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

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

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?

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?