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Stream: learning: questions

Topic: classifying topos of local rings

sarahzrf (Feb 19 2021 at 01:42):

The nlab claims that the Zariski topos classifies local rings, but the category of presheaves on the same site should classify general commutative rings, im pretty sure. But as far as i can tell, the category of points of a subtopos is in general a full subcategory of the category of points of the overall topos, so that means that whatever is classified by the Zariski topos should be a full subcategory of CRing. That is, the Zariski topos thinks that morphisms between the things it classifies are arbitrary ring homomorphisms, not just invertibility-reflecting ring homomorphisms.

sarahzrf (Feb 19 2021 at 01:44):

i guess this is an arguably reasonable interpretation of a "category of local rings", but it does mean that the lax slice over the Zariski topos is not a direct generalization of the usual notion of "locally ringed locale"

sarahzrf (Feb 19 2021 at 01:45):

so:

1. does my reasoning here check out, or is there some reason ive missed why the Zariski topos actually does give rise to local ring homomorphisms?
2. if what ive said is correct, then is there a nice topos that does classify local rings with the correct morphisms?

John Baez (Feb 19 2021 at 02:22):

Hi! Long time no see!

Theorem VIII.6.3 in Sheaves in Geometry and Logic says that the Zariski topos classifies local rings and arbitrary ring homomorphisms between them. Actually you have to read the proof to find where they admit it's giving a full subcategory of the commutative ring objects, but they do say it.

sarahzrf (Feb 19 2021 at 02:30):

how about (2), then?

John Baez (Feb 19 2021 at 04:42):

That's above my pay grade: I don't know enough algebraic geometry to know in which circumstances invertiblity-reflecting morphisms of local rings count as the "correct" kind of morphisms, and I don't know a way to get a classifying topos for local rings and those morphisms. I can just guess that if you want to force your morphisms of local rings to be local, you probably need to treat the maximal ideal as extra structure somehow rather than just an extra property.

Peter Arndt (Feb 19 2021 at 10:15):

I don't think there is such a topos in itself. A first try to realize John's idea could be to introduce a relation symbol $\mathfrak{m}$ and add the axiom $\mathfrak{m}(x) \Leftrightarrow \exists y: (1-x)y=1$, which forces $\mathfrak{m}$ to be interpreted as the maximal ideal of the local ring. But the definition of morphism of such strutcures says that "belonging to $\mathfrak{m}$" has to be _preserved_, not reflected. I don't see how this backwards condition, of being reflected, could be realized with just a single geometric theory.

Peter Arndt (Feb 19 2021 at 10:17):

What you can do, however, is set up a classifying topos of local homomorphisms of local rings: The theory would be two-sorted, each sort should be endowed with the structure and properties of a local ring object and there should be a single map between them. To ask that that map reflects the maximal ideal is no problem. Let's call the classifying topos of this theory $Set[LocHom]$.

Peter Arndt (Feb 19 2021 at 10:24):

You have two forgetful maps from that new topos to the classifying topos of local rings; one selecting the source and one selecting the target. You have a map going the other way selecting the identity morphism of local rings. And from two models of the theory of a local homomorphism of local rings one can produce a new one by composing (if source and target fit). This gives a category object in the category of toposes, $Set[LocHom] \rightrightarrows Set[LocRing]$. And the class of geometric morphisms from some topos into this category object (i.e. pairs of geom. morphism compatible with everything in sight, you know...) should now yield a category. Starting from the category of sets you should get the category of local rings and local homomorphisms.

Peter Arndt (Feb 19 2021 at 10:25):

I am not sure how natural transformations are to interact with all that.

Peter Arndt (Feb 19 2021 at 10:26):

I could imagine that @Matthias Hutzler knows if I said something wrong, and whether there are references for that.

Morgan Rogers (he/him) (Feb 19 2021 at 11:13):

Peter Arndt said:

What you can do, however, is set up a classifying topos of local homomorphisms of local rings: The theory would be two-sorted, each sort should be endowed with the structure and properties of a local ring object and there should be a single map between them. To ask that that map reflects the maximal ideal is no problem. Let's call the classifying topos of this theory $Set[LocHom]$.

When you say it's "no problem", how do you see this working? I don't immediately see a way of talking about reflection (in geometric logic) in the signature you've set up.

