FINITE LOCALIZATIONS AND THE CLASSICAL ADAMS SPECTRAL SEQUENCE

PAUL SHICK

January 18, 2001

The goal of this note is to present some of the relationship between some “old-fashioned” constructions in homological algebra and the recent finite localization functors of Mahowald-Sadofsky, Ravenel, Miller, et al ([7], [14], [9].) The main result verifies a conjecture of Mahowald and Sadofsky (for finite spectra,) yielding a somewhat geometric interpretation of the so-called “v_{n}-periodic Ext-groups.” It is a pleasure to thank Mark Hovey, Mark Mahowald, John Palmieri and Hal Sadofsky for helpful conversations and correspondence.

The homological algebra referred to above is concerned with

Ext_{A}^{s,t}(ℤ/p, ℤ/p),

the E_{2} term of the classical Adams spectral sequence for π_{∗}(S^{0}) for any prime p, where A is the mod p Steenrod algebra. One way to study this is by using appropriate finite subalgebras. For each n ≥ 0, let A(n) denote the finite subHopf algebra of the Steenrod algebra generated by

Sq^{0}, Sq^{1}, …, Sq^{2^{n}}

if p = 2 and by

β, P^{1}, …, P^{p^{n−1}},

if p odd. Then the Hopf algebra cohomology

H^{∗}(A(n)) = Ext_{A(n)}(ℤ/p, ℤ/p)

is of interest for several reasons:

• Since lim_{\overleftarrow{n}}Ext_{A(n)}(ℤ/p, ℤ/p) = Ext_{A}(ℤ/p, ℤ/p), we get complete

information about our goal.

• For each n ≥ 0, Ext_{A(n)} is effectively computable by “finite” methods.

• These Ext-groups can be used to study the “chromatic” approach to homotopy in the setting of the classical Adams spectral sequence. In particular, Ext_{A(n)} is closely related to the “periodic” spectra BP ⟨n⟩, as shown below.

For each prime p, the Brown-Peterson spectrum BP is well-known, with H^{∗}(BP) = A//E(Q_{0}, Q_{1}, …), where Q_{n} is the n^{th} Milnor generator of the mod p Steenrod algebra. The associated Johnson-Wilson spectrum is BP ⟨n⟩,, with H^{∗}(BP ⟨n⟩) = A//E_{n} where we use the notation E_{n} = E(Q_{0}, Q_{1}, …, Q_{n}), and H^{∗}(−) denotes cohomology with ℤ/p coefficients. So we can calculate π_{∗}(BP ⟨n⟩) from the classical Adams spectral sequence:

E_{2}^{s,t} = Ext_{A}^{s,t}(H^{∗}(BP ⟨n⟩), ℤ/p)
=  Ext_{A}^{s,t}(A//E_{n}, ℤ/p)
≅  Ext_{E_{n}}(ℤ/p, ℤ/p)
≅  ℤ/p[q_{0}, q_{1}, …, q_{n}],

where the class q_{i} ∈ Ext^{1,2p^{i}−1}. Since these generators are concentrated in even degrees, we conclude that E_{2} = E_{∞} here. Further, because the E_{2} generator q_{0} corresponds to the Bockstein operation, it represents the element p ∈ π_{0}(BP ⟨n⟩). So we see easily that

π_{∗}(BP ⟨n⟩) ≅ ℤ_{(p)}[v_{1}, …, v_{n}],

where |v_{i}| = 2p^{i} − 2. The periodic version of the Johnson-Wilson spectrum is E(n), with

E(n)_{∗} = ℤ_{(p)}[v_{0}, v_{1}, …, v_{n}, v_{n}^{−1}].

The inclusion i : E_{n} ↪ A(n) induces a restriction homomorphism in cohomology

i^{∗} : Ext_{A(n)}(ℤ/p, ℤ/p) → Ext_{E_{n}}(ℤ/p, ℤ/p).

We use this to see clearly the link between BP ⟨n⟩ and Ext_{A(n)}. The main result of [15] is the following.

Theorem 1. For p odd, there exists a polynomial subalgebra

ℤ/p[v_{0}, v_{1}^{p^{n}}, v_{2}^{p^{n−1}}, …, v_{n}^{p}] ⊂ Ext_{A(n)}(ℤ/p, ℤ/p),

where the generators restrict to the obvious classes in H^{∗}(E_{n}).

The proof uses a careful analysis of the Cartan-Eilenberg spectral sequence for the extension

𝔽_{p} → P_{n} → A(n)_{∗} → E^{n} → 𝔽_{p}

where P_{n} is the truncated polynomial algebra on the ξ_{i}s and E^{n} is the exterior algebra on the τ_{i}s.

