SEPARABLE EXTENSIONS IN TENSOR-TRIANGULAR GEOMETRY AND GENERALIZED QUILLEN STRATIFICATION

PAUL BALMER

Abstract. We exhibit a link between the Going-Up Theorem in commutative algebra and Quillen Stratification in modular representation theory. To this effect, we study the continuous map induced on spectra by a separable extension of tensor-triangulated categories. We determine the image of this map and bound the cardinality of its fibers by the degree of the extension. We then prove a weak form of descent, “up-to-nilpotence”, which allows us to generalize Quillen Stratification to equivariant derived categories.

Nous exhibons un lien entre le théorème du “going-up” en algèbre commutative et le théorème de stratification de Quillen en théorie des représentations modulaires. Dans ce but, nous étudions l’application continue induite sur les spectres par une extension séparable de catégories triangulées tensorielles. Nous en déterminons l’image et bornons le cardinal de ses fibres par le degré de l’extension. Nous prouvons alors une forme faible de descente, “à nilpotence près”, qui nous permet de généraliser la stratification de Quillen à d’autres catégories dérivées équivariantes.

Contents

Introduction 1 1. Statement of results 2 2. Recalling tt-rings and their degree 5 3. Descent up-to-nilpotence 8 4. Applications 13 References 16

Introduction

There are rich and growing connections between commutative algebra and modular representation theory, notably via homological methods. Some of these connections can be built using tensor-triangulated categories. Recall that tensortriangulated categories also appear in several other settings, like motivic theory, equivariant stable homotopy theory, or Kasparov’s KK-theory of C*-algebras, for instance. This framework is the backdrop of tensor-triangular geometry, or ttgeometry for short, see [Bal10].

Date: 2016 February 13. 2010 Mathematics Subject Classification. 18E30, 20J05, 13B24, 55U35. Key words and phrases. Separable, tt-category, descent, nilpotence, Quillen stratification;

Séparable, tt-catégorie, descente, nilpotence, stratification de Quillen.

Research supported by NSF grant DMS-1303073.

In the present work, we use tt-geometry to connect two classical and well-known results, namely the Going-Up Theorem in commutative algebra and Quillen’s Stratification Theorem in modular representation theory. Let us first remind the reader :

Theorem (Going-Up). Let R ⊂ A be an integral extension of commutative rings, let 𝔮 be a prime in A and 𝔭′ a prime in R containing 𝔮 ∩ R; then there exists a prime 𝔮′ in A containing 𝔮 such that 𝔮′ ∩ R = 𝔭′. Further, Spec(A) → Spec(R) is surjective and a weak form of injectivity holds, known as “Incomparability” : if 𝔮 ⊆ 𝔮′ are two primes in A such that 𝔮 ∩ R = 𝔮′ ∩ R then 𝔮 = 𝔮′.

Theorem (Quillen Stratification [Qui71]). Let G be a finite group and 𝕜 be a field of positive characteristic p dividing the order of G. Let V_{G} := Proj(H^{•}(G, 𝕜)) be the projective support variety of G. Then there is a canonical homeomorphism

(0.1)  colim_{H∈Or(G, Abelem)}V_{H} ⟶^{∼} V_{G}

where Or(G, Abelem) is the full subcategory of the orbit category of G on the elementary abelian p-subgroups of G (see Definition 4.6 if necessary).

It is not obvious to the naked eye why the above two results should be related. Let us observe the tip of the iceberg : The Going-Up Theorem forces the rings A and R to have the same Krull dimension; similarly Quillen Stratification forces the Krull dimension of V_{G} to be the maximum of the dimensions of the V_{H}, i.e. the p-rank of G minus one – an important application of [Qui71].

Here, we prove an analogue of Going-Up in tt-geometry and show that it specializes to Quillen Stratification when applied to modular representation theory. This result illustrates the connections between the two subjects and the depth of tt-geometry. To go beyond unification and connection, we also prove some new results, namely we extend Quillen Stratification to any tensor-triangulated category receiving the derived category of G.

1. Statement of results

To do tt-geometry, one needs a tt-category 𝒦. (Here, “tt” is short for “tensortriangular” or “tensor-triangulated”, as appropriate.) In commutative algebra, we use 𝒦 = D^{perf}(R), the homotopy category of bounded complexes of finitely generated projective R-modules. In modular representation theory, we use 𝒦 = stab(𝕜G), the stable category of finitely generated 𝕜G-modules modulo the projective ones.

To recover spaces like the affine scheme Spec(R) in the first example and the support variety V_{G} in the second, we use the spectrum Spc(𝒦) of a tt-category 𝒦. This fundamental tool of tt-geometry was introduced in [Bal05] as the universal topological space in which objects x of 𝒦 admit reasonable supports supp(x) ⊆ Spc(𝒦); see Rem. 2.6. The spectrum can also be constructed by means of prime ideals 𝒫 in 𝒦. By [Bal05], it recovers the Zariski spectrum Spc(D^{perf}(R)) ≅ Spec(R) and the support variety Spc(stab(𝕜G)) ≅ V_{G}. This unification is the key to relate affine schemes and support varieties via tt-geometry. See more in Remark 2.7.

To exhibit a first analogy between Going-Up and Quillen Stratification, let us observe that both situations involve not only one tt-category but actually two (or more). In commutative algebra, it is rather obvious: We have the ring homomorphism R → A, hence an extension-of-scalars D^{perf}(R) → D^{perf}(A) which is a tt-functor. In modular representation theory, we also have tt-functors Res_{H}^{G} : stab(𝕜G) → stab(𝕜H) given by restriction from the group G to its subgroups H ≤ G, for instance the elementary abelian ones. Another key step in our discussion is to understand those restriction functors stab(𝕜G) → stab(𝕜H) as a form of “extension-of-scalars”, similar to the extension-of-scalars of commutative algebra.

This is made possible thanks to a good notion of “ring” in a general tt-category 𝒦, inspired by commutative algebra but flexible enough to be useful in representation theory as well. These good rings are the “tt-rings” of [Bal14] : An associative and unital ring object A ⊗ A ⟶^{μ} A in 𝒦 is called a tt-ring if it is commutative and separable, in the classical sense (i.e. the multiplication μ has a section σ : A → A ⊗ A which is A-linear on both sides). Separability of A guarantees that the category A - Mod_{𝒦} of good-old A-modules in 𝒦 remains triangulated, by [Bal11] (see Remark 2.14). Commutativity of A allows us to equip A - Mod_{𝒦} with a tensor ⊗_{A}. In short, if A is a tt-ring in a tt-category 𝒦, then A - Mod_{𝒦} is again a tt-category and extension-of-scalars F_{A} : 𝒦 ⟶ A - Mod_{𝒦} is a tt-functor.

By [Bal15] the restriction functor Res_{H}^{G} : stab(𝕜G) → stab(𝕜H) is actually such an extension-of-scalars with respect to the particular tt-ring A_{H}^{G} := 𝕜(G/H), with multiplication extending 𝕜-linearly the rule γ · γ = γ and γ · γ′ = 0 for all γ ≠ γ′ in G/H (see Constr. 4.1). More precisely, there is an equivalence

(1.1)  A_{H}^{G} - Mod_{𝒦(G)} ≅ 𝒦(H)

for 𝒦(G) = stab(𝕜G) and of course 𝒦(H) = stab(𝕜H); under this equivalence, extension-of-scalars F_{A_{H}^{G}} becomes isomorphic to restriction Res_{H}^{G}. In particular, the induced map on spectra recovers the usual map on support varieties V_{H} → V_{G}. This recasting of restriction as an extension-of-scalars is already true for derived categories, i.e. the above (1.1) holds if 𝒦(G) means D^{b}(𝕜G), etc. See details in [Bal15, Part I]. Furthermore we show in [BDS14] that such results hold quite generally for equivariant tt-categories, way beyond representation theory.

In general, for any tt-category 𝒦 and any tt-ring A in 𝒦, the tt-functor F_{A} : 𝒦 → A - Mod_{𝒦} induces a continuous map on spectra

(1.2)  φ_{A} := Spc(F_{A}) : Spc(A - Mod_{𝒦} ) ⟶ Spc(𝒦) .

It is clearly important to study this map φ_{A} in general, independently of Going-Up or Quillen Stratification. Here is our main tt-geometric result (see details below):

1.3. Theorem (Descent-up-to-nilpotence, see Thm. 3.19). Let 𝒦 be a tt-category and let A be a tt-ring in 𝒦. Suppose that A has finite degree and that A detects ⊗-nilpotence of morphisms. Then we have a coequalizer of topological spaces

(1.4)  \xymatrix{Spc(A^{⊗2} - Mod_{𝒦} ) \ar[r]^{φ_{1}} \ar[r]_{φ_{2}} & Spc(A - Mod_{𝒦} ) \ar[r]^{φ_{A}} & Spc(𝒦)}

where φ_{1} and φ_{2} are induced by the two obvious homomorphisms A [figure]A ⊗ A.

The degree of a tt-ring A has also been introduced in [Bal14] but can easily be considered as a black-box here. It is the natural measure of the “size” of the tt-ring A. It will allow us to prove some results by induction on the degree, via Theorem 2.19. Having finite degree is a mild hypothesis which holds for all tt-rings in all standard tt-categories of compact objects in algebraic geometry, homotopy theory or modular representation theory, by [Bal14, § 4]. See more in Section 2.

