∞-Lie theory (higher geometry)
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The moduli stack $\mathcal{M}_{FG}$ of all formal groups. Often meant are 1-dimensional commutative formal groups-
Let $L = \pi_\bullet MU$ be the Lazard ring.
Write $G^+$ for the group scheme given on a ring $R$ by
There is a canonical action of $G^+$ on $Spec(L)$. The quotient stack of this action is the moduli stack of (1d commutative) formal groups
(e.g. Lurie, lecture 11, def. 2)
The moduli stack of formal groups $\mathcal{M}_{FG}$ 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.
Write $\overline{\mathbb{F}_{\mathrm{p}}}$ for the algebraic closure of $\mathbb{F}_p$.
The stratum $\mathcal{M}_{FG}^n$ can be identified with the homotopy quotient $Spec (\overline{\mathbb{F}}_{\mathrm{p}})// \mathbb{G}$, where the group $\mathbb{G}$ is the Morava stabilizer group.
This is (Lurie 10, lect. 19, prop. 1) See also the beginning of Lurie 10, lect 21.
The deformation theory around these strata is Lubin-Tate theory.
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:
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 | $\mathbb{S}$ |
Niko Naumann, Comodule categories and the geometry of the stack of formal groups, Advances in Mathematics 215 (2007) pp 569-600, doi:10.1016/j.aim.2007.04.007, arXiv:math/0503308.
Brian D. Smithling, On the moduli stack of commutative, 1-parameter formal Lie groups (arXiv:0708.3326)
Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 11 Formal groups (pdf)
Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 14 Classification of formal groups (pdf)
Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 19 Morava stabilizer groups (pdf)
Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 21 Lubin-Tate theory (pdf)
On quasicoherent sheaves over $\mathcal{M}_{fg}$:
Paul Goerss, Realizing Families of Landweber Exact Homology Theories (arXiv:0905.1319)
Paul Goerss, Quasi-coherent sheaves on the moduli stack of formal groups (arXiv:0802.0996)