For the prime 2, the situation is more difficult. The following is proved in [8].

Theorem 2. There exists a polynomial subalgebra

ℤ/p[v_{0}, v_{1}^{N_{1}}, v_{2}^{N_{2}}, …, v_{n}^{N_{n}}] ⊂ Ext_{A(n)}(ℤ/2, ℤ/2),

where the generators restrict to the obvious classes in H^{∗}(E_{n}).

The proof uses results of Lin [3] and Wilkerson [16] that show that the restriction homomorphism i^{∗} is onto in infinitely many positive degrees. Note that these generators are only defined up to cosets for p = 2. Using a spectral sequence based on a Koszul-type resolution of A(n)//A(n − 1), we showed that the exponent N_{n} could be identified:

v_{n}^{2^{n+1}} ∈ Ext_{A(n)}(ℤ/2, ℤ/2).

The other exponents were more mysterious. In answer to a recent question of W. Singer, we have the following.

Theorem 3. For all i ≥ n + 1, there exists r_{i} such that

v_{i}^{2^{r_{i}}k} ∈ Ext_{A(n)}(ℤ/2, ℤ/2),

for all k ≥ 1.

Proof: Consider the May-Ivanovskii spectral sequence for H^{∗}(A(n)). This collapses at a finite E_{r}, by the results of Bajer and Sadofsky ([1]). The element v_{i} corresponds to h_{i,0} in the E_{1} term. Because the differentials are derivations, we have room for only finitely many:

v_{i} ↦ a_{1}
v_{i}^{2} ↦ a_{2}
⋮
v_{i}^{2^{r_{i}}} ↦ 0.

Thus the appropriate powers of the elements are cycles in the spectral sequence. Mapping to the same spectral sequence for H^{∗}(E_{n}) shows that these elements are never boundaries. This completes the proof of the theorem.

We use these polynomial subalgebras in H^{∗}(A(n)) to define “v_{n}- periodic” Ext, v_{n}^{−1}Ext_{A}(ℤ/p, ℤ/p), as follows:

Ext_{A}(ℤ/p, ℤ/p) ⋮  ⋮_{Ext_{A(k+1)}(ℤ/p, ℤ/p)}  v_{n}^{−1}Ext_{A(k+1)}(ℤ/p, ℤ/p)_{Ext_{A(k)}(ℤ/p, ℤ/p)}  v_{n}^{−1}Ext_{A(k)}(ℤ/p, ℤ/p) ⋮  ⋮_{Ext_{A(n)}(ℤ/p, ℤ/p)}  v_{n}^{−1}Ext_{A(n)}(ℤ/p, ℤ/p)._{u}_{u}  _{u}_{w}_{u}  _{u}_{w}_{u}  _{u}_{u}  _{u}_{w}

So we define

v_{n}^{−1}Ext_{A}(ℤ/p, ℤ/p) := lim_{\overleftarrow{k}}\{v_{n}^{−1}Ext_{A(k)}(ℤ/p, ℤ/p)\}.

Taking the inverse limit, we obtain a “localization” map

f_{n} : Ext_{A}(ℤ/p, ℤ/p) ⟶ v_{n}^{−1}Ext_{A}(ℤ/p, ℤ/p),

defining v_{n}-periodic and v_{n}-torsion in Ext_{A}(ℤ/p, ℤ/p) in terms of the kernel of f_{n}.

The following is proved in [15] and [8].

Theorem 4. If a class a ϵ Ext_{A}(ℤ/p, ℤ/p) is v_{n}-periodic, then a is also v_{n+k}-periodic for all k ≥ 0. Equivalently, if a is v_{n}-torsion, then a is also v_{i}-torsion for all i ≤ n.

Note that this sets up a “chromatic filtration” of Ext_{A}ℤ/p, closely related to that subsequently defined by Palmieri ([10]).

We wish to investigate the relationship between this “v_{n}-periodic Ext” and the finite or telescopic localizations that have been the subject of much study in recent years. Such functors have been considered at least partly as a consequence of calculations that show that the Telescope Conjecture ([13]) is most likely false ([14],[6],) providing a construction which agrees with the geometrically defined telescope, but which preserves the desirable functorial properties of the usual Bousfield localization.