The assumption that A detects ⊗-nilpotence is the important condition of the theorem. It means that if a morphism f in 𝒦 satisfies A⊗f =0 then f^{⊗n} =0 for some n ≥ 1. This property of the tt-ring A will be called nil-faithfulness. Faithfulness (A ⊗ f = 0 ⇒ f = 0) is a special case of interest, as we shall discuss below. Proposition 3.15 provides further equivalent characterizations of nil-faithfulness, like for instance that A generates 𝒦 as a thick ⊗-ideal, i.e. supp(A) = Spc(𝒦).

The strength of (1.4) will be discussed below, when we return to examples. For the moment, simply note that it tells us that φ_{A} is onto and that two points of Spc(A - Mod_{𝒦} ) have the same image only if they are the image of the same point under φ_{1} and φ_{2}. This result is as good as it gets in full tt-generality. In fact, Theorem 1.3 is the culmination of a series of results of independent interest, proved in Section 3 and summarized in Theorem 1.5 below. They illustrate how our study of φ_{A} in tt-geometry mirrors integral extensions in commutative algebra.

1.5. Theorem. Let A be a tt-ring in a tt-category 𝒦 and consider φ_{A} as in (1.2).

(a) The image of φ_{A} equals the support of A, that is, Im(φ_{A}) = supp(A)

in Spc(𝒦). It will follow that φ_{A} is onto if and only if A is nil-faithful. Suppose furthermore that A has finite degree d.

(b) Going-Up : For every 𝒬 ∈ Spc(A - Mod_{𝒦} ) and every 𝒫′ ∈ \overline{\{φ_{A}(𝒬)\}} there exists 𝒬′ ∈ \overline{\{𝒬\}} such that φ_{A}(𝒬′) = 𝒫′.

(c) Incomparability : If 𝒬′ ∈ \overline{\{𝒬\}} in Spc(A - Mod_{𝒦} ) are such that φ_{A}(𝒬) = φ_{A}(𝒬′) then 𝒬 = 𝒬′.

(d) The fibers of φ_{A} are finite and discrete, with at most d points. The result implies that Spc(A - Mod_{𝒦} ) and supp(A) have the same Krull dimension. If A is nil-faithful then Spc(A - Mod_{𝒦} ) and Spc(𝒦) have the same Krull dimension.

As we shall see, the proofs of these results are quite different from their commutative algebra counterparts. The main obstacle to the tt-generalization of classical techniques is the absence of a tt-quotient, i.e. the tt-equivalent of the ring R/I for an ideal I ⊂ R. (Verdier quotients 𝒦/𝒥 lead to open subschemes, not to closed ones.) In particular, there is no “residue field” in general tt-geometry, at least for the moment, and this makes the description of the fibers of φ_{A} particularly difficult.

To conclude our discussion, we still want to explain why Theorem 1.3 is called “descent-up-to-nilpotence” and understand nil-faithfulness in examples.

Conceptually, the category A - Mod_{𝒦} should be approached as a simplification of the original category 𝒦. In some sense, it represents a “finite étale extension of 𝒦”. This idea goes hand-in-hand with Grothendieck’s theory of descent, i.e. the problem of reconstructing the original category 𝒦 from A - Mod_{𝒦} and data over A^{⊗2} and A^{⊗3}. By [Bal12], if A was not just nil-faithful but really faithful (A ⊗ f =0 ⇒ f =0) then A would satisfy descent in 𝒦, which roughly says that

\xymatrix{𝒦 \ar[r] & A - Mod_{𝒦} \ar[r] \ar[r] & A^{⊗2} - Mod_{𝒦}}

is an “equalizer of categories” (plus cocycle conditions over A^{⊗3}). In that case, it is easier to prove, and even easier to believe, that the contravariant Spc(−) turns the latter “equalizer” into the coequalizer of spaces (1.4). The remarkable fact about Theorem 1.3 is that (1.4) is a coequalizer even when A is not faithful but only nil-faithful, despite failure of descent in this situation! This explains the idea of “descent-up-to-nilpotence”.

To fully appreciate the difference between faithful and nil-faithful, i.e. between descent and descent-up-to-nilpotence, we turn to our example of modular representation theory, where we will also better understand the coequalizer (1.4).

Let us test the distinction between faithful and nil-faithful on the tt-rings A_{H}^{G} = 𝕜(G/H) that we discussed above, see (1.1). We assume for simplicity that G is a p-group, where p = char(𝕜) > 0. Then consider in 𝒦 = stab(𝕜G), the tt-ring

A := 𝕜(G/H_{1}) × ⋯ × 𝕜(G/H_{n}) for subgroups H_{1}, …, H_{n} ≤ G .

This A is faithful only if one of the H_{i} equals G, which is sadly restrictive since A - Mod_{𝒦} contains a copy of 𝒦 in that case. On the other hand, this same A is nil-faithful much more often, most famously if H_{1}, …, H_{n} are the elementary abelian p-subgroups of G. In fact, nil-faithfulness of this particular tt-ring A is Serre’s old theorem on the vanishing of the Bocksteins [Ser65]; see Thm. 4.3. In other words, descent happens only in the trivial case where 𝒦 is a summand of A - Mod_{𝒦} but descent-up-to-nilpotence happens much more often. In fact, in that case, our descent-up-to-nilpotence Theorem 1.3 recovers Quillen Stratification.

Although our goal is not to give a new proof of Quillen’s result, we hope that recovering such a major result as an example could help the reader grasp the power of Theorem 1.3. To provide new applications, we extend Quillen Stratification to every tt-category receiving the derived category. This is Theorem 4.10 below :

1.6. Theorem. Let p be a prime and G a finite group with p dividing |G|. Let 𝒦(G) be a tt-category which admits a tt-functor Φ : D^{b}(𝔽_{p}G) ⟶ 𝒦(G), where 𝔽_{p} is the finite field with p elements. For every subgroup H ≤ G, consider the tt-ring A_{H} = Φ(𝔽_{p}(G/H)), image in 𝒦(G) of the complex 𝔽_{p}(G/H) ∈ D^{b}(𝔽_{p}G) concentrated in degree zero. Define 𝒦(H) = A_{H} - Mod_{𝒦(G)} the corresponding category of modules. Then the tt-functors F_{A_{H}} : 𝒦(G) ⟶ 𝒦(H) induce a homeomorphism

(1.7)  colim_{H∈Or(G, Abelem)} Spc(𝒦(H)) ⟶^{∼} Spc(𝒦(G)) .

The reason for the notation 𝒦(H) is that, in most examples, the category 𝒦(G) is defined for all finite groups G together with an equivalence A_{H} - Mod_{𝒦(G)} ≅ 𝒦(H), as in (1.1). As we show in [BDS14], such “monadicity” results happen in most equivariant settings, way beyond representation theory. They give (1.7) its full flavor since in that case the left-hand category 𝒦(H) is not merely a category of modules in 𝒦(G) but the true object of study over the subgroup H. To give an example for which Spc(𝒦(G)) is not known, one could take 𝒦(G) to be the bounded derived category of equivariant vector bundles over a variety with G-action. In that case, Quillen Stratification is new. See Example (4.14).

The organization of the paper should now be clear from the table of content.

2. Recalling tt-rings and their degree

2.1. Convention. We use the word triangulated in the sense of [Bal11, § 5], or Künzer [Kün07] and Maltsiniotis [Mal06]. This is a mild strengthening of Verdier’s traditional definition in which we require octahedra to satisfy the analogue of the morphism axiom. One can also require higher triangles but we do not need them here. All examples (stable homotopy categories) are triangulated in that sense.

2.2. Definition. A tt-category 𝒦 is an essentially small, idempotent-complete tensortriangulated category. Hence 𝒦 is triangulated, all idempotent endomorphisms in 𝒦 split, and 𝒦 is equipped with a symmetric monoidal structure ⊗ : 𝒦×𝒦 ⟶ 𝒦 such that x ⊗ − : 𝒦 → 𝒦 is exact for all objects x ∈ 𝒦. We denote by 𝟙 ∈ 𝒦 its ⊗-unit. Throughout this paper, 𝒦, ℒ, … denote tt-categories. A tt-functor F : 𝒦 → ℒ is a functor which is both exact and monoidal, in such a way that the two natural isomorphisms from F (x ⊗ Σy) to ΣF (x ⊗ y) agree (using in different order the compatibility of F with ⊗, of F with Σ, and of ⊗ with Σ) and similarly on the left.

2.3. Example. Let X be a quasi-compact and quasi-separated scheme (e. g. a noetherian scheme or an affine scheme) then the derived category 𝒦 = D^{perf}(X) of perfect complexes [BGI71] is a tt-category. For X = Spec(R), this is the tt-category D^{perf}(R) of the introduction.

2.4. Example. Let G be a finite group and 𝕜 a field of positive characteristic p dividing |G|. Then the derived category D^{b}(𝕜G) := D^{b}(𝕜G – mod) and the stable category stab(𝕜G) of finitely generated 𝕜G-modules are tt-categories and the wellknown (Buchweitz-Rickard) quotient D^{b}(𝕜G) ↠ stab(𝕜G) is a tt-functor. See [Ric89].

