Motivic homotopy theory of group scheme actions

TL;DR

This paper develops an equivariant motivic homotopy theory for algebraic group actions over schemes, proving representability of equivariant K-theory and establishing key theorems like purity and blow-up for finite abelian groups.

Contribution

It introduces an unstable equivariant motivic homotopy category for algebraic groups and proves foundational theorems including K-theory representability and homotopical purity.

Findings

Equivariant algebraic K-theory is representable in the new homotopy category.

Established homotopical purity and blow-up theorems for finite abelian groups.

Developed a framework for equivariant motivic homotopy theory over schemes.

Abstract

We define an unstable equivariant motivic homotopy category for an algebraic group over a Noetherian base scheme. We show that equivariant algebraic $K$-theory is representable in the resulting homotopy category. Additionally, we establish homotopical purity and blow-up theorems for finite abelian groups.