nLab Moore spectrum

Definition

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:

1. $\pi_{\lt 0} S G = 0$;

2. $\pi_0(S G) = G$;

3. $H_{\gt 0}(S G,\mathbb{Z}) = 0$.

Here $H \mathbb{Z}$ is the Eilenberg-MacLane spectrum of the integers and $H_\ast(S G,\mathbb{Z})= \pi_{\ast}(S G \wedge H \mathbb{Z})$.

Properties

Bousfield localization at Moore spectra

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.

Proposition

For $A_1$ and $A_2$ two abelian groups then the following are equivalent

1. the Bousfield localizations at their Moore spectra are equivalent

   $$L_{S A_1} \simeq L_{S A_2} \,;$$

2. $A_1$ and $A_2$ have the same type of acyclicity, meaning that

   1. every prime number $p$ is invertible in $A_1$ precisely if it is in $A_2$;

   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

  $$L_{S A} \simeq L_{S \mathbb{Z}[I^{-1}]} \,,$$

  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

  $$L_{S A }\simeq L_{S(\oplus_q \mathbb{F}_q ) } \,,$$

  where the direct sum is over all cyclic groups of order $q$ for $q$ a prime not invertible in $A$.

Serre’s theorem

• For $\mathbb{Q}$ the rational numbers there is an equivalence $S \mathbb{Q} \stackrel{\simeq}{\longrightarrow} H \mathbb{Q}$ between the Moore spectrum and the Eilenberg-MacLane spectrum. (e.g. Banagl 10, p. 6)

Related concepts

• Eilenberg-MacLane spectrum

• Bousfield localization of spectra

References

• 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

• Aldridge Bousfield, section 2 of The localization of spectra with respect to homology , Topology vol 18 (1979) (pdf)

See also

• Markus Banagl, Rational generalized intersection homology theories, Homology, Homotopy and Applications, 12 1 (2010) 157-185 [pdf, euclid]