Let $E$ be a ring spectrum and $F$ a connective spectrum. Then we have a convergent Atiyah-Hirzebruch spectral sequence $H\_s(F,E\_t) \Rightarrow E\_{s+t}(F)$. Suppose now that $F$ is also a ring spectrum. Then does the multiplication on the $E^2$ page induce multiplications on all subsequent pages, and do they agree with the multiplication on $E\_\ast(F)$?


Bizarrely, the only reference I can find for the multiplicative properties of the Atiyah-Hirzebruch spectral sequence is [another MO question](https://mathoverflow.net/questions/225579/multiplicative-structure-of-the-atiyah-hirzebruch-leray-serre-spectral-sequence), and there is only treated the case of the cohomological AHSS where $F$ is a space, rather than the homological spectral sequence where $F$ is a ring spectrum.



Multiplicativity of the homology Atiyah-Hirzebruch spectral sequence for a ring spectrum
You can give a proof of multiplicativity by using that the smash product preserves connectivity. Here is a sketch proof.


(EDIT: Denis Nardin pointed me towards [this reference by Dugger](https://arxiv.org/abs/math/0305173). This reference shows that I was a little too cavalier about pairings in the homotopy category vs. lifting them to the stable category, and I've tried to make some adjustments accordingly.)


For any $n$, let $\tau\_{\geq n} E$ be the $(n-1)$-connected cover of $E$, assembling into the Whitehead tower
$$
\dots \to \tau\_{\geq 2} E \to \tau\_{\geq 1} E \to \tau\_{\geq 0} E \to \tau\_{\geq -1} E \to \dots \to E.
$$
The associated graded consists of the spectra $\tau\_{\geq n} E / \tau\_{\geq n+1} E = K(\pi\_n E, n)$.


The smash product of an $(n-1)$-connected object with an $(m-1)$-connected object is $(n+m-1)$-connected, and so the composite maps
$$
\tau\_{\geq n} E \wedge \tau\_{\geq m} E \to E \wedge E \to E
$$
lift, essentially uniquely up to homotopy, to $\tau\_{\geq n+m} E$. This means that, in the homotopy category, the tower $\{\tau\_{\geq n} E\}$ forms a *filtered algebra*. Because these lifts are essentially unique, this multiplication can also be lifted in a more structured one, lifting the Whitehead tower to a filtered algebra in spectra. (To be precise, a filtered algebra is a lax symmetric monoidal functor from the symmetric monoidal poset $(\Bbb Z, \geq, +)$ to spectra.)


The smash product is symmetric monoidal. Therefore, if $F$ is a ring spectrum the tower $\{(\tau\_{\geq n} E) \wedge F\}$ is also a filtered algebra, and the associated graded consists of the spectra $K(\pi\_n E, n) \wedge F$ because the smash product preserves cofibers. Therefore, the associated-graded spectral sequence starts with the homology of $F$ with coefficients in the homotopy groups of $E$: this constructs the Atiyah-Hirzebruch spectral sequence, except that the indexing is slightly different (the $E\_1$-term is a reindexed $E\_2$-term of the Atiyah-Hirzebruch SS).


Thus, this boils down to the following assertion:



> 
> If $\{\dots \to R^{-1} \to R^0 \to R^1 \to \dots\}$ is a (commutative / associative / unital) filtered algebra, then the associated-graded spectral sequence with
>  $$
> E^1\_{p,q} = \pi\_{p+q}(R^p / R^{p-1})
> $$
>  is multiplicative (and commutative / associative / unital).
> 
> 
> 


This is a little more standard. Dugger's reference that I linked to above gives a proof; he also says that he "found the existing literature extremely frustrating," which I don't think will surprise many people.


