On Atiyah-Segal completion for T-equivariant Hermitian K-theory

TL;DR

This paper establishes an Atiyah-Segal type completion theorem for T-equivariant Hermitian K-theory of schemes with trivial T-action using derived completion, advancing the understanding of equivariant Hermitian K-theory.

Contribution

It introduces a derived completion approach to prove an Atiyah-Segal completion analogue for T-equivariant Hermitian K-theory, a step towards a broader theorem.

Findings

Derived completion proves the Atiyah-Segal completion for Hermitian K-theory.

Applicable to schemes with trivial T-action containing 1/2 and satisfying the resolution property.

Lays groundwork for a more general Atiyah-Segal completion theorem in Hermitian K-theory.

Abstract

We show how derived completion can be used to prove an analogue of Atiyah-Segal completion for the $T$-equivariant Hermitian K-theory of a scheme $X$ with a trivial $T$-action, containing $\tfrac{1}{2}$ and satisfying the resolution property, where $T$ is a split torus of rank $t$. This result is an important first step towards a more general Atiyah-Segal completion theorem for Hermitian K-theory.