Exponentials and logarithms on Witt rings
type: Item
instance of: scholarly article
title: Exponentials and logarithms on Witt rings
zbMATH Open document ID: 0638.10012
published in: Pacific Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: The author considers rings of the form \(R={\mathbb{Z}}[G]/K\), where G is an abelian group of exponent p, p being a prime, which are supposed to be abstract Witt rings in the meaning of \textit{M. Knebusch}, \textit{A. Rosenberg} and \textit{R. Ware} [Am. J. Math. 94, 119-155 (1972; Zbl 0248.13030)]. Under the assumption that for all \(x\in {\mathbb{Z}}[G]\), \(x\in K\) implies \(x^ p/p\in K\), he shows the existence of a system of divided powers [in the meaning of \textit{P. Berthelot}, Cohomologie cristalline des schémas de caractéristique \(p>0\) (Lect. Notes Math. 407) (1974; Zbl 0298.14012)] defined on the torsion part of the fundamental ideal of \(\bar M=M/K\). It allows the definition of \(\exp: (\bar M^ 2)_{tor}\to 1+(\bar M^ 2)_{tor}\) and \(\log: 1+(\bar M^ 2)_{tor}\to (\bar M^ 2)_{tor}.\) These maps occur to be group isomorphisms and inverse to each other. As an application of this, the author describes the group of units of R of finite order. The results cause special interest in the case \(p=2\), since they relate to the theory of quadratic forms.
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E16
zbMATH DE Number: 4039984
zbMATH Keywords: exponential map
zbMATH Keywords: logarithmic map
zbMATH Keywords: abstract Witt rings
zbMATH Keywords: system of divided powers
zbMATH Keywords: group of units
author: Murray Marshall
reviewed by: Andrzej Sładek
MaRDI profile type: Publication
full work available at URL: pjm.1987.127.127
OpenAlex ID: W2090917404
DOI: 10.2140/PJM.1987.127.127
description: scientific article; zbMATH DE number 4039984