A canonical lift of Frobenius in Morava $E$-theory

We prove that the $p$th Hecke operator on the Morava $E$-cohomology of a space is congruent to the Frobenius $\mathrm{mod} \: p$. This is a generalization of the fact that the $p$th Adams operation on the complex $K$-theory of a space is congruent to the Frobenius $\mathrm{mod} \: p$. The proof implies that the $p$th Hecke operator may be used to test Rezk’s congruence criterion.

Keywords

Morava $E$-theory, Frobenius, Hecke operator

2010 Mathematics Subject Classification

55N20

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Received 26 May 2016

Received revised 28 October 2016

Published 7 November 2018