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

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 ${\mathrm{\Lambda }}_G={\mathbb{Z}}_p[G][[T]]$, 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.