# There aren’t that many Morava $E$–theories

Kiran Luecke  and  Eric Peterson

###### Abstract.

Let $k$ be a perfect field of characteristic $p$. Associated to any
(1-dimensional, commutative) formal group law of finite height $n$ over $k$
there is a complex oriented cohomology theory represented by a spectrum
denoted $E(n)$ and commonly referred to as Morava $E$-theory. These spectra
are known to admit $E_{\infty}$-structures, and the dependence of the
$E_{\infty}$-structure on the choice of formal group law has been well studied
(cf. [GH], [R], [L], Section 5, [PV]). In this note we show that the
underlying homotopy type of $E(n)$ is independent of the choice of formal
group law.

##  1\. Introduction

In this section we collect some standard results in chromatic homotopy theory
that will be used in the proof. A combined reference for (almost) all of them
is [P]. All rings in this note are commutative, graded, and concentrated in
even degree. Rings that are commonly ungraded (e.g. a perfect characteristic
$p$ field $k$ and its $p$-typical Witt vectors $W(k)$) are viewed graded rings
concentrated in degree zero.

######  Definition 1.

In light of the above, define a formal group law over a ring $R$ to be a
(graded) ring map $MU_{*}\rightarrow R$. Recall that $MU_{*}$ is the
coefficient ring of the complex bordism spectrum $MU$, and
$MU_{*}\simeq\mathbb{Z}[a_{1},a_{2},...]$, $|a_{i}|=2i$. Define an ungraded
formal group law over $R$ to be a (graded) ring map $\mathrm{Laz}\rightarrow
R$, $\mathrm{Laz}\simeq\mathbb{Z}[b_{1},b_{2},...]$, $|b_{i}|=0$.

######  Theorem 2.

([Lan], cf. also [H]) Let $f\colon\thinspace MU_{*}\rightarrow R$ be a formal
group law over a ring $R$. Let $\mathcal{M}_{MU}$ be the stack associated to
the the Hopf algebroid $(MU_{*},MU_{*}MU)$. It admits a canonical map
$\operatorname{Spec}MU_{*}\rightarrow\mathcal{M}_{MU}$. Then the assignment

$X\mapsto MU_{*}X\otimes_{MU_{*}}R$

defines a homology theory if and only if the composite

$\operatorname{Spec}R\xrightarrow{f}\operatorname{Spec}MU_{*}\rightarrow\mathcal{M}_{MU}$

is flat.

######  Theorem 3.

([A]) Every homology theory is represented by a spectrum, which is unique up
to homotopy equivalence. Every morphism of homology theories is represented by
some morphism of representing spectra. This morphism of spectra is well-
defined up to phantom maps.

######  Theorem 4.

([HS]) Let $MU_{*}\rightarrow E_{*}$ and $MU_{*}\rightarrow F_{*}$ be two
Landweber exact formal group laws and let $E$ and $F$ be spectra representing
the homology theories $E_{*}X=MU_{*}X\otimes_{MU_{*}}E_{*}$ and
$F_{*}X=MU_{*}X\otimes_{MU_{*}}F_{*}$. There are no phantom maps $E\rightarrow
F$.

######  Theorem 5.

([LT]) Let $k$ be a perfect field of characteristic $p$. Let
$f\colon\thinspace\mathrm{Laz}\rightarrow k$ be an ungraded formal group law
of height $n$. Define the ring

$LT(k)=W(k)[[u_{1},u_{2},\ldots,u_{n-1}]][\beta^{\pm}],\quad|\beta|=2.$

There is a Landweber exact formal group law

$LT(f)\colon\thinspace MU_{*}\rightarrow LT(k)$

satisfying the following:

  1. (1)

$LT(f)$ is a universal deformation of $f$ up to $\star$–isomorphism.

  2. (2)

If $g\colon\thinspace\mathrm{Laz}\rightarrow k$ is an ungraded formal group
law that is isomorphic to $f$, then $LT(g)$ can be chosen to be isomorphic to
$LT(f)$.

  3. (3)

If $t\colon\thinspace k\rightarrow k^{\prime}$ is a map of perfect
characteristic $p$ fields and $T\colon\thinspace LT(k)\rightarrow
LT(k^{\prime})$ denotes the map which is $W(t)\colon\thinspace W(k)\rightarrow
W(k^{\prime})$ on coefficients and sends $u_{i}$ to $u_{i}$ and $\beta$ to
$\beta$, then $LT(t\circ f)$ can be chosen such that $T\circ LT(f)=LT(t\circ
f)$.

######  Theorem 6.

([Laz] Theorem IV) All ungraded formal group laws of a fixed height $n$ over
an algebraically closed field of positive characteristic are isomorphic.

