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I added a little bit to maximal ideal (first, a first-order definition good for commutative rings, and second a remark on the notion of scheme, adding to what Urs wrote about closed points).
The second bit is almost a question to myself: I don’t feel I really grok the notion of scheme (why it’s this and not something slightly different that’s the natural definition, the Tao if you like). In particular, it’s where fields – simple objects in the category of commutative rings – make their entrance in the notion of covering by affine opens that I don’t feel I really understand.
In particular, it’s where fields – simple objects in the category of commutative rings – make their entrance in the notion of covering by affine opens that I don’t feel I really understand.
Isn’t this something like a valuative criterion for surjectivity? You don’t need to use fields, here.
If one has the notion of open map (and so open embedding), such as in Lurie’s ’geometries’ (of course, with precursors in work of Toën-Vezzosi and others that I can’t recall, probably Joyal, Moerdijk etc) then you can define a notion of scheme using epimorphisms of sheaves. The trick is that the notion of ’open embedding’ in this abstract sense really is an open embedding at the level of locally ringed space, which is perhaps slightly more mysterious. Or, one can see it as a special case of an open map of locally ringed toposes, but I guess this reduces to localic toposes, and so perhaps linking the open maps that help define (or are defined from) the Zariski topology with open maps of locales might be a good way to go.
This would link up the ’modern’ ie Buffalo lectures-style functor of points definition of a scheme (Zhen Lin has written about this in a nice way, I think on MathOverflow) with locale theory, rather than with locally ring spaces. This might generalise nicely to other settings, such as rigs etc, or even log schemes. I suppose one should also think about the sheaf of local rings on the locale, and in a constructive setting, this might get interesting. Certainly the result you’ve done at prime ideal theorem would imply that the locale has a point precisely when the appropriate ideals exist etc etc.
Now I’m off to bed, so sorry if the above is a bit rambling!
Thanks, David. I’m following some of what you’ve written, which sounds interesting, but unfortunately I don’t recognize all the allusions.
However, you’re quite right that it’s the “Buffalo lectures” style presentation that I want to understand better. My bringing up simple objects (fields, if the base theory is commutative rings) was based on a vague memory of something Lawvere wrote in connection with the functor of points POV, which I may as well copy here now, for easy access and to give me something to stare at later:
… The acceptance of the view that, for non-algebraically-closed k, the appropriate base topos consists not of pure sets but rather of sheaves on just the simple objects in A, has in fact many simplifying conceptual and technical advantages; for example this base (in some sense due to Galois!) is at least qd in the sense of Johnstone, and even atomic Boolean in the sense of Barr.
(Note to self: ’qd’ stands for “quotients of decidable”; see for example locally decidable topos.)
Also for easy reference: pdf scan of Buffalo lectures. (However, it boggles my mind that a document of such historical significance should be photocopied in such a slapdash manner. Ever try reading music when the notes are cut off like this? It’s really pretty annoying.)
It so happens that for defining schemes, you can use field-valued points to detect surjectivity. But I think the right way of doing it is to use sheaf-theoretic epimorphisms. I have some slides discussing a general approach to manifold-like notions, and with some luck, I’ll soon have a write-up which also discusses the Joyal–Moerdijk approach.
Aha! Thanks so much, Zhen Lin – those slides look incredibly useful for me (I’ve just scanned through them quickly). To be compared with the general discussion of open and etale maps in toposes by Joyal and Moerdijk here, perhaps?
Also somewhat related, perhaps: the article Gaeta topos could use some serious love.
Indeed, there is something to be said about the Joyal–Moerdijk axioms. The main point to make is this: the class of local homeomorphisms (in my sense) is very nearly a class of étale morphisms (in their sense) except for the descent axiom. This is actually a good thing, because there are too many étale quotients. Roughly speaking, this corresponds to the difference between a scheme and an algebraic space, or the difference between a localic topos and an étendu.
Yes, sorry for being a bit cryptic. I was going to spell out the more abstract definition, but made a (wrong!) guess you’d seen this already and so would be teaching you to suck eggs, as it were. As it stands, I think the locale/constructive view is not done, or at least would be interesting to look at.
Surely there is a physical copy of the Buffalo notes that can be rescanned? Maybe ask on the categories mailing list to see if anyone has them?
David, I don’t know what you mean by “suck eggs” here, but whatever insights you have that you could bring to bear on the article scheme would probably be worthwhile, if you have time. It’s not in a great state, I don’t think.
Sorry, it probably sounds really weird. See https://en.wikipedia.org/wiki/Teaching_grandmother_to_suck_eggs. I’ll have a look at what I could add to scheme, but Zhen Lin has a much deeper understanding than I do.
(I didn’t think you actually meant it like this, but I’ve never heard another use of the phrase. Live and learn!)
