A note on the thick subcategory theorem

Alain Jeanneret University of Bern CH-3012 Bern

              jeanner@math-stat.unibe.ch

Peter S. Landweber ^{∗} Rutgers University New Brunswick, New Jersey 08903 USA

             landwebe@lagrange.rutgers.edu

Douglas C. Ravenel ^{†} University of Rochester Rochester, New York 14627 USA

              drav@troi.cc.rochester.edu

July 14, 1994

1 Introduction

In this paper we will discuss an algebraic version (Theorem 1.6) of the thick subcategory theorem of Hopkins-Smith [HS] (Theorem 1.4). The former is stated as Theorem 3.4.3 in [Rav92], but the proof given there is incorrect. (A list of errata for [Rav92] can be obtained by email from the third author.)

First we recall the nilpotence theorem in its p-local version. Let BP be the Brown-Peterson spectrum at the prime p, which satisfies:

π_{∗}(BP) ≅ BP_{∗} ≅ Z_{(p)}[v_{1}, v_{2}, ⋯], |v_{i}| = 2(p^{i} − 1).

Theorem 1.1 (Nilpotence theorem) [DHS88]

(i) Let R be a p-local ring spectrum. The kernel of the BP Hurewicz homomorphism BP_{∗} : π_{∗}(R) ⟶ BP_{∗}(R) consists of nilpotent elements.

∗Partially supported by the National Science Foundation

†Partially supported by the National Science Foundation

(ii) Let f : F ⟶ X be a map from a p-local finite spectrum to an arbitrary spectrum. If BP ∧ f is null homotopic, then f is smash nilpotent; i.e. the i-fold smash product f^{(i)} = f ∧ ⋯ ∧ f is null for i sufficiently large.

(iii) Let ⋯ ⟶ X_{n} ⟶^{f_{n}} X_{n+1} ⟶^{f_{n+1}} X_{n+2} ⟶ ⋯ be a sequence of p-local

spectra with X_{n} c_{n}-connected. Suppose that c_{n} ≥ mn + b for some m and b. If BP_{∗}f_{n} = 0 for all n then hocolimX_{n} is contractible.

The Baas-Sullivan theory of bordism with singularities allows one to define ring spectra K(n) and P (n) for 0 < n < ∞ satisfying [Rav86]:

π_{∗}(K(n)) ≅ K(n)_{∗} ≅ F_{p}[v_{n}, v_{n}^{−1}]
π_{∗}(P (n)) ≅ P (n)_{∗} ≅ F_{p}[v_{n}, v_{n+1}, ⋯]

as BP_{∗}-algebras. We also set P (0) = BP and K(0) = HQ, the rational Eilenberg-Mac Lane spectrum. K(n) is known as the n^{th} Morava K-theory at the prime p. The following corollary of the nilpotence theorem will be proved in §2. This is stated in [Rav92] as Corollary 5.1.5, but again the proof given there is incorrect.

Corollary 1.2 Let W, X and Y be p-local finite spectra and f : X ⟶ Y. Then W ∧ f^{(k)} is null homotopic for k ≫ 0 if K(n)_{∗}(W ∧ f) = 0 for all n ≥ 0.

Now let CP_{0} be the homotopy category of finite p-local spectra and let CP_{n} ⊂ CP_{0} be the full subcategory of K(n − 1)_{∗}-acyclics. In [Rav84] it was shown that the CP_{n} fit into a sequence:

⋯ ⊂ CP_{n+1} ⊂ CP_{n} ⊂ ⋯ ⊂ CP_{0}.

Moreover all the inclusions are strict [Mit85].

Definition 1.3 A full subcategory C of CP_{0} is thick if:

(i) An object weakly equivalent to an object in C is in C.

(ii) If X ⟶ Y ⟶ Z is a cofibration in CP_{0} and two of {X, Y, Z} are in C then so is the third.

(iii) A retract of an object in C is in C.

Corollary 1.2 is the form of the nilpotence theorem needed to prove the thick subcategory theorem (see §5.3 of [Rav92]):

Theorem 1.4 (Thick subcategory theorem) If C is a thick subcategory of CP_{0}, then there exists an integer k such that C = CP_{k}.

