Completion theorem for equivariant $K$-theory

TL;DR

This paper proves an algebraic analogue of the Atiyah-Segal completion theorem for smooth projective schemes in equivariant K-theory, but shows it does not hold universally for non-projective schemes.

Contribution

It establishes the completion theorem for algebraic equivariant K-theory of smooth projective schemes and identifies its limitations in non-projective cases.

Findings

Completion theorem verified for smooth projective schemes

Failure of the theorem in general for non-projective schemes

Provides insights into algebraic equivariant K-theory structure

Abstract

In this paper, we study the algebraic analogue of the topological Atiyah-Segal completion theorem. We verify this completion theorem for the algebraic equivariant $K$-theory of smooth projective schemes. We also show that the completion theorem fails in general for smooth non-projective schemes.