BOUSFIELD CLASSES OVER A p-GROUP

MARK HOVEY AND JOHN H. PALMIERI

Abstract. Extending work of Benson, Carlson, and Rickard on thick subcategories of finitely generated modules over a p-group, we completely describe the partially ordered set of Bousfield classes of all modules over a p-group: it is isomorphic to the poset of subsets of the prime ideal spectrum of the group cohomology algebra.

1. Introduction

In this note, we classify the Bousfield classes for modules over a p-group. Bousfield classes were defined by Bousfield [Bou79] in stable homotopy theory, so in order to make sense of our classification, we first recall what it means to do stable homotopy theory over a group.

Given a finite group G and a field k, we define a category 𝒞(kG) as follows: its objects are unbounded chain complexes of projective kG-modules, and its morphisms are chain homotopy classes of maps. This is a stable homotopy category in the sense of [HPS97]—see [HPS97, 9.5]. One should keep in mind the following analogies:

• The tensor product of chain complexes plays the role of the smash product, so we will write X ∧ Y for X ⊗ Y.

• The trivial module k (or its projective resolution, which we denote by S) plays the role of the sphere spectrum. As an example of this analogy, in the appropriate contexts, k and the sphere spectrum are each the unit of the smash product. Also, when working over a p-group, k is the only simple module, and so one can “build” every object of 𝒞(kG) out of S, in the same way that one can build every spectrum out of the sphere spectrum.

• Given two kG-modules M and N with projective resolutions X and Y, then the graded homotopy classes of maps from X to Y, written [X, Y]_{∗}, is isomorphic to Ext_{kG}^{∗}(M, N); hence the group cohomology Ext_{kG}^{∗}(k, k) corresponds to π_{∗}S^{0}.

• The direct sum of objects in 𝒞(kG) corresponds to the wedge of spectra, so

we use ∨ to mean ⊕. This category is most interesting, from a stable homotopy theoretic point of view, if the characteristic of k divides the order of G.

We also define the stable category of kG-modules, StMod(kG), as follows: the objects are kG-modules, and the morphisms \underline{Hom} are given by

\underline{Hom}_{kG}(M, N) = Hom_{kG}(M, N)/ ≃,

Date: March 18, 1998.

1991 Mathematics Subject Classification. 20C05, 20J05, 18E30, 18G35, 55P42.

where f ≃ g if f − g factors through a projective module. This is also a stable homotopy category, and it can be obtained from 𝒞(kG) by Bousfield localization (of a particularly nice sort—a finite, hence smashing, localization). See [HPS97, 9.6]. As a result, one can often translate results between the two categories 𝒞(kG) and StMod(kG), so one should work in whichever is more convenient. From the stable homotopy theoretic point of view, 𝒞(kG) is more convenient than StMod(kG), because the “homotopy groups of the sphere” is group cohomology in the former—a Noetherian k-algebra—and is Tate cohomology in the latter—usually non-Noetherian, and in general not as pleasant a ring as H^{∗}(G; k). On the other hand, the objects of StMod(kG) are kG-modules, and hence are more familiar than the objects of 𝒞(kG).

For the remainder of the paper, we assume that G is a finite p-group, and k is an algebraically closed field of characteristic p. With these hypotheses, the trivial module k is the only simple module, and hence both 𝒞(kG) and StMod(kG) are monogenic stable homotopy categories [HPS97, 1.1.4]; in other words, there is only one sort of “sphere object” from which every other object is built. (For a general finite group G, each simple kG-module would give a different sort of sphere.) Since the group cohomology algebra of a finite group is Noetherian, then 𝒞(kG) is a Noetherian stable homotopy category. See [HPS97, Section 6] for results about general Noetherian stable homotopy categories.

