groups-rings-fields

[crystal turtle]

uhh

[topaz solar]

I started with Artin

[crystal turtle]

Serre is a fields medalist

[topaz solar]

And Weibel but we don’t talk about that

[steel light]

Ooh shiiii

[crystal turtle]

His "A course on arithmetic" starts by defining the p-adics as an inverse limit

[steel light]

Wtf is an inverse limit LOL?

[topaz solar]

I forget if I have Serre in the archive

[crystal turtle]

So if that's an indication of how his other books are, they're probably not beginner friendly

[topaz solar]

Btw you’re on your last year right Ryx?

[crystal turtle]

hom alg?

[crystal turtle]

yeh

[topaz solar]

Make sure you make good use of those institutional accesses

[crystal turtle]

sotrue

[topaz solar]

The whole Springer catalogue

[crystal turtle]

but I think I'll be going to Toronto for masters, I imagine they have institutional access as well lol

[topaz solar]

Well

Hopefully lmfao

[crystal turtle]

assuming, ya know, if I don't get rejected

[steel light]

UofT

[topaz solar]

Going to work under Piper I see

[steel light]

Tterra has words for you

[crystal turtle]

yeah well it's best in canada

[steel light]

Yeah lols

[crystal turtle]

also they just hired 2 new homotopical algebraists

which my supervisor was telling me about 👀 that should help my prospects

[steel light]

That’s what you’re interested in?

What is homotopical algebra? I only know homotopy theory in the context of topology

Algebra of homotopy groups?

I’m sure it’s more than that

[crystal turtle]

uhh I mean I'd also like to learn AG this year, but that's kinda an artificial interest as of now since I don't know it

[steel light]

We can learn it together ;)

[crystal turtle]

Essentially, it's a generalization of homotopical methods that people were using in algebraic topology

to "categorize" it, we don't really speak as much about homotopy groups (because uhhh might not make sense in other contexts)

but rather, as with anything categorical really, we characterize stuff in terms of maps. In particular, there are three types of maps that are of interests

cofibrations, fibrations, and weak equivalences

A model structure on a (bicomplete) category is 3 distinguished classes of morphisms, as above, subject to a few conditions that uhhhhhhh

I will not motivate

[steel light]

Model :gross:

What is it about AT that makes it so connected to category theory

[crystal turtle]

But it's modelled after those of Hurewicz (co)fibrations and homotopy equivalences (which is one model structure on Top), and that of relative cell complexes, Serre fibrations, and weak homotopy equivaleces (which is the main model structure of interest on Top)

[steel light]

The association of algebraic objects to topological ones?

[crystal turtle]

beyond that uhhh it's a bit harder to motivate briefly

[steel light]

And then you have maps between those two things

[crystal turtle]

categories have a lot of roots in AT actually

[steel light]

I know

Just don’t know why

[crystal turtle]

Mac Lane and Eilenberg both did AT

[steel light]

Who? 😭

[crystal turtle]

the people who first defined categories

[steel light]

I see

Can you give a brief motivation?

Something to illustrate why they’re best friends

[crystal turtle]

oh you mean AT and cats huh

[steel light]

Yes

[crystal turtle]

I thought you meant ML and eilenberg

[steel light]

LOL

[crystal turtle]

🏳️‍🌈

[topaz solar]

Well, Seifert-van Kampen

[crystal turtle]

Uhh well I'm pretty sure that Mac Lane had once said categories were introduced, not to study functors, but rather to study natural transformations

[topaz solar]

I’m not gonna try and state that theorem normally

[crystal turtle]

As for what in particular, I'm not sure. But before that people had some notions of what functors were

Uhh but I guess it's just that, when we pass from a topological space to an algebraic structure, it's usually functorial, which means that it acts on both objects and morphisms, in a coherent way

[topaz solar]

Also, consider, homotopy groups and homology

Those are functors

[crystal turtle]

In particular, homotopy groups and homology yeah

[topaz solar]

And there’s a natural transformation there

[crystal turtle]

(even many topological constructions are functorial as well: stone-cech and alexandroff compactification, ...)

[steel light]

I have a lot of definitions to learn :^)

[crystal turtle]

But I'll be honest, I don't actually know specifically what it was that they had in mind

granted I also have not taken an AT course lol

[topaz solar]

I don’t care much for historical motivations

But I don’t care for history in general

[steel light]

I’m tempted to skim over to Ch3 just to see what I’m in for

[crystal turtle]

uhh and then people like Daniel Kan and Grothendieck came along and went "this shits pretty cool lmao" and start applying cats to everything

Quillen came along and said "everything is homotopy theory"

and then scholze's doing whatever the fuck he's doing idk

[steel light]

Aw fuck man I saw tensor products in Chapter 2 😭 scared

There’s so much stuff on modules, darn lol

I’m just scared because of the tensor stuff

Made 0 sense to me last time I tried to learn it

[crystal turtle]

tensors aren't too bad once you figure out their universal property

But they will show up a good amount throughout the textbook, so you do need to know them

[steel light]

Yeah

They’re in AG too?

[crystal turtle]

Dunno

[topaz solar]

Tao, Scholze, Shelah are in worlds of their own

[summer path]

ok stupid question, why does $a \equiv 1 \mod I$ and $I \subset J$ ideals imply that $a \equiv 1 \mod J$?

[cloud walrus]

most likely to :3c

[crystal turtle]

what does it mean for a \equiv 1 mod I to holds?

[summer path]

a = 1 + I

[crystal turtle]

and what does this mean, in terms of elements of your ring?

[summer path]

a-1 in I

[crystal turtle]

Right

[summer path]

im braindead

[crystal turtle]

So a - 1 = x for some x in I

hence for some x in J

[summer path]

it's past my bedtime

[crystal turtle]

same

[summer path]

but i need to finish reviewing my notes from 2 days ago

but im sleepy

[crystal turtle]

[summer path]

,ti 863145419034722315

[cloud walrus]

This user hasn't set their timezone! Ask them to set it using ,ti --set.

