Power Operations in Morava E-Theory of Flat Ring Spectra

TL;DR

This paper explores the algebraic structures and power operations in Morava E-theory, specifically analyzing the $T(n)$-algebra structure on the zeroth homotopy of certain $K(n)$-local $E_n$-algebras derived from flat ring spectra.

Contribution

It provides a detailed description of the $T(n)$-algebra structure on $pi_0$ of $K(n)$-local tensor products involving Morava $E$-theory and flat ring spectra, linking Frobenius maps and power operations.

Findings

$pi_0$ admits a $T(n)$-algebra structure encoding power operations.

The Frobenius map induces a $delta$-ring structure on $pi_0 R$.

The $T(n)$-structure captures the algebraic essence of power operations in this context.

Abstract

Let $E_n$ be Morava $E$-theory of height $n$. Let $R$ be a $p$-adically flat commutative ring spectrum. Then the Tate-valued Frobenius map endows $\pi_0 R$ with the structure of a $\delta$-ring. On the other hand, we may form the $K(n)$-completed tensor product $L_{K(n)}(R \otimes E_n)$, which is a $K(n)$-local $E_n$-algebra. Then $\pi_0(L_{K(n)}(R \otimes E_n)) = LT_n \widehat{\otimes} \pi_0 R$ admits the structure of an algebra over the monad $\mathbb{T}(n)$ defined by Rezk. The $\mathbb{T}(n)$-algebra structure encodes the power operations of $L_{K(n)}(R \otimes E_n)$. In this paper we describe the $\mathbb{T}(n)$-algebra structure on $\pi_0(L_{K(n)}(R \otimes E_n))$.