GENERALIZED TENSOR IDEMPOTENTS AND THE TELESCOPE CONJECTURE

PAUL BALMER AND GIORDANO FAVI

Abstract. We transpose Rickard’s construction of idempotent representations to the general context of tensor triangulated categories. We connect our construction to the Telescope Conjecture and prove that the latter is of local nature. As an illustration, we extend the affine Telescope Theorem of Neeman to arbitrary noetherian schemes. We also develop supports for non-compact objects.

Contents

Introduction 1 1. All hypotheses on the table 3 2. Bousfield localization and smashing ideals 4 3. Tensor idempotents 7 4. Inflating and Telescope 11 5. Tensor idempotents and geometry 13 6. Rickard map and Telescope Conjecture 17 7. Residue objects and supports for big objects 20 References 24

Introduction

Tensor triangular geometry is the study of tensor triangulated categories, in algebraic geometry, homotopy theory, modular representation theory and beyond; see [2, 3]. This abstract framework allows the transposition of techniques and ideas from one area to another. For instance, the basic idea of gluing has been abstracted from usual algebraic geometry to tensor triangular geometry in [7] and then applied to modular representation theory in [6] and [4].

In the present paper, we proceed in the opposite direction : We extend a technique from modular representation theory, namely idempotent representations in the sense of Rickard [33], to the general framework of tensor triangulated categories and give an application in algebraic geometry.

But let us first remind the reader of Rickard’s insight [33]. Let G be a finite group and k a field. Consider V_{G} =^{def.} Proj(H^{•}(G, k)) the

associated projective support variety, see [8, Vol. II, Chap. 5]. For every closed subset W ⊂ V_{G}, Rickard tells us how to construct two possibly infinite-dimensional kG-modules, E(W) and F (W), which are tensor idempotents, E(W) ⊗ E(W) ≃ E(W) and F (W) ⊗ F (W) ≃ F (W), modulo projective modules. More precisely he constructs an exact triangle E(W) → k → F (W) → ΣE(W) in the stable module category Stab(kG) = kG – Mod /kG – Proj, such that tensoring with F (W) realizes localization away from W. The latter means that the functor F (W) ⊗ − : Stab(kG) ⟶ Stab(kG) is the Bousfield localization of Stab(kG) with respect to the smashing subcategory Stab_{W} (kG) := ⟨stab_{W} (kG)⟩. Here, ⟨stab_{W} (kG)⟩ is the localizing subcategory of Stab(kG) generated by the thick ⊗-ideal subcategory

Date: October 30, 2010.

1991 Mathematics Subject Classification. 18E30, 20C20, 14F05.

Key words and phrases. Bousfield localization, Telescope Conjecture, Mayer-Vietoris exact triangle, triangulated category, triangular geometry, monoidal category.

The first author is supported by NSF grant DMS-0969644.

(0.1)  stab_{W} (kG) := { M ∈ stab(kG) | V_{G}(M) ⊂ W }

of those finite dimensional representations whose projective support variety V_{G}(M) is contained in our chosen closed subset W. This interplay between the subcategory stab(kG) of compact objects (finite dimensional representations) and the big tensor triangulated category Stab(kG) is a central theme in this whole subject.

Rickard proves E(W_{1}∩W_{2}) ≃ E(W_{1})⊗E(W_{2}) and F (W_{1}∪W_{2}) ≃ F (W_{1})⊗F (W_{2}) and he constructs two Mayer-Vietoris exact triangles in Stab(kG)

\xymatrix{E(W_{1} ∩ W_{2}) \ar[r] & E(W_{1}) ⊕ E(W_{2}) \ar[r] & E(W_{1} ∪ W_{2}) \ar[r] & ΣE(W_{1} ∩ W_{2})}
\xymatrix{F (W_{1} ∩ W_{2}) \ar[r] & F (W_{1}) ⊕ F (W_{2}) \ar[r] & F (W_{1} ∪ W_{2}) \ar[r] & Σ F (W_{1} ∩ W_{2}) .}

All this is very useful in modular representation theory.

As Rickard already indicates, his techniques are reminiscent of the topologists’ homotopy theoretic methods involved in Brown representability. Nowadays, it is well known how such techniques can be generalized to big enough triangulated categories 𝒯, e.g. compactly generated ones; see Hovey-Palmieri-Strickland [22] or Neeman [31], for instance. For us, 𝒯 will be a compactly generated tensor triangulated category, in the precise sense of Hypotheses 1.1 below. We denote by 𝒯^{c} ⊂ 𝒯 the subcategory of compact objects. For example, (Stab(kG))^{c} = stab(kG).

The above is classical by now. Nonetheless, until recently, one essential ingredient in Rickard’s work did not have a clear generalization to any compactly generated tensor triangulated category 𝒯: It is the projective support variety V_{G}. This is a fundamental question since V_{G} contains the closed subsets W which parametrize Rickard’s idempotents. Similarly, the above Mayer-Vietoris results involve some ambient space in which to take union and intersection of W_{1} and W_{2}. In fact, in the above construction, the only place where V_{G} plays an essential role is in the definition of the thick ideal stab_{W} (kG) of compact objects, as in (0.1) above. In this definition, one could actually use any “support datum” (Rem. 5.3) instead of V_{G}(−) and still obtain thick ideals. See more on this in Remark 5.26 below. However, assuming we want the best theory right away, we should try to use a notion of support which catches all thick ideals of compact objects immediately. This is exactly what the spectrum of [2] does for us.

Recall from [2] that for a tensor triangulated category 𝒦, like here 𝒦 = 𝒯^{c}, the spectrum Spc(𝒦) is a topological space in which every object x ∈ 𝒦 has a support supp(x) ⊂ Spc(𝒦) and such that one can parametrize all thick ideals in 𝒦 via W ↦ 𝒦_{W} := { x ∈ 𝒦 | supp(x) ⊂ W }, where W ⊂ Spc(𝒦) runs through so-called Thomason subsets (Def. 5.7). In the example of 𝒯 = Stab(kG), we have 𝒦 = 𝒯^{c} = stab(kG) and its spectrum coincides with V_{G} in such a way that supp(M) = V_{G}(M), see [2, Cor. 5.10]. (And V_{G} is noetherian, so every closed subset is Thomason.)

In short, a conceptual generalization of Rickard’s framework to all possible 𝒯 is provided by tensor triangular geometry of 𝒯^{c}, following [2].

Using our approach, we then prove the complete generalization of Rickard’s results to any compactly generated tensor triangulated category 𝒯 (as in Hyp. 1.1) : For every Thomason subset W ⊂ Spc(𝒯^{c}) we construct two ⊗-idempotents e(W) ≃ e(W) ⊗ e(W) and f(W) ≃ f(W) ⊗ f(W) in 𝒯, together with an exact triangle e(W) → 𝟙 → f(W) → Σe(W), such that f(W) ⊗ − : 𝒯 ⟶ 𝒯 realizes Bousfield localization with respect to 𝒯_{W} := ⟨(𝒯^{c})_{W} ⟩, i.e. localization away from W. (Here 𝟙 ∈ 𝒯 is the unit of the tensor structure on 𝒯.) This is done in Section 5. We establish generalized Mayer-Vietoris triangles in Theorem 5.18.

There are as many applications of this construction as there are compactly generated tensor triangulated categories 𝒯. This goes even beyond the areas mentioned above and does apply to motivic theory or noncommutative topology (equivariant KK-theory of C^{∗}-algebras) for instance; see Dell’Ambrogio [19]. We do not pursue this direction here. Instead, we connect our construction to the so-called Telescope Conjecture, which is the property that every smashing subcategory of 𝒯 be generated from its compact part; see Def. 4.2. Some categories 𝒯 have this property, some don’t. It is notoriously unclear whether the stable homotopy category 𝒯 = SH of topological spectra has the Telescope property or not. See Remark 4.3.

Here, we prove that the Telescope property is of local nature relatively to open covers of Spc(𝒯^{c}); see a precise statement in Theorem 6.6. As a corollary, we give a quick proof that the Telescope property holds for the derived category of a noetherian scheme (Alonso et al. [1, Thm. 5.8]) by simply reducing it to the affine case proven by Neeman [30]; see Corollary 6.8.

Another thing one can do with tensor idempotents is create residue objects κ(𝒫) ∈ 𝒯 for 𝒫 ∈ Spc(𝒯^{c}) in the spectrum of the compact part of 𝒯. This is done in Section 7 and allows the construction of support for big objects, i.e. subsets Supp(t) ⊂ Spc(𝒯^{c}) for any t ∈ 𝒯. This generalizes the first definition of supports V_{G}(M) for possibly non finite-dimensional representations M, given in [9].

The structure of the paper should now be clear from the table of contents.

1. All hypotheses on the table

1.1. Hypotheses. Throughout the paper, 𝒯 stands for a (rigidly-) compactly generated tensor triangulated category. The poetically inclined reader might prefer to call 𝒯 a “unital algebraic stable homotopy category”, as in [22]^{1}. We prefer the above somewhat more explicit terminology. We denote by

𝒦 = 𝒯^{c}

the subcategory of rigid and compact objects. We now unfold the details.

We assume familiarity with the notion of triangulated category and we denote by Σ : 𝒯 → 𝒯 the suspension functor. We assume that 𝒯 has arbitrary (small) coproducts ⊔_{i∈I}t_{i}. An object c ∈ 𝒯 is compact if, for every set {t_{i}}_{i∈I} of objects of 𝒯, the natural map ⊕_{i∈I}Hom(c, t_{i}) → Hom(c, ⊔_{i∈I}t_{i}) is an isomorphism. In [22], compact objects are called small. We assume that 𝒯 is compactly generated, that is, the subcategory of compact objects 𝒦 = 𝒯^{c} is essentially small (has a set of isomorphism classes) and, most importantly, if an object t ∈ 𝒯 is such that Hom(c, t) = 0 for every c ∈ 𝒦 then t = 0. See also Remark 2.3 below.

1The Brown representability hypothesis [22, Def. 1.1.4. (e)] is redundant by [31, Thm. 8.3.3].

Finally, let us discuss tensor and rigidity. We assume that 𝒯 admits a closed symmetric monoidal structure (see for instance [22, App. A])

⊗ : 𝒯 × 𝒯 ⟶ 𝒯 .

Again, we remind the reader. We have natural isomorphisms x ⊗ y ≅ y ⊗ x, which for our present purposes we may treat as identities, and a unit 𝟙 for the tensor product: 𝟙⊗x ≅ x. Moreover, there exists an adjoint \underline{hom} : 𝒯^{op} ×𝒯 ⟶ 𝒯 such that Hom(x ⊗ y, z) ≅ Hom(x, \underline{hom}(y, z)). Both functors, ⊗ and \underline{hom}, are also assumed exact in each variable. An object x ∈ 𝒯 is called rigid (or strongly dualizable) if for every y ∈ 𝒯, the natural map \underline{hom}(x, 𝟙)⊗y → \underline{hom}(x, y) is an isomorphism. We also require that 𝟙 is compact and that every compact object is rigid, which is equivalent to say that compact objects coincide with rigid objects; see [22, Thm. 2.1.3 (d)]. So, the subcategory 𝒦 of rigid-compact objects is essentially small, rigid (meaning that all its objects are so) and ⊗-triangulated.

1.2. Examples. We refer to [22, Ex. 1.2.3] for justifications.

(1) 𝒯 = SH, the stable homotopy category of topological spectra. Here 𝒦 = SH^{fin}, the stable homotopy category of finite spectra. See [28] for details.

(2) 𝒯 = D(R – Mod), the unbounded derived category of a commutative ring R (see [26, Example 5.8]). Here 𝒦 = K^{b}(R – proj). More generally 𝒯 = D(X) := D_{Qcoh}(O_{X}), the derived category of unbounded complexes of O_{X}-modules with quasi-coherent homology over a quasi-compact and quasi-separated scheme X. (Quasi-separated means that quasi-compact open subsets are stable under pairwise intersection, i.e. form an open basis. This is equivalent to say that X is spectral, see Rem. 5.11.) By Bondal and van den Bergh [14, Thm. 3.1.1], 𝒦 = 𝒯^{c} is the category D^{perf}(X) of perfect complexes over X, i.e. complexes which are locally quasi-isomorphic to bounded complexes of vector bundles; see SGA 6 [12] or [36]. It is shown in [13, Cor. 5.5] that the natural functor D(Qcoh(X)) → D(X) is an equivalence when X is moreover separated, which the reader can assume if D(Qcoh(X)) sounds more familiar. The tensor structure is standard; detailed references are collected in [17, § 1] for instance.

(3) 𝒯 = Stab(kG), the stable category of all kG-modules, for G a finite group (scheme). Here 𝒦 = stab(kG), the stable category of finite dimensional kGmodules.

2. Bousfield localization and smashing ideals

2.1. Definition. Recall that a triangulated subcategory 𝒮 ⊂ 𝒯 is called thick if it is stable under taking direct summands. It is called localizing if it is stable under arbitrary coproducts. It is colocalizing if it is stable under arbitrary products. (Co)localizing subcategories are thick. Finally, 𝒮 is called ⊗-ideal if 𝒯 ⊗ 𝒮 ⊂ 𝒮.

