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12:58 AM
I contradicted twin primes, bro
They can't find anything wrong with it. It's a miracle!
Yes, the 5 people who have viewed it since you posted it 10 minutes ago haven't found anything wrong with it.
It's in the style of Furstenberg
as a topology is sort of formed, but we haven't used that fact
@Ultradark you there, mon?
1:16 AM
I’m here
Check out my post
Q: The set $X$ of all $A \subset \Bbb{Z}$ such that $A \cap (A + 2)$ is finite contains the primes?

Shine On You Crazy DiamondLet $X = \{ A \subset \Bbb{Z} : A \cap (A + 2) = $ a finite set $\}$ where $A + 2 = \{ a + 2 : a \in A\}$. $X$ is closed under taking arbitrary intersections. Proof. Let $A_i \in X, i \in I$, where $I$ is an arbitrary index set. Then $(\bigcap_i A_i ) \cap (\bigcap_i A_i + 2) = (\bigcap_i A_i)...

it uses ideals
as we were talking about
$(n\Bbb{Z} + 2 ) \cap n \Bbb{Z} = \varnothing$ when $n \gt 2$.
Hi everyone.
I have a soft question.
The direct product operation of finite groups up to isomorphism forms a commutative monoid.
Do you think it's possible to take this as multiplicative and form a ring?
In other words, is there an additive operation on finite groups?
1:33 AM
@Semiclassical Read the Call of Cthulhu last night, finally
It's very hard to get spooked by it because the plot idea is literally everywhere nowadays
I suggest the temple and the rats in the walls by Lovecraft
But I can imagine how original it was back then
@Alessandro I'll try it out
They're not part of the old gods cycle
I used to really like Lovecraft, I have a nice hardcover brick volume with his complete works, but now I haven't read anything by him in years
1:37 AM
bit.ly/33UwCAk a very potent, meaningful, obviously not racist poem by Lovecraft
yeah we're aware
lol nah I just wanted to post that because it's so silly
I was going to say that his distinct revulsion to non-European cultures in Call of Cthulhu itself doesn't help that much in reading it
It's quite known that he had some racist opinions, but that doesn't mean you can't appreciate his works without xenophobic passages
(I know, I was just being a dick)
1:42 AM
Similarly studying Teichmüller theory doesn't make you a nazi supporter
or basically any number theory lol
I disagree with people who try to justify it as "it was early 1900 america, everyone was racist" though
Idk, why do you disagree with that?
I mean that doesn't justify being racist
@AlessandroCodenotti yeah, totally. but now that i read call of cthulhu, i think the "horror of xenophobia" plays a key part, much like Joseph Conrad's "Heart of Darkness"
1:46 AM
ofc not, but at the same time, it's hard to imagine how one would act in such a situation; can you imagine being the ONLY person you know who doesn't believe in something that 99% of your country believes?
(maybe 99% is an exaggeration, but you know what I mean)
that the mysterious cult of ancient ages is lurking in voodoo cults of Africa or India, of which Europeans know nothing about, etc
its plays a part in building the horror
@Balarka I like that, it shows how big a part ignorance plays in the racist psyche
how are we defining "racism" here?
yeah, but also without the political connotations, it does build up a genuine sense of horror :)
Xenophobia I guess
1:49 AM
@BalarkaSen I see your point, for sure he knew his audience, whether or not he actually believed in those opinions
idk I guess so, just "hatred of the other"
@Alessandro Yeah
It's as John Carpenter would say, "right-wing horror" (horror of the "others") in its most honest form
in any case, Call of Cthlulhu is cool af
so much shorter than I expected it to be somehow lol
1:50 AM
yeah I like it
but found hard to REALLY like it because I know the plot from so many horror literature, movies and videogames
Uncle Tom's Cabin; or, Life Among the Lowly, is an anti-slavery novel by American author Harriet Beecher Stowe. Published in 1852, the novel had a profound effect on attitudes toward African Americans and slavery in the U.S. and is said to have "helped lay the groundwork for the Civil War".Stowe, a Connecticut-born teacher at the Hartford Female Seminary and an active abolitionist, featured the character of Uncle Tom, a long-suffering black slave around whom the stories of other characters revolve. The sentimental novel depicts the reality of slavery while also asserting that Christian love can...
