NILPOTENCE THEOREMS VIA HOMOLOGICAL RESIDUE FIELDS

PAUL BALMER

Abstract. We prove nilpotence theorems in tensor-triangulated categories using suitable Gabriel quotients of the module category, and discuss examples.

1. Introduction

For the average Joe, and the median Jane, the Nilpotence Theorem refers to a result in stable homotopy theory, conjectured by Ravenel and proved by Devinatz, Hopkins and Smith in their famous work on chromatic theory [DHS88, HS98]. One form of the result says that a map between finite spectra which gets annihilated by all Morava K-theories must be tensor-nilpotent. Under Hopkins’s impetus [Hop87], these ideas soon expanded beyond topology. Neeman [Nee92] in commutative algebra and Thomason [Tho97] in algebraic geometry proved nilpotence theorems for maps in derived categories of perfect complexes, using ordinary residue fields instead of Morava K-theories. Benson, Carlson and Rickard [BCR97] led the charge into yet another area, namely modular representation theory of finite groups, where the appropriate ‘residue fields’ turned out to be shifted cyclic subgroups, and later π-points [FP07]. As further areas kept joining the fray, expectations rose of a unified treatment applicable to every tensor-triangulated category in Nature. In this vein, Mathew [Mat17, Thm. 4.14 (b)] proved an abstract nilpotence theorem via E_{∞}-rings in ∞-categories over the field ℚ. However, this rationality assumption is a severe restriction, incompatible with the chromatic joys of topological Joe and the positive characteristic tastes of modular Jane. Here, we prove abstract nilpotence theorems, integrally and without ∞-categories. For instance, Corollary 4.5 says:

1.1. Theorem. Let f: x → y be a morphism in an essentially small, rigid tensortriangulated category 𝒦. If we have h̄(f) = 0 for every homological residue field h̄: 𝒦 → 𝒜̄, then there exists n ≥ 1 such that f^{⊗n} = 0 in 𝒦.

We need to explain the homological residue fields that appear in this statement. Their purpose is to encapsulate the key features of Morava K-theories, ordinary residue fields, shifted cyclic subgroups, etc, from an abstract point of view. In other words, when first meeting a tensor-triangulated category 𝒦, we would like to extract its ‘residue fields’ without knowing intimate details about 𝒦, as we are used to extract residue fields κ(𝔭) = R_{𝔭}/𝔭R_{𝔭} from any commutative ring R, without much knowledge about R beyond its propensity to harbor prime ideals 𝔭 ∈ Spec(R). We investigated this question of ‘tensor-triangular fields’ in the recent joint work [BKS19], with Krause and Stevenson. Although the present article can be read independently, we refer to that prequel for motivation, background, justification, and a couple of lemmas. In retrospect, our nilpotence theorems further validate the ideas introduced in [BKS19].

Date: 2019 January 22.

2010 Mathematics Subject Classification. 18E30 (20J05, 55U35).

Key words and phrases. Homological residue field, tt-geometry, module category, nilpotence.

Author supported by NSF grant DMS-1600032.

In a nutshell, as we do not know how to produce residue fields within triangulated categories, we consider instead homological tensor-functors to abelian categories

h̄ = h̄_{ℬ} : 𝒦 ↪ mod-𝒦 ↠ (mod-𝒦)/ℬ

composed of the Yoneda embedding h: 𝒦 ↪ mod-𝒦 into the Freyd envelope of 𝒦 (Remark 2.6) followed by the Gabriel quotient Q_{ℬ} : mod-𝒦 ↠(mod-𝒦)/ℬ with respect to any maximal ⊗-ideal Serre subcategory ℬ. Thus Theorem 1.1 can be rephrased as follows:

If a morphism f: x → y in 𝒦 is annihilated by h̄_{ℬ} : 𝒦 → (mod-𝒦)/ℬ for every maximal Serre ⊗-ideal ℬ ⊂ mod-𝒦, then f is ⊗-nilpotent in 𝒦.

At first, it might be counter-intuitive to only invoke maximal ⊗-ideals ℬ, instead of some kind of more general ‘prime’ ⊗-ideals of mod-𝒦, but we explain in Section 3 why this notion covers all points of the triangular spectrum Spc(𝒦) of 𝒦, not just the closed points. We also explain in Remark 3.10 how the above homological residue fields correspond to the local constructions proposed in [BKS19, § 4].

As a matter of fact, in the examples, there exist alternative formulations of the nilpotence theorem. And the same holds here. Most notably, if our triangulated category 𝒦 sits inside a ‘big’ one, 𝒦 ⊂ 𝒯, as the compact objects 𝒦 = 𝒯^{c} of a socalled ‘rigidly-compactly generated’ tensor-triangulated category 𝒯 (Remark 4.6), we expect a nilpotence theorem for maps f: x → Y with compact source x ∈ 𝒦 but arbitrary target Y in 𝒯. This flavor of nilpotence theorem is Corollary 4.7.

In order to handle such generalizations, we consider the big Grothendieck category 𝒜 := Mod-𝒦 of all right 𝒦-modules (Notation 2.5), not just the subcategory of finitely presented ones that is the Freyd envelope 𝒜^{fp} = mod-𝒦. When 𝒦 = 𝒯^{c}, the big category 𝒯 still admits a so-called ‘restricted-Yoneda’ functor h: 𝒯 → Mod-𝒦 (Remark 4.6). Every maximal Serre ⊗-ideal ℬ ⊂ 𝒜^{fp} generates a localizing (Serre) ⊗-ideal ⟨ℬ⟩ of 𝒜 and we can consider the corresponding ‘big’ Gabriel quotient 𝒜̄ := 𝒜/⟨ℬ⟩. Composing with restricted-Yoneda, we obtain a homological ⊗- functor h̄_{ℬ} : 𝒯 ⟶ 𝒜̄ on the ‘big’ category 𝒯, extending the one on 𝒦:

[figure: 𝒦 = 𝒯^{c}; h; ⋂; ^{h̄^{ℬ}}; 𝒜^{fp} = mod-𝒦; Q_{ℬ}; ⋂; 𝒜̄^{fp} = (mod-𝒦)/ℬ; ⋂; 𝒯; h; _{h̄_{ℬ}}; 𝒜 = Mod-𝒦; Q_{ℬ}; 𝒜̄ = (Mod-𝒦)/⟨ℬ⟩ .]

Thanks to [BKS17, Prop. A.14], the image h(Y) of every object Y in the big category 𝒯 remains ⊗-flat in the module category 𝒜 = Mod-𝒦, meaning that the functor h(Y) ⊗ −: 𝒜 → 𝒜 is exact. In fact, this ⊗-flatness plays an important role in the proof of the nilpotence theorem. In particular, Corollary 4.2 tells us:

1.2. Theorem. Let f: h(x) → F be a morphism in 𝒜 = Mod-𝒦, for x ∈ 𝒦. Suppose that the 𝒦-module F is ⊗-flat and that Q_{ℬ} (f) = 0 in 𝒜̄ = 𝒜/⟨ℬ⟩, for every maximal Serre ⊗-ideal ℬ ⊂ mod-𝒦. Then f is ⊗-nilpotent in Mod-𝒦.

All these statements are corollaries of our most general Nilpotence Theorem 4.1, which further involves a ‘parameter’ à la Thomason [Tho97], i.e. a closed subset W ⊆ Spc(𝒦) of the spectrum on which we test the vanishing of f.

Finally, in Section 5, we classify those homological residue fields in examples. For brevity, let us pack three theorems into one:

1.3. Theorem. There exists a surjection ϕ: Spc^{h}(𝒦) ↠ Spc(𝒦) from the set of maximal Serre ⊗-ideals ℬ ⊂ mod-𝒦 to the triangular spectrum of 𝒦. Moreover, it is a bijection for each of the following tensor-triangulated categories 𝒦:

(a) Let X be a quasi-compact and quasi-separated scheme and 𝒦 = D^{perf}(X) its derived category of perfect complexes. (Corollary 5.11.)

(b) Let G be a compact Lie group and 𝒦 = SH(G)^{c} the G-equivariant stable homotopy category of finite genuine G-spectra. In particular, this holds for 𝒦 = SH^{c} the usual stable homotopy category. (Corollary 5.10.)

(c) Let G be a finite group scheme over a field k and 𝒦 = stab(kG) its stable module category of finite-dimensional kG-modules modulo projectives. (Example 5.13.)

