/sci/ - Science & Math

Question 1 is an easy warm up.

Errata to keep in mind https://mathoverflow.net/questions/42241/errata-for-atiyah-macdonald
None identified in chapter 1 at least

What sort of timeframe are you thinking for each thread OP?

>>10090674
in. I just got CA-Eisenbud last week

>>10090900
I don't know, the beginning 2 chapters are pretty basic, so perhaps a week each. Depends on how much traction and other people are doing the exercises

>>10090906
I also have Eisenbud at hand but it's a bit too long and has too much, but I'm also liking how it reads.

These took quite a bit, a lot of induction going on, but still nothing unelementary

>>10090959

>>10090964
Note that this is the important Gauss lemma. You may have seen it in an earlier course in less generality as: if a polynomial is irreducible over the integers, then it is irreducible over the rationals. Somehow, augmenting your ring by passing to the field of fractions doesn't add roots to polynomials. This latter fact can be proved using this lemma. It is also an important part of the proof that R is a UFD implies R[x] is a UFD.

Exercise 3 is just by induction so it is skipped. Exercise 4 really exacerbates the importance of doing previous exercises, since otherwise I wouldn't have known where to start. Also, the criterion of Proposition 1.9 is often very useful when dealing with the Jacobson radical. The property is called quasi regularity, and it is not just exclusive to commutative rings.

>>10091160
This one took me a bit, since I couldn't think of non-Noetherian non-integral domains, but ultimately there's an obvious answer

>>10090948
I think even Eisenbud himself would be surprised if anyone actually _read_ a majority of his CA book.
It's intended as a reference work, but in a difference sense than the usual meaning of "mega concise compilation of results for experts".
Eisenbud is for looking up good explanations of things when the explanation in your study source is shit or nonexistant.

The questions are starting to speed up, which is certainly welcome.

Note: in question 8, one has to apply Zorn's lemma in an unintuitive way: there is an importance distinction in the direction of the subset symbol and what are typically minimal elements / lower bounds, in the parlance of Zorn's lemma they are technically maximal elements / upper bounds.

>>10091359
forgot file

Did every implication since they were all pretty easy

I'm pretty tired so I'll continue tomorrow. Please keep the thread alive, perhaps if you can by posting your own solutions or questions

>>10090674
Most of chapter 1 looks like review and with my undergrad behind me a lot of this looks easy (not to imply that it is, I certainly would've found it difficult a few years ago). I may follow along.
On a sidenote, I have a category theory background but I was never really interested in algebra. Are there any interesting applications of commutative algebra in pure mathematics or is it all just real-world physics stuff?

man I feel for you. This thread is too good for /sci/

>>10091485
>Are there any interesting applications of commutative algebra in pure mathematics
The majority of people only learn it for algebraic geometry, because commutative algebra tools are essentially a requirement to do anything.
Commutative ring theory also shows up quite a bit in algebraic number theory, usually in a slightly more prosaic form.

Neat thread. I might post some stuff once my midterms are done.

keep it up!

>>10090674
I'm taking commutative algebra/Algebraic geometry next semester. I definitely need this, but I have an analytic number theory exam this Thursday. I'll get to work on this as soon as that exam is over, anon

But commutative shit is usually the trivial stuff lmao

Interpretation is a quality of a man in sequence. a mind cannot alter a condition made prior to the initial reaction of the possible qualities some may hold to an argument as a consequence of pejorative nonsense. That makes initially less easy to qualify an argument that doesnt stand the predicate. If a conjugate form holds a vowel in it, the likelohood that the variable in that vowel is a number value we can use immediately means that a spectrum is likely involved but not for the sake of remanding authority. Ghe attempt is an encapsulation of thought. Guessing a word oug of a number is like guessing this is a conspiracy not just fact and also allowing yourself to guess that out like its right. Sudoku is okay its within the to 10 digot doctrine so to speak but it isnt able to actually help people sleep at night. You can guess this way there, dont so it here.

bump for interest

>>10091485
As the other anon said, CA is in the heart of algebraic geometry, and as a consequence of algebraic number theory. I really doubt it has much application in physics, but I don't know much about it.

Chapter 1 is mostly review yeah. There may be some new concepts for some but it's not particularly challenging or out-of-the-box. However, as I mentioned, doing the exercises is the heart of the book and some of them require more originality and other fields, as you may see in the later exercises in chapter 1.

