The trivial fiber topology and framed motives over the integers

Abstract

This paper introduces the trivial fiber topology on schemes. For one-dimensional base schemes, we use it to describe fibrant replacements in the stable motivic homotopy category and motivic infinite loop spaces. We also extend the Garkusha-Panin and Voevodsky strict ${\mathbb{A}}^1$-invariance theorems to one-dimensional base schemes. The trivial fiber topology plays a central role in the proof of refined localization results for motivic homotopy categories. Moreover, we extend Morel's ${\mathbb{A}}^1$-connectivity theorem on Nisnevich sheaves of stable motivic homotopy groups. These results open new vistas for computations of motivic invariants over deeper base schemes of arithmetic interest.

To appear in Memoirs of the EMS