We caution that the above does not give a new proof of the nilpotence theorems known in those examples, except perhaps for modular representation theory (c), as we discuss further in Remark 5.14. Indeed, the above results rely on existing classification results, which themselves often rely on a form of nilpotence theorem. These results should rather be read as a converse to our nilpotence theorems via homological residue fields: If a collection of homological residue fields detects nilpotence then that collection contains all homological residue fields (Theorem 5.4).

Acknowledgments: I am very thankful to Henning Krause and Greg Stevenson for their precious support, during my stay at Bielefeld University and later during the preparation of this paper. I also thank Beren Sanders for useful discussions.

2. Background and notation

2.1. Hypothesis. Throughout the paper, we denote by 𝒦 an essentially small tensortriangulated category and by 𝟙 ∈ 𝒦 its ⊗-unit. We often assume 𝒦 rigid, in the sense recalled in Remark 2.4 below. See details in [Bal05, § 1] or [Bal10, § 1].

2.2. Examples. Such 𝒦 include the usual suspects: in topology 𝒦 = SH^{c} the stable homotopy category of finite spectra; in algebraic geometry 𝒦 = D^{perf}(X) = D_{Qcoh}(O_{X}-Mod)^{c} the derived category of perfect complexes over a scheme X which is assumed quasi-compact and quasi-separated (e.g. a noetherian, or an affine one); in modular representation theory 𝒦 = stab(kG) = Stab(kG)^{c} the stable category of finite-dimensional k-linear representations of a finite group G over a field k of characteristic dividing the order of G. But there are many more examples: equivariant versions, categories of motives, KK-categories of C^{∗}-algebras, etc, etc.

2.3. Remark. We use the triangular spectrum Spc(𝒦) = { 𝒫 ⊂ 𝒦 | 𝒫 is a prime } where a proper thick ⊗-ideal 𝒫 ⊊ 𝒦 is called a (triangular) prime if x⊗y ∈ 𝒫 implies x ∈ 𝒫 or y ∈ 𝒫. The support of an object x ∈ 𝒦 is the closed subset supp(x) := { 𝒫 ∈ Spc(𝒦) | x ∉ 𝒫 } = { 𝒫 ∈ Spc(𝒦) | x is non-zero in 𝒦/𝒫 }. These are exactly the so-called Thomason closed subsets of Spc(𝒦), i.e. those closed subsets Z ⊆ Spc(𝒦) with quasi-compact open complement Spc(𝒦) ∖ Z, by [Bal05, Prop. 2.14].

2.4. Remark. Rigidity of 𝒦 will play an important role in the proof of the main Theorem 4.1. Rigidity means that every object x ∈ 𝒦 is strongly-dualizable, hence admits a dual x^{∨} ∈ 𝒦 with an adjunction Hom_{𝒦} (x ⊗ y, z) ≅ Hom_{𝒦} (y, x^{∨} ⊗ z). In particular, the unit-counit relation forces x ⊗ η_{x}: x → x ⊗ x^{∨} ⊗ x to be a split monomorphism where η_{x}: 𝟙 → x^{∨} ⊗ x is the unit of the adjunction. This implies that x is a direct summand of a ⊗-multiple of x^{⊗n} for any n ≥ 1. It follows that if a map f satisfies (f ⊗ x)^{⊗n} = 0 then f^{⊗n} ⊗ x = 0 as well.

2.5. Notation. The Grothendieck abelian category Mod-𝒦 of right 𝒦-modules, i.e. additive functors M: 𝒦^{op} → Ab, receives 𝒦 via the Yoneda embedding, denoted

h : 𝒦 ↪ Mod-𝒦 = Add(𝒦^{op}, Ab)
x ↦ x̂ := Hom_{𝒦} (−, x)
f ↦ f̂.

2.6. Remark. Let us recall some standard facts about 𝒦-modules. See details in [BKS17, App. A]. By Day convolution, the category 𝒜 = Mod-𝒦 admits a tensor ⊗: 𝒜 × 𝒜 → 𝒜 which is colimit-preserving (in particular right-exact) in each variable and which makes h: 𝒦 → 𝒜 a tensor functor: { \atop x ⊗ y} ≅ x̂ ⊗ ŷ. Hence h preserves rigidity, so x̂ will be rigid in 𝒜 whenever x is in 𝒦. Moreover, the object x̂ ∈ 𝒜 is finitely presented projective and ⊗-flat in 𝒜. The tensor subcategory

𝒜^{fp} = mod-𝒦 ⊂ Mod-𝒦 = 𝒜

of finitely presented objects is itself abelian and is nothing but the classical Freyd envelope of 𝒦, see [Nee01, Chap. 5]. Recall that h: 𝒦 ↪ mod-𝒦 is the universal homological functor out of 𝒦, and that a functor from a triangulated category to an abelian category is homological if it maps exact triangles to exact sequences. Every object of 𝒜 is a filtered colimit of finitely presented ones. In short, 𝒜 is a locally coherent Grothendieck category.

2.7. Remark. Given a Serre subcategory ℬ ⊆ 𝒜^{fp} we can form ⟨ℬ⟩, or \overrightarrow{ℬ}, the localizing subcategory of 𝒜 generated by ℬ. The subcategory ⟨ℬ⟩ is the smallest Serre subcategory containing ℬ and closed under coproducts; it consists of all (filtered) colimits in 𝒜 of objects of ℬ. For instance ⟨𝒜^{fp}⟩ = 𝒜 and it follows that if ℬ is ⊗-ideal in 𝒜^{fp} then so is ⟨ℬ⟩ in 𝒜. We denote the corresponding Gabriel quotient [Gab62] by

Q_{ℬ} : 𝒜 ↠ 𝒜/⟨ℬ⟩.

We recall that ⟨ℬ⟩ is also locally coherent with ⟨ℬ⟩^{fp} = ℬ and so is the quotient 𝒜̄ with 𝒜̄^{fp} ≅ 𝒜^{fp}/ℬ. When ℬ is ⊗-ideal then 𝒜̄ inherits a unique tensor structure turning Q_{ℬ} : 𝒜 ↠ 𝒜̄ into a tensor functor, which preserves ⊗-flat objects. All this remains true without assuming 𝒦 rigid. See details in [BKS19, § 2].

2.8. Remark. For 𝒦 rigid, consider the special case of the adjunction Hom_{𝒦} (x, y) ≅ Hom_{𝒦} (𝟙, x^{∨}⊗y). Under this isomorphism, if f: x → y corresponds to g: 𝟙 → x^{∨}⊗y then for n ≥ 1 the morphism f^{⊗n}: x^{⊗n} → y^{⊗n} corresponds to g^{⊗n}: 𝟙 → (x^{∨} ⊗ y)^{⊗n} under the analogous isomorphism Hom_{𝒦} (x^{⊗n}, y^{⊗n}) ≅ Hom_{𝒦} (𝟙, (x^{⊗n})^{∨} ⊗ y^{⊗n}) ≅ Hom_{𝒦} (𝟙, (x^{∨} ⊗ y)^{⊗n}). In particular, f is ⊗-nilpotent if and only if g is. Note that the above observation only uses that x is rigid in a tensor category and does not use that y itself is rigid. We can therefore also use this trick for any morphism f: x̂ → M in the module category 𝒜 = Mod-𝒦, as long as x comes from 𝒦.

We shall need the following folklore result about modules and localization:

2.9. Proposition. Let 𝒥 ⊂ 𝒦 be a thick ⊗-ideal and let q: 𝒦 → ℒ be the corresponding Verdier quotient 𝒦 ↠ 𝒦/𝒥 or its idempotent completion 𝒦 → (𝒦/𝒥)^{♮}. Consider the left Kan extension q_{!} : Mod-𝒦 ⟶ Mod-ℒ, left-adjoint to restriction q^{∗}: Mod-ℒ ⟶ Mod-𝒦 along q. Then q_{!} is a localization identifying Mod-ℒ as the Gabriel quotient of Mod-𝒦 by Ker(q_{!}) = ⟨h(𝒥)⟩ the localizing subcategory generated by h(𝒥). The localization q_{!} restricts to a localization q_{!} : mod-𝒦 ↠ mod-ℒ on finitely presented objects, identifying mod-ℒ as the quotient of mod-𝒦 by Ker(q_{!})^{fp} which is also the Serre envelope of h(𝒥) in mod-𝒦.

