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Stream: learning: questions

Topic: Equivalence of two definitions of a scheme

Valery Isaev (Jan 16 2025 at 07:05):

There are two definitions of a scheme: as a ringed space and as a sheaf on $\mathrm{CRing}^\mathrm{op}$. It is obvious how to go from the former to the latter. It is less clear to me how to construct a ringed space from a sheaf. All the references that I found are either in French or don't contain the actual construction.

Does anybody have a reference in English for this, or maybe could give a quick description of this construction here?

Damiano Mazza (Jan 16 2025 at 08:46):

These notes by Sam Raskin are the closest thing I have found to something developing algebraic geometry entirely using the functorial perspective (i.e., schemes are defined from the start as certain functors $\mathbf{CRing}\to\mathbf{Set}$, locally ringed spaces are never even introduced). Unfortunately it does not describe explicitly how to build a scheme-as-locally-ringed-space from a scheme-as-functor, but I think it gives enough material to do that oneself, maybe with some help from the definitions given on the nLab. I don't have time to do it now but I'd be happy to try it myself later!

Josselin Poiret (Jan 16 2025 at 13:45):

Valery Isaev said:

There are two definitions of a scheme: as a ringed space and as a sheaf on $\mathrm{CRing}^\mathrm{op}$.

do you mean resp. locally ringed space and locally affine zariski sheaf or something more general?

Valery Isaev (Jan 16 2025 at 13:53):

Yes, at least that, but I was also wondering about generalizations, e.g., whether we can construct some space starting with an arbitrary Zariski sheaf. I thought I could try to generalize this myself (maybe realizing it is impossible in the process), but I cannot find anywhere even the basic case.

Josselin Poiret said:

Valery Isaev said:

There are two definitions of a scheme: as a ringed space and as a sheaf on $\mathrm{CRing}^\mathrm{op}$.

do you mean resp. locally ringed space and locally affine zariski sheaf or something more general?

Yes, at least that, but I was also wondering about generalizations, e.g., whether we can construct some space starting with an arbitrary Zariski sheaf. I thought I could try to generalize this myself (maybe realizing it is impossible in the process), but I cannot find anywhere even the basic case.

Ali Caglayan (Jan 16 2025 at 14:09):

Zhen Lin's thesis has quite a bit on this topic: https://www.repository.cam.ac.uk/items/22303859-8bb1-4558-bc82-08cb869c38f4 He also has another note on this topic: https://zll22.user.srcf.net/talks/2015-01-29-FunctorOfPoints.pdf

I tried to answer this question myself a few years ago and I don't recall finding a satisfactory answer. People who use functor-of-points at the Zariski level are typically interested in things like group schemes, but they are rarely interested in the locally ringed space. Demazure and Gabriel have a book in French and English on this point of view. The terminology is a bit funny but Z-functors correspond to sheaves in the Zariski topology IIRC.

There are then algebraic gometers who are interested in starting with a prebuilt category of Schemes and putting funny topolgoies on that to make more general scheme-looking objects. There is a lot of literature on this kind of investigation, but almost nobody thinks about locally ringed spaces at this point. A natural question to ask is: Can you do this from CRing? I think the answer is generally no, only Zariski and etale schemes can be built that way, the other topologies need a category of schemes to begin with. Here is an atlas of topologies for example: https://pbelmans.ncag.info/topologies-comparison/ I can't find a link, but there are some surveys on existing ones.

Ali Caglayan (Jan 16 2025 at 14:11):

A general impression I recived when I asked working algebraic geometers about the functor-of-points view is that they mostly view it as a one-way-street. I have also heard the same claims that "these appraoches are equivalent" but I have yet to see anybody sit down and do scheme theory with Zariski sheafs. If I am wrong, please let me know. I am dying to know!

Oscar Cunningham (Jan 16 2025 at 14:58):

The category $\mathcal{C}$ of sheaves on $\mathbf{CRing}^\mathrm{op}$ is a topos. So for any scheme $X$, the over category $\mathcal{C}/X$ is also a topos. Any topos has a 'localic reflection', the locale of subterminal objects. Is the localic reflection of $\mathcal{C}/X$ the underlying locale of the original scheme?