Finally, we finish the less interesting question section.

Finally, we finish the less interesting question section

>>10092076
she bumps my interest if you know what I mean

>>10091360
Forgot to mention that the set is non empty since maximal ideals exist and are prime, which is important to note

Simple results are nice to do over your morning coffee desu

15 was somewhat trivial but it is important to define more algebro-geometric concepts.

I don't know what they mean by "draw" SpecA here, but it is not hard to think of pictures of inclusions. I didnt include an explanation for SpecZ[x] but it is clear that any other combination would produce a quotient ring with zero divisors, or make it the zero ring.

On second thought, (iv) was more trivial than what I thought

>>10090674
Keep at it OP. I'm studying algebraic geometry and algebraic number theory right now and have been referring back to this book a lot. I've done most of the exercises in the first three chapters but might join once you get to 4.

As for drawing Spec, there are a few pictures I've seen in several AG books: Spec C[x] is a "complex line", with a closed point (x - a) for each complex number, and a generic point (0). Spec R[x] is similar, and it even includes closed points for complex numbers corresponding to irreducible quadratics (x^2 - ax - b), but with complex conjugates glued together (since they are roots of the same irreducible polynomial.) Spec Z is a line/curve with a point for each prime, and the whole curve is the generic point Z. Spec Z[x] should be two-dimensional somehow, with one dimension corresponding to primes and the other to irreducible polynomials, and so on. (I'd need to think more about what this last one should look like, but I am sure it is 2 dimensional, since Z[x] is two-dimensional as a ring.)

>>10092565
the other two, (vi) and (vii) are pretty trivial so I wont do them since I've got to go, I'll come back tomorrow.

>>10092585
thanks dude, may take a while for chapter 4 to come, maybe you can help a bit around. Also, I see it's very much related to the Krull dimension, isn't it?
I'm doing a PhD in AG and ANT too btw, but first year and I've got much to learn about algebra since I did my undergrad focusing mostly on algebraic topology.

I think this type of thread is great. Thanks for your contribution.

>>10092585
Schemes are really strange. How do you draw the generic point?

To say a little more about the spectrum of Z[x], remember that there exists a unique map Z->Z[x], hence a unique map of affine schemes u:Spec(Z[x])->Spec(Z) and the fibers of this map, which are subschemes, are the affine lines u^{-1}(p) \iso Spec(F_p[x]) where F_p is the prime field of char p or Q when p=0 (and the first half happens for any scheme).
Still no clue on how to draw this.

Fuck off Reddit fags
Mods please ban

>>10093396
>he posts on reddit

>>10093396
T B F the math subreddit is pretty alright as far as reddit goes

Math undergrad here, doing Dummit & Foote up to and including rings and PIDs. Would love to read through this after I finish my course. Is there an archived version of this somewhere (or will there be) that one might access later?

Thanks for an awesome thread!

>>10093527
Just google "4chan archive" and you'll find what you're looking for
>>10093396
Eh, considering this thread is primarily about learning the basics of the subject and doing exercises I don't think getting more involvement is a bad idea. I mean, most of /sci/ is trash, how can the math subreddit be any worse?

>>10093162
I think it might help to draw what the other closed subschemes (corresponding to ideals of Z[x]) are, and how they intersect (corresponding to how the ideals add in Z[x]). For each prime p of Z you have the generic point (p), corresponding to a (unique reduced irreducible) subscheme. These are as you said the fibers of the map [math] \operatorname{Spec} \mathbb Z[x] \to \operatorname{Spec} \mathbb Z [/math], so they are disjoint and their union is the whole space. Each of these is isomorphic to the "line" [math] \operatorname{Spec} \mathbb F_p[x] [/math]

Now consider the generic points of the form (f) for an irreducible polynomial [math] f \in \mathbb Z[x] [/math], and how these should intersect the lines corresponding to each prime. For something like [math] x - a [/math], the ideal [math] (p, x - a) [/math] is maximal, and has residue field [math] \mathbb F_p [/math]. For something more interesting like [math] x^2 + 1 [/math], it depends on the prime. [math] \mathbb Z[x]/(2, x^2 + 1) = \mathbb F_2[x]/(x^2 + 1) = \mathbb F_2[x]/(x + 1)^2 [/math], so we should maybe draw this like the (x^2 + 1) is tangent to (2)? This is the only prime where this happens. Otherwise, x^2 + 1 remains irreducible or splits into distinct factors, depending on if p is congruent to 1 or 3 mod 4. This suggests x^2 + 1 intersects (p) transversely, either at a single [math] \mathbb F_p(\sqrt{-1}) [/math]-valued point, or at two [math] \mathbb F_p [/math]-valued points.

