Hypercomplete étale framed motives and comparison of stable homotopy groups of motivic spectra and étale realizations over a field

Abstract

For any base field and integer $l$ invertible in $k$, we prove that ${\Omega }_{{\mathbb{G}}_m}^{\infty }$ and ${\Omega }_{{\mathbb{P}}^1}^{\infty }$ commute with hyper étale 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 étale analog of the framed motives theory developed by Garkusha and Panin. Using Bachman's hypercomplete étale \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 \[Δ^\mathrm{op}\mathrm{Sm}_k\to \mathrm{Spt}^{\mathbb{G}_m^{-1}}_{\mathbb{A}^1,\acute{e}t}(\mathrm{Sm}_k)\xrightarrow{Ω^\infty_{\mathbb{G}_m}} \mathrm{Spt}_{\acute{e}t,\hat{n}}(\mathrm{Sm}_k).\] The first applications include the hypercomplete étale stable motivic connectivity theorem and an étale local isomorphism \[π^{\mathbb{A}^1,\mathrm{Nis}}_{i,j}(E)\simeqπ^{\mathbb{A}^1,\acute{e}t}_{i,j}(E)\] for any $l$-complete effective motivic spectra $E$, and $j\ge 0$. 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.