nLab Lubin-Tate formal group

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

• Wikipedia, Lubin-Tate formal group law

For the role in chromatic homotopy theory see

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