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Lubin-Tate formal group in nLab

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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

R≔W(k)[[v 1,⋯,v n−1]] R \coloneqq W(k)[ [ v_1, \cdots, v_{n-1} ] ]

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

There is a canonical morphism

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

whose kernel is the maximal ideal

ker(p)≃(p,v 1,⋯,v n−1), 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.

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 July 31, 2023 at 07:02:26. See the history of this page for a list of all contributions to it.