Let 𝒮 be either of these categories, or indeed any monogenic stable homotopy category. Given an object X of 𝒮, we say that an object W is X-acyclic if X∧W = 0 in 𝒮. (In StMod(kG), this means that the module X ⊗ W is projective; in 𝒞(kG), this means that the chain complex X ⊗W is chain homotopy equivalent to the zero chain complex.) We define the Bousfield class of X, ⟨X⟩, to be the collection of X-acyclic objects. We partially order the Bousfield classes by reverse inclusion: we say that ⟨X⟩ ≥ ⟨Y ⟩ if X ∧W = 0 implies Y ∧W = 0. Note that the sphere object S has the maximal Bousfield class (it is the unit of the smash product, and hence has no nonzero acyclics), and the zero object has the minimal Bousfield class. Given a set {X_{α}} of objects in 𝒮, we define ⋁⟨X_{α}⟩ to be ⟨⋁ X_{α}⟩; one can see that this is well-defined, and also that it is the least upper bound, with respect to this partial ordering, of the classes ⟨X_{α}⟩. Indeed, the collection of acyclics for the wedge is the intersection of the collections of acyclics for each object. The smash product of two Bousfield classes is defined similarly: ⟨X⟩ ∧ ⟨Y ⟩ = ⟨X ∧ Y ⟩; it is a lower bound for ⟨X⟩ and ⟨Y ⟩, but need not be the greatest lower bound.

We let B be the collection of all Bousfield classes. In stable homotopy theory, an understanding of B would be a very useful, and also very hard; for instance, such an understanding would encompass a classification of thick subcategories of the finite objects—knowing the thick subcategories of finites classifies the Bousfield classes which contain finite objects. The classification of thick subcategories in the ordinary stable homotopy category, due to Hopkins and Smith [HS, Hop87], is itself a deep result, and a complete description of B would greatly generalize this.

Let Spec be the prime ideal spectrum of the (graded) commutative Noetherian kalgebra Ext_{kG}^{∗}(k, k) = π_{∗}S. Since 𝒞(kG) is a Noetherian stable homotopy category, for each prime ideal 𝔭 in Spec, one can construct an object κ(𝔭) of 𝒞(kG). These objects have several nice properties; we give a construction and the properties in Section 2. In [BCR96], Benson, Carlson, and Rickard define the variety V_{G}(M) associated to any kG-module M, with projective resolution X, to be the following subset of Spec:

V_{G}(M) = V_{G}(X) = {𝔭 ∈ Spec | κ(𝔭) ∧ X ≠ 0}.

One can extend this to define varieties for any object of 𝒞(kG), in the obvious way.

These three authors have also given a classification of the thick subcategories of finitely generated kG-modules in [BCR97]. Let stmod(kG) be the full subcategory of StMod(kG) consisting of finitely generated kG-modules; their result says that the thick subcategories of stmod(kG) are in one-to-one correspondence with the set of nonempty subsets of Spec which are closed under specialization (T ⊆ Spec is closed under specialization if whenever 𝔭 ∈ T and 𝔭 ⊆ 𝔮, then 𝔮 ∈ T; i.e., T is a union of Zariski closed sets). More precisely, a thick subcategory 𝒞 corresponds to the set ⋃_{X∈𝒞} V_{G}(X), and a subset T of Spec corresponds to the thick subcategory of all finite objects X so that V_{G}(X) ⊆ T.

One can work in 𝒞(kG) instead of StMod(kG) and get a similar result, with the thick subcategories of “finite objects” corresponding to all subsets of Spec which are closed under specialization, not just the nonempty subsets. (See [HPS97, 9.6.8].) This theorem bears a striking resemblance to Hopkins’ classification in [Hop87] of thick subcategories of finite objects in the unbounded derived category of a Noetherian ring (see [Nee92] for a corrected version, and see [Tho97] for a generalization to non-Noetherian rings). In particular, thick subcategories of finite objects in the derived category of R are in one-to-one correspondence with subsets of Spec(R) which are closed under specialization, where the correspondence is as above. It would be very interesting to see an explicit bijection between the thick subcategories of finitely generated kG-modules and the thick subcategories of finite objects in the derived category of the Noetherian ring Ext_{kG}^{∗}(k, k).

Here is our main result.

Theorem 1.1. The partially ordered set B of Bousfield classes of 𝒞(kG) is isomorphic to the Boolean algebra of all subsets of Spec. More precisely, for any object X, we have

⟨X⟩ = ⋁_{𝔭∈V_{G}(X)}⟨κ(𝔭)⟩.

This generalizes the (simpler) computation that

⟨S⟩ = ⋁_{𝔭∈Spec}⟨κ(𝔭)⟩.

This formula holds in any Noetherian stable homotopy category—see [HPS97, 6.1.9]. One can also see that κ(𝔭) ∧ κ(𝔮) = 0 if 𝔭 ≠ 𝔮. Together with Strickland, we conjecture in [HPS97, 6.1.3] that the theorem holds in any Noetherian stable homotopy category, as well.