Proof. The fact that q_{!} is a localization follows from [Kra05, Thm. 4.4 and § 3]. The left Kan extension q_{!}(M) is defined as colim_{α: x̂→M} q_{!}(x̂) with q_{!}(x̂) = q̂(x). To identify the kernel of q_{!} : Mod-𝒦 → Mod-ℒ, since h(𝒥) ⊆ Ker(q_{!})^{fp} is clear, it suffices to show Ker(q_{!}) ⊆ ⟨h(𝒥)⟩. For every M ∈ Ker(q_{!}), using that q̂(x) is finitely presented projective for all x ∈ 𝒦 together with faithfulness of Yoneda ℒ ↪ Mod-ℒ, one shows that every morphism α: x̂ → M with x ∈ 𝒦 factors via a morphism β̂: x̂ → ŷ where q(β) vanishes in 𝒦/𝒥, meaning that the morphism β: x → y in 𝒦 factors via an object of 𝒥. In short, every morphism x̂ → M factors via an object in h(𝒥) which implies that M belongs to the localizing subcategory ⟨h(𝒥)⟩. The ⊗-properties are then easily added onto this purely abelian picture. □

3. Homological primes and homological residue fields

Let 𝒦 be a tensor-triangulated category as in Hypothesis 2.1.

3.1. Definition. A (coherent) homological prime for 𝒦 is a maximal proper Serre ⊗- ideal subcategory ℬ ⊂ 𝒜^{fp} = mod-𝒦 of the Freyd envelope of 𝒦. The homological residue field corresponding to ℬ is the functor constructed as follows

[figure: h̄_{ℬ} = Q_{ℬ} ∘ h:; 𝒦; h; 𝒜 = Mod-𝒦; Q_{ℬ}; 𝒜̄(𝒦; ℬ) := \frac{Mod-𝒦}{⟨ℬ⟩}; x; x̂; x̄ .]

The functor h̄_{ℬ} : 𝒦 → 𝒜̄(𝒦; ℬ)^{fp} is a (strong) monoidal homological functor (Remark 2.6), that lands in the finitely presented subcategory 𝒜̄(𝒦; ℬ)^{fp} ≅ (mod-𝒦)/ℬ. By construction, the tensor-abelian category 𝒜̄(𝒦; ℬ)^{fp} has only the trivial Serre ⊗-ideals, 0 and 𝒜̄(𝒦; ℬ)^{fp} itself. These homological residue fields are truly the same as the homological ⊗-functors constructed in [BKS19, § 4], up to a little paradigm change that we explain in Remark 3.10 below.

3.2. Remark. Since 𝒦 is essentially small, its Freyd envelope, mod-𝒦, admits only a set of Serre subcategories. So we can apply Zorn to construct homological primes and homological residue fields as soon as 𝒦 ≠ 0. Contrary to what happens with commutative rings, these maximal Serre ⊗-ideals are not only picking up ‘closed points’ as one could first fear. In fact, they ‘live’ above every prime of the triangular spectrum Spc(𝒦) of 𝒦 (Remark 2.3). First, let us explain the relationship.

3.3. Proposition. Let ℬ be a homological prime with homological residue field h̄_{ℬ} : 𝒦 → 𝒜̄(𝒦; ℬ). Then 𝒫(ℬ) := Ker(h̄_{ℬ} ) = h^{−1}(ℬ) is a triangular prime of 𝒦.

Proof. Since Yoneda h: 𝒦 → mod-𝒦 is homological and (strong) monoidal, the preimage 𝒫(ℬ) = h^{−1}(ℬ) is a proper thick ⊗-ideal of 𝒦. To see that 𝒫(ℬ) is prime, let x, y ∈ 𝒦 with x ⊗ y ∈ 𝒫(ℬ) and x ∉ 𝒫(ℬ) and let us show that y ∈ 𝒫(ℬ). Consider the ⊗-ideal 𝒞 = { M ∈ mod-𝒦 | x̂ ⊗ M ∈ ℬ }. It is Serre by flatness of x̂ and the assumption x ∉ 𝒫(ℬ) implies ℬ ⊆ 𝒞 ≠ mod-𝒦. Therefore 𝒞 = ℬ by maximality of ℬ and we get y ∈ h^{−1}(𝒞) = h^{−1}(ℬ) = 𝒫(ℬ). □

3.4. Remark. We can push the analogy with the triangular spectrum Spc(𝒦) a little further by considering the set Spc^{h}(𝒦) of all homological primes:

Spc^{h}(𝒦) = { ℬ ⊂ mod-𝒦 | ℬ is a maximal Serre ⊗-ideal } .

We call it the homological spectrum of 𝒦 and equip it with a topology having as basis of closed subsets the following subsets supp^{h}(x), one for every x ∈ 𝒦:

supp^{h}(x) := { ℬ ∈ Spc^{h}(𝒦) | x̂ ∉ ℬ } = { ℬ ∈ Spc^{h}(𝒦) | x̄ ≠ 0 in 𝒜̄(𝒦; ℬ) }.

One can verify that this pair (Spc^{h}(𝒦), supp^{h}) is a support data on 𝒦, in the sense of [Bal05]. Hence there exists a unique continuous map

ϕ: Spc^{h}(𝒦) → Spc(𝒦)

such that supp^{h}(x) = ϕ^{−1}(supp(x)) for every x ∈ 𝒦. The explicit formula for ϕ ([Bal05, Thm. 3.2]) shows that ϕ is exactly the map ℬ ↦ h^{−1}(ℬ) of Proposition 3.3. We prove in Corollary 3.9 below that this comparison map is surjective, at least for 𝒦 rigid. In fact, there are many known examples where ϕ is bijective. See Section 5.

3.5. Definition. It will be convenient to say that a homological prime ℬ ∈ Spc^{h}(𝒦) lives over a given subset W ⊆ Spc(𝒦) of the triangular spectrum if the prime 𝒫(ℬ) = h^{−1}(ℬ) of Proposition 3.3 belongs to W. By extension, we shall also say in that case that the corresponding homological residue field h̄ = h̄_{ℬ} lives over W.

In order to show surjectivity of ϕ, we derive from Proposition 2.9 the following:

3.6. Corollary. Let 𝒥 ⊂ 𝒦 be a thick ⊗-ideal. With notation as in Proposition 2.9, there is an inclusion-preserving one-to-one correspondence 𝒞 ↦ (q_{!})^{−1}(𝒞) between (maximal) Serre ⊗-ideals 𝒞 of mod-ℒ and the (maximal) Serre ⊗-ideals ℬ of mod-𝒦 which contain h(𝒥); the inverse is given by ℬ ↦ q_{!}(ℬ). If 𝒞 corresponds to ℬ then the residue categories are canonically equivalent 𝒜̄(𝒦; ℬ) ≅ 𝒜̄(ℒ; 𝒞) in such a way that the following diagram commutes up to canonical isomorphism

[figure: 𝒦; h; ^{h̄^{ℬ}}; q; mod-𝒦; q_{!}; Mod-𝒦; Q_{ℬ}; q_{!}; 𝒜̄(𝒦; ℬ); ≅; ℒ; h; _{h̄_{𝒞}}; mod-ℒ; Mod-ℒ; Q_{𝒞}; 𝒜̄(ℒ; 𝒞) .]

Proof. Standard ‘third isomorphism theorem’ about ideals in a quotient. □

3.7. Remark. In the notation of Remark 3.4, the above Corollary 3.6 can be rephrased as saying that, for ℒ = 𝒦/𝒥 or (𝒦/𝒥)^{♮}, the map Spc^{h}(q_{!}): 𝒞 ↦ (q_{!})^{−1}𝒞 yields a homeomorphism between Spc^{h}(ℒ) and the subspace { ℬ ∈ Spc^{h}(𝒦) | 𝒫(ℬ) ⊇ 𝒥 } of Spc^{h}(𝒦). In other words, the following commutative diagram is cartesian :

\xymatrix{Spc^{h}(ℒ) \ar@{^{(}->}[r]^{Spc^{h}(q_{!})} \ar[d]_{ϕ} & Spc^{h}(𝒦) \ar[d]^{ϕ} \\ Spc(ℒ) \ar@{^{(}->}[r]^{Spc(q)} & Spc(𝒦) .}

In the terminology of Definition 3.5, the homological primes (or the residue fields) of ℒ = 𝒦/𝒥 or ℒ = (𝒦/𝒥)^{♮} canonically correspond to those of 𝒦 which live ‘above’ the subset W (𝒥) = { 𝒫 ∈ Spc(𝒦) | 𝒥 ⊆ 𝒫 } ≅ Spc(ℒ) of Spc(𝒦).

