Verschiebung and Frobenius operators
type: Item
instance of: scholarly article
title: Verschiebung and Frobenius operators
zbMATH Open document ID: 0706.13021
DOI: 10.1007/BF02881112
author: K. Varadarajan
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1990-01-01T00:00:00Z
review text: The necklace ring Nr(A) of a commutative ring A introduced by \textit{N. Metropolis} and \textit{G.-C. Rota} [Adv. Math. 50, 95-125 (1983; Zbl 0545.05009)] consists of infinite vectors \(a=(a_ i)\) with addition defined componentwise and convolution product \((a_ i)*(b_ j)=(c_ n)\), \(c_ n=\sum_{[i,j]=n}(i,j)a_ ib_ j \), \([i,j]=lcm(i,j)\), \((i,j)=\gcd (i,j)\). The set of formal power-series \(1+tAt\) allows a transfer \(\gamma:\;Nr(A)\to 1+tAt\) provided \({\mathbb{Q}}\subset A\). On Nr(A) and on \(1+tAt\) operators \(V_ r\) and \(F_ r\) have been defined independently. The author shows that \(\gamma\) is natural with respect to these operators, i.e., \(\gamma (V_ r(a))=V_ r(\gamma (a))\) and \(\gamma (F_ r(a))=F_ r(\gamma (a))\), a fact the proof of which was not available in the literature in the manner described in this paper.
Mathematics Subject Classification ID: 13K05
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 13A35
zbMATH DE Number: 4158795
zbMATH Keywords: ring of Witt vectors
zbMATH Keywords: Verschiebung
zbMATH Keywords: Frobenius
zbMATH Keywords: necklace ring
zbMATH Keywords: formal power-series
reviewed by: J. Neggers
MaRDI profile type: Publication
cites work: Q5555033
cites work: The Burnside ring of the infinite cyclic group and its relations to the necklace algebra, \(\lambda\)-rings, and the universal ring of Witt vectors
cites work: La théorie des classes de Chern
cites work: Q3901604
cites work: Witt vectors and the algebra of necklaces
cites work: Q3216657
cites work: Aperiodic rings, necklace rings, and Witt vectors
cites work: Aperiodic rings, necklace rings and Witt vectors. II
full work available at URL: https://doi.org/10.1007/bf02881112
OpenAlex ID: W2013441074
description: scientific article; zbMATH DE number 4158795