$\mathbf{A}_{\text {inf}}$ has uncountable Krull dimension

Heng Du

$\mathbf{A}_{\text {inf}}$ has uncountable Krull dimension

Number Theory 2025-12-19 v4 Commutative Algebra

Authors: Heng Du

Abstract

Let $\mathcal{O}_E$ be a complete discrete valuation ring and $R$ be a perfect ring in characteristic $p$ , we also assume $R$ is a complete valuation ring whose valuation group is of rank one and non-discrete, we prove the Krull dimension of the ring $W_{\mathcal{O}_E}(R)$ of $\mathcal{O}_E$ -Witt vectors over $R$ is at least the cardinality of the continuum.

Cite

@article{arxiv.2002.10358,
  title  = {$\mathbf{A}_{\text {inf}}$ has uncountable Krull dimension},
  author = {Heng Du},
  journal= {arXiv preprint arXiv:2002.10358},
  year   = {2025}
}

Comments

8 pages; the chain of subsets we constructed in our first version might not be a chain of ideals. We managed to fix the proof using a completely different argument. Published version

$\mathbf{A}_{\text {inf}}$ has uncountable Krull dimension

Abstract

Cite

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