Manuscripta Mathematica 84 (1994), 193-198.

A REMARK ON

THE GENERALIZED

SMASHING CONJECTURE

Bernhard Keller

Using one of Wodzicki’s examples of H-unital algebras [14] we exhibit a ring whose derived category contains a smashing subcategory which is not generated by small objects. This disproves the generalization to arbitrary triangulated categories of a conjecture due to Ravenel [8, 1.33] and, originally, Bousfield [2, 3.4].

1. Statement of the conjecture

Keller

We refer to [7] for a nicely written analysis of the following setup: Let S be a triangulated category [13] admitting arbitrary (set-indexed) coproducts. An object X ∈ S is small if the functor Hom (X,?) commutes with arbitrary coproducts. We denote the full subcategory on the small objects of S by S^{b}. We suppose that S^{b} is equivalent to a small category. A full subcategory of S is localizing if it is a triangulated subcategory in the sense of Verdier which is closed under forming coproducts with respect to S. We suppose that S is generated by S^{b}, i.e. coincides with its smallest localizing subcategory containing S^{b}.

A localizing subcategory R ⊂ S is smashing if the inclusion R → S admits a right adjoint commuting with arbitrary coproducts. Suppose that R is generated by small objects. Since S^{b} is equivalent to a small category, the small generators of R may be assumed to form a set. Hence R is smashing by Brown’s representability theorem [3]. The “generalized smashing conjecture” states the converse (which is disproved below):

Every smashing subcategory is generated by small objects.

Remarks. a) I thank D. Ravenel for pointing out the following facts: The “generalized smashing conjecture” is not the generalization of Ravenel’s Smashing Conjecture [8, 10.6], but rather of his conjecture [8, 1.33] due originally to Bousfield [2, 3.4]. This latter conjecture is now known to be false due to the failure of the telescope conjecture [8, 10.5]. The proof of this involves very hard homotopy theory (cf. [10] for an outline of the argument). More information on the conjectures of [8] is to be found in [9].

b) The quotient functor j^{∗} : S → S/R admits a right adjoint j_{∗} iff the inclusion functor i_{∗} : R → S admits a right adjoint i^{!}, cf. [13]. One easily checks that in this case, the functor j_{∗} commutes with arbitrary coproducts iff the functor i^{!} does. This leads to an equivalent formulation of the smashing conjecture where the inclusion functor is replaced by the quotient functor.

2. An example

Let A be a ring with unit and DA the (unbounded) derived category [13] of the category of (right, unitary) A-modules. We identify A-modules with complexes concentrated in degree 0. The unbounded derived category was studied in [12],[1],[5]. It has arbitrary coproducts. An object of DA is small iff it is isomorphic to a perfect complex (=finite complex of finitely generated projective modules) [11]. Moreover, DA is generated by the right A-module A. Hence S = DA satisfies the above assumptions.

Let I be a two-sided ideal of A and R ⊂ DA the localizing subcategory generated by the right A-module I. Suppose that • we have Tor_{i} (A/I, A/I) = 0 for all i > 0 and

• the ideal I is contained in the Jacobson radical of A.

Proposition. The subcategory R → DA is smashing but R contains no non-zero small object of DA.

Note that if I satisfies both conditions and is moreover finitely generated, then we have I = 0, by Nakayama’s lemma. In particular, no noetherian ring contains a non-trivial ideal satisfying both conditions. This is not surprising since at least for a commutative noetherian ring R the “generalized smashing conjecture” is true, as follows from the algebraic counterpart [6] of Hopkins–Smith’s theorem on the classification of thick subcategories [4] (cf. [9] for a comprehensive account).

Now let k be a field and l an integer ≥ 2. Consider the (nonnoetherian) algebra

B = k[t, t^{l^{−1}}, t^{l^{−2}}, …] = ⋃_{n=0}^{∞} k[t^{l^{−n}}]

and its augmentation ideal J ⊂ B, which is generated by t, t^{l^{−1}}, t^{l^{−2}}, …. This algebra is Wozicki’s example 3 of [14, 4.7]. He proved in [loc. cit.] that J is H-unital. Since B is the augmented algebra obtained from J by adjoining a unit, this means that Tor_{i}^{B} (k, k) = 0 for all i > 0 (cf. section 3 of [loc. cit.]). Now let A be the localization of B at J and let I be the maximal ideal of A. Localization preserves the vanishing of the Tor and I equals the Jacobson radical of A. Thus I satisfies both conditions.

