Functoriality of the canonical fractional Galois ideal

TL;DR

This paper extends the fractional Galois ideal to broader Galois extensions, explores its naturality, and discusses applications in non-commutative Iwasawa theory, aiming to deepen understanding of number field invariants.

Contribution

It generalizes the fractional Galois ideal to arbitrary Galois extensions and proves its naturality properties under extension changes.

Findings

Extended the fractional Galois ideal to non-abelian Galois extensions.

Proved naturality properties under canonical extension changes.

Discussed applications to non-commutative Iwasawa algebras.

Abstract

The fractional Galois ideal of [Victor P. Snaith, Stark's conjecture and new Stickelberger phenomena, Canad. J. Math. 58 (2) (2006) 419--448] is a conjectural improvement on the higher Stickelberger ideals defined at negative integers, and is expected to provide non-trivial annihilators for higher K-groups of rings of integers of number fields. In this article, we extend the definition of the fractional Galois ideal to arbitrary (possibly infinite and non-abelian) Galois extensions of number fields under the assumption of Stark's conjectures, and prove naturality properties under canonical changes of extension. We discuss applications of this to the construction of ideals in non-commutative Iwasawa algebras.