groups-rings-fields

[next obsidian]

Because of dominant

I got that

[latent anvil]

right

[next obsidian]

So there’s a clear map right?

At least I think

[latent anvil]

right, look at f on stalks

[shy bluff]

@olive mirage I dont' know what a canonical form for finitely generated abelian groups is

[next obsidian]

I’m almost a little worried the map doesn’t extend to the field of fractions

Wouldn’t the map have to be injective?

[latent anvil]

it induces O_Y,η -> O_X,ξ

[next obsidian]

In order to do that

[latent anvil]

Field of fractions is functorial, right?

[next obsidian]

Yeah but you need to take fields of fractions

Is it?

[shy bluff]

Also, second quick question, what does a set of representatives look like? Say for example for the conjugacy classes of S_4?

[latent anvil]

No alex you don't

need to take field of fractions

The local ring is the function field

Right?

[next obsidian]

Oh I’m dumb

Nvm

[latent anvil]

Isnt that the previous problem?

[next obsidian]

Wait no

Yes

Oh!

My god

Nvm

I’m so dumb yeah

Ffs

[latent anvil]

But yeah field of fractions isn't functorial I'm dumb

I think

[next obsidian]

Dude omg I’m dumb

[latent anvil]

Lmaooo

[next obsidian]

No I don’t see how it could be

[latent anvil]

yeah

[next obsidian]

You’d need objectivity to do it in the obvious way

[latent anvil]

Right

[next obsidian]

Maybe in like Ring_mono to Ring

Or Field I guess

Wait no!

Mono isn’t even injective

[latent anvil]

Yeah I agree now

[next obsidian]

Lmao

[latent anvil]

It all works out

mono is injective

epi isn't surjective

[next obsidian]

Oh sure yeah

Cool

But yeah I’m dumb

Uggggghhhh

[latent anvil]

So you get a map from the residue field of the kernel

like φ : B -> A induces a map κ(ker φ) -> Frac(A)

Right?

if you do the obvious thing

[next obsidian]

Ihhh

[latent anvil]

And for φ mono you get a map on the fraction fields

[robust plover]

[next obsidian]

Wut

[latent anvil]

?

[next obsidian]

How do you get a map from function field of kernel

Everything goes to 0

[latent anvil]

Mod out by the kernel

That gives an injection B/ker φ -> A -> Frac(A)

[next obsidian]

Yeah

[latent anvil]

We can extend maps into a field to the field of fractions

[next obsidian]

Yes

[latent anvil]

So we get Frac(B/ker φ) -> Frac(A)

[next obsidian]

Wait what is kapp(H)

I think I might be mega dumb here

Oh...

[latent anvil]

κ(p) = Frac(B/p), right?

[next obsidian]

Yeah

Dude I am peanut brain rn

[latent anvil]

Lmaooo

[next obsidian]

Ffs

I think I have more ideas now

Thank you, I have been like

[latent anvil]

okay so we know how the field extension works

[next obsidian]

Not doing any math

Because I have been out of ideas on this

[latent anvil]

I feel like the number of sheets should be the cardinality of the fiber of η

err, the degree of the extension

[next obsidian]

Sheets?

[latent anvil]

Not number of sheets

[next obsidian]

Oh lmfao

[latent anvil]

I was thinking of covering spaces lol

[next obsidian]

Maybe

I thought you were talking about spectral sequences

Like with the pages

[latent anvil]

lmaooooo

[next obsidian]

Lmaooo

[latent anvil]

Absolutely not

[next obsidian]

Okay, I gotta do something for a bit

Tfw shamrock who said he isn’t doing AG is better at it than you

[latent anvil]

Oh wait, f is finite type

[next obsidian]

Yeah this is mondo important

[latent anvil]

So maybe we can get the extension to be finite type

yee

[next obsidian]

Yeah that’s the goal

[latent anvil]

does finite type imply finite type on stalks?

[next obsidian]

Then do Zariski’s lemma

[latent anvil]

It should right?

Morally? Lol

[next obsidian]

Yeah I wasn’t sure, but I imagine

The thing is I didn’t pursue it

[latent anvil]

well take a cover by affine with the finite type assumption

[next obsidian]

Since I thought I had to take FOF

[latent anvil]

Take the cover element containing your point

[next obsidian]

Because I was peanut brain

[latent anvil]

take the same generators

Now you're looking at localization

And it's easy

I think

[next obsidian]

Idk, I think generic finiteness might matter here

I feel like you need a finite cover or something

[latent anvil]

It probably will later, but I think this proves the extension is finite

You have a finite cover

[next obsidian]

By?

[latent anvil]

It's finite type, not locally finite type

Right?

[next obsidian]

Oh

Yeah

[latent anvil]

But also I don't see why that's relevant

[next obsidian]

Self oof time

[latent anvil]

I think it's just true that a locally finite type map of schemes is finite type on stalks

[next obsidian]

Maybe

[latent anvil]

By going to affines and thinking about localization

[next obsidian]

Anyway let’s say it is tho

Let’s say we know the thing is like finite extension

[latent anvil]

Yup

[next obsidian]

Now you want an open that’s dense

So like

Any open nbd of the generic point will be dense

[latent anvil]

Isn't any open set dense?

