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I am splitting off Zariski topology from Zariski site, in order to have a page for just the concept in topological spaces.
So far I have spelled out the details of the old definition of the Zariski topology on $\mathbb{A}^n_k$ (here).
Hm, there’s a mismatch between section 1 and (edit: the beginning of) section 2.
It looks like section 2 wants to be more about the maximal ideal spectrum, but that’s not quite right either since in order to have maximal ideals in $k[x_1, \ldots, x_n]$ align with points of $k^n$, we need to assume $k$ is algebraically closed (cf. Nullstellensatz).
Probably it would be a good idea to mention Galois connections and the like, since knowing we are dealing with adjunctions helps simplify some proofs for those who are categorically literate.
Todd,
I am meaning to bring some classical material into place as used in standard undergraduate courses (and indeed for that usage). That’s why I do want to have the consideration of the naive points of $k^n$ at the beginning, and that’s why I do want proofs for the categorically illiterate.
But my arrangement of the sub-sections is meant to allow for this naive part to be just the subsection “2. On affine spaces”, to be followed up by the subsection “3. On affine varieties” which should then (and now does begin to) discuss prime spectra and maximal ideals in more generality. In its Examples-subsub-section it should turn back and look at the naive picture of section 2 from this more sophisticated perspective (not done yet).
The perspective of Galois connections is discussed tersely at Nullstellensatz. We could just copy it over. Alternatively, it might be good to use the occasion to re-write this discussion in a more inviting pedagogical way, that would help make those readers categorically literate who are not already, hence help explain this perspective to people who are not already familiar with it.
In order to clarify I have expanded the statement of the plan of the entry here. I have to leave it at that for the moment. Will come back to this entry later this month.
For the definition of the Zariski topology on affine space (currently Def 2.1), I think you need to take a finite set of polynomials to get the closed sets. The basic open sets arise from a single polynomial, no?
Wikipedia also says an arbitrary set of polynomials, but if that is true, any set $X\subset \mathbb{A}^1$ can be closed, for instance, by taking the set $\{x-a\mid a\in X\}$ of polynomials.
Urs: that sounds like a very reasonable plan; thanks.
David: it comes to the same thing because the polynomial ring is Noetherian, but on general grounds you’d want an arbitrary set of polynomials because the closed sets need to be closed under arbitrary intersections. Keep in mind that the correspondence between ideals and closed sets is contravariant, and takes a join of ideals to the intersection of closed sets. The set of polynomials you named generates the improper ideal (the entire ring); the corresponding closed set is empty.
@Todd
Aha, this is the kind of pedagogical point that might be worth amplifying. And the fact that one takes the ideal generated by the set, not the set itself.
And the fact that one takes the ideal generated by the set, not the set itself.
One may take either, but it comes down to the same.
Presently this is mentioned in the proof here.
I need to run now. Feel invited to edit entry.
Okay, I have expanded further:
To the section Properties on affine varieties I added statement and proof that
corollary: Zariski topology on prime spectra is sober
I tried to state accurately which classicality assumptions we need for the conclusions, but it wouldn’t hurt if the constructive experts among you checked. Thanks to Todd for weaking my AC assumption for point 1 above to ultrafilter principle, I have put that in.
Then I also expanded the Idea section further, trying to provide some of the pedagogy that David R was requesting in #7. I added the non-Hausdorffness in an Example (here) and mentioned that in the Idea-section, too.
Finally I did copy over from Nullstellensatz the paragraph on the abstract Galois connection perspective and gave it a stubby subsection here. I would like this paragraph to be expanded a bit more to make it more inviting to the reader.
Did some further minor edits here and there. But there is much room left to further expand.
One thing that could (probably should) be done is give Galois connection more examples, including the type of example we’re discussing now. It’s such a ubiquitous concept that undergraduates ought really to be familiar with the basic pattern and all the basic consequences. But for some reason, throughout undergraduate and beginning graduate education, that same basic rigmarole seems to get repeated on each and every separate occasion, starting from scratch and as if in ignorance of all other such occasions. At least, that was my experience.
But here I guess we could really hammer it home, because we have at least three notions of Zariski topology going on here, so three examples of Galois connection (all closely related of course) can be given.
