Michiel Hazewinkel 1 CWI Direct line: +31-20-5924204 POBox 94079 Secretary: +31-20-5924233 1090GB Amsterdam Fax: +31-20-5924166 E-mail:
[email protected] original version: 06 October 2007 revised version: 20 April 2008 Witt vectors. Part 1 1 by Michiel Hazewinkel CWI POBox 94079 1090GB Amsterdam The Netherlands Table of contents 1. Introduction and delimitation ??? 1.4 Historical motivation for the p-adic Witt polynomials, 1.8 Historical motivation for the (generalized) Witt polynomials, 1.11 Representability of the Witt vector functor. Symmetric functions, 1.14 The plethora of structures on Symm, 1.15 Initial sources of information 2. Terminology ??? 3. The p-adic Witt vectors. More historical motivation ??? 4. Teichmüller representatives ??? 5. Construction of the functor of the p-adic Witt vectors ??? 5.1 The p-adic Witt polynomials, 5.2 ‘Miracle’ of the p-adic Witt polynomials, 5.4 Congruence formulas for the p-adic Witt polynomials, 5.9 Witt vector addition and multiplication polynomials, 5.14 Existence theorem for the p-adic Witt vectors functor, 5.16 Ghost component equations, 5.19 Ideals and topology for Witt vector rings, 5.21 Teichmüller representatives in the Witt vector case, 5.22 Multiplication with a Teichmüller representative, 5.25 Verschiebung on the p-adic Witt vectors, 5.27 Frobenius on the p-adic Witt vectors, 5.37 Taking p-th powers on the p-adic Witt vectors, 5.40 Adding ‘disjoint’ Witt vectors, 5.42. Product formula, 5.44 f p Vp = [p] 6. The ring of p-adic Witt vectors over a perfect ring of characteristic p.