SPLITTING TOWER AND DEGREE OF TT-RINGS

PAUL BALMER

Abstract. After constructing a splitting tower for separable commutative ring objects in tensor-triangulated categories, we define and study their degree.

Introduction

Let 𝒦 be a tensor-triangulated category (tt-category, for short). Denote its tensor by ⊗ : 𝒦 × 𝒦 ⟶ 𝒦 and its ⊗-unit by 𝟙. Let A be a ring object in 𝒦, that is, an associative monoid μ : A ⊗ A → A with unit η : 𝟙 → A. We want to study the degree of such a ring object under the assumption that A is what we call a tt-ring, i.e. is commutative and separable. We focuss on tt-rings because their Eilenberg-Moore category A - Mod_{𝒦} of A-modules in 𝒦 remains a tt-category and extension-of-scalars F_{A} : 𝒦 ⟶ A - Mod_{𝒦} is a tt-functor (a fact which also explains the terminology : tt-rings preserve tt-categories). See Section 1.

In practice, tt-rings appear in commutative algebra as finite étale algebras and in representation theory of finite groups as the amusing algebras A = 𝕜(G/H) associated to subgroups H < G; see [Bal12]. In the latter case, if 𝒦 = 𝒦(G) is the derived or the stable category of the group G over a field 𝕜, then A - Mod_{𝒦} is nothing but the corresponding category 𝒦(H) for the subgroup. These two sources already provide an abundance of examples. Furthermore, the topological reader will find tt-rings among ring spectra, equivariant or not.

Let us contemplate the problem of defining a reasonable notion of degree, i.e. an integer deg(A) measuring the size of the tt-ring A in a general tt-category 𝒦. When working over a field 𝕜, it is tempting to use dim_{𝕜}(A). When A is a projective separable R-algebra over a commutative ring R then its rank must be finite ([DI71]) and provides a fine notion of degree for A viewed in the tt-category of perfect complexes D^{perf}(R) = K^{b}(R – proj). However, tt-geometry covers more than commutative algebra. Unorthodox separable algebras already emerge in representation theory for instance, as the above A = 𝕜(G/H). In D^{b}(𝕜G – mod), one can still forget the G-action and take dimension over 𝕜 as a possible degree – which of course yields the index [G: H] in that example – but one step further, in the stable category 𝒦 = stab(𝕜G), dimension over 𝕜 becomes a slippery notion whereas the tt-ring A = 𝕜(G/H) remains equally important. Such questions become even harder in general stable homotopy categories (see [HPS97]) where there is simply no ground field 𝕜 to deal with in the first place. So a good concept of degree in the broad generality of tt-geometry requires a new idea.

Date: 2013 December 2.

1991 Mathematics Subject Classification. 18E30, 13B40, 20J05, 55U35.

Key words and phrases. Separable, tensor triangulated category, degree.

Research supported by NSF grant DMS-1303073.

Our solution relies on the following Splitting Theorem 2.1, which echoes a classical property of usual separable rings (see [CHR65]). Note that such a result is completely wrong for non-separable rings, already with B = A[X] for instance.

Theorem. Let f : A → B and g : B → A be homomorphisms of tt-rings such that g∘f = id_{A}. Then there exists a tt-ring C and a ring isomorphism ( \begin{matrix}g \\ ∗ \end{matrix}) : B →^{∼} A×C.

Using this theorem, we construct (Definition 3.1) a tower of tt-rings and homomorphisms A =: A^{[1]} → A^{[2]} → ⋯ → A^{[n]} → ⋯ such that after extending scalars to A^{[n]} our A splits as the product of 𝟙 × ⋯ × 𝟙 (n times) with A^{[n+1]}. The degree of A is defined to be the last n such that A^{[n]} ≠ 0 (Definition 3.4).

We prove a series of results which show that this concept of degree behaves according to intuition and provides a reasonable invariant. In basic examples, we recover expected values, like [G: H] in the case of 𝕜(G/H) in D^{b}(𝕜G – mod). In the stable category however, deg(𝕜(G/H)) can be smaller than [G : H]. In the extreme case of H < G strongly p-embedded, we even get deg(𝕜(G/H)) = 1 in stab(𝕜G); see Example 4.6. We prove in Section 4 that the degree is finite for every tt-ring in the derived category of perfect complexes over a scheme, in the bounded derived category of a finite dimensional cocommutative Hopf algebra and in the stable homotopy category of finite spectra.

It is an open question whether the degree must always be finite, at least locally. Several aspects of this work extend to non-triangulated additive tensor cate-

gories. This is discussed in Remark 3.13.

In [Bal13], our degree theory will be used to control the fibers of the map Spc(A - Mod_{𝒦} ) → Spc(𝒦). We shall notably reason by induction on the degree, thanks to the following Theorem 3.6 :

Theorem. Let A be a tt-ring of finite degree d in 𝒦. Then in the tt-category of A-modules, we have an isomorphism of tt-rings F_{A}(A) ≃ 𝟙 × A^{[2]} where the tt-ring A^{[2]} has degree d − 1 in A - Mod_{𝒦} .

Convention : All our tt-categories are essentially small and idempotent complete.

1. The tt-category of A-modules

We quickly list standard properties of the Eilenberg-Moore category A - Mod_{𝒦} of A-modules in 𝒦, see [EM65], [ML98, Chap. VI] or [Bal11].

