${\mathbb{P}}^1$ Homotopy Theory of Schemes

Abstract

We study the ${\mathbb{P}}^1$-local motivic homotopy theory under the Nisnevich topology on $S{m}_S$. We show that although the homotopy category ${\mathcal{H}}^{{\mathbb{P}}^1}(-)$ admits standard schematic functorialities, it fails six-functor gluing, thereby obstructing a full six-functor formalism. On the unstable side, we prove that the unstable ${\mathbb{P}}^1$-motivic localization preserves Nisnevich connectivity, despite the absence of an interval structure on ${\mathbb{P}}^1$. Over perfect fields, we then establish the ${\mathbb{P}}^1$-invariance of the higher unstable homotopy group sheaves ${\pi }_i^{{\mathbb{P}}^1}$ ($i\ge 1$) using the Bloch-Ogus-Gabber theorem. On the stable side, we prove the stable ${\mathbb{P}}^1$-connectivity theorem over fields. As a consequence, we obtain a homotopy $t$-structure on the stable $\infty$-category of ${\mathbb{P}}^1$-invariant Nisnevich spectra, and deduce the strict ${\mathbb{P}}^1$-invariance of stable ${\mathbb{P}}^1$-homotopy group sheaves.