Exponentials and logarithms on Witt rings
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.
scientific article; zbMATH DE number 4039984