$E_\infty$ Structures from Frobenius Lifts

Definitions and mathematical statement

Let $\mathbb S$ denote the sphere spectrum. Following the supplied flatness condition, a spectrum $A$ is flat over $\mathbb S$ if its abelian group $R=\pi_0(A)$ is flat over ${\mathbb{Z}}=\pi_0(\mathbb S)$, and the maps induced by the sphere action are isomorphisms \[ R\otimes_{{\mathbb{Z}}}\pi_n(\mathbb S)\xrightarrow{\ \cong\ }\pi_n(A) \qquad(n\in{\mathbb{Z}}). \] Here flatness of $R$ means that $R\otimes_{{\mathbb{Z}}}-$ preserves exact sequences of abelian groups.

Equip $R$ with a commutative ring structure extending its given abelian group structure. For each prime $p$, write \[ \widehat R_p=\varprojlim_{k\geq1}R/p^kR, \] and prescribe a Frobenius lift $\varphi_p:R\to\widehat R_p$. Under the usual ring-theoretic interpretation, this is a ring homomorphism satisfying \[ \rho_p(\varphi_p(r))=r^p+pR \qquad(r\in R), \] where $\rho_p:\widehat R_p\to R/pR$ is the canonical projection. The supplied excerpt does not explicitly define completion or Frobenius lift; these formulas state the conventional interpretation used in this draft. The unit convention and any additional compatibility conditions between different primes require confirmation from the full source.

An $E_\infty$ ring structure means a multiplication and unit on a spectrum with the coherent commutativity and associativity encoded by an $E_\infty$ operad. The supplied Proposition 7 associates to a flat $E_\infty$ ring spectrum a canonical map \[ A\longrightarrow (A^{\otimes p})^{tC_p} \longrightarrow A^{tC_p}\simeq A_p^{\wedge}. \] Here $\otimes$ is the smash product over $\mathbb S$, $C_p$ acts on $A^{\otimes p}$ by cyclic permutation, and $(-)^{tC_p}$ denotes the Tate construction. In the displayed target $A^{tC_p}$, the action on $A$ is trivial. The source describes the resulting maps on $\pi_0$ as the Frobenius lifts above; its completion identification is retained as a supplied assertion.

Problem. For every flat spectrum $A$, characterize precisely the data \[ \left(R,\text{ commutative ring structure},(\varphi_p)_{p\text{ prime}}\right) \] that are induced by an $E_\infty$ ring structure on $A$. More explicitly, determine necessary and sufficient conditions for there to exist an $E_\infty$ ring spectrum $B$ and an equivalence of underlying spectra $e:B\simeq A$ such that $\pi_0(e)$ transports both the ring structure and every induced Frobenius lift to the prescribed data on $R$. For realizable data, determine how much of the $E_\infty$ structure is determined by those data, including whether the extension is unique.

The uniqueness target remains unresolved at the level of formulation: the supplied record does not specify whether it asks for equivalence classes of unmarked ring spectra, equivalences compatible with the identification with $A$, or a contractible space of extensions. These are distinct assertions and are not identified here.

As supplied background, Theorem 9 states that $H_\infty$ structures on a flat spectrum are fully determined by the commutative ring on $\pi_0$ and its Frobenius lifts. This is recorded as the source's partial result, without asserting that every prescribed family is realizable or that it determines an $E_\infty$ structure. The original Question 8 asks broadly how much of the $E_\infty$ structure is determined; the registry's reconstructed statement explicitly asks for an existence characterization and uniqueness.

Short English statement

For a spectrum flat over the sphere, determine which commutative ring structures on its zeroth homotopy group, together with a Frobenius lift at every prime, come from an $E_\infty$ ring structure. Determine also the extent of uniqueness when such a structure exists. The source supplies an $H_\infty$ partial result, while the precise meaning of uniqueness and some conventions require clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006652$, title: $E_\infty$ Structures from Frobenius Lifts; exact source problem code catalogue entry. The record attributes the question to an Oberwolfach workshop report, gives proposed year 2026, and names no individual proposer.
• Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, pp. 541--610, DOI 10.4171/owr/2026/9: https://doi.org/10.4171/owr/2026/9. The supplied excerpt identifies Question 8, Proposition 7, and Theorem 9. It attributes the observation underlying Proposition 7 to Nikolaus--Scholze.
• Carmeli--Nikolaus--Yuan, Maps between spherical group rings: https://arxiv.org/abs/2405.06448. Listed in the supplied literature sources; its specific bearing on the question is not detailed in the record.
• Nikolaus--Scholze, On topological cyclic homology: https://doi.org/10.4310/ACTA.2018.v221.n2.a1. Listed in the supplied literature sources and associated by the excerpt with the Frobenius construction.
• Supplied literature assessment checked on 2026-08-22: the report is said to determine $H_\infty$ structures from the ring and Frobenius lifts while leaving extension and uniqueness of $E_\infty$ structures unresolved. The registry labels the problem partially_solved. This is supplied dated curation, not an independently verified assessment. The record also labels its reconstructed statement corrected_verified and says it was checked against the official report; that verification claim is supplied metadata and has not been independently checked here.