SPECTRA, SPECTRA, SPECTRA – TENSOR TRIANGULAR SPECTRA VERSUS ZARISKI SPECTRA OF ENDOMORPHISM RINGS

PAUL BALMER

Abstract. We construct a natural continuous map from the triangular spectrum of a tensor triangulated category to the algebraic Zariski spectrum of the endomorphism ring of its unit object. We also consider graded and twisted versions of this construction. We prove that these maps are quite often surjective but far from injective in general. For instance, the stable homotopy category of finite spectra has a triangular spectrum much bigger than the Zariski spectrum of ℤ. We also give a first discussion of the spectrum in two new examples, namely equivariant KK-theory and stable 𝔸^{1}-homotopy theory.

Contents

Introduction 1 1. Basics 6 2. Morphisms and actions 7 3. Graded homomorphisms and central localization 10 4. Local tensor triangulated categories 14 5. From triangular spectrum to Zariski spectra 15 6. Locally ringed space structures 18 7. Two criteria for surjectivity 20 8. Some examples, old and new 24 9. Spectra of topological spectra 26 10. Spectra of motivic spectra 29 References 32

Introduction

Algebraic geometers, stable topologists, finite group representation theorists, motivic theorists, noncommutative geometers and many other mathematicians have triangulated categories in common : The derived category of sheaves of modules over a scheme, the stable homotopy category of topological spectra, the derived category or the stable category of representations of a finite group or finite group scheme, the various motivic derived categories, Morel and Voevodsky’s stable 𝔸^{1}-homotopy category, equivariant KK-theory or E-theory of C^{∗}-algebras are famous examples. In several cases, a tensor structure is also available and is especially well-behaved on the triangulated subcategory of compact objects. In the above examples, this leads us to focus on perfect complexes, finite topological spectra, finite dimensional representations, geometric motives, etc.

Date: May 29, 2010.

1991 Mathematics Subject Classification. 18E30, (14F05, 19K35, 20C20, 55P42, 55U35).

Key words and phrases. Tensor triangular geometry, spectra.

Research supported by NSF grant 0654397.

This profusion of examples motivates the study of tensor triangulated categories per se. Emphasizing the geometric aspects of this unified theory leads us to a subject called tensor triangular geometry, to which the present paper belongs.

To explain these ideas, let us denote by 𝒦 one of our tensor triangulated categories (say, of compact objects), by ⊗ : 𝒦 × 𝒦 ⟶ 𝒦 the symmetric tensor x ⊗ y = y ⊗ x and by 11 ∈ 𝒦 the ⊗-unit : 11 ⊗ x = x.

Our geometric study was started in [1] with the definition of a topological space

Spc(𝒦)

called the spectrum of 𝒦 (see also Def. 1.3 below). We call it the triangular spectrum here, to avoid confusion with other meanings of the word “spectrum”. This fundamental space Spc(𝒦) is the canvas on which to draw tensor triangular geometry. For instance, every object x ∈ 𝒦 has a support, supp(x) ⊂ Spc(𝒦), which is a closed subset behaving nicely with respect to exact triangles and tensor product. The pair (Spc(𝒦), supp) is indeed the best such pair, for it has a universal property : In the language of [1, Thm. 3.2], (Spc(𝒦), supp) is “the final support datum”.

Although a young subject, tensor triangular geometry has already produced some interesting results. For instance, in [2, Thm. 2.11], it is proved that if an object x ∈ 𝒦 has a disconnected support, i.e. supp(x) = Y_{1} ∪ Y_{2} with Y_{1} and Y_{2} closed and disjoint, then the object is decomposable accordingly : x ≃ x_{1} ⊕ x_{2} with supp(x_{i}) = Y_{i}. This neat result has applications, like the gluing technique of Balmer–Favi [6] and its representation theoretic incarnations in Balmer–Benson– Carlson [5] and in [4]. Another application, this time to algebraic K-theory of schemes, can be found in [3].

In all applications though, the crucial anchor point is the computation of the triangular spectrum Spc(𝒦) in the first place. Without this knowledge, abstract results of tensor triangular geometry are difficult to translate into concrete terms. It is therefore a major challenge to compute the spectrum Spc(𝒦) in as many examples as possible, or at least to provide some information about that space when the full determination of Spc(𝒦) lies beyond reach of the community’s current forces.

Actually, by [1, Thm. 4.10, Thm. 5.2] and [9, Cor. 5.2], we know that the information about 𝒦 contained in the space Spc(𝒦) is exactly equivalent to the so-called classification of thick triangulated ⊗-ideals of 𝒦. Existing such classifications allow us to compute Spc(𝒦) and, conversely, given the space Spc(𝒦), we obtain a classification of thick triangulated ⊗-ideals by assigning to suitable subsets Y ⊂ Spc(𝒦) the thick triangulated ⊗-ideal 𝒦_{Y} := { a ∈ 𝒦 | supp(a) ⊂ Y } of 𝒦.

Using pre-existing classifications of thick ⊗-ideals, some triangular spectra have been computed in [1] and [9] : In algebraic geometry, the spectrum of the derived category of perfect complexes 𝒦 = D^{perf}(X) over a quasi-compact and quasiseparated scheme X reconstructs the scheme X itself; this uses Thomason’s classification, initiated by Hopkins and Neeman in the affine case. In modular representation theory, the spectrum of the stable category 𝒦 = kG – stab is the projective support variety V_{G}(k) = Proj(H^{•}(G, k)), by Benson-Carlson-Rickard’s classification, generalized to finite group schemes by Friedlander-Pevtsova. See [1] for details and references about these statements. Recently, Krishna [17] identified spectra of perfect complexes over Deligne–Mumford stacks as coarse moduli spaces.

Yet, the precursor of all these classifications is to be found in topology, in the celebrated work of Hopkins and Smith [15]. Since the reformulation of this original result in terms of a triangular spectrum has not been written down so far, we provide it in Section 9, en passant; see (0.1) further down in this introduction.

Such classifications are usually rather non-trivial. In noncommutative topology little is known beyond the very first cases, recently treated by Dell’Ambrogio in [11] (see Ex. 8.7 below). In 𝔸^{1}-homotopy theory, the classification is not known and the question is essentially wide open. It would be very interesting to use the theory of triangular spectra in order to compute the spectrum directly, by independent techniques, and then deduce the classification by [1, Thm. 4.10]. This is actually sometimes possible. It works for instance in commutative algebra, as we shall see in Remark 8.4. But more generally, in the new areas were the classification might be too hard for now, any information about the relevant triangular spectra provides an indication of the complexity of the triangulated categories involved. Such results are among the objectives of the present paper, as we shall see below.

Let us stress the point that, to this day, all known homeomorphisms between Spc(𝒦) and a more explicit topological space X come from the universal property of the pair (Spc(𝒦), supp) as a final support datum. Such maps X → Spc(𝒦) are always going from a candidate-spectrum X towards Spc(𝒦).

The main purpose of the present article is to construct continuous maps out of Spc(𝒦), into topological spaces X of “spectral” nature : \xymatrix{Spc(𝒦) \ar@{-->}[r] & X} .

Indeed, by many aspects, the triangular spectrum Spc(𝒦) resembles the Zariski spectrum of a commutative ring; first naively, since points of Spc(𝒦) are sophisticated prime ideals, see Def. 1.3. But more rigourously, the space Spc(𝒦) is always spectral in the sense of Hochster [14], i.e. it has a basis of quasi-compact opens and any irreducible closed has a unique generic point. Hochster proved that this characterizes spectra of commutative rings among topological spaces. So, we might hope for a nice little commutative ring R such that Spc(𝒦) = Spec(R). Since the endomorphism ring of the unit, End_{𝒦} (11), provides a natural commutative ring for each tensor triangulated category 𝒦, it is legitimate to ask whether the triangular spectrum Spc(𝒦) can be related to the Zariski spectrum Spec(End_{𝒦} (11)). Alternatively, one can consider the homogeneous spectrum of the graded endomorphism ring End𝒦^{•} (11) = Hom_{𝒦} (11, Σ^{•}(11)). Our main construction establishes such links :

Theorem (Thm. 5.3 and Cor. 5.6). There exist two continuous maps

ρ_{𝒦} : Spc(𝒦) ⟶ Spec (End_{𝒦} (11))  and  ρ𝒦^{•} : Spc(𝒦) ⟶ Spec^{h}(End𝒦^{•} (11))

which are natural in the tensor triangulated category 𝒦.

A direct consequence is that one can produce open covers of the triangular spectrum Spc(𝒦) by taking pre-images of open covers of the above Zariski spectra, even before knowing Spc(𝒦). This will be investigated in subsequent work.

It is interesting to inspect the various incarnations of the maps ρ_{𝒦} and ρ𝒦^{•} in the areas of application of tensor triangular geometry mentioned above. These maps are always non-trivial but seem especially informative in algebraic examples. In fact, the maps ρ_{𝒦} and ρ𝒦^{•} are often surjective (yet not always; see Ex. 8.3) :

Theorem (Thm. 7.13). Assume that 𝒦 is connective, i.e. Hom_{𝒦} (Σ^{i}(11), 11) = 0 for all i < 0. Then the map ρ_{𝒦} : Spc(𝒦) ⟶ Spec(End_{𝒦} (11)) is surjective.

There is also a rather mild sufficient condition for surjectivity in the graded case :

Theorem (Thm. 7.3 and Cor. 7.4). If the graded ring End𝒦^{•} (11) is coherent, for instance if it is noetherian, then both maps ρ𝒦^{•} : Spc(𝒦) ⟶ Spec^{h}(End𝒦^{•} (11)) and ρ_{𝒦} : Spc(𝒦) ⟶ Spec(End_{𝒦} (11)) are surjective.

On the other hand, these maps ρ_{𝒦} and ρ𝒦^{•} are not injective in general, already in algebraic geometry, that is, for 𝒦 = D^{perf}(X) over a scheme X. For instance, in the case of projective n-space X = ℙ_{k}^{n}, we have Spc(𝒦) ≅ ℙ_{k}^{n} but End_{𝒦} (11) = End𝒦^{•} (11) = k. However, such counter-examples are rather weak for the following two reasons. First, when X = Spec(A) is affine, the map ρ𝒦^{•} = ρ_{𝒦} is actually a homeomorphism (Prop. 8.1), inverse to the one Spec(A) →^{∼} Spc(𝒦) given in [1, Cor. 5.6]. So, in algebraic geometry, ρ𝒦^{•} is at least “locally injective”. Secondly, in examples like the above X = ℙ_{k}^{n}, one can in fact obtain injectivity of ρ𝒦^{•} if we twist the graded endomorphism ring by a line bundle ; see Remark 8.2.

In topology, for 𝒦 = SH^{fin} the (Spanier-Whitehead) stable homotopy category of finite spectra, the triangular spectrum Spc(SH^{fin}) and the surjective continuous map ρ_{SH}_{fin} : Spc(SH^{fin}) ⟶ Spec(ℤ) look as follows (see Corollary 9.5) :

(0.1)  [figure: 𝒫_{2,∞}; 𝒫_{3,∞}; ⋯; 𝒫_{p,∞}; ⋯; Spc(SH^{fin}) =; ρ_{SH_{fin}}; ⋮; ⋮; ⋮; 𝒫_{2,n+1}; 𝒫_{3,n+1}; ⋯; 𝒫_{p,n+1}; ⋯; 𝒫_{2,n}; 𝒫_{3,n}; ⋯; 𝒫_{p,n}; ⋯; ⋮; ⋮; ⋮; 𝒫_{2,1}; 𝒫_{3,1}; ⋯; 𝒫_{p,1}; ⋯; _{SH{fin \atop tor}}; Spec(ℤ) =; 2ℤ; 3ℤ; ⋯; pℤ; ⋯; (0)]

Here, p runs through prime numbers and n through positive integers. The notation in this result (the announced reformulation of Hopkins-Smith’s classification) is explained in Section 9. The map ρ_{SH}_{fin} is the obvious vertical projection. Note that its non-injectivity is more serious than before. Indeed, here, injectivity even fails locally, that is, for SH_{(p)}^{fin} the localization of SH^{fin} at a prime p, despite SH_{(p)}^{fin} being a local tensor triangulated category (in the sense of Section 4).

This simply indicates that endomorphism rings of the unit cannot control the whole triangulated category 𝒦 and that Spc(𝒦) is a richer invariant.

At this stage, I do not know a good general criterion which could guarantee the (local) injectivity of the map ρ_{𝒦} : Spc(𝒦) → Spec(End_{𝒦} (11)) or its twisted graded version ρ𝒦^{•} . Such a criterion should probably work only for “algebraic enough” triangulated categories 𝒦 and seems to be an interesting research challenge.

Finally, here are the announced applications to 𝔸^{1}-homotopy theory and noncommutative topology, where the triangular spectrum is not known yet but some partial information can be obtained by the techniques of this paper :

Theorem (Cor. 10.1). Let 𝒦^{𝔸^{1}} := (SH_{F}^{𝔸^{1}})^{c} be the subcategory of compact objects in the stable 𝔸^{1}-homotopy category SH_{F}^{𝔸^{1}} over a perfect field F of characteristic different from 2. Let GW(F) be the Grothendieck-Witt ring of quadratic forms over F. Then there is a surjective continuous map ρ_{𝒦^{𝔸^{1}}} : Spc(𝒦^{𝔸^{1}}) ⟶ Spec(GW(F)).

This uses among other things our first criterion for surjectivity (connectivity). The Zariski spectrum Spec(GW(F)) is classical and can be found in Remark 10.2.

Theorem (Cor. 8.8). Let G be a compact Lie group and KK^{G} be the category of separable G-C^{∗}-algebras with morphisms given by Kasparov’s equivariant KKtheory (see Meyer [23]). Let R(G) := K_{0}^{G}(ℂ) be the complex representation ring of G. Then there is a continuous surjection ρ_{KK^{G}} : Spc(KK^{G}) ⟶ Spec(R(G)).

This uses among other things our second criterion for surjectivity (noetherianity). The Zariski spectrum Spec(R(G)) is classical and can be found in Segal [31, § 3].

The structure of the paper is the following. After quickly recalling from [1] a few facts about tensor triangulated categories and their spectrum in the one-page Section 1, we consider the central ring R_{𝒦} = End_{𝒦} (11) in Section 2. As in any symmetric monoidal category, the ring R_{𝒦} acts on any object of 𝒦. We show in Theorem 2.15 that the smallest thick triangulated ⊗-ideal ⟨cone(f)⟩ containing the cone of an endomorphism f : 11 → 11 coincides with the subcategory of objects x ∈ 𝒦 on which f is nilpotent. In Section 3, we use these techniques to construct localizations S^{−1}𝒦 of any tensor triangulated category 𝒦, with respect to any multiplicative subset S ⊂ R_{𝒦} . We also provide graded versions of these constructions.

Section 4 has a broader importance for tensor triangular geometry, for we introduce the notion of local tensor triangulated category. For instance, a rigid (Def. 1.5) tensor triangulated category 𝒦 is local if and only if

(0.2)  x ⊗ y = 0  implies  x = 0 or y = 0 .

Following commutative algebra, one might naively call tensor triangulated categories satisfying (0.2) “integral” or “domains”. The correct terminology, namely “local”, is justified by the fact that Spc(𝒦) is a local topological space (Prop. 4.2) and comforted by the fact that a commutative ring A is local if and only if D^{perf}(A) is local in the above sense. A key result is Theorem 4.5, which says :

Theorem. If 𝒦 is a local tensor triangulated category, then R_{𝒦} is a local ring and R𝒦^{•} = End𝒦^{•} (11) is a local graded ring.

It is then easy to construct the maps ρ_{𝒦} and ρ𝒦^{•} explicitly in Section 5. Somewhat independent of the rest, Section 6 recalls from [1] the sheaf of rings O_{𝒦} on the spectrum Spc(𝒦). This yields a locally ringed space Spec(𝒦) := ( Spc(𝒦), O_{𝒦} ) and our maps ρ_{𝒦} and ρ𝒦^{•} become morphisms of locally ringed spaces. Incidentally, the example of 𝒦 = SH^{fin} also shows that the locally ringed space Spec(𝒦) is not a scheme in general. We prove our surjectivity results in Section 7. The last three sections are dedicated to examples, from algebraic geometry, modular representation theory and KK-theory in Section 8, from topological stable homotopy theory in Section 9 and finally from stable 𝔸^{1}-homotopy theory in Section 10.

