# nLab Lubin-Tate formal group

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Higher algebra

higher algebra

universal algebra

## Theorems

#### $\infty$-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# Contents

## Definition

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.

## Properties

### As the universal deformation

The Lubin-Tate theorem asserts that the Lubin-Tate formal group $\overline{f}$ is the universal deformation of $f$. See at Lubin-Tate theorem.

### As inducing Morava E-theory

The Lubin-Tate formal group is Landweber exact and hence induces a complex oriented cohomology theory. This is Morava E-theory, see there for more details.

## References

For a general review see

For the role in chromatic homotopy theory see

Last revised on June 11, 2022 at 16:33:19. See the history of this page for a list of all contributions to it.