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In the examples section of extensive category, it is stated that the category of affine schemes is infinitary extensive.
For all I know, I was the one who stuck in that example. But is that statement actually true? I’m having trouble seeing it.
If $S$ is a commutative ring over $R$ (by which I mean under $R$ (-:), does the functor $S \otimes_R -: CAlg_R \to CAlg_R$ preserve arbitrary cartesian products? Because it seems that’s what we basically need for the statement to be true.
Just for the record: of course $S \otimes_R - : CAlg_R \to CAlg_R$ cannot be expected to preserve arbitrary products in general, i.e., products in $CRing$ are not stable under pushout. In the opposite category of affine schemes, this means coproducts are not stable under pullback, so affine schemes are not infinitary (l)extensive. Which leads me to say: I’m going to fix the nLab entry.
The lowly details of the proof of the first sentence in the preceding paragraph are perhaps best left to the reader’s discretion, but if anyone cares, my own demonstration can be found here.
Interesting! Looks like the error was probably a result of confusion at the time we were settling on the terminology “finitary extensive” and “infinitary extensive”.
I presume that the category of not-necessarily-affine schemes is infinitary extensive. If so, that means that infinite coproducts of affine schemes are no longer affine. Is that right?
That’s right: the disjoint union of infinitely many copies of $\operatorname{Spec} k$ ($k$ a field) is not affine because, well, its underlying space is not quasicompact.
First of all, Mike: congratulations on the birth of Arthur! And hope mom (Megan, right?) is doing well.
I suppose that the category of schemes is infinitary extensive, but I don’t know where the details might have been written down. Unless someone knows, I might take a crack at it – although I’m a little sketchy on limits and colimits of schemes and how they behave. It would obviously be good to know and have recorded.
Come to think of it, infinitary extensivity of the category of schemes should be conceptually obvious, given the observations that this is a full subcategory of the category of functors $CRing \to Set$ that is closed under finite limits and small coproducts (finite limits is a well-known result, and small coproducts follows very quickly from the Demazure-Gabriel functorial definition of scheme). I don’t see any set-theoretic difficulties getting in the way of that.
I don’t think it is literally closed under small coproducts: you have to apply Zariski sheafification, if I remember correctly. Basically this is because not all affine schemes are connected.
Hm.. need to think on this. But may need some time…
Thanks Todd! Yes, Megan is doing well, thanks for asking.
Could it be a similar full subcategory of the category of presheaves on connected affine schemes? Relatedly, is the category of affine schemes the free finite-coproduct completion of the subcategory of connected ones?
The two questions are closely related, I think. The answer to the second question is no: there are affine schemes that are not the disjoint union of their connected components. For example, $\operatorname{Spec} \mathbb{F}_2^{\mathbb{N}}$ is (homeomorphic to) the ultrafilter space $\beta \mathbb{N}$, which is totally disconnected.
Okay, I’m back. Thanks for pointing out #7, Zhen: that makes sense. So I should say instead that schemes are closed under finite limits and small coproducts of Zariski sheaves, and then the argument I had tried before should apply (since Zariski sheaves form an infinitary lextensive category, so must any full subcategory closed under finite limits and small coproducts).
Interesting! I had no idea there were affine schemes that looked like that topologically.
But there’s stuff here that I’ve never been too clear on, and that I’d love to get sorted out.
In #6, I blithely referred to the category of functors $CRing \to Set$. Implicitly I’m taking the functorial approach to schemes advanced by Demazure-Gabriel; if we follow the nLab’s description of this in the article scheme, then we consider $CRing^{op}$ as a large site, and form the category of sheaves with respect to the big Zariski topology, which is subcanonical. The category of schemes is by definition the full subcategory of sheaves that admit (small?) coverings by open affine subsheaves. (The nLab article doesn’t say “small” with regard to coverings; it’s just “family”, whatever that is supposed to mean.)
It looks as if the category of Zariski sheaves wants to be what some people call the (big) Zariski topos, but due to size considerations, it’s hard for me to see it as a Grothendieck topos. Actually, the nLab seems just a bit waffly on this: in the article Zariski site, the idea section refers to sheaves on a class-sized site $CRing^{op}$, and the definition section starts with this site, but then reverts to the small site of finitely presented rings with the inherited topology. I’ve seen people use one or the other, and I’ve never been too clear on the relationship between the sheaf categories or why people routinely invoke the class-sized site. (I see John Baez and Mike Shulman had similar concerns in 2009, and I didn’t see the issues truly clarified in that discussion.)
