Hypercomplete \'etale framed motives and comparison of stable homotopy groups of motivic spectra and \'etale realizations over a field

TL;DR

This paper develops an $l$-complete hypercomplete étale motivic homotopy theory, establishing new comparison theorems between motivic and étale stable homotopy groups over fields, extending classical results and providing new proofs.

Contribution

It introduces an $l$-complete hypercomplete étale analog of framed motives, proving recognition principles and comparison theorems that unify motivic and étale homotopy groups over fields.

Findings

Proves étale stable motivic connectivity theorem.

Establishes étale local isomorphism of motivic homotopy groups.

Provides new proofs of classical comparison isomorphisms.

Abstract

For any base field and integer $l$ invertible in $k$, we prove that $\Omega^\infty_{\mathbb{G}_m}$ and $\Omega^\infty_{\mathbb{P}^1}$ commute with hyper \'etale sheafification $L_{\acute{e}t}$ and Betti realization through infinite loop space theory in motivic homotopy theory. The central subject of this article is an $l$-complete hypercomplete \'etale analog of the framed motives theory developed by Garkusha and Panin. Using Bachman's hypercomplete \'etale \RigidityTheorem and the $\infty$-categorical approach of framed motivic spaces by Elmanto, Hoyois, Khan, Sosnilo, Yakerson, we prove the recognition principle and the framed motives formula for the composite functor \[\Delta^\mathrm{op}\mathrm{Sm}_k\to \mathrm{Spt}^{\mathbb{G}_m^{-1}}_{\mathbb{A}^1,\acute{e}t}(\mathrm{Sm}_k)\xrightarrow{\Omega^\infty_{\mathbb{G}_m}} \mathrm{Spt}_{\acute{e}t,\hat{n}}(\mathrm{Sm}_k).\] The first applications include the hypercomplete \'etale stable motivic connectivity theorem and an \'etale local isomorphism \[\pi^{\mathbb{A}^1,\mathrm{Nis}}_{i,j}(E)\simeq\pi^{\mathbb{A}^1,\acute{e}t}_{i,j}(E)\] for any $l$-complete effective motivic spectra $E$, and $j\geq 0$. Furthermore, we obtain a new proof for Levine's comparison isomorphism over $\mathbb C$, $\pi_{i,0}^{\mathbb{A}^1,\mathrm{Nis}}(E)(\mathbb{C})\cong \pi_i(Be(E))$, and Zargar's generalization for algebraically closed fields, that applies to an arbitrary base field.