BROWN-COMENETZ DUALITY AND THE ADAMS SPECTRAL SEQUENCE

MARK MAHOWALD AND CHARLES REZK

Abstract. We show that the class of p-complete connective spectra with finitely presented cohomology over the Steenrod algebra admits a duality theory related to Brown-Comenetz duality. This construction also produces a full-plane version of the classical Adams spectral sequence for such spectra, which converges to the homotopy groups of a “finite” localization.

1. Introduction

In the paper [3], Brown and Comenetz introduced a notion of duality into stable homotopy. In [4] Hopkins and Gross showed that this notion in certain situations is closely connected with Spanier-Whitehead duality. In this note we wish to explore this connection and investigate it in connection with Adams spectral sequence considerations. In particular, we study a class of spectra which we call fp-spectra (Section 3). These are connective, p-complete spectra whose mod p cohomology is finitely presented over the Steenrod algebra; that is, the cohomology of such a spectrum is described by a finite set of generators together with a finite set of relations. This class of spectra includes the Johnson-Wilson spectra BP ⟨n⟩, connective K-theories, and the “higher” connective K-theory spectrum eo_{2}. The class of fpspectra also includes some objects whose Bousfield L_{n}-localizations are the same as L_{n} localizations of finite complexes, at least in some cases. A classical example is the connective image-of-J spectrum, whose L_{1}-localization is L_{1}S^{0}. It follows from calculations of Shimomura and Yabe that a −1-connective cover of L_{2}S^{0} at primes p ≥ 5 is also an fp-spectrum (Proposition 3.7).

We show that the category of fp-spectra admits a notion of duality (Theorem 8.11). This duality is related to both Brown-Comenetz duality and Spanier-Whitehead duality. The dual W X of an fp-spectrum X will be defined to be the Brown-Comenetz dual of the fiber of the map X → L_{n}^{f}X to the “finite localization” of X (for n sufficiently large). The dual W X is itself an fp-spectrum. This duality is related to Spanier-Whitehead duality through its action on cohomology, in the following sense. If H^{∗}X ≈ A^{∗} ⊗_{A^{∗}(n)} M where A^{∗}(n) ⊂ A^{∗} is a finite sub-Hopf algebra of the Steenrod algebra, and M is a finite A^{∗}(n) module, then H^{∗}W X ≈ A^{∗} ⊗_{A^{∗}(n)} M̌, where M̌ ≈ hom_{𝔽_{p}}(M, 𝔽_{p}) is the “Spanier-Whitehead dual” of M as a finite module over A^{∗}(n).

Because of this duality, the L_{n}^{f}-localization of an fp-spectrum is quite computable. We show that there is a full-plane spectral sequence computing π_{∗}L_{n}^{f}X, with E_{2}-term a “Tate cohomology” of H^{∗}X as a module over the Steenrod algebra (Proposition 6.3 and Theorem 7.1). In all cases we know of L_{n}^{f}X ≈ L_{n}X for an fp-spectrum X.

Date: May 15, 1998.

1.1. Organization of the paper. In Section 2 we discuss modules and comodules which are finitely presented over the Steenrod algebra. In Section 3 we define the notion of fp-spectra, and give examples. In Section 4 we discuss a duality functor for finitely presented comodules over the Steenrod algebra, which is related to the action of Brown-Comenetz duality on Eilenberg-Mac Lane spectra discussed in Section 5. In Section 6 we note that an fp-spectrum admits a tower associated to a spectral sequence whose E_{2}-term is the “Tate cohomology” of the homology of the spectrum, and in Section 7 show that such a tower realizes the localization functor L_{n}^{f}. In Section 8 we describe the duality theory of fp-spectra. In Section 9 we calculate some examples.

1.2. Notation. In this paper we work at one prime p at a time. We let A^{∗} denote the mod p Steenrod algebra, and A_{∗} denote the dual mod p Steenrod algebra.

Unless otherwise indicated, all vector spaces, modules, and comodules in this paper are graded. If V is a graded vector space over 𝔽_{p}, then V̌ denotes the linear dual hom(V, 𝔽_{p}). If V is a left comodule over a graded Hopf algebra B, then V̌ is taken to be a left comodule over B, via the canonical anti-automorphism χ of B.

When dealing with graded objects, we use the following sign convention: a sign is introduced whenever two symbols of odd degree are commuted.

2. Finitely presented modules and comodules over the Steenrod algebra

A module M over the mod p Steenrod algebra A^{∗} is called finitely presented if it fits in an exact sequence of modules

A^{∗} ⊗ V_{1} → A^{∗} ⊗ V_{0} → M → 0

where V_{i} for i = 0, 1 are finite dimensional graded 𝔽_{p}-vector spaces. Likewise, a comodule N over the dual mod p Steenrod algebra A_{∗} is called finitely presented if it fits in an exact sequence of comodules

0 → N → A_{∗} ⊗ V_{0} → A_{∗} ⊗ V_{1}

where V_{i} for i = 0, 1 are finite dimensional graded 𝔽_{p}-vector spaces. Because all finitely presented modules and comodules are of finite type, we can pass easily between comodule and module language by taking vector space duals.

The Steenrod algebra A^{∗} is a union of finite-dimensional sub-Hopf algebras. For example, A^{∗} = ⋃_{n} A^{∗}(n), where A^{∗}(n) ⊂ A^{∗} is a finite dimensional sub-Hopf algebra of the Steenrod algebra, generated as an algebra by { Sq^{2^{i}} | i ≤ n + 1 } if p = 2 and by { β, P^{p^{i}} | i ≤ n } if p is odd. Recall that A^{∗} is free as an A^{∗}(n)-module.

Lemma 2.1.

1. A module M over A^{∗} is finitely presented if and only if it is of the form M ≈ A^{∗} ⊗_{E} N for some finite dimensional sub-Hopf algebra E ⊂ A^{∗} and some finite dimensional E-module N.

2. Every map f: M → M′ of finitely presented A-modules is of the form f ≈ A^{∗} ⊗_{E} g: A^{∗} ⊗_{E} N → A^{∗} ⊗_{E} N′ for some finite dimensional sub-Hopf algebra E ⊂ A^{∗} and some map g: N → N′ of finite dimensional E-modules.

Proof. Any finite sub-Hopf algebra E ⊂ A^{∗} is contained in A^{∗}(n) for some n ≥ 0, whence part 1 is [12, Ch. 13, Prop. 2(a)]. Part 2 follows by similar arguments.

Proposition 2.2.

1. The kernel and cokernel of a map of finitely presented A^{∗}-modules are finitely presented.

2. A retract of a finitely presented A^{∗}-module is finitely presented.

3. If 0 → M′ → M → M′′ → 0 is a short exact sequence in which M′ and M′′ are finitely presented, then M is also finitely presented.

Proof. Since A^{∗} is A^{∗}(n)-free, the functor A^{∗} ⊗_{A^{∗}(n)} (−) is exact, and hence part 1 follows from (2.1).

To prove part 2, note that a retract N of M is the kernel of an idempotent self-map e: M → M. Hence part 2 follows from part 1.

The proof of part 3 is a standard result about finitely presented modules over any ring.

Proposition 2.3. Suppose M is an A^{∗}-module and F is a finite A^{∗}-module. Then M is finitely presented if and only if M ⊗ F is.

Proof. It is immediate from (2.2) that M ⊗ F is finitely presented if M is, since F admits a finite filtration whose subquotients are copies of 𝔽_{p}.

Suppose M ⊗ F is finitely presented. Since F is a finite module, we can choose a “pinch” map π: F → Σ^{d}𝔽_{p} to a “bottom-dimensional cell” of F, and we can write i: F̄ → F for the kernel of π. Then there is an exact sequence

M ⊗ F̄ ⊗ F ⟶^{1⊗i⊗π} M ⊗ F ⟶^{1⊗π} M → 0

which exhibits M as a cokernel of a map between finitely presented modules, and the result follows from (2.2).

Remark 2.4. Note that (2.1), (2.2), and (2.3) dualize to similar statements about finitely presented comodules. We will not state the dual form of these results, although we will make use of them in what follows.

2.5. Homological algebra for finitely presented comodules. Henceforth we concentrate on finitely presented comodules. We let M_{fp} denote the category of finitely presented comodules over A_{∗}. By (2.2) we see that M_{fp} is an abelian category.