Nonempty

[next obsidian]

I mean like

[latent anvil]

I guess we're saying the same thing lol

[next obsidian]

Image is dense

[latent anvil]

Ahh yeah

[next obsidian]

Sorry that was very not clear

Oh wait

That’s doesn’t matter I think

The problem want an open dense subset of Y

But that just means nonempty like you said lmao

[latent anvil]

Right lol

[next obsidian]

Maybe like

Take stalks at each point in the fiber

Such that something good

[latent anvil]

lol

[next obsidian]

Take the opens from that

Union

Or something idk

[latent anvil]

¯\_(ツ)_/¯

You said you needed to do something else

[next obsidian]

God, I said I need to do something

Yeah I’m gonna go do that

[latent anvil]

hmm I may have learned some algebraic geometry last year

possibly

[bleak abyss]

Mood

[latent anvil]

Still seems unlikely

ugh I gotta work through Hartshorne now so I can keep flexing on magician

[olive mirage]

btw, I really like Liu's algebraic geometry text

that's the one my profs were really advocates for

[bleak abyss]

I've heard good things about it modulo his treatment of cohomology

Tbh when I do Vakil's class I'm thinking of TeXing notes of my own lol

[latent anvil]

dangerous game there

[olive mirage]

yeah, I'm usually looking for "Who gives me the right three sentence intuition for this"

which is not what Liu does, btw. But Liu does the technical stuff in sufficient generality to apply it\

[latent anvil]

Honestly one thing that I'm realizing might have hurt me with Hartshorne is that it doesn't have a ton of examples

Maybe they were in the exercises I didn't do

But I don't feel like I can write down very many interesting schemes ig

[bleak abyss]

I can write down one for every ring

[latent anvil]

hmm sounds sus

I don't believe you

[hot lake]

think about the cardinality of that

so many schemes

[latent anvil]

(yeah I wasn't counting affine ones as "interesting" lol)

[olive mirage]

yeah, I think a rich stable of examples you care about is essential in algebraic geometry

and most algebraic geometry texts are allergic to examples

[latent anvil]

When I think about like, schemes that aren't affine I'm pretty much just looking at projective space and the line with a doubled origin

there's all this complicated machinery for gluing

Baked into the definition of a scheme

And I don't really see the point

[olive mirage]

I mean, it is helpful to note that for curves, they're all affine plus some stuff

so you really are getitng nearly the whole picture by thinking about affines

if X is a curve, take the field of rational functions on X, and then the places of that form the points of a smooth projective variety.

(this may not be the kind of thing that makes you happy, but I find it reassuring)

(it also shows you that the notion of a "point at infinity" is nonsense)

[latent anvil]

Sorry, I don't know what the places of a field are

and I happen to like my point at infinity thank you very much

[hot lake]

you can define nonarchimedean norms on a field

these induce discrete valuation rings, and then places

[olive mirage]

the places are the valuations, like Q, for example, has ap lace for each prime, and the archimedean absolute value

(which, not at all coincidentally, corresponds to the points of Spec(Z))

though the analogy is a little messy, and I think people oversell the "Spec(Z) is a curve" stuff

[latent anvil]

ahhh

[olive mirage]

note I"m using "variety" here in the sense of "special type of scheme"

[latent anvil]

yup

[olive mirage]

because if you do the whole thing I suggested, you definitely get fuzzy points, like the valuation corresponding to (x^2+1) if that happens to be irreducible over your field

[latent anvil]

So how is the original curve related to this new thing built out of places?

[olive mirage]

(Spec(RealNumbers[x]) is a great example to understand geometrically what is happening so you have better intuition for e.g. Spec(F_p[x])

so, are you familiar with divisors?

[latent anvil]

no, sorry

I dropped out of my ag course before we got to curves and divisors and stuff

didn't understand schemes well enough

[olive mirage]

So, this is something that shows up in a lot of places, but if you have a function, then the divisor of the function is a formal sum of the zeroes of the function minus the poles of the function

[latent anvil]

I feel like this showed up in my complex analysis course

not under that name

or stated explicitly

[olive mirage]

it definitely did. Divisors, like most of algebraic geometry, is just a very good fake of complex analysis.

[latent anvil]

lol

[olive mirage]

but the elements of your function field are the meromorphic functions on your curve

and the valuations let you calculate the divisors of them

which lets you calculate the sheaf of functions and all that jazz

[latent anvil]

ugh this is the thing about AG. Every time someone talks to me about it it sounds really interesting and cool, but then when I actually try to learn it I just get stuck on nitty gritty details about schemes

I guess that's the price of the theory

[olive mirage]

But a nice example of this is if you consider that ugly nodal cubic, Spec(k[t^2,t^3]) and you do this process, the fraction field is definitely k(t)

so then you get P^1

and so this even kills all the singularities for you

[hot lake]

so by only looking at fields you lose some information

[olive mirage]

(and yeah @latent anvil with you 100%, I definitely spend a lot of my life feeling that way when drudging through algebraic geometry stuff)

Well, lose or gain, depending on your view. If you want to study singularities of curves, yes, you lost something

but this means every curve is automatically and canonically associated with a smooth projective curve which it is birational to.

and for most applications, if you are looking at a singular curve, what you actually want is the smooth projective thing it is a singular model of.

[latent anvil]

ahh it kind of makes sense that this new projective curve will be birational to the original one

Since we built it up out of the function field

[next obsidian]

Yeah Shamrock the promise land exists

Once you finally get to riemann roch

And some coho

The world opens up

[olive mirage]

(note this all falls apart terribly once you get to surfaces)

[next obsidian]

You just have to go through an entire quarter of bullshit to get there

[olive mirage]

I mean, it is incredibly frustrating how long it takes to get to things like dimension in algebraic geometry that are intuitive and basic

[next obsidian]

I meaaaan

Dimension is intuitive

In the great cases

But can go really fuckly in the not great cases

[olive mirage]

Well, I assume no one does algebraic geometry for any reason other than counting solutions to polynomials over finite fields

so dimension there isn't messy

[next obsidian]

A completely harmless assumption

Haha

Finite fields scare me

Besides F_p for a prime p

and even those

[olive mirage]

(but one consequence of that function field stuff is that every curve is birational to the zero set of a two variable polynomial)

[next obsidian]

Like F_2

shudder

[olive mirage]

which element of F_2 scares you?