One version of this sort of functor is due to Mahowald and Sadofsky, who define the “telescopic localization” of a spectrum. To explain their construction, we need to recall some results that follow from the Nilpotence Theorem, most of which are due to Hopkins and Smith ([4].) We recall that a finite spectrum X is said to be of type n if K(n − 1)_{∗}(X) = 0 and K(n)_{∗}(X) ≠ 0, where K(j) is the j^{th} mod p Morava K-theory, with K(j)_{∗} = 𝔽_{p}[v_{j}, v_{j}^{−1}]. Such a type n finite complex always has a “v_{n}-self map:

v : Σ^{i}X → X,

which induces an isomorphism in K(n)_{∗}(X). For X of type n, the v_{n}- telescope of X is

Tel_{v_{n}}(X) = v_{n}^{−1}X = lim_{⟶}(X → Σ^{−i}X → Σ^{−2i}X → ⋯)

for any v_{n} self map v : Σ^{i}X → X. All such v_{n}-telescopes of X are easily seen to be homotopy equivalent, so that Tel_{v_{n}}(X) is well-defined.

We further recall some basics of Bousfield’s theory. A spectrum X is said to be E-local if

[W, X]_{∗} = 0 whenever E_{∗}W = 0.

The Bousfield localization of X with respect to E is a map

L : X → L_{E}X,

where L_{E} is E-local and E_{∗}L is an isomorphism. Two spectra X and Y are Bousfield equivalent if

X ∧ W ≃ ∗ ⟺ Y ∧ W ≃ ∗.

Two spectra are said to be in the same Bousfield class if they are Bousfield equivalent. Bousfield localization with respect to the periodic Johnson-Wilson spectrum E(n) has been so important to the “chromatic” approach to topology that the notation

L_{n}X := L_{E(n)}X

has become standard. See [13] for the origins of this school. Note that L_{n}X = L_{(K(0)∨···∨K(n))}X for all n ≥ 0.

Mahowald and Sadofsky base their definition of telescopic localization on the following observation: for all n ≥ 0, for any finite complex X of type n, the Bousfield class of Tel_{v_{n}}(X) depends only on n. We hereafter denote any such telescope by T_{n}. For any spectrum X (finite or not,) we define the n^{th} telescopic localization of X as

L_{n}^{f}X := L_{(T_{0}∨T_{1}∨···∨T_{n})}X.

One can show easily that if X is finite type n, then

Tel_{v_{n}}(X) ≃ L_{T_{n}}X ≃ L_{n}^{f}X.

The definition of L_{n}^{f}X above can be shown to agree with the finite localization functors L_{n}^{f}, given by Ravenel ([14]) and Miller ([9].) Ravenel’s definition involves constructing an appropriate spectrum L_{n}^{f}S^{0} as a limit of certain finite complexes, then defining

L_{n}^{f}X = L_{L_{n}^{f}S^{0}}X,

for all spectra X. Miller’s definition follows the original ideas behind usual Bousfield localization with respect to the periodic Johnson-Wilson spectrum E(n), considering the cofiber of

⋁_{(W finite, E(n)_{∗}W =0)} W → X.

Further, one can show that L_{n}^{f} is smashing. That is, L_{n}^{f}X = (L_{n}^{f}S^{0})∧X, for all spectra X.

The Telescope Conjecture ([13]) predicts that for any finite complex of type n, the localization map

T el_{v_{n}}X⟶^{L}L_{n}X

should be an equivalence. At the time the conjecture was proposed, it was thought that only the localization functors L_{n} were smashing. Since finite localization with respect to E(n) agrees with the v_{n}- telescope, some have reformulated the Telescope Conjecture as “In the stable homotopy category, all smashing localizations are finite.” (See [5].)

Mahowald and Sadofsky found a more geometric construction of L_{n}^{f}X. For an ordered index I = (i_{0}, i_{1}, …, i_{n}), let

M(I) = M(p^{i_{0}}, v_{1}^{i_{1}}, …, v_{n}^{i_{n}})

denote any choice of finite spectrum with

BP_{∗}(M(p^{i_{0}}, v_{1}^{i_{1}}, …, v_{n}^{i_{n}})) = BP_{∗}/(p^{i_{0}}, v_{1}^{i_{1}}, …, v_{n}^{i_{n}}),

a so-called “generalized Moore space.” With the usual order on these indices, we have maps

f_{J}^{I} : M(I) → Σ^{|I−J|}M(J)

commuting with the projection onto the top cell, whenever J ≤ I, where

|I| = i_{1}2(p − 1) + i_{2}2(p^{2} − 1) + ⋯ + i_{n}2(p^{n} − 1).

