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moduli stack of formal groups

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Idea

The moduli stack FG\mathcal{M}_{FG} of all formal groups. Often meant are 1-dimensional commutative formal groups-

Definition

Let L=π MUL = \pi_\bullet MU be the Lazard ring.

Write G +G^+ for the group scheme given on a ring RR by

G +(R){gR[[x]]|g(t)=b 1t+b 2t 2+withb 1R ×}. G^+(R) \coloneqq \{g\in R[ [x] ] \vert g(t) = b_1 t + b_2 t^2 + \cdots \; with\; b_1 \in R^\times \} \,.

There is a canonical action of G +G^+ on Spec(L)Spec(L). The quotient stack of this action is the moduli stack of (1d commutative) formal groups

fg=(Spec(L))/G +. \mathcal{M}_{fg} = (Spec(L))/G^+ \,.

(e.g. Lurie, lecture 11, def. 2)

Properties

Chromatic height stratification

The moduli stack of formal groups FG\mathcal{M}_{FG} admits a natural stratification whose open strata are labeled by a natural number called the height of formal groups.

The complex oriented cohomology theories associated to these formal groups by the Landweber exact functor theorem accordingly also inherit such an integer label, called chromatic filtration. Studying this is the topic of chromatic homotopy theory.

tower diagram/filteringspectral sequence of a filtered stable homotopy type
filtered chain complexspectral sequence of a filtered complex
Postnikov towerAtiyah-Hirzebruch spectral sequence
chromatic towerchromatic spectral sequence
skeleta of simplicial objectspectral sequence of a simplicial stable homotopy type
skeleta of Sweedler coring of E-∞ algebraAdams spectral sequence
filtration by support
slice filtrationslice spectral sequence

Morava stabilizer group

Write 𝔽 p¯\overline{\mathbb{F}_{\mathrm{p}}} for the algebraic closure of 𝔽 p\mathbb{F}_p.

The stratum FG n\mathcal{M}_{FG}^n can be identified with the homotopy quotient Spec(𝔽¯ p)//𝔾Spec (\overline{\mathbb{F}}_{\mathrm{p}})// \mathbb{G}, where the group 𝔾\mathbb{G} is the Morava stabilizer group.

This is (Lurie 10, lect. 19, prop. 1) See also the beginning of Lurie 10, lect 21.

Deformation theory

The deformation theory around these strata is Lubin-Tate theory.

Relation to moduli of elliptic curves and tori

Inside the moduli stack of formal groups sit, in that order, that of cubic curves, the moduli stack of elliptic curves, the moduli stack of tori.

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

coveringby of level-n structures (modular curve)
*=Spec()\ast = Spec(\mathbb{Z})\toSpec([[q]])Spec(\mathbb{Z}[ [q] ])\to ell¯[n]\mathcal{M}_{\overline{ell}}[n]
structure group of covering /2\downarrow^{\mathbb{Z}/2\mathbb{Z}} /2\downarrow^{\mathbb{Z}/2\mathbb{Z}} SL 2(/n)\downarrow^{SL_2(\mathbb{Z}/n\mathbb{Z})} (modular group)
moduli stack 1dTori\mathcal{M}_{1dTori}\hookrightarrow Tate\mathcal{M}_{Tate}\hookrightarrow ell¯\mathcal{M}_{\overline{ell}} (M_ell)\hookrightarrow cub\mathcal{M}_{cub}\to fg\mathcal{M}_{fg} (M_fg)
of1d toriTate curveselliptic curvescubic curves1d commutative formal groups
value 𝒪 Σ top\mathcal{O}^{top}_{\Sigma} of structure sheaf over curve Σ\SigmaKUKU[[q]]KU[ [q] ]elliptic spectrumcomplex oriented cohomology theory
spectrum Γ(,𝒪 top)\Gamma(-, \mathcal{O}^{top}) of global sections of structure sheaf(KO \hookrightarrow KU) = KR-theoryTate K-theory (KO[[q]]KU[[q]]KO[ [q] ] \hookrightarrow KU[ [q] ])(Tmf \to Tmf(n)) (modular equivariant elliptic cohomology)tmf𝕊\mathbb{S}

References

On quasicoherent sheaves over fg\mathcal{M}_{fg}:

Revised on December 13, 2017 09:52:51 by Anonymous (46.19.85.151)