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Let $S[q^{-1}]$ denote the homotopy colimit of the sequence $S\overset{q}\to S\overset{q}\to\cdots$ where $S$ denotes the sphere spectrum. Is it then the case that $S[q^{-1}]$ is the Moore spectrum of $\mathbb{Z}[q^{-1}]$? This seems intuitively obvious, and I imagine I'd like to make the argument: $[S,H\mathbb{Z}\wedge S[q^{-1}]]\cong[S,H(\mathbb{Z}[q^{-1}])]\cong \mathbb{Z}[q^{-1}], $ but I'm not sure how to get the first isomorphism. I believe to some this may be a standard fact or trivially true (unless of course it's false) but I'm interested in how to rigorously prove such a thing, i.e. what is the interaction between the functor $H:Rng\to Spectra$ with various algebraic operations?

Thanks!

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    lol yeah - I am crazy2014-07-11

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