Acknowledgments : I warmly thank Aravind Asok, Mark Blunk, Theo Bühler, Ivo Dell’Ambrogio, Fabien Morel, Amnon Neeman, Manuel Ojanguren, Tobias Peter, Greg Stevenson and Charles Walter for precious discussions and comments. I also thank the referee for suggesting some improvements.

1. Basics

Convention 1.1. Unless stated otherwise, all categories are assumed essentially small and all subcategories are assumed full and closed under isomorphism.

Definition 1.2. A tensor triangulated category (𝒦, ⊗, 11) is the data of a triangulated category 𝒦 and of a “compatible” symmetric monoidal structure

⊗ : 𝒦 × 𝒦 ⟶ 𝒦

with unit 11 ∈ 𝒦 (for symmetric monoidal ; see Mac Lane [21, § XI.1]). We denote by σ_{a,b} : a ⊗ b →^{∼} b ⊗ a the symmetry isomorphism (switch). The compatibility means that ⊗ should be exact in each variable. See further references in [1, Def. 1.1].

Definition 1.3. In [1], we define the spectrum, Spc(𝒦), of a tensor triangulated category 𝒦 as the set of proper thick triangulated ⊗-ideals 𝒫 ⊊ 𝒦 which are prime, i.e. such that a ⊗ b ∈ 𝒫 ⇒ a or b ∈ 𝒫. We define for all a ∈ 𝒦 the open subsets U(a) := { 𝒫 ∈ Spc(𝒦) | a ∈ 𝒫 }, which form a basis of the topology on Spc(𝒦). The support of an object a ∈ 𝒦 is the closed complement of U(a) :

supp(a) = { 𝒫 ∈ Spc(𝒦) | a ∉ 𝒫 } ⊂ Spc(𝒦) .

Remark 1.4. Let 𝒥 ⊂ 𝒦 be a thick triangulated ⊗-ideal (that is, 𝒥 ⊂ 𝒦 is a non-empty full subcategory, stable in 𝒦 under taking cones, direct summands and tensor product with any object of 𝒦). The Verdier quotient 𝒦/𝒥 ([32, § II.2]) becomes a ⊗-triangulated category, with q : 𝒦 ↠ 𝒦/𝒥 being ⊗-triangulated. The spectrum of the quotient Spc(𝒦/𝒥) is naturally homeomorphic to the subspace { 𝒫 ∈ Spc(𝒦) | 𝒫 ⊃ 𝒥 } of Spc(𝒦), via 𝒬 ↦ q^{−1}(𝒬) ; see [1, Prop. 3.11].

Definition 1.5. We say that the tensor triangulated category is rigid if there exists an exact functor D : 𝒦^{op} → 𝒦 and natural isomorphisms

Hom_{𝒦} (a ⊗ b, c) ≃ Hom_{𝒦} (a, D(b) ⊗ c) .

This says that \underline{hom}(b, c) := D(b) ⊗ c is an internal hom functor. In [2], [6] we wrote “strongly closed” instead of “rigid” but the latter terminology seems more common.

Notation 1.6. For a collection E ⊂ 𝒦 of objects, we denote by ⟨E⟩ the smallest thick triangulated ⊗-ideal containing E.

Proposition 1.7. Let E ⊂ 𝒦 be a class of objects in our ⊗-triangulated category 𝒦. (a) For a ∈ ⟨E⟩ and for b ∈ 𝒦, we have a ⊗ b ∈ ⟨E ⊗ b⟩.

(b) If ⟨E⟩ = 𝒦, then for every n ≥ 1 we have ⟨{ x^{⊗n} | x ∈ E }⟩ = 𝒦 as well.

Proof. (a) : The subcategory { x ∈ 𝒦 | x ⊗ b ∈ ⟨E ⊗ b⟩ } is a thick triangulated ⊗-ideal containing E, hence containing ⟨E⟩, hence containing a.

(b) : There exists x_{1}, …, x_{m} ∈ E such that 11 ∈ ⟨x_{1}, …, x_{m}⟩. So, we can assume E = {x_{1}, …, x_{m}} finite. From (a), we get x_{i} ∈ ⟨E ⊗ x_{i}⟩ ⊂ ⟨E(2)⟩, where E(r) := { y_{1} ⊗ ⋯ ⊗ y_{r} | y_{1}, …, y_{r} ∈ E } for any r ≥ 2. Hence 11 ∈ ⟨x_{1}, …, x_{m}⟩ ⊂ ⟨E(2)⟩. Repeating this, we get 11 ∈ ⟨E(r)⟩ for all r ≥ 2. Now since E is finite, for any given n ≥ 1, there exists r ≥ 1 big enough such that E(r) ⊂ ⟨{ x^{⊗n} | x ∈ E }⟩. Hence 11 ∈ ⟨{ x^{⊗n} | x ∈ E }⟩ and therefore 𝒦 = ⟨{ x^{⊗n} | x ∈ E }⟩. □

2. Morphisms and actions

Definition 2.1. For a tensor triangulated category (𝒦, ⊗, 11), we define its central ring R_{𝒦} to be the endomorphism ring of its unit

R_{𝒦} := End_{𝒦} (11) .

It is commutative since composition coincides with tensor product :

Proposition 2.2. For all a, b ∈ 𝒦, the group Hom_{𝒦} (a, b) is a left R_{𝒦} -module via (f, g) ↦ f ⊗ g, for f ∈ R_{𝒦} and g ∈ Hom_{𝒦} (a, b), using the identifications 11 ⊗ a ≅ a and 11 ⊗ b ≅ b. This left action coincides with the right action (g, f) ↦ g ⊗ f defined analogously. We denote this action by f ·g. With this structure, composition Hom_{𝒦} (b, c) × Hom_{𝒦} (a, b) → Hom_{𝒦} (a, c), (g, h) ↦ g ∘ h, becomes R_{𝒦} -bilinear.

Proof. Let us check that f ⊗ g = g ⊗ f. Consider the commutative diagram :

\xymatrix{ & 11 ⊗ a \ar[r]^{f⊗g} \ar[dd]^{σ_{11,a}}_{≅} & 11 ⊗ b \ar@{-->}[dr]^{≅} \ar[dd]^{σ_{11,b}}_{≅} \\ a \ar[ur]^{≅} \ar[dr]_{≅} &  &  & b . \\  & a ⊗ 11 \ar[r]^{g⊗f} & b ⊗ 11 \ar@{-->}[ur]_{≅}}

The left and right triangles commute by axiom of symmetric monoidal categories [21, XI.1.(4), p. 252]. The upper composition from a to b is what we called f ⊗g and the lower one is g ⊗ f. The rest of the proof is left to the reader (or see [11, 2.1.1]). □

Remark 2.3. Proposition 2.2 justifies the term “central ring” for the ring R_{𝒦} .

Definition 2.4. An object u ∈ 𝒦 is invertible if u ⊗ v ≃ 11 for some v ∈ 𝒦. When 𝒦 is rigid (Def. 1.5), v is necessarily isomorphic to D(u).

Example 2.5. When u is invertible, so is Σ^{i}(u) for every i ∈ ℤ. In particular, Σ^{i}(11) is invertible for every i ∈ ℤ, with inverse Σ^{−i}(11).

Notation 2.6. For a morphism f : 11 → u and for an object a ∈ 𝒦, we denote by

f_{⇂a} : a ⟶ u ⊗ a

the morphism f ⊗ id_{a} (composed with a ≅ 11 ⊗ a). This applies in particular to u = 11, in which case, f_{⇂a} is nothing but f · id_{a} in the notation of Proposition 2.2.

Remark 2.7. Let u ∈ 𝒦 be invertible. Then f ↦ f ⊗ id_{u} induces an isomorphism Hom_{𝒦} (a, b) →^{∼} Hom_{𝒦} (a ⊗ u, b ⊗ u) for all a, b ∈ 𝒦. In particular, there exists a central unit ϵ_{u} ∈ R_{𝒦}^{×} such that the switch σ_{u,u} : u ⊗ u →^{∼} u ⊗ u is equal to (ϵ_{u})_{⇂u⊗u} = ϵ_{u} · id_{u⊗u}. Note that (σ_{u,u})^{2} = id_{u⊗u} implies (ϵ_{u})^{2} = 1.

Definition 2.8. We say that an object x ∈ 𝒦 has central switch if there exists a unit ϵ = ϵ(x) ∈ R_{𝒦}^{×} such that σ_{x,x} = ϵ_{⇂x⊗x} : x ⊗ x → x ⊗ x. Remark 2.7 shows that invertible objects have central switch.

Remark 2.9. Assume that x and y have central switch. Then x ⊗ y also does. Indeed, for x_{1}, x_{2}, y_{1}, y_{2} ∈ 𝒦 the switch σ_{x_{1}⊗y_{1},x_{2}⊗y_{2}} decomposes as follows :

\xymatrix{x_{1} ⊗ y_{1} ⊗ x_{2} ⊗ y_{2} \ar[r]^{σ_{x_{1}⊗y_{1},x_{2}⊗y_{2}}} \ar[d]_{1⊗σ_{y_{1},x_{2}} ⊗1} & x_{2} ⊗ y_{2} ⊗ x_{1} ⊗ y_{1} . \\ x_{1} ⊗ x_{2} ⊗ y_{1} ⊗ y_{2} \ar[r]^{σ_{x_{1},x_{2}} ⊗σ_{y_{1},y_{2}}} & x_{2} ⊗ x_{1} ⊗ y_{2} ⊗ y_{1} \ar[u]_{1⊗σ_{x_{1},y_{2}} ⊗1}}

For x_{1} = x_{2} = x and y_{1} = y_{2} = y, the bottom morphism is central (= ϵ(x) · ϵ(y)) and the two vertical morphisms are mutual inverses. So x ⊗ y has central switch : ϵ(x ⊗ y) = ϵ(x) · ϵ(y) in R_{𝒦} . In particular, x^{⊗n} has central switch for all n ≥ 1.

Lemma 2.10. Let f : a → x and g : b → x be two morphisms and assume that x has central switch (Def. 2.8). Then there exists an isomorphism τ : b ⊗ a →^{∼} a ⊗ b such that g ⊗ f = (f ⊗ g) ∘ τ. Similarly, for f′ : x → a′ and g′ : x → b′ there exists

an isomorphism τ′ : a′ ⊗ b′ →^{∼} b′ ⊗ a′ such that g′ ⊗ f′ = τ′ ∘ (f′ ⊗ g′).

Proof. This is elementary. Let ϵ ∈ R_{𝒦}^{×} such that σ_{x,x} = ϵ_{⇂x⊗x}. Since the diagram

\xymatrix{b ⊗ a \ar[r]^{g⊗f} \ar[d]_{σ_{b,a}}^{≅} & x ⊗ x \ar[d]^{σ_{x,x}=ϵ_{⇂x⊗x}}_{≅} \\ a ⊗ b \ar[r]^{f⊗g} & x ⊗ x}

commutes, and since ϵ ∈ R_{𝒦} is “central” (Prop. 2.2), we can take τ = ϵ · σ_{b,a}. □

Lemma 2.11. Let g : x → y be a morphism. Let [figure: a; k; b; l; c; m; Σa] be a distinguished triangle and suppose that the following diagram commutes

\xymatrix{x ⊗ a \ar[r]^{id_{x} ⊗k} \ar[d]_{0} & x ⊗ b \ar[r]^{id_{x} ⊗l} \ar[d]_{g⊗id_{b}} & x ⊗ c \ar[r]^{id_{x} ⊗m} \ar[d]_{0} & Σ(x ⊗ a) \ar[d]^{0} \\ y ⊗ a \ar[r]_{id_{y} ⊗k} & y ⊗ b \ar[r]_{id_{y} ⊗l} & y ⊗ c \ar[r]_{id_{y} ⊗m} & Σ(y ⊗ a) .}

Then the morphism g ⊗ g ⊗ id_{b} : x ⊗ x ⊗ b ⟶ y ⊗ y ⊗ b is equal to zero.

Proof. From the commutative diagram, we get (a) (g ⊗ id_{b}) ∘ (id_{x} ⊗k) = 0,

(b) g ⊗ l = (id_{y} ⊗l) ∘ (g ⊗ id_{b}) = 0. By (a) and the exact triangle of the first row, there exists h : x⊗c → y ⊗b such that g ⊗ id_{b} = h ∘ (id_{x} ⊗l). Hence g ⊗ g ⊗ id_{b} = g ⊗ (h ∘ (id_{x} ⊗l)) = (id_{y} ⊗h) ∘ (g ⊗ id_{x} ⊗l) and this is zero because g ⊗ id_{x} ⊗l = 0. To check this last claim, note that (b) implies 0 = id_{x} ⊗g ⊗ l : x ⊗ x ⊗ b ⟶ x ⊗ y ⊗ c and switch the first two factors, that is, precompose by σ_{x,x} ⊗ id_{b} and postcompose by σ_{x,y} ⊗ id_{c}. □

Proposition 2.12. Let f : x → y be a morphism. Then the objects a ∈ 𝒦 for which there exists n ≥ 1 with f^{⊗n} ⊗ id_{a} = 0 (as a morphism x^{⊗n} ⊗ a ⟶ y^{⊗n} ⊗ a) form a thick triangulated ⊗-ideal of 𝒦.

Proof. Thick and ⊗-ideal is easy; triangulated follows from Lemma 2.11. □

The following is well-known for x = y = 11.

Proposition 2.13. Let f : x → y be a morphism and assume that x and y have central switch (Def. 2.8) ; e.g. x and y invertible. Then (f ⊗ f) ⊗ id_{cone(f)} = 0.

Proof. Consider a distinguished triangle x ⟶^{f} y ⟶^{f_{1}} cone(f) ⟶^{f_{2}} Σ(x) and the commutative diagram

\xymatrix{x ⊗ x \ar[r]^{id_{x} ⊗f} \ar[d]_{f⊗id_{x}} & x ⊗ y \ar[r]^{id_{x} ⊗f_{1}} \ar[d]_{f⊗id_{y}} & x ⊗ cone(f) \ar[r]^{id_{x} ⊗f_{2}} \ar[d]_{f⊗id_{cone(f)}} & Σ(x ⊗ x) \ar[d]^{Σ(f⊗id_{x})} \\ y ⊗ x \ar[r]_{id_{y} ⊗f} & y ⊗ y \ar[r]_{id_{y} ⊗f_{1}} & y ⊗ cone(f) \ar[r]_{id_{y} ⊗f_{2}} & Σ(y ⊗ x) .}

We claim that the diagonal morphisms of the second and third squares vanish : (a) (id_{y} ⊗f_{1}) ∘ (f ⊗ id_{y}) = 0 and

(b) Σ(f ⊗ id_{x}) ∘ (id_{x} ⊗f_{2}) = 0. Since y has central switch, we can apply Lemma 2.10 to our f and to g := id_{y} to obtain an isomorphism τ : y ⊗ x →^{∼} x ⊗ y such that (f ⊗ id_{y}) ∘ τ = id_{y} ⊗f. Then (a) is easily checked by composing on the right by the isomorphism τ and using that f_{1} ∘ f = 0. Similarly, for (b), using the second statement of Lemma 2.10.

By (a) and (b), we can now replace the above morphism of distinguished triangles (f ⊗ id_{x} , f ⊗ id_{y} , f ⊗ id_{cone(f)} ) by the morphism (0, 0, f ⊗ id_{cone(f)}). The result then follows from Lemma 2.11 (modulo obvious rotations). □

Proposition 2.14. Let f : x → y be a morphism in 𝒦 and suppose that ⟨x, y⟩ = 𝒦; e.g. x or y invertible. If a ∈ 𝒦 is such that f ⊗ id_{a} = 0 then a ∈ ⟨cone(f)⟩.

Proof. Tensor the distinguished triangle x ⟶^{f} y → cone(f) → Σ(x) with a and use the assumption f ⊗ id_{a} = 0 to deduce that (y ⊗ a) ⊕ Σ(x ⊗ a) ≃ cone(f) ⊗ a. This shows that x⊗a and y ⊗a belong to ⟨cone(f)⟩. Therefore ⟨x⊗a, y ⊗a⟩ ⊂ ⟨cone(f)⟩. Finally, from 11 ∈ ⟨x, y⟩ we deduce a ∈ ⟨x ⊗ a, y ⊗ a⟩ by Proposition 1.7 (a). □

Theorem 2.15. Assume that x, y ∈ 𝒦 have central switch (Def. 2.8) and that ⟨x, y⟩ = 𝒦; e.g. x and y invertible. Let f : x → y be a morphism in 𝒦. Then the thick triangulated ⊗-ideal generated by its cone, ⟨cone(f)⟩, coincides with the subcategory of those objects a ∈ 𝒦 for which f^{⊗n} ⊗ id_{a} = 0 for some n ≥ 1.

