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

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 $$k(n) \wedge X$$.)

• 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 $H$. – Tyler Lawson May 6 at 19:19
• @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!) – Nicholas Kuhn May 6 at 19:59

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.