Spectral sequences in unstable higher homotopy theory and applications to the coniveau filtration

TL;DR

This paper extends spectral sequence theory into the unstable motivic homotopy setting, introducing cohomotopy with supports and generalizing classical theorems to better understand $A^1$-homotopy sheaves and related structures.

Contribution

It develops an unstable spectral sequence framework and cohomotopy theory with supports, generalizing classical results to the unstable motivic homotopy context.

Findings

Extended spectral sequences to unstable motivic homotopy theory.

Generalized Bloch-Ogus-Gabber theorem for unstable cohomology.

Applied theory to motivic homotopy and étale homotopy types.

Abstract

With the aim of understanding Morel's result on the $\mathbb{A}^1$-homotopy sheaves over a field, we extend the theory of unstable spectral sequences of Bousfield and Kan in the $\infty$-categorical setting. With this natural extension, parallel to the classical formalism of cohomology theory with supports, we introduce the notion of cohomotopy theory with supports. We extend the Bloch-Ogus-Gabber theorem for Cohomology theory with supports to that of unstable setting, in order to obtain unstable Gersten (or Cousin) resolutions associated with the coniveau filtration, under suitable assumptions. We apply this theory to motivic homotopy, Nisnevich-local torsors and Artin-Mazur \'etale homotopy types.