Before we state an algebraic version of Theorem 1.4 let us fix some notation. Let BP_{0} be the abelian category of BP_{∗}(BP)-comodules finitely presented as BP_{∗}-module [Lan76]. A typical object in BP_{0} is BP_{∗}(X) for X in CP_{0}. We denote by BP_{k} the full subcategory of BP_{0} whose objects M satisfy v_{k−1}^{−1}M = 0 (we set v_{0} = p). Results of Johnson-Yosimura [JY80] (see also [Lan79] for a more algebraic proof) show that:

⋯ ⊂ BP_{k+1} ⊂ BP_{k} ⊂ ⋯ ⊂ BP_{0}.

Definition 1.5 Let A be an abelian category. A full subcategory C of A is thick if it satisfies the following condition: If

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0

is a short exact sequence in A, M belongs to C if and only if M′ and M′′ belong to C. (It means that C is stable under subobjects, quotient objects and extensions.)

The classification of the thick subcategories of BP_{0} is now the following; see §3 for the proof.

Theorem 1.6 (Algebraic thick subcategory theorem) If C is a thick subcategory of BP_{0}, then there exists an integer k such that C = BP_{k}.

Let us conclude the introduction with some remarks.

• Theorem 3.4.2 of [Rav92] is the analog of Theorem 1.6 stated in a different category, CΓ, which is defined in terms of MU rather than BP.

• The BP -homology functor, BP_{∗}(·) : CP_{0} ⟶ BP_{0} sends the category CP_{k} into BP_{k}. This comes from the fact [Rav84] that if X ∈ CP_{0} then

K(n)_{∗}(X) = 0 ⟺ v_{n}^{−1}BP_{∗}(X) = 0.

• Theorem 1.6 can be generalized to the abelian category of P (n)_{∗}(P (n))- comodules, finitely presented over P (n)_{∗}, which we denote by P(n). Similarly as for BP_{0} we can define the subcategories P(n)_{k} and prove the following.

Theorem 1.7 If C is a thick subcategory of P(n), then there exists an integer k ≥ n such that C = P(n)_{k}.

A further generalization of Theorem 1.6 can be obtained in the following setting. Let E_{∗} be a commutative P (n)_{∗}-algebra such that E_{∗} ⊗_{P (n)_{∗}} − is an exact functor on P(n). In [Lan76] the second author gave sufficient conditions for exactness. (The necessity of these conditions was shown by Rudyak in [Rud86].) Define

E_{∗}(E) = E_{∗} ⊗_{P (n)_{∗}} P (n)_{∗}(P (n)) ⊗_{P (n)_{∗}} E_{∗};

It can be made into a Hopf algebroid by extending the structure maps for P (n)_{∗}(P (n)). Moreover E_{∗}(E) is a flat E_{∗}-module because P (n)_{∗}(P (n)) is a flat P (n)_{∗}-module and if N is a E_{∗}-module then

E_{∗}(E) ⊗_{E_{∗}} N ≅ E_{∗} ⊗_{P (n)_{∗}} (P (n)_{∗}(P (n)) ⊗_{P (n)_{∗}} N) .

If M is an object of P(n) then E_{∗} ⊗_{P (n)_{∗}} M is an E_{∗}(E)-comodule via the E_{∗}-extension of the composite:

M ⟶ P (n)_{∗}(P (n)) ⊗_{P (n)_{∗}} M
↓
E_{∗}(E) ⊗_{P (n)_{∗}} M  ⟶ E_{∗}(E) ⊗_{E_{∗}} (E_{∗} ⊗_{P (n)_{∗}} M) .

Let E be the category whose objects are E_{∗} ⊗_{P (n)_{∗}} M with M ∈ P(n) and morphisms E_{∗} ⊗ f : E_{∗} ⊗ M_{1} ⟶ E_{∗} ⊗ M_{2} with f : M_{1} ⟶ M_{2} in P(n); then E is an abelian category equipped with an exact functor:

E_{∗} ⊗_{P (n)_{∗}} − : P(n) ⟶ E.

