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Let $k(n)$ be the $n$th connective Morava K-theory, with $k(n)_* = \mathbb F_p[v]$ where $|v| = 2p^n-2$. If $X$ is a space or a spectrum (assumed bounded below), one can compute $k(n)_*(X)$ using either the classical Adams spectral sequence or the even more classical Atiyah-Hirzebruch spectral sequence.

Both spectral sequences are spectral sequences of modules over $k(n)_*$. (In the ASS the bidegree of $v$ is $(1,2p^n-1)$.) Both spectral sequences start with $k(n)_* \otimes H_*(X;\mathbb F_p)$, at $E_1$ for the ASS, and $E_2$ for the AHSS. Both have first possible nontrivial differential given by the formula $ d(x) = vQ_n(x)$, where $Q_n$ is the $n$th Milnor primitive in the Steenrod algebra (acting on homology by going down in degree by $2p^n-1$).

So it seems that these must really be the same spectral sequence. Is this true? Is this a fact in the literature? (I am a tad bothered by the fact that the AHSS arises from an increasing filtration of $k(n) \wedge X$ while the ASS arises from a decreasing filtration of $k(n) \wedge X$.)

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    $\begingroup$ Hi Nick, I think this is true. The AHSS for $E_* X$ can either use a cellular filtration of X (starting at $E_2$) or the Postnikov filtration of E (starting at $E_1$), and "shearing" the spectral sequence accounts for the increasing/decreasing difference -- I think this is in Appendix B of Greenlees-May's "Generalized Tate cohomology". The result should then follow from the fact that the Postnikov tower for k(n) is an Adams tower, because mutliplication by $v_n$ becomes null after smashing with $H$. $\endgroup$ May 6, 2020 at 19:19
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    $\begingroup$ @TylerLawson: this is perfect! And that Appendix is well written and definitive. (And I am pretty sure Peter taught me this in grad school, though that was awhile ago!) $\endgroup$ May 6, 2020 at 19:59
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    $\begingroup$ The comparison of the exact couples (and spectral sequences) derived from the cellular and Postnikov filtrations goes back to: Maunder, C. R. F. The spectral sequence of an extraordinary cohomology theory. Proc. Cambridge Philos. Soc. 59 (1963), 567–574. $\endgroup$ Jul 1, 2021 at 19:47
  • $\begingroup$ In terms of publication dates, the Adams spectral sequence (Comment. Math. Helv., 1958) predates the Atiyah-Hirzebruch spectral sequence (PSPM III, 1961). Is there an earlier reference for the AHSS? $\endgroup$ Jul 1, 2021 at 19:52
  • $\begingroup$ @JohnRognes I've always thought of the paper you mention as the `source' of the AHSS. So I guess you are right, the Adams SS is curiously older than the AHSS. The Atiyah-Hirzebruch paper is also introducing us to generalized cohomology theories. Just a year later, in 1962, Adams is happily computing with the AHSS in his Vector Fields on Spheres paper! $\endgroup$ Jul 2, 2021 at 1:45

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Tyler's comment answers my question. A bit more detail: the Postnikov tower of $k(n)$ is an Adams resolution, because the `bottom class' map $k(n) \rightarrow H\mathbb F_p$ is onto in mod $p$ cohomology; indeed $H^*(k(n);\mathbb F_p) = A_p//E(Q_n)$.

The appendix by Greenlees and May has the details that two spectral sequences converging to $\pi_*(Y \wedge X)$, one coming from filtering $Y$ by its Postnikov tower and the other by filtering $X$ by its skeleta, agree.

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