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I would like to understand the connection between $p$-adic completion and the strong convergence of a spectral sequence. Precisely, suppose $E^2_{s,t}\implies G_{s+t}$ is a first quadrant strongly convergent spectral sequence (of abelian groups) that comes from some Postnikov tower, say. What is the obstruction to having a strongly convergent spectral sequence after $p$-adic completion of both sides? If the groups are finitely generated, then this amounts to tensoring by $\mathbb{Z}_p$ which is flat...so everything is fine. What about other cases?

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  • $\begingroup$ Have you looked at Boardman's "Conditionally convergent spectral sequences"? I recommend it for everyone who ever thought they might want to know anything about a spectral sequence. $\endgroup$ Dec 15, 2015 at 9:30
  • $\begingroup$ The filtration $* \to HZ \to HZ \to HQ$ of $HQ$ gives a spectral sequence with $E^2_{0,0} = Z$ and $E^2_{2,-2} = Q/Z$, converging to $G = Q$. After $p$-completion you get a spectral sequence with $E^2_{0,0} = Z_p$ and $E^2_{2,-1} = Z_p$, with $d^2_{2,-1}$ an isomorphism, converging to $0$. Does this count as "having a strongly convergent spectral sequence after $p$-completion" for your purposes? $\endgroup$ Jan 27, 2018 at 23:32

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