TATE COHOMOLOGY LOWERS CHROMATIC BOUSFIELD CLASSES

MARK HOVEY AND HAL SADOFSKY

December 7, 1994

Abstract. Let G be a finite group. We use the results of [5] to show that the Tate homology of E(n) local spectra with respect to G produces E(n − 1) local spectra. We also show that the Bousfield class of the Tate homology of L_{n}X (for X finite) is the same as that of L_{n−1}X.

To be precise, recall that Tate homology is a functor from G-spectra to G-spectra. To produce a functor P_{G} from spectra to spectra, we look at a spectrum as a naive G-spectrum on which G acts trivially, apply Tate homology, and take G-fixed points. This composite is the functor we shall actually study, and we’ll prove that ⟨P_{G}(L_{n}X)⟩ = ⟨L_{n−1}X⟩ when X is finite.

When G = Σ_{p}, the symmetric group on p letters, this is related to a conjecture of Hopkins and Mahowald (usually framed in terms of Mahowald’s functor RP_{−∞}(−)).

1. Introduction

We briefly recall the spectra that occur in Lin’s proof of the Segal conjecture for the group Z/(2). Embed Z/(p) into S^{1} and look at the pullback of the tautological (complex) line bundle over BS^{1} as a bundle over BZ/(p). Call this bundle ξ.

We denote by P_{−2k} the spectrum given by the Thom spectrum (BZ/(p))^{−kξ} when p = 2, or when p is odd the summand of that spectrum corresponding to BΣ_{p}. (We refer the reader to [18] for the definition of a Thom spectrum associated to a virtual bundle). P_{−2k} has a cell in every dimension ≥ −2k and is the spectrum freqeuently called RP_{−2k}^{∞} when p = 2. When p is odd, P_{−2k} has a cell in every dimension congruent to 0 or −1 modulo q = 2p − 2 and ≥ −2k. P_{−2k} is the same as the spectrum denoted P_{−2k}^{∞} in [17], and constructed there by James periodicity rather than via Thom spectra.

Lin’s theorem [10] (Gunarwardena’s theorem when p > 2 [1]) states that

lim_{⟵}_{k} (P_{−2k} ∧ X) = Σ^{−1}X_{p}ˆ (1)

when X is a finite spectrum. This inverse system of spectra (with some minor alterations) is also what is used to define the root invariant (see [11] for p = 2, or more generally, [17]). As shorthand, we write P_{−∞}(X) for lim_{⟵}_{k} (P_{−2k} ∧ X).

1991 Mathematics Subject Classification. 55P60, 55P42; Secondary 55N91.

The authors were partially supported by the NSF.

Mahowald and Ravenel [12] have conjectured a relationship between chromatic periodicity and Mahowald’s root invariant. There is a related conjecture by Hopkins and Mahowald that is more closely related to our concerns in this paper. Denote Bousfield localization with respect to E(n) by L_{n}. They conjecture that

P_{−∞}(L_{n}X) = Σ^{−2}L_{n−1}X_{p}ˆ ∨ Σ^{−1}L_{n−1}X_{p}ˆ (2)

for X finite. (A word to the experts: this conjecture is connected with Hopkins’s chromatic splitting conjecture (see [7]) and at the time (2) was conjectured the chromatic splitting conjecture was in too simple a form. In light of its current corrected form as in [7], (2) is probably also too optimistic, though We expect it is true as stated when X has type n − 1.)

Greenlees and May put the P_{−∞} construction in a more general context in [4]. There they define the Tate G spectrum associated to a G spectrum X, t_{G}(X). We are not concerned with equivariant spectra here, but we use t_{G} to construct a functor from (ordinary) spectra to spectra. We abuse notation and write i_{∗} for the functor which is the composite of the inclusion of ordinary spectra into the category of naive G-spectra (as the objects on which G acts trivially) with the left adjoint of the forgetful functor from G-spectra to naive G-spectra. So i_{∗} is a functor from ordinary spectra to G-spectra. Our other functor is the G-fixed point functor, (−)^{G} which goes from G-spectra to spectra. We refer the reader to [9] for details. We define

P_{G}(X) = t_{G}(i_{∗}X)^{G}.

Then [4, 16.1] shows that

P_{Σ_{p}}(X) = P_{−∞}(ΣX).

(the left hand side needs to be localized at p when p is odd).

Henceforth we will only be concerned with the case where G is a finite group. We now have a family of functors, one for each finite group G. Our main theorem is the following.

Theorem 1.1. Let X be a finite spectrum. Then ⟨P_{G}(L_{n}X)⟩ = ⟨L_{n−1}X⟩

Here ⟨X⟩ means the Bousfield class of the spectrum X as given in [2].