a must read^
the whole idea of cosmic horror, of revelation of the sort that would make reality fall apart at the seams, feels very akin to Borges somehow
to be specific I am thinking of Tlon, Uqbar, Orbis Tertius
apparently Borges thought Lovecraft was a parody of Poe lol
to which, well, which horror fiction author isn't?
yeah, Poe was a giant
I remember The Pit and The Pendulum being the first serious fiction in English I ever read
blew my mind lmao
2:03 AM
@BalarkaSen hi
how's galois theory
Good, haven't thought about anything new yet though
Will tell you if something occurs
2:11 AM
Shadow over Innsmouth and The Music of Erich Zann are also great Lovecraft classics, the latter at under 20 pages
short is good, but not too short
average Borgest fictions are like 5 pages so i disagree!
@Thorgott I'll check these out
For sale: Baby shoes, never used.
shortest story^
on a tangent, I read some Sam Beckett short stories a while back
they're pretty damn good
I think Imagination Dead Imagine has to be the shortest novel ever written or something
not counting Hemingway's baby shoes, that's a prank
2:26 AM
i love how all these three pranksters, Hemingway, Joyce and Beckett, knew each other very well
best crossover i know of
Have you read Uncle Tom's Cabin?
@ÍgjøgnumMeg Thanks! I understood.
there's a poetry collection by James Joyce called "chamber music"
he claimed the name is derived from the sound of urinating on a chamber pot
@skullpatrol I haven't actually
2:43 AM
On the subject of Lovecraft, try "At the Mountains of Madness".
Yeah I have that in my list as well
I'll try it out
2 hours later…
4:19 AM
@LeakyNun I was about to ask if I have two field extensions $K, L$ of $F$, and an $F$-automorphism $L \to L$, does this naturally extend to an $F$-automorphism $KL \to KL$? But then I figured $K = \Bbb Q(2^{1/4})$ and $L = \Bbb Q(2^{1/2})$ is a counterexample.
Non-normality is annoying
@Thorgott I've read surprisingly little actual Lovecraft but Music of Erich Zann is great
The automorphism $\sigma : \Bbb Q(\sqrt{2}) \to \Bbb Q(\sqrt{2})$ given by $\sigma(\sqrt{2}) = -\sqrt{2}$ does not extend, because the only nontrivial $\Bbb Q$-automorphism of $\Bbb Q(2^{1/4})$ sends $2^{1/4}$ to $-2^{1/4}$, which sends $\sqrt{2}$ to itself.
I wonder what composite of fields correspond to in the covering space analogy. If $Y \stackrel{p}{\to} X \stackrel{q}{\leftarrow} Z$ are two covering spaces, the composite would perhaps be the smallest covering space which covers both $Y$ and $Z$?
So exactly the pullback diagram $$\require{AMScd}\begin{CD}E @>>> Y \\ @VVV @V{p}VV \\ Z @>{q}>> X\end{CD}$$
It's $Y \times_X Z$.
Nah, that actually corresponds to $K \otimes_F L$ (because $\text{Spec} K \times_{\text{Spec} F} \text{Spec} L \cong \text{Spec}(K \otimes_F L)$). I know that's a field iff $[KL, F] = [K, F][L, F]$, though, when $K/F$ and $L/F$ are both finite
In that case $K \otimes_F L \cong KL$, I think.
4:48 AM
Yeah, that should be right. $[KL, F] = [K, F][L, F]$ implies $K \cap L = \{0\}$, because otherwise choose an $F$-basis $\{\alpha_1, \cdots, \alpha_k\}$ for $K \cap L$, extend that to a basis $\{\alpha_1, \cdots, \alpha_k, \gamma_1, \cdots, \gamma_{n-k}\}$ of $K$ and a basis $\{\alpha_1, \cdots, \alpha_k, \beta_1, \cdots, \beta_{m-k}\}$ of $L$.
