GLUING TECHNIQUES IN TRIANGULAR GEOMETRY

PAUL BALMER AND GIORDANO FAVI

Abstract. We discuss gluing of objects and gluing of morphisms in triangulated categories. We illustrate the results by producing, among other things, a Mayer-Vietoris exact sequence involving Picard groups.

Contents

Introduction 1 1. Recalling tensor triangular geometry 4 2. Formal Mayer-Vietoris covers 8 3. Mayer-Vietoris long exact sequence 10 4. Gluing of objects 14 5. Mayer-Vietoris in tensor triangular geometry 16 6. Picard groups 19 7. Excision 23 References 24

Introduction

Tensor triangular geometry is the geometry of tensor triangulated categories. Heuristically, this contains at least algebraic geometry and the geometry of modular representation theory but it also appears in many other areas of mathematics, as recalled in the introduction of [1].

We will denote by 𝒦 a triangulated category (with suspension T : 𝒦 →^{∼} 𝒦) equipped with a tensor product, that is, an exact symmetric monoidal structure ⊗ : 𝒦 × 𝒦 ⟶ 𝒦; see more in Section 1. Two key examples to keep in mind appear respectively in algebraic geometry, as 𝒦 = D^{perf}(X), the derived category of perfect complexes over a quasi-compact and quasi-separated scheme X (e.g. a noetherian scheme), and in modular representation theory, as 𝒦 = kG - stab, the stable category of finite dimensional representations modulo projective ones, for G a finite group and k a field of characteristic p > 0, typically dividing the order of the group.

In [1], the spectrum, Spc(𝒦), of such categories is introduced. It is the universal topological space in which one can define supports, supp(a) ⊂ Spc(𝒦), for objects a ∈ 𝒦 in a reasonable way. In the above two examples, this spectrum is respectively isomorphic to the scheme X itself and to the projective support variety Proj H^{•}(G, k).

Date: December 7, 2006.

1991 Mathematics Subject Classification. 18E30, 20C20.

Key words and phrases. Mayer-Vietoris, gluing, triangulated category, triangular geometry.

Research supported by Swiss National Science Foundation, grant 620-66065.

One fundamental construction of [1] is the presheaf of triangulated categories, U ↦ 𝒦(U), which associates to an open U ⊂ Spc(𝒦) a tensor triangulated category 𝒦(U) defined as follows. Consider Z = Spc(𝒦)∖U the closed complement of U and consider the thick subcategory 𝒦_{Z} ⊂ 𝒦 of those objects a ∈ 𝒦 with supp(a) ⊂ Z, i.e. those objects which ought to disappear on U. Then, the category

𝒦(U) := { \atop 𝒦/𝒦_{Z}}

is defined as the idempotent completion of the Verdier quotient 𝒦/𝒦_{Z}. Localization 𝒦 ↠ 𝒦/𝒦_{Z} followed by idempotent completion 𝒦/𝒦_{Z} ↪ 𝒦(U) yields a restriction functor ρ_{U} : 𝒦 → 𝒦(U). In the scheme example, it is an important theorem of Thomason [10] that for a quasi-compact open U ⊂ X and for 𝒦 = D^{perf}(X), the above 𝒦(U) is equivalent to D^{perf}(U). This is one reason for working with idempotent complete categories. Another reason is a key result of [2] which says that if 𝒦 is idempotent complete and if the support of an object of 𝒦 decomposes into two connected components then the object itself decomposes into two direct summands accordingly, see Theorem 1.8 below.

The present paper deals with the following type of questions. Suppose that Spc(𝒦) is covered by two quasi-compact open subsets Spc(𝒦) = U_{1} ∪ U_{2} and consider the commutative diagram of restriction functors :

(1)  \xymatrix{ & 𝒦 \ar[r] \ar[dl] & 𝒦(U_{1}) =: 𝒦_{1} \ar[d] \\ 𝒦_{2} := 𝒦(U_{2}) \ar[rr] &  & 𝒦(U_{1} ∩ U_{2}) =: 𝒦_{12} .}

Question : Is the global category 𝒦 obtained by “gluing” 𝒦_{1} and 𝒦_{2} over 𝒦_{12} ?

This is a very natural question but it is known to be tricky, already in algebraic geometry. Indeed, it is easy to find non-zero morphisms f : a → b in 𝒦 = D^{perf}(X) such that f_{|U_{1}} = 0 and f_{|U_{2}} = 0 for an open cover X = U_{1} ∪ U_{2} . Over X = ℙ_{k}^{1} an example is the morphism f : O(2) → T (O) which is the third one in the exact triangle associated to the non-split Koszul exact sequence O ↣ O(1)⊕O(1) ↠ O(2) ; take for U_{1} and U_{2} two affine subsets. (For an exact sequence of vector bundles E′ ↣ E ↠ E′′ over a scheme X, the corresponding morphism f : E′′ → T (E′) is zero in D^{perf}(X) if and only if the sequence splits.) This example also shows that the phenomenon is not pathological but observable in very common situations.

Still, the problem admits the following solution, to be found in our main results :

Theorem (Mayer-Vietoris for morphisms, see Cor. 5.8). In the above situation (1), given two objects a, b ∈ 𝒦, there exists a long exact sequence :

⋯ Hom_{𝒦_{12}} (T a, b) →^{∂} Hom_{𝒦} (a, b) → Hom_{𝒦_{1}} (a, b) ⊕ Hom_{𝒦_{2}} (a, b) → Hom_{𝒦_{12}} (a, b) →^{∂} ⋯

The connecting homomorphism ∂ : Hom_{𝒦_{12}} (T a, b) → Hom_{𝒦} (a, b) is defined in Construction 3.1. The other homomorphisms are the obvious restrictions and differences thereof.

Theorem (Gluing of two objects, see Cor. 5.10). In the above situation (1), given two objects a_{1} ∈ 𝒦_{1} and a_{2} ∈ 𝒦_{2} and an isomorphism σ : a_{1} →^{∼} a_{2} in 𝒦_{12} , there exists an object a ∈ 𝒦 which becomes isomorphic to a_{i} in 𝒦_{i} for i = 1, 2. Moreover, this gluing is unique up to (possibly non-unique) isomorphism.

We can extend the above result to three open subsets and three objects, at the cost of possibly losing uniqueness of the gluing :

Corollary (Gluing of three objects, see Cor. 5.11). Let Spc(𝒦) = U_{1} ∪ U_{2} ∪ U_{3} be a cover by quasi-compact open subsets. Consider three objects a_{i} ∈ 𝒦(U_{i}) for i = 1, 2, 3 and three isomorphisms σ_{ij} : a_{j} →^{∼} a_{i} in 𝒦(U_{i} ∩ U_{j}) for 1 ≤ i < j ≤ 3. Suppose that the cocycle relation σ_{12} ∘ σ_{23} = σ_{13} is satisfied in 𝒦(U_{1} ∩ U_{2} ∩ U_{3}). Then there exists an object a ∈ 𝒦, isomorphic to a_{i} in 𝒦(U_{i}) for i = 1, 2, 3.

In general, we do not know if this gluing is possible with more than three open subsets. Nevertheless, in Theorem 5.13, we give elementary conditions under which the gluing is possible for arbitrary covers.

Of course, there is a gigantic literature on Mayer-Vietoris long exact sequences. We do not doubt that the reader considers such ideas as basic mathematics and does not expect us to survey former Mayer-Vietoris results here. Let us simply mention, in the framework of algebraic geometry, that our results recover Thomason’s Mayer-Vietoris long exact sequences for D^{perf}(X), see [9], which themselves already generalized Mayer-Vietoris long exact sequences in Zariski cohomology.

We then apply the above main results to obtain an exact sequence involving Picard groups. For us, the Picard group, Pic(𝒦), is the set of isomorphism classes of invertible objects in 𝒦, with the tensor product as multiplication. In algebraic geometry, Pic(D^{perf}(X)) is well known to be the usual Picard group of X up to possible shifts, see Prop. 6.4. On the other hand, Pic(kG - stab) is nothing but the group of endotrivial representations, which is one of the fundamental invariants of modular representation theory. In the next statement, we denote by 𝔾_{m}(𝒦) = Hom_{𝒦} (1, 1)^{×} the abelian group of automorphisms of the ⊗-unit object 1 ∈ 𝒦.

Theorem (Mayer-Vietoris for Picard groups, see Thm. 6.7). Let Spc(𝒦) = U_{1} ∪ U_{2} with U_{i} quasi-compact. See (1). Then there is half a long exact sequence :

[figure: ⋯; Hom_{𝒦(U_{1}∩U_{2})} (T 1, 1); 1+∂; 1+∂; 𝔾_{m}(𝒦); 𝔾_{m}(𝒦(U_{1})) ⊕ 𝔾_{m}(𝒦(U_{2})); 𝔾_{m}(𝒦(U_{1} ∩ U_{2})); δ; δ; Pic(𝒦); Pic(𝒦(U_{1})) ⊕ Pic(𝒦(U_{2})); Pic(𝒦(U_{1} ∩ U_{2})) .]

To the left, we have the Mayer-Vietoris long exact sequence, the homomorphism ∂ is as before and the non-labelled morphisms are again the obvious restrictions and (multiplicative) differences of restrictions. The new homomorphism

δ : 𝔾_{m}(𝒦(U_{1} ∩ U_{2})) → Pic(𝒦)

assigns to a unit σ ∈ 𝔾_{m}(𝒦(U_{1} ∩ U_{2})) the invertible object obtained by gluing two copies of the objects 1 ∈ 𝒦(U_{1}) and 1 ∈ 𝒦(U_{2}) along σ : 1 →^{∼} 1 in 𝒦(U_{1} ∩ U_{2}).

It would be very interesting to continue this sequence to the right, say, with Brauer groups of Azumaya algebras. Although this is still work in progress, the authors do not know yet whether such an extension is possible in general. Neither do we know what the Brauer group of 𝒦 = kG - stab should be, for instance.

In fact, in modular representation theory, applying the above results to 𝒦 = kG - stab gives us a way to construct endotrivial kG-modules from any Čech 𝔾_{m}- cocycle over the projective support variety Proj H^{•}(G, k), as long as the involved cover has at most three open pieces. In particular, the map δ of the last result might be of interest to representation theorists and we do not know if it has been studied, even in special cases. Dave Benson and Jon Carlson suggested a possible link with the recent article [5]. This is investigated in a subsequent paper [3].

Using the conditional gluing of more than three objects, we obtain the following result (Thm. 6.8), which relates invertible modules over the spectrum Spc(𝒦) and invertible objects in 𝒦. See more comments in Remark 6.9.

Theorem. Suppose that Hom_{𝒦(U)} (T 1, 1) = 0 for every quasi-compact open subsets U ⊂ Spc(𝒦). Then, gluing induces an injective homomorphism from the first Čech cohomology of Spc(𝒦) with coefficients in 𝔾_{m} into the Picard group of 𝒦

Ȟ^{1}(Spc(𝒦), 𝔾_{m}) ↪ Pic(𝒦) .

