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I was talking to an ex-Adelaide student now at Oxford about some technicalities they were trying to track down regarding locally ringed spaces. I checked locally ringed topological space, and found the Stacks Project reference was out of date. I replaced it with a link to the specific tag for the definition, at least.
I added a reformulation of the locality condition which doesn’t refer to points.
Out of curiosity, what technicalities did they want to track down?
Ingo,
Something in EGA about fibred products of locally ringed spaces that referred to a proof in the first edition of Bourbaki’s Algèbre Chapter 8. I found that Martin B linked on MO to some (now missing) notes. They would be handy I’m sure.
Isn’t it an even better formulation to just say that it is a “local ring” in the internal language of the topos of sheaves?
Maybe. But this student may wish to do something different. We were really just tracking down a reference chain to see what the algebraic “meat” of the construction is.
That said, is the construction of products and pullbacks of locally ringed spaces, seen via the topos lens, easier? The hard nut at the core of this is that the pushout of local rings in the category of rings is not local anymore. One needs to reflect it back into the category of local rings. Even in the case of affine schemes one can’t get away from this, methinks.
I’ve never thought about that question. But I would expect that it would be easier to construct the pushout of local rings in the internal language than to have to deal explicitly with stalks or sections. By “reflect” you don’t mean a literal category-theoretic reflection, do you? I didn’t think local rings were reflective in the category of arbitrary rings; for one thing they aren’t even a full subcategory, are they?
Indeed, I’d prefer to just say “local ring” in the internal language.
And indeed, constructing fiber products $X \times_Z Y$ in the category of locally ringed locales is quite nice using the internal language. First, you construct the fiber product in the category of ringed locales. This is done in the naive way – take the fiber product $|X| \times_{|Y|} |Z|$ of the underlying locales, pull back the structure sheaves, and take their tensor product: $\mathcal{A} \coloneqq \pi_X^{-1}\mathcal{O}_X \otimes_{(...)^{-1}\mathcal{O}_Z} \pi_Y^{-1}\mathcal{O}_Y$. This ring will in general not be local [*].
Then, to obtain the fiber product in the category of locally ringed locales, construct, from the internal point of view of $Sh(|X| \times_{|Y|} |Z|)$, the spectrum of $\mathcal{A}$ (in a constructively sensible way, as a locale). Externally, this will result in a locale $P$ over $|X| \times_{|Y|} |Z|$ which is equipped with a local sheaf of rings.
However, we are not quite done yet: There are morphisms $P \to X$ and $P \to Y$ as required for a fiber product, but these are only morphisms of ringed locales, not of locally ringed locales. To fix this, we have to restrict to a certain sublocale of $P$ – the greatest sublocale $P'$ where $P' \to X$ and $P' \to Y$ are morphisms of locally ringed locales [**]. This sublocale, together with the ring $\mathcal{A}|_{P'}$, is the desired fiber product.
[*] Even in the easiest case – that all rings involved are fields – locality is not preserved: The rings $\mathbb{R}$ and $\mathbb{C}$ are local, and the homomorphism $\mathbb{R} \to \mathbb{C}$ is local, but $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \cong \mathbb{C} \otimes_{\mathbb{R}} \mathbb{R}[X]/(X^2+1) \cong \mathbb{C}[X]/(X^2+1) \cong \mathbb{C} \times \mathbb{C}$ is not.
[**] This sublocale can be explicitly described. Recall that $P = Spec(\mathcal{A})$ is the classifying locale of the theory of filters in $\mathcal{A}$ (a filter is a constructively sensible substitute for the complement of a prime ideal). The sublocale $P'$ is the classifying locale of the theory of those filters $F$ in $\mathcal{A}$ which enjoy the following special property: If $x \otimes 1 \in F$, where $x : \pi_X^{-1}\mathcal{O}_X$, then $x$ is invertible in $\pi_X^{-1}\mathcal{O}_X$; and similarly with $\mathcal{O}_Y$ instead of $\mathcal{O}_X$. It’s also possible to explicitly write down the frame of opens of this sublocale.
Thanks both. I asked early on in the discussion if spaces involved were sober, and it’s possible they are not, so I’m not sure using locales will.help.
Mike: You are right that the non-full subcategory of local rings is not reflective in the category of arbitrary rings. [***]
However, the situation is better if we allow for rings over arbitrary locales. The non-full subcategory of local rings over arbitrary locales is reflective in the category of arbitrary rings over arbitrary locales. The reflector maps a possibly non-local ring $\mathcal{A}$ over some locale $X$ to the structure sheaf of the spectrum of $\mathcal{A}$ (constructed inside $Sh(X)$).