Let us prove an analogue of an old tensor-triangular friend [Bal05, Lem. 2.2]:

3.8. Lemma. Suppose that 𝒦 is rigid. Let 𝒮 ⊂ 𝒦 be a ⊗-multiplicative class of objects (i.e. 𝟙 ∈ 𝒮 ⊇ 𝒮 ⊗ 𝒮) and let ℬ_{0} ⊂ 𝒜^{fp} = mod-𝒦 be a Serre ⊗-ideal which avoids 𝒮, that is, ℬ_{0} ∩ h(𝒮) = ∅. Then there exists ℬ ⊂ 𝒜^{fp} a maximal Serre ⊗-ideal such that ℬ_{0} ⊆ ℬ and ℬ still avoids 𝒮.

Proof. By Zorn, there exists ℬ maximal among the Serre ⊗-ideals which avoid 𝒮 and contain ℬ_{0}. So we have ℬ ⊇ ℬ_{0} and ℬ ∩ h(𝒮) = ∅ and we are left to prove that ℬ is plain maximal in 𝒜^{fp}. Consider ℬ′ := { M ∈ 𝒜^{fp} | x̂ ⊗ M ∈ ℬ for some x ∈ 𝒮 }. Since 𝒮 is ⊗-multiplicative and each x̂ is ⊗-flat, the subcategory ℬ′ ⊊ 𝒜^{fp} is a Serre ⊗-ideal avoiding 𝒮 and containing ℬ_{0}. By maximality of ℬ among those, the relation ℬ ⊆ ℬ′ forces ℬ = ℬ′. In particular, M = ker(η̂_{x}: ˆ𝟙 → x̂^{∨} ⊗ x̂) belongs to ℬ for every x ∈ 𝒮 since x̂ ⊗ M = 0 by rigidity (Remark 2.4). Let us show that ℬ ⊂ 𝒜^{fp} is a maximal Serre ⊗-ideal by showing that a strictly bigger Serre ⊗-ideal 𝒞 ⊋ ℬ of 𝒜^{fp} must be 𝒜^{fp} itself. Since ℬ is maximal among those avoiding 𝒮 and containing ℬ_{0}, such a strictly bigger 𝒞 cannot avoid 𝒮. Therefore 𝒞 contains some x̂ for x ∈ 𝒮 and, by the above discussion, we also have ker(η̂_{x}: ˆ𝟙 → x̂^{∨} ⊗ x̂) ∈ ℬ ⊆ 𝒞. So in the exact sequence 0 → ker(η̂_{x}) → ˆ𝟙 → x̂^{∨} ⊗ x̂ we have ker(η̂_{x}) and x̂ in 𝒞. This forces ˆ𝟙 ∈ 𝒞 by Serritude and therefore 𝒞 = 𝒜^{fp} as wanted. □

3.9. Corollary. Suppose that 𝒦 is rigid. Then the map ℬ ↦ 𝒫(ℬ) from homological primes to triangular primes as in Proposition 3.3 (i.e. the comparison map ϕ: Spc^{h}(𝒦) → Spc(𝒦) of Remark 3.4) is surjective. That is, every triangular prime 𝒫 ∈ Spc(𝒦) is of the form 𝒫 = h^{−1}(ℬ) for some maximal Serre ⊗-ideal ℬ in mod-𝒦.

Proof. Consider the quotient 𝒦/𝒫 (or its idempotent completion 𝒦_{𝒫} := (𝒦/𝒫)^{♮}). The map Spc(q): Spc(𝒦/𝒫) ↪ Spc(𝒦) sends 0 to q^{−1}(0) = 𝒫. So, by Corollary 3.6 applied to 𝒥 = 𝒫, it suffices to prove the result for 𝒫 = 0. In that case, 𝒦 is local, meaning that 0 is a prime: x⊗y = 0 ⇒ x = 0 or y = 0. We conclude by Lemma 3.8 for ℬ_{0} = 0 and 𝒮 = 𝒦 ∖ {0} which is ⊗-multiplicative because 𝒦 is local. (^{1}) □

3.10. Remark. There are a few differences between our approach to homological residue fields and the treatment in [BKS19, § 4]. First, the whole [BKS19] is written for a ‘big’ (i.e. rigidly-compactly generated) tensor-triangulated category 𝒯 and the modules are taken over its rigid-compact objects 𝒦 := 𝒯^{c}. This restriction is unimportant, certainly as far as most examples are concerned.

1 The argument was already used in [BKS19, Cor. 4.9], which in turn inspired Lemma 3.8.

Another difference is that, in loc. cit., we focussed on a local category in the sense that Spc(𝒦) is a local space, i.e. has a unique closed point ℳ = 0. We then considered quotients of the module category 𝒜 ↠ 𝒜/⟨ℬ⟩ for ℬ ⊆ 𝒜^{fp} maximal among those which meet 𝒦 trivially, i.e. ℬ ∩ h(𝒦) = {0}. This property means that the homological prime ℬ lives above the closed point of Spc(𝒦) in the sense of Definition 3.5. Equivalently, it means that the functor h̄_{ℬ} : 𝒦 → 𝒜̄(𝒦; ℬ) is conservative, i.e. detects isomorphisms. All these properties are reminiscent of commutative algebra, where the residue field of a local ring R is indeed conservative on perfect complexes and maps the unique prime of the field to the closed point of Spec(R).

Continuing the analogy with commutative algebra, when dealing with a global (i.e. not necessarily local) category 𝒦, we can analyze it one prime at a time. For each 𝒫 ∈ Spc(𝒦) we can consider the local category 𝒦_{𝒫} = (𝒦/𝒫)^{♮}. By Corollary 3.6 we can identify the homological primes 𝒞 of this local category 𝒦_{𝒫} with a subset of those of the global category. Requesting that the local prime 𝒞 lives ‘above the closed point’ of Spc(𝒦_{𝒫} ) as we did in [BKS19] amounts to requesting that the corresponding global prime ℬ = (q_{!})^{−1}(𝒞) lives exactly above the point 𝒫 in Spc(𝒦).

In other words, in Definition 3.1 we are considering all homological residue fields h̄_{ℬ} : 𝒦 → 𝒜̄(𝒦; ℬ) at once but we can also regroup them according to the associated triangular primes h^{−1}(ℬ), in which case we obtain the constructions of [BKS19] for the local category 𝒦_{𝒫} .

4. The Nilpotence Theorems

In this section, we assume 𝒦 rigid.

Let us prove Theorem 1.2, in a strong form ‘with parameter’. Recall that an object F in a tensor abelian category 𝒜 is ⊗-flat if F ⊗ −: 𝒜 → 𝒜 is exact.

4.1. Theorem. Let 𝒦 be an essentially small, rigid tensor-triangulated category (Hypothesis 2.1) and W ⊆ Spc(𝒦) a closed subset (the ‘parameter’). Let f: x̂ → F be a morphism in 𝒜 = Mod-𝒦 satisfying the following hypotheses:

(i) The source of f comes from an object x ∈ 𝒦 via Yoneda, as indicated above. (ii) Its target F is ⊗-flat in 𝒜. (iii) The morphism f vanishes in every homological residue field over W in the

following sense: For every maximal Serre ⊗-ideal ℬ ⊂ mod-𝒦 living over W (Definition 3.5) we have Q_{ℬ} (f) = 0 in 𝒜̄(𝒦; ℬ) = (Mod-𝒦)/⟨ℬ⟩.

Then:

(a) There exist an object s ∈ 𝒦 such that supp(s) ⊇ W and an integer n ≥ 1 such that f^{⊗n} ⊗ ŝ = 0 in 𝒜.

(b) For any object s as above, if we let Z = supp(s) ⊇ W, then for every z ∈ 𝒦_{Z} = { z ∈ 𝒦 | supp(z) ⊆ Z } there exists m ≥ 1 with f^{⊗m} ⊗ ẑ = 0.

Proof. By Remark 2.8, we can and shall assume that x = 𝟙. So f: ˆ𝟙 → F. Consider

𝒮 := { s ∈ 𝒦 | supp(s) ⊇ W } .

This is a ⊗-multiplicative class of objects of 𝒦 since supp(s_{1} ⊗ s_{2}) = supp(s_{1}) ∩ supp(s_{2}). Since W is closed and {supp(s)}_{s∈𝒦} is a basis of closed subsets, we have ∩_{s∈𝒮} supp(s) = W. On the other hand, consider the following subcategory of finitely presented 𝒦-modules

ℬ_{0} := { M ∈ 𝒜^{fp} | f^{⊗n} ⊗ M = 0 in 𝒜 for some n ≥ 1 }.