Then you can make an $F$-generating set of $KL$ with less than $nm = [K, F][L, F]$ elements by taking mutual products of $\alpha_i, \beta_j, \gamma_k$.
With $mn - k^2$ elements.
Then choose some primitive element $\alpha \in K$, so $K \cong F(\alpha) \cong F[x]/(f(x))$. Then $K \otimes_F L \cong L[x]/(f(x)) \cong L(\alpha)$.
But if $\{\beta_1, \cdots, \beta_m\}$ is an $F$-basis for $L$, then as $\{1, \alpha, \cdots, \alpha^n\}$ is an $F$-basis for $K$, jointly they form an $F$-basis for $KL$, which shows $KL \cong F \langle \beta_1, \cdots, \beta_m, 1, \alpha, \cdots, \alpha^n \rangle = L(\alpha)$.
I guess I used the primitive element theorem, so separability, somewhere. Unsure how to argue cleanly without that.
OK, same stuff. If $K = F(\alpha_1, \cdots, \alpha_n)$, then $K \cong F[x_1, \cdots, x_n]/(f_1(x_1), \cdots, f_n(x_n))$ where $f_i$ is the minimal polynomial of $\alpha_i$.
Then $K \otimes_F L = L[x_1, \cdots, x_n]/(f_1(x_1), \cdots, f_n(x_n))$, which is still a field because of given hypothesis, $K \otimes_F L \cong L(\alpha_1, \cdots, \alpha_n)$ which is in turn $KL$.
For completion: $[KL, F] = [K, L][L, F]$ implies $K \otimes_F L$ is a field because an irreducible polynomial over $F$ which has a root in $K$ cannot be reducible in $L$, otherwise $[KL, L] \leq [K, F]$. Then you run the same argument as before.
I think disconnected covers correspond to product of fields. For example, if you take the double cover $S^2 \to \Bbb{RP}^2$ and do fibered product of it with itself, $S^2 \to \Bbb{RP}^2 \leftarrow S^2$, then you get the disconnected $2$-sheeted cover $S^2 \times \{0, 1\} \to S^2$.
Similarly, $\Bbb C \otimes_{\Bbb R} \Bbb C \cong \Bbb C \times \Bbb C$.
Can we classify finite dimensional $F$-algebras? They are not always product of field extensions of $F$ because of weird things like $F[x]/(x^n)$.
If $A$ is a finite dimensional $F$-algebra then $A$ has finitely many prime ideals, because $A \otimes_F F$ is a finite $F$-vector space, and since Spec of that is fiber of the map $\text{Spec} A \to \text{Spec} F$ over $(0)$, this implies this is a finite-to-one map. So $A$ has finitely many prime ideals, and $A/\text{nil}(A)$ is a product of finite extensions of $F$ by CRT.
That's all I can say I suppose.
5:31 AM
I know less than I should about compositum of fields
You always have the consolation of knowing more than me in anything about algebra :)
If I take $K = \Bbb F_p(t^{1/p})$ over $\Bbb F_p(t)$, and do $K \otimes_{\Bbb F_p(t)} K$, what do I get? $\Bbb F_p(t^{1/p})[x]/(x^p - t) = \Bbb F_p(t^{1/p})[x]/((x - t^{1/p})^p)$.
A local algebra like $F[x]/(x^n)$
This is the sort of stuff that never occurs in covering spaces; it's like covering $S^2$ by $S^2 \times (-\varepsilon, \varepsilon)$.
A cover with a multiplicity on the sheet
If we stick to characteristic 0, every finite extension $K/F$ is separable, so $K \otimes_F L = L[x]/(f(x))$ where $f$ is the minimal defining polynomial for $F$, would never have multiplicities in the factor, so we can safely say $K \otimes_F L$ is a product of field extensions of $L$.
Separable field extensions are like connected covering spaces. Field extensions are like connected covering spaces with "multiplicities on sheets"
Product of field extensions are like disconnected covering spaces
So in general a finite dimensional $F$-algebra is like a non-necessesarily-connected covering space with multiplicites on the sheets.