Note that, since the embedding is not full, the reflection of a ring which already happens to be local is not in general isomorphic to that ring.
[***] One can show that a ring $A$ admits a universal homomorphism $A \to A'$ to a local ring $A'$ if and only if $A$ contains exactly one prime ideal. In this case, the ring $A$ is already local and the universal localization is given by $A$ itself. (In constructive mathematics, one should probably rephrase “contains exactly one prime ideal” as “every element of $A$ is either nilpotent or invertible”.)
Ingo: that is all very cool. You should record it somewhere on the nLab.
@Mike
you don’t mean a literal category-theoretic reflection, do you?
Sure, take that as an informal notion.
Re 11: Sure, I’ll do that. I’m currently in the very last steps of finishing my PhD thesis and will then incorporate some topics of interest into the nLab.
I’m confused about the definition of morphism of locally ringed space in terms of the comorphism $f^\sharp : \mathcal{O}_Y \to f_*\mathcal{O}_X$ being a map of local rings. At first glance, this doesn’t make sense to me because I see no reason a priori why the direct image $f_* : \operatorname{Sh}X\to\operatorname{Sh}Y$ should even send local rings to local rings, since having a maximal ideal is a geometric geometrical property. I believe the correct definition is rather that the transpose $f_\sharp : f^*\mathcal{O}_Y \to \mathcal{O}_X$ should send local rings to local rings; this typechecks because $f^*$ is an inverse image functor, and so it makes sense to restrict a local ring along it.
Furthermore, I believe it is not difficult to check that this definition is the one that actually corresponds to the usual definition involving stalks. Reformulate “$f_\sharp$ reflects invertible elements” as “the subsheaf of invertible elements of $f^*\mathcal{O}_Y$ is the pullback of the subsheaf of invertible elements of $\mathcal{O}_X$ along $f_\sharp$”. As $\operatorname{Sh}X$ has enough points, this is equivalent to asking for each stalk of this square to be a pullback square in $\mathbf{Set}$. But this is somewhat obvious, since invertibility of sections is stalk-local and the stalk $(f^*\mathcal{O}_Y)_x$ is the stalk $\mathcal{O}_{Y,f x}$.
Can someone confirm if this is correct? thanks!
Checking the reference https://stacks.math.columbia.edu/tag/01HA I agree the definition you mention doesn’t make sense. Even the definition of locally ringed space relies on the condition on stalks, so one would expect the definition of morphism to connect back to this.
OK thanks David, I’ll fix it.
I am just following with half an eye, being busy elsewhere.
But if I see correctly, the previous definition is the usual one for ringed spaces, where it is trivially equivalent to the other definition, by adjointness. The distinction only shows up here in the locally ringed situation.
A comment along these lines would be useful to add to the entry here.
That’s a good point, Urs — but there is a kind of related confusion on the page for comorphisms, which I have commented on here: https://nforum.ncatlab.org/discussion/16204/comorphism/#Comment_108448
I’m a little worried I’m missing something…
opening a textbook… p 247-248 in
essentially agrees with what the entries here say/said.
opening the next best textbook… p. 55 in
says it more the other way around :-)
(Sorry, I am not really looking into it, just bringing up some references.)
@jon what does it mean for a map of sheaves of rings to be a map of local rings? IIRC the structure sheaf of a locally ringed space is not a sheaf of local rings, and it’s only the stalks that are local rings.
@DavidRoberts It actually is a local ring object in the internal sense. It happens that you can test this stalkwise since the topos has enough points…
@jon in the internal logic? This should be noted on the page as a reminder. And there should be a definition of what it means to be a map of local-rings-defined-in-internal-logic (perhaps it’s obvious, but as written it looked like a category error)
I don’t think it would be any more of a category error than speaking about a morphism of groups in a category with finite products. Local rings are a geometric theory, so they make sense in any topos. So I would not agree that there is any need for a comment about internal local rings, but certainly it would not be bad to include one.
I have adjusted and expanded the wording of the two examples currently in the entry.
The example concerning schemes (here) I have expanded out as follows:
Historically, schemes are thought of as locally ringed spaces and this application of the notion to algebraic geometry motivated much of its development. (However, already Grothendieck (1973) pointed out that it is often more frutiful to view schemes instead via their functor of points, see at functorial geometry for more.)
And in the example concerning smooth manifolds (here), after
This inclusion of smooth manifolds into locally ringed spaces is fully faithful. For a proof, see Lucas Braune’s comment at Math.SE:511604.
I have appended this remark:
(In fact, already the embedding of smooth manifolds into formal duals of R-algebras is fully faithful, which means that all smooth manifolds are even “affine schemes”, in a sense.)
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