Update to the notes: I will probably post a new version of The Rising Sea notes after our quarter ends. By the end of two quarters of the three-quarter sequence this year, we will reach the end of the chapter on curves, and there are only around ten known issues in the notes I want to fix up until that point. (But there are a triple-digit number of issues afterwards, which I hope to make serious progress on next quarter!)
(i) I’d like to give explicit references to examples of what behavior can go seriously wrong in the non-Noetherian setting, and long ago I’d scribbled some notes, but now I can’t remember what the actual references were to. Can anyone point me to where these are in the literature (because I’m sure I’ve seen them)? I had thought they were in the stacks project, but couldn’t easily find them there. The right person to ask is the incomparable Johan de Jong, and I can email him later, but I thought I may as well ask here first.
(a) There is a projective flat morphism where the fiber dimension is not locally constant.
(b) There is a finite flat morphism where the degree of the fiber is not locally constant.
(c) There is a projective flat morphism where the fibers are curves, and the arithmetic genus is not locally constant.
(d) There is a projective morphism for which the pushforward of coherent sheaves are not always coherent.
Behind these might be the famous “five-pound” counter-examples in the stacks project (aka tag 05LB, and now you can never forget its tag), based on a ring R with an ideal I such that R/I is flat but not projective. It is easy to describe R and I: R are the infinitely differentiable functions on the real numbers, and I are those functions that vanish in an open neighborhood of zero.
These examples are really unimportant in general in some sense, but I like having a strong sense of where the boundaries of civilization are, and what kinds of monsters actually live beyond those boundaries.
(ii) In return, I’ll give a sample new brief section. I realize now that I’ve been teaching and understanding the Jordan-Holder theorem wrong in group theory (way back in “introduction to group theory”). I’m now going to teach it in a way that naturally leads you to the notion of length, and simple objects in more general categories (without using those words), in a more natural way. One day I would like to write it up as a “bedtime story” in the style of what I did for exact sequences here. But I here is the write-up in a short section in the current version of The Rising Sea. I want it to be friendly and easy (for someone reading The Rising Sea, not someone seeing group theory the first time).
Incidentally, we traditionally teach/present the groups/rings/fields/modules class/material in the order “groups, then rings, then perhaps some modules or fields, then move from there”. I think I will next try to do it: “you know linear algebra, so let’s define a field, and vector space together, then start the course with abelian groups and their actions (leading to quotients etc., ie experience with the abelian category package extending vector spaces), then rings, then a touch of modules (and quotients etc), then UFDs, PIDs, Smith normal form from which we get classification of finitely generated abelian groups. Only then, to general groups, actions, quotients (with added weirdness). I would have to not give up any “canonical” material in the class (Math 120 at Stanford) in order to set them up for the next class, but I think I could do it. Unsurprisingly, I like Paolo Aluffi‘s approach (his homepage has a picture of the tree in which he lives); he told me last week he also does largely follows this path, although he actually does rings first. (It seems logically harder than doing abelian groups first, but conceptually I think he is right that it is easier, because people will already have an excellent intuition for the integers to build on.) He does this in his book Algebra: Notes from the Underground, which is in keeping with the excellent philosophy of his book Algebra: Chapter 0.
Math 216: Foundations of Algebraic Geometry 
March 11, 2023 at 1:03 pm
Rotman’s Advanced Modern Algebra (part 1) begins with commutative rings, then groups (followed by Galois theory) and then modules …
March 11, 2023 at 2:22 pm
I think the main reason I don’t like the rings before groups approach is that at the point people are learning these for the first time, they also have to unlearn the high school idea that algebra is just about complex numbers. For example, if you give them maybe the symmetric group and ask them to solve g^2=1, they’ll frequently answer +/-1. I agree that rings are more familiar and they’ll have more intuition about them, but one of the key things they have to learn for the first time is that one should be able to work and compute despite not having intuition.
March 12, 2023 at 7:18 am
When I was taught group theory during high school at an enrichment program, I recall weak interest because I didn’t see the point of it. Fields and rings have more interesting elementary properties, applications and opportunities for calculation, but still introduce a diversity of interpretations for familiar symbols with some familiar axioms.
March 12, 2023 at 2:54 pm
You make an essential point that I have to remember to keep central in my mind. If you don’t know why you care (or perhaps more important, know why you are interested), you won’t learn nearly as quickly. This is a strong argument in favor of doing rings first, even before abelian groups.
I still offer for discussion (without yet strong evidence and hence without strong personal conviction) the possibility of doing *abelian groups* before rings. I think we don’t think of abelian groups as a totemic idea on the same level as “rings, fields, groups” in part because the name suggests that they are just a subcase of the more important notion of group. So I guess I’m suggesting an ordering along the following lines.