We also give a result about complex oriented v_{n}-periodic spectra.

Theorem 1.2. If E is Landweber exact and v_{n}-periodic then P_{Z/(p)}E is Landweber exact and v_{n−1}-periodic. It follows generally that ⟨P_{G}(E)⟩ = ⟨E(n − 1)⟩.

Our proofs rely on [5, Theorem 1.1] which implies that P_{G}(K(n)) ≃ ∗. We also use two other results that are relatively well known. We use Ravenel’s Proposition 1.34 from [14]:

⟨X⟩ = ⟨C(f)⟩ ∨ ⟨T el(f)⟩ (3)

where f is a self map of X, C(f) is the cofiber and T el(f) is the mapping telescope. Finally, we use a theorem of Hopkins and Ravenel from [16] to show that P_{G}(K(n)) ≃ ∗ implies P_{G}(L_{n}X) ≃ ∗ when X is finite type n.

Using the interesting results of Mahowald and Shick in [13] one can show that P_{Z/(2)}(T el(X)) ≃ ∗ where X is finite type n and T el(X) is the infinite mapping telescope of X under a v_{n} map. One can use this to deduce our theorems in the special case G = Z/(2). Chun-Nip Lee has done this independently [8].

2. There is a finite type n spectrum X with L_{n}X K(n)-nilpotent

Recall that X is said to be E-prenilpotent if L_{E}X is E nilpotent; that is if L_{E}X can be built up from cofibrations in a finite number of stages with cofibers retracts of spectra of form E ∧ Z. By [16, 8.3] there is a finite type 0 spectrum Y that is L_{n}BP -prenilpotent.

By [16, Lemma 8.1.4]

⟨L_{n}BP ⟩ = ⟨v_{n}^{−1}BP ⟩

which is in turn equal to ⟨E(n)⟩ by [14]. So L_{n}Y is L_{n}BP -nilpotent, which gives a sequence of spectra

∗ = Y_{0} → Y_{1} → Y_{2} → ⋯ → Y_{r} = L_{n}Y

such that cofiber(Y_{i−1} → Y_{i}) is a retract of L_{n}BP ∧ Z_{i} for some Z_{i}. Now let M be a finite type n spectrum with

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

and such that M is a ring spectrum. (See [3] for the existence of such ring spectra.) Then BP ∧ M = BP/(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}), so by [15, Theorem 1],

L_{n}BP ∧ M = v_{n}^{−1}BP ∧ M.

But v_{n}^{−1}BP ∧M is made out of finitely many cofibrations with cofiber v_{n}^{−1}BP/I_{n} = B(n). Now by [19, Remark 6.19], B(n) = K(n) ∧ B for some B, so it follows that L_{n}BP ∧ M is K(n)-nilpotent.

From this we see that L_{n}BP ∧Z_{i}∧M is K(n)-nilpotent, and deduce the following lemma.

Lemma 2.1. There is a finite type n spectrum F with L_{n}F K(n)-nilpotent.

Proof. Take F = Y ∧ M.

3. P_{G}(L_{n}X ∧ F) ≃ ∗ if F is type n.

We recall from [5] that t_{G}(i_{∗}K(n)) ≃ ∗ as a G-spectrum. It follows that P_{G}(K(n)) ≃ ∗. We record the following lemma.

Lemma 3.1. If X is K(n)-nilpotent then P_{G}(X) ≃ ∗.

Proof. First note that since P_{G} takes cofibrations to cofibrations, it suffices to prove that P_{G}(K(n) ∧ Z) ≃ ∗ for any Z. But P_{G}(R) is a ring spectrum when R is a ring spectrum, and P_{G}(N) is a module spectrum over P_{G}(R) if N is a module spectrum over R [4, Proposition 3.5]. It follows that P_{G}(K(n) ∧ Z) ≃ ∗.

Remark: The same proof shows that t_{G}(X) ≃ ∗ equivariantly if X is i_{∗}K(n)- nilpotent in the category of G-spectra.

Corollary 3.2. If F is finite type n, then P_{G}(L_{n}X ∧ F) ≃ ∗ for any spectrum X.

Proof. Let C be the category of finite spectra F such that P_{G}(L_{n}X ∧ F) ≃ ∗ for all spectra X. C is a thick subcategory in the sense of [6]. It follows that if C ∩ C_{n} ≠ ϕ then C_{n} ⊆ C. We recall that if a spectrum Y is K(n)-nilpotent, so is X ∧ Y for any spectrum X. Since L_{n}X ∧ F = X ∧ L_{n}F (L_{n} is smashing) by Lemma 3.1 and Lemma 2.1, C ∩ C_{n} ≠ ϕ.

