Characterising properties of commutative rings using Witt vectors

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.