For completeness, we give in Section 7 the following variant of Mayer-Vietoris :

Theorem (Excision, see Thm. 7.1). Let Y ⊂ U ⊂ Spc(𝒦). Assume that Y is closed with quasi-compact complement and that U is open and quasi-compact. Then the restriction functor 𝒦 → 𝒦(U) induces an equivalence between the subcategories of objects supported on Y, that is, 𝒦_{Y} ⟶^{∼} 𝒦(U)_{Y}.

The referee of the first version of this article suggested we give more formal proofs, postponing as long as possible the assumption that 𝒦 carries a tensor product. We therefore rewrote the paper with the “formal Mayer-Vietoris” language of Sections 2, 3 and 4. These sections can be read without reference to tensor triangular geometry, that is, without assuming 𝒦 has a tensor structure. Some readers will benefit from this gain of generality, despite a little loss in geometric intuition. Then, tensor triangular geometry really enters the game in Section 5.

1. Recalling tensor triangular geometry

We survey the main concepts and results of [1] and [2]. Standard notions about triangulated categories can be found in Verdier [11] or Neeman [8].

Definitions 1.1. A tensor triangulated category (𝒦, ⊗, 1) is an essentially small triangulated category 𝒦 with a symmetric monoidal structure ⊗ : 𝒦 × 𝒦 ⟶ 𝒦, (a, b) ↦ a⊗b. We have in particular a⊗b ≅ b⊗a and 1⊗a ≅ a for the unit 1 ∈ 𝒦. We assume moreover that the functors a ⊗ − and − ⊗ b are exact for every a, b ∈ 𝒦 and that the usual diagram

\xymatrix{T (a) ⊗ T (b) \ar[r]^{≅} \ar[d]_{≅} & T (T (a) ⊗ b) \ar[d]^{≅} \\ T (a ⊗ T (b)) \ar[r]_{≅} & T^{2}(a ⊗ b)}

anti-commutes. We use T : 𝒦 →^{∼} 𝒦 to denote the translation (suspension). A prime ideal 𝒫 ⊊ 𝒦 is a proper subcategory such that (1)-(4) below hold true :

(1) 𝒫 is a full triangulated subcategory, i.e. 0 ∈ 𝒫, T (𝒫) = 𝒫 and if a, b ∈ 𝒫 and if a → b → c → T (a) is a distinguished triangle in 𝒦 then c ∈ 𝒫;

(2) 𝒫 is thick, i.e. if a ⊕ b ∈ 𝒫 then a, b ∈ 𝒫; (3) 𝒫 is a ⊗-ideal, i.e. if a ∈ 𝒫 then a ⊗ b ∈ 𝒫 for all b ∈ 𝒦;

(4) 𝒫 is prime, i.e. if a ⊗ b ∈ 𝒫 then a ∈ 𝒫 or b ∈ 𝒫.

A subcategory 𝒥 ⊂ 𝒦 satisfying (1), (2) and (3) is called a thick ⊗-ideal.

The spectrum Spc(𝒦) is the set of primes 𝒫 ⊂ 𝒦. The support of an object a ∈ 𝒦 is defined as the subset supp(a) = {𝒫 ∈ Spc(𝒦) | a ∉ 𝒫} ⊂ Spc(𝒦). The complements U(a) = {𝒫 ∈ Spc(𝒦) | a ∈ 𝒫} of these supports form a basis {U(a)}_{a∈𝒦} of the so-called Zariski topology on the spectrum.

Theorem 1.2 ([1] Thm. 3.2). Let 𝒦 be a tensor triangulated category. We have (i) supp(0) = ∅ and supp(1) = Spc(𝒦). (ii) supp(a ⊕ b) = supp(a) ∪ supp(b). (iii) supp(T a) = supp(a). (iv) supp(a) ⊂ supp(b) ∪ supp(c) for any distinguished a → b → c → T (a).

(v) supp(a ⊗ b) = supp(a) ∩ supp(b). Moreover, (Spc(𝒦), supp) is universal for these properties.

Notation 1.3. Let Y ⊂ Spc(𝒦). We denote by 𝒦_{Y} the full subcategory 𝒦_{Y} := {a ∈ 𝒦 | supp(a) ⊂ Y } of those objects supported on Y.

Definition 1.4. We call a tensor triangulated category (𝒦, ⊗, 1) strongly closed if there exists a bi-exact functor hom : 𝒦^{op} × 𝒦 ⟶ 𝒦 with natural isomorphisms

(2)  Hom_{𝒦} (a ⊗ b, c) ≅ Hom_{𝒦} (a, hom(b, c))

and such that all objects are strongly dualizable, i.e. the natural morphism

(3)  D(a) ⊗ b →^{∼} hom(a, b)

is an isomorphism for all a, b ∈ 𝒦, where we denote by D(a) the dual D(a) := hom(a, 1) of an object a ∈ 𝒦. More details can be found in [7, App. A], for instance. It follows from (3) that D^{2}(a) ≅ a for all a ∈ 𝒦; see for instance [7, Thm. A.2.5 (b)].

Proposition 1.5 ([2] Cor. 2.5). Let 𝒦 be a strongly closed tensor triangulated category and let a ∈ 𝒦 be an object. Then supp(a) = ∅ if and only if a = 0.

Proposition 1.6 ([2] Cor. 2.8). Let 𝒦 be a strongly closed tensor triangulated category. Suppose that the supports of two objects do not meet : supp(a)∩supp(b) = ∅. Then there is no non-trivial morphism between them : Hom_{𝒦} (a, b) = 0.

Proposition 1.7. Let 𝒦 be a strongly closed tensor triangulated category. A morphism f : a → b in 𝒦 is an isomorphism if and only if it is an isomorphism in 𝒦/𝒫 for all 𝒫 ∈ Spc(𝒦).

Proof. This easily follows from the fact that a morphism f in a triangulated category is an isomorphism if and only if cone(f) = 0. Thus if f is an isomorphism in 𝒦/𝒫 we have that cone(f) ∈ 𝒫. If this is true for all 𝒫 ∈ Spc(𝒦) we have that supp(cone(f)) = ∅ which implies that cone(f) = 0 by Proposition 1.5. □

Theorem 1.8 ([2] Thm. 2.11). Let 𝒦 be a strongly closed tensor triangulated category. Assume that 𝒦 is idempotent complete. Then, if the support of an object a ∈ 𝒦 can be decomposed as supp(a) = Y_{1} ∪ Y_{2} for disjoint closed subsets Y_{1}, Y_{2} ⊂ Spc(𝒦), with each open complement Spc(𝒦) ∖ Y_{i} quasi-compact, then the object itself can be decomposed as a direct sum a ≃ a_{1} ⊕ a_{2} with supp(a_{i}) = Y_{i} for i = 1, 2.

Remark 1.9. Recall that an additive category 𝒦 is idempotent complete (or pseudoabelian or karoubian) if all idempotents of all objects split, that is, if e ∈ Hom_{𝒦} (a, a) with e^{2} = e then the object a decomposes as a direct sum a ≃ a′ ⊕ a′′ on which e becomes ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ), that is, a ≅ Im(e)⊕Ker(e). One can always “idempotent complete” an additive category 𝒦 ↪ 𝒦̃. If 𝒦 is triangulated, its idempotent completion 𝒦̃ inherits a unique triangulation such that 𝒦 ↪ 𝒦̃ is exact, see more in [4].

The rest of the paper heavily relies on the next set of definitions :

Definitions 1.10. Let 𝒦 be an idempotent complete strongly closed tensor triangulated category. Let U be a quasi-compact open subset of Spc(𝒦), and let us denote by Z = Spc(𝒦)∖U its closed complement. We denote by ℒ(U) = 𝒦/𝒦_{Z} the Verdier localization with respect to 𝒦_{Z} (which can be realized by keeping the same objects as 𝒦 and by inverting all morphisms whose cone belongs to 𝒦_{Z} , by means of calculus of fractions). We denote by 𝒦(U) = { \atop ℒ(U)} its idempotent completion. We have a fully faithful cofinal morphism ℒ(U) ↪ 𝒦(U). (Some authors say dense instead of cofinal, like in [9]. This means that every object a of the big category is a direct summand of an object of the small one, for instance a ⊕ T a. See [4].)

For U = Spc(𝒦), by Proposition 1.5, we have ℒ(U) = 𝒦 = 𝒦̃ = 𝒦(U) since we assume 𝒦 idempotent complete. If U ⊂ V we denote by ρ_{U,V} : ℒ(V) → ℒ(U) the localization functor and we also denote by

ρ_{U,V} : 𝒦(V) → 𝒦(U)

the induced functor, that we call the restriction functor from V to U. When V = Spc(𝒦), we simply write ρ_{U} : 𝒦 → 𝒦(U) for ρ_{U,V}.

For two objects a, b of 𝒦 we denote by

Hom_{𝒦(U)} (a, b) := Hom_{ℒ(U)} (ρ_{U}(a), ρ_{U}(b)) = Hom_{𝒦(U)} (ρ_{U}(a), ρ_{U}(b))

the set of morphisms between ρ_{U}(a) and ρ_{U}(b) in ℒ(U) or equivalently in its idempotent completion 𝒦(U) ; for simplicity, we might speak of morphisms between a and b in 𝒦(U), or simply morphisms between a and b over U.

Proposition 1.11. For U ⊂ Spc(𝒦) quasi-compact and open, the restriction functor ρ_{U} : 𝒦 → 𝒦(U) induces a homeomorphism Spc(𝒦(U)) →^{∼} U, under which supp(ρ_{U}(a)) = U ∩ supp(a), for any object a ∈ 𝒦.

Proof. In fact, by [1] Cor. 3.14, Spc(𝒦(U)) = Spc(𝒦/𝒦_{Z}) and by loc. cit. Prop. 3.11, the localization functor induces a homeomorphism between Spc(𝒦/𝒦_{Z}) and the subspace V := {𝒫 ∈ Spc(𝒦) | 𝒦_{Z} ⊂ 𝒫} of Spc(𝒦). So, it suffices to check that V = U. The last equality supp(ρ_{U}(a)) = U ∩ supp(a) will then be a general fact about the functor Spc(−), see loc. cit. Prop. 3.6.

Let 𝒫 ∈ Spc(𝒦). By the classification of thick ⊗-ideals, loc. cit. Thm. 4.10, we have 𝒫 ∈ V, i.e. 𝒦_{Z} ⊂ 𝒫, if and only if Z = supp(𝒦_{Z}) ⊂ supp(𝒫) =^{def.} ∪_{a∈𝒫} supp(a). By taking complements, this is equivalent to ∩_{a∈𝒫} U(a) ⊂ U, where U(a) = Spc(𝒦) ∖ supp(a) = {𝒬 ∈ Spc(𝒦) | a ∈ 𝒬}. Tautologically, ∩_{a∈𝒫} U(a) = {𝒬 ∈ Spc(𝒦) | 𝒫 ⊂ 𝒬}. The latter set is contained in U if and only if 𝒫 ∈ U : one direction is trivial and the other one uses that Z is specialization closed, see loc. cit. Prop. 2.9. So, 𝒫 ∈ V if and only if 𝒫 ∈ U, as was left to check. □

Remark 1.12. The above result cannot hold without assuming U quasi-compact since Spc(𝒦) is quasi-compact for any 𝒦. It is used above to insure Z = supp(𝒦_{Z}).

* * *

We end this Section with some general facts about triangulated categories.

