Non-representable six-functor formalisms

TL;DR

This work extends motivic homotopy theory to the realm of algebraic stacks by comparing the classical Nisnevich-based and extended NL-based constructions, proving their equivalence on NL-stacks, and developing a robust six-functor formalism for non-representable morphisms. It introduces a non-representable relative purity theory, ambidexterity, and formal Thom twist machinery, built via the category of correspondences and deformation-theoretic methods, and shows these structures extend to higher derived stacks with nil-invariance. The results yield a unified framework for motivic invariants on stacks, including a K-theory–Thomulus link via the Borel J-homomorphism and a Grothendieck–Verdier duality theory, enabling new duality and localization statements in motivic settings. Overall, the paper broadens the applicability of motivic six-functor formalisms from schemes to stacks and derived stacks, providing tools for computing and understanding motivic invariants in a broader geometric context.

Abstract

In this article, we study the properties of motivic homotopy category $\mathcal{SH}_{\operatorname{ext}}(\mathcal{X})$ developed by Chowdhury and Khan-Ravi for $\mathcal{X}$ a Nis-loc Stack. In particular, we compare the above construction with Voevodsky's original construction using NisLoc topology. Using the techniques developed by Liu-Zheng and Mann's notion of $\infty$-category of correspondences and abstract six-functor formalisms, we also extend the exceptional functors and extend properties like projection formula, base change and purity to the non-representable situation.

Theorems & Definitions (153)

• Theorem 1: Comparison \ref{['Sec.5:_cl_and_ext_SH_pointwise_equivalence']}
• Theorem 2: Non-Representable Purity
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Proposition 2.4
• Remark 2.5
• Theorem 2.6: Alper, Laumon2000
• proof
• proof : Proof of Proposition \ref{['smoothnisquassep']}