As A is separable (i.e. μ : A ⊗ A → A has a section σ as A, A-bimodules) the category A - Mod_{𝒦} admits a unique triangulation such that both extensionof-scalars F_{A} : 𝒦 ⟶ A - Mod_{𝒦} , x ↦ A ⊗ x, and its forgetful right adjoint U_{A} : A - Mod_{𝒦} ⟶ 𝒦 are exact; see [Bal11]. Also, A - Mod_{𝒦} is equivalent to the idempotent completion of the Kleisli category A – Free_{𝒦} of free A-modules, see [Kle65]. Objects of A – Free_{𝒦} are the same as those of 𝒦, denoted F_{A}(x) for every x ∈ 𝒦, and morphisms Hom_{A}(F_{A}(x), F_{A}(y)) := Hom_{𝒦} (x, A⊗y). Denote by f̄: F_{A}(x) → F_{A}(y) the morphism in A – Free_{𝒦} corresponding to f : x → A ⊗ y in 𝒦.

As our tt-ring A is furthermore commutative, there is a tensor structure −⊗_{A} − : A - Mod_{𝒦} × A - Mod_{𝒦} ⟶ A - Mod_{𝒦} making F_{A} : 𝒦 ⟶ A - Mod_{𝒦} a tt-functor. Indeed, one can define ⊗_{A} on the Kleisli category by F_{A}(x) ⊗_{A} F_{A}(y) := F_{A}(x ⊗ y) and f̄⊗_{A} ḡ = \overline{(μ ⊗ 1 ⊗ 1)(23)(f ⊗ g)} if f : x → A⊗x′ and g : y → A⊗y′, as below :

\xymatrix{x ⊗ y \ar[r]^{f⊗g} & A ⊗ x′ ⊗ A ⊗ y′ \ar[r]^{(23)} & A ⊗ A ⊗ x′ ⊗ y′ \ar[r]^{μ⊗1⊗1} & A ⊗ x′ ⊗ y′.}

Idempotent completion then yields ⊗_{A} on A - Mod_{𝒦} . One can also describe ⊗_{A} on all modules directly. First only assume that A is separable with a chosen A, Abimodule section σ : A → A ⊗ A of μ. Let (x_{1}, ϱ_{1}) and (x_{2}, ϱ_{2}) be right and left A-modules in 𝒦 respectively. Then the following endomorphism v in 𝒦

(1.1)  \xymatrix{v : x_{1} ⊗ x_{2} \ar[r]^{1⊗η⊗1} & x_{1} ⊗ A ⊗ x_{2} \ar[r]^{1⊗σ⊗1} & x_{1} ⊗ A ⊗ A ⊗ x_{2} \ar[r]^{ϱ_{1}⊗ϱ_{2}} & x_{1} ⊗ x_{2}}

is an idempotent : v ∘ v = v. Hence one can define x_{1} ⊗_{A} x_{2} := Im(v) as the corresponding direct summand of x_{1} ⊗ x_{2}. We obtain a split coequalizer in 𝒦:

\xymatrix{x_{1} ⊗ A ⊗ x_{2} \ar[r]^{ϱ_{1}⊗1} \ar[r]_{1⊗ϱ_{2}} & x_{1} ⊗ x_{2} \ar@{->>}[r]^{v} & Im(v) = x_{1} ⊗_{A} x_{2}}

as in the traditional definition of ⊗_{A}. When A is commutative, left and right Amodules coincide and one induces an A-action on x_{1} ⊗_{A} x_{2} from the usual formula on x_{1} ⊗ x_{2}. One verifies that this coincides with the tensor constructed above.

1.2. Proposition (Projection formula). Let A be a tt-ring in 𝒦. For all y ∈ 𝒦 and x ∈ A - Mod_{𝒦} , we have a natural isomorphism U_{A}(x ⊗_{A} F_{A}(y)) ≅ U_{A}(x) ⊗ y in 𝒦.

Proof. By construction of ⊗_{A}, it suffices to prove the existence of such an isomorphism for x ∈ A – Free_{𝒦} , which is natural in x in that category (and in y too but that is easy). So, let x = F_{A}(z) for some z ∈ 𝒦. Then U_{A}(x ⊗ F_{A}(y)) = U_{A}(F_{A}(z) ⊗_{A} F_{A}(y)) = U_{A}(F_{A}(z ⊗ y)) = A ⊗ (z ⊗ y) ≅ (A ⊗ z) ⊗ y = U_{A}(x) ⊗ y. This looks trivial but the point is that this isomorphism is natural with respect to morphisms f̄: x = F_{A}(z) → F_{A}(z′) = x′ in A – Free_{𝒦} for f : z → A ⊗ z′ in 𝒦 (not just natural in z). This is now an easy verification. □

1.3. Remark. For two ring objects A and B, the ring object A × B is A ⊕ B with component-wise structure. The ring object A⊗B has multiplication (μ_{1} ⊗μ_{2}) (23) : (A ⊗ B)^{⊗2} ⟶ A ⊗ B and obvious unit. The opposite A^{op} is A with μ^{op} = μ (12). The enveloping ring A^{e} is A ⊗ A^{op}. Left A^{e}-modules are just A, A-bimodules.

If A and B are separable, then so are A × B, A ⊗ B and A^{op}. Conversely, if A × B is separable then so are A and B (restrict the section “σ” to each factor).

1.4. Remark. Let h : A → B be a homomorphism of tt-rings in 𝒦 (i.e. h is compatible with multiplications and units). We also say that B is an A-algebra or a tt-ring over A. Then idempotent-complete the functor F_{h} : A – Free_{𝒦} ⟶ B – Free_{𝒦} defined on objects by F_{h}(F_{A}(x)) = F_{B}(x) and on morphisms by F_{h}(f̄) = \overline{(h ⊗ 1) ∘ f}. Alternatively, equip B with a right A-module structure via h and define for every A-module x ∈ A - Mod_{𝒦} , its extension F_{h}(x) = B ⊗_{A} x equipped with the left B-module structure on the B factor. Both define the same tt-functor F_{h} : A - Mod_{𝒦} ⟶ B - Mod_{𝒦} and the following diagram commutes up to isomorphism :

\xymatrix{ & 𝒦 \ar[dl]_{F_{A}} \ar[dr]^{F_{B}} \\ A - Mod_{𝒦} \ar[rr]^{F_{h} ≅ B⊗_{A}−} &  & B - Mod_{𝒦} .}

Furthermore, if k : B → C is another homomorphism then F_{kh} ≅ F_{k} ∘ F_{h}.