Note that ℬ_{0} is a Serre ⊗-ideal. This uses that F is ⊗-flat in 𝒜 and was already proved in [BKS19, Lemma 4.17]. In particular, when we prove that ℬ_{0} is closed under extension, if 0 → M_{1} → M_{2} → M_{3} → 0 is an exact sequence in 𝒜^{fp} and if f^{⊗n_{1}} ⊗ M_{1} = 0 and f^{⊗n_{3}} ⊗ M_{3} = 0 then we show that f^{⊗(n_{1}+n_{3})} ⊗ M_{2} = 0. This is the place where nilpotence is needed, as opposed to mere vanishing.

If ℬ_{0} meets h(𝒮) we obtain the conclusion of part (a). Suppose ab absurdo, that ℬ_{0} ∩ h(𝒮) = ∅. By Lemma 3.8 there exists a homological prime ℬ ∈ Spc^{h}(𝒦) containing ℬ_{0} and still avoiding 𝒮. The latter property ℬ ∩ h(𝒮) = ∅ means that the triangular prime h^{−1}(ℬ) belongs to the subset { 𝒫 ∈ Spc(𝒦) | 𝒫 ∩ 𝒮 = ∅ } = { 𝒫 ∈ Spc(𝒦) | s ∉ 𝒫, ∀ s ∈ 𝒮 } = ∩_{s∈𝒮} supp(s) = W, as we proved above from the definition of 𝒮. So ℬ lives over W and we trigger hypothesis (iii) for that ℬ, namely that we have Q_{ℬ} (f) = 0 in 𝒜̄(𝒦; ℬ).

Consider now the kernel of f: ˆ𝟙 → F in 𝒜 and the commutative diagram:

\xymatrix{0 \ar[r] & ker(f) \ar[r]^{j} \ar[d]_{f⊗1} & ˆ𝟙 \ar[r]^{f} \ar[d]_{f} & F \ar[d]_{f⊗1} \\ 0 \ar[r] & F ⊗ ker(f) \ar[r]^{1⊗j} & F \ar[r]^{1⊗f} & F ⊗ F}

whose first row is the obvious one; the diagram is obtained by tensoring that first row with f: ˆ𝟙 → F itself (on the left) and using that ˆ𝟙 is the ⊗-unit in 𝒜. It is essential here that the source of f is ˆ𝟙 (which we achieved through rigidity), and not some random object. Indeed, the diagonal of the left-hand square is now f ∘ j = 0. Since the lower row is exact (F ⊗-flat again), we conclude that f ⊗ ker(f) = 0. We cannot jump to the conclusion that ker(f) ∈ ℬ since ker(f) is not finitely presented. However, ker(f) ↣ ˆ𝟙 is a sub-object of a finitely presented object, hence it is the union of its finitely presented subobjects as in [BKS17, Lemma 3.9], i.e.

ker(f) = colim_{{M ↣ ker(f) \atop M∈𝒜^{fp}}} M .

For any such i: M ↣ ker(f) with M ∈ 𝒜^{fp}, we have a commutative square obtained by tensoring i: M ↣ ker(f) with f: ˆ𝟙 → F:

[figure: M; i; f⊗1; ker(f); f⊗1=0; F ⊗ M; 1⊗i; F ⊗ ker(f) .]

Note that the bottom map remains a monomorphism because F is ⊗-flat. The vanishing of the right-hand vertical map, proved above, gives us f ⊗ M = 0, which means M ∈ ℬ_{0} ⊆ ℬ. It follows that ker(f) is a colimit of objects M ∈ ℬ and therefore belongs to ⟨ℬ⟩. Applying the exact functor Q_{ℬ} : 𝒜 ↠ 𝒜̄ = 𝒜/⟨ℬ⟩ to the morphism f, we have just proved that Q_{ℬ} (f) has trivial kernel, i.e. it is a monomorphism Q_{ℬ} (ˆ𝟙) ↣ Q_{ℬ} (F) in 𝒜̄. But this monomorphism Q_{ℬ} (f) is also zero by assumption (iii) on f that we triggered in the first part of the proof. This forces Q_{ℬ} (ˆ𝟙) = 0 and thus ˆ𝟙 ∈ ℬ, a contradiction.

This finishes the proof of part (a). To deduce (b), we use the fact that ℬ_{0} = { M ∈ 𝒜^{fp} | f^{⊗m} ⊗ M = 0 for some m ≥ 1 } is a Serre ⊗-ideal, as we saw above, by [BKS19, Lem. 4.17]. Therefore h^{−1}(ℬ_{0}) is a thick ⊗-ideal of 𝒦 and so the fact that h^{−1}(ℬ_{0}) contains s implies that it contains the whole thick ⊗-ideal of 𝒦 generated by s, namely exactly 𝒦_{Z} where Z = supp(s), see [Bal05, § 4]. □

We can then deduce the form announced in Theorem 1.2:

4.2. Corollary. Let f: x̂ → F be a morphism in 𝒜 = Mod-𝒦, with x ∈ 𝒦 and with F ⊗-flat in 𝒜. Suppose that for every homological prime ℬ ⊂ mod-𝒦 we have Q_{ℬ} (f) = 0 in 𝒜̄(𝒦; ℬ). Then there exists m ≥ 1 such that f^{⊗m} = 0 in 𝒜.

Proof. Apply Theorem 4.1 for W = Spc(𝒦). Thus the subset Z ⊇ W of Part (b) must be Z = Spc(𝒦), and we can take z = 𝟙. □

4.3. Remark. We stress that the ‘parameter’ W in Theorem 4.1 is more flexible than the ‘parameter’ in [Tho97, Thm. 3.8], where the closed subset W is supposed to be the support of some object, i.e. a Thomason closed subset (Remark 2.3). In that case, any object s with supp(s) = W will do, as follows from Theorem 4.1 (b).

Of course, the easiest source of ⊗-flat objects in Mod-𝒦 is the Yoneda embedding:

4.4. Corollary. Let f: x → y be a morphism in 𝒦 and W ⊆ Spc(𝒦) be a closed subset such that h̄_{ℬ} (f) = 0 for every homological residue field corresponding to a homological prime ℬ living above W (Definition 3.5). Then there exists a Thomason closed subset Z ⊇ W (Remark 2.3) with the property that for every z ∈ 𝒦 such that supp(z) ⊆ Z there exists n ≥ 1 with f^{⊗n} ⊗ z = 0 in 𝒦. In particular, this holds for some z with supp(z) = Z ⊇ W.

Proof. This is Theorem 4.1 for F = ŷ, which is ⊗-flat, combined with faithfulness of Yoneda h: 𝒦 ↪ 𝒜 to bring the conclusion back into 𝒦. □

This specializes to the flagship Nilpotence Theorem 1.1:

4.5. Corollary. Let f: x → y be a morphism in 𝒦 such that h̄(f) = 0 for every homological residue field h̄ of 𝒦. Then there exists n ≥ 1 with f^{⊗n} = 0 in 𝒦.

Proof. Corollary 4.4 for W = Spc(𝒦), hence Z = Spc(𝒦), and z = 𝟙. □

4.6. Remark. Many of our examples of tensor-triangulated categories 𝒦, if not all, appear as the compact-rigid objects 𝒦 = 𝒯^{c} in some compactly-rigidly generated ‘big’ tensor-triangulated category 𝒯. See [BKS19, Hyp. 0.1]. In that case, we have a restricted-Yoneda functor which extends h: 𝒦 ↪ 𝒜 = Mod-𝒦 to the whole of 𝒯:

\xymatrix{𝒦 = 𝒯^{c} \ar@{^{(}->}[r]^{h} \ar@{^{(}->}[d] & 𝒜^{fp} = mod-𝒦 \ar@{^{(}->}[d] \\ 𝒯 \ar[r]^{h} & 𝒜 = Mod-𝒦 \\ X \ar@{|->}[rr] &  & X̂ := Hom_{𝒯} (−, X) .}

Note that h: 𝒯 → 𝒜 is not faithful anymore (it kills the so-called phantom maps). However, it is faithful for maps out of a compact, by the usual Yoneda Lemma, that is, Hom_{𝒯} (x, Y) → Hom_{𝒜} (x̂, Ŷ) is bijective as soon as x ∈ 𝒦. We prove in [BKS17, Prop. A.14] that every Ŷ remains ⊗-flat in Mod-𝒦, even for Y ∈ 𝒯 non-compact.

For every homological prime ℬ ∈ Spc^{h}(𝒦) we can still compose restricted-Yoneda 𝒯 → 𝒜 with Q_{ℬ} : 𝒜 ↠ 𝒜̄(𝒦; ℬ). We obtain a well-defined homological residue field on the whole ‘big’ category 𝒯, that we still denote

h̄_{ℬ} : 𝒯 ⟶ 𝒜̄(𝒦; ℬ) .