Let \overline{M}(I) denote the fiber of the projection onto the top cell

M(I)⟶^{p}S^{|I|+n+1},

So we have a cofiber sequence

S^{n+|I|}⟶^{g_{I}}\overline{M}(I) → M(I),

for all I.

Theorem 5. (Mahowald-Sadofsky, [7]). The map

S^{0}⟶^{g_{I}}lim_{\overrightarrow{I}}Σ^{−n−|I|}\overline{M}(I)

is L_{n}^{f} localization.

This is a vast generalization of earlier work of Davis, Mahowald and Miller on limits of stunted projective spaces ([2].) Since L_{n}^{f} is smashing, it gives a construction of L_{n}^{f}X for all X. However, this looks inherently incalculable.

The surprising part of the Mahowald-Sadofsky program is their construction of an Adams spectral sequence converging to π_{∗}(L_{n}^{f}X). The construction is quite technical, requiring a modified Adams tower and a lot of work with “stably finite” A-modules and spectra (ala Palmieri-Sadofsky [12].) In the end, they identify the E_{2} term of the spectral sequence in the following manner.

First, they work in category of comodules over the coalgebra A_{∗}. Let St denote the category of chain complexes of A_{∗}-comodules, with G_{•} an injective resolution of ℤ/p = H_{∗}(S^{0}). For any spectrum X,

E_{2}^{clASS}(X) = Ext_{A_{∗}}(ℤ/p, H_{∗}(X)) ≅ [G_{•}, N_{•}],

where N_{•} = injective resolution of H_{∗}(X) and [−, −] denotes chain homotopy classes of maps in St. This category is explored in great detail in [11].

The “finite localization” Adams spectral sequence uses a modified Adams resolution for the finite spectrum \overline{M}(I) (for appropriate I) and then takes a direct limit, resulting in a spectral sequence converging to π_{∗}(L_{n}^{f}X). Mahowald and Sadofsky identify the resulting E_{2} term as

E_{2}^{L_{n}^{f}ASS}(X) ≅ [G_{•}, L_{J}(N_{•})],

where L_{J} denotes a certain chain complex localization that looks like an algebraic analog of the “finite localization” of Miller, et al. In particular, one can choose J_{•} = the finite chain complex given by a modified Adams resolution for M(U^{I}), for some particular I, a finite complex of type n + 1.

The connection between the ideas of Mahowald and Sadofsky, among others, and the homological algebra presented earlier is the following conjecture.

Conjecture 6. (L_{n}^{f}-Ext Conjecture, Mahowald-Sadofsky [7]) For X any spectrum (in the usual stable homotopy category,)

E_{2}^{L_{n}^{f}ASS}(X) ≅ v_{n}^{−1}Ext_{A}(H^{∗}X, ℤ/p),

where

v_{n}^{−1}Ext_{A}(ℤ/p, ℤ/p) := lim_{\overleftarrow{k}}\{v_{n}^{−1}Ext_{A(k)}(ℤ/p, ℤ/p)\},

as above.

Mahowald and Sadofsky cite some evidence for the conjecture. They note that it’s true for n = 0, 1 by direct computation for X = S^{0} (then use the fact that L_{n}^{f} is smashing to extend to all spectra.) Further, they prove that the conjecture holds for X a finite spectrum of type n, with the added condition of H^{∗}X having appropriate P_{s}^{t}-homology groups vanish. However, the conjectured isomorphism is between an object defined as a direct limit (the E_{2}^{L_{n}^{f}ASS}) and an inverse limit (the v_{n}-periodic Ext-group,) so one is uninclined to be optimistic about the conjecture.

Here the finite localization L_{n}^{f} is a spectrum level construction, (i. e. in the usual stable homotopy category,) with analogues in several appropriate algebraic settings. Note that we have a number of options on how to view the functor L_{n}^{f}, from the work of Miller, Ravenel, Mahowald and Sadofsky:

• As a direct limit (colimit) of cofibers built from type n + 1 finites.

• As a Bousfield localization w.r.t. L_{n}^{f}S^{0}, which can be constructed as a direct limit of \overline{M}(I)s.

• As a finite localization away from a type n + 1 finite.

• As a finite localization with respect to E(n) or K(≤ n), where E(n) is the periodic Johnson-Wilson spectrum and K(≤ n) is a shorthand notation for K(0) ∨ ⋯ ∨ K(n).

• As Bousfield localization with respect to T (≤ n), a wedge of v_{i}- telescopes of finite type i complexes, with i ≤ n.

Again, each spectrum level construction has an appropriate counterpart in appropriate algebraic settings.