2.2. Notation. If ℰ is a class of objects of 𝒯, we will denote by ⟨ℰ⟩ the smallest localizing triangulated subcategory of 𝒯 which contains all objects of ℰ. Similarly ⟨ℰ⟩^{⊗} for the smallest localizing ⊗-ideal.

2.3. Remark. In Hypotheses 1.1, the fact that 𝒯 is compactly generated can be expressed as ⟨𝒦⟩ = 𝒯. Consequently, if ℰ ⊂ 𝒯 satisfies 𝒦⊗ℰ ⊂ ⟨ℰ⟩ (typically ℰ ⊂ 𝒦 ⊗-ideal inside 𝒦, that is 𝒦 ⊗ ℰ ⊂ ℰ), then ⟨ℰ⟩ is ⊗-ideal in 𝒯 and ⟨ℰ⟩ = ⟨ℰ⟩^{⊗}.

We now recall standard results about localization of triangulated categories. Most of this material can be found in the now classical reference [31] or in the monograph [22]. For a recent comparative survey, see Krause [26].

Recall the Verdier localization q : 𝒯 ⟶ 𝒯/𝒮 of a triangulated category 𝒯 by a thick subcategory 𝒮, see [31, § 2.1]. One pitfall with Verdier’s construction is that the quotient 𝒯/𝒮, in which morphisms are equivalence classes of fractions, might have proper classes of morphisms, not sets, i.e. 𝒯/𝒮 might be a “large” category. Sometimes, the functor q : 𝒯 ⟶ 𝒯/𝒮 has an adjoint, which is automatically faithful, hence realizes 𝒯/𝒮 as a subcategory of 𝒯, avoiding the above pitfall. This happens in Bousfield localization, as we now recall.

2.4. Definition. A Bousfield localization functor on 𝒯 is a pair (L, λ) where L : 𝒯 → 𝒯 is an exact functor and λ : Id_{𝒯} → L is a natural transformation such that Lλ : L → L^{2} is an isomorphism and Lλ = λL. A morphism of localization functors (L, λ) → (L′, λ′) on the same category 𝒯 is a natural transformation ℓ : L → L′ such that ℓ ∘ λ = λ′. For a localization functor (L, λ), we have Im(L) = {t ∈ 𝒯 | λ_{t} is an isomorphism}. (For a functor F, we denote by Im(F) its essential image.) These are often called L-local objects.

The dual notion of colocalization functor is also important for us : (Γ, γ) is a colocalization functor on 𝒯 if (Γ^{op}, γ^{op}) is a localization functor on 𝒯^{op}, etc.

2.5. Definition. If 𝒜, ℬ are classes of objects in 𝒯 we write 𝒜⊥ℬ if Hom(a, b) = 0 for all a ∈ 𝒜 and b ∈ ℬ. For a class 𝒞 of objects of 𝒯 we consider its right orthogonal 𝒞^{⊥} = {t ∈ 𝒯 | Hom(c, t) = 0 for all c ∈ 𝒞} and its left orthogonal ^{⊥}𝒞 = {t ∈ 𝒯 | Hom(t, c) = 0 for all c ∈ 𝒞}. Both ^{⊥}𝒞 and 𝒞^{⊥} are thick subcategories of 𝒯. Moreover, ^{⊥}𝒞 is always localizing and 𝒞^{⊥} is always colocalizing.

2.6. Theorem (Bousfield localization). Let 𝒯 be a triangulated category and 𝒮 be a thick subcategory. Then the following statements are equivalent :

(i) There exists a localization functor (L, λ) : 𝒯 → 𝒯 with 𝒮 = Ker(L). (ii) There exists a colocalization functor (Γ, γ) : 𝒯 → 𝒯 with 𝒮 = Im(Γ). (iii) Each t ∈ 𝒯 fits in an exact triangle t′ → t → t′′ → Σt′ with t′ ∈ 𝒮 and t′′ ∈ 𝒮^{⊥}. (iv) The Verdier quotient functor q : 𝒯 → 𝒯/𝒮 exists and has a right adjoint.

(v) The composition 𝒮^{⊥} → 𝒯 → 𝒯/𝒮 is an equivalence.

Assume that (i)–(v) hold. Then 𝒮^{⊥} = Im(L) = Ker(Γ) and ^{⊥}(𝒮^{⊥}) = 𝒮. Moreover, the triangle in (iii) is functorial in t and is the unique one with these properties (up to unique isomorphism of triangles being the identity on t). We denote it by

(2.7)  Δ_{𝒮}(t) := ( Γ_{𝒮}(t) → t → L_{𝒮}(t) → ΣΓ_{𝒮}(t) ) .

Proof. See [31, Chap. 9] or [22, Chap. 3] or [26, § 4] (for this and even more). □

2.8. Definition. A localizing subcategory 𝒮 satisfying the equivalent conditions of Thm. 2.6 is called a Bousfield subcategory of 𝒯. The localization exact triangle for 𝒮 will refer to Δ_{𝒮} of (2.7) above, featuring the (co)localization functors Γ_{𝒮} and L_{𝒮}.

2.9. Remark. If 𝒮 ⊂ 𝒮′ are Bousfield subcategories of 𝒯 then there is a unique morphism of triangles (ϵ, id, φ) : Δ_{𝒮} → Δ_{𝒮′}. In the following (solid) diagram

[figure: Γ_{𝒮}(t); γ; ϵ; t; λ; L_{𝒮}(t); φ; ΣΓ_{𝒮}(t); Σϵ; Γ_{𝒮′} (t); γ′; t; λ′; L_{𝒮′} (t); ΣΓ_{𝒮′} (t),]

we have λ′ ∘ γ ∈ Hom(Γ_{𝒮}(t), L_{𝒮′} (t)) = 0 since Γ_{𝒮}(t) ∈ 𝒮 ⊂ 𝒮′ ⊥ (𝒮′)^{⊥} ∋ L_{𝒮′} (t). Using standard properties of exact triangles, we get the existence of ϵ such that γ′ϵ = γ, and then of φ making the diagram commute. These ϵ and φ are unique since another choice would differ by a map in Hom(ΣΓ_{𝒮}(t), L_{𝒮′} (t)) = 0. This yields a unique morphism of localization functors φ : L_{𝒮} → L_{𝒮′}.

The equivalence 𝒯/𝒮 ≅ 𝒮^{⊥} of Thm. 2.6 (v) allows us to discuss 𝒯/𝒮 inside 𝒯, without calculus of fractions. As an interlude, we illustrate this with the gluing technique, originally treated without Bousfield localization in [7].

2.10. Definition (See [7, Def. 2.1]). We say that two thick subcategories 𝒮_{1}, 𝒮_{2} of a triangulated category 𝒯 are in formal Mayer-Vietoris situation if 𝒮_{1}⊥𝒮_{2} and 𝒮_{2}⊥𝒮_{1}, i.e. Hom(𝒮_{1}, 𝒮_{2}) = Hom(𝒮_{2}, 𝒮_{1}) = 0. Then 𝒮_{1} ∩ 𝒮_{2} = 0. We denote by 𝒮_{1} ⊕ 𝒮_{2} the thick subcategory whose objects are {t ∈ 𝒯 | t ≃ s_{1} ⊕ s_{2} for some s_{1} ∈ 𝒮_{1}, s_{2} ∈ 𝒮_{2}}. We want to express the weak cartesian nature of the following diagram :

\xymatrix{𝒯 \ar[r] \ar[d] & 𝒯/𝒮_{1} \ar[dr] \\ 𝒯/𝒮_{2} \ar[rr] &  & 𝒯/(𝒮_{1} ⊕ 𝒮_{2}) .}

2.11. Theorem (Gluing of objects [7, Thm. 4.3]). Let 𝒯, 𝒮_{1} and 𝒮_{2} be as above. Set 𝒮_{12} = 𝒮_{1} ⊕ 𝒮_{2}. Assume that 𝒯/𝒮_{i} has small Hom sets, for i ∈ {1, 2, 12} (e.g. 𝒮_{i} Bousfield). Let t_{1} ∈ 𝒯/𝒮_{1} and t_{2} ∈ 𝒯/𝒮_{2} be two objects and σ : t_{1} →^{∼} t_{2} an isomorphism in 𝒯/𝒮_{12}. Then there exists an object t ∈ 𝒯 and isomorphisms t ≃ t_{i} in 𝒯/𝒮_{i} for i = 1, 2, compatible with σ in 𝒯/𝒮_{12}. The object t so obtained is unique up to possibly non unique isomorphism and is called a gluing of t_{1} and t_{2} along σ.

Alternative proof in the Bousfield case : For i ∈ {1, 2, 12}, we further suppose that 𝒮_{i} is Bousfield (Def. 2.8) with localization functors L_{i} : 𝒯 → 𝒯. By Thm. 2.6 (vi), we can identify 𝒯/𝒮_{i} with 𝒮_{i}^{⊥} = Im(L_{i}) and view the isomorphism σ : t_{1} → t_{2} in 𝒯/𝒮_{12} as an isomorphism σ′ : L_{12}(t_{1}) →^{∼} L_{12}(t_{2}) in 𝒯. By Rem. 2.9, the inclusion 𝒮_{i} ⊂ 𝒮_{12} induces morphisms of localizations φ_{12,j} : L_{j} → L_{12} for j = 1, 2. Now form a weak pull-back in 𝒯 of φ_{12,2} and σ′ ∘ φ_{12,1} as follows

[figure: t; α_{1}; α_{2}; ⌟; L_{1}(t_{1}); φ_{12,1}; L_{12}(t_{1}); σ′; ≃; L_{2}(t_{2}); φ_{12,2}; L_{12}(t_{2})]

From this construction, it is an exercise to see that t has the properties of a gluing. Compare [6, § 6]. The desired isomorphisms correspond to L_{1}(α_{1}) and L_{2}(α_{2}). □

Bousfield localization also yields an alternative proof of the following :

2.12. Theorem (Gluing of morphisms [7, Thm. 3.5]). Under the same hypotheses as Thm. 2.11, one can also glue morphisms in 𝒯: Given objects s, t ∈ 𝒯 and morphisms f_{i} : s → t in 𝒯/𝒮_{i} for i = 1, 2, which agree in 𝒯/𝒮_{12}, there is a (non necessarily unique) morphism f : s → t in 𝒯 whose image in 𝒯/𝒮_{i} is equal to f_{i}, for i = 1, 2.

So far, we have not used the tensor structure on 𝒯. Here it comes.

2.13. Theorem (Smashing localization). Let (𝒯, ⊗, 𝟙) be a ⊗-triangulated category

and let 𝒮 be a Bousfield subcategory (Def. 2.8). Let Δ_{𝒮} : [figure: Γ_{𝒮}; γ; Id; λ; L_{𝒮}; ΣΓ_{𝒮}] be the localization triangle for 𝒮, see (2.7). Now, suppose in addition that 𝒮 is a ⊗-ideal (Def. 2.1). Then the following conditions are equivalent:

(i) The subcategory 𝒮^{⊥} is also a ⊗-ideal.

(ii) There is an isomorphism of functors L_{𝒮} ≃ L_{𝒮}(𝟙) ⊗ −.

(iii) There is an isomorphism of functors Δ_{𝒮} ≃ Δ_{𝒮}(𝟙) ⊗ −.

Proof. This is also standard. We drop the subscripts _{𝒮} from the notation. Assume that (i) holds. For every t in 𝒯 consider the exact triangle Δ(𝟙) ⊗ t:

Γ(𝟙) ⊗ t ⟶ t ⟶ L(𝟙) ⊗ t ⟶ Σ Γ(𝟙) ⊗ t.

Since 𝒮 is ⊗-ideal, we have Γ(𝟙) ⊗ t ∈ 𝒮. Since we assume 𝒮^{⊥} ⊗-ideal, we also have L(𝟙) ⊗ t ∈ 𝒮^{⊥}. By uniqueness of the triangle Δ(t) we get (iii). Clearly (iii)⇒(ii). To show (ii)⇒(i), let t ∈ 𝒮^{⊥} = Im(L) and t′ ∈ 𝒯 arbitrary. Then t ⊗ t′ ≃ L(t) ⊗ t′ ≃ L(𝟙) ⊗ t ⊗ t′ ≃ L(t ⊗ t′) hence t ⊗ t′ ∈ Im(L) = 𝒮^{⊥}. □

2.14. Remark. The isomorphisms in Thm. 2.13 (ii)-(iii) are unique, if they are the identity morphism on the t in the second term of Δ_{𝒮}(t). We leave this to the reader.

2.15. Definition. A ⊗-ideal Bousfield subcategory 𝒮 of 𝒯 is called a smashing ideal if 𝒮 satisfies the equivalent conditions of Theorem 2.13. We will denote by 𝕊(𝒯) the class of all smashing ideals of 𝒯 and order it by inclusion ⊂.

2.16. Remark. Under Hypotheses 1.1, this 𝕊(𝒯) is indeed a set by Krause [25, Thm. 4.9]. Also, the notion of smashing subcategory is sometimes formulated without ⊗-structure, by asking the localization functor L_{𝒮} to preserve arbitrary coproducts, see [26, Prop. 5.5.1]. These two notions coincide here, see [22, Def. 3.3.2].

