Characterising properties of commutative rings using Witt vectors
Rubén Muñoz--Bertrand
TL;DR
This work analyzes how properties of a commutative ring $R$ are reflected in its ring of $p$-typical Witt vectors $W(R)$, with a focus on Noetherianity and $p$-torsion. It establishes a precise domain-theoretic correspondence, proving that $W(R)$ is a domain exactly when $R$ is a domain of characteristic $p$, and extends this to normal and completely normal domains in relation to $R$ being perfect. A central contribution is the complete classification of Noetherian Witt vector rings: $W(R)$ is Noetherian iff $R$ is zero or a Noetherian perfect ring of characteristic $p$, and further characterizations are given for when such $W(R)$ are DVRs. The introduction of preduced rings provides a robust framework to characterize $p$-torsion freeness of $W(R)$ and, via the de Rham–Witt complex, links to torsion phenomena in crystalline cohomology; the paper also discusses coherence, showing that with $p$ invertible, $W(R)$ is coherent exactly when $R$ is uniformly coherent. Overall, the results offer a cohesive approach to understanding how Witt vectors encode structural properties of $R$ and their implications for $p$-adic cohomology theories.
Abstract
We give equivalences between given properties of a commutative ring, and other properties on its ring of Witt vectors. Amongst them, we characterise all commutative rings whose rings of Witt vectors are Noetherian. We define a new category of commutative rings called preduced rings, and explain how it is the category of rings whose ring of Witt vectors has no $p$-torsion. We then extend this characterisation to the torsion of the de Rham-Witt complex.