Lubin-Tate theory

The moduli stack of formal groups $\mathcal{M}_{FG}$ admits a natural stratification whose open strata $\mathcal{M}^n_{FG}$ are labeled by a natural number called the *height of formal groups*.

The deformation theory around these strata is *Lubin-Tate theory*.

The universal Lubin-Tate deformation ring of a formal group of height $n$ induces, via the Landweber exact functor theorem a complex oriented cohomology theory, a localization of this is $n$th Morava E-theory $E(n)$.

Let $k$ be a perfect field and fix a prime number $p$.

Write $W(k)$ for the ring of Witt vectors and

$R \coloneqq W(k)[ [ v_1, \cdots, v_{n-1} ] ]$

for the ring of formal power series over this ring, in $n-1$ variables; called the *Lubin-Tate ring*.

There is a canonical morphism

$p \;\colon\; R \longrightarrow k$

whose kernel is the maximal ideal

$ker(p) \simeq (p,v_1, \cdots, v_{n-1})
\,,$

This induces (…) for every formal group $f$ over $k$ a deformation $\overline{f}$ over $R$. This is the *Lubin-Tate formal group*.

The Lubin-Tate formal group $\overline{f}$ is the universal deformation of $f$ in that for every infinitesimal thickening $A$ of $k$, $\overline{f}$ induces a bijection

$Hom_{/k}(R,A) \stackrel{\simeq}{\longrightarrow} Def(A)$

between the $k$-algebra-homomorphisms from $R$ into $A$ and the deformations of $A$.

(e.g. Lurie 10, lect 21, theorem 5)

Lecture 21 of

- Jacob Lurie,
*Chromatic Homotopy Theory*, Lecture notes 2010 (web), Lecture 21*Lubin-Tate theory*(pdf)