[next obsidian]

The one that makes it char 2

Haha

[olive mirage]

so the 1! haha

I looove finite fields

[next obsidian]

Fields without 2 🤢

[olive mirage]

F_2 has a 2

that 2 just happens to be 0

Here is a fun problem: Prove that no ring has exactly 5 units

[latent anvil]

weird

so the group of units is cyclic

Choose a generator ζ

[next obsidian]

Proof 5 is prime QED

[latent anvil]

I want to say ζ will generate a field

[next obsidian]

:^)

[latent anvil]

No

[olive mirage]

I know several rings with a prime number of units

[next obsidian]

It’s a joke

[latent anvil]

I disagree with myself

[next obsidian]

Is it in any way shape / form

Wait

Finite ring

[latent anvil]

Maybe we should look at the characteristic of our ring A

?

[next obsidian]

Anything is a sum of two units right?

Or am I being dumb

[latent anvil]

why finite?

[next obsidian]

I feel like I proved that

Because the argument involved counting

[latent anvil]

anything is a unit or a zero divisor

[next obsidian]

Something like that was on our algebra midterm I feel

[latent anvil]

Are all the zero divisors contained in the Jacobson radical or something?

[next obsidian]

But I forget if it was unit or whatnot

[latent anvil]

Sounds fake

[next obsidian]

Hold up let me get problem

[latent anvil]

Anyways our ring isn't finite

So we have a map Z[x] -> A picking out ζ

[next obsidian]

Lol I’m dumb

Finite field, sum of two squares

[latent anvil]

The image will also have 5 units

[next obsidian]

[latent anvil]

Wait doesn't that mean the image is Z[x]/(x^5 - 1)?

Which has too many units

Idk if that was correct

Alex check what I did or else

🔫

[next obsidian]

Umm

Why is that the kernel

But that kinda seem right ngl

[latent anvil]

ζ satisfies it right?

[next obsidian]

Yeah

But Brendan here’s question

[latent anvil]

oh sorry

I'm wrong

[next obsidian]

Why specifically 5?

[latent anvil]

It'll be the elephant

[next obsidian]

Wait also oh yeah

Zeta isn’t a root of unity

[latent anvil]

Z[x]/(x^4 + x^3 + x^2 + 1)

?

[next obsidian]

Or is it

[latent anvil]

It kind of is

[next obsidian]

Wait

Wait

I said nothing

[latent anvil]

ζ^5 = 1

[next obsidian]

I swear it wasn’t me

I didn’t know any better

[latent anvil]

post a screenshot of the notes app

Make the apology good or I'll cancel you

[next obsidian]

Yeah why specifically 5 tho?

You know what

[latent anvil]

because there's 5 units

I chose it to generate the group of units

[olive mirage]

I'm not sure why you assumed that you could get rid of the zeta-1?

[latent anvil]

Wym zeta?

[next obsidian]

This wouldn’t happen with 3. Because 3 is great. No I mean why does this argument work for 5 but not any random ass prime

[latent anvil]

The image will contain ζ-1

[olive mirage]

I only get to Z[x]/<x^5-1>

I think you assumed something about no zero divisors to get further

[latent anvil]

Oh I see

Yeah you're right

Mb

So we get a map but it might not be injective?

[olive mirage]

haha I don't get any points for seeing things in a problem I gave you haha

[latent anvil]

hmm this is less good

So wlog we're looking at quotients of Z[x]/(x^5 - 1)

and x generates the group of units

[olive mirage]

Excellent progress

(and good eye to go straight for the universal map from Z[x])

[latent anvil]

oh Z[x] is a nice ring

hmm not that nice nvm

[next obsidian]

Shamrock is just good at algebra

[latent anvil]

I wanted to list off the ideals containing x^5 - 1

[next obsidian]

Lol

[olive mirage]

you are missing only one key idea, although you alluded to it earlier

[latent anvil]

Characteristic?

hmm I don't really see how to determine that easily

Or what it tells us

[olive mirage]

What can you say about the unit group of a ring of, say, characteristic 0

[next obsidian]

Isn’t it infinite?

No

[latent anvil]

No

[next obsidian]

Whoops

[olive mirage]

(supposing it is finite)

[latent anvil]

it's even

[next obsidian]

I was thinking about <1> under addition

[latent anvil]

if not in char 2?

i think so, you can negate units

And there's no fixed points

[next obsidian]

What about 1?

Oh yeah

Umm

[latent anvil]

So wlog char A = 2

so our ideal contains 2

[next obsidian]

The ideal contains 2?

[latent anvil]

We're looking at a quotient of F2[x]/(x^5 - 1)

that's pretty small

The kernel

So we start with some abstract ring

then we get a map from Z[x]

Right?

I showed that that ring has to have char 2

Which means the kernel of the unique map from Z is (2)

So the map from Z[x] contains 2 in its kernel

by looking at the image we can reduce to looking at quotients of F2[x]/(x^5 - 1)

And F2[x] is a very nice ring

The ideals containing (x^5 - 1) are just that, (1), (x-1), and (x^4 + x^3 + x^2 + x + 1)

right?

[next obsidian]

I have no clue tbh. I hate finite fields

[latent anvil]

okay so from the beginning, assume A is a ring with 5 units

[next obsidian]

I don’t see why you can’t factor the elephant tho

[latent anvil]

oh I think I've just done that before?

In like Galois theory

Might be wrong though

Hmm ig this is over F2

[next obsidian]

Idk I hate F2

[latent anvil]

It has no roots

[next obsidian]

So that’s why I’m concerned

[latent anvil]

It has to factor as a product of quadratics

x^2 + x + 1 is the only irreducible quadratic right?

[olive mirage]

it is

[next obsidian]

Uhhhhh

Oh okay well there’s proof? I guess? Does it factor as that squared?

