For $G$ an abelian group, then the Moore spectrum $S G$ (often $M G$) of $G$ is the spectrum characterized by having the following homotopy groups:
$\pi_{\lt 0} S G = 0$;
$\pi_0(S G) = G$;
$H_{\gt 0}(S G,\mathbb{Z}) = 0$.
Here $H \mathbb{Z}$ is the Eilenberg-MacLane spectrum of the integers and $H_\ast(S G,\mathbb{Z})= \pi_{\ast}(S G \wedge H \mathbb{Z})$.
A basic special case of $E$-Bousfield localization of spectra is given by $E = S A$ the Moore spectrum of an abelian group $A$. For $A = \mathbb{Z}_{(p)}$ this is p-localization and for $A = \mathbb{F}_p$ this is p-completion.
For $A_1$ and $A_2$ two abelian groups then the following are equivalent
the Bousfield localizations at their Moore spectra are equivalent
$A_1$ and $A_2$ have the same type of acyclicity, meaning that
every prime number $p$ is invertible in $A_1$ precisely if it is in $A_2$;
$A_1$ is a torsion group precisely if $A_2$ is.
(Bousfield 79, prop. 2.3) recalled e.g. in (VanKoughnett 13, prop. 4.2).
This means that given an abelian group $A$ then
either $A$ is not torsion, then
where $I$ is the set of primes invertible in $A$ and $\mathbb{Z}[I^{-1}] \hookrightarrow \mathbb{Q}$ is the localization at these primes of the integers;
or $A$ is torsion, then
where the direct sum is over all cyclic groups of order $q$ for $q$ a prime not invertible in $A$.
Frank Adams, Part III.6 of Stable homotopy and generalised homology, 1974
Stefan Schwede, section II.6.3 of Symmetric spectra, 2012 (pdf)
Lecture notes include
Neil Strickland, p. 9 of An introduction to the category of spectra (pdf)
Paul VanKoughnett, Spectra and localization, 2013 (pdf)
John Rognes, section 4.6 of The Adams spectral sequence, 2012 (pdf)
Discussion in the context of Bousfield localization of spectra is in
See also
Last revised on November 14, 2023 at 20:10:52. See the history of this page for a list of all contributions to it.