Remark: Since L_{n}F = L_{K(n)}F when F is finite type n, one might ask when P_{G}(L_{K(n)}X) ≃ ∗. While we don’t know the most general answer, this does not hold in general for X finite. Using the methods of this paper, one can easily check that if X is finite, ⟨P_{G}(L_{K(n)}X)⟩ = ⟨L_{n−1}X⟩.

4. P_{G}(L_{n}X) is E(n − 1)-local.

We use equation (3) inductively. We get

⟨P_{G}L_{n}X⟩ = ⟨p^{−1}P_{G}L_{n}X⟩ ∨ ⟨P_{G}L_{n}X ∧ M(p^{i_{0}})⟩
= ⟨p^{−1}P_{G}L_{n}X⟩ ∨ ⟨v_{1}^{−1}P_{G}L_{n}X ∧ M(p^{i_{0}})⟩ ∨ ⋯

∨⟨v_{n−1}^{−1}P_{G}L_{n} ∧ M(p^{i_{0}}, …, v_{n−2}^{i_{n−2}})⟩ ∨ ⟨P_{G}L_{n}X ∧ M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}})⟩.

Now since the P_{G}L_{n}X ∧ M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}) ≃ ∗ by Corollary 3.2, we get

⟨P_{G}L_{n}X⟩ = ⟨p^{−1}P_{G}L_{n}X⟩ ∨ ⟨v_{1}^{−1}P_{G}L_{n}X ∧ M(p^{i_{0}})⟩ ∨ ⋯
∨⟨v_{n−1}^{−1}P_{G}L_{n} ∧ M(p^{i_{0}}, …, v_{n−2}^{i_{n−2}})⟩.

Since P_{G}L_{n}X is an L_{n}S^{0} module, it follows that

v_{j}^{−1}P_{G}L_{n}X ∧ M(p^{i_{0}}, …, v_{j−1}^{i_{j−1}}) = P_{G}L_{n}X ∧ v_{j}^{−1}L_{n}M(p^{i_{0}}, …, v_{j−1}^{i_{j−1}})
= (P_{G}L_{n}X) ∧ L_{j}M(p^{i_{0}}, …, v_{j−1}^{i_{j−1}}).

Since ⟨L_{j}M(p^{i_{0}}, …, v_{j−1}^{i_{j−1}})⟩ = ⟨K(j)⟩, we see that

⟨P_{G}L_{n}X⟩ ≤ ⟨K(0)⟩ ∨ ⋯ ∨ ⟨K(n − 1)⟩ = ⟨E(n − 1)⟩. (4)

Now since L_{n}X is a L_{n}S^{0}-module, P_{G}L_{n}X is a P_{G}L_{n}S^{0} module. But P_{G}L_{n}S^{0} is self local since it is a ring spectrum [14], so by equation (4) P_{G}L_{n}S^{0} is E(n − 1)- local, hence so is P_{G}L_{n}X.

To finish the proof of Theorem 1.1 it remains to show the inequality in equation (4) is actually an equality when X = S^{0}. In section 6 we use Theorem 1.2 to do this.

5. P_{G} Landweber exact v_{n}-periodic theories.

In this section we prove Theorem 1.2. We will suppose that E is a complex oriented homology theory. Without loss of generality, we assume that E is p-local, since we will be applying the functor P_{Z/(p)} which gives the same value on X as on X_{(p)}. Then we can assume E is oriented by a map from BP, so that we can consider v_{i} as an element of E_{∗}. We remind the reader that I_{j} = (p, v_{1}, …, v_{j−1}), and that for BP (and hence for any spectrum oriented from BP),

[p](x) = px +_{F} v_{1}x^{p} +_{F} v_{2}x^{p^{2}} +_{F} ⋯ +_{F} v_{i}x^{p^{i}} +_{F} ⋯

where +_{F} is the sum in the formal group law on E_{∗}.

We begin by remarking that for complex oriented E

π_{∗}P_{Z/(p)}E = E_{∗}((x))/([p](x))

where |x| = −2, E_{∗}((x)) denotes the ring of Laurent series over E_{∗} which have only finitely many terms involving negative powers of x, and [p](x) is the p-series. It follows that when [p](x) is not a zero divisor, we have a short exact sequence

E_{∗}((x))⟶^{·[p](x)}E_{∗}((x)) → π_{∗}P_{Z/(p)}E.

We now assume that E is v_{n}-periodic Landweber exact. We define v_{n}-periodic almost as in [5, Definition 1.3]; E is v_{n}-periodic if v_{n} is a unit on E_{∗}/I_{n} and in addition E_{∗}/I_{n} ≠ 0.