2.17. Remark. Let 𝒯 be a ⊗-triangulated category and 𝒮 be a smashing ideal of 𝒯. Then there is an inclusion-preserving bijection between the smashing ideals of 𝒯 which contain 𝒮 and the smashing ideals of 𝒯/𝒮, given by 𝒮′ ↦ 𝒮′/𝒮 ⊂ 𝒯/𝒮.

3. Tensor idempotents

Instead of smashing ideals (Def. 2.15) we can equivalently consider ⊗-idempotents.

3.1. Proposition. Let 𝒯 be a ⊗-triangulated category and let Δ : [figure: e; γ; 𝟙; λ; f; Σe] be an exact triangle. Then the following are equivalent :

(i) γ ⊗ id_{e} is an isomorphism e ⊗ e →^{∼} e. (ii) e ⊗ f = 0. (iii) λ ⊗ id_{f} is an isomorphism f →^{∼} f ⊗ f. Moreover, in that case, Ker(− ⊗ f) = Im(− ⊗ e) and Ker(− ⊗ e) = Im(− ⊗ f).

Proof. Triangles Δ ⊗ e and Δ ⊗ f give (i)⇔(ii) and (ii)⇔(iii), respectively. The moreover part can be read on Δ ⊗ t for arbitrary t ∈ 𝒯. □

3.2. Definition. An exact triangle Δ : [figure: e; γ; 𝟙; λ; f; Σe] satisfying the equivalent conditions of Prop. 3.1 will be called an idempotent triangle. Moreover, γ : e → 𝟙 will be called a left idempotent and λ : 𝟙 → f a right idempotent. A morphism of idempotent triangles Δ → Δ′ is a morphism of triangles of the form (ϵ, id_{𝟙}, φ) :

(3.3)  [figure: e; γ; ϵ; 𝟙; λ; f; φ; Σe; Σϵ; e′; γ′; 𝟙; λ′; f′; Σe′.]

Morphisms of left and right idempotents are defined in the obvious similar way, keeping only the square with ϵ and φ respectively. We denote by

𝔻(𝒯), 𝔼(𝒯) and 𝔽(𝒯)

the collection of isomorphism classes of idempotent triangles, of left idempotents and of right idempotents respectively. We denote by [−] these isomorphism classes. All three 𝔻(𝒯), 𝔼(𝒯) and 𝔽(𝒯) carry a partial order induced by morphisms. For instance, [Δ] ≤ [Δ′] if and only if there exist a morphism of idempotent triangles (ϵ, id, φ) : Δ → Δ′. We will often simply write Δ ≤ Δ′, e ≤ e′ or f ≤ f′.

3.4. Remark. Following Proposition 3.1 and its proof, one sees that the (partially ordered) classes 𝔻(𝒯), 𝔼(𝒯) and 𝔽(𝒯) are isomorphic via the forgetful maps 𝔻(𝒯) → 𝔼(𝒯), 𝔽(𝒯), whose inverses come from completing morphisms into exact triangles. We now prove that they also agree with the class of smashing ideals 𝕊(𝒯), see Def. 2.15.

3.5. Theorem (Compare [22, Lem. 3.1.6]). Let (𝒯, ⊗, 𝟙) be a ⊗-triangulated category. All four partially ordered classes above, 𝕊(𝒯), 𝔻(𝒯), 𝔼(𝒯) and 𝔽(𝒯), are isomorphic. More precisely :

(a) For every smashing ideal 𝒮 ⊂ 𝒯, the triangle Δ_{𝒮}(𝟙) defined in (2.7) is an idempotent triangle, which will be denoted

(3.6)  Δ(𝒮) := ( [figure: e(𝒮); γ; 𝟙; λ; f(𝒮); Σe(𝒮)] ) .

Up to unique isomorphism of idempotent triangles, Δ(𝒮) is characterized by the following three properties : Δ(𝒮) is exact, e(𝒮) ∈ 𝒮 and f(𝒮) ∈ 𝒮^{⊥}.

(b) For every idempotent triangle Δ : [figure: e; γ; 𝟙; λ; f; Σe] , the functor L_{f} defined by − ⊗ f : 𝒯 → 𝒯 is a localization functor, whose kernel Ker(− ⊗ f) is a smashing ideal, denoted 𝒮(Δ). We have 𝒮(Δ) = Ker(−⊗f) = Im(−⊗e) whereas 𝒮(Δ)^{⊥} = Ker(− ⊗ e) = Im(− ⊗ f).

The constructions 𝒮 ↦ Δ(𝒮) and Δ ↦ 𝒮(Δ), described in (a) and (b), are orderpreserving, mutually inverse bijections between 𝕊(𝒯) and 𝔻(𝒯).

Proof. For (a), since 𝒮 and 𝒮^{⊥} are ⊗-ideals, Γ_{𝒮}(𝟙)⊗L_{𝒮}(𝟙) ∈ 𝒮∩𝒮^{⊥} = 0. We conclude by Proposition 3.1. For (b), the functor L_{f} = − ⊗ f : 𝒯 → 𝒯 is a localization with λ : Id → L_{f} induced by λ : 𝟙 → f. It is clear that L_{f} satisfies Theorem 2.13 (ii). Hence 𝒮(Δ) := Ker(− ⊗ f) is a smashing ideal. We claim that f ∈ 𝒮(Δ)^{⊥}. Indeed, if α : t → f is a morphism with t ∈ 𝒮(Δ), then tensoring α with λ : 𝟙 → f gives the commutative square

[figure: t; α; t ⊗ f = 0; f; ≃; f ⊗ f]

which implies α = 0. Thus f ∈ 𝒮(Δ)^{⊥} and e ∈ 𝒮(Δ). In short, Δ : \xymatrix{e \ar[r] & 𝟙 \ar[r] & f \ar[r] & Σe} must be the unique exact triangle with these properties, i.e. Δ ≃ Δ_{𝒮(Δ)}(𝟙). It is now easy to see that Δ ↦ 𝒮(Δ) and 𝒮 ↦ Δ(𝒮) are inverse bijections and that the latter is order-preserving, see Rem. 2.9. Finally, we show that Δ ↦ 𝒮(Δ) is order-preserving. Let (ϵ, id, φ) : Δ → Δ′ be a morphism of idempotent triangles as in (3.3) above. Applying e ⊗ − to (3.3) and using that e ⊗ f = 0 shows that id_{e} ⊗λ′ : e → e ⊗ f′ = L_{f′}(e) is zero. This implies L_{f′}(e) = 0 by universality of e → L_{f′}(e) (or just by applying − ⊗ f′), hence e ∈ Ker(− ⊗ f′). We get as wanted 𝒮(Δ) = Im(− ⊗ e) ⊂ Ker(− ⊗ f′) = 𝒮(Δ′). □

3.7. Corollary. Let (𝒯, ⊗, 𝟙) be a ⊗-triangulated category.

(a) Let Δ = (e → 𝟙 → f → Σe) be an idempotent triangle. Then we have Im(− ⊗ e) = Ker(− ⊗ f) ⊥ Im(− ⊗ f) = Ker(− ⊗ e).

(b) If [Δ] ≤ [Δ′] then there exists a unique morphism of idempotent triangles Δ → Δ′. Similarly for left idempotents and for right idempotents.

(c) If two right idempotents f and f′ are isomorphic as objects of 𝒯, via an isomorphism α : f →^{∼} f′, then there is a unique isomorphism of right idempotents φ : f →^{∼} f′ (possibly different from α). Same with left idempotents and idempotent triangles, mutatis mutandis.

Proof. We use Theorem 3.5 throughout : (a) is clear from Ker(− ⊗ f) = 𝒮(Δ) ⊥ 𝒮(Δ)^{⊥} = Im(− ⊗ f). For (b), if Δ ≤ Δ′, we have 𝒮(Δ) ⊂ 𝒮(Δ′) and uniqueness of Δ → Δ′ is Remark 2.9. For (c), use that Ker(− ⊗ f) = Ker(− ⊗ f′) in 𝕊(𝒯). □

3.8. Remark. For 𝒯 compactly generated, 𝔻(𝒯), 𝔼(𝒯) and 𝔽(𝒯) are sets (Rem. 2.16).

3.9. Remark. Any ⊗-triangulated functor q : 𝒯 → 𝒯′ induces three obvious compatible maps 𝔻(𝒯) → 𝔻(𝒯′), 𝔼(𝒯) → 𝔼(𝒯′) and 𝔽(𝒯) → 𝔽(𝒯′) and consequently a fourth one 𝕊(𝒯) → 𝕊(𝒯′) by the bijection of Thm. 3.5. If moreover 𝒯′ is of the form 𝒯/𝒮 for some smashing subcategory 𝒮 of 𝒯 then 𝕊(𝒯) → 𝕊(𝒯/𝒮) is nothing but the map of Remark 2.17. We leave this verification to the reader.

3.10. Notation. When e → 𝟙 → f → Σe is an idempotent triangle, we use ^{⊥}f to mean e and similarly e^{⊥} to mean f. For any right idempotent f, we thus have (^{⊥}f)^{⊥} = f and f ⊗^{⊥}f = 0 as well as Im(− ⊗^{⊥}f) = Ker(− ⊗ f).

3.11. Proposition. Let 𝒯 be a ⊗-triangulated category. Then the four isomorphic partial orders (𝕊(𝒯), ⊂), (𝔻(𝒯), ≤), (𝔼(𝒯), ≤) and (𝔽(𝒯), ≤) are lattices, i.e. any finite subset has a supremum and an infimum. The tensor product provides the meet ∧ (pairwise infimum) on 𝔼(𝒯) and the join ∨ (pairwise supremum) on 𝔽(𝒯). Explicitly, for any pair of left idempotents, we have e ∧ e′ = e ⊗ e′ and for any pair of right idempotents we have f ∨ f′ = f ⊗ f′. On 𝕊(𝒯), this becomes : If 𝒮 and 𝒮′ are smashing ideals, then 𝒮 ∧ 𝒮′ = 𝒮 ∩ 𝒮′ whereas 𝒮 ∨ 𝒮′ = Ker(− ⊗ f(𝒮) ⊗ f(𝒮′)).

Proof. Tensor clearly preserves left and right idempotents. If α : f → f′′ and α′ : f′ → f′′ are morphisms of right idempotents then α ⊗ α′ : f ⊗ f′ → f′′ ⊗ f′′ ≃ f′′ is the unique (Cor. 3.7) morphism of right idempotents making the following diagram commute

\xymatrix{𝟙 \ar[r]^{λ} \ar[d]_{λ′} & f \ar[d]_{1⊗λ′} \ar[ddr]^{α} \\ f′ \ar[r]^{λ⊗1} \ar[drr]_{α′} & f ⊗ f′ \ar[dr]_{α⊗α′} \\  &  & f′′.}

This shows that − ⊗ − gives coproducts of right idempotents, hence joins in 𝔽(𝒯). Dually, − ⊗ − gives products of left idempotents, hence meets in 𝔼(𝒯). The bijections 𝔻(𝒯) ≃ 𝔼(𝒯) ≃ 𝔽(𝒯) ≃ 𝕊(𝒯) of Thm. 3.5 yield joins and meets everywhere. One checks that Im(−⊗e)∧Im(−⊗e′) = Im(−⊗e⊗e′) = Im(−⊗e)∩Im(−⊗e′). □

3.12. Corollary (Compare [33, Lem. 7.2]). If e ≤ e′ in 𝔼(𝒯), then e⊗(e′)^{⊥} = 0 and there is a unique isomorphism of left idempotents e ⊗ e′ ≅ e. Similarly, if f ≤ f′ in 𝔽(𝒯), then (^{⊥}f) ⊗ f′ = 0 and f ⊗ f′ ≅ f′. (See Notation 3.10.)

Proof. e ≤ e′ implies e∧e′ = e and ∧ is given by ⊗ on left idempotents by Prop. 3.11. Hence the unique isomorphism of left idempotents e ⊗ e′ ≅ e by Cor. 3.7. It follows that e ⊗ (e′)^{⊥} ≅ e ⊗ e′ ⊗ (e′)^{⊥} = 0. The dual is dual. □

We can now state and prove an abstract version of Rickard’s Mayer-Vietoris triangles, see [33, Thm. 8.1]. The idea of the proof is the same as in loc. cit.

3.13. Theorem (Abstract Mayer-Vietoris triangles; see Rouquier [34, Prop. 5.10]). Let e, e′ ∈ 𝔼(𝒯) be left idempotents and f, f′ ∈ 𝔽(𝒯) be right idempotents in any ⊗-triangulated category 𝒯. Then there exist exact triangles in 𝒯

(a) e ⊗ e′ → e ⊕ e′ → e ∨ e′ → Σ(e ⊗ e′) ,

(b) f ∧ f′ → f ⊕ f′ → f ⊗ f′ → Σ(f ∧ f′) .