2.5. Notation. We write x ≤ y when x is a direct summand of y. A triangulated subcategory 𝒥 ⊆ 𝒦 is called a thick ⊗-ideal if x ≤ y ∈ 𝒥 ⇒ x ∈ 𝒥 and if 𝒦 ⊗ 𝒥 ⊆ 𝒥. We denote by ⟨ℰ⟩ the thick ⊗-ideal generated by a collection of objects ℰ ⊆ 𝒦.

2.6. Remark. We do not need much tt-geometry here but we simply recall that the spectrum Spc(𝒦) of a tt-category 𝒦 consists of all prime thick ⊗-ideals 𝒫 ⊊ 𝒦, i.e. such that x ⊗ y ∈ 𝒫 implies x ∈ 𝒫 or y ∈ 𝒫. The topology of Spc(𝒦) has an open basis composed of all subsets 𝒰(x) := { 𝒫 | x ∈ 𝒫 }, for x ∈ 𝒦. The closed complement supp(x) := { 𝒫 | x ∉ 𝒫 } is the support of x in Spc(𝒦).

2.7. Remark. Determining tt-spectra is an ongoing enterprise in several fields of mathematics, with direct link to the classifications of thick ⊗-ideals. This program originated in stable homotopy theory [HS98], and extended to algebraic geometry [Hop87, Nee92, Tho97] and modular representation theory [BCR97, FP07] prior to [Bal05]. More recent progress has been made in noncommutative topology [Del10], for Artin-Tate motives [Pet13], for noncommutative motives [DT12], for group rings over commutative algebras [Ste13], for category algebras [Xu14] and for Lie superalgebras [BKN14]. Still, computing Spc(𝒦) remains an open challenge for many important tt-categories 𝒦, like equivariant derived categories or equivariant stable homotopy categories [GM95].

One basic tt-result that we shall need a few times is the following :

2.8. Lemma (Existence Lemma [Bal05, Lem. 2.2]). Let 𝒦 be a tt-category, 𝒥 ⊆ 𝒦 a thick ⊗-ideal and S ⊂ 𝒦 a ⊗-multiplicative collection of objects (𝟙 ∈ S ⊇ S ⊗ S) such that S ∩ 𝒥 = ∅. Then there exists 𝒫 ∈ Spc(𝒦) such that 𝒥 ⊆ 𝒫 and 𝒫 ∩ S = ∅.

2.9. Notation. Given n objects x_{1}, …, x_{n} in 𝒦 and a permutation π ∈ S_{n}, we also denote by π : x_{1} ⊗ ⋯ ⊗ x_{n} →^{∼} x_{π(1)} ⊗ ⋯ ⊗ x_{π(n)} the induced isomorphism.

2.10. Definition. A ring object A in 𝒦 is a monoid (A , μ : A ⊗ A → A , η : 𝟙 → A), i.e. admits associative multiplication μ : A ⊗ A → A and two-sided unit η. It is commutative if μ ∘ (12) = μ. A morphism of (commutative) ring objects, or simply a homomorphism, is a morphism f : A → B in 𝒦, compatible with multiplications and units. We also say that B is an A-algebra or that B is a ring over A.

2.11. Notation. For two ring objects A and B, the ring object A × B is A ⊕ B with component-wise structure. On the other hand, the ring object A ⊗ B has multiplication (μ_{1} ⊗ μ_{2}) ∘ (23) : (A ⊗ B)^{⊗2} ⟶ A ⊗ B and obvious unit.

2.12. Definition. A ring object A is called separable if there exists a morphism σ : A → A ⊗ A such that μσ = id_{A} and (1 ⊗ μ) ∘ (σ ⊗ 1) = σμ = (μ ⊗ 1) ∘ (1 ⊗ σ) : A^{⊗2} → A^{⊗2}. This simply means that A is projective as A, A-bimodule.

2.13. Remark. If A and B are separable then so are A × B and A ⊗ B.

2.14. Remark. A (left) A-module is a pair (x, ρ) where x ∈ 𝒦 and where the socalled action ρ : A ⊗ x → x satisfies the usual associativity and unit conditions. We denote by A - Mod_{𝒦} the category of A-modules with A-linear morphisms and by F_{A} : 𝒦 → A - Mod_{𝒦} the functor F_{A}(y) = (A ⊗ y, μ ⊗ 1), etc. For details see [EM65] or [Bal11].

By [Bal11], when A is separable, the category of A-modules in 𝒦 remains triangulated in such a way that both the extension-of-scalars functor F_{A} : 𝒦 ⟶ A - Mod_{𝒦} and the forgetful functor U_{A} : A - Mod_{𝒦} ⟶ 𝒦 are exact. We really mean “modules in the homotopy category” here, not “homotopy category of modules”, thanks to separability! The key fact is that the co-unit ϵ : F_{A}U_{A} → Id_{A - Mod_{𝒦}} is split surjective. In particular, every A-module x is a direct summand of a free one : x ≤ F_{A}U_{A}(x).

The following simple definition [Bal14] plays a central role in our theory.

2.15. Definition. A ring object A is a tt-ring if A is commutative and separable.

The terminology is chosen so that if A is a tt-ring in a tt-category 𝒦 then A - Mod_{𝒦} remains a tt-category. See details in [Bal14], notably about the tensor structure ⊗_{A} over A on the category A - Mod_{𝒦} . The projection formula [Bal14, Prop. 1.2] says that U_{A}(x ⊗_{A} F_{A}(y)) ≅ U_{A}(x) ⊗ y for all x ∈ A - Mod_{𝒦} and y ∈ 𝒦.

2.16. Remark. In commutative algebra, an R-algebra A will not be a tt-ring in D^{perf}(R) in general, unless A is finite étale over R. In that case, the tt-category D^{perf}(A) coincides with A - Mod_{D^{perf} (R)} by [Bal11, Cor. 6.6]. Thus the reader might prefer to think of our extensions-of-scalars F_{A} : 𝒦 → A - Mod_{𝒦} along tt-rings as finite étale morphisms.

2.17. Remark. Suppose that 𝒦 = 𝒯^{c} consists of the compact objects of a compactly generated (big) tensor-triangulated category 𝒯. Then any smashing localization of 𝒯 is in fact an extension-of-scalars with respect to a tt-ring A in 𝒯, which is moreover an idempotent. Conversely, any ring object such that μ : A ⊗ A → A is an isomorphism yields a smashing localization. This provides another class of examples of separable extensions. In this paper, we restrict attention to compact tt-rings A ∈ 𝒦, in order to speak of supp(A), thus excluding smashing localizations. Yet, some of our results might extend to “big” tt-rings A ∈ 𝒯 with a suitable notion of “big support”.

2.18. Remark. We refer to [Bal14] for the notion of the degree of a tt-ring A. We use it as a black box here, emphasizing that we do not know any tt-ring of infinite degree. (Even the big smashing rings of Remark 2.17 actually have degree 1.) In particular the “equivariant” tt-rings 𝕜(G/H) of [Bal15] are of finite degree [G: H] in D^{b}(𝕜G). Furthermore, the degree cannot increase along tt-functors. The key result allowing induction on the degree is the following [Bal14, Thm. 3.6] :

2.19. Theorem. Let A be a tt-ring of finite degree d in a tt-category 𝒦. Then there exists a tt-ring C and a homomorphism g : A⊗A → C such that ( \begin{matrix}μ \\ g \end{matrix}) : A⊗A →^{∼} A×C is an isomorphism. Moreover, C is of degree d − 1 as a tt-ring in A - Mod_{𝒦} , when viewed as an A-algebra through g ∘ (1 ⊗ η) : A → C.

3. Descent up-to-nilpotence

3.1. Notation. Given a tt-ring A in a tt-category 𝒦, we have a continuous (and spectral) map φ_{A} := Spc(F_{A}) : Spc(A - Mod_{𝒦} ) ⟶ Spc(𝒦), as in (1.2). It is defined by φ_{A}(𝒫) = F_{A}^{−1}(𝒫) for every prime 𝒫 ⊂ A - Mod_{𝒦} . See [Bal05, § 3]. See Remark 2.14 for the functors F_{A} : 𝒦 ⇄ A - Mod_{𝒦} : U_{A}.

3.2. Lemma. For every thick ⊗-ideal 𝒥_{0} ⊆ 𝒦, the thick subcategory U_{A}^{−1}(𝒥_{0}) of A - Mod_{𝒦} is ⊗-ideal and equals the thick ⊗-ideal ⟨F_{A}(𝒥_{0})⟩ generated by F_{A}(𝒥_{0}).

