@JonathanBeardsley What does S[X] mean here?

does anyone have feelings about the following books: artin's "algebra", nicholson's "introduction to abstract algebra", and lanski's "concepts in abstract algebra"? i am teaching an undergrad course in the fall, and need to decide on a book soon. i'm mostly between artin and nicholson; i like nicholson slightly better for reasons that i can't articulate, but then again artin is a classic and it seems like it'd be harder to go wrong with that.

(i already ruled out dummit & foote, as i think it is too encyclopedic and dense for a typical budding algebraist.) one thing that's a little weird about artin is that he spends a lot of time on stuff other than groups, for the sake of examples; groups are introduced in chapter 2, and then revisited in chapters 6 and 7. i'm not sure how i feel about that; i don't know why, but part of me feels like "a first course in abstract algebra" should focus primarily on groups.

i should say: i am also very happy to hear broader input about teaching abstract algebra, including (in particular!) contradictions to the vague feelings i've expressed

[i hope this isn't too off-topic, this is just my go-to place for math-related discussion. if you want to connect back to homotopy theory, well, i suppose i'm teaching about pointed 1-groupoids!]

@JonathanBeardsley For non-simply connected take $Y=BG$ for $G$ finite group, $X=Z=pt$. Do you have a reference for the statement about the EMSS youre thinking about? Since rationally everything okay it suffices to show that the EMSS converges strongly to the $p$-completion in this case i guess...

@OmarAntolín-Camarena ah just suspension spectrum

@SaalHardali so, this actually gets at something I'm moderately confused about, which is when (strong?) convergence of a Bousfield-Kan spectral sequence, or EMSS, implies anything about the actual homotopy limit of the cosimplicial object

But the reference I'm thinking of is the proof of Lemma 17 on page 7 of this paper, where it's stated without proof: pdfs.semanticscholar.org/e389/…

I always thought that "strong convergence" only has a meaning when a filtered group is specified. I guess the natural choice would be the group $\pi_*(Tot X)$ with the decreasing filtration by the $Ker(pi_*(Tot X) \to Tot_n(X))$. Then if this filtration is hausdorff and complete then strong convergence implies that the SS converges to $\pi_*(Tot X)$ in the simplistic sense.

To have a strong conclusion I would guess you'd need to hange the filtered group to something else (perhaps something like $X \times_Y Z$ in this case...).

"simplistic sense" doesn't mean anything formal just that nothing can really go wrong in this situation and the only thing left to solve is the extension problems.

I don't know when the eilenberg moore spectral sequence converges but I think that if for example $E \to B$ is 1-connected then the cosimplicial space $Cobar(\ast,E,B)= X$ satisfies taht the connectivity of the fiber of $Tot_n(X) \to Tot_{n-1}(X)$ grows (at least) linearly with $n$ so I wiuld imagine that the $Tot(X)=fib(E\toB)$ but i'm not sure that what i said is enough

(one needs to check that a certain $lim^1$ vanishes, I think this part is not extremely difficult to figure out but I also have a very heavy talk soon so I'll leave that for someone else to answer)

@AaronMazel-Gee I have no experience with the books you asked about, but have always been fond of Allan Clark's Elements of Abstract Algebra which is short and sweet (and as Dover book, inexpensive too!).

What's $H\mathbb{F}_p \otimes_{MU} H \mathbb{F}_p$? is it known?

It’s exterior on generators in every odd degree.

@DylanWilson hmmm, why that?

A silly question: for a spectrum X with G-action, the homotopy fixed point spectral sequence could be derived either by the Postnikov tower of X, or the skeleton filtration of BG. I wonder whether they are known to coincide?

obviously not. sorry for the annoyance

I don't know a reference off the top of my head, but I believe that the methods in this answer should work in this case too (the given reference covers the case of the trivial action)

I just learned from an old question of @JonathanBeardsley that the spectra $X(n)$ are related to formal group $n$-chunks just as $MU$ is related to formal groups. Where can I read about this?