The image of the subcategory P(n)_{k}, written E_{k}, satisfies:

⋯ ⊂ E_{k+1} ⊂ E_{k} ⊂ ⋯ ⊂ E_{n} = E.

We are no longer claiming that the inclusions are strict. The thick subcategories of E can be described as follow:

Theorem 1.8 If C is a thick subcategory of E, then there exists an integer k ≥ n such that C = E_{k}.

It should be emphasized that under the above assumption on E_{∗}, the functor E_{∗}⊗_{P (n)_{∗}}P (n)_{∗}(·) is a homology theory [Lan76] taking its values in the category E as far as finite spectra are concerned.

2 The proof of Corollary 1.2

Let D : CP_{0} ⟶ CP_{0} be the anti-equivalence induced by the Spanier-Whitehead duality [Ada74]. If X ∈ CP_{0} and Y is any spectrum, the graded group [X, Y]_{∗} is isomorphic to π_{∗}(DX ∧Y). We say that the maps f : Σ^{n}X ⟶ Y and f̂: S^{n} ⟶ DX ∧ Y are adjoint if they correspond to each other under the above isomorphism of groups. In particular the adjoint of the identity X ⟶ X is a map e : S^{0} ⟶ DX ∧ X. Recall that X^{(i)} is a notation for the i-fold smash product X ∧ ⋯ ∧ X.

Set R = DW ∧W, a ring spectrum whose unit is e and whose multiplication is the composite

R ∧ R = DW ∧ W ∧ DW ∧ W ⟶^{DW ∧De∧W} DW ∧ S^{0} ∧ W = R.

The map f : X ⟶ Y is adjoint to f̂: S^{0} ⟶ DX ∧ Y and W ∧ f is adjoint to the composite

S^{0} \underrightarrow{_{f}} DX ∧ Y ⟶^{e∧DX∧Y} R ∧ DX ∧ Y,

which we denote by g. Set F = R ∧ DX ∧ Y. The map W ∧ f^{(i)} is adjoint to the composite

S^{0} ⟶^{g^{(i)}} F^{(i)} = R^{(i)} ∧ DX^{(i)} ∧ Y^{(i)} ⟶ R ∧ DX^{(i)} ∧ Y^{(i)},

the latter map being induced by the multiplication in R.

We want to show that W ∧ f^{(k)} is null for large k; by adjointness it suffices to prove that g^{(k)} is null for large k. The second statement of Theorem 1.1 implies that we only need to show that BP ∧ g^{(i)} is null for large i, so we can take k to be an appropriate multiple of i. Let T_{i} = R ∧ DX^{(i)} ∧ Y^{(i)} and let T be the direct limit of

S^{0} ⟶^{g} T_{1} ⟶^{T_{1}∧_{f}} T_{2} ⟶^{T_{2}∧_{f}} T_{3} ⟶ · · · .

The desired conclusion will follow from showing that BP ∧ T is contractible.

At this point we need to use the theory of Bousfield classes. Recall that the Bousfield class of a spectrum X (denoted ⟨X⟩) is the collection of spectra Z for which X ∧ Z is not contractible. In [Rav84] it was shown that

⟨BP ⟩ = ⟨K(0)⟩ ∨ ⟨K(1)⟩ ∨ ⋯ ∨ ⟨K(n)⟩ ∨ ⟨P (n + 1)⟩.

By assumption, K(n) ∧ T is contractible for all n. Therefore it suffices to show that P (m) ∧ T is contractible for large m.

Since we are concerned only with finite spectra, we have for large enough m:

K(m)_{∗}(W ∧ f) = K(m)_{∗} ⊗_{F_{p}} H_{∗}(W ∧ f; F_{p})
P (m)_{∗}(W ∧ f) = P (m)_{∗} ⊗_{F_{p}} H_{∗}(W ∧ f; F_{p}).

Our hypothesis implies that both of these homomorphisms are trivial, so the smash product P (m) ∧ T is contractible as required.