This remains a homological tensor-functor. Compare [BKS19, Thm. 1.6]. Note that we may use these functors to define a support for big objects in 𝒯, as will be investigated elsewhere.

We can finally unpack our nilpotence theorems in that special case:

4.7. Corollary. Let 𝒯 be a rigidly-compactly generated ‘big’ tensor-triangulated category and 𝒦 = 𝒯^{c} as in Remark 4.6. Let f: x → Y be a morphism in 𝒯 with x ∈ 𝒦 compact and Y arbitrary.

(a) Suppose that h̄_{ℬ} (f) = 0 in 𝒜̄(𝒦; ℬ) for every homological residue field h̄_{ℬ} of Remark 4.6, for every homological prime ℬ ⊂ mod-𝒦 of Definition 3.1. Then we have f^{⊗n} = 0 in 𝒯 for some n ≥ 1.

(b) Suppose that h̄_{ℬ} (f) = 0 in 𝒜̄(𝒦; ℬ) for every homological residue field over a closed subset W ⊆ Spc(𝒦) (Definition 3.5). Then there exists a Thomason closed subset Z ⊆ Spc(𝒦) such that Z ⊇ W and such that for every z ∈ 𝒦_{Z} = { z ∈ 𝒦 | supp(z) ⊆ Z } we have f^{⊗n} ⊗ z = 0 for some n ≥ 1. In particular, this holds for some z with supp(z) = Z ⊇ W (Remark 2.3).

Proof. This follows from Theorem 4.1 applied to F = Ŷ, together with the partial faithfulness of restricted-Yoneda explained in Remark 4.6. □

5. Examples

For this section, we keep the setting of Remark 4.6, that is, 𝒯 is a ‘big’ tensortriangulated category generated by its subcategory 𝒦 = 𝒯^{c} of rigid-compact objects.

5.1. Remark. We recall some of the tools developed in [BKS19, § 3]. Let ℬ be a proper Serre ⊗-ideal in 𝒜^{fp} = mod-𝒦. Picking an injective hull of the unit ¯𝟙 in the quotient 𝒜̄(𝒦; ℬ) yields a (pure-injective) object E_{ℬ} in 𝒯 such that

(5.2)  ⟨ℬ⟩ = Ker(Ê_{ℬ} ⊗ −) = { M ∈ Mod-𝒦 | Ê_{ℬ} ⊗ M = 0 } .

In fact the object E_{ℬ} is a weak ring in 𝒯, i.e. it comes with a map η_{ℬ} : 𝟙 → E_{ℬ} such that E_{ℬ} ⊗ η_{ℬ} : E_{ℬ} → E_{ℬ} ⊗ E_{ℬ} is a split monomorphism. A retraction E_{ℬ} ⊗ E_{ℬ} → E_{ℬ} of this monomorphism can be viewed as a (non-associative) multiplication on E_{ℬ} for which η_{ℬ} : 𝟙 → E_{ℬ} is a right unit. In any case, one important property of η_{ℬ} is that it cannot be ⊗-nilpotent, for otherwise E_{ℬ} would be a retract of zero, hence zero, forcing ℬ = 𝒜^{fp}.

Another important feature of the objects E_{ℬ} , for ℬ maximal, is the following:

5.3. Proposition. For distinct ℬ ≠ ℬ′ in Spc^{h}(𝒦) we have E_{ℬ} ⊗ E_{ℬ′} = 0.

Proof. This is already in [BKS19, Cor. 4.12], at least in the local setting. Let us recall the idea, which is easy. In 𝒜^{fp} = mod-𝒦, the kernel Ker(Ê_{ℬ} ⊗ Ê_{ℬ′} ⊗ −) is a Serre ⊗-ideal containing both ℬ and ℬ′, which are maximal. If that kernel was a proper subcategory it would then be equal to both ℬ and ℬ′, thus forcing the forbidden ℬ = ℬ′. So this kernel is not proper, i.e. it contains the unit ˆ𝟙. This reads Ê_{ℬ} ⊗ Ê_{ℬ′} = 0. Now restricted-Yoneda h: 𝒯 → Mod-𝒦 is a conservative ⊗-functor, hence E_{ℬ} ⊗ E_{ℬ′} = 0 as claimed. □

Let us now prove the converse to Corollary 4.7.

5.4. Theorem. Let 𝒯 be a ‘big’ tensor-triangulated category with 𝒦 = 𝒯^{c}, as in Remark 4.6. Consider a family F ⊆ Spc^{h}(𝒦) of points in the homological spectrum. Suppose that the corresponding functors h̄_{ℬ} : 𝒯 → 𝒜̄(𝒦; ℬ) collectively detect ⊗- nilpotence in the following sense: If f: x → Y in 𝒯 is such that x ∈ 𝒯^{c} and h̄_{ℬ} (f) = 0 for all ℬ ∈ F, then f^{⊗n} = 0 for n ≫ 1. Then we have F = Spc^{h}(𝒦).

Proof. Suppose that F ≠ Spc^{h}(𝒦). Then there exists ℬ′ ∈ Spc^{h}(𝒦) which does not belong to F. By Proposition 5.3 we have E_{ℬ} ⊗E_{ℬ′} = 0 for all ℬ ∈ F. Consider the map η_{ℬ′} : 𝟙 → E_{ℬ′} as in Remark 5.1. We have therefore proved that h̄_{ℬ} (η_{ℬ′} ) = 0 for all ℬ ∈ F for the obvious reason that its target object, E_{ℬ′} , goes to zero along all h̄_{ℬ} . On the other hand, we have seen in Remark 5.1 that η_{ℬ′} cannot be ⊗-nilpotent. Hence a proper family F ⊊ Spc^{h}(𝒦) cannot detect ⊗-nilpotence. □

5.5. Remark. The proof shows that it is enough to assume the property that the family {h̄_{ℬ} }_{ℬ∈F} detects the vanishing of objects Y ∈ 𝒯. Compare Remark 5.12.

Here is the picture we will observe in several examples:

5.6. Theorem. Let 𝒯 be a ‘big’ tensor-triangulated category and 𝒦 = 𝒯^{c} as in Remark 4.6. Suppose given for every point 𝒫 ∈ Spc(𝒦) of the triangular spectrum, a coproduct-preserving, homological and (strong) monoidal functor

H_{𝒫} : 𝒯 → 𝒜_{𝒫}

with values in a tensor-abelian category 𝒜_{𝒫} and satisfying the following properties :

(1) For each 𝒫, the target 𝒜_{𝒫} is a locally coherent (Remark 2.6) Grothendieck category with colimit-preserving tensor; the subcategory 𝒜_{𝒫}^{fp} of finitely presented objects is simple in the sense that its only Serre ⊗-ideals are 0 and 𝒜_{𝒫}^{fp} ≠ 0.

(2) The functor H_{𝒫} : 𝒯 → 𝒜_{𝒫} maps compacts to finitely presented: H_{𝒫} (𝒦) ⊆ 𝒜_{𝒫}^{fp}. Furthermore, it maps every X ∈ 𝒯 to a ⊗-flat object in 𝒜_{𝒫} . Finally the thick ⊗-ideal Ker(H_{𝒫} ) ∩ 𝒦 = { x ∈ 𝒦 | H_{𝒫} (x) = 0 } is equal to 𝒫.

(3) The family {H_{𝒫} }_{𝒫∈Spc(𝒦)} detects ⊗-nilpotence of maps f: x → Y in 𝒯, with x ∈ 𝒦 compact: If H_{𝒫} (f) = 0 for all 𝒫 then f^{⊗n} = 0 for n ≫ 1.

Then the comparison map ϕ: Spc^{h}(𝒦) → Spc(𝒦) of Proposition 3.3 is a bijection.