Proposition 2.6. The dual Steenrod algebra A_{∗}, viewed as a A_{∗}-comodule, is both projective and injective in M_{fp}, and M_{fp} has enough projectives and injectives.

Proof. It is clear that A_{∗} is injective in the full category of A_{∗} comodules, and hence A_{∗} is injective in M_{fp} and there are enough injectives in M_{fp}. To prove that A_{∗} is projective, consider a surjection M → M′ of finitely presented comodules. By (2.1), this map is extended up from a surjection N → N′ of finite A_{∗}(n)-comodules. Since hom_{A_{∗}}(A_{∗}, M) ≈ hom_{A_{∗}(n)}(A_{∗}, N) and A_{∗} is A_{∗}(n)-free, any map A_{∗} → M′ can be lifted to a map to M. Furthermore, we can always produce enough maps from A_{∗} to a finitely presented comodule, and thus M_{fp} has enough projectives.

Remark 2.7. It is known that the Steenrod algebra is injective as an A^{∗}-module over itself [12, p. 201]. It would be interesting to know whether A_{∗} is projective as a comodule over itself, without the restriction to the finitely presented category.

Given a finitely presented comodule M, one can define its Tate cohomology as follows. By (2.6) we can choose injective and projective resolutions

0 → M → C_{0} → C_{1} → C_{2} → …

and

… → C_{−3} → C_{−2} → C_{−1} → M → 0

by finitely generated free A_{∗}-comodules. By gluing the ends together, we obtain an unbounded complex

… → C_{−2} → C_{−1} → C_{0} → C_{1} → …,

which in each degree is injective in the category of comodules. For s ∈ ℤ define

H_{Tate}^{s}(M) = H^{s} [hom_{A_{∗}}(𝔽_{p}, C_{•})] .

Of course, if we can write M ≈ A_{∗} ⊗_{A_{∗}(n)} N for some A_{∗}(n)-comodule N, and we choose resolutions 0 → N → D_{•} and D_{−•} → N → 0 of N by finite free A_{∗}(n)- comodules, then we see that

H_{Tate}^{s}(M) ≈ H^{s} [hom_{A_{∗}(n)}(𝔽_{p}, D_{•})] .

These groups are the same as morphisms in the stable category of A_{∗}(n)-modules, as is shown in [7, Sec. 9.6].

3. fp-spectra

In this section we define the notion of fp-spectra, and produce several examples. Recall that we work in the category of p-local spectra.

We first note the following theorem of Mitchell.

Theorem 3.1 (Mitchell). [13] For each n there exists a non-trivial finite complex F such that H_{∗}F is A_{∗}(n)-free.

If X is a spectrum, say that π_{∗}X is finite if π_{k}X = 0 for all but finitely many k ∈ ℤ, and is a finite group otherwise.

Proposition 3.2. Suppose X is a connective, p-complete spectrum. Then the following are equivalent.

1. H_{∗}X is finitely presented as a comodule over the Steenrod algebra.

2. H_{∗}X ≈ A_{∗} ⊗_{A_{∗}(n)} M for some n ≥ 0 and some finite A_{∗}(n)-comodule M.

3. There exists a non-trivial finite complex F such that X ∧ F is a finite wedge of suspensions of mod p Eilenberg-Mac Lane spectra.

4. There exists a non-trivial finite complex F such that π_{∗}(X ∧ F) is finite.

Proof. The equivalence of 1 and 2 is just (2.1). Likewise, 4 is immediate given 3.

To show that 2 implies 3, we let F be as in (3.1), with H_{∗}F free over A_{∗}(n). Thus H_{∗}(X ∧ F) is free over the Steenrod algebra on a finite set of generators, whence X ∧ F is a wedge of mod-p Eilenberg-Mac Lane spectra H𝔽_{p}.

To show that 4 implies 1, note that if π_{∗}Y is finite for a connective spectrum Y, then Y can be built from finitely many copies of H𝔽_{p}, whence H_{∗}Y is finitely presented by (2.2). Thus if π_{∗}(X ∧ F) is finite, then H_{∗}(X ∧ F) ≈ H_{∗}X ⊗ H_{∗}F is finitely presented, and hence H_{∗}X is finitely presented by (2.3).

We call a spectrum X an fp-spectrum if it is connective, p-complete, and satisfies any of the four equivalent statements of (3.2). Let C denote the class of all fp-spectra. This class includes the Eilenberg-Mac Lane spectra Hℤ/p^{n} and Hℤ_{p}, the p-completed Johnson-Wilson spectrum BP ⟨n⟩, which has π_{∗}BP ⟨n⟩ ≈ ℤ_{p}[v_{1}, …, v_{n}], and connective Morava K-theories. Non-trivial suspension spectra, and in particular finite complexes, are not fp-spectra.

Recall that a finite complex F is of type n if K(0)_{∗}F ≈ ⋯ ≈ K(n − 1)_{∗}F ≈ 0 and K(n)_{∗}F ≉ 0, where K(m) denotes the mth Morava K-theory. Define the fp-type of an fp-spectrum X by

fptype(X) = min{(type(F) − 1) such that π_{∗}(X ∧ F) is finite}.

By the thick subcategory theorem [6], if fptype(X) = n then π_{∗}(X ∧ F) is finite for all F of type > n. Thus, fptype(H𝔽_{p}) = −1, fptype(Hℤ_{p}) = 0, and fptype(BP ⟨n⟩) = n. Furthermore, if H_{∗}X ≈ A_{∗} ⊗_{A_{∗}(n)} M, then fptype(X) ≤ n.

Let C_{n} denote the class of fp-spectra of type ≤ n. Then C_{n} is a subcategory of the category of spectra, and C_{n} ⊂ C_{n+1}.

Proposition 3.3. The classes C and C_{n} for n ≥ −1 are thick subcategories of the homotopy category of spectra.

Proof. That C is a thick subcategory is an immediate consequence of criterion 4 of (3.2). Alternately, this follows from criterion 1 of (3.2) together with (2.2). The proof for C_{n} is similar.

For example, C_{−1} is the class of all spectra X with π_{∗}X finite; as a thick subcategory it is generated by H𝔽_{p}. Likewise, C_{0} is the class of all fp-spectra which are finite Postnikov towers; as a thick subcategory it is generated by Hℤ_{p}. The class C_{1} contains the p-completed connective K-theory spectra bo and bu, along with their connective covers. Thus C_{1} contains the image of J spectrum, since J ≈ fib(bo → bspin). The class C_{2} contains eo_{2}, the connective version of the “higher real Ktheory” spectrum EO_{2} of Hopkins and Miller.

3.4. L_{n}-localization and fp-spectra. Let L_{n} denote Bousfield localization with respect to the wedge K(0) ∨ ⋯ ∨ K(n) of Morava K-theories. A spectrum W is L_{n}-local if L_{n}W ≈ W.

Proposition 3.5. Let W be an L_{n}-local spectrum such that for each k ∈ ℤ the homotopy group π_{k}W has the form

π_{k}W ≈ F_{k} ⊕ ℤ_{p}^{⊕a_{k}} ⊕ (ℚ/ℤ_{(p)})^{⊕b_{k}} ⊕ ℚ_{p}^{⊕c_{k}},

where F_{k} is a finite p-group, a_{k} = 0 = c_{k} for all sufficiently small k ≪ 0, and b_{k} = 0 = c_{k} for all sufficiently large k ≫ 0.

Then there exists a map f: X → W such that X is an fp-spectrum of fp-type n and L_{n}X → L_{n}W ≈ W is a weak equivalence.

Proof. Consider the connected cover Y = W (−N, …, ∞), where N is chosen so that a_{k} = c_{k} = 0 for k < −N. If F is a finite complex with bottom cell in dimension 0 and top cell in dimension d, then the map Y ∧ F → W ∧ F is an isomorphism on π_{k} for k > d − N, as can be seen by comparing the Atiyah-Hirzebruch spectral sequences computing Y_{∗}F and W_{∗}F.

If F is a type (n + 1) complex, then W ∧ F ≈ ∗, and so Y ∧ F has non-trivial homotopy in only a finite range of dimensions (−N, …, d−N), and each homotopy group is finite. Thus we have found a connective spectrum Y which satisfies criterion 4 of (3.2), and furthermore L_{n}Y ≈ W, since the fiber of Y → W is coconnective with torsion homotopy and thus is killed by L_{n}.

