The Chromatic Filtration and Morava $E$-Theory in Spectral Geometry¶

Job: develop the chromatic filtration of $\mathrm{Sp}$ (the $v_n$-periodic layers, the chromatic convergence), the Morava $E$-theory $E_n$ (the Lubin–Tate deformation, the stabilizer group, the homotopy fixed points), the $K(n)$-local category and the Morava $E_2$-term, and the spectral-geometric / arithmetic interpretation (the Lubin–Tate tower, the local Langlands) — with the $n=1$ Morava $E$-theory / $p$-adic $K$-theory / the image of $J$ as the worked example.

Core¶

The chromatic filtration (Ravenel)¶

The chromatic filtration of the stable homotopy category by the $v_n$-periodic spectra (the Morava $K$-theory $K(n)$ and the Johnson–Wilson $E(n)$): the chromatic tower $$ \cdots\to L_{E(n+1)}\mathrm{Sp}\to L_{E(n)}\mathrm{Sp}\to L_{E(n-1)}\mathrm{Sp}\to\cdots, $$ the Bousfield localization with respect to $E(n)$ (the $v_n$-periodic localization), with the chromatic convergence theorem (Ravenel): $$ \mathrm{Sp}=\varprojlim_n L_{E(n)}\mathrm{Sp}, $$ the stable homotopy category is the inverse limit of the chromatic localizations. The chromatic fracture squares: $L_{E(n)}\mathrm{Sp}$ is the pullback of $L_{K(n)}\mathrm{Sp}$ and $L_{E(n-1)}\mathrm{Sp}$ over $L_{K(n)}L_{E(n-1)}\mathrm{Sp}$ (the fracture / the gluing of the chromatic layer $n$ and the lower layers).

Chromatic layer	Bousfield class	Localization
$v_0$ (rational)	$K(0)=H\mathbb{Q}$	$L_{H\mathbb{Q}}\mathrm{Sp}$ (the rational stable homotopy)
$v_1$ (the $p$-adic $K$-theory)	$K(1)$	$L_{K(1)}\mathrm{Sp}$ (the $K(1)$-local, the $J$-homomorphism image)
$v_n$ (Morava $K(n)$)	$K(n)$	$L_{K(n)}\mathrm{Sp}$ (the $v_n$-periodic layer)
$E(n)$ (Johnson–Wilson)	$E(n)$	$L_{E(n)}\mathrm{Sp}$ (the $v_0,\dots,v_n$-local)
Chromatic convergence	$E(\infty)=\bigvee_n K(n)$	$\mathrm{Sp}=\varprojlim_n L_{E(n)}\mathrm{Sp}$

The chromatic spectral sequence (the $E_2$-page the Morava $E_2$-term, converging to the stable stems). The relation to the algebraic cobordism chromatic picture: the $\mathrm{MGL}$-based chromatic filtration is the motivic analog.

Morava $E$-theory and the Lubin–Tate deformation¶

The Morava $E$-theory $E_n$ (the Lubin–Tate deformation of a height-$n$ formal group $\Gamma_n$ over $\mathbb{F}_{p^n}$): the universal deformation ring $$ R_n=W(\mathbb{F}{p^n})[[u_1,\dots,u, $$ the even-periodic }]]\quad\text{(the Lubin–Tate ring)$\mathbb{E}_\infty$-ring $E_n$ with $\pi_0E_n=R_n$ and $\pi_*E_n=R_n[u^{\pm}]$ (the periodicity $u\in\pi_2E_n$, $|u|=(2)$). The Morava stabilizer group $$ \mathbb{G}n=\mathrm{Aut}(\Gamma_n)\rtimes\mathrm{Gal}(\mathbb{F}_p) $$ (the automorphisms of the formal group }/\mathbb{F$\Gamma_n$, the profinite group acting on $E_n$ via the $\mathbb{E}_\infty$-ring automorphisms — the Hopkins–Miller theorem: $E_n$ has a canonical $\mathbb{E}_\infty$-structure with the $\mathbb{G}_n$-action).

