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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 6th 2013
    • (edited Aug 6th 2013)

    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 SS is a commutative ring over RR (by which I mean under RR (-:), does the functor S R:CAlg RCAlg RS \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.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 7th 2013

    Just for the record: of course S R:CAlg RCAlg RS \otimes_R - : CAlg_R \to CAlg_R cannot be expected to preserve arbitrary products in general, i.e., products in CRingCRing 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.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeAug 7th 2013

    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?

    • CommentRowNumber4.
    • CommentAuthorZhen Lin
    • CommentTimeAug 7th 2013

    That’s right: the disjoint union of infinitely many copies of Speck\operatorname{Spec} k (kk a field) is not affine because, well, its underlying space is not quasicompact.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 7th 2013

    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.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 7th 2013

    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 CRingSetCRing \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.

    • CommentRowNumber7.
    • CommentAuthorZhen Lin
    • CommentTimeAug 7th 2013

    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.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 7th 2013

    Hm.. need to think on this. But may need some time…

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeAug 7th 2013

    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?

    • CommentRowNumber10.
    • CommentAuthorZhen Lin
    • CommentTimeAug 7th 2013

    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, Spec𝔽 2 \operatorname{Spec} \mathbb{F}_2^{\mathbb{N}} is (homeomorphic to) the ultrafilter space β\beta \mathbb{N}, which is totally disconnected.

    • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 7th 2013

    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).

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeAug 8th 2013

    Interesting! I had no idea there were affine schemes that looked like that topologically.

    • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 8th 2013

    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 CRingSetCRing \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 opCRing^{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 opCRing^{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 VV and UU, with UV\mathbb{N} \in U \in V, calling members of UU “small sets”. (One might suppose they mean that VV (with its membership relation) is assumed to satisfy ZFC, as does UU 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 MM, which will be the supporting site for things like the Zariski topology. In other words, they will consider presheaves and sheaves on M opM^{op}, where presheaf means a functor MSetM \to Set to the category of sets (that live as elements in VV).

    They wave their hands a bit, essentially saying they don’t really need universes. Since they want [M,Set][M, Set] to be locally small, they don’t want MM 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 RR, every model with cardinality less than or equal to Card(R) 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.

    • CommentRowNumber14.
    • CommentAuthorZhen Lin
    • CommentTimeAug 8th 2013
    • (edited Aug 8th 2013)

    Yes, there are some set-theoretic issues at play.

    1. The functor of points of a scheme is a sheaf on the large Zariski site. (In fact, it is even a sheaf on the large fpqc site.) I think it is small (i.e. a colimit of a small diagram of representables) but I could be wrong.
    2. The large Zariski site is not essentially small in the sense of Johnstone, i.e. it is not Morita-equivalent to a small site… or at least it is not obviously so. Certainly what is true is that the cardinality of the non-trivial principal localisations of an integral domain AA is bounded below by the cardinality of AA itself, and there are integral domains of arbitrarily large cardinality.
    3. One can certainly look at gros Zariski sites that are not large. For example, for M=CRing fpM = {CRing}_{fp}, the category of locally finitely presentable schemes embeds as a full subcategory of the topos of Zariski sheaves on MM. The general principle is that one can always get the schemes that are “locally MM”.
    • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 8th 2013

    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.

    • CommentRowNumber16.
    • CommentAuthorZhen Lin
    • CommentTimeAug 8th 2013
    • (edited Aug 8th 2013)

    Edit. The argument below is not quite correct…


    No, there is no such a site. Suppose MM is a small full subcategory of CRingCRing such that the Yoneda representation CRing op[M,Set]{CRing}^{op} \to [M, Set] is fully faithful. Since MM is small and CRingCRing is complete, we get a left adjoint, making CRing op{CRing}^{op} a reflective subcategory of [M,Set][M, Set]. But then CRing op{CRing}^{op} would be an l.f.p. category, and we know that is impossible because CRingCRing itself is l.f.p. but not a preorder category.

    • CommentRowNumber17.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 8th 2013
    • (edited Aug 8th 2013)

    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 EE is a infinitary extensive pretopos, and that FEF \to E is a full embedding with a left exact left adjoint. Is it true that FF is an infinitary extensive pretopos?

    • CommentRowNumber18.
    • CommentAuthorZhen Lin
    • CommentTimeAug 8th 2013
    • (edited Aug 9th 2013)

    Oops, what I said there was not quite correct: CRing op{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 CRingCRing, 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][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 AA, the class of Zariski-covering sieves on SpecASpec 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 FF can be computed by applying the reflector to the corresponding construction in EE, so any good properties those have in EE are inherited by FF.

    • CommentRowNumber19.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 8th 2013

    Okay, great – thanks again, Zhen. I’ve now amended the relevant example at extensive category, referring back to your last comment.

    • CommentRowNumber20.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 9th 2013

    @Zhen Lin - so (CRing op,Zar)(CRing^{op},Zar) satisfies WISC?

    • CommentRowNumber21.
    • CommentAuthorZhen Lin
    • CommentTimeAug 9th 2013

    Yes, that looks right. Every Zariski-covering sieve contains one that is generated by a finite set of principal affine open subschemes.

    • CommentRowNumber22.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 9th 2013

    Conversely, the fpqc site doesn’t. The question is, where is the transition?

    • CommentRowNumber23.
    • CommentAuthorZhen Lin
    • CommentTimeAug 9th 2013
    • (edited Aug 9th 2013)

    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 {SpecBSpecA}\{ \operatorname{Spec} B \to \operatorname{Spec} A \} where BB is finitely presented over AA. The fpqc topology has no such finiteness requirement.

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