In order to get a p-complete spectrum, it suffices to replace any copies of ℚ/ℤ_{(p)} or ℚ_{p} in the homotopy of Y by a finite torsion group or a copy of ℤ_{p} respectively; by hypothesis there are only finitely many such copies to worry about. Note that [Σ^{−i}HA, Hℚ/ℤ_{(p)}] is a finite torsion group if A is a finitely generated ℤ_{p}-module and i > 0; thus by induction on the Postnikov tower of Y we can see that we can always find a map

Y → ⋁_{i=1}^{r} Σ^{n_{i}}Hℚ/ℤ_{(p)}

which is surjective on homotopy, and so that the fiber X of this map is p-complete. Then since L_{n}Hℚ/ℤ_{(p)} ≈ ∗ we see that X is the desired spectrum.

Remark 3.6. In view of Conjecture (7.3) and (8.9), it seems likely that the converse of (3.5) should hold. That is, if X is an fp-spectrum with fptype(X) = n, then we expect that π_{k}L_{n}X has the form given in (3.5).

It is interesting to know when the L_{n}-localization of a finite complex F is also the L_{n}-localization of an fp-spectrum X. One can say the following.

Proposition 3.7. If F is the p-completion of a finite complex, then L_{n}F is the L_{n}-localization of an fp-spectrum X in the following cases,

1. n = 0 or n = 1 (at any prime),

2. n = 2 if p ≥ 5, or

3. for any n and any prime p if F is a type n complex.

Proof. It suffices to show in each case that the hypotheses of (3.5) are satisfied.

The cases n = 0 and n = 1 are well known. In fact, for n = 0 take X = Hℤ_{p} ∧F, and for n = 1 take X = J ∧ F, where J is the connective image-of-J spectrum.

If n is arbitrary, but F is a type n complex, then the hypotheses of (3.5) hold. This is a consequence of the fact that the cohomology of the Morava stabilizer algebra is a finitely generated algebra (see [15, Thm. 6.2.10]), together with Hopkins and Ravenel’s demonstration of a horizontal vanishing line at the E_{∞}-term of the Adams-Novikov spectral sequence of L_{n}F (see [16, Section 8.3]). These imply that π_{k}F is finite for all k.

When n = 2 and p ≥ 5, one can take a spectrum of the form Y ∧ F, where Y is an fp-spectrum such that L_{2}Y ≈ L_{2}S^{0}. We show that the hypotheses of (3.5) hold for L_{2}S^{0} at p ≥ 5.

First, we note that the hypotheses of (3.5) hold for L_{2}M(p), the localization of the mod p Moore space. This is a consequence of calculations of Shimomura [18], as we explain below. There is a diagram

\xymatrix{L_{2}M(p) \ar[r] \ar[d] & L_{1}M(p) \ar[r] \ar[d] & L_{2}M(p, v_{1}^{∞}) \ar[d]^{∼} \\ L_{K(2)}M(p) \ar[r] & L_{1}L_{K(2)M(p)} \ar[r] & L_{2}M(p, v_{1}^{∞})}

in which the left-hand square is a pull-back square; this is because all the objects in it are L_{2}-local, and the square is a pull-back after smashing with K(0)∨K(1)∨K(2).

Since L_{1}M(p) ≈ L_{1}(v_{1}^{−1}M(p)) ≈ L_{2}(v_{1}^{−1}M(p)), the top row is a cofiber sequence, and thus so is the bottom row.

Shimomura computes the E_{2}-term of the Adams-Novikov spectral sequence for L_{2}M(p, v_{1}^{∞}). Using Shimomura’s calculation one may (with careful analysis) read off that each group π_{k}L_{2}M(p, v_{1}^{∞}) must be finite; see the presentation of the results of this calculation given in [17]. Since π_{k}L_{1}M(p) is known to be finite, this shows that π_{k}L_{2}M(p) must be finite.

Hovey and Strickland [8, Thm. 15.1] actually carry out the careful analysis to show that each group π_{k}L_{K(2)}M(p), k ∈ ℤ, is finite, so we will derive what we need from their results. To derive the finiteness of π_{k}L_{2}M(p), it suffices to show that L_{1}L_{K(2)}M(p) has finite homotopy groups. This spectrum is equivalent to v_{1}^{−1}L_{K(2)}M(p), since the v_{1}-self map of M(p) is trivial on K(2)_{∗}M(p). It happens that π_{∗}L_{K(2)}M(p) decomposes as a finite sum of copies 𝔽_{p}[v_{1}] plus a summand which is v_{1}-torsion. Thus π_{∗}v_{1}^{−1}L_{K(2)}M(p) is a finite sum of copies of the form 𝔽_{p}[v_{1}^{±}], which is clearly a finite group in each dimension.

When n = 2 and p ≥ 5, the hypotheses of (3.5) hold for L_{2}S^{0}; this is a consequence of the above remarks together with the calculation of Shimomura and Yabe [19] of π_{∗}L_{2}S^{0} at p ≥ 5. They show that the homotopy of L_{2}S^{0} consists of a free summand in dimension 0, summands of the form ℚ/ℤ_{(p)} in stems −3, −4, and −5, together with a summand T consisting of non-infinitely divisible torsion. The above remarks on the finiteness of π_{k}L_{2}M(p) imply that the summand T of π_{∗}L_{2}S^{0} is finite in each stem.

We are led to make the following conjectures.

Conjecture 3.8. The hypotheses of (3.5) hold for any L_{n}-localization of a finite complex, for any n ≥ 0.

Conjecture 3.9. For every finite complex F there exists an fp-spectrum X such that L_{n}F ≈ L_{n}X.

Of course, Conjecture (3.8) implies Conjecture (3.9) as we have shown above. There is also reason to believe that Conjecture (3.9) would imply Conjecture (3.8); see section (7.2).

3.10. Adams towers for fp-spectra. For a spectrum X we can construct an Adams tower. This is a tower of spectra … → X_{s+1} → X_{s} → … with X_{0} = X and with fiber sequences X_{s+1} → X_{s} ⟶^{k_{s}} Σ^{−s}HV_{s}, where k_{s} is injective on mod p homology; hence there is a resolution

0 → H_{∗}X → H_{∗}HV_{•},

and an Adams spectral sequence with E_{2}^{s,t} = Ext_{A_{∗}}^{s,t}(𝔽_{p}, H_{∗}X).

If X is an fp-spectrum, we can choose an Adams tower in which each V_{s} is a finite dimensional graded vector space. We call such an fp-Adams resolution. In fact, if H_{∗}X ≈ A ⊗_{A_{∗}(n)} M for some finite A_{∗}(n)-module M, then the chain complex H_{∗}HV_{•} is induced from a resolution of M by A_{∗}(n)-comodules.

Given an Adams tower {X_{s}}, we may produce another tower

… → X_{0}^{s} → X_{0}^{s−1} → … → X_{0}^{1} → X_{0}^{0} → ∗

by taking cofibers X_{s+1} → X → X_{0}^{s}. The fibers in this tower are Σ^{−s}HV_{s} → X_{0}^{s} → X_{0}^{s−1}. Note that in the E_{2}-term of the spectral sequence for this tower, each horizontal line E_{2}^{s,∗} is finite dimensional, and thus the spectral sequence satisfies the complete convergence condition of [2, p. 263]. Thus the homotopy inverse limit holim_{s} X_{0}^{s} does not depend on the choice of fp-Adams resolution, and by a result of Bousfield [1, Prop. 5.8 and Thm. 6.6] is equivalent to X, since X is p-complete and connective.

4. Duality for finitely presented comodules

In this section we describe a duality functor Ĩon the category of finitely presented comodules. This functor was essentially introduced by Brown and Comenetz in [3]. Our interest in this functor stems from the fact that it generalizes the notion of “Spanier-Whitehead duality” of finite comodules over A_{∗}(n), in which a comodule is dual to its vector space dual. It will be needed in later sections to study duality on fp-spectra.

4.1. Construction of functors J̃ and Ĩ. To motivate the construction, note that if X is a p-complete spectrum and H denotes the mod p Eilenberg-Mac Lane spectrum, then the graded vector space [H, X]_{∗} is in a natural way a module over the Steenrod algebra. This object is approximated by the edge map [H, X]_{∗} → hom_{A_{∗}}(H_{∗}H, H_{∗}X) of the Adams spectral sequence. This algebraic approximation itself admits an action by the Steenrod action, and we will call this module J̃(H_{∗}X). The vector space dual to J̃(H_{∗}X) will be Ĩ(H_{∗}X), which admits a coaction by the dual Steenrod algebra.

