Commutative algebra - 2017-05-07

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Xam

00:31

Just for the record: If $D$ is condensed and $S$ is a multiplicatively closed subset of $D$ such that $0\not in S$, then $D_S$ is condensed

7 hours later…

Martin Sleziak

07:16

I did not notice this room until now. Let's hope it will last at least as long as the previous commutative algebra rooms.

Eric Stucky

62 days so far :P

9 hours later…

Xam

16:40

@EricStucky would you like to help me with the statement I posted above?

Eric Stucky

17:15

oh!

I thought you were telling it to the room as an interesting fact :P

yeah sure

let me get some paper

@Xam

So, for the record, condensed means that all ideals I and J satisfy $IJ=\{ij\}$, right?

and $D_S$ means the localization, i.e. $\{d/s \in\text{Frac}(D): d\in D, s\in S\}$?

Xam

17:34

@EricStucky haha xd

you're right

Eric Stucky

okay :)

so obviously the goal is to turn prime ideals $I'$ and $J'$ of $D_S$ into ideals $I$ and $J$ of $D$.

Xam

so this is my idea, take two arbitrary ideals $I,J$ in $D_S$, then there are $I', J'$ in $D$ such that $I'^{e}=I$ and $J'^{e}=J$

yes

Eric Stucky

okay lol I'll use your prime notation >.<

what does ^e mean?

Xam

if $\alpha\in IJ$, then we can write $\alpha=xt$ such that $x\in IJ$

$I'^{e}$ means the extension of $I$ using the canonical map $D\rightarrow D_S$

Eric Stucky

okie

Xam

17:37

r\mapsto r/1

Let me write my idea and then you can tell me if is right, ok?

Eric Stucky

okie

Xam

As I said, for $I,J$ in $D_S$ there are $I',J'$ in $D$ such that that $I'^{e}=I$ and $J'^{e}=J$, so if $\alpha\in IJ$, then $\alpha\in I'^{e}J'^{e}$

So we can write $\alpha=x/s$ for some $x\in IJ$

But since $D$ is condensed there are $a\in I$, $b\in J$ such that x=ab

so $\alpha=x/s=ab/s=(a/s)(b/1)$ and $a/s\in I$, $b/1\in J$

Therefore $alpha$ can be written as the product of two elements, one of $I$ and the other of $J$

Since both are arbitrary, $D_S$ is condensed

I'm using one property about extensions of ideals, namely $I^{e}J^{e}=(IJ)^{e}$

Eric Stucky

Right, that's at line 1 to 2, yes?

Xam

Yes, $\alpha\in I'^{e}J'^{e}=(I'J')^{e}$

So for at least one representation of $\alpha$ we can take the numerator to be in $I'J'$

Eric Stucky

right

Xam

17:47

And then we use that $D$ is condensed

Eric Stucky

Yeah it looks pretty airtight :)

Xam

So from there follows that $\alpha=(a/s)(b/1)$ with $a/s\in I$, $b/1\in J$$

Thank you for reviewing it.

Actually, there is a more general result that says that an overring of a condensed domain is condensed.

Eric Stucky

Right, I mean you basically have the proof here, right?

just replace $^e$ with an injective homomorphism

you may need to include an extra line about exactly why $x$ exists in that context.

Xam

Yeah, something like that

But I only proved the case for localizations because that's enough to prove a beautiful characterization of Bezout domains among GCD-domains

D is a Bezout domain iff D is a condensed GCD domain

Eric Stucky

hmm, interesting :)

Xam

17:56

I don't know but maybe you found this paper: books.google.com.pe/…

Eric Stucky

no I hadn't

but it does appear to be about that >.<

#usefultitle

Xam

In that paper are all these results about condensed domains.

I found the link of the Canadian mathematical bulletin like a moth ago, but now I can't :(

I only get the link of google books u.u

Eric Stucky

heh

Xam

Oh, actually that's not the right paper

That paper is the continuation.

Eric Stucky

(showering, brb)

Xam

18:00

But I found the paper!!!

Finally: cms.math.ca/openaccess/cmb/v26/cmb1983v26.0106-0114.pdf

And the continuation: cms.math.ca/openaccess/cmb/v28/cmb1985v28.0098-0102.pdf

Eric Stucky

18:17

wow I was really looking forward to another #usefultitle but no that first one is worthless

Xam

That's why I couldn't find the paper

The author didn't use the word condensed in the title xd

Eric Stucky

>.<

Xam

But that's the paper where condensed domains are defined and studied :)

Eric Stucky

sounds good

(also, there's someone in math chat rn named Simple and apparently != SBA...)

Xam

Do you wanna know how I got to that paper?

I might seen that user

Eric Stucky

18:20

well it looks like it's open access journal?

Xam

I mean the story about how did I discover condensed domains

Eric Stucky

oicic

sure

Xam

You know those kind of domains only appear in papers and not textbooks

Eric Stucky

it seems so

Xam

Well, I found them when I read this answer: math.stackexchange.com/a/21447/133781

Eric Stucky

18:21

ah yes, our good friend Bill :P

Xam

I was looking for product of ideals and characterizations of bezout domains

And then I got to that answer by the great Bill Dubuque

I have to say that he has included or commented about lot of papers in his nice answers

And the only upvote that he has in that answer is mine xD

Eric Stucky

haha yeah

Xam

Well that's the whole story xd

Eric Stucky

Xam

Actually is only one part of the whoole story hehe

Would you like to listen about it?

Or read it? xd

Eric Stucky

18:28

sure

Xam

Well, it turns out that I'm studying ring theory by myself and I also writing notes about it

So I was studying about factorization in integral domains

I used the notes of Pete L. Clark and he talks about Bezout domains, GCD-domains, etc

But he doesn't give characterizations of Bezout domains among GCD-domains, for example

He just included this result: D is a valuation ring iff D is a local Bezout domain

So I decided that would be nice to look for some characterizations of Bezout domains

And there I was searching through diferente papers and also searching here on MSE

Until I found that paper about condensed domains

And I found the characterization of Bezout domains among GCD-domains that I wrote previously

But it turned out that the above result uses another interesting characterization: Bezout domain iff condensed Prufer domain

So I had to study a little about Prufer domains and well my notes have extended more than I though xd

And that's it :)

I hope you're not bored xd

Eric Stucky

nah, I enjoy listening to people's journeys around the masthscape

or reading, if they're not terribly long :P

Xam

Haha oks :)

Eric Stucky

are the notes hand-written? do you have a binder for them, maybe?

Xam

Yes, there are hand-written at the moment, but I plan to write them using latex

Probably around august

Rn I don't have enough time :/

Eric Stucky

18:46

idk I have found a lot of value in keeping my notes on paper. Maybe someday I'll scan them so they don't burn down. But having the physical copies, I occasionally run into them during random life activities

Xam

Yeah sure, I mean I won't destroy my physical notes, but I also would like to have them online

Specially bc I'm writing my notes in spanish, which is my native language.

Eric Stucky

Right, I guess what I'm saying is that I wouldn't go through all the effort of actually TeXing them

I mean, okay that's not true for me at the moment, because they make great blog content... but after the blog stops I don't think I'll do that anymore :P

wait really?

damn mate wouldn't have guessed

Xam

Mmm but, how would you share your notes? Or you don't want that?

Eric Stucky

dropbox?

well, SAGE, actually

Xam

Oh, that's sounds interesting

Well, I have to go. Things to do at home u.u xd

Eric Stucky

18:52