Witt vectors.
(Wittvektoren.)

*(German)*Zbl 0883.13022This 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 constitutes the largest part of the paper. The need for a cohomology theory of varieties over \(k\) (a perfect field of characteristic \(p\)) with coefficients in a limit of rings with \(p^n\)-torsion, is explained. As a basic example abelian varieties are considered. Then cristalline cohomology and the problems that arise when one considers Galois representations associated with it are discussed (the case of elliptic curves is taken as an example). Finally a survey of the work of Fontaine, Messing, Faltings, Kato and Hyodo on the general problem and the different functorial isomorphism is given.

The paper under review can be recommended, it is a very nicely written “homage” for Witt’s work. An English translation by B. and Ch. Deninger will appear in Witt’s collected works.

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 constitutes the largest part of the paper. The need for a cohomology theory of varieties over \(k\) (a perfect field of characteristic \(p\)) with coefficients in a limit of rings with \(p^n\)-torsion, is explained. As a basic example abelian varieties are considered. Then cristalline cohomology and the problems that arise when one considers Galois representations associated with it are discussed (the case of elliptic curves is taken as an example). Finally a survey of the work of Fontaine, Messing, Faltings, Kato and Hyodo on the general problem and the different functorial isomorphism is given.

The paper under review can be recommended, it is a very nicely written “homage” for Witt’s work. An English translation by B. and Ch. Deninger will appear in Witt’s collected works.

Reviewer: Jan Van Geel (Gent)

##### MSC:

13K05 | Witt vectors and related rings (MSC2000) |

14F30 | \(p\)-adic cohomology, crystalline cohomology |

13A35 | Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure |

01A60 | History of mathematics in the 20th century |

14-03 | History of algebraic geometry |

14L15 | Group schemes |