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Moore spectrum

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Definition

For GG an abelian group, then the Moore spectrum SGS G (often MGM G) of GG is the spectrum characterized by having the following homotopy groups:

  1. π <0SG=0\pi_{\lt 0} S G = 0;

  2. π 0(SG)=G\pi_0(S G) = G;

  3. H >0(SG,)=π >0(SGH)=0H_{\gt 0}(S G,\mathbb{Z}) = \pi_{\gt 0}(S G \wedge H \mathbb{Z}) = 0.

Here HH \mathbb{Z} is the Eilenberg-MacLane spectrum of the integers.

Properties

Bousfield localization at Moore spectra

A basic special case of EE-Bousfield localization of spectra is given by E=SAE = S A the Moore spectrum of an abelian group AA. For A= (p)A = \mathbb{Z}_{(p)} this is p-localization and for A=𝔽 pA = \mathbb{F}_p this is p-completion.

Proposition

For A 1A_1 and A 2A_2 two abelian groups then the following are equivalent

  1. the Bousfield localizations at their Moore spectra are equivalent

    L SA 1L SA 2; L_{S A_1} \simeq L_{S A_2} \,;
  2. A 1A_1 and A 2A_2 have the same type of acyclicity, meaning that

    1. every prime number pp is invertible in A 1A_1 precisely if it is in A 2A_2;

    2. A 1A_1 is a torsion group precisely if A 2A_2 is.

(Bousfield 79, prop. 2.3) recalled e.g. in (VanKoughnett 13, prop. 4.2).

This means that given an abelian group AA then

  • either AA is not torsion, then

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

    where II is the set of primes invertible in AA and [I 1]\mathbb{Z}[I^{-1}] \hookrightarrow \mathbb{Q} is the localization at these primes of the integers;

  • or AA is torsion, then

    L SAL S( q𝔽 q), L_{S A }\simeq L_{S(\oplus_q \mathbb{F}_q ) } \,,

    where the direct sum is over all cyclic groups of order qq for qq a prime not invertible in AA.

Serre’s theorem

References

Lecture notes include

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, vol. 0(0), 2010, pp.1{28 (pdf)

Last revised on March 8, 2016 at 04:51:12. See the history of this page for a list of all contributions to it.