5:49 AM
@BalarkaSen the term is "etale algebra" or something like that
Yeah, it makes a lot of sense. I vaguely know the term from Szamuely, which I skimmed through once
@BalarkaSen isn't it a big example in Galois category
it's like etale morphism <-> covering space
or something like that
Something like that, but I think you know this better than I do
I am a beginner struggling to understand field extensions hah
Let me try to get this off my chest anyway, at the cost of sounding pretentious:
If $K/L$ is a field extension, we think of the map $f : \text{Spec} L \to \text{Spec} K$. Choose a geometric basepoint $x : \text{Spec} \overline{K} \to \text{Spec} K$ and a neighborhood of $x$, which would be an etale neighborhood, a map of schemes $U : X \to \text{Spec} K$ along with a geometric point $x_U : \text{Spec} \overline{K} \to X$, such that the triangle involving $x$ and $x_U$ commutes. Say $X = \text{Spec} A$, so $A$ is a $K$-algebra.
The lift of this neighborhood is the fibered product $U \times_{\text{Spec} K} \text{Spec} L \cong \text{Spec} A \otimes_K L$.
So it would be really nice if $A \otimes_K L$ broke off as a product of $L$-algebras. Then we can say $f$ satisfies the evenly covered property, just like classical covering space theory
But if $A$ is an etale algebra, like you say, it's a product of finite separable extensions of $K$, $A \cong K_1 \times \cdots \times K_n$. Then $A \otimes_K L$ is also a product of finite separable extensions of $L$, by doing the tensor product termwise, and using the fact about separability and tensor product I said earlier.
It's like if you diagrammatize everything happening in classical covering space theory it translates word by word to Galois theory if you know what are the right words
Very fascinating, but about time I get back to concrete math instead of philosophy :P
5:58 AM
@BalarkaSen so you get things like
Jun 17 '18 at 19:45, by MatheinBoulomenos
$K/F$ is separable iff $K \otimes_F \overline{F}$ is reduced
this is easy to see if $K/F$ is simple
i.e. $K = F(\alpha)$
It's easy to see that if $K/F$ is not separable, then $K \otimes_F \overline{F}$ is not reduced, right? We can take an element $\alpha \in K$ whose minimal polynomial does not split into distinct linear factors over $\overline{F}$. Then $F(\alpha)$ is a simple subextension which is not separable, so $F(\alpha) \otimes_F \overline{F}$ gives a subalgebra which is not reduced.
oh nice!
K/F is a direct limit of simple extensions... lol
I didn't see this
@LeakyNun Is it? $K/F$ count be uncountable dimensional, not?
It's not like it's algebraic
6:12 AM
$K = \bigcup_{a \in K} F(a)$
but I just realized that this isn't directed
Yes, I got confused. Thanks.
(there's a generalization of primitive element theorem that says if $b$ is separable then $F(a,b)$ is simple; unfortunately no such assumption exists in our situation)
Oh cool. I'll think about that when I get to primitive element theorem.
6:14 AM
so it's a union of simple extensions nevertheless, if that has any categorical meaning
If $K/F$ is finite, separable does imply $K \otimes_F \overline{F}$ is reduced. I'm thinking why it's true for infinite.
because... :P
Your union idea works?
let's say K/F is algebraic
(is $K(x)/K$ separable?)
Algebraic is fine
It's a direct limit of finite algebraics, as you pointed out
6:16 AM
indeed it is
Oh, algebraic is in-built in separable, right?
My bad
This is what I get when I talk before reading. Back to Dummit-Foote.
you can generalize separability using that very equivalence
$K(x) \otimes_K \overline{K} = \overline{K}(x)$ is reduced so $K(x)/K$ is "separable"
@LeakyNun I can't help but point out that if $K/F$ is finite (always a safe assumption to have when comparing with covering spaces, I think), $K \otimes_F \overline{F} \cong \prod \overline{F}$ where there are $[K : F]$ many factors. So a geometric point $\text{Spec} \overline{F} \to \text{Spec} F$ lifts to $[K : F]$ many geometric points $\text{Spec} \overline{F} \to \text{Spec} K$
The "constant fiber property" of covering maps, along with establishing the degree of the field extension as degree of the cover.
