Strict homotopy invariance via compactified homotopies and correspondences, and fibres of essentially smooth schemes over one-dimensional base schemes
Andrei Druzhinin
TL;DR
The work extends Voevodsky's strict $\mathbb A^1$-invariance beyond base-field assumptions by developing compactified correspondences and focused one-dimensional base constructions. It proves that for an $\mathbb A^1$-invariant quasi-stable framed presheaf $F$ over a field $k$, the cohomology functors satisfy $H^n_\mathrm{Nis}(\mathbb A^1_k\times X,F_\mathrm{Nis})\cong H^n_\mathrm{Nis}(X,F_\mathrm{Nis})$ (and the corresponding Zariski/Nisnevich compatibilities), using a two-stage strategy that moves cohomology classes to infinity and then contracts them via compactified homotopies. The paper introduces compactified $d$-dimensional framed correspondences over one-dimensional bases, proving a universal endo-correspondence and a contracting one-dimensional homotopy that together yield vanishing results for cohomologies on generic fibres; these feed into analogs of Gersten and Nisnevich conjectures for Cousin complexes and their acyclicity on generic fibres. As a culmination, strict homotopy invariance is established in greater generality and the Cousin complex framework is shown to be acyclic, enabling new applications to square-homotopy theories and reciprocity-sheaf contexts. Overall, the results broaden the applicability of motivic homotopy theory to non-perfect base fields and higher-dimensional bases, with potential impact on stability phenomena and structural computations in motivic homotopy categories.
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.