@TedShifrin well, it also depends what one calls the chain rule. for better or worse, I always tend to think of it in terms of stuff like $\frac{d}{dx}=\frac{du}{dx}\frac{\partial}{\partial u}+\frac{dv }{dx}\frac{\partial}{\partial v}$
Well, Semiclassic, getting them to do the single-variable chain rule to do Frenet calculations for non-arclength-parametrized curves usually flunked one or two students every time I taught it.
Beyond the usual homology/cohomology stuff, I did wander into some Ext stuff when I was exploring one of the topics Griffiths suggested I work on for a thesis. I did something else.
I am skeptical of categories but I think the triangulated category axioms are a triumph of that kind of thinking. They thought really hard about how you do homological algebra in practice and then pulled out precisely how we think about it
And then even better it allows us to focus our thinking. If those are the only axioms we use, then those are the only axioms we have to think about
you don't really need the fancy approach to homological algebra via derived categories and triangulated categories for most things, it just makes certain things more clear on an abstract level and you can do stuff like reason about derived functors when there are not necessarily enough injectives. Just throwing resolutions at everything does seem ad-hoc in comparision
Mike: That's too much baggage for the average calculus student. I can talk about "good" linear approximation without the linear algebra. I just think it's not the ideal course for a mature math student.
@Semiclassic: My undergrad diff geo is meant to be naive. No need for the derivative as a linear map or the fancy chain rule, except perhaps in one place.
it's independent of rep theory, but you need some commutative algebra beforehand (not sure how much is in D&F). I'd do it after rep theory, though as I think it's harder
I took a chemistry thermo my freshman year, Semiclassic. The math wasn't that difficult, although obviously I had less trouble than most of the chem majors.
the basics of representation theory can be appreciated with just linear alegebra and basic group theory, though at some point knowing some module theory over non-commutative rings becomes really handy
@KasmirKhaan the easiest book on commutative algebra I studied from is probably Atiyah-Macdonald, but I'm not sure if that's right for you. There are not much examples and the important part of the book are really the exercises which are of varying difficulty
I think of commutative algebra, especially when you assume that everything is Noetherian as the basic case where everything works out nicely and isn't too difficult. To me, when you globalize it as in alg geo or drop assumptions so you deal with non-Noetherian rings or non-commutative rings (or both) is when stuff gets really hard
On an orthogonal note, the proof that number of paths on an $n \times n$ grid going from $(0, 0)$ to $(n, n)$ using only Up or Right as available moves and staying above the diagonal for all time is equal to the Catalan number is so tricky.
I need a framework for the whole discipline which I was never able to build. What do you care about? Why do you care? What tools do you havr for these? How do those tools affect the way you think about the objects?
@MikeMiller maybe one could say that commutative algebra abstracts some structures that appear in fundamental objects of algebraic number theory, algebraic geometry, invariant theory and elsewhere. It's hard to care about, say Dedekind domains unless you've seen them in a course on algebraic number theory or you see their relation to the group law on an elliptic curve
The reasons to care is related to the applicability I think (unless you're a crazy algebraist like me who just loves to learn about rings even without any motivation). The most basic example is perhaps the structure theorem for f.g. modules over a PID which is on the border between linear algebra and commutative algebra. It allows an elegant treatment of some advanced linear algebra, classifies finite abelian groups and is used all over the place when you deal with Dedekind domains in ANT
Other examples are e.g. the Noetherian hypothesis which allows an easy treatment of primary decomposition and stuff like the Hilbert basis theorem implies geometrically that varieties are cut out by finitely many polynomials
I could go on, but I think you get the idea. Some structural insight on special classes of commutative rings gives interesting results in other fields. As for the fundamental tools, I'd say that the two fundamental tools in a modern treatment of commutative algebra are localizations and module theory (I'm no expert, this is just my personal impression)
Localizations allow us to restrict our attention to certain subsets of the spectrum, so if you think of a commutative rings as a ring of functions on the affine scheme, this is like the analog of restricting functions. This allows us to do an algebraic analog of a partition of unity argument.
(An "algebraic partition of unity" is just a finite set of elements that generates the unit ideal, but if you work out what this means for thecorresponding subsets of the spectrum, it becomes clear why this is called partition of unity)
even more important perhaps than partitions of unity are localizations at prime ideals, these are the stalks of the affine scheme, so it's a bit like a tangent space on a manifold which is the stalk of the sheaf of $C^\infty$ functions
Localization of a prime ideal is a standard argument because local rings are easier, just like linear algebra is nicer than differential geometry. (But probably not that much easier) I don't remember who wrote it, but I read somewhere that the parts of commutative algebra which are well understood are often the parts which have been succesfully reduced to local rings
I'd say the other fundamental tool is module theory, so instead of studying a ring directly, we study who it acts on abelian groups. So we can associate to each ring a really nicely behaved category that reflects some properties of the ring and for commutative rings even completely determines the ring (Morita-equivalent rings have isomorphic centers)
I guess this is justifies that a lot of objects that come up naturally are modules. If $R$ is a ring, then each ideal is a submodule, the quotient by the ideal is a submodule, if $f: R \to S$ is a ring homomorphism, it makes $S$ into a $R$-module and in the commutative case, you easily also get the $R$-module structure on Hom-Sets where on the modules is a $R$-module (this is more difficult for non-commutative rings)
This means that one can apply results on modules in a quite flexible way
Of course there are also examples in othe disciplines, e.g. in diff geo the sections of a vector bundle are a module over the ring of smooth functions (even a module sheaf over the ringed space.) Or representations are modules over group algebras, but that's mostly noncommutative, so yeah.
but the idea behind studying modules over a ring to understand the ring itself is the same idea why you would study group actions or linear representations to understand a group, you pick a nice category and study how your object can act on that category
@MikeMiller Does that make commutative algebra a bit less mysterious?
It was definitely a nice explanation and I very much appreciate it, but I think I'd heard most of that before. I think I need to get my hands dirty at some point to really understand this
Guys, I got a 48/50 on my representation theory homework! He took of two points because I assumed the result that if a vector space is irreducible, then its dual V* is also irreducible :(
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