Matthias Hutzler (Feb 19 2021 at 11:34):

Peter Arndt said:

the axiom $\mathfrak{m}(x) \Leftrightarrow \exists y: (1-x)y=1$ [...] forces $\mathfrak{m}$ to be interpreted as the maximal ideal of the local ring.

Wait a second. The maximal ideal of a local ring contains precisely the non-invertible elements. But $x$ non-invertible is not at all equivalent to $1-x$ invertible! :smile:

Matthias Hutzler (Feb 19 2021 at 11:37):

Also, introducing a new relation symbol and then immediately forcing it to be equivalent to some geometric formula never changes the models (or model homomorphisms) of the theory in any topos -- it is an equivalence extension. (We might call such theory extensions "extensions by definitions".)

Matthias Hutzler (Feb 19 2021 at 11:45):

Also, being a local ring homomorphism actually means that the maximal ideal is preserved (equivalently, that invertibility (which is always preserved) is reflected). The fact that there are non-local homomorphisms of local rings tells us that non-invertibility can not be expressed as a geometric formula in the theory of local rings. (Whereas invertibility can of course be expressed: $\exists y: xy=1$.)

Matthias Hutzler (Feb 19 2021 at 11:46):

I hope this is the kind of comments you wanted, @Peter Arndt! :big_smile:

Matthias Hutzler (Feb 19 2021 at 11:52):

Morgan Rogers (he/him) said:

When you say it's "no problem", how do you see this working? I don't immediately see a way of talking about reflection (in geometric logic) in the signature you've set up.

If $f : A \to B$ is the function symbol for the ring homomorphism we want to be local, we can add this axiom:
$(\exists (y : B).\, f(x)y = 1) \vdash_{x : A} (\exists (y : A).\, xy = 1)$.

Matthias Hutzler (Feb 19 2021 at 12:04):

Regarding @sarahzrf's original question: There is a closely related Question on MathOverflow: What is the theory of local rings and local ring homomorphisms?.
(Recall that every Grothendieck topos is the classifying topos for some geometric theory and every geometric theory has a classifying topos.)

Matthias Hutzler (Feb 19 2021 at 12:17):

And I now fully agree with the accepted answer there by Achilleas K! (Is that @Achilleas Kryftis?) (Upshot: there can't be a classifying topos / geometric theory of local rings with only the local homomorphisms.)

Matthias Hutzler (Feb 19 2021 at 12:37):

Here is another way to see it: If $\mathcal{E}_1$ classifies local rings (with all homomorphisms) and $\mathcal{E}_2$ classifies local rings (with local homomorphisms) then we have two universal (in two slightly different senses) local rings $R_1$ in $\mathcal{E}_1$ and $R_2$ in $\mathcal{E}_2$. This implies that we have geometric morphisms $f : \mathcal{E}_1 \to \mathcal{E}_2$ and $g : \mathcal{E}_2 \to \mathcal{E}_1$ with $f^* R_2 \cong R_1$ and $g^* R_1 \cong R_2$. But then also $f^*g^* R_1 \cong R_1$ and $g^*f^* R_2 \cong R_2$, so these are actually quasi-inverse geometric morphisms and our toposes are equivalent.

Morgan Rogers (he/him) (Feb 19 2021 at 12:47):

Why should there be mappings in both directions? Surely one of them will be a local ring in a more restricted sense, and so the other will not qualify as an instance of it.

Morgan Rogers (he/him) (Feb 19 2021 at 13:02):

Matthias Hutzler said:

And I now fully agree with the accepted answer there by Achilleas K! (Is that Achilleas Kryftis?) (Upshot: there can't be a classifying topos / geometric theory of local rings with only the local homomorphisms.)

I don't agree with that answer. It's true that a geometric theory determines its morphisms, and that if we know the models of a theory in every topos then there is at most one topos up to equivalence representing it. However, the name "local ring" isn't the same as a formal determination of the objects in every topos, and we can modify a theory while retaining the same objects/data as models in (for example) Set.

For example: the topos classifying the theory of decidable objects is distinct from the one classifying the theory of objects. These theories have the same models in any Boolean topos, but the former has morphism consisting of monomorphisms rather than arbitrary morphisms.

Morgan Rogers (he/him) (Feb 19 2021 at 13:04):

So Achilleas' answer doesn't exclude the possibility of a geometric theory whose category of models in Set is exactly the theory of local rings and local ring homomorphisms.

