v(2.1) (Some of these definitions are in my note, or many other texts availables elsewhere)
Let $X$ be a spectrum (in homotopy theory), such as the sphere spectrum $\S=\Sigma^\infty S^0$. We often want to know the stable homotopy groups $\pi_n(X):= [\Sigma^n\S, X]$.
Given another spectrum $E$, representing some nice cohomology, such as $H\Z$ for integral cohomology, or $M\Z_{(p)}$ for $p$-local Moore spectrum.
Definition 1.1
A $E$-acyclic spectrum $X$, is a spectrum such that $E\wedge X\simeq 0$.
A $E$-equivalence $f:X\to Y$, is a morphism such that $f:E\wedge X\to E\wedge Y$
A $E$-local spectrum $X$, is a spectrum such that the following equivalent conditions are satisfied
1) for any $E$-equivalence $f:Y\to Z$, $[f,X]:[Z, X]\to[Y,X]$ is an isomorphism
2) for any $E$-acyclic spectrum $Y$, $[Y,X]\cong 0$.
Some cohomology of interests are
Definition 1.2
A complex oriented cohomology is a spectrum $E$, such that $E^2(\CP^\infty)\to E^2(\CP^1)\cong E^0$ is surjective. A complex orientation is a lift $t_E\in E^2(\CP^∞)$ of $1_E$.
Using Atiyah-Hirzebruch Spectral Sequence to compute $E^*(\CP^n)$, we have $E^*(\CP^n)\cong E^*[[t_E]]/(t_E^{n+1})$, and $\lim^1 E^*(\CP^n)$ vanishes so
Theorem 1.3
For a complex oriented cohomology $E$, and a complex orientation $t_E\in E^2(\CP^∞)$, then \[ E^*(\CP^∞)\cong E^*[[t_E]] \]
There is a group structure on $\CP^∞$ induced from the tensor product of complex line bundles (note, $BU(1)\simeq \CP^∞$). Therefore, given a complex oriented cohomology $(E,t_E)$, there is then a cogroup structure on $E^*[[t_E]]$. We refer to a cogroup structure on the formal power series ring over $E^*$ as a formal group law over $E^*$. This is determined by a formal power series $F(x,y)$ determined by the image of $t_E$.
A formal group law $x+_F y:=F(x,y)$ over $R$ determines a sequence of power series \[ [n]_F t:=t+_F t+_F…+_F t \] $n$ times.
For a prime $p$, we let $v_n\in R$ denote the coefficient in front of $t^{p^n}$ term in $[p]_F t$.
Let $MU$ denote the Thom spectrum of universal complex vector bundle.
Theorem 1.4
There is a complex orientation $t_{MU}$ on $MU$, and let $E$ be a commutative ring spectrum, there is bijection \[ \text{Ring maps $MU\to E$}/\sim \cong \text{complex orientation of $E$} \]
Therefore, not only $MU$ is complex oriented, it is universal among all the complex orientation over multiplicative cohomology.
A classic theorem of Quillen says.
Theorem 1.5
The ring $MU^*\cong \Z[a_1, a_2,…]$ is the Lazard ring, classifying formal group laws and the formal group law $F_{MU}$ over $MU^*$ is the universal formal group law, such that the ring map determining the complex orientation over $E$ determines the formal group law over $E$.
Therefore, if we have a formal group law $F$ over a graded ring $R_*$, there is then an unique graded ring map $MU_*\to R_*$. We have a functor in graded $R_*$-modules \[ R_*(X):=MU_*(X)\otimes _{MU_*} R_* \] This is not necessarily a homology. However, under certain circumstances, this determines a homology.
Theorem 1.6
$R_*(-)$ is a homology if there is a prime $p$, and associated $v_n$ in $MU_*$ such that $(p, v_1,v_2,…)$ is a regular sequence in $R_{*}$, i.e. $v_n$ is not a zero divisor of $R_*/(p,v_1,…,v_{n-1})R_*$.
We refer to this condition as Landweber flat
Let $\Mfg$ be the (fpqc) moduli stack of formal groups over $\spec\Z$. As it turns out
Theorem 1.7
A graded ring map $MU_*\to R_*$ determining a formal group law is Landweber flat iff the classifying map $\spec(R_*)\to \Mfg$ is flat.
A multiplicative cohomology $E$ is even periodic if there exists $u\in E^2$ such that $\times u:E^*\to E^{*+2}$ is an isomorphism, and $E^1\cong 0$. A even periodic cohomology is clearly complex orientable.