$\Bbb F_p(t^{1/p}) \otimes_{\Bbb F_p(t)} \overline{\Bbb F_p(t)} = \overline{\Bbb F_p(t)}[x]/(x^p)$
$K$ is finite separable :P
Because we were thinking about separable stuff before
The geometric fiber is exactly the pullback $\text{Spec}(\overline{F}) \times_{\text{Spec} F } \text{Spec} K = \text{Spec}(K \otimes_F \overline{F})$.
@LeakyNun But yeah your example illustrates that the extension $\Bbb F_p(t^{1/p}) \to \Bbb F_p(t)$ is like an infinitisimal neighborhood of $0$ in $\Bbb C \to \Bbb C$, $z \mapsto z^p$
The geometric fiber contains only one geometric point, but with multiplicity $p$
6:26 AM
what does $\overline{K} \otimes_K \overline{K}$ look like
If $K$ is a perfect field, that's going to be a product of $\overline{K}$'s as well, right?
Direct limit should behave well with tensor product.
But I don't think it's useful to compare infinite extensions with covering spaces; isn't taking stuff for finite etale coverings and then inverse limiting how Grothendieck constructed the etale fundamental group? It's not a great analogy to compare $\overline{K}$ with the universal cover of $K$, I think, albeit meaningful.
But I don't entirely understand and treading on muddy waters
OK I am actually leaving now because I can't resist myself to rant on Grothendieck's Galois theory without understanding classical Galois theory, and that's bad for me
lol ok
7:10 AM
@Balarka this sounds a lot like you are reading Szamuely's book
1 hour later…
8:12 AM
Hey, someone can help me with some discrete math? need to prove some function is onto
8:53 AM
Let $x \in \mathbb{R}^n$ be a fixed vector and $U$ a subspace. How can we show that
$$\left \{y\in \mathbb{R}^n\mid x-y\in U\right \}=x+U$$
@ÍgjøgnumMeg I skimmed it once, but I plan to read it seriously soon.
9:21 AM
@Balarka saaaame I'd very much like to
I have a copy, just haven't gotten round to it
Can the 256 cellular automata by Stephen Wolfram be expressed through addition, subtraction and multiplication?
It is possible for rule 30 and it seems that it is also possible for the rest 255 cellular automatas.
2 hours later…
12:38 PM
@Shootforthemoon Well, you can derive the implicit equation in the neighborhood where the implicit function exists and solve for its derivative. However, the derivative of the implicit function will usually depend on the implicit function itself, which you usually don't have a handy expression for (otherwise, you wouldn't need to apply the IFT), so this information doesn't help you very much.
In one dimension, you could integrate this derivative, but note that you would then have expressed the function as an integral over an expression involving itself, which also isn't handy. Even finding primitives in higher dimensions, which is where you usually apply the IFT, is a much more difficult task, but calculating an implicit function like that would be akin to solving a differential equation, which, in general, can be very hard.
For example take $F(x,y)=x+y+y^5$. For fixed $x$, this increases monotonically in $y$ from $-\infty$ to $\infty$, so for each $x$, there is exactly one $y=g(x)$, such that $F(x,y)=0$. You can check this implicit function $g$ is differentiable using the IFT. Then you have $g'(x)=-\frac{1}{1+5g(x)^4}$. This tells you, for example, that $g$ is monotonically decreasing, which is good to know, but I think it will be difficult to reconstruct $g$ from this.
However, it should be noted that you usually know the value of the implicit function at one point, namely your initial point and if your original function is analytic, then the implicit function will be analytic as well (this is a more difficult theorem), so in that case you can theoretically expand the implicit function into a Taylor series around the initial point, which may or may not help you.
1:36 PM
@Thorgott Thank you very much for the detail and the help given!
Do you want to post this as an answer?
2:20 PM
I've posted a more detailed version of the above as an answer, hope that helps!
