nLab Goerss-Hopkins-Miller theorem

Idea

The Goerss-Hopkins-Miller theorem (Goerss-Hopkins A, Goerss-Hopkins B) also called now the Goerss-Hopkins-Miller-Lurie theorem (Lurie (Survey), Lurie (Representability)) as such establishes that the assignment to each elliptic curve of its elliptic cohomology theory lifts in a coherent way to an assignment of the representing spectra regarded as E-∞ rings. In summay this makes for a structure sheaf $\mathcal{O}_{\mathcal{M}^{der}}$ of the moduli stack of elliptic curves in derived algebraic geometry, realizing that moduli stack as a derived scheme $\mathcal{M}^{der}$ in E-∞ geometry (“spectral geometry”) This is sketched in (Lurie) and discussed in detail in (Behrens 13).

In detail the theorem utilizes and establishes more general tools concerning the obstruction theory of lifting formal group laws to cohomology theories and homotopy associative/commutative spectra to A-∞ rings and to E-∞ ring structure, respectively.

In the succinct and suggestive form of (Lurie, theorem 1.1) the Goerss-Hopkins-Miller theorem leads the existence of the lift $\mathcal{O}_{\mathcal{M}^{top}}$ in the diagram

$$\array{ && E_\infty Rings \\ &{}^{\mathllap{\mathcal{O}_{\mathcal{M}^{der}}}}\nearrow& \downarrow^{\mathrlap{Brown\;representation}} \\ \mathcal{M}(R) &\longrightarrow& MCTs } \,,$$

where the bottom map sends (derived) elliptic curves over a ring $R$ to their associated elliptic multiplicative cohomology theory (“MTC”) and where the right vertical map sends a spectrum to the cohomology theory which it represents.

Hopkins-Miller theorem

For $n \in \mathbb{N}$, consider the category whose objects are pairs consisting of a perfect field $k$ of finite characteristic and of a formal group of height $n$, and whose morphisms are base change of fields (as rings).

The Hopkins-Miller theorem says that there is a functorial assignment $(k,\Gamma) \mapsto E_{k,\Gamma}$ to such data of an A-∞ ring spectrum $E_{k,\Gamma}$ such that

1. $\pi_2 E_{k,\Gamma}$ contains a unit and $\pi_{2n+1} E_{k,\Gamma} = 0$, hence in particular $E_{k,\Gamma}$ is complex orientable;

2. the corresponding formal group law over $\pi_0 E_{k,\Gamma}$ is the universal deformation of $(k,\Gamma)$.

In this form this appears as (Rezk 97, theorem 2.1).

Goerss-Hopkins theorem

(…)

this appears reproduced as (Behrens 13, theorem 4.2)

Lurie theorem

By (Lurie (Survey), theorem 4.1), the spectral Deligne-Mumford stack refinement $(\mathcal{M}_{ell}, \mathcal{O}^{top})$ of the moduli stack of elliptic curves is the moduli stack of derived elliptic curves, in that there is a natural equivalence in E-∞ rings $A$ of the form

$$Hom(Spec(A), (\mathcal{M}_{ell},\mathcal{O}^{der})) \simeq E(A) \,,$$

where on the left we have maps of structured (∞,1)-toposes and on the right the ∞-groupoid of derived elliptic curves over $A$.

This is based on the Artin-Lurie representability theorem (Lurie (Survey), prop. 4.1, Lurie (Representability)).

Related concepts

Substructure of the moduli stack of curves and the (equivariant) cohomology theory associated with it via the Goerss-Hopkins-Miller-Lurie theorem:

covering					by of level-n structures (modular curve)
	$\ast = Spec(\mathbb{Z})$	$\to$	$Spec(\mathbb{Z}[ [q] ])$	$\to$	$\mathcal{M}_{\overline{ell}}[n]$
structure group of covering	$\downarrow^{\mathbb{Z}/2\mathbb{Z}}$		$\downarrow^{\mathbb{Z}/2\mathbb{Z}}$		$\downarrow^{SL_2(\mathbb{Z}/n\mathbb{Z})}$ (modular group)
moduli stack	$\mathcal{M}_{1dTori}$	$\hookrightarrow$	$\mathcal{M}_{Tate}$	$\hookrightarrow$	$\mathcal{M}_{\overline{ell}}$ (M_ell)	$\hookrightarrow$	$\mathcal{M}_{cub}$	$\to$	$\mathcal{M}_{fg}$ (M_fg)
of	1d tori		Tate curves		elliptic curves		cubic curves		1d commutative formal groups
value $\mathcal{O}^{top}_{\Sigma}$ of structure sheaf over curve $\Sigma$	KU		$KU[ [q] ]$		elliptic spectrum				complex oriented cohomology theory
spectrum $\Gamma(-, \mathcal{O}^{top})$ of global sections of structure sheaf	(KO $\hookrightarrow$ KU) = KR-theory		Tate K-theory ($KO[ [q] ] \hookrightarrow KU[ [q] ]$)		(Tmf $\to$ Tmf(n)) (modular equivariant elliptic cohomology)		tmf		$\mathbb{S}$

References

An introduction to and survey of the Goerss-Hopkins-Miller-Lurie theorem is in

• Paul Goerss, Topological modular forms (after Hopkins, Miller, and Lurie) Séminaire BOURBAKI Mars 2009 61ème année, 2008-2009, no 1005(2009)(arXiv:0910.5130)

which has grown out of

• Topological Algebraic Geometry - A Workshop.

• pdf

Details on the Hopkins-Miller theorem are in

• Charles Rezk, Notes on the Hopkins-Miller theorem, 1997 (pdf)

and a survey is also in

• Akhil Mathew, Hopkins-Miller theorem I

The Goerss-Hopkins theorem is laid out in

• Paul Goerss, Michael Hopkins, Moduli spaces of commutative ring spectra, in Structured ring spectra, London

  Math. Soc. Lecture Note Ser., vol. 315, Cambridge Univ. Press, Cambridge, 2004, pp. 151-200. (pdf, doi:10.1017/CBO9780511529955.009)

• Paul Goerss, Michael Hopkins, Moduli problems for structured ring spectra (pdf)

The actual details of the application of the Goerss-Hopkins-Miller theorem to the construction of $\mathcal{O}_{\mathcal{M}^{der}}$ (and hence of tmf) is in

• Mark Behrens, Notes on the construction of $tmf$, 2013 (pdf)

An exposition with indications of further developments in (infinity,1)-category theory is in

• Aaron Mazel-Gee, Goerss-Hopkins obstruction theory for $\infty$-categories, November 2013 (pdf)

See also at survey of elliptic curves – Gluing of curves to a spectrum

A more abstract formulation of this is sketched in

• Jacob Lurie, A Survey of Elliptic Cohomology

with technical details in a sequence of articles, including

• Jacob Lurie, Moduli problems for ring spectra (pdf)

• Jacob Lurie, Representability Theorems