This material strikes me as simple enough to teach undergraduates within the space of a lecture. Starting with any relation between sets $X, Y$, say $E \hookrightarrow X \times Y$, we get a pair of maps (called a “Galois connection”) between power sets $P(X), P(Y)$, as follows. Let $E(x, y)$ abbreviate the formula $(x, y) \in E$. Define $V_E: P(X) \to P(Y)$ by $V_E(S) = \{y \in Y: \forall_{x: X} \; x \in S \Rightarrow E(x, y)\}$, and similarly define $I_E: P(Y) \to P(X)$ by $I_E(T) = \{x \in X: \forall_{y: Y} \; y \in T \Rightarrow E(x, y)\}$. Two basic facts:
$V_E$ and $I_E$ are contravariant, i.e., if $S \subseteq S'$, then $V_E(S') \subseteq V_E(S)$. This is obvious: the larger $S$ is, the more conditions that are placed on $y$ in order to belong to $V_E(S)$, and so the smaller $V_E(S)$ will be.
We have the basic adjunction law: $T \subseteq V_E(S)$ iff $S \subseteq I_E(T)$. This is because both these conditions are equivalent to the condition $S \times T \subseteq E$.
Here’s an easy consequence: $V_E$ takes unions to intersections, and similarly for $I_E$. In proving this, we’ll use a ubiquitous lemma (Yoneda): in a poset such as $P(Y)$, we have that $A = B$ iff for all $C$, $C \leq A$ iff $C \leq B$. (Proof left to reader!) So:
$\array{ T \subseteq V_E(\bigcup_{i \in I} S_i) & iff & \bigcup_{i: I} S_i \subseteq I_E(T) \\ & iff & \forall_{i: I} S_i \subseteq I_E(T) \\ & iff & \forall_{i: I} T \subseteq V_E(S_i) \\ & iff & T \subseteq \bigcap_{i: I} V_E(S_i) }$and we conclude $V_E(\bigcup_{i: I} S_i) = \bigcap_{i: I} V_E(S_i)$ “by Yoneda”.
Some more easy consequences:
$S \subseteq I_E \circ V_E(S)$ for $S \in P(X)$ (immediate from the adjunction law and $V_E(S) \subseteq V_E(S)$), and similarly $T \subseteq V_E \circ I_E(T)$ for $T \in P(Y)$.
Applied to sets $S$ of the form $I_E(T)$, we see $I_E(T) \subseteq I_E \circ V_E \circ I_E(T)$. But applying the contravariant map $I_E$ to the inclusion $T \subseteq V_E \circ I_E(T)$, we also have $I_E \circ V_E \circ I_E(T) \subseteq I_E(T)$. Thus $I_E(T) = I_E \circ V_E \circ I_E(T)$ for $T \in P(Y)$. Similarly, $V_E(S) = V_E \circ I_E \circ V_E(S)$ for $S \in P(S)$.
From 2. it follows further that the function $I_E \circ V_E: P(S) \to P(S)$. is idempotent. It is also covariant. These facts together with 1. means $I_E \circ V_E$ is a closure operator on $P(S)$. (More abstractly, this is an example of a monad.) Similarly $V_E \circ I_E$ is a closure operator on $P(Y)$.
If $S \in P(X)$ is in the image of $I_E: P(Y) \to P(X)$, then from 2. it follows that $S$ is closed, i.e., equals its closure $I_E \circ V_E(S)$, and that in fact the closed elements of $P(S)$ are exactly the elements of $im(I_E)$. Similarly, the closed elements in $P(T)$ under the operator $V_E \circ I_E$ are exactly the elements of $im(V_E)$.
It now follows easily from 4. that the contravariant function $V_E: im(I_E) \to im(V_E)$ is inverse to the function $I_E: im(V_E) \to im(I_E)$, i.e., the closed elements on each side are in bijection with the closed elements on the other. Such a pair of mutually inverse contravariant maps between posets is called a Galois correspondence.
Closed elements are closed under intersections. If $\{T_i \in P(Y)\}_{i: I}$ is a collection of elements closed under the operator $K = V_E \circ I_E$, then by 1. it is automatic that $\bigcap_{i: I} T_i \subseteq K(\bigcap_{i: I} T_i)$, so it suffices to prove the reverse inclusion. But since $\bigcap_{i: I} T_i \subseteq T_i$ for all $i$ and $K$ is covariant and $T_i$ is closed, we have $K(\bigcap_{i: I} T_i) \subseteq K(T_i) \subseteq T_i$ for all $i$, and $K(\bigcap_{i: I} T_i) \subseteq \bigcap_{i: I} T_i$ follows.
