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Stream: learning: questions

Topic: schemes

John Baez (Nov 06 2021 at 16:49):

Suppose

$F : \mathsf{CommRing} \to \mathsf{Set}$

is the functor of points associated to some scheme. I imagine any such functor will be completely determined by its restriction to some subcategory $C \subseteq \mathsf{CommRing}$. Is there some famous subcategory that does the job and is small enough to be interesting? (I don't want to hear $C = \mathsf{CommRing}$, or $C$ is the full subcategory containing all free commutative rings.)

I'm hoping some nice result like this will follow from $F$ being a sheaf with respect to the Zariski topology.

Fawzi Hreiki (Nov 06 2021 at 19:50):

If you only need schemes locally of finite presentation, then you can take finitely presented rings.

Fawzi Hreiki (Nov 06 2021 at 19:51):

Restricting to connected rings (no nontrivial idempotents) gives you extensive sheaves (which schemes are).

Fawzi Hreiki (Nov 06 2021 at 20:02):

You can't represent the category of Zariski sheaves as a presheaf category since the theory of local rings is inherently existential

John Baez (Nov 06 2021 at 21:03):

It sounds like your answer to my question might be "connected rings".

John Baez (Nov 06 2021 at 21:04):

But you didn't exactly come out and say that the functor of points for any scheme is determined by its action on the full subcategory of connected rings. Is that true?

David Michael Roberts (Nov 06 2021 at 23:09):

On any lextensive category with a superextensive topology, a representable sheaf should be determined by its restriction to the connected objects, no?

Fawzi Hreiki (Nov 06 2021 at 23:46):

Yes

Fawzi Hreiki (Nov 06 2021 at 23:47):

The functor of points for any scheme (or any Zariski sheaf for that matter) is determined by its action on connected rings

Zhen Lin Low (Nov 07 2021 at 00:49):

No, that presupposes that all objects in an extensive category are disjoint unions of connected objects. This is not true for schemes, nor for topological spaces.

Fawzi Hreiki (Nov 07 2021 at 07:56):

But isn’t every ring uniquely a direct product of connected rings?

Zhen Lin Low (Nov 07 2021 at 07:57):

Even if that were so, infinite direct products of rings are not turned into infinite disjoint unions of schemes.

Jens Hemelaer (Nov 07 2021 at 08:19):

There is a morphism of schemes
$\bigsqcup_{i \in \beta(\mathbb{N})} \mathrm{Spec}(\mathbb{C}) \longrightarrow \mathrm{Spec}(\prod_{k \in \mathbb{N}} \mathbb{C} )$
which is a bijection on closed points. In both schemes, all points are closed, and they are given by ultrafilters, i.e. elements of $\beta(\mathbb{N})$.
If $X$ is any scheme that is a finite disjoint union of connected components, then each morphism $X \to \mathrm{Spec}(\prod_{k \in \mathbb{N}} \mathbb{C} )$ will lift in a unique way to a morphism $X \to \bigsqcup_{i \in \beta(\mathbb{N})} \mathrm{Spec}(\mathbb{C})$. Each connected component gets mapped to a single point.

So the subcategory $C$ definitely has to contain affine schemes that cannot be written as a finite disjoint union of connected components. This is another reason why you cannot take $C$ to be the category of connected rings.

As @Fawzi Hreiki already mentioned, the solution is to restrict to schemes of finite presentation. For example, if you work with schemes that are of finite presentation over a field $k$, then the scheme is uniquely determined by its functor of points on the connected rings.

Jens Hemelaer (Nov 07 2021 at 11:30):

Another less drastic solution would be to restrict to (locally) Noetherian schemes. Because every (locally) Noetherian scheme can be Zariski-covered by connected Noetherian affine schemes, it follows that a scheme like that will be completely determined by its functor of points on connected Noetherian rings.

Fawzi Hreiki (Nov 07 2021 at 11:40):

Just out of curiosity, how often does one actually deal with schemes which are not locally of finite presentation in practice?