Lemma 1.13. Let 𝒦 be a triangulated category. Then for every distinguished triangle in which one object decomposes into two direct summands

[figure: a; s; b; ^{( \begin{matrix}α \\ β \end{matrix} )}; c_{1} ⊕ c_{2}; ^{(^{γ δ} )}; T a]

there exist two objects, d and e, and four distinguished triangles :

[figure: d; α_{0}; b; α; c_{1}; α_{2}; T d; a; δ_{0}; d; δ_{1}; c_{2}; δ; T a]
[figure: e; β_{0}; b; β; c_{2}; β_{2}; T e; a; γ_{0}; e; γ_{1}; c_{1}; γ; T a]

where α_{2} = T δ_{0} γ, δ_{1} = β α_{0} , β_{2} = T γ_{0} δ and γ_{1} = α β_{0}. Moreover, the morphism s factors as s = α_{0} δ_{0} = β_{0} γ_{0}.

In particular, we have cone(α) ≃ cone(δ) and cone(β) ≃ cone(γ).

Proof. We will prove the existence of the first two triangles, the other two are obtained symmetrically (c_{1} ⊕ c_{2} ≃ c_{2} ⊕ c_{1}). The triangles are obtained by applying the Octahedron Axiom to the equality (1 0)( \begin{matrix}α \\ β \end{matrix} ) = α as displayed below:

[figure: c_{1}; α_{2}; 0; c_{1} ⊕ c_{2}; (1 0); (γ δ); T c_{2}; _{( \begin{matrix}0 \\ 1 \end{matrix} )}; ·; T δ; ·; b; _{( \begin{matrix}α \\ β \end{matrix} )}; α; T d; −T α_{0}; ·; −T δ_{1}; T a; −T s; ·; T δ_{0}]

□

Definition 1.14. We say that a commutative square as follows is a weak push-out

\xymatrix{a \ar[r]^{f} \ar[d]_{g} & b \ar[d]^{h} \\ c \ar[r]_{k} & d}

if there exists a distinguished triangle [figure: a; ^{( \begin{matrix}f \\ g \end{matrix} )}; b ⊕ c; ^{(^{−h k} )}; d; ℓ; T a] for some morphism ℓ : d → T (a). This is justified since (d, h, k) satisfies the universal property of the push-out of f and g but without uniqueness of the factorization. Since such a square is then also a weak pull-back, we call it weakly bicartesian.

2. Formal Mayer-Vietoris covers

The referee sagaciously suggested that we make clear how our proofs only depend on Theorem 1.8, which is indeed the keystone to Mayer-Vietoris. In this logic, we now give an abstract version of our results, only assuming the conclusions of Theorem 1.8, without necessarily carrying the tensor structure around.

Definition 2.1. Let 𝒦 be an idempotent complete triangulated category. A formal Mayer-Vietoris cover of 𝒦 is the data of two thick triangulated subcategories 𝒥_{1} and 𝒥_{2} of 𝒦 such that Hom_{𝒦} (c_{1}, c_{2}) = Hom_{𝒦} (c_{2}, c_{1}) = 0 for every pair of objects c_{1} ∈ 𝒥_{1} and c_{2} ∈ 𝒥_{2}. This implies in particular that 𝒥_{1} ∩ 𝒥_{2} = 0.

Remark 2.2. The subcategories 𝒥_{1} and 𝒥_{2} do not really “cover” 𝒦 (e.g., 𝒥_{1} = 𝒥_{2} = 0 define a “formal cover”). In fact, the cover is rather realized by the corresponding restrictions (see Def. 1.10), i.e. the localization functors 𝒦 → 𝒦/𝒥_{1} and 𝒦 → 𝒦/𝒥_{2} followed by the idempotent completions. See Construction 2.4.

The following is a proto-Theorem 1.8.

Lemma 2.3. Given a formal Mayer-Vietoris cover of 𝒦, the full subcategory 𝒥_{1}⊕𝒥_{2} of 𝒦 on the objects of the form c_{1} ⊕ c_{2} where c_{1} ∈ 𝒥_{1} and c_{2} ∈ 𝒥_{2}, is a thick triangulated subcategory of 𝒦.

Proof. See the proof of [2, Thm. 2.11]. We sketch it for the convenience of the reader. To see that 𝒥_{1} ⊕ 𝒥_{2} is a triangulated subcategory, it suffices to show that the cone of any morphism f : c_{1} ⊕ c_{2} → d_{1} ⊕ d_{2} with c_{i} , d_{i} ∈ 𝒥_{i} also belongs to 𝒥_{1} ⊕ 𝒥_{2}. This follows from the fact that f must be diagonal by assumption. To see that 𝒥_{1} ⊕ 𝒥_{2} is a thick subcategory of 𝒦, take a direct summand of an object c_{1} ⊕ c_{2} ∈ 𝒥_{1} ⊕ 𝒥_{2} that we can describe as the image, Im(e), of some (projection) idempotent e = e^{2} ∈ End_{𝒦} (c_{1} ⊕ c_{2}). By assumption, e must be diagonal, that is, e = ( \begin{matrix}e_{1} & 0 \\ 0 & e_{2} \end{matrix} ) with e_{i} = e_{i}^{2} on c_{i}. Since 𝒦 is idempotent complete, c_{i} ≅ Im(e_{i}) ⊕ Ker(e_{i}) for i = 1, 2. Since 𝒥_{i} is thick, Im(e_{i}) ∈ 𝒥_{i} and our direct summand Im(e) ≅ Im(e_{1}) ⊕ Im(e_{2}) belongs to 𝒥_{1} ⊕ 𝒥_{2} as was to be shown. □

Note that the above result is wrong if 𝒦 is not idempotent complete, as shown in the counter-example [2, Ex. 2.13], because 𝒥_{1} ⊕ 𝒥_{2} needs not be thick in 𝒦.

Construction 2.4. Given a formal Mayer-Vietoris cover of 𝒦 (Def. 2.1), let us define 𝒥_{12} := 𝒥_{1} ⊕ 𝒥_{2}. We can consider the Verdier quotients ℒ_{1} = 𝒦/𝒥_{1} , ℒ_{2} = 𝒦/𝒥_{2} , ℒ_{12} = 𝒦/𝒥_{12} and the respective idempotent completions 𝒦_{1} = ℒ̃_{1} , 𝒦_{2} = ℒ̃_{2} and 𝒦_{12} = ℒ̃_{12} (see Rem. 1.9). We have the following commutative diagram :

(4)  [figure: 𝒦; ρ_{1}; ρ_{2}; ρ_{1}; ρ_{2}; 𝒦_{1}; ρ_{12}; ℒ_{1}; ρ_{12}; ℒ_{2}; ρ_{21}; ℒ_{12}; 𝒦_{2}; ρ_{21}; 𝒦_{12} ,]

where ↠ indicates a Verdier localization and ↪ a fully faithful cofinal embedding.

Remark 2.5. Of course, the outer square of (4) should be thought of as the formal version of the square (1) of the Introduction.

Definition 2.6. For any i ∈ {1, 2, 12}, we say that a morphism s : a → b in 𝒦 is a 𝒦_{i}-isomorphism if s becomes an isomorphism in the localization 𝒦_{i} (or equivalently in ℒ_{i}). The morphism s is a 𝒦_{i}-isomorphism if and only if cone(s) ∈ 𝒥_{i}.

Remark 2.7. In a formal Mayer-Vietoris situation, a morphism which is both a 𝒦_{1}- and a 𝒦_{2}-isomorphism must be an isomorphism since its cone belongs to 𝒥_{1}∩𝒥_{2} = 0.

Notation 2.8. Like in Definitions 1.10, for two objects a, b ∈ 𝒦, we abbreviate

Hom_{𝒦_{1}} (a, b) :=Hom_{𝒦_{1}} (ρ_{1}(a), ρ_{1}(b)) = Hom_{ℒ_{1}} (ρ_{1}(a), ρ_{1}(b))
Hom_{𝒦_{2}} (a, b) :=Hom_{𝒦_{2}} (ρ_{2}(a), ρ_{2}(b)) = Hom_{ℒ_{2}} (ρ_{2}(a), ρ_{2}(b))
Hom_{𝒦_{12}} (a, b) :=Hom_{𝒦_{12}} (ρ_{12}ρ_{1}(a), ρ_{12}ρ_{1}(b)) = Hom_{ℒ_{12}} (ρ_{12}ρ_{1}(a), ρ_{12}ρ_{1}(b))

the groups of homomorphisms in the various localizations, see (4). More generally, we tend to drop the mention of the restriction functors when it improves readability.

Definition 2.9. Again, as in Definitions 1.10, for objects a, b ∈ 𝒦 and for i ∈ {1, 2, 12}, we refer to a morphism in Hom_{𝒦_{i}} (a, b) as a morphism between a and b in 𝒦_{i}. By construction of Verdier localizations, any morphism a → b in 𝒦_{i} is

the equivalence class of a (left) fraction f s^{−1} = ( \xymatrix{a & x \ar[l]_{s} \ar[r]^{f} & b}) in 𝒦, where s : x → a is a 𝒦_{i}-isomorphism and f : x → b is any morphism in 𝒦. The equivalence relation on such fractions is given by amplification f s^{−1} = (ft) (st)^{−1} for any 𝒦_{i}-isomorphism t : y → x.

Lemma 2.10. In a formal Mayer-Vietoris situation (4), every 𝒦_{12}-isomorphism s : a → b can be factored as s = s_{1} ∘ s_{2} where s_{i} is a 𝒦_{i}-isomorphism for i = 1, 2.

Proof. By hypothesis we have that cone(s) ∈ 𝒥_{12} = 𝒥_{1} ⊕ 𝒥_{2}. Thus cone(s) may be written as cone(s) ≃ c_{1} ⊕ c_{2} where c_{i} ∈ 𝒦_{i}. Now use Lemma 1.13 which tells that s = α_{0} δ_{0} and that cone(α_{0}) ≃ c_{1} and cone(δ_{0}) ≃ c_{2}. □

Remark 2.11. One can actually prove that the above factorization is essentially unique but we shall not use this fact below.

We now give two useful lemmas about weakly bicartesian squares (Def. 1.14).

Lemma 2.12. Consider a weakly bicartesian square in 𝒦:

\xymatrix{a \ar[r]^{f} \ar[d]_{g} & b \ar[d]^{h} \\ c \ar[r]_{k} & d .}

Then f is a 𝒦_{i}-isomorphism if and only if k is.

Proof. There exists a distinguished triangle \xymatrix{T^{−1}d \ar[r] & a \ar[r]^{^{( \begin{matrix}f \\ g \end{matrix} )}} & b ⊕ c \ar[r]^{^{(^{−h k} )}} & d .} By Lemma 1.13, cone(f) ≃ cone(k) and the result follows. □

Lemma 2.13. In a formal Mayer-Vietoris situation (4), consider a commutative square in 𝒦:

\xymatrix{a \ar[r]^{s_{1}} \ar[d]_{t_{2}} & b \ar[d]^{s_{2}} \\ c \ar[r]_{t_{1}} & d .}

Assume that s_{i} and t_{i} are 𝒦_{i}-isomorphisms for i = 1, 2. Then the square is weakly bicartesian.