Morgan Rogers (he/him) (Feb 19 2021 at 13:30):

In fact, since invertibility is automatically preserved, I think we can adapt the construction of the theory of decidable objects for this purpose!
Let's start with what @Peter Arndt suggested, taking the single-sorted theory of rings and adding a unary relation $\mathfrak{m}$, along with a short-hand $\mathfrak{i}$ where $\mathfrak{i}(x) \dashv \vdash_{x:A} (\exists y:A) xy = 1$. Then let's add the axioms
$\top \vdash_{x:A} \mathfrak{m}(x) \vee \mathfrak{i}(x)$ and
$\mathfrak{m}(x) \wedge \mathfrak{i}(x) \vdash_{x:A} \bot$.
A $\mathbf{Set}$-model of this theory is still a local ring, but the morphisms have to respect the relations, so preserve the maximal ideal, as required.

Morgan Rogers (he/him) (Feb 19 2021 at 13:36):

This already causes problems for spectra of rings, though, as remarked in another answer to that same MO question. The structure sheaf on the Zariski topos of $\mathbb{Z}$ (an internal local ring in the usual sense) apparently doesn't have $R = R^* \cup \mathfrak{m}_R$, so is not a model of the theory I just described.

Morgan Rogers (he/him) (Feb 19 2021 at 13:46):

Okay I may have misquoted that example; the quoted paper is this one, which does construct a local ring counterexample over the Zariski topos of $\mathbb{Z}$, but I don't think it's the structure sheaf. Also, that paper is using the condition "$x \in \mathfrak{m}$ if and only if $\forall y \, \mathfrak{i}(1-xy)$", which is subtly but importantly different from what I wrote.
In any case, there will be work to do to verify that the things which you want to ensure are local rings in other toposes still are!

Matthias Hutzler (Feb 19 2021 at 14:31):

Of course it makes a big difference whether we only want our topos (or theory) to have the right category of points (or Set-based models) or we want it to classify the right things (have the right models in any Grothendieck topos).
I think we can call the theory you described the theory of local rings with "decidable invertibility", right? And I agree that it has the desired category of Set-based models. :smile:

Morgan Rogers (he/him) (Feb 19 2021 at 14:33):

But how do you decide what the "right models in any Grothendieck topos" are ahead of time? If two theories give the same models in the subclass of Grothendieck toposes you care about, there is nothing a priori that says which is the "right" one.

Matthias Hutzler (Feb 19 2021 at 14:37):

I think what I had in mind (unconsciously for a while) is the pseudofunctor from Grothendieck toposes to categories that assings to any topos $\mathcal{E}$ the category of local rings in $\mathcal{E}$ (a non-full subcategory of the category of all rings in $\mathcal{E}$). We can ask: Is this pseudofunctor representable (up to equivalence)? This is how I would have made the question "Is there a classifying topos of local rings?" precise.

Morgan Rogers (he/him) (Feb 19 2021 at 14:39):

Ah that makes some sense, but it relies on being able to formally say what a local ring is, so it ends up being circular!

Matthias Hutzler (Feb 19 2021 at 14:46):

Oh, I think I see what you mean. But I think it is "clear" what local rings and local ring homomorphism are in any topos, by expressing "invertibility is reflected" in the internal language, as above.

Morgan Rogers (he/him) (Feb 19 2021 at 14:53):

That certainly sounds like a reasonable definition of morphism of local rings once we know which the local rings are, but I'll stick to my position that the fact that the classical definition of a local ring, "having a unique maximal ideal", is not directly expressible in geometric logic means that the objects are flexible, as is the exact axiomatisation of what their homomorphisms are. If what you suggested above gives @sarahzrf a theory with the property she was looking for, then great!

Matthias Hutzler (Feb 19 2021 at 14:54):

Matthias Hutzler said:

I think it is "clear" what local rings and local ring homomorphism are in any topos

(Note however that it doesn't even matter, what exactly @sarahzrf meant by "the correct morphisms" of local rings, as long as it is a proper subset of all ring homomorphisms between local rings that contains all isomorphisms.)

Matthias Hutzler (Feb 19 2021 at 14:56):

Morgan Rogers (he/him) said:

the objects are flexible

I was assuming that we want the same objects as what is classified by the big Zariski topos. This is how I read the question. :smile:

Matthias Hutzler (Feb 19 2021 at 14:59):

Morgan Rogers (he/him) said:

If what you suggested above gives sarahzrf a theory with [...]

Just to be clear: There is no geometric theory for the "pseudofunctor of local rings".

