The algebraic $K$-theory of Green functors

TL;DR

This paper develops computational tools, including a spectral sequence, to study the higher algebraic K-theory of Green functors, providing explicit calculations for specific cases and introducing the concept of Green meadow.

Contribution

It introduces a spectral sequence for G-Green functors, computes K-theory for specific Green functors, and defines Green meadow structures with freeness results.

Findings

Spectral sequence converges to algebraic G-theory of G-Green functors.

Complete calculation of K-theory for constant C2-Green functor over field with two elements.

Freeness of projective modules over G-Green meadow under mild conditions.

Abstract

In this paper we develop computational tools to study the higher algebraic $K$-theory of Green functors. We construct a spectral sequence converging to the algebraic $\mathbb{G}$-theory of any $G$-Green functor, for $G$ a cyclic $p$-group. From the spectral sequence we deduce a complete calculation of the algebraic $K$-theory of the constant $C_2$-Green functor associated to the field with two elements, and a calculation of the $p$-completion of the algebraic $K$-theory of the constant $G$-Green functor associated to the integers when $G$ is a cyclic $p$-group. Additionally, we introduce the notion of a Green meadow to abstract the Green functor structure underlying clarified Tambara fields, and show, under mild conditions, that every finitely generated projective module over a $G$-Green meadow is free when $G$ is a cyclic $p$-group. This gives a computation of $K_0$ for such Green functors.