[latent anvil]

Yes

[olive mirage]

(it does not)

[latent anvil]

So we've narrowed it down to four rings

the zero ring has too few units

[olive mirage]

(x^2+x+1)^2=x^4+x^2+1 because of squaring in char 2 etc

[next obsidian]

Oh yeah

[latent anvil]

F2[x]/(x-1) = F2 has too few units

F2[x]/(elephant) is F16 I think?

It's a finite field anyways

[next obsidian]

F2[x] has how many units?

[latent anvil]

?

I'm not looking at that ring

[olive mirage]

and note that F_16 has a unit group of order 15, which is divisibe by 5, which is a nice little sanity check there

[latent anvil]

It has 2 though I think

[olive mirage]

(I think just 1 unit in F_2[x])

[next obsidian]

I mean the ideal could be (1) right?

So don’t you have to account for it?

[latent anvil]

Sorry yeah zeta

that gives the zero ring

We're quotienting by it

[next obsidian]

Tfw

[olive mirage]

(no no, you're good)

[next obsidian]

You right lmfaooo

So...

[latent anvil]

finally F2[x]/(x^5 - 1) uhh

can we CRT?

[next obsidian]

The final

[latent anvil]

Yeah we can

[next obsidian]

Why not?

[latent anvil]

They're both irreducible

so they generate coprime ideals

since we're in a PID

So the last ring is F2 × F16

which doesn't work

[next obsidian]

which has 31 units?

[latent anvil]

It has 15 units still

[next obsidian]

Errr

30

Errr

Yes

😶

[latent anvil]

(1, 0)^(-1) time

[next obsidian]

Yeah

[latent anvil]

That was a fun problem zeta!

[olive mirage]

haha I looooove algebra problems, I have many

[latent anvil]

I may put it on my algebra problem sets next year...

[olive mirage]

it is not so hard to go from there to showing exactly which odd numbersa re the number of units in a ring

[latent anvil]

Yeah so odd implies char 2

[next obsidian]

Wait uhhh

[latent anvil]

Oh but it might not be cyclic

[next obsidian]

Are you going to run 398 again?

[latent anvil]

Maybe

Idk

I haven't decided

[olive mirage]

(though you need to throw around the word semi simple, which is not a word I'm very happy when shows up)

[latent anvil]

I'm bummed about not getting 33X

[next obsidian]

Okay. If u are let me know I can help again if u like

[latent anvil]

But ig I'll have more time

[next obsidian]

Not getting 33X is a bummer

But I sort of felt Logan would get it

[latent anvil]

yeah, it went to Logan though and he's great

Students will have a good time

[olive mirage]

are you an undegrad, grad student, prof shamrock?

[latent anvil]

undergrad

[next obsidian]

Same as me

Although he might graduate a year early apparently

[latent anvil]

we're mostly taking the same courses

[next obsidian]

We go to the same uni and everything 🤗

[latent anvil]

magician and I

[next obsidian]

Although next year not really 😦

[olive mirage]

anyway, if you want a massive pile of algebra problems let me know

[latent anvil]

Loll I would love that

you could take analysis with me

and then we would

[next obsidian]

I wanna do manifolds this

I neeeeeeeeeees it

[latent anvil]

Yeah manifolds is pretty great

[old nimbus]

😦

[latent anvil]

Zeta pm me with them or post a link or something

[latent anvil]

Ty

I was going to start doing research stuff

After my break

This is more fun

lol

[next obsidian]

I’m gonna get so good at manifolds you’ll be so jealous. You’ll be like woaaah you solved the tubular neighborhood problem so quickly woaaah

[olive mirage]

if you decide you want more, I have all manner of old exams and homework sets I've given grad/undergard courses, so if you want to use them pedagogically I'm happy to share those

[next obsidian]

Except I won’t since you’re mad nasty at manifolds apparently lol

[olive mirage]

(though I won't post that in public channel, but you can PM)

[latent anvil]

yeah np

I might hit you up for that

[next obsidian]

Wow that’s a lot of problems

[latent anvil]

so for context about what I was saying with problem sets

I started an algebra study group in my first year

and ran it for students this year

[next obsidian]

I was in it first year!

[latent anvil]

The undergrad algebra courses here are pretty slow

[olive mirage]

(This packet was originally constructed to help students study for quals when I was a grad student. Though since then I've often used it to write quals)

[latent anvil]

students in honors analysis usually don't like them because of the pace

But it's hard to skip directly to grad algebra

[next obsidian]

Yeah it looked like quals type stuff Zeta

[latent anvil]

anyways, I write problem sets for this unofficial course thing

And organize it

[olive mirage]

oh that's cool

[latent anvil]

http://untyped.me

[olive mirage]

But yeah, I think there are a lot of problems in there that are naturally sort of intriguing

[latent anvil]

My problem sets are on here

If you want to trade

Lol

[olive mirage]

I am always happy to steal more problems

so this is your syllabus for your study group?

[latent anvil]

Yup

it might need to be reworked

[olive mirage]

I was going to say "this doen'st seem so slow" haha

[latent anvil]

There were some difficulties this year

Yeah the issue is that I organized the study group in my first year primarily for myself

So I could learn algebra

And I had a little bit of background with group theory already

so the original schedule and problem sets were written to accommodate that (also the person running it was insane and had no concept of difficulty)

[next obsidian]

OMG

He was so psycho omg

[latent anvil]

bffs tho

[next obsidian]

Now he’s at Berkeley studying ANT

So

Yeah he’s so sweet

He baked us cookies when we studied for math tests in analysis

Since he was our analysis TA

[olive mirage]

Yeah, my problem set is definitely designed with the idea that you can tease hints at people to guid them through it if they get stuck

[latent anvil]

galaxy brain algebra course:

Give them that problem set packet

meet twice a week

that's it

no structure to the meetings

just work on the problems and learn as needed

[next obsidian]

Lol

[latent anvil]

oh yeah I forgot to brag about the other cool thing with my study group

[olive mirage]

hahaha, nah, if I'm doing grad algebra I'm not giving up having a captive audience to listen to me talk about algebra for hours 😛

[latent anvil]

Students give lectures

Lollll

which is pretty fun

[next obsidian]

Mine were great

[olive mirage]

I was guessing that from the guest lecture thing

[latent anvil]

Yours were looooong

[next obsidian]

And went on for 30 minutes longer

I got kind of fucked tho

[latent anvil]

30 minutes???