1.5. Remark. For A a tt-ring in 𝒦, there is a one-to-one correspondence between (i) A-algebras in 𝒦, i.e. homomorphism h : A → B of tt-rings in 𝒦, and (ii) tt-rings B̄ in A - Mod_{𝒦} .

The correspondence is the obvious one : To every tt-ring B̄ = (B̄, μ̄, η̄) in A - Mod_{𝒦} , associate B := U_{A}(B̄) and h := U_{A}(η̄). The ring structure on B is given by

B ⊗ B = U_{A}(B̄) ⊗ U_{A}(B̄) ↠^{v} U_{A}(B̄ ⊗_{A} B̄) →^{μ̄} U_{A}(B̄) = B and η_{B} : 𝟙 →^{η_{A}} A →^{h} B.

Conversely, if h : A → B is a homomorphism, then one can use h to equip B̄ := B with an A-module structure and verify that μ : B ⊗B → B respects the idempotent v of (1.1), hence defines μ̄ : B ⊗_{A} B → B. Then B is separable in 𝒦 (with section σ of μ) if and only if B̄ is separable in A - Mod_{𝒦} (with section v σ of μ̄).

We tacitly use this dictionary below. If we need to distinguish the A-algebra B in 𝒦 from the tt-ring B̄ in A - Mod_{𝒦} , we shall write U_{A}(B̄) for the former.

Under this correspondence, if ℒ := A - Mod_{𝒦} , there is an equivalence B - Mod_{𝒦} ≅ B̄ - Mod_{ℒ} such that the following diagram commutes up to isomorphism :

[figure: 𝒦; F_{A}; F_{B}; ℒ = A - Mod_{𝒦}; FB̄; B - Mod_{𝒦}; ≃; B̄ - Mod_{ℒ} .]

On Kleisli categories, it maps F_{B}(x) to FB̄(F_{A}(x)) for every x ∈ 𝒦 and follows the sequence of isomorphisms Hom_{B}(F_{B}(x), F_{B}(y)) ≅ Hom_{𝒦} (x, B ⊗ y) ≅ Hom_{A}(F_{A}(x), B ⊗ y) ≅ Hom_{A}(F_{A}(x), B̄ ⊗_{A} F_{A}(y)) ≅ HomB̄(FB̄F_{A}(x), FB̄F_{A}(y)) on morphisms. Idempotent completion does the rest.

1.6. Remark. Let F : 𝒦 → ℒ be a tt-functor. Let A be a tt-ring in 𝒦 and let B := F (A) its image in ℒ. Then B is also a tt-ring and there exists a tt-functor F̄ : A - Mod_{𝒦} ⟶ B - Mod_{ℒ}

(1.7)  \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}}}

such F̄F_{A} = F_{B}F and U_{B}F̄ = F U_{A}. Explicitly, for every A-module (x, ϱ), we have

F̄(x, ϱ) = (F (x), F (ϱ)) where B ⊗ F (x) = F (A) ⊗ F (x) ≅ F (A ⊗ x) ⟶^{F (ϱ)} F (x). On

morphisms, F̄(f) = F (f). The “Kleislian” description of F̄ is equally easy.

2. Splitting theorems

We will iteratively use the following splitting result :

2.1. Theorem. Let f : A → B and g : B → A be homomorphisms of tt-rings in 𝒦 such that g ∘ f = id_{A}. Then there exists a tt-ring C and a ring isomorphism h : B →^{∼} A × C such that pr_{1} h = g. Consequently, C becomes an A-algebra, via pr_{2} h f. Moreover, if C′ is another A-algebra and h′ : B →^{∼} A × C′ is another A-algebra isomorphism such that pr_{1} h = g, then there exists an isomorphism of A-algebras ℓ : C →^{∼} C′ such that h′ = (1 × ℓ) h.

We start with a couple of additive lemmas.

2.2. Lemma. Let B be a ring object, B_{1} and B_{2} two B, B-bimodules and h : B →^{∼} B_{1} ⊕ B_{2} an isomorphism of B, B-bimodules. Then B_{1} and B_{2} admit unique structures of ring objects such that h becomes a ring isomorphism B →^{∼} B_{1} × B_{2}.

Proof. Write the given isomorphisms h = ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) : B ⟶^{∼} B_{1} ⊕B_{2} and h^{−1} = (^{k_{1}} ^{k_{2}} ) : B_{1} ⊕ B_{2} ⟶^{∼} B. If h is to be a ring isomorphism, we must have for i = 1, 2 that the multiplication μ_{i} : B_{i} ⊗ B_{i} → B_{i} is given by μ_{i} = h_{i} μ (k_{i} ⊗ k_{i}) and the unit η_{i} : 𝟙 → B_{i} by η_{i} = h_{i} η. Hence uniqueness. Conversely, let us see that these formulas provide the wanted ring structures. Let ρ : B ⊗ B_{2} ⟶ B_{2} be the left B-action on B_{2}. By left B-linearity of k_{2} : B_{2} → B, we have μ (1 ⊗ k_{2}) = k_{2} ρ : B ⊗ B_{2} → B. Note that h h^{−1} = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) implies h_{i}k_{j} = 0 when i ≠ j. Therefore h_{1} μ (1 ⊗ k_{2}) = h_{1} k_{2} ρ = 0. Similarly, B, B-linearity of k_{1} and k_{2} gives h_{i} μ (1 ⊗ k_{j}) = 0 and h_{i} μ (k_{j} ⊗ 1) = 0 when i ≠ j. So the bottom square of the following diagram commutes, in which the top one commutes by definition :

