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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