Recall that if we regard the dual Steenrod algebra A_{∗} as a left-comodule over itself, then each element a ∈ A^{∗} of the Steenrod algebra induces a map A_{∗} → A_{∗} of left A_{∗}-comodules via

a · z = ∑(−1)^{|a||z|}z′⟨z′′, a⟩,

where z ∈ A_{∗}, ∑ z′ ⊗ z′′ is the diagonal of z in A_{∗} ⊗ A_{∗}, and ⟨z, a⟩ represents the usual pairing of A_{∗} and A^{∗}. In fact, there is an isomorphism of algebras

hom_{A_{∗}}(A_{∗}, A_{∗}) ≈ A^{∗}. (4.2)

We reserve the notation a · z for this action. Note that this action gives A_{∗} the structure of a left A^{∗}-module; however, this is not identical to the usual left action of the Steenrod algebra on H_{∗}H.

We define a functor J̃ from left A_{∗}-comodules to left A^{∗}-modules by

J̃(M) = hom_{A_{∗}}(A_{∗}, M).

This has a natural right A^{∗}-action induced by pre-composition of comodule maps, using (4.2), which is made into a left A^{∗}-action using χ; if z ∈ A_{∗}, a ∈ A^{∗}, and f ∈ J̃(M), the left action can be written

(a · f)(z) = (−1)^{|a||f|}f(χa · z).

Note that if M is finitely generated then J̃(M) is bounded below (where we use cohomological grading for J̃(M)).

We define a functor Ĩ from finitely generated left comodules to left comodules by

Ĩ(M) = (J̃(M))ˇ ≈ hom_{A_{∗}}(A_{∗}, M)ˇ .

Since J̃(M) was bounded below, Ĩ(M) is bounded below, and thus receives a left comodule structure in the usual way. This structure is characterized as follows: if u ∈ Ĩ(M) and u ↦ ∑ u′ ⊗ u′′ ∈ A_{∗} ⊗ Ĩ(M) is the comodule action on u, and if a ∈ A^{∗} and f ∈ J̃(M), then

⟨u, a · f⟩ = ∑(−1)^{|a||u′′|}⟨u′, a⟩⟨u′′, f⟩.

In comparison, Brown and Comenetz [3] define a functor c_{p} on the category of A^{∗}-modules; their functor is defined by

c_{p}(M) ≈ hom_{A^{∗}}(M, A^{∗}),

with an appropriate A^{∗}-action. Thus, our Ĩ is just a comodule version of their c_{p}.

4.3. Action of J̃ and Ĩ on free comodules. Let M_{free} ⊂ M_{fp} denote the full subcategory of comodules of the form A_{∗} ⊗ V, where V is a finite vector space. We want compute the action of J̃ and Ĩ on this subcategory.

The following describes the category M_{free}.

Proposition 4.4. For finite V and W, hom_{A_{∗}}(A_{∗}⊗V, A_{∗}⊗W) ≈ A^{∗}⊗hom(V, W), where a ⊗ σ ∈ A^{∗} ⊗ hom(V, W) corresponds to the morphism of comodules

z ⊗ v ↦ (−1)^{|σ||z|}a · z ⊗ σ(v), z ∈ A_{∗}, a ∈ A^{∗}, v ∈ V, σ ∈ hom(V, W),

and composition of maps is given by (a ⊗ σ) ∘ (b ⊗ τ) = (−1)^{|σ||b|}ab ⊗ στ.

Proof. Straightforward.

Proposition 4.5. There is a natural isomorphism

J̃(A_{∗} ⊗ V) ≈ hom_{A_{∗}}(A_{∗}, A_{∗} ⊗ V) ≈ A^{∗} ⊗ V

of left A^{∗}-modules, where a ⊗ v ∈ A^{∗} ⊗ V corresponds to the map defined by

z ↦ (−1)^{|z||v|}χa · z ⊗ v, z ∈ A_{∗}, a ∈ A^{∗}, v ∈ V.

Given a map a ⊗ σ: A_{∗} ⊗ V → A_{∗} ⊗ W of comodules, the induced map J̃(a ⊗ σ): A^{∗} ⊗ V → A^{∗} ⊗ W sends b ⊗ v ↦ (−1)^{|b|(|a|+|σ|)}b χa ⊗ σ(v).

Proof. Straightforward.

Corollary 4.6. There is a natural isomorphism

Ĩ(A_{∗} ⊗ V) ≈ (A^{∗} ⊗ V)ˇ ≈ A_{∗} ⊗ V̌

of left-comodules. The induced map Ĩ(a ⊗ σ): A_{∗} ⊗ W̌ → A_{∗} ⊗ V̌ sends

z ⊗ w̌ ↦ (−1)^{|σ||z|}χa · z ⊗ σ̌(w̌), z ∈ A_{∗}, a ∈ A^{∗}, w̌ ∈ W̌ ,

where σ̌ ∈ hom(W̌ , V̌) is the adjoint to σ ∈ hom(V, W). In other words, Ĩ(a ⊗ σ) = χa ⊗ σ̌.

Remark 4.7. In terms of bases v_{j} and w_{i} for V and W we can view maps A_{∗} ⊗V → A_{∗} ⊗ W as matrices (a_{ij}) with entries in the Steenrod algebra, acting by

∑_{j} z_{j} ⊗ v_{j} ↦ ∑_{i,j} a_{ij} · z_{j} ⊗ w_{i}.

Hence, the induced map Ĩ(a_{ij}): A_{∗} ⊗W̌ → A_{∗} ⊗V̌ corresponds to the matrix (χa_{ji}) in terms of the dual bases v̌_{j} and w̌_{i} of V̌ and W̌.

4.8. Properties of the duality functor.

Proposition 4.9. The functor Ĩ restricts to a functor Ĩ: M_{fp}^{op} → M_{fp}, and is exact on M_{fp}. Furthermore, there is a natural isomorphism M → ĨĨM for objects in M_{fp}.

Proof. That Ĩ is exact on the subcategory of finitely presented comodules follows from (2.6). By (4.6) the functor Ĩ takes finitely generated free comodules to the same, and thus by exactness Ĩ(M_{fp}) ⊂ M_{fp}.

One can construct a natural isomorphism M → ĨĨM when M ≈ A_{∗} ⊗ V, since by (4.6) ĨĨ(A_{∗} ⊗ V) is tautologically isomorphic to A_{∗} ⊗ V. This isomorphism extends by exactness to all finitely presented comodules.

Proposition 4.10. Let M be a finite A_{∗}(n)-comodule. Then

Ĩ(A_{∗} ⊗_{A_{∗}(n)} M) ≈ A_{∗} ⊗_{A_{∗}(n)} Σ^{a(n)}M̌

as left comodules, where a(n) is the dimension of the “top cell” of A_{∗}(n).

Proof. There is an exact sequence

0 → M → A_{∗}(n) ⊗ V ⟶^{(a_{ij})} A_{∗}(n) ⊗ W

of A_{∗}(n)-comodules, where a_{ij} ∈ A^{∗}(n). After applying vector space duals we can identify the resulting sequence with

0 ← Σ^{−a(n)}M̌ ← A_{∗}(n) ⊗ V̌ ⟵^{(χa_{ji})} A_{∗}(n) ⊗ W̌

by “Poincaré duality” of A_{∗}(n) [12, Ch. 12.2]. The result follows from (4.7) after extending up to A_{∗}.

5. Brown-Comenetz duality

Recall that the functor

X ↦ hom(π_{0}X, ℚ/ℤ)

is a generalized cohomology theory satisfying the wedge axiom, and hence is represented by a spectrum I. We write IY = F(Y, I) for the function spectrum, whence IY is the spectrum representing the functor

X ↦ IY^{0}(X) = hom(Y_{0}X, ℚ/ℤ).

The spectrum IY is called the Brown-Comenetz dual of Y.

We write DX ≈ F(X, S^{0}) for the Spanier-Whitehead dual of X. Note that if X is any spectrum and F is a finite complex, then the natural map IX ∧DF → I(X ∧F) is an equivalence.