[figure: (B_{1}⊕B_{2}) ⊗ (B_{1}⊕B_{2}); ≅; μ_{B_{1}×B_{2}}; h^{−1}⊗h^{−1}; ≃; B_{1} ⊕ B_{2}; (B_{1}⊗B_{1}) ⊕ (B_{1}⊗B_{2}) ⊕ (B_{2}⊗B_{1}) ⊕ (B_{2}⊗B_{2}); _{( \begin{matrix}μ_{1} & 0 & 0 & 0 \\ 0 & 0 & 0 & μ_{2} \end{matrix} )}; _{(_{k_{1}⊗k_{1} k_{1}⊗k_{2} k_{2}⊗k_{1} k_{2}⊗k_{2}} )}; B_{1} ⊕ B_{2}; B ⊗ B; μ; B; h=( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ); ≃]

Hence h : B →^{∼} B_{1}⊕B_{2} is an isomorphism of objects-equipped-with-multiplications. Since B is associative and unital, B_{1} and B_{2} must have the same properties. □

2.3. Lemma. Let C and C′ be ring objects and ( \begin{matrix}1 & 0 \\ s & ℓ \end{matrix}) : 𝟙 × C →^{∼} 𝟙 × C′ a ring isomorphism. Then s = 0 and ℓ is a ring isomorphism.

Proof. Let us denote by (C, μ, η) and (C′, μ′, η′) the structures. Clearly ℓ is an isomorphism of objects. From the fact that ( \begin{matrix}1 & 0 \\ s & ℓ \end{matrix}) preserves the structures it follows

that η′ = s + ℓη and that ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & μ′ \end{matrix} ) ( ( \begin{matrix}1 & 0 \\ s & ℓ \end{matrix}) ⊗ ( \begin{matrix}1 & 0 \\ s & ℓ \end{matrix}) ) = ( \begin{matrix}1 & 0 \\ s & ℓ \end{matrix}) ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & μ \end{matrix} ), giving in

particular μ′(s⊗ℓ) = 0 and μ′ (ℓ⊗ℓ) = ℓμ. Composing the former with (1⊗ℓ^{−1}η′) : 𝟙 ⊗ 𝟙 → 𝟙 ⊗ C, we get 0 = μ′(s ⊗ ℓ)(1 ⊗ ℓ^{−1}η′) = μ′(s ⊗ η′) = μ′(1_{C′} ⊗ η′)s = s and therefore ℓη = η′. Hence ℓ preserves multiplication and unit. □

Proof of Theorem 2.1. Via the morphism g : B → A, we can equip A with a structure of B, B-bimodule, so that g becomes B^{e}-linear. Since B^{e} = B ⊗ B^{op} is separable, the category B^{e} - Mod_{𝒦} of B, B-bimodules is triangulated in such a way that U_{B^{e}} : B^{e} - Mod_{𝒦} ⟶ 𝒦 is exact. Choose a distinguished triangle over g in B^{e} - Mod_{𝒦} say C → B →^{g} A →^{z} ΣC. Forgetting the B^{e}-action, since g is split by f in 𝒦, we see that U_{B^{e}}(z) = 0. Since U_{B^{e}} is faithful, z is also zero in the triangulated category B^{e} - Mod_{𝒦} , which in turn yields an isomorphism h : B →^{∼} A ⊕ C of B, Bbimodules such that pr_{1} h = g. By Lemma 2.2, A and C can be equipped with ring structures so that h is a ring isomorphism. We are left to verify that this new ring structure on A is indeed the original one. This follows from the fact that g : B → A is a split epimorphism which is a homomorphism from B to A with both structures (the original one by hypothesis and the new one because h = ( \begin{matrix}g \\ ∗ \end{matrix}) is a homomorphism). Note that C is separable by Remark 1.3. Finally, for uniqueness of C as A-algebra, with the notation of the statement, we obtain an isomorphism k := h′ ∘ h^{−1} : 𝟙 × C →^{∼} 𝟙 × C′ in A - Mod_{𝒦} such that pr_{1} k = pr_{1} which means that k has the form ( \begin{matrix}1 & 0 \\ s & ℓ \end{matrix}) and we conclude by Lemma 2.3. □

2.4. Theorem. Let A be a tt-ring in 𝒦. Then there exists a ring isomorphism h : A ⊗ A →^{∼} A × A′ for some tt-ring A′ in such a way that pr_{1} h = μ. Moreover, the A-algebra A′ is unique up to isomorphism with this property, where A ⊗ A is considered as an A-algebra on the left (via the homomorphism 1 ⊗ η : A → A ⊗ A).

Proof. Apply Theorem 2.1 to the tt-ring B = A ⊗ A with g = μ : A ⊗ A → A and f = 1_{A} ⊗ η = F_{A}(η) : A → A ⊗ A. □

2.5. Remark. From the isomorphism ( \begin{matrix}μ \\ ∗ \end{matrix}) : A ⊗ A →^{∼} A ⊕ A′, we see that A′ ≃ Σ^{−1}cone(μ) ≃ cone(1_{A} ⊗ η) ≃ A ⊗ cone(η) in 𝒦. Also supp(A′) ⊆ supp(A).

