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I've asked this question https://math.stackexchange.com/questions/1407451/about-the-relation-between-the-categories-textsch-textlrs-and-text on math.stackexchange , however I don't think I will receive any response there (because of the current activity in my post). Therefore I'm asking it here. If the question seems inconvenient because of the excess of questions, I can split this question into other ones (let me know).

I've been trying to find some useful categorical facts about the category of schemes, locally ringed spaces and ringed spaces (that I shall denote by $\text{Sch}$, $\text{LRS}$ and $\text{RS}$ respectively). The motivation is that I'm trying to compute some (co)limits explicitly in the category of schemes.

There's this excellent answer here https://math.stackexchange.com/questions/102973/on-limits-schemes-and-spec-functor , but I still have some doubts.

More precisely, I want to know about the following assertions:

1) Is the category of locally ringed spaces (co)complete? The answer is yes by prop 1.6 in Demazure and Gabriel's "Groupes Algébriques" (I didn't notice that they proved the general case and not just the case of filtered colimits when I posted this question, sorry)

In the answer cited above, the references implies the existence of cofiltered limits and filtered colimits, however as I understand the notion of filtered in these cases is restricted to the case where the index category is a poset.

2)Is the category of ringed spaces (co)complete?

3)What can be said about the underlying topological space of the (co)limit of locally ringed spaces? (Is it the (co)limit of the topological spaces?)

4)What can be said about the underlying topological space of the (co)limit of ringed spaces? (Is it the (co)limit of the topological spaces)

5)What can be said about the underlying topological space of the colimit of schemes? (Is it the (co)limit of the topological spaces)

Obviously, the underlying topological space of the pullback of schemes is not the pullback of the topological spaces (for instance, $\text{Spec} (\mathbb{C}) \times_{\text{Spec} (\mathbb{R})}\text{Spec} (\mathbb{C}) \cong \text{Spec} (\mathbb{C}\times\mathbb{C})$ by $a \otimes z \mapsto (az, a\overline{z})$), but the case of push outs seems to be true.

6) Are (co)limits preserved under the inclusions $\text{Sch} \hookrightarrow\text{LRS} \hookrightarrow \text{RS}$?

The inclusion $\text{LRS} \hookrightarrow \text{RS}$ preserves colimits since it's a left adjoint (see below)

7) For each forgetful functor $U : \mathcal{C} \rightarrow \mathcal{D}$, where $\mathcal{C}$ and $\mathcal{D}$ are equal to $\text{Sch}$, $\text{LRS}$ and $\text{RS}$ (for all possible coherent substitutions), are there adjoint functors?

8) For each inclusion $\mathcal{C} \hookrightarrow \mathcal{D}$, where $\mathcal{C}$ and $\mathcal{D}$ are equal to $\text{Sch}$, $\text{LRS}$ and $\text{RS}$ (for all possible coherent substitutions), are there adjoint functors?

According to http://arxiv.org/abs/1103.2139 [Cor. 6], the inclusion $\text{LRS} \hookrightarrow \text{RS}$ have a right adjoint given by localization of the terminal prime system.

Thanks in advance.

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user40276
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1 Answers1

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Edited to incorporate Marc's comments below:

1) The cocompleteness of LRS is Prop 1.6 in Demazure-Gabriel's Groupes algebriques. Completeness is proved in http://arxiv.org/abs/1103.2139, Corollary 5.

2) Yes. The category of ringed spaces is (co)fibered over the category of topological spaces (which is (co)complete) and has (co)complete fibers. EDIT: To answer the OPs question, I can't think of a reference offhand but it's not too bad to prove straight from the definitions. Take your diagram upstairs, form the (co)limit downstairs, choose (co)cartesian lifts of the projection/inclusion maps from/to your (co)limit, form the (co)limit in the fiber, and you're good.

4) The forgetful functor RS → Top preserves limits and colimits. See (2), in this situation (co)limits are constructed via their projections to the underlying topological space.

3) The proof of Demazure-Gabriel's Prop 1.6 shows that the inclusion $LRS\subset RS$ preserves colimits (even better: creates them).

5) See (6).

6) Colimits are not preserved by the inclusion $Sch \subset LRS$, see Emerton's comment here: Colimits of schemes. Colimits are preserved by $LRS \subset RS$ as stated above in (3). The inclusion $Sch \subset LRS$ preserves finite limits by results in section 5.1 of Demazure-Gabriel.

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    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. – Marc Hoyois Aug 31 '15 at 22:21
  • @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) – user40276 Sep 01 '15 at 05:52
  • 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). – user40276 Sep 01 '15 at 05:57
  • @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? – user40276 Sep 01 '15 at 06:29
  • 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…). – user40276 Sep 01 '15 at 06:46
  • 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! – user40276 Sep 01 '15 at 06:50
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    @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... – Marc Hoyois Sep 01 '15 at 14:37
  • @MarcHoyois: whoops! I've added your amendments and made this community wiki so anyone else can correct as they see fit. – Dylan Wilson Sep 01 '15 at 15:31
  • @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 ... – user40276 Sep 03 '15 at 21:46
  • (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. – user40276 Sep 03 '15 at 21:47
  • @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)? – user40276 Sep 03 '15 at 21:52
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    @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. – Marc Hoyois Sep 04 '15 at 01:49
  • @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. – user40276 Sep 04 '15 at 02:45