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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
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.
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
$\pi_2 E_{k,\Gamma}$ contains a unit and $\pi_{2n+1} E_{k,\Gamma} = 0$, hence in particular $E_{k,\Gamma}$ is complex orientable;
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).
(…)
this appears reproduced as (Behrens 13, theorem 4.2)
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
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)).
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 | $\mathbb{S}$ |
An introduction to and survey of the Goerss-Hopkins-Miller-Lurie theorem is in
which has grown out of
Details on the Hopkins-Miller theorem are in
and a survey is also in
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
An exposition with indications of further developments in (infinity,1)-category theory is in
See also at survey of elliptic curves – Gluing of curves to a spectrum
A more abstract formulation of this is sketched in
with technical details in a sequence of articles, including
Last revised on November 17, 2020 at 13:08:49. See the history of this page for a list of all contributions to it.