3. Splitting tower and degree

3.1. Definition. We define the splitting tower of a tt-ring A

A^{[0]} → A^{[1]} → A^{[2]} → ⋯ → A^{[n]} → A^{[n+1]} → ⋯

as follows : We start with A^{[0]} = 𝟙, A^{[1]} = A and η : A^{[0]} → A^{[1]}. Then for n ≥ 1 we define A^{[n+1]} = (A^{[n]})′ in the notation of Theorem 2.4 applied to the tt-ring A^{[n]} in the tt-category A^{[n−1]} - Mod_{𝒦} (see Remark 1.5). Equivalently, A^{[n+1]} is characterized as A^{[n]}-algebra by the existence of an isomorphism of A^{[n]}-algebras

(3.2)  h : A^{[n]} ⊗_{A^{[n−1]}} A^{[n]} →^{∼} A^{[n]} × A^{[n+1]}

such that pr_{1} h = μ, where A^{[n]} ⊗_{A^{[n−1]}} A^{[n]} is an A^{[n]}-algebra via the left factor. This tower {A^{[n]}}_{n≥0} is well-defined up to isomorphism.

3.3. Remark. By Remark 2.5, supp(A^{[n+1]}) ⊆ supp(A^{[n]}), and if A^{[n]} = 0 for some n then A^{[m]} = 0 for all m ≥ n. Also, by construction, if we consider A^{[n]} as a tt-ring in A^{[n−1]} - Mod_{𝒦} , we have (A^{[n]})^{[m]} ≅ A^{[n+m−1]} for all m ≥ 1.

3.4. Definition. We say that A has finite degree d if A^{[d]} ≠ 0 and A^{[d+1]} = 0. In that case, we write deg(A) = d or deg_{𝒦} (A) = d if we need to stress the category. If A^{[n]} ≠ 0 for all n ≥ 0, we say that A has infinite degree.

3.5. Example. For A = 𝟙 × 𝟙, we have A ⊗ A ≃ A × A. Hence A^{[2]} = A = A^{[1]}. If one was to compute A^{[2]} ⊗ A^{[2]} one would get A^{[2]} × A again and misreading Definition 3.4 could lead to the false impression that A^{[3]} is A again and that all A^{[n]} are equal. This is not the way to compute A^{[3]} ! One needs to compute A^{[2]} ⊗_{A^{[1]}} A^{[2]} = A ⊗_{A} A = A = A^{[2]} × 0 and therefore A^{[3]} = 0. So, the tt-ring 𝟙 × 𝟙 has degree 2. In (3.2), it is important to perform the tensor over A^{[n−1]}.

An immediate gain of having a numerical invariant like the degree is the possibility of making proofs by induction. This will be applied in [Bal13] using the Splitting Theorem 2.4, under the following form :

3.6. Theorem. Let A be a tt-ring of finite degree d in a tt-category 𝒦. Then we have a ring isomorphism F_{A}(A) ≃ 𝟙_{A} × A^{[2]} and deg(A^{[2]}) = d − 1 in A - Mod_{𝒦} .

Proof. Since A^{[2]} = A′, this is simply Theorem 2.4 with A-algebras replaced by ttrings in A - Mod_{𝒦} (see Remark 1.5 if necessary), together with the observation that (A^{[2]})^{[n]} = A^{[n+1]} for all n ≥ 1, which gives deg_{A - Mod_{𝒦}} (A^{[2]}) = deg_{𝒦} (A) − 1. □

Before showing in Section 4 that many tt-rings have finite degree, let us build our understanding of this deg(A) ∈ ℕ ∪ {∞}, starting with functorial properties.

3.7. Theorem. Let A be a tt-ring in 𝒦.

(a) Let F : 𝒦 → ℒ be a tt-functor. Then for every n ≥ 0, we have F (A)^{[n]} ≃ F (A^{[n]}) as tt-rings. In particular, deg(F (A)) ≤ deg(A).

(b) Let F : 𝒦 → ℒ be a tt-functor. Suppose that F is “ weakly conservative on supp(A)”, i.e. for x ∈ 𝒦_{supp(A)} if F (x) = 0 then x^{⊗m} = 0 for m ≥ 0; for instance, if F is just conservative. Then deg(F (A)) = deg(A).

(c) Suppose that B ∈ 𝒦 is a tt-ring such that supp(B) ⊇ supp(A); for instance, if supp(B) = Spc(𝒦). Then deg(A) is equal to the degree of F_{B}(A) in B - Mod_{𝒦} .

(d) Suppose 𝒦 is local and that B ∈ 𝒦 is a non-zero tt-ring. Then the degree of A in 𝒦 is equal to the degree of F_{B}(A) in B - Mod_{𝒦} .

Proof. To prove (a) by induction on n, simply apply F to (3.2), which characterizes the splitting tower A^{[∗]}. So, if deg(A) < ∞ then F (A)^{[deg(A)+1]} ≃ F (A^{[deg(A)+1]}) = 0 and deg(F (A)) ≤ deg(A). For (b), recall that A^{[n]} ∈ 𝒦_{supp(A)} for every n ≥ 1 (Rem. 3.3). As deg(F (A)) =: d < ∞ implies F (A^{[d+1]}) ≃ F (A)^{[d+1]} = 0 we get by weak-conservativity of F that A^{[d+1]} is ⊗-nilpotent, hence zero (every ring object is a direct summand of its ⊗-powers via the unit). This A^{[d+1]} = 0 means deg(A) ≤ d = deg(F (A)), which finishes (b). Then (c) follows since supp(B) ⊇ supp(A) implies that F_{B} : 𝒦 → B - Mod_{𝒦} is weakly conservative on 𝒦_{supp(A)}. Indeed, if supp(x) ⊆ supp(A) and F_{B}(x) = 0 then B ⊗ x = U_{B}F_{B}(x) = 0 and ∅ = supp(B ⊗ x) = supp(B) ∩ supp(x) = supp(x), which implies that x is ⊗- nilpotent. For (d), recall that a tt-category is local if x ⊗ y = 0 implies that x or y is ⊗-nilpotent. Hence for the non-zero tt-ring B, the functor F_{B} : 𝒦 → B - Mod_{𝒦} is weakly conservative on the whole of 𝒦 and we can apply (b). □

Let us now describe the local nature of the degree. Recall that for every prime 𝒫 ∈ Spc(𝒦), the local category 𝒦_{𝒫} = (𝒦/𝒫)^{♮} at 𝒫 is the idempotent completion of the Verdier quotient 𝒦/𝒫, hence comes with a tt-functor q_{𝒫} : 𝒦 ↠ 𝒦/𝒫 ↪ 𝒦_{𝒫} .

