MarcHoyois: whoops! I've added your amendments and made this community wiki so anyone else can correct as they see fit. | |
| Sep 1, 2015 at 15:30 | history | edited | Dylan Wilson | CC BY-SA 3.0 |
added 368 characters in body; Post Made Community Wiki
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| Sep 1, 2015 at 14:37 | comment | added | Marc Hoyois | @user40276 I should have said that LRS ⊂ RS creates colimits. This is not the statement of Prop 1.6 but that's how it's proved. For limits it's clear for instance that $Spec(k)\times Spec(k')=\emptyset$ in LRS if $k$ and $k'$ are fields with different characteristics. I did mean section 5.1 for the statement that Sch ⊂ LRS preserves finite limits, which is also proved in the paper you linked to, but I don't think it's as formal as you suggest. It's definitely not true that any map from an affine scheme factors through an affine open... | |
| Sep 1, 2015 at 6:50 | comment | added | user40276 | Ops! This is trivial! In the case $\mathscr{F}$ is represented by a scheme the maps going to an affine chart will factor through it hence it suffices to restrict to the case where $\mathscr{F}$ is affine and in this case $(A, 1_A)$ is initial in the opposite category! | |
| Sep 1, 2015 at 6:46 | comment | added | user40276 | Oh, but there's a problem it's not clear that the open immersions $\text{Spec} (A) \hookrightarrow \mathscr{F}$ are initial in all the morphisms from $\text{Spec} (A)$ to $\mathscr{F}$. I think it's sufficient to restrict to the case where the topological space of the image of a given morphism $\text{Spec} (A) \rightarrow \mathscr{F}$ is fixed, in this case open immersions should be initial (topologically this is clear, but the underlying sheaf of the image may be a little weird…). | |
| Sep 1, 2015 at 6:29 | comment | added | user40276 | @MarcHoyois Thanks for your comment. However I think you mean proposition 4.1 in section 1.4 instead of section 5.1. The point is that the right functor that they define picking the presheaf from locally ringed spaces to rings $\mathscr{F} \mapsto |\mathscr{F}| = colim d_{\mathscr {F}}$ with $ d_{\mathscr {F}}$ given by $(A, \rho) \mapsto A$ where $\rho \in \mathscr{F} (A)$ is the usual gluing by affine schemes that produces a scheme in the case $\mathscr{F}$ is already representable by a scheme. Furthermore the Yoneda embedding preserve limits hence the limits are preserved. Am I right? | |
| Sep 1, 2015 at 5:57 | comment | added | user40276 | Thanks for your answer. But, as I understand the inclusion $\text{LRS} \hookrightarrow \text{RS}$ is a left adjoint and, therefore, preserves colimits. So Marc comment is indeed consistent. Furthermore, I could not follow the answer of 2). Do you have any reference for this case? As I understand the category of schemes and locally ringed spaces are fibered over topological spaces (since there's a pullback functor). | |
| Sep 1, 2015 at 5:52 | comment | added | user40276 | @MarcHoyois Why the inclusion $\text{LRS} \hookrightarrow \text{RS}$ preserves colimits follows from prop 1.6. As I understand this is indeed true, because this inclusion is a left adjoint, however prop 1.6 just says that the category of locally ringed spaces is cocomplete (although, unfortunately he uses the term "inductive limit" but, now I'm seeing that he really proves the general case). Furthermore, why "of course not limits". Is there a trivial example where this fails? Indeed cofiltered limits with affine transitions of schemes preserves the topological space (somewhere in EGA IV) | |
| Aug 31, 2015 at 22:21 | comment | added | Marc Hoyois | In your last example, a pullback of fields will be a field... In fact LRS ⊂ RS preserves colimits, though of course not limits. This is Prop 1.6 in "Groupes algébriques" by Demazure and Gabriel. On the other hand, Sch ⊂ LRS preserves finite limits (section 5.1 in loc. cit.). I think it also preserves cofiltered limits with affine transition morphisms. | |
| Aug 31, 2015 at 16:27 | history | answered | Dylan Wilson | CC BY-SA 3.0 |