The most convenient framework in which to explore these ideas is given in Palmieri’s opus [11]. As above, let St denote the homotopy category whose objects are cochain complexes of injective graded comodules over the dual Steenrod algebra A_{∗}, with morphisms cochain homotopy classes of graded maps of complexes. Let St(i) denote the obvious analog of St over the dual of the finite Hopf algebra A(i). In these categories, the sphere S (actually S^{∗,∗}, but we’ll suppress the bigrading whenever possible) is an injective resolution of ℤ/p, which we denote by G(∞)^{•} ∈ St, as above, or by G(k)^{•} ∈ St(k). For a cochain complex X^{•} ∈ St, which is a resolution of a comodule M, the homotopy groups are just

π_{∗∗}(X^{•}) = [G(∞)^{•}, X^{•}]_{∗} = Ext_{A_{∗}}^{∗,∗}(ℤ/p, M),

For a cochain complex X^{•} ∈ St(k),

π_{∗∗}(X^{•}) = [G(k)^{•}, X^{•}]_{∗} = Ext_{A(k)_{∗}}^{∗,∗}(ℤ/p, M).

We’ll attempt as much as possible to be careful about which category we’re working in, to avoid confusion about what π_{∗∗}(−) means. In particular, we adopt the following convention: for a cochain complex over A_{∗}, X^{•} ∈ St, we’ll use X_{k}^{•} to denote X^{•} thought of as an A(k)_{∗}-resolution. Further following Palmieri’s notation, let HA(i)^{•} denote an injective resolution of A_{∗}□_{A(i)_{∗}}ℤ/p, so that π_{∗∗}(HA(i)^{•}) = Ext_{A(i)_{∗}}(ℤ/p, ℤ/p). We also have St(k) analogs of HA(i)^{•}, for k ≥ i, which are not just given by taking A_{∗}□_{A(i)_{∗}}ℤ/p, as A(k)-modules. Instead, in St(i), HA(i)_{i}^{•} = G(i)^{•}, and in St(k), HA(i)_{k}^{•} is an injective resolution of A(k)_{∗}□_{A(i)_{∗}}ℤ/p, so that the homotopy groups in St(k) are again just Ext_{A(i)_{∗}}(ℤ/p, ℤ/p).

There are obvious analogues of the L_{n}^{f} functor in both St and St(k), for all k, which we’ll denote by L_{n}^{f} and L_{n,k}^{f}, respectively. In particular, if X is a spectrum (in the usual stable category) and C^{•} is an A_{∗}- resolution of H_{∗}X, then

π_{∗∗}(L_{n}^{f}C^{•}) = E_{2}^{L_{n}^{f}ASS}(X).

With this form of Palmieri’s notation, we can define v_{n}^{−1}X^{•}, for any X^{•} ∈ St, in several ways:

v_{n}^{−1}X^{•} : = lim_{\overleftarrow{k}}(v_{n}^{−1}HA(k) ∧ X_{k}^{•})

and

v_{n}^{−1}X^{•} : = lim_{\overleftarrow{k}}(v_{n}^{−1}HA(k)) ∧ X^{•} = (v_{n}^{−1}S^{0}) ∧ X^{•}.

For finite X^{•}s, these two definitions agree, as shown in [11]. A third possibility is to remember that in St(k), the sphere S_{k}^{0} = HA(k)^{•} = G(k)^{•} has a v_{n} self map, so that for any spectrum X_{k}^{•} ∈ St(k), we can form the v_{n}-telescope T el_{v_{n}}(X_{k}^{•}), using the self map

S^{|v_{n}^{L}|} ∧ X⟶^{v_{n}^{L}}S^{0} ∧ X = X.

So for any spectrum X^{•} ∈ St, we have another possible definition:

v_{n}^{−1}X^{•} : = lim_{\overleftarrow{k}}T el_{v_{n}}(X_{k}^{•}).

Lemma 7. For a finite spectrum X^{•} ∈ St, the three definitions of v_{n}^{−1}X^{•} agree.

Proof: For finite X^{•}s, the first two definitions agree, by Palmieri’s work. For any finite X^{•}, we have

v_{n}^{−1}X^{•} := lim_{\overleftarrow{k}}v_{n}^{−1}(A□_{A(k)}X_{k}^{•})
= lim_{\overleftarrow{k}}A□_{A(k)}(Tel_{v_{n}}X_{k}^{•})
= lim_{\overleftarrow{k}}HA(k)^{•} ∧ (Tel_{v_{n}}X_{k}^{•})
≃  lim_{\overleftarrow{k}}(Tel_{v_{n}}X_{k}^{•})
=  lim_{\overleftarrow{k}}v_{n}^{−1}X^{•}

since lim_{\overleftarrow{k}}HA(k)^{•} = S^{0} = G(∞)^{•} in St.