David Michael Roberts (Nov 07 2021 at 11:44):

Aha, I was worried about the definition of 'connected', my intuition was trying to tell me something. Thanks for picking that up @Zhen Lin Low

Jens Hemelaer (Nov 07 2021 at 11:54):

Fawzi Hreiki said:

Just out of curiosity, how often does one actually deal with schemes which are not locally of finite presentation in practice?

For example $\mathrm{Spec}(\mathbb{Q})$ is not locally of finite presentation over $\mathrm{Spec}(\mathbb{Z})$. Or if you are working over $\mathbb{C}$ then $\mathrm{Spec}(\mathbb{C}(t))$ is not locally of finite presentation.

John Baez (Nov 07 2021 at 23:06):

Thanks, everybody! So it seems my original question (as opposed to various modified questions) has received no (correct) answer better than the category "free commutative rings".

John Baez (Nov 07 2021 at 23:08):

I asked: what is a subcategory $C \subseteq \mathsf{CommRing}$ such that the functor of points for any scheme,

$F : \mathsf{CommRing} \to \mathsf{Set}$

is determined by its restriction to $C$?

John Baez (Nov 07 2021 at 23:10):

I'm pretty sure we can use the full subcategory of free commutative rings, but this answer makes no use of the fact that $F$ is the functor of points of a scheme; it'd be true for any functor $F : \mathsf{CommRing} \to \mathsf{Set}$.

John Baez (Nov 07 2021 at 23:10):

Maybe knowing that $F$ is the functor of points of an (arbitrary) scheme doesn't help at all!

John Baez (Nov 07 2021 at 23:11):

That's interesting in itself, so thanks. I also learned a lot of other things....

Zhen Lin Low (Nov 07 2021 at 23:11):

Actually, I don't think the category of free commutative rings works either.

John Baez (Nov 07 2021 at 23:20):

Okay... yeah, now my argument for that seems to be dissolving into the mist. Thanks.

John Baez (Nov 07 2021 at 23:20):

So, right now it's a question whose best known answer is completely boring and unhelpful.

Fawzi Hreiki (Nov 07 2021 at 23:47):

There is the notion of a small presheaf which just means a presheaf which is a small colimit of representables. I’d imagine that the functor of points for any scheme (as defined classically) is small.

This is not an answer to your question though because the domain category is still all rings but it’s just a certain class of functors.

Jens Hemelaer (Nov 08 2021 at 08:27):

John Baez said:

I asked: what is a subcategory $C \subseteq \mathsf{CommRing}$ such that the functor of points for any scheme,

$F : \mathsf{CommRing} \to \mathsf{Set}$

is determined by its restriction to $C$?

Any such category $C$ will contain very pathological affine schemes (otherwise the schemes you build out of them will be 'nice' as well). Categories $C$ that work are:

• the category of non-noetherian rings
• the category of rings with infinitely many idempotents
• the category of rings with cardinality $\geq \kappa$, for some cardinal $\kappa$.

I admit that these are not very helpful solutions to the problem. Also, they work for any functor, not only for a functor of points of a scheme.

Zhen Lin Low (Nov 08 2021 at 09:05):

I think none of those work for all functors, but only for the stupid reason that you can't get $\{ 0 \}$ as a retract of anything. If you restrict to functors that preserve the terminal object then I think it works.

Jens Hemelaer (Nov 08 2021 at 09:39):

Thanks, I agree. So that's one small advantage of working with the functor of points of a scheme: you know that $F$ sends the trivial ring to a singleton, so it isn't a problem if $C$ does not contain the trivial ring.

Morgan Rogers (he/him) (Nov 08 2021 at 11:41):

John Baez said:

Suppose

$F : \mathsf{CommRing} \to \mathsf{Set}$

is the functor of points associated to some scheme. I imagine any such functor will be completely determined by its restriction to some subcategory $C \subseteq \mathsf{CommRing}$. Is there some famous subcategory that does the job and is small enough to be interesting?

