Man, SGA is awesome....

I think that resolution in Prop 4.6 is using some finite generation hypothesis -- the standard cosimplicial object would be Hom_R(R[GxGx...xG],M), but since the domain is finitely generated free this can be identified with Hom_R(R[GxGx...xG],R) tensor M.

Aha, I see what you're saying.

Tyler to the rescue again!

Wait, I think that's what Saul said

Yeah.... I think you're right.

Well, I mean, not in so many words.

Tyler, can I ask you a question?

Sure.

Say we want to work out C-comodule structures on X, for some coring C. If I take the cosimplicial object C-->CxC-->CxCxC-->... shouldn't Hom(X,C^\bullet x X) compute such structures?

It seems like the maps in the cosimplicial object are precisely the ones determining associativity and unitality.

Where C^\bullet x X is the aforementioned cosimplicial object tensored with X

Actually....

I've made a mistake.

Well, you can certainly find comodule structures there

I want a particular case.

The case where C=Rx_S R

You kind of want to extend things by constructing a cosimplicial structure on C^\bullet x X.

Hmmmmm

So.... a resolution of the cosimplicial object?

As in, a cosimplicial, cosimplicial object

Well, it's like the cosimplicial structure on C^\bullet x X gives you some of the maps in the cosimplicial structure you want

but it doesn't give you, e.g., the map X -> C x X for example

Yeah. sorry. so that was my mistake

and so you need to somehow "finish it off"

if i take $C = R\otimes_S R$ and I'm working over $R$-modules, then when I smash with $X$ (over $R$!) I get $X\to X\otimes_S R\to X\otimes_S R\otimes_S R\to\cdots$

Sure.

So, if I hom $X$ into this I get a cosimplicial object $Hom(X,X)\to Hom(X,X\otimes_S R)\to\cdots$

And if I sort of negate this bottom piece by demanding that it's only the indentity or something.

So this is going to give me the necessary structure maps for being an $R\otimes_S R$-comodule over $R$

Hey @CraigWesterland! Not Toda brackets, but in Massey products this paper by Isaksen shows what can go wrong if your brackets only contain 0: arxiv.org/pdf/1309.4637v1.pdf

But.... I mean, suppose $S=\mathbb{S}$ and $R=MU$. Then I've basically got to compute a pretty complicated BK-sseq?

I mean... I feel like that's too basic. Something needs to be said about... homotopy. Or something.

MATH!

i know, right??

lawl

Anyone have a digital copy of Bousfield's "homotopy spectral sequences and obstructions"?

I'm at school, but for some reason Springer won't recognize this, even after I log in through the library.

I e-maeiled you

thanks!

okay, so to compute the BKSS for a cosimplicial space, i first need to know the homotopy levelwise?

it is also now in the dropbox

and this gives me a cosimplicial group

whose "cohomotopy" i now need to compute?

or i guess sort of a graded cosimplicial group.

When working with ring spectra (with working here, I just mean using them as objects for generalized cohomology theories in somewhat rudimentary terms) how important is it to know details on how the smash product is constructed?

or should one treat it more like a blackbox?

I'd say blackbox it for now

I still don't know.

Something about linear isometries operad.

Blech.

are there any general properties of ring spectra one should know about? For example, it bothers me somewhat that I haven't done any concrete calculations with ring spectra as one does for say rings, but maybe that isn't possible to do in the same way here?

Maybe it's important to know about what doesn't work for ring spectra. For instance, taking the cone on multiplication by 2 on the sphere spectrum does not give you a ring spectrum back.

that is quite good to know, yes

Things like that. Hm.

And then you get into a whole maze when you want to start talking about ring spectra with various levels of homotopy coherent multiplications.

yeah, that should be related to operads, right?

(I don't know much about operads)

Yea, exactly.

Maybe a good thing to check out is the back of Ravenel's orange book where he shows how, for nice ring spectra, you can get so-called Hopf algebroids from the pair (R, R\wedge R) and so forth.

If you're going to learn chromatic homotopy theory, this is a good thing to be familiar with.

Ravenel's orange book = his green book?

that is not green anymore

Ah, no. So the orange one, in my mind, is the smaller one.

Nilpotence and Periodicity in Stable Homotopy

or something.

right

I will check it out

The orange book is still orange!

Yeah.

The green book is either the green book or the red book.

I didn't know about this book - seems fantastic since I want to learn about chromatic homotopy theory

Be sure to get the errata from Ravenel's website, that book has a lot of mistakes

Or, for me, "The Music of the Spheres"

Are there any other good resources besides those of Ravenel?