This is pretty sketchy but it feels about right, and parallels familiar ideas from number theory.

Great idea, OP. I wish I had the time to follow along diligently, but I might download the pdf and read along with the threads. Bumping.

>>10093527
You might be able to follow along anyway if you're interested. From how you worded it it sounds like you mean you're doing group theory now, and will do ring theory at the end of the semester? If that's the case you don't really need to wait for much. Most of the interesting stuff in group theory becomes trivial for abelian groups, which is what we're mostly concerned with in commutative algebra. The first chapter defines all the basic concepts like rings, ideals, modules, etc. Of course, it is pretty brief compared to what you would get from an intro to algebra course.

(Also, be warned: "ring" in this book and this thread always means "commutative ring with unity", and all ring homomorphisms send 1 to 1, contrary to what D&F says)

>>10093527
Fuck off reddit

Back on the grind

>>10090674
Do you plan to follow some kind of schedule, like one chapter a week?

>>10095418
I didn't have any schedule planned, but I can take suggestions. Some chapters (and in particular, the exercises) are much more than one can reasonably do in a week unless theyre putting full attention on it. Since chapter 1 is mostly review, I think it should be alright to do it in a week, even though there's very many exercises. Chapter 2/3 are much longer and have many exercises, so i dont expect it to take less than so.

>>10095432
That's good to hear, I have seen you were already posting exercise 20 and thought you were going to go speedy. I will try to catch up during the weekend.

Welp that took a while, doesn't help that I have a massive hangover

>>10095884

>>10096159

I'm gonna skip the next 3 since im not too interested in boolean rings, but i welcome anyone to try

A very nice and interesting exercise. Unfortunately, most has already been solved for you.

Again, exercise 27 has nothing to solve, unless you would try to attempt on your own to prove Hilbert's Nullstellensatz.

28 actually shows (with a bit more work on functoriality property) that the category of affine varieties is contravariantly equivalent to the category of finitely generated k-algebras.

This finalises all exercises of Chapter 1. I won't be posting much again until there are more replies about it, then I might start the next thread on Chapter 2

Final graveyard bump

Welp, I guess there's not really much interest. Perhaps something easier (like first algebra/analysis course) would have worked better, but this is directed to very few individuals, few of whom can put much effort

>>10098991
Good initiative anon! Unfortunately I have to spend most of my time with preparation for my own abstract algebra lectures. But thanks for the book!

>>10098991
>>10099011
Sorry I wasn't able to participate in the first thread, I'm kind of busy on my end with fellowship applications, grading midterms, and the like.

>>10090674
I would absolutely love to participate but I am busy with other stuff right now. I'll see if I can catch up and join you guys next week. How far along into the book are you?

>>10099378
Well, I'm going over the second chapter on my own, but I won't be posting solutions in this thread at least.

>>10099404
Why so butthurt?

>>10099521
cos this thread is dedicated to chapter 1 i guess

>>10099530
Why kill a thread at 70 posts? Just continue here, i'll tag along after midterms.

>>10099532
yeah but nobody else is posting. it's pretty time consuming if im doing it only for myself

>>10099557
What is? You'll do the exercises anyway, posting them here won't take much time. At least 3 people are interested here, just not right now.

>>10099569
oh right, very reassuring

>>10099404
OK, I am going to go through the 1st chapter fast Monday (I am sadly busy with family bullshit tomorrow) and start with the 2nd chapter Tuesday.

>>10099575
Well, you do you. But why come here in the first place when you know it's full of larping undergrads and then be butthurt that only few people show mild interest?