Proof. Since A is separable, we have z ≤ F_{A}U_{A}(z) for every z ∈ A - Mod_{𝒦} (Remark 2.14). Now, if y ∈ U_{A}^{−1}(𝒥_{0}) and x ∈ 𝒦 then U_{A}(y) ∈ 𝒥_{0} and since 𝒥_{0} is ⊗-ideal, 𝒥_{0} ∋ x ⊗ U_{A}(y) ≃ U_{A}(F_{A}(x) ⊗_{A} y). So F_{A}(x) ⊗_{A} y ∈ U_{A}^{−1}(𝒥_{0}). Hence for every z ∈ A - Mod_{𝒦} , we have U_{A}^{−1}(𝒥_{0}) ∋ F_{A}(U_{A}(z))⊗_{A} y ≥ z⊗_{A} y hence z⊗_{A} y ∈ U_{A}^{−1}(𝒥_{0}) as wanted. So, U_{A}^{−1}(𝒥_{0}) is ⊗-ideal. Let us show that U_{A}^{−1}(𝒥_{0}) is the smallest thick ⊗-ideal containing F_{A}(𝒥_{0}). On the one hand, U_{A}F_{A}(x) ≃ A ⊗ x belongs to 𝒥_{0} for every x ∈ 𝒥_{0}. So, F_{A}(𝒥_{0}) ⊆ U_{A}^{−1}(𝒥_{0}) and therefore ⟨F_{A}(𝒥_{0})⟩ ⊆ U_{A}^{−1}(𝒥_{0}) by the above discussion. Conversely, if z ∈ U_{A}^{−1}(𝒥_{0}) then F_{A}(U_{A}(z)) ∈ F_{A}(𝒥_{0}) belongs to ⟨F_{A}(𝒥_{0})⟩, hence so does every direct summand of F_{A}(U_{A}(z)), like z itself. □

3.3. Lemma. Let 𝒫 ∈ Spc(A - Mod_{𝒦} ) and let s ∈ A - Mod_{𝒦} be an object such that U_{A}(s) ∈ φ_{A}(𝒫). Then s ∈ 𝒫.

Proof. U_{A}(s) ∈ φ_{A}(𝒫) = F_{A}^{−1}(𝒫) reads 𝒫 ∋ F_{A}(U_{A}(s)) ≥ s and 𝒫 is thick. □

3.4. Theorem. Let A be a tt-ring in a tt-category 𝒦. Then :

(a) For every x ∈ 𝒦, we have φ_{A}^{−1}(supp(x)) = supp(F_{A}(x)) in Spc(A - Mod_{𝒦} ).

(b) Let S ⊂ A - Mod_{𝒦} be a ⊗-multiplicative collection of objects and consider the closed subset 𝒵(S) = ∩_{s∈S} supp(s) in Spc(A - Mod_{𝒦} ). Then its image is closed too; more precisely φ_{A}(𝒵(S)) = 𝒵(U_{A}(S)) where U_{A}(S) = { U_{A}(s) | s ∈ S } (not necessarily ⊗-multiplicative). In particular, φ_{A} is a closed map.

(c) For all y ∈ A - Mod_{𝒦} , we have φ_{A}(supp(y)) = ∩_{n≥1} supp(U_{A}(y^{⊗n})) in Spc(𝒦).

(^{1})

(d) The image of φ_{A} : Spc(A - Mod_{𝒦} ) → Spc(𝒦) is exactly the support of A.

Proof. Part (a) is a general property of Spc(F) for any tt-functor F, see [Bal05, Prop. 3.6]. For (b), note that 𝒵(S) = { 𝒫 | S ∩ 𝒫 = ∅ }. It follows from Lemma 3.3 that φ_{A}(𝒵(S)) ⊆ 𝒵(U_{A}(S)). Conversely, suppose that 𝒫_{0} ∈ Spc(𝒦) is such that U_{A}(s) ∉ 𝒫_{0} for all s ∈ S (∗) and consider the thick ⊗-ideal 𝒥 := U_{A}^{−1}(𝒫_{0}) in A - Mod_{𝒦} (Lemma 3.2). Consider also the ⊗-multiplicative subset T := S ⊗_{A} { F_{A}(x) | x ∈ 𝒦∖𝒫_{0} } in A - Mod_{𝒦} . We claim that T ∩ 𝒥 = ∅. Indeed, suppose ab absurdo, that there exists s ∈ S and x ∈ 𝒦∖𝒫_{0} such that s⊗_{A}F_{A}(x) ∈ 𝒥 = U_{A}^{−1}(𝒫_{0}). Then 𝒫_{0} ∋ U_{A}(s⊗_{A} F_{A}(x)) ≃ U_{A}(s)⊗x. Since x ∉ 𝒫_{0} and 𝒫_{0} is prime, this implies that U_{A}(s) ∈ 𝒫_{0} which contradicts (∗). By the Existence Lemma 2.8, there exists a prime 𝒫 ⊂ A - Mod_{𝒦} such that 𝒫 ⊇ 𝒥 and 𝒫 ∩ T = ∅. These properties imply

1 When 𝒦 is rigid, or A has finite degree, one can prove φ_{A}(supp(y)) = supp(U_{A}(y)).

(1) 𝒫 ⊇ U_{A}^{−1}(𝒫_{0}), (2) 𝒫 ∩ S = ∅ (3) 𝒫 ∩ F_{A}(𝒦∖𝒫_{0}) = ∅. Since U_{A}^{−1}(𝒫_{0}) = ⟨F_{A}(𝒫_{0})⟩ by Lemma 3.2, the first relation above implies 𝒫_{0} ⊆ F_{A}^{−1}(𝒫). Combining with (3), which implies F_{A}^{−1}(𝒫) ⊆ 𝒫_{0}, we get 𝒫_{0} = F_{A}^{−1}(𝒫) = φ_{A}(𝒫). Finally, (2) reads 𝒫 ∈ 𝒵(S) hence 𝒫_{0} = φ_{A}(𝒫) ∈ φ_{A}(𝒵(S)) as wanted.

For (c), let S = {y^{⊗n}, n ≥ 1} so that 𝒵(S) = supp(y). By (b), we deduce φ_{A}(supp(y)) = ∩_{n≥1} supp(U_{A}(y^{⊗n})).

Part (d) follows from (c) applied to y = 𝟙_{A} since U_{A}(𝟙_{A}) = A. □

3.5. Theorem (Going-Up). Let A be a tt-ring in 𝒦. Let 𝒬 ∈ Spc(A - Mod_{𝒦} ) be a point and 𝒫 := φ_{A}(𝒬) its image in Spc(𝒦). Let 𝒫′ ∈ \overline{\{𝒫\}} be a point in the closure of 𝒫. Then there exist 𝒬′ ∈ \overline{\{𝒬\}} in the closure of 𝒬 such that φ_{A}(𝒬′) = 𝒫′. In cash : if 𝒫′ ⊆ 𝒫 = F_{A}^{−1}(𝒬) then there exists 𝒬′ ⊆ 𝒬 such that F_{A}^{−1}(𝒬′) = 𝒫′.

Proof. This follows by Theorem 3.4 (b) with S = (A - Mod_{𝒦} )∖𝒬. Indeed, \overline{\{𝒬\}} = 𝒵(S) for that particular S (see [Bal05, Prop. 2.9]) and 𝒫′ ∈ 𝒵(U_{A}(S)) since for every s ∉ 𝒬, Lemma 3.3 implies that U_{A}(s) ∉ φ_{A}(𝒬) = 𝒫 ⊇ 𝒫′ so U_{A}(s) ∉ 𝒫′. □

3.6. Remark. Let F : 𝒦 → ℒ be a tt-functor, A a tt-ring in 𝒦 and B := F (A) the image tt-ring in ℒ. Then there exists a tt-functor F̄ : A - Mod_{𝒦} ⟶ B - Mod_{ℒ} such that F̄F_{A} = F_{B}F and U_{B}F̄ = F U_{A} (see [Bal14, Rem. 1.6]) :

(3.7)  \xymatrix{𝒦 \ar[r]^{F} \ar[d]_{F_{A}} & ℒ \ar[d]_{F_{B}} \\ A - Mod_{𝒦} \ar[u]_{U_{A}} \ar[r]^{F̄} & B - Mod_{ℒ} \ar[u]_{U_{B}}}

In two places below, we shall apply this construction to F = F_{A} itself. To avoid confusion, let us say that A_{1}, A_{2} are two tt-rings in 𝒦 (later to be A_{1} = A_{2} = A). In the above notation, we set A = A_{2}, ℒ = A_{1} - Mod_{𝒦} and F = F_{A_{1}}. So the ring B = F_{A_{1}}(A_{2}) in ℒ has underlying ring A_{1} ⊗ A_{2} in 𝒦. By [Bal14, Rems. 1.4 and 1.5], we have an identification B - Mod_{ℒ} ≅ (A_{1} ⊗ A_{2}) - Mod_{𝒦} under which the two functors F_{B} and F̄ become the two functors A_{1} - Mod_{𝒦} → (A_{1} ⊗ A_{2}) - Mod_{𝒦} and A_{2} - Mod_{𝒦} → (A_{1} ⊗ A_{2}) - Mod_{𝒦} associated to the two homomorphisms 1 ⊗ η_{2} : A_{1} → A_{1} ⊗ A_{2} and η_{1} ⊗ 1 : A_{2} → A_{1} ⊗ A_{2}.

