The Nisnevich Motive of an Algebraic Stack

TL;DR

This paper constructs the Nisnevich motive of algebraic stacks, providing a presentation for quotient stacks and establishing the equivalence of motivic cohomology with known Chow groups.

Contribution

It introduces a method to define the motive of algebraic stacks in the Nisnevich topology and relates motivic cohomology to existing Chow group theories.

Findings

Motivic cohomology of quotient stacks matches Edidin-Graham-Totaro Chow groups.

Provides a simplicial scheme presentation for stacks that are Nisnevich locally quotient.

Establishes a framework for studying algebraic stacks via motives in the Nisnevich topology.

Abstract

We construct the motive of an algebraic stack in the Nisnevich topology. For stacks which are Nisnevich locally quotient stacks, we give a presentation of the motive in terms of simplicial schemes. We also show that for quotient stacks the motivic cohomology agrees with the Edidin-Graham-Totaro Chow groups with integer coefficients.