Verschiebung and Frobenius operators
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.
scientific article; zbMATH DE number 4158795