Hi all
I'm not very confident with matrix stuff
I have a determinant that is of the form det(I+epsilon * A)
@Thorgott Wow, I am really obliged to you
I'm considering my system at some "meso"-scale meaning epsilon is not yet infinitesimal but rather it is small, so I'd like to use the secnd order expansion in epsilon
Which should be quite useful, my question is in in regard to the applicability based on the assumptions of the matrices and the scalar
In some answers it is referred that it is important what the field properties are of the variables considered
Field with characteristic 0 for example in another answer:
A: Series expansion of the determinant for a matrix near the identity

SpencerNote that the determinant of a matrix is just a polynomial in the components of the matrix. This means that the series in question is finite. This also means that the functions $f_1,f_2,\dots,f_N$ are polynomials. We start by computing the term which is first order in $\epsilon$. Let $A_1, A...

So my question is in regard to the applicability of the idea that the second order expansion in epsilon is (tr^2 A - tr (A^2) )/ 2
As in, for a general n by n matrix
No field charactersitic required in the consideration
2:36 PM
Well, in characteristic you would be dividing by $0$, so that certainly doesn't work
A class in programming is a blueprint for making objects, which are like custom types that are made in terms of other predefined types along with functions
A universe which obeys the simulation hypothesis will mean all types are composed of some fixed collection of predefined types
Such is sufficient to enable emergence as new behaviour results from combining arrangement of these types
However, there may be types that are inherently unspecifiable in terms of code. There existence of at least one of these will disprove the simulation hypothesis
2:53 PM
can anyone prove this?
Q: $ \dfrac{d}{dz} \left \{ (1-z^2)P'(z) \right\} + \left\{ \beta - \dfrac{m^2}{1-z^2} \right\}p(z) = 0 $ Conversion of the DE into other forms.

Abhas Kumar SinhaI guess it's a bit related to physics, but the doubt is in the mathematical part. I was just watching some stuff on Quantum Mechanics for the solution of the Hydrogen atom. The lecture ends at.. $$ \dfrac{d}{dz} \left \{ (1-z^2)P'(z) \right\} + \left\{ \beta - \dfrac{m^2}{1-z^2} \right\}p(z) =...

In this answer : math.stackexchange.com/a/2212554/377528, it says to compute the cohomology of $0 \to \Bbb Z_m \to \Bbb Z_m \to 0$. What does this explicitly mean ?
Nevermind I got it
3:29 PM
Anyone know of a good book on type theory + implementation details (code)
Like making a basic proof assistant
3:43 PM
@Ultradark let's code some stuff, mon
4:07 PM
@Thorgott I definitely liked The Music of Erich Zann, although I felt similar to how Balarka described in regard to not finding it scary (and the reason why). I think the 'can't find the place ever again' theme has been copied one million times.
4:36 PM
@TedE Hi!
Hey there!
How have you been?
@ShineOnYouCrazyDiamond Is your name inspired by Rihanna's Diamonds?
I'm well, just recapping some math I did awhile ago right now.
I wish I had typed up notes on some of this old stuff when I knew it well.
What stuff in particular are you recapping?
4:39 PM
Cech cohomology atm.
You're Ceching your Cech cohomology knowledge for potential flaws?
What about you? What are you working on?
Bad Balarka :P.
There are more than potential flaws :).
I have to check a few things that might be obvious (or not, or they might be false even, but they were claimed in talks):
1) intersections of (probably open) affine subschemes of a separated scheme are also affine
2) Cech agrees with standard (zariski) sheaf cohomology when quasicompact and separated (which I think relied on the above being true)
I am studying for exams.
I guess you are an algebraic geometer of a sort?
I never really ended up understanding what separated actually means, though I understand it's trying to enforce some sort of Hausdorffness
Well, I wouldn't put myself into any camp yet really.
4:46 PM
(As in, the classical example of the line with two origins scheme not being separated)
I haven't captured the true meaning of it either yet.
We could try to work that out now if you want.
I guess $\Bbb C^2$ with two origins (two copies of $X = \text{Spec} \Bbb C[x, y]$ glued to each other by the identity map along the subscheme $X \setminus \{0\}$) is an example where intersection of two affine subschemes is not affine
Isn't Hausdorff = separated?
Or do you mean not in a strictly topological sense?
