I know very little about algebraic topology, and more about $k$-linear stable $\infty$-categories (i.e. homological algebra).


Given an abelian group $A$, there is the Eilenberg-Mac Lane spectrum $HA$, for which: $\pi\_{< 0}(\cdot) = 0$, $\pi\_0 (\cdot) = A$, $\pi\_{>0}(\cdot) = 0$. There is also the Moore spectrum $SA$, for which $\pi\_{<0}(\cdot) = 0$, $\pi\_{=0} (\cdot)= A$, $H\_{> 0} (\cdot , \mathbb{Z}) = 0$.


I more-or-less have some feeling for $HA$ (under Dold-Kan, it is simply $A$, so one simply gets connected deloopings $BA$, $BBA$, etc. making the infinite loop space structure explicit).


I have no feeling what-so-ever about $SA$.


But reading about Bousfield localization, it seems that most the standard examples are with $SA$. How to understand this for a novice? Say, for $p$-localization and $p$-completion, how do I understand that I "want" to use the Moore spectrum and not the Eilenberg-Mac Lane spectrum?


Or, at least, what is the rationale for introducing Moore spectra?


Thank you



Intuition - difference between Moore spectrum and Eilenberg-Mac Lane spectrum
I'm not sure I can answer to the general question, but I can explain why the Moore spectrum $\mathbb{S}/p$ shows up in the discussion of Bousfield localizations.


This is just the cofiber of multiplication by $p$ from $\mathbb{S}$ to itself. Hence saying that for a spectrum $X$ we have $X\wedge \mathbb{S}/p=0$ means simply that multiplication by $p$ acts on $X$ as an equivalence. So $X$ is $\mathbb{S}[1/p]$-local iff it is $\mathbb{S}/p$-acyclic. From standard localization arguments (plus the fact that $\mathbb{S}/p$ is a finite spectrum) you get the fracture square relating $\mathbb{S}[1/p]$-localization and $p$-completion (i.e. the "complementary localization" of $\mathbb{S}[1/p]$-localization).


So, essentially, the reason it shows up is because the Moore spectrum is the layer of the diagram
$$ \mathbb{S} \xrightarrow{p} \mathbb{S}\xrightarrow{p}\mathbb{S}\xrightarrow{p}\mathbb{S}\xrightarrow{p}\cdots $$
that computes $\mathbb{S}[1/p]$-localization. From this it is clear that $\mathbb{S}/p^n$ is also going to be relevant to the study of $p$-completion (since these are just the "higher" layers of the tower), and from there to
$$\mathbb{S}/p^\infty = \mathbb{S}\mathbb{Q}\_p/\mathbb{Z}\_p = \mathrm{colim}\,\mathbb{S}/p^n$$
and
$$\mathbb{S}\mathbb{Q}/\mathbb{Z}=\bigvee\_p \mathbb{S}/p^\infty$$
is just a small step.


