The Smash Product Theorem (Lecture 31)

April 22, 2010
In this lecture, we will apply the results of Lecture 30 to prove (part of) the smash product theorem.
We begin by summarizing the results of the previous lecture. Fix a ring spectrum E (which we take to be astructured ring spectrum for simplicity). For every specturm X, we let X • denote the cosimplicial spectrum[n] 7→ E ⊗n+1 ⊗ X.
Proposition 1. Let E be a ring spectrum and X an arbitrary spectrum. Suppose that there exists an integers ≥ 1 such that, for every finite spectrum F , the E-based Adams spectral sequence {Erp,q , dr } for X ⊗ F hasEsp,q ' 0 for p ≥ s. Then the Adams tower {Totn X • } is equivalent, as a pro-object, to a constant tower.
In the situation of Proposition 1, we have an equivalence of pro-spectra {Totn X • } ' X 0 for somespectrum X 0 . We saw in the last lecture that X 0 ' Tot(X • ) can be identified with the localization LE X.
Note that if the hypotheses of Proposition 1 are satisfied, then for every other spectrum Y , the tower{Totn (X ⊗ Y )• } ' {Totn X • ⊗ Y } is pro-equivalent to the constant tower with value X 0 ⊗ Y . It follows thatthe canonical map(LE X) ⊗ Y → LE (X ⊗ Y )is a homotopy equivalence.
Proposition 2. Let E be a p-local ring spectrum, and suppose that there exists a finite p-local spectrum Xof type 0 which satisfies the hypotheses of Proposition 1. Then LE is a smashing localization.
Proof. Let T be the collection of p-local finite spectra X such that, for every p-local spectrum Y , the canonicalmap (LE X) ⊗ Y → LE (X ⊗ Y ) is an equivalence. It is clear that T is thick. If T contains a finite p-local
spectrum of type 0, then the thick subcategory theorem implies that T contains every finite p-local spectrum;in particular, S(p) ∈ T so that LE S(p) ⊗ Y ' LE (Y ) for all Y and therefore LE is smashing.
Let us now specialize to the case where E is Morava E-theory E(n). The spectrum E(n) is complexoriented, and the map Spec π∗ E → MFG × Spec Z(p) is a faithfully flat cover of the open substack M≤nFG
classifying formal groups of height ≤ n. For every spectrum X, let FX denote the associated quasi-coherent
sheaf on MFG . The E2 -term of the E(n)-based Adams-Novikov spectral sequence for X is given by thecohomology of the chain complex
E(n)∗ (X) → (E(n) ⊗ E(n))∗ X → (E(n) ⊗ E(n) ⊗ E(n))∗ X → · · · ,which computes the cohomology of M≤nFG with coefficients in the quasi-coherent sheaves F Σk X for varying k.
Consequently, we obtain the following:
Proposition 3. Suppose that there exists a finite p-local spectrum X of type 0 and an integer s0 ≥ 1 with thefollowing property: for every finite spectrum F , the cohomology groups Hs (M≤nFG ; F X⊗F ) vanish for s ≥ s0 .
Then the localization LE(n) is smashing.