Proof. Let λ : 𝟙 → f and λ′ : 𝟙 → f′ be in 𝔽(𝒯). By Prop. 3.11, there is a commutative square of right idempotents as in the outer square of the following diagram :

[figure: f ∧ f′; i; i′; v; f; p; d; q; q′; ⌟; f′; p′; f ⊗ f′,]

which we complete by introducing the weak pull-back d of p = id_{f} ⊗λ′ and p′ = λ ⊗ id_{f′}, as well as a corner morphism v : f ∧ f′ → d making the diagram commute. Let us prove that c := cone(v) is zero. First note that all the objects f, f′, f ∧f′ and f ⊗f′ become zero when we tensor them by^{⊥}(f ∧f′) = ^{⊥}f ⊗^{⊥}f′. Thus d⊗^{⊥}(f ∧f′) = 0 and c ⊗^{⊥}(f ∧ f′) = 0 as well. It now suffices to prove that c ⊗^{⊥}(f ∧ f′) ≃ c, that is, c ∈ Im(− ⊗^{⊥}(f ∧ f′)) = Ker(− ⊗ (f ∧ f′)) = Ker(− ⊗ f) ∩ Ker(− ⊗ f′), using Proposition 3.11 again.

Observe that id_{f} ⊗p′ = id_{f} ⊗λ ⊗ id_{f′} is an isomorphism since f is idempotent. Applying f ⊗ − to the above weak pull-back gives another weak pull-back, from which we deduce that id_{f} ⊗q is an isomorphism as well. On the other hand, the same is true for i by Corollary 3.12 applied to f ∧ f′ ≤ f, that is, id_{f} ⊗i is an isomorphism. By 2-out-of-3, we see that v has the same property : it becomes an isomorphism under f ⊗ −. This implies that f ⊗ c ≃ 0, that is, c ∈ Ker(− ⊗ f). Symmetrically, c ∈ Ker(− ⊗ f′) hence c ∈ Ker(− ⊗ f) ∩ Ker(− ⊗ f′) as wanted.

So, c ≃ 0 and v is an isomorphism. This gives the exact triangle in (b). The proof for the triangle in (a) is dual. □

3.14. Remark. The proof gives more precisely two weakly (bi) cartesian squares

[figure: e ⊗ e′ = e ∧ e′; e; e′; e ∨ e′; and; f ∧ f′; f; f′; f ∨ f′ = f ⊗ f′]

in 𝒯, where all the maps are morphisms of idempotents, as given by Proposition 3.11. This characterizes those maps uniquely by Corollary 3.7.

Always walking in Rickard’s path (see [33, Cor. 8.2] and its proof) we deduce a generalized version of a celebrated theorem of Jon Carlson (see [18]).

3.15. Corollary (Generalised Carlson Theorem). Let 𝒮, 𝒮′ be smashing ideals in 𝒯.

(a) If 𝒮 ∩ 𝒮′ = 0, then we have e(𝒮 ∨ 𝒮′) ≃ e(𝒮) ⊕ e(𝒮′) and 𝒮 ∨ 𝒮′ = 𝒮 ⊕ 𝒮′, that is, any t ∈ 𝒮 ∨ 𝒮′ is decomposable as t ≃ s ⊕ s′ where s ∈ 𝒮 and s′ ∈ 𝒮′.

(b) If 𝒮 ∨ 𝒮′ = 𝒯, then f(𝒮 ∩ 𝒮′) ≃ f(𝒮) ⊕ f(𝒮′) and (𝒮 ∩ 𝒮′)^{⊥} = 𝒮^{⊥} ⊕ (𝒮′)^{⊥}.

Proof. When 𝒮∩𝒮′ = 0, we have e(𝒮∧𝒮′) = e(0) = 0 and e(𝒮∨𝒮′) ≃ e(𝒮)⊕e(𝒮′) by the Mayer-Vietoris triangle in Thm. 3.13 (a). For t ∈ 𝒮∨𝒮′, we have t ≃ t⊗e(𝒮∨𝒮′), hence t ≃ (t ⊗ e(𝒮)) ⊕ (t ⊗ e(𝒮′)) ∈ 𝒮 ⊕ 𝒮′, which gives (a). Part (b) is dual. □

4. Inflating and Telescope

So far, we have used the triangular structure of 𝒯 (for localization) and the ⊗- structure (for smashing and idempotents). We now throw in the assumption that 𝒯 is also compactly generated. The main device throughout the following chapters is what we informally call inflation, that is the assignment of the smashing ideal ⟨𝒞⟩ ⊂ 𝒯 (Notation 2.2) to every thick ideal 𝒞 ⊂ 𝒦 of compact objects.

4.1. Theorem (Miller, Neeman). Under Hypotheses 1.1, let 𝒞 be a thick ⊗-ideal of 𝒦 = 𝒯^{c}. Then, we have :

[figure: ⟨𝒞⟩; ⟨𝒦⟩ = 𝒯; 𝒯/⟨𝒞⟩; 𝒞; 𝒦; 𝒦/𝒞]

(a) ⟨𝒞⟩ is a smashing ⊗-ideal of 𝒯 and ⟨𝒞⟩^{c} = ⟨𝒞⟩ ∩ 𝒦 = 𝒞.

(b) 𝒯/⟨𝒞⟩ has small hom sets and is a compactly generated ⊗-triangulated category in the sense of Hypotheses 1.1.

(c) 𝒦/𝒞 fully faithfully embeds into the compact objects of 𝒯/⟨𝒞⟩ and the thick closure of 𝒦/𝒞 is exactly (𝒯/⟨𝒞⟩)^{c}.

Proof. (a) is Miller’s theorem, see [29] or [22, Thm. 3.3.3]. Parts (b) and (c) are Neeman’s results [31, Cor. 4.4.3, Thm. 4.4.9]. This is also nicely done in [26, Thm. 5.6.1]. In order to fulfil our Hypotheses 1.1, 𝒯̄ := 𝒯/⟨𝒞⟩ must also carry a closed ⊗- structure. The tensor on 𝒯̄ is induced by that of 𝒯 since ⟨𝒞⟩ is a ⊗-ideal. Since 𝒯̄ is compactly generated it follows that this ⊗-structure is closed as well. Explicitly, let r : 𝒯̄ ⟶ ⟨𝒞⟩^{⊥} ⊂ 𝒯 be a right-adjoint right-inverse of the localization functor q : 𝒯 ⟶ 𝒯̄. Then the tensor on 𝒯̄ is given by x ⊗ y = q(r(x) ⊗ r(y)) and the internal hom can be given by (x, z) ↦ q(\underline{hom}(r(x), r(z))). Indeed, we have

𝒯̄(x ⊗ y, z) = 𝒯̄(q(r(x) ⊗ r(y)), z) ≅ 𝒯(r(x) ⊗ r(y), r(z)) ≅ 𝒯(r(y), \underline{hom}(r(x), r(z)))
≅ 𝒯(r(y), rq(\underline{hom}(r(x), r(z)))) ≅ 𝒯̄(y, q(\underline{hom}(r(x), r(z))))

where the ≅ at the beginning of the second row uses the fact that \underline{hom}(𝒯, ⟨𝒞⟩^{⊥}) ⊂ ⟨𝒞⟩^{⊥}, which holds since ⟨𝒞⟩ is ⊗-ideal. Since q is a ⊗-functor, it preserves rigid objects by [27, Prop. 1.9]. Therefore, by (c), all compact objects of 𝒯̄ are rigid. By (c) again, the unit 𝟙𝒯̄ = q(𝟙_{𝒯}) remains compact in 𝒯̄. □

4.2. Definition. Under Hypotheses 1.1, we say that the Telescope Conjecture holds for 𝒯 if every smashing ⊗-ideal 𝒮 of 𝒯 is inflated, i.e. 𝒮 = ⟨𝒞⟩ where 𝒞 is a thick ⊗-ideal of 𝒦 (necessarily equal to 𝒮 ∩ 𝒦). We often just write “TC holds for 𝒯”.

4.3. Remark. The Telescope Conjecture, also known as the Smashing Conjecture, was first formulated in topology for 𝒯 = SH, see Bousfield [15, 3.4] and Ravenel [32, 1.33], where it remains open. See a modern approach in Krause [25]. The conjecture fails for general compactly generated triangulated categories, see Keller [24].

4.4. Proposition. Under Hypotheses 1.1, let 𝒮 ⊂ 𝒯 be a smashing ideal. If TC holds for 𝒯 then TC holds for 𝒯/𝒮.

Proof. Since TC holds for 𝒯, we have 𝒮 = ⟨𝒞⟩ for 𝒞 := 𝒮 ∩ 𝒦 ⊂ 𝒦 and we know that 𝒯/⟨𝒞⟩ is compactly generated by Thm. 4.1. Let now 𝒮̄ be a smashing ⊗-ideal of 𝒯/𝒮. It is easy to see that there is a smashing ⊗-ideal 𝒮′ of 𝒯, that contains 𝒮, whose image under q : 𝒯 → 𝒯/𝒮 is 𝒮̄ (Rem. 2.17). Since TC holds for 𝒯, we have that 𝒮′ = ⟨𝒞′⟩ for 𝒞′ = 𝒮′ ∩ 𝒦. Then 𝒮̄ = q(⟨𝒞′⟩) = ⟨q(𝒞′)⟩ is easy to check. □

Recall from Prop. 3.11 that the join 𝒮 ∨ 𝒮′ is defined to be Ker(− ⊗ f(𝒮) ⊗ f(𝒮′)) and is the smallest smashing ideal containing both smashing ideals 𝒮 and 𝒮′.

4.5. Lemma. Under Hypotheses 1.1, let 𝒞, 𝒞′ be thick ⊗-ideals of 𝒦 = 𝒯^{c}. Then

(a) ⟨𝒞⟩ ∩ ⟨𝒞′⟩ = ⟨𝒞 ∩ 𝒞′⟩.

(b) If 𝒞′′ ⊂ 𝒦 is the thick ⊗-ideal generated by 𝒞 and 𝒞′, then ⟨𝒞′′⟩ = ⟨𝒞⟩∨⟨𝒞′⟩.

(c) If 𝒞 ⊥ 𝒞′ (Def. 2.5) then ⟨𝒞⟩ ⊥ ⟨𝒞′⟩.

(d) If 𝒞 ⊥ 𝒞′ and 𝒞′ ⊥ 𝒞 then ⟨𝒞 ⊕ 𝒞′⟩ = ⟨𝒞⟩ ⊕ ⟨𝒞′⟩

Proof. The non-trivial inclusion ⟨𝒞⟩ ∩ ⟨𝒞′⟩ ⊂ ⟨𝒞 ∩ 𝒞′⟩ in (a) uses the equality

(4.6)  ⟨𝒞⟩ = \{ t ∈ 𝒯 | \begin{matrix}for every f : x → t with x ∈ 𝒦, there exists c ∈ 𝒞 \\ and a factorization & [figure: x; f; t; c] \end{matrix}\}

(see [31, Thm. 4.3.3]) and the rigid ⊗-structure on 𝒦. Indeed, using (4.6) twice, we reduce to show that any morphism g : c → c′ in 𝒦 with c ∈ 𝒞 and c′ ∈ 𝒞′ factors by an object of 𝒞 ∩ 𝒞′. If h : 𝟙 → \underline{hom}(c, 𝟙) ⊗ c′ is the adjoint of g, then g factors as

\xymatrix{c \ar[rr]^{g} \ar[dr]_{𝟙⊗h} &  & c′ \\  & c ⊗ \underline{hom}(c, 𝟙) ⊗ c′ \ar[ur]_{ϵ⊗𝟙}}

where ϵ is the counit of the adjunction. Of course c ⊗ \underline{hom}(c, 𝟙) ⊗ c′ ∈ 𝒞 ∩ 𝒞′. This gives (a). Both inclusions in (b) are very easy. For (c), note that (4.6) implies 𝒞 ⊂ ^{⊥}⟨𝒞′⟩ and then use that ^{⊥}⟨𝒞′⟩ is localizing. Finally, the non-trivial inclusion ⟨𝒞 ⊕ 𝒞′⟩ ⊂ ⟨𝒞⟩⊕⟨𝒞′⟩ in (d) comes from the fact that ⟨𝒞⟩⊕⟨𝒞′⟩ is localizing (triangulated is the issue). In fact, ⟨𝒞⟩ ⊕ ⟨𝒞′⟩ = ⟨𝒞⟩ ∨ ⟨𝒞′⟩ by Corollary 3.15 (a). □

5. Tensor idempotents and geometry

We quickly recall elements of tensor triangular geometry from [2], [3] and [7].

5.1. Definition (See [2]). Let (𝒦, ⊗, 𝟙) be an essentially small ⊗-triangulated category. A prime ideal 𝒫 ⊊ 𝒦 is a proper thick ⊗-ideal such that a ⊗ b ∈ 𝒫 forces a ∈ 𝒫 or b ∈ 𝒫. The spectrum Spc(𝒦) is the set of prime ideals 𝒫 ⊂ 𝒦. The support of an object a ∈ 𝒦 is defined as the subset supp(a) = {𝒫 ∈ Spc(𝒦) | a ∉ 𝒫}. The complements U(a) = {𝒫 ∈ Spc(𝒦) | a ∈ 𝒫} of these supports form a basis {U(a)}_{a∈𝒦} of the so-called Zariski topology on the spectrum.

5.2. Examples. Here are the spectra Spc(𝒦) for the standard 𝒦 = 𝒯^{c} of Exas 1.2.

(1) For 𝒦 = SH^{fin}, one has Spc(𝒦) = {SH_{tor}^{fin}} ∪ { 𝒫_{p,n} | p prime and 1 ≤ n ≤ ∞ }, where 𝒫_{p,n} is the kernel of the n^{th} Morava K-theory at p for n < ∞, where 𝒫_{p,∞} = ∩_{n≥1}𝒫_{p,n} = Ker(SH^{fin} → SH_{(p)}^{fin}) and where SH_{tor}^{fin} is the category of torsion spectra (which is also 𝒫_{p,0} for all p). The main reference is Hopkins-Smith [21], with explanations in [5, § 9].