3.8. Theorem. Let A be a tt-ring in 𝒦. Suppose that q_{𝒫}(A) has finite degree in the local tt-category 𝒦_{𝒫} for every point 𝒫 ∈ Spc(𝒦). Then A has finite degree and

deg(A) = max_{𝒫∈Spc(𝒦)} deg(q_{𝒫}(A)) = max_{𝒫∈supp(A)} deg(q_{𝒫}(A)) .

Proof. There exists, for every 𝒫 ∈ Spc(𝒦), an integer n_{𝒫} ≥ 1 such that q_{𝒫}(A^{[n_{𝒫} ]}) = (q_{𝒫}(A))^{[n_{𝒫} ]} = 0. Hence 𝒫 belongs to the open 𝒰(A^{[n_{𝒫} ]}) := Spc(𝒦) − supp(A^{[n_{𝒫} ]}). Putting all those open subsets together, we cover Spc(𝒦). But the spectrum is always quasi-compact and 𝒰(A^{[n]}) ⊆ 𝒰(A^{[n+1]}), hence there exists n ≥ 0 such that 𝒰(A^{[n]}) = Spc(𝒦). This means A^{[n]} = 0, i.e. d := deg(A) < ∞. By Theorem 3.7 (a) we have d = deg(A) ≥ max_{𝒫∈Spc(𝒦)} deg(q_{𝒫}(A)) ≥ max_{𝒫∈supp(A)} deg(q_{𝒫}(A)). Since A^{[d]} ≠ 0 there exists 𝒫 ∈ supp(A^{[d]}) ⊆ supp(A) with 0 ≠ q_{𝒫}(A^{[d]}) ≃ (q_{𝒫}(A))^{[d]} hence deg(q_{𝒫}(A)) ≥ d = deg(A), wrapping up all above inequalities into equalities. □

We now discuss the link between the degree and the trivial tt-ring 𝟙.

3.9. Theorem. Let A be a tt-ring in 𝒦. Suppose 𝒦 ≠ 0.

(a) For every n ≥ 1, we have deg(𝟙^{×n}) = n.

(b) For every n ≥ 1 we have F_{A^{[n]}}(A) ≃ 𝟙^{×n} × A^{[n+1]} as tt-rings in A^{[n]} - Mod_{𝒦} .

(c) If deg(A) < ∞ then B := A^{[deg(A)]} is non-zero and we have in B - Mod_{𝒦}

(3.10)  F_{B}(A) ≃ 𝟙^{× deg(A)} .

(d) If a tt-functor F : 𝒦 → ℒ is weakly conservative on 𝒦_{supp(A)}, see Thm. 3.7 (b), e. g. if F is conservative, and if F (A) ≃ 𝟙^{×d} in ℒ then deg(A) = d.

(e) Let B be a tt-ring such that F_{B}(A) ≃ 𝟙^{×d} as tt-rings in B - Mod_{𝒦} . Suppose either that supp(B) ⊇ supp(A), or that 𝒦 is local and B ≠ 0. Then d = deg(A).

We need another additive lemma, whose naive proof (with a permutation) fails.

3.11. Lemma. Let A = 𝟙^{×n}. Then there exists an isomorphism h : F_{A}(A) = A ⊗ A →^{∼} A × A^{×(n−1)} of A-algebras such that pr_{1} h = μ.

Proof. To keep track of the various copies of 𝟙, write A = ⊕_{i=1}^{n}𝟙_{i} and A^{×(n−1)} = ⊕_{ℓ=1}^{n−1} ⊕_{i=1}^{n} 𝟙_{iℓ} where 𝟙_{i} = 𝟙_{iℓ} = 𝟙 for all i, ℓ. Then A⊗A = ⊕_{i,j}𝟙_{i} ⊗𝟙_{j}. Define h by mapping the summand 𝟙_{i} ⊗𝟙_{i} = 𝟙 identically to 𝟙_{i} ↪ A and 𝟙_{i} ⊗𝟙_{j} = 𝟙 identically to 𝟙_{ij} ↪ A^{×(n−1)} when i ≠ j and j ≤ n − 1 but mapping 𝟙_{i} ⊗ 𝟙_{n} = 𝟙 diagonally to ⊕_{ℓ=1, ℓ≠i}^{n−1}𝟙_{iℓ} ↪ A^{×(n−1)} for all i < n. Verifications are now an exercise. □

Proof of Theorem 3.9. We prove (a) by induction on n. The result is clear for n = 1. If A = 𝟙^{×n} for n ≥ 2 then Lemma 3.11 gives A^{[2]} ≃ A^{×(n−1)} ≃ 𝟙_{A}^{×(n−1)} in A - Mod_{𝒦} . By induction hypothesis applied to the tt-category A - Mod_{𝒦} we get deg(A^{[2]}) = n − 1 hence the result by definition of the degree. For (b), we need to prove that there are A^{[n]}-algebra isomorphisms A^{[n]} ⊗A ≃ A^{[n]} ×⋯×A^{[n]} ×A^{[n+1]} (with n factors A^{[n]}). This is an easy induction on n, applying A^{[n+1]} ⊗_{A^{[n]}} − and using (3.2) at each stage. Formula (3.10) follows since A^{[deg(A)+1]} = 0. Parts (d) and (e) follow from (a) and Theorem 3.7 (b)-(d). □

3.12. Corollary. Suppose that 𝒦 is local and that A, B ∈ 𝒦 are two tt-rings of finite degree. Then A×B and A⊗B have finite degree with deg(A×B) = deg(A)+deg(B) and deg(A ⊗ B) = deg(A) · deg(B).