This is for schemes, not topological spaces.
4:49 PM
@BalarkaSen Do you mean glued along identity map on $X\backslash\{(0)\}$ or on $X\backslash\{(x,y)\}$?
(<-- knows nothing about schemes) maybe it is equivalent in the Zariski topology?
By 0 I of course mean the ideal (x, y)
@anakhro Zariski is totally nonHausdorff it implies nothing about the underlying topology really
@anakhro Schemes have more than just a topology, they have a sheaf of rings whose stalks are local rings (really more has to be said to make this precise, since really this is just a 'local' scheme)
4:50 PM
The forgetful functor Sch -> Top is very badly behaved
That's right, the Zariski-open sets are all very big, right?
I forgot about that.
@BalarkaSen What are your two affines here?
@TedE You don't need open in 1 iirc
@TedE The two copies of $\mathrm{Spec}\Bbb C[x,y]$
4:52 PM
@TedE So $Z = X \times_{X \setminus 0} X$. The two affines are $X \hookrightarrow Z$
They intersect at $X \setminus 0$, which is not isomorphic to an affine scheme
Oh yes, of course, funnily enough I proved that ages ago using cech cohomology lol
If that was affine, then by leray vanishing theorem it would have trivial higher cech cohomology, but it doesn't
That looks like massive overkill
It's an exercise in hartshorne
You can also do it by considering $X\backslash\{(x,y)\}\to Spec(\Gamma(X\backslash\{(x,y)\}))$
I guess
I shouldn't talk about algebraic geometry, I know nothing about it
I really doubt you need this. You can argue $\Bbb C^2 \setminus 0$ is not affine by noting that the inclusion map $\Bbb C^2 \setminus 0 \hookrightarrow \Bbb C^2$ is an isomorphism on the structure sheaf by the Hartogs extension theorem
4:57 PM
@TedE What isn't though
@BalarkaSen That doesn't tell you that it's not isomorphic to an affine scheme though
I meant serres vanishing theorem
That can be argued in a completely elementary way by computing the regular functions on $\Bbb C^2\setminus 0$ by hand
@Alessandro Since it's an isomorphism on global sections, if it was an affine scheme, then the inclusion had to be an isomorphism
That's impossible
@BalarkaSen That was my 'you can also do this by considering' argument
Similar argument should push through for $\text{Spec} \Bbb C[x, y]$ minus $(x, y)$
4:59 PM
Hmm I think I don't understand your argument
Why doesn't it apply to $\Bbb C\setminus 0\hookrightarrow\Bbb C$?
Because Hartogs doesn't apply
There are regular functions on C \ 0 which doesn't extend to C
Oh ok I see now
@TedE Ah yeah I missed it when I was frowning at your Cech cohomology overkill
Haha, I just had that overkill in mind because I am recapping Cech cohomology
Hey @TedShifrin!
Howdy, a @Balarka, demonic @Alessandro, other @TedE
5:02 PM
Hi @TedShifrin!
Hi @AlessandroCodenotti
Ah, I was trying to ping @TedS
I suggest a western style duel because this chat is too small for two Teds
Multiple Teds are a pain.
<--- stakes a claim to prior existence (5+ years)
@AlessandroCodenotti I know, I was just joking about that
@TedShifrin It's okay, they can ping me with it
5:04 PM
So I see Hartogs was mentioned. :)
In the algebraic category, far easier than the analytic Hartogs to your disappointment
@BalarkaSen I think you can't think of separatedness as being 'topological' solely, since it is a relative notion
Is this the same Hartogs of the Hartogs number in set theory?
in the analytic setting, it holds for any compact subset. So I forget how that's handled in the algebraic category.
yeah thats so surprising
5:06 PM
Like your example above isn't $\Bbb Z$-separated, but maybe it is separated over itself or something
@TedE Yeah I don't know this stuff
$S$-separated means the diagonal $X \to X \times_S X$ is a closed immersion?
