# About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$

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.

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.

• 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 1 '15 at 5:52
• 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 1 '15 at 5:57
• @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 1 '15 at 14:37
• @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 4 '15 at 1:49