THE LUBIN-TATE MODULI SPACE IN HOMOTOPY THEORY

DANIEL G. DAVIS

Abstract. This short note is an introduction to how the Lubin-Tate moduli space appears in modern homotopy theory. It is written for algebraic geometers and number theorists, who are familiar with this moduli space and who are interested in learning about its use in homotopy theory. Note: All unnumbered references are to my note “Finite CW-complexes and the chromatic tower,” available at my homepage.

I begin with a description of the Goerss-Hopkins-Miller theorem, which is the main way that the Lubin-Tate moduli space appears in homotopy theory, and, more specifically, chromatic stable homotopy theory. Most of the description below comes from [5] (see also [8]).

Let FG be the category of pairs (k, Γ), where k is an algebraic extension of 𝔽_{p} and Γ is a formal group law of height n over k. Let Alg_{c} denote the category of complete local ℤ_{p}-algebras. Let W (k) be the ring of Witt vectors with coefficients in k. Then there is a functor

Ψ: FG^{op} → Alg_{c}, (k, Γ) ↦ E(k, Γ)_{0},

where

E(k, Γ)_{0} = W (k)[[u_{1}, ⋯, u_{n−1}]],

the ring that corepresents ⋆-isomorphism classes of deformations.

The Goerss-Hopkins-Miller theorem lifts the functor Ψ into what we call point-set level “homotopy theory.” To be precise, there is a functor

Ψ̂: FG^{op} → E_{∞}, (k, Γ) ↦ E(k, Γ),

such that:

(1) E(k, Γ) is the spectrum [pg. 2] representing the generalized cohomology theory E(k, Γ)^{∗}(−) [pg. 2].

(2) Let E(k, Γ)_{∗} be the ℤ-graded coefficient ring of E(k, Γ) [pg. 2]. Then, in degree zero,

E(k, Γ)_{0} = Ψ(k, Γ), and E(k, Γ)_{∗} = E(k, Γ)_{0}[u, u^{−1}],

where the degree of u is −2, so that E(k, Γ)_{∗} is two-periodic.

(3) There is a “graded” Lubin-Tate theory (see [2, §1]) such that E(k, Γ)_{∗} corepresents the expected functor:

(lifts_{Γ}^{⋆})_{∗}(A) ≅ Hom_{W (k)−algebras}^{cts}(E(k, Γ)_{∗}, A).

(4) E_{∞} is the category of commutative S-algebras. To say that the spectrum E(k, Γ) is a commutative S-algebra is basically to say that E(k, Γ) is a commutative ring object in the category of spectra.

(5) E(k, Γ) is complex orientable with formal group law G(k, Γ) over E(k, Γ)_{0}, such that G(k, Γ) is the universal deformation of (k, Γ).

(6) Associated to (k, Γ) is a 1-morphism of stacks

x(k, Γ): Spec(k) → M_{FGL},

where M_{FGL} is the moduli stack of formal group laws.

(7) By functoriality, Ψ̂ induces an action by the automorphism group Aut_{FG}(k, Γ) on the spectrum E(k, Γ), and this group action induces the action of Aut_{FG}(k, Γ) on the complete ring W (k)[[u_{1}, ⋯, u_{n−1}]].

(8) In particular, when (k, Γ) = (𝔽_{p^{n}}, Γ_{n}), where Γ_{n} is the Honda formal group law satisfying [p]_{Γ_{n}}(x) = x^{p^{n}}, then

E(𝔽_{p^{n}}, Γ_{n}) = E_{n},

the Lubin-Tate spectrum. Recent work by Devinatz and Hopkins [3], together with my Ph.D. dissertation, shows that the action of

G_{n} = Aut_{FG}(𝔽_{p^{n}}, Γ_{n}) = Aut_{𝔽_{p^{n}}}(Γ_{n}) ⋊ Gal(𝔽_{p^{n}}/𝔽_{p})

on E_{n} is a continuous action, and the G_{n}-homotopy fixed points of this continuous action is the K(n)-local sphere L_{K(n)}S^{0} [pp. 3-4].

I make a remark about the history of the above theorem. This theorem was originally proved by Haynes Miller and Hopkins (see [8]), except that their version replaced the target of Ψ̂ with A_{∞}, instead of E_{∞}, where A_{∞} is basically the category of associative ring objects in the category of spectra. Then Goerss and Hopkins, in [6], did the upgrade to E_{∞}.

Reference [8] is an excellent introduction to how the theory of formal group laws appears in homotopy theory. The theory of stacks also turns out to be very important in chromatic theory. [5, §2.6] discusses Ψ̂ from this perspective; [5] is the best introduction to how the theory of stacks relates to this area.

In fact, the algebraic geometry of stacks turns out to be very helpful in understanding chromatic theory. [7, Table 1] gives a dictionary between sheaf theory and stable homotopy theory, and currently there is a lot of interest in understanding how this dictionary unlocks and simplifies complex pieces of chromatic theory. For example, the Chromatic Convergence Theorem [described in Theorem 2.1; due to Hopkins and Ravenel] is one of the central theorems of the theory, and, as mentioned in [4, pg. 2], this theorem can be deduced from a familiar phenomenon in algebraic geometry that uses the ideas of coherent sheaves and the moduli stack of p-typical formal group laws.

Also, there has been a tremendous amount of interest in the work of Hopkins and his collaborators on how the algebraic geometry of elliptic curves and the corresponding moduli stack M_{ell} [5, 1.5] relate to homotopy theory. The two ICM talks by Hopkins (in 1994 and 2002) give a good introduction to this; see also [1], a primary reference.

Briefly, there is a category of “etale A_{∞}-elliptic spectra,” each of which is canonically associated to an elliptic curve, and the homotopy limit of spectra over this category is the spectrum tmf, which is currently the best candidate for the spectrum that represents elliptic cohomology. The spectrum tmf is called the spectrum of “topological modular forms,” because it is a topological realization of the algebraic ring of modular forms. In his 2002 ICM talk, Hopkins begins with homotopy theory and ends up talking about arithmetic gadgets like unimodular lattices, the Atkin operator, and the Ramanujan τ function. Also, it is my understanding that the topology of tmf, due to work by Hopkins and others, yields new results in number theory.

References

[1] M. Ando, M. J. Hopkins, and N. P. Strickland. Elliptic spectra, the Witten genus and the theorem of the cube. Invent. Math., 146(3):595–687, 2001.

[2] Ethan S. Devinatz and Michael J. Hopkins. The action of the Morava stabilizer group on the Lubin-Tate moduli space of lifts. Amer. J. Math., 117(3):669–710, 1995.

[3] Ethan S. Devinatz and Michael J. Hopkins. Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups. Topology, 43(1):1–47, 2004.

[4] Paul Goerss, Hans-Werner Henn, Mark Mahowald, and Charles Rezk. A resolution of the K(2)-local sphere. Available at http://www.math.nwu.edu/^{∼}pgoerss, 2002.

[5] Paul G. Goerss. (Pre-)sheaves of Ring Spectra over the Moduli Stack of Formal Group Laws. Manuscript, available at the website http://www.math.nwu.edu/^{∼}pgoerss.

[6] Paul G. Goerss and Michael J. Hopkins. Moduli spaces of commutative ring spectra. manuscript, available at the website http://www.math.nwu.edu/^{∼}pgoerss, 2003.

[7] M. J. Hopkins and B. H. Gross. The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory. Bull. Amer. Math. Soc. (N.S.), 30(1):76–86, 1994.

[8] Charles Rezk. Notes on the Hopkins-Miller theorem. In Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997), pages 313– 366. Amer. Math. Soc., Providence, RI, 1998.