@TedE Yeah, that's understandable. I think there is more to the Cosmic Horror aesthetic than just the literal horror though. Ideas present in Lovecrafts work that may have been novel at the time have been replicated time and time again since, but I find there to be little fiction that is actually Lovecraftian as a whole. To me, reading Lovecraft is not as much about the horror as it is about diving into the unique worlds and headspaces he creates, exploring the boundaries of our existence.
Yes, where $X$ is an $S$-scheme and that fibre product is over the structure map both times, i.e. the structure map is separated
@TedS I asked Georges about an analytic vs algebraic question in this comment today
Hopefully there's a good answer and I am not making a fool of myself
@Thorgott Is some sense, that's why I read math :-).
5:09 PM
If you think about $X$ being separated as $\Delta:X\to X\times X$ being a closed immersion, where the product is categorical then being $S$-separated is being separated in $\mathsf{Sch}/S$
He's a good one to ask, a @Balarka.
@AlessandroCodenotti That's a restatement yes, since that's just saying we can denote the fiber product over the structure map to $S$ as a genuine product, as long as we intend for it to occur in the category of $S$-schemes
(This is just a complicated way to say that $\mathsf{Sch}$ is the same as schemes over $\mathrm{Spec}(\Bbb Z)$ and that the product in $\mathsf{Sch}/S$ is a fibered product in $\mathsf{Sch}$)
I know, I just meant that being separated can't really be totally topological, since it's a relative notion
It's like you can take a (absolutely) singular scheme over itself (with the identity map) and it's smooth over itself
Cthulhus lair was described as Non-Euclidean, after all
5:12 PM
@Thorgott That was a good touch
@TedE That's a good point
I hope Cthulhus are hyperbolic
Very trippy
@Thorgott I think my favorite part of Lovecraftian horror is that he describes it in such a way that you have your own mental image, and your own mental image can typically be much more horrifying than anything they describe.
@BalarkaSen When chtulhus gather in a group they have solvable word problem
Describing the horrible creatures as moving in ways that are self contradicting
5:15 PM
@Alessandro ayup
its very hard to imagine cthulhus differently after seeing them depicted as tentacle faced gentlemen through time immemorial tho
Yeah, that is definitely an important aspect. "Pickman's Model" is an entire story about an incredibly horrific painting, yet it's very minimal in actually describing what's being depicted.
Yep, that's actually exactly the one I had in mind :D
I thought that one was actually genuinely scary, and I haven't really seen it copied
Mainly the twist at the end
Lovecraft seems very disapproving of futurism and cubism in Call of Cthulhu
What do you mean by that?
I don't know what cubism is I guess
he ascribes the horrific pattern of the bas relief to cubism and futurism
or something like that
he passingly mentions it but maybe im biased because i know Lovecraft hated the vorticists
5:21 PM
Oh, I don't remember that
Yeah, Pickman's Model is really cool. I guess it's lesser known than his most popular ones. Finding otherness in oneself, coupled with descent into insanity, is another great theme recurrent in lots of Lovecrafts writing
the major people involved with the BLAST magazine were Ezra Pound and T S Eliot, both of whom Lovecraft has been known to mock
Also, I don't think having a clear image of Cthulhu as it is always perpetuated is that detrimental. Lovecrafts descriptions are, in a way, designed to transcend your understanding, they let you know that you don't know. The contrast between the little concrete that you have and the inconcrete that is implied is, as Ted alluded to earlier, where a lot of the horror lies.
@AlessandroCodenotti For what it's worth (and this doesn't imply that it really is an essential assumption), I found that Hartshorne says: Let $X$ be a separated scheme over an affine base. Then for $U,V$ affine open subsets of $X$, $U\cap V$ is also affine.
Makes sense, I was thinking about separated as separated over $\Bbb Z$ earlier
5:25 PM
So you think working specifically over $Spec(Z)$ removes the openness assumption on $U$ and $V$?
This might be nonsense, but I think if that were true, then since separated is preserved under base change, that might imply the openness of $U,V$ is irrelevant in general (for affine base as in the exercise I allude to above)
No openness is actually needed
I was wrong
Ahh ok, cool
I guess there's no reason why the intersection of two (arbitrary, i.e. not necessarily open or closed) affine subsets should even be a scheme?
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