I went and got lunch

[next obsidian]

I still maintain in winter I got the longest one

[latent anvil]

and you were still going

after an hour of talking already

[next obsidian]

Okay yes

But by spring one was good

I wrote the write up

And Thomas said “this is a good write up”

And I was like yesssss

[latent anvil]

I kind of want to start another study group

[next obsidian]

For

[latent anvil]

Good question

There are many things

[next obsidian]

If you do ANT

[latent anvil]

The problem is that I like manifolds

a lot

[next obsidian]

I’ll bite the bullet and join

: (

[latent anvil]

lmaooo

get the gang back together

[next obsidian]

Blood pact

[latent anvil]

get Thomas to TA over zoom

[olive mirage]

These are good problem sets, and I've seen some nice ones here I had not seen befroe

[next obsidian]

Zoph is typing

[latent anvil]

Thanks!

I really like them

[next obsidian]

My favorite

Is the one about S^G

[latent anvil]

The quality gets dodgy at the second half of spring

[next obsidian]

It’s so crazy

[latent anvil]

Corona made things awkward

[olive mirage]

haha yeah Well if you want a lot of Galois Theory problems, there is no shortage

although problem one is "Prove everything about Finite fields"

[next obsidian]

Aiyayay

[latent anvil]

These were originally written by an algebraic number theory person who has a finite group theory hobby

[next obsidian]

Yeah all the ring theory was just number theory lol

[latent anvil]

Which I think is a great combination for intro group theory and Galois theory

But the ring theory section had to be reworked...

[next obsidian]

We also had fun and tried to use the word involution as much as possible for the week 1 hw

[latent anvil]

Omg I forgot about fixed point free automorphisms

They're so cute

[next obsidian]

Our TA randomly used it and we thought it was funny

[olive mirage]

I really dig finite group theory, it's an excuse to do combinatorics. And ring theory is mostly pretty ugly

[next obsidian]

Ring theory kinda sucks, but then comm alg is actually kinda cool

Actually wait

[latent anvil]

ring theory thinks you're ugly too

[next obsidian]

I like ring theory too

Or at least I maintained I did

Lol

[olive mirage]

it is commutative algebra I don't like. I like all the polynomial rings over fields stuff.

[latent anvil]

oh zeta you might find this interesting: https://math.berkeley.edu/~tb65536/Commuting_Probability.pdf

[next obsidian]

Ohhhh that’s so cool

[latent anvil]

there was a problem on those problem sets about commuting probability

which we were very excited to put on there

[next obsidian]

Shamrock

[latent anvil]

because it's cool

[next obsidian]

Show him the secret problem

[latent anvil]

oh lol

http://untyped.me/problem_sets/winter10.pdf

[next obsidian]

I have it memorized, the number

[latent anvil]

1004913?

[next obsidian]

Yeah

[latent anvil]

if i remember correclty

okay im gonna dip

[next obsidian]

Also, the fact he emailed them and they got back!

[latent anvil]

what?

[next obsidian]

Thomas emailed one of them about the origins of that problem

And they replied

[latent anvil]

oh hih

I forgot that

[next obsidian]

He posted it in slack I think

[latent anvil]

aight I gotta go think about operads

[next obsidian]

Aight

Cya

[olive mirage]

this problem looks familiar

[next obsidian]

It’s cool

[olive mirage]

I think this is in Dummit and Foote?

or something similar

the probability two elements commute paper is awesome

[next obsidian]

It is

It’s a problem, but this dude put it on our like hw for week 5 lmao

Absolutely insane haha

You can also solve it via rep theory using induced characters

[olive mirage]

something something tensor product disappears in a puff of smoke

[next obsidian]

Wut

For groups??

Maybe with abelian groups haha

[olive mirage]

well, a character is equivalaent to a module over the group ring FG

[next obsidian]

Ohhhh

Sure

[olive mirage]

and the induced character is what you get by doing the change of base tensor product of an FH module to an FG module

[next obsidian]

Ah okay, yeah I never learned induced character

We stopped right before that in our class

And I don’t really care for rep theory

[olive mirage]

which is on the one hand easy to say and on the other hand horrible haha

[next obsidian]

I just don’t get rep theory haha

[olive mirage]

haha and yet to a lot of people the whole point of algebraic geometry is to produce representations of Qbar

[next obsidian]

Anytime I have to do anything related to semisimodules or rep theory I have to crack open my textbook

Yeah I think someone at my uni does that for her research

Julia Petsova I believe

[olive mirage]

Yeah, I really do not know why, but modules do not... how do you youths say... vibe with me 😛

[next obsidian]

Like she has some papers doing some rep theory stuff over schemes, I don’t get it at all

I like modules

They’re cool

[olive mirage]

I love group actions and vector spaces

[next obsidian]

Also how do you do AG

[olive mirage]

and a module is just like those two had a baby

[next obsidian]

Haha

A big thing is sheaves of modules haha

[olive mirage]

I don't really think of those as modules, intuitively

they're functions

and I get functions

[next obsidian]

Oh wow

I am like

Damn why would this be true...

