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

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.