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

@misc{wang2026spherical,
  title = {{Spherical Completeness, Coherence, and GCD Properties of Formal Power Series and Witt Vector Rings}},
  author = {Yiding Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.04897},
  eprint = {2608.04897},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.04897},
}

TY  - UNPB
TI  - Spherical Completeness, Coherence, and GCD Properties of Formal Power Series and Witt Vector Rings
AU  - Yiding Wang
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2608.04897
UR  - https://arxiv.org/abs/2608.04897
AB  - Let K be a complete nonarchimedean valued field with v(K^×)=R, and let V=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.
ER  -