Spectral sequences in unstable higher homotopy theory and applications to the coniveau filtration
Frédéric Déglise, Rakesh Pawar
TL;DR
This work develops an unstable analogue of the Bousfield–Kan spectral sequence theory inside ∞-categories, introducing cohomotopy theory with supports and unstable Gersten/Cousin resolutions to study coniveau filtrations. It constructs unstable coniveau exact couples, derives unstable spectral sequences, and proves a degeneracy criterion that yields E2-collapse under suitable boundedness and Gabber-type hypotheses. The authors extend Grothendieck–Bloch–Ogus methods to non-abelian settings, establish homotopy-Cohen–Macaulay properties for certain group-sheaves, and apply these ideas to motivic homotopy, Nisnevich torsors, and Artin–Mazur étale homotopy types, including an étale Gabber effaceability result. The framework unifies and generalizes unstable coniveau theory beyond A1-homotopy, enabling new computations and adelic formulas for torsors, with potential for an unstable motivic Adams spectral sequence. Overall, the paper provides a robust, internally consistent machinery for unstable homotopical filtrations with broad applications in algebraic geometry and hands-on tools for managing non-abelian cohomotopical data.
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 étale homotopy types.