Strict homotopy invariance via compactified homotopies and correspondences, and fibres of essentially smooth schemes over one-dimensional base schemes

Abstract

We develop the technique of compactified correspondences and homotopies over one-dimensional base schemes, and illuminate the perfectness and the inverting of characteristic assumptions from the celebrating Voevodsky's strict homotopy invariance theorem and its framed correspondences generalisation over an arbitrary base field. The assumption in this crucial theorem for Voevodsky's motives theory %over a field was kept from the origins of the study, and came later into more modern theory of framed motives by Garkusha-Panin. Applying the technique, we obtain also analogs of Gersten and Nisnevich conjectures for Cousin complexes of generalised motivic cohomotopies over a field, and acyclicity of Cousin complexes on generic fibres of essentially smooth local schemes over one-dimensional base schemes.