Comparison of power operations in Morava \(E\)-theories (Q1688687)

The paper under review concerns chromatic homotopy theory, more specifically some transchromatic phenomena between consecutive heights \(n+1\) and \(n\). Fixing a prime \(p\), let \(E_n\) denote Morava \(E\)-theory at height \(n\). Let \(\mathbb{S}_n\) denote the Morava stabilizer group and \(\mathbb{G}_n = \mathrm{Gal}(\mathbb{F}_{p^n} / \mathbb{F}_p) \ltimes \mathbb{S}_n\) the extended Morava stabilizer group. In previous work [\textit{T. Torii}, Math. Z. 266, No. 4, 933--951 (2010; Zbl 1203.55004)], the author proved that for a finite spectrum \(X\), the cohomology \(E_n^* X\) as an \(E_n^*\)-module with \(\mathbb{G}_n\)-action (compatible with the \(\mathbb{G}_n\)-action on \(E_n^*\)) can be functorially recovered from \(E_{n+1}^* X\) as an \(E_{n+1}^*\)-module with compatible \(\mathbb{G}_{n+1}\)-action. This paper provides a refined analogue of that statement taking into account power operations. Those naturally act on the homotopy of any \(K(n)\)-local commutative (or merely \(H_{\infty}\)) \(E_n\)-algebra, in particular on the \(E_n\)-cohomology of a space. Here, the author focuses on the degree \(0\) part. The main result is that for a finite CW complex \(X\), the cohomology \(E_n^0 X\) as a commutative \(E_n^0\)-algebra equipped with power operations and compatible \(\mathbb{G}_n\)-action can be functorially recovered from \(E_{n+1}^0 X\) as a commutative \(E_{n+1}^0\)-algebra with power operations and compatible \(\mathbb{G}_{n+1}\)-action. Among other ingredients, the construction uses work of Rezk on the algebra of power operations for Morava \(E\)-theory [\textit{C. Rezk}, Homology Homotopy Appl. 11, No. 2, 327--379 (2009; Zbl 1193.55010)] and work of Strickland on formal groups [\textit{N. Strickland}, J. Pure Appl. Algebra 121, No. 2, 161--208 (1997; Zbl 0916.14025)]. As a side note, the reviewer would love to see a comparison between the author's generalized Chern character maps and Stapleton's transchromatic character maps [\textit{N. Stapleton}, Algebr. Geom. Topol. 13, No. 1, 171--203 (2013; Zbl 1300.55011)].