Liang, Dingli; (2025) On Non-Noetherian Iwasawa Theory. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

In this thesis, we investigate two non-Noetherian rings of arithmetic interest: the p-adic completed group ring Zp[[ZNp ]], where ZNp denotes the direct product of countably infinitely many copies of Zp, and the integral completed group ring Z[[G]] associated to compact p-adic Lie groups. We further study the module theory over these rings and explore arithmetic applications of the resulting algebraic structures. For the first ring Zp[[ZNp]], we establish a general structure theorem for finitely presented torsion modules over a class of commutative rings that need not be Noetherian. This theorem is then applied to the study of the Weil-étale cohomology groups of Gm for curves over finite fields. A particularly striking outcome is that we prove an Iwasawa Main Conjecture under mild assumptions. As an application, we show that the inverse limit, taken with respect to norm maps, of the p-primary parts of degree-zero divisor class groups can only form a finitely generated Zp[[ZNp]]-module under a small class of ZNp-extensions. For the second ring Z[[G]], we study its coherence properties. We prove that for every compact p-adic Lie group G of rank d, the ring Z[[G]] is not coherent, but is d + 3-coherent. This result contributes to a better understanding of the homological behavior of modules over this non-Noetherian Iwasawa algebra.

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