Proof. Let 𝒥 := { a ∈ 𝒦 | ∃ n ≥ 1 s.t. f^{⊗n} ⊗ id_{a} = 0 } the subcategory in question. By Proposition 2.12, 𝒥 is a thick triangulated ⊗-ideal and by Proposition 2.13 we have cone(f) ∈ 𝒥, hence ⟨cone(f)⟩ ⊂ 𝒥. Let us now check the other inclusion.

Let a ∈ 𝒦 be such that f^{⊗n}⊗id_{a} = 0. Then a ∈ ⟨cone(f^{⊗n})⟩ by Proposition 2.14 applied to f^{⊗n} : x^{⊗n} → y^{⊗n}. Note that ⟨x^{⊗n}, y^{⊗n}⟩ = 𝒦 by Proposition 1.7 (b).

So, it remains to prove that ⟨cone(f^{⊗n})⟩ ⊂ ⟨cone(f)⟩ for all n ≥ 1. We do so by induction on n. Suppose that n ≥ 1 and that ⟨cone(f^{⊗n})⟩ ⊂ ⟨cone(f)⟩. Then the octahedron axiom applied to the relation f^{⊗(n+1)} = (f ⊗ id_{y^{⊗n}}) ∘ (id_{x} ⊗f^{⊗n}) shows that cone(f^{⊗(n+1)}) ∈ ⟨cone(f), cone(f^{⊗n})⟩ ⊂ ⟨cone(f)⟩, which gives the result. □

Proposition 2.16. Let x, y ∈ 𝒦 be as in Theorem 2.15, e.g. x and y invertible. Let f : x → y and n ≥ 1. Then ⟨cone(f)^{⊗n}⟩ = ⟨cone(f^{⊗n})⟩ = ⟨cone(f)⟩.

Proof. Note that x^{⊗n} and y^{⊗n} still satisfy the hypotheses of Theorem 2.15 by Remark 2.9 and Proposition 1.7 (b). It is then direct from Theorem 2.15 that ⟨cone(f)⟩ = ⟨cone(f^{⊗n})⟩. Obviously, cone(f)^{⊗n} ∈ ⟨cone(f)⟩ hence it suffices to prove that cone(f) ∈ ⟨cone(f)^{⊗n}⟩. By induction (and Prop. 1.7 (a)), it suffices to treat the case n = 2. Consider the composition

\xymatrix{x ⊗ x ⊗ cone(f) \ar[r]^{f⊗1⊗1} & y ⊗ x ⊗ cone(f) \ar[r]^{1⊗f⊗1} & y ⊗ y ⊗ cone(f) .}

Since the cone of the first map, cone(f) ⊗ x ⊗ cone(f), and the cone of the second, y ⊗ cone(f) ⊗ cone(f), both belong to ⟨cone(f)^{⊗2}⟩, the octahedron axiom tells us that the cone of the composition also belongs to ⟨cone(f)^{⊗2}⟩. But that composition is zero by Proposition 2.13 hence its cone is Σ(x^{⊗2} ⊗ cone(f)) ⊕ (y^{⊗2} ⊗ cone(f)). By definition, ⟨cone(f)^{⊗2}⟩ is thick, so we have proved :

⟨x^{⊗2} ⊗ cone(f) , y^{⊗2} ⊗ cone(f)⟩ ⊂ ⟨cone(f)^{⊗2}⟩ .

Proposition 1.7 (b) and (a) gives us 11 ∈ ⟨x^{⊗2}, y^{⊗2}⟩ and cone(f) = 11 ⊗ cone(f) ∈ ⟨x^{⊗2} ⊗ cone(f) , y^{⊗2} ⊗ cone(f)⟩ ⊂ ⟨cone(f)^{⊗2}⟩, as wanted. □

3. Graded homomorphisms and central localization

Notation 3.1. Let us fix u ∈ 𝒦 an invertible object. For any pair of objects a, b ∈ 𝒦, and any i ∈ ℤ, we denote by

Hom_{𝒦}^{i} (a, b) := Hom_{𝒦} ( a , u^{⊗i} ⊗ b) .

When we need to stress the dependency on u, we can write Hom_{𝒦,u}^{i} (a, b). The above notation is commonly used with u = Σ(11), for which Hom_{𝒦}^{i} (a, b) ≅ Hom_{𝒦} (a, Σ^{i}(b)). We denote by Hom𝒦^{•} (a, b) the ℤ-graded abelian group ⊕_{i∈ℤ}Hom_{𝒦}^{i} (a, b). We have an obvious composition :

Hom_{𝒦}^{j} (b, c) × Hom_{𝒦}^{i} (a, b)  ⟶^{⋄}  Hom_{𝒦}^{i+j}(a, c)
( b ⟶^{g} u^{⊗j} ⊗ c , a ⟶^{f} u^{⊗i} ⊗ b )  ⟼  \xymatrix{a \ar[rr]^{g⋄f} \ar[dr]_{f} &  & u^{⊗(i+j)} ⊗ c \\  & u^{⊗i} ⊗ b \ar@{-->}[ur]_{id_{u}^{⊗i} ⊗g}}

which coincides with the usual composition when i = 0.

We shall mostly use the case u = Σ(11) but the above flexibility can be useful in general, as illustrated in algebraic geometry in Remark 8.2.

Definition 3.2. We denote by R𝒦^{•} (or R𝒦,u^{•} ) the graded group Hom𝒦^{•} (11, 11), which is a unital associative ℤ-graded ring with respect to ⋄. We call it the graded central ring of 𝒦 (with respect to u ∈ 𝒦). Note that R_{𝒦}^{0} is just the central ring R_{𝒦} = End_{𝒦} (11) of Section 2, independently of u.

Generalizing Proposition 2.2, we have the following compatibility between ⋄ and ⊗. Recall the central unit ϵ_{u} ∈ R_{𝒦} = R_{𝒦}^{0} such that the switch σ_{u,u} : u ⊗ u →^{∼} u ⊗ u is given by multiplication by ϵ_{u}. (See Rem. 2.7.) Of course, we have (ϵ_{u})^{2} = 1 and the reader can simply assume that ϵ_{u} = ±1, as it is often the case.

Proposition 3.3. Let u ∈ 𝒦 be an invertible object. With the above notation :

(a) For a, b ∈ 𝒦, the graded abelian group Hom𝒦^{•} (a, b) carries both a structure of left and of right graded module over the graded central ring R𝒦^{•} , defined via the tensor product ⊗. Explicitly, for f ∈ R_{𝒦}^{i} and g ∈ Hom_{𝒦}^{j} (a, b), we set

\begin{matrix}f · g := ( \xymatrix{a \ar[r]^{f⊗g} & u^{⊗i} ⊗ u^{⊗j} ⊗ b ≅ u^{⊗(i+j)} ⊗ b} ) & and \\ g · f := ( \xymatrix{a \ar[r]^{g⊗f} & u^{⊗j} ⊗ b ⊗ u^{⊗i} \ar[r]^{id ⊗σ}_{≅} & u^{⊗j} ⊗ u^{⊗i} ⊗ b ≅ u^{⊗(i+j)} ⊗ b} ) . \end{matrix}

(b) When a = b = 11, both actions coincide with composition f · g = f ⋄ g.

(c) Composition ⋄ : Hom𝒦^{•} (b, c) × Hom𝒦^{•} (a, b) ⟶ Hom𝒦^{•} (a, c) is bilinear with respect to both the left action and the right action, in an ϵ_{u}-graded way, that is, for f ∈ R_{𝒦}^{i} , g ∈ Hom_{𝒦}^{j} (a, b) and h ∈ Hom_{𝒦}^{k} (b, c), we have

\begin{aligned}& (f · h) ⋄ g = f · (h ⋄ g) = (ϵ_{u})^{i·k} h ⋄ (f · g) \\ and & h ⋄ (g · f) = (h ⋄ g) · f = (ϵ_{u})^{i·j} (h · f) ⋄ g . \end{aligned}

(d) The left and right actions coincide in an ϵ_{u}-graded way, that is, for f ∈ R_{𝒦}^{i} and g ∈ Hom_{𝒦}^{j} (a, b), we have

f · g = (ϵ_{u})^{i·j} g · f .

(e) In particular, the ℤ-graded central ring R𝒦^{•} is ϵ_{u}-commutative.

Sketch of Proof. Consider the external product

Hom_{𝒦}^{i} (a, b) × Hom_{𝒦}^{j} (c, d) ⟶^{⊠} Hom_{𝒦}^{i+j}(a ⊗ c, b ⊗ d)

defined for f ∈ Hom_{𝒦}^{i} (a, b) and g ∈ Hom_{𝒦}^{j} (c, d) by the formula

[figure: a ⊗ c; f⊗g; f⊠g; u^{⊗i} ⊗ b ⊗ u^{⊗j} ⊗ d; 1⊗σ⊗1; u^{⊗i} ⊗ u^{⊗j} ⊗ b ⊗ d; u^{⊗i+j} ⊗ b ⊗ d .]

This product is associative, natural and ϵ_{u}-commutative, in the obvious sense. Statements (a)-(e) can be deduced from these properties. (Compare Proposition 2.2.) Further verifications are left to the reader, who can find some details in [11, § 2.1.2]. □

Remark 3.4. We are going to need some elementary graded commutative algebra over ϵ-commutative ℤ-graded rings. Let R^{•} = ⨁_{i∈ℤ} R^{i} be a unitary and associative ℤ-graded ring. We denote by R^{hom} := ∪_{i∈ℤ}R^{i} the subset of homogeneous elements. We say that R^{•} is ϵ-commutative if there exists ϵ ∈ R^{0} such that ϵ^{2} = 1 and

(3.1)  f · g = ϵ^{ij} · g · f

for all homogeneous f ∈ R^{i} and g ∈ R^{j}. Hence the subset R^{even} := ∪_{i∈ℤ}R^{2i} is central in R^{•}, and in particular so is the unit ϵ.

Let S ⊂ R^{hom} be a central multiplicative subset (1 ∈ S and S · S ⊂ S, typically S ⊂ R^{even}). Then the localization S^{−1}R^{•} is graded as usual by setting (S^{−1}R^{•})^{i} := \{ \frac{f}{s} | f ∈ R^{i+j}, s ∈ S ∩ R^{j} \}. If we need to invert a more general multiplicative subset T ⊂ R^{hom}, hypothesis (3.1) allows us to do it by inverting the central subset S := { t^{2} | t ∈ T } instead.

We define the homogeneous spectrum, Spec^{h}(R^{•}), as the set of proper homogeneous ideals 𝔭^{•} of R^{•} which are prime (a · b ∈ 𝔭^{•} ⇒ a ∈ 𝔭^{•} or b ∈ 𝔭^{•}; this can be checked on homogeneous elements only). The Zariski topology has as closed subsets the V (I^{•}) := { 𝔭^{•} ∈ Spec^{h}(R^{•}) | I^{•} ⊂ 𝔭^{•} } for homogeneous ideals I^{•} ⊂ R^{•}. The principal open subsets D(s) := { 𝔭^{•} ∈ Spec^{h}(R^{•}) | s ∉ 𝔭^{•} } define a basis of the topology, for s running through R^{hom}, or just through R^{even} since D(s) = D(s^{2}). We define a structure sheaf O^{•} of ϵ-commutative ℤ-graded rings on Spec^{h}(R^{•}) by setting O^{•}(D(s)) = R^{•}[\frac{1}{s}] := S^{−1}R^{•} where S = \{ s^{2i} | i ∈ ℕ \} ⊂ R^{even}.

When ϵ = 1, the reader can find in [20] all results we use below. We also use this reference even for ϵ ≠ 1. Unfortunately, the literature does not seem to have an ϵ-commutative equivalent of [20] and we will follow here the usual practice of leaving to the reader to extend the proofs in the ϵ-commutative case.

Here, we need to associate to a homogeneous prime 𝔭^{•} ⊂ R^{•} a localization R^{•} → S^{−1}R^{•} such that S^{−1}R^{•} has a unique maximal homogeneous ideal (i.e. S^{−1}R^{•} is a local graded ring) in such a way that this maximal ideal maps to our 𝔭^{•} under the natural map (defined as usual)

Spec^{h}(S^{−1}R^{•}) ↪ Spec^{h}(R^{•}) .

This can be realized by localization with respect to the central multiplicative subset

(3.2)  S_{𝔭^{•}} := {s ∈ R^{even} such that s ∉ 𝔭^{•}} .

Indeed, a homogeneous prime ideal 𝔮^{•} ⊂ R^{•} disjoint from S_{𝔭^{•}} is contained in 𝔭^{•} (otherwise, there is a homogeneous t ∈ 𝔮^{•} with t ∉ 𝔭^{•}, but then t^{2} ∈ S_{𝔭^{•}} ∩ 𝔮^{•} = ∅).

Construction 3.5. Let us return to our tensor triangulated category 𝒦. Consider now S ⊂ R𝒦^{•} a multiplicative subset of homogeneous elements (1 ∈ S and S·S ⊂ S). When ϵ_{u} ≠ 1, we further require S ⊂ R_{𝒦}^{even}. So, S is central in R𝒦^{•} and acts centrally on any graded module Hom𝒦^{•} (a, b). For instance, for each 𝔭^{•} ∈ Spec^{h}(R𝒦^{•} ), we can take S = S_{𝔭^{•}} as defined in (3.2) above.

We can construct as above the ℤ-graded ϵ_{u}-commutative ring S^{−1} R𝒦^{•} . Then, for any a, b ∈ 𝒦, we can construct the S^{−1} R𝒦^{•} -module S^{−1}Hom𝒦^{•} (a, b). Let us denote by (S^{−1}Hom𝒦^{•} (a, b))^{0} its degree zero part. Composition of morphisms (⋄) being R𝒦^{•} -bilinear, it induces a well-defined homomorphism

(S^{−1}Hom𝒦^{•} (b, c))^{0} × (S^{−1}Hom𝒦^{•} (a, b))^{0} ⟶ (S^{−1}Hom𝒦^{•} (a, c))^{0}.

This defines a new category, that we denote S^{−1}𝒦, with the same objects as 𝒦 and

Hom_{S^{−1}𝒦} (a, b) = (S^{−1}Hom𝒦^{•} (a, b))^{0}.

The natural homomorphism Hom_{𝒦} (a, b) ⟶ (S^{−1}Hom𝒦^{•} (a, b))^{0} defines a functor

q_{S} : 𝒦 ⟶ S^{−1}𝒦 .

We now show that S^{−1}𝒦 is a Verdier localization of 𝒦, in the obvious way. (Compare Hovey et al. [16, Thm. 3.3.7] under additional hypotheses about 𝒦.)

Theorem 3.6. Let us keep notations as above. Consider 𝒥 = ⟨cone(s)|s ∈ S⟩ the thick triangulated ⊗-ideal generated by the cones of elements of S. Consider the Verdier localization q : 𝒦 ↠ 𝒦/𝒥. Then there is an equivalence α : S^{−1}𝒦 ⟶^{∼} 𝒦/𝒥 such that α ∘ q_{S} = q. In particular, S^{−1}𝒦 can be equipped with the structure of a tensor triangulated category such that q_{S} : 𝒦 ⟶ S^{−1}𝒦 is a morphism of tensor triangulated categories.

We need two preliminary results. The first one is a generalization of Theorem 2.15. Recall from Notation 2.6 the shorthand f_{⇂a} = f ⊗ id_{a} for f ∈ R_{𝒦}^{even} and an object a ∈ 𝒦.

Proposition 3.7. With notation as in 3.5 and 3.6, we have 𝒥 = { c ∈ 𝒦 | ∃ s ∈ S such that s_{⇂c} = 0 }.

Proof. Let 𝒥′ := { c ∈ 𝒦 | ∃ s ∈ S such that s_{⇂c} = 0 }. Proposition 2.14 implies 𝒥′ ⊂ 𝒥. To check the converse, first note that since S is multiplicative and since multiplication and composition coincide on R𝒦^{•} , we have 𝒥′ = { c ∈ 𝒦 | ∃ s ∈ S and n ≥ 1 such that (s^{⊗n}) ⊗ id_{c} = 0 }. It is then easy to deduce from Proposition 2.12 that 𝒥′ is thick, triangulated and ⊗-ideal. By Proposition 2.13, we have cone(s) ∈ 𝒥′ for all s ∈ S. Therefore 𝒥 = ⟨cone(s)|s ∈ S⟩ ⊂ 𝒥′. □

Lemma 3.8. With notation as in 3.5 and 3.6, a morphism k : b → x in 𝒦 has its cone in 𝒥 if and only if there exists s ∈ S of some degree d ∈ ℤ and l, m ∈ Hom_{𝒦}^{d} (x, b) such that l ∘ k = s_{⇂b} in Hom_{𝒦}^{d} (b, b) and k ⋄ m = s_{⇂x} in Hom_{𝒦}^{d} (x, x).