(I’m going to continue the discussion in the next comment, since comments have a space restriction.)
The entry Zariski topology mentions “fixed points of Galois connection” in a long related discussion. I can not find the definition of what is a fixed point of a Galois connection neither in fixed point nor in Galois connection nor am sure what it is. E.g. it is confusing to have to ask: There are two posets, in which one one defines the fixed points ?
Todd: in point 2 some LaTeX parsed incorrectly.
Here are the examples relevant to us.
Of course we already know a lot about varieties $V = V_E(S)$ just on the basis of the preceding generalities; for example, varieties are closed under arbitrary intersections. We get a little more in this specific case by exploiting the ring structure of $k[x_1, \ldots, x_n]$: we can prove that varieties are also closed under finite unions (making them the closed sets of a topology). Namely, the empty union $\emptyset$ is $V(1)$ (the variety associated with the constant polynomial $1$), and for binary unions $V_E(S) \cup V_E(S')$, we can use 2. above to replace the sets $S$ and $S'$ by the ideals $I = I_E \circ V_E(S)$ and $I' = I_E \circ V_E(S')$, and then we claim $V_E(I) \cup V_E(I') = V(I \cdot I')$ where $I \cdot I'$ is the ideal consisting of finite sums of elements of the form $f g$ with $f \in I$ and $g \in I'$.
+– {: .proof}
Applying the contravariant operator $V_E$ to the inclusions $I \cdot I' \subseteq I$ and $I \cdot I' subseteq I'$ (which are clear since $I, I'$ are ideals), we derive $V_E(I) \subseteq V_E(I \cdot I')$ and $V_E(I') \subseteq V(I \cdot I')$, so the inclusion $V_E(I) \cup V_E(I') \subseteq V(I \cdot I')$ is automatic.
In the other direction, to prove $V(I \cdot I') \subseteq V_E(I) \cup V(I')$, suppose $x \in V(I \cdot I')$ and that $x$ doesn’t belong to $V(I)$. Then $f(x) \neq 0$ for some $f \in I$. For every $g \in I'$, we have $f(x)g(x) = (f \cdot g)(x) = 0$ since $f \cdot g \in I \cdot I'$ and $x \in V_E(I \cdot I')$. Now divide by $f(x)$ to get $g(x) = 0$ for every $g \in I'$, so that $x \in V_E(I')$. =–
Here is our second example:
Of course it need not be the case that all maximal ideals $M$ are given by points in this way; for example, the ideal $(x^2 + 1)$ is maximal in $\mathbb{R}[x]$ but is not given by evaluation at a point because $x^2 + 1$ doesn’t vanish at any real point. However, if the ground field $k$ is algebraically closed, then every maximal ideal of $k[x_1, \ldots, x_n]$ is given by evaluation at a point $a = (a_1, \ldots, a_n)$. This result is not completely obvious; it is sometimes called the “weak Nullstellensatz”.
Looking a little more closely at this example, for a subset $T \subseteq MaxIdeal(k[x_1, \ldots, x_n])$ we calculate
$I_E(T) = \{f \in k[x_1, ldots, x_n]: \forall_{M \in MaxIdeal} M \in S \Rightarrow f \in M\} = \bigcap_{M \in S} M$which is an ideal, since the intersection of any collection of ideals is again an ideal. (However, not all ideals are given as intersections of maximal ideals, a point to which we will return in a moment.)
As in the previous example, sets $S \subseteq k^n$ that are closed under the operator $V_E \circ I_E: P(k^n) \to P(k^n)$ form a topology. The proof is virtually exactly the same as before: they are closed under arbitrary intersections by our earlier generalities, and they are closed under finite unions by the similar reasoning: $V_E(S) = V_E(I)$ where $I = I_E \circ V_E(S)$ is an ideal, so there is no loss of generality in considering $V_E(I)$ for ideals $I$, and $V_E(I) \cup V_E(I') = V_E(I \cdot I')$. If $M \in V_E(I \cdot I')$ (meaning $I \cdot I' \subseteq M$) but $M$ doesn’t belong to $V_E(I)$, i.e., $f \notin M$ for some $f \in I$, then for every $g \in I'$ we have $f g \in M$. Taking the quotient map $\pi: R \to R/M$ to the field $R/M$, we have $\pi(f g) = \pi(f)\pi(g) = 0$, and since $\pi(f) \neq 0$ we have $\pi(g) = 0$ for every $g \in I'$, hence $M \in V_E(I')$.
