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am starting power operation, but nothing there yet except references
Gave power operation an Idea-section, based on Charles Rezk’s old MO comment (see the references).
By remark 2.2.9 in Jacob Lurie’s Rational and p-adic Homotopy Theory, the power operations in multiplicative cohomology are the refinement of the p-derivation/Fermat quotient/Frobenius homomorphism-map $(-)^p$ from arithmetic geometry to E-infinity arithmetic geometry.
In view of what it says at Borger’s arithmetic geometry – Motivation this is a conceptually very satisfying state of affairs. I have therefore added brief pointers to this to all the above entries, at the relevant places.
made that an MO question
Added an example provided by Akhil Mathew on power operations on $K(1)$-local $KU$-algebras, and their relation to the Lambda-ring-idea.
If your looking for an $E_{\infty}$ version of lamba rings and the $\mathbb{Z}/ \mathbb{F}_1$ relation, wouldn’t there need to be a spectrum-like form of the latter. I see Toën and Vaquié have $\mathbb{S}_1$,
l’anneau en spectres à un élément,
the ring spectrum with one element in Under Spec Z.
Maybe Morava and Santhanam are heading in this direction in Power operations and absolute geometry:
This is a sketch and summary of work in progress on the relation of power operations in homotopy theory (ie over $Spec S^0$) to those in absolute geometry (ie over $Spec \mathbb{F}_1$). In its current form it is nothing but a tissue of conjecture! Comments and suggestions are welcome.
There’s a construction of spaces which
… are just B-comonad coalgebras in the category of commutative $S^0$ -algebras [analogous to Borger’s interpretation [1] of $\mathbb{F}_1$-objects as algebras with a coaction of the $\Lambda$-ring comonad].
They end with a comment on $\beta$-rings which crop up in Guillot’s Adams operations in cohomotopy:
We study a collection of operations on the cohomotopy of any space, with which it becomes a “beta-ring”, an algebraic structure analogous to a lambda-ring. In particular, this ring possesses Adams operations, represented by maps on the infinite loop space of the sphere spectrum. We compute their effect in homotopy on the image of J, and in mod 2 cohomology. The motivation comes from the interpretation of the symmetric group as the general linear group of the “field with one element”, which leads to an analogy between cohomotopy and algebraic K-theory.
A good deal of this article may be considered as a survey of the theory of beta-rings.
This last reference appeared in an answer to this MO question.
Thanks! That looks interesting. You should post this as an answer to my MO question.
I have added pointers to these articles to the References-sections of the entries power operation and Borger’s absolute geometry.
Will have to look at this in more detail another time.
And Charles Rezk’s ICM talk Isogenies, power operations, and homotopy theory.
Thanks! I had been looking for these slides at some point, but didn’t find them. Have added the pointer here now.
I added a link to the associated article in #8.
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