Looking this up on Stacks Project, "the functor of points associated to some scheme", say $X$, is a representable functor $T \mapsto \mathrm{Hom}_{\mathrm{Sch}}(T,X)$. Representable functors preserve a lot of structure, in this instance (since we're considering this as a functor on CommRing) we're looking at preservation of all limits. So any subcategory of CommRing which generates everything under limits is good enough.

If there were cofree rings (if rings were coalgebraic over sets), we would have a canonical choice, since every ring would be a regular subring of a cofree ring. Jens mentioned some classes which are sufficiently large to guarantee that any ring can be found as a nice subring. Perhaps there are some embedding theorems for rings that you can use ("every commutative ring embeds into a ring having property X") to whittle down the subcategory a little; even better if you can find a result of the form "every commutative ring embeds into a product of rings having property X". All such results I can find with a quick search have caveats, though, so the best I can do to start you off is to the class of rings where every element is either invertible or a zero divisor, since the "total quotient" of any ring $R$ has this property and contains $R$ as a subring. There might a product decomposition you can implement based on the behaviour of the zero divisors and units. It's not clear to me whether every ring is a nice enough subring of its total quotient for this to be a valid answer, I'll let someone else work that out :grinning_face_with_smiling_eyes: .

Zhen Lin Low (Nov 08 2021 at 11:44):

The functor $\operatorname{Spec} : \textbf{CRing}^\textrm{op} \to \textbf{Set}$ does not preserve arbitrary colimits. So for a general scheme, the functor of points does not preserve all limits. (If it did it would be affine.)

Morgan Rogers (he/him) (Nov 08 2021 at 11:45):

But Spec isn't an instance of the type of functor under discussion, since there isn't a universal "point of a scheme"

Morgan Rogers (he/him) (Nov 08 2021 at 11:48):

Ah I see what you're saying: the representable functor will preserve colimits of schemes, but these don't coincide with (duals of) limits of rings?

Zhen Lin Low (Nov 08 2021 at 11:48):

Yes.

Morgan Rogers (he/him) (Nov 08 2021 at 11:54):

Ah well. The general strategy remains: work out what structure/constructions your functor preserves, find a subcategory which generates CRing under those constructions (I think we're still okay on finite limits? In which case my suggestion could still be a starting point).

Zhen Lin Low (Nov 08 2021 at 12:00):

No. There is an equivalence relation in the category of affine schemes that has a coequaliser that isn't preserved by the inclusion into the category of schemes. Namely, the standard presentation of the projective line as a gluing of two copies of the affine line.

Zhen Lin Low (Nov 08 2021 at 12:30):

It feels like there's a lot of wishful thinking about this topic. The long and short of it is that the category of schemes is a nasty category containing nasty objects. I tried quite hard to find a uniform way of constructing both the category of schemes and the category of manifolds but I didn't find anything much better than a more or less direct translation of the traditional chart-gluing construction (after eliminating the need to use the scaffolding provided by an underlying locale). You _can_ view it as a free cocompletion, but giving a precise statement requires pages of auxiliary definitions.

Fawzi Hreiki (Nov 08 2021 at 12:40):

Except the difference is that with manifolds the site is genuinely small

Fawzi Hreiki (Nov 08 2021 at 12:41):

Whereas with schemes there are all these compromises that need to be made to get a small site

Zhen Lin Low (Nov 08 2021 at 12:46):

I think that is basically because manifolds are finite-dimensional, whereas schemes can be infinite-dimensional – some of the nasty objects I alluded to above. Anyway, the universal property of this scheme-cocompletion construction is good enough to let you enlarge the site so in some sense the choice of site becomes irrelevant.

Fawzi Hreiki (Nov 08 2021 at 13:31):

So does the category of rings have a small condense subcategory?

Zhen Lin Low (Nov 08 2021 at 14:26):

I don't get what you mean by "so" but the answer is no. For any small subcategory you choose you can find a ring $A$ such that $\textbf{CRing} (A, B)$ is empty or trivial for all $B$ in your subcategory: take $A$ to be a field of cardinality greater than any ring in your chosen subcategory.