1. Abs of Steel
2. Rings of Power
3. Fields of Fire
4. Groups of Suspicious-Looking Individuals
(I couldn’t really think of anything better for the last one, I’m afraid…)
But I’m a little stuck for an answer to your implicit question: how will I convince someone that they should care about Abs? So this suggests moving through them as quickly as possible, en route to rings (not fully highlighting how important they are, but still subconsciously giving them philosophy). I should ask myself: what do I want people to know about Abs before moving to rings? And it should be as little as possible. Offhand: definitions, maps, kernels/cokernels, quotients (which is hard, as it is for vector spaces), actions of abs (and in particular the orbit-stabilizer theorem), and what goes by the name of isomorphism theorems. (I don’t like the numbering of the isomorphism theorems.)
The orbit-stabilizer theorem discussion leads directly to the most important isomorphism theorem, the first isomorphism theorem (and also ties to the difficulty of quotients).
The third isomorphism theorem is then also worth discussing after that, and important.
I want to make the fourth isomorphism theorem (the “lattice isomorphism theorem”) into an “observation”.
Perhaps do the Jordan-Holder theorem in the way I described above, and from there, the second isomorphism theorem just turns up. This also connects to prime factorization, and opens the question about classification of finitely generate abelian groups.
Then move on to rings! Depending on the audience, I could plausibly do that in 3-4 weeks. And then I’d want part of rings to be easier and faster (e.g. ideals, quotienting by ideals; quick introduction to modules).
I wish I had a punchline theorem (even just a “fun fact”) for the Ab discussion. The best I can think of offhand: Fermat’s little theorem, a^{\phi(n)} \equiv 1 mod n. Maybe toss in for fun in a problem set the group law on an elliptic curve, and then using it to get lots of solutions to diophantine problems?
Thanks @chaikens and @allenknutson for those comments!
March 13, 2023 at 1:23 pm
For motivating groups, I start with graphs (e.g. Petersen drawn two different ways). Then graph isomorphisms. Then graph automorphisms. Then automorphism groups.
If I wanted to do abelian groups first I’d probably start with the Chinese Remainder Theorem.
March 31, 2023 at 5:06 pm
About isomorphisms and automorphisms — ooh, very categorical, sneaking in some sophisticated thinking!
About the Chinese Remainder Theorem: that’s an excellent idea. It is compelling, can be told very early, and then keeps coming up. This is a winner. I have to think about it a bit.
March 13, 2023 at 2:25 pm
More old recollections.. The course for high schoolers used Rotman’s group theory book, and a big deal was made of Cayley’s theorem. Years later, in grad school, I felt a much greater appreciation of the subject when Michael Artin introduced early on general actions, symmetry and the 17 wallpaper groups.
March 31, 2023 at 5:04 pm
I remain kind of unimpressed by Cayley’s Theorem. I probably shouldn’t say this in public, but when I teach group theory, I have consistently called it “Cayley’s Stupid Theorem”.
March 16, 2023 at 9:31 am
Interesting. The rectangular approach to Jordan–Holder is just the unpacking of the usual proof where you consider the intersection of the highest nontrivial elements in two filtrations and then proceed by induction. I’ve found that such “unpacked” proofs are illuminating once the student understands what’s going on. Until then, they might get lost in the indices. The pictures are nice though.
March 31, 2023 at 5:03 pm
For me, the pictures are the point — they tell you the proof, and then there is no induction — the entire thing is there at once. It is indeed as you say just an unpacking of the usual proof (or perhaps a pictoral representation of the usual proof). Perhaps it will be very enlightening to a certain kind of person, and very unenlightening to another kind of person. I find it hard to tell, since of course I am only one type of person!
March 26, 2023 at 9:16 pm
I like fields arising as lecture 1 of both first-year linear algebra and analysis (helping show the unity of mathematics). After that I like groups before rings, since beyond the phenomenon of noncommutativity, abstract algebra happens there with one operation instead of two.
It is easy to motivate group theory: group live to act on things, so we have isometry groups of metric spaces, the general linear group (with the determinant from linear algebra 1 being one example of a homomorphism), the symmetric group acting on combinatorial structures and so on.
March 31, 2023 at 5:01 pm
I fully agree with your point on fields.
With groups vs. rings, I’m still undecided. I wonder whether the concept of a group seems simpler to people like you and me, but elements of a ring are “things” — people feel like they know what integers and polynomials are. But elements of a group (thought appropriately, as you say, as coming from action on things) is a surprising change of perspective the first time you see it. You are abstracting away the things acting, from the thing acted upon. The elements of a group are ghosts of symmetries of things once you forget what the things are.
