Integral models for spaces via the higher Frobenius

Abstract

We give a fully faithful integral model for simply connected finite complexes in terms of E ∞ \mathbb {E}_{\infty } -ring spectra and the Nikolaus–Scholze Frobenius. The key technical input is the development of a homotopy coherent Frobenius action on a certain subcategory of p p -complete E ∞ \mathbb {E}_{\infty } -rings for each prime p p . Using this, we show that the data of a simply connected finite complex X X is the data of its Spanier-Whitehead dual, as an E ∞ \mathbb {E}_{\infty } -ring, together with a trivialization of the Frobenius action after completion at each prime. In producing the above Frobenius action, we explore two ideas which may be of independent interest. The first is a more general action of Frobenius in equivariant homotopy theory; we show that a version of Quillen’s Q Q -construction acts on the ∞ \infty -category of E ∞ \mathbb {E}_{\infty } -rings with “genuine equivariant multiplication,” which we call global algebras. The second is a “pre-group-completed” variant of algebraic K K -theory which we call partial K K -theory. We develop the notion of partial K K -theory and give a computation of the partial K K -theory of F p \mathbb {F}_p up to p p -completion.

On the strict Picard spectrum of commutative ring spectra

“…This is a generalization of the notion of p-perfect commutative ring spectra from [Yua23]. We are unaware of examples of perfect commutative ring spectra that are not -perfect.…”

On the Strict Picard Spectrum of Commutative Ring Spectra

“…Remark 4.7. This is a generalization of the notion of p-perfect commutative ring spectra from [Yua22]. We are unaware of examples of perfect commutative ring spectra that are not p-perfect.…”

“…(cf. [Yua22,Example 6.14], [Mao20, Proposition 2.7]) Let κ be a perfect ring of characteristic p. Then:…”

The simplicial coalgebra of chains under three different notions of weak equivalence

“…Furthermore, Mandell [28] proved a similar statement for finite type nilpotent spaces using the E ∞ -dg-algebra of singular cochains and also obtained an intrinsic description of the essential image of this functor. In another direction and using different techniques, Yuan [50] recently described a full and faithful integral model for finite type nilpotent spaces up to weak homotopy equivalence using the E ∞ -ring spectrum of spherical cochains.…”