Peter Arndt (Feb 19 2021 at 15:28):

Thanks, @Matthias Hutzler, for setting me straight above! I first defined the maximal ideal as the set of non-invertible elements, and then tried to get rid of the negation. But as you showed, that was not only wrong, but bound to fail in principle. Defining the set of invertibles, as Morgan did, is a better way. But then we need reflection instead of preservation, which is not expressible...

Peter Arndt (Feb 19 2021 at 15:35):

Matthias Hutzler said:

Also, introducing a new relation symbol and then immediately forcing it to be equivalent to some geometric formula never changes the models (or model homomorphisms) of the theory in any topos -- it is an equivalence extension. (We might call such theory extensions "extensions by definitions".)

I agree with the statement on models, but not the one on homomorphisms. An additional relation symbol, even if definable in terms of the other symbols, changes the notion of morphism. On the extreme end (now in traditional model theory, let's say), if you define a relation symbol for absolutely every formula, the only morphisms are elementary embeddings.

Peter Arndt (Feb 19 2021 at 15:37):

But so what about that category object in toposes? Do you know of any place where that appears in the literature?

Matthias Hutzler (Feb 19 2021 at 15:48):

Peter Arndt said:

An additional relation symbol, even if definable in terms of the other symbols, changes the notion of morphism.

Not if it is definable by a geometric formula, I think. Truth of geometric formulas is preserved by model homomorphisms.

Matthias Hutzler (Feb 19 2021 at 16:01):

Peter Arndt said:

But so what about that category object in toposes? Do you know of any place where that appears in the literature?

That looks correct to me! I don't think I have seen it anywhere before.

Peter Arndt (Feb 19 2021 at 16:30):

Matthias Hutzler said:

Not if it is definable by a geometric formula, I think. Truth of geometric formulas is preserved by model homomorphisms.

Ah, of course - I have actually even used that in other places! :silence: Thanks! :+1:

Jens Hemelaer (Feb 19 2021 at 19:04):

sarahzrf said:

i guess this is an arguably reasonable interpretation of a "category of local rings", but it does mean that the lax slice over the Zariski topos is not a direct generalization of the usual notion of "locally ringed locale"

I'm surprised by this, but as @John Baez already confirmed, the morphisms in the lax slice are indeed only morphisms of ringed toposes and not of locally ringed toposes.

There is this paper by Lurie in which he defines the category of locally ringed spaces in a more topos-theoretic way (see for example Remark 2.5.12). Suppose you have locally ringed spaces $f : \mathbf{Sh}(X) \to \mathcal{E}$ and $g : \mathbf{Sh}(Y) \to \mathcal{E}$, where $X$ and $Y$ are topological spaces and $\mathcal{E}$ is the Zariski topos. Then a morphism of ringed spaces is a geometric morphism $h : \mathbf{Sh}(X) \to \mathbf{Sh}(Y)$ together with a natural transformation $\alpha : h^* g^* \Rightarrow f^*$, with the additional property that
$h^*g^*(A[1/a]) \simeq h^*g^*(A) \times_{f^*(A)} f^*(A[1/a])$
for any localization $A \to A[1/a]$ of finitely generated commutative rings.

We have in particular that $f^*(\mathbb{Z}[t])$ and $g^*(\mathbb{Z}[t])$ are the local ring objects in $\mathbf{Sh}(X)$ respectively $\mathbf{Sh}(Y)$, these are usually called $\mathcal{O}_X$ respectively $\mathcal{O}_Y$. It then makes sense to use the notation $\mathcal{O}_X^* = f^*(\mathbb{Z}[t,t^{-1}])$ and $\mathcal{O}_Y^* = g^*(\mathbb{Z}[t,t^{-1}])$. As one example of the condition above we then get the pullback diagram: $h^*(\mathcal{O}_Y^*) \simeq h^*(\mathcal{O}_Y) \times_{\mathcal{O}_X} \mathcal{O}_X^*$, so "invertible elements are reflected".
Maybe this last pullback diagram is enough in the sense that it implies the other pullback diagrams as well.

Lurie's definition uses an additional structure on the Zariski topos, by specifying which ring maps are localizations (he calls these the "admissible maps"). It would be interesting to see if this additional structure is really needed, or if everything can be reformulated without referring to localizations.

sarahzrf (Feb 20 2021 at 09:35):

wow, i did not expect this many replies

sarahzrf (Feb 20 2021 at 09:35):

a birthday present, i suppose