##  2\. The Proof

######  Lemma 7.

Let $R$ be a ring with two Landweber exact formal group laws
$e,f\colon\thinspace MU_{*}\rightarrow R$, and let $E$ and $F$ be the spectra
corresponding to $e$ and $f$ under the Landweber exact functor theorem. If
there is a ring extension $u\colon\thinspace R\rightarrow S$ which is split as
an $R$-module map and over which the formal group laws $u\circ e$ and $u\circ
f$ are isomorphic and Landweber exact, then $E$ and $F$ have the same homotopy
type.

###### Proof.

First we show that $E\simeq F$ if and only if the map $\eta\colon\thinspace
E_{*}\rightarrow F_{*}E$ (induced by $1\wedge
id\colon\thinspace\mathbb{S}\wedge E\rightarrow F\wedge E$) is split as a map
of $F_{*}$-modules. Note that $E_{*}$ is canonically an $F_{*}$ module by the
equality $E_{*}=R=F_{*}$. We claim that $F_{*}E$ is flat over $F_{*}$. Indeed,
interpret Landweber exactness as flatness of maps to $\mathcal{M}_{MU}$ (cf.
Theorem 2). Consider the following pullback diagram

${\operatorname{Spec}(F_{*}E)}$${\operatorname{Spec}(E_{*})}$${\operatorname{Spec}(F_{*})}$${\mathcal{M}_{MU}.}$$\scriptstyle{e}$$\scriptstyle{f}$

The right vertical map is flat by the assumption that $e$ is Landweber exact,
hence so is the left vertical map as flat maps are preserved under pullback,
i.e., $F_{*}E$ is flat over $F_{*}$. It follows that $(F\wedge
E)_{*}X=F_{*}X\otimes_{F_{*}}F_{*}E$. For an $F_{*}$-module map
$\xi\colon\thinspace F_{*}E\rightarrow F_{*}$ consider, for each spectrum $X$,
the diagram

${E_{*}X}$${(F\wedge
E)_{*}X=F_{*}X\otimes_{F_{*}}F_{*}E}$${F_{*}X\otimes_{F_{*}}F_{*}.}$$\scriptstyle{(1\wedge
id)\circ(-)}$$\scriptstyle{id\otimes\xi}$

The diagonal composite $t_{\xi}\colon\thinspace E_{*}X\rightarrow F_{*}X$
defines a morphism of homology theories. Applying Brown representability for
maps of homology theories (cf. Theorem 3) and the absence of phantom maps
between Landweber exact homotopy types (cf. Theorem 4) produces a map of
spectra $T_{\xi}\colon\thinspace E\rightarrow F$ well-defined up to homotopy.
The assignment $\xi\mapsto T_{\xi}$ defines a map
$\varphi\colon\thinspace\mathsf{Modules}_{F_{*}}(F_{*}E,F_{*})\rightarrow
F^{*}E$. Now suppose a splitting of $\eta$ exists and call it $s$. Then there
is a map $\varphi(s)\colon\thinspace E\rightarrow F$ which on homotopy groups
(set $X=\mathrm{pt}$ above) is the composite of $\eta$ and $s$, which is the
identity map, so $E\simeq F$. In the other direction, if $E\simeq F$ then
$F_{*}E\simeq E_{*}E$, so it suffices to split $(1\wedge
id)_{*}\colon\thinspace E_{*}\rightarrow E_{*}E$, which is split by the
multiplication map $E\wedge E\rightarrow E$.

Now we show that $\eta\colon\thinspace E_{*}\rightarrow F_{*}E$ is split if
there is a ring extension $u\colon\thinspace R\rightarrow S$ which is split as
an $R$-module map, and such that the formal group laws $u\circ e$ and $u\circ
f$ are isomorphic and Landweber exact. Write $g:=u\circ e$ and $h:=u\circ f$.
Assume they are isomorphic and Landweber exact. Let $G$ and $H$ be the spectra
associated to $g$ and $h$. Then we have the following commutative diagram of
$F_{*}(=R)$-modules

${S=G_{*}}$${H_{*}G=S\otimes_{MU_{*}}MU_{*}MU\otimes_{MU_{*}}S}$${R=E_{*}}$${F_{*}E=R\otimes_{MU_{*}}MU_{*}MU\otimes_{MU_{*}}R.}$$\scriptstyle{1\otimes
1\otimes id}$$\scriptstyle{u}$$\scriptstyle{1\otimes 1\otimes
id}$$\scriptstyle{u\otimes id\otimes u}$

