2
$\begingroup$

Related question: [1] ring-valued points of locally ringed spaces

There is a natural functor from locally ringed spaces to presheaves on affine schemes: $$ \begin{aligned} F \colon \text{LRS} &\to \text{Psh}(\text{Aff}) \\ X &\mapsto X(\_) \end{aligned} $$

As Martin showed in [1], this functor is not full. He asks whether it is also not faithful.

I want to pose two related questions.

Q1 Is there a non-scheme locally ringed space $X$, such that $X(\_)$ is representable by a scheme? (In other words, are there an LRS $X$, and a scheme $Y$, such that $X \not\cong Y$, but $F(X) \cong F(Y)$.?)

Q2 Is the functor $F$ essentially surjective? (In other words, are all presheaves $\mathcal{F}$ in $\text{Psh}(\text{Aff})$ “representable” by locally ringed spaces?)

My gut feeling is that the answer to Q1 is “yes” and the answer to Q2 is “no”. I do not really have a good feeling where to look for an answer (but the Christmas break brought me a severe cold, which might keep me from seeing trivial (counter)examples).

Improve this question
$\endgroup$
3
  • $\begingroup$ 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? $\endgroup$
    – user76758
    Commented Jan 3, 2014 at 22:20
  • $\begingroup$ @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...) $\endgroup$
    – jmc
    Commented Jan 3, 2014 at 22:41
  • $\begingroup$ 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. $\endgroup$
    – user76758
    Commented Jan 4, 2014 at 0:37

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .