groups-rings-fields

[next obsidian]

Then xyz is also order 2

Since xy = z

(Just pick z such that that’s true)

Right so yeah if we map -1 to f, the element of order 2

i to x, j to y, k to z

This is well-defined

There’s a few presentations but im using the one that’s

Generated by -1,i,j,k with (-1)^2 = 1

i^2 = j^2 = k^2 = ijk = -1

So we get a well-defined map, and we only need subjectivity or injectivity to conclude its bijevtive

And i think injectivity is immediate

We only need consider where -i,-j,-k go

But that goes to x^-1,y^-1,z^-1

So it’s injective

Then we’re done?

[carmine fossil]

Guess so

[next obsidian]

Dope

Do you have any objections to any step in the proof?

[carmine fossil]

What about p^3 in general

No

[next obsidian]

Idk about p^3

p^2 is fine

Hmm

Okay so 3^3 has 5 groups

3 abelian, 2 not

The kon abelian is some

C9 semi C3

And uh the “prime cube order group U(3,3)”

So maybe it’s always just 5 groups??

https://kconrad.math.uconn.edu/blurbs/grouptheory/groupsp3.pdf

Hey look at this lol

Keith Conrad

I’ve never tackled general p^3

[carmine fossil]

Do we have a more powerful thing than the semi direct product?

[next obsidian]

¯_(ツ)_/¯

Ask me again once I go through more of my finite group theory book haha

But apparently yes there’s only 5

And Keith says the descriptions are different for even and odd p

But at the end he notes the number is not uniform for p^4

[round oxide]

Is abstract algebra hard?

[viscid pewter]

it's abstract

i find it pretty tough

[golden pasture]

hard is relative

there will always be areas that seem super hard to you

until you build your way to the area

abstract != hard

[marsh fractal]

so this is my lecturers method of factorising to find gcds, I havent seen a method like this before, any thoughts?

[carmine fossil]

It's the Euclidean algorithm

[marsh fractal]

Ok this is a standard method?

or most efficient etc?

[carmine fossil]

Kind of standard

[marsh fractal]

ok!

Id like to understand the process

theres a pattern but its just kind of strange

Ok I see theres a lot on it on the interwebs

ty!

[marsh fractal]

One more question on this: why do they conclude that x+1 is the gcd and not 2x+3? they give the reason : 4/11 is a unit, but 11/4 is also a unit

[nova plank]

The gcd is the last nonzero remainder

But they are dropping the 11/4 to make it monic

Because an associate of a common divisor is also a common divisor

[marsh fractal]

ok so they choose it just because its monic

why is that necessary?

[nova plank]

Possibly you defined the gcd to be monic

Otherwise there are infinitely many gcd's

So it's standard to use the monic one as the gcd

[marsh fractal]

Ok I see

and why do they use the dot in the last line?

instead of normal (invisible) multiplication

[nova plank]

Just to be clear, 4/11 (2x + 3) is not in the conversation to be the GCD at all.

[marsh fractal]

ah ok

yes

[nova plank]

Because it doesn't look nice to have a number following a bracket

[marsh fractal]

ok but is it still a gcd?

[nova plank]

Is what?

[marsh fractal]

just not a nice one to choose

4/11 (2x + 3)

[nova plank]

No

It's not a common divisor at all

[marsh fractal]

ah ok yes that does make more sense, because they swapped them around

[nova plank]

You look at the REMAINDERS

[marsh fractal]

YES

that is a quotient

[nova plank]

11/4 (x+1) is the last nonzero remainder. So its monic associate is the GCD

[marsh fractal]

ok

great thanks

but why the dot still ?

[nova plank]

This. Just because it looks better.

[marsh fractal]

Ah ok

And i guess the dot is assumed to be normal multiplication of rationals

[nova plank]

It's multiplication of polynomials

[marsh fractal]

ok nvm kind of redundant, ty!!

[nova plank]

Np

[round oxide]

So are all of you doing engineering here

It really scares me lol

I hardly understand any of the problems sent here

They just started groups in uni

And vector space

[chilly ocean]

i think most of the people doing engineering don't post in these channels

[bronze trench]

hey people, I'm having the tiniest bit of trouble with a homework exercise

it's a very concrete thing, the rest of the homework is pretty much done 😄

So I proved that for an R-module M and m \in M there's always a maximal submodule N of M relative to the condition m \not\in N

ok maybe not that clear, it needs not be a maximal submodule, it's maximal among those with that property

I now want to find that for a concrete module and an arbitrary element, and in this case the module is a ring viewed as a module over itself. Don't want to go into much detail because a general approach may be all I need. If anyone has seen this before or has any ideas I'll gradly hear and discuss 😄

also all modules are left modules, and rings are unital

and in the concrete example it's actually a commutative ring

[golden pasture]

concrete example just take vector spaces or modules over PID, they are pretty simple to work with🤔

[bronze trench]

no no, I do have a concrete example presented to me to solve 😂

[golden pasture]

ahhh

[bronze trench]

I just didn't want to say it yet because I'd benefit from a general thought process 😄

Like a true mathematician I want the most general thing I can xD

[golden pasture]

modules get super ugly very quickly in the general case

[bronze trench]

Yeah ok you're right. I'll show my concrete example, probably better 😂

[twilit pawn]

so just to be clear, the question is "given a module M and an element m in M, is there a submodule N maximal among those which don't contain m?"

or am i misunderstanding

[bronze trench]

Yes but Zorn's lemma deals with that kinda easily

[twilit pawn]

oh you want to construct one

[golden pasture]

i think the question is more is there a general way to construct these for as large of a class of modules as possible

[bronze trench]

now it's a concrete module and I want to find such submodules

[twilit pawn]

you want a technique for constructing it (when it's possible), i gotchu

[bronze trench]

yep my idea is that but your point about modules being messy is a good point xD

[golden pasture]

so like non-f.g. modules over a noncommutative ring would be the most ugly case to consider

[bronze trench]

So for the concrete example, the ring is Z_(2) as a module over itself, this being the localisation at the prime ideal generated by 2

This is what I actually need to solve

so actually this reduces the problem a lot

as a commutative ring and a module over itslef this is just ideals

and not any more fancy things

[golden pasture]

id reckon you’ll quickly run into set theoretic problems and can really only show existance

[uncut girder]

Z_(2) has all its ideals being powers of (2)

[golden pasture]

tbh anything non f.g. + using zorns to prove have a decent chance to run into these problems where you know it exists but there is no explicit construction

[twilit pawn]

yeah my gut is that if you have to use zorn's lemma, there shouldn't be a general technique

[bronze trench]

wait really?

[twilit pawn]

yeah basically what cat said

[golden pasture]

yup (more info it's known as a discrete valuation ring)

[bronze trench]

yeah I get you. I'm way more into the existance things, constructing is not my cup of tea xD

[golden pasture]

so the way you show that for Z_(2) is like

you know Z is a pid

so ideals are (n)

[next obsidian]

DVR pog

[bronze trench]

wait then this is not that hard I think, if I use that info

[uncut girder]

When you localize you get rid of ideals

[golden pasture]

but localizing at (2) 'kills' all ideals that are not subset of (2)

[bronze trench]

ah fuck me that's right

[golden pasture]

so you are left with (2^n)

more generally if you have a dedekind domain and localize at any prime you get a dvr

[bronze trench]

you know I'm really not into AG or NT so the parts of algebra used in those aren't all that known to me as I haven't taken courses on those

this is a general algebra course btw

[uncut girder]

That's fine

[golden pasture]

ye a lot of things seem super unmotivated without nt or ag tbh

[next obsidian]

Ari tbh

When I think DVR

I literally think primes are 0 and then a maximal one

Which definitely is not a complete description lol

[bronze trench]

yeah lol I had a "commutative algebra" class I totally failed, but I learnt stuff about local rings so I should have thought about it

[golden pasture]

a morally sufficient one tbh

[twilit pawn]

existence proofs which are not constructive never tell you anything about how to construct (obviously). so you shouldn't expect to be able to construct in a specific example unless you can "cheat" and already know what the answer should be in that case, because of some intuition or understanding about the specific situation. or you can construct if the proof that used axiom of choice doesn't actually need the axiom in your specific example.

that's like, how i think about constructing things very very generally

[golden pasture]

a lot of examples are basically ag examples tbh

[maiden ocean]

comm alg is cool it motivates itself

[golden pasture]

cope

[solemn rain]

comm alg is cool

[maiden ocean]

Yes? And?

[solemn rain]

its like

[next obsidian]

It definitely doesn’t

[solemn rain]

it like holds both ANT and AG ig

which is cool

[next obsidian]

Comm alg made me want to castrate myself

[bronze trench]

I failed it because I was super underprepared and also know 0 AG so examples and motivation didn't make a lot of sense

[next obsidian]

Then I did AG

And now it’s cool

[solemn rain]

emmy noether is so bae omg i want to call her mummy so bad

[golden pasture]

doing ag nt and com alg simultaneously

[maiden ocean]

hurb

[golden pasture]

wtf momen

go to chill

[maiden ocean]

begone with u

[solemn rain]

sorry lmao

[next obsidian]

That’s a dan

But also

Same

[golden pasture]

smh simp

[solemn rain]

but yea comm alg is useful for like ag nt

[next obsidian]

Yeah but mo2men

That’s the point

[solemn rain]

but idk about motivation

[next obsidian]

Moth is saying it’s cool by itself

[bronze trench]

yep thanks for the help I think I got it solved in my head already, just gotta write it down 😄

[solemn rain]

why

[next obsidian]

We’re saying no it’s pretty shite without motivation

[bronze trench]

But also

non commutative is way cooler

[maiden ocean]

idk i have no motivation but AM is still fun

[golden pasture]

well

[next obsidian]

🥱

[solemn rain]

the only cool algebra thats useful by itself obv is group theory

[next obsidian]

Moth

[uncut girder]

It's taking me several passes to grok CA

[golden pasture]

non commutative is suppper painful to deal with

[next obsidian]

Stockholm syndrome

[maiden ocean]

Yes?

[golden pasture]

i get non commutative is like

[maiden ocean]

And?

[golden pasture]

have a feeling of more general

but also less useful

when doing nt

[maiden ocean]

non com never feels more general to me

[solemn rain]

? when do you have to deal with noncommutative stuff?

[maiden ocean]

it just feels like less structure

[golden pasture]

or normal ag

[solemn rain]

i never dealt with any noncommutative anything

[next obsidian]

Look if you want to like CA this is the Chad way

Get totally totally fucked by AG by not knowing it

[golden pasture]

like all groups are basic noncommutative

[bronze trench]

try and put your socks on after your shoes and you realise pretty quickly that the world is not commutative

[next obsidian]

See some proofs hat go “this is just ___ theorem from CA”

[maiden ocean]

I think I will not do that hurb

[next obsidian]

Then go and do CA

[golden pasture]

noncommutative rings are rarer

[solemn rain]

yea but i mean

[next obsidian]

And suddenty it is so much cooler and enjoyable

[golden pasture]

cuz they are so nasty

[solemn rain]

they werent bad ig

[maiden ocean]

Chmonkey is trying to subliminal message me into being an AGcel

[next obsidian]

And then you learn a lot of CA

Then when you do AG it becomes much easier

[solemn rain]

AG is the biggest part of math

[uncut girder]

AGcel

[solemn rain]

right?

[golden pasture]

ya

[uncut girder]

First time I heard that word

[golden pasture]

you literally need to learn ag and ca simultaneously

[bronze trench]

I have ptsd with GA

[next obsidian]

I think CA should come first to not make AG hell BUT

[bronze trench]

AG*

[next obsidian]

It cannot be learnt first until AG Hs fucked you a bit

Because it’s too boring

[sour plume]

"AG is the biggest part of math" i don't get the algebraic geometry boner that the math world has

[golden pasture]

HAHAHAHA

ikr

[solemn rain]

@sour plume same

[next obsidian]

One must accept Hartshorne destroying you once

[sour plume]

i studied in germany and i learnt absolutely nothing about AG

[next obsidian]

Then learn CA

[uncut girder]

At my school its 1 sem of CA then 1 sem of AG

[solemn rain]

maybe ur an analyst

[sour plume]

yeah idk i like smooth manifolds

[golden pasture]

idk i just

[next obsidian]

Lart honestly me too

[golden pasture]

cant learn ca without ag

[next obsidian]

I mean I’m interested in it so I just talk about it a lot

[golden pasture]

i also cant learn ag without ca

[solemn rain]

i think the only non dead math fields now are AG and number theory

right?

[next obsidian]

LOL

[sharp sonnet]

lmao

[golden pasture]

wtf no

[sour plume]

l o l

[sharp sonnet]

PDE is 50% of math

[golden pasture]

literally

[solemn rain]

lmfao

[golden pasture]

analysis

is a whole new world

[sour plume]

fam let me tell you

[sharp sonnet]

also only internet has this huge AG boner

[sour plume]

the math world is a lot bigger than what you see

[golden pasture]

go to arxiv

[next obsidian]

It’s present in math depts too depending where you’re at

[sour plume]

https://mathseminars.org/ like, look at all the garbage that people study here

don't get me wrong

NONE of it is interesting

[solemn rain]

what do you do if ur interested in something

that is dead

[sour plume]

but it's more than just AG

[solemn rain]

in the math community

[sharp sonnet]

you don't

[chilly ocean]

explains the three algebraic geometry seminars going on at once in my uni

[next obsidian]

It’s kind of hard to find a dead subject

[solemn rain]

is algebraic geometry abstract

[next obsidian]

L o l

[sharp sonnet]

it's easy to find dead subjects

[golden pasture]

i feel like internet have a lot of AG cuz like it has a shit ton of prereq

compass and ruler constructions

dead

[sharp sonnet]

because they are either done or not interesting anymore

[golden pasture]

=euclidean geometry

[solemn rain]

point-set topology

😦

[next obsidian]

No it’s cuz all AG people never go outside since we’re too busy doing AG

[sour plume]

"interesting" is a garbage term

[next obsidian]

So we migrate to the internet

[sharp sonnet]

point-set still exists

[golden pasture]

point set isnt rlly dead

[sour plume]

i have a lot of very interesting questions that no one cares about

[sharp sonnet]

but interest is becoming less

[sour plume]

just because all the people who cared about it some time ago

[golden pasture]

yea

[solemn rain]

what about AT

[sour plume]

are dead now

[sharp sonnet]

just because interest in set theory is decreasing

[golden pasture]

AT is defo not dead

[sharp sonnet]

AT is big

[golden pasture]

it is very very very not dead

[next obsidian]

K-theory

[golden pasture]

AT is hugeeeee

[uncut girder]

Compass and ruler constructions may be dead but multi fold origami constructions?

[solemn rain]

i thought they only ahve

like 1 problem left in AT

and thats it

[sharp sonnet]

origami was never really alive

[golden pasture]

[next obsidian]

L o l

[golden pasture]

tf

[solemn rain]

the sphere groups of whatever

[golden pasture]

mo2men

[next obsidian]

L o l

[sharp sonnet]

there are like 2 people doing origami

and some others for didactic reasons

[next obsidian]

Who is NC

[golden pasture]

where are you getting your info from

AT has many active areas

[solemn rain]

who is nc

[golden pasture]

dm

[sour plume]

why do people care about origami math

[next obsidian]

Wtf is origami math

[sharp sonnet]

you can formalize origami

[solemn rain]

origami?

[sharp sonnet]

and then do math

[next obsidian]

Meme

[solemn rain]

the like paper game?

[sour plume]

yes but why would you

this just feels like chmess

[uncut girder]

Why not

[sharp sonnet]

i know someone who does it for didactic reasons

[solemn rain]

coool

[sharp sonnet]

i.e. teach it in school

as babies first formalizing something

and intro proofs

[sour plume]

oookaaaay i think i can get behind that

[sharp sonnet]

it has practical uses in like protein folding and solar sails apparently

but 🤷

[uncut girder]

The point is origami is is more powerful than straight edge an compass

[sour plume]

i have to say

[uncut girder]

You can solve cubic equations with origami but cant with straight edge and compass

[sour plume]

i don't believe mathematicians anymore when they say "this has practical use in"

until i see it

[sharp sonnet]

and analysis is more powerful than either

you don't care about this because of power

[uncut girder]

It's kinda cool

[sharp sonnet]

just because its a question to ask

[uncut girder]

Why do any math @sour plume

[sour plume]

idk pays the bills and it's the least amount of suffering

[uncut girder]

LOL

[next obsidian]

This is to get funding for your projects. Uhh... something quantum

[sharp sonnet]

all of math was actually invented to solve diophantine equations

[next obsidian]

Yes my derived algebraic geometry Hs applications to squints uhhhh chemistry

[golden pasture]

honestly agreed

[uncut girder]

All of math serve number theory, that is where the true primitive questions lie

[next obsidian]

Loch I once solved a Diophantine equation to do something for an algebra midterm

Then realized I was considering rational solutions

And cried

Then I did something else to solve it which wasn’t fucking stupid

[sour plume]

for example, some time ago i happened to stumble upon this short little wikipedia article: https://en.wikipedia.org/wiki/Double_affine_Hecke_algebra

and it just states "Infinitesimal Cherednik algebras have significant implications in representation theory, and therefore have important applications in particle physics and in chemistry."

and without any explanation this feels like such a reach

[golden pasture]

[sour plume]

because yes, sure, representation theory IS related to particle physics and chemistry

[sharp sonnet]

"give funding please"

[sour plume]

but i think making that connection from these strange algebras to REAL LIFE

is probably at least three papers

[uncut girder]

That is a reach

Also nothing is relevant to anything

[sour plume]

if it's even possible

[maiden ocean]

false loch

all of math was invented to please me

[golden pasture]

moth are you interested in reading jech with me

[maiden ocean]

whats that on?

like wut field

[next obsidian]

F_7

[maiden ocean]

stop talking

[golden pasture]

jech set theory

[next obsidian]

This reminds me of the first messages in this channel

An old mo2men account asking

What’s a set

[maiden ocean]

oh hurb

[next obsidian]

What’s a group

[golden pasture]

XD

[sharp sonnet]

it's a good question

what is a set

[maiden ocean]

me

i am a set

[sour plume]

that is true

[golden pasture]

anything that is morally a set

is a set

[next obsidian]

Is there a set of all moths?

[maiden ocean]

yes

thats also me

im the last of my kind : (

[sour plume]

where does your motivation to study set theory come from cat

[golden pasture]

sad!

[uncut girder]

You are the set containing yourself

[next obsidian]

Set theory 🥱

[golden pasture]

i just want to poke into as many fields of math now lol

see what is most interesting

[sharp sonnet]

it's not set theory

[next obsidian]

then don’t do set theory

[maiden ocean]

ari is traumatized from doing 1.5 years of analysis ors omething

and wants to make sure she doesnt miss out on more stuff

[golden pasture]

[sharp sonnet]

you can't take a single step into set theory without being bogged down by formalism

[golden pasture]

look i used to do frickin crypto formalisms

how to formalize security

that was uh

something

[sour plume]

i imagine understanding set theory is pretty cool, but i just kinda doubt that it's useful for anything else

the cost/benefit of learning set theory seems pretty meh

[golden pasture]

what is cost benefit analysis

fake news tbh

[viscid pewter]

something something transfinite inductions

[uncut girder]

Why does everything have to have a use @sour plume

[next obsidian]

People who care about set issues are the kind of ppl to use condoms and wear masks when they go outside

[golden pasture]

if it exists i would be becoming doctor

[sour plume]

i don't wanna talk you out of this though, do what makes you happy

[golden pasture]

unironically useful occasionally

[sharp sonnet]

well, the thing is if you do anything else, you have a lot of side fields you can go into

[next obsidian]

Transfinite induction is cool

[golden pasture]

by set theory i mean like getting stabbed by formalism

ye

[sharp sonnet]

if you spend all your time on set theory

[sour plume]

@uncut girder well, not just in a capitalist sense, but also in the sense of wanting to become a good mathematician, i think there are good paths and less ideal paths

[sharp sonnet]

you can basically do set theory

[next obsidian]

Projective module over local are free 😎

[golden pasture]

who knows maybe ill be one of those hott memers in my 30s

[next obsidian]

L O L

[golden pasture]

maybe ill be useful and go to med school

[uncut girder]

@sour plume that depends on current fashion trends, there are no objectively good paths or objectively bad paths, and you'll get different answers from different people

[golden pasture]

or be even more useless and become musician

[sour plume]

the fashion trends are what decides about your future though

you can be the best mathematician in the world in one specific area

if no one cares about that area, they're not going to fund you

so idk i think there's some strategic elements there

[next obsidian]

I am Euclidean geometry master

Pls gibe monee

[golden pasture]

idk if i want to actually do math 24/7 in academia ngl i like how i am now just being super chill

[uncut girder]

You're kinda assuming people learn math to get hired as a mathematician, but that's not always the case

[sharp sonnet]

well, you have to do something for a living

[sour plume]

idk it feels like people who go on math servers in their free time are pretty interested in doing math for all eternity

[golden pasture]

im perfectly ok with being a music teacher

[sour plume]

(or they want their homework done)

but i'm assuming things, that's true

[sharp sonnet]

(or both)

[sour plume]

i want my very complicated homework done, too

[sharp sonnet]

yes, same

but i cba to ask

[sour plume]

is #groups-rings-fields the new #chill

[uncut girder]

Yes

#groups-rings-fields for cool kids only

[golden pasture]

chill is such a disaster at times

why do i procrastinate there

[sour plume]

idk seems like a nice place in general

i've been on worse discord servers

[next obsidian]

Have you been around when certain people started having Israel-Palestine “debates”

[chilly ocean]

chmonkey chill

[next obsidian]

Lol

[golden pasture]

most of my discord servers are like work servers or music servers lol so this is p degen for me

[next obsidian]

F you Godel why do you keep doing this to me

[sour plume]

politics discussions are just too fun tho

idk i think my brain is poisoned

[next obsidian]

I have a simple tensor product question I should probably know, but here goes

If we have a map like $f\colon M\to N$, given another module $L$ denote by $f_L$ the induced map $M\otimes L\to N\otimes L$

[cloud walrus]

Chmonkey:

[next obsidian]

If $L$ is flat by looking at the exact sequence $0\to \ker f\to M\to F\to \text{im} f\to 0$ after tensoring with $L$ we get that $0\to \ker f\otimes L\to M\otimes L\to \text{im} f\otimes L\to 0$ is exact

[cloud walrus]

Chmonkey:

[next obsidian]

The map $M\otimes L\to \text{im}f\otimes L$ is just $f_L$ so this says that we identify im $f\otimes L$ with im $f_L$

However $f_L$ works on simple tensors by simply mapping $m\otimes l$ to $f(m)\otimes l$, so I feel like by looking at simple tensors we get that im $f\otimes L = \text{im}f_L$ even if $L$ isn't flat.

[cloud walrus]

Chmonkey:

Chmonkey:

[sour plume]

That did make me wonder, too, at first, but this seems fine, since tensoring with another module is always right-exact

[next obsidian]

Yeah I thought so

For the statement about kernels like this it makes sense I need L flat

[sour plume]

Ye probably

[next obsidian]

but Matsumura says something like "since __ is flat" for the statement about the image

which made me wonder

And he doesn't tend to do unnecessary things

[sturdy marsh]

What's the map from F to im f

[next obsidian]

Just $f$

[sour plume]

(yeah the big F is a typo i think)

[next obsidian]

Wait did I ever type a big F? haha

[sour plume]

I dunno, I think your argument is fine, the image isn't more complicated than that

[next obsidian]

Oh wait oops

HAHA

I see yeah that F is an accident

[sturdy marsh]

F

[next obsidian]

it shouldn't be a term there

Idk where it came from lol

So like

both are generated by elements of the form f(m) (x) l

so they should be equal I think

I'll just chalk this one up to a weird moment Matsumura is doing something unnecessary

[sour plume]

everybody does a poopy in panty sometimes

[next obsidian]

😦

[bronze trench]

update on my modules thing, I asked my prof and he lets me do whatever the fuck I want provided I give references so using DVR results it is 😂

[next obsidian]

Pogchamp

I need to properly learn DVR stuff lol, instead of this ad hoc knowledge I have from them popping up from time to time

[bronze trench]

I don't know a whole lot I literally went to Atiyah and saw a proposition of interest so I will invoke that, prove my thing is a DVR and then I have what I want

because I just want to see that in Z_(2) all ideals are powers of the maximal ideal and I don't feel like doing it "properly" so I'll prove it's a DVR and say that's true for all DVRs

[next obsidian]

Lol

I think it being a DVR is immediate too from like

One of the 7 million equivalent conditions to be a DVR

[bronze trench]

yeah that's it, I'll invoke that, it's indeed one of the equivalent conditions

[next obsidian]

Maybe like local PID with maximal principal?

I think that’s one of them

Or maybe you don’t even need PID

[bronze trench]

eh maybe but I gave the valuation as it's kinda obvious

[next obsidian]

Like Noetherian local with principal maximal?

Lol

[sturdy marsh]

Just learned a pretty cool trick

A retract of a representable functor from the category of schemes is representable

[next obsidian]

Tf is a retract of a functor

[sturdy marsh]

A is a retract of B if you have $A \rightarrow B \rightarrow A$, such that the composite is identity

[next obsidian]

Huh

[cloud walrus]

Brofibration:

[golden pasture]

inverse?

[next obsidian]

Not quite m

What’s the source of A?

[sturdy marsh]

Scheme

[next obsidian]

Actually it has to be the other category

Lmao

[bronze trench]

ah fuck I can't use atyiah since it has conditions on the ring I don't want to prove, I'll find another book without those xD

I can't cite the wikipedia article 😦

[sturdy marsh]

But anyway, using that trick you can construct the hilbert scheme of curves on a surface as a subscheme of a grassmanian

[next obsidian]

🥴

Moduli space

[bronze trench]

you're literally making up words

this server humbles me so much lol

[sturdy marsh]

Mumford left that trick as an excercise

[next obsidian]

Give me a quarter before I can talk to you about that

LEL

Geometry of schemes?

No way right

Or wait

[sturdy marsh]

lectures on curves on an algebraic surface

[next obsidian]

Did Mumford do that

Figured

Haha

[sturdy marsh]

geometry of schemes is by eisenbud

and harris

[next obsidian]

No that’s eisenbud who did geometry

Mumford did red book

[sturdy marsh]

that too

and abelian varieties

[next obsidian]

Right

Do you know any algebraic group stuff?

[sturdy marsh]

nope

[next obsidian]

I think I want to learn some of it next year

[sturdy marsh]

learned a wee bit for GIT

[next obsidian]

Apparently Milne made a like introductory, scheme theoretic algebraic groips book recently

Like very recently

I might try that out

My prof talked about G_m actions in our intro course because he’s irresponsible and suddnely I want to understand

And apparently Proj is something something some quotient by a G_m action on the Spec or some shit

[sturdy marsh]

Mukai has some basic stuff on algebraic groups

[next obsidian]

Which I guess mimicked the classical construction of Projective space as the quotient by the group of units

[sturdy marsh]

stuff on reductive algebraic groups, hilbert's theorem on finite generation of invariants, etc.

[next obsidian]

Mukai, I’ve heard the name but don’t recall what the books called

[sturdy marsh]

intro to invariantsand moduli

invariants and*

[next obsidian]

Gotcha

Geometric invariant theory stuff?

[sturdy marsh]

yup

[next obsidian]

Too much to learn

For now i

Hartshorne

😂🔫

[sturdy marsh]

the other books are a lot more fun tho

and you actually get to see schemes in action

[next obsidian]

But... background

[sturdy marsh]

well you can build background as needed

you dont need to do every single hartshorne problem before learning other AG

[next obsidian]

I feel I should do some cohomology too tho

[sturdy marsh]

ye

[next obsidian]

Before I do other shit

Haha

[sturdy marsh]

cohomology is important

but that's all you need to get started

[next obsidian]

🥴

[sturdy marsh]

I didnt get the point of a lot of the stuff while reading just hartshorne

moduli stuff is cool as you really need schemes to do some of it

[next obsidian]

Well good thing I learn that next quarter

:^)

In theory

[sturdy marsh]

is the class just on moduli stuff or anything in particular?

[next obsidian]

Uhhh let me find the like course description

The primary goal of this coarse is to understand and establish the following statement: the moduli space parameterizing stable curves of genus g is represented by an irreducible, smooth and proper Deligne-Mumford stack with a projective coarse moduli space. Assuming the background of a first course in algebraic geometry, we will begin by introducing the language of algebraic spaces and algebraic stacks. Using this language, we will then proceed to construct the moduli space of stable curves as a projective variety.
Topics at a glance:
• Grothendieck topologies and sites • Categories fibered in groupoids
• Descent
• Algebraic spaces and stacks
• Deligne-Mumford stable curves
• Semistable reduction for curves: properness
• Irreducibility
• Existence of a coarse moduli space: the Keel-Mori theorem • Projectivity of moduli

[sturdy marsh]

noice

[next obsidian]

So idk how schemey it’ll be haha

[sturdy marsh]

stacks are cool

there's a sheaf of cohomology theories on the moduli stack of elliptic curves

[next obsidian]

...

[sturdy marsh]

a sheaf of spectra

but yeah

cool stuff

[sturdy marsh]

but the schemey stuff that I was talking about is Grothendieck's existence theorem

and deformation theory

[next obsidian]

How long have you been doing all this for lol

[sturdy marsh]

oh just this quarter

Im reading mumford

[next obsidian]

I meant AG

[chilly ocean]

chmonkey in the first 2 weeks you been in this server you sent about 8k msgs. You've been here for about 5 months and you sent almost 43k messages overall

[next obsidian]

Yeah Godel I know HAHA

[chilly ocean]

blocked

[next obsidian]

:(

He actually did block me I can’t react to his messages

[sturdy marsh]

bout the same as you ig

[next obsidian]

😂😂😂😂

[sturdy marsh]

started hartshorne last winter

[next obsidian]

[sturdy marsh]

Probably will redo hartshorne next summer

turns out I didnt get a lot of it properly lmao

[next obsidian]

[sturdy marsh]

it looks like I wont be doing any AG for the rest of the year after this quarter

😦

[next obsidian]

:(

time for me to catchup so you don’t smoke me when we apply to grad schools

[sturdy marsh]

There's plenty of people out there to smoke all of us

[next obsidian]

Don’t remind me

[sturdy marsh]

some dudes have been doing AG starting freshman year lmao

[next obsidian]

[chilly ocean]

what year are u guys

[sturdy marsh]

3

[chilly ocean]

undergrad?

[sturdy marsh]

yesh

[golden pasture]

niceee

which uni has ag i need to know

/s but recommendation would be nice

[uncut girder]

Which uni has NT I need to know

[sturdy marsh]

most of them?

[uncut girder]

NO

[next obsidian]

Yeah haha

[uncut girder]

@sturdy marsh that's a fkin lie

[chilly ocean]

why does everyone at 3rd year knows like 10 times more shit than I do

[uncut girder]

Many schools do not have Nt

[sturdy marsh]

what do you mean by NT

[uncut girder]

Number theory

[next obsidian]

Where there’s AG there’s NT

[sturdy marsh]

im pretty sure almost every place has some nt

[uncut girder]

@next obsidian false

[next obsidian]

I think he meant what flavor

[sturdy marsh]

yes

[next obsidian]

Analytic, algebraic

[uncut girder]

"Some" as in 1 faculty remember

[next obsidian]

Arithmetic

[uncut girder]

I'd like to go somewhere that has all flavors of NT so I can explore

[next obsidian]

Considering analytic NT, that’s kind of a dan moment

[sturdy marsh]

I thought I wanted to do analytic NT when I was starting out lmao

[uncut girder]

@next obsidian you're wrong

[next obsidian]

Dan

[uncut girder]

Analytic number theory is cool

[sturdy marsh]

it is

but me know 0 analysis

[uncut girder]

Combinatorial number theory is cool

Algebraic number theory is cool

Arithemtic geometry is cool

I cant choose

[sturdy marsh]

algebraic number theory is dope

I thought it was boring af

until I took a class on it

but anyway, NT is super common, a lot of places should have some people working in NT

it's probably the most common out of NT, AG, AT

but back to the trick, if A is a retract of h_M, then we have $A \rightarrow h_M \rightarrow A$ such that the composition is identity

which gives us an idempotent $h_M \rightarrow A \rightarrow h_M$

which corresponds by Yoneda to a map $M \rightarrow M$

and the equalizer of that map and the identity gives us the representing object for A

this was a part of a proof that mumford left to the reader (in a weirder setting )

a couple of friends and I spent ~6 hrs trying to figure it out and gave up

fml

found the answer on mathoverflow

[next obsidian]

🥴

[sturdy marsh]

seems to be a pretty useful trick

works on any category with equalizers

[fierce perch]

@next obsidian where is that class?

Is it by alper? Is it the one that’s supposed to be online?

[latent anvil]

Chmonkey did this lmao

started summer quarter of his freshman year

[sturdy marsh]

rip

ive got a bad feeling about grad school admissions

doing more AG might not get me into a grad school

but at least I will understand more memes

[bronze trench]

only legit reason to study math

[sturdy marsh]

yes that and to get a girlfriend/SO

heard that girls go GAGA for AG

pun intended

[bronze trench]

well my gf kinda flinches at every math related thing I say so I don't really know about that... 😂

[sturdy marsh]

that must be the best pun ive made in a while lmao

[round oxide]

Is it okay if i ask what major yall doing?

Just curious

[sturdy marsh]

math

[bronze trench]

I'm in a master's in (pure) math after doing a double major in math and physics

[round oxide]

You like maths?

[sturdy marsh]

nope, in it for the money /s

[bronze trench]

I like maths yeah, but I like a lot of things. Maths is more like an obsession really ig

[round oxide]

Idk how someone manages a double major in maths and phy

Doesnt phy defy laws of maths occasionally

[bronze trench]

well I started just with physics but ended up liking math more so switched to a double major at some point

and I get what you mean in that physics is not that rigorous, but it's usually not wrong either 😂

and it's a different thing, like you can like both football and volleyball and someone asks "how can you play both? Don't volleyball players use their hands?"

It's a non issue as they are different things 🙂

[round oxide]

I think its great

I dont think i could do it

I already hate physics

And abstract algebra

And discrete maths

And electronics

[sturdy marsh]

when I was in high school I also wanted to do physics; physicists are wayy better at advertising their field

[bronze trench]

I mean kinda

[sturdy marsh]

I thought math was boring af lmao

[bronze trench]

in HS I kinda never knew what math or physics actually were

[sturdy marsh]

yup

[round oxide]

Ohh in mine I opted for maths physics and advanced chemistry in HS

[bronze trench]

so physics also seemed a bit boring tbh, just applying formulas and stuff... I just had a good teacher who motivated me 🙂

[round oxide]

With c++

[bronze trench]

then I got real maths and physics subjects

and liked physics a lot. But fell in love with maths 😄

[thorn delta]

To show that Z_5[x]/(x^2 + x + 1) is a field, i showed that x^2 + x + 1 is irreducible by plugging in different values from Z_5 to show that it has no roots, and therefore no linear factors. Its not too much work, but is there an easier way to see that x^2 + x + 1 is irreducible?

Also, I reduced elements of Z_5[x] modulo (x^2 + x + 1) and its not really obvious to me what field this quotient is isomorphic to? If f = f_0 + f_1x + .... + f_nx^n is a polynomial, then it reduces to f_0 + f_1x + f_2x^2 and using that x^2 \equiv -x-1 you can get f_0 - f_2 + (f_1 - f_2)x. Still not really obvious to me how a field comes out of this

[dim escarp]

Putting in 01234 is easiest to calc irred I think

Also, you calculate the degree of the field extension

• which you already got

[thorn delta]

Also, you calculate the degree of the field extension
hm, what does that have to do with anything?

[dim escarp]

You know, there are unique finite field of given size up to isomorphism

Size gives you all information what the field is

[thorn delta]

ig naively, i'd say Z_5[x]/(x^2 + x + 1) is the field of order 5^3.

[dim escarp]

?

Not 5^2?

[thorn delta]

Hm, yea okay 5^2

[light shoal]

Could someone help me with dilation

I’m pretty, it’s for homework

Could I send you a picture of my problem and you just explain how I get the answer

don’t give me the answer

It’s for 10th grade

[thorn delta]

probably wrong channel

#prealg-and-algebra or #precalculus

[light shoal]

What do you mean, it’s for geometry

Geometry

[dim escarp]

It's still

[thorn delta]

unless its algebraic geometry, then #geometry-and-trigonometry

[dim escarp]

Oh right, #geometry-and-trigonometry

This place is for university classes

[light shoal]

Ohh

Sorry

[chilly ocean]

[dim escarp]

I mean real geometry is much advanced so it's better to mistakenly be here

It's not

Hm

[next obsidian]

Yes this is correct

[fair shard]

I haven't learned much ring theory yet but I'm studying it very soon, is there a classification of prime polynomials?

[next obsidian]

Nooooo way

If you’re over specific rings maybe but

In general absolutely not

I wish there were life would be so much easier

[fair shard]

ok let's say like integer coefficients

is that a good thing

then is there?

[next obsidian]

I don’t think so

[fair shard]

actually the more I think of it the more messed up this seems to be

like I'm confusing myself

if I suck at group theory will I also suck at the rest of algebra

[next obsidian]

¯_(ツ)_/¯

I mean I feel other algebra has a different feel but

sucking at X is never a good sign you’ll do well at things related to X

But it’s not a sure fire sign

[fair shard]

I think just cause I'm new to algebra

it's a lot for ke

me

there are so many patterns

like

it's lovely

but overwhelming at the moment

hard to get intuition and insight as to why the structures of some things are so perfect

like it seems like you can define a group in any way "let S be the set of all a in G such that a is in love with x, then S is a subgroup of G" stuff like thay

[dim escarp]

Wow, is group theory lovely?

Interesting

[fair shard]

if H is a normal subgroup of G and G/H is generated by a certain set of cosets, let's just say Ha,Hb,Hc does that mean G is generated by a,b,and c?

I don't feel like that should be the case

[sturdy marsh]

it isnt

[fair shard]

but I saw it as reasoning in one proof

[sturdy marsh]

take H = G

[fair shard]

yeah ok

does it work for G/H isomorphic to Z/2Z?

oh

ok fine here is the specific example

[dim escarp]

Isn't G/H generated by a, b and c then tho

[sturdy marsh]

yes

[fair shard]

OHHHH omg I'm dumb

I just realized

[dim escarp]

Eh?

[fair shard]

G isn't generated by a, b,c

it is generated by H,a,b,and c

easily

[sturdy marsh]

yes

[fair shard]

and THAT is what the proof explained

[dim escarp]

Good that you figured it out

I still struggle figuring out what proofs mean

[sturdy marsh]

there's a lot of proofs of that kind lol

you understand the proof locally

but have no clue on what it's doing

[dim escarp]

Hahaha true

But what I mean was that, sometimes I don't get why parts of the proof holds

and realize that my prof later come back and say "oh ok this was wrong but you can show this this way"

[chilly ocean]

Anyone know a simple video / site that explain Quaternion for a dingus like me?

[dim escarp]

Do you mean Q8 group?

[marsh fractal]

So a field of fractions essentially just allows for division?

[thorn delta]

its a generalization of the construction of Q from Z

[marsh fractal]

ok cool and then multiplication by the inverse of elements within them create the field that the ring is embedded in?

like Z to Q

ok awesome thanks

that makes the sense

ok yes i have heard much of this anticipated localization on here

[next obsidian]

[maiden ocean]

have you seen the first isomorphism theorem

[cinder bone]

yes

[maiden ocean]

so we have that

if f is a homomorphism

Z x Z/ker(f) = Im f

correct?

[cinder bone]

yep

[maiden ocean]

so there are 3 things you need to verify

1. that f is a homomorphism
2. what is the image
3. what is the kernel

[cinder bone]

and after that

?

[maiden ocean]

then you are done

hurb

[cinder bone]

not sure about 5 or 7 order

[carmine fossil]

Hint:||if G/Z(G) is cyclic,G is abelian||

[cinder bone]

yeah I get that if the order of Z(G) is 5 or 7

then Z(G) is cyclic

[maiden ocean]

mhm

|Z(G)| has to divide |G| right?

[cinder bone]

yeah so it has to be 1,5,7,35

Can’t be 35

[maiden ocean]

lol that was like a leading question drake hahaha

yeah

[cinder bone]

if G/Z(G) is noncyclic then G is nonabelian right

maybe?

[next obsidian]

I mean yes

If G is abelian Z(G) = G

So that quotient is trivial is cyclic

[cinder bone]

yeah idk then

cus subgroups of 5,7,1 are all cyclic subgroups

and similarly G/Z(G) would also be cyclic if it has an order of 5,7

which means G would abelian

which contradicts

so Z(G) is the identity??

[maiden ocean]

mhm

[cinder bone]

ok word

🙂

[gray light]

just kind of a sanity check on this question

if |G| = p^3, how can |G| = p??

[maiden ocean]

uh

yeah i

i think it means |Z(G)| = p

[gray light]

ahh ok that makes more sense

[maiden ocean]

also |G| = p would imply G cyclic and thus abelian so

lol

[gray light]

yea must just be a typo, thanks

[chilly ocean]

follows by looking at G/Z(G)

[chilly ocean]

Hey guys, so the problem is prove H = (a b c d) | ab-bc = 1 is a subgroup of GL (2,R)

This is the solution, I can follow it up until the last like 2 lines.
Ok, H = det (AB^-1), i dont really know where the det B^-1 comes from, but ok sure, det(AB^-1) =1, np.
But how does that prove H is a subgroup of GL(2,R)?

[wind parrot]

It says it at the top of your image, a nonempty subset H is a subgroup if and only if a,b in H implies ab^-1 in H

[gray light]

ive seen that called the "one step subgroup test" if you wanna research further

[paper flint]

Yes, this is the one-step subgroup test.

As an exercise, you can prove why the one-step subgroup test indeed works.

[chilly ocean]

anyone have a good little resource for that?

[paper flint]

Gallian's Contemporary Abstract Algebra, Chapter 3 has both the theorem and proof.

I could look up for something online if you like?

[chilly ocean]

Yeah i know that, thats linear algebra

No i get that, im just wondering how det * det^-1 proves H is a subgroup in GL (2,R)

[chilly ocean]

yes plz 🙂

[paper flint]

Okay.

https://proofwiki.org/wiki/One-Step_Subgroup_Test

[gray light]

This proof seems wrong. Am I assuming η is 1-1 when I say η(g^x) = η(e)? Sorry if my handwriting is bad

[chilly ocean]

@paper flint oh, so that's just like a rule?

[maiden ocean]

i mean it can be proved

its not axiomatic

[paper flint]

Kinda like a theorem

Or a theorem

[gray light]

it follows from the def of subgroup

[paper flint]

Yeppp

[chilly ocean]

oh ok, then that's simple enough. thanks everyone ❤️

[gray light]

@chilly ocean confused, if (gN)^x is the identity, is |gN| not x?

ohhh i see

smallest integer st g*n = e. i dont know for sure that x is smallest

but either way

yea. thanks

yeah, makes total sense from there. i always feel hesitant if i prove more than what is being asked

[next obsidian]

Does anyone have just a hint towards how to prove this? I don’t want a full solution (I could look one up in the back of the book haha), altho maybe just a hint might even give it all away. Anyway

If A is a subset of B with both integral domains such that their fields of fractions are equal, then if B is faithfully flat over A then A = B

So what I’ve done is written out an exact sequence 0 -> A -> B -> C -> 0 with C the cokernel of the inclusion. I want to show C = 0 implying A = B. The first thing that stands out is that C (x)_A B = 0 if and only if C = 0 by faithful flatness, but this doesn’t seem to use the assumption on field of fractions.

I’ve shown that B (x)_A A_(0) is actually the field of fractions of B using the assumption that Frac(B) = Frac(A), from which it follows that C (x)_A A_(0) = 0, but I don’t see how this uses the fact that B is faithfully flat.

So it’s like I have two things which are kind of obvious both of which use one of the two hypotheses I have, but they don’t use the other, and if I try to combine the two by tensoring by both B and A_(0) I don’t get anything useful I think

[fierce perch]

Maybw use that localization are flat

And do stuff locally

[next obsidian]

Like reduce to the local case?

Hmmm... that might work

Oh duh

I was worried about the field of fractions equality not being preserved but the field of fractions of localizations are just the normal field of fractions

And maybe B_m is still faithfully flat over A_m?

I know it’s certainly flat over it

[fierce perch]

You will have faithfull flatness ocer all prime ideals

[next obsidian]

I’m not sure how you show you have the faithful part of that

Perhaps I sort of see it actually?

The idea in my head involves tensoring by the product of all A_p though haha

At once

[fierce perch]

I mean over one prime ideal at a time

Because flatness is preserved by base change

I was thinking something like it might work since the localizations sit inside the fraction field

But i havent seen this before

[next obsidian]

Right, but if I wanted to just reduce to the local case I’d need to show that B_p is faithfully flat over A_p

At least if I interpreted it correctly

Or that B_P is with P actually a prime of B and p = P\cap A

I know that flatness is preserved by localizing at a prime but I don’t think the faithfulness is

Presumably you’d need to show the fraction field business implies this but I’m not really sure how I’d do that

[fierce perch]

Faithfullness should be easy to show by surjective spec

[next obsidian]

Hmmm

I think you’re right

[fierce perch]

Because localizations on prime ideals give you local rings

[next obsidian]

Yeah you’re totally right

Okay cool so we can just reduce to the local case where I think it should be a lot easier

Thanks

[chilly ocean]

Hey guys, a classmate and I are really lost on this. "How do you find the normal subgroup of a dihedral group?"

[thorn delta]

like, all of them, or just a normal subgroup?

[golden pasture]

subgroups of index 2 are normal

hence you know of one

[chilly ocean]

well likfe for example, we're trying to do D4

From what we understand that there's 2 different ways depending on if n is even or odd.

[sturdy marsh]

yup

[thorn delta]

are you asking about D4 or Dn?

[chilly ocean]

What im thinking is that if it's even, the amount of normal subgroups is n+2. Is that right?

@thorn delta well, D4 is just an example. it would be nice to know in general just so that we can do it with any n, since we have an exam today

[thorn delta]

the only way I can think of in general is to compute the conjugacy classes and pick out the unions that are subgroups. The conjugacy classes are different depending on whether n is even or odd

[chilly ocean]

ok, how would you do that?

[thorn delta]

pick elements of Dn and conjugate them with other elements. For example, taking Dn = <r,s : r^n = s^2 = (rs)^2 = 1> you can do
r^i s (s) sr^{-i} = r^{2i} s and
r^i s (r^{2k}s) sr^{-i} = r^{2i-2k} s
so {s, r^2s, r^4s, ....} is a conjugacy class when n is even and {s, rs, r^2s, r^3s, ...} is a conjugacy class when n is odd assuming i haven't messed up somewhere.

when n is even, all that's left to figure out are the conjugacy classes containing elements of the form r^{2k + 1} s and elements of the form r^k. When n is odd, all that's left are the conjugacy classes containing elements of the form r^k.

[chilly ocean]

kk thank youuuu

[thorn delta]

np

[dim escarp]

O no I forgot all conjugacy class stuff dih

[sturdy marsh]

Does anyone know how did projective/injective resolutions first come up?

[maiden ocean]

what do you mean exactly

like historically?

[sturdy marsh]

yeah

I'm pretty sure they were around before people cared about derived functors and cohomology

they were also around before cofibrant/fibrant replacement became a big deal ig?

[maiden ocean]

no idea but im guessing cartan and eilenberg had something to do with it

[sturdy marsh]

that's a good guess

yeah they definitely existed before Quillen

Even hilbert cared about free resolutions

so even before cartan-eilenberg

[maiden ocean]

monkaS

yeah im not sure

lets revive eilenberg and ask him

[sturdy marsh]

actually you may be right

hilbert probably only cared about free resolutions to keep track of relations

'resolving' objects probably started off while trying to understand extensions of groups

either that or cones and cocones in topology

[vital quail]

i refer to the latter as "nes"

[next obsidian]

Syzgies

[chrome hinge]

Hello there!

Im struggling with proving that Z[pi] (where p is prime in Z) is not a unique factorization domain

By Z[pi] i mean the set of elements of the form a + ipb with a,b integers

Does anyone know if any prime in Z is prime in Z[pi] ??

[latent anvil]

Can you say -p^2 = (pi)^2 = (-p)p?

I don't think pi and p are associate

[chrome hinge]

Hmm

But is p irreducible?

I just showed that 1+ip is irreducible

[chrome hinge]

They arent indeed

[latent anvil]

hmm, I think p is irreducible but maybe not

I didn't check it

[chrome hinge]

I think so too

But i havent proved it yet

Actually...

If p=ab then |a||b|=p and that happens only if either a or b has norm 1

And then either a or b is a unity

So p should be irreducible... 🤔

The same would apply to ip, and then we're done

You agree? :D

Man ive been hours in this proof

Much appreciated your help

[latent anvil]

Np

the only units are 1 and -1, since |a+ipb|^2 = a^2 + p^2 b^2 >= p^2 > 1 if b ≠ 0

so units are real

and they have to have modulus 1 so they're 1 or -1

Then clearly p and -p aren't ip, so we have two distinct factorizations

[chrome hinge]

Exactly

I just finished the proof

Its just the example you showed is amazing

I hadnt thought of it

[fierce perch]

@sturdy marsh mayer and hurewitz