Proof. Consider the weak push-out (e, u_{1}, u_{2}) of s_{1} and t_{2} and some morphism v : e → d induced by weak push-out of s_{2} and t_{1} :

\xymatrix{a \ar[r]^{s_{1}} \ar[d]_{t_{2}} & b \ar[d]_{u_{2}} \ar[dr]^{s_{2}} \\ c \ar[r]^{u_{1}} \ar[rr]_{t_{1}} & e \ar@{-->}[r]^{v} & d .}

By Lemma 2.12, u_{i} is a 𝒦_{i}-isomorphism for i = 1, 2. By 2-out-of-3, v is both a 𝒦_{1}- and a 𝒦_{2}-isomorphism, hence an isomorphism (see Remark 2.7). □

3. Mayer-Vietoris long exact sequence

Construction 3.1. Consider a formal Mayer-Vietoris situation (Def. 2.1) and two objects a, b ∈ 𝒦. We define a homomorphism

∂ : Hom_{𝒦_{12}} (a, b) ⟶ Hom_{𝒦} (a, T (b))
g ⟼ ∂(g)

as follows. Let f s^{−1} = ( \xymatrix{a & x \ar[l]_{s} \ar[r]^{f} & b}) be a fraction representing a morphism g ∈ Hom_{𝒦_{12}} (a, b). The cone of the 𝒦_{12}-isomorphism s belongs to 𝒥_{12} = 𝒥_{1} ⊕ 𝒥_{2}. So, we can chose a distinguished triangle

[figure: x; s; a; ^{( \begin{matrix}α \\ β \end{matrix} )}; c_{1} ⊕ c_{2}; ^{(^{γ δ} )}; T x]

where c_{i} ∈ 𝒥_{i} for i = 1, 2. Note that γ α + δ β = 0, that is, γ α = −δ β. Define now

(5)  ∂(g) := T (f) ∘ γ α = −T (f) ∘ δ β .

This is a morphism in Hom_{𝒦} (a, T (b)), independent of the choices, see Theorem 3.5.

Remark 3.2. Since T is an equivalence, the above construction also induces :

\xymatrix{Hom_{𝒦_{12}} (T (a), b) \ar[r]^{∂} & Hom_{𝒦} (T (a), T (b)) \ar[r]_{≅}^{T^{−1}} & Hom_{𝒦} (a, b)}

and we also denote this homomorphism by ∂, since no confusion should follow.

Explicitly, for a morphism g = ( \xymatrix{T (a) & x \ar[l]_{s} \ar[r]^{f} & b}) in Hom_{𝒦_{12}} (T (a), b), choose

any distinguished triangle [figure: a; ^{( \begin{matrix}α \\ β \end{matrix} )}; c_{1} ⊕ c_{2}; ^{(^{γ δ} )}; x; s; T (a)] with c_{i} ∈ 𝒥_{i} for i = 1, 2 ; then we have ∂(g) = f ∘ γ α = −f ∘ δ β ∈ Hom_{𝒦} (a, b).

Remark 3.3. Note that the above definition of ∂ is asymmetric in the subcategories 𝒥_{1} and 𝒥_{2} because of the sign involved in the definition of ∂(g), see (5). In other words, if we switch 𝒥_{1} and 𝒥_{2} we would get the opposite homomorphism : −∂.

Remark 3.4. One can also define ∂ : Hom_{𝒦_{12}} (a, b) → Hom_{𝒦} (a, T b) by means of right fractions instead of left fractions. We leave it to the reader to show, using Lemma 1.13, that these two definitions agree for judicious choices of signs. The original definition (5) of ∂ is obviously natural in the object b. Naturality in the object a easily follows from this equivalent construction of ∂ via right fractions.

Theorem 3.5. Consider a formal Mayer-Vietoris cover of an idempotent complete triangulated category 𝒦 (Def. 2.1) and consider the categories of Diagram (4). Then, for every pair of objects a, b ∈ 𝒦 there is a natural long exact sequence

[figure: ⋯; Hom_{𝒦_{12}} (T a, b); ∂; ∂; Hom_{𝒦} (a, b); _{( \begin{matrix}ρ_{1} \\ ρ_{2} \end{matrix} )}; Hom_{𝒦_{1}} (a, b) ⊕ Hom_{𝒦_{2}} (a, b); _{(_{−ρ_{12} ρ_{21}})}; Hom_{𝒦_{12}} (a, b); ∂; ∂; Hom_{𝒦} (a, T b); ⋯]

where the connecting homomorphism ∂ is defined as in Construction 3.1.

Proof. First, we have to check that the definition of ∂(f s^{−1}) given in 3.1 does not depend on the choice of the objects c_{i} ∈ 𝒥_{i} such that cone(s) ≃ c_{1} ⊕c_{2}. This is easy, for other c_{i} must be isomorphic to the chosen ones : c_{1} ⊕ c_{2} ≃ c′_{1} ⊕ c′_{2} and c_{i} ∈ 𝒥_{i} forces c_{i} ≃ c′_{i} for i = 1, 2, by the assumption Hom_{𝒦} (c_{i}, d_{j}) = 0 for i ≠ j. The isomorphism c_{1} ≃ c′_{1} disappears in the composition γ ∘ α and a fortiori in ∂(f s^{−1}).

Then, we have to check that ∂(f s^{−1}) only depends on the class of the fraction f s^{−1}. To see this, let t : y → x be a 𝒦_{12}-isomorphism and let (ft) (st)^{−1} =

( \xymatrix{a & y \ar[l]_{st} \ar[r]^{ft} & b}) be the amplified fraction. We have to show that ∂((ft) (st)^{−1}) = ∂(f s^{−1}). The morphism st fits in a distinguished triangle

[figure: y; st; a; ^{( \begin{matrix}α′ \\ β′ \end{matrix} )}; d_{1} ⊕ d_{2}; (γ′ δ′); T y]

where d_{i} ∈ 𝒥_{i}. Comparing the triangles for s and for st yields the diagram

(6)  [figure: y; st; t; a; ^{( \begin{matrix}α′ \\ β′ \end{matrix} )}; d_{1} ⊕ d_{2}; ∃ ϵ; (γ′ δ′); T y; T t; x; s; a; _{( \begin{matrix}α \\ β \end{matrix} )}; c_{1} ⊕ c_{2}; (γ δ); T x]

for some morphism ϵ. But since Hom_{𝒦} (d_{i}, c_{j}) = 0 for i ≠ j by the formal Mayer-Vietoris assumption, we have that ϵ = ( \begin{matrix}ϵ_{1} & 0 \\ 0 & ϵ_{2} \end{matrix} ). Now compute

∂ ((ft) (st)^{−1}) =^{def.} T (f) T (t) γ′ α′ =^{(6)} T (f) γ ϵ_{1} α′ =^{(6)} T (f) γ α =^{def.} ∂ (f s^{−1}) .

This proves that ∂ is well-defined. The fact that ∂ does not depend on the amplification of the fraction also shows that in order to prove that ∂ is a group homomorphism it suffices to see that ∂((f + g) s^{−1}) = ∂(f s^{−1}) + ∂(g s^{−1}), which is immediate from the definition of ∂.

We now prove that the sequence is exact. It is easy to see that all consecutive compositions are zero. (Recall the notation for the restriction functors ρ_{i} for i ∈ {1, 2, 12, 21} from Construction 2.4.) For instance, for i = 1, 2, we have ρ_{i}(∂(f s^{−1})) = 0 because ∂(f s^{−1}) factors via c_{i} ∈ 𝒥_{i} which becomes zero in 𝒦_{i}. To see that ∂ ∘ ρ_{12} = 0, we check that ∂(f s^{−1}) = 0 if s is a 𝒦_{1}-isomorphism. In this case, c_{2} = 0 and ∂(f s^{−1}) factors via c_{2} . One proves ∂ ∘ ρ_{21} = 0 similarly.

Exactness at Hom_{𝒦_{1}} (a, b)⊕Hom_{𝒦_{2}} (a, b) : Let (f_{1}, f_{2}) ∈ Hom_{𝒦_{1}} (a, b)⊕Hom_{𝒦_{2}} (a, b)

such that ρ_{12}(f_{1}) = ρ_{21}(f_{2}). Write f_{i} = ( \xymatrix{a & x_{i} \ar[l]_{s_{i}} \ar[r]^{g_{i}} & b}). Then there exist an object x and 𝒦_{12}-isomorphisms t_{i} : x → x_{i} such that the diagram

(7)  \xymatrix{ & x_{1} \ar[dl]_{s_{1}} \ar[dr]^{g_{1}} \\ a & x \ar[u]_{t_{2}} \ar[d]^{t_{1}} & b \\  & x_{2} \ar[ul]^{s_{2}} \ar[ur]_{g_{2}}}

is commutative in 𝒦. By Lemma 2.10 we know that every 𝒦_{12}-isomorphism factors as a 𝒦_{1}-isomorphism followed by a 𝒦_{2}-isomorphism (and viceversa) so that we may choose t_{2} to be a 𝒦_{2}-isomorphism, up to possibly amplifying the fraction g_{1} s_{1}^{−1} without changing the morphism f_{1} in 𝒦_{1}. Similarly, we can assume t_{1} is a 𝒦_{1}-isomorphism. By Lemma 2.13, the left “square” of (7) is weakly bicartesian. Therefore (weak push-out), g_{1} and g_{2} induce a morphism f ∈ Hom_{𝒦} (a, b) such that f ∘ s_{i} = g_{i} for i = 1, 2. Hence f = g_{i} s_{i}^{−1} = f_{i} in 𝒦_{i} as wanted.

Exactness at Hom_{𝒦} (a, b) : Let f ∈ Hom_{𝒦} (a, b) be such that ρ_{i}(f) = 0 in ℒ_{i} = 𝒦/𝒥_{i} for i = 1, 2. This means that f factors through objects c_{i} ∈ 𝒥_{i} as follows :

\xymatrix{ & c_{1} \ar[dr]^{f_{1}} \\ a \ar[rr]^{f} \ar[ur]^{α} \ar[dr]_{β} &  & b . \\  & c_{2} \ar[ur]_{f_{2}}}

Take now x the weak push-out of α and β. By construction of the weak push-out (Def. 1.14), we have a distinguished triangle as in the first row of the diagram below. Since (^{f_{1}} ^{−f_{2}} ) · ( \begin{matrix}α \\ β \end{matrix} ) = f − f = 0, there exists a morphism h : x → b as follows :

\xymatrix{a \ar[r]^{^{( \begin{matrix}α \\ β \end{matrix} )}} & c_{1} ⊕ c_{2} \ar[r]^{^{(^{γ δ} )}} \ar[dr]_{_{(_{f_{1} −f_{2}} )}} & x \ar@{-->}[d]^{h} \ar[r]^{s} & T a \\  &  & b .}

We obtain a morphism h s^{−1} = ( \xymatrix{T a & x \ar[l]_{s} \ar[r]^{h} & b}) ∈ Hom_{𝒦_{12}} (T a, b). By Construction 3.1 and Remark 3.2, we have ∂(h s^{−1}) = h γ α = f_{1} α = f.