Proof. By Thm. 3.9 (c), there exists two tt-rings Ā ≠ 0 and B̄ ≠ 0 such that FĀ(A) ≃ 𝟙^{× deg(A)} and FB̄(B) ≃ 𝟙^{× deg(B)}. Let then C̄ = Ā⊗B̄. Extending scalars from Ā and from B̄ to C̄ gives FC̄(A × B) ≃ 𝟙^{×(deg(A)+deg(B))} and FC̄(A ⊗ B) ≃ 𝟙^{×(deg(A)·deg(B))}. Finally C̄ ≠ 0 since 𝒦 is local and apply Theorem 3.9 (e). □

3.13. Remark. It will be clear to the interested reader that several arguments, mostly the early ones of Section 2, only depend on the property that split epimorphisms in 𝒦 admit a kernel (a property which holds when 𝒦 is triangulated, regardless of idempotent-completeness). The reader interested in using the degree in that generality will easily adapt our definition. However, all results which involve Spc(𝒦), or the support supp(A), or the local categories 𝒦/𝒫, as well as the geometric applications in [Bal13], only make sense when 𝒦 is triangulated. It is nonetheless interesting to be able to speak of the degree in the generality of, say, the category of abelian groups, for instance.

4. Examples

We start by quickly discussing tt-rings of minimal degree (beyond deg(0) = 0).

4.1. Proposition. Let A be a tt-ring with deg(A) = 1, i.e. such that μ : A⊗A → A is an isomorphism. Then A ⊗ − : 𝒦 → 𝒦 is a (very special) Bousfield localization with F_{A} : 𝒦 → A - Mod_{𝒦} as (Verdier) localization. Also, Spc(A - Mod_{𝒦} ) is homeomorphic to the open and closed subset supp(A) of Spc(𝒦). If 𝒦 is rigid, this further implies a decomposition 𝒦 = 𝒦_{1} × 𝒦_{2} under which A ≅ (𝟙, 0).

Proof. Since μ is an isomorphism, then so are its two right inverses η ⊗ 1 and 1 ⊗ η : A → A^{⊗2} and these inverses coincide. So, L := A ⊗ − is a Bousfield localization (ηL = Lη is an isomorphism). Let C → 𝟙 →^{η} A → Σ(C) be an exact triangle on η. Since, A ⊗ η is an isomorphism, we have A ⊗ C = 0. Therefore Spc(𝒦) = supp(A) ⊔ supp(C) hence supp(A) is open and closed. Since every object x ∈ 𝒦 fits in an exact triangle C ⊗ x → x → A ⊗ x → Σ(C ⊗ x), it is standard to show that the kernel of A ⊗ − is exactly the thick ⊗-ideal 𝒥 := ⟨C⟩ generated by C and that F_{A} induces an equivalence 𝒦/𝒥 ⟶^{∼} A - Mod_{𝒦} . Hence Spc(F_{A}) induces a homeomorphism Spc(A - Mod_{𝒦} ) ≅ { 𝒫 ∈ Spc(𝒦) | 𝒥 ⊆ 𝒫 } = { 𝒫 | C ∈ 𝒫 } = 𝒰(C) = supp(A). When 𝒦 is rigid, supp(A) ∩ supp(C) = ∅ forces furthermore Hom_{𝒦} (A, ΣC) = 0, in which case the above triangle splits : 𝟙 ≃ A ⊕ C. This gives the wanted decomposition, where 𝒦_{1} = A ⊗ 𝒦 and 𝒦_{2} = C ⊗ 𝒦. □

We want to show that the degree is finite in examples. Our main tool is :

4.2. Theorem. Suppose that 𝒦 admits a conservative tt-functor F : 𝒦 → ℒ into a tt-category ℒ such that every object of ℒ is isomorphic to a sum of suspensions of 𝟙_{ℒ} . (^{1}) Then every tt-ring in 𝒦 has finite degree. More precisely, if F (A) ≃ ⊕_{i=k}^{ℓ}Σ^{i}𝟙^{r_{i}} for r_{k}, …, r_{ℓ} ∈ ℕ then deg(A) = ∑_{i=k}^{ℓ} r_{i}.

