A $\Lambda$-structure on a commutative ring $R$ is a ring endomorphism wich restricts to the $p$-Frobenius homomorphism after localizing at $(p)$. One may think of this as a "flow" $\Phi \colon Spec(R) \longrightarrow Spec(R)$ in arithmetic geometry, which lifts the Fermat p-derivation on the base $Spec(\mathbb{Z})$. If we allowed ourselves to denote derivations as endomorphisms, then with slight but very suggestive abuse of notation we have the picture

$$ "\array{ Spec(R) &\stackrel{\Phi + p \cdot \partial_p^{\Phi}}{\longrightarrow}& Spec(R) \\ \downarrow && \downarrow \\ Spec(\mathbb{Z}) &\stackrel{(-)^p = id + p\cdot \partial_p}{\longrightarrow}& Spec(\mathbb{Z}) } " $$

See on the $n$Lab at Borger's absolute geometry -- Motivation for more on what I have in mind here, following ideas famously promoted by James Borger and Alexadru Buium.

I would like to know if there is a sensible generalization of this from arithmetic geometry to $E_\infty$-arithmetic geometry, hence from commutative rings $R$ to $E_\infty$-rings.

Via discussion which is clearly articulated for instance starting from remark 2.2.9 in Jacob Lurie's DAGXIII Rational and p-adic homotopy theory, the $E_\infty$-analog of "this" are the power operations in multiplicative cohomology theory.

I am a little shaky on some details though. Therefore my question: what would be the good generalization of the concept of $\Lambda$-rings to $E_\infty$-algebra in the sense of Frobenius lifts and with an eye towards absolute geometry, as above? Can one say anything?

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    $\begingroup$ It's not really clear what it means to "lift Frobenius" in the setting of $E_\infty$-rings, because it is not clear what "Frobenius" means. There are relatively few $E_\infty$-rings which are "characterstic $p$" (those that exist must have underlying spectra which are products of Eilenberg-Mac Lane spectra). Further, there is no natural Frobenius endomorphism on the class of char $p$ $E_\infty$-ring spectra. $\endgroup$ – Charles Rezk Aug 17 '14 at 7:27
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    $\begingroup$ Thanks, Charles! I was hoping you would see this, despite being busy at ICM. Below in the comments to Akhil Mathew's answer it seems that Elden Elmanto is thinking of some general concept of Frobenius lifts for at least some class of $E_\infty$-rings. (?) The story of $\Lambda$-rings in view of $\mathbb{F}_1$ suggests that what matters for having a Frobenius-like endomorphism is not characteristic $p$ as such, but reduction to a maximal ideal. From this analogy I would expect $E_\infty$-analogs of Frobenius at each prime for $K(n)$-local spectra. Any chance for that? $\endgroup$ – Urs Schreiber Aug 18 '14 at 9:58
  • $\begingroup$ As David Corfield kindly points out to me elsewhere: explicit comments/speculation on relation between power operations and Borger-style absolute geometry is in Morava-Santhanam 12 (ncatlab.org/nlab/show/power+operation#MoravaSanthanam12) referring to closely related discussion in Guillot 06 (ncatlab.org/nlab/show/power+operation#Guillot06) $\endgroup$ – Urs Schreiber Aug 21 '14 at 10:00

Although I don't have much to say here, perhaps the following example is worth pointing out. Let $R$ be a $K(1)$-local $E_\infty$-ring under ($p$-adic) complex $K$-theory $KU$. Then there exists a power operation $\theta: \pi_0 R \to \pi_0 R$ such that:

  • $\psi(x) \stackrel{\mathrm{def}}{=} x^p + p \theta(x)$ defines a ring homomorphism from $\pi_0 R \to \pi_0 R$.
  • $\theta$ satisfies all the identities needed to make $\psi$ a ring-homomorphism after "division by p." For instance, $\psi(x+y) = \psi(x) + \psi(y)$ implies that $$\theta(x+y) = \theta(x) + \theta(y) + \frac{x^p - y^p - (x+y)^p}{p},$$ where the last term is an integral polynomial in $x,y$ and is interpreted as such.

$\theta$ is the basic power operation for $K(1)$-local $E_\infty$-ring spectra, as explained in these notes of Hopkins, and the algebraic structure it gives is called a "$\theta$-algebra." (These also seem to be called $p$-derivations by algebraists.)

Note in particular that $\psi$ is a lift of the Frobenius. There are generalizations of $\psi, \theta$ at higher chromatic levels, too, and there (as I understand) a modular interpretation of the resulting algebraic structure in this paper of Rezk.

  • $\begingroup$ This is excellent, thank you. I have taken the liberty of recording a version of your reply here: ncatlab.org/nlab/show/power+operation#OnK1LocalKUAlgebras $\endgroup$ – Urs Schreiber Aug 15 '14 at 19:23
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    $\begingroup$ Just to make the correspondence between power operations in Morava $E$-theory and Frobenius lifts precise we have the theorem of Strickland that says: if $G$ is the formal group associated to a Morava $E$-theory, then Frobenius lifts (which corresponds to degree $p^k$ subgroups of $G$) are classified by maps into $E^0(B\Sigma_{p^r})/I_{tr}$ where $I_{tr}$ is the transfer ideal. So, for example, the map $\psi$ above is corresponds to a universal map $KU^0 \rightarrow KU^0(B\Sigma_p)/I_{tr} \simeq KU^0$. $\endgroup$ – Elden Elmanto Aug 15 '14 at 23:57
  • $\begingroup$ Thanks! You mean Strickland 98 (ncatlab.org/nlab/show/power+operation#Strickland98). Also, you seem to think of some established theory of Frobenius lifts in this case, maybe you could provide further pointers to the literature for that? $\endgroup$ – Urs Schreiber Aug 18 '14 at 7:18

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