Exactness at Hom_{𝒦_{12}} (a, b) : Let f s^{−1} = ( \xymatrix{a & x \ar[l]_{s} \ar[r]^{f} & b}) be a morphism in 𝒦_{12} such that ∂(f s^{−1}) = 0. As in Construction 3.1, choose a distinguished triangle

[figure: x; s; a; ^{( \begin{matrix}α \\ β \end{matrix} )}; c_{1} ⊕ c_{2}; ^{(^{γ δ} )}; T x]

with c_{i} ∈ 𝒥_{i}. The assumption ∂(f s^{−1}) = 0 reads T (f) γ α = 0. Now apply Lemma 1.13 to the above triangle to produce objects d, e ∈ 𝒦 and morphisms α_{0}, β_{0}, γ_{0} and δ_{0} satisfying all the conclusions of Lemma 1.13, which the reader is encouraged to have at hand. Recall in particular that α_{2} = T (δ_{0}) γ.

Claim : There exists a distinguished triangle of the form

(8)  [figure: b; γ α; T a; ^{( \begin{matrix}T δ_{0} \\ −T γ_{0} \end{matrix} )}; T d ⊕ T e; T b .]

Indeed, the composition α_{2} γ_{1} = T (δ_{0}) γ γ_{1} = 0 yields an octahedron (the triangles over γ_{1} and α_{2} are taken from the conclusions of Lemma 1.13) :

[figure: T d; _{( \begin{matrix}1 \\ 0 \end{matrix} )}; −T α_{0}; c_{1}; α_{2}; γ; T b; −T α; ·; −T (γα); ·; e; γ_{1}; 0; T d ⊕ T e; (0 1); ·; ζ; T a; −T γ_{0}; ·; _{( \begin{matrix}φ \\ −T γ_{0} \end{matrix} )}]

for some morphisms φ and ζ. Note in particular the distinguished triangle

[figure: b; γ α; T a; ^{( \begin{matrix}φ \\ −T γ_{0} \end{matrix} )}; T d ⊕ T e; ζ; T b .]

To obtain Triangle (8), we still need to “replace” φ by T δ_{0}. Observe that we have (T (δ_{0}) − φ) ∘ γ = α_{2} − α_{2} = 0. By the distinguished triangle over γ, there exists a morphism h : T e → T d such that T δ_{0} − φ = h T γ_{0}. Using this equality we get an isomorphism of triangles

[figure: b; γ α; T a; ^{( \begin{matrix}φ \\ −T γ_{0} \end{matrix} )}; T d ⊕ T e; ζ; ≃; _{( \begin{matrix}1 & −h \\ 0 & 1 \end{matrix} )}; T b; b; γ α; T a; _{( \begin{matrix}T δ_{0} \\ −T γ_{0} \end{matrix} )}; T d ⊕ T e; ζ′; T b]

for ζ′ := ζ · ( \begin{matrix}1 & h \\ 0 & 1 \end{matrix} ). So, the lower triangle is distinguished and fulfills the Claim.

Using Triangle (8) and the assumption T (f) ∘ γ α = ∂(f s^{−1}) = 0 yields a factorization of T f as follows :

\xymatrix{a \ar[r]^{γ α} & T x \ar[dr]_{T f} \ar[r]^{^{( \begin{matrix}T δ_{0} \\ −T γ_{0} \end{matrix} )}} & T d ⊕ T e \ar@{-->}[d]^{_{(_{T f_{1} T f_{2}} )}} \ar[r] & T a \\  &  & T b}

for some morphisms f_{1} : d → b and f_{2} : e → b. This reads f = f_{1} δ_{0} − f_{2} γ_{0}. Using the triangles of Lemma 1.13, it is easy to see that α_{0}, γ_{0} are 𝒦_{1}-isomorphisms and that β_{0} and δ_{0} are 𝒦_{2}-isomorphisms. Consider now the morphisms f_{1} α_{0}^{−1} =

( \xymatrix{a & d \ar[l]_{α_{0}} \ar[r]^{f_{1}} & b}) and f_{2} β_{0}^{−1} = ( \xymatrix{a & e \ar[l]_{β_{0}} \ar[r]^{f_{2}} & b}) in 𝒦_{1} and 𝒦_{2} respectively.

When restricted to 𝒦_{12} they clearly satisfy f_{1} α_{0}^{−1} − f_{2} β_{0}^{−1} = (f_{1}δ_{0}) (α_{0}δ_{0})^{−1} − (f_{2}γ_{0}) (β_{0}γ_{0})^{−1} = f s^{−1}. The last equation uses the relation s = α_{0}δ_{0} = β_{0}γ_{0} from Lemma 1.13 and the above relation f = f_{1} δ_{0} − f_{2} γ_{0}. □

4. Gluing of objects

Definition 4.1. Consider a formal Mayer-Vietoris cover of an idempotent complete triangulated category 𝒦, by means of two subcategories 𝒥_{1} and 𝒥_{2} (see Def. 2.1). Recall Diagram (4). Consider two objects a_{1} ∈ 𝒦_{1} and a_{2} ∈ 𝒦_{2} and an isomorphism σ : a_{1} →^{∼} a_{2} in 𝒦_{12}. A gluing of the objects a_{i} along the isomorphism σ is an object a ∈ 𝒦 and two isomorphisms f_{i} : a →^{∼} a_{i} in 𝒦_{i} for i = 1, 2 such that σ f_{1} = f_{2} in 𝒦_{12}. An isomorphism of gluings α : (a, f_{1}, f_{2}) →^{∼} (a′, f′_{1}, f′_{2}) is an isomorphism α : a →^{∼} a′ in 𝒦 such that f′_{i} α = f_{i} in 𝒦_{i} for all i = 1, 2.

We first prove the gluing of objects without idempotent completions.

Lemma 4.2. In a formal Mayer-Vietoris situation (Def. 2.1)

\xymatrix{𝒦 \ar[r]^{ρ_{1}} \ar[d]_{ρ_{2}} & ℒ_{1} \ar[d]^{ρ_{12}} \\ ℒ_{2} \ar[r]_{ρ_{21}} & ℒ_{12} ,}

two objects a_{1} ∈ ℒ_{1}, a_{2} ∈ ℒ_{2} with an isomorphism σ : ρ_{12}(a_{1}) →^{∼} ρ_{21}(a_{2}) in ℒ_{12} always admit a gluing (Def. 4.1).

Proof. The isomorphism σ can be represented by a fraction \xymatrix{a_{1} & x \ar[l]_{s} \ar[r]^{t} & a_{2}} where s, t both are 𝒦_{12}-isomorphisms. By Lemma 2.10, s and t factor as s = s_{1} s_{2} and t = t_{2} t_{1} where s_{i}, t_{i} are 𝒦_{i}-isomorphisms, see the upper part of Diagram (9). Now complete this diagram by taking the weak push-out of s_{2} and t_{1}:

(9)  \xymatrix{a_{1} &  & x \ar[ll]_{s} \ar[dl]^{s_{2}} \ar[rr]^{t} \ar[dr]_{t_{1}} &  & a_{2} \\  & y \ar[ul]^{s_{1}} \ar[dr]_{u_{1}} &  & z \ar[ur]_{t_{2}} \ar[dl]^{u_{2}} \\  &  & a}

Applying Lemma 2.12, u_{i} is a 𝒦_{i}-isomorphism. The object a is then isomorphic to a_{i} in 𝒦_{i} via f_{1} := s_{1} ∘ u_{1}^{−1} and f_{2} := t_{2} ∘ u_{2}^{−1} respectively; the relation σ f_{1} = f_{2} is satisfied in ℒ_{12} because of the commutativity of (9). □

Theorem 4.3. Assume given a formal Mayer-Vietoris cover of an idempotent complete triangulated category 𝒦 (Def. 2.1) and consider the categories constructed in 2.4. Then, given two objects a_{i} ∈ 𝒦_{i} for i = 1, 2 and an isomorphism σ : ρ_{12}(a_{1}) →^{∼} ρ_{21}(a_{2}) in 𝒦_{12}, there exists a gluing (Def. 4.1). This gluing is unique up to (possibly non-unique) isomorphism of gluings.

Proof. Since ℒ_{i} is cofinal in 𝒦_{i}, we have a_{i} ⊕ T (a_{i}) ∈ ℒ_{i} for i = 1, 2 (see [4]). Obviously σ⊕T σ gives an isomorphism ρ_{12}(a_{1}⊕T a_{1}) →^{∼} ρ_{21}(a_{2}⊕T a_{2}) in 𝒦_{12} hence in ℒ_{12} since ℒ_{12} → ℒ̃_{12} = 𝒦_{12} is fully faithful. By Lemma 4.2 there exists an object b ∈ 𝒦 and isomorphisms f_{i} : ρ_{i}(b) → b_{i} in ℒ_{i} such that (σ ⊕T σ)∘ρ_{12}(f_{1}) = ρ_{21}(f_{2}). Consider now, for each i = 1, 2 the idempotent π_{i} : ρ_{i}(b) → ρ_{i}(b) in ℒ_{i} defined by :

\xymatrix{ρ_{i}(b) \ar[r]^{f_{i}} \ar[rrr]_{π_{i}} & b_{i} \ar[r]^{^{( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} )}} & b_{i} \ar[r]^{f_{i}^{−1}} & ρ_{i}(b) ,}

where ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) on b_{i} = a_{i} ⊕ T (a_{i}) is the projection on a_{i}. Now, since the diagram

[figure: ρ_{12}(b); ρ_{12}(f_{1}); ρ_{12}(b_{1}); σ⊕T σ; ^{( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} )}; ρ_{12}(b_{1}); σ⊕T σ; ρ_{12}(f_{1})^{−1}; ρ_{12}(b); ρ_{21}(b); ρ_{21}(f_{2}); ρ_{21}(b_{2}); _{( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} )}; ρ_{21}(b_{2}); ρ_{21}(f_{2})^{−1}; ρ_{21}(b)]

is commutative in ℒ_{12}, we have that ρ_{12}(π_{1}) = ρ_{21}(π_{2}). We can now apply formal Mayer-Vietoris for morphisms (Thm. 3.5) to π_{1} and π_{2} to show that there exists an endomorphism π : b → b such that ρ_{i}(π) = π_{i} for i = 1, 2. We would like π to be an idempotent, like its restrictions ρ_{i}(π) to ℒ_{i} for i = 1, 2. Consider h = π^{2} − π. Since h = 0 in ℒ_{i} it factors in 𝒦 through objects c_{i} ∈ 𝒥_{i} as follows :

\xymatrix{b \ar[rr]^{h} \ar[dr] &  & b \ar[rr]^{h} \ar[dr] &  & b . \\  & c_{1} \ar[ur] &  & c_{2} \ar[ur]}

Since Hom_{𝒦} (c_{1}, c_{2}) = 0 by the formal Mayer-Vietoris assumption, we have h^{2} = 0. Then, by a standard trick, p := π + h − 2πh satisfies p^{2} = p and still has the property ρ_{i}(p) = π_{i} since ρ_{i}(h) = 0. So p : b → b is an idempotent lifting π_{i} for i = 1, 2. Now, our category 𝒦 is idempotent complete by hypothesis, so b splits as b ≃ Im(p) ⊕ Ker(p). Setting a = Im(p) gives the desired object with the property a ≃ a_{i} in 𝒦_{i}. Further details are left to the reader.

