TITLE: Are the AHSS and Adams spectral sequence the same when computing connective Morava K-theory of a space?
QUESTION [18 upvotes]: 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 $k(n) \wedge X$.)

REPLY [12 votes]: 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.