Morava E-Theory and Morava K-Theory (Lecture 22)

April 27, 2010
Let k be a perfect field of characteristic p, and suppose we are given a formal group law f of heightn over k. In the last lecture, we saw that the universal deformation of f is classified by the Lubin-Tatering R = W (k)[[v1 , . . . , vn−1 ]. We note that this deformation of f over R is Landweber-exact: the sequencev0 = p, v1 , . . . , vn−1 is regular by construction, and vn has invertible image in R/(v0 , v1 , . . . , vn−1 ) ' k byvirtue of our assumption that the original formal group law f has height n.
Using results of previous lectures, we can construct an even periodic spectrum E(n) with π∗ E(n) 'W (k)[[v1 , . . . , vn−1 ]][β ±1 ], where β has degree 2. The cohomology theory E(n) (which really depends not
only on n, but on a choice of field k and a formal group of height n over k) is called Morava E-theory. It isalso sometimes called Lubin-Tate theory or completed Johnson-Wilson theory.
Associated to E(n) is a Bousfield localization functor LE(n) . Note that a spectrum X satisfies LE(n) X = 0
if and only if X is E(n)-acyclic: that is, the homology groups E(n)∗ (X) ' MP∗ (X) ⊗L R vanish. We canassociate to X a quasi-coherent sheaf FX on MFG × Spec Z(p) . The vanishing of E(n)∗ (X) is equivalent to
the requirement that both FX and FΣ(X) (and therefore FΣk X for every integer k) are supported on theclosed substack M≥n+1⊆ MFG × Spec Z(p) . This is one sense in which LE(n) “behaves like” restriction toFGthe open substack M≤nFG ⊆ MFG × Spec Z(p) . This suggests that LE(n) should be a smashing localization.
This is indeed the case:Theorem 1 (Smash Product Theorem). The localization LE(n) is smashing: that is, it preserves directsums.
We will prove Theorem 1 later in this course.
Our next goal is to introduce a homotopy theoretic counterpart to the mechanism of restricting to thenclosed substack MnFG ⊆ M≤nFG . This is more subtle, since MFG is not flat over MFG , so we cannot proceed
via Landweber’s theorem.
Fix a prime p, and consider the p-local complex bordism spectrum MU(p) . This complex bordism spectrum has the structure of an E∞ -ring. In particular, there is a good theory of (structured) MU(p) -modules,and a relative smash product (M, N ) 7→ M ⊗MU(p) N .
i

We have π∗ MU(p) ' L(p) ' Z(p) [t1 , t2 , . . .], where we may assume that vi = tp −1 for each i > 0. By
convention, we set t0 = p ∈ π0 MU(p) .
For each integer k, let M (k) denote the cofiber of the map Σ2k MU(p) → MU(p) given by multiplication
by tk .
Lemma 2. Each M (k) admits a unital and homotopy associative multiplication (in the category of MU(p) module spectra).
Proof. We fix the unit of M (k) to be the evident map u : MU(p) → M (k). The smash product M (k) ⊗MU(p)1M (k) can be realized as the total homotopy cofiber of the commutative diagramΣ4k MU(p)tk/ Σ2k MU(p)tktkΣ2k MU(p)/ MU(p) .

tk

We let K denote the total cofiber of an analogous diagram, where we replace the upper left hand cornerwith the zero spectrum. In other words, K is the cofiber of the map(tk ,tk )(Σ2k M (k))2 → M (k).
There is an evident map α : K → M (k). To define an MU(p) -linear multiplication on M (k) (having u as aunit) is equivalent to factoring u as a compositionβγ

K → M (k) ⊗MU(p) M (k) → M (k)in the setting of MU(p) -modules.
To produce such a factorization, it suffices to show that the composition
αker(β) → K → M (k)is nullhomotopic. Note that ker(β) can be identified with the desuspension of the total cofiber of the squareΣ4k M (k)

/0
0
/ 0;

that is, we have ker(β) ' Σ4k+1 M (k), and the relevant obstruction lives in π4k+1 M (k) ' (Lp /tk )4k+1 ' 0.
We now show that the multiplication γ is homotopy associative (in the setting of MU(p) -modules). We
have two natural multiplication maps
f, g : X = M (k) ⊗MU(p) M (k) ⊗MU(p) M (k) → M (k).
We wish to prove that the difference f − g is nullhomotopic. Note that X can be described as the total
cofiber of a cube/ Σ4k MU(p)Σ6k MU(p)
MMM
MMM
MMM
MMM
MMM
MMM
M&M&4k/Σ MU(p)Σ2k MU(p)Σ4k MU(p)
MMM
MMM
MMM
M&

Σ2k MU(p)/ Σ2k MU(p)
MMM
MMM
MMM
MM& / MU(p) .

Let Y be the total cofiber of an analogous diagram obtained by replacing the upper left corner by zero. By
construction, the difference f − g is nullhomotopic on Y , so that f − g factors as a composition
X → X/Y ' Σ6k+3 MU(p) → M (k).
Since π6k+3 M (k) ' (L(p) /tk )6k+3 ' 0, the second map is nullhomotopic so that f ' g as desired.
2Remark 3. The multiplication on M (k) constructed above is not unique: the same argument shows thatthe collection of such multiplications forms a torsor P for the group π4k+2 M (k).
The torsor P has a canonical action of the permutation group Σ2 (which acts on the smash productM (k) ⊗MU(p) M (k). If p 6= 2, then H1 (Σ2 ; π2k+2 M (k)) ' 0 so that P has a Σ2 -fixed point. This means that
we can choose the multiplication on M (k) to be homotopy commutative when p 6= 2.
Remark 4. By continuing the analysis of Lemma 2, one can show that M (k) admits the structure of an
A∞ -algebra over MU(p) .
Definition 5. Fix a prime N
number p and an integer n > 0. We let K(n) denote the smash product (overMU(p) ) of MU(p) [vn−1 ] with k6=pn −1 M (k). The spectrum K(n) is called Morava K-theory.
Using Lemma 2, we see that K(n) has the structure of a homotopy associative MU(p) -algebra; if p 6= 2,
we can even assume that K is homotopy commutative.
A simple calculation shows that the homotopy groups of K(n) are given byπ∗ K(n) ' (π∗ MU(p) )[vn−1 ]/(t0 , t1 , . . . , tpn −2 , tpn , . . .) ' Fp [vn±1 ],where vn has degree 2(pn − 1).
We have a map of ring spectrum MU(p) → K(n), giving a complex orientation on K(n). This determines
a formal group law over the ring π∗ K(n) ' Fp [vn±1 ], which has height exactly n.
Warning 6. When p = 2, the Morava K-theory spectra generally do not admit homotopy commutative ringstructures. Nevertheless, the theory of complex orientations makes sense in this setting: though K(n) itself
is not homotopy commutative, the cohomology rings K(n)∗ (X) are commutative for many important spaces(like CP∞ , BU (n), and so forth) since they are given by MU∗ (X) ⊗L Fp [vn±1 ].
Warning 7. Our construction of the ring spectra M (k) (and therefore the Morava K-theories K(n) ) involvea number of arbitrary choices. We will later see that, as a spectrum, K(n) does not depend on these choices.
We let LK(n) denote the localization with respect to the Morava K-theory K(n). We will later see that
LK(n) behaves like completion along the locally closed substack MnFG ⊆ MFG .

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