Monochromatic Layers (Lecture 34)

April 27, 2010
Fix a prime number p. To any spectrum X, we can associate its chromatic tower· · · → LE(2) X → LE(1) X → LE(0) X.
If X is a finite p-local spectrum, then the chromatic convergence theorem tells us that the homotopy limitof this tower is X. In particular, we can associate to X the chromatic spectral sequence {Erp,q , dr }, whereE1p,∗ is given by the homotopy groups of the homotopy fiber of the map LE(p) X → LE(p−1) X. (In fact, theproof of the chromatic convergence theorem tells us that this spectral sequence converges in a strong sense:
for example, the chromatic filtration on each homotopy group πn X is finite). This motivates the following:
Definition 1. For each spectrum X, we let Mn (X) denote the homotopy fiber of the map LE(n) X →LE(n−1) X. We will refer to Mn (X) as the nth monochromatic layer of X.
The essential features of Mn (X) are captured by the following definition:
Definition 2. A spectrum X is monochromatic of height n if it is E(n)-local and E(n − 1)-acyclic.
Remark 3. For any spectrum X, we have a map of E(n)-local spectra LE(n) X → LE(n−1) X which induces
an isomorphism on E(n − 1)-homology. It follows that the fiber Mn (X) is monochromatic of height n.
Conversely, if X is monochromatic of height n, then LE(n) X ' X and LE(n−1) X ' 0, so that X ' Mn (X).
Example 4. Let X be a finite p-local spectrum of type ≥ n. Then LE(n) X is monochromatic of height n.
To see this, it suffices to observe that E(n − 1)∗ LE(n) X ' E(n − 1)∗ X ' 0.
Notation 5. Let Mn denote the collection of all spectra which are monochromatic of height n. Since LE(n)is a smashing localization, we see that Mn is closed under homotopy colimits. We say that that an object
X ∈ Mn is compact if, for every filtered diagram {Yα } of objects of Mn , the induced map
lim Map(X, Yα ) → Map(X, lim Yα )−→−→is a homotopy equivalence.
Example 6. Let X be a finite p-local spectrum of type ≥ n. Then LE(n) X is a compact object of Mn . To
see this, we note that if {Yα } is a filtered diagram in Mn , then we have
Map(LE(n) X, lim Yα ) ' Map(X, lim Yα ) ' lim Map(X, Yα ) ' lim Map(LE(n) X, Yα ).
−→−→−→−→
Our next goal is to establish a converse to Example 6. The essential observation is the following:
Proposition 7. Let X be a spectrum which is monochromatic of height n. Then X can be written as afiltered colimit lim Xα , where each Xα is the E(n)-localization of a finite spectrum of type ≥ n.
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Proof. We have a cofiber sequence
X 0 → X → Ltn−1 X,
where X 0 is a filtered colimit of p-local finite spectra of type ≥ n. This induces a cofiber sequence
LE(n) X 0 → LE(n) X → LE(n) Ltn−1 X.
Since X ∈ Mn we have LE(n) X ' 0, and since LE(n) is smashing we conclude that LE(n) X 0 is a filteredcolimit of E(n)-localizations of finite p-local spectra of type ≥ n. It will therefore suffice to show thatLE(n) Ltn−1 X ' 0; that is, that Ltn−1 X is E(n)-acyclic. Since E(n) is Landweber exact, it will suffice to showthat Ltn−1 X is MU-acyclic. In the last lecture, we saw thatMU∗ Ltn−1 X ' MU∗ LE(n−1) X,
and the right hand side vanishes since X is assumed to be E(n − 1)-acyclic.
Corollary 8. An object X ∈ Mn is compact if and only if it is a retract of LE(n) Y for some finite spectrum
Y of type ≥ n.
Proof. Write X as a filtered colimit of spectra Xα of the form LE(n) Yα . Since X is compact, the identity
map X → lim Xα factors through some Xα , so that X is a retract of LE(n) Yα .
−→Corollary 9. The homotopy theory Mn is compactly generated: that is, every object of Mn can be obtained
as a filtered colimit of compact objects of Mn .
We want to draw attention to a crucial features of the compact objects of Mn . First, we state a slightly
stronger version of the periodicity theorem of Lecture 27:
Theorem 10. Let X be a finite p-local spectrum of type ≥ n. Then there exists a vn -self map f : Σk X → XNwhere k = 2(pn − 1)pN for N  0, which acts by multiplication by vnp on K(n)∗ X.
Corollary 11. Let X be a compact object of Mn . Then X is periodic. More precisely, for N  0, there isN
na homotopy equivalence X ' Σ2p (p −1) X.
Proof. According to Corollary 8, we can assume that X is a retract of LE(n) Y for some finite p-local spectrum
Y of type ≥ n. Let f : Σk Y → Y be the vn -self map of Theorem 10, where k = 2pN (pn − 1). Then theNaction of f on K(n)∗ LE(n) Y ' K(n)∗ Y is given by vnp . It follows that the composite map
ff 0 : Σk X → Σk LE(n) Y → LE(n) Y → X
Ninduces multiplication by vnp on K(n)∗ X; in particular, it is bijective. Since f 0 is also bijective on K(m)∗ Xfor m < n (since these groups vanish), we conclude that the homotopy fiber of f 0 is K(m)-acyclic for m ≤ n
and therefore E(n)-acyclic. Since X is E(n)-local, the homotopy fiber of f 0 is also E(n)-local and therefore
trivial; this proves that f 0 is an equivalence Σk X ' X.
If X is a general monochromatic spectrum of height n, then X is a filtered colimit of compact objects
Xα , each of which is periodic of some period 2(pn − 1)pNα . The exponent Nα generally depends on α, so that
X itself is not periodic. Nevertheless, elements of the homotopy of X are organized into “periodic families”:
that is, any class x ∈ πk X is given by an element in some πk Xα , which in turn determines elements ofπk+2m(pn −1)pNα X for all m ∈ Z. This is the motivation for the term “chromatic homotopy theory”: thechromatic tower of a spectrum X is like a prism, which separates X into “monochromatic layers” Mn (X)each of which exhibit a sort of generalized 2(pn − 1)-fold periodicity.
We conclude with a few remarks relating the monochromatic category Mn with the K(n)-local homotopy
category.

