Geometric models for fibrant resolutions of motivic suspension spectra

TL;DR

This paper develops geometric models for motivic spectra, specifically constructing fibrant replacements of suspension spectra using pairs of smooth ind-schemes, advancing the understanding of motivic homotopy theory.

Contribution

It introduces explicit geometric models for fibrant replacements of motivic suspension spectra within the framework of framed motives.

Findings

Constructed geometric models for $oldsymbol{ ext{P}^1}$-spectra.

Provided explicit models for positively motivically fibrant replacements.

Established equivalences with existing motivic spectra models.

Abstract

We construct geometric models for the $\mathbb P^1$-spectrum $M_{\mathbb P^1}(Y)$, which computes in Garkusha-Panin's theory of framed motives \cite{GP14} a positively motivically fibrant $\Omega_{\mathbb P^1}$ replacement of $\Sigma_{\mathbb P^1}^\infty Y$ for a smooth scheme $Y\in \Sm_k$ over a perfect field $k$. Namely, we get the $T$-spectrum in the category of pairs of smooth ind-schemes that defines $\mathbb P^1$-spectrum of pointed sheaves termwise motivically equivalent to $M_{\mathbb P^1}(Y)$.