Proof. Consider a distinguished triangle over k : b → x and, for any s ∈ S of degree d, the commutative (plain) diagram :

\xymatrix{b \ar[r]^{k} \ar[d]_{s_{⇂b}} & x \ar[r]^{k_{1}} \ar[d]^{s_{⇂x}} \ar@{-->}[dl]_{l} \ar@{-->}[dl]^{m} & cone(k) \ar[r]^{k_{2}} \ar[d]^{s_{⇂cone(k)}} & Σ(b) \ar[d]^{Σ(s_{⇂b})} \\ u^{⊗d} ⊗ b \ar[r]_{1⊗k} & u^{⊗d} ⊗ x \ar[r]_{1⊗k_{1}} & u^{⊗d} ⊗ cone(k) \ar[r]_{1⊗k_{2}} & Σ(u^{⊗d} ⊗ b) .}

If cone(k) ∈ 𝒥, by Proposition 3.7, we can choose s ∈ S such that the third vertical morphism s_{⇂cone(k)} vanishes. Then (1 ⊗ k_{1}) ∘ s_{⇂x} = 0 and s_{⇂b} ∘ Σ^{−1}(k_{2}) = 0 and the existence of l and m with l ∘ k = s_{⇂b} and (1 ⊗ k) ∘ m = s_{⇂x} are standard facts in triangulated categories. Conversely, if such morphisms l and m exist for some s ∈ S, then s_{⇂cone(k)} ∘ k_{1} = 0 and (1 ⊗ k_{2}) ∘ s_{⇂cone(k)} = 0. By Lemma 2.11, this implies (s^{2})_{⇂cone(k)} = s⊗s⊗id_{cone(k)} = 0. Since s^{2} ∈ S as well, we have just proved cone(k) ∈ 𝒥 by Proposition 3.7. □

Proof of Theorem 3.6. We define the functor α : S^{−1}𝒦 → 𝒦/𝒥 as the identity on objects. Let a, b ∈ 𝒦. For any s ∈ S, we have cone(s_{⇂b}) = cone(s)⊗b ∈ 𝒥. Therefore q(s_{⇂b}) is an isomorphism in 𝒦/𝒥 and we have a natural homomorphism

α_{a,b} : (S^{−1}Hom𝒦^{•} (a, b))^{0} ⟶ Hom_{𝒦/𝒥} (a, b)
\frac{f}{s} ⟼ q(s_{⇂b})^{−1} ∘ f .

It is easy to deduce from Lemma 3.8 that α_{a,b} is an isomorphism of abelian groups. Indeed, Lemma 3.8 shows that any fraction in 𝒦/𝒥 can be amplified to have denominator in S. Explicitly, a morphism a → b in 𝒦/𝒥 can be represented by a fraction k^{−1}g as in the upper row of the following diagram :

\xymatrix{a \ar[r]^{g} & x \ar@{-->}[d]_{l} & b \ar[l]_{k} \ar@{-->}[dl]^{s_{⇂b}} \\  & u^{⊗d} ⊗ b}

with cone(k) ∈ 𝒥. Lemma 3.8 gives us a morphism l ∈ Hom_{𝒦}^{d} (x, b) such that l∘k = s_{⇂b} for some s ∈ S. Then the morphism k^{−1} g is equal to α_{a,b}(\frac{lg}{s}). Injectivity of α_{a,b} follows from Lemma 3.8 in a similar way. □

Corollary 3.9. Let 𝔭^{•} ∈ Spec^{h}(R𝒦^{•} ). The localization 𝒦_{𝔭^{•}} := S_{𝔭^{•}}^{−1}𝒦 where S_{𝔭^{•}} = { s ∈ R_{𝒦}^{even} | s ∉ 𝔭^{•} } has graded central ring End𝒦_{𝔭^{•}}^{•} (11) isomorphic to (R𝒦^{•} )_{𝔭^{•}}. □

Corollary 3.10. Let S ⊂ End_{𝒦} (11) be a multiplicative subset of the central ring R_{𝒦} . Then the Verdier quotient S^{−1}𝒦 of 𝒦 by the thick triangulated ⊗-ideal ⟨cone(s) | s ∈ S⟩ has the same objects as 𝒦 and morphisms Hom_{S^{−1}𝒦} (a, b) = S^{−1}Hom_{𝒦} (a, b).

Proof. This can be checked as above or deduced from Theorem 3.6 (using any invertible object u), using that for a graded R^{•}-module M^{•} and for S ⊂ R^{0}, we have (S^{−1}M^{•})^{0} = S^{−1}(M^{0}). □

Corollary 3.11. Let 𝔭 ∈ Spec(R_{𝒦} ). Then the localization 𝒦_{𝔭} := S_{𝔭}^{−1}𝒦 where S_{𝔭} := R_{𝒦} ∖𝔭 has central ring End_{𝒦_{𝔭}} (11) isomorphic to (R_{𝒦} )_{𝔭}. □

4. Local tensor triangulated categories

Definition 4.1. A tensor triangulated category 𝒦 is local if Spc(𝒦) is a local topological space, that is, if any open cover Spc(𝒦) = ⋃_{i∈I} U_{i} is trivial, in that there exists i ∈ I such that U_{i} = Spc(𝒦).

Proposition 4.2. The following are equivalent :

(i) The tensor triangulated category 𝒦 is local (Def. 4.1).

(ii) The space Spc(𝒦) has a unique closed point.

(iii) The category 𝒦 has a unique minimal prime.

(iv) The ideal \sqrt[⊗]{0} ⊂ 𝒦 of ⊗-nilpotent objects is prime (and is the minimal one).

(v) For any objects a, b ∈ 𝒦, if a ⊗ b = 0 then a or b is ⊗-nilpotent. If 𝒦 is moreover rigid (Def. 1.5), then the above are further equivalent to :

(vi) If a ⊗ b = 0 then a = 0 or b = 0. (In this case, 0 is the minimal prime.)

Proof. (i)⇒(ii) : If 𝒫 and 𝒫′ were distinct closed points, then the cover Spc(𝒦) = ( Spc(𝒦) − {𝒫}) ∪ ( Spc(𝒦) − {𝒫′}) would contradict the local nature of Spc(𝒦).

(ii)⇒(i) : Recall that any non-empty closed subset of Spc(𝒦) contains a closed point, by [1, Cor. 2.12]. So, if there is only one closed point ℳ, any open subset U ⊂ Spc(𝒦) which contains ℳ has an empty closed complement, i.e. U = Spc(𝒦).

(ii)⇔(iii) : By [1, Prop. 2.9], the closure of a point 𝒫 ∈ Spc(𝒦) is described by \overline{\{𝒫\}} = \{ 𝒬 ∈ Spc(𝒦) | 𝒬 ⊂ 𝒫 \}. Hence closed points are just minimal primes.

(iii)⇔(iv) : Immediate from ⋂_{𝒫∈Spc(𝒦)} 𝒫 = \sqrt[⊗]{0} ; see [1, Cor. 2.4].

(iv)⇔(v) : Just reformulation since a ∈ \sqrt[⊗]{0} if and only if a^{⊗n} ∈ \sqrt[⊗]{0} for some n. Finally, when 𝒦 is rigid, \sqrt[⊗]{0} = 0 ; see [2, Cor. 2.5]. □

Example 4.3. For any prime 𝒫 ∈ Spc(𝒦) the category 𝒦/𝒫 is local. Indeed, in 𝒦/𝒫, the ideal 0 = 𝒫/𝒫 is a prime.

The reader’s intuition from commutative algebra might be puzzled by (iii) or (vi), in Proposition 4.2, which look more like 𝒦 being “integral”. Yet, Definition 4.1 is the conceptual one and the following example should lift any doubt.

Example 4.4. For a commutative ring A, the category K^{b}(A – proj) is local if and only if A is local. Indeed, Spc(K^{b}(A – proj)) is homeomorphic to Spec(A) via

Spec(A) ⟶^{∼} Spc(K^{b}(A – proj))
𝔭  ⟼ 𝒫(𝔭) := { M_{•} ∈ K^{b}(A – proj) | (M_{•})_{𝔭} ≃ 0 }

by [1, Cor. 6.3 (a)] and [9, Thm. 8.5]. Now, Spec(A) has a unique closed point if and only if A is local. The subtlety is that the above map is inclusion-reversing : 𝔭 ⊂ 𝔮 ⇒ 𝒫(𝔭) ⊃ 𝒫(𝔮).

Theorem 4.5. If the tensor triangulated category 𝒦 is local (Def. 4.1), then R𝒦^{•} = End𝒦,u^{•} (11) is a local graded ring. In particular, R_{𝒦} = End_{𝒦} (11) is a local ring.

Proof. We have to prove that R𝒦^{•} admits a unique maximal homogeneous ideal, which must then be the ideal generated by the non-invertible homogeneous elements. Since the product of a non-invertible with any other element remains noninvertible, it suffices to check that the sum of two non-invertible elements of same degree remains non-invertible. Let f, g ∈ R_{𝒦}^{d} , for some d ∈ ℤ, be such that f + g is invertible and let us prove that f or g is. In Notation 2.6, consider the morphism

(f + g)_{⇂cone(f)⊗cone(g)} : cone(f) ⊗ cone(g) ⟶ u^{⊗d} ⊗ cone(f) ⊗ cone(g) .

We claim that this morphism is both invertible and nilpotent in the graded ring End𝒦^{•} (cone(f)⊗cone(g)). The former is clear since f+g is invertible by assumption. To prove nilpotence, since f and g ϵ_{u}-commute, it suffices to show that both f and g are nilpotent on cone(f) ⊗ cone(g). By Proposition 3.3, (f_{⇂cone(f)⊗cone(g)})^{2} = f ⊗ f ⊗ id_{cone(f)} ⊗ id_{cone(g)} = 0 by Proposition 2.13, and similarly for g. Hence for some n ≥ 1, the morphism (f + g)_{⇂cone(f)⊗cone(g)}^{n} is both invertible and zero. This forces cone(f) ⊗ cone(g) = 0. But 𝒦 is local, so by Proposition 4.2 (v), we have cone(f)^{⊗n} = 0, or cone(g)^{⊗n} = 0, for n ∈ ℕ big enough. By Proposition 2.16, we deduce cone(f) = 0 or cone(g) = 0, i.e. f or g is an isomorphism.

The second statement follows : For any R^{•} local graded ring, R^{0} is local. □

Example 4.6. The converse to Theorem 4.5 does not hold. Consider 𝒦 = D^{perf}(X) for a non-local scheme X. It can nevertheless happen that End_{𝒦} (11) = Γ(X, O_{X}) is local, say, when O_{X} does not have many global sections, e.g. for X = ℙ_{k}^{n}.

5. From triangular spectrum to Zariski spectra

Let 𝒦 be a tensor triangulated category. As before, we tacitly fix an invertible object u ∈ 𝒦, like u = Σ(11), for instance. Recall the graded central ring R𝒦^{•} = End𝒦^{•} (11) = Hom_{𝒦} (11, u^{⊗•}) of Section 3.

Definition 5.1. Let 𝒫 ∈ Spc(𝒦) be a triangular prime. We define ρ𝒦^{•} (𝒫) ⊂ R𝒦^{•} as the following homogeneous ideal :

ρ𝒦^{•} (𝒫) := ⟨f ∈ R_{𝒦}^{hom} | cone(f) ∉ 𝒫⟩ .

Remark 5.2. Observe that ρ𝒦^{•} “reverses” inclusions : If 𝒫 ⊂ 𝒬 are two primes of 𝒦,

then ρ𝒦^{•} (𝒫) ⊃ ρ𝒦^{•} (𝒬) in R𝒦^{•} .

Theorem 5.3. Let 𝒦 be a tensor triangulated category. Let u ∈ 𝒦 be an invertible object and recall the graded central ring R𝒦^{•} = R𝒦,u^{•} of Definition 3.2. Then (a) Let 𝒫 ∈ Spc(𝒦). The subset ρ𝒦^{•} (𝒫) ⊂ R𝒦^{•} is a homogeneous prime ideal,

equal to the preimage under the localization homomorphism R𝒦^{•} → R𝒦/𝒫^{•} of the maximal homogeneous ideal of R𝒦/𝒫^{•} .

(b) The map ρ𝒦^{•} : Spc(𝒦) → Spec^{h}(R𝒦^{•} ) is continuous. More precisely, for any s ∈ R𝒦^{•} homogeneous, the preimage of the principal open D(s) ⊂ Spec^{h}(R𝒦^{•} ) is the open U(cone(s)) ⊂ Spc(𝒦) ; see Def. 1.3.

(c) The above defines a natural transformation ρ^{•} between the contravariant functors 𝒦 ↦ Spc(𝒦) and 𝒦 ↦ Spec^{h}(R𝒦^{•} ) from the category of tensor triangulated categories (with tensor triangulated functors respecting the chosen invertible objects) to the category of topological spaces.

Proof. Consider the localization functor q : 𝒦 ⟶ ℒ := 𝒦/𝒫. Choose q(u) as invertible in ℒ. The functor q gives us in particular a ring homomorphism f ↦ q(f)

R𝒦^{•} = Hom𝒦^{•} (11_{𝒦} , 11_{𝒦} ) \underrightarrow{q} Homℒ^{•} (11_{ℒ} , 11_{ℒ} ) = Rℒ^{•} .

Since the category ℒ is local (Ex. 4.3) we know by Theorem 4.5 that Rℒ^{•} is a local graded ring, with maximal ideal 𝔪^{•} := ⟨f ∈ R_{ℒ}^{hom} | f is not invertible⟩. For a morphism f in 𝒦, since q : 𝒦 ⟶ ℒ is exact, we have cone(q(f)) ≃ q(cone(f)) and for an object x ∈ 𝒦, we have q(x) = 0 if and only if x ∈ 𝒫. Therefore ρ𝒦^{•} (𝒫) = ⟨f ∈ R_{𝒦}^{hom} | q(f) is not invertible⟩ = q^{−1}(𝔪^{•}). So, ρ𝒦^{•} (𝒫) is the image of 𝔪^{•} by the map Spec^{h}(q) : Spec^{h}(Rℒ^{•} ) → Spec^{h}(R𝒦^{•} ). Hence (a).

For (b), unfolding the definitions, we see that 𝒫 ∈ (ρ𝒦^{•} )^{−1}(D(s)) if and only if cone(s) ∈ 𝒫, that is, 𝒫 ∈ U(cone(s)) see Definition 1.3.

Finally, for (c), let F : 𝒦 → 𝒦′ be a morphism of tensor triangulated categories, that is, an exact ⊗-functor. We have the fixed invertible objects u in 𝒦 and u′ = F (u) in 𝒦′. Let us also denote by F : R𝒦^{•} → R𝒦′^{•} , f ↦ F (f), the induced ring homomorphism. We need to prove the commutativity of the following diagram :

\xymatrix{Spc(𝒦′) \ar[r]^{Spc F} \ar[d]_{_{ρ{• \atop 𝒦′}}} & Spc(𝒦) \ar[d]^{_{ρ{• \atop 𝒦}}} \\ Spec^{h}(R𝒦′^{•} ) \ar[r]_{Spec^{h} F} & Spec^{h}(R𝒦^{•} ) .}

Let 𝒫′ ∈ Spc(𝒦′) and recall that (Spc F)(𝒫′) = F^{−1}(𝒫′). For f ∈ R_{𝒦}^{hom}, unfolding the definitions, we have f ∈ ρ𝒦^{•} ((Spc F)(𝒫′)) ⇔ cone(f) ∉ (Spc F)(𝒫′) = F^{−1}(𝒫′) ⇔ F (cone(f)) ∉ 𝒫′. Now, F being exact, F (cone(f)) = cone(F (f)) and the condition becomes equivalent to cone(F (f)) ∉ 𝒫′ ⇔ F (f) ∈ ρ𝒦′^{•} (𝒫′) ⇔ f ∈

F^{−1}(ρ𝒦′^{•} (𝒫′)) = (Spec^{h} F)(ρ𝒦′^{•} (𝒫′)). □

Theorem 5.4. Let S ⊂ R𝒦^{•} be a central homogeneous multiplicative subset and let q : 𝒦 → S^{−1}𝒦 be the corresponding localization (see Thm. 3.6). Then the diagram

\xymatrix{Spc(S^{−1}𝒦) \ar@{^{(}->}[rr]^{Spc(q)} \ar[dr]_{_{ρ{• \atop S^{−1}𝒦}}} &  & Spc(𝒦) \ar[d]^{_{ρ{• \atop 𝒦}}} \\  & Spec^{h}(RS^{−1}𝒦^{•} ) = Spec^{h}(S^{−1} R𝒦^{•} ) \ar@{^{(}->}[r] & Spec^{h}(R𝒦^{•} ) ,}

commutes and is cartesian : Spc(S^{−1}𝒦) ≅ { 𝒫 ∈ Spc(𝒦) | S ∩ ρ𝒦^{•} (𝒫) = ∅ }.