Hopkins' so-called "COCTALOS" notes are good, I think.

But maybe not super accessible at first go.

Yeah. Eric made an exhaustive list I really like

math.berkeley.edu/~ericp/teaching/Fall2013/msri

As well as Lurie's class notes on chromatic homotopy theory, though again, you sort of have to understand Lurie-ese to understand his notes.

Hey guys, can I think of Aut(M) as something like a colimit over all groups which act on M?

I learned chromatic homotopy from Lurie's notes

I have some background in algebraic geometry and formal geometry, but I am not sure that it fits in with Lurie's approach

but you have to be aware that he's sweeping a lot of stuff under the rug

Yeah.

i recently learned the following fact: every spectrum can be written as the colimit of an ind-system of desuspensions of suspension spectra, and the smash product of two spectra so expressed is the colimit of the smash product of the two systems

i really appreciate how this points out where the cross-interference in, like, HF2 ^ HF2 comes from

it's also a sentence a model category neanderthal can still parse

indeed, for a spectrum $E$ indexed on the natural numbers can be written $E = colim_n Sigma^{\infty - n} E_n$

isn't that neat

it is

i don't think i really had had that thought in my head before, like, june

also it's worth pointing out that this is the smash product defined in switzer, he's just reluctant to say the words 'ind-system' or 'homotopy colimit', and so it takes him a few pages to spit it out

It is a bit of a shame that there isn't that many (any) modern textbooks using these kinds of words that simplify it a whole lot

it's sort of understandable; it's easy to be flippant about definitions in a chatroom, but when writing a textbook you're under some obligation to be thorough, and if you thoroughly express the above idea you end up writing switzer's book without much deviation

the disappointing thing about switzer is that it's more encyclopedic than it is conversational. you won't find these sorts of summarizing sentences anywhere in there

but that's different from simplification

True

Switzer seems to me to be the best introduction for someone that don't know a lot of Stable homotopy theory and wants to learn some

(along with Adam's book)

i like it a lot, though i've met people who don't

Of course, more exercises would be good

Cotor is the Romanian word for the stalk of a plant.

Also, tor is the Romanian word for the costalk of a plant.

Lol.

Somehow I have not played minefield in a long time.

it's very meditative

Anyone have a lot of experience with computing Bousfield Kan spectral sequences? (i.e. FML)

@Tedar: I am equally an amatuer but if you want some words on modern approaches to the smash product then Rezk's Notes on the Hopkins--Miller theorem and the start of Hopkins Notes on Elliptic Cohomology are both pleasingly short and informative

I also tremendously enjoyed these notes of miller as an introduction to chromatic homotopy: www-math.mit.edu/~hrm/papers/cobordism.pdf

i'd forgotten about these notes

good notes

I just learned what a voxel is.

:)

I totally cannot parse Kochman's definition of a higher Toda bracket. Aggh

for a little while voxels were popular for animating figures in RTS/RTT games, which is where i learned the word

there was some chrome app called voxelwright that looked cool

but then i realized all it is is just sticking blocks on stuff

very very slowly....

just finished proctoring a midterm

blerg

now. what now.

Kathryn Hess thinks the right sort of cohomology theory for Hopf-galois extensions should be some kind of cotor.

Gonna have to figure that one out, haha

i think there should be a book on homotopy theory that uses \infty-categories as a total black-box

everything you want to say you can say there

haha....and that just ticked off @CPM enough to leave the catroom

*chatroom

man, if only this were a catroom

@TriThangTran - Yo!

@Tedar absolutely black-box it; this is what i was most of all referring to back a few days ago

Great, it seems kinda painful to go into it in full detail

@JonBeardsley i feel like the deeper issue is that we don't know abstractly what "ideals" of ring spectra should be

yeah don't

Well... I'm not sure that idea is even... like... meaningful.

(go into full detail)

it's not going to help you compute anything anyways, afaik

jon, what do you mean? there "should" be a notion of ideal, i'd expect

if you can think of any important properties that I should know of, tell me ^^

really? i'm not sure.

i think the key property you'd want is that they're the (additive) fiber of a ring map

I don't think I understand that sentence fully, but I will look into it

oh, don't. the point is that nobody knows how to abstractly characterize which maps I --> R of a spectrum into a ring spectrum ought to have that the cofiber R/I is again a ring spectrum

at least, i think that's the state of affairs

Oh, got it somewhat then

Is there any way to "scheme-theorise" the notion of (commutative) ring spectrums? Maybe that is kinda what lurie does

I found these notes really helpful: math.stanford.edu/~carym/stable.pdf

I think some people in here are probably friends with the author

@Tedar yes.