Morava $E$-theory data	Description
The height-$n$ FGL $\Gamma_n$	over $\mathbb{F}_{p^n}$, the Honda formal group (height $n$)
Lubin–Tate ring $R_n$	$W(\mathbb{F}_{p^n})[[u_1,\dots,u_{n-1}]]$ (the universal deformation)
$E_n$	the even-periodic $\mathbb{E}_\infty$-ring, $\pi_0E_n=R_n$, $\pi_*E_n=R_n[u^{\pm}]$
Stabilizer group $\mathbb{G}_n$	$\mathrm{Aut}(\Gamma_n)\rtimes\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)$ (profinite)
Homotopy fixed points $E_n^{h\mathbb{G}_n}$	the Devinatz–Hopkins / Goerss–Hopkins–Miller construction
$K(n)$-local category $\mathrm{Sp}_{K(n)}$	the Bousfield localization w.r.t. $K(n)$

The $K(n)$-local category and the Morava $E_2$-term¶

The $K(n)$-local stable homotopy category $\mathrm{Sp}_{K(n)}$ (the Bousfield localization with respect to the Morava $K$-theory $K(n)$, the category where the chromatic layer $n$ lives). The Morava $E_2$-term $$ E_2=\mathrm{Ext}_{\mathbb{G}_n}(E_n^{,},E_n^{,}) $$ (the continuous cohomology of the stabilizer group $\mathbb{G}_n$ with coefficients in $E_n^{*,*}$, the Shimura / the Morava $E_2$), converging to the $K(n)$-local stable stems $\pi_*L_{K(n)}\mathbb{S}$.

$K(n)$-local / Morava $E_2$	Status
$n=0$ (rational)	$\pi_*L_{K(0)}\mathbb{S}=\mathbb{Q}$ (the rational stems)
$n=1$ ($p$-adic $K$-theory, the $J$-homomorphism)	computed (the Adams–Novikov $E_2$ at $n=1$, the image of $J$)
$n=2$ (the Kervaire invariant, Hill–Hopkins–Ravenel)	computed (the HHR / the $C_2$-motivic, the Kervaire-one resolution)
Higher $n$	open (the Morava $E_2$ the continuous cohomology of $\mathbb{G}_n$)

The grossoids / the height-$n$ periodicity (the $v_n$ self-map, the Smith–Toda / the periodicity theorem — the $v_n$-periodic spectra exist for $n=0,1,2,\dots$ with the $v_n$ self-map of degree $2(p^n-1)$).

The spectral geometry of the Morava $E$-theory¶

The spectral DM stack of the Morava $E$-theory (the Lubin–Tate deformation space as a spectral formal neighborhood of the height-$n$ point in the moduli of formal groups $\mathcal{M}_{\mathrm{FG}}$): the étale local structure of $\mathcal{M}_{\mathrm{FG}}$ at the height-$n$ point is $\mathrm{Spec}\,R_n$ (the Lubin–Tate deformation), and the $E_n$ is the even-periodic $\mathbb{E}_\infty$-ring over $\mathrm{Spec}\,R_n$.

The $\mathbb{G}_n$-equivariant spectral geometry (the homotopy fixed points as the quotient stack $[\mathrm{Spec}\,E_n/\mathbb{G}_n]$ in the spectral / equivariant sense): the descent spectral sequence $$ E_2=H^(\mathbb{G}_n,E_n^{,})\Rightarrow\pi_(E_n^{h\mathbb{G}_n}) $$ (the $\mathbb{G}_n$-equivariant descent of $\mathrm{Spec}\,E_n\to[\mathrm{Spec}\,E_n/\mathbb{G}_n]$, the spectral-sequence $E_2$ the cohomology of the stack). The chromatic spectral sequence as a descent spectral sequence (the Morava $E_2$ the cohomology of the quotient stack).