(2) For 𝒦 = D^{perf}(X), one recovers Spc(𝒦) ≅ X itself. See [2, Thm. 6.3].

(3) For 𝒦 = stab(kG), one has Spc(𝒦) ≅ Proj(H^{•}(G, k)), which is the projective support variety V_{G} of G over k. See [2, Thm. 6.3] as well.

5.3. Remark. The pair (Spc(𝒦), supp) is a support datum in the sense of [2, Def. 3.1], i.e. supp(0) = ∅, supp(𝟙) = Spc(𝒦), supp(a ⊕ b) = supp(a) ∪ supp(b), supp(Σa) = supp(a), supp(c) ⊂ supp(a) ∪ supp(b) for every exact triangle a → b → c → Σa and finally supp(a ⊗ b) = supp(a) ∩ supp(b). All this for a, b, c ∈ 𝒦.

5.4. Remark. We have \overline{\{𝒫\}} = \{ 𝒬 ∈ Spc(𝒦) | 𝒬 ⊂ 𝒫 \} by [2, Prop. 2.9].

5.5. Definition. Let Y ⊂ Spc(𝒦). We denote by 𝒦_{Y} the thick ⊗-ideal of 𝒦

(5.6)  𝒦_{Y} := {a ∈ 𝒦 | supp(a) ⊂ Y }

of objects supported on Y. It is radical, that is, it contains a^{⊗n} only if it contains a.

5.7. Definition. For a topological space X, a subset Y of the form Y = ∪_{i∈I}Y_{i} where all Y_{i} are closed and have quasi-compact complement, is called a Thomason subset of X (or dual open). We write Th(X) for the set of Thomason subsets of X.

5.8. Examples. In Examples 1.2, when 𝒦 = D^{perf}(X) for X a noetherian scheme, or when 𝒦 = stab(kG) for G a finite group, then a Thomason subset Y ⊂ Spc(𝒦) is simply a specialization-closed one, meaning y ∈ Y ⇒ \overline{\{y\}} ⊂ Y for all y ∈ Y, or equivalently, Y is a union of closed subsets. Compare Proposition 7.13 below. For the other running example, 𝒦 = SH^{fin}, a Thomason subset is a union of closed subsets of the form \overline{\{𝒫_{p,n}\}} for n finite. One excludes \overline{\{𝒫_{p,∞}\}} = \{𝒫_{p,∞}\} because its open complement is not quasi-compact. See more in Section 7.

5.9. Theorem (Classification Theorem, [2, Thm. 4.10]). Let 𝒦 be a ⊗-triangulated category as above. Let ℝ(𝒦) be the set of all thick radical ⊗-ideals of 𝒦. Then there is a bijection Th(Spc(𝒦)) → ℝ(𝒦) mapping every Thomason subset Y ⊂ Spc(𝒦) to 𝒦_{Y}, see (5.6). Its inverse is given by 𝒥 ↦ supp(𝒥) := ∪_{a∈𝒥} supp(a).

5.10. Remark. When 𝒦 is rigid (as it is for our 𝒦 = 𝒯^{c}), every thick ⊗-ideal is automatically radical by [3, Prop. 4.2]. So “radical” is irrelevant for us here.

5.11. Remark. The space Spc(𝒦) is always spectral in the sense of Hochster [20], i.e. it has a basis of quasi-compact open subsets and every irreducible subset of X has a unique generic point (in particular X itself is assumed quasi-compact and it is T_{0}). Note that every quasi-compact open subset U of a spectral space X is again spectral. To each spectral space X one can associate its Hochster dual X^{∗} which is again a spectral space. The underlying space of X^{∗} is the set X itself, while the dual-open subsets are given by the collection of what we called Thomason subsets of X (Def. 5.7). Hochster proves that (X^{∗})^{∗} = X in [20, Prop. 8].

5.12. Definition. Under Hypotheses 1.1, let U be a quasi-compact open subset of Spc(𝒦) and Z its closed complement. We set

𝒯(U) := 𝒯/⟨𝒦_{Z}⟩  and  𝒦(U) = 𝒯(U)^{c}.

We call 𝒯(U) the category 𝒯 on U. We will denote by res_{U} : 𝒯 → 𝒯(U) the corresponding localization functor. The assignment U ↦ 𝒯(U) is a “presheaf” of triangulated categories on Spc(𝒦). By Theorem 4.1, 𝒯(U) is a compactly generated ⊗-triangulated category whose subcategory of compact objects 𝒦(U) is the idempotent completion of 𝒦/𝒦_{Z}. We have Spc(𝒦(U)) ≅ U by [7, Prop. 1.11].

5.13. Remark. If 𝒯 = D(X) for a quasi-compact quasi-separated scheme X, see Ex. 1.2 (2), and if U is a quasi-compact open of X ≃ Spc(D^{perf}(X)), then the above category 𝒯(U) is nothing but D(U). Indeed, D(U) ≅ D(X)/ D_{Z}(X), where D_{Z}(X) is the subcategory of D(X) consisting of complexes supported on Z = X ∖ U; see Thomason-Trobaugh [36] or Jørgensen [23, Thm. 1]. The key point is now that ⟨D_{Z}^{perf}(X)⟩ = D_{Z}(X), as proven in Rouquier [34, Thm. 6.8] for instance.

5.14. Theorem. Under Hypotheses 1.1, let Y, Z ⊂ Spc(𝒦) be disjoint Thomason subsets. Then ⟨𝒦_{Y} ⟩ and ⟨𝒦_{Z}⟩ are in a formal Mayer-Vietoris situation (Def. 2.10). Hence we have ⟨𝒦_{Y} ⟩ ⊕ ⟨𝒦_{Z}⟩ = ⟨𝒦_{Y} ⊕ 𝒦_{Z}⟩ = ⟨𝒦_{Y ∪Z}⟩.

Proof. Since 𝒦 is rigid, [3, Cor. 2.8 and Thm. 2.11] give 𝒦_{Y} ⊥ 𝒦_{Z} ⊥ 𝒦_{Y} and 𝒦_{Y ∪Z} = 𝒦_{Y} ⊕ 𝒦_{Z}. The result follows from Lemma 4.5 (c) and (d). □

5.15. Corollary. If Spc(𝒦) = U_{1} ∪ U_{2} with U_{i} quasi-compact open, i = 1, 2, then

\xymatrix{𝒯 \ar[r] \ar[d] & 𝒯(U_{1}) \ar[dr] \\ 𝒯(U_{2}) \ar[rr] &  & 𝒯(U_{1} ∩ U_{2}) .}

satisfies gluing of objects and morphisms as in Theorems 2.11 and 2.12. □

5.16. Definition. Under Hypotheses 1.1, let Y ⊂ Spc 𝒦 be a Thomason subset. Set

𝒯_{Y} := ⟨𝒦_{Y} ⟩ =^{(5.6)} ⟨{ a ∈ 𝒦 | supp(a) ⊂ Y }⟩ ⊂ 𝒯 .

By Thm. 4.1, 𝒯_{Y} is a smashing ideal of 𝒯. We define e(Y) := e(𝒯_{Y}) and f(Y) := f(𝒯_{Y}). We call these the (left and right) tensor idempotents associated to the subset Y ⊂ Spc(𝒦). As usual, they are characterized by the exact triangle

(5.17)  Δ(Y) := ( \xymatrix{e(Y) \ar[r] & 𝟙 \ar[r] & f(Y) \ar[r] & Σe(Y)} )

such that e(Y) ∈ 𝒯_{Y} and f(Y) ∈ (𝒯_{Y})^{⊥}. We have e(Y) ⊗ f(Y) = 0 and 𝒯_{Y} = Im(− ⊗ e(Y)) = Ker(− ⊗ f(Y)); see Thm. 3.5.

The following result generalizes Prop. 6.2, Thm. 7.5 and Thm. 8.1 in [33].

5.18. Theorem (Geometric Mayer-Vietoris triangles). Under Hypotheses 1.1, let Y_{1}, Y_{2} ⊂ Spc(𝒦) be Thomason subsets. There are unique isomorphisms of left idempotents

e(Y_{1} ∩ Y_{2}) ≅ e(Y_{1}) ⊗ e(Y_{2})  and  e(Y_{1} ∪ Y_{2}) ≅ e(Y_{1}) ∨ e(Y_{2})

in 𝒯. Similarly, there are unique isomorphisms of right idempotents

f(Y_{1} ∩ Y_{2}) ≅ f(Y_{1}) ∧ f(Y_{2})  and  f(Y_{1} ∪ Y_{2}) ≅ f(Y_{1}) ⊗ f(Y_{2}) .

Finally, there exist exact triangles in 𝒯:

e(Y_{1} ∩ Y_{2}) → e(Y_{1}) ⊕ e(Y_{2}) → e(Y_{1} ∪ Y_{2}) → Σ(e(Y_{1} ∩ Y_{2})) and
f(Y_{1} ∩ Y_{2}) → f(Y_{1}) ⊕ f(Y_{2}) → f(Y_{1} ∪ Y_{2}) → Σ(f(Y_{1} ∩ Y_{2})).

Proof. First note that by definition, 𝒦_{Y_{1}∩Y_{2}} = 𝒦_{Y_{1}} ∩ 𝒦_{Y_{2}}. By Lemma 4.5 (a), it follows that ⟨𝒦_{Y_{1}∩Y_{2}}⟩ = ⟨𝒦_{Y_{1}}⟩ ∩ ⟨𝒦_{Y_{2}}⟩ = ⟨𝒦_{Y_{1}}⟩ ∧ ⟨𝒦_{Y_{2}}⟩. On the other hand, by the Classification Theorem 5.9, the thick ⊗-ideal of 𝒦 generated by 𝒦_{Y_{1}} and 𝒦_{Y_{2}} must be 𝒦_{Y_{1}∪Y_{2}}, hence ⟨𝒦_{Y_{1}∪Y_{2}}⟩ = ⟨𝒦_{Y_{1}}⟩ ∨ ⟨𝒦_{Y_{2}}⟩ by Lemma 4.5 (b). We have proved :

(5.19)  𝒯_{Y_{1}∩Y_{2}} = 𝒯_{Y_{1}} ∧ 𝒯_{Y_{2}}  and  𝒯_{Y_{1}∪Y_{2}} = 𝒯_{Y_{1}} ∨ 𝒯_{Y_{2}} .

Proposition 3.11 translates this into the four stated isomorphisms of idempotents, which can then be used in the abstract Mayer-Vietoris triangles of Thm. 3.13. □

5.20. Remark. Following-up on Remark 3.14, we have more precisely constructed two weakly (bi) cartesian squares in 𝒯:

[figure: e(Y_{1} ∩ Y_{2}); e(Y_{1}); e(Y_{2}); e(Y_{1} ∪ Y_{2}); and; f(Y_{1} ∩ Y_{2}); f(Y_{1}); f(Y_{2}); f(Y_{1} ∪ Y_{2})]

in which all the morphisms are the unique morphisms of idempotents induced by the four inclusions : Y_{1} ∩ Y_{2} ⊂ Y_{i} and Y_{i} ⊂ Y_{1} ∪ Y_{2} for i = 1, 2.

5.21. Remark. Let ρ : Spc(𝒦) → X be a map to a spectral topological space X, see Rem. 5.11. Assume that ρ is spectral (i.e. is continuous for the given topologies and for their dual). Then this yields left and right idempotents in 𝒯 by sending any Thomason subset W ⊂ X to e(ρ^{−1}(W)) and f(ρ^{−1}(W)). Mayer-Vietoris triangles in this setting are immediate consequences of Theorem 5.18 and the trivial fact that W ↦ ρ^{−1}(W) commutes with union and intersection.

5.22. Example. Following [33, § 6], we can describe the idempotents e(Y) and f(Y) when Y ⊂ Spc(𝒦) is principal in the following sense. Let ζ : 𝟙 → Σ^{d}(𝟙) be a homogeneous element of degree d ∈ ℤ in the graded central ring

R𝒦^{•} := End^{•}(𝟙) = Hom_{𝒦} (𝟙, Σ^{•}𝟙)

using notation of [5]. Then the closed subset V (ζ) = { 𝔭 | ζ ∈ 𝔭 } in the homogeneous spectrum Spec^{h}(R𝒦^{•} ) defines a closed subset (ρ^{•})^{−1}(V (ζ)) ⊂ Spc(𝒦), where ρ^{•} : Spc(𝒦) → Spec^{h}(R𝒦^{•} ) is the spectral map defined in [5, Def. 5.1]. By [5, Thm. 5.3], this closed subset is simply supp(cone(ζ)) ⊂ Spc(𝒦), the support of the cone of the chosen ζ. Let Δ(ζ) := Δ(supp(cone(ζ))) be the corresponding idempotent triangle

(5.23)  Δ(ζ) = ( \xymatrix{e(ζ) \ar[r] & 𝟙 \ar[r] & f(ζ) \ar[r] & Σe(ζ)} )

as in (5.17) for Y = supp(cone(ζ)). Here is an alternative description of f(ζ).