So we can think about v_{n}^{−1}X^{•} in several ways, all of which agree if X is a finite spectrum. Of course, we’re really interested in

v_{n}^{−1}Ext_{A}(M, ℤ/p) := lim_{\overleftarrow{k}}v_{n}^{−1}Ext_{A(k)}(M_{k}, ℤ/p)

= lim_{\overleftarrow{k}}π_{∗∗}(v_{n}^{−1}HA(k)^{•} ∧ X_{k}^{•})

= π_{∗∗}((lim_{\overleftarrow{k}}v_{n}^{−1}HA(k)) ∧ X_{k}^{•})

= π_{∗∗}(v_{n}^{−1}S^{0} ∧ X^{•}),

where X^{•} is an injective A_{∗}-resolution of some finite comodule M. (We’re restricting to finites here to make everything in sight agree.)

Mahowald and Sadofsky’s conjecture would follow from the following:

Conjecture 8. For any X^{•} ∈ St,

L_{n}^{f}X^{•} ≅ v_{n}^{−1}X^{•}.

One might also conjecture that this isomorphism holds at the St(k) level, asking if

L_{n,k}^{f}X_{k}^{•} ≅ v_{n}^{−1}X_{k}^{•}???

for all X_{k}^{•} ∈ St(k). Looking at this question forces one to think about all sorts of interesting phenomena.

First, we have the following result, probably originally due to Mark Hovey (unpublished):

Lemma 9. In St(k), v_{n}^{−1}X_{k}^{•} is just finite localization of X_{k}^{•} away from the cofiber of the v_{n} element in the sphere, HA(k)/v_{n}.

This is just “algebraic localization” of the unit in the category St(k), so the proof is done in Hovey-Palmieri-Strickland ([5].) Here’s a brief sketch. The category St(k) is a unital algebraic stable homotopy category (in HPS terms,) with sphere S = HA(k) = G(k). There’s a power of v_{n} in π_{∗∗}(HA(k)), which we’ll denote by v_{n} to avoid unnecessary complications. We have the usual cofibration HA(k)⟶^{v_{n}}HA(k) → HA(k)/v_{n}. Recall that finite localization away from HA(k)/v_{n} is a smashing finite localization with acyclics generated by HA(k)/v_{n}. Note that the functor v_{n}^{−1}π_{∗∗}(−) is a homology theory on St(k), since it preserves coproducts and is exact as a functor of π_{∗∗}(HA(k))-modules. So

v_{n}^{−1}π_{∗∗}(X_{k}^{•}) = v_{n}^{−1}π_{∗∗}(HA(k) ∧ X_{k}^{•}) = v_{n}^{−1}π_{∗∗}(L_{(away HA(k)/v_{n})}X_{k}^{•}).

Further, [HA(k)/v_{n}, L_{(away HA(k)/v_{n})}X_{k}^{•}] = 0 for any X_{k}^{•} ∈ St(k), so that multiplication by v_{n} is an isomorphism on π_{∗∗}(L_{(away HA(k)/v_{n})}X_{k}^{•}). Hence

v_{n}^{−1}π_{∗∗}(L_{(away HA(k)/v_{n})}X_{k}^{•}) = π_{∗∗}(L_{(away HA(k)/v_{n})}X_{k}^{•}).

This completes the proof of the Lemma.

Recall that in St(k) or St, L_{n,k}^{f} (resp. L_{n}^{f}) is finite localization away from an injective resolution of F (n + 1), a finite complex of type n+1. This is easy to see, since the finitely-K(≤ n)-locals are exactly the finitely-F (n + 1)-acyclics. (Recall that in the category St, we replace K(≤ n) with its couterpart, Z(≤ n), where

Z(≤ n) := ⋁_{d≤n} Z(d),

with Z(d) is a localized version of the cochain complex resolving the comodule A/(u_{d}), with u_{d} the d^{th} P_{t}^{s} element in the obvious ordering.)

We recall that finite localization away from C is the same as finite localization away from D if and only if C and D generate the same thick subcategories. But the thick subcategories generated by HA(k)/v_{n} and F (n + 1) are quite different, since the times p map acts nonnilpotently on HA(k)/v_{n} but not on F (n + 1).