Ohhh it’s true for modules

Haha

[olive mirage]

like sure, C([0,1]) is a C(R) module, but I'm not inclined to think of it that way

[next obsidian]

That’s fair

Wait

Oh yeah, I would more naturally think of it as an R-module

But maybe R embeds into C(R) like, faithfully as constant functions

So C(R) might just be a strict expansion, but I’m too tired to think about it lol

I think it does haha

Idk, thinkin of stuff as functions isn’t really@how I go about it

Like for schemes people talk about like the functions being ideals or something and the primes the points

And evaluating is quotienting??

I don’t get that at all and don’t bother thinking about it that way haha

[olive mirage]

like, thinking of the germ of a function makes way more sense to me than thinking about localizing at a maximal ideal

[next obsidian]

Oh I am complete opposite

Then again I haven’t ever seen a definition of germ, but to me that is just the image of the function in the stalk lmao

In the case where your sheaf is like C^infinity functions or something

[olive mirage]

the germ is tantamount to the Taylor series if your function is holomorphic

[next obsidian]

Is it?

Huh

I guess that makes sense given that germ to my knowledge is supposed to be like it’s local behavior

[olive mirage]

yeah, so the germ is really some kind of (inverse?) limit over open sets taht contain your point

[next obsidian]

That’s just how you construct the stalks haha

[olive mirage]

ahhh, but complex geometry is so rigid that knowing the local behavior at one point dictates the entire global behavior

[next obsidian]

Yes

Identity theorem

[olive mirage]

Yeah, so that's why a germ is equivalent to a Taylor series

[next obsidian]

It’s like the super gluing axiom haha

[olive mirage]

(note that real functions are horrible creatures, so this is not at all true if you look at, say, real manifolds)

[next obsidian]

Sure

Real functions suck

Even C^infinity ones

[olive mirage]

but Algebraic Geometry is just Complex Analysis

(nothing like real analysis, basically)

[next obsidian]

:^)

My uh frehsman honors analysis prof is cited in Hartshorne

For a paper he did with his advisor Kodaira

But back then AG was just several complex variables haha

[olive mirage]

there is a big theorem of Serre that goes by the somewhat comical name GAGA, that says that after you develop all this machinery, algebraic geometry over C is exactly isomorphic to analytic geometry over C

[next obsidian]

He just retired at the end of this year

Yeah GAGA is big

I’ve heard of it

A friend read it

And his only comments were

“The results should not follow from the machinery”

He basically said that like

[olive mirage]

but the way I always frame it is that C is the answer in the back of the book for algebraic geometry that told us we got the answer right

[next obsidian]

For how relatively basic the machinery was

[olive mirage]

and gives us hope that the results will be meaningful elsewhre

[next obsidian]

The strength of the results were unbelievable

Is GAGA for C or C^n?

[olive mirage]

well, manifolds over C include C^n and such

[next obsidian]

Ah okay, I wasn’t sure if analytic geometry over C included that

[olive mirage]

Yeah, it does.

[next obsidian]

I see

[olive mirage]

(so it isn't just for curves)

[next obsidian]

I want to learn some several complex variables eventually

It seems fun

But I need to redo my riemann surfaces stuff

My class fell apart due to COVID

So I learned like nothing

[olive mirage]

several complex variables is an area I know almost nothing about that I wish I knew more of

I am going to really make myself learn Griffiths and Harris some day 😛

[next obsidian]

I’d like to understand riemann surfaces eventually too lol

But from the small amount of stuff I did

[olive mirage]

yeah, understanding Riemann Surfaces is a must

and Miranda is THE book for it

[next obsidian]

All of it is about understanding LFTs and uhhh, fucking groups of automorphisms

[olive mirage]

it does Riemann Surfaces and algebraic geometry together, and is amazing

[next obsidian]

Fuchsian groups and stuff

We used “a concise course” and I highly disagree with that choice of text

We did not use it the other two quarters

And we black boxed stuff so I didn’t feel like I actually understood it

[olive mirage]

I wish my first semester of algebraic geometry had been Miranda, but instead I only found the book after spending years learning the stuff

oh you would hate me as a lecturer, I blackbox everything haha

[next obsidian]

The exercises were very hard

And nothing like anything we did before, and none of it used tiemann surfaces

It was all about groups of automorphisms so it was really just topological group stuff

[olive mirage]

Algebraic geometry as a subject does not have good exercises

[next obsidian]

So there was this disconnect where

The stuff you learn and the stuff I did were not at all connected

[olive mirage]

and this extends to Riemann surface stuff

[next obsidian]

Algebraic Geometry as a discipline sort of sucks

I’m inclined to parrot what people say that it’s just very hard and...

But all math is hard

Yet they manage to teach it in a less shitty way lmao

But I’m just stuck in the grindstone, so Hartshorne is always by my side

[olive mirage]

forming an area into an easily digestible story takes time

and is especially difficult when the area is constantly shifting focus

[next obsidian]

Fair enough, it’s just I feel that the mindset of the people doing it is a bit off

Any other subject I’ve seen people actively recommend you start with easier stuff with normal exercises

Which build up, and have some computation, builds the story up gradually

[olive mirage]

yeah, I think in Algebraic Geometry they should really make you do the classical stuff in a better way first.

[next obsidian]

It seems most people just tell you to go do every Hartshorne exercise from the get go and to go duck off

[olive mirage]

haha yeah, I think "Don't look at Hartshorne" was good advice that I got

but I have very few algebraic geometry sources I enthusiastically recommend

[next obsidian]

I’m already stuck with it 😔

I feel like the subject is sort of, this self feeding loop tho which sort of bred this issue

I think modern day some people are trying to change it like Vakil

[olive mirage]

yeah, I do like his notes, and wish they existed when I was a student

[next obsidian]

But AG suffers that issue where all the terminology becomes so fucked it’s all but impenetrable even for those in neighboring fields

I feel like there’s sort of a “big boys club” mentality

Which I hope to see be eroded

[olive mirage]

haha neighboring fields? it is impentrable to me and I'm published in it 😛

but yeah, I think taht mentality bled over from number theory, and especially the style of people like Grothendeik who really loved abstraction

[next obsidian]

Idk, I didn’t think I’d ever do AG but UW just sucks you into it

[olive mirage]

You've heard of the Grothendeik prime, right?

