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arXiv · 2608.04897

Spherical Completeness, Coherence, and GCD Properties of Formal Power Series and Witt Vector Rings

Yiding Wang
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Abstract

Let $K$ be a complete nonarchimedean valued field with $v(K^\times)=\mathbf R$, and let $V=\mathcal O_K$. We prove that $K$ is spherically complete if and only if $V[[T]]$ is coherent, and that this is also equivalent to $V[[T]]$ being a GCD domain. If $K$ is perfect of characteristic $p$, the same characterization holds for the Witt vector ring $W(V)$. Thus, this settles the previously unresolved full-real-value-group case in the coherence problems for both formal power series and Witt vector rings. In particular, this result also gives affirmative answers to Questions~9 and~10 of Anderson--Kang--Park. The proof combines a coherence criterion for complete rings with a spherically complete valuation quotient and a uniform construction of non-finitely generated intersections of two principal ideals from an empty ball chain.

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Yiding Wang. 2026-08-05. Spherical Completeness, Coherence, and GCD Properties of Formal Power Series and Witt Vector Rings. https://arxiv.org/abs/2608.04897

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