Fitting Ideals of Projective Limits of Modules over Non-Noetherian Iwasawa Algebras

TL;DR

This paper extends the understanding of Fitting ideals and projective limits from classical Iwasawa algebras to more general, non-Noetherian settings, with applications in geometric Iwasawa theory and related fields.

Contribution

It generalizes previous results on Fitting ideals to non-Noetherian Iwasawa algebras with broader coefficient rings, addressing recent developments in geometric and function field Iwasawa theories.

Findings

Generalization of Fitting ideal results to non-Noetherian Iwasawa algebras.

Application to geometric Iwasawa theory and function fields.

Framework for future number theoretic applications.

Abstract

In \cite{grku1}, Greither and Kurihara proved a theorem about the commutativity of projective limits and Fitting ideals for modules over the classical equivariant Iwasawa algebra $\Lambda_G=\mathbb{Z}_p[G][[T]]$, where $G$ is a finite, abelian group and $\Bbb Z_p$ is the ring of $p$--adic integers, for some prime $p$. In this paper, we generalize their result first to the Noetherian Iwasawa algebras $\mathcal O[[T_1, T_2, \dots, T_n]]$ and, most importantly, to non-Noetherian algebras $\mathcal O[[T_1, T_2, \dots, T_n, \dots]]$ of countably many generators, with more general rings of coefficients $\mathcal O$. The latter generalization is motivated by the recent work of Bley--Popescu on the Geometric Equivariant Iwasawa Conjecture for function fields, as well as by the emerging Iwasawa theory of Taelman class--modules associated to Drinfeld modules, where the Iwasawa algebras are not Noetherian, of the type described above. A sample application of our results to non--Noetherian geometric Iwasawa theory is given in Appendix B. Further number theoretic applications will be given in an upcoming paper.