Proof. The diagram commutes by naturality of ρ^{•} (Thm. 5.3 (c)). As in Theorem 3.6, let 𝒥 = ⟨cone(s) | s ∈ S⟩, so that S^{−1}𝒦 = 𝒦/𝒥. We know that Spc(S^{−1}𝒦) is homeomorphic, via Spc(q), to { 𝒫 ∈ Spc(𝒦) | 𝒥 ⊂ 𝒫 }; see Remark 1.4. By definition of 𝒥, we have 𝒥 ⊂ 𝒫 if and only if cone(s) ∈ 𝒫 for all s ∈ S, which is equivalent to s ∉ ρ𝒦^{•} (𝒫) for all s ∈ S, by Definition 5.1. □

Remark 5.5. For each ℤ-graded ϵ-commutative ring R^{•}, we have a continuous map (−)^{0} : Spec^{h}(R^{•}) ⟶ Spec(R^{0}), 𝔭^{•} ↦ 𝔭^{0} = 𝔭^{•} ∩ R^{0}. This is natural in R^{•}, with respect to homogeneous ring homomorphisms.

Note that this map is surjective. Indeed, let 𝔭 ∈ Spec(R^{0}). We can localize at 𝔭 (invert S = R^{0} ∖ 𝔭) and reduce to the case where R^{0} is local with maximal ideal 𝔭. Then any homogeneous prime 𝔮^{•} containing the proper ideal 𝔭 · R^{•} (e.g. a maximal one) will satisfy 𝔮^{•} ∩ R^{0} = 𝔭: One inclusion since 𝔮^{•} contains 𝔭 and the other one since 𝔭 is maximal.

Composing ρ𝒦^{•} with the map (−)^{0} yields the following corollary.

Corollary 5.6. Let 𝒦 be a tensor triangulated category and let 𝒫 ∈ Spc(𝒦). We define ρ_{𝒦} (𝒫) ⊂ R_{𝒦} = End_{𝒦} (11) as the following subset :

(5.1)  ρ_{𝒦} (𝒫) := { f ∈ R_{𝒦} | cone(f) ∉ 𝒫 } .

(This does not rely on the choice of an invertible object u ∈ 𝒦.) Then we have : (a) For each 𝒫 ∈ Spc(𝒦), the subset ρ_{𝒦} (𝒫) ⊂ R_{𝒦} is a prime ideal.

(b) The map ρ_{𝒦} : Spc(𝒦) → Spec(R_{𝒦} ) is continuous, natural in 𝒦, and the following diagram commutes :

\xymatrix{Spc(𝒦) \ar[r]^{^{ρ{• \atop 𝒦}}} \ar[dr]_{_{ρ_{𝒦}}} & Spec^{h}(R𝒦^{•} ) \ar[d]^{(−)^{0}} \\  & Spec(R_{𝒦} ) .}

(c) Let S ⊂ R_{𝒦} be a multiplicative subset and let q : 𝒦 → S^{−1}𝒦 be the corresponding localization (see Cor. 3.10). Then the commutative diagram

\xymatrix{Spc(S^{−1}𝒦) \ar@{^{(}->}[rr]^{Spc(q)} \ar[dr]_{_{ρ_{S^{−1}𝒦}}} &  & Spc(𝒦) \ar[d]^{_{ρ_{𝒦}}} \\  & Spec(R_{S^{−1}𝒦} ) = Spec(S^{−1} R_{𝒦} ) \ar@{^{(}->}[r] & Spec(R_{𝒦} )}

is cartesian, i.e. Spc(S^{−1}𝒦) ≅ { 𝒫 ∈ Spc(𝒦) | S ∩ ρ_{𝒦} (𝒫) = ∅ }.

Proof. The diagram in (b) commutes by definition ; see (5.1). Therefore (a) and the rest of (b) follow. Only (c) requires some comment. Indeed, it follows from Theorem 5.4 and the following facts : For any graded ring R^{•} and any multiplicative subset S ⊂ R^{0}, we have (S^{−1}R^{•})^{0} = S^{−1}R^{0} and the commutative diagram

[figure: Spec^{h}(S^{−1}R^{•}); (−)^{0}; Spec^{h}(R^{•}); (−)^{0}; Spec(S^{−1}R^{0}); Spec(R^{0})]

is cartesian. Indeed, for 𝔭^{•} ∈ Spec^{h}(R^{•}) we have 𝔭^{•} ∩ S = ∅ ⇔ 𝔭^{0} ∩ S = ∅. □

Remark 5.7. Not only are the spaces Spc(𝒦), Spec^{h}(R𝒦^{•} ) and Spec(R_{𝒦} ) spectral in the sense of Hochster [14] but Theorem 5.3 (b) shows that ρ𝒦^{•} and ρ_{𝒦} are spectral maps, namely the preimage of a quasi-compact open is quasi-compact.

6. Locally ringed space structures

The spectrum Spc(𝒦) of a tensor triangulated category 𝒦 can be turned into a ringed space, as explained in [1, § 6]. We recall the definition, introduce a graded version of it, and then compare these structure sheaves with the ones on Spec(R_{𝒦} ) and on Spec^{h}(R𝒦^{•} ) ; see Rem. 3.4. More important than the sheaf of rings, the fundamental geometric feature is the following “presheaf” of triangulated categories.

Throughout this section, 𝒦 is a rigid (Def. 1.5) tensor triangulated category. We use this assumption to deduce that any thick triangulated ⊗-ideal 𝒥 of 𝒦 is radical, that is x^{⊗n} ∈ 𝒥 ⇒ x ∈ 𝒥 (see [1, Prop. 2.4]).

Construction 6.1. Let U ⊂ Spc(𝒦) be a quasi-compact open with closed complement Z. Define 𝒦_{Z} := { a ∈ 𝒦 | supp(a) ⊂ Z } to be the thick triangulated ⊗-ideal of 𝒦 supported outside U. Then define the tensor triangulated category “𝒦 on U”

(6.1)  𝒦(U) := (𝒦/𝒦_{Z})^{♮}

as the idempotent completion (−)^{♮} of the Verdier quotient 𝒦/𝒦_{Z}. This quotient is the localization S^{−1}𝒦 with respect to S = { s : a → b | supp(cone(s)) ⊂ Z }. By construction we have a natural functor

q_{U} : 𝒦 ⟶ 𝒦(U) .

It sends in particular the unit 11 = 11_{𝒦} ∈ 𝒦 to the unit 11_{𝒦(U)} , which we simply denote 11_{U}. We will consider endomorphism rings of these unit objects. The above idempotent completion is harmless for the sequel since the inclusion 𝒦 ↪ 𝒦^{♮} is fully faithful but (6.1) is the right definition of 𝒦(U) in view of geometric examples ; see more in [1, Rem. 6.2] including the reference to Thomason’s localization theorem.

Remark 6.2. The space Spc(𝒦) has a basis of quasi-compact open subsets ; see [1, Rem. 2.7 and Prop. 2.14]. We tacitly use this fact everywhere by describing our presheaves on such opens only. This is enough for the associated sheaves anyway. We focus on quasi-compact opens when defining 𝒦(U) because Spc(𝒦(U)) ≅ U if and only if U is quasi-compact ; see [1, Cor. 2.15] and [6, Prop. 1.11].

Lemma 6.3. Let 𝒫 ∈ Spc(𝒦) and let a, b ∈ 𝒦. There is a natural isomorphism colim_{U∋𝒫} Hom_{𝒦(U)} (a, b) →^{∼} Hom_{𝒦/𝒫} (a, b), where the colimit is taken over the quasi-

compact open subsets of Spc(𝒦) containing the point 𝒫.

Proof. Since we are considering a, b ∈ 𝒦, and since idempotent completion is fully faithful, we have for every U that Hom_{𝒦(U)} (a, b) = Hom_{𝒦/𝒦_{Z}} (a, b) where Z = Spc(𝒦) ∖ U. A morphism in the localization 𝒦/𝒦_{Z} is the equivalence class of a fraction a ←^{s} x → b where supp(cone(s)) ⊂ Z, the equivalence being with respect to amplification, as usual. Letting U shrink among the quasi-compact opens containing 𝒫, it is easy to check that the stated colimit amounts to the equivalence classes of fractions a ←^{s} x → b where supp(cone(s)) ⊂ Spc(𝒦)∖{𝒫}, which is equivalent to cone(s) ∈ 𝒫. Such fractions precisely describe the morphisms in 𝒦/𝒫. □

Definition 6.4. For each quasi-compact open U ⊂ Spc(𝒦), define the commutative ring pO_{𝒦} (U) and the ϵ_{u}-commutative ℤ-graded ring pO𝒦^{•} (U) by

pO_{𝒦} (U) := R_{𝒦(U)} = End_{𝒦(U)} (11_{U})  and  pO𝒦^{•} (U) := R𝒦(U)^{•} .

For the right-hand graded ring (Def. 3.2), some invertible object u ∈ 𝒦 is assumed fixed beforehand, for instance u = Σ(11) and we use its image in 𝒦(U). These pO_{𝒦} and pO𝒦^{•} are partially-defined presheaves on Spc(𝒦), only defined on an open basis ; see Remark 6.2. Their associated sheaves on the space Spc(𝒦) are denoted

O_{𝒦} := p̃O_{𝒦}  and  O𝒦^{•} := p̃O𝒦^{•} .

Finally, we denote the ringed spaces thus obtained by

Spec(𝒦) := ( Spc(𝒦), O_{𝒦} )  and  Spec^{•}(𝒦) := ( Spc(𝒦), O𝒦^{•} ) .

Note that they are both defined on the same space, namely the spectrum of 𝒦.

Proposition 6.5. Let 𝒫 ∈ Spc(𝒦). The stalk of O𝒦^{•} and of pO𝒦^{•} at 𝒫 are naturally isomorphic to R𝒦/𝒫^{•} . In particular, the stalk of O_{𝒦} and of pO_{𝒦} at 𝒫 are naturally isomorphic to R_{𝒦/𝒫} .

Proof. It follows directly from Lemma 6.3 applied to a = 11 and b = u^{⊗i} that the stalk at 𝒫 of the presheaf U ↦ R_{𝒦(U)}^{i} is R_{𝒦/𝒫}^{i} for all i ∈ ℤ. This gives the result for the presheaves pO_{𝒦} and pO𝒦^{•} , hence for the associated sheaves O_{𝒦} and O𝒦^{•} . □

Corollary 6.6. The ringed space Spec(𝒦) is a locally ringed space.

Proof. By Proposition 6.5, it suffices to prove that R_{𝒦/𝒫} is a local ring. This holds by Theorem 4.5 since the category 𝒦/𝒫 is local (see Ex. 4.3). □

Remark 6.7. The author doesn’t know whether the concept of “locally graded ringed space” exists but the proof of Corollary 6.6 applies to Spec^{•}(𝒦) as well, showing that the stalks of O𝒦^{•} are local in the graded sense.

Remark 6.8. We now want to show that the continuous maps ρ_{𝒦} : Spc(𝒦) → Spec(R_{𝒦} ) and ρ𝒦^{•} : Spc(𝒦) → Spec^{h}(R𝒦^{•} ) of Section 5 can be upgraded into morphisms of locally ringed spaces.

Recall that a morphism ρ : (X, O_{X}) → (Y, O_{Y}) of ringed spaces consists of a continuous map ρ : X → Y together with ring homomorphisms r_{U} : O_{Y} (U) → O_{X}(ρ^{−1}(U)) for all open U ⊂ Y, in a compatible way with the restrictions. As usual, it is enough to do so on a basis of the topology of the base space Y.

Lemma 6.9. For every homogenous s ∈ R_{𝒦}^{even} there is a natural isomorphism between the sections of the sheaf OSpec^{h}(R_{𝒦}^{•} )^{•} over the principal open D(s) of Spec^{h}(R𝒦^{•} ) and the sections of the presheaf pO𝒦^{•} over the preimage (ρ𝒦^{•} )^{−1}(D(s)) = U(cone(s)). They are both naturally isomorphic to R𝒦^{•} [\frac{1}{s}].

There is an analogous obvious ungraded statement for s ∈ R_{𝒦} mutatis mutandis.

Proof. By definition, OSpec^{h}(R_{𝒦}^{•} )^{•}(D(s)) = R𝒦^{•} [\frac{1}{s}]. On the other hand, we know by

Theorem 5.3 (b) that the preimage of D(s) under ρ𝒦^{•} : Spc(𝒦) → Spec^{h}(R𝒦^{•} ) is U(s) := U(cone(s)). Finally pO𝒦^{•} (U(s)) = R𝒦(U(s))^{•} can be computed by Theorem 3.6 applied to S = { s^{i} | i ≥ 0 }. (This uses 𝒦 rigid to prove that 𝒦_{supp(cone(s))} is indeed ⟨cone(s)⟩ and not just \sqrt[⊗]{⟨cone(s)⟩}.) This yields R𝒦^{•} [\frac{1}{s}] as well. Hence the result. The ungraded statement follows by contemplating degree zero only. □

Construction 6.10. For every s ∈ R_{𝒦}^{even}, let U(s) := U(cone(s)) = (ρ𝒦^{•} )^{−1}(D(s)) ⊂ Spc(𝒦) and define the ring homomorphism

r_{D(s)} : OSpec^{h}(R_{𝒦}^{•} )^{•}(D(s)) ⟶ O𝒦^{•} (U(s))

as the composite of the isomorphism OSpec^{h}(R_{𝒦}^{•} )^{•}(D(s)) ⟶^{∼} pO𝒦^{•} (U(s)) of Lemma 6.9 followed by the (sheafification) morphism pO𝒦^{•} → O𝒦^{•} on U(s). This construction is compatible with restriction and defines a morphism of ringed spaces

(6.2)  (ρ𝒦^{•} , r) : Spec(𝒦) ⟶ Spec^{h}(R𝒦^{•} ) .

Concentrating on the degree zero part (hence allowing s ∈ R_{𝒦} only), we obtain a morphism of ringed spaces (ρ_{𝒦} , r) : Spc(𝒦) → Spec(R_{𝒦} ).

Proposition 6.11. The above morphisms are morphisms of locally ringed spaces. Moreover, we have the following properties :

(a) For a prime 𝒫 ∈ Spc(𝒦), let 𝔭 = ρ_{𝒦} (𝒫) ∈ Spec(R_{𝒦} ) and 𝔭^{•} = ρ𝒦^{•} (𝒫) ∈ Spec^{h}(R𝒦^{•} ). The induced homomorphisms on stalks are the natural ones, (R_{𝒦} )_{𝔭} → R_{𝒦/𝒫} and (R𝒦^{•} )_{𝔭^{•}} → R𝒦/𝒫^{•} , given by localization at 𝒫.

(b) If ρ_{𝒦} or ρ𝒦^{•} is a homeomorphism on the underlying spaces, it is automatically an isomorphism of ringed spaces.

Proof. We prove the graded version. The ungraded version can be proved similarly. For readability, let us abbreviate ρ := ρ𝒦^{•} : Spc(𝒦) → Spec^{h}(R𝒦^{•} ) on spaces.

Localization q_{𝒫} : 𝒦 → 𝒦/𝒫 induces a ring homomorphism R𝒦^{•} → R𝒦/𝒫^{•} . By Theorem 5.3 (a), the preimage under this homomorphism of the maximal ideal of R𝒦/𝒫^{•} is precisely the ideal 𝔭^{•} = ρ(𝒫) ⊂ R𝒦^{•} . Hence, the induced morphism of local rings ℓ : (R𝒦^{•} )_{𝔭^{•}} → R𝒦/𝒫^{•} is local (the image of the maximal ideal of the source is contained in the maximal ideal of the target). We claim that this morphism ℓ : (R𝒦^{•} )_{𝔭^{•}} → R𝒦/𝒫^{•} is precisely the one induced stalkwise by the morphism of ringed spaces Spec^{•}(𝒦) → Spec^{h}(R𝒦^{•} ) above. Recall Proposition 6.5. An element \frac{f}{s} ∈ (R𝒦^{•} )_{𝔭^{•}} defines an element of End𝒦(U(s))^{•} (11_{U(s)}) whose class in 𝒦/𝒫 is precisely ℓ(\frac{f}{s}). This yields Part (a). Hence our morphism of ringed spaces is stalkwise local, so it is a morphism of locally ringed spaces.

