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Stream: theory: algebraic topology

Topic: Adams spectral sequence and p-completion

Cloudifold (Jul 07 2024 at 02:05):

if we replace the spectra $X$,$Y$ in the $H\mathbb{F}_p$-Adams spectra sequence

$$Ext_{A}^{*,*} (H\mathbb{F}_p^{*} X, H\mathbb{F}_p^{*} Y) \cong E_2^{*,*} \Longrightarrow \widehat{[X,Y]}_p$$

by their $p$-completion $\widehat{X}_p$, $\widehat{Y}_p$. Will the Adams spectra sequence still hold?

Kevin Carlson (Sep 17 2024 at 20:47):

I'm having trouble parsing the question--isn't that just substituting variables that run over all spectra with some particular spectra? Or are you trying to only substitute on the left and not on the right?

Brendan Murphy (Sep 23 2024 at 02:27):

I'm also not sure what the question is, but I would expect it to be true if X, Y are connective (after suspending some number of times) and untrue otherwise (this condition ensures that p-completion calculates H F_p-localization)

Brendan Murphy (Sep 23 2024 at 02:34):

Well, I guess it's possible that the difference between the two could get corrected as you go far enough out in the spectral sequence, or something? But in that case I don't know