There is a natural double-dual map X → IIX. If Y is a spectrum such that each homotopy group π_{k}Y is finite, then Y ≈ IIY via this map. Thus, given such a Y and given any spectrum X, there is a natural isomorphism

[X, Y] ≈ [IY, IX].

5.1. Eilenberg-Mac Lane spectra. Let H ≈ H𝔽_{p} denote the mod p Eilenberg-Mac Lane spectrum. Then IH ≈ H. In fact, by the universal coefficient theorem,

IH^{∗}X ≈ hom(H_{∗}X, ℚ/ℤ) ≈ hom(H_{∗}X, 𝔽_{p}) ≈ H^{∗}X.

Lemma 5.2. Under the above identification the map

I: [H, H]_{∗} → [IH, IH]_{∗} ≈ [H, H]_{∗}

sends a ∈ A^{∗} to χa.

Proof. This is [3, Thm. 1.9(d)].

More generally, let V denote a finite dimensional graded 𝔽_{p}-vector space, and let V̌ = hom(V, 𝔽_{p}) denote its vector space dual. Let HV denote the generalized Eilenberg-Mac Lane spectrum with π_{∗}HV = V, whence HV_{∗}X ≈ H_{∗}X ⊗ V.

Proposition 5.3. There is an equivalence IHV ≈ HV̌, and we have isomorphisms

H^{∗}(IHV) ≈ J̃(H_{∗}HV)

of A^{∗}-modules and

H_{∗}(IHV) ≈ Ĩ(H_{∗}HV)

of A_{∗}-comodules which are natural in HV.

Proof. This is immediate from (4.6) and (5.2); alternatively, it follows from [3, Thm. 1.3].

5.4. Algebraic approximation. We note that the functor Ĩ of Section 4 serves as an algebraic “approximation” to H_{∗}IX, at least when H_{∗}X has finitely presented homology.

Proposition 5.5. For each spectrum X with finitely presented homology, there is a map

ι_{X}: Ĩ(H_{∗}X) → H_{∗}IX

which is natural in X. Furthermore, this map is an isomorphism when π_{∗}X is finite.

Compare with [3, Thm. 1.13].

Proof. Given X we can choose an Adams resolution

X → HV_{0} → HV_{1} → …,

in which V_{0} and V_{1} are finite dimensional vector spaces, and the sequence

0 → H_{∗}X → H_{∗}HV_{0} → H_{∗}HV_{1}

is exact. Applying I to the first diagram gives maps

IX ← HV̌_{0} ← HV̌_{1}

and a sequence

H_{∗}IX ← Ĩ(H_{∗}HV_{0}) ← Ĩ(H_{∗}HV_{1}),

not necessarily exact. We let ι_{X} be the induced map

ι: Ĩ(H_{∗}X) = Cok (Ĩ(H_{∗}HV_{1}) → Ĩ(H_{∗}HV_{0})) → H_{∗}IX.

To see that ι_{X} is independent of the choice of resolution and is natural, use a map between Adams resolutions.

The map ι is by construction an isomorphism for X = H𝔽_{p}. By the exactness property of (4.9), we see that ι_{X} is an isomorphism for any X in the thick subcategory generated by H𝔽_{p}, which are precisely the X with finite homotopy.

6. Geometric realization of Tate cohomology

In this section we note that one can construct for each fp-spectrum a ℤ-indexed Adams tower; this is a tower which extends both above and below X, whose layers are finite mod p generalized Eilenberg-Mac Lane spectra, and which leads to a spectral sequence whose E_{2}-term is the Tate cohomology of H_{∗}X. We give several constructions, starting with the most general.

6.1. Construction of the tower.

Lemma 6.2. Let X be an fp-spectrum. Then

[H, X]_{∗} ≈ hom_{A_{∗}}(A_{∗}, H_{∗}X) ≈ J̃(H_{∗}X).

Proof. Choose an fp-Adams tower {X_{s}} for X. Then there is a spectral sequence E_{1}^{s,t} = [H, HV_{s}]_{t} ⟹ [H, X]_{t−s}. We claim that

1. E_{2}^{s,t} ≈ 0 for s > 0, so that E_{2}^{0,t} ≈ E_{∞}^{∗,t}, and

2. E_{2}^{0,t} ≈ hom_{A_{∗}}(A_{∗}, H_{∗}X). Since X is connective and p-complete, the first claim implies that the spectral sequence converges, and thus E_{2}^{0,t} ≈ [H, X]_{t}.

To prove the claim, recall that the resolution 0 → H_{∗}X → H_{∗}HV_{s} is extended up from a resolution 0 → M → C(s) of A_{∗}(n)-modules. Now

[H, HV_{s}]_{∗} ≈ hom_{A_{∗}}(A_{∗}, A_{∗} ⊗_{A_{∗}(n)} C(s)) ≈ hom_{A_{∗}(n)}(A_{∗}, C(s)).

Since as an A_{∗}(n)-comodule A_{∗} ≈ A_{∗}(n) ⊗ A_{∗}//A_{∗}(n) we see that A_{∗} is projective as an A_{∗}(n)-comodule, and thus the sequence

0 → hom_{A_{∗}}(A_{∗}, H_{∗}X) → hom_{A_{∗}}(A_{∗}, H_{∗}HV_{•})

is exact as desired.

Thus, given an fp-spectrum X, we may construct a tower “realizing” the Tate cohomology by constructing fp-spectra X_{−s} for s ≥ 0 inductively as follows. First, let X_{0} = X. Since H_{∗}X_{−s} ≈ A_{∗} ⊗_{A_{∗}(n)} M(−s) for some finite module M(−s), we can choose a surjection C(−s − 1) → M(−s) from a free A_{∗}(n)-comodule. This may be extended to a surjection A_{∗} ⊗_{A_{∗}(n)} C(−s − 1) → A_{∗} ⊗_{A_{∗}(n)} M(−s), which in turn by the above lemma is realized by a map of spectra Σ^{s}HV_{−s−1} → X_{−s}. If we take the cofiber

Σ^{s}HV_{−s−1} → X_{−s} → X_{−s−1}

we get another fp-spectrum X_{−s−1}; iteration produces an infinite sequence … → X_{−s} → X_{−s−1} → . . .. If we put this sequence together with an fp-Adams tower for X, we get a tower {X_{s}}_{s∈ℤ}, which we call a ℤ-indexed Adams tower for X.

Note that given a map f: X → Y of fp-spectra and given ℤ-indexed Adams towers {X_{s}} and {Y_{s}}, we can extend f to a map of towers, by the “dual” of the usual argument, using (6.2). In particular, given any two ℤ-indexed Adams towers for X we can produce a map between them.

For such a tower, let X̂ = hocolim_{s→∞} X_{−s}. There is a full-plane Adams-type spectral sequence E_{2}^{s,t} ≈ H_{Tate}^{s,t}(H_{∗}X) ⟹ π_{t−s}X̂ approximating the homotopy of X̂, where H_{Tate}^{s,t} is the Tate cohomology of (2.5).

Given such a tower {X_{s}}, write X_{p}^{q} = cof(X_{q+1} → X_{p}) for −∞ < p ≤ q < ∞. Then let X_{∞}^{−1} = cof(X → X̂) ≈ hocolim_{s→∞} X_{−s}^{−1}. A straightforward convergence argument shows that the homotopy spectral sequence of this colimit converges, and hence X_{∞}^{−1} does not depend on the choice of the ℤ-indexed tower.

The above remarks are summarized in the following proposition.

Proposition 6.3.

1. For each fp-spectrum X there exists a ℤ-indexed Adams tower {X_{s}}.

2. Given ℤ-indexed Adams towers {X_{s}} and {Y_{s}} for X and Y, and a map f: X → Y, there exists a map {f_{s}: X_{s} → Y_{s}} of towers which extends f.

3. The colimit X̂ = hocolim_{s→∞} X_{−s} of a ℤ-indexed Adams tower depends only on X, and not on the choice of tower.

6.4. Alternate construction of the tower. Let X be an fp-spectrum, and let F be a finite complex such that H_{∗}(X) ⊗ H_{∗}(F) is a free A_{∗}-comodule. Suppose also a map S^{0} → F which is injective on homology. Such an F always exists; if H_{∗}X is an extended A_{∗}(n)-comodule, let F be a Mitchell complex of type n as in (3.1), and S^{0} → F the inclusion of a bottom cell.