@Drew arnav knows Cary

How does the notion of prime ideals generalize to ring spectra?

anyway, Hovey has sort of classified exactly those maps, @AaronMazel-Gee

(if it does)

It's scheme theory from a functor of points POV

rather than sets of prime ideals.

yeah cary's my bro

there's an enormous amount to be said here, @Tedar

oh cool jon, i didn't know that

cary and i started the xkcd seminar together: math.berkeley.edu/~aaron/xkcd

@AaronMazel-Gee yeah, though i think there have been mumblings that it's not really the right idea. they're called smith ideals, so, really i should say that it's jeff smith's ideals

I read a lot of those notes before I met you guys. So, err, thanks I guess!

but hovey wrote the most understandable thing about them

Yeah, I get that it is more to the picture

we met the summer before i began grad school. eric quickly joined as soon as we met when grad school started

@Drew you're welcome!! i'm glad they were helpful

@Tedar the so-called "fields" of stable homotopy theory are the Morava K-theories.

as well HF for F a field

butt they are not E_\infty-ring spectra, which introduces complications

so one could attempt to say something like a "maximal ideal" of a ring spectrum R is a map I-->R whose cofiber is one of those fields.

correct.

right

one of my long-term goals is to formulate a coherent notion of DAG across all operads, to whatever extent this hasn't already been done

What do you mean by that?

i'm aware e.g. of john francis's thesis

well for starters there's E_n for 1<n<\infty

can you talk about generic points etc.?

so, here is one gloriously simplifying feature of DAG for E_\infty-ring spectra:

well.... you might run into problems considering that alg. geometry over non-commutative discrete rings isn't even really well developed...

it is (equivalent to) The Right Thing to define Spec of an E_\infty-ring spectrum to be a space with a sheaf of E_\infty-rings, namely the space Spec(\pi_0(R)), with the sheaf given by the various localizations R[f^{-1}]

@JonBeardsley i feel strongly that once everything is sufficiently categorical, then you get the right notions

maybe. i dunno.

sorry, totally don't mean to be the negative guy

haha don't worry about it

I am quite curious if Lurie's DAG will have any applications to arithmetic geometry since I have mostly done that before - it would be interesting . But once again, I know nothing about DAG apart from some buzzwords

looking forward to see what you come up with :-)

@Tedar what do you mean when you say arithmetic geomtry? anything in particular?

apparently lurie+gaitsgory proved some things, the tamagawa number conjectures or something like that

using DAG

Well, consider cohomology theories for schemes of finite type over Spec Z and the conjectures related to L-functions / zeta-functions. There have been many attempt sand dreams to find "right" cohomology theories for proving these conjectures

i.e. motivic stuff?

are motivic cohomology theories related here?

right

Yeah

or maybe i should say e.g. rather than i.e.

@AaronMazel-Gee: the thought of someone rage quitting a maths chatroom over that suggestion is delightful

hah.

;o)

so there is Weil-étale cohomology (see the work of Baptiste Morin, I remember that he said something he stumbled upon probably would be an (inf,1)-topos ) , and the dream of (the very, very) conjectural Deninger Cohomology

hey guys, where can i find a general categorical definition of "cotor"?

@Tedar that sounds pretty interesting.

@JonBeardsley isn't it just the derived functor of the cotensor product of comodules?

I guess so.

=P

Just look at what Ravenel does

Yeah, I guess so.

I'm trying to figure out what that should mean, for, say, spectra which are comodules over another spectrum.

derive the shit outta those dudes

to be less precise, you need a model structure on either co/chain complexes of comodule-spectra or perhaps co/simplicial comodule-spectra

Yeah but.... ugh. This never really is clear to me. Like, what that means.

@AaronMazel-Gee yeah. that sounds right.

is cotensoring a left or a right adjoint? that will determine which way to take what i said

I think so too :) I know that some people , to realize these different theories are trying to generalize scheme theory in some sense for arithmetic schemes

but this is just very conjectural

Cotor is a right derived functor

k thx

haha glad there's someone who actually knows what the hell they're talking about

i'm just being a categorical d-wad

but, so, okay.... so if I want to get Cotor(X,Y)

I should like... replace something with something.

lol

then cotensor

then take cohomology.

yup

that's how you compute any derived functor at all

well yes.

