The Witt filtration of Lubin-Tate deformation rings

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 $\ge 2$. We show that ${\mathbb{W}}_p$ splits naturally from ${\mathbb{W}}_E$. A key property of ${\mathbb{W}}_E$ is the isomorphism ${\pi }_{0}E\approx {\mathbb{W}}_E({\pi }_{0}E\mathrm{/}\mathfrak{m})$, 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 ${\pi }_{0}E\mathrm{/}(p)$. We use this to give a new proof of cofreeness.