@aaaaaaaaaaaaaaa well i guess i have a very specific thing that someone was talking to me about, and i'm not sure i'm understanding it correctly. maybe i'll just write it down and let whomever feels like it (if anyone...) say whatever they feel like
I feel like I have decent low-level grip on deformation theory, but there's a certain kind of deformation functor I'm trying to understand, which is the following: given a ring map (quotient map?) $\phi:R\to k$, we can ask for lifts of $\phi$ along maps $A\to k$, where $A$ is some kind of Artin $R$-algebra with residue field $k$. So the value of the functor of interest on an augmented Artin $R$-algebra over $k$ should be the set of such lifts. Is this a "deformation functor?"
Or is there something similar that maybe I should be talking about but that I'm screwing up the details of?
I think what this is supposed to capture is "deformations of $Spec(k)$ inside of $Spec(R)$" or something?
user351585
12:22 AM
I don't think I have anything to say about that situation that isn't on the first page of Schlessinger's paper, but if you haven't looked at Schlessinger's paper, take a look at the first page--a prorepresentable deformation functor isn't quite the same as what you're describing (you would need to complete R at the kernel of phi), but it's pretty close.
MathOverflow first came online on September 28, 2009!
Let's celebrate five years of MO!
I bought a delicious cake from the best bakery in the area to have with dinner tonight. I'm sure you will also find your own delectable way to celebrate this anniversary.
Some years ago, I asked community m...
@JonathanBeardsley , @skd: I have been thinking a bit more about this question on complex orientations, and the closest thing I found in the literature is section 3 of a certain preprint "Constructions of the Complex Cobordism Spectrum by Attaching Coherently Multiplicative Cells in Odd Degrees" by someone you might know ;) The idea there is to attach an E_1-cell to S in each odd degree, to get MU, and the way you attach these E_1-cells is determined by looking at Thom spectra of $\Omega SU(n)$...
user351585
3:48 PM
...for each n. Here is my question: the element in $\pi_{2n-1}(X(n))$ that you cone off, how far does it pull back in the sequence $\pi_*(X(1)) \rightarrow \pi_*(X(2)) \rightarrow ...$? I see that for n=2 it actually pulls back to $\pi_*(X(1))$, and I think this happens again at n=4.
user351585
3:59 PM
Oh never mind, I got it, that pulling-back is basically a 2-primary fluke
user351585
Oh maybe not. OK I would like to know an answer to that question, especially after localization at an odd prime
@aaaaaaaaaaaaaaa Hm. Not sure, and, I'm not sure that the way I was thinking about these things is immediately amenable to answering this question, but I might be wrong.
user351585
@JonathanBeardsley I am beginning to think that, p-locally, many of these elements pull back to earlier stages in that sequence, and that this pulling-back gives you a nice result on E_1-complex orientations--but there are parts of the argument that I'm not sure of. I'm writing up some notes now, and I'll send them to you and you can see what you think
@aaaaaaaaaaaaaaa I look forward to it! I'm currently staring at this diagram involving $\Omega SU(n)$ that I've got etched into my brain from my thesis, trying to see if there's obvious way to get a handle on it, but maybe it's more a computational question.
user351585
@JonathanBeardsley Do you have a similar formula for the p-local spectra T(n) given by attaching E_1-cells?
@aaaaaaaaaaaaaaa Unfortunately no, because I'm not sure how the splittings that give you the $T(n)$ work out with these cell attachments. Certainly the formulas all hold for $X(n)$ localized at $p$ though.
it seems however that if $n\neq p^k-1$ for some prime $p$ then $\chi_n$ is 0.
At least, I remember sort of coming to this conclusion in a conversation with Doug Ravenel, although there are some things about I'm slightly anxious about.
user351585
Hmm, definitely the component of \chi_n supported on the E_1 0-skeleton of X(n) has to be zero if $n\neq p^k-1$, but I would expect that \chi_p might nontrivially attach an E_1-cell to the E_1 3-cell in X(2), for example. But that sort of thing, how the splitting of p-local X(p^n) into suspensions of T(n) is related to your E_1-cell attachment formula, that's what I'm trying to understand at the moment
MathOverflow first came online on September 28, 2009!
Let's celebrate five years of MO!
I bought a delicious cake from the best bakery in the area to have with dinner tonight. I'm sure you will also find your own delectable way to celebrate this anniversary.
Some years ago, I asked community m...
