nLab Lubin-Tate formal group

Contents

Context

Stable Homotopy theory

Higher algebra

\infty-Lie theory

∞-Lie theory (higher geometry)

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 kk be a perfect field and fix a prime number pp.

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

RW(k)[[v 1,,v n1]] R \coloneqq W(k)[ [ v_1, \cdots, v_{n-1} ] ]

for the ring of formal power series over this ring, in n1n-1 variables; called the Lubin-Tate ring.

There is a canonical morphism

p:Rk p \;\colon\; R \longrightarrow k

whose kernel is the maximal ideal

ker(p)(p,v 1,,v n1), ker(p) \simeq (p,v_1, \cdots, v_{n-1}) \,,

This induces (…) for every formal group ff over kk a deformation f¯\overline{f} over RR. This is the Lubin-Tate formal group.

Properties

As the universal deformation

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