Let $MP:=MU[u^{\pm}]\cong \bigvee_{n\in \mathbb{Z}} \Sigma^{2n}MU$, this is even periodic. Then $MP_0\cong MU_*$. Then a Landweber flat $R$ (ungraded) determines an homology $RP_*(-):=MP_*(-)\otimes_{MP_0}R$.
From now we assume everything is $p$-local, then there is a open substack filtration $\Mfg^{\leq n}\subseteq \Mfg$ determining the $p$-local formal groups of height $\leq n$, the complement of $Mfg^{\geq n}$ the formal groups of height $\geq n$ i.e. formal group laws such that and $v_{m}=0$ for $m\leq n-1$.
The open substack $\Mfg^{\leq n}$ is determined by an affine cover \[\spec(\Z_{(p)}[v_1,…,v_n^{\pm}]\to \Mfg^{\leq n} \]
This determines the Johnson-Wilson spectrum $E(n)$ such that $E(n)_*=\Z_{(p)}[v_1,…,v_n^{\pm}][u^\pm]$.
The $E(n)$-localization, denotes $L_n$ determines a tower \[ id\to …\to L_n\to L_{n-1}\to …L_1\to L_0 \simeq L_{H\mathbb{Q}} \]
Theorem 1.8
Lur10 For finite spectrum $X$, $X\simeq \holim_n L_n X$.
The closed substack $\Mfg^n$ of formal groups of height exactly $n$, i.e. height $\geq n$ such that $v_n$ is invertible. There is an affine cover $\spec(\Fp[v_n^{\pm}])
This is not flat over $Mfg$, but there is still a spectrum $K(n)$, the Morava K-theory, such that \[K(n)_*\cong \Fp[v_n^{\pm}][u^\pm]\].
Theorem 1.9
For any spectrum $X$, there is a homotopy pullback diagram
\begin{array}{ccc} L_n X & \longrightarrow & \LKn X \end{array} \begin{array}{ccc} \downarrow && \downarrow \end{array} \begin{array}{ccc} L_{n-1} X & \longrightarrow & L_{n-1}\LKn X \end{array}
Therefore, it is important to understand $LKn \S$, as it is the first step to recover $\pi_* \S$.
Let $\LT_n$ be the Lubin-Tate space of deformation of the Honda formal group law, a height $n$ formal group law $\Gamma_n$ over $\Fpn$. This space lives over $\widehat{\Mfg^{n}}$, the infinitesimal formal neighbourhood of $\Mfg^{n}$.
The space $\LT_n$ is discrete, and it is represented by a profinite ring $W(\Fpn)[[v_1,…,v_{n-1}]]$ and an universal deformation of $\Gamma_n$, where $W(\Fpn)$ is the Witt vectors ring of $\Fpn$, which is local with residue field $\Fpn$, and there is a spectrum $E_n$ such that \[ (E_n)_{*}=W(\Fpn)[[v_1,…,v_{n-1}]][u^\pm] \]
Moreover, let $\Gn$ denote the automorphism group of $\Gamma_n$ over $\Fp$, which is $Gal(\Fpn/\Fp)\ltimes P_n$ where $P_n$ is the automorphism of $\Gamma_n$ over $\Fpn$ (we follow the notation used in [OAI26]). Let $\O_{D_n}$ denote the endomorphism ring of $\Gamma_n$ over $\Fpn$, with its fraction ring, a division algebra $D_n$ and a valuation on $D_n$, then $P_n=\O_{D_n}^\times$.
We have $\LT_n//\Gn\simeq \widehat{\Mfg^{n}}$. The group $P_n$ (hence $\Gn$) is profinite determined by $\lim_a P_n/U_a$, where $U_a$ is the closed subgroup of $P_n$ of automorphisms of $\Gamma_n$ that is equivalent to the identity modulo $T^{p^{a+1}}$. $\Gn$ acts continuously on $(E_n)_*$.
Theorem 1.10
For a spectrum $X$, there is a spectral sequence \[ E^{s,t}_2=\Hcts^{s}(\Gn, (E_n)_t X)\implies \pi_{t-s}\LKn X \]which strongly converges for CW spectra.
We will refer to this spectral sequence DHSS.
The problem of concern is the following of Hovery-Strickland
Conjecture A
For every $n$, $\pi_* \LKn\S$ is $\Z_p$-finitely generated for each grade.