3 The proof of Theorem 1.6

The proof of Theorem 1.6 is a consequence of the filtration theorem of Landweber, namely Theorem 3.1 [Lan73] Each object M ∈ BP_{0} has a filtration

0 = M_{s} ⊂ ⋯ ⊂ M_{1} ⊂ M_{0} = M

in the category BP_{0}, so that for 0 ≤ i ≤ s − 1 the quotient M_{i}/M_{i+1} is stably isomorphic to BP_{∗}/I_{n_{i}} in BP_{0}, where I_{n_{i}} = (p, v_{1}, ⋯, v_{n_{i}−1}) are invariant prime ideals of BP_{∗}. (Stably isomorphic means isomorphic after a dimension shift.)

For M ∈ BP_{0} define Spec(M) = {m ≥ 1: v_{m−1}^{−1}M = 0} ⋃{0} (set as usual v_{0} = p). If M ≠ 0 then Spec(M) is a finite subset of N and is of the form:

Spec(M) = {0, 1, ⋯, N_{M}}

with N_{M} ≥ 0.

Let C be a thick subcategory of BP_{0}. Define an integer k by:

⋂_{M∈C} Spec(M) = {0, 1, ⋯, k}.

From the definition of k, one has C ⊂ BP_{k} and C ⊄ BP_{k+1}. Let M in C be such that

v_{k−1}^{−1}M = 0 and v_{k}^{−1}M ≠ 0,

and let

0 = M_{s} ⊂ ⋯ ⊂ M_{1} ⊂ M_{0} = M

be a Landweber filtration of M. As C is thick and M ∈ C, all the M_{i}’s belong to C as well as all the quotients M_{i}/M_{i+1} ≅ BP_{∗}/I_{n_{i}}.

Localization being an exact functor, all the v_{k−1}^{−1}M_{i} are null and hence v_{k−1}^{−1}M_{i}/M_{i+1} ≅ v_{k−1}^{−1}BP_{∗}/I_{n_{i}} = 0. Therefore

n_{i} ≥ k for 0 ≤ i ≤ s − 1.  (3.2)

On the other hand, v_{k}^{−1}M ≠ 0 implies the existence of a j for which v_{k}^{−1}BP_{∗}/I_{n_{j}} ≠ 0, which forces

n_{j} ≤ k for some j, 0 ≤ j ≤ s − 1.  (3.3)

From (3.2) and (3.3) we obtain that n_{j} = k for some j, 0 ≤ j ≤ s−1, hence BP_{∗}/I_{k} ∈ C. Now it is fairly easy to prove by induction that BP_{∗}/I_{k+l} ∈ C for all l ≥ 0. Consider the exact sequence in BP_{0}

0 ⟶ BP_{∗}/I_{k+l} ⟶^{v_{k+l}} BP_{∗}/I_{k+l} ⟶ BP_{∗}/I_{k+l+1} ⟶ 0

where the first morphism is multiplication by v_{k+l}. The subcategory C being thick, BP_{∗}/I_{k+l} ∈ C implies BP_{∗}/I_{k+l+1} ∈ C.

We are now ready to show the inclusion BP_{k} ⊂ C. Let N be an object in BP_{k} and 0 = N_{s} ⊂ ⋯ ⊂ N_{1} ⊂ N_{0} = N be a Landweber filtration of N. We have seen that v_{k−1}^{−1}N = 0 implies n_{i} ≥ k for all 0 ≤ i ≤ s − 1 with, as usual, n_{i} such that N_{i}/N_{i+1} ≅ BP_{∗}/I_{n_{i}}. By downward induction on i we prove that N_{i} ∈ C. This works as follows.

First N_{s} = 0 ∈ C. Second, the short exact sequence in BP_{0}

0 ⟶ N_{i+1} ⟶ N_{i} ⟶ BP_{∗}/I_{n_{i}} ⟶ 0

is such that N_{i+1} ∈ C (by the inductive assumption) and BP_{∗}/I_{n_{i}} ∈ C as n_{i} ≥ k. From the thickness of C we obtain that N_{i} ∈ C. For i = 0 we have N ∈ C and so BP_{k} = C, as required.

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