Part (b) is then easy to deduce from Lemma 6.9. Indeed, assuming the two spaces are the same, the presheaf pO𝒦^{•} agrees with the sheaf OSpec^{h}(R_{𝒦}^{•} )^{•} on principal opens. Consequently its sheafification O𝒦^{•} is exactly OSpec^{h}(R_{𝒦}^{•} )^{•} as claimed. □

So much for structure sheaves. We now return to spectra.

7. Two criteria for surjectivity

We want to give conditions for the maps ρ𝒦^{•} : Spc(𝒦) → Spec^{h}(R𝒦^{•} ) and ρ_{𝒦} : Spc(𝒦) → Spec(R_{𝒦} ) to be surjective. First, we reduce to the case of R𝒦^{•} local.

Proposition 7.1. Let 𝔭^{•} ∈ Spec^{h}(R𝒦^{•} ) be a homogeneous prime. Consider the localization ℒ := 𝒦_{𝔭^{•}} = S_{𝔭^{•}}^{−1}𝒦 of Corollary 3.9. We know that Rℒ^{•} ≅ (R𝒦^{•} )_{𝔭^{•}} is a local graded ring. Suppose that the maximal homogeneous ideal of Rℒ^{•} belongs to the image of ρℒ^{•} . Then 𝔭^{•} belongs to the image of ρ𝒦^{•} .

Proof. By naturality of ρ^{•}, the following diagram commutes :

[figure: Spc(ℒ); _{ρ{• \atop ℒ}}; Spc(𝒦); _{ρ{• \atop 𝒦}}; Spec^{h}(Rℒ^{•} ) = Spec^{h}((R𝒦^{•} )_{𝔭^{•}}); Spec^{h}(R𝒦^{•} ) .]

Since the lower map sends the maximal ideal of Rℒ^{•} to 𝔭^{•}, we get the result. □

The above can be combined with the following :

Proposition 7.2. Let 𝒦 be a tensor triangulated category such that R𝒦^{•} is local with maximal homogeneous ideal 𝔪^{•}. Then the following conditions are equivalent :

(i) There exists a prime 𝒫 ∈ Spc(𝒦) such that ρ𝒦^{•} (𝒫) = 𝔪^{•}.

(ii) For every n ≥ 1 and every homogeneous f_{1}, …, f_{n} ∈ 𝔪^{•}, the product of their cones cone(f_{1}) ⊗ ⋯ ⊗ cone(f_{n}) is non-zero in 𝒦.

Proof. Suppose that ρ𝒦^{•} (𝒫) = 𝔪^{•} and let f_{1}, …, f_{n} ∈ R𝒦^{•} be homogeneous elements such that cone(f_{1})⊗⋯⊗cone(f_{n}) = 0 ∈ 𝒫. Since 𝒫 is prime, this implies cone(f_{i}) ∈ 𝒫 for some 1 ≤ i ≤ n. By Definition 5.1 this means f_{i} ∉ ρ𝒦^{•} (𝒫), hence f_{i} ∉ 𝔪^{•}.

Conversely, suppose that S := { cone(f_{1}) ⊗ ⋯ ⊗ cone(f_{n}) | n ≥ 1, f_{1}, …, f_{n} ∈ 𝔪^{•} homogeneous } ∪ {11} does not contain zero. This S is ⊗-multiplicative by construction. By Zorn, more precisely by [1, Lemma 2.2 applied to 𝒥 = 0], there exists a prime 𝒫 ∈ Spc(𝒦) such that 𝒫 ∩ S = ∅. Hence, for every homogeneous f ∈ 𝔪^{•}, we have cone(f) ∈ S, so cone(f) ∉ 𝒫, that is f ∈ ρ𝒦^{•} (𝒫), by Definition 5.1. We have shown 𝔪^{•} ⊂ ρ𝒦^{•} (𝒫) and this is enough since 𝔪^{•} is maximal. □

We now turn to the situation where R𝒦^{•} is coherent. We shall say that a graded ring is coherent if every finitely generated homogeneous ideal is finitely presented. Of course, a noetherian ring is coherent.

For this part, we choose the invertible object u = Σ(11). That is, we have

Hom𝒦^{•} (a, b) = Hom_{𝒦} (a, Σ^{•}b)

for any a, b ∈ 𝒦 and in particular, R𝒦^{•} = Hom_{𝒦} (11, Σ^{•}11).

Theorem 7.3. Let 𝒦 be a tensor triangulated category such that R𝒦^{•} is coherent (e.g. noetherian). Then the map ρ𝒦^{•} : Spc(𝒦) → Spec^{h}(R𝒦^{•} ) is surjective.

We prove this result below. By surjectivity of (−)^{0} : Spec^{h}(R𝒦^{•} ) → Spec(R_{𝒦} ), see Rem. 5.5, and since ρ_{𝒦} = (−)^{0} ∘ ρ𝒦^{•} , see Cor. 5.6 (b), we immediately have :

Corollary 7.4. Let 𝒦 be a tensor triangulated category such that R𝒦^{•} is coherent (e.g. noetherian). Then the map ρ_{𝒦} : Spc(𝒦) → Spec(R_{𝒦} ) is surjective. □

The key application of the coherence assumption is the following proposition. Recall that a graded module is coherent if it is finitely presented and if any finitely generated graded submodule is finitely presented as well.

Proposition 7.5. Suppose that R𝒦^{•} is coherent (e.g. noetherian) and local, with maximal homogeneous ideal 𝔪^{•}. Let f ∈ 𝔪^{•} be homogeneous and let a ∈ 𝒦 be an object such that Hom𝒦^{•} (11, a) is non-zero and coherent as graded R𝒦^{•} -module. Then Hom𝒦^{•} (11, a ⊗ cone(f)) is non-zero and coherent as well.

Proof. Let d ∈ ℤ be the degree of f. We have a distinguished triangle

[figure: a; f_{⇂a}; Σ^{d}(a); a ⊗ cone(f); Σa]

which induces a long exact sequence under the homological functor Hom_{𝒦} (11, −). This long exact sequence can be wrapped up (⊕) into a periodic exact sequence

(7.1)  [figure: Hom𝒦^{•} (11, a); ·f; (d); Hom𝒦^{•} (11, a); Hom𝒦^{•} (11, a ⊗ cone(f)); (1)]

of graded R𝒦^{•} -modules, where (d) and (1) indicate homogeneous R𝒦^{•} -linear homomorphisms of degree d and 1 respectively. (Here we used our choice of u = Σ(11).) If, ab absurdo, the lower module in (7.1) is zero, it means that f acts surjectively on the coherent, hence finitely generated R𝒦^{•} -module Hom𝒦^{•} (11, a). Since f ∈ 𝔪^{•}, by the graded version of Nakayama’s Lemma (see [20, 1.10]), we would have Hom𝒦^{•} (11, a) = 0, which is excluded. Hence the lower module is non-zero as claimed. The fact that it remains coherent is immediate : It fits in a short exact sequence with the cokernel and the kernel of the horizontal map ·f of (7.1), which are both coherent. See [10, Prop. 1.2 and comments before Prop. 1.1]. □

Proof of Theorem 7.3. Since the localization of a coherent graded ring remains coherent (extend [12, Thm. 3] to graded-commutative), we can use Propositions 7.1 and 7.2 to assume that R𝒦^{•} is local and coherent and reduce the proof to showing that cone(f_{1}) ⊗ ⋯ ⊗ cone(f_{n}) ≠ 0 for any homogeneous f_{1}, …, f_{n} ∈ 𝔪^{•}. By Proposition 7.5, we even have the stronger fact that

Hom𝒦^{•} (11, cone(f_{1}) ⊗ ⋯ ⊗ cone(f_{n})) ≠ 0

by induction on n ≥ 1. (Take of course a = 11 when n = 1.) □

* * *

We now prove surjectivity of ρ_{𝒦} : Spc(𝒦) → Spec(R_{𝒦} ) for another class of tensor triangulated categories.

Definition 7.6. Let us say that 𝒦 is connective if Hom_{𝒦} (11, Σ^{i}(11)) = 0 for all i > 0.

Remark 7.7. Maybe surprisingly, this property is self-dual : If 𝒦 is connective then 𝒦^{op} is connective as well. This comes from the fact that the suspension in 𝒦^{op} must be Σ^{−1}. In other words, the results below cannot be applied merely under the “complementary” assumption that Hom_{𝒦} (11, Σ^{i}(11)) = 0 for i < 0, by claiming duality. Applying the following results to 𝒦^{op} gives the same results.

As in Propositions 7.1 and 7.2, we have reductions to the local case :

Proposition 7.8. Let 𝔭 ∈ Spec(R_{𝒦} ) be a prime. Consider the localization ℒ := 𝒦_{𝔭} of Corollary 3.11. We know that the ring R_{ℒ} = (R_{𝒦} )_{𝔭} is local. Suppose that the maximal ideal of R_{ℒ} belongs to the image of ρ_{ℒ} . Then 𝔭 belongs to the image of ρ_{𝒦} .

Proof. Easy exercise on the naturality of ρ, as in the proof of Proposition 7.1. □

Proposition 7.9. Let 𝒦 be a tensor triangulated category whose central ring R_{𝒦} is local with maximal ideal 𝔪. Then the following conditions are equivalent :

(i) There exists a prime 𝒫 ∈ Spc(𝒦) such that ρ_{𝒦} (𝒫) = 𝔪.

(ii) For every n ≥ 1 and every f_{1}, …, f_{n} ∈ 𝔪, the product of their cones cone(f_{1}) ⊗ ⋯ ⊗ cone(f_{n}) is non-zero in 𝒦.

Proof. Same proof as Proposition 7.2. □

Lemma 7.10. Let 𝒦 be connective. Let f_{1} , …, f_{n} ∈ R_{𝒦} . Let c = cone(f_{1}) ⊗ ⋯ ⊗ cone(f_{n}) ∈ 𝒦 be the product of their cones. Then, we have :

(a) Hom_{𝒦} (11, Σ^{i}(c)) = 0 for all i > 0.

(b) There is a natural isomorphism R_{𝒦} /⟨f_{1} , …, f_{n}⟩ →^{∼} Hom_{𝒦} (11, c).

Proof. By induction on n ≥ 0. For n = 0, that is for c = 11, statement (a) is the connectivity assumption and (b) is trivial. Consider d = cone(f_{1})⊗⋯⊗cone(f_{n−1}) and the distinguished triangle

[figure: d; (f_{n})_{⇂d}; d; c; Σ(d)]

obtained by tensoring d with the triangle defining cone(f_{n}). Consider the long exact sequence obtained by applying Hom_{𝒦} (11, −) to the above triangle (and Prop. 2.2) :

⋯ → Hom_{𝒦} (11, Σ^{i}d) ⟶^{·f_{n}} Hom_{𝒦} (11, Σ^{i}d) → Hom_{𝒦} (11, Σ^{i}c) → Hom_{𝒦} (11, Σ^{i+1}d) → ⋯

for i ∈ ℤ. The induction hypothesis (for d) applied to this long exact sequence around i > 0 gives (a) ; applied around i = 0, it gives (b). □

Proposition 7.11. Let 𝒦 be a connective tensor triangulated category and let I ⊊ R_{𝒦} be a proper ideal. Then for any f_{1} , …, f_{n} ∈ I the product of their cones does not vanish : cone(f_{1}) ⊗ ⋯ ⊗ cone(f_{n}) ≠ 0.

Proof. By Lemma 7.10 (b), we have Hom_{𝒦} (11, cone(f_{1}) ⊗ ⋯ ⊗ cone(f_{n})) ≠ 0. □

Corollary 7.12. Let 𝒦 be a connective tensor triangulated category such that R_{𝒦} is local with maximal ideal 𝔪. Then there exists 𝒫 ∈ Spc(𝒦) such that ρ_{𝒦} (𝒫) = 𝔪.

Proof. It suffices to check condition (ii) of Proposition 7.9 and this follows from Proposition 7.11 applied to I = 𝔪. □

Theorem 7.13. Let 𝒦 be a connective tensor triangulated category (Def. 7.6). Then the map ρ_{𝒦} : Spc(𝒦) ⟶ Spec(R_{𝒦} ) is surjective.

Proof. This is immediate from Proposition 7.8 and Corollary 7.12 once we know that 𝒦_{𝔭} remains connective for any 𝔭 ∈ Spec(R_{𝒦} ). This is obvious by construction of 𝒦_{𝔭} = S_{𝔭}^{−1}𝒦 (see Cor. 3.11), since S_{𝔭} = R_{𝒦} ∖𝔭 sits in degree zero. □

Corollary 7.14. Let 𝒦 be a connective tensor triangulated category such that R_{𝒦}^{<0} = ⊕_{i<0}Hom_{𝒦} (11, Σ^{i}11) is a nil-ideal. (In more topological notations, we are assuming that any element f : Σ^{j}11 → 11 is zero for j < 0 and nilpotent for j > 0.) Then Spec^{h}(R𝒦^{•} ) ≅ Spec(R_{𝒦} ) via (−)^{0}. In particular, ρ𝒦^{•} : Spc(𝒦) → Spec^{h}(R𝒦^{•} ) coincides with ρ_{𝒦} and is surjective.

Proof. We have the commutative diagram of Corollary 5.6 (b) and we know by Theorem 7.13 that ρ_{𝒦} is surjective. So, it suffices to show that (−)^{0} : Spec^{h}(R𝒦^{•} ) → Spec(R_{𝒦} ), 𝔭^{•} ↦ 𝔭^{0} is a bijection. Since R_{𝒦}^{<0} is a homogeneous nil-ideal, it belongs to all homogeneous prime ideals. Hence, we have 𝔭^{•} = R_{𝒦}^{<0} ⊕𝔭^{0} for all 𝔭^{•} ∈ Spec^{h}(R𝒦^{•} ) and the bijection follows easily. □

8. Some examples, old and new

Proposition 8.1. Let A be a commutative ring and let 𝒦 = K^{b}(A – proj) be the tensor triangulated category of perfect complexes. Then ρ_{𝒦} : Spc(K^{b}(A – proj)) → Spec(A) is a homeomorphism, inverse to the one given in Example 4.4.

Proof. Recall that the homeomorphism Spec(A) →^{∼} Spc(K^{b}(A – proj)) is given by 𝔭 ↦ 𝒫(𝔭) = { M_{•} ∈ K^{b}(A – proj) | (M_{•})_{𝔭} = 0 in K^{b}(A_{𝔭} – proj) }. Let 𝔭 ∈ Spec(A). We need to show that ρ_{𝒦} (𝒫(𝔭)) = 𝔭. This is easy : For f : 11 → 11, i.e. for f ∈ A, we have by Definition of ρ_{𝒦} in (5.1) that f ∈ ρ_{𝒦} (𝒫(𝔭)) ⇔ cone(f) ∉ 𝒫(𝔭) ⇔ the image of f : 11 → 11 is not invertible in K^{b}(A_{𝔭} – proj) ⇔ the image of f in A_{𝔭} is not invertible ⇔ f ∈ 𝔭. □

Remark 8.2. For a non-affine scheme X, this result fails for 𝒦 = D^{perf}(X), simply because there aren’t always enough global sections, as can be checked for X = ℙ_{k}^{n}. However, in this case, or more generally if X = Proj(A^{•}) is a projective scheme for A^{•} such that the natural map A^{d} → Γ(X, O(d)) is an isomorphism for large d (see for instance [13, Ex. 2.5.9, p. 125]) and such that A^{0} = Γ(X, O_{X}), then Spec^{h}(A^{•}) = Spec^{h}(Γ(X, O(•))). In this case, we can choose u = O(1) as invertible object in 𝒦 and the map ρ𝒦^{•} : Spc(D^{perf}(X)) ⟶ Spec^{h}(A^{•}) induces a homeomorphism onto X. This is similar to the above, using [1, Cor. 5.6], and is left to the reader.

I’m thankful to Ch. Walter for the following example of non-surjectivity of ρ_{𝒦} .

