# Algebraic spaces as locally ringed spaces

Let $S$ be a scheme (although I am more than happy to have $S=\text{Spec}(k)$ for a field $k$) and $\mathsf{AlgSp}/S$ the category of algebraic spaces over $S$.

Does there exist an embedding $\varphi:\mathsf{AlgSp}/S\hookrightarrow \mathsf{LRS}$ (where $\mathsf{LRS}$ is the category of locally ringed spaces) such that we have a natural equivalence $$X\cong \text{Hom}_\text{LRS}(-,\varphi(X))$$ in the category of presheaves on $\mathsf{Sch}/S$?

It would also be desirable to have that if $X$ is finite type over $\mathbb{C}$, that there is a map $X^\text{an}\to \varphi(X)$ which is initial amongst maps from $X^\text{an}$ to elements of $\varphi(X)$.

• +1 for being more than happy with $Spec(k)$ :-)
– M.G.
Feb 5 '16 at 9:14
• I think you might be looking for stacks.math.columbia.edu/tag/04KL, but you'll need locally ringed toposes, not locally ringed spaces to make this work. I need to think about a nice counterexample, but one piece of intuition is that non-quasicompact non-quasiseparated algebraic spaces have a very bad notion of points. Feb 5 '16 at 9:39
• @pbelmans Thank you for your comment! I was aware of this actually, and wanted actual locally ringed space. The intuition you gave is roughly the intuition I have as well. Feb 5 '16 at 9:43

Probably you want your functor $\varphi$ to restrict to the "identity functor" on the full subcategory of algebraic spaces that are schemes (which doesn't seem to be a purely formal consequence of your hypotheses). And to be "reasonable" you likely want such a hypothetical functor $\varphi$ to carry open immersions to open immersions and etale maps to flat maps.

Under these very mild assumptions (without which $\varphi$ would be rather weird and hard to use) the answer is no even in the quasi-separated case, where one has a good theory of "associated topological space".

To discover examples showing this cannot be done (as you must surely be expecting anyway), let's first address what must be the actual motivation: to give a characterization of analytification of algebraic spaces locally of finite type over $\mathbf{C}$ (and non-archimedean fields) in terms of an "initial object" property akin to that for such schemes mapping to locally ringed spaces. (At the end of the question, I think you meant to write "maps from analytic spaces to $\varphi(X)$".) That would be a "replacement" for the usual definition of $X^{\rm{an}}$, perhaps felt to be a bit ad hoc, as $U^{\rm{an}}/R^{\rm{an}}$ for an etale chart $R \rightrightarrows U$ in schemes for $X$.

As you likely know, the existence or not of the quotient $U^{\rm{an}}/R^{\rm{an}}$ is independent of the etale chart $R \rightrightarrows U$ for $X$ and assuming existence that quotient is (uniquely up to unique isomorphism) independent of the chart and uniquely functorial in such an $X$ (in a sense made precise in section 2.2 of http://arxiv.org/abs/0706.3441, for example).

To give an "initial object" characterization of $X^{\rm{an}}$ it is more natural (as pbelmans has noted) to use locally ringed topoi (or a broader notion that replaces local rings with henselian local rings) and not to try to force onself into the setting of locally ringed spaces, but that would then kill the motivation for the question posed.

Note that for such algebraic spaces $X$ over $\mathbf{C}$ (or non-archimedean fields $k$), $X^{\rm{an}}$ exists only if the quasi-compact diagonal morphism $\Delta_{X/\mathbf{C}}$ (or $\Delta_{X/k}$) is an immersion; this is remarked without proof in Ch. I, 5.17ff. in D. Knutson's book on algebraic spaces. The necessity of this diagonal condition is proved in Theorem 2.3.4 of the above arxiv link, whereas sufficiency is shown to fail badly over non-archimedean fields beyond the separated case in section 3 of loc. cit. The sufficiency of the diagonal condition over $\mathbf{C}$ (proved as Ch. I, Prop. 5.18 in D. Knutson's book on algebraic spaces) is a bit more interesting in view of the necessity (i.e., that one needs some non-trivial diagonal hypothesis for the existence).

So to speak of $X^{\rm{an}}$ as an analytic space you need to assume the diagonal of $X$ is an immersion (this property is usually called "locally separated", somewhat unfortunate terminology); as you know, this is stronger than quasi-separatedness. Of course, one could get around that by enlarging the scope of what is meant by a (complex) analytic space, to go a bit beyond the framework of locally ringed spaces, but that would again somewhat kill the motivation for the question.

This finally brings us to examples showing that no reasonable $\varphi$ satisfying the additional mild properties mentioned above, and even if we limit attention to quasi-separated algebraic spaces of finite type over a field $k$. Consider a "locally separated" (in the sense of diagonal being an immersion) smooth geometrically connected algebraic space of dimension 2 built by modifying a smooth surface along a geometrically integral curve by "replacing" the curve with a $2:1$ finite etale cover from another geometrically integral curve. See Example 3.1.1 of the arxiv link above for a discussion of why such examples made over any non-archimedean field cannot be analytified in the sense of non-archimedean geometry (a la Tate, Berkovich, or even Huber, all by the same obstruction), whereas when built over $\mathbf{C}$ they can be analytified (so it is most fun with such examples over $\mathbf{Q}$, which communicates to both $\mathbf{Q}_p$ and $\mathbf{C}$).

By using an etale chart presentation for such a surface $S$ as in Example 3.1.1 mentioned above, the hypotheses that $\varphi$ carries open immersions to open immersions and etale maps to flat maps enables one to deduce (by some elementary considerations that I prefer not to write out here) that $\varphi(S)$ must have as its underlying topological space that of the algebraic space $S$ (not just as a set!) and its structure sheaf must be the sheaf of rings $A$ whose value on an open set $\Omega \subset |S|$ is $O_S(U)$ where $U \subset S$ is the unique Zariski-open subset such that $\Omega = |U|$. That is, $\varphi(S)$ has to be the naive ringed space (in fact locally ringed) attached to the algebraic space $S$ and its associated topological space. Concretely, if $R \rightrightarrows U$ is an etale scheme chart for $S$ then $\varphi(S)$ is the quotient $U/R$ in the category of locally ringed spaces.

The assumed main property of $\varphi$ (and the argument I omitted which identifies exactly what $\varphi(S)$ must be) implies that the "obvious" morphism of locally ringed spaces $f:S \rightarrow \varphi(S)$ has the property that for any scheme $T$ a morphism of locally ringed spaces $T \rightarrow \varphi(S)$ uniquely factors through $f$. But consider the generic point $\eta$ of the distinguished geometrically integral curve $C$ in the construction of $S$; that curve is a 2:1 cover of another curve $C_0$ such that the local stalk of $O_{\varphi(S)}$ at $\eta \in |S| = |\varphi(S)|$ has residue field $k(C_0)$ whereas $O_S$ has local henselian stalk at $\eta$ with residue field $k(C)$. So we have a natural map $T := {\rm{Spec}}(k(C_0)) \rightarrow \varphi(S)$ of locally ringed spaces, and this does not factors through $f:S \rightarrow \varphi(S)$. Indeed, if it were to factor then the resulting map $T \rightarrow S$ would have to hit exactly $\eta \in |S|$, and so would factor through the closed subscheme $C \subset S$, sandwiching $k(C) = O_{C, \eta}$ between $k(C_0)$ and $k(C_0)$, forcing $[k(C):k(C_0)] = 1$, a contradiction.

• Thank you for your nice answer. :) I will certainly think hard on it! Feb 17 '16 at 1:43