Proof. By Theorem 3.7 (b), it suffices to prove that every tt-ring A in ℒ has finite degree d = ∑_{i=k}^{ℓ} r_{i}, where A ≃ ⊕_{i=k}^{ℓ}Σ^{i}𝟙^{r_{i}} as objects in ℒ. First, let B = A^{[d+1]}. Then, by Theorem 3.9 (b), we have F_{B}(A) ≃ 𝟙^{d+1} ⊕ x in B - Mod_{ℒ} . On the other hand F_{B}(A) ≃ ⊕_{i=k}^{ℓ}Σ^{i}𝟙^{r_{i}}. Therefore there is a split monomorphism 𝟙^{d+1} ↣ ⊕_{i=k}^{ℓ}Σ^{i}𝟙^{r_{i}} in B - Mod_{ℒ} which can be described by a split injective d×(d+1)- matrix with coefficients in the graded-commutative ring S^{•} = EndB^{•}(𝟙_{B}). This is impossible (by mapping to a graded residue field of S^{•}) unless S^{•} = 0, that is B = 0 meaning A^{[d+1]} = 0. Hence deg(A) ≤ d. Now, replace B by A^{[deg(A)]} and reason as above. We now have isomorphisms F_{B}(A) ≃ 𝟙^{deg(A)} and F_{B}(A) ≃ ⊕_{i=k}^{ℓ}Σ^{i}𝟙^{r_{i}} in B - Mod_{ℒ} with B ≠ 0. The isomorphism 𝟙^{deg(A)} ≃ ⊕_{i=k}^{ℓ}Σ^{i}𝟙^{r_{i}} forces (periodicities Σ^{i}𝟙 ≃ 𝟙 in B - Mod_{𝒦} whenever r_{i} ≠ 0 and) deg(A) = ∑_{i=k}^{ℓ} r_{i}. □

4.3. Corollary. Let X be a quasi-compact and quasi-separated scheme (e. g. an affine or a noetherian scheme). Then every tt-ring in D^{perf}(X) has finite degree.

Proof. By Theorem 3.8, we can assume that X = Spec(R) with (R, 𝔪) local. Then, the functor D^{perf}(R) → D^{perf}(k) to the residue field k = R/𝔪 is conservative. □

1Such an ℒ is sometimes called a “field” but the author finds this definition too restrictive.

Also note that the existence of such a functor F forces 𝒦 to be local.

4.4. Example. Let A be a separable commutative R-algebra which is projective as R-module (and finitely generated by [DI71, Prop. II.2.1]). Since A is R-flat, we can view it as the “same” tt-ring in D^{perf}(R). Then its degree can be computed in every residue field, hence deg(A) coincides with the rank of A as R-module.

4.5. Corollary. Let ℍ be a finite dimensional cocommutative Hopf algebra over a field 𝕜. Then every tt-ring in the bounded derived category D^{b}(ℍ – mod) of finitely generated ℍ-modules (with ⊗ = ⊗_{𝕜}) has finite degree.

Proof. Apply Theorem 4.2 to the fiber functor D^{b}(ℍ – mod) → D^{b}(𝕜). □

4.6. Example. For any finite group G, all tt-rings in D^{b}(𝕜G – mod) have finite degree. For every subgroup H ≤ G, the tt-ring A = 𝕜(G/H) has finite degree deg(A) = dim_{𝕜}(A) = [G: H] in D^{b}(𝕜G – mod). Hence A has also finite degree in stab(𝕜G) ≅ \frac{D^{b}(𝕜G – mod)}{D^{perf} (𝕜G)} by Theorem 3.7 (a). However, if H < G is a strongly p-embedded

subgroup then F_{A} ≅ Res_{H}^{G} is an equivalence stab(𝕜G) ⟶^{∼} stab(𝕜H) and η_{A} : 𝟙 →^{∼} A is an isomorphism, hence deg(A) = 1 in stab(𝕜G). (Example : p = 2 and C_{2} < S_{3}.)

4.7. Example. Let H_{1}, H_{2} be two non-conjugate cyclic subgroups of order p in G (for instance two non-conjugate symmetries in D_{8} for p = 2) and consider A_{i} = 𝕜(G/H_{i}) in 𝒦 = stab(𝕜G) as above. Then, by the Mackey formula A_{1} ⊗ A_{2} = 0. Consequently they have disjoint support and therefore both formulas of Corollary 3.12 fail in this case, showing the importance of our assumption that the category be local. Yet, one can still deduce global formulas via Theorem 3.8.

4.8. Corollary. In the stable homotopy category 𝒦 = SH^{fin} of finite (topological) spectra, every tt-ring has finite degree.

Proof. First note that the result is true in the localizations SH_{ℚ}^{fin} ≅ D^{b}(ℚ – mod) and SH_{p}^{fin}, at zero and at each prime p. For the latter, it suffices to apply Theorem 4.2 to homology with coefficients in ℤ/p, which is conservative on SH_{p}^{fin} and takes values in D^{b}(ℤ/p – mod). Now, if A is a tt-ring in SH^{fin}, then there exists m ≥ 1 such that A^{[m]} goes to zero in SH_{ℚ}^{fin} (since its degree is finite there). Replacing A by A^{[m]}, we can assume that A itself maps to zero in SH_{ℚ}^{fin}, that is, A is torsion. But then A is non-zero in SH_{p}^{fin} for only finitely many primes p. Therefore we can find n big enough so that A^{[n]} = 0 everywhere. Hence A^{[n]} = 0. □

Acknowledgments : It is my pleasure to thank an anonymous referee for several useful suggestions, including the one discussed in Remark 3.13 and a simplification of the proof of Corollary 4.8.

References

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[Bal12] Paul Balmer. Stacks of group representations. J. Eur. Math. Soc. (JEMS), preprint 2012. 36 pages, to appear.

[Bal13] Paul Balmer. Separable extensions in tt-geometry and generalized Quillen stratification. Preprint, 14 pages, 2013.

[CHR65] S. U. Chase, D. K. Harrison, and Alex Rosenberg. Galois theory and Galois cohomology of commutative rings. Mem. Amer. Math. Soc. No., 52:15–33, 1965.

[DI71] F. DeMeyer and E. Ingraham. Separable algebras over commutative rings. Lecture Notes in Mathematics, Vol. 181. Springer-Verlag, Berlin, 1971.

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

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[Kle65] Heinrich Kleisli. Every standard construction is induced by a pair of adjoint functors. Proc. Amer. Math. Soc., 16:544–546, 1965.

[ML98] Saunders Mac Lane. Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1998.

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