In passing from 𝒞(kG) to StMod(kG), one localizes away from the thick subcategory generated by the object kG; in other words, one kills off this thick subcategory. The Bousfield class ⟨kG⟩ is the unique minimal nonzero class in B, and as a result we have the following.

Corollary 1.2. The Bousfield lattice in StMod(kG) is isomorphic, as a poset, to the lattice of all nonempty subsets of Spec, with the correspondence as in the theorem.

We also have these corollaries.

Corollary 1.3. The smash product gives the greatest lower bound in B.

Corollary 1.4. Every Bousfield class ⟨X⟩ is complemented: there is a class ⟨Y ⟩ so that ⟨X⟩ ∨ ⟨Y ⟩ = ⟨S⟩ and ⟨X⟩ ∧ ⟨Y ⟩ = ⟨0⟩.

An object R of 𝒞(kG) is a ring object if there is a product map R ∧ R ⟶ R and a unit map S ⟶ R making the appropriate associativity and unital diagrams commute. Because each κ(𝔭) is Bousfield equivalent to a ring object, we have the following.

Corollary 1.5. Every object in 𝒞(kG) is Bousfield equivalent to a ring object.

We say that a ring object R of 𝒞(kG) detects nilpotence if it satisfies the following condition:

• For every ring object T, an element α ∈ Ext_{kG}^{∗}(k, T) is nilpotent if and only

if R_{∗}α ∈ Ext_{kG}^{∗}(k, R ∧ T) is nilpotent. For instance, Carlson’s extension in [Car81] of the Quillen-Venkov nilpotence theorem [QV72] says that the elementary abelian p-subgroups of G detect nilpotence. (Rather, the associated ring objects in 𝒞(kG), namely the projective resolutions of k[G/E], do.) One can see in this case that

⟨S⟩ = ⋁_{E}⟨k[G/E]⟩.

(See [HPS97, 9.6.10].) Indeed, this is just a restatement of Chouinard’s theorem [Cho76]. Whenever an object detects nilpotence, then it detects every ring object. Since every object is Bousfield equivalent to a ring object, we have the following.

Corollary 1.6. If a ring object of 𝒞(kG) detects nilpotence, then it is Bousfield equivalent to the sphere object, S.

All of this is much simpler than the situation with the ordinary stable homotopy category. In that setting, there is no known complete description of B, but the following is known:

• There are nonzero spectra X with X ∧X = 0; hence ∧ cannot be the greatest lower bound. The standard example is when X = I is the Brown-Comenetz dual of the sphere; see [Bou79, 2.5].

• There are spectra whose Bousfield classes are not complemented; ⟨I⟩ is an example, as is ⟨HF_{p}⟩. See [Bou79, 2.7].

• There are spectra which are not Bousfield equivalent to ring spectra, such as I. (If R is a ring spectrum, then one can show that ⟨R⟩ = ⟨R⟩ ∧ ⟨R⟩.)

• There are nontrivial nilpotence theorems. For example, the spectrum BP detects nilpotence, as does the wedge of the Morava K-theories, ⋁ K(n). Their Bousfield classes satisfy

⟨S^{0}⟩ > ⟨BP ⟩ > ⋁⟨K(n)⟩.

In summary, the situation over a p-group is much simpler than in the ordinary stable homotopy category. We would be interested in seeing examples of stable homotopy categories which are more complicated than 𝒞(kG) but in which one can still compute everything; as the examples get more complicated, one may find new approaches to use for the ordinary stable homotopy category. The categories 𝒞(A) for infinite-dimensional Hopf algebras A, as well as the derived categories for non-Noetherian rings, are likely candidates.

Despite the classification of Theorem 1.1, the discussion of Bousfield classes for p-groups is not completely settled. Given an object X, its Bousfield class is the collection of X-acyclic objects, and this collection of objects is an example of a localizing subcategory: a thick subcategory which is closed under coproducts.

Conjecture 1.7. Every localizing subcategory is a Bousfield class. Hence there is a bijection between the set of localizing subcategories and the set of subsets of Spec.

This conjecture holds in the unbounded derived category of a Noetherian ring— see [Nee92]—and given the apparent formal similarities between that category and 𝒞(kG), this conjecture seems likely. See [HPS97, Section 6] for a discussion of this and related issues.

2. Proofs

Given the point of view in [HPS97, Section 6] and the results in [BCR96, Section 10], Theorem 1.1 is easy to prove.