For uniqueness, suppose that (a, f_{1}, f_{2}) and (a′, f′_{1}, f′_{2}) are two gluings. By formal Mayer-Vietoris for morphisms (Thm. 3.5) the morphisms f_{1}^{−1} ∘ f′_{1} and f_{2}^{−1} ∘ f′_{2} glue into a morphism a′ → a which must be an isomorphism (Rem. 2.7). □

5. Mayer-Vietoris in tensor triangular geometry

Definition 5.1. Let 𝒦 be an idempotent complete strongly closed tensor triangulated category (see Section 1). We say that we are in a Mayer-Vietoris situation if the spectrum of 𝒦 is covered by two quasi-compact open subsets Spc(𝒦) = U_{1} ∪U_{2}. We shall denote by Z_{i} = Spc(𝒦) ∖ U_{i} the closed complements for i = 1, 2.

We now have the key to the results of the previous sections :

Theorem 5.2. In a Mayer-Vietoris situation as above, the thick subcategories 𝒥_{1} := 𝒦_{Z_{1}} and 𝒥_{2} := 𝒦_{Z_{2}} define a formal Mayer-Vietoris cover of 𝒦 (Def. 2.1) and 𝒥_{12} = 𝒥_{1} ⊕ 𝒥_{2} coincides with 𝒦_{Z_{1}∪Z_{2}}.

Proof. This is Proposition 1.6 and Theorem 1.8. □

Remark 5.3. We do not really need to have U_{1} and U_{2} open and it would be enough to assume that they are arbitrary intersections of quasi-compact open subsets. Indeed, the key result taken from [2], Theorem 1.8, holds in this generality. Therefore, everything below holds in similar generality. We stick to open pieces because this is closer to the reader’s understanding of a Mayer-Vietoris framework.

We can now apply the notions and results of Section 2, 3 and 4.

Definition 5.4. Recall the important Definitions 1.10. With the simplified notation ρ_{i} = ρ_{U_{i}} and ρ_{ij} = ρ_{U_{i}∩U_{j} , U_{i}} for the restriction functors, we have the following commutative diagram

(10)  [figure: 𝒦; ρ_{1}; ρ_{2}; ρ_{1}; ρ_{2}; 𝒦(U_{1}); ρ_{12}; ℒ(U_{1}); ρ_{12}; ℒ(U_{2}); ρ_{21}; ℒ(U_{1} ∩ U_{2}); 𝒦(U_{2}); ρ_{21}; 𝒦(U_{1} ∩ U_{2}) ,]

which is exactly Diagram (4) here. Recall that ↠ stands for a Verdier localization and ↪ for a fully faithful cofinal embedding.

Rephrasing Definition 2.6 yields :

Definition 5.5. Let U ⊂ Spc(𝒦) be a (quasi-compact) open with closed complement Z. A morphism s : a → b in 𝒦 is called a U-isomorphism if it is an isomorphism in ℒ(U), or equivalently in 𝒦(U). This is also equivalent to saying that cone(s) belongs to 𝒦_{Z} which also reads supp(cone(s)) ∩ U = ∅.

Corollary 5.6. In a Mayer-Vietoris situation, suppose that s : a → b is a U_{1} ∩ U_{2}- isomorphism. Then s may be factored as s = s_{1} s_{2} where s_{i} is a U_{i}-isomorphism.

Proof. Lemma 2.10 (and Theorem 5.2). □

Corollary 5.7. In a Mayer-Vietoris situation, consider a commutative diagram :

\xymatrix{a \ar[r]^{s_{1}} \ar[d]_{t_{2}} & b \ar[d]^{s_{2}} \\ c \ar[r]_{t_{1}} & d .}

If s_{i} and t_{i} are U_{i}-isomorphisms for i = 1, 2, the square is weakly bicartesian.

Proof. Lemma 2.13 (and Theorem 5.2). □

Corollary 5.8 (Mayer-Vietoris for morphisms). In a Mayer-Vietoris situation (Def. 5.4), for each pair of objects a, b ∈ 𝒦, the homomorphisms ∂ of Construction 3.1 and Remark 3.2 fit in a natural long exact sequence

\xymatrix{⋯ \ar[r]^{∂} & Hom_{𝒦} (a, b) \ar[r] & Hom_{𝒦(U_{1})} (a, b) ⊕ Hom_{𝒦(U_{2})} (a, b) \ar[r] & Hom_{𝒦(U_{1}∩U_{2})} (a, b) \ar[r]^{∂} & ⋯}

where the other homomorphisms are the restrictions and differences of restrictions.

Proof. Apply Theorem 3.5 (and Theorem 5.2). Of course, Construction 3.1 should be applied to 𝒥_{1} = 𝒦_{Z_{1}} and 𝒥_{2} = 𝒦_{Z_{2}}. □

* * *

Let us now discuss the gluing of objects. It is convenient to fix the following standard terminology, generalizing Definition 4.1.

Definition 5.9. Let Spc(𝒦) = U_{1} ∪ … ∪ U_{n} be a cover by quasi-compact open subsets. Consider objects a_{i} ∈ 𝒦(U_{i}) and isomorphisms σ_{ji} : a_{i} →^{∼} a_{j} in 𝒦(U_{i} ∩ U_{j}) such that σ_{ki} = σ_{kj} σ_{ji} in 𝒦(U_{i} ∩U_{j} ∩U_{k}) for 1 ≤ i, j, k ≤ n. A gluing of the objects a_{i} along the isomorphisms σ_{ij} is an object a ∈ 𝒦 and n isomorphisms f_{i} : a →^{∼} a_{i} in 𝒦(U_{i}) such that σ_{ji} f_{i} = f_{j} in 𝒦(U_{i} ∩U_{j}) for all 1 ≤ i, j ≤ n. An isomorphism of

gluings f : (a, f_{1}, …, f_{n}) →^{∼} (a′, f′_{1}, …, f′_{n}) is an isomorphism f : a →^{∼} a′ in 𝒦 such

that f′_{i} f = f_{i} in 𝒦(U_{i}) for all i = 1, …, n. (As before, we temporarily dropped the restriction functors for readability.)

Corollary 5.10 (Gluing of two objects). In a Mayer-Vietoris situation (Def. 5.4)

\xymatrix{𝒦 \ar[r]^{ρ_{1}} \ar[d]_{ρ_{2}} & 𝒦(U_{1}) \ar[d]^{ρ_{12}} \\ 𝒦(U_{2}) \ar[r]_{ρ_{21}} & 𝒦(U_{1} ∩ U_{2}) ,}

given two objects a_{i} ∈ 𝒦(U_{i}) for i = 1, 2 and an isomorphism σ : ρ_{12}(a_{1}) →^{∼} ρ_{21}(a_{2}) in 𝒦(U_{1} ∩ U_{2}), there exists a gluing (Def. 5.9), which is unique up to (possibly non-unique) isomorphism.

Proof. Apply Theorem 4.3 (and Theorem 5.2). □

Corollary 5.11 (Gluing of three objects). Let Spc(𝒦) = U_{1} ∪ U_{2} ∪ U_{3} be a cover by quasi-compact open subsets. Consider three objects a_{i} ∈ 𝒦(U_{i}) for i = 1, 2, 3 and three isomorphisms σ_{ij} : a_{j} →^{∼} a_{i} in 𝒦(U_{i} ∩ U_{j}) for 1 ≤ i < j ≤ 3 satisfying the cocycle relation σ_{12} ∘ σ_{23} = σ_{13} in 𝒦(U_{1} ∩ U_{2} ∩ U_{3}). Then they admit a gluing.

Proof. Note that V := Spc(𝒦(U_{1} ∪ U_{2})) = U_{1} ∪ U_{2} by Proposition 1.11. Using gluing of two objects (Cor. 5.10), we can glue a_{1} and a_{2} into an object b ∈ 𝒦(V). Using Mayer-Vietoris for morphisms (Cor. 5.8) for the cover of V ∩ U_{3} by U_{1} ∩ U_{3} and U_{2} ∩ U_{3} , we can construct a (possibly non-unique) isomorphism between b and a_{3} in 𝒦(V ∩ U_{3}). By gluing of two objects (Cor. 5.10) for the cover of Spc(𝒦) given by V and U_{3} , we can now glue b and a_{3} into an object of 𝒦. □

Remark 5.12. As the above proof shows, the problem that arises with three open subsets is that the isomorphism between the objects b ∈ 𝒦(V) and a_{3} ∈ 𝒦(U_{3}) on V ∩ U_{3} is not unique. Various choices are parameterized by Hom_{𝒦(U_{1}∩U_{2})} (T a_{1}, a_{2}) but we were not able to prove that two such choices yield isomorphic gluings and we tend to believe that this is wrong in general. Nevertheless, here is a case where the gluing works for several open subsets. This applies in particular to vector bundles (concentrated in degree zero) in 𝒦 = D^{perf}(X) and hence recovers the standard gluing of vector bundles in algebraic geometry, for instance.

Theorem 5.13 (Connective gluing of several objects). Let Spc(𝒦) = U_{1} ∪ … ∪ U_{n} be a cover by quasi-compact open subsets for n ≥ 2. Consider objects a_{i} ∈ 𝒦(U_{i}) and isomorphisms σ_{ji} : a_{i} →^{∼} a_{j} in 𝒦(U_{i} ∩ U_{j}) satisfying the cocycle condition σ_{kj}σ_{ji} = σ_{ki} in 𝒦(U_{i} ∩ U_{j} ∩ U_{k}) for 1 ≤ i, j, k ≤ n. Assume moreover the following Connectivity Condition : For any i = 2, …, n and for any quasi-compact open V ⊂ U_{i}, we suppose that :

(11)  Hom_{𝒦(V)} (T a_{i}, a_{i}) = 0 .

(It suffices to have it for the V ⊂ U_{i} which are unions of intersections of U_{1}, …, U_{n}.) Then there exists a gluing (Def. 5.9), which is unique up to unique isomorphism.

Proof. We prove the result by induction on n. Let us first establish the n = 2 case. By gluing of two objects (Cor. 5.10) we only need to prove the uniqueness of the isomorphism. To see this, it suffices to prove that for two gluings a, a′ ∈ 𝒦, two (iso)morphisms g, g′ : a → a′ which agree in 𝒦(U_{1}) and 𝒦(U_{2}) are equal. By Mayer-Vietoris for morphisms (Cor. 5.8), it suffices to show that Hom_{𝒦(U_{1}∩U_{2})} (T a, a′) = 0 which follows from the Connectivity Condition (11) and from a ≃ a′ ≃ a_{2} in 𝒦(U_{2}) .

Suppose n ≥ 3 and the result known for n−1. Define V = U_{1} ∪…∪U_{n−1}. Since V is quasi-compact, we know by Proportion 1.11 that Spc(𝒦(V)) = V and we can apply the induction hypothesis to construct a gluing b ∈ 𝒦(V) with isomorphisms

g_{i} : b →^{∼} a_{i} in 𝒦(U_{i}) for i = 1, …, n − 1, such that σ_{ij}g_{j} = g_{i} for all 1 ≤ i, j ≤ n − 1.