Proof. The core of the proof amounts to the kernel of each H_{𝒫} defining an element of Spc^{h}(𝒦). More precisely, let us fix 𝒫 ∈ Spc(𝒦) for the moment and denote by

Ĥ_{𝒫} : 𝒜 ⟶ 𝒜_{𝒫}

the unique coproduct-preserving exact functor that extends H_{𝒫} to 𝒜 = Mod-𝒦, that is, such that the following diagram commutes :

(5.7)  \xymatrix{𝒯 \ar[dr]_{H_{𝒫}} \ar[r]^{h} & 𝒜 = Mod-𝒦 \ar[d]^{Ĥ_{𝒫}} \\  & 𝒜_{𝒫} .}

The existence of such an Ĥ_{𝒫} was established by Krause [Kra00, Cor. 2.4]. It is not hard to show that Ĥ_{𝒫} is also monoidal. At the very least, for every M ∈ 𝒜, we can find g: Y → Z in 𝒯 such that M = im(ĝ) and then exactness of Ĥ_{𝒫} gives

(5.8)  Ĥ_{𝒫} (X̂ ⊗ M) ≅ H_{𝒫} (X) ⊗ Ĥ_{𝒫} (M)

for every X ∈ 𝒯 and M ∈ 𝒜. Consider now the kernel of Ĥ_{𝒫} in 𝒜. The exact functor Ĥ_{𝒫} : 𝒜 → 𝒜_{𝒫} preserves coproducts and finitely presented objects because of (2). It then follows by a general property of abelian categories that Ker(H_{𝒫} ) is generated by its finitely presented objects (see [BKS19, Prop. A.6]). In other words, if we define a Serre ⊗-ideal of 𝒜^{fp} as follows

ℬ(𝒫) := Ker(Ĥ_{𝒫} ) ∩ 𝒜^{fp} = { M ∈ mod-𝒦 | Ĥ_{𝒫} (M) = 0 }

then we have Ker(Ĥ_{𝒫} ) = ⟨ℬ(𝒫)⟩. The fact that 𝒜_{𝒫} ≠ 0 tells us that ℬ := ℬ(𝒫) is proper. We claim that it is maximal. Let ℬ′ be a strictly larger Serre ⊗-ideal

ℬ = ℬ(𝒫) ⊊ ℬ′ ⊆ 𝒜^{fp}.

We want to prove that ℬ′ = 𝒜^{fp}. Choose M ∈ ℬ′ which does not belong to ℬ. Let us now invoke the objects E_{ℬ} and E_{ℬ′} of 𝒯, as in Remark 5.1, so that

⟨ℬ⟩ = Ker(Ê_{ℬ} ⊗ −)  and  ⟨ℬ′⟩ = Ker(Ê_{ℬ′} ⊗ −) .

We then have Ê_{ℬ′} ⊗ M = 0 because M ∈ ℬ′. Hence by (5.8), we have H_{𝒫} (E_{ℬ′} ) ⊗ Ĥ_{𝒫} (M) = 0. On the other hand, we have Ĥ_{𝒫} (M) ≠ 0 because M ∉ ℬ. Since H_{𝒫} (E_{ℬ′} ) is ⊗-flat in 𝒜_{𝒫} , we can consider the Serre ⊗-ideal

Ker(H_{𝒫} (E_{ℬ′} ) ⊗ −) ∩ 𝒜_{𝒫}^{fp}

of 𝒜_{𝒫}^{fp}. We just proved that it contains a non-zero object, namely Ĥ_{𝒫} (M). By the ‘simplicity’ of 𝒜_{𝒫}^{fp}, we get that Ker(H_{𝒫} (E_{ℬ′} ) ⊗ −) ∩ 𝒜^{fp} must be the whole of 𝒜_{𝒫}^{fp}. This means that H_{𝒫} (E_{ℬ′} ) = 0, or in other words, Ê_{ℬ′} ∈ Ker(Ĥ_{𝒫} ) = ⟨ℬ⟩ = Ker(Ê_{ℬ} ⊗ −). We have thus proved

(5.9)  Ê_{ℬ} ⊗ Ê_{ℬ′} = 0 .

Consider now the exact sequence in 𝒜 associated to the morphism η_{ℬ} : 𝟙 → E_{ℬ}

0 → I_{ℬ} → ˆ𝟙 ⟶^{η̂_{ℬ}} Ê_{ℬ} .

Since E_{ℬ} ⊗ η_{ℬ} is a split monomorphism, we have I_{ℬ} ∈ Ker(Ê_{ℬ} ⊗ −) = ⟨ℬ⟩ ⊆ ⟨ℬ′⟩ and therefore I_{ℬ} ⊗ Ê_{ℬ′} = 0. Combined with (5.9) we see from the above exact sequence that ˆ𝟙 is also killed by Ê_{ℬ′} , that is Ê_{ℬ′} = 0, or ℬ′ = 𝒜^{fp} as claimed.

In summary, we have now shown that ℬ(𝒫) = Ker(Ĥ_{𝒫} ) ∩ 𝒜^{fp} belongs to the homogeneous spectrum Spc^{h}(𝒦). By the last assumption in (2), we see that ϕ(ℬ(𝒫)) = 𝒫. Finally, we need to relate the functor H_{𝒫} with the homological residue field h̄_{ℬ} for ℬ := ℬ(𝒫). This is now easy. From (5.7) and the fact that ⟨ℬ⟩ = Ker(Ĥ_{𝒫} ) we can further factor Ĥ_{𝒫} by first modding out this kernel. We obtain a unique exact functor H̄_{𝒫} : 𝒜̄(𝒦; ℬ) → 𝒜_{𝒫} making the right-hand triangle in the following diagram commute :

\xymatrix{𝒯 \ar[dr]_{H_{𝒫}} \ar[r]^{h} \ar[rr]^{^{h̄^{ℬ}}} & 𝒜 \ar[d]^{Ĥ_{𝒫}} \ar@{->>}[r]^{Q_{ℬ}} & 𝒜̄(𝒦; ℬ) = mod-𝒦/⟨ℬ⟩ \ar[dl]^{H̄_{𝒫}} \\  & 𝒜_{𝒫} .}

The left-hand triangle was already in (5.7). The top ‘triangle’ commutes by definition of h̄_{ℬ} : 𝒯 → 𝒜̄(𝒦; ℬ). Expanding the notation ℬ = ℬ(𝒫), we have factored each H_{𝒫} : 𝒯 → 𝒜_{𝒫} via a homological residue field h̄_{ℬ(𝒫)} as follows

H_{𝒫} = H̄_{𝒫} ∘ h̄_{ℬ(𝒫)} .

We now claim that the family

F := { ℬ(𝒫) | 𝒫 ∈ Spc(𝒦) }

satisfies the hypothesis of Theorem 5.4, in other words that the family of functors {h̄_{ℬ(𝒫)} }_{𝒫∈Spc(𝒦)} detects ⊗-nilpotence of maps f: x → Y in 𝒯 with x ∈ 𝒦 compact. Indeed if h̄_{ℬ(𝒫)} (f) = 0 then H_{𝒫} (f) = H̄_{𝒫} ∘h̄_{ℬ(𝒫)} (f) = 0 by the above factorization. If this holds for all 𝒫, we conclude by (3) that f^{⊗n} = 0 for n ≫ 1. So Theorem 5.4 tells us that this family F = { ℬ(𝒫) | 𝒫 ∈ Spc(𝒦) } is the whole Spc^{h}(𝒦).

In conclusion, we have constructed a set-theoretic section

σ: Spc(𝒦) → Spc^{h}(𝒦), 𝒫 ↦ ℬ(𝒫)

of ϕ: Spc^{h}(𝒦) ↠ Spc(𝒦) and we just proved that im(σ) = F = Spc^{h}(𝒦). In other words, the surjection ϕ admits a surjective section, i.e. ϕ is a bijection. □

We can now use known nilpotence-detecting families in examples, to prove that ϕ: Spc^{h}(𝒦) → Spc(𝒦) is bijective.

5.10. Corollary. Let 𝒯 = SH be the stable homotopy category and 𝒦 = SH^{c}. Then ϕ: Spc^{h}(𝒦) → Spc(𝒦) is a bijection. More generally, let G be a compact Lie group and 𝒯 = SH(G) the G-equivariant stable homotopy category of genuine G-spectra, and 𝒦 = SH(G)^{c}. Then ϕ: Spc^{h}(𝒦) → Spc(𝒦) is a bijection.

Proof. In the case of 𝒯 = SH, this relies on [DHS88, HS98]. As explained in [Bal10, § 9], the spectrum consists of points 𝒫(p, n) for each prime number p and each ‘chromatic height’ 1 ≤ n ≤ ∞, with the collision 𝒫(0) := 𝒫(p, 1) = SH^{tor} for all p. This prime 𝒫(0) is the kernel of rational homology Hℚ ⊗ −: SH^{c} → D^{b}(ℚ) ≅ ℚ- Mod_{•}. The other primes 𝒫(p, n) for n ≥ 2 are given as the kernels of Morava K-theory K(p, n − 1)_{•}, which are homological functors

K(p, n)_{•}: SH → 𝒜_{p,n} := 𝔽_{p}[v_{n}^{±1}]- Mod_{•}

for 1 ≤ n < ∞ with v_{n} of degree 2(p^{n} − 1), and K(p, ∞)_{•}: SH → 𝔽_{p}- Mod_{•} is mod-p homology. The target categories are graded modules over (graded) fields and the Morava K-theories satisfy Künneth formulas, which amounts to say that they are monoidal functors when 𝒜_{p,n} is equipped with the graded tensor product.