5.24. Theorem. Under Hypotheses 1.1, let ζ : 𝟙 → Σ^{d}(𝟙), d ∈ ℤ. Consider the homotopy colimit hocolim_{i} Σ^{id}(𝟙) (see [31, Def. 1.6.4]) of the following sequence :

𝟙⟶^{ζ}Σ^{d}(𝟙)⟶^{ζ}Σ^{2d}(𝟙) ⟶ ⋯ ⟶ Σ^{id}(𝟙)⟶^{ζ}Σ^{(i+1)d}(𝟙) ⟶ ⋯

Then 𝟙 ⟶ hocolim_{i} Σ^{id}(𝟙) is a right idempotent isomorphic to 𝟙 → f(ζ).

Proof. Let Y := supp(cone(ζ)). Let ζ^{∞} : 𝟙 → hocolim_{i} Σ^{id}(𝟙) =: h be the map of

the statement and choose an exact triangle [figure: g; 𝟙; ζ^{∞}; h; Σg] . It is easy to check that ζ ⊗ id_{h} : h →^{∼} Σ^{d}(h) is an isomorphism, using [31, Lem. 1.7.1] for instance. Hence cone(ζ) ⊗ h = 0. Then the localizing subcategory Ker(− ⊗ h) ⊂ 𝒯 contains cone(ζ), hence also contains ⟨cone(ζ)⟩ = ⟨𝒦_{Y} ⟩ = 𝒯_{Y} ∋ e(Y). So, e(Y) ⊗ h = 0, that is, h ∈ Ker(− ⊗ e(Y)) = (𝒯_{Y})^{⊥}. From the commutative diagram

[figure: 𝟙; ζ; 𝟙; ζ^{2}; 𝟙; ζ^{i−1}; 𝟙; ζ^{i}; 𝟙; ζ^{i+1}; ⋯; 𝟙; ζ^{∞}; Σ^{d}(𝟙); ζ; Σ^{2d}(𝟙); Σ^{(i−1)d}(𝟙); ζ; Σ^{id}(𝟙); ζ; Σ^{(i+1)d}(𝟙); ⋯; h]

one proves easily that cone(ζ^{∞}) belongs to ⟨{ cone(ζ^{i}) | i ∈ ℕ }⟩ = ⟨cone(ζ)⟩ = 𝒯_{Y}. Hence g ∈ 𝒯_{Y}. In short, we have an exact triangle g → 𝟙 → h → Σg with g ∈ 𝒯_{Y} and h ∈ (𝒯_{Y})^{⊥}. This characterizes the triangle Δ(𝒯_{Y}) = Δ(Y) = Δ(ζ), see (3.6), (5.17), (5.23). In particular, h ≅ f(ζ) as claimed. □

5.25. Remark. When (the spectrum of) the ring R𝒦^{•} = End^{•}(𝟙) is noetherian, one can extend the above definition of e(ζ) and f(ζ) to any closed W ⊂ Spec^{h}(R𝒦^{•} ) by choosing ζ_{1}, …, ζ_{r} such that W = ∩_{i=1}^{r}V (ζ_{i}) and setting e(W) = e(ζ_{1}) ⊗ ⋯ ⊗ e(ζ_{r}) and f(W) = f(ζ_{1}) ∧ ⋯ ∧ f(ζ_{r}) or equivalently by setting directly e(W) = e(Y) and f(W) = f(Y) where Y = (ρ^{•})^{−1}(W) is the preimage of W in Spc(𝒦) under the continuous map ρ^{•} : Spc(𝒦) → Spec^{h}(R𝒦^{•} ) of [5]. These coincide; see Rem. 5.21.

5.26. Remark. There are other generalizations of Rickard’s idempotents on the market, mostly using the spectrum of some ring acting on 𝒯, instead of our Spc(𝒯^{c}). For instance, Hovey et al. [22, Def. 6.1.6] consider supports in the spectrum of the endomorphism ring of the unit of 𝒯. See also Benson-Iyengar-Krause [11] for a graded version. Actually, any support datum on 𝒦 = 𝒯^{c} (see Rem. 5.3), say {σ(a) ⊂ X}_{a∈𝒦} in some space X, allows an ad hoc definition of the subcategory 𝒦_{W} as { a ∈ 𝒦 | σ(a) ⊂ W } for W ⊂ X. One could then apply the machinery to 𝒯_{W} := ⟨𝒦_{W} ⟩. Yet, it is a theorem that the best support datum on 𝒦 is the one provided by Spc(𝒦); see [2, Thm. 3.2]. Any variation is at most as good as this one.

Specifically, Theorem 5.24 shows how the tensor idempotents e(ζ) and f(ζ) coming from the homogenous spectrum of the central graded ring R𝒦^{•} = End𝒦^{•} (𝟙) are just special cases of our general construction. Note that Spec^{h}(R𝒦^{•} ) fails to recover the spectrum Spc(𝒦) for general 𝒯, as illustrated in topology; see [5, § 9]. So, Spc(𝒦) is strictly better than Spec^{h}(R𝒦^{•} ).

Another variation on the theme would be to consider support data {σ(t) ⊂ X}_{t∈𝒯} on the whole of 𝒯 and to define 𝒯_{W} boldly as { t ∈ 𝒯 | σ(t) ⊂ W }. This is simply useless unless we can decide when 𝒯_{W} is smashing.

In conclusion, the balanced path seems to follow Rickard’s original idea, rooting back to Miller’s theorem in topology, i.e. to create smashing subcategories of 𝒯 by inflating thick subcategories of 𝒯^{c} and parametrizing the latter with some subsets of some topological space, the best choice for such a space being Spc(𝒯^{c}).

6. Rickard map and Telescope Conjecture

6.1. Definition. Under Hypotheses 1.1, recall that Th(Spc 𝒦) is the set of Thomason subsets of the spectrum of the subcategory of compact objects 𝒦 = 𝒯^{c} (Def. 5.7) and that 𝔻(𝒯) is the set of idempotent triangles in 𝒯 (Def. 3.2). We define the Rickard map of 𝒯 as the assignment of Definition 5.16

Δ : Th(Spc 𝒦) ⟶ 𝔻(𝒯)
Y ⟼ Δ(Y) = ( \xymatrix{e(Y) \ar[r] & 𝟙 \ar[r] & f(Y) \ar[r] & Σe(Y)} )

of an idempotent triangle Δ(Y) to every Thomason subset Y ⊂ Spc(𝒦). Recall that Δ(Y) is the localization triangle of the smashing ideal 𝒯_{Y} := ⟨𝒦_{Y} ⟩ inflated from the thick ideal 𝒦_{Y} = { a ∈ 𝒦 | supp(a) ⊂ Y } of compact objects supported on Y. So, Δ(Y) is an exact triangle, with left idempotent e(Y) in 𝒯_{Y} and right idempotent f(Y) such that − ⊗ f(Y) : 𝒯 → 𝒯 gives localization modulo 𝒯_{Y}.

6.2. Proposition. Under Hypotheses 1.1, the Rickard map Th(Spc 𝒦) ⟶^{Δ} 𝔻(𝒯) is always injective. It is surjective if and only if the Telescope Conjecture holds for 𝒯.

Proof. The Rickard map Δ is the composition of the following three maps :

Th(Spc 𝒦) ⟶^{∼} ℝ(𝒦) ,  ℝ(𝒦) ⟶ 𝕊(𝒯)  and  𝕊(𝒯) ⟶^{∼} 𝔻(𝒯)
Y  ⟼ 𝒦_{Y}  𝒞 ⟼ ⟨𝒞⟩  𝒮 ⟼ Δ(𝒮) .

Here, ℝ(𝒦) is the set of (radical) thick ⊗-ideals of 𝒦, as in the Classification Theorem 5.9, which also gives the bijectivity of the first map; see also Rem. 5.10. The second map is the “inflation” of Theorem 4.1 (a), which also provides injectivity : actually a retraction 𝕊(𝒯) → ℝ(𝒯) is given by 𝒮 ↦ 𝒮 ∩ 𝒦. The surjectivity of the second map is exactly the Telescope property of Definition 4.2. Finally, the last map is the bijection of Theorem 3.5. □

The above simple result is the key to the understanding of the Telescope Conjecture in a local way. But first, let us express the naturality of our Rickard map.

6.3. Theorem. Let 𝒯 and 𝒯′ be compactly generated tensor triangulated as in Hypotheses 1.1 and let 𝒦 = 𝒯^{c} and 𝒦′ = (𝒯′)^{c}. Let F : 𝒯 → 𝒯′ be a ⊗-triangulated functor such that F (𝒦) ⊂ 𝒦′ and such that F preserves small coproducts. (Equivalently, F has a right adjoint which itself preserves small coproducts, see [26, Prop. 5.3.1 and Lem. 5.4.1].) Then the induced map Φ := Spc(F) : Spc(𝒦′) → Spc(𝒦) induces another map Th(Spc 𝒦) → Th(Spc 𝒦′), defined by Y ↦ Φ^{−1}(Y), that we simply denote F. Moreover the following diagram commutes :

\xymatrix{Th(Spc 𝒦) \ar[r]^{Δ} \ar[d]_{F} & 𝔻(𝒯) \ar[d]^{F} \\ Th(Spc 𝒦′) \ar[r]^{Δ} & 𝔻(𝒯′) .}

Proof. The continuous map Φ = Spc(F) : Spc(𝒦′) → Spc(𝒦) is spectral (cf. Rem. 5.21) which amounts to say that the preimage under Φ of a quasi-compact open is quasi-compact; to check this, recall from [2, Prop. 2.14] that the quasicompact open subsets of Spc(𝒦) are the U(a) = Spc(𝒦) ∖ supp(a), for a ∈ 𝒦, and from [2, Prop. 3.6] the first equality below (the second equality is just the notation) :

(6.4)  supp(F (a)) = Φ^{−1}(supp(a)) = F (supp(a)) .

It follows that the preimage under Φ of every Thomason subset of Spc(𝒦) is still Thomason; see Def. 5.7. Hence the map denoted F : Th(Spc 𝒦) → Th(Spc 𝒦′) and given by Y ↦ Φ^{−1}(Y) is well-defined. Note that F : Th(Spc 𝒦) → Th(Spc 𝒦′) preserves inclusion. The obvious map F : 𝔻(𝒯) → 𝔻(𝒯′) sends an idempotent triangle Δ = (e → 𝟙 → f → Σe) to F (Δ) := (F (e) → 𝟙 → F (f) → ΣF (e)), which is still idempotent since F (e) ⊗ F (f) ≃ F (e ⊗ f) = F (0) = 0; see Def. 3.2.

Let us prove the last statement. Let Y ⊂ Spc(𝒦) be a Thomason subset and

Δ(Y) = ( \xymatrix{e(Y) \ar[r] & 𝟙 \ar[r] & f(Y) \ar[r] & Σe(Y)} )

the associated idempotent triangle (5.17). Let 𝒥′ ⊂ 𝒦′ the thick ⊗-ideal of 𝒦′ generated by F (𝒦_{Y}). By (6.4), 𝒥′ ⊂ 𝒦′_{F (Y)}. Therefore supp(𝒥′) ⊂ F (Y). Conversely, for any a ∈ 𝒦_{Y}, we have F (a) ∈ 𝒥′ hence supp(𝒥′) ⊃ supp(F (a)) = F (supp(a)) by (6.4) again. Hence F (Y) = F (supp(𝒦_{Y})) ⊂ ∪_{a∈𝒦_{Y}} F (supp(a)) ⊂ supp(𝒥′). In conclusion, supp(𝒥′) = F (Y), which proves that 𝒥′ = 𝒦′_{F (Y)}, by the Classifi-

cation Theorem 5.9. Therefore, in 𝒯′, we have ⟨F (𝒦_{Y})⟩ = ⟨𝒦′_{F (Y)}⟩ =^{def.} 𝒯′_{F (Y)}. Hence F (𝒯_{Y}) = F (⟨𝒦_{Y} ⟩) ⊂ ⟨F (𝒦_{Y})⟩ = 𝒯′_{F (Y)}. In particular, F (e(Y)) ∈ 𝒯′_{F (Y)}. On the other hand, 𝒯′_{F (Y)} = ⟨𝒦′_{F (Y)}⟩ = ⟨F (𝒦_{Y})⟩ ⊂ ⟨F (⟨𝒦_{Y} ⟩)⟩ = ⟨F (𝒯_{Y})⟩ = ⟨F (Im(− ⊗ e(Y)))⟩ ⊂ ⟨Im(− ⊗ F (e(Y)))⟩ = Im(− ⊗ F (e(Y))) since the latter is a localizing ideal. Putting everything together, we have 𝒯′_{F (Y)} = Im(− ⊗ F (e(Y))). So, F (e(Y)) is the left idempotent generating the smashing ideal 𝒯′_{F (Y)}, i.e. it is e(𝒯′_{F (Y)}), which is denoted e(F (Y)) in 𝒯′. Consequently, F (Δ(Y)) = Δ(F (Y)). □

6.5. Corollary. Under Hypotheses 1.1, let U ⊂ Spc(𝒦) be quasi-compact open. Recall the localization res_{U} : 𝒯 ⟶ 𝒯(U) of Definition 5.12. Recall that (𝒯(U))^{c} = 𝒦(U) and that Spc(𝒦(U)) ≅ U. Then the following diagram commutes

\xymatrix{Y ∈ \ar@{|->}[d] & Th(Spc 𝒦) \ar[d] \ar[r]^{Δ} & 𝔻(𝒯) \ar[d]^{_{res_{U}}} \\ Y ∩ U ∈ & Th(U) ≅ Th(Spc 𝒦(U)) \ar[r]^{Δ} & 𝔻(𝒯(U))}

Proof. This is a special case of Theorem 6.3. Indeed, the homeomorphism U ≅ Spc(𝒦(U)) is such that supp(res_{U}(a)) = supp(a) ∩ U, see [7, Prop. 1.11]. So the map res_{U} : Th(Spc 𝒦) → Th(Spc 𝒦(U)) of Theorem 6.3 is indeed Y ↦ Y ∩ U. □

We now state the main result of this section which says that in order to prove the Telescope Conjecture it is enough to prove it locally.

