D\'{e}vissage Hermitian Theory

TL;DR

This paper establishes dévissage theorems in Hermitian K-theory, enabling computations of GW groups for regular local rings with 1/2, extending classical K-theory results to Hermitian contexts.

Contribution

It proves dévissage theorems for Hermitian K-theory and related spectra, providing new computational tools for GW groups of regular local rings.

Findings

Proved dévissage theorems for GW spaces and spectra.

Computed GW groups for regular local rings with 1/2 in the ring.

Extended classical dévissage results to Hermitian K-theory.

Abstract

We prove D\'{e}vissage theorems for Hermitian $K$ Theory (or $GW$ theory), analogous to Quillen's D\'{e}vissage theorem for $K$-theory. For abelian categories ${\mathscr A}:=({\mathscr A}, ^{\vee}, \varpi)$ with duality, and appropriate abelian subcategories ${\mathscr B}\subseteq {\mathscr A}$, we prove D\'{e}vissage theorems for ${\bf GW}$ spaces, $G{\mathcal W}$-spectra and ${\mathbb G}W$ bispectra. As a consequence, for regular local rings $R$ with $1/2\in R$, we compute the ${\mathbb G}W$ groups ${\mathbb G}W^{[n]}_k(Spec{R})$ forall $k, n\in {\mathbb Z}$, where $n$ represent the translation.