The upshot of all this is that in St(k)

L_{n,k}^{f}M_{k}^{•} ≠ v_{n}^{−1}M_{k}^{•},

for at least one injective A(k)_{∗}-resolution M_{k}^{•}.

This does NOT mean that the L_{n}^{f}-Ext conjecture is false, only that the most obvious analog in the A(k)-world can’t be true. Further, this counterexample M_{k}^{•} seems likely to be an infinite spectrum in St(k), as we will see below.

The following Lemma shows how to use information on the St(k) level to help us address the actual conjecture.

Lemma 10. For all finite X^{•} ∈ St,

L_{n}^{f}X^{•} = lim_{\overleftarrow{k}}L_{n,k}^{f}X_{k}^{•}.

Note that, at least on the surface, we’ve somehow interchanged a direct and an inverse limit, which seems woefully optimistic. However, there is a simple way to think about this: First, we show that the functor

L′(X^{•}) : = lim_{\overleftarrow{k}}L_{n,k}^{f}X_{k}^{•}

is a localization functor on the category St. Then we show that it agrees with L_{n}^{f}. One’s initial inclination in approaching such a question might be to use some sort of thick subcategory argument, but there’s no known characterization of thick subcategories of St. However, it’s still easy to compare L′ and L_{n}^{f} in St.

First, we need to make sure that L′ is even defined. In particular, for a complex X^{•} ∈ St, we need to know that the map given by the following diagram makes sense:

X_{k+1}^{•}  X_{k}^{•} L_{n,k+1}^{f}X_{k+1}^{•}  L_{n,k}^{f}X_{k}^{•}^{forget} w_{u_{L}}  _{u_{L}}_{^{???} w}

The localizations on the bottom row take place in different categories, so there’s no obvious map, at least on the surface. However, we recall that the Mahowald-Sadofsky theorem shows that L_{n}^{f}X^{•} can be computed by taking lim_{\overrightarrow{I}}X^{•} ∧ \overline{M(I)}, for M(I) appropriate finite complexes

of type n + 1. In the St(k) setting, we can just choose these complexes as their images under the forgetful functor St → St(k). So it’s clear that the map in question is just the forgetful functor applied both to X_{k+1}^{•} and to the \overline{M}s.

Recall that a functor L on a stable homotopy category is a localization if

• The natural transformation L to L^{2} is an equivalence.

• [LX, LY]_{∗} → [X, LY]_{∗} is an iso.

• LX = 0 implies that L(X ∧ Y) = 0 for all Y.

For our functor L′ above, we know that on the St(k) level, all three conditions are satisfied by L_{n,k}^{f}. In particular, it’s easy to see that the inverse limit of the isomorphisms L_{n,k}^{f} ≅ (L_{n,k}^{f})^{2} shows that L′ → (L′)^{2} is an equivalence. The second condition is more troublesome, because the homomorphism induced from X^{•} → L′X^{•},

[L′X^{•}, L^{•}Y^{•}] → [X′, L′Y^{•}]

is an inverse limit of St(k) maps on the second variable and a direct limit of St(k) maps on the first variable. However, our observation that L_{n,k}^{f} can be described as smashing with a direct limit of \overline{M}_{k}s shows that

(L′X^{•})_{k} = L_{n,k}X_{k}^{•},

where, as usual, D_{k}^{•} is the image of the complex D^{•} under the forgetful functor, St → St(k). So the desired homomorphism

[L′X^{•}, L^{•}Y^{•}] → [X′, L′Y^{•}]

can be thought of either as the direct limit (on k) of

[L_{n,k}^{f}X_{k}^{•}, (L^{•}Y^{•})_{k}]_{St(k)} → [X_{k}^{•}, (L^{•}Y^{•})_{k}]_{St(k)},

which are all isomorphisms, or as the inverse limit (on j) of

[(L′X^{•})_{j}, L_{n,j}^{f}Y_{j}^{•}]_{St(j)} →, [X_{j}^{•}, L_{n,j}^{f}Y_{j}^{•}]_{St(j)}

which are also all isomorphisms. Finally, since L_{n,k}^{f} is smashing, so is the inverse limit L′, so that if L′X^{•} = 0, then smashing with Y_{k}^{•} still yields zero. Hence L is indeed a localization.

How do we show that L′ agrees with L_{n}^{f}? We recall that finite localization away from C is the same as finite localization away from D if and only if C and D generate the same thick subcategories. Now L_{n}^{f} is finite localization away from F (n + 1), a finite complex of type n + 1 in St. But L′ = lim_{\overleftarrow{k}}L_{n,k}^{f} is clearly finite localization away from

F = lim_{\overleftarrow{k}}F_{k}(n + 1), an inverse limit of finite type n + 1 in the categories

St(k). For appropriate choices of F_{k}(n + 1), this inverse limit is exactly an F (n + 1) in St. This completes the proof of the lemma.