3. Proof of the proposition

We keep the assumptions preceding the proposition. We refer to [12], [1], [5] for the definition and the basic properties of the unbounded left derived functor ⊗L_{A} of the tensor product over A. In particular, this functor commutes with arbitrary coproducts. The proposition is immediate from the two following lemmas.

Lemma 1. The functor X ↦ X ⊗L_{A} I is right adjoint to the inclusion R → S = DA.

Proof. Let X be an object of DA. Consider the triangle

X ⊗L_{A} I → X → X ⊗L_{A} (A/I) → Σ(X ⊗L_{A} I).

We will show that the object X ⊗L_{A} I belongs to R and that the object X ⊗L_{A} (A/I) is R-local, i.e. for each object R ∈ R we have Hom (R, X ⊗L_{A} A/I) = 0. The assertion of the lemma is immediate from the Hom-sequence associated with the triangle.

For the generator X = A of DA, the object A ⊗L_{A} I = I clearly belongs to R. Since ?⊗L_{A}I commutes with arbitrary coproducts, the object X ⊗L_{A} I belongs to R for arbitrary X ∈ DA. Now we claim that the morphism R ⊗L_{A} I → R is invertible for R ∈ R. Indeed, since ? ⊗L_{A} I commutes with arbitrary coproducts, it is enough to check this for X = I. By the above triangle, we only have to show that I ⊗L_{A} A/I = 0. This is clear from the triangle

I ⊗L_{A} (A/I) → A/I → (A/I) ⊗L_{A} (A/I) → Σ(I ⊗L_{A} (A/I))

since the morphism A/I → (A/I) ⊗L_{A} (A/I) is invertible by the assumption. To prove that X ⊗L_{A} (A/I) is R-local, let R ∈ R and Y ∈ DA. We have A/I →^{∼} (A/I) ⊗L_{A} (A/I) and thus the morphism Y ⊗L_{A} (A/I) → (Y ⊗L_{A} (A/I)) ⊗L_{A} (A/I) is invertible as well. Now if f : R → Y ⊗L_{A} (A/I) is a morphism, then by the diagram

Y ⊗L_{A} (A/I) →^{∼} (Y ⊗L_{A} (A/I)) ⊗L_{A} (A/I)
f ↑  ↑ f ⊗L_{A} (A/I)
R  →  R ⊗L_{A} (A/I)

we have f = 0 since R⊗L_{A}(A/I) = 0 by the invertibility of R⊗L_{A}I → R.

Lemma 2. If R ∈ DA is small and belongs to R, then R = 0.

Proof. We may assume that R is a perfect complex. Since R belongs to R, the morphism R ⊗L_{A} I → R is invertible (see the proof of lemma 1). So R ⊗L_{A} (A/I) →^{∼} R ⊗_{A} (A/I) is acyclic. On the other hand, R ⊗_{A} (A/I) is a right bounded complex of projective A/I-modules. Hence it is null-homotopic. We will deduce that R is null-homotopic. We proceed by induction on the length of R. If R = R^{0} is concentrated in degree 0, then R^{0} is a finitely generated projective A-module with R^{0} ⊗(A/I) = 0. Hence R^{0} = 0 by Nakayama’s lemma. For general R we may assume that R = 0 for i > 0. Then d^{−1} : R^{−1} → R^{0} induces a split surjection R^{−1} ⊗ (A/I) → R^{0} ⊗ (A/I). Since R^{−1} and R^{0} are finitely generated projective, Nakayama’s lemma implies that d^{−1} is itself a split surjection. Therefore R is homotopy equivalent to the truncated complex

R′ = (… R → R^{i+1} → … → R^{−2} → Ker d^{−1} → 0 → …).

By the induction hypothesis, R′ is null-homotopic.

Acknowledgments

I am grateful to A. Neeman for his help and encouragment. I thank D. Ravenel for his detailed comments on a first version of this paper, and in particular for explaining the Smashing Conjecture to me and pointing out an error in a previous example.

References

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Bernhard Keller U.F.R. de Mathématiques U.R.A. 748 du CNRS Université Paris 7 2, place Jussieu, 75251 Paris Cedex 05, France keller@mathp7.jussieu.fr