3.8. Lemma. Let F : 𝒦 → ℒ be a tt-functor, A a tt-ring in 𝒦 and B = F (A) as in Remark 3.6. Then diagram (3.7) yields a commutative square of spectra :

(3.9)  [figure: Spc(𝒦); Spc(ℒ); φ=Spc(F); 𝒬; ∋; 𝒫; ∈; Spc(A - Mod_{𝒦} ); φ_{A}=Spc(F_{A}); Spc(B - Mod_{ℒ} ) .; φ̄=Spc(F̄); φ_{B}=Spc(F_{B})]

Let 𝒫 ∈ Spc(A - Mod_{𝒦} ) and 𝒬 ∈ Spc(ℒ) such that φ_{A}(𝒫) = φ(𝒬). Then there exists 𝒬′ ∈ Spc(B - Mod_{ℒ} ) such that φ̄(𝒬′) ∈ \overline{\{𝒫\}} (i.e. φ̄(𝒬′) ⊆ 𝒫) and φ_{B}(𝒬′) = 𝒬.

Proof. Consider the ⊗-multiplicative S ⊂ B - Mod_{ℒ} given by S = { F̄(s) | s ∈ A - Mod_{𝒦} , s ∉ 𝒫 }. We claim that 𝒬 ∩ U_{B}(S) = ∅. Indeed, if 𝒬 ∋ U_{B}(F̄(s)) ≃ F (U_{A}(s)) for some s ∉ 𝒫 then U_{A}(s) ∈ F^{−1}(𝒬) = φ(𝒬) = φ_{A}(𝒫) hence s ∈ 𝒫 by Lemma 3.3, which contradicts the choice of s. In other words, 𝒬 ∈ 𝒵(U_{B}(S)) and by Theorem 3.4 (b) applied to the tt-ring B in the tt-category ℒ, there exists 𝒬′ ∈ 𝒵(S) such that φ_{B}(𝒬′) = 𝒬. The property 𝒬′ ∈ 𝒵(S) means that F̄((A - Mod_{𝒦} )−𝒫)∩𝒬′ = ∅ which means that φ̄(𝒬′) = F̄^{−1}(𝒬′) ⊆ 𝒫, as claimed. □

3.10. Theorem. Let A ∈ 𝒦 be a tt-ring of finite degree d ≥ 1 (see Remark 2.18).

(a) Incomparability : Let 𝒫, 𝒫′ ∈ Spc(A - Mod_{𝒦} ) be two primes such that 𝒫 ⊆ 𝒫′ (i.e. 𝒫 ∈ \overline{\{𝒫′\}}) and φ_{A}(𝒫) = φ_{A}(𝒫′). Then 𝒫 = 𝒫′.

(b) The space Spc(A - Mod_{𝒦} ) has the same Krull dimension as supp(A) – the latter as a subspace of Spc(𝒦).

(c) The fibers of the map φ_{A} : Spc(A - Mod_{𝒦} ) → Spc(𝒦) are discrete (have Krull dimension zero) and contain at most d points.

Proof. Note right away that (b) follows from part (a), Theorem 3.4 (d) and Going-Up Theorem 3.5. Similarly, (a) implies the discretion of the fibers in (c). So let us prove Incomparability and the bound on the cardinality of the fibers by induction on d = deg(A). For d = 0, we have A = 𝟙 and there is nothing to prove. This being essentially a convention, let us discuss the next case. For d = 1, we use [Bal14, Prop. 4.1] which tells us that the multiplication μ : A ⊗ A →^{∼} A is an isomorphism and F_{A} is simply a (Bousfield, smashing) localization, in such a way that the map φ_{A} : Spc(A - Mod_{𝒦} ) → Spc(𝒦) is the inclusion of an open and closed component of Spc(𝒦) corresponding to supp(A); again in this case the results are straightforward. Suppose deg(A) > 1 and the result true for tt-rings of degree d−1.

Let us do a preparation useful for (a) and (c). By Theorem 2.19, there exists a ring isomorphism h = ( \begin{matrix}μ \\ g \end{matrix}) : A ⊗ A →^{∼} A × C such that C has degree d − 1, when viewed as tt-ring in A - Mod_{𝒦} . To be careful, let us write C̄ ∈ A - Mod_{𝒦} for C viewed as an A-module via g (1 ⊗ η), so that C = U_{A}(C̄). See [Bal14, Rem. 1.5], which also recalls the canonical equivalence C̄ - Mod_{ℒ} ≅ C - Mod_{𝒦} where ℒ := A - Mod_{𝒦} in such a way that FC̄ ∘ F_{A} ≅ F_{C}. Consider now the left-hand diagram of tt-rings below (defining the two homomorphisms g_{i} : A → C for i = 1, 2) :

\xymatrix{𝟙 \ar[r]^{η} \ar[d]_{η} & A \ar[d]^{η⊗1} \ar[ddr]^{_{( \begin{matrix}1 \\ g_{2} \end{matrix} )}} &  & 𝒦 \ar[r]^{F_{A}} \ar[d]_{F_{A}} & A - Mod_{𝒦} \ar[d]^{F_{2}} \ar[ddr]^{_{( \begin{matrix}Id \\ G_{2} \end{matrix} )}} \\ A \ar[r]^{1⊗η} \ar[drr]_{_{( \begin{matrix}1 \\ g_{1} \end{matrix} )}} & A ⊗ A \ar[dr]^{h}_{≃} &  & A - Mod_{𝒦} \ar[r]^{F_{1}} \ar[drr]_{_{( \begin{matrix}Id \\ G_{1} \end{matrix} )}} & (A ⊗ A) - Mod_{𝒦} \ar@{-->}[dr]^{H}_{≃} \\  &  & A × C &  &  & A - Mod_{𝒦} ×C - Mod_{𝒦}}

These homomorphisms induce the above right-hand diagram of tt-functors, which in turn induces the following commutative diagram of topological spaces :

(3.11)  [figure: Spc(𝒦); Spc(A - Mod_{𝒦} ); φ_{A}; Spc(A - Mod_{𝒦} ); φ_{A}; Spc(A^{⊗2} - Mod_{𝒦} ); φ_{1}; φ_{2}; ≃; (id ψ_{1}); Spc(A - Mod_{𝒦} ) ⊔ Spc(C - Mod_{𝒦} ); (id ψ_{2})]

where φ_{i} = Spc(F_{i}) and ψ_{i} = Spc(G_{i}), for i = 1, 2. Note that the upper-left square in (3.11) is nothing but (3.9) applied to F = F_{A} itself. (See the explanations in the second half of Remark 3.6.) By the construction of C and the discussion above, the functor G_{1} : A - Mod_{𝒦} → C - Mod_{𝒦} ≅ C̄ - Mod_{ℒ} is just an extension-of-scalar FC̄ with respect to the tt-ring C̄ in A - Mod_{𝒦} , which has degree d−1. We can therefore apply the induction hypothesis to the map ψ_{1}, i.e. we can assume that (a) and (c) hold for ψ_{1}.

Before jumping into the proof of (a), note that since φ_{A} = F_{A}^{−1}(−) commutes with arbitrary intersection, an easy application of Zorn’s Lemma shows that any φ_{A}-fiber admits a minimal prime inside any given 𝒫. So, we can assume 𝒫 minimal in its φ_{A}-fiber : If 𝒫̃ ⊆ 𝒫 and φ_{A}(𝒫̃) = φ_{A}(𝒫) then 𝒫̃ = 𝒫. By Lemma 3.8, there exists 𝒬′ ∈ Spc(A^{⊗2} - Mod_{𝒦} ) such that φ_{1}(𝒬′) ∈ \overline{\{𝒫\}} and φ_{2}(𝒬′) = 𝒫′. In the decomposition of Spc(A^{⊗2}) as Spc(A) ⊔ Spc(C), we have two possibilities. Either 𝒬′ ∈ Spc(A), in which case (since φ_{1} = φ_{2} = id on the Spc(A) part) we get 𝒫′ = φ_{2}(𝒬′) = 𝒬′ = φ_{1}(𝒬′) ∈ \overline{\{𝒫\}}, meaning 𝒫′ ⊆ 𝒫, and we have 𝒫 = 𝒫′ as wanted. Or 𝒬′ ∈ Spc(C). In that case, by Going-Up Theorem 3.5 applied to φ_{2} (i.e. to extension-of-scalar F_{2}), we deduce from φ_{2}(𝒬′) = 𝒫′ ⊇ 𝒫 the existence of 𝒬 ⊆ 𝒬′ such that φ_{2}(𝒬) = 𝒫. Since 𝒬 belongs to \overline{\{𝒬′\}}, it is also in the Spc(C) part. Now φ_{1}(𝒬) ⊆ φ_{1}(𝒬′) ⊆ 𝒫 and since φ_{A}φ_{1} = φ_{A}φ_{2}, the point φ_{1}(𝒬) belongs to the same φ_{A}-fiber as 𝒫. Since 𝒫 is minimal in its fiber, we must have φ_{1}(𝒬) = φ_{1}(𝒬′) = 𝒫. In short, we have 𝒬 ⊆ 𝒬′ in Spc(C) such that ψ_{1}(𝒬) = ψ_{1}(𝒬′). As explained above, we can apply the induction hypothesis to ψ_{1}, hence deduce 𝒬 = 𝒬′. Taking the image under φ_{2} finally gives 𝒫 = 𝒫′, as wanted.