See [Rav92]. The reader can now verify Conditions (1)–(3) of Theorem 5.6. The crucial (3) is the original Nilpotence Theorem [DHS88].

For 𝒯 = SH(G) and 𝒦 = SH(G)^{c}, the description of the set Spc(𝒦) and the relevant nilpotence theorem was achieved for finite groups in [BS17], and more recently for arbitrary compact Lie groups in [BGH18]. Specifically, there is exactly one prime 𝒫(H, p, n) = (Φ^{H})^{−1}(𝒫(p, n)) in Spc(𝒦) for every conjugacy class of closed subgroups H ≤ G and for every ‘chromatic’ prime 𝒫(p, n) as above; here Φ^{H}: SH(G) → SH is the geometric H-fixed point functor, which is tensortriangulated. The relevant homology theories are simply these Φ^{H} composed with the non-equivariant Morava K-theories. So Conditions (1)–(2) in Theorem 5.6 are easy to verify. The relevant nilpotence theorem giving us (3) can be found in [BGH18, Thm. 3.12] (or [BS17, Thm. 4.15] for finite groups, a result also obtained earlier by N. Strickland). □

5.11. Corollary. Let X be a quasi-compact and quasi-separated scheme and 𝒯 = D(X) the derived category of O_{X}-modules with quasi-coherent homology. Here 𝒦 = D^{perf}(X) is the category of perfect complexes, the spectrum Spc(𝒦) ≅ |X| is the underlying space of X and the map ϕ: Spc^{h}(𝒦) → Spc(𝒦) is a bijection.

Proof. The homological functors of Theorem 5.6 are simply the classical residue fields κ(x) ⊗_{O_{X}}^{L} −: D(X) → D(κ(x)) ≅ κ(x)- Mod_{•} at each x ∈ |X|, where of course κ(x) is the residue fields of the local ring O_{X,x}. Here, the relevant nilpotence theorem is due to Thomason [Tho97, Thm. 3.6]. □

5.12. Remark. There is a simpler proof of the above when X is noetherian, following the pattern of the next example. It is worth noting that when X is not noetherian, even for |X| = ∗, the residue fields do not detect vanishing of objects. See [Nee00].

5.13. Example. Let G be a finite group scheme over a field k and 𝒯 = Stab(kG) the category of k-linear representations of G modulo projectives. See [BIKP18]. Here 𝒦 = stab(kG) is the stable category of finite-dimensional representations modulo projectives, Spc(𝒦) ≅ Proj(H^{•}(G, k)) is the so-called projective support variety, and the map ϕ: Spc^{h}(𝒦) → Spc(𝒦) is again bijective. The method of proof is different, for there is no known homology theories capturing the points of Spc(𝒦) and satisfying a Künneth formula. Indeed, points are detected by equivalence classes of so-called π-points, following [FP07], but these functors are not monoidal!

Instead, we can use the fact that localizing subcategories of 𝒯 are classified by subsets of Spc(𝒦) in this case, a non-trivial result that can be found in [BIKP18, § 10]. In such situations, the property E_{ℬ} ⊗ E_{ℬ′} = 0 isolated in Proposition 5.3, for ℬ ≠ ℬ′ in Spc^{h}(𝒦) can be used to show that ϕ: Spc^{h}(𝒦) → Spc(𝒦) is injective. Indeed, if ϕ(ℬ) = ϕ(ℬ′) =: 𝒫, we can use minimality of the localizing category 𝒯_{𝒫} supported at the point 𝒫 to show that E_{ℬ} ⊗ E_{ℬ′} = 0 forces E_{ℬ} = 0 or E_{ℬ′} = 0 which is absurd. This argument can already be found in [BKS19, Cor. 4.26].

5.14. Remark. We note that the above does not use a nilpotence theorem for 𝒯 = Stab(kG). To the best of the author’s knowledge, there is no such result in the literature, the probable reason being that π-points (or shifted cyclic subgroups) are not monoidal. Thanks to the present work, we now know that there exists for every point 𝒫 ∈ Spc(stab(kG)) ≅ Proj(H^{•}(G, k)) a unique homological tensor functor

h̄_{ℬ(𝒫)} : Stab(kG) → 𝒜̄(𝒦; ℬ(𝒫))

to a ‘simple’ Grothendieck tensor category (in the sense of (1) in Theorem 5.6), whose kernel on compacts 𝒦 = 𝒯^{c} = stab(kG) is exactly 𝒫. And we know that the family {h̄_{ℬ(𝒫)} }_{𝒫∈Spc(𝒦)} detects tensor-nilpotence.

5.15. Remark. In view of the avalanche of examples where ϕ: Spc^{h}(𝒦) → Spc(𝒦) is a bijection, it takes nerves of steel not to conjecture that this property holds for all tensor-triangulated categories.

References

[Bal05] Paul Balmer. The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math., 588:149–168, 2005.

[Bal10] Paul Balmer. Spectra, spectra, spectra – tensor triangular spectra versus Zariski spectra of endomorphism rings. Algebr. Geom. Topol., 10(3):1521–1563, 2010.

[BCR97] David J. Benson, Jon F. Carlson, and Jeremy Rickard. Thick subcategories of the stable module category. Fund. Math., 153(1):59–80, 1997.

[BGH18] Tobias Barthel, John Greenlees, and Markus Hausmann. On the Balmer spectrum of

         compact Lie groups. Preprint https://arxiv.org/abs/1810.04698, 2018.

[BIKP18] Dave Benson, Srikanth B. Iyengar, Henning Krause, and Julia Pevtsova. Stratification for module categories of finite group schemes. J. Amer. Math. Soc., 31(1):265–302, 2018.

[BKS17] Paul Balmer, Henning Krause, and Greg Stevenson. The frame of smashing tensorideals. Math. Proc. Cambridge Philos. Soc., to appear, preprint 2017. https://arxiv.

         org/abs/1701.05937.

[BKS19] Paul Balmer, Henning Krause, and Greg Stevenson. Tensor-triangular fields: ruminations. Selecta Math. (N.S.), 25(1):25:13, 2019.

[BS17] Paul Balmer and Beren Sanders. The spectrum of the equivariant stable homotopy category of a finite group. Invent. Math., 208(1):283–326, 2017.

[DHS88] Ethan S. Devinatz, Michael J. Hopkins, and Jeffrey H. Smith. Nilpotence and stable homotopy theory. I. Ann. of Math. (2), 128(2):207–241, 1988.

[FP07] Eric M. Friedlander and Julia Pevtsova. Π-supports for modules for finite group schemes. Duke Math. J., 139(2):317–368, 2007.

[Gab62] Pierre Gabriel. Des catégories abéliennes. Bull. Soc. Math. France, 90:323–448, 1962.

[Hop87] Michael J. Hopkins. Global methods in homotopy theory. In Homotopy theory (Durham, 1985), volume 117 of LMS Lect. Note, pages 73–96. Cambridge Univ. Press, 1987.

[HS98] Michael J. Hopkins and Jeffrey H. Smith. Nilpotence and stable homotopy theory. II. Ann. of Math. (2), 148(1):1–49, 1998.

[Kra00] Henning Krause. Smashing subcategories and the telescope conjecture – an algebraic approach. Invent. Math., 139(1):99–133, 2000.

[Kra05] Henning Krause. Cohomological quotients and smashing localizations. Amer. J. Math., 127(6):1191–1246, 2005.

[Mat17] Akhil Mathew. Residue fields for a class of rational E_{∞}-rings and applications. J. Pure Appl. Algebra, 221(3):707–748, 2017.

[Nee92] Amnon Neeman. The chromatic tower for D(R). Topology, 31(3):519–532, 1992.

[Nee00] Amnon Neeman. Oddball Bousfield classes. Topology, 39(5):931–935, 2000.

[Nee01] Amnon Neeman. Triangulated categories, volume 148 of Annals of Mathematics Studies. Princeton University Press, 2001.

[Rav92] Douglas C. Ravenel. Nilpotence and periodicity in stable homotopy theory, volume 128 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1992.

[Tho97] R. W. Thomason. The classification of triangulated subcategories. Compositio Math., 105(1):1–27, 1997.

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

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