Thus the fixed elements of $V_E \circ I_E$ on one side of the Galois correspondence are the closed sets of a topology. The fixed elements of $I_E \circ V_E$ on the other side are a matter of interest; in the case where $k$ is algebraically closed, they are the radical ideals of $k[x_1, \ldots, x_n]$ according to the “strong Nullstellensatz.
(To be continued.)
Yes, I’m repeating stuff that has been said hundreds of thousands of times. But what I’m really trying to do is generate text for possible use in nLab articles.
Thanks, Todd, for going through the trouble of producing such a text!
Zoran, “Galois correspondence” is another name for “adjunction between posets”, and then there is the general concept of fixed point of an adjunction.
Re #14, I was just wondering if anyone would like to save time by extracting some of that material. But maybe it wouldn’t save time.
Actually, I say “Galois connection” for an adjunction between posets, and “Galois correspondence” for an adjoint equivalence between posets.
Okay, I have put this material into the entry Zariski topology in the section In terms of Galois connections.
First I made a sub-sub-section Background on Galois connections with the material from Todd’s #10. I reformatted this, creating numbered environments, adding hyperlinks, and trying to decompose it into bite-sized bits, with cross-references between them.
Then I ran out of steam and time.
So for the moment I did a blind copy-and-paste of the remaining material from Todd’s #13 to two further sub-sub-sections:
Thanks again to Todd!!
That was awfully nice of you, Urs; thanks! I’m having more than a usual number of interruptions today which accounts for slowness of continuations.
Getting back to David C.: Simon’s posts (I only looked at the 4th though) do look valuable as pedagogy and could be used somewhere in the nLab. Indeed, the best way to introduce categorical ideas to beginners is often just to do the $\mathbf{2}$-enriched case, where so many complications can be ignored. And often: don’t even bother telling them you’re doing category theory, until they get a little older. Once they begin to get good at this type of simple algebra, they will gain a certain power and facility, which even grown mathematicians often seem to lack in these matters.
15: thanks, in the meantime somebody changed the link to fixed point into the link into the more appropriate link to fixed point of an adjunction. I knew well before what is a Galois connection (though, sorry, never liked them) but did not have an idea where to look for the definition of a fixed point in that case. Todd: thanks for a great work here.
Another example of a Galois connection on the arxiv today: abstract stability.
Todd #17: You say “I say”. Is that terminology standard?
Mike: shouldn’t I say “We say” instead? (Yes, I believe “Galois connection for an adjunction is standard. Isn’t it?)
I have now merged the first example from Todd’s #13 into the entry, here.
(So I copied it over, then added numbered environmens, adopted the notation a little to fit with the rest of the entry, added cross-pointers to previous definitions/propositions that the example refers to or which it re-proves.)
Didn’t do it for the second example in #13 yet.
Re #21, the paper there
[GS17a] Moritz Groth and Michael Shulman. Abstract stabilization: the universal absolute.
sounds like you’re picking up on the Hegelian vibe.
Todd, I meant that I never learned to distinguish between a Galois connection meaning an adjunction and a Galois correspondence meaning an equivalence. As you can see, in the paper with Moritz we used the two words interchangeably to mean an adjunction. But it could be standard, in some circles at least, and just that neither of us had encountered that standardization before. Did you learn it as “standard”?
Mike: okay, interesting. I had thought that was the standard, yes. And I can say it is a standard according to Wikipedia, and I thought this also was the convention used in for example Paul Taylor’s book, although I can’t find my copy at the moment to double-check that.
That being said, the notion of Galois connection seems to be due to Oystein Ore (who spells it ’connexion’), and actually he’s like you: he uses ’connexion’ and ’correspondence’ pretty interchangeably for the adjunction concept in his article. He calls the correspondence or connexion ’perfect’ if it’s an equivalence. Encyclopedia of Mathematics doesn’t even use the term ’connection’; it’s just ’correspondence’.