Consider the intersection W := V ∩ U_{n}, which is covered by n − 1 quasi-compact subsets W = (U_{1} ∩ U_{n}) ∪ … ∪ (U_{n−1} ∩ U_{n}). In the category 𝒦(W), we have two objects b and a_{n} (i.e. their restrictions, of course) which are isomorphic in 𝒦(U_{i} ∩ U_{n}) for i = 1, …, n − 1 in a compatible way with the σ_{ij}. By uniqueness of the gluing for n − 1, there exists a unique isomorphism σ : b →^{∼} a_{n} on V ∩ U_{n} such that σ_{in} σ = g_{i} for i = 1, …, n − 1. By the n = 2 case, we obtain the wanted gluing a ∈ 𝒦 of b and a_{n}, unique up to unique isomorphism. Details are left to the careful reader, who will note that uniqueness of the isomorphism σ (at stage n − 1) is essential for uniqueness of the gluing a (at stage n).

In the above induction, we needed that if the tuple (U_{1}, …, U_{n}; a_{1}, …, a_{n}) satisfies the Connectivity Condition (11) for n, then :

• the tuple (U_{1} , …, U_{n−1} ; a_{1}, …, a_{n−1}) satisfies (11) for n − 1,

• the tuple (U_{1} ∩ U_{n} , …, U_{n−1} ∩ U_{n}; a_{1}, …, a_{n−1}) satisfies (11) for n − 1,

• the 4-uple (U_{1} ∪ … ∪ U_{n−1} , U_{n}; b, a_{n}) satisfies (11) for n = 2, for any object b. These are easy to check. The last one comes from the assumption i > 1 in (11). □

6. Picard groups

The next definition is an elementary fact for line bundles in algebraic geometry and its generalization to (closed) symmetric monoidal category roots back to the French geometer Jacques II de Chabannes (1470-1525). See also [6] and further references therein.

Definition 6.1. An object a ∈ 𝒦 is called invertible if there exists an object b such that a ⊗ b ≃ 1. By adjunction, see Def. 1.4, an object a ∈ 𝒦 is invertible if and only if the evaluation map η : Da ⊗ a → 1 is an isomorphism.

Lemma 6.2. An object a in 𝒦 is invertible if and only if it is invertible in 𝒦/𝒫 for all 𝒫 ∈ Spc(𝒦).

Proof. The evaluation map η : Da ⊗ a → 1 is an isomorphism by Proposition 1.7. □

Definition 6.3. Here we call Picard group, Pic(𝒦), the set of isomorphism classes of invertible objects in 𝒦. The tensor product ⊗ : 𝒦 × 𝒦 → 𝒦 makes Pic(𝒦) into an abelian group with unit the class of 1. (The reader should be aware that some authors can call Picard group something different, like sometimes the set of all auto-equivalences of 𝒦.)

The following, essentially obvious result is well-known. We give a short proof for the convenience of the reader.

Proposition 6.4. Let X be a scheme and consider D^{perf}(X) its derived category of perfect complexes. Then there is a split short exact sequence of abelian groups

0 → Pic(X) → Pic(D^{perf}(X)) → C(X; ℤ) → 0

where C(X; ℤ) stands for the group of locally constant functions from X to ℤ.

Proof. We first describe Pic(D^{perf}(X)) where X = Spec(R) is the spectrum of a local ring (R, 𝔪). In this case, any object of D^{perf}(R) is isomorphic to a so-called minimal complex of the form

C = \xymatrix{⋯ \ar[r] & R^{ℓ_{i}} \ar[r]^{d_{i}} & R^{ℓ_{i−1}} \ar[r] & ⋯}

where, for all i, the differential d_{i} is a matrix with coefficients in 𝔪. If C is invertible in D^{perf}(R) so is C̄, its image under the functor D^{perf}(R) → D^{perf}(R/𝔪). But all the differentials of C̄ are 0 and the relation C ⊗ D ≃ R, for some complex D, hence C̄ ⊗ D̄ ≃ R/𝔪, shows that the complex C must be R concentrated in some degree, i.e. there exists n_{0} = n_{0}(C; R) such that ℓ_{i} = 1 if i = n_{0} and ℓ_{i} = 0 otherwise.

For a global X, the map Pic(D^{perf}(X)) → C(X; ℤ) is now easily defined: for an invertible complex C ∈ D^{perf}(X) and for x ∈ X denote by C_{x} its image in D^{perf}(O_{X,x}). The function f_{C} : X → ℤ is then defined by x ↦ n_{0}(C_{x}; O_{X,x}). The rest of the proof is straightforward : a perfect complex which is locally trivial is quasi-isomorphic to its homology in degree zero, which must be a line bundle. □

Remark 6.5. The result is the same if one works with bigger derived categories instead of D^{perf}(X). See for instance Fausk’s paper [6]. This might look more general than the above Proposition but one should keep in mind that invertible objects in such big categories are necessarily compact, just by abstract non-sense, see [7, Prop. A.2.8] for instance. So, the above argument already contains most of the relevant algebraic geometry for this problem.

Definition 6.6. Define 𝔾_{m}(𝒦) = Hom_{𝒦} (1, 1)^{×} to be the group of invertible elements of the (commutative) ring Hom_{𝒦} (1, 1).

Theorem 6.7. In a Mayer-Vietoris situation (Def. 5.4), there is an exact sequence of abelian groups

[figure: ⋯; Hom_{𝒦(U_{1}∩U_{2})} (T 1, 1); 1+∂; 1+∂; 𝔾_{m}(𝒦); 𝔾_{m}(𝒦(U_{1})) ⊕ 𝔾_{m}(𝒦(U_{2})); 𝔾_{m}(𝒦(U_{1} ∩ U_{2})); δ; δ; Pic(𝒦); Pic(𝒦(U_{1})) ⊕ Pic(𝒦(U_{2})); Pic(𝒦(U_{1} ∩ U_{2})) .]

The homomorphism ∂ is as in Construction 3.1 and the unlabelled homomorphisms are the restrictions and the (multiplicative) differences thereof.

The homomorphism δ : 𝔾_{m}(𝒦(U_{1} ∩ U_{2})) → Pic(𝒦) is defined by gluing two copies of 1 by means of Corollary 5.10. Explicitly, it can be described as follows :

Let σ = ts^{−1} : \xymatrix{1 & x \ar[l]_{s} \ar[r]^{t} & 1} in 𝔾_{m}(𝒦(U_{1} ∩ U_{2})), where s and t are U_{1} ∩ U_{2}- isomorphisms ; by Corollary 5.6, there exist factorizations s = s_{1} s_{2} and t = t_{2} t_{1} where s_{i} and t_{i} are U_{i}-isomorphisms ; then δ(σ) is defined as the isomorphism class of the weak push-out p ∈ 𝒦 of x_{1} and x_{2} over x

(12)  \xymatrix{1 &  & x \ar[ll]_{s} \ar[dl]^{s_{2}} \ar[rr]^{t} \ar[dr]_{t_{1}} &  & 1 \\  & x_{1} \ar@{-->}[ul]^{s_{1}} \ar[dr]_{u_{1}} &  & x_{2} \ar@{-->}[ur]_{t_{2}} \ar[dl]^{u_{2}} \\  &  & p .}

Proof. First note that the homomorphism 1+∂ : Hom_{𝒦(U_{1}∩U_{2})} (T 1, 1) → Hom_{𝒦} (1, 1) defined by g ↦ 1 + ∂(g) lands in 𝔾_{m}(𝒦). Indeed for any g ∈ Hom_{𝒦(U_{1}∩U_{2})} (T 1, 1) one has ∂(g) ∘ ∂(g) = 0, since ∂(g) : 1 → 1 is zero in 𝒦(U_{i}) and hence factors via some object of 𝒦_{Z_{i}} for i = 1, 2 and since Hom_{𝒦} (𝒦_{Z_{1}}, 𝒦_{Z_{2}}) = 0 by Proposition 1.6. So, 1 + ∂(g) is invertible with inverse 1 − ∂(g).

The connecting homomorphism δ : 𝔾_{m}(𝒦(U_{1} ∩ U_{2})) → Pic(𝒦) produces an object p ∈ 𝒦, see Diagram (12), which is isomorphic to 1 in 𝒦(U_{1}) via s_{1} u_{1}^{−1} and in 𝒦(U_{2}) via t_{2} u_{2}^{−1}, in a compatible way with σ in 𝒦(U_{1} ∩U_{2}). The object p is then the gluing of two copies of 1 along the isomorphism σ in 𝒦(U_{1} ∩ U_{2}). Such a gluing is unique up to isomorphism by Corollary 5.10, and this gluing only depends on the map σ, and not on the various choices (s, t, s_{1}, s_{2}, t_{1}, t_{2}, p). So, the map δ is well-

defined and we now check that it is a group homomorphism. Take \xymatrix{1 & x \ar[l]_{s} \ar[r]^{t} & 1}

and \xymatrix{1 & x′ \ar[l]_{s′} \ar[r]^{t′} & 1} two units in 𝔾_{m}(𝒦(U_{1} ∩ U_{2})). With the same notations as above, factor these morphisms and perform the corresponding weak push-outs :

\xymatrix{1 &  & x \ar[ll]_{s} \ar[dl]^{s_{2}} \ar[rr]^{t} \ar[dr]_{t_{1}} &  & 1 & 1 &  & x′ \ar[ll]_{s′} \ar@{-->}[dl]^{s′_{2}} \ar[rr]^{t′} \ar@{-->}[dr]_{t′_{1}} &  & 1 \\  & x_{1} \ar[ul]^{s_{1}} \ar[dr]_{u_{1}} &  & x_{2} \ar[ur]_{t_{2}} \ar[dl]^{u_{2}} &  &  & x′_{1} \ar@{-->}[ul]^{s′_{1}} \ar@{-->}[dr]_{u′_{1}} &  & x′_{2} \ar@{-->}[ur]_{t′_{2}} \ar@{-->}[dl]^{u′_{2}} \\  &  & p &  &  &  &  & p′}

In a symmetric monoidal category, the composition of two morphisms between the unit object is equal to their tensor product. Hence we tensor together the two above diagrams to obtain the following one :

\xymatrix{1 &  & x ⊗ x′ \ar[ll]_{s⊗s′} \ar[dl]^{s_{2}⊗s′_{2}} \ar[rr]^{t⊗t′} \ar[dr]_{t_{1}⊗t′_{1}} &  & 1 \\  & x_{1} ⊗ x′_{1} \ar[ul]^{s_{1}⊗s′_{1}} \ar[dr]_{u_{1}⊗u′_{1}} &  & x_{2} ⊗ x′_{2} \ar[ur]_{t_{2}⊗t′_{2}} \ar[dl]^{u_{2}⊗u′_{2}} \\  &  & p ⊗ p′}

By Corollary 5.7, the above middle square is weakly bicartesian as well. Hence, p ⊗ p′ = δ(σ ⊗ σ′) = δ(σ σ′). This shows that δ is an group homomorphism.

(Recall the restriction functors ρ_{i} and ρ_{ij} from Definition 5.4.) It is straightforward from the above definition of δ that ρ_{i} ∘ δ = 0 for i = 1, 2. To see that δ ∘ ρ_{12} = 0, for instance, one can assume that s_{2} = id and t_{2} = id_{1} in (12), in which case u_{2} must also be an isomorphism, i.e. p ≃ x_{2} = 1. The other compositions are clearly 0 (keeping in mind that 0 means 1 or 1 in the multiplicative groups 𝔾_{m} and Pic). The exactness of the left-hand side of the sequence up to 𝔾_{m}(𝒦) follows from Mayer-Vietoris for morphisms (Cor. 5.8) applied at a = b = 1. It remains to check the exactness of the sequence at four spots.