>>10099593
I wanted this a place to share, not for me to do everything. So I'm putting all the effort, and some dickhead comes and says, "why don't you keep putting all that effort for me? There's a chance I might follow eventually"

It's the equivalent of me starting a charity so people can eat, but nobody comes and the food I bought rots, then some bum comes up to me, hey, I'm full right now, but you keep buying food, I might drop it at some point next week

>>10099606
Calm down with your self-righteousness. You pick a time when yuropoors have midterms, burgers are too lazy, what do you want?
14.11. is my last midterm (model theory), then i'll have about 4 weekends.

>>10099606
I am going to give you a helping hand starting next week anon, I promise.

>>10099606
bruh this is /sci/ and you're going through a graduate math course. 80% of this board is calc 1 fags. The people that are at the level to study this are probably busy with real life

>>10099606
I am the guy who said would try to do some during the weekend(10095418, 10095474), but I spent the day reinstalling artixlinux (lived with the OS on the pendrive for two months, now it slowed down and the applications freeze because it can't load the stuff), so I ended up lazing off. I will try to do some work tomorrow, I promess (and sorry for being a disappointment).

>>10099642
I loved model theory two years ago, the setup was boring but the results (compactness, skolem, types, saturation) opened my eyes. It is somehow sad that you cannot do things like topology with FOL, and after the initial hype for infinitesimal analysis (a la Robinson) I grew a tad tired of it. But for algebra it seems a very nice tool.

>>10098991
Sorry, I'm interested but I've got a real analysis exam this week.

Bump Function

>>10090775
>>10090959
>>10090964
>>10090972
Done the first few and got basically the same proofs, gonna start working on the rest and check against the ones your presented. If I get something different I'll post it. Next thread I'll try posting answers early one if I'm not too bogged down with work. So far the questions aren't too bad but they do seem to build on each other quite quickly

>>10099606
>I wanted this a place to share, not for me to do everything
Did you really expect to find a significant audience of people on /sci/ who
>are capable and interested in studying commutative algebra
>have the time to dedicate to it
>but aren't already doing a course or reading on it

>>10100319
they start getting interesting once you hit the more topological and geometric setting in my opinion, other than just pure algebraic manipulation

>>10091927
0 IQ post.

>>10100362
Yeah, a good chunk of the first half boil down to using a few basic tricks that he flat out tells you how to do.

Graveyard Bump

>>10101354
I'm at page 2. Is it correct so far? Some exercise will come soon enough, I promexx.

iii) as CA guy, the easy induction is
(x_1 + x_n) = (x_1) + .. + (x_{n-1}) + (x_n) =
= (x_1) + .. + (x_{n-1}') = ... = (x_1').

23 is 11 + words.

>>10101543
23.iv was incomplete, I got a question on 25.

>>10097389
So all the exercises are somehow finished. I didn't show 24 because I have seen it in a logic course.
I recall that it was highly tedious (very long computations), so that the teacher preferred to give the Stone isomorphism and used that to prove BA <-> BR.
The 25 is immediate but it is interesting to complete the duality, or determine the topological properties of Spec(A) such that A is Boolean.
Maybe for the next chapter you want to split the proofs between whoever wants to partecipate? I'm interested but really slow. But I'm sure that it is better for you to give up on this thread, people will just troll or slow you down.

>>10101678
Well, we can keep going to chapter 2 in this thread, and I won't do all of them.

>>10101687
Can I ask where are you from? Are you in the ALGANT program?

>>10101696
No, I'm a first year PhD student at a very nice uni, but I think I got in by accident

>>10101699
Aha, accident or not that's good.

back with some chapter 2

>>10101858

>>10101858
I read up to the beginning of tensor product, but I hot stuck to the proof of the determinant trick.
What does it means to multiply the left hand side by the adjoint matrix of \psi_j?

>>10101960
It's the "classical adjoint." The adjoint of A is a matrix A* such that A*A = (det A) I. This formula holds over any commutative ring. (Since it holds over Z[x1,...x_{n^2}], by universal property nonsense).

I've seen this trick come up several times in relation to integral extensions, when you have a system of equations linear equations and you want to relate that to a polynomial.

>>10101960
>>10101974
I should add that each entry of A* is a polynomial in the entries of A as well.

>>10101974
So what I am getting confused about, the matrix he is talking about is the one with entries \psi_ij,
which has entries in A[\phi] c End(M)?