For each j ≤ n, we know that E_{∗}/I_{j} → v_{j}^{−1}E_{∗}/I_{j} is injective by the hypothesis of Landweber exactness. It follows that E_{∗}((x))/I_{j} → v_{j}^{−1}E_{∗}((x))/I_{j} is injective also. Now [p](x) is a unit in v_{j}^{−1}E_{∗}((x))/I_{j} since it is a power series with leading term v_{j}x^{p^{j}}, which is a unit. It follows that

E_{∗}((x))/I_{j} → v_{j}^{−1}E_{∗}((x))/I_{j}⟶^{·[p](x)}v_{j}^{−1}E_{∗}((x))/I_{j}

is injective, hence

E_{∗}((x))/I_{j}⟶^{·[p](x)}E_{∗}((x))/I_{j}

is also.

We examine the diagram of short exact sequences below, in which the bottom row is the cokernel of the map between the top two rows.

0 ⟶ E_{∗}((x))/I_{j} ⟶^{·[p](x)} E_{∗}((x))/I_{j} ⟶ (π_{∗}P_{Z/(p)}E)/I_{j} ⟶ 0
·v_{j} ↓  ↓^{·v_{j}}  ↓^{·v_{j}}
0 ⟶ E_{∗}((x))/I_{j} ⟶^{·[p](x)} E_{∗}((x))/I_{j} ⟶ (π_{∗}P_{Z/(p)}E)/I_{j} ⟶ 0
↓  ↓  ↓
0 ⟶ E_{∗}((x))/I_{j+1} ⟶^{·[p](x)} E_{∗}((x))/I_{j+1} ⟶ (π_{∗}P_{Z/(p)}E)/I_{j+1} ⟶ 0

By the snake lemma applied to the first two rows (together with the observation that the first two vertical maps are injective) we see that v_{j} is injective on (π_{∗}P_{Z/(p)}E)/I_{j}.

We can also see that ·[p](x) is not a unit on E_{∗}((x))/I_{j} unless v_{j} is a unit on E_{∗}, therefore (π_{∗}P_{Z/(p)}E)/I_{j} ≠ 0 unless j = n, and this last observation tells us that v_{n−1} is a unit on (π_{∗}P_{Z/(p)}E)/I_{n−1}.

We conclude that P_{Z/(p)}E is Landweber exact, and that π_{∗}(P_{Z/(p)}E)/I_{n−1} ≠ 0 while π_{∗}(P_{Z/(p)}E)/I_{n} = 0. It follows by using (3) as before (see [7, Corollary 1.12], that

⟨P_{Z/(p)}E⟩ = ⟨E(n − 1)⟩.

By using the maps of complex oriented ring spectra

E → P_{G}E → P_{Z/(p)}E

(when Z/(p) ⊆ G) we also deduce that

⟨P_{G}E⟩ = ⟨E(n − 1)⟩.

6. Proof of Theorem 1.1.

We recall from section 4 that v_{j}^{−1}P_{G}L_{n}S^{0} ∧ M(p^{i_{0}}, …, v_{j−1}^{i_{j−1}}) has the Bousfield class of either a point or of K(j). So to show it has the class of K(j), we need only show that it is not contractible.

We pick i_{0}, …, i_{j−1} so that M(p^{i_{0}}, …, v_{j−1}^{i_{j−1}}) is a ring spectrum. Then we observe that the map

S^{0} → v_{j}^{−1}P_{G}L_{n}S^{0} ∧ M(p^{i_{0}}, …, v_{j−1}^{i_{j−1}}) → v_{j}^{−1}P_{Z/(p)}L_{n}S^{0} ∧ M(p^{i_{0}}, …, v_{j−1}^{i_{j−1}}) →

v_{j}^{−1}P_{Z/(p)}E(n) ∧ M(p^{i_{0}}, …, v_{j−1}^{i_{j−1}})⟶^{=}v_{j}^{−1}P_{Z/(p)}E(n)/(p^{i_{0}}, …, v_{j−1}^{i_{j−1}}) →

v_{j}^{−1}P_{Z/(p)}E(n)/(p, …, v_{j−1})

is the unit of the ring spectrum v_{j}^{−1}P_{Z/(p)}E(n)/(p, …, v_{j−1}). This is non-zero if j < n by Theorem 1.2. So none of the intervening spectra are contractible either.

For arbitrary finite X (instead of S^{0}) just smash with X. Note that ⟨−⟩, P_{G}(−) and localization commute with smashing with a finite spectrum.

Remark: The same proof can be iterated to draw the obvious conclusions about P_{G}^{k}(L_{n}S^{0}).

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Department of Mathematics, M.I.T., Cambridge, MA 02139 E-mail address: hovey@math.mit.edu

Department of Mathematics, Johns Hopkins University, Baltimore MD 21230 E-mail address: hs@math.jhu.edu