Injectivity and Surjectivity of the Dress Map

TL;DR

This paper investigates the conditions under which the Dress map from the Burnside ring to the Grothendieck-Witt ring is injective or surjective in the context of finite Galois extensions, revealing key algebraic properties.

Contribution

It characterizes injectivity and surjectivity of the Dress map in terms of quadratic extensions and the field's quadratic closure, providing new insights into the structure of Galois extensions.

Findings

Dress map is injective iff $L=k( oot{ ot 2} exists ext{sum of squares}$

Dress map is surjective iff $k$ is quadratically closed in $L$

Results give necessary conditions for faithfulness and fullness of the Heller-Ormsby functor

Abstract

For a nontrivial finite Galois extension $L/k$ (where the characteristic of $k$ is different from 2) with Galois group $G$, we prove that the Dress map $h_{L/k}: A(G) \to GW(k)$ is injective if and only if $L=k(\sqrt{\alpha})$ where $\alpha$ is not a sum of squares in $k^\times$. Furthermore, we prove that $h_{L/k}$ is surjective if and only if $k$ is quadratically closed in $L$. As a consequence, we give strong necessary conditions for faithfulness of the Heller-Ormsby functor $c_{L/k}^* : \text{SH}_G\to \text{SH}_k$, as well as strong necessary conditions for fullness of $c_{L/k}^*$.