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wrote something at Nullstellensatz, prompted by this old MO comment by Lawvere.
The interesting thing is that “pieces have points” implies that $\Gamma : \mathcal{E} \to \mathcal{S}$ reflects initial objects, which is perhaps the more expected version of the Nullstellensatz.
Feel invited to edit the entry further.
I have added some more-or-less classical algebraic geometry material to Nullstellensatz, with a strategic reduction to the case of uncountable algebraically closed fields by standard applications of Łoś’s theorem. At some point I plan to actually write up Łoś’s theorem at ultraproduct.
A finishing touch in the proof of the (classical) weak Nullstellensatz, with a lemma on the cardinality of an ultrapower.
The proof of existence of points in the case of uncountable algebraically closed field is a bit of a “dirty trick” (due to Emil Artin, perhaps), but I find it quite memorable, and overall the proof is fairly short. Other proofs seem to me a little ponderous by comparison.
How far does the general abstract formulation take us towards the (classical) weak Nullstellensatz? Is there a more nPOV-like approach?
Thanks for your response, David. I’ll think about it some more, but if you mean a more nPOV-like approach to the proof, then my current impression is that the classical Nullstellensatz, which can be rephrased as saying that (for any field $k$) finitely generated $k$-algebras that are fields must be algebraic over $k$, is not at all difficult, and I am coming to see that the proof I wrote out is actually not such a “dirty trick” after all but is pretty conceptual. Maybe it can be made to look more conceptual, but I still think it’s easy.
It seems to me that nPOV as I’ve seen it exposed thus far (the general abstract formulation) considers the Nullstellensatz not as a goal, but (part of) an axiomatic starting point for a general “science of cohesion” as Lawvere might put it. One verifies the Nullstellensatz axiom however one may in particular cases such as the classical AG setting, in ways particular to those cases, but I’m not seeing how that nPOV could be used to simplify the proof in a particular case like AG a la Weil.
Thanks, Todd.
Still I suppose we could do for this article like we do at Galois theory, that is, show that the abstract formulation, there in the setting of sheaves over the small etalé site of a field, corresponds to the classical theory.
Yes, that makes sense. But unless I missed something, maybe you mean not “what we do” but “what we started to do” at Galois theory, since the relevant subsection seems to be still in an unfinished state.
Oh yes.
The existence of nonprincipal ultrafilters is, of course, dependent on classical logic and some amount of the axiom of choice. Is a similar Łoś-like trick available constructively? (Actually, I’m not quite sure what the correct statement of the algebraic Nullstellensatz is constructively.)
Is a similar Łoś-like trick available constructively?
Maybe, but already there is choice used before that point, where an ideal is contained in a maximal ideal.
I’m not quite sure what the correct statement of the algebraic Nullstellensatz is constructively.
I think it’s a very good question. The assertion is that a point exists, and my gut feeling is that’s hard to prove constructively without a major reworking (and perhaps restriction) of the statement. Might be worth asking at MO.
Possibly the Nullstellensatz is just false constructively, and we should just consider Spec of a ring to be a locale rather than a space?
I’ve always thought of the Nullstellensatz as fundamentally non-constructive. It’s a great example of something which is actually largely useless in practise, but is of fundamental significance in justifying the foundation of algebraic geometry upon commutative algebra. I think it is not surprising that the axiom of choice enters for this kind of statement, and it illustrates how a non-constructive axiom can still be useful even for a constructivist, in showing that, as long as one does not think that the axiom of choice is actually disprovable, then the ’logical game’ of including it helps us to see that we will not lose anything by founding algebraic geometry upon commutative algebra, even if it might be difficult in a particular case to construct the relevant commutative ring.
Actually isn’t the point of Hakim’s thesis that one should think of the (locally ringed) topos associated to the ring as its spectrum? I can’t recall offhand if this is localic or not.
In the early 70s Joyal has been working on a constructive theory of spectrum using distributive lattices. I think this is largely unpublished but has left some traces in the Johnstone 1977 - and Joyal paper referenced at Cole’s theory of spectrum as well as in the Wraith paper referenced at Barr’s theorem. See also
In Stone Spaces, Johnstone defines the Zariski spectrum of a ring $R$ to be the locale whose opens are the radical ideals of $R$. (Well, actually he defines it in a more roundabout way, then proves it’s equivalent to that.) It then follows that its points are the prime ideals of $R$, but in the absence of AC it may not be spatial. He then goes on to construct a sheaf of rings over this locale.
Coincidentally, I just picked up the volume^{1} containing the article The spectrum as a functor, which points out that making the construction $A \mapsto Spec A$ functorial is nontrivial (they are using something called the ’hull kernel topology’, but this is apparently the same as the Zariski topology).
Their solution is that one can produce an easy functor $Id\colon Ring^op \to INF^\uparrow$ (unital rings, not necc. commutative I guess, and unit-preserving maps), where the latter category is that of complete lattices with maps that preserve arbitrary infs and directed sups. The lattice $Id(R)$ is that of two-sided ideals, and morphisms are given by taking the inverse image under the ring map. The same construction works for closed two-sided ideals in topological algebras.
The same one with Scott is not always sober, by Johnstone ↩
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