Yeah, exactly

but... i dunno... i'm way more okay with this for, say, chain complexes of modules.

the whole point is to make it homotopy-invariant in the ambient model category

yeah....

a left quillen functor only preserves equivalences between cofibrant objects -- it doesn't preserve equivalences in general

so hess and others have worked out model category structures on categories of comodules

so what do you do, you replace your objects by cofibrant ones, and then you apply your functor. and then by what i said, this is homotopy-invariant!

this took me a long time to realize, but it's the entire point of derived functors

Well, yeah, I get that.

it's "the closest homotopy-invariant approximation"

ok good

But I don't get like, how to actually do this and then compute something.

I also don't know what a cofibrant replacement of... I dunno... MU would even look like.

lol\

well it depends on your model structure

yeah.

argh.

is the cobar or bar or something always a good replacement?

or rather... always a valid replacement.

it's a good bet, yes

never a good one....

you gotta make sure it's co/fibrant

but it's pretty co/free, which makes it a good bet

I'm still confused about what category you're taking cotor in

what's a "good replacement", is that a technical term?

haha, nooooo

nothing i'm saying is remotely technical

comodule-spectra, it sounds like

yes.

C-comodules for some coring in spectra.

co-ring

yeah. =P

I like the sound of coring though

coring always makes me think of apples

(which i love)

haha. i love apples too.

but it also makes me pronounce it wrong, which is a nicht-nicht

i think i like anjou pears more tho.

or like, d'anjou, or whatever.

@Tedar - This question is somewhat relevant: mathoverflow.net/questions/15687/…

but not too ripe.

So do you know that category has enough projectives\injectives?

@Drew how does knowing that it's right derived help me? does it tell me something about which side i need to (co)fibrantly replace?

oh, ummm, i'll have to look at hess and shipley's paper more i guess.

but i believe it does.

Meh, I dunno. I only think about this from the point of view of homological algebra I guess

i mean, that's fine. why is it relevant in that case whether it's right or left derived?

arxiv.org/pdf/1205.3979.pdf

@Drew you always have (co)fibrant replacement

I don't think it matters

Sure. I guess I'm not thinking too much about cofibrant replacement. I'm just thinking about cotor from the point of view of homological algebra

yeah. so, i guess, right. as long as i've got a model category, i should be alright.

Hm. I think I'm getting somewhere.

u guys are the bee's knees

hopefully someday someone will come in here and have a question that I can answer... instead of the other way around.

it really blew my mind when I realized that taking cw-approximation and projective resolution are the same thing

lol

i mean.... yeah. i dunno. i feel like i realized all this shit pretty on, but from a purely theoretical viewpoint. i had to give a talk on model categories, i dunno, my second year (i didn't know much topology when i got to graduate school, lol), and i realized like, oh shit, cofibrant replacement = projective resolution = "cell object" etc

but... what the hell to DO with that when your objects aren't modules....

is kind of overwhelming.

you figure out what cofibrant things look like

Haha. Yeah....

that's being incredibly glib about it

Or fibrant.

I mean I get it. I guess what I'm saying is I know all the stupid glib remarks to make... I'm actually trying to compute something.

But for the most part, when it comes to computing things people seem to just ignore this sort of stuff.

So, it's really confusing.

@JonBeardsley I got the impression that that was because "take the (co)bar complex" often works. Which is I guess the point of that comment from Hess-Shipley you made earlier

Yeah. I guess so. Coming up with better resolutions seems to be rather difficult.

well, better functorial resolutions maybe

yeah

@JonBeardsley left/right is incredibly important! this is just what i said previously. what matters is whether your functor of interest (e.g. cotensoring with some fixed comodule) is a left or a right adjoint

Maybe this is completely wrong. But if you actually want to compute these thing, there is probably going to be a spectral sequence Cotor(X_*,Y_*) -> Cotor(X,Y), in analogue with EKMM

that sounds reasonable. i think the question is whether the homotopy of a resolution is a resolution of the homotopy

what's the EKMM story?

@AaronMazel-Gee well, if you have something specific you want to compute, functoriality isn't that big a deal, it doesn't seem

@AaronMazel-Gee can you say more about this? I mean, what does the right/left-ness control?

In EKMM you get spectral sequence Tor^{R_*}(X_*,Y_*) -> Tor^{R}(X,Y)

Where they define Tor_n^R(X,Y) = \pi_n(X \wedge_R Y)

for example