are there elements associated to the chromatic v_n in the actual stable homotopy groups of spheres?

Beyond like v_1 is related to the alpha family and v_2 the beta family?

hrm.maybe.

i guess that's pretty good, haha.

yeah.

one of the periodicity theorems is that a spectrum lies in the category of K(n)-acyclics iff it has a v(n+1)-self-map. since the sphere is rationally nontrivial, it itself doesn't have vn elements in its homotopy groups

yeah.

but it does have elements which are themselves interrelated by things that look like vn maps. a type n spectrum is built out of some cells and some attaching maps, and the fact that it carries a vn-self-map is a statement about some kind of interrelationship between those attaching maps and vn

Oh nice

hm yah.

which is made most precise by the chromatic ss

yeah.

why are completion and localization sometimes the same in the category of spectra?

that's what i wanna know....

that i don't know

i actually feel a little guilty each time i introduce an adams SS to someone, i say those words and then think oh gosh don't press me about this

hahah, what do you mean?

here you are pressing me about it

well, no no, don't talk about the ASS. like, what is it that you don't want to talk about?

i've been thinking about whether or not one can define a topology on "spec(S)" (whatever that is) by looking at ring spectra S--->R for which the R-ANSS converges.

i.e. thinking of that as a definition of "flat" cover

but then, i'm not sure... like, there are some weird things going on, for instance, what precisely the ANSS converges to.

because if it's the R-localization, then that sort of makes sense... but the R-completion... well, wtf?

the distinction between localization and completion, and when i mean which

aha

but either way.... the frustrating thing is, there don't really appear to be any "universal" covers for the sphere spectrum

in the sense of, descending a sheaf from spec(E) to to Spec(S) really only works if that sheaf is already E-local

but perhaps being E-local is a way of saying that the sheaf lives only over Spec(E), so we're getting all of it.

i.e. its "support" is contained inside of E

er. Spec(E) (over Spec(S))

gonna write to jack about his truncations remark

which is that?

this cl.ly/image/433z2g3R471L

nice!

awwwww yeah. i'm booked for the conference in Louisiana.

y'all should go.

Make sure to take notes of Andrew Salch's talk!

Hahah. Yeah yeah.

Then the world will know how awful my handwriting is.

is there an easy expression for the mod 2 homology of $K(F_2, n)$ as a subalgebra of the dual Steenrod algebra?

isn't there some fiber sequence that space sits in....

i'm trying to remember.

dunno if that would help.

agh. i dunno.

really dumb question: how do the mod-two homology of K(Z/2,n) and of HZ/2 compare?

on cohomology there's a surjection $H^* HF_2 \to \Sigma^{-n} H^* K(F_2, n)$ - I think you're just quotienting out by the monomials with too much excess

so it should be an injection on homology

hm

ah, but it can't be an inclusion of subrings

because $H_* HF_2$ is polynomial but the multiplication on $H_* RP^\infty$ is trivial

this is nice mathoverflow.net/q/146187/1094

@JonBeardsley how do you get full-size links to questions, like in chat.stackexchange.com/transcript/message/11856055#11856055