First we construct the objects κ(𝔭). Given any thick subcategory 𝒜 of finite objects in a monogenic stable homotopy category 𝒮, there is a functor L_{𝒜}^{f} : 𝒮 ⟶ 𝒮, called finite localization away from 𝒜, which satisfies the following:

• There is a natural transformation η: 1 ⟶ L_{𝒜}^{f} .

• The induced map L_{𝒜}^{f} ⟶ L_{𝒜}^{f} ∧ L_{𝒜}^{f} , obtained by smashing the natural transformation η with L_{𝒜}^{f} , is an equivalence.

• For any object X, the natural map X ∧ L_{𝒜}^{f} S ⟶ L_{𝒜}^{f} X is an isomorphism.

• For a finite object X, L_{𝒜}^{f} X = 0 if and only if X is in 𝒜.

• Given two thick subcategories 𝒜 ⊆ ℬ, there is a natural transformation L_{𝒜}^{f} ⟶ L_{ℬ}^{f} , compatible with the other structure.

These functors were constructed by Miller [Mil92] in the ordinary stable homotopy category; see [HPS97, 3.3.3] for the general approach. One can use these functors to produce new categories or to produce new objects in existing categories. For example, given the category 𝒞(kG), let 𝒜 be the thick subcategory generated by kG, and consider the full subcategory of 𝒞(kG) of objects of the form L_{𝒜}^{f} X. This category is equivalent to StMod(kG)—see [HPS97, 9.6.4]. As a second example, in the category StMod(kG), for each thick subcategory 𝒜, applying the functor L_{𝒜}^{f} to the trivial module k gives an idempotent module L_{𝒜}^{f} k; see [Ric97] for another construction of these modules.

Given a Noetherian stable homotopy category and a prime ideal 𝔭 of the homotopy groups of spheres, define thick subcategories

(1) 𝒜(𝔭) = {X finite | (π_{∗}X)_{𝔮} = 0 for all 𝔮 ≤ 𝔭},

(2) 𝒜(< 𝔭) = {X finite | (π_{∗}X)_{𝔮} = 0 for all 𝔮 < 𝔭}.

Here, π_{∗}X denotes the set of graded maps from the sphere object to X; the subscript 𝔮 denotes localization at the indicated ideal. Since 𝒜(𝔭) ⊆ 𝒜(< 𝔭), there is a natural transformation between the associated finite localization functors. We define κ(𝔭) to be the fiber of L_{𝒜(𝔭)}^{f} S ⟶ L_{𝒜(<𝔭)}^{f} S. These objects have also been also called M_{𝔭}S; one can show that they are Bousfield equivalent to ring objects—see [HPS97, 6.0.8, 6.1.8].

Proposition 2.1. In the category 𝒞(kG), the objects κ(𝔭) have the following properties:

(a) ⟨S⟩ = ⋁_{p}⟨κ(𝔭)⟩.

(b) If p ≠ q, then κ(𝔭) ∧ κ(𝔮) = 0.

(c) κ(𝔭) ∧ X ∧ Y = 0 if and only if κ(𝔭) ∧ X = 0 or κ(𝔭) ∧ Y = 0.

(d) Hence ⟨κ(𝔭)⟩ is a minimal nonzero Bousfield class.

Proof. Parts (a) and (b) are formal; they hold because the homotopy groups of the sphere (i.e., the group cohomology ring) form a Noetherian ring: see [HPS97, 6.1.9].

Part (c) is the content of [BCR96, 10.3 and 10.8]. For part (d), assume that ⟨0⟩ < ⟨X⟩ ≤ ⟨κ(𝔭)⟩. Then by (b), X ∧ κ(𝔮) = 0 when

q ≠ p, so smashing the equality in part (a) with X gives

⟨X⟩ = ⋁_{q}⟨X ∧ κ(𝔮)⟩ = ⟨X ∧ κ(𝔭)⟩.

Since ⟨X ∧κ(𝔭)⟩ = ⟨X⟩ > ⟨0⟩, then part (c) implies that ⟨κ(𝔭)⟩ = ⟨X ∧κ(𝔭)⟩ = ⟨X⟩. This proves (d).

The theorem now follows from [HPS97, 3.7.5].

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Department of Mathematics, Wesleyan University, Middletown, CT 06459

E-mail address: hovey@member.ams.org

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556

E-mail address: palmieri@member.ams.org