Graham Manuell (Jan 16 2025 at 18:19):

Oscar Cunningham said:

The category $\mathcal{C}$ of sheaves on $\mathbf{CRing}^\mathrm{op}$ is a topos. So for any scheme $X$, the over category $\mathcal{C}/X$ is also a topos. Any topos has a 'localic reflection', the locale of subterminal objects. Is the localic reflection of $\mathcal{C}/X$ the underlying locale of the original scheme?

Unfortunately, I believe not. I think it ends up being bigger.

Ali Caglayan (Jan 16 2025 at 19:55):

A scheme isn't exactly a sheaf however. It is something like a sheaf with some extra data like being covered by representables in a suitable way.

John Baez (Jan 16 2025 at 21:29):

I thought that was just an extra property, not extra structure. (To me 'data' = 'structure', which we need to insist is preserved by morphisms.)

Martti Karvonen (Jan 17 2025 at 11:00):

Oscar Cunningham said:

The category $\mathcal{C}$ of sheaves on $\mathbf{CRing}^\mathrm{op}$ is a topos. So for any scheme $X$, the over category $\mathcal{C}/X$ is also a topos. Any topos has a 'localic reflection', the locale of subterminal objects. Is the localic reflection of $\mathcal{C}/X$ the underlying locale of the original scheme?

Don't we need to restrict to finitely presentable rings or some other suitable (essentially) small subcategory? I thought that taking sheaves on a large site won't result in a topos in general.

Oscar Cunningham (Jan 17 2025 at 11:34):

Right, the safest way to define it is as $\mathrm{Sh}(\mathbf{CRing}_{\mathrm{fp}}^\mathrm{op})$. But my understanding is that since the finitely presentable commutative rings do in fact generate all commutative rings by colimits, we get the same category if we just do $\mathrm{Sh}(\mathbf{CRing}^\mathrm{op})$. The sheaves on a large site aren't a topos in general, but it's safe to do in this case because we have this nice small subcategory.

David Corfield (Jan 17 2025 at 13:52):

For anyone with the time, the discussion on the Secret Blogging Seminar of Algebraic geometry without prime ideals is quite something. An incredibly impressive array of mathematicians get stuck into the pedagogical value of teaching the functor-of-points approach.

Ali Caglayan (Jan 17 2025 at 14:45):

Could this "equivalence" be a case of "fully faithful essentially surjective" hiding the inverse? I don't come across many equivalences of categories that don't have good descriptions of their inverses, but I have a feeling that this might be such a case which is why we are struggling to describe it.

Mike Shulman (Jan 17 2025 at 16:40):

I haven't read any of the references mentioned here, but I always assumed the construction was basically to start with some covering by representables, as Ali mentioned is assumed to exist when defining schemes as functors, inspect the way in which the representables are "glued together", and then glue together the corresponding affine schemes (qua locally ringed spaces) in the "same way". In other words, we shouldn't look for a general way of turning a sheaf into a ringed space; instead we should think of both definitions of scheme as the closure of $\mathrm{CRing}^{op}$ under good colimits in two different larger categories (sheaves or locally ringed spaces, respectively), which turn out to be equivalent because those two categories are both "sufficiently" freely generated by $\mathrm{CRing}^{op}$.

Josselin Poiret (Jan 21 2025 at 09:22):

there probably is a more direct way by looking at all closed k-points of your scheme and building the space that way. Using this method, you don't have to show that all possible coverings by affine schemes give the same gluing viewed as a locally ringed space

Damiano Mazza (Jan 21 2025 at 13:08):

How about non-empty schemes having no closed point? (I have absolutely no idea about how to construct them, but I've heard they exist).

Josselin Poiret (Jan 21 2025 at 15:17):

ah sorry, i meant point, not necessarily closed

John Baez (Jan 21 2025 at 17:23):

Theorem 4.6 here contains an example:

• Karl Schwede, Gluing schemes and a scheme without closed points.