Demazure-Gabriel (i.e., their Introduction to Algebraic Geometry and Algebraic Groups), to which the nLab refers, do this: they assume two fixed universes $V$ and $U$, with $\mathbb{N} \in U \in V$, calling members of $U$ “small sets”. (One might suppose they mean that $V$ (with its membership relation) is assumed to satisfy ZFC, as does $U$ with the inherited membership relation, but they don’t actually say.) Then they call a small commutative ring with identity a “model”, and denote the category of models by $M$, which will be the supporting site for things like the Zariski topology. In other words, they will consider presheaves and sheaves on $M^{op}$, where presheaf means a functor $M \to Set$ to the category of sets (that live as elements in $V$).
They wave their hands a bit, essentially saying they don’t really need universes. Since they want $[M, Set]$ to be locally small, they don’t want $M$ to be “too large”. On the other hand, they want to be able to freely apply a whole bunch of commutative algebra constructions to models: residue class fields, rings of fractions, completions… They say it would be enough to assume that for any model $R$, every model with cardinality less than or equal to $Card(R)^\mathbb{N}$ is also a model (notice that finitely presented rings will not suit that purpose!), but justify the approach they do take by saying that many mathematicians “are accustomed to universes by now”, and also that they would like to use freely direct limits in their category of models.
Anyway, it looks like some of this could use some sorting out.
Yes, there are some set-theoretic issues at play.
Let me try asking a concrete question: is there a small Zariski site such that schemes are recognized as a full subcategory of locally affine Zariski sheaves?
Looking at what Demazure-Gabriel say, the answer seems to be a tentative ’yes’, but it’s somewhat hard for me to be sure.
Edit. The argument below is not quite correct…
No, there is no such a site. Suppose $M$ is a small full subcategory of $CRing$ such that the Yoneda representation ${CRing}^{op} \to [M, Set]$ is fully faithful. Since $M$ is small and $CRing$ is complete, we get a left adjoint, making ${CRing}^{op}$ a reflective subcategory of $[M, Set]$. But then ${CRing}^{op}$ would be an l.f.p. category, and we know that is impossible because $CRing$ itself is l.f.p. but not a preorder category.
Thanks very much, Zhen: that’s of course a very convincing argument.
In particular, sheaves on the large Zariski site cannot possibly form a Grothendieck topos. So what sort of category do they form? Cartesian closure, for instance, looks highly doubtful to me. How about exactness properties? Do they form an infinitary extensive pretopos? (Perhaps this is easy; I haven’t thought hard about it.)
Edit: A more general question is this. Suppose that $E$ is a infinitary extensive pretopos, and that $F \to E$ is a full embedding with a left exact left adjoint. Is it true that $F$ is an infinitary extensive pretopos?
Oops, what I said there was not quite correct: ${CRing}^{op}$ has to be accessibly embedded in order to get a contradiction. But in any case the point is that to get a small Zariski site we would have to find a small codense subcategory of $CRing$, which seems like an odd thing to have. I still think that the category of all schemes does not embed into the category of sheaves on any small Zariski site.
As for reflective subcategories of infinitary-pretoposes: while it is true that $[CRing, Set]$ is always an infinitary-pretopos, it’s not obvious whether or not Zariski sheaves form a reflective subcategory! (See, for instance, [Waterhouse, 1975].) But in fact they do, because of the following fact: for each commutative ring $A$, the class of Zariski-covering sieves on $Spec A$ (partially ordered by inclusion) has a small coinitial cofiltered family. So the colimit (which is indexed over the opposite of the class of Zariski-covering sieves) in the Grothendieck plus construction exists and preserves finite limits.
So we are in the situation you describe. Such a full subcategory is indeed an infinitary-pretopos: small colimits and finite limits in $F$ can be computed by applying the reflector to the corresponding construction in $E$, so any good properties those have in $E$ are inherited by $F$.
Okay, great – thanks again, Zhen. I’ve now amended the relevant example at extensive category, referring back to your last comment.
@Zhen Lin - so $(CRing^{op},Zar)$ satisfies WISC?
Yes, that looks right. Every Zariski-covering sieve contains one that is generated by a finite set of principal affine open subschemes.
Conversely, the fpqc site doesn’t. The question is, where is the transition?
The étale and fppf topologies satisfy WISC, at least over a locally noetherian base scheme. This is basically because they are defined by adjoining to the Zariski topology certain (finite) families of the form $\{ \operatorname{Spec} B \to \operatorname{Spec} A \}$ where $B$ is finitely presented over $A$. The fpqc topology has no such finiteness requirement.
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