Fitting Ideals of Projective Limits of Modules over Non-Noetherian Iwasawa Algebras
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913151368298496 |
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| author | Popescu, Cristian D. Yin, Wei |
| author_facet | Popescu, Cristian D. Yin, Wei |
| contents | 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 $Λ_G=\mathbb{Z}_p[G][[T]]$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_11562 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fitting Ideals of Projective Limits of Modules over Non-Noetherian Iwasawa Algebras Popescu, Cristian D. Yin, Wei Commutative Algebra Number Theory 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 $Λ_G=\mathbb{Z}_p[G][[T]]$, 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. |
| title | Fitting Ideals of Projective Limits of Modules over Non-Noetherian Iwasawa Algebras |
| topic | Commutative Algebra Number Theory |
| url | https://arxiv.org/abs/2409.11562 |