Automorphisms of perfect power series rings (Q1663528)

Let \(R\) be a perfect ring of characteristic \(p,\) \(S=R[\![X]\!]\) the ring of formal power series with coefficients in \(R\) and \(m\) the ideal of \(S\) consisting of series with zero constant term. Let \(S'\) be the \(t\)-adic completion of \(R[t^{1/p},t^{1/p^{2}},\cdots].\) We denote by Aut\(^{\mathrm{cts}}_R(S)\) the group of continuous \(R\)-linear automorphisms of \(S\) preserving \(m.\) In the paper under review, the author shows that the map Aut\(^{\mathrm{cts}}_R(S)\rtimes\mathbb{Z}\) \(\rightarrow\) Aut\(^{\mathrm{cts}}_R(S')\) taking \(n\in \mathbb{Z}\) to the map \(t\mapsto t^{p^n}\) is an isomorphism of groups. This answers a question of Jared Weinstein.