Unfortunately it is quite complicated. The earlier parts of this paper seem to have a lot of useful insights that are easier to understand.

Valery Isaev (Jan 23 2025 at 10:23):

I checked the book by Demazure and Gabriel and I think I got the main gist. First, we define the notion of an open subfunctor of a functor $F : \mathbf{CRing} \to \mathbf{Set}$. I think the set of such functors form a frame, which gives us the underlying locale of the scheme. Then, for every such subfunctor $G$, the set $\mathbf{Set}^\mathbf{CRing}(G,U)$ (where $U$ is the forgetful functor) is naturally a ring. Next, if $F$ is a sheaf, this collection of rings should be a sheaf of rings. So, we get our ringed space/locale.

There is a set-theoretic issue though: the locale we constructed is large in general. I'm not entirely sure how to fix this problem. Maybe if the original functor $F$ is covered by a small set of affine schemes, it is enough to enforce this locale to be essentially small.

Valery Isaev (Jan 23 2025 at 10:26):

Oh, I just noticed that the ring $\mathbf{Set}^\mathbf{CRing}(G,U)$ is also large in general, and I don't know how to fix that.

David Michael Roberts (Jan 23 2025 at 10:40):

Isn't U representable? Not sure how this might help, though

Valery Isaev (Jan 23 2025 at 13:08):

It is; not sure either :smile:

Damiano Mazza (Jan 23 2025 at 13:25):

No representability doesn't help. However, as observed in this (unanswered) MO question, $G$ being a small copresehaf is enough. I think that @Valery Isaev is right, being covered by representable functors must be the key here: since the covering family is small, $G$ should end up being small and everything should be fine (but I don't know for sure, I never worked this out in detail).

Morgan Rogers (he/him) (Jan 23 2025 at 13:33):

You said $F$ should be a sheaf; if you impose this condition earlier, you can use the fact that sheaves on $\mathbf{CRing}^{\mathrm{op}}$ are determined by their values on finitely presentable rings (if I'm not getting something backwards here), so size issues are contained.

Damiano Mazza (Jan 23 2025 at 15:14):

I don't follow. $\mathbb R$ (say) is certainly a colimit of finitely presentable rings, but the objects in this colimit do not form a covering family in the Zariski topology.

Damiano Mazza (Jan 23 2025 at 15:18):

Sorry, in fact, it's even worse: colimits in $\mathbf{CRing}$ become limits in $\mathbf{CRing}^{\mathrm{op}},$ so they have really nothing to do with covering families. I am confused, I don't see how you can determine the image of a non-finitely-presentable ring by knowing only the images of the finitely presentable ones...

Morgan Rogers (he/him) (Jan 23 2025 at 19:23):

So yes, I was getting things backwards.

John Baez (Jan 23 2025 at 19:35):

Valery Isaev said:

I checked the book by Demazure and Gabriel and I think I got the main gist. First, we define the notion of an open subfunctor of a functor $F : \mathbf{CRing} \to \mathbf{Set}$. I think the set of such functors form a frame, which gives us the underlying locale of the scheme. Then, for every such subfunctor $G$, the set $\mathbf{Set}^\mathbf{CRing}(G,U)$ (where $U$ is the forgetful functor) is naturally a ring. Next, if $F$ is a sheaf, this collection of rings should be a sheaf of rings. So, we get our ringed space/locale.

That's really fascinating! I've been writing a paper about functors $F: \mathsf{CRing} \to \mathsf{Set}$, so this is the sort of thing I should study.

I don't care about size issues when I'm just thinking idly about topics, but you've got me curious: to Demazure and Gabriel mention the size problem you mention? Do they overlook it? Or does this problem not affect their claims?

Valery Isaev (Jan 23 2025 at 20:36):

John Baez said:

Valery Isaev said:

I checked the book by Demazure and Gabriel and I think I got the main gist. First, we define the notion of an open subfunctor of a functor $F : \mathbf{CRing} \to \mathbf{Set}$. I think the set of such functors form a frame, which gives us the underlying locale of the scheme. Then, for every such subfunctor $G$, the set $\mathbf{Set}^\mathbf{CRing}(G,U)$ (where $U$ is the forgetful functor) is naturally a ring. Next, if $F$ is a sheaf, this collection of rings should be a sheaf of rings. So, we get our ringed space/locale.

