The Morel-Voevodsky Construction over Algebraic Stacks

TL;DR

The paper extends the Morel-Voevodsky stable motivic homotopy theory from schemes to algebraic stacks by defining $\mathbf{SH}(\mathscr{X})$ on the smooth-Nisnevich site and proving descent and functoriality. It proves that $\mathbf{SH}(\mathscr{X})$ agrees with stack models $\mathbf{SH}_{\triangleleft}(\mathscr{X})$ and $\mathbf{SH}_{\mathrm{ext}}(\mathscr{X})$, and develops framed motivic spectra with a reconstruction theorem. Rigidity in étale motivic homotopy is extended to stacks, and Hoyois reconstruction plus a universal six-functor formalism via cocomplete coefficient systems is established, including étale realisation that commutes with six operations. Applications to Deligne-Mumford stacks, coarse-space morphisms, and quotient stacks interpret motivic sheaves with group actions and enable descent arguments in the stack context, broadening motivic homotopy theory to stacks.

Abstract

In this article, we give a construction of the (un-)stable motivic homotopy category of an algebraic stack in the spirit of Morel-Voevodsky. We prove that this new construction agrees with the stable motivic homotopy category defined by Chowdhury and D'Angelo. As an application, we extend Bachmann's spectral rigidity theorem to algebraic stacks. Moreover, we extend the construction of the framed motivic homotopy category to algebraic stacks and prove Hoyois' Reconstruction Theorem in this setting. Finally, we discuss an extension of the formalism of cocomplete coefficient systems à la Drew-Gallauer to algebraic stacks.

Theorems & Definitions (85)

• Definition 1.1
• Example 1.2
• Theorem 1.3
• Proposition 1.4
• proof
• Definition 2.1
• Remark 2.2
• Remark 2.5
• Proposition 2.6
• proof