The Lubin–Tate / Morava $E$-theory in arithmetic geometry¶

The Lubin–Tate tower (the tower of the deformations of $\Gamma_n$, the Rapoport–Zink / the Lubin–Tate spaces in arithmetic geometry), and the Harris–Taylor / the local Langlands via the Lubin–Tate spaces:

Arithmetic / Langlands	Morava $E$-theory
Lubin–Tate tower (Rapoport–Zink)	the deformations of $\Gamma_n$ (the Lubin–Tate spaces)
Local Langlands (Harris–Taylor)	the Lubin–Tate tower realizes the local Langlands correspondence
Fargues–Fontaine curve	the $p$-adic geometric framework (the Morava $E$-theory / the prismatic via the BMS)
Morava $E$-theory as a cohomology theory of the Lubin–Tate spaces	the relation to the crystalline / the prismatic cohomology

See it¶

$n=1$ Morava $E$-theory and the $p$-adic $K$-theory / the Adams–Novikov $J$-homomorphism. The height-1 formal group $\Gamma_1$ over $\mathbb{F}_p$ is the multiplicative formal group $\hat{\mathbb{G}}_m$ (the height-1 FGL, the $p$-series $[p]_F=(x^p-x)/(\text{unit})$ or $[p]_F=p x+\dots$ depending on the convention, height 1). The Lubin–Tate deformation $E_1$ is the $p$-adic $K$-theory: $$ E_1=K_p^\wedge\quad(p\text{-completed }K\text{-theory}),\qquad \pi_0E_1=\mathbb{Z}p,\qquad \piE_1=\mathbb{Z}_p[u^{\pm}], $$ the periodicity $K_p^\wedge\simeq\Sigma^2K_p^\wedge$ (the Bott periodicity, $u\in\pi_2$). The stabilizer group is $$ \mathbb{G}_1=\mathrm{Aut}(\Gamma_1)\rtimes\mathrm{Gal}(\mathbb{F}_p/\mathbb{F}_p)=\mathbb{Z}_p^\times\quad(p\text{-adic units, acting via the Adams operation }\psi^k), $$ the $p$-adic units acting on $K_p^\wedge$ via the Adams operation $\psi^k$ (the $k\in\mathbb{Z}_p^\times$ acting by $\psi^k$ on the $K$-theory). The homotopy fixed points $$ E_1^{h\mathbb{G}_1}=(K_p^\wedge)^{h\mathbb{Z}_p^\times} $$ (the Devinatz–Hopkins construction, the $p$-adic $K$-theory with the $\mathbb{Z}_p^\times$-action). The $K(1)$-local stable stems $\pi_*L_{K(1)}\mathbb{S}$ are computed by the Morava $E_2$-term $$ E_2=H^(\mathbb{Z}p^\times,\mathbb{Z}_p)=\mathrm{Ext}_p), $$ recovering the }_p^\times}(\mathbb{Z}_p,\mathbb{Zimage of $J$ (the $J$-homomorphism $\pi_*SO\to\pi_*^{\mathrm{top}}\mathbb{S}$, the Adams–Novikov $E_2$-term at $n=1$, the $p$-torsion in the stable stems from the $J$-homomorphism). The key point: the height-1 Morava $E$-theory is the $p$-adic $K$-theory, and the $K(1)$-local stable stems are the image of $J$ — the prototypical chromatic / Morava computation, the base case for the chromatic filtration. The $J$-homomorphism $\pi_{2k-1}SO\to\pi_{2k-1}^{\mathrm{top}}\mathbb{S}$ has image the $p$-torsion in $\pi_{2k-1}^{\mathrm{top}}\mathbb{S}$ of order $p^{v_p(k)+1}$ (the Adams–Novikov formula), exactly the $K(1)$-local / the Morava $E_2$ at $n=1$.

Check¶

Verify that the height-1 Morava $E$-theory $E_1$ is the $p$-adic $K$-theory, and identify the stabilizer group action with the Adams operations.

