Algebraic K-theory of quotient stacks

TL;DR

This paper establishes key properties of equivariant algebraic K-theory for schemes with affine group actions, including localization, descent, and invariance under vector bundle morphisms, advancing the theoretical framework of equivariant K-theory.

Contribution

It proves fundamental properties like localization, excision, and descent for equivariant K-theory, and demonstrates invariance under vector bundle morphisms, extending the theoretical understanding of equivariant algebraic K-theory.

Findings

Proves localization, excision, and descent properties for equivariant K-theory.

Shows invariance of equivariant K-theory with finite coefficients under vector bundle morphisms.

Establishes nil-invariance and other key properties for equivariant homotopy K-theory.

Abstract

We prove some fundamental results like localization, excision, Nisnevich descent and the Mayer-Vietoris property for equivariant regular blow-up for the equivariant K-theory of schemes with an affine group scheme action. We also show that the equivariant K-theory with finite coefficients is invariant under equivariant vector bundle morphisms. We show that the equivariant homotopy K-theory is invariant under equivariant vector bundle morphisms and satisfies all the above properties, including nil-invariance.