6.6. Theorem. Under Hypotheses 1.1, suppose that the spectrum Spc(𝒦) = ⋃_{i∈I} U_{i} is covered by quasi-compact open U_{i}. Then TC holds for 𝒯(U_{i}) for all i ∈ I if and only if TC holds for 𝒯.

Proof. We have already seen ⇐ in Proposition 4.4. For the opposite implication, quasi-compactness of Spc(𝒦) and a simple induction argument reduce the proof to the case Spc 𝒦 = U_{1} ∪ U_{2}. By Corollary 6.5, we have a commutative cube of sets :

(6.7)  [figure: Th(Spc 𝒦); Δ; Th(U_{1}); Δ; 𝔻(𝒯); 𝔻(𝒯(U_{1})); Th(U_{2}); Δ; Th(U_{12}); Δ; 𝔻(𝒯(U_{2})); 𝔻(𝒯(U_{12}))]

where U_{12} := U_{1} ∩ U_{2}, where the maps Δ : Th → 𝔻 are the Rickard maps, where the maps in the back are of the form Y ↦ Y ∩ U_{i} for i ∈ {1, 2, 12} and where the maps in the front are the obvious localizations.

We will show that the two faces in the front and in the back are pullbacks. For the square involving Thomason subsets, it suffices to show that in a spectral

space X, covered by two quasi-compact open X = U_{1} ∪ U_{2}, a subset Y ⊂ X is Thomason if Y ∩ U_{i} is Thomason for i = 1, 2. Since all spaces involved are spectral we can consider the dual topologies (Rem. 5.11). Then, both subsets Y ∩ U_{i} are open in the closed subspaces U_{i}^{∗} of X^{∗} = U_{1}^{∗} ∪U_{2}^{∗} hence Y is open in X^{∗} as wanted.

The square involving idempotents is a pullback by a gluing argument in triangulated categories. We leave the gluing of idempotents as an exercise for we do not really need it below but we check uniqueness. That is, we check that if two idempotent triangles Δ = (e → 𝟙 → f → Σe) and Δ′ = (e′ → 𝟙 → f′ → Σe′) in 𝔻(𝒯) are isomorphic over U_{1} and U_{2}, that is, in 𝒯(U_{i}) for i = 1, 2, then they are already isomorphic in 𝒯. By uniqueness of the isomorphism (Cor. 3.7), the two isomorphisms res_{U_{i}}(Δ) →^{∼} res_{U_{i}}(Δ′) in 𝒯(U_{i}), i = 1, 2, must be equal in 𝒯(U_{12}). Therefore, by Theorem 2.12, there exists an isomorphism φ : f →^{∼} f′, as objects of 𝒯. But Corollary 3.7 implies that Δ ≃ Δ′ as idempotent triangles.

So, the commutative cube (6.7) has cartesian front and back faces. We can now conclude the proof. Suppose that TC holds for both 𝒯(U_{1}) and 𝒯(U_{2}). Then the Rickard map Δ : Th(Spc 𝒦) → 𝔻(𝒯) is surjective by an easy diagram chase in the cube (6.7), using the bijectivity of Δ : Th(U_{i}) → 𝔻(𝒯(U_{i})) for i = 1, 2 and the injectivity only on U_{12}; see Proposition 6.2. □

As an application of Theorem 6.6, we deduce the Telescope Conjecture for noetherian schemes, from the affine case proven by Neeman [30]. Let then X be a noetherian scheme and let 𝒯 = D(X) = D_{Qcoh}(O_{X}), as in Ex. 1.2 (2). (Recall that D(X) ≅ D(Qcoh(X)) when X is moreover separated.) We have seen that 𝒯 is compactly generated and that 𝒦 = 𝒯^{c} = D^{perf}(X). Moreover we know that Spc(D^{perf}(X)) ≃ X, see [2, Cor. 5.6] for X (topologically) noetherian, or [16, Thm. 8.5] in maximal generality, i.e. for X quasi-compact and quasi-separated.

6.8. Corollary (See Alonso et al. [1, Thm. 5.8]). Let X be a noetherian scheme. Then the Telescope Conjecture holds for D(X) = D_{Qcoh}(O_{X}). Consequently every smashing subcategory of D(X) is of the following form, for a unique specialization closed subset Y ⊂ X:

D_{Qcoh,Y} (O_{X}) := { E ∈ D_{Qcoh}(O_{X}) | E_{x} ≃ 0 in D(O_{X,x}) for all x ∈ X ∖ Y } .

Proof. One can cover Spc(𝒦) ≃ X = ∪_{i∈I}U_{i} by (quasi-compact) affine open subschemes U_{i} = Spec(R_{i}), with R_{i} noetherian. By Rem. 5.13 we have that 𝒯(U_{i}) is equivalent to D(U_{i}) ≅ D(R_{i}). Now by a result of Neeman [30, Cor. 3.4] we know that TC holds for each D(R_{i}). We conclude by Theorem 6.6 that TC holds for 𝒯 = D(X). The classification of thick ⊗-ideals in 𝒯^{c} = D^{perf}(X) is due to Thomason [35, Thm. 3.15], as in Thm. 5.9 above. Hence the second part. □

7. Residue objects and supports for big objects

We keep 𝒯 compactly generated ⊗-triangulated as in Hypotheses 1.1 and Spc(𝒦) the spectrum of the subcategory of compact objects 𝒦 = 𝒯^{c}. We now define residue objects and supports for not necessarily compact objects, generalizing [10].

7.1. Remark. In Section 5, we constructed a left and a right idempotent e(Y) and f(Y) fitting in an exact triangle Δ(Y) : e(Y) → 𝟙 → f(Y) → Σe(Y), for every Thomason subset Y ⊂ Spc(𝒦); see (5.17). We can combine left and right to produce

(7.2)  g(Y, Z) := e(Y) ⊗ f(Z)

in 𝒯, for any pair of Thomason subsets Y, Z ⊂ Spc(𝒦). This is neither a right, nor a left idempotent but it is a ⊗-idempotent : g(Y, Z) ⊗ g(Y, Z) ≃ g(Y, Z).

7.3. Notation. Let us write (−)^{c} for the complement of subsets in Spc(𝒦).

7.4. Lemma. Under Hypotheses 1.1, let Y, Y′, Z, Z′ ∈ Th(Spc 𝒦).

(a) Suppose that Y ∩ Z^{c} = Y′ ∩ Z^{c}, then g(Y, Z) ≅ g(Y′, Z).

(b) Suppose that Y ∩ Z^{c} = Y ∩ (Z′)^{c}, then g(Y, Z) ≅ g(Y, Z′).

Proof. Replacing Y′ by Y ∪ Y′, we easily reduce (a) to the case Y ⊂ Y′. Hence ⟨𝒦_{Y} ⟩ ⊂ ⟨𝒦_{Y′}⟩ and we have a (unique) morphism of idempotent triangles

[figure: e(Y); γ; ϵ; 𝟙; λ; f(Y); φ; Σe(Y); Σϵ; e(Y′); γ′; 𝟙; λ′; f(Y′); Σe(Y′) .]

Since Y ∪ Z = (Y ∩ Z^{c}) ∪ Z = (Y′ ∩ Z^{c}) ∪ Z = Y′ ∪ Z, Theorem 5.18 gives an isomorphism f(Y) ⊗ f(Z) ≅ f(Y ∪ Z) = f(Y′ ∪ Z) ≅ f(Y′) ⊗ f(Z) which must be φ ⊗ id_{f(Z)} (Cor. 3.7). Applying − ⊗ f(Z) to the above diagram, it follows that ϵ ⊗ id_{f(Z)} is also an isomorphism e(Y) ⊗ f(Z) →^{∼} e(Y′) ⊗ f(Z) as wanted in (a). The proof of (b) is similar. □

7.5. Corollary. Up to isomorphism in 𝒯, the object g(Y, Z) = e(Y) ⊗ f(Z) only depends on the subset Y ∩ Z^{c} in Spc(𝒦). □

7.6. Remark. Let X be a (spectral) topological space. We could call A ⊂ X tropbeau if it is the intersection of a Thomason subset and the complement of a Thomason subset : A = Y ∩ Z^{c}, with Y, Z ∈ Th(X). Every Thomason is trop-beau. By Corollary 7.5, for A trop-beau, we can define a ⊗-idempotent g(A) in 𝒯 as

(7.7)  g(A) := e(Y) ⊗ f(Z)

for any choice of Y, Z ∈ Th(Spc 𝒦) such that A = Y ∩Z^{c}. It is easy to check that if A, A′ ⊂ Spc(𝒦) are trop-beau subsets then so is A∩A′ and g(A∩A′) ≅ g(A)⊗g(A′) in 𝒯. Indeed, say A = Y ∩ Z^{c} and A′ = Y′ ∩ (Z′)^{c} with Y, Y′, Z, Z′ ∈ Th(Spc 𝒦). Then A ∩ A′ = (Y ∩ Y′) ∩ (Z ∪ Z′)^{c} and Theorem 5.18 gives us that g(A ∩ A′) = e(Y ∩ Y′) ⊗ f(Z ∪ Z′) ≅ e(Y) ⊗ e(Y′) ⊗ f(Z) ⊗ f(Z′) ≅ g(A) ⊗ g(A′).

7.8. Lemma (See [30, Lem. 1.4]). For 𝒫 ∈ Spc(𝒦), the following are equivalent :

(i) The open Spc(𝒦) ∖ \overline{\{𝒫\}} is quasi-compact. (ii) There exists s ∈ 𝒦 such that \overline{\{𝒫\}} = supp(s). (iii) \overline{\{𝒫\}} is a Thomason subset of Spc(𝒦). If this holds, then

(a) The singleton \{𝒫\} is trop-beau (Rem.7.6). More precisely \{𝒫\} = \overline{\{𝒫\}}∩(Y_{𝒫} )^{c} where Y_{𝒫} := { 𝒬 ∈ Spc(𝒦) | 𝒫 ⊄ 𝒬 } is the Thomason subset such that 𝒦_{Y_{𝒫}} = 𝒫, that is, Y_{𝒫} = supp(𝒫); see the Classification Theorem 5.9.

(b) For any s ∈ 𝒦 such that supp(s) = \overline{\{𝒫\}}, we have s ⊗ e(\overline{\{𝒫\}}) ≃ s.

(c) The idempotent g(\{𝒫\}) = e(\overline{\{𝒫\}}) ⊗ f(Y_{𝒫} ) (see (7.7)) is non-zero.

Proof. The equivalence (i)⇔(ii) is [2, Prop. 2.14], (i)⇒(iii) is trivial and (iii)⇒(i) is easy, as in the proof of Prop. 7.13 (iv)⇒(v) below. Now, suppose (i)–(iii) hold. The equality \{𝒫\} = \overline{\{𝒫\}} ∩ (Y_{𝒫} )^{c} follows from \overline{\{𝒫\}} = \{ 𝒬 | 𝒬 ⊂ 𝒫 \} (Rem. 5.4) and from (Y_{𝒫} )^{c} = { 𝒬 | 𝒫 ⊂ 𝒬 }. Moreover, it is easy to check that Y_{𝒫} = ∪_{a∈𝒫} supp(a) = supp(𝒫), as in Theorem 5.9. So Y_{𝒫} is Thomason and we get (a). For (b), since s ∈ 𝒦_{supp(s)} = 𝒦\overline{\{𝒫\}} , we have that s ∈ ⟨𝒦\overline{\{𝒫\}} ⟩ = Im(−⊗e(\overline{\{𝒫\}})). Now, for (c), suppose

ab absurdo that g(\{𝒫\}) = 0. Then 0 = s⊗g(\{𝒫\}) = s⊗e(\overline{\{𝒫\}})⊗f(Y_{𝒫} ) ≃ s⊗f(Y_{𝒫} ) by (b). This implies s ∈ Ker(− ⊗ f(Y_{𝒫} )) = ⟨𝒦_{Y_{𝒫}} ⟩ = ⟨𝒫⟩. So, s ∈ 𝒦 ∩ ⟨𝒫⟩ = 𝒫 (Thm. 4.1 (a)). The latter reads 𝒫 ∉ supp(s) which contradicts supp(s) = \overline{\{𝒫\}}. □

7.9. Definition. Under Hypotheses 1.1, a prime 𝒫 ∈ Spc(𝒦) is called visible if it satisfies the equivalent conditions (i)–(iii) of Lemma 7.8. We define the residue object at 𝒫 as the ⊗-idempotent g({𝒫}) of (7.7), that is :

(7.10)  κ(𝒫) := g(\overline{\{𝒫\}}, Y_{𝒫} ) =^{def.} e(\overline{\{𝒫\}}) ⊗ f(Y_{𝒫} ) .