Example 8.3. Let (A, 𝔪) be a regular local ring and U = Spec(A) ∖ {𝔪} be its punctured spectrum. Let 𝒦 = D^{perf}(U). When d = dim_{Krull}(A) ≥ 2, we have R_{𝒦} = A and Spec(R_{𝒦} ) = Spec(A) which is strictly bigger than U = Im(ρ_{𝒦} ). In fact, R𝒦^{•} ≃ A ⊕ R_{𝒦}^{d−1} where R_{𝒦}^{d−1} = H^{d−1}(U, O_{U}) is an injective envelope of the residue field A/𝔪 and this non-connective R𝒦^{•} isn’t coherent either.

Remark 8.4. The above proof of Proposition 8.1 uses the classification of thick triangulated ⊗-ideals, hidden in [1, Cor. 5.6]. Yet our Theorem 7.13 proves surjectivity of ρ_{𝒦} : Spc(𝒦) ⟶ Spec(A) without the classification. It is therefore interesting to find a direct proof of injectivity, for this would yield a new proof of classification, via [1, Thm. 4.10]. Actually such direct proofs of the injectivity of ρ_{𝒦} do exist in this case. There is a rather elementary such proof, using only perfect complexes, but it is a little long to include here. Another proof, due to Amnon Neeman, uses bigger categories but can be sketched as follows. The general machinery developed above allows us to reduce to the case of (A, 𝔪) local and to show that the preimage in Spc(𝒦) of the maximal ideal consists only of the ideal zero. To do that, it suffices to show that for any non-zero object X ∈ 𝒦 = D^{perf}(A) there exists f_{1}, …, f_{r} ∈ 𝔪 such that Kos(f) ∈ ⟨X⟩, where Kos(f) := cone(f_{1})⊗⋯⊗cone(f_{r}). One can reduce to A noetherian local. Then in the big derived category D(A), the complex X ⊗^{L} κ is a finite sum of shifts of the residue field κ := A/𝔪. Then, using the t-structure on D(A), one can show by induction that any bounded complex with finite length homology belongs to ⟨κ⟩ and hence to the localizing triangulated subcategory ⟨X⟩_{loc} generated by X in D(A). In particular, if f = (f_{1}, …, f_{r}) is a sequence generating 𝔪, we have Kos(f) ∈ ⟨X⟩_{loc} ∩𝒦. Since the object X ∈ 𝒦 = (D(A))^{c} is compact in the compactly generated D(A), we can use a general result of triangulated categories, see Neeman [27, Thm. 4.4.9], which implies here that ⟨X⟩_{loc} ∩ 𝒦 = ⟨X⟩ and we are done.

* * *

Let us switch to modular representation theory. The following result is new although known to some experts (announced for instance in talks by Julia Pevtsova, joint work with Paul Smith; see also Benson et al. [7]) :

Proposition 8.5. Let G be a finite group (scheme) and k be a field. Consider 𝒦 = D^{b}(kG – mod) with ⊗ = ⊗_{k}. Its graded central ring R𝒦^{•} (Def. 3.2) is H^{•}(G, k) and the map ρ𝒦^{•} : Spc(D^{b}(kG – mod)) → Spec^{h}(H^{•}(G, k)) is a homeomorphism.

Proof. Let 𝒥 = K^{b}(kG – proj) ⊂ 𝒦. By Rickard [30], the quotient 𝒦/𝒥 is equivalent to the stable category kG – stab, whose spectrum is described in [1, Cor. 5.10] as

φ : Proj(H^{•}(G, k)) ⟶^{∼} Spc(kG – stab)
𝔭^{•}  ⟼ 𝒫(𝔭^{•}) := { M | Ann_{H^{•}(G,k)}(H^{•}(G, M)) ⊄ 𝔭^{•} } .

We then have the following picture :

[figure: Spc(𝒦/𝒥) = Spc(kG – stab); Spc(𝒦); _{ρ𝒦^{•}}; {0}; Proj(H^{•}(G, k)); ≃; φ; Spec^{h}(H^{•}(G, k)); {H^{+}(G, k)}]

We claim that this diagram commutes. Indeed, the graded ring H^{•}(G, k) is local with maximal ideal H^{+}(G, k). We claim that 𝒦 is local as well. This is obvious since we have a forgetful ⊗-functor 𝒦 → D^{b}(k) whose kernel is zero (exactness of a complex of kG-modules being a property of the underlying complex of vector spaces). So 0 ∈ Spc(𝒦) and 𝒦 is local by Proposition 4.2. It is obvious that ρ𝒦^{•} (0) = H^{+}(G, k) since no homogeneous element of positive degree can be invertible. Then the righthand square of the above diagram commutes. To check that the left-hand square also commutes, consider a homogeneous prime 𝔭^{•} ∈ Proj(H^{•}(G, k)) and a homogeneous element f ∈ H^{d}(G, k) for some d ≥ 0. Let q : 𝒦 ⟶ kG – stab be the localization functor. Then, f ∈ ρ𝒦^{•} (q^{−1}(𝒫(𝔭^{•}))) ⟺ cone(f) ∉ q^{−1}(𝒫(𝔭^{•})) ⟺ q(cone(f)) ∉ 𝒫(𝔭^{•}) ⟺ Ann_{H^{•}(G,k)}(H^{•}(G, q(cone(f)))) ⊂ 𝔭^{•} ⟺ 𝔭^{•} ∈ V (Ann_{H^{•}(G,k)}(H^{•}(G, q(cone(f))))) = V (f). The last equality comes from the fact that q(cone(f)) = cone(q(f)) is the suspension of the so-called Carlson module L_{f}, whose support is known to be V (f) ; see [8, Prop. II.5.9.1, p. 186]. In short, we have f ∈ ρ𝒦^{•} (q^{−1}(𝒫(𝔭^{•}))) ⟺ f ∈ 𝔭^{•}, which proves ρ𝒦^{•} (q^{−1}(φ(𝔭^{•}))) = 𝔭^{•}, as wanted.

Finally, we claim that Spc(𝒦) = Spc(𝒦/𝒥) ∪ {0}. This follows from the fact that every non-zero ⊗-ideal 𝒫 ⊂ 𝒦 contains kG. Indeed, for any M_{•} ∈ D^{b}(kG – mod), we have by Frobenius reciprocity that kG ⊗ M_{•} ≃ ind_{e}^{G} ∘ res_{e}^{G}(M_{•}) and since res_{e}^{G}(M_{•}) ∈ D^{b}(k – mod), the complex res_{e}^{G}(M_{•}) is isomorphic to a complex of vector spaces with zero differentials. So, if M_{•} ≠ 0 then kG⊗M_{•} has a direct summand isomorphic to kG. Hence if 0 ≠ 𝒫 ∈ Spc(𝒦) then 𝒥 = K^{b}(kG – proj) = ⟨kG⟩ ⊂ 𝒫, that is, 𝒫 belongs to the image of Spc(𝒦/𝒥) ↪ Spc(𝒦) ; see Remark 1.4.

We have shown that the above diagram commutes, that the top row describes a partition Spc(𝒦) = Spc(𝒦/𝒥) ∪ {0} formed by the closed point {0} and its open complement Spc(𝒦/𝒥). Since ρ𝒦^{•} is a bijection on both parts, we see that ρ𝒦^{•} is bijective. Since ρ𝒦^{•} is continous, it only remains to check that it is open. This is very easy. The only open subset of Spc(𝒦) which contains {0} is the whole Spc(𝒦). Any other open of Spc(𝒦) will be in Spc(𝒦/𝒥), hence will have an open image in Proj(H^{•}(G, k)), which is itself open in Spec^{h}(H^{•}(G, k)). □

Remark 8.6. In this example of 𝒦 = D^{b}(kG – mod), like for the case of K^{b}(A – proj), the general results of Section 7 actually provide a proof of the surjectivity of ρ𝒦^{•} : Spc(D^{b}(kG – mod)) ⟶ Spec^{h}(H^{•}(G, k)) without using classification results. Indeed, H^{•}(G, k) is noetherian by the Venkov-Evens Theorem (see [8, Thm. II.4.2.1, p. 126]) so we can apply Theorem 7.3. It would be interesting to prove injectivity directly, in the spirit of Remark 8.4. Again, such a direct proof would provide a new proof of classification, via general tensor triangular theory [1, Thm. 4.10].

* * *

Section 7 also applies to the following more analytic example.

Example 8.7. Let G be a compact (Lie) group and KK^{G} the tensor triangulated category which has separable G-C^{∗}-algebras as objects, equivariant Kasparov’s KKtheory as morphisms and tensor given by the minimal tensor product with diagonal G-action. Details can be found in Meyer’s survey [23] or in Dell’Ambrogio’s thesis [11, Chapter 5]. Here, the ring R_{KK^{G}} = KK_{0}^{G}(ℂ, ℂ) = K_{0}^{G}(ℂ) =: R(G) is the Grothendieck group of continuous complex representations of G, following Segal [31]. Dell’Ambrogio proves by other techniques that ρ_{𝒦^{G}} : Spc(𝒦^{G}) → Spec(R(G)) is split surjective when G is finite and where 𝒦^{G} is the triangulated subcategory of KK^{G} generated by the unit. He also proves that the map ρ_{𝒦^{G}} : Spc(𝒦^{G}) → Spec(R(G)) is a homeomorphism for G trivial and conjectures that this holds for all finite groups. With Corollary 7.4 we can generalize the surjectivity from finite groups to any compact group. This is also a new result.

Corollary 8.8. Let G be a compact Lie group. With the above notation, the continuous map ρ_{KK^{G}} : Spc(KK^{G}) → Spec(R(G)) is surjective. Consequently, so is ρ_{𝒦^{G}} : Spc(𝒦^{G}) → Spec(R(G)).

Proof. By Bott periodicity Spec^{h}(RKK^{G}^{•}) = Spec(R_{KK^{G}}). Also, KK_{1}^{G}(ℂ, ℂ) = K_{1}^{G}(ℂ) = R(G) ⊗ K_{1}(ℂ) = 0 ; see [28, Rem. 2.8.5]. So, R_{KK^{G}}^{i} = 0 for i odd. It is a result of Segal [31, Cor. 3.3] that R(G) is noetherian for any compact Lie group and consequently RKK^{G}^{•} is noetherian as well. Corollary 7.4 then gives the result. The last statement follows since ρ_{KK^{G}} factors via ρ_{𝒦^{G}} , using 𝒦^{G} ↪ KK^{G}. □

9. Spectra of topological spectra

Let us consider the tensor triangulated category 𝒦 = SH^{fin}, the topological stable homotopy category of finite spectra ; see [22]. The tensor product is induced by the smash product and the unit 11 = S^{0} is the sphere spectrum. We have R_{𝒦} = ℤ and R𝒦^{•} is connective with nilpotent R_{𝒦}^{<0}. So, by Corollary 7.14, we can focus on ρ_{𝒦} .

Let us denote the set of prime numbers by ℙ ⊂ ℕ = {0, 1, 2, 3, …}. Let p ∈ ℙ and let SH_{(p)}^{fin} be the localization of SH^{fin} at p, as in Corollary 3.11 applied to 𝔭 = pℤ.

We recall the following major result from Hopkins and Smith [15] ; see also [29].

Theorem 9.1 (Hopkins-Smith). Let p ∈ ℙ be a prime. Recall that Morava Ktheories define a collection of functors K_{p,n} : SH_{(p)}^{fin} ⟶ 𝒱_{p,n} into ⊗-categories 𝒱_{p,n} of free graded modules, with index n ∈ ℕ. All functors K_{p,n} satisfy a Künneth formula, i.e. they are ⊗-functors. Let 𝒞_{p,n} := Ker(K_{p,n}). Then,

0 =: 𝒞_{p,∞} ⊊ ⋯ ⊊ 𝒞_{p,n} ⊊ 𝒞_{p,n−1} ⊊ ⋯ ⊊ 𝒞_{p,1} ⊊ 𝒞_{p,0} ⊊ SH_{(p)}^{fin}

are all the thick triangulated ⊗-ideals of SH_{(p)}^{fin}. Moreover, ∩_{n∈ℕ} 𝒞_{p,n} = 0.

Corollary 9.2. Every proper thick triangulated ⊗-ideal of SH_{(p)}^{fin} is prime. In particular, SH_{(p)}^{fin} is a local tensor triangulated category (see Def. 4.1).

Proof. Indeed, since K_{p,n} satisfies a Künneth formula and since the graded tensor product V_{1} ˆ⊗V_{2} of two free graded modules cannot vanish unless one of the factors is zero, Ker(K_{p,n}) is prime for every n ≥ 0. Hence so is their intersection 𝒞_{p,∞} = 0. Then SH_{(p)}^{fin} is local by Proposition 7.8 (iii). □

Example 9.3. For n = 0, the Morava K-theory K_{p,0} is rational (singular) homology : K_{p,0} = H_{∗}(−, ℚ) and 𝒞_{p,0} is therefore the subcategory of torsion finite spectra localized at p. Note that the preimage of 𝒞_{p,0} in SH^{fin} under the localization functor SH^{fin} ⟶ SH_{(p)}^{fin} is the subcategory of torsion finite spectra, SH_{tor}^{fin} , that is, the kernel of rational homology. In particular, it is independent of p.

Proposition 9.4. Let p ∈ ℙ be a prime. The central ring R_{SH}{fin \atop (p)} = ℤ_{(p)} is the ring

of integers localized at p. The map ρ := ρ_{SH}{fin \atop (p)} : Spc(SH_{(p)}^{fin}) ⟶ Spec(ℤ_{(p)}) maps 𝒞_{p,n} to the maximal ideal pℤ_{(p)} for every n > 0 and maps 𝒞_{p,0} to 0.

Proof. By definition, see (5.1), the prime ideal ρ(𝒞_{p,0}) consists of all f ∈ R_{SH}{fin \atop (p)} =

ℤ_{(p)} whose cone does not belong to 𝒞_{p,0}, i.e. is not torsion. So, ρ(𝒞_{p,0}) = { f ∈ ℤ_{(p)} | f is not an isomorphism rationally } = 0.

As ρ is inclusion-reversing (Rem. 5.2), and as pℤ_{(p)} is maximal, in order to prove ρ(𝒞_{p,n}) = pℤ_{(p)} for all n ≥ 1, it suffices to prove it for n = 1. The Morava K-theory K_{p,1} is a direct summand of mod p topological K-theory. Hence K_{p,1}(S^{0} ⟶^{p} S^{0}) is the zero map. Therefore, K_{p,1}(cone(p)) = 11 ⊕ 11(1) in 𝒱_{p,1} and in any case it is not zero. So cone(p) ∉ Ker(K_{p,1}) = 𝒞_{p,1}, i.e. p ∈ ρ(𝒞_{p,1}) by definition ; see (5.1). Therefore pℤ_{(p)} ⊂ ρ(𝒞_{p,1}) and we have equality since pℤ_{(p)} is the maximal ideal. □

Corollary 9.5. The spectrum of SH^{fin} is the following topological space. We also describe the surjective continuous map ρ := ρ_{SH}_{fin} : Spc(SH^{fin}) ⟶ Spec(ℤ).

(9.1)  [figure: 𝒫_{2,∞}; 𝒫_{3,∞}; ⋯; 𝒫_{p,∞}; ⋯; Spc(SH^{fin}) =; ρ_{SH_{fin}}; ⋮; ⋮; ⋮; 𝒫_{2,n+1}; 𝒫_{3,n+1}; ⋯; 𝒫_{p,n+1}; ⋯; 𝒫_{2,n}; 𝒫_{3,n}; ⋯; 𝒫_{p,n}; ⋯; ⋮; ⋮; ⋮; 𝒫_{2,1}; 𝒫_{3,1}; ⋯; 𝒫_{p,1}; ⋯; _{SH{fin \atop tor}}; Spec(ℤ) =; 2ℤ; 3ℤ; ⋯; pℤ; ⋯; (0)]

In the above picture, a line indicates that the higher prime is in the closure of the lower one (specialization). We have more precisely :

(a) The preimage of the dense point of Spec(ℤ) has only one point, ρ^{−1}(0) = {SH_{tor}^{fin}}, namely the subcategory of torsion spectra SH_{tor}^{fin} = Ker(H_{∗}(−, ℚ)).