For (c), let 𝒫_{0} ∈ Spc(𝒦) and consider its fiber φ_{A}^{−1}(𝒫_{0}) ⊆ Spc(A - Mod_{𝒦} ). We need to show that |φ_{A}^{−1}(𝒫_{0})| ≤ d. If it is empty, we are done. Otherwise, pick 𝒫 ∈ φ_{A}^{−1}(𝒫_{0}). Consider Diagram 3.11 again. We are going to define an injection φ_{A}^{−1}(𝒫_{0})∖{𝒫} ↪ ψ_{1}^{−1}(𝒫) ⊆ Spc(C). This is sufficient since by induction hypothesis, the fibers of ψ_{1} have at most d − 1 points. For every 𝒫′ ∈ φ_{A}^{−1}(𝒫_{0}) with 𝒫′ ≠ 𝒫, there exists by Lemma 3.8 a prime 𝒬′ ∈ Spc(A^{⊗2} - Mod_{𝒦} ) such φ_{2}(𝒬′) = 𝒫′ and φ_{1}(𝒬′) ⊆ 𝒫. Since φ_{A}φ_{1} = φ_{A}φ_{2}, the prime φ_{1}(𝒬′) goes to 𝒫_{0} under φ_{A} and therefore φ_{1}(𝒬′) = 𝒫 by Incomparability (a). Under the decomposition of Spc(A^{⊗2}) as Spc(A) ⊔ Spc(C), we cannot have 𝒬′ ∈ Spc(A) for then the relations 𝒫 = φ_{1}(𝒬′) and 𝒫′ = φ_{2}(𝒬′) force 𝒫 = 𝒫′ (since φ_{1} = id and φ_{2} = id on the Spc(A) part). Hence 𝒬′ belongs to the Spc(C) part and φ_{1}(𝒬′) = 𝒫 now reads ψ_{1}(𝒬′) = 𝒫. Therefore, we have constructed for every 𝒫′ ∈ φ_{A}^{−1}(𝒫_{0})∖{𝒫} a prime 𝒬′ ∈ ψ_{1}^{−1}(𝒫) such that ψ_{2}(𝒬′) = 𝒫′. The latter relation shows that different 𝒫′ have different 𝒬′. Hence the wanted injection φ_{A}^{−1}(𝒫_{0})∖{𝒫} ↪ ψ_{1}^{−1}(𝒫). □

3.12. Corollary. Let A ∈ 𝒦 be a tt-ring of finite degree and let F : 𝒦 → ℒ be a tt-functor. Let B = F (A) in ℒ. Then the induced commutative square (3.9)

[figure: Spc(𝒦); Spc(ℒ); φ; 𝒬; ∋; 𝒫; ∈; Spc(A - Mod_{𝒦} ); φ_{A}; Spc(B - Mod_{ℒ} ); φ̄; φ_{B}]

is weakly cartesian in the following sense : For every 𝒫 ∈ Spc(A - Mod_{𝒦} ) and 𝒬 ∈ Spc(ℒ) such that φ_{A}(𝒫) = φ(𝒬) there exists 𝒬′ ∈ Spc(B - Mod_{ℒ} ) such that φ̄(𝒬′) = 𝒫 and φ_{B}(𝒬′) = 𝒬.

Proof. Choose 𝒬′ as in Lemma 3.8, i.e. with φ̄(𝒬′) ⊆ 𝒫 and φ_{B}(𝒬′) = 𝒬. Then φ_{A}(φ̄(𝒬′)) = φ(φ_{B}(𝒬′)) = φ(𝒬) = φ_{A}(𝒫). So φ̄(𝒬′) = 𝒫 by Theorem 3.10 (a). □

3.13. Remark. The above square is not cartesian in general. For instance, using ℂ ⊗_{ℝ} ℂ ≃ ℂ × ℂ, we see that for 𝒦 = D^{b}(ℝ) and A = ℂ, and thus ℒ = D^{b}(ℂ), the square has one point in three corners but two points in the bottom-right corner.

3.14. Theorem. Let A be a tt-ring of finite degree. Then we have a coequalizer

\xymatrix{Spc(A^{⊗2} - Mod_{𝒦} ) \ar[r]^{φ_{1}} \ar[r]_{φ_{2}} & Spc(A - Mod_{𝒦} ) \ar[r]^{φ_{A}} & supp(A)}

where φ_{i} is the map induced by extension-of-scalars F_{i} : A - Mod_{𝒦} ⟶ A^{⊗2} - Mod_{𝒦} along the two obvious homomorphisms f_{i} : A → A⊗A, i.e. f_{1} = 1⊗η and f_{2} = η⊗1.

Proof. Let W be the coequalizer of φ_{1} and φ_{2}. Since φ_{A}φ_{1} = φ_{A}φ_{2}, the map φ_{A} induces a continuous map φ̄ : W → supp(A). By Theorem 3.4 (d), the map φ̄ is surjective. It is injective by Corollary 3.12 for F = F_{A}. (See the explanations in the second half of Remark 3.6 again.) Finally φ̄ is a closed map since φ_{A} : Spc(A - Mod_{𝒦} ) → Spc(𝒦) is a closed map by Theorem 3.4 (b). □

We now want to discuss conditions to replace the above supp(A) by Spc(𝒦).

3.15. Proposition. Let A be an object in 𝒦. The following are equivalent : (i) It has maximal support : supp(A) = Spc(𝒦). (ii) The thick ⊗-ideal generated by A is the whole 𝒦; in symbols : ⟨A⟩ = 𝒦. If A is a (separable) ring object then the above properties are also equivalent to :

(iii) In some (hence in every) distinguished triangle J →^{ξ} 𝟙 →^{η} A →^{ζ} ΣJ over the

unit map η, the morphism ξ is ⊗-nilpotent. (iv) Extension-of-scalars F_{A} : 𝒦 → A - Mod_{𝒦} is nil-faithful : If a morphism f in

𝒦 is such that F_{A}(f) = 0 (i.e. A ⊗ f = 0) then f is ⊗-nilpotent.

(v) The image of the forgetful functor U_{A} : A - Mod_{𝒦} → 𝒦 generates 𝒦 as a thick ⊗-ideal; in symbols : ⟨U_{A}(A - Mod_{𝒦} )⟩ = 𝒦.

Proof. (i) ⟺ (ii) is standard, see [Bal05, Cor. 2.5]. (ii)⇒(iii) follows from the observation that { x ∈ 𝒦 | ξ ⊗ x is ⊗-nilpotent } is a thick ⊗-ideal of 𝒦 which contains A since η ⊗ A is a split monomorphism. (iii)⇒(iv) : If f : x → y is such that A ⊗ f = 0 then (η ⊗ y)f = 0 and therefore f factors through ξ ⊗ y which is ⊗-nilpotent. (iv)⇒(iii) is clear. (iii)⇒(ii) is well-known : by the Octahedron axiom cone(ξ^{⊗n}) ∈ ⟨cone(ξ)⟩ = ⟨A⟩ and if ξ^{⊗n} = 0 : J^{⊗n} → 𝟙^{⊗n} then 𝟙 ∈ ⟨cone(ξ^{⊗n})⟩. To show (ii) ⟺ (v), it suffices to note that ⟨U_{A}(A - Mod_{𝒦} )⟩ = ⟨A⟩ in general. Since A = U_{A}(𝟙_{A}), only the inclusion ⊆ requires verification. If (x, ϱ) is an Amodule then ϱ : A ⊗ x → x is split by η ⊗ x, hence ⟨A⟩ ∋ x = U_{A}(x, ϱ). □

3.16. Definition. Let us say that an object A ∈ 𝒦 is nil-faithful if for every morphism f in 𝒦, we have that A ⊗ f = 0 implies f^{⊗n} = 0 for some n ≥ 0. For a tt-ring A this is equivalent to any of the properties of Proposition 3.15, see (iv).

3.17. Corollary. Let F : 𝒦 → ℒ be a tt-functor. If A is a nil-faithful tt-ring in 𝒦 then so is F (A) in ℒ.

Proof. Condition (iii) in Proposition 3.15 is preserved by tt-functors. □

3.18. Remark. This Corollary really uses the tensor. For instance, if A only generates 𝒦 as (thick) triangulated category, it is not necessarily true of F (A) in ℒ.

3.19. Theorem (Descent-up-to-nilpotence). Let A be a nil-faithful tt-ring of finite degree. With notation of Theorem 3.14, we have a coequalizer of topological spaces :

\xymatrix{Spc(A^{⊗2} - Mod_{𝒦} ) \ar[r]^{φ_{1}} \ar[r]_{φ_{2}} & Spc(A - Mod_{𝒦} ) \ar[r]^{φ_{A}} & Spc(𝒦) .}

Proof. In Theorem 3.14, use that supp(A) = Spc(𝒦) by Proposition 3.15. □

4. Applications

Let G be a finite group and G−sets the category of finite left G-sets. Let 𝕜 be a field of positive characteristic p dividing |G|, typically just the prime field 𝔽_{p} = ℤ/p.

