In Section 11 of their paper https://arxiv.org/pdf/1802.03261, Bhatt-Morrow-Scholze discuss the polynomial algebra over the sphere spectrum. I'm wondering whether its possible to define a notion of formal power series over the sphere spectrum and, if so, where this is discussed?
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1$\begingroup$ The formal power series ring $\mathbb S[\![t]\!]$ is just the $t$-completion of the polynomial ring $\mathbb S[t]$: the $t$-completion functor $D(\mathbb S[t])\to D(\mathbb S[t])$ carries a canonical lax symmetric monoidal, thus it sends an $E_\infty$-algebra to an $E_\infty$-algebra. $\endgroup$– Z. MCommented Oct 7 at 21:18
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$\begingroup$ @Z.M What's a reference for this kind of completion? $\endgroup$– onefishtwofishCommented Oct 7 at 23:24
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$\begingroup$ Lurie's Spectral Algebraic Geometry, and more precisely, section 7.3.5. For this particular formal power series, you even have $\mathbb S[\![t]\!]=\lim_n\mathbb S[t]/t^n$ where each $\mathbb S[t]/t^n$ is a monoidal ring. $\endgroup$– Z. MCommented Oct 8 at 8:40
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$\begingroup$ @Z.M Thank you I appreciate your reply. $\endgroup$– onefishtwofishCommented Oct 8 at 13:53
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