Here are some solutions from chap. 2

>>10101987
That's correct. Everything here is in the ring End(A). In particular it's in the subring A[phi], which is commutative, even though End(A) isn't. (I'm not 100% sure how this trick works in the non-commutative case).

>>10102013
Yes, just what I got. Thanks.

>>10101687
Sounds fair, no reason to make a new thread for each chapter.

Starting with the (easy) exercises. Kinda wish Q2 gave less hint, would have been more exciting

>>10101354
>>10101356
I gotta ask, how do you do diagrams in LaTeX? I can never get them to work.

>>10103054
Look up a package called tikz-cd. This is the most "standard" way to do commutative diagrams.

>>10103063
Thanks, I've needed to add a few diagrams to some notes I'm typing up but couldn't get it to work properly. Actually, there's a lot of LaTeX that I can almost get working but not quite at the moment. Especially when I tried making a poster in LaTeX.

this is a pretty good thread desu familia

Sorry anon, my schedules has changed and I will be very busy in the next couple of weeks. I'll only be free late in the evening and pretty tired for anything much by that point, so I won't be able to catch up with you in due time because I would be progressing at a much slower pace (just 1 hour or so after dinner at best).
I had your thread bookmarked though and figured I'd give you a final bump. Maybe someone else will tag along.

>>10103836
I'll try keeping up but I need to seriously go over ring theory again, it's been about 4 years since I took abstract algebra and I'm rusty to the point that I need more than a "review". It's not that the problems are hard it's just that I know later on I'll probably need to strengthen my foundations. I'll do my best to keep up till then.

I have been banned on 4chan from my student internet probably because of some faggot so I'm gonna find it a bit harder to post...

Also the ban is for a year so...

>hand written notes
Aren't you meant to be fluent at Latex by grad school for math?

>>10104070
I'd rather spend more time working on the exercises than typing shit up on latex, of which there's no point in doing

Am I missing something here? You only require the last module to be finitely generated. I was thinking something along the lines of kerf perhaps being infinitely generated being a problem, but I'm pretty sure that any elements generating M" will generate M"/kerf

>>10104103
M''=M/ker f and not viceversa?

>>10104103
The map from M' to M has to be injective, right? So I think because of that you have to enforce M' to be finitely generated, otherwise you'll have problems with the generators of M.

>>10104234
Fuck I'm retarded. Not the most rigorous proof but it fits the space

wew appeal actually went through

I feel like I'm being dumb here. How does nakayama lemma implies this? I've tried looking it up but it seems so obvious to every post i've seen about it. What I've tried:

If [math]I\subset \text{Jac}(R)[/math], then by Nakayama, [math]I^2=I\implies I=0[/math], so we can assume that [math]I\not\subset \text{Jac}(R)[/math] and hence if [math]0\neq x\in I[/math], there exists some [math]y\in R[/math] such that [math]1-xy[/math] is not a unit, say, [math]1-xy=r[/math]. Then, multiplying by [math]x[/math] we get [math]rx=x-x^2y\iff x(1-r)=x^2y[/math].

But this doesn't take me anywhere...

>>10104936
oh wait... instead of multiplying by x, just rearrange the equation

>>10104524
What pen are you using anon? Bic crystal?

>>10104936
Isn't this kind of straight forward? Nakayama's lemma is basically what the hint is telling you, and since you can multiply (1-x) by x to get that x(1-x)=0 which also means that (1-x)(1-x)=(1-x)-x(1-x)=(1-x) so that (1-x) is an idempotent in the ideal I. Now it just remains to be shown that this generates the whole ideal, but since we're dealing with commutative rings it's easy to show that a principal ideal generated by (1-x) satisfies the desired properties. You probably already answered the question but I just typed this up to see if anyone could point out if my logic was flawed or not, still learning the subject as well.

Two very similar exercises that could ultimately have been solved by the same lemma in (homological?) algebra

>>10104946
yeah, maybe it's not of the crystal variety though

>>10105008
yeah, that's exactly what I'm not understanding...

where do you get that x(1-x)=0 equation from??

>>10105045
Maybe I'm mistaken, but since I is finitely generated then we can apply Nakayama's lemma
https://en.wikipedia.org/wiki/Nakayama%27s_lemma#Statement
which means that there exists x in the ring R s.t. e=1-x in the ideal I and xI=0. That last statement means that elements of the ideal multiplied by x are 0 right? So since 1-x is in I then x(1-x)=0 right? Or am I being an idiot here?