[next obsidian]

My analysis prof said for some reason a ton of his students just go and do AG for their PhD

Yeah lmao

I don’t know it, I wanna say 42?

Or something lol

[olive mirage]

57

[next obsidian]

Oh okay

Divisible by 3 lol

Not as egregious as being even

[olive mirage]

yes, not quite that bad haha

but not as unbad as 91

[next obsidian]

Is that not prime?

Does 7 divide it

Yeah 7*13

Tricky asshole

I thought to myself “okay if anything divides it it’s gotta be 7”

That’s the trickiest one among low numbers haha

[olive mirage]

91 is the only number less than 100 that will trick you

if you can identify squares, evens, mutliples of 3 5 and 11

[next obsidian]

Yeah, I’m incluned to say some multiples of 3 look prime until you just add the digits

Like if you had only 100 milliseconds to answer I think I’d say 93 is prime

But at least there’s an easy test lol

[shy bluff]

What is the centre of an action?

[next obsidian]

I’m not familiar with the term

[olive mirage]

can we see the context?

[next obsidian]

What’s the context

[hot lake]

it should be defined somewhere before

[next obsidian]

I have never seen that term, and to be honest google comes up with nothing. What textbook are you using?

[hot lake]

ctrl+f center

[next obsidian]

It certainly isn’t a standard definition, so if it isn’t defined earlier on the problem set or something send an email to your prof

[shy bluff]

He hasnt' responded yet

[next obsidian]

It can’t be like the set of elements x such that for all g, gx = xg since a group action ai priori only has one side

[shy bluff]

Dummit and Foote

[bleak abyss]

Probably think of an action as a homomorphism G->Sym(X)

[next obsidian]

Ohhhhhh

[shy bluff]

Sym?

But he uh skips over a lot of stuff

[next obsidian]

Symmetric group on X

[bleak abyss]

That's my guess

[next obsidian]

But wait

What would the center be then though?

[shy bluff]

Adjoint action = conjugation?

[next obsidian]

I think so?

[bleak abyss]

The set of g such that ghx = hgx for all h

[shy bluff]

Like we have

[bleak abyss]

So phi(g) is in the center of phi(G)

[next obsidian]

Perhaps they mean the stabilizer?

[shy bluff]

But also

Idk

[latent anvil]

C^infty functions are great

[shy bluff]

A lot of the uh things here are not very well defined?

[latent anvil]

modules are great

you're all heathans

[bleak abyss]

Who dissed on them?

[shy bluff]

I.e, he never explained what a semidirect product is but it's in the homework

[latent anvil]

you know what else is great? modules over C^infty(M)

zeta and magician did up above

[bleak abyss]

And thus shamrock just discovered sheaf theory

[latent anvil]

ban them plz

[shy bluff]

wait while I'm asking questions, what does a "Set of representatives" look like

[next obsidian]

I didn’t shit talk modules

[olive mirage]

haha I'm very happy with modules of functions

[next obsidian]

In what context?

Normally like

[shy bluff]

Because we have this

[next obsidian]

Ah

So a conjugacy class is formed by an equivalence relation right?

A representative is one element in a given equivalence class

[shy bluff]

But the onyl thing that we have regarding "set of representatives is

[bleak abyss]

A set of representatives is just a fundamental domain 🙃

[next obsidian]

So it just means one element from each conjugate class

[shy bluff]

Hrm

Can you show me what you mean in say S_4?

[bleak abyss]

• Grothendieck probably

[latent anvil]

oh also magician, sandor said he always tells his students to do several complex variables/complex manifolds

and no one ever does

[shy bluff]

So like for D_4 you'd have r is a representative?

[latent anvil]

be the change he wishes to see in the world

[next obsidian]

Well

[bleak abyss]

Lmao

[next obsidian]

Let’s go with a quotient group first

[shy bluff]

ok

[bleak abyss]

There's a guy in the actual UW who does K3 surfaces

[shy bluff]

UW?

[bleak abyss]

He's fairly young (first year as an assistant prof, previously he was a Vakil postdoc)

[latent anvil]

who is this?

[next obsidian]

So like what’s an explicit example

[bleak abyss]

Michael Kemeny

[next obsidian]

Umm Take Z/4Z

[shy bluff]

Sure

[latent anvil]

i learned how K3 surfaces were named

and it's kind of funny

[next obsidian]

Your elements are like 0,1,2,3 right up to equivalence

[bleak abyss]

https://www.math.wisc.edu/~kemeny/homepage.html his stuff is basically complex geometry and it seems cool

K3 surfaces, syzygies, moduli stuff

[next obsidian]

Then you have a copy of Z/2Z in ther with {0,2} right?

[shy bluff]

yes

wait

What do you mean a copy

[next obsidian]

It’s iso to Z/2Z

[shy bluff]

ok yes

[next obsidian]

Then the quotient is Z/2Z right? Or at least isomorphic to it?

[shy bluff]

quotient of?

[next obsidian]

But as elements, if H = {0,2}

Umm Z/4Z / H

[shy bluff]

ah ok

[next obsidian]

For ease call Z/4Z G

So elements of G/H are of the form g + H

Where g + H = g’ + H if g^-1g’ is in H?