(b) For each prime p ∈ ℙ the preimage ρ^{−1}(pℤ) consists of the countable collection of 𝒫_{p,n} := q_{p}^{−1}(𝒞_{p,n}) where q_{p} : SH^{fin} → SH_{(p)}^{fin} is the localization functor and where 1 ≤ n ≤ ∞. When n < ∞, we have 𝒫_{p,n} = Ker(K_{p,n} ∘ q_{p}) = { x ∈ SH^{fin} | K_{p,n}(q_{p}(x)) = 0 }, where K_{p,n} is the n^{th} Morava K-theory at p. Finally, 𝒫_{p,∞} = Ker(q_{p}) = { x ∈ SH^{fin} | q_{p}(x) = 0 in SH_{(p)}^{fin} }.

(c) SH_{tor}^{fin} is the unique dense point of Spc(SH^{fin}). For each p ∈ ℙ and each 1 ≤ n ≤ ∞, the closure of \{𝒫_{p,n}\} is \overline{\{𝒫_{p,n}\}} = \{ 𝒫_{p,m} | n ≤ m ≤ ∞ \}. The closed points of Spc(SH^{fin}) are exactly the 𝒫_{p,∞} = Ker(q_{p}) for all p ∈ ℙ.

(d) For an object x ∈ SH^{fin}, we have supp(x) = Spc(SH^{fin}) if and only if x is not torsion. When x ∈ SH_{tor}^{fin}, then

supp(x) = ⋃_{{p∈ℙ \atop s.t. q_{p}(x)≠0}} \overline{\{𝒫_{p,n(x,p)}\}}

where the union involves only finitely many p ∈ ℙ and where the integer 1 ≤ n(x, p) < ∞, sometimes called the “type” of q_{p}(x), see [29, Def. 1.5.3], is the smallest integer n such that q_{p}(x) ∉ 𝒫_{p,n}.

(e) The proper non-empty closed subsets of Spc(SH^{fin}) are all possible finite unions of the following subsets : \overline{\{𝒫_{p,n}\}} = \{ 𝒫_{p,m} | n ≤ m ≤ ∞ \} for all p ∈ ℙ and all 1 ≤ n ≤ ∞.

Proof. By Thm. 7.13, the map ρ : Spc(SH^{fin}) → Spec(ℤ) is surjective. So, to describe the set Spc(SH^{fin}), it suffices to describe the preimages ρ^{−1}(0) and ρ^{−1}(pℤ) for p ∈ ℙ. Let us start with the preimage of the generic point (0) ∈ Spec(ℤ). By Cor. 5.6 (c) for S = ℤ ∖ (0), we see that the preimage of (0) is the image of Spc(SH_{(0)}^{fin}) in Spc(SH^{fin}), that is, the image of Spec(q_{0} : SH^{fin} → SH_{(0)}^{fin}). But SH_{(0)}^{fin} = SH^{fin} ⊗ℚ is just D^{b}(ℚ) and q_{0} : SH^{fin} → SH^{fin} ⊗ℚ ≃ D^{b}(ℚ) is (total) rational homology (see [22, Thm. 8.1.7, p. 113]). Since D^{b}(ℚ) has only {0} as prime ideal, the preimage ρ^{−1}(0) consists only of Ker(q_{0}) = SH_{tor}^{fin}. This proves (a).

Let now p ∈ ℙ be a prime. By Cor. 5.6 (c) again but this time for S = ℤ ∖ pℤ, we see that the fiber above pℤ is the image in Spc(SH^{fin}) of the part of Spc(SH_{(p)}^{fin}) which maps to pℤ_{(p)} under ρ_{SH}{fin \atop (p)}. By the Hopkins-Smith Theorem 9.1 and by Proposition 9.4, we know that this fiber is exactly the collection 𝒞_{p,n} for 1 ≤ n ≤ ∞, whose images in SH^{fin} are the announced 𝒫_{p,n} for 1 ≤ n ≤ ∞. This proves (b).

The inclusion 𝒫_{p,n} ⊂ SH_{tor}^{fin} is immediate from SH_{tor}^{fin} = q_{p}^{−1}(𝒞_{p,0}) and from the inclusions 𝒞_{p,n} ⊂ 𝒞_{p,0}. Then (c) is easy to check from the definitions, recalling that in any tensor triangulated category 𝒦, the closure of a point 𝒫 ∈ Spc(𝒦) is described by \overline{\{𝒫\}} = \{ 𝒬 ∈ Spc(𝒦) | 𝒬 ⊂ 𝒫 \}; see [1, Prop. 2.9].

Now, let x ∈ SH^{fin}. If x is not torsion, i.e. x ∉ SH_{tor}^{fin}, then for every p ∈ ℙ and n ∈ ℕ ∪ {∞} we have x ∉ 𝒫_{p,n} as well, since 𝒫_{p,n} ⊂ SH_{tor}^{fin}. Hence by Def. 1.3, we

have supp(x) =^{(def)} { 𝒫 | x ∉ 𝒫 } = Spc(SH^{fin}). On the other hand, if x is torsion, then it is well-known that q_{p}(x) ≠ 0 for only finitely many p ∈ ℙ. (By Prop. 3.7, there exists m ∈ ℤ, m ≠ 0, such that m_{⇂x} = 0 and then q_{p}(x) = 0 for all p prime to m.) On the other hand, since 0 = 𝒞_{p,∞} = ∩_{n≥0}𝒞_{p,n}, when q_{p}(x) ≠ 0 we have an integer n = n(x, p) ∈ ℕ, n ≥ 1, such that q_{p}(x) ∈ 𝒞_{p,n−1} ∖ 𝒞_{p,n}. Hence x ∈ 𝒫_{p,n(p)−1} ∖ 𝒫_{p,n(p)}. This proves (d).

Finally, by definition of the topology of Spc(𝒦) (Def. 1.3) any closed subset is an intersection of supports, supp(x), for x ∈ 𝒦. It is easy to check that they are the announced ones. The only subtlety is the appearance of the closed point {𝒫_{p,∞}} which is not the support of any object but which is ∩_{n≥1}\overline{\{𝒫_{p,n}\}}. □

Remark 9.6. We observe that Spc(SH^{fin}) is not a noetherian topological space since, for any p ∈ ℙ, the non-empty family of closed subsets \{ \overline{\{𝒫_{p,n}\}} | 0 ≤ n < ∞ \} has no minimal element, its intersection being {𝒫_{p,∞}}. In particular, Spc(SH^{fin}) cannot be realized as the homogeneous spectrum Spec^{h}(R^{•}) of a noetherian graded ring R^{•}.

Proposition 9.7. The locally ringed space Spec(SH^{fin}) is not a scheme. Nor is any of the local ones Spec(SH_{(p)}^{fin}), for any prime p.

Proof. Let p ∈ ℙ be a prime. We know by Corollary 9.2 that SH_{(p)}^{fin} is a local category and its spectrum Spc(SH_{(p)}^{fin}) is a local topological space. Hence, if the ringed space Spec(SH_{(p)}^{fin}) was a scheme, it would be the spectrum of the local ring of global sections of its structure sheaf, which is just ℤ_{(p)}. Obviously, the spectrum of SH_{(p)}^{fin} is much bigger than Spec(ℤ_{(p)}) = {(0), (p)}. The global result follows, as a scheme remains a scheme locally. □

Remark 9.8. The category SH_{(p)}^{fin} only has trivial invertible objects, so even the twisted version of ρ𝒦^{•} cannot be injective. In conclusion, the triangular spectrum Spc(𝒦) considerably differs from the algebraic spectra Spec(R_{𝒦} ) or Spec^{h}(R𝒦^{•} ) in general. This of course indicates that Spc(𝒦) is a better, finer invariant than both Spec^{h}(R𝒦^{•} ) and Spec(R_{𝒦} ).

10. Spectra of motivic spectra

For this section, let 𝒦 = 𝒦^{𝔸^{1}} := (SH_{F}^{𝔸^{1}})^{c} be the subcategory of compact objects in the Morel-Voevodsky stable 𝔸^{1}-homotopy category SH_{F}^{𝔸^{1}} over a field F ; see [33, 24]. This category can be defined over a more general basis but the computation of End_{𝒦} (11), which we use below, is only performed for a field so far. The category 𝒦^{𝔸^{1}} is connective in the sense of Definition 7.6, i.e. Hom_{𝒦} (11, Σ^{i}(11)) = 0 for i > 0, see [33, Thm. 4.14], or [25, § 6], or [26, Cor. 4.3.3]. The beautiful computation of End_{𝒦} (11) is due to Morel, see [25, Thm. 6.2.1] :

(10.1)  End_{𝒦^{𝔸^{1}}} (11) ≅ GW(F) .

This holds at least when the field F is perfect and of characteristic different from two, which we assume from now on. The ring GW(F) is the Grothendieck-Witt ring of quadratic forms over F, that is, the group completion of the abelian semi-ring of isometry classes of non-degenerate quadratic forms over F, with orthogonal sum as addition and tensor product as multiplication. This ring is classical and its quotient W(F) = GW(F)/h by the 2-dimensional hyperbolic plane h = ⟨1, −1⟩ is the even more classical Witt ring ; see Lam [18]. Applying our Theorem 7.13, we get :

Corollary 10.1. The map ρ_{𝒦} : Spc(𝒦^{𝔸^{1}}) → Spec(GW(F)) is surjective. □

Remark 10.2. The latter spectrum Spec(GW(F)) is well-known, at least for the Witt group part. Let us remind the reader.

There is always a ring homomorphism, rank : GW(F) → ℤ, induced by the rank of quadratic forms. For instance, rank : GW(F) ≅ ℤ is an isomorphism for F quadratically closed, like F = ℂ. On the other hand, if there exists an ordering α on F (equivalently, if −1 is not a sum of squares in F) there is another ring homomorphism, sgn_{α} : GW(F) → ℤ, induced by the signature of quadratic forms (for a ∈ F^{×}, we have sgn_{α}(⟨a⟩) = 1 if a > 0 with respect to the ordering α and sgn_{α}(⟨a⟩) = −1 otherwise). Using this, one can show for instance that Spec(GW(ℝ)) is equal to two copies of Spec(ℤ) attached together at the point 2ℤ.

More generally, the above surjective ring homomorphisms, rank : GW(F) → ℤ and sgn_{α} : GW(F) → ℤ, yield copies of Spec(ℤ) as closed subsets of Spec(GW(F)).

(a) For q ∈ Spec(ℤ) define the ideal 𝔮_{q} := rank^{−1}(q). The ideal I := 𝔮_{2} = { x ∈ GW(F) | rank(x) is even } is called the fundamental ideal.

(b) For every ordering α on F (if any) and for every prime p ∈ Spec(ℤ), define

the ideal 𝔭_{α,p} := (sgn_{α})^{−1}(p). One has I = 𝔭_{α,2} for every α and there is no other redundancy in the above list :

Spec(GW(F)) = { 𝔮_{q} | q ∈ Spec(ℤ) } ∪ ⋃_{α ordering} { 𝔭_{α,p} | p ∈ Spec(ℤ) ∖ {2} } .

Here is a picture of Spec(GW(F)) :

(10.2)  [figure: □; □; 𝔮_{5}; □; 𝔮_{q}; ∘; 𝔮_{0}; □; 𝔮_{3}; □; 𝔭_{α,5}; □; □; ⋯; □; 𝔭_{α,3}; ∘; 𝔭_{α,0}; □; 𝔭_{α,p}; ⋯; □; □^{I}; _{...}; ⋯; □; □; ⋯; □; 𝔭_{α′,3}; ∘; 𝔭_{α′,0}; □; 𝔭_{α′,p′}; □; ∘; □; □; 𝔭_{α′,5}; □; □; □]

The lines ∘ □ and ∘ □ indicate inclusions ∘ ⊂ □, that is, □ is in the closure of ∘. These are the only inclusions. The dotted part, beyond I, appears only when F has at least one ordering otherwise Spec(GW(F)) = Spec(ℤ). This copy of Spec(ℤ), coming via the rank, is the only one which is always present.

To prove this, first recall the spectrum of W(F) = GW(F)/h where h = ⟨1, −1⟩, from Harrison and Lorenz-Leicht [19]. It can be found in [18, § VIII.7, p. 277] and corresponds exactly to the dotted part of Figure (10.2), including I. It just remains to describe Spec(GW(F)[1/h]) and this is immediate from GW(F)[1/h] ≃ ℤ[1/2]. The latter is induced by the rank and uses the equation ⟨a⟩ ⊗ h = ⟨a, −a⟩ = ⟨1, −1⟩ = h in GW(F) (which holds in any characteristic). Note that the induced topology on the subspace { 𝔭_{α,0} | α ordering on F } is homeomorphic to the socalled Harrison topology on the set of orderings ; see [18, Cor. 7.6, p. 279]. Being a Boolean space, it is not discrete unless it is finite.

Remark 10.3. We now wonder what lives “above” this space Spec(GW(F)), that is, how much bigger is the spectrum of our 𝒦^{𝔸^{1}} = (SH_{F}^{𝔸^{1}})^{c}, compared to Spec(GW(F)). The first answer is coming from SH^{fin}. Let us assume that F ⊂ ℂ has a complex embedding. Then we have two ⊗-triangulated functors

SH^{fin} ⟶ 𝒦^{𝔸^{1}} →^{Re} SH^{fin} ,

the first one being the constant one and the second being complex realization. Their composition is the identity. So, there is a copy of Spc(SH^{fin}), as described in Section 9, embedded inside Spc(𝒦^{𝔸^{1}}) via Spc(Re). On the level of central rings, this corresponds to the copy of Spec(ℤ) in Spec(GW(F)) given by the rank. This shows that the fibers of ρ_{𝒦} are at least as big as in topology.

As should be expected, there is even more here than in topology :

Proposition 10.4. The split inclusion Spc(Re) : Spc(SH^{fin}) → Spc(𝒦^{𝔸^{1}}) is not surjective. Its image is contained in the proper closed subset supp(cone(η)), where η ∈ Hom_{𝒦} (11, 𝔾_{m}^{⊗−1}) is the algebraic Hopf map. This closed subset is also supp(ℙ^{2}).

Proof. The algebraic Hopf map is given by the natural map 𝔸_{F}^{2} ∖{0} → ℙ_{F}^{1} and the facts that 𝔸_{F}^{2} ∖ {0} ≅ Σ(𝔾_{m}^{⊗2}) and ℙ_{F}^{1} ≅ Σ(𝔾_{m}). See for instance Morel [25, § 6.1]. Its complex realization Re(η) is the topological Hopf map, which is ⊗-nilpotent. However, Morel proves that the algebraic η is not ; see [25, Cor. 6.4.5, p. 258]. We then use two general results :

(a) For any ⊗-triangulated functor F : 𝒦 → ℒ, if we denote by φ = Spc(F) : Spc(ℒ) → Spc(𝒦) the induced map and if x ∈ 𝒦, then supp(F (x)) = φ^{−1}(supp(x)) in Spc(ℒ) ; see [1, Prop. 3.6]. Hence φ(supp(F (x))) ⊂ supp(x).

(b) By Theorem 2.15 above, for any ⊗-invertible object u ∈ 𝒦, a morphism f : 11 → u is ⊗-nilpotent if and only if ⟨cone(f)⟩ = 𝒦. The latter is equivalent to supp(cone(f)) = Spc(𝒦) ; see [1, Cor. 2.5].

In our case, (b) and the nilpotence of Re(η) imply that supp(Re(cone(η))) = Spc(SH^{fin}). Hence by (a), we get Im(Spc(Re)) ⊂ supp(cone(η)), as announced. The latter is smaller than Spc(𝒦^{𝔸^{1}}) by (b) and the non-nilpotence of the algebraic η. Finally cone(η) = ℙ^{2} can be seen from the Mayer-Vietoris square

\xymatrix{ & 𝔸^{2} − \{0\} \ar[dl]_{η} \ar[d] \ar[r] & 𝔸^{2} ≃ ∗ \ar[d] \\ ℙ^{1} & ℙ^{2} − \{0\} \ar[l]_{≃} \ar[r] & ℙ^{2} .}

(All schemes are pointed, e.g. at the points corresponding to [1 : 1 : 1] ∈ ℙ^{2}.) □

Remark 10.5. There is also a continuous map ρ𝒦^{•} : Spc(𝒦^{𝔸^{1}}) → Spec^{h}(R𝒦^{𝔸^{1}} ,u^{•} ) obtained from § 5 with u = 𝔾_{m} as ⊗-invertible. Morel proves that R𝒦^{𝔸^{1}} ,u^{•} ≅ KMW_{•}(F) is the Milnor-Witt K-theory of the field F ; see more in [25, § 6, p. 251].

Further determination of Spc ((SH_{F}^{𝔸^{1}})^{c}), including the question of the surjectivity of this ρ𝒦^{•} , is a challenging endeavor in which the reader is welcome to engage.

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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