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 $k(n) \wedge X$.)



Are the AHSS and Adams spectral sequence the same when computing connective Morava K-theory of a space?
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.


