AI 中文总结
本文刻画了完备非阿基米德赋值域上形式幂级数环与Witt向量环的凝聚性、GCD性质与球面完备性的等价关系，解决了相关问题并肯定回答了Anderson等人的两个问题。
AI 中文摘要
设$K$为满足$v(K^\times)=\mathbf{R}$的完备非阿基米德赋值域，$V=\mathcal{O}_K$。我们证明：$K$是球面完备的当且仅当$V[[T]]$是凝聚环，且这等价于$V[[T]]$是GCD整环。若$K$是特征为$p$的完美域，则该刻画对Witt向量环$W(V)$同样成立。由此，本文解决了形式幂级数与Witt向量环凝聚性问题中此前未解决的全实值群情形。特别地，该结果还对Anderson–Kang–Park的问题9和问题10给出了肯定回答。证明结合了完备环的凝聚准则、球面完备赋值商，以及从空球链构造两个主理想的非有限生成交的统一方法。
英文摘要
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.
Comments12 pages, no figures
