2026/08/05 by Yiding Wang
Mathematics · #math.AC #msc:13F30 #msc:13F25 #msc:13F35 #msc:12J25
12 pages, no figures
arxiv created 2026/08/05 · arxiv updated 2026/08/06
Let K be a complete nonarchimedean valued field with v(K^×)=\mathbf R, and let V=\mathcal OK. 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.