Regarding my claim about locales, of course people have thought of it before. See the extended quote by Lurie at scheme as a locally affine structured (infinity,1)-topos. I think, though that one could flesh it out, like as has been done at prime ideal theorem
Re #10, that’s a link to the “mobile” wikipedia site, which looks weird for those of us reading this on a real computer.
@Mike, alright, I edited it.
I have added statement and proof here that the axiom of choice implies that every proper ideal is contained in a maximal ideal.
One strictly needs this (namely choice and in fact also excluded middle) for deducing that Zariski closed points are maximal ideals, right? (The other direction works without.)
Strictly speaking, I think you can get by with a little less: the prime ideal theorem, which requires just the ultrafilter principle (and not excluded middle? I’d have to double-check).
I can expand on this if need be, but the closure of an element $p \in spec(R)$ is $\{q \in spec(R): p \subseteq q\}$, so to say $p$ is Zariski-closed means that the only prime ideal including it is $p$. I claim then that $p$ is maximal, meaning that for any proper ideal $I$ with $p \subseteq I$ we have $I \subseteq p$. But by the prime ideal theorem, we have $I \subseteq q$ for some prime $q$, and then $p \subseteq q$ which implies $q = p$ by our closure condition, and we are done.
By the way, the wording of Proposition 2.2 (the word “then”) suggests to me that it is being viewed as a corollary to Proposition 2.1 which required Zorn. But it’s easy to see without Zorn that maximal ideals are prime, right? An ideal $I$ of $R$ is maximal iff $R/I$ is a field, and is prime iff $R/I$ is an integral domain. But since fields are integral domains, maximals must be primes.
the wording of Proposition 2.2 (the word “then”)
You mean where it has
In classical mathematics then:
?
I wanted to be on the safe side, the proof I came up with used excluded middle somewhere. But if we don’t need it, we should remove that clause.
I need to run now to catch a train. Maybe you could fill in the proof quickly?
(I can’t right away; sorry. I’d have to think carefully whether PEM slips in somewhere, and don’t have time now. Maybe Toby knows right away.)
Depending on what you mean by the ultrafilter principle, it implies excluded middle; or so I recall, but I can't find the reference. I think that it's something like, the Boolean prime ideal theorem implies excluded middle, but the ultrafilter theorem for filters in a power set might not; and without excluded middle (making a power set into a Boolean algebra) there is no reason why these should be equivalent. (And I don't think that we ever decided whether the ultrafilter theorem might imply excluded middle by some other argument after all. I do know a simple argument that equality of real numbers is decidable if nonstandard analysis works, but that's not the same thing, and I'm not sure precisely what goes into making nonstandard analysis work anyway.)
As for whether maximal ideals are prime, this depends on what you mean by a maximal ideal. The best developed constructive ring theory is about rings equipped with apartness relations, and it's the complements of the ideals that you really want to use. See antiideal for this. There we have that a minimal (or antimaximal) antiideal $A$ of a ring $R$ satisfies
$\forall\, p \in A,\; \exists\, q \in R,\; \forall r \in A,\; p q + r \neq 1 \;\wedge\; q p + r \neq 1 ,$which we are told ‘is constructively stronger than being prime and minimal among proper antiideals’. I'm not sure why that is, but I must have gotten it from the reference there (Mines, Richman, and Ruitenberg). Anyway, this implies (but does not outright state) that being minimal among proper antiideals is not enough to make an antiideal prime; and since antiideals behave better than ideals, it strongly suggests that being maximal among proper ideals is not enough to make an ideal prime. But I don't know any specific counterexamples.
Note that if you're going to talk about integral domains and fields, then you're implicitly using this rings-with-apartness concept. In a field, an element is invertible iff it is apart from $0$; in an integral domain, a product is apart from $0$ if (hence iff) all of its factors are. (And if you know what's apart from $0$, then you know what's apart from anything, via subtraction.) Yes, you could state these differently, but then you're either going to get something too weak (can't prove reasonable theorems) or too strong (can't prove common examples); see field. When forming a quotient ring, you need an antiideal to know what apartness means in the quotient, and then the definition of minimal antiideal above is exactly what guarantees (in the commutative case) that the quotient ring is a field. (Similarly, a prime antiideal is exactly one whose quotient is an integral domain.)
Depending on what you mean by the ultrafilter principle, it implies excluded middle; or so I recall, but I can’t find the reference.
I’d like to hear more if you can find the reference.
When we were discussing this a few years back (Toby), the most I came up with was an assertion by John Bell that UP implies De Morgan but not PEM. The discussion was removed from Tychonoff theorem but has been archived here (as you probably remember).
Todd, that's exactly what I was looking for! So the Boolean Prime Ideal Theorem implies the De Morgan Law, but not full Excluded Middle. But without Excluded Middle, the Ultrafilter Theorem is still not the same as the Boolean Prime Ideal Theorem, so I don't know where that stands.
That fact (BPIT implies DML) should be recorded somewhere on the lab.
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