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Suppose $A$ is a commutative ring. By a "lift to the sphere" I mean a commutative ring spectrum $\mathbb{S}_A$ such that $A \simeq \mathbb{S}_A\otimes_{\mathbb{S}} \mathbb{Z}$ as commutative ring spectra, where $\mathbb{S}$ is the sphere spectrum. For which $A$ does this exist, and can it be made functorial?

If $A$ is a polynomial algebra over $\mathbb{Z}$, then the spherical polynomial rings are such lifts. More generally if $A = \mathbb{Z}[G]$ for abelian group $G$, then it lifts to $\mathbb{S}[G]$. Another example in the $p$-complete setting is when $A = W(k)$ is the $p$-typical Witt vectors (e.g. $k$ is a char. $p$ finite field, in which case we have the spherical Witt vectors).

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    $\begingroup$ @QiaochuYuan You cannot do this a priori: it is unclear how you could lift a simplicial diagram of polynomial rings to a simplicial diagram of ring spectra (I guess that you could lift edges, but not higher coherent homotopies). $\endgroup$
    – Z. M
    Commented Jul 16 at 8:41
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    $\begingroup$ There are known obstructions. If I understand correctly, for every prime $p$, there is a $\delta$-structure on $A$, coming from the Tate diagonal of $\mathbb S_A$ and the Segal conjecture. You should consult Thomas Nikolaus and Achim Krause — they thought about this (cf. Achim's talk at IHÉS in the K-theory workshop last summer). $\endgroup$
    – Z. M
    Commented Jul 16 at 8:43
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    $\begingroup$ @QiaochuYuan I see what you mean (I'm also a beginner in this subject so don't trust me too much). My first suspicion is that maps of polynomial rings cannot always be lifted to $E_\infty$ maps between their spherical counterparts. (They can always be lifted to $E_1$ maps but I don't think it is necessarily $E_\infty$, since the free $E_\infty$-algebra in one generator is not $\mathbb{S}[x]$ but something quite complicated.) $\endgroup$
    – atticusw
    Commented Jul 16 at 21:37
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    $\begingroup$ The construction for Witt vectors of finite fields was effectively given in "Schwänzl, R.; Vogt, R. M.; Waldhausen, F. Adjoining roots of unity to $\mathbb{E}_\infty$ ring spectra in good cases—a remark. Homotopy invariant algebraic structures (Baltimore, MD, 1998), 245–249, Contemp. Math., 239, Amer. Math. Soc., Providence, RI, 1999", and these authors also proved that $\mathbb{Z}[i]$ cannot be lifted to the sphere. $\endgroup$
    – John Rognes
    Commented Jul 22 at 15:50
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    $\begingroup$ You can lift polynomial rings ($\mathbb{S}[x_1,\ldots,x_n]$ can be written as suspension spectrum of the monoid $\mathbb{N}^n$), but not all maps between them. For example already $\mathbb{S}[x] $ does not admit an $E_\infty$ endomorphism taking $x\mapsto x+1$. $\endgroup$
    – Achim Krause
    Commented Jul 22 at 17:58

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