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The Witt filtration of Lubin-Tate deformation rings

Charles Rezk

Abstract

This note is a meditation on a generalization $\mathbb{W}_E$ of the classical p-typical Witt vectors $\mathbb{W}_p$, which arises (geometrically) from isogenies of deformations of formal groups, or (topologically) from the theory of power operations on Morava $E$-theory. For formal groups of height $1$ we have $\mathbb{W}_E=\mathbb{W}_p$, but the $\mathbb{W}_E$ are richer when height is $\geq 2$. We show that $\mathbb{W}_p$ splits naturally from $\mathbb{W}_E$. A key property of $\mathbb{W}_E$ is the isomorphism $π_0E\approx \mathbb{W}_E(π_0E/\mathfrak{m})$, the ``cofreeness of the Morava $E$-theory'' proved by Burklund, Schlank, and Yuan. This isomorphism determines a natural ``Witt filtration'' on $π_0 E$. We describe how this Witt filtration interpolates between the $p$-adic filtration and a geometric filtration on $π_0E/(p)$. We use this to give a new proof of cofreeness.

The Witt filtration of Lubin-Tate deformation rings

Abstract

This note is a meditation on a generalization of the classical p-typical Witt vectors , which arises (geometrically) from isogenies of deformations of formal groups, or (topologically) from the theory of power operations on Morava -theory. For formal groups of height we have , but the are richer when height is . We show that splits naturally from . A key property of is the isomorphism , the ``cofreeness of the Morava -theory'' proved by Burklund, Schlank, and Yuan. This isomorphism determines a natural ``Witt filtration'' on . We describe how this Witt filtration interpolates between the -adic filtration and a geometric filtration on . We use this to give a new proof of cofreeness.
Paper Structure (32 sections, 53 theorems, 72 equations)

This paper contains 32 sections, 53 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Conventions
  4. Acknowledgements
  5. Lubin-Tate-Witt vectors
  6. Multiplicative section
  7. Representability of $\mathbb{W}_E$
  8. Witt filtration
  9. The associated graded of the Witt filtration
  10. Ghost components
  11. The ghost component tower for perfect $\kappa$-algebras
  12. Base-change along Frobenius
  13. The maps $\rho_d$
  14. The maps $\overline{\rho}_d$
  15. Multiplication by $p$ in the Witt filtration
  16. ...and 17 more sections

Key Result

Theorem 1.2

The $\mathcal{O}$-algebra structure map $\mathcal{O}\rightarrow \mathbb{W}_E(\kappa)$ is an isomorphism.

Theorems & Definitions (111)

  • Example 1.1: Lubin-Tate-Witt vectors at height 1
  • Theorem 1.2: Cofreeness of the Lubin-Tate ring burklund-schlank-yuan-chromatic-nullstellensatz*Thm. 3.4
  • Theorem
  • Theorem
  • Remark 1.3
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • ...and 101 more