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Power Operations in Morava E-Theory of Flat Ring Spectra

Yuval Lotenberg

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 $π_0 R$ with the structure of a $δ$-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 $π_0(L_{K(n)}(R \otimes E_n)) = LT_n \widehat{\otimes} π_0 R$ 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 $π_0(L_{K(n)}(R \otimes E_n))$.

Power Operations in Morava E-Theory of Flat Ring Spectra

Abstract

Let be Morava -theory of height . Let be a -adically flat commutative ring spectrum. Then the Tate-valued Frobenius map endows with the structure of a -ring. On the other hand, we may form the -completed tensor product , which is a -local -algebra. Then admits the structure of an algebra over the monad defined by Rezk. The -algebra structure encodes the power operations of . In this paper we describe the -algebra structure on .
Paper Structure (24 sections, 47 theorems, 95 equations)

This paper contains 24 sections, 47 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Background and Main Results
  3. Motivation
  4. Conventions
  5. Organization & Proof Outline
  6. Acknowledgments
  7. Preliminaries
  8. $\delta$-Rings
  9. Formal Groups and Deformations
  10. $p$-adically Flat Spectra
  11. Morava E-theory
  12. Proper Tate Construction
  13. Construction of the Functor $\mathrm{Alg}_{\delta} \to \operatorname{Sh}(\operatorname{Def}, \mathrm{Alg})$
  14. Proper Tate Construction
  15. Exactness & Multiplicativity
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $(R,\delta)$ be a $p$-complete torsion-free $\delta$-ring, with associated lift of Frobenius $\psi(x) = x^p+ p\delta(x)$ for $x \in R$. Let $B \in \operatorname{Sh}(\operatorname{Def}, \mathrm{Alg})$ be image of $R$ under the composition of functors Then for a deformation $G$ over $S$, we have $B_S(G) = S \,\widehat{\otimes}\,R$ (completed with respect to the maximal ideal of $S$), and to

Theorems & Definitions (107)

  • Example 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Definition 2.1: Joyal85
  • Definition 2.2
  • Definition 2.3: Homomrophism, Isogeny
  • Definition 2.4: Deformation of Formal Group
  • Definition 2.5: Defomration of Frobenius
  • ...and 97 more