I am following the nLab notes on the Adams spectral sequence, as they seem to be the most detailed I can find. That being said, I am still struggling to understand many of the steps. I have explained in detail where ambiguity arises in the notes, and I have summarzied my questions at the very end in a list.
In Theorem 2.33, it is asserted that under suitable coniditons on a commutative ring spectrum $E$ and spectra $X$ and $Y$, we have isomorphisms of $\mathbb Z$-graded abelian groups $$[X,E\wedge Y]_*\to\mathrm{Hom}^*_{E_*(E)}(E_*(X),E_*(E\wedge Y)),$$ where $[X,Y]_n:=[\Sigma^nX,Y]=[S^n\wedge X,Y]$, and $\mathrm{Hom}_{E_*(E)}^n(N_*,M_*):=\mathrm{Hom}_{E_*(E)}(N_{*-n},M_*)$ denotes the abelian group of strictly degree preserving comodule homomorphisms from the shifted comodule $N_{*-n}$ to $M_*$.
The proof of this proposition really only considers the degree $0$ case, and it only actually shows that under suitable conditions, the map $$E_*(-):[X,E\wedge Y]\to\mathrm{Hom}_{E_*(E)}^0(E_*(X),E_*(E\wedge Y)),\qquad f\mapsto E_*(f)$$ is an isomorphism of abelian groups. It never actually explains what this arrow does to maps of the form $\Sigma^nX\to E\wedge Y$ for $n\neq0$. So how do we actually get the $\mathbb Z$-graded isomorphisms?
This is not too bad, supposing we have natural isomorphisms in $E_*(E)\text-\mathbf{CoMod}^\mathbb{Z}$ $$t^n_X:E_{*-a}(X)\to E_*(\Sigma^nX),$$ we can get the desired maps $[X,E\wedge Y]_n\to\mathrm{Hom}^n_{E_*(E)}(E_*(X),E_*(E\wedge Y))$ as the compositions $$[\Sigma^nX,E\wedge Y]\xrightarrow{E_*(-)}\mathrm{Hom}_{E_*(E)}(E_*(\Sigma^nX),E_*(E\wedge Y))\xrightarrow{{(t^n_X)}^*}\mathrm{Hom}_{E_*(E)}(E_{*-n}(X),E_*(E\wedge Y)).$$ Note that there are lots of ways to define these $t^n$'s. For example, we can define $t^n_X$ to be the composition $$E_{*-n}(X):=[S^{*-n},E\wedge X]\xrightarrow{-\wedge S^n}[S^{*-n}\wedge S^n,E\wedge X\wedge S^n]\xrightarrow{{(\phi_{*-n,n})}^*}[S^*,E\wedge X\wedge S^n]\xrightarrow{E\wedge\tau}[S^*,E\wedge S^n\wedge X]=E_*(\Sigma^nX),$$ where here for $n,m\in\mathbb Z$, $\phi_{n,m}:S^{n+m}\xrightarrow\cong S^{n}\wedge S^{m}$ is the canonical isomorphism, and $\tau$ denotes the braiding isomorphism in the stable homotopy category. It is straightforward to check that this construction of $t^n_X$ yields a natural $\mathbb Z$-graded isomorphism of left $E_*(E)$-comodules.
Then, in Theorem 2.34, this proposition is applied to characterize the $E_1$ page of the $E$-Adams spectral sequence. Namely, by the proposition, we have isomorphisms $$E_1^{s,t}(X,Y):=[X,E\wedge Y_s]_{t-s}\xrightarrow{E_*(-)}\mathrm{Hom}_{E_*(E)}^{t-s}(E_*(X),E_*(E\wedge Y_s)).$$ Furthermore, we have a differential $d_1:E_1^{s,t}(X,Y)\to E_1^{s+1,t}(X,Y)$, i.e., it is a map $$d_1:[X,E\wedge Y_s]_{t-s}\to[X,E\wedge Y_{s+1}]_{t-s-1},$$ which is induced by maps $h:E\wedge Y_s\to \Sigma^1Y_{s+1}$ and $g:Y_{s+1}\to W_{s+1}$. As above, the notes never actually explicitly define what this differential is, instead they say $d_1$ is the map $$[X,g\circ h]_*:[X,E\wedge Y_s]_*\to[X,E\wedge Y_{s+1}]_{*-1},$$ when in reality $[X,g\circ h]_*$ would simply be a map $$[X,E\wedge Y_s]_*\to[X,\Sigma^1(E\wedge Y_{s+1})]_*.$$ Obviously we know $[X,\Sigma^1(E\wedge Y_{s+1})]_*$ is isomorphic to $[X,E\wedge Y_{s+1}]_{*-1}$, but that isomorphism is certainly not unique, and there are lots of ways to construct it.
Finally, Theorem 2.34 asserts (but does not prove) that under this identification of the $E_1$-page $$E_1^{s,t}(X,Y)\cong\mathrm{Hom}_{E_*(E)}^{t-s}(E_*(X),E_*(E\wedge Y_s)),$$ that the differential $d_1:E_1^{s,t}(X,Y)\to E_1^{s+1,t}(X,Y)$ is given by "$\mathrm{Hom}_{E_*(E)}(E_*(X),E_*(g\circ h))$". In other words, the following diagram commutes
I have spent hours trying to figure out what the correct interpretation of all the maps in this diagram are, and chase elements around to show the diagram commutes, and nothing I have tried has worked. Thus I am completely stuck
So, the questions I would like answered are:
- What exactly does the differential $d_1:[X,E\wedge Y_s]_*\to[X,E\wedge Y_{s+1}]_{*-1}$ in the $E$-Adams spectral sequence do to elements of nonzero degree?
- What exactly does the map of $\mathbb Z$-graded abelian groups $E_*(-):[X,Y]_*\to\mathrm{Hom}^*_{E_*(E)}(E_*(X),E_*(Y))$ do to elements of nonzero degree?
- How do we show that the above assignment is functorial with respect to elements of nonzero degree? In other words, how can I show that the above map $E_*(-)$ sends the differential $d_1$ to the pushforward of $E_*(g\circ h)$? I.e., how exactly should we define $E_*(g\circ h)$, and how do we show the above diagram commutes?
Bonus: I am able to construct everything in a way that works perfectly as expected if I can find natural isomorphisms of $E_*(E)$-comodules $t^n_X:E_{*-n}(X)\to E_*(\Sigma^nX)$ which are coherent, in that they make the following diagrams commute for all spectra $X$ and integers $n$ and $m$
(In fact, I really only need the right diagram to commute for arbitrary $m$ and $n=1$ here). The $t^n$'s I have constructed above make the left diagram commute, but not the right. Sadly, I can not find any $t^n$'s which work. (I have tried basically everything I can think of). Does anyone know how to do this?
EDIT: After looking into this more, I have become increasingly convinced that maybe Theorem 2.34 in the nLab page for the Adams spectral sequence is not true at all, at least the part which characterizes the differential under the given isomorphisms. Namely, I believe such a characterization would require $t^n$'s as above which satisfy the given coherence conditions, but no $t^n$'s can, as one will always run into a sign error as a result of the graded commutativity of the stable homotopy category. I would love to be proven wrong about this.