# Contents

## Idea

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 $E(n)$.

## Lubin-Tate formal group

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

###### Definition

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.

## Lubin-Tate theorem

###### Theorem

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 $A$.

## References

Lecture 21 of

Last revised on November 18, 2013 at 07:39:48. See the history of this page for a list of all contributions to it.