Everyone who’s thought about the L_{n}^{f}-Ext conjecture seems to have believed the following:

Conjecture 11. (Local conjecture) For any cochain complex X^{•} ∈ St,

v_{n}^{−1}X^{•} = lim_{\overleftarrow{i}}v_{n}^{−1}HA(i) ∧ X^{•}

is L_{n}^{f}-local in St. Similarly, for any X^{•} ∈ St(k), v_{n}^{−1}X^{•} is L_{n}^{f}-local there.

In particular, Palmieri states that the Local Conjecture is true for all finite X^{•} ∈ St. In light of this, it’s hard to see why the St(k) analogue wouldn’t also be true, until one gets into the guts of the proof.

Theorem 12. The Local Conjecture (11) is true for any finite X^{•} ∈ St.

Proof: The easiest way to see this is to use the formulation of L_{n}^{f} as finite localization with respect to E(n). (As above, in the St setting, of course, we replace the spectrum E(n) (where E(n)_{∗} = ℤ_{(p)}[v_{1}, …, v_{n}^{±1}]) by Z(≤ n), a wedge of the periodic versions of the complexes which resolve the A-modules A/(u_{d}), with d ≤ n, where u_{d} is Palmieri’s notation for the P_{t}^{s}’s ordered appropriately.) Let T be any finite E(n)-acyclic. Consider [T, v_{n}^{−1}X^{•}]_{∗}. Intuitively, we think of such a finite T as being E(n)-acyclic in St because it has a vanishing line of slope less than or equal to 1/(|v_{n}| + 1). More precisely, there must be such a vanishing line in some E_{r} of the Adams spectral sequence for T, so we conclude that

[T, v_{n}^{−1}X^{•}]_{∗} = [S^{0}, DT ∧ v_{n}^{−1}X^{•}]_{∗} = [S^{0}, v_{n}^{−1}(DT ∧ X^{•})]_{∗} = 0,

where the second equality follows from the centrality of v_{n}-self maps. Without finiteness on T, we don’t have the needed vanishing line. Further, if X were not finite, we don’t have the centrality needed to conclude that the homotopy classes of maps must add up to zero.

The result is almost certainly not true for a general X^{•} ∈ St. Also, in the St(k) world, we don’t have such vanishing lines, because of the h_{0}-towers in Ext. There seems to be no clear St(k) analogue of this result.

Here’s why this Local conjecture is the key point:

Lemma 13. If X_{k}^{•} ∈ St(k) is a finite spectrum for k > n, then

L_{n,k}^{f}v_{n}^{−1}X_{k}^{•} ≅ L_{n,k}^{f}X_{k}^{•}.

Further, the obvious analogue holds in St itself:

L_{n}^{f}v_{n}^{−1}X^{•} ≅ L_{n}^{f}X^{•}.

Proof. For finite X_{k}^{•} ∈ St(k),

v_{n}^{−1}X_{k}^{•} = v_{n}^{−1}HA(k) ∧ X_{k}^{•} = v_{n}^{−1}G(k)^{•} ∧ X_{k}^{•},

is just a telescope. Since the maps in the telescope are just multiplication by some power of v_{n}, these maps are just the identity in L_{n}^{f}X_{k}^{•}. Thus the finite localization map factors through the telescope, as usual:

X_{k}^{•}  v_{n}^{−1}X_{k}^{•} L_{n,k}^{f}X_{k}^{•} w_{[ [ [ [[]} u

and clearly L_{n,k}^{f}v_{n}^{−1}X_{k}^{•} ≅ L_{n,k}^{f}X_{k}^{•} in St(k). If X^{•} ∈ St, taking an inverse limit of these diagrams for X_{k}^{•} and using Lemma 7 completes the proof.

Note that this lemma, along with the Local theorem above, implies the following: for all finite X^{•} ∈ St, the localization map

v_{n}^{−1}X^{•}⟶^{loc}L_{n}^{f}X^{•},

is an isomorphism in St, since

v_{n}^{−1}X^{•} = L_{n}^{f}v_{n}^{−1}X^{•} = L_{n}^{f}X^{•},

where the first equality follows from the Local Conjecture and the second from Lemma 13. However, if this were true in the St(k) category for all X^{•} ∈ St(k), this would directly contradict Hovey’s observation.

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