Witt vectors
type: Item
instance of: scholarly article
title: Witt vectors
zbMATH Open document ID: 0883.13022
author: Günter Harder
published in: Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV)
publication date: 1998-01-19T00:00:00Z
review text: This paper is a very beautiful survey of Ernst Witt's work on cyclic extensions and algebras of degree \(p^n\) in characteristic \(p\). The basic construction is now known as ``Witt vectors''. Witt's construction is explained starting from its roots (as one finds them in the introduction of Witt's original paper), the starting point being the Artin-Schreier theory of cyclic field extensions of degree \(p\) in characteristic \(p\). The structural properties of the Witt ring allow to describe cyclic extensions of degree \(p^n\) in a very clear way. Harder explains how Witt came to his construction, the universal formulas for the addition and multiplication of Witt vectors follow from the fact that the ring of Witt vectors over \(\mathbb{F}_{p^N}\), \(W(\mathbb{F}_{p^N})\), is constructed so that it is isomorphic to the valuation ring of the (unique) unramified extension of degree \(N\) of \(\mathbb{Q}_p\), the idea being that this ring must be constructed from its residue field \(\mathbb{F}_{p^N}\) in a natural way. In the section: ``Die ersten Anwendungen'', the author discusses the paper of H. L. Schmid and Witt in which they study the maximal unramified abelian extension of exponent \(p^n\) of a function field in one variable over a perfect field of characteristic \(p\). The problem is explained very nicely starting from the expontent \(p\) case (due to Hasse and Witt). The key role of the Witt vectors (for the general case) is made clear. In two subsequent parts of the paper the author explains the fundamental importance of Witt's work for modern algebra and arithmetical algebraic geometry. In a first section the theory of commutative affine algebraic groups is discussed. The structure theorems for unipotent commutative algebraic groups and for commutative finite group schemes annihilated by a power of \(p\) are given. In the final section it is made clear that Witt rings are basic building blocks for many constructions in modern arithmetic geometry. This section …
reviewed by: Jan Van Geel
Mathematics Subject Classification ID: 13K05
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 01A60
Mathematics Subject Classification ID: 14-03
Mathematics Subject Classification ID: 14L15
zbMATH DE Number: 988008
zbMATH Keywords: Witt vectors
zbMATH Keywords: cristalline cohomology
zbMATH Keywords: finite group schemes
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 988008