An example of this is how groups tend to be introduced: Definition: a group is a set with a binary operation with blah blah blah. The mature perspective you describe ends up coming later, and instead you just have a definition coming out of nowhere, for an audience that has never had this happen to them before.
That’s why I’m proposing (at least for the sake of argument) abelian groups first. It has the advantage you say of just having one operation. And it is part of the way to formalizing rings. The downside is that they seem like kind of an uninteresting thing — I can’t think of a way to make them super compelling.
March 31, 2023 at 11:23 pm
But they should have had this happen to them before: week 1 of year 1 was “a field is a set with two binary operations such that …” in two different courses; one of them continues to “an ordered field is …”, “an ordered field is complete if …”. The other continues to “a vector space over the field F is a set with two binary operations …” so by the time year 2 rolls around and you say “a group is a set with …” it shouldn’t be news.
In most North American universities we aren’t be this rigorous from the start, and also need some concrete algebra at the start of the group theory course — at least as far as unique factorization and modular arithmetic in Z, so on second thought maybe rings first might make sense.
April 1, 2023 at 1:18 pm
Lior, this conversation really feels like a discussion with myself — these are precisely the things I’m going back and forth with in my own head. Your last paragraph is what is slightly winning the tug of war in my mind: I want to start with something concrete. When making a new definition (*especially* earlier in undergrad mathematics, but actually, everywhere) I prefer to go: example-example-example-definition, rather than definition-example-example, because I would rather say “here are some things where there is something in common, so now we abstract away what is going on”, which teaches abstraction (by example). I like saying “you already know what an X is, you just haven’t given it a name”. So with fields, I like the fact that you pointed out: they are already happy with R and C (and perhaps have thought about features of both that give them individually advantages), and then perhaps Q, and then we abstract then notion of field. And with rings we have examples ready to go as well, which are interesting.
After sleeping on it, I’m really taken by Allen Knutson’s idea of the chinese remainder theorem turning up early and often. It grows up into factoring integers, factoring abelian groups, classification of finitely generated abelian groups, composition series, to simple groups, etc etc. When I was in high school I thought of it as a fairly low-key fact, unlike some fancier things in number theory.
April 3, 2023 at 2:51 am
Incidentally my son and I made a video for SoME2 connecting the Chinese Remainder Theorem to partial fractions (and from there, to Jordan canonical form).
March 26, 2023 at 9:18 pm
Regarding Ex. 6.5.C: isn’t it more natural to separately prove for subobjects (intersect a composition series with the subobject) and then for quotients by subobjects (take composition series of subobject and extent to the full object)?
March 31, 2023 at 5:09 pm
The nice thing about the pictoral approach is that you can do both at once (with a table with three rows). But perhaps I should include a remark along the lines of “(You can do the subobject and quotient object cases separately if you wish.)”
May 21, 2023 at 1:39 am
Richard Borcherds has a very similar approach to Jordan-Hölder in his group theory videos here:
May 24, 2023 at 8:18 am
I like the graphical lay out of Jordan-Holder. I’ve drawn pictures like this, but I hadn’t worked out the axiomatization of all possible pictures, which is really clarifying. It’s also good preparation for Schubert calculus: If you consider the case where your ground ring is a field, this is the recipe to assign a permutation to a pair of complete flags.
Could I make a notational suggestion? You call your modules M_i, M’_j and M_{i,j}. If you called them X_i, Y_j and Z_{i,j}, not only would they be more visually distinctive, but you could make it mnemonic — X_i on the x-axis and Y_j on the y-axis. This would be particularly nice because you use the variables i, i’, j and j’ later, and it surprised me that i and i’ went with M while j and j’ go with M’.
June 12, 2023 at 6:40 pm
Thanks David!
About the Schubert calculus: you won’t be surprised that this is how I came to think about this.
And about the new notation: I’m sold, and I’ve made the change. (In the process, I noticed a number of small mistakes.) I had earlier learned from you the importance of wisely choosing names and notation, and this very much applies here.
June 15, 2023 at 2:46 pm
Could you please say more on the connection with Schubert calculus? Maybe giving an explicit example? I don’t understand how this argument relates to it.
January 22, 2024 at 5:44 pm
At risk of just confusing the issue, I’ll just say the following. If you have a vector space V over a field, and two different flags F_\bullet and G_\bullet on V, and you make Jordan-Holder table for F_\bullet \cap G_\bullet, then the shape of the table is precisely given by the Schubert cell F_\bullet lives in, with relation to G_\bullet (or equivalently/inversely, the Schubert cell G_\bullet lives in with respect to F_\bullet). That point of view is what led to my paper “A geometric Littlewood-Richardson rule”, and you’ll see similar pictures there.