>>10105055
huh, I had only seen the statement 2 which is the one in Atiyah Macdonald, although I'm guessing they're all equivalent.

>>10105062
Yeah, you basically have to use the definition of the Jacobson radical which is probably what he wanted you to do. Then using the fact that a finitely generated Ideal is also a finitely generated module of a ring A you've basically got what the hint is telling you.

>>10105074
yeah, my issue was that every answer I saw was "by nakayama, there is such an element x" and then continues with the proof, as if it was extremely obvious. But it wasn't extremely obvious parting from the atiyah's Nakayama lemma (still obvious though, but not extremely, something that would merit at least 1 sentence), which was confusing me to hell and back because i thought i was missing something completely.

>>10105090
>which was confusing me to hell and back because i thought i was missing something completely.
Well that's the point of this thread after all, to learn together and try to fix each others misunderstandings.

16, which is essentially the universal property of direct limits in an abelian category

I just saw ghis but I'm interested. Will look at it tonight (i.e. in 12 hours).
How do I approach this - do I just read the post? What's the intention behind the scribblings, in relation to the book. Are you doing a QandA of sorts?

>>10105605
Well, what I had planned was for us each individually read a chapter, and perhaps if there's something in the book that we don't understand, to ask each other for insight. Then to also do the exercises in a similar fashion. I've been posting my solutions to the exercises so other people might have solutions to compare to, or perhaps so that others might tell me if I'm wrong somewhere.

However, chapters 1+2 are relatively straightforward standard material (perhaps a first meeting with the tensor product and exact sequences for some), so it's less likely that someone might have a question about them.

Bump

bump for interest

These last few questions remind me of my times reading Aluffi's algebra book. Something tells me that a thread like this for category theory would have been more popular if only for the larping categorytheoryfags

>>10107372
There's something satisfying about doing something the messy way first and then doing it all by pushing universal properties around. The other day when we were proving some lemma giving criteria for a functor to be representable, my algebraic geometry professor described it as a "magical bookkeeping device."

To be honest though I don't think a thread like that would go very well past the first chapter. If I tried to learn category theory before doing any algebra, algebraic topology, algebraic geometry, etc. I don't think I would have gotten much out of it.

>>10107412
To me it's the other way around. I find it so much more satisfying when something doesn't just work because of a clever map, but because the universe decided it was meant to be (or something of the like, I'm not the best at expressing myself).

And yeah, perhaps it wouldnt work past the first chapter, but I could expect it to last several threads where half the pics are anime avatars

>>10107454
>but I could expect it to last several threads where half the pics are anime avatars
Don't remind me of what happened to /mg/
>>10107412
Category theory is fairly cool at times but I do think it sometimes it's better to get down to the nitty gritty to understand the moving components.

A satisfying chase

>>10107680

>>10108843
An exhausting question honestly, mostly because A/M just glance over the fact that you have to first prove regular products commute with the direct limit.

All that typing is therapeutic in a way. Anyways, Q. 21 shows us how direct limits can exist even in Ring, which is interesting as some important ring constructions (i.e. localization) can be seen as direct limits

>>10109180
Also to note: exercise 13 in chapter 1 is essentially constructing the algebraic closure as the direct limit, and you can see it by applying exercise 17 in chapter 2

This is the last answer to chapter 2 that I'm posting. 23 is trivial and I don't know about Tor yet although I will eventually get to it. These last few questions are a lot of symbol pushing, but they do define important notions. Direct limits are also called in category theory "colimits", so you can expect that this idea has a dual notion.

Anyone that knows is welcome to post answers of the rest of the stuff. I'll be passing to Chapter 3 soon but doesn't mean you can't post about previous chapters

>>10104070
Unless you're proficient LateX usually takes up way too much time, though I do know people who can take live LaTeX notes.

>>10109562
I have the solutions written up from when I did them this summer, although they are a bit messy. I may upload them tomorrow.

Tor is intuitive if you're familiar with homology, e.g. from algebraic topology. It takes the right-exact sequence you get from tensoring and turns it into a long exact sequence, using the same algebraic idea as the long exact sequence in homology.

>>10109562

>that tidy penmanship

Just slit your throat anon, go fuck yourself. Only leave behind these beautiful leaves.