[shy bluff]

What's g'

[next obsidian]

That’s like, the element wise construction you probably have seen

Just any other element of G

g,g’ in G

Like in our specific case

0 + H = 2 + H

Since the difference of 2 and 0 is in H = {0,2}

[shy bluff]

sure

[next obsidian]

I mean, if it doesn’t make sense that’ll be big so it sure means yes then I can move in

But if it means “uh, idk sure” the next part won’t make sense

[shy bluff]

I understand what G/H is

[next obsidian]

Okay

So in that case 0 + H and 2 + H are the same object right?

They’re equal?

[shy bluff]

Yea

[next obsidian]

A representative of that object is the specific form you write it in

So in this case

0 + H is a representative of that element

As is 2 + H

You can choose to write that element multiple ways, but a specific way you choose to write it is a representative

That’s why when you define a function from a quotient you have to make sure it’s well defined

[shy bluff]

Oh but 0 + H and 2+H are the same objects

[next obsidian]

Since you normally write it in terms of “a representstive” but you have to check that if you took a different one the output is the same

Yeah

But for example if I made a function that sends x + H to just x

It isnt well defined

Because then 0 + H goes to 0

And 2 + H goes to 2

But they’re equal!

So they have to go to the same thing

[shy bluff]

I think I see

[next obsidian]

So for conjugacy classes

[shy bluff]

So then say for example 1 + H is a different object from 0 + H or 2 + H

[next obsidian]

In this case it is

Given G = Z/4Z and the H I made

[shy bluff]

Yea, for Z/4Z and H = {0, 2}

[next obsidian]

Yup

Oh so like let’s take a set of representatives for G/H

You have 4 choices

{0 + H, 1 + H}

Or {2 + H, 1 + H}

Or {0 + H, 3 + H}

Or finally {2 + H, 3 + H}

so you expressed G/H as a set, you accounted for all (2) elements it has

By providing one representative of each equivalence class

[shy bluff]

Oooh ok

[next obsidian]

So conjugacy classes is the same thing

Pick one element from each conjugacy class

So you have to compute what the conjugacy classes are

[shy bluff]

So it's ilke how for modular arithmetic you have like, say if it's mod 4 then you have just {0, 1, 2, 3} to pick from right?

[next obsidian]

Yeah

I mean teeechnically

You could pick 5

But that’s the same as 1

So why bother?

There’s a most natural set of representatives

That’s {0,1,2,3}

[shy bluff]

So in this case I want to find all the things in that can possibly be mapped to in S_8?

And then pick 1 to represent each one?

[next obsidian]

No so

You know what a conjugacy class is right?

A set where like all the elements are conjugate to one another

You need to pick one of them from each one

Buuuut in the case of symmetric groups conjugacy classes are exactly the same as the like, sizes of the orbits

[shy bluff]

Oh wait it's not asking for equivalence classes

Ok yea idk what a conjugacy class is

😔

[next obsidian]

Umm

Pg 123 of D&F

And then page 125 is specifically about S_n

[shy bluff]

O ok

Thank

[next obsidian]

So you can use that to figure out what the conjugacy classes are

[shy bluff]

I will read through this and then if I still have questions come back

[next obsidian]

And then to compute the size of the conjugacy classes

Is combinatorics

[shy bluff]

Thank you!

Oh wait ok this bit I think I understand; partitions iand combinatorices are something that I understand

[next obsidian]

That’s good then!

[shy bluff]

o_o nubre of conjugacy classes of S_n equals the number of partitions of n.... that means 22 conjugacy classes of this lmao

Oh so there's a bijection between partitions and conjugacy classes

[next obsidian]

Yeah

But you can just write trivial examples for each conjugacy class

I mean, not trivial

[shy bluff]

Yea

[next obsidian]

But like just got (123)(45)

Or whatever

[shy bluff]

Yea

[next obsidian]

So it’ll just be tedious to write out

[shy bluff]

But uh still lmao that is a lot

Unexpectedly a lot

[next obsidian]

The issue is justifying the size

I don’t think D&F does it, but it’s all combinatorics

[shy bluff]

Justifying the size?

[next obsidian]

I just don’t think it’ll be fun to compute it 22 fucking times

The problem asks for the size of the conjugacy classes too

[shy bluff]

Yea computing the size is fine

But uh

Oh god

Compuet it 22 times wtf

[next obsidian]

Yeah, not very fun

It’s a lot of !

GL, I did this for like... S_6? But that’s way less

[shy bluff]

D&F gives an explicit formula tho

[next obsidian]

Uhhh

Isn’t that only for a cycle?

Like an m-cycle

[shy bluff]

oh wait

yea

[next obsidian]

You can have something of the form (123)(45)

And that isn’t

You have to calculate how many of any given cycle type there can be which is annoying

I forget the formula off the top of my head

There isn’t a really nice way to put it

It’s like n!

Then you divide by how many cycles of every size you have

Since (123)(456) = (456)(123)

Then divide it by like the product of the sizes of cycles since (123) = (312) = (231)

Then you do something else I totally forget lol

[shy bluff]

https://groupprops.subwiki.org/wiki/Conjugacy_class_size_formula_in_symmetric_group that does not look fun

[next obsidian]

It isn’t too bad once you do a few

But justifying it is annoying and it’s hard to write in a closed form

[shy bluff]

thanks prof

[half nebula]

So I'm a bit stuck on this one here. How does a have an order of 9 if the group has an order of 27?

[latent anvil]

Why do you think a having order 9 is a problem?

Also, a^9 = e does not mean a has order 9

It means it has order dividing 9 (i.e. 1, 3, 9)

[oblique river]

@half nebula the group doesn't have an element of order 27. If it did, it would be cyclic, but the problem assumes the group isn't cyclic

[half nebula]

Ooooooooh okay.

So the elements for it can't be order 27, so they have to be order 1, 3, or 9.

[kindred rivet]

...