$\mathbf{A}_{\text{inf}}$ is infinite dimensional

TL;DR

This paper proves that the ring of Witt vectors over a certain class of valuation rings has infinite Krull dimension, revealing complex structural properties of these algebraic objects.

Contribution

It establishes that the ring  of Witt vectors over perfect valuation rings of characteristic p has infinite Krull dimension, a new insight into their algebraic structure.

Findings

 has infinite Krull dimension

Witt vectors over perfect valuation rings exhibit complex structural properties

Advances understanding of algebraic properties of valuation rings and Witt vectors

Abstract

Given a perfect valuation ring $R$ of characteristic $p$ that is complete with respect to a rank-$1$ nondiscrete valuation, we show that the ring $\mathbf{A}_{\text{inf}}$ of Witt vectors of $R$ has infinite Krull dimension.