Lubin-Tate theory



The moduli stack of formal groups FG\mathcal{M}_{FG} admits a natural stratification whose open strata FG n\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 nn induces, via the Landweber exact functor theorem a complex oriented cohomology theory, a localization of this is nnth Morava E-theory E(n)E(n).

Lubin-Tate formal group

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.

Lubin-Tate theorem


The Lubin-Tate formal group f¯\overline{f} is the universal deformation of ff in that for every infinitesimal thickening AA of kk, f¯\overline{f} induces a bijection

Hom /k(R,A)Def(A) Hom_{/k}(R,A) \stackrel{\simeq}{\longrightarrow} Def(A)

between the kk-algebra-homomorphisms from RR into AA and the deformations of AA.

(e.g. Lurie 10, lect 21, theorem 5)


Lecture 21 of

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