Form a fiber sequence F̄ → S^{0} → F, so that by Spanier-Whitehead duality we get a dual fiber sequence DF → S^{0} → DF̄. We obtain a ℤ-indexed Adams tower with

X_{s} = \begin{cases}X ∧ F̄^{(s)} & if s ≥ 0 \\ X ∧ (DF̄)^{(−s)} & if s < 0, \end{cases}

where the maps in the tower are induced by fiber sequences X∧F̄^{(s+1)} → X∧F̄^{(s)} → X ∧ F̄^{(s)} ∧ F and X ∧ DF̄^{(s)} ∧ DF → X ∧ DF̄^{(s)} → X ∧ DF̄^{(s+1)}.

In those cases when there actually exists a complex F with H_{∗}F ≈ A_{∗}(n), and H_{∗}X is an extended A_{∗}(n)-module, then this complex realizes the unbounded chain complex obtained by gluing together the bar complex for M and the co-bar complex for M as an A_{∗}(n)-comodule.

6.5. An explicit construction for bo. Recall that H^{∗}bo ≈ A^{∗}/A^{∗}(Sq^{1}, Sq^{2}). Consider the minimal Adams tower {bo^{s}} for bo at p = 2; write H^{∗}bo^{s} ≈ A^{∗} ⊗_{A^{∗}(1)} M(s).

Let R(n) denote the cofiber of P^{n} → S^{0} for n ≥ 0; thus R(0) ≈ S^{0}. Recall from [9] that

M(4s) ≈ H^{∗}R(8s).

We define a map R(8s + 8) → R(8s) as follows. Let p: R(8s + 8) → ΣP_{8s+1}^{8s+8} denote the pinch map obtained by taking the quotient of R(8s) → R(8s + 8). Then p ∘ (16) ∼ 0, where (16) is the degree 16 map on R(8s + 8) → R(8s + 8), and so (16) lifts to a map f_{s}: R(8s + 8) → R(8s). The map f_{s} has degree 16 on the bottom cell.

Likewise, we may consider the Spanier-Whitehead dual map

f_{−s} = Df_{s−1} : DR(8s − 8) → DR(8s).

This map has degree 16 on the top cell. Note that f_{−1} : S^{0} → DR(8s), so we can put all the f_{s} for s ∈ ℤ together into a ℤ-indexed tower

… ⟶^{f_{2}} R(16) ⟶^{f_{1}} R(8) ⟶^{f_{0}} R(0) ≈ S^{0} ⟶^{f_{−1}} DR(8) ⟶^{f_{−2}} DR(16) ⟶^{f_{−3}} ….

Proposition 6.6. There is a ℤ-indexed Adams tower {bo^{s}} for bo with

bo^{4s} ≈ \begin{cases}bo ∧ R(8s) & if s ≥ 0, \\ bo ∧ DR(−8s) & if s < 0, \end{cases}

and the map bo^{4s+4} → bo^{4s} is bo ∧ f_{s}.

Proof. Set bo^{4s} as above, and construct a the start of a minimal Adams tower bo^{4s+3} → bo^{4s+2} → bo^{4s+1} → bo^{4s} over bo^{s}. It is not hard to see that R(8s + 8) → bo ∧ R(8s) ≈ bo^{4s} lifts to bo^{4s+3}, so that we can extend to a map bo ∧ R(8s + 8) → bo^{4s+3}; likewise, DR(8s − 8) → bo ∧ DR(8s) ≈ bo^{−4s} lifts to bo^{−4s+3}, so we can extend to a map bo ∧ DR(8s − 8) → bo^{−4s+3}. The resulting tower is an Adams tower.

Here is a chart which presents the Adams E_{2}-term for bo ∧ DR(16).

[figure]
A chart for Ext_{A(1)}^{s,t}(H^{∗}(DR(16)), ℤ/2) for s < 16.

7. Finite localization

There exists a functor L_{n}^{f} and natural map X → L_{n}^{f}X, called finite localization. It is Bousfield localization with respect to the wedge Tel(0) ∨ ⋯ ∨ Tel(n), where Tel(k) denotes the v_{k}-telescope on some chosen type k finite complex. The functor L_{n}^{f} is characterized by the following properties [11].

1. The fiber C_{n}^{f}X = fib(X → L_{n}^{f}X) is a homotopy colimit of some diagram of type-(n + 1) finite complexes.

2. There are no essential maps from a type (n + 1) finite complex to L_{n}^{f}X (i.e., L_{n}^{f}X is L_{n}^{f}-local).

3. L_{n}^{f}X is smashing; i.e., L_{n}^{f}X ≈ X ∧ L_{n}^{f}S^{0}. Note also that 1 and 2 imply that L_{n}^{f}X ≈ X if X is L_{n}^{f}-local. Also, if X is a type n finite complex, then L_{n}^{f}X ≈ v^{−1}X, where v: Σ^{d}X → X is a v_{n}-self map of X.

Theorem 7.1. Let X be an fp-spectrum with fptype(X) ≤ n, and let {X_{s}} be a ℤ-indexed Adams tower for X, with colimit X̂. Then L_{n}^{f}X ≈ X̂.

Proof. Consider the map i: X → X̂. It suffices to show that 1. L_{n}^{f}(i) is an equivalence, and

2. X̂ is L_{n}^{f}-local.

Claim 1 follows immediately from the fact that L_{n}^{f}H𝔽_{p} ≈ ∗ and that L_{n}^{f} is smashing, since X̂ is obtained from X by attaching mod p Eilenberg-Mac Lane spectra.

To prove claim 2, we must show that X̂ admits no essential maps from a type- (n + 1) finite complex F, or equivalently, that X̂ ∧ F ≈ ∗ for any such complex. This follows from the following:

3. For any finite complex F, {X_{s} ∧ F } is a ℤ-indexed Adams tower for X ∧ F, and X̂ ∧ F ≈ hocolim X_{−s} ∧ F.

4. If Y is an fp-spectrum with π_{∗}Y finite, then Ŷ ≈ ∗ for any ℤ-indexed Adams tower for Y.

The proof of claim 3 is straightforward.

To prove claim 4, apply Brown-Comenetz duality to the sequence

Y = Y_{0} → Y_{−1} → Y_{−2} → … → Ŷ.

This produces a tower

IŶ → … → IY_{−2} → IY_{−1} → IY_{0} = IY

which is easily seen to be an Adams tower for IY by (5.5). Since IY is connective and p-complete and π_{∗}IY is finite, its Adams spectral sequence converges, and IŶ ≈ ∗, whence Ŷ ≈ ∗.

7.2. Relation between L_{n}^{f} and L_{n}. Let X → L_{n}X denote Bousfield localization with respect to K(0) ∨ ⋯ ∨ K(n). There is a natural map t_{n}: L_{n}^{f}X → L_{n}X; this map is an equivalence for X ≈ S^{0}, and hence for all X, if and only if the Telescope Conjecture holds. This conjecture is true for n = 0 and n = 1, and is believed to be false for n ≥ 2.

It is reasonable to ask whether t_{n} is an equivalence when X is an fp-spectrum. We note that t_{n} is an equivalence in the following cases, which include all cases we know of.

1. Since BP ⟨n⟩ is a BP -module spectrum, one can compute L_{n}BP ⟨n⟩ using the chromatic tower method of [14, Sec. 6]. (We would like to thank Hal Sadofsky for pointing this out to us.) This calculation shows in particular that the fiber of BP ⟨n⟩ → L_{n}BP ⟨n⟩ is coconnective with torsion homotopy; thus the fiber is killed by L_{n}^{f}, and hence t_{n} is an equivalence on L_{n}BP ⟨n⟩, and is in fact an equivalence on the thick subcategory of C generated by BP ⟨n⟩.

2. For any fp-spectrum X obtained by the procedure of the proof of (3.5) the map t_{n}: L_{n}^{f}X → L_{n}X is an equivalence, since by construction the fiber of X → L_{n}X is coconnective with torsion homotopy.

We make the following conjecture.

Conjecture 7.3. The map t_{n}: L_{n}^{f}X → L_{n}X is an equivalence for all fp-spectra X.

