Motivic spectral Mackey functors

TL;DR

This paper establishes an equivalence between G-equivariant motivic stable homotopy categories and stabilized motivic G-spaces with finite étale transfers, introducing new norm constructions and extending homotopy t-structures to DM-stacks.

Contribution

It introduces an equivalence of categories under certain conditions, constructs motivic norms for stacks, and extends the homotopy t-structure to DM-stacks, advancing equivariant motivic homotopy theory.

Findings

Equivalence of G-equivariant motivic categories and stabilized motivic G-spaces.

Construction and analysis of norms in motivic homotopy theory of stacks.

Extension of homotopy t-structure to DM-stacks.

Abstract

We show that if G is a finite constant group acting on a scheme X such that the order of G is invertible in the residue fields of X, then the G-equivariant motivic stable homotopy category of X is equivalent to the stabilization of the category of motivic G-spaces with finite \'etale transfers over X at the trivial representation sphere. Along the way we obtain several results of independent interest, among them: we construct and study norms in the motivic homotopy theory of stacks, and we extend the homotopy t-structure to DM-stacks and establish some favorable properties.