The height-1 formal group $\Gamma_1$ over $\mathbb{F}_p$ is the multiplicative formal group $\hat{\mathbb{G}}_m$ (the formal completion of $\mathbb{G}_m$, the FGL $F(x,y)=x+y+xy$, the height-1 FGL with $[p]_F=(1+x)^p-1\equiv x^p\pmod{p}$). The Lubin–Tate universal deformation is the $p$-adic multiplicative group, the formal completion of $\mathbb{G}_m$ over $\mathbb{Z}_p$, and the even-periodic $\mathbb{E}_\infty$-ring with this formal group is $K_p^\wedge$ (the $p$-completed complex $K$-theory, $\pi_0=\mathbb{Z}_p$, $\pi_*=\mathbb{Z}_p[u^{\pm}]$). The stabilizer group $\mathbb{G}_1=\mathrm{Aut}(\hat{\mathbb{G}}_m/\mathbb{F}_p)=\mathbb{F}_p^\times\rtimes\mathrm{Gal}(\mathbb{F}_p/\mathbb{F}_p)=\mathbb{Z}_p^\times$ (the $p$-adic units, the automorphisms of $\hat{\mathbb{G}}_m$), acting on $K_p^\wedge$ by the Adams operation $\psi^k$ (the $k\in\mathbb{Z}_p^\times$ acting by $\psi^k$ on the $K$-theory, $\psi^k(u)=k\cdot u$ in $\pi_2$). So $E_1=K_p^\wedge$ with the $\mathbb{Z}_p^\times$-action via the Adams operations.

Explain why the chromatic convergence $\mathrm{Sp}=\varprojlim_n L_{E(n)}\mathrm{Sp}$ is a convergence (a limit), not a splitting (a direct sum), and what the fracture squares do.

The chromatic convergence says the stable homotopy category is the inverse limit of the chromatic localizations $L_{E(n)}\mathrm{Sp}$ — a convergence (the limit, the recovery of $\mathrm{Sp}$ from the chromatic layers), not a splitting (the layers are not independent: there are the fracture / the gluing data). The fracture squares are the splitting of $L_{E(n)}\mathrm{Sp}$ as the pullback of $L_{K(n)}\mathrm{Sp}$ (the $v_n$-periodic layer) and $L_{E(n-1)}\mathrm{Sp}$ (the lower layers) over $L_{K(n)}L_{E(n-1)}\mathrm{Sp}$ — the gluing of the layer $n$ with the lower layers. So the chromatic tower is a tower of localizations (the convergence), and the fracture squares are the gluing of the layers (not a direct sum / splitting, but a pullback / gluing). The $E(n)$-local is the gluing of $K(0),\dots,K(n)$ via the fracture, and the convergence is the limit over $n$.

Pitfalls¶

The chromatic convergence is the chromatic tower (the convergence $\mathrm{Sp}=\varprojlim_n L_{E(n)}\mathrm{Sp}$, the limit, not the splitting — the fracture squares are the gluing, not the direct sum). Do not state the chromatic convergence as a direct-sum splitting.
The Morava $E$-theory $E_n$ is the Lubin–Tate deformation (the universal deformation of the height-$n$ FGL $\Gamma_n$, the Lubin–Tate ring $R_n$), not an arbitrary height-$n$ theory — the universality is the key.
The $\mathbb{G}_n$-homotopy fixed points (Devinatz–Hopkins) is the continuous / the profinite homotopy fixed points, not the naive fixed points — the profinite / continuous version is essential (the $\mathbb{G}_n$ is a profinite group, the action is continuous).
The $K(n)$-local category is the $v_n$-periodic layer (the $K(n)$-localization is not the same as the $E(n)$-localization — the $E(n)$ is the gluing of $K(0),\dots,K(n)$, the $K(n)$ is the single layer $n$). Do not conflate $L_{K(n)}$ and $L_{E(n)}$.
The Lubin–Tate / local Langlands connection (Harris–Taylor) is via the Lubin–Tate tower (the Rapoport–Zink spaces, the tower of deformations), not directly the Morava $E$-theory — the Morava $E$-theory is a cohomology theory of the Lubin–Tate spaces, the connection to the local Langlands is via the tower (the Rapoport–Zink / the Harris–Taylor), not a direct geometric equivalence.
The Fargues–Fontaine curve is the $p$-adic geometric framework (the relation to the Morava $E$-theory / the prismatic is the BMS / the cyclotomic connection, not a direct geometric equivalence) — the curve is a $p$-adic avatar, the Morava $E$-theory a chromatic / homotopy-theoretic object.
The $n=2$ Kervaire invariant (Hill–Hopkins–Ravenel) is via the $C_2$-motivic / $\mathbb{R}$-motivic Adams (see motivic Adams), not directly the Morava $E_2$ — the HHR uses the $C_2$-equivariant / $\mathbb{R}$-motivic framework, the Morava $E_2$ at $n=2$ is related but the HHR is the $C_2$-motivic computation.