That's really fascinating! I've been writing a paper about functors $F: \mathsf{CRing} \to \mathsf{Set}$, so this is the sort of thing I should study.

I don't care about size issues when I'm just thinking idly about topics, but you've got me curious: to Demazure and Gabriel mention the size problem you mention? Do they overlook it? Or does this problem not affect their claims?

They fix a universe and mostly work with rings living in it. So, essentially they work with a small category of rings. It doesn't seem they say much beyond that.

John Baez (Jan 24 2025 at 00:09):

Okay, that's fine with me.

David Michael Roberts (Jan 24 2025 at 02:20):

Damiano Mazza said:

No representability doesn't help. However, as observed in this (unanswered) MO question, $G$ being a small copresehaf is enough. I think that Valery Isaev is right, being covered by representable functors must be the key here: since the covering family is small, $G$ should end up being small and everything should be fine (but I don't know for sure, I never worked this out in detail).

Yes, I would have assumed that using smallness of G is essential here, and one would hope that a subsheaf of a small sheaf is small (modulo sorting out opposites, sheaves/cosheaves etc)

Damiano Mazza (Jan 24 2025 at 07:49):

Valery Isaev said:

They fix a universe and mostly work with rings living in it. So, essentially they work with a small category of rings. It doesn't seem they say much beyond that.

I don't have a copy of the book with me but I remember their treatment of size issues being a little more elaborate than that. I remember them introducing universes and explicitly using several of them in their definitions. This MO question says that they "use three nested universes". As a matter of fact, I don't think you can escape some kind of universe-juggling if you want to formally reconstruct schemes from coprehseaves. As you noticed, at some point you must handle things that in general are not small. Of course one then argues that, in the special case of interest, these things are actually small, but how do you make this argument go through if those general things don't exist in the first place?

Damiano Mazza (Jan 24 2025 at 08:16):

(Notice that the MO question I linked to asks precisely whether one can do everything using only one universe. The answer seems to be "yes" if one wants to develop the functorial approach independently, and the solution seems to be "use small functors". But the answer is not so clear as to whether one can still formally relate the functorial approach with the locally-ringed-sheaf approach. The answer was accepted but, if you look at the comments, there seems to be an unresolved technical problem with one part of the answer).

Valery Isaev (Jan 24 2025 at 19:15):

Damiano Mazza said:

Valery Isaev said:

They fix a universe and mostly work with rings living in it. So, essentially they work with a small category of rings. It doesn't seem they say much beyond that.

I don't have a copy of the book with me but I remember their treatment of size issues being a little more elaborate than that. I remember them introducing universes and explicitly using several of them in their definitions. This MO question says that they "use three nested universes". As a matter of fact, I don't think you can escape some kind of universe-juggling if you want to formally reconstruct schemes from coprehseaves. As you noticed, at some point you must handle things that in general are not small. Of course one then argues that, in the special case of interest, these things are actually small, but how do you make this argument go through if those general things don't exist in the first place?

They require only two universes, but they do not really need the second one. They introduce it probably to avoid talking about classes. They even mention themselves that the second universe is not really needed.

Now, their definition of schemes is based on Z-functors, which are functors from the category of commutative rings living in the small universe to the category of sets in the large universe. I don't see they discuss functors from large rings to sets anywhere.

Damiano Mazza (Jan 24 2025 at 19:57):

Ok, I misremembered then! And I guess that my worries about needing universes are unfounded. I'd still be curious to understand why they use a category of large sets. If they say that they don't really need it, maybe this choice makes simpler something which would otherwise be more tedious or complicated. I wonder what it is.

John Baez (Jan 24 2025 at 23:51):

I can easily imagine someone using a category of large sets to avoid mentioning classes. Classes are a bit annoying in some approaches to set theory, since you kinda want everything to be a set.