ODDBALL BOUSFIELD CLASSES

AMNON NEEMAN

Abstract. We produce a commutative ring R, whose spectrum, Spec(R), is the one point space. Nevertheless, there is a huge set of distinct Bousfield classes in its derived category.

Contents

0. Introduction 1 1. Many Bousfield classes 2 References 5

0. Introduction

Let 𝒯 be a triangulated category, closed with respect to all small coproducts. A full subcategory 𝒮 ⊂ 𝒯 is called localising if it is triangulated, and contains all 𝒯–coproducts of its objects. If 𝒯 has a smash product and E is an object of 𝒯, the subcategory 𝒯_{E} defined by

𝒯_{E} = {x ∈ 𝒯 | x ∧ E = 0}

is a localising subcategory. For many 𝒯’s, the subcategories 𝒯_{E} form a partially ordered set, with respect to the partial ordering given by inclusion. [The fact that this is a set and not a class is a theorem of Ohkawa’s; see [5]]. This is called the set of Bousfield classes. One can, and usually does, identify the points of this partially ordered set with equivalence classes of objects E ∈ 𝒯, where E ≃ E′ if 𝒯_{E} = 𝒯_{E′}. The equivalence class of E is denoted ⟨E⟩.

Now let 𝒯 be the homotopy category of spectra. One of the puzzling observations in homotopy theory, is that the poset of Bousfield classes in 𝒯 is very strange; see Bousfield’s [1]. Ravenel [6] made a number of illuminating conjectures, most of which have now been proved. One, the telescope conjecture, turns out to be very false.

Following ideas of Hopkins [2] and Waldhausen, one draws a parallel between the situation in the homotopy category, and the derived categories of commutative rings. This has now been studied very extensively, in great systematic generality, in the book of Hovey, Palmieri and Strickland [3]. If R is a commutative ring and D(R) is its derived category, and if all the conjectures work out beautifully, then Bousfield classes in D(R) should correspond to subsets of Spec(R). One example in which the conjectures are all true, is when D(R) is the derived category of a Noetherian, commutative ring R. See [4], and also Thomason’s [7].

We produce here an example, of a ring with exactly one prime ideal. That is, the spectrum is the one–point space. But it so turns out, that there are many Bousfield classes. The collection of all Bousfield classes has a cardinality at least that of the Stone-Čech compactification of the integers.

The author would like to thank the referee for pointing out a couple of inaccuracies, and more importantly, for suggesting expository improvements.

1. Many Bousfield classes

Let k be a field. For all integers i > 1, i ∈ ℤ, we will let x_{i} be an indeterminate. Let R be the following commutative ring:

R = \frac{k[x_{2}, x_{3}, x_{4}, ⋯]}{(x_{2}^{2}, x_{3}^{3}, x_{4}^{4}, ⋯)}.

In the ring R, let m be the ideal generated by {x_{i}, i > 1}. Clearly, m is a maximal ideal of R. Equally clearly, every element of m is nilpotent. Every element of m is, after all, a polynomial in finitely many of the x_{i}, each of which is nilpotent. It follows that m is the nilradical of the ring R. Every prime ideal of R contains m. Since m is maximal, it is the unique prime ideal in R. The spectrum, Spec(R) = {m}, is the one–point space.

Now let D(R) be the derived category of R; it is the category of all chain complexes of R–modules. A localising subcategory is a subcategory closed under coproducts and triangles. If E is an object of D(R), one can form the localising subcategory D(R)_{E}, whose objects are the objects of D(R) with a vanishing derived tensor product with E.

Definition 1.1. Let ℕ be the set of integers ≥ 1. Let P_{0} be the set of functions f : ℕ ⟶ ℕ such that, ∀j ∈ ℕ, f(j) ≤ j. We make P_{0} into a partially ordered set by setting

f ≤ g  if  f(j) ≤ g(j), ∀j ≫ 1.

Define f ∼ g if f ≤ g and g ≤ f. Let P = P_{0}/ ∼.

Construction 1.2. Let ℕ be the set of integers ≥ 1. Let f : ℕ ⟶ ℕ, f(j) ≤ j be an element of P_{0}. We consider the R–module

E_{f} = \frac{R}{(x_{2}^{f(2)}, x_{3}^{f(3)}, x_{4}^{f(4)}, ⋯)}.

Lemma 1.3. If f ∼ g in P, then the Bousfield classes satisfy ⟨E_{f}⟩ = ⟨E_{g}⟩.

Proof: Since f ∼ g, it follows that for all j ≫ 1, f(j) = g(j). Pick an integer N ∈ ℕ, so that if j > N, then f(j) = g(j). Define h : ℕ ⟶ ℕ by

h(j) = \{ \begin{matrix}j & if j ≤ N \\ f(j) = g(j) & if j > N \end{matrix}

It clearly suffices to prove that ⟨E_{f}⟩ = ⟨E_{h}⟩ and ⟨E_{g}⟩ = ⟨E_{h}⟩. The two statements being interchangeable, we prove ⟨E_{f}⟩ = ⟨E_{h}⟩.

But now observe that

E_{f} = E_{h} ∧ {R/x_{1}^{f(1)}} ∧ {R/x_{2}^{f(2)}} ∧ ⋯ ∧ {R/x_{N}^{f(N)}}.

So it is enough to notice that, whenever x ∈ R is nilpotent, ⟨E⟩ = ⟨E ∧ {R/x}⟩. □

Lemma 1.4. If f ≤ g in P_{0}, then the Bousfield classes satisfy ⟨E_{g}⟩ ≤ ⟨E_{f}⟩.