This conjecture is of interest, because it would give information on how badly the Telescope Conjecture fails, assuming it does fail. Namely, suppose F is a type n finite complex and v^{−1}F is its v_{n}-telescope; then if the Telescope Conjecture fails for n and Conjecture (7.3) holds, it follows that π_{k}(v^{−1}F) is an infinite group for some k ∈ ℤ, whereas π_{k}L_{n}F is finite for all k. To see this, we argue as follows; if each π_{k}(v^{−1}F) were a finite group, then the proof of (3.5) would apply to show that there exists an fp-spectrum X with L_{n}^{f}X ≈ v^{−1}F. However, Conjecture (7.3) would then imply that v^{−1}F ≈ L_{n}X ≈ L_{n}F.

Conjecture (7.3) also would imply, using (8.9) of the next section, that the two conjectures (3.8) and (3.9) discussed in section 3 are in fact equivalent statements.

8. Duality

8.1. Duality for fp-spectra of type less than n. Let

W_{n}X = IC_{n}^{f}X,

where C_{n}^{f} = fib(X → L_{n}^{f}X). Thus W_{n} is a contravariant functor from spectra to spectra. Since L_{n}^{f} is smashing,

W_{n}X ≈ F(X ∧ C_{n}^{f}S^{0}, IS^{0}) ≈ F(X, W_{n}S^{0}).

That is, W_{n}S^{0} is a dualizing complex for W_{n}. Furthermore, there is a natural map X → W_{n}W_{n}X; this map is adjoint to the evaluation map X ∧ F(X, W_{n}S^{0}) → W_{n}S^{0}.

Note that W_{n} vanishes on L_{n}^{f}-local spectra. Also, if π_{∗}X is finite, then L_{n}^{f}X ≈ ∗ and we have W_{n}X ≈ IX.

Recall that C_{n} denotes the category of fp-spectra with fp-type ≤ n.

Theorem 8.2. Let X be an object in C_{n}.

1. There is a natural isomorphism H_{∗}W_{n}X ≈ Ĩ(H_{∗}X).

2. W_{n}X is in C_{n}.

3. The natural map X → W_{n}W_{n}X is an equivalence.

From this we obtain the following.

Corollary 8.3. The functor W_{n} restricts to a functor W_{n}: C_{n}^{op} → C_{n}, which is an equivalence of categories.

Proof. The corollary follows from the fact that the functor W_{n} is “self-adjoint”; that is,

W_{n}: S^{op} ⇄ S : (W_{n})^{op}

is a pair of adjoint functors, where S represents the homotopy category of spectra. Part 2 of (8.2) says that W_{n} carries C_{n}^{op} into C_{n}, and part 3 says that the restriction of W_{n} to C_{n} gives an adjoint equivalence W_{n}: C_{n}^{op} ⇄ C_{n} : (W_{n})^{op} of categories.

Proof of Theorem 8.2. If X is an fp-spectrum, choose a ℤ-indexed Adams tower {X_{s}}, and as in (6.1) write X_{q}^{p} = cof(X_{p+1} → X_{q}). Then

C_{n}^{f}X ≈ Σ^{−1} hocolim_{s→∞} X_{−s}^{−1}

by (7.1), and thus

W_{n}X ≈ Σ (holim_{s→∞} IX_{−s}^{−1}) . (8.4)

Write H_{∗}X ≈ A_{∗} ⊗_{A_{∗}(n)} M. We can construct the bottom part of the ℤ-indexed tower to realize any projective resolution of M by finite A_{∗}(n)-modules, e.g., by the resolution dual to the minimal resolution 0 → M̌ → C_{•} of M̌ by finite A_{∗}(n)- modules. Then (8.4) immediately implies that W_{n}X is connective and p-complete, since the tower {I(X_{−s}^{−1})}_{s≥1} must necessarily be the tower associated with an Adams tower for W_{n}X which realizes C_{•}, and so the connectivity of I(X_{−s}^{−1}) is bounded below by a fixed N for all s ≥ 1.

Note that for any Y we have that H𝔽_{p} ∧ L_{n}^{f}Y ≈ ∗, whence the map H_{∗}C_{n}^{f}Y → H_{∗}Y is an isomorphism, so that there exists by (5.5) a natural comparison map

ι^{n}(Y): Ĩ(H_{∗}Y) ⟶^{∼} Ĩ(H_{∗}C_{n}^{f}Y) → H_{∗}W_{n}Y.

We want to show that ι^{n}(X) is an isomorphism.

Since X has a ℤ-indexed tower, there is a sequence

… → HV_{−2} → HV_{−1} → X

which induces an exact sequence

H_{∗}HV_{−2} → H_{∗}HV_{−1} → H_{∗}X → 0

on homology. Applying W_{n} to this sequence gives a sequence

W_{n}X → W_{n}HV_{−1} → W_{n}HV_{−2} → …

which corresponds to an Adams resolution for W_{n}X and hence induces an exact sequence

0 → H_{∗}W_{n}X → H_{∗}W_{n}HV_{−1} → H_{∗}W_{n}HV_{−2}.

It is clear that the comparison maps ι^{n}(HV_{−s}) are isomorphisms, and so ι^{n}(X) is an isomorphism, since Ĩ is exact. Thus we have proved parts 1 and 2 of the proposition.

Part 3 follows from parts 1 and 2, together with (8.7), to be proved below.

Remark 8.5. If X is a spectrum which might not be p-complete, and X_{p} denotes its p-completion, then one can show via an arithmetic square argument that C_{n}^{f}X ≈ C_{n}^{f}X_{p}, and hence W_{n}X ≈ W_{n}X_{p}. In particular, if X_{p} is an fp-spectrum, we may conclude that W_{n}W_{n}X ≈ X_{p}.

We still owe the reader one more fact.

Proposition 8.6. Suppose X and W_{n}X have finitely presented homology. Then there is a commutative square

\xymatrix{Ĩ(H_{∗}W_{n}X) \ar[r]^{ι^{n}(W_{n}X)} \ar[dr] \ar[d]^{Ĩι^{n}(X)} & H_{∗}W_{n}W_{n}X \\ ĨĨ(H_{∗}X) & H_{∗}X \ar[l]_{∼} \ar[u]}

Corollary 8.7. If X and W_{n}X have finitely presented homology and the maps

ι^{n}(X): Ĩ(H_{∗}X) → H_{∗}W_{n}X,  and  ι^{n}(W_{n}X): Ĩ(H_{∗}W_{n}X) → H_{∗}W_{n}W_{n}X

are isomorphisms, then the the map X → W_{n}W_{n}X induces an isomorphism in homology.

Proof of Proposition 8.6. Choose resolutions X → HC_{0} → HC_{1} and W_{n}X → HD_{0} → HD_{1}. This leads to a sequence

HĎ_{1} → HĎ_{0} → W_{n}W_{n}X → HC_{0} → HC_{1}.

The diagonal map in the diagram is the induced map

Cok (H_{∗}HĎ_{1} → H_{∗}HĎ_{0}) → Ker (H_{∗}HC_{0} → H_{∗}HC_{1}) .

It is straightforward to check commutativity of the diagram.

8.8. A finiteness result.

Proposition 8.9. Let X be an fp-spectrum with fptype(X) = n, and let Y = L_{n}^{f}X be its finite localization. Then for each k ∈ ℤ the homotopy group π_{k}Y has the form

π_{k}Y ≈ F_{k} ⊕ ℤ_{p}^{⊕a_{k}} ⊕ (ℚ/ℤ_{(p)})^{⊕b_{k}} ⊕ ℚ_{p}^{⊕c_{k}},

where F_{k} is a finite p-group, a_{k} = 0 = c_{k} for all sufficiently small k ≪ 0, and b_{k} = 0 = c_{k} for all sufficiently large k ≫ 0.

Proof. There is a fiber sequence C_{n}^{f}X → X → L_{n}^{f}X, and the Brown-Comenetz dual W_{n}X of C_{n}^{f}X is an fp-spectrum by (8.2). Thus X is connective with π_{k}X ≈ F_{k}⊕ℤ_{p}^{m_{k}}, and C_{n}^{f}X is coconnective with π_{k}C_{n}^{f}X ≈ F′_{k}⊕(ℚ/ℤ_{(p)})^{n_{k}}. The image and coimage of the connecting map π_{k}C_{n}^{f}X → π_{k}X can only be a finite torsion group; thus, to prove the result for π_{k}L_{n}^{f}X we need to show that, in a group extension of the form

0 → F ⊕ ℤ_{p}^{m} → M → F′ ⊕ (ℚ/ℤ_{(p)})^{n} → 0,

where F and F′ are finite p-groups, that M is as described in the statement of the proposition.