Refs¶

Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory (Annals 1992); the chromatic convergence
Lubin–Tate, "Formal moduli for one-parameter formal Lie groups" (BAMS 1966); Hopkins–Miller (the $\mathbb{E}_\infty$-structure on $E_n$)
Devinatz–Hopkins, "Homotopy fixed point spectra for closed profinite subgroups" (2004); Goerss–Hopkins, "Moduli spaces of commutative ring spectra" (2005)
Hill–Hopkins–Ravenel, "On the nonexistence of elements of Kervaire invariant one" (Annals 2016)
Harris–Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties (Annals 2001); Fargues–Fontaine, Courbes et fibrés vectoriels (2018)

Next¶

SAG index · Foundations · Elliptic Cohomology & TMF · Derived Moduli & Deformation · Spectral GAGA & Descent · Motivic / algebraic cobordism · Motivic Adams · Langlands

Chromatic layer	Bousfield class	Localization
\(v_0\) (rational)	\(K(0)=H\mathbb{Q}\)	\(L_{H\mathbb{Q}}\mathrm{Sp}\) (the rational stable homotopy)
\(v_1\) (the \(p\)-adic \(K\)-theory)	\(K(1)\)	\(L_{K(1)}\mathrm{Sp}\) (the \(K(1)\)-local, the \(J\)-homomorphism image)
\(v_n\) (Morava \(K(n)\))	\(K(n)\)	\(L_{K(n)}\mathrm{Sp}\) (the \(v_n\)-periodic layer)
\(E(n)\) (Johnson–Wilson)	\(E(n)\)	\(L_{E(n)}\mathrm{Sp}\) (the \(v_0,\dots,v_n\)-local)
Chromatic convergence	\(E(\infty)=\bigvee_n K(n)\)	\(\mathrm{Sp}=\varprojlim_n L_{E(n)}\mathrm{Sp}\)

Morava \(E\)-theory data	Description
The height-\(n\) FGL \(\Gamma_n\)	over \(\mathbb{F}_{p^n}\), the Honda formal group (height \(n\))
Lubin–Tate ring \(R_n\)	\(W(\mathbb{F}_{p^n})[[u_1,\dots,u_{n-1}]]\) (the universal deformation)
\(E_n\)	the even-periodic \(\mathbb{E}_\infty\)-ring, \(\pi_0E_n=R_n\), \(\pi_*E_n=R_n[u^{\pm}]\)
Stabilizer group \(\mathbb{G}_n\)	\(\mathrm{Aut}(\Gamma_n)\rtimes\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)\) (profinite)
Homotopy fixed points \(E_n^{h\mathbb{G}_n}\)	the Devinatz–Hopkins / Goerss–Hopkins–Miller construction
\(K(n)\)-local category \(\mathrm{Sp}_{K(n)}\)	the Bousfield localization w.r.t. \(K(n)\)

\(K(n)\)-local / Morava \(E_2\)	Status
\(n=0\) (rational)	\(\pi_*L_{K(0)}\mathbb{S}=\mathbb{Q}\) (the rational stems)
\(n=1\) (\(p\)-adic \(K\)-theory, the \(J\)-homomorphism)	computed (the Adams–Novikov \(E_2\) at \(n=1\), the image of \(J\))
\(n=2\) (the Kervaire invariant, Hill–Hopkins–Ravenel)	computed (the HHR / the \(C_2\)-motivic, the Kervaire-one resolution)
Higher \(n\)	open (the Morava \(E_2\) the continuous cohomology of \(\mathbb{G}_n\))

The Chromatic Filtration and Morava \(E\)-Theory in Spectral Geometry¶

Core¶

The chromatic filtration (Ravenel)¶

Morava \(E\)-theory and the Lubin–Tate deformation¶

The \(K(n)\)-local category and the Morava \(E_2\)-term¶

The spectral geometry of the Morava \(E\)-theory¶

The Lubin–Tate / Morava \(E\)-theory in arithmetic geometry¶

See it¶

Check¶

Pitfalls¶

Refs¶

Next¶