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

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 ${\mathbb{Z}}_p$ 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.

arXiv comment31 pages