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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 24th 2017

created Tate diagonal

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeJul 24th 2017

And that can be seen as showing $\mathbb{S}^{t C_p}$ as the $p$-adic completion of $\mathbb{S}$, and so similarly for spectra bounded below (p. 68).

Perhaps this answers your question at the thread on cycltomic spectra, as to whether there’s anything special about $\mathbb{S}$ with trivial circle-action. It seems not, but they wanted to bring attention to this case as it’s used again through the article.