Homotopy Theory

as functors valued in sets?

well, i'm trying to decide that. the point is.... $\mathbb{G}_a'$ is a group scheme

12:40 AM

You already know G_a' is a group scheme, or you want it to be a group scheme?

no i mean it is

and the higher n is, the more ways it can be

You also need to specify what the group operation is on R[x]/x^n

Otherwise you are just giving us a (pre)sheaf of sets

for instance, there is a cogroup structure on $R[x]/(x^2)$ given by $f(x)\to f(x+y)$

hmmmmm okay.... so..... maybe they're cogroup schemes

If the group structure turns out to be just R^n,+, then you are just dealing with G_a^n

so, $Spec(R[x]/(x^2)$ is a group scheme

yessss

12:43 AM

No, not quite

but! there are more ways than just additivity to do it right!

oh.....

The Hopf algebra structure on a ring is what you get from turning around the arrows on an affine group scheme

yeah

This is not the same as defining a pointwise group structure on a scheme

And where by pointwise I mean functor-of-points, i.e. a lift Aff^op --> Grp of the sheaf of sets

underlying the scheme

i mean. there are two things going on here.

i'm pretty sure that using the unique 1-bud structure, we know that $Spec(R[x]/(x^2))$ is a group scheme, yes?

12:45 AM

1-bud?

ummmmmm, so a truncation of a formal group law

But Spec(R[x]/x^2) is not the same as R |-> R[x]/x^2

oh no absolutely not

the latter should have a cogroup structure

So the question is, which one do you need?

the latter

and i didn't entirely know that until i was forced to clarify to you what the hell i was talking about, haha

12:46 AM

Since I came in late, may I ask why and what for?

if you have a representable functor valued in cogroups you are looking at a cogroup object in the category not a group object even though the functor is contravariant

yeah exactly

that's what i just realized

i was getting my arrows backwards

:-)))))))

i'm super happy

You have lots of chins

:P

yeah, all i do is sit in this chatroom and eat cheezits

@davidroberts the short of it is, i'm trying to think about building formal group laws up from what are called n-buds, or formal group law k-chunks, from a deformation theory perspective

and i guess i'd like to think about it as something like.... deforming cogroup structures on a certain scheme

Hmm, sounds interesting!

12:50 AM

yeah. i'm sort of pleased with it. Michel Lazard basically proved that any n-bud can be deformed into an n+1-bud, but he didn't do it from this perspective

So it's working from the bottom up, rather than starting from the Lazard ring?

yeah exactly

and Lazard actually did that originally

but.... i guess he didn't know deformation theory, haha

well actually, the cohomology theory that the obstructions live in was developed later by lubin and tate

and they used it to do some very cool stuff

So my question is, given an elliptic curve, can you deform it to give an explicit fgl one level up?

And I mean a specific elliptic curve from Weierstrass form

well, ummmmmm, i think such things always give you formal group laws, right?

when you say one level up, are you referring to height?

I mean chromatically

12:53 AM

yeah, so, that's an awesome question. I think that other people have worked out that elliptic curves always give you FGL's of either height 1 or 2

yes according to ordinary or supersingular

yeah

I ask because people, when talking about cohomologies from fgls say: G_a -> HR, G_m -> ku (or similar, can't remember connective or otherwise)

@will so are elliptic curves which are ordinary, are they... smooth?

Then E -> elliptic cohomologies

12:54 AM

all elliptic curves are smooth by definition

for E a curve

oh okay, good!

And then what next?

haha

It's all quotients of BP and so on.

12:54 AM

it's just that in characteristic p the p-torsion of an elliptic curve can be either (Z/pZ) or (Z/pZ)^2

(as sets)

Shimura varieties or something

aha. okay, that's very helpful, thankyou

sorry

I would like to see an example of a cohomology theory induced by a different fgl

0 or Z/pZ

@Will that makes more sense

12:55 AM

and so, an elliptic curve is an affine group scheme, and i get an FGL by look at an infinitesimal neighborhood of the identity?

YEo

the formal group essentially only sees what is RIGHT BY the origin

Er, yes

in the case where it's Z/pZ

you have it spread out a bit

so in a sense you only have on Z/pZ factor that has collapsed

if you have 0

hmmmmmm, okay

12:56 AM

the scheme still has degree p^2

but it's all concentrated around the origin

right

and the formal neighbourhood of the origin sees all of that

interesting

so the height is 2

:)

12:57 AM

Nice

so @DavidRoberts, there are for instance the Honda formal group laws

which are related to, if i'm not mistaken (and i often am), MoravaK-theory

Ah, ok.

How far up the chromatic tower are they?

arbitrary height

Hmm ok

is that right @will? you probably know better than i

12:58 AM

But say you're given a curve, can you deform it to a Honda fgl at height 3?

i don't think so

I don't know, are they associated to higher rank L functions or what?

(that was for you @Jon)

Have a look at www.math.uiuc.edu/~rezk/hopkins-miller-thm.pdf

yeah i know. considering that these chats are saved, i'm not sure i want to say a lot more

hahaha

but yeah, that's the place to look @davidroberts

Example 3.8

1:02 AM

i'm sorry, Honda FGLS give you Morava E-theory

apologies, but I've got to go now

later

1:34 AM

the thing with everything being on record is that you can't trick people into thinking the chat was active and they need to join in

speak for yourself

i am v. tricky

wait what

i don't understand the trick i might polay

*play

people can tell whether or not you were lying

b/c they can see when the last person who talked was

hm

you can't just start a conversation mid-way through to get a reaction of someone who just joined

1:36 AM

yeah, beardsley

yeah, i guess i haven't spent enough times in chat rooms to understand this particular trick

i guess

now would have been the perfect opportunity for instance

like when eric joined?

instead he can alleviate the confusion by scrolling up

1:42 AM

hah hah!

we were just discussing our favorite homotopies, eric

i see

homotopoii

yeah. and cogroup schemes

Eric

when david comes back: you can't perform an p-primary infinitesimal deformation to change the height of a formal group law

you can do noninfinitesimal things, but then examples are obvious

is the Honda formal group law something obvious

Eric

1:45 AM

no, it certainly is not

there are papers by charles rezk and a student of tyler lawson which give explicit models for the honda formal group laws in characteristics 2 and 3 and at height 2 using accidentally available models of elliptic curves

the general case is abysmal

i forget tyler's student's name. yi fei? idk

ok thanks

it exists

technically

Eric

haha, good stasheff email