Proof: Since ∀j ≫ 1, we have f(j) ≤ g(j), it follows that ∀j ≫ 1, we must have g(j) = max (f(j), g(j)). Replacing g(j) by max (f(j), g(j)) does not change ⟨E_{g}⟩, by Lemma 1.3. We may therefore assume that, for all j ≥ 1, f(j) ≤ g(j).

But now E_{g} is an R–algebra, and E_{f} is an E_{g} algebra. It follows that E_{f} is a retract of E_{f} ∧ E_{g}. If E_{g} ∧ X = 0, then so is E_{f} ∧ E_{g} ∧ X, and hence the retract E_{f} ∧ X. Thus ⟨E_{g}⟩ ≤ ⟨E_{f}⟩. □

Corollary 1.5. There is an order–reversing map of partially ordered sets

ψ : P ⟶ B,

where P = P_{0}/ ∼ is the set of Definition 1.1, and B is the poset of Bousfield classes. The map ψ takes f to ψ(f) = ⟨E_{f}⟩. The fact that ψ is well–defined is Lemma 1.3, and the fact that it is order–reversing is Lemma 1.4.

Proposition 1.6. The map ψ of Corollary 1.5 is injective.

Proof: Let f and g be two functions ℕ ⟶ ℕ. Suppose that f ≁ g. That is, for infinitely many i ∈ ℕ, f(i) ≠ g(i). Interchanging f and g if necessary, we may assume that, for infinitely many i ∈ ℕ, f(i) < g(i). I assert that there exists an object t ∈ D(R)_{E_{f}} − D(R)_{E_{g}}.

Choose an infinite sequence of increasing integers

2 ≤ i_{1} < i_{2} < i_{3} < ⋯

so that f(i_{n}) < g(i_{n}) ≤ i_{n}. We need to construct an object t ∈ D(R), and then prove t ∈ D(R)_{E_{f}} − D(R)_{E_{g}}. We define t to be the mapping telescope (the homotopy colimit) of the sequence

R ⟶^{x_{i_{1}}^{f(i_{1})}} R ⟶^{x_{i_{2}}^{f(i_{2})}} R ⟶^{x_{i_{3}}^{f(i_{3})}} ⋯

Then t ∧ E_{f} is the mapping telescope of the sequence

E_{f} ⟶^{x_{i_{1}}^{f(i_{1})}} E_{f} ⟶^{x_{i_{2}}^{f(i_{2})}} E_{f} ⟶^{x_{i_{3}}^{f(i_{3})}} ⋯

Now on E_{f}, multiplication by x_{i_{n}}^{f(i_{n})} vanishes. The maps in the sequence are all zero, and hence so is the telescope. We deduce t ∧ E_{f} = 0.

On the other hand, t ∧ E_{g} is the telescope of the sequence

E_{g} ⟶^{x_{i_{1}}^{f(i_{1})}} E_{g} ⟶^{x_{i_{2}}^{f(i_{2})}} E_{g} ⟶^{x_{i_{3}}^{f(i_{3})}} ⋯

All the finite composites take 1 ∈ E_{g} to a non–zero element, namely x_{i_{1}}^{f(i_{1})}x_{i_{2}}^{f(i_{2})} ⋯ x_{i_{n}}^{f(i_{n})}. The colimit of the sequence above is non–zero. Hence t ∧ E_{g} ≠ 0. □

Remark 1.7. It is now easy to see that there are at least 2^{ℵ_{0}} distinct Bousfield classes. To see that there are more, one can slightly refine the argument above. Given a subset S ⊂ ℕ, one can define a function f_{S} : ℕ ⟶ ℕ by the rule

f_{S}(i) = \{ \begin{matrix}i & if i ∉ S \\ 1 & if i ∈ S \end{matrix}

Suppose we are given an ultrafilter U in ℕ. That is, U is a family of subsets S ⊂ ℕ, closed under finite intersection and majorization, and maximal with these properties. Define

E_{U} = ∐_{S∈U} E_{f_{S}}.

A slight modification of the argument above, which we leave as an exercise, shows that, given two non–principal ultrafilters U and V, we have D(R)_{E_{U}} ≠ D(R)_{E_{V}}. This proves that there are at least as many Bousfield classes, as points in the Stone–Čech compactification of the integers.

Of course, it is possible to Bousfield localise with respect to each of the homology theories E_{U}. The mind boggles, trying to imagine what these localisations might be.

References

[1] A.K. Bousfield, The boolean algebra of spectra, Comm. Math. Helv. 54 (1979), 368–377.

[2] Michael J. Hopkins, Global methods in homotopy theory, Homotopy Theory— Proceedings of the Durham Symposium 1985, London Math. Soc. Lecture Notes Series, vol. 117, Cambridge University Press, 1987, pp. 73–96.

[3] Mark Hovey, John H. Palmieri, and Neil P. Strickland, Axiomatic stable homotopy theory, vol. 128, Memoirs AMS, no. 610, Amer. Math. Soc., 1997.

[4] Amnon Neeman, The the chromatic tower for D(R), Topology 31 (1992), 519– 532.

[5] Tetsusuke Ohkawa, The injective hull of homotopy types with respect to generalised homology functors, Hiroshima Math. J. 19 (1989), 631–639.

[6] Douglas C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. of Math. 106 (1984), 351–414.

[7] Robert W. Thomason, The classification of triangulated subcategories, Compositio Math. 105 (1997), 1–27.

School of Mathematical Sciences, John Dedman Building, The Australian National University, Canberra, ACT 0200, AUATRALIA

  E-mail address: Amnon.Neeman@anu.edu.au