# Localizing Ring Spectra at p as Bousfield Localization

I have learned the basic "procedure" for localizing a ring spectrum $R$ at a prime $p$, i.e. smashing with the $p$-local sphere $S_{(p)}$. Is this equivalent to applying $L_{S_{(p)}}$, the Bousfield localization functor, to $R$?

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I have, since beginning to write this question, received a simple answer in the affirmative which I have yet to check. However, I still thought it an interesting question to have documented. –  Jon Beardsley Jan 30 '12 at 17:48
some localizations are "smashing" some aren't. –  Sean Tilson Jan 30 '12 at 20:48
Thanks @Sean, I had heard that term before, but until you used it in this way, and I had thought of this particular issue, I didn't understand its relevance! Do you happen to know a good reference for localizations of spectra? All the references I have are only on spaces. Can I assume all basic theorems lift to spectra? –  Jon Beardsley Jan 30 '12 at 22:19
I think things work out better in spectra, although I am not sure. I think the best to do would be looking at what Ravenel has written about Bousfield localization. He has something in the orange book and then also in the paper where the Ravenel conjectures appear. Also, completion in topology is a Bousfield localization, I believe. The idea is that it should have the same effect on stable homotopy groups as applying the ordinary algebraic construction to rings. Also, Boardman has a little brief thing in his paper CCSS. –  Sean Tilson Jan 31 '12 at 1:02
It turns out that Bousfield has a paper called "Localization of spectra with respect to homology" which is rather nice. –  Jon Beardsley Jan 31 '12 at 17:23
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