Suppose I have a Kan complex X, and suppose that for every a,b \in X, the Homspace ho(X)(a,b) is the point. Does this condition tell me anything about the homotopy type of X? What if I had some stronger conditions, such as say the mapping spaces in X, considered as an \infty-cstegory, all were contractible. Would this tell me anything?

@Dedalus To say that $ho(X)(a,b) = \ast$ for all $a,b$ is to say that $X$ is connected and $\pi_0(\Omega X) = \pi_1(X)$ is trivial. To say that $Map_X(a,b)$ is contractible is to say that $X$ is connected and $\Omega X$ is contractible, i.e. that $X$ is contractible.

Oh, great. That was what I was hoping for. Since X is connected, is it enough to just assume that Map_X(a,a) is contractible?

Yeah, since every two objects are equivalent, all mapping spaces are the same

More generally if $X$ is a Kan complex, $\mathrm{Map}_X(a,b)$ is the pathspace from $a$ to $b$, i.e. the space of all paths from $a$ to $b$

And $hX$ is just the fundamental groupoid of $X$ (i.e. its 1-truncation $P_1X$)

@SaalHardali use the Kunneth spectral sequence. Alternatively, use that HF_p=MU\otimes_{S^0[x_1,x_2,...]}S^0 and juggle some tensor products around.

**S^0[x_0,...]

@DylanWilson Where does this second equation come from and what do you mean by $S^0[x_0,...]$? (in any case I guess the kunneth spectral sequence is pretty easy). What I really want is the comultiplication on this guy. I'll try to think about it by myself more, thanks.

@JonathanBeardsley I have a question which is related to this bousfield kan spectral sequence which confuses me too but its a drastically simplified situation so there should be a simple answer.

Question: Suppose $X_n$ is an inverse system of spectra s.t. $lim \pi_{\ast}(X_n)=0$ (1) what are the things that go wrong that would make $\pi_{\ast}(lim X_n) \ne 0$? (2) What does one need to check to make sure non of these happen?

There's an fiber sequence $F_k \to lim X_n \to X_k$ for every $k$ so you could take for every $k$ the homotppy groups of this to get an inverse system of long exact sequences. If all the boundaries vanish you get a $lim^1$ exact sequence showing that all you need to do is check that $lim^1 \pi_{\ast} (F_n )=0$.

When the boundary maps are not 0 I don't know how to proceed...

(lets assume for the moment that $X_n$ are all connective, probably if their not it becomes terrifyingly complicated).

This gets at the question of how to deal with derived functors applied to long exact sequences (in this case its the derived limit functor).

@TimCampion this is stated in hopkins' chapter "from spectra to stacks" in the tmf book. you could prove the claim there by knowing what MU_*(X(n)) is as a subcomodule of MU_*(MU): the two stacks (moduli of n-buds and the stack associated to X(n)) live over M_fg, and since there's a map between them over M_fg, they're isomorphic if and only if they are so after pulling back to Spec(Lazard ring). knowing the comodule structure tells you that this is indeed the case.

@SaalHardali i think what's being said is that HF_p is MU/(p, v_1, ...), where dylan's x_i goes to v_i

@skd I see, it makes sense, thanks.

@skd Thanks!

@SaalHardali sorry, HF_p is BP mod (p, v_1, ...), and replacing BP with MU means replacing (p, v_1, ...) with the ideal (p, x_1, x_2, ...) inside pi_* MU = Z[x_1, x_2, ...], where |x_i| = 2i

@TimCampion it's in Ravenel's "green book"

@SaalHardali all the generators are primitive, so that determines the comultiplication. S^0[x_0,...] is the smash product of a bunch of free A_infty algebras on generators in degree 2i. (This can be upgraded to an E_2-ring but I don’t think you need that for this argument).

(Sorry I’m being so terse- I’m just typing from my phone so it’s hard to give more in depth answers)