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Cofreeness of the Lubin-Tate deformation ring

Charles Rezk

Abstract

We give a proof of the cofreeness of the Lubin-Tate deformation ring, by generalizing earlier results by Matt Ando and Yifei Zhu about $\mathsf{H}_\infty$-orientations to the context of power operations for Morava $E$-theory.

Cofreeness of the Lubin-Tate deformation ring

Abstract

We give a proof of the cofreeness of the Lubin-Tate deformation ring, by generalizing earlier results by Matt Ando and Yifei Zhu about -orientations to the context of power operations for Morava -theory.
Paper Structure (18 sections, 13 theorems, 43 equations)

This paper contains 18 sections, 13 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. $\mathsf{H}_\infty$-orientation of Morava $E$-theory
  3. Power operations in Morava $E$-theory and $\mathsf{H}_\infty$-maps
  4. Cofreeness and its proofs
  5. Idea of the proof
  6. Comparison to the $\mathsf{H}_\infty$-argument
  7. Structure of this paper
  8. Frobenius lifting
  9. Quasicoherent sheaves on deformations
  10. Deformations
  11. Trivial deformations
  12. Frobenius morphisms
  13. Quasicoherent sheaves
  14. Frobenius congruence
  15. The $p$th power map
  16. ...and 3 more sections

Key Result

Theorem 1.2

Restriction along $\Sigma^{\infty-2}\mathbb{CP}^\infty\rightarrow MU$ and projection along $\mathcal{O}\rightarrow \kappa$ induce a bijection That is, every coordinate of the formal group $\Gamma/\kappa$ lifts uniquely to a coordinate on $\Gamma^\mathrm{univ}/\mathcal{O}$, which is furthermore realized by a unique $\mathsf{H}_\infty$-map $MU\rightarrow E$.

Theorems & Definitions (40)

  • Theorem 1.2: Ando ando-isogenies-and-power-ops; Zhu zhu-norm-coherence
  • Theorem 1.5: Cofreeness of $\mathcal{O}$
  • Theorem 1.6: Cofreeness of $\mathcal{O}(A)$
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3: Frobenius lifting
  • proof
  • Corollary 2.4
  • proof
  • Proposition 3.2
  • ...and 30 more