Since the right vertical map is split (as a map of $F_{*}$-modules), splitting
the bottom horizontal map (as a map of $F_{*}$-modules) is equivalent to
splitting the diagonal map (as a map of $F_{*}$-modules). Moreover, the top
horizontal map is split (as an $S=H_{*}$-algebra (and hence $F_{*}$-algebra)
map in fact) by first choosing an isomorphism of $g$ and $h$, which induces an
$H_{*}$-algebra isomorphism $H_{*}G\simeq G_{*}G$ and then using the
multiplication map $G\wedge G\rightarrow G$ to split $G_{*}\rightarrow
G_{*}G$. Therefore splitting the diagonal map is equivalent to splitting the
left vertical map (as a map of $F_{*}$-modules). ∎

######  Lemma 8.

At height $n<\infty$ the homotopy type of Morava $E$-theory depends only on
the choice of a perfect characteristic $p$ field. In other words, let $k$ be a
perfect characteristic $p$ field and let $E(n)_{1}$ and $E(n)_{2}$ be two
Morava $E$-theory spectra corresponding to two arbitrary height $n<\infty$
ungraded formal group laws $e_{1}$ and $e_{2}$ over $k$. Then as spectra,
$E(n)_{1}\simeq E(n)_{2}$.

###### Proof.

It suffices to check the conditions of the lemma. Let $\overline{k}$ be the
algebraic closure of $k$. Let $R$ and $S$ be the (2-periodic) Lubin-Tate
deformation rings

$R:=W(k)[[u_{1},...,u_{n-1}]][\beta^{\pm}]$
$S:=W(\overline{k})[[u_{1},...,u_{n-1}]][\beta^{\pm}].$

Let $e$ and $f$ be formal group laws over $R$ universally deforming $e_{1}$
and $e_{2}$. Then $e$ and $f$ are Landweber exact (cf. Theorem 5) and the
corresponding spectra are the Morava $E$-theory spectra $E(n)_{1}$ and
$E(n)_{2}$ (cf. [P] Definition 3.5.4). Let $u\colon\thinspace R\rightarrow S$
be the map that extends coefficients by $W(k)\rightarrow W(\overline{k})$,
sends $u_{i}$ to $u_{i}$, and $\beta$ to $\beta$. Then the formal group laws
$g:=u\circ e$ and $h=u\circ f$ over $S$ are isomorphic and Landweber exact
(again, cf. Theorem 5), since they universally deform the isomorphic formal
group laws $u\circ e$ and $u\circ f$ over $\overline{k}$ (cf. Theorem 6).
Finally, the map $u\colon\thinspace R\rightarrow S$ is split as an $R$-module
map, because the map $W(k)\rightarrow W(\overline{k})$ is split as a
$W(k)$-module map (cf. [S]). ∎

##  3\. Acknowledgements

We would like to thank Rodrigo Marlasca Aparicio for posting the Mathoverflow
question “Does the spectrum of Morava E-theory depend only on height?” [Ap]
which prompted this note. We would also like to thank Will Sawin for providing
us with the statement we needed regarding the splitting of $p$-typical Witt
vectors (in private communication as well as the Mathoverflow post [S]).

##  4\. References

[A] Adams, J.F. A variant of E. H. Brown’s representability theorem, Topology,
Volume 10, Issue 3, 1971, Pages 185-198, ISSN 0040-9383,

[Ap] Aparicio, R. Mathoverflow question.
https://mathoverflow.net/questions/412686/does-the-spectrum-of-morava-e-
theory-depend-only-on-height?noredirect=1&lq=1

[GH] Goerss, P. and M. Hopkins. Moduli Spaces of Commutative Ring Spectra

[H] Hohnhold, H. The Landweber exact functor theorem, Math. Surveys Monogr.,
vol. 201, Amer. Math. Soc., Providence, RI, 2014, pp. 35–45. MR 3328537

[HS] Hovey, M. and Strickland, N. P. Morava K-theories and localisation, Mem.
Amer. Math. Soc., 139 (1999), no. 666.

[Lan] Landweber, P. Homological Properties of Comodules over MU∗(MU) and
BP∗(BP). American Journal of Mathematics, Volume 98, No. 3, pp. 591-610, 1976

[Laz] Lazard, M. Sur les groupes de Lie formels a un parametre. Bull. Soc.
Math. France, 83:251-274, 1955.

[LT] Lubin, J. and Tate, J. Formal moduli for one-parameter formal LIe groups.
Bull. Soc. Math. France, 94:49-59, 1966.

[L] Lurie, J. Elliptic Cohomology II: Orientations.

[P] Peterson, E. Formal geometry and bordism operations. Cambridge studies in
advanced mathematics. 177, 2019.

[PV] Pstragowski P. and Vankoughnett, P. Abstract Goerss-Hopkins Theory

[R] Rezk, C. Notes on the Hopkins-Miller Theorem

[S] Sawin, W. Mathoverflow answer.
https://mathoverflow.net/questions/413773/splitting-the-witt-vectors-of-
overline-mathbbf-p