[summer path]

[crystal turtle]

is that me :3

[summer path]

yes

[crystal turtle]

it is 11:55 pm here :o

[summer path]

,ti

[cloud walrus]

The current time for tubularcat is 11:56 PM (EDT) on Tue, 29/08/2023.

[summer path]

[crystal turtle]

omg twinsies!

[summer path]

[crystal turtle]

:o bladewood is also EST?

triplets even

[summer path]

waw nakayama lemma proof is so short

but so useful

[steel light]

Uh

So AM claim that a nonzero commutative ring is a field

[chilly ocean]

[steel light]

Wait no it’s a bit more than that I keep forgetting lol

I think I’m going to approach it from the perspective of

A nonzero commutative ring A is a field iff the only ideals in A are (0) and (1)

The former is important because fields cannot have nonzero zero divisors

The latter is important because fields must have multiplicative inverses for their nonzero elements

[steel light]

This is gonna take a while to get used to

I’ve actually never encounter the equivalence method of proving equivalent statements

I’ve heard of it but this is the first time I’ve seen it in practice

Equivalence method being 1 iff 2 iff 3

[rocky cloak]

So (0) and (1) are always ideals for any ring, the notable thing is that these are the only ideals

[steel light]

Yes

I understand why

[rocky cloak]

Fun fact, it works for noncommutative as well. I.e if A is a ring, then the only left ideals of A are (0) and (1) iff A is a skew-field / division ring

[steel light]

Agh I already forgot what a skew field is

Oh

Ring with multiplicative inverses

[rocky cloak]

Just field but possible noncommutative

[steel light]

But it’s different from field bc it’s not necessarily commutative

Fuck

Sniped

Unlucky

[steel light]

I’ve noticed a pattern among noncommutative algebraic structures in that most definitions often just define things in terms of left operations

Left multiplication, left composition, bla bla

Left measure I think is something I heard

Is this just convention? Do the same arguments extend to right operations and we just choose to only define the left case for conciseness?

[rocky cloak]

Yes, for any left-thing you can define an analogous right-thing.

For any ring you can define it's opposite ring by simply reversing multiplication. This exchanges left things and right things, so there really isn't any difference.

You just pick one side to work with

Of course there are many interesting results that involve both sides. Like we talked about earlier if (0) and (1) are the only left ideals, then they are also the only right ideals.

[coral spindle]

There is a reason that we pick the left specifically, though

You've probably seen a result similar to Cayley's theorem for groups – every ring is isomorphic to a ring of functions where the multiplication operation is composition. We can therefore sort of see the action of a ring as being similar to applying a function. So since we apply functions on the left in usual notation, we also apply ring elements on the left.

[delicate orchid]

Liberals

[coral spindle]

Yes, also communism

It happens from time to time that algebraists prefer to write actions on the right. In that case you will see them also write function application on the right too! In James & Liebeck's book on representation theory, they 'drive on the other side of the road' in this way, and so they write xf instead of f(x).

So @steel light there's a nice tidbit

[last spoke]

where the fuck are you

Wait I forgot time works, I'm dumb

I thought this was recent

[delicate orchid]

Is applying them on the left really the standard? I read fg as “f then g” so applying on the left is pre-composition to me

[coral spindle]

That's a little unusual I think! Usually I write fg to mean f \circ g and this is what I most often see written too

So "f after g" rather than "f then g"

[delicate orchid]

And also you get the nice visual of $fg$ corresponding to $X \overset{f}{\longrightarrow} Y \overset{g}{\longrightarrow} Z$

[cloud walrus]

Wew Lads Tbh

[delicate orchid]

Yeah but who actually know what f \circ g actually means

Is it f(g(x)) or g(f(x))?? Who knows!

[coral spindle]

^ here's an algebraist who would prefer writing xf

[steel light]

OOOOO This is good!!!

Great example

[steel light]

Yeah I’ve seen this notation!

[steel light]

I read it as f of g

So it’s always implicit that we’re composing the element on the right

[delicate orchid]

cringe

[steel light]

I’m just a poor old high school Algebra 2ist what can I say

[delicate orchid]

Hmm yes the action of Hom(-, X) on morphisms should be phi(-)... that definitely makes sense to me!

[steel light]

Words!

[topaz solar]

Definitely standard in most fields right?

[delicate orchid]

yeah the non-soyjak ones

[karmic moat]

hmm yes indeed

[south patrol]

Bruh fg is definitely g then f

[karmic moat]

i swear ive seen some books use fg for f(g) and some for g(f)

it’s very annoying

[coral spindle]

Excerpt from James & Liebeck

Not saying this is standard, but some books use this convention rather than the usual one.

[coral shale]

did i see xfg

[solar vessel]

no

[coral shale]

[crystal turtle]

[last spoke]

Same. It's 2am here

[steel light]

AM has so many exercises and a lot of them look good

Should I just pull from the end of chapter exercises or look online

[delicate orchid]

just do the end of chapter ones

[elder wave]

skip the tor ones

[formal ermine]

chap 1 exercises may interest you as they dive into the very beginning of ag

[lethal dune]

wow ppl understoond Atiya-MacDonald just from the acronym

[formal ermine]

wait till you hear about decawavtag

[lethal dune]

what is that?

[crystal turtle]

why is this a :timo: moment

[formal ermine]

[lethal dune]

ry_ gang

my comment was unrelated to

[formal ermine]

this never happened

[jovial quiver]

Does anyone know how to show that tensor product commutes with direct sum using ONLY universal properties, without referencing the constructions AT ALL? I have been stuck on this for some time lmao.

in Mod_A of course

[formal ermine]

what have you tried so far

[jovial quiver]

let me write it down give me a few minutes

[crystal turtle]

good ol' left adjoints preserve colimits

[jovial quiver]

ah so true

[crystal turtle]

that it's entirely categorical means you can do it with just the universal properties (but that does not mean i remember the proof whatsoever )

[jovial quiver]

ok well I can just look up a proof of left adjoints preserve colimits now if I get stuck. Thanks!

[night onyx]

Show directly that (X \tensor Y) + (X \tensor Z) together with the bilinear map (x, y + z) -> (x \tensor y) + (x \tensor z) satisfies the universal property, so for any bilinear map f : X x (Y + Z) -> W there is a unique linear map f* : (X \tensor Y) + (X \tensor Z) -> W such that f(x, y + z) = f* ((x tensor y) + (x tensor z))

[formal ermine]

let em cook

[wet zodiac]

of course people know the AM acronym in an algebra channel

[void cosmos]

anyone?

[rocky cloak]

I'm not completely sure what you're asking. The picture doesn't mention R being semisimple as a ring, it only talks about semisimple modules

[void cosmos]

yea mb

a semisimple module being a direct sum of simple submodules

thats the definiton in my text

i was just aksing if the defintion in my text matches with the one expected for the test

u get me?

[rocky cloak]

This indeed the standard definition.

[void cosmos]

yes but for rings

R need not be left artinian

i dont know shit but like i thought maybe the definitions used in the test paper were diff

so i was asking

u get me

[rocky cloak]

A ring is usually defined to be semisimple if it is semisimple as a module

This is not equivalent to J(R)=0

Which was your definition(?)

But it's equivalent for artinian rings

[void cosmos]

just having J(R) = 0

thats it

while i noticed for other sources it had this + left artinian

so it would suck that on the test i get some problem

involving semisimple rings that needs that artinian property

etc

[rocky cloak]

Z satisfies J(Z)=0 for example

But the name semisimple isn't very fitting for Z...

[delicate orchid]

what the deuce

[crystal turtle]

Common Z win

[void cosmos]

okay fuck it

i will just fuck it on the test

like

just write a note

that semisimple is left artinian + J(R) = 0

free minimal ideal ig

wont hurt

[rocky cloak]

I mean, you can always just... Like... Ask the professor

[void cosmos]

the professor said the text used is hungerford

[next obsidian]

No, that doesn’t make sense

[void cosmos]

yea i dont like asking professors honestly especially if i can spot a problem that needs it being left artinian

but yeah i will

[topaz solar]

I’d ask em

Swallow your pride idk

[summer path]

Or ask your ta I guess

[topaz solar]

That too

[void cosmos]

IFJHQWAIHJNIAF

argh

wait tf

R cant be a maximal ideal of itself right?

[next obsidian]

No

[void cosmos]

okay i have been stuck at this problem for like 30 mins

can u just check this proof for me?

R is a ring with 1

TFAE: 1) Jac(R) is a maximal left ideal. 2) the sum of any two non units is a non unit

3)... whatever but foir now

1 --> 2

or basically without like wasting much time

does Jac(R) being maximal imply that any non unit contained in Jac(R) ?

cuz like u go a is a non unit , (a) is in some maximal left ideal but Jac(R) is a subset of that maximal left ideal but Jac(R) is maximal tho ..?

[next obsidian]

Jac(R) is the intersection of all maximal left ideals so for it to be maximal itself means there’s only a single maximal left ideal

Okay GG

[void cosmos]

yea

thats what i had in mind but i thought this was too stupidi

or idk why

so yeah and then 1-->2 follows

tysm

[chilly ocean]

What's a good contemporary book on algebraic curves? I am mostly looking for an introductory book.

[void cosmos]

fulton

take it from a beginner like me

also #book-recommendations

[chilly ocean]

Honestly, it looks perfect. Thanks for the recommendation

[void cosmos]

okay

so

wait wtf

[noble saddle]

tbh the first chapter of Silverman is a great place to start

[void cosmos]

i need help here

croqueta if ur typing a questoin then ping me

react to this

wihtout deleting ur whoel shit

lol

cuz i have been stuck

on a problem

for like an hour

ok

gl

[rotund aurora]

Pete L Clark has notes from the valuation POV which are short http://alpha.math.uga.edu/~pete/8320_2020.pdf (available from his webpage)

For more analytical approaches, either Kirwan or Rick Miranda (Miranda's book talks about Riemann surfaces since the beginning)

[void cosmos]

okay suppose R is a ring with 1

and suppose the sum of any two non units is a non unit

prove that Jac(R) is a left maximal ideal

which means its the only maximal ideal

what i got to was that the set of all non-units is now an ideal

and maximal s fuck

as fuck*

i need to show that this ideal is the same as the jacobson radical

[rotund aurora]

haha thanks for offering help, but I was responding to fellow username0000 🙂

[void cosmos]

it's impossible for me

literally impossible

i will never make it

[void cosmos]

np

if R was artinian then i would be fucking done

left artinian*

[noble saddle]

You need to show that a non-unit is in every maximal ideal. Let x be a non-unit, m a maximal ideal. If not in m, x is invertible mod m (why?). Does this jive with your assumptions?

[topaz solar]

Jac(R) is contained in the set of non units then yes?

Is there any other feasible way you can have a maximal ideal

[void cosmos]

i got to that

but then i realized

if x is a unit in R/M

does not imply

it being a unit in R

that was what i wanted to go for

[noble saddle]

write down exactly what it means

[topaz solar]

It kinda does here

[void cosmos]

how

wtf

[topaz solar]

Since ya know

[void cosmos]

Z/5Z and Z?

[topaz solar]

It’s not a non-unit

Therefore

Unit

[void cosmos]

wait

[topaz solar]

Like

[noble saddle]

like. literally write the definition of "x is a unit in A/m"

[void cosmos]

there exists y ssuch that xy is in M

[noble saddle]

wrong

[topaz solar]

No

[void cosmos]

xy-1 is in M

[south patrol]

assuming a strong enough logic

[void cosmos]

mb

[topaz solar]

But like

If m contains everything that isn’t a unit

Then

Anything that isn’t in it is a unit

[noble saddle]

in other words, xy + m = 1 for some m in M . So you have two things adding up to a unit

[void cosmos]

oh u mean the coset itself

yes

[noble saddle]

no

[void cosmos]

thats what u did

[noble saddle]

there is an element m such that xy + m is literally equal to 1

[void cosmos]

xy+M = M

?

i meant

u pre imaged

the projection

[topaz solar]

That’s kinda unnecessary here since m is literally {nonunits in R}

[void cosmos]

no

m is not this

m is some maximal ideal

[noble saddle]

M is not literally non-units. M is any maximal ideal

[void cosmos]

ssuch that a non unit is not there

or yeah

mb mb

im just tilted

tbh

i got stuck p hard

here

osrry

[topaz solar]

Well, you know the nonunits form a maximal ideal

[noble saddle]

yes, and now is trying to show the reverse inclusion

[topaz solar]

So you need a non-nonunit in your m to not be contained in it

This is an issue

[void cosmos]

how did u get that

[topaz solar]

That’s how quotients work though yes

[south patrol]

M is maximal

[topaz solar]

I would not worry about some arbitrary maximal ideal to show inclusions

[noble saddle]

To clarify for everyone, Let I be the set of nonunits. Then since I is maximal, I contains the Jacobson radical J. The reverse inclusion is to show that that I is contained in all maximal ideals

[void cosmos]

yes this is what im trying to do

[topaz solar]

I’d just say there is nothing you can put in a nontrivial ideal

[void cosmos]

now please drgigca

[topaz solar]

Except the nonunits

[void cosmos]

when u tlak about something in the quotient

[topaz solar]

Done

[void cosmos]

make it capital or like

bar or something

[noble saddle]

yes but dragon keeps saying \dumb shit and I'm trying to set expectations

[void cosmos]

so i know what ur talking about

[topaz solar]

dumb stuff

[void cosmos]

chill he was trying to help

idk u two are friends or what

bantering or whatever

but chill

[topaz solar]

I have never seen this guy before

[void cosmos]

okay chill

i apologize on his behalf

[noble saddle]

you need to show that for any maximal ideal M, we have I contained in M.

if an element x of your ring is not in M, then it is invertible in R/M

which literally means xy + m = 1 for some y in R, m in M

can x be in I?

[topaz solar]

Or, M not contained in nonunits J => x in M \setminus J

[crystal turtle]

[topaz solar]

Which has to be a unit

That’s literally it

[void cosmos]

okay so at first you had xy + M = M correct?

inside the quotient

as cosets

[noble saddle]

no

[topaz solar]

1+M

[void cosmos]

oh fuck

1

1

1ji34e9io12qjemqiwdqawnf

[topaz solar]

He’s saying you get literally 1

Not a coset

Nothing coset here

[void cosmos]

i interchanged

between 0 and 1

3 times

in this convo

or 2

[topaz solar]

Ikr

[void cosmos]

..

please just say yes to this:

you got xy + M = 1 + M ---> xy+m = 1+m' for some m,m' in M so u get xy+m'' = 1 for some m''

in M

[topaz solar]

Yes

[void cosmos]

is this what u were saying

okay

[topaz solar]

A very unnecessary route but yes

[void cosmos]

i literally had this argument in my head but then i just went xy = 0 an then just skipped this

[crystal turtle]

equivalently, xy - 1 in M (or xy - 1 = m for some m in M)

[void cosmos]

for my functional analaysis professor who went " do it in your head first before writing on paper "

nvm i will get banned..

ty guys sm

[topaz solar]

Alternatively

Nonunits are a maximal ideal

You showed this

[void cosmos]

yes

this was the first part of the problem ig

[topaz solar]

Any other maximal ideal has to contain something in the complement, yes?

That’s only units

Ideals containing units don’t tend to be maximal

[void cosmos]

yea yea obv

haha

yeah

i got you

[topaz solar]

but oh who cares just dumb stuff amirite

[void cosmos]

yea just forget about it it's discord

[crystal turtle]

so if everything that's a nonunit forms an ideal it's necessarily maximal ezpz

[noble saddle]

oh I misread what you posted earlier

[topaz solar]

mhm

[void cosmos]

yeah now kiss

lmfao

anyways

did you guys see this b4

or are u just 10x smarter than me

[crystal turtle]

I have never worked with noncommutative rings

[void cosmos]

like it took me like 30 mins of trial and erorr

[crystal turtle]

they don't exist for me

[void cosmos]

just to get to this argument and then i tossed it

cuz i went xy =0

in R/M

[topaz solar]

I haven’t had alg classes and I’m blitzing D&F atm

But LEM saves the day often

[void cosmos]

no i meant seeing this exact problem

[long nebula]

is moamen learning about local rings

[noble saddle]

matrices don't exist for you? :c

[topaz solar]

This is the problem okay bro had earlier about k[[x]] essentially

[crystal turtle]

nope

[topaz solar]

But I’ve seen local rings before ye

[void cosmos]

no actually im going to be learning about semisimple ones now

[topaz solar]

[long nebula]

I have no idea what those are

[topaz solar]

Same

[crystal turtle]

Not this exact problem but probably something similar

[noble saddle]

okay but local rings are a million times more important

[void cosmos]

isnt J(R) being maximal

just R being local

so ig im dealign with them now

[topaz solar]

Yes

[void cosmos]

lol

also nakayamas lemma deals with them

iog

ig*

[crystal turtle]

well nakayamas lemma is much more general

[void cosmos]

anyways i feel so stupid and i dont want to do any more math

i will go play tf2 or something

[crystal turtle]

based

[void cosmos]

yea true

[long nebula]

Moamen, if you want a fun problem, if R is a comm. ring with 1, then for any two elements r,s in R, does

rR = sR <=> r = us for some unit u
hold?

[void cosmos]

i have been dealing with this shit textbook for like a month now , once i see R is comm its trivial

haha jk

[topaz solar]

What book

[void cosmos]

hungerford

algebra

[void cosmos]

honestly i just feel stupid rn cuz i have been trying for like 30 mins for this problem and 3 people just 1 shot it

sorta like ur deadlifting some heavy ass weight and some guy just curls it

[long nebula]

this problem is p hard

so don't feel dumb if you decide to do it and it takes you a long time lol

[topaz solar]

[void cosmos]

it does

it looks very technical too

if u catch my drift

[crystal turtle]

this is how I feel looking at some of the exercises in AM lmao

[void cosmos]

yea

AM has hard ones tho

[topaz solar]

See that’s the power of slaying dragons for ya frfr

[void cosmos]

im happy i did one problem tho

it's an easy one but im stupid so relatives

[void cosmos]

😠

[crystal turtle]

better than I have done today

[crystal turtle]

[void cosmos]

it was if R = n/m n is an integer and m is odd then J(R) is the same but with n being even

[topaz solar]

[void cosmos]

yea king math fucking sucks literally

like no joke

[topaz solar]

What did he mean by this

[topaz solar]

I tried emailing a professor (who knows if he’ll even open it)

[void cosmos]

i swear if i were to restart my life i would have done like

motorsports or something

cs is too boring honestly

[crystal turtle]

oh I finished packing today to move tomorrow (almost at least)

just no math

[void cosmos]

fuck math

[last spoke]

Agreed

[crystal turtle]

I would rather not

[void cosmos]

math is literally just " where did this come from " over and over again

hahahaha

jk

[topaz solar]

or worse, reads it and judges it

[void cosmos]

it depends

on what u sent him/her

haha

if it's this problem of yours

then they will block you asap

[topaz solar]

Another prof was like “hey email him”

[void cosmos]

oh then yeah its gucci

[topaz solar]

And specifically called him out by name

Threw him under the bus

[void cosmos]

yea then dont worry about it

[topaz solar]

Was about it though

[void cosmos]

well the thing is

[solar vessel]

about what

[void cosmos]

u probably emailed it to the wrong person and then he referenced the correct one

[topaz solar]

Though about giving some lower bounds on min poly degree of elements

[void cosmos]

so ur gucci too 😄

[void cosmos]

nudes

[topaz solar]

Nah I was talking in person to the one who said to email him

[void cosmos]

oh

yeah

it's really nice hto

that you can talk to people who do math

irl

[topaz solar]

This is because I assume they already dislike me so I don’t worry about annoying them

[void cosmos]

great tactic

can lead to self esteem issues tho

haha

[crystal turtle]

me constantly thinking my supervisor is annoyed of me

[topaz solar]

Bro has a supervisor

[crystal turtle]

for summer research yessir

[summer path]

how does it feel to be supervised

[delicate orchid]

my supervisor loves me as does everyone else

[void cosmos]

damn thats hot

[topaz solar]

Not the funding committee

[delicate orchid]

they don't exist

[void cosmos]

damn

[topaz solar]

Based

[summer path]

[crystal turtle]

lmao

[void cosmos]

i wonder if i could get accepted in a (not trashcan) program

damn

it would be so funny'

[topaz solar]

Apply idk

[void cosmos]

but also be inspiring you know

[crystal turtle]

apply

[void cosmos]

cuz im literally a brainlet

[summer path]

ap

[void cosmos]

i will once i get those rec letters

[crystal turtle]

do it

[delicate orchid]

they let me in

[void cosmos]

thats why im doing these exams tho

[void cosmos]

yea ur 10x smarter + ur a math major

[crystal turtle]

:3c are you grad student

[summer path]

yes :3c

[delicate orchid]

ah true I forgot you're not a maths major

[void cosmos]

yeah

but ik it happens that non math majors get in

its the question of where

haha

[crystal turtle]

[summer path]

are you?

[crystal turtle]

no last year of UG

[summer path]

[crystal turtle]

[void cosmos]

is there any like math course that is expected from a grad school applicant other than algebra analysis and topology?

[delicate orchid]

numerial methods with applications to fluid dynamics

[void cosmos]

yeah im out

[delicate orchid]

if you don't have that you ain't getting in

[summer path]

depends on what kind of program you're going into

[delicate orchid]

who up lax-ing they friedrichs

[summer path]

but generally you want something from algebra, analysis, and topology, and then more stuff is just nicer for the application and for when you get into grad school, i think

[summer path]

but also that

[void cosmos]

yea i am p sure i can do a test in all 3 of these (at a grad lvl)

and do fairly "good" in them

under supervision of professors each

i am studying for the algebra one rn

wish me luck

[summer path]

do note that different programs may do different things for their quals

[delicate orchid]

I would definitely fail the latter two

[void cosmos]

yea thats what im trying to aim for

to study my non-interests up to the point i can pass aqual in them

and also have my application show that

( which is impossible )

anyways back to algebra cuz this is isnt chill

what is the injective hull of Z_24?

[delicate orchid]

as a Z-module?

[void cosmos]

yesir

ik the answer but i dont understand it

it was just to break the discussion

its some kind of direct limit group

that i need to know of

[delicate orchid]

so the injective Abs are divisible groups hmmm

you can probably do this without explicitly computing the maximal essential extenstion but now I want to know what the maximal essential extenstion is

[void cosmos]

god i love sheaves

like food itself

is such an art

[topaz solar]

Same

[delicate orchid]

ah fuck it - so we need a group such that for all H <= G H \cap Z_{24} = {0} => H = {0} with no proper essential extensions

whatever this thing is has to be infinite obviously

[void cosmos]

this was a problem in a past paper of my exam

and i dont know shit about injective hulls yet

XD

so good luck

[delicate orchid]

yeah me too

[void cosmos]

yea who does

lmfao

[delicate orchid]

sharp you got any ideas

[topaz solar]

Same

[delicate orchid]

guess that's a no

[void cosmos]

do u want

a hint

to the naswer

i will just yell the group name literally

but its just not that group

[delicate orchid]

what

[void cosmos]

prufer group

[crystal turtle]

waht the fuck is an injectie hull

[delicate orchid]

dual of a projective hull

[crystal turtle]

so enlightening

thanks chief

[delicate orchid]

oh is it something stupid like 2^{\infty} x 3^{\infty}

[void cosmos]

what

2^infinity is infinity bro

222...?

[delicate orchid]

the group you muppet

[void cosmos]

what group is 2^infinity

[delicate orchid]

the prufer group at the prime 2

[void cosmos]

yea

it is

congrats

[delicate orchid]

LMFAOOOOOOOOOOOOO

[void cosmos]

u get a medal

[delicate orchid]

I hate that so much

[void cosmos]

yea skill issue

[delicate orchid]

I guess it makes sense - the prufer groups are like

the "prime numbers" of divisible groups

[void cosmos]

yea its literally the answer hahahahaha

but yeah they look cool

[crystal turtle]

do you mean that infinity is an even number?

[void cosmos]

so i will lern about them

[crystal turtle]

[void cosmos]

yea cuz infinity/2 is infinity which is an integer ofc

[topaz solar]

Icky

[delicate orchid]

does the sequence {1,3,5,...} only contain odd numbers

[void cosmos]

obv

guys

no joke

[delicate orchid]

but it converges to infinity in the one point compactification so infinity must be odd

[void cosmos]

can u guys give an example of "math crankery"

[crystal turtle]

collatz

[void cosmos]

that someone really takes serious

[crystal turtle]

and gtbot

oh

seriously

[void cosmos]

no i meant like

some wrong stuff

[long nebula]

there are some infinite naturals which are even and some which are odd :^)

[sly crescent]

Is the group of rational points on the unit circle isomorphic to the direct product of every Prüfer group?

[delicate orchid]

oh now that is a fun question

[topaz solar]

Squaring the circle

[void cosmos]

i actually know the answer to this

[topaz solar]

Pi is rational

[void cosmos]

yea ofc some geometrical stuff

like the proof that every triangle is iscocless

this geometric stuff

seriously do you guys believe in pics?

for math

seris

no joke

[crystal turtle]

seris

[delicate orchid]

lemme just $p^{\infty} \cong {z \in \bC | z^{p^n} = 1, n \in \mathbb{N}}$

yeah I buy it

[cloud walrus]

Wew Lads Tbh

[topaz solar]

I don’t know prüfer groups or injective/projective hulls at all

[void cosmos]

this is ||true||

[delicate orchid]

I got the quantifier a bit wrong but who cares

clearly whatever this is will be abelian so I can safely just mash together all of the mfs

and say that z^m = 1 then prime factor m blah blah blah

cool

[topaz solar]

gg

[crystal turtle]

wp

[topaz solar]

I need to learn more algebruh

[void cosmos]

yea but you forgot one thing

which is ||HAHAHAHAHAHAHHAAH IM SO SICK IN THE BRAIN||

[delicate orchid]

bossman u doin alright?

[topaz solar]

No

[crystal turtle]

no

[void cosmos]

no

[delicate orchid]

glad to hear it!

[void cosmos]

this spoiler shit

is so fun

i swear

they should make like a seen thing

like logs whoever clicked on it

and sends it to you

[delicate orchid]

If you think about it Q is the prufer group for the prime 0

[void cosmos]

yea this is ||please make me happy god im so sad||

[delicate orchid]

If you wanted an example of maths crankery

[sly crescent]

And Q is also the field with 0 elements

[topaz solar]

Mochizuki

[topaz solar]

Or ya know my garbage problem

[delicate orchid]

It’s like how every char 0 field has Q in it and every char p field has F_p in it

But instead of characteristic it’s torsion

[topaz solar]

What are prüfer groups? Prime models for divisible group things?

[delicate orchid]

Dunno what a model is but probably

[topaz solar]

As in, they embed into em all of the appropriate type

[crystal turtle]

Prüfer groups are just ||I can't fucking take it anymore I'm going insane||

[steel light]

How can a local ring not have only 0 as the maximal ideal?

[topaz solar]

Because they have nonzero elements?

[delicate orchid]

I’m not too familiar with how it works but I equate “divisible group with p-torsion” to “contains or is related to p^{infty}”

[noble saddle]

0 is only maximal in a field

[delicate orchid]

Depending on ur definition it can

I’d definitely say that fields are local

[crystal turtle]

Exercise: in a local ring, the unique maximal ideal is precisely the set of elements which are not invertible

[topaz solar]

So can I embed prüfer groups of the appropriate kind in any group with that divisibility/torsion stuff?

[crystal turtle]

Or equivalently, everything not in the max ideal is a unit

[void cosmos]

hmmmmmm HAHAHDISAHDIASNCISANIQWHGFIWH

[crystal turtle]

Same

[delicate orchid]

I’m not gonna say yes or no cause I’m not qualified to

But probably

[topaz solar]

Aight so it’s not an absolutely insane statement

[void cosmos]

hello sir thank you forrecommending this exercise on a scale of 1-10 how would you rate the difficulty

1- being a problem moamen could solve

10- being grh

0

HAHDFISAICNAIN

[delicate orchid]

Boss man…

[proud spindle]

me learning group theory

[crystal turtle]

You just did it king

[void cosmos]

HAHAHAHAHA

[topaz solar]

Q a prüfer group?

[crystal turtle]

We don't need to rank everything on a scale

[void cosmos]

im literally the funniest person alive i swear to god

like no joke

[delicate orchid]

No that was me making things up a little bit

[topaz solar]

Ok hm

Q has some prime model stuff ofc

(Prime field obv)

[delicate orchid]

I gave a nice description of them before

[topaz solar]

Which is what I meant

[steel light]

Am I just going to go back confused about definitions not fully understanding or internalizing their nuances until I do some cancer exercises

[delicate orchid]

Yup

[crystal turtle]

You need to do exercises

[topaz solar]

Ok yeah that shouldn’t be prime?

[crystal turtle]

There's how we learn chief

[topaz solar]

At least not the way I was think

[steel light]

Yes but like sometimes definitions just make sense

But like here I have to keep going back, every time I see a statement about maximal ideals it makes me uncomfortable

[crystal turtle]

Once you get to more advanced definitions

Uhh

[topaz solar]

I don’t write out exercises but do em mentally

[crystal turtle]

They ain't so obvious anymore

[topaz solar]

But also im cracked so

[steel light]

Especially when they involve quotient rings I’m still having trouble visualizing what those look like

Like I know it’s a set of cosets

[topaz solar]

Doing the simpler exercises forces familiarity

[noble saddle]

they don't, you have to do exercises

[steel light]

but like…why…where muh examples…

[topaz solar]

Can you visualize quotient group?

[crystal turtle]

It's AM LMAO

[delicate orchid]

No

[topaz solar]

I need to go back and do some exercises on stability

[steel light]

no lol

[crystal turtle]

Also hmmm

[toxic zephyr]

idk i just think of Z/nZ and that's pretty much it 🙃

[topaz solar]

Ok so maybe look at quotient groups a bit, particularly of abelian groups

[crystal turtle]

Maybe this is why we were suggesting getting a solid foundation in abs alg first

That might help

[steel light]

I think I just stop bitching and wait until exercises

[delicate orchid]

I’m not joking I don’t know what people mean by this I just picture the equations in my head

[steel light]

Nah

[topaz solar]

And noting that ring quotients are Ab quotients

[long nebula]

what nix said

[topaz solar]

Fair

[crystal turtle]

I don't care for visualization usually

[delicate orchid]

Obviously I have good intuition for them but that’s not a visualisation

[topaz solar]

Im used to absurdity but

[crystal turtle]

I just intuit it

[long nebula]

I just imagine the group elements as boxes and then they all smush together

[topaz solar]

The “visualization” I speak of is rather conceptualizatiom

[void cosmos]

@rocky cloak sorry for pinging you , do you rememeber when you helped me do this proof: if R is left artinian then J(R) is nilpotent? iirc u said that if R is lft arttinian the its left noetehrian so J is f.g so we get nakayama? my question is doesnt left artinian --> left noetherian imply the ring having identity? or is this also true for nonunital rings?

[long nebula]

Groups are just m×n grids

[sly crescent]

You just merge elements together in a certain way

[long nebula]

Quotient groups are m×1 grids

[noble saddle]

doesn't D&F have a picture of quotient groups that is basically how everyone thinks of it?

[long nebula]

:^)

[topaz solar]

Not an image in genuine, but imagery of how it acts

Like music visualizers

[void cosmos]

you mean the natural covering something something transformation

no

[crystal turtle]

What does an aleph_0 x aleph-omega grid look like :^)

[noble saddle]

also I picture quotient rings as closed subvarieties not as quotient groups

[long nebula]

This is fairly straightforward

[void cosmos]

i think localizations are what make quotients so good tbh

[topaz solar]

R x R grid

[toxic zephyr]

another good quotient group is like R=R[x]/(x^2+1) which is basically just a+bx+I where x^2=-1. i just think "you can always subtract off the ideal"

[void cosmos]

yea

my NT prof went more like : what remainders can you get when dividing by this

[long nebula]

Annihilating left ideals are my favorite

[void cosmos]

math is literally like art - the bad stuff about art

like everyone has his own picture or idea of something but they are all right

[steel light]

Z/nZ makes sense to me too but nothing else does :( like besides just definitionally. Nothing else makes sense intuitively. When I think quotient I think of topology, the notion of “gluing together” (identifying as the same thing) objects that are “like” (equivalent) and then the space you get under that. Is this the same thing but with ring addition instead of an equivalence relation?

[void cosmos]

isntead in art

everyone is stupid

lmfao

[long nebula]

uh

what

[long nebula]

You glue together the stuff in the ideal yes

And this induces gluing together on the rest of the terms

[void cosmos]

i mean like everyone has some idea aswell but they are not necessairly the same most of hte time just diff from the artist intended

like the curtains were blue kind of shit

[long nebula]

So for example in Z/3Z, you glue together 0, 3, 6, 9, etc

[noble saddle]

Z is like an affine line, and Z/nZ is like a 0 dimensional subset of a line

[steel light]

But what does it mean to glue something together under addition? 😭

[long nebula]

this also forces you to glue together 1, 4, 7, etc

[void cosmos]

identifying them as one

u glue them

[long nebula]

idk what you mean by under addition

[toxic zephyr]

i didn't like quotient stuff at first, but they're so great. you can just like... cut stuff off of everything and it's fine. simplifies so much.

[long nebula]

I just mean you glue them together

quotient just means "imma ignore this"

[topaz solar]

The equivalence relation is exactly a subalgebra (subgroup, subring, etc) of X x X

[steel light]

Yeah but that makes sense to me because it’s the same thing like saying x ~ y iff x mod n = y mod n right?

[void cosmos]

yea feather is actually a surgery theory expert so what u said just doesnt make sense 🙂

[long nebula]

yes, that is a quotient

[steel light]

right

[topaz solar]

This is exactly the statement that it is compatible with the algebra structure

[long nebula]

feather I think you just need to compute some quotients and then you'll get it

[noble saddle]

but also, Z isn't the model ring. Something like C[x] is easier to picture quotients, ideals, etc

[topaz solar]

Z[x]

[void cosmos]

i think identifying stuff and equivalence classes and relations are probably the most important idea in math right?

[noble saddle]

Z[x] is a plane tbh

[toxic zephyr]

it's satisfying to have like 20x^20+32345x^15+9999x^9+12321245x^2+41424x^2+1+(x^2+1) and be like, "hey, it's just 1"

[steel light]

but how does that work with ring addition, isn’t that what a coset is? For R/m (Ring, mideal), the coset of r in R is r + m. How is that “gluing”? What like objects are we considering as the same thing?

[long nebula]

Ignoring stuff is the most important idea in math

I ignore everything

[crystal turtle]

what

[long nebula]

we're considering everything in m the same

[steel light]

I think so too

[crystal turtle]

Oh yes. So enlightening

[noble saddle]

Reduce mod p you get F_p[x], which is like a line over F_p. Z[x] bundles those up

[long nebula]

and this also implies you have to glue some other stuff due to the ring structure

[crystal turtle]

What a wonderful picture. It teaches me so much. Great geometric intuition

[long nebula]

just like how saying 0=3 in Z/3Z forces you to also say 1=4

[topaz solar]

Algebraic “”geometry””

[void cosmos]

yo guys i think geometry is literally just about going from A to B to something that is induced from A to something that is induced from B ( IM NOT JOKING LITERALLY )

[steel light]

Why is r there then?

[void cosmos]

like functoriality

[crystal turtle]

Pretty much

[long nebula]

Everything in m becomes 0

[noble saddle]

If you ignore the squiggles, I think it's not so bad

[south patrol]

No

[topaz solar]

Model theory is AG but without fields

[south patrol]

Geometry is about going from A -> B to smth from B to smth from A

[topaz solar]

And horrible

[steel light]

what

[void cosmos]

true literally

[long nebula]

try thinking about this just in terms of the definition

[void cosmos]

u catch my drift

[toxic zephyr]

it's just like a+m=a'+m means a-a' in m. so a'=a+m' where m' is any elt of m

[crystal turtle]

Contravariance my beloved

[void cosmos]

its literally functoriality

thats it

thats geometry

no angles no lines no nothing

[toxic zephyr]

if they're equal in the quotient ring, they're just off by an elt of m

[crystal turtle]

It's all sheaves in the end what can I say

[toxic zephyr]

sounds like gluing to me

[long nebula]

alternatively, think about this using Z/3Z as an example first

[topaz solar]

I hate the idea of having to draw Spec Z[x] like that picture, but those lines are like the uhhh basis closed sets I think?

From the prime ideals

[noble saddle]

there are more basis closed sets not pictured, but roughly yes

[crystal turtle]

I hate having to draw pictures

[void cosmos]

yoo

[topaz solar]

Well yeah they aren’t drawing literally every irreducible polynomial

Or every prime

[noble saddle]

you said lines, I wanted to clarify that there are curves that are not the lines

[void cosmos]

showing that the sphere U {a point} is never a manifold is very good with pictures

or the sphere with hair

its an exercise in Tu

its so cool

[noble saddle]

not being pedantic

[topaz solar]

Ye, those should be the uhh idea of the ones generated by the primes, which puts primes in Z on the bottom (the line bundle he referred to)

And the right side is the polynomial-y ones

[noble saddle]

but the point is that Z[x] is like studying a fiber bundle. To understand polynomials with integer coefficients, we can do honest 1-dimensional geometry over a field and try to patch together

[steel light]

Oh

[topaz solar]

I always mix up the basis set directions

[steel light]

I

okay

LOL

[topaz solar]

Spectrum of Boolean ring my beloved

[void cosmos]

yea just consider the sheaf that acts on fries and pizza

and the geometry untangles

[steel light]

I see what you mean now by everything in m becomes 0

[void cosmos]

okay funs over im going to sleep ( watch netflix )

thank you guys for the math help

[crystal turtle]

god i gotta start reading more when I move into my new place

and working more idk

[topaz solar]

Same

Iron sharpens iron 👀

[long nebula]

Yeah, so glue everything in m together

Unlike in topology, this also necessitates gluing other stuff together

So glue it

[steel light]

Still not gonna see it well until I compute other quotient rings lol

Can you give me nontrivial examples that aren’t too complicated and aren’t Z

[topaz solar]

R[x]/(x^2+1)

[steel light]

LOL

[crystal turtle]

Q/Z

so intuitive

[solar vessel]

that ain't an ideal

[topaz solar]

That’s not a ring quotient u nerd

[crystal turtle]

fuck

dumbass moment

[steel light]

So do you always “mod” by whatever you’re quotienting by after you take a coset (is that English?)

[solar vessel]

anyway

[topaz solar]

Z[x]/(x^2)

[topaz solar]

That’s what taking a coset does

[steel light]

Not to me

[delicate orchid]

Yeah this is a good one

[steel light]

To me it just adds together an element and another set

[topaz solar]

[delicate orchid]

if x in I then x + I = 0 + I

[steel light]

How does that involve anything about mod or quotienting like how I know it topologically

[delicate orchid]

Who cares about topologies

[topaz solar]

Because what wew said

Topology quotient is cosets too btw

[solar vessel]

I would suggest not having your idea of ring quotients depend on quotient topologies

[noble saddle]

cosets are just a technicality, you're basically just saying I have a symbol x satisfying the equation x^2 + 1 = 0

[steel light]

Yeah I need to lose this

[noble saddle]

don't actively think about cosets

[steel light]

Just don’t know how

[topaz solar]

Give ‘em the disjoint topology

Gg

[solar vessel]

it's a new thing

imagine it's a new word

[long nebula]

although tbh it's probably good if you explicitly list out some similarities and some differences

they serve the same general purpose

[long nebula]

the what

[delicate orchid]

Boss means discrete

[topaz solar]

You heard me

[solar vessel]

anyway

[topaz solar]

I meant discrete yes

[long nebula]

tfw set quotient not ring quotient

[solar vessel]

how are you with quotient groups?

[delicate orchid]

Discrete is kind of disjoint cause it’s uhhh hausdorf

[steel light]

I am not with quotient groups

I just recognize the notation

[topaz solar]

Cosets are imaginary elements anyhow

[steel light]

lmao

[long nebula]

feather, calculate C^× / R^>0 as groups

[delicate orchid]

Hmmm orbit category not concrete you go bye bye now!

[noble saddle]

cosets suck all my dudes work only with 1st isomorphism theorem

[rotund aurora]

Now I think the "geometry" part in "algebraic geometry" is actual geometry

[topaz solar]

Imaginary as in

[steel light]

what does this even mean

[topaz solar]

https://en.m.wikipedia.org/wiki/Imaginary_element#:~:text=An imaginary element a%2Fφ,θ(x%2C b).

[steel light]

I don’t understand what you’re asking