A canonical lift of Frobenius in Morava \(E\)-theory (Q1625466)

Let $E$ be a Morava $E$-spectrum, and $Q_{\sigma}$ be the linear endomorphism defined by $\sigma\in \Gamma$, given by the Dyer-Lashof algebra for $E$. On the other hand let $T_{p}$ be the $p$-th Hecke operator defined by \textit{M. Ando} [Duke Math. J. 79, No. 2, 423--485 (1995; Zbl 0862.55004)]. The author proves that there is a canonical lift of $\sigma$ to a class $\overline{\sigma}\in\Gamma/p$ such that $Q_{\sigma_{can}}=T_{p}$. As a corollary, the author proves the following Corollary 1.3: Let $X$ be a space such that $E^{0}(X)$ is torsion-free. There exists a canonical operation $\theta:E^{0}(X)\to E^{0}(X)$ such that, for all $x\in E^{0}(X)$, $T_{p}(x)=x^{p}+p\theta(x)$.