However, I do like the idea of using ’correspondence’ for the equivalence. First, why multiply terms needlessly for the same concept? (why indeed, Oystein?), and second, the word ’correspondence’ is used in the phrase “bijective correspondence” and so it should be easy to remember – a Galois correspondence would be a bijective correspondence between posets that is contravariant.
why multiply terms needlessly for the same concept?
Why indeed? We have perfectly good terms “adjunction” and “equivalence” already, why bother with this “Galois” stuff? (-: It doesn’t really make any sense to name “adjunctions of posets” after Galois just because Galois theory involves a particular adjunction between posets; if we were going to name it after anyone we ought to call it an “Ore connection” if he was the first to study them abstractly. My tongue is in my cheek, of course, but I always have wondered. Presumably the term “Galois connection” predates the term “adjunction”, but why continue to use it now?
Well, to me the phrase “Galois connection” mainly connotes a contravariant adjunction between posets $P, Q$. You could just say ’adjunction’, with the normal meaning of covariant functors, but I tend not to like this because of the symmetry-breaking: you have to choose which of $P, Q$ you’re going to attach the $^{op}$ to, and if it’s $P$ say, you have to remember which is the right adjoint: is it in the direction $P^{op} \to Q$ or the direction $Q \to P^{op}$ (or similarly, on which of these posets do we get a coclosure operator)?
So for me “Galois connection” says just a tiny bit more than adjunction: it implies a certain array of choices of presentation based on symmetry and convenience. I’m so used to thinking about that that “Galois connection” just gets my neurons firing a certain way and saves me from having to think.
On the other hand, I personally would never use the phrase “monotone Galois connection” – there I’d say “adjunction” instead, unhesitatingly. For me, “Galois connection” should imply antitonicity or contravariance of the maps involved.
True. But we also have dual adjunctions.
28, 29: Pareigis uses at some places the notion of adjunction between a covariant and a contravariant functor, calling it “adjoint on the right” in one of the two imaginable cases. For example in his paper with Morris on formal schemes, proposition 1.1 doi pdf. I did not check if he has this notion in his category textbook.
I have added to the Idea-section of Galois connection some of these remarks. But one of you should expand further. Also, maybe further discussion of Galois connections as such should be had in its own thread.
I’ll just make one more comment on terminology.
why multiply terms needlessly for the same concept?
The context for my making that remark in the first place was wondering why Oystein Ore chose to give two names for the same concept within the same article. (At least, any subtle difference of meaning between them is undetectable to me.) That’s what I don’t understand.
It is practically inevitable that in the course of history, various names will be given to concepts by different people under different circumstances. The term “Galois connection” has long been established and people do use it (including me, and it seems Mike as well), and it wouldn’t be right for the nLab not to mention it and any nPOV insights that may be brought to bear, including the connection with adjunctions. Another example would be closure operator and monad. The concept of monad is more general than closure operator, and the exact same remark about “multiplying terms needlessly” would seem to apply there too – but then so would the point I’m making now, which is that the nLab has an important descriptive function to fulfill in surveying the language of mathematics. (And BTW I still do use the term “closure operator”.)
So I’m really not sure what the takeaway from Mike’s comments should be. We shouldn’t be saying “Galois connection” from now on?
As I said, my tongue was in my cheek; sorry if I sounded too serious. (-: I agree with everything in #32.
It did somewhat annoy me when I first learned that something with a grandiose name like “Galois connection” was really just a (contravariant) adjunction between posets. I usually prefer to just say “adjunction” or “equivalence” even in the posetal case. I do think there is something interesting in the specific origin of such adjunctions between powersets by way of a relation between sets as in #10, and I’m more tempted to use the phrase “Galois connection” in that case (as in my paper with Moritz). If I was making any real point at all — which I doubt — it might be that I’d rather not have to learn to remember to distinguish between “Galois connection” and “Galois correspondence”; in the case of an equivalence I’d rather just say “equivalence”.
something with a grandiose name like “Galois connection” was really just
In this case what bothers me is not that it is a grandiose term for a simplistic concept (that’s a problem for, say, “hyperdoctrine”) but that its reference to Galois theory is quite uninformative as to the nature of the concept.
Mike: I thought the “tongue-in-cheek” was about the suggestion to call it “Ore connection” instead. Sorry for misunderstanding.
Urs: I do agree that ’Galois’ isn’t very helpful. All in all, Ore didn’t do such a hot job here on nomenclature. :-)
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