7.11. Example. Let p be a prime number and 𝒯 = SH_{(p)} the p-local stable homotopy category. As in Exa. 5.2 (1), the primes of 𝒦 = SH_{(p)}^{fin} are exactly

𝒞_{∞} = 0 ⊂ ⋯ ⊂ 𝒞_{n} ⊂ 𝒞_{n−1} ⊂ ⋯ ⊂ 𝒞_{2} ⊂ 𝒞_{1} ⊂ 𝒞_{0}

where 𝒞_{n} = (𝒫_{p,n})_{(p)} ⊂ SH_{(p)}^{fin} is the kernel of the n^{th} Morava K-theory at p, for n ≥ 0, see [5, Thm. 9.1]. There is a unique closed point {𝒞_{∞}} but there is no s ∈ SH_{(p)}^{fin} such that supp(s) = {𝒞_{∞}} for this would mean s ∈ ∩_{n≥1}𝒞_{n} and the latter intersection is zero, contradicting 𝒞_{∞} ∈ supp(s) which reads s ∉ 𝒞_{∞} = 0. In other words, the prime 𝒫 = 𝒞_{∞} is not visible in the sense of Definition 7.9.

7.12. Example. Let R be a local ring and 𝒯 = D(R) as in Ex. 1.2 (2). Under Spec(R) ≃ Spc(𝒦), the maximal ideal 𝔪 ⊂ R corresponds to the triangular prime 𝒫 = 0. So, e(Y_{𝒫} ) ∈ ⟨𝒫⟩ = 0 and f(Y_{𝒫} ) = 𝟙. Hence, κ(𝒫) = e(\overline{\{𝒫\}}) is a left idempotent. It is therefore not to be confused with the residue field R → R/𝔪.

7.13. Proposition. Let X be a spectral topological space (Rem. 5.11). The following are equivalent :

(i) X is noetherian (any non-empty family of closed subsets has a minimal one). (ii) Every open subset of X is quasi-compact. (iii) Thomason subsets of X (Def. 5.7) coincide with specialization closed ones. (iv) For every x ∈ X, the closed subset \overline{\{x\}} is Thomason.

(v) For every x ∈ X, the open subset X ∖ \overline{\{x\}} is quasi-compact.

Proof. (i)⇔(ii) is an easy exercise. To see (ii)⇒(iii), note that a Thomason subset is always specialization closed. Conversely, if Y = ∪_{y∈Y} \overline{\{y\}}, it is Thomason since every X ∖ \overline{\{y\}} is quasi-compact by (ii). The implication (iii)⇒(iv) is trivial. For (iv)⇒(v), if \overline{\{x\}} = ∪_{i∈I}Y_{i} with X ∖ Y_{i} quasi-compact, then x belongs to some Y_{i} hence \overline{\{x\}} = Y_{i} and X ∖\overline{\{x\}} = X ∖Y_{i} is quasi-compact. Let us prove the non-trivial part : (v)⇒(ii). Let F = { Z ⊂ X closed | X ∖ Z is not quasi-compact }. Suppose, ab absurdo, that F ≠ ∅. Let C ⊂ F be totally ordered with respect to inclusion and let Z_{0} = ∩_{Z∈C}Z. Then the open cover X ∖ Z_{0} = ∪_{Z∈C}(X ∖ Z) either has no finite subcover and therefore X ∖ Z_{0} is not quasi-compact, or has a finite subcover in which case X ∖Z_{0} = X ∖Z for some Z ∈ C (since C is totally ordered) and X ∖Z_{0} is therefore not quasi-compact again. In both cases, Z_{0} ∈ F. So Z_{0} is a lower bound for C in F. By Zorn, it follows that F has a minimal element for inclusion. Since X is spectral, this minimal element must be non-empty (X is quasi-compact) and irreducible (the intersection of two quasi-compact opens remains quasi-compact). Hence this minimal element is of the form \overline{\{x\}}. But \overline{\{x\}} ∈ F contradicts (v). □

7.14. Corollary. Every prime 𝒫 in Spc(𝒦) is visible (Def. 7.9) if and only if the spectrum Spc(𝒦) is a noetherian topological space. □

7.15. Example. Inspecting the examples of 1.2, we get :

(1) Spc(SH^{fin}) is not noetherian as already observable p-locally, see Ex. 7.11.

(2) Spc(D^{perf}(X)) ≃ X is noetherian if the scheme X is (topologically) noetherian.

(3) Spc(stab(kG)) ≃ V_{G} is always noetherian, see [8, Vol. II, Thm. 4.2.1].

From now on, we assume Spc(𝒦) noetherian, i.e. every 𝒫 is visible. So we have residue objects κ(𝒫) = e(\overline{\{𝒫\}}) ⊗ f(Y_{𝒫} ) in 𝒯 for all 𝒫 ∈ Spc(𝒦).

7.16. Definition. Under Hypotheses 1.1, suppose that Spc(𝒦) is noetherian. For any object t ∈ 𝒯 we define its (big) support as follows

Supp(t) = {𝒫 ∈ Spc(𝒦) | t ⊗ κ(𝒫) ≠ 0} ,

which we temporarily distinguish from the already defined supp(t) when t ∈ 𝒯^{c}.

7.17. Proposition. Under Hypotheses 1.1, assume that Spc(𝒦) is noetherian. Then the assignment t ↦ Supp(t) satisfies the following properties:

(a) For every compact object x ∈ 𝒦 one has Supp(x) = supp(x).

(b) Supp(0) = ∅, Supp(𝟙) = Spc(𝒦).

(c) Supp(⊔_{i∈I}t_{i}) = ∪_{i∈I} Supp(t_{i}) for any set {t_{i}}_{i∈I} of objects of 𝒯.

(d) Supp(Σt) = Supp(t).

(e) Supp(t) ⊂ Supp(t′) ∪ Supp(t′′) for any exact triangle t → t′ → t′′ → Σt.

(f) Supp(t ⊗ t′) ⊂ Supp(t) ∩ Supp(t′).

Proof. For point (a) we have to prove that for a prime 𝒫 and for a compact object x one has x ∈ 𝒫 ⟺ x ⊗ κ(𝒫) = 0. If x ∈ 𝒫 then x ∈ ⟨𝒫⟩ = 𝒯_{Y_{𝒫}} and thus x ⊗ f(Y_{𝒫} ) = 0 already. Whence x ⊗ κ(𝒫) = x ⊗ e(\overline{\{𝒫\}}) ⊗ f(Y_{𝒫} ) = 0. Conversely, suppose that x ⊗ κ(𝒫) = 0. Since 𝒫 is visible, there exists s ∈ 𝒦 such that supp(s) = \overline{\{𝒫\}}, see Lemma 7.8, whose part (b) yields

0 = x ⊗ s ⊗ κ(𝒫) = x ⊗ s ⊗ e(\overline{\{𝒫\}}) ⊗ f(Y_{𝒫} ) = x ⊗ s ⊗ f(Y_{𝒫} ).

This implies that x ⊗ s ∈ 𝒯_{Y_{𝒫}} ∩ 𝒦 = ⟨𝒦_{Y_{𝒫}} ⟩ ∩ 𝒦 = 𝒦_{Y_{𝒫}} = 𝒫 (using Thm. 4.1 (a) again and Y_{𝒫} = supp(𝒫)). But s ∉ 𝒫 since 𝒫 ∈ supp(s), so x ∈ 𝒫 by definition of a prime ideal. This finishes (a). The rest is straightforward. □

7.18. Proposition. Under Hypotheses 1.1, suppose Spc(𝒦) noetherian. Let Y ⊂ Spc(𝒦) be a Thomason subset (i.e. specialization closed). Then we have

Supp(t ⊗ e(Y)) = Supp(t) ∩ Y and Supp(t ⊗ f(Y)) = Supp(t) ∩ Y^{c}

for every t in 𝒯. In particular we have Supp(e(Y)) = Y and Supp(f(Y)) = Y^{c}. More generally if A ⊂ Spc(𝒦) is trop-beau (Rem. 7.6) then Supp(g(A)) = A.

Proof. Notice first that if 𝒫 ∈ Y then \overline{\{𝒫\}} ⊂ Y, since Y is specialization closed. Thus, by Thm. 5.18 we have e(\overline{\{𝒫\}}) ⊗ e(Y) ≃ e(\overline{\{𝒫\}}). This implies that t ⊗ e(Y) ⊗ κ(𝒫) ≃ t ⊗ κ(𝒫), for any t in 𝒯. Thus Supp(t ⊗ e(Y)) ∩ Y = Supp(t) ∩ Y. It remains to show Supp(t ⊗ e(Y)) ⊂ Y. Since Supp(t ⊗ e(Y)) ⊂ Supp(t) ∩ Supp(e(Y)), it is enough to show that Supp(e(Y)) ⊂ Y. Let us prove Y^{c} ⊂ (Supp(e(Y))^{c}. Since Y is specialization closed, if 𝒫 ∉ Y then Y ⊂ \{ 𝒬 | 𝒫 ∉ \overline{\{𝒬\}} \} = \{ 𝒬 | 𝒫 ⊄ 𝒬 \}, see Rem. 5.4. This reads Y ⊂ Y_{𝒫} , see Lemma 7.8. Hence e(Y) ⊗ e(Y_{𝒫} ) ≃ e(Y) and, since e(Y_{𝒫} ) ⊗ f(Y_{𝒫} ) = 0, we get e(Y) ⊗ κ(𝒫) ≃ e(Y) ⊗ e(Y_{𝒫} ) ⊗ e(\overline{\{𝒫\}}) ⊗ f(Y_{𝒫} ) = 0. The latter means 𝒫 ∉ Supp(e(Y)) as wanted. This gives the first desired equality. The other equality is obtained in a similar manner, interchanging e’s and f’s. □

7.19. Corollary. Under Hypotheses 1.1, suppose Spc(𝒦) noetherian. Let t ∈ 𝒯 and 𝒫 ∈ Spc(𝒦). Let t_{𝒫} = t ⊗ f(Y_{𝒫} ) = L_{⟨𝒫⟩} (t) the localization of t at ⟨𝒫⟩. Then Supp(t_{𝒫} ) = Supp(t) ∩ {𝒬 ∈ Spc(𝒦) | 𝒫 ⊂ 𝒬} in Spc(𝒦/𝒫) ≅ { 𝒬 | 𝒫 ⊂ 𝒬 }. □

7.20. Remark. Proposition 7.18 is the first indication of a so-called ⊗-theorem for supports, see [10], that is, the hope to have Supp(t⊗t′) = Supp(t)∩Supp(t′) for all t, t′ in 𝒯. By Proposition 7.18, the ⊗-theorem holds when one of the two objects is e(Y) or f(Y). We want to prove another case of this ⊗-theorem.

7.21. Lemma. Let s, s′ in 𝒯 generate the same localizing ⊗-ideal of 𝒯, that is, ⟨s⟩^{⊗} = ⟨s′⟩^{⊗}. Then, Supp(t ⊗ s) = Supp(t ⊗ s′) for all t ∈ 𝒯. In particular, Supp(s) = Supp(s′).

Proof. We claim that for every v ∈ 𝒯 we have v ⊗ s = 0 if and only if v ⊗ s′ = 0. Indeed, Ker(v ⊗ −) is a localizing ⊗-ideal and therefore contains s if and only if it contains ⟨s⟩^{⊗}. The claim follows from ⟨s⟩^{⊗} = ⟨s′⟩^{⊗}. Now, for 𝒫 ∈ Spc(𝒦), apply this claim to v = t ⊗ κ(𝒫). This gives Supp(t ⊗ s) = Supp(t ⊗ s′) as wanted. □

7.22. Theorem (Half ⊗-Theorem). Under Hypotheses 1.1 with Spc(𝒦) noetherian, let t be in 𝒯 and x ∈ 𝒦 be a compact object. Then

Supp(t ⊗ x) = Supp(t) ∩ supp(x).

Proof. Since ⟨x⟩^{⊗} = ⟨𝒦_{supp x}⟩ = ⟨e(supp x)⟩^{⊗}, the above lemma yields Supp(t ⊗ x) = Supp(t ⊗ e(supp x)) = Supp(t) ∩ supp(x) by Proposition 7.18. □

Acknowledgments : We thank Ivo Dell’Ambrogio and Henning Krause for precious discussions. We also thank Greg Stevenson, Tobias Peter and the referee for comments on an earlier version. The second author also warmly thanks Family Regenass–Baader at the Florastrasse Institute in Basel for support and inspiration.

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Paul Balmer, UCLA Mathematics Department, Los Angeles, CA 90095-1555, USA

   E-mail address: balmer@math.ucla.edu
   URL: http://www.math.ucla.edu/∼balmer

Giordano Favi, Department Mathematics, Universität Basel, 4051 Basel, Switzerland; also : Mathematics Section, EGG, EPFL, 1015 Lausanne, Switzerland

   E-mail address: giordano.favi@gmail.com
   URL: http://www.math.unibas.ch/∼giordano