The Witt filtration of Lubin-Tate deformation rings

TL;DR

This paper explores a generalized Witt vector construction related to formal groups and Morava E-theory, revealing a natural filtration that bridges p-adic and geometric structures, and offers a new proof of cofreeness.

Contribution

It introduces a Witt filtration for Lubin-Tate deformation rings, generalizes classical Witt vectors, and connects algebraic and geometric filtrations in Morava E-theory.

Findings

Witt filtration interpolates between p-adic and geometric filtrations.

Witt vectors split naturally from the generalized Witt vectors.

Provides a new proof of the cofreeness property of Morava E-theory.

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 $\pi_0E\approx \mathbb{W}_E(\pi_0E/\mathfrak{m})$, the ``cofreeness of the Morava $E$-theory'' proved by Burklund, Schlank, and Yuan. This isomorphism determines a natural ``Witt filtration'' on $\pi_0 E$. We describe how this Witt filtration interpolates between the $p$-adic filtration and a geometric filtration on $\pi_0E/(p)$. We use this to give a new proof of cofreeness.