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To make things more concrete, let us assume that X is an even finite p-local spectrum: that is, a finite
p-local spectrum whose homology groups H∗ (X; Z(p) ) are free Z(p) -modules concentrated in even degrees.
This is equivalent to saying that X admits a finite cell decomposition, where each cell is an even suspensionof S(p) . Such a spectrum is always of type 0, provided that it is nonzero. For such a spectrum, the AtiyahHirzebrucn spectral sequence for computing (MU(p) )∗ (X) degenerates: that is, MU(p) ⊗X is a free module
over MU(p) (on generators corresponding to some basis for H∗ (X; Z(p) )). It follows that FX is a vector
bundle on MFG × Spec Z(p) , and that for any other spectrum F we have(MUp )∗ (X ⊗ F ) ' π∗ ((MU(p) ⊗X) ⊗MU(p) (MU(p) ⊗F )) ' (MU(p) )∗ (X) ⊗MU(p) (MU(p) )∗ F.
On the moduli stack MFG , we deduce that the canonical map of quasi-coherent sheaves
FX ⊗ FF → FX⊗Fis an isomorphism after localization at p. We conclude the following:
Proposition 4. Suppose that there exists a nonzero finite even p-local spectrum X and an integer s0 with the≤n≤nsfollowing property: for every quasi-coherent sheaf G on M≤nFG , the cohomology groups H (MFG ; (F X | MFG )⊗G)
vanish for s ≥ s0 . Then the localization functor LE(n) is smashing.
We can attack this problem using the filtration of M≤n
FG by height. Suppose we have chosen an even≥k,≤nfinite p-local spectrum X. For each k ≤ n, let MFG
denote the closed substack of M≤n
FG classifying formalgroups which have height ≥ k. Let us attempt to prove that, for every quasi-coherent sheaf G on M≥k,≤n,FG≥k,≤n
the groups Hs (M≥k,≤n;
(F|M)
⊗G)
vanishforlarges.
Theideaistousedescending
inductiononk.
XFGFG
≥k,≤n
Note that M≥k+1,≤n
can be regarded as a closed substack of MFG: it is the zero locus of vk , which weFG
pk −1
regard as a section of ω(here ω is the line bundle on MFG given by assigning to each formal group thedual of its Lie algebra). In particular, multiplication by vn induces a map of sheaves
kG → G ⊗ω p −1
≥k+1,≤n. We may therefore assume, bywhose kernel and cokernel are supported on the closed substack MFG
our inductive hypothesis, that vn induces an isomorphismk≥k,≤n≥k,≤n
Hs (M≥k,≤n; (FX | M≥k,≤n) ⊗ G) → Hs (MFG; (FX | MFG) ⊗ G ⊗ω p −1 )FGFGfor sufficiently large s. It follows that for large s, we have an isomorphismk≥k,≤n≥k,≤nHs (M≥k,≤n; (FX | M≥k,≤n) ⊗ G) ' lim Hs (MFG; (FX | MFG) ⊗ G ⊗ω (p −1)m );FGFG−m→here the latter group can be identified with the cohomology of FX ⊗ G on the open substack of M≥k,≤nFG≥k+1,≤ncomplementary to MFG: this is the moduli stack of formal groups of height exactly k. We are therefore
reduced to the following:
Proposition 5. Suppose that there exists a nonzero finite even p-local spectrum X and an integer s0 withthe following property: for 0 ≤ k ≤ n and every quasi-coherent sheaf G on MkFG , the cohomology groupsHs (MkFG ; (FX | MkFG ) ⊗ G) vanish for s ≥ s0 .
The vanishing condition appearing in Proposition 6 is automatic when k = 0, since quasi-coherent sheaves
on M0FG ' BGm have no higher cohomology. Assume that k > 0, and choose a formal group law f (x, y) ofheight k over Fp . Let Gk denote the automorphism group of f (as a formal group law over Fp ), regarded asa profinite group. We have a pullback diagram of algebraic stacks
BGk × Spec Fp/ Mk
Spec Fp/ Spec Fp .
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FG

This implies:(a) Every quasi-coherent sheaf F on MkFG determines an Fp -vector space V equipped with a continuous
action of Gk .
(b) We have a canonical isomorphism H∗ (MkFG ; F) ⊗Fp Fp ' H∗ (Gk ; V ).
Consequently, we are reduced to proving the following:
Proposition 6. Suppose that there exists a nonzero finite even p-local spectrum X and an integer s0 with thefollowing property: for 1 ≤ k ≤ n, if we let V denote the representation of the profinite group Gk associatedto FX , then Hs (Gk ; V ⊗ W ) ' 0 for s ≥ s0 and any continuous representation W of Gk . Then LE(n) is asmashing localization.
Let us now indicate briefly why it is plausible that the hypothesis of Proposition 6 should be satisfied.
Fix 1 ≤ k ≤ n. Recall that Gk can be described as the group of units in the ring End(f ), which is anoncommutative valuation ring of rank k 2 over Zp . In particular, it is a p-adic Lie group. Consequently, ona sufficiently small open subgroup of Gk , the group structure on Gk closely approximates the (commutative)
22group structure on Zkp . If M is a discrete p-torsion module over Zkp , then the profinite group cohomology
22Zkp with coefficients in M agrees with the ordinary group cohomology of Zk with coefficients in M , andtherefore vanishes in degrees larger than k 2 . Using this, Lazard shows that there is an open subgroup U ⊆ Gk
such that Hs (U ; M ) vanishes for s > k 2 and any Gk -module M . The same result does not necessarily holdwhen U = Gk . However, an argument of Serre shows that if Gk is not of finite cohomological dimension,then it must contain an element of order p. However, elements of order p are well-understood: these are
pth roots of unity in the division algebra D = End(f )[p−1 ], and they exist only when the rank k of D isdivisible by the degree (p − 1) of the field extension Qp (ζp ). In particular, if p > n + 1, then the profinitegroups {Gk }1≤k≤n have finite cohomological dimension and the hypotheses of Proposition 6 are satisfied for
X = S(p) .
When p ≤ n, we need to work harder. In this case, some of the groups Gk do contain elements of orderp. However, each Gk contains at most a single conjugacy class of subgroups V having order p: this follows
from the Skolem-Noether theorem. In this case, the subgroup V can be regarded as the “obstruction” to Gkbeing of finite cohomological dimension: one can show that the cohomology groups H∗ (Gk ; M ) are bounded
if the subgroup V acts freely on M . It therefore suffices to choose a spectrum X such that the associatedrepresentation V of Gk is free over V . This requires some representation-theoretic constructions which wewill not pursue further.

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