4.1. Construction. Let 𝒞_{ℤ}(ℤG) be the category of those finitely generated left ℤGmodules which are free as ℤ-modules. It is a tensor category with ⊗_{ℤ} and diagonal G-action. Consider for every X ∈ G−sets the permutation ℤG-module ℤX and, following [Bal15], define on it a ℤ-bilinear multiplication μ_{X} by setting x · x = x for all x ∈ X and x·x′ = 0 when x ≠ x′. This gives a ring object \underline{A}(X) = (ℤX, μ_{X}, η_{X}) in the category 𝒞_{ℤ}(ℤG) with unit η_{X} : ℤ → ℤX given by 1 ↦ ∑_{x∈X} x. For every G-map f : X → Y, define \underline{A}(f) : \underline{A}(Y) → \underline{A}(X) by y ↦ ∑_{x∈f^{−1}(y)} x. For instance, \underline{A}(X → ∗) = η_{X}. The ring object \underline{A}(X) is separable via σ : \underline{A}(X) → \underline{A}(X) ⊗ \underline{A}(X) given by x ↦ x ⊗ x. (Note that this map does not come from G−sets.)

Consequently, for every tensor functor Φ : 𝒞_{ℤ}(ℤG) ⟶ 𝒦 to a tt-category 𝒦, for instance the standard 𝒞_{ℤ}(ℤG) ⟶ D^{b}(𝕜G) or 𝒞_{ℤ}(ℤG) ⟶ stab(𝕜G), the composite functor A := Φ ∘ \underline{A} : G−sets^{op} ⟶ 𝒦 provides tt-rings A(X) in 𝒦 and ring homomorphisms A(f) : A(Y) → A(X) for every f : X → Y in G−sets.

4.2. Example. For X = G/H an orbit, our tt-ring A_{H}^{G} = 𝕜(G/H) is such an A(X).

4.3. Theorem (Serre’s vanishing of Bocksteins, gone tt). Consider as above A : G−sets^{op} ⟶ D^{b}(𝕜G) given by X ↦ A(X) = 𝕜X. Let^{2} Abelem(G) be the collection of elementary abelian p-subgroups of G (or only the maximal ones, or representatives up to conjugacy). Then the following tt-ring in D^{b}(𝕜G)

A_{elem} := ∏_{H∈Abelem(G)} A(G/H)

is nil-faithful in the sense of Definition 3.16 (and Proposition 3.15).

Proof. Let 𝒦 = D^{b}(𝕜G). Since A(G/H) ⊗ − ≃ Res_{H}^{G} is faithful when [ G : H] is prime to p, we reduce to G a p-group. By induction on |G| it suffices to show that for every p-group G which is not elementary abelian, the tt-ring

A := ∏_{H≨G} A(G/H)

is nil-faithful in D^{b}(𝕜G). By Proposition 3.15 (v) again, it suffices to prove that ⟨A⟩ = D^{b}(𝕜G). Now, Serre’s Theorem [Ser65, Prop. 4] precisely says that there are proper subgroups (of index p) H_{1}, …, H_{m} of G such that the product β(z_{1}) ⋯ β(z_{m}) vanishes in H^{2m}(G, 𝕜), where β(z_{i}) ∈ H^{2}(G, 𝕜) is the Bockstein element associated to z_{i} : G ↠ G/H_{i} ≃ ℤ/p. As an element of Ext_{𝕜G}^{2}(𝕜, 𝕜) ≅ H^{2}(G, 𝕜), this Bockstein is given by an exact sequence of the form

2In French Picardie, “abelem” is a clever contraction for “abélien élémentaire”.

(4.4)  β(z_{i}) = ( 0 → 𝕜 → 𝕜(G/H_{i}) → 𝕜(G/H_{i}) → 𝕜 → 0 ) .

If we also denote by β(z_{i}) : 𝕜 → 𝕜[2] the corresponding element in Hom_{𝒦} (𝕜, 𝕜[2]) ≅ Ext_{𝕜G}^{2}(𝕜, 𝕜) then its cone is the complex ⋯ → 0 → 𝕜(G/H_{i}) → 𝕜(G/H_{i}) → 0 → ⋯ appearing above, “in the middle”. In particular, cone(β(z_{i})) ∈ ⟨𝕜(G/H_{i})⟩ = ⟨A(G/H_{i})⟩. Therefore, an easy application of the octahedron (or just Yoneda splice) shows that the cone of our product β(z_{1}) ⋯ β(z_{m}) = β(z_{1}) ⊗ ⋯ ⊗ β(z_{m}) belongs to ⟨A(G/H_{1}), …, A(G/H_{m})⟩ ⊂ ⟨A⟩. Since this product is zero and since cone(0 : 𝕜 → 𝕜[2m]) = 𝕜[1] ⊕ 𝕜[2m] we get 𝕜 ∈ ⟨A⟩ as wanted. □

4.5. Remark. I am thankful to Raphaël Rouquier for simplifying my earlier proof of Theorem 4.3, which involved the following, perhaps interesting, observation. The exact sequence (4.4) is the cone of an isomorphism ψ_{i} : J_{i} →^{∼} J_{i}^{∗}[−2] in D^{b}(𝕜G), where J_{i} = (⋯ 0 → 𝕜 → 𝕜(G/H_{i}) → 0 ⋯) fits in the distinguished triangle J_{i} →^{ξ_{i}} 𝕜 →^{η_{i}} A(G/H_{i}) → ΣJ_{i} as in Proposition 3.15. A direct computation shows that the Bockstein β(z_{i}) is equal to the composite ξ_{i}[2] ∘ ψ_{i}^{−1} ∘ ξ_{i}^{∗}. In other words, our morphism ξ_{i} appears “twice” in the corresponding Bockstein.

The analogue of Theorem 4.3 also holds for 𝒦 = stab(𝕜G) by Corollary 3.17.

4.6. Definition. Recall that the orbit category Or(G) of the group G has objects indexed by subgroups H ≤ G, thought of as the corresponding orbit, i.e. morphisms are given by Mor_{Or(G)}(H, K) = Mor_{G−sets}(G/H, G/K). We write Or(G, Abelem) for the full subcategory of Or(G) on those H which are elementary abelian p-groups.

4.7. Remark. In [Car00], Carlson already noted that Serre’s theorem implies that the support variety V_{G} = Proj(H^{•}(G, 𝕜)) is covered by the images of the V_{H}

(4.8)  V_{G} = ⋃_{H∈Abelem(G)} Res^{∗} V_{H} .

This equality is part of Quillen’s Stratification Theorem (0.1), which says that :

(4.9)  V_{G} ≅ colim_{H∈Or(G,Abelem)} V_{H} .

In terms of cohomology rings, (4.8) says that H^{•}(G, 𝕜) ⟶ ∏_{H} H^{•}(H, 𝕜) has nilpotent kernel whereas the colimit-version (4.9) is saying moreover that elements in lim_{H} H^{•}(H, 𝕜) have some p^{r}-power coming from H^{•}(G, 𝕜). See for instance [Ben98, Cor. II.5.6.4]. Another way to see how the colimit-version (4.9) is more informative than (4.8) is to note that (4.8) does not prevent V_{G} from being a point, for instance, whereas (4.9) allowed Quillen to compute the Krull dimension of V_{G}.

We now want to show that descent-up-to-nilpotence (Theorem 3.19) yields a generalization of Quillen’s Stratification Theorem, in its strong form (4.9).

Quillen’s Stratification Theorem (4.9) is the case 𝒦 = stab(𝕜G) of the following :

4.10. Theorem. Let 𝒦 be a tt-category and let Φ : D^{b}(𝔽_{p}G) ⟶ 𝒦 be a tt-functor. For every subgroup H ≤ G consider the tt-ring A_{H} = Φ(𝔽_{p}(G/H)) in 𝒦 and define 𝒦(H) = A_{H} - Mod_{𝒦} the corresponding category of modules. Then the collection of tt-functors F_{A_{H}} : 𝒦 = 𝒦(G) ⟶ 𝒦(H) induces a homeomorphism

(4.11)  φ̄ : colim_{H∈Or(G,Abelem)} Spc(𝒦(H)) ⟶^{∼} Spc(𝒦) .

Proof. Following Construction 4.1, we have a functor A(−) : G−sets^{op} → 𝒦, taking values in tt-rings and homomorphisms and whose restriction to Or(G) recovers the ring A(G/H) = Φ(𝔽_{p}(G/H)) = A_{H} of the statement. Since A_{G} = Φ(𝕜) = 𝟙_{𝒦} , since G/G is the final G-set and since Spc(−) is contravariant, we indeed have a canonical continuous map φ̄ as in (4.11) such that precomposed with the canonical map Spc(𝒦(H)) → colim_{H} Spc(𝒦(H)) gives Spc(F_{A_{H}}). Consider the ttring A := ∏_{H∈Abelem(G)} A_{H}, which is the image in 𝒦 of the tt-ring A_{elem} of Serre’s Theorem 4.3. Since A_{elem} was nil-faithful in D^{perf}(𝔽_{p}G), Corollary 3.17 implies that A is also nil-faithful in 𝒦. By descent-up-to-nilpotence (Theorem 3.19), we have a coequalizer of spaces

(4.12)  \xymatrix{Spc(A^{⊗2} - Mod_{𝒦} ) \ar[r]^{φ_{1}} \ar[r]_{φ_{2}} & Spc(A - Mod_{𝒦} ) \ar[r]^{φ_{A}} & Spc(𝒦)}

where the maps φ_{i} are induced by f_{i} : A → A⊗A given by f_{1} = 1⊗η and f_{2} = η⊗1. Let us identify A_{H} ⊗ A_{K} = Φ(𝕜(G/H)) ⊗ Φ(𝕜(G/K)) ≅ Φ(𝕜(G/H × G/K)) = A(G/H × G/K), so that the components A_{H} → A_{H} ⊗ A_{K} and A_{K} → A_{H} ⊗ A_{K} of f_{1} and f_{2} : A → A ⊗ A are simply the images under A : G−sets^{op} → 𝒦 of the two projections pr_{1} : G/H ×G/K → G/H and pr_{2} : G/H ×G/K → G/K, respectively.

