Decomposing the Diagonals of Invariant Fields

TL;DR

This paper explores the decomposition of diagonals as a method to analyze the rationality of invariant fields under finite group actions, introducing new tools and results linking group invariants and Sylow subgroups.

Contribution

It develops an open Chow zero concept, connects invariant fields to Galois extensions, and proves Sylow-type theorems for diagonal decomposition and division algebra centers.

Findings

Nontrivial unramified cohomology implies non-rationality of invariant fields.

Established Sylow properties for decomposition of the diagonal.

Proved Sylow theorems for centers of generic division algebras.

Abstract

This work begins the process of using the decomposition of the diagonal as a tool for studying the rationality of invariant fields of finite groups $G$. Our ground field must be characteristic 0 because of the use we make of Bertini theorems. The steps we take are, first, defining and studying an "open" version of Chow zero. Second, we use this to translate our study to that of a Chow group of $G$ Galois extensions. We prove a "Sylow" property and thereby yield a connection between the invariants of $G$ and that of its Sylow subgroups. In particular, we show that if $G$ is a finite group with $p$ Sylow subgroup $P$, $V$ is a faithful $G$ module, and $F(V)^P$ has nontrivial unramified cohomology, then $F(V)^G$ is not retract rational. Finally, we prove Sylow type theorems for decomposition of the diagonal and the centers of generic division algebras.