nLab Lubin-Tate theory

Idea

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

Lubin-Tate formal group

Let $k$ be a perfect field and fix a prime number $p$.

Definition

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.

Lubin-Tate theorem

Theorem

The Lubin-Tate formal group $\overline{f}$ is the universal deformation of $f$ in that for every infinitesimal thickening $A$ of $k$, $\overline{f}$ induces a bijection

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

between the $k$-algebra-homomorphisms from $R$ into $A$ and the deformations of $A$.

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

Related concepts

• Lubin-Tate space

• Lubin-Tate formal group

• chromatic homotopy theory

• Morava E-theory

References

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

On the topological Hochschild homology of the Lubin-Tate ring spectrum via factorization homology:

• Geoffroy Horel, Higher Hochschild homology of the Lubin-Tate ring spectrum, Algebr. Geom. Topol. 15 (2015) 3215-3252 [arXiv:1311.2805, doi:10.2140/agt.2015.15.3215]