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Proposition 12. The constructions
X 7→ LK(n) X
Y 7→ Mn (Y )determine mutually inverse equivalences between the homotopy category of monochromatic spectra of heightn and the homotopy category of K(n)-local spectra.
We first recall a fact we proved earlier:Lemma 13. Let X be an E(n − 1)-local spectrum. Then K(n)∗ X ' 0.
Proof. Since LE(n−1) is smashing, K(n) ⊗ X is E(n − 1)-local. It will therefore suffice to show that K(n) ⊗ Xis E(n − 1)-acyclic; that is, that E(n − 1) ⊗ K(n) ⊗ X ' 0. This is clear, since E(n − 1) ⊗ K(n) is a complex
orientable spectrum whose formal group has height < n and exactly n, and therefore E(n−1)⊗K(n) ' 0.
Proof of Proposition 12. We argue that both composite functors are the identity. First, fix a monochromatic
spectrum X of height n. We wish to show that X ' Mn (LK(n) X). Since LK(n) X is K(n)-local, it is E(n)local; thus Mn (LK(n) X) can be identified with the homotopy fiber F of the map LK(n) X → LE(n−1) LK(n) X.
Since X is monochromatic, LE(n−1) X ' 0 so there is a canonical map α : X → F . We claim that α isan equivalence. Since X and F are both E(n)-local, it will suffice to show that α induces an isomorphism
K(m)∗ X → K(m)∗ F for m ≤ n. If m < n, then both groups vanish. If m = n, we are reduced to proving
thatK(n) ⊗ X → K(n) ⊗ LK(n) X → LE(n−1) LK(n) X
is a fiber sequence. This follows from the observation that the first map is an equivalence and the third term
vanishes (Lemma ??).
Now let Y be a K(n)-local spectrum. Then Y is E(n)-local, so that Mn (Y ) is the homotopy fiber ofthe map Y → LE(n−1) Y . We wish to prove that the map Mn (Y ) → Y exhibits Y as a K(n)-localizationof Mn (Y ). Since Y is K(n)-local, it suffices to show that this map is a K(n)-equivalence; that is, thatK(n)∗ LE(n−1) Y ' 0; this also follows from Lemma ??.
Corollary 14. The K(n)-local stable homotopy category is compactly generated; its compact objects are
precisely the retracts of spectra of the form LK(n) X, where X is a finite spectrum of type ≥ n.
Warning 15. For a general finite spectrum X, the localization LK(n) X is not a compact object of theK(n)-local stable homotopy category. For example, if n > 0, then the K(n)-local sphere LK(n) S is not acompact object of the K(n)-local stable homotopy category.

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