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 $k(n) \wedge X$.)

  • 11
    $\begingroup$ 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
  • 1
    $\begingroup$ @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
  • 1
    $\begingroup$ 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

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.