Geometric models for fibrant resolutions of motivic suspension spectra

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 $M_{{\mathbb{P}}^1}(Y)$.