Bit off topic but - my algebra skills are non-existent (issues in upper schools).

Where or what textbook(s) should I start with to give me a basic understanding and allow me to not be too retarded.

>>10109789
I know a bit about Ext_1 that I learnt in my algebraic topology class but since we didnt use tensor products, I didnt learn Tor (and I didnt learn about Ext_n) so it would be too much for a digression for me right now, but I'll be getting into homological algebra right after commutative algebra

>>10110015
This is a very different type of algebra. If you're referring to the type of algebra "solve for x when x^2-4=3x+1" then you should probably go on Khan Academy, or perhaps Gelfand's Algebra followed by Axler's precalculus

>>10110107
cheers

>>10109789
Do you have a good reference for learning the homological algebra side of things? I have to blast through Atiyah for winter break since I'm taking intro to Algebraic Geometry

>>10110408
I have no idea what the "best" reference is. I learned it as part of an algebra class using Dummit and Foote, plus the professor's notes for that class.

>>10110408
Lang might also do you well.

>>10110797
Thanks! I'll look around for a reference. I've tried reading Lang a few times and just didn't click with me. I'll sit down and put a serious effort into Lang and Atiyah. I got 6 weeks of break so I'll have plenty of time to get up to speed

>>10110408

Weibel's 'Introduction to Homological Algebra' or Cartan-Eilenberg's 'Homological Algebra' function well enough, though it is usually a better idea to learn homological methods in a particular context, say, in algebraic geometry/topology upon encountering derived functor (co)homologies for sheaves such as H^i from global sections, or in the case of modules, global Ext^i from Hom or Tor from tensor products

Very simple questions, with Q.24 's primary significance in establishing a criterion for flatness by way of Tor vanishing. Technically, the free (projective) resolution in the first part is a flat resolution, and so Tor may be calculated as well in terms of flat resolutions

Also interesting: Q. 24 is important in that it ties in with the fact that the projective dimension of a flat A-module M that is also finitely generated is 0, and so by Auslander-Buchsbaum, we understand that the depth of M gives us the Krull dimension of A

>>10110408
Not him, but have you seen the Stacks project? Chapters (there are single pdf for each of them)
(10) commutative algebra,
(12) homological algebra.
Also Eisenbud commutative algebra and Weibel, but they are too much for a 6 weeks break.

>>10111708
I'm not at a super high powered University, we'll be using Fulton's algebraic curves. It's just I've done analysis for the past three and a half years with very little algebra done so I need to get up to speed

>>10111740
Fulton does not uses homology, as far as I remember. Also the fist chapters from the stacks project are not high powered at all (in fact, it tries to go from the ground to the heaven trough a very long road).
Not that I want to sell you the book, this was intended as an explanation of why I mentioned it.

>>10111708
Unpopular opinion but I think the Stacks project works solely as a reference and not something you learn from.

>>10112294
Looking around I think I might just pick up the rising sea, it starts off with category theory and does a bunch of the arrow diagram stuff

posting some chapter 3 now

>>10112783

>>10112787

>>10112790

how many hours are you putting into this a day? you are progressing fast

>>10112951
a lot of time, but I take too many breaks honestly, mostly because of lack of sleep makes it hard for me to concentrate.

>>10112542
Vakil's notes are more than you need for Fulton's book. I'm not 100% sure but I don't think Fulton uses schemes at all. Make sure not to get too buried in algebra that you miss the geometric motivation! That being said, Vakil's notes are a good place to go if you want to know what all the abstract nonsense is about.

>>10113005
I went through the algebraic curve stuff on Riemann surfaces with Terry Tao in the spring. It's not my first venture into the subject, but I definitely need to go through the abstract nonsense

Q. 25, 26 here. I'll try to post the remainder of Ch. 2 for completion's sake. Simple, as the strategies are already given, though A-M gloss over the symmetry of Tor, which isn't obvious but not hard to prove

Bump

Consider two points in Z_n,
x=(x1, x2)
y=(y1, y2)

Is there any significance to
x1 + x2 + y1 + y2
?

I mean that in the sense that the determinant expression
x1 * y2 - x2 * y1
would be significant.

This sum popped up in my studies of propositional logic (in a boolean algebra mod2).

>>10093396
based