Exactness at 𝔾_{m}(𝒦(U_{1})) ⊕ 𝔾_{m}(𝒦(U_{2})): This is again immediate from Mayer-Vietoris for morphisms (Cor. 5.8) recalling that a local isomorphism is an isomorphism (Prop. 1.7).

Exactness at 𝔾_{m}(𝒦(U_{1} ∩ U_{2})): Let σ = ( \xymatrix{1 & x \ar[l]_{s} \ar[r]^{t} & 1} ) in 𝔾_{m}(𝒦(U_{1} ∩ U_{2})) be such that δ(σ) ≃ 1 in 𝒦. This means that one can find a diagram of the form

\xymatrix{1 &  & x \ar[ll]_{s} \ar[dl]^{s_{2}} \ar[rr]^{t} \ar[dr]_{t_{1}} &  & 1 \\  & x_{1} \ar[ul]^{s_{1}} \ar[dr]_{u_{1}} &  & x_{2} \ar[ur]_{t_{2}} \ar[dl]^{u_{2}} \\  &  & 1}

see (12). One then sees two morphisms, namely σ_{1} = u_{1} s_{1}^{−1} ∈ 𝔾_{m}(𝒦(U_{1})) and σ_{2} = u_{2} t_{2}^{−1} ∈ 𝔾_{m}(𝒦(U_{2})), such that σ_{2}^{−1} ∘ σ_{1} = σ in 𝔾_{m}(𝒦(U_{1} ∩ U_{2})).

Exactness at Pic(𝒦): Let p be an invertible object in 𝒦 such that ρ_{i}(p) ≃ 1 for

i = 1, 2. Thus there exist U_{i}-isomorphisms \xymatrix{1 & y_{i} \ar[l]_{s_{i}} \ar[r]^{t_{i}} & p}. Performing the weak pull-back of s_{1} and s_{2} one obtains the diagram

\xymatrix{1 &  & y \ar[dl]_{u_{2}} \ar[dr]^{u_{1}} &  & 1 \\  & y_{1} \ar[ul]^{t_{1}} \ar[dr]_{s_{1}} &  & y_{2} \ar[ur]_{t_{2}} \ar[dl]^{s_{2}} \\  &  & p}

and this defines ( \xymatrix{1 & y \ar[l]_{t_{1} u_{2}} \ar[r]^{t_{2} u_{1}} & 1} ) ∈ 𝔾_{m}(𝒦(U_{1} ∩ U_{2})). The image of this morphism under δ is clearly isomorphic to p by construction, see (12), the middle square in the above diagram being weakly bicartesian.

Exactness at Pic(𝒦(U_{1})) ⊕ Pic(𝒦(U_{2})): This follows from gluing of two objects (Cor. 5.10) and from invertibility being a local property (see Lemma 6.2). □

Theorem 6.8. Let 𝒦 be an idempotent complete strongly closed tensor triangulated category. Suppose that Hom_{𝒦(U)} (T 1, 1) = 0 for every quasi-compact open subsets U ⊂ Spc(𝒦). Then there exists a unique sheaf 𝔾_{m} on Spc(𝒦), such that 𝔾_{m}(U) = 𝔾_{m}(𝒦(U)) when U ⊂ Spc(𝒦) is quasi-compact open. Moreover, there exists an injective homomorphism from the first Čech cohomology of Spc(𝒦) with coefficients in 𝔾_{m} into the Picard group of 𝒦

α : Ȟ^{1}(Spc(𝒦), 𝔾_{m}) ↪ Pic(𝒦)

which sends a 𝔾_{m}-cocycle σ to the unique gluing of copies of 1 along the isomorphisms over the pairwise intersections given by σ, as described in Theorem 5.13.

Proof. We first prove by induction on n the following

Claim : Given a cover of a quasi-compact subset V ⊂ Spc(𝒦) by n ≥ 2 quasicompact open subsets, V = U_{1} ∪ … ∪ U_{n}, and given morphisms f_{i} : 1 → 1 in 𝒦(U_{i}) such that f_{i} = f_{j} in 𝒦(U_{i} ∩ U_{j}) for 1 ≤ i, j ≤ n, there exists a unique morphism f : 1 → 1 in 𝒦(V) such that f = f_{i} in 𝒦(U_{i}) .

Replacing 𝒦 by 𝒦(V), we can assume that V = Spc(𝒦) (see Prop. 1.11). Now, for n = 2, this is Mayer-Vietoris for morphisms (Cor. 5.8). Note that uniqueness follows from Hom_{𝒦(U_{1}∩U_{2})} (T 1, 1) = 0. The induction on n is then easy : To construct f, glue the n − 1 first morphisms f_{i} into a morphism g : 1 → 1 in 𝒦(U_{1} ∪ … ∪ U_{n−1}) and show that it agrees with f_{n} in 𝒦((U_{1} ∪ … ∪ U_{n−1}) ∩ U_{n}) – this uses uniqueness for n − 1 ; then apply the n = 2 case to glue g and f_{n} into a global f. To prove uniqueness of f, proceed similarly, using uniqueness for n − 1 and for n = 2 again. Hence the Claim.

Then the existence of the sheaf 𝔾_{m} is immediate from the claim and from the fact that quasi-compact open subsets form a basis of the topology of Spc(𝒦) by [1, Rem. 2.7 and Prop. 2.14]. For the same reason and because of quasi-compactness of Spc(𝒦), to define the homomorphism α, it suffices to consider 𝔾_{m}-cocycles over finite covers of Spc(𝒦) by quasi-compact open subsets. In this situation, the gluing is guaranteed by Theorem 5.13. Hence α is well-defined.

Finally, injectivity of α is easy. Indeed, given a 𝔾_{m}-cocyle σ over a cover Spc(𝒦) = U_{1} ∪ … ∪ U_{n} with every U_{i} quasi-compact open, the gluing a ∈ Pic(𝒦) comes with isomorphisms f_{i} : a →^{∼} 1 in each 𝒦(U_{i}), compatible with the σ(U_{i} ∩ U_{j}) as usual. Now, if a = 1, the latter compatibilty means that the Čech boundary of the 0-cochain defined by the f_{i} ∈ 𝔾_{m}(U_{i}) is nothing but σ, that is, σ = 0 in Ȟ^{1}(Spc(𝒦), 𝔾_{m}). □

Remark 6.9. Note that the condition Hom_{𝒦(U)} (T 1, 1) = 0 does not hold in general, for instance in modular representation theory, i.e. for 𝒦 = kG - stab. For instance, for k = 𝔽_{2} and G = ℤ/2, we even have T 1 ≃ 1.

Remark 6.10. When the condition Hom_{𝒦(U)} (T 1, 1) = 0 holds for every quasicompact open U ⊂ Spc(𝒦) and when Spc(𝒦) happens to be a scheme, Theorem 6.8 gives an injective homomorphism Pic(Spc(𝒦)) ↪ Pic(𝒦). In the case of 𝒦 = D^{perf}(X) for X a scheme, this homomorphism is the one of Proposition 6.4.

7. Excision

For later use, we state the next result in greater generality than in the Introduction. See Remark 5.3. In the following statement, the reader can as well consider the case of A and B reduced to a singleton, i.e. U open and Y closed.

Theorem 7.1 (Excision). Let 𝒦 be an idempotent complete strongly closed tensor triangulated category and let Y ⊂ U ⊂ Spc(𝒦). Assume that Y = ∪_{α∈A}Y_{α} with every Y_{α} closed with quasi-compact complement and assume that U = ∩_{β∈B}U_{β} with every U_{β} open and quasi-compact. Then the restriction functor ρ : 𝒦 → 𝒦(U) induces an equivalence between the respective subcategories of objects supported on Y :

𝒦_{Y} ⟶^{∼} 𝒦(U)_{Y} .

Proof. Remark first of all that Spc(𝒦(U)) ≅ U by Proposition 1.11 (stated for A = {∗} but whose proof generalizes verbatim to the present situation).

Let us see that the functor ρ : 𝒦_{Y} → 𝒦(U)_{Y} is full. Given a, b ∈ 𝒦_{Y} and a frac-

tion \xymatrix{a & x \ar[l]_{s} \ar[r]^{f} & b} with s a U-isomorphism, we have supp(cone(s))∩supp(T a) ⊂ supp(cone(s)) ∩ U = ∅, so Hom_{𝒦} (a, cone(s)) = 0 by Proposition 1.6. So, any distinguished triangle starting with the morphism s must have zero in second place, i.e. the morphism s is a split epimorphism, say s ∘ u = id_{a} for some morphism u : a → x. Amplifying the fraction f s^{−1} by u shows that this morphism f s^{−1} is equal to (the restriction of) the morphism fu : a → b.

Let us see that the functor ρ : 𝒦_{Y} → 𝒦(U)_{Y} is faithful. Let f : a → b be a morphism in 𝒦_{Y} such that ρ(f) = 0, that is, there exists a U-isomorphism s : x → a such that fs = 0. As above, s must be a split epimorphism, hence f = 0.

Let us see that the functor ρ : 𝒦_{Y} → 𝒦(U)_{Y} is essentially surjective. Let b ∈ 𝒦(U)_{Y}. There is an object a ∈ 𝒦 such that ρ(a) = b ⊕ T (b). We have supp(a)∩U = supp_{U}(b)∪supp_{U}(T b) = supp_{U}(b) ⊂ Y. So, if we call Z = Spc(𝒦)∖U the complement of U, we have proved that supp(a) ⊂ Y ∪ Z. Since Y ∩ Z = ∅, we know by Theorem 1.8 that a ≃ c ⊕ d with supp(c) ⊂ Y and supp(d) ⊂ Z. But then ρ(a) = ρ(c) and we have found an object c ∈ 𝒦_{Y} such that ρ(c) = b ⊕ T (b). Now, consider the idempotent of b ⊕ T (b) corresponding to the projection on b. Since ρ is fully faithful, there exists a corresponding idempotent on the object c, which then decomposes accordingly, one factor going to b, as was to be shown. □

Remark 7.2. As an exercise, one can reformulate and prove Excision in the framework of formal Mayer-Vietoris covers (Def. 2.1).

Remark 7.3. If needed, the reader can establish the following assertion : Given a point 𝒫 ∈ Spc(𝒦), the “local” category 𝒦/𝒫, or rather its idempotent completion, is equivalent to the colimit of the categories 𝒦(U), over the quasi-compact open neighborhoods U ∋ 𝒫.

Acknowledgments : We thank Jacques Thévenaz for a counter-example, which prevented us from losing too much time on some wild dream of ours. The second author thanks the FIM of the Mathematics Department at the ETH Zurich for impartial partial support. We thank the referee for requesting the generalizations to non-tensorial situations.

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Paul Balmer, Department Mathematics, ETH Zurich, 8092 Zurich, Switzerland

E-mail address: balmer@math.ethz.ch
URL: http://www.math.ethz.ch/∼balmer

Giordano Favi, Department Mathematics, Universität Basel, 4051 Basel, Switzerland

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