In G−sets, for any two subgroups H, K ≤ G, we have a Mackey isomorphism

∐_{[g]∈H\G/K} β_{g} : ∐_{[g]∈H\G/K} G/(H^{g} ∩ K) →^{∼} G/H × G/K

where the map β_{g} : G/(H^{g} ∩ K) ⟶ G/H × G/K is [x] ↦ ([xg^{−1}], [x]). As usual, this map is non-canonical since it involves the choice of g ∈ G for each double class [g] ∈ H\G/K. Still, we get an isomorphism (first in D^{b}(𝔽_{p}G) and then in 𝒦) :

∏_{{H,K∈Abelem(G) \atop [g]∈H\G/K}} A(β_{g}) : A ⊗ A = ∏_{H,K∈Abelem(G)} A_{H} ⊗ A_{K} ⟶^{∼} ∏_{{H,K∈Abelem(G) \atop [g]∈H\G/K}} A_{H^{g}∩K} .

Replacing the decompositions for A and A ⊗ A in (4.12), we obtain a coequalizer

[figure: ∐_{{H,K∈Abelem(G) \atop [g]∈H\G/K}} Spc(A_{H^{g}∩K}); ψ_{1}; ψ_{2}; ∐_{H∈Abelem(G)} Spc(A_{H}); ∐ φ_{A_{H}}; Spc(𝒦)]

where the maps ψ_{1} : Spc(A_{H^{g}∩K}) → Spc(A_{H}) and ψ_{2} : Spc(A_{H^{g}∩K}) → Spc(A_{K}) are induced by the composition of the above maps f_{i} : A [figure]A ⊗ A with the maps A(β_{g}) : A_{H} ⊗ A_{K} → A_{H^{g}∩K} on components. Since all maps come through A : G−sets^{op} → 𝒦, we can compute these compositions in G−sets already and get

[figure: G/H; G/(H^{g} ∩ K); β_{g}; pr_{1} β_{g} = α_{g}; pr_{2} β_{g} = α_{1}; G/H × G/K; _{pr_{1}}; _{pr_{2}}; G/K]

where α_{g} : G/L → G/L′ denotes [x] ↦ [xg^{−1}] whenever ^{g}L ≤ L′. So, for every H, K ≤ G and g ∈ G, we have

(4.13) ψ_{1}_{| Spc(A_{H^{g} ∩K})} = Spc(A(α_{g}))  and  ψ_{2}_{| Spc(A_{H^{g} ∩K})} = Spc(A(α_{1})) .

We can now compare our coequalizer with the colimit as follows :

[figure: ∐_{{H,K∈Abelem(G) \atop [g]∈H\G/K}} Spc(A_{H^{g}∩K}); ψ_{1}; ψ_{2}; ∐_{H∈Abelem(G)} Spc(A_{H}); ⊔ φ_{A_{H}}; π; Spc(𝒦); π̄; colim_{H∈Or(G,Abelem)} Spc(A_{H}); φ̄]

where π is the quotient map and φ̄ is the above continuous map such that φ̄π = ⊔_{H}φ_{A_{H}}. By (4.13), the two maps ψ_{1} and ψ_{2} are the images of maps α_{g} and α_{1} in Or(G, Abelem) via the functor H ↦ Spc(A_{H}). Therefore by the colimit property π ∘ ψ_{1} = π ∘ ψ_{2}. Hence, by the coequalizer property, there is a continuous map π̄ : Spc(𝒦) → colim_{H} Spc(A_{H}) such that π̄ ∘ (⊔φ_{A_{H}}) = π, as in the above diagram. This π̄ is therefore the inverse of φ̄ since π and ⊔φ_{A_{H}} are epimorphisms. □

4.14. Example. Let X be a scheme over 𝕜 on which our finite group G acts. Let 𝒦(G) = D_{G}^{b}(VB_{X}) be the bounded derived category of G-equivariant vector bundles on X. Then it receives D^{b}(𝕜G) = D_{G}^{b}(Spec(𝕜)) and A_{H} - Mod_{K(G)} ≅ D_{H}^{b}(VB_{X}) for every subgroup H ≤ G as can be readily verified from the restriction-coinduction adjunction. Alternatively, see [BDS14, § 5]. Theorem 4.10 tells us that the spectrum of D_{G}^{b}(VB_{X}) is the colimit of the spectra of D_{H}^{b}(VB_{X}) over H in Or(G, Abelem). Furthermore, the Krull dimension of Spc(D_{G}^{b}(VB_{X})) is the maximum of the Krull dimensions of Spc(D_{H}^{b}(VB_{X})) among the elementary abelian p-subgroups H ≤ G.

Acknowledgements : I would like to thank Serge Bouc, Ivo Dell’Ambrogio, John Greenlees, Alexander Merkurjev, Mike Prest, Raphaël Rouquier and Jacques Thévenaz for helpful discussions. I am also thankful to two anonymous referees for their valuable comments.

References

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

[Bal10] Paul Balmer. Tensor triangular geometry. In International Congress of Mathematicians, Hyderabad (2010), Vol. II, pages 85–112. Hindustan Book Agency, 2010.

[Bal11] Paul Balmer. Separability and triangulated categories. Adv. Math., 226(5):4352–4372, 2011.

[Bal12] Paul Balmer. Descent in triangulated categories. Math. Ann., 353(1):109–125, 2012.

[Bal14] Paul Balmer. Splitting tower and degree of tt-rings. Algebra Number Theory, 8(3):767– 779, 2014.

[Bal15] Paul Balmer. Stacks of group representations. J. Eur. Math. Soc. (JEMS), 17(1):189– 228, 2015.

[BDS14] Paul Balmer, Ivo Dell’Ambrogio, and Beren Sanders. Restriction to finite index subgroups as étale extensions in topology, KK-theory and geometry. Algebr. Geom. Topol., 2014. Preprint, 17 pages, to appear.

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

[Ben98] David J. Benson. Representations and cohomology I & II, volume 30 & 31 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1998.

[BGI71] Pierre Berthelot, Alexander Grothendieck, and Luc Illusie, editors. SGA 6 : Théorie des intersections et théorème de Riemann-Roch. Springer LNM 225. 1971.

[BKN14] Brian Boe, Jonathan Kujawa, and Daniel Nakano. Tensor triangular geometry for classical Lie superalgebras. Preprint, 2014.

[Car00] Jon F. Carlson. Cohomology and induction from elementary abelian subgroups. Q. J. Math., 51(2):169–181, 2000.

[Del10] Ivo Dell’Ambrogio. Tensor triangular geometry and KK-theory. J. Homotopy Relat. Struct., 5(1):319–358, 2010.

[DT12] Ivo Dell’Ambrogio and Gonçalo Tabuada. Tensor triangular geometry of noncommutative motives. Adv. Math., 229(2):1329–1357, 2012.

[EM65] Samuel Eilenberg and John C. Moore. Adjoint functors and triples. Illinois J. Math., 9:381–398, 1965.

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

[GM95] J. P. C. Greenlees and J. P. May. Equivariant stable homotopy theory. In Handbook of algebraic topology, pages 277–323. North-Holland, Amsterdam, 1995.

[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.

[Kün07] Matthias Künzer. Heller triangulated categories. Homology, Homotopy Appl., 9(2):233–

320, 2007.

[Mal06] Georges Maltsiniotis. Catégories triangulées supérieures. Preprint available online at

         http://people.math.jussieu.fr/$\sim$maltsin/ps/triansup.ps, 2006.

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

[Pet13] Tobias J. Peter. Prime ideals of mixed Artin-Tate motives. Journal of K-theory, 11(2):331–349, 2013.

[Qui71] Daniel Quillen. The spectrum of an equivariant cohomology ring. I, II. Ann. of Math. (2), 94:549–572; ibid. (2) 94 (1971), 573–602, 1971.

[Ric89] Jeremy Rickard. Derived categories and stable equivalence. J. Pure Appl. Algebra, 61(3):303–317, 1989.

[Ser65] Jean-Pierre Serre. Sur la dimension cohomologique des groupes profinis. Topology, 3:413– 420, 1965.

[Ste13] Greg Stevenson. Disconnecting spectra via localization sequences. J. K-Theory, 11(2):324–329, 2013. Appendix to: Module categories for group algebras over commutative rings, by D. Benson, H. Krause and S. Iyengar.

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

[Xu14] Fei Xu. Spectra of tensor triangulated categories over category algebras. Arch. Math. (Basel), 103(3):235–253, 2014.

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