It is easy to reduce to the case when F = 0 = F′. Then extensions are classified by elements of Ext((ℚ/ℤ_{(p)})^{n}, ℤ_{p}^{m}) ≈ hom(ℤ_{p}^{n}, ℤ_{p}^{m}); if A ∈ hom(ℤ_{p}^{n}, ℤ_{p}^{m}) classifies the extension then

M ≈ Cok (ℤ_{p}^{n} ⟶^{(A,I)} ℤ_{p}^{m} ⊕ ℚ_{p}^{n})

where I: ℤ_{p}^{n} → ℚ_{p}^{n} is the standard inclusion. It follows that M ≈ ℤ_{p}^{m}/ im A ⊕ ℚ_{p}^{n}/ ker A; this can be shown by choosing a map B: ℤ_{p}^{m} → ℚ_{p}^{n} such that (I − BA): ℤ_{p}^{n} → ℚ_{p}^{n} projects to the kernel of A ⊗ ℚ, in which case there is an exact sequence

0 → ℤ_{p}^{n} ⟶^{(A,I)} ℤ_{p}^{m} ⊕ ℚ_{p}^{n} ⟶^{(x,y)→(x,y−Bx)} ℤ_{p}^{m}/ im A ⊕ ℚ_{p}^{n}/ ker A → 0

which realizes the splitting of M. Now the result follows from the fact that ℤ_{p}^{m}/ im A ≈ F ⊕ℤ_{p}^{a} and ℚ_{p}^{n}/ ker A ≈ (ℚ/ℤ_{(p)})^{b}⊕ℚ_{p}^{c}, where F is a finite p-group.

8.10. Duality for fp-spectra. Recall that there exist natural transformations L_{n+1}^{f}X → L_{n}^{f}X, and hence natural transformations W_{n}X → W_{n+1}X. We define

W X = hocolim_{n→∞} W_{n}X.

If X is an fp-spectrum with fptype(X) = n, then (7.1) shows that L_{m}^{f}X ≈ L_{n}^{f}X for m ≥ n, and thus W_{m}X ≈ W_{n}X for m ≥ n. This, together with (8.3) and the fact that C = ⋃_{n} C_{n} is the homotopy category of all fp-spectra, gives

Theorem 8.11. The functor W induces an equivalence of categories W: C^{op} → C, and H_{∗}W X ≈ Ĩ(H_{∗}X) for all X in C.

9. Calculations

In this section we compute W X in several cases, and thus implicitly compute L_{n}^{f}X (and L_{n}X, by (7.2)) for sufficiently large n.

Lemma 9.1. If X is a ring spectrum and Y is an X-module spectrum, then W_{n}Y is also an X-module spectrum.

Proof. This is a formal consequence of the fact that W_{n}Y ≈ F(Y, W_{n}S^{0}).

Proposition 9.2. If X is an fp-spectrum which is a ring spectrum, H_{∗}X is selfdual as a finitely-presented comodule (i.e., Ĩ(H_{∗}X) ≈ Σ^{d}H_{∗}X for some d), and H_{∗}X does not split over the Steenrod algebra, then W X ≈ Σ^{d}X.

Proof. Choose a map S^{d} → W X which hits the bottom homology class, dual to the unit in H_{∗}X. By (9.1) this map extends to a map Σ^{d}X → W X of X-module spectra, and this map is necessarily an isomorphism on mod p homology, and hence an equivalence.

Corollary 9.3. We have that

1. W BP ⟨n⟩ ≈ Σ^{e(n)}BP ⟨n⟩, where e(n) = 2\frac{p^{n+1}−1}{p−1} − (n + 1),

2. W bu ≈ Σ^{4}bu (at all primes),

3. W bo ≈ Σ^{6}bo (at all primes), and

4. W eo_{2} ≈ Σ^{23}eo_{2} at p = 2 and at p = 3.

Proof. The only case which needs comment is 4. In this case it can be derived from the following facts [5]. At p = 2, eo_{2} ∧ F ≈ BP ⟨2⟩, where F is a certain finite complex with H^{∗}F ≈ DA^{∗}(1), the “double” of A^{∗}(1). At p = 3, eo_{2} ∧ (S^{0} ∪_{α} e^{4} ∪_{2α} e^{8}) ≈ BP ⟨2⟩ ∨ Σ^{8}BP ⟨2⟩.

Remark 9.4. In each of the above examples, we can read off π_{∗}L_{n}X from our knowledge of the homotopy of X. In particular, in each case there is a wide “gap” between the first copy of ℤ_{p} and the last copy of ℚ/ℤ_{(p)} in the homotopy of L_{n}X; if W X ≈ Σ^{d}X, then

π_{s}X ≈ \begin{cases}ℤ_{p} & if s = 0, \\ 0 & if 1 − d < s < 0, and \\ ℚ/ℤ_{(p)} & if s = 1 − d. \end{cases}

There is a convenient heuristic for reading off the expected size of the “gap” in π_{∗}L_{n}X for ring spectra of the above type. If π_{∗}X ⊗ ℚ ≈ ℚ_{p}[x_{1}, …, x_{n}], then the size of the gap is ∑_{i=1}^{n}(|x_{i}| + 1). For example, π_{∗}eo_{2} ⊗ ℚ ≈ ℚ_{p}[x_{8}, x_{12}] at p = 2 or 3, so the gap is (8 + 1) + (12 + 1) = 22. For BP ⟨n⟩ the gap is the same as the dimension of the Toda complex V (n), should it exist.

Recall from (4.10) that if H_{∗}X ≈ A_{∗} ⊗_{A_{∗}(n)} M, then Ĩ(H_{∗}X) ≈ A_{∗} ⊗_{A_{∗}(n)} Σ^{d}M̌, where d is the dimension of the “top cell” of A_{∗}(n).

Proposition 9.5. Let J_{p} denote the connective image-of-J spectrum completed at the prime p.

1. For p odd, W J_{p} ≈ Σ^{3}J_{p}.

2. For p = 2, there is a cofiber sequence Σ^{3}H𝔽_{2} → J_{2} ∧ (S^{0} ∪_{2} e^{1} ∪_{η} e^{3}) → W J_{2}.

Proof. At an odd prime, J_{p} is the fiber of any map BP ⟨1⟩ → Σ^{q}BP ⟨1⟩, q = 2(p−1), which sends the cohomology generator ι ∈ H^{q}Σ^{q}BP ⟨1⟩ to P^{1}ι ∈ H^{q}BP ⟨1⟩. At p = 2, J_{2} is the fiber of any map bo → bspin which sends the cohomology generator ι_{bspin} ∈ H^{4}bspin to Sq^{4}ι_{bo} ∈ H^{4}bo [10]. In either case, the map in cohomology is induced from a map of A^{∗}(2)-modules.

We leave the odd prime case to the reader. Suppose p = 2, and let F = S^{0} ∪_{2} e^{1} ∪_{η} e^{3}. Since bspin ≈ Σ^{7}bo ∧ DF and H^{∗}(F ∧ DF) is a direct sum over the Steenrod algebra of a spherical class in dimension 0 with a free A^{∗}(1) module on one generator, we see that J ∧ F fits in a fiber sequence

J ∧ F → bo ∧ F → Σ^{7}bo ∨ Σ^{4}H𝔽_{2}.

We can kill the copy of H𝔽_{2} by taking the evident cofiber Σ^{3}H𝔽_{2} → J ∧ F → C, so that we obtain a cofiber sequence

C → bo ∧ F → Σ^{7}bo.

On applying W we get a fiber sequence

Σ^{−1}bo → Σ^{6}bo ∧ DF → W C → bo ⟶^{f} Σ^{7}bo ∧ DF.

We can compute the action on f on cohomology, since by (4.10) it is induced from a map of A^{∗}(2)-modules. The computation, which is straightforward, shows that the bottom class of Σ^{7}bo ∧ DF hits Sq^{4}ι_{bo} ∈ H^{4}bo, and thus W C ≈ J.

Corollary 9.6 (Hopkins).

I(L_{1}S^{0}) ≈ L_{1}(S^{−1} ∪_{2} e^{0} ∪_{η} e^{2}).

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Department of Mathematics, Northwestern University, Evanston, IL 60208

E-mail address: mark@math.nwu.edu
E-mail address: rezk@math.nwu.edu