Motivic Homotopy Theory of Algebraic Stacks

TL;DR

This paper extends motivic homotopy theory from schemes to a broad class of algebraic stacks, establishing a six functor formalism and verifying key properties like homotopy invariance and localization.

Contribution

It introduces a framework for motivic homotopy theory of algebraic stacks using $$-categories and the enhanced operation map, expanding the scope of the theory.

Findings

Established six functor formalism for algebraic stacks

Proved properties like homotopy invariance, localization, and purity

Included examples such as quotient stacks and moduli stacks

Abstract

The aim of this paper is to extend the definition of motivic homotopy theory from schemes to a large class of algebraic stacks and establish a six functor formalism. The class of algebraic stacks that we consider includes many interesting examples: quasi-separated algebraic spaces, local quotient stacks and moduli stacks of vector bundles. We use the language of $\infty$-categories developed by Lurie. Morever, we use the so-called 'enhanced operation map' due to Liu and Zheng to extend the six functor formalism from schemes to our class of algebraic stacks. We also prove that six functors satisfy properties like homotopy invariance, localization and purity.