I'm reading Switzer's "Algebraic Topology", which talks about the (homology) ASS in chapter 19. His ASS (where he puts $S^0$ in the first slot) either converges to $\pi_n(Y)$, or to $\pi_n(Y)/\cap_{s\geq 0}F^{s,n+s}$ (which happens e.g. when our theory $E$ is the spectrum $H\mathbb{F}_p$ representing $H^*(-;\mathbb{F}_p)$).

Switzer doesn't talk about this at all, but I've seen/heard before that one can also present the ASS as converging to something denoted $[L_EX, L_EY]$. I don't know much about the functor $L_E:Top\rightarrow Top$, but I believe it is characterized as follows: If $E_*X=0$ then we say that $X$ is *$E$-acyclic*. Now, a space $Y$ is called *$E$-local* if whenever $X$ is $E$-acyclic, $[X,Y]=0$. Then, the *$E$-localization* $L_EY$ of $Y$ is an initial object in the category of $E$-local spaces under $Y$.

So here is what I'm wondering. The next result after the ASS is the following Proposition 19.11: If $\iota_*:\pi_q(S^0)\rightarrow \pi_q(E)$ is an isomorphism for $q\leq 0$ and an epimorphism for $q=1$ and if $\pi_q(Y)=0$ for $q\lt N$ for some $N\in \mathbb{Z}$, then $A^{s,t}=D^s=0$ for all $s,t$, and hence the Adams spectral sequence converges to $\pi_*(Y)$. (Here $\iota:S^0\rightarrow E$ is the unit map.) This seems very much like a fact about localization in disguise -- presumably it'd be saying something about $L_EY$, perhaps that $Y=L_EY$. Except that I don't know what $L_ES^0$ is, which I'd imagine should be the first thing anyone would try to figure out.

So I guess my questions are:

- Do I have the right characterization of localization?
- Assuming the answer to (a) is yes (or I guess even if it isn't), what is this proposition saying about localization?
- Replacing $S^0$ with an arbitrary $X$, in Switzer's setup we can hope for the ASS to give us $[X,Y]/\cap_{s\geq 0}F^{s,*+s}$ -- how can we pass between this quotient and $[L_EX,L_EY]$? Is there anything more geometric to say than that the denominator is the subgroup of maps inducing 0 in homology?