Picard Groups, Weight Structures,

and (noncommutative) Mixed Motives

Mikhail Bondarko and Gonçalo Tabuada^{1}

Received: January 4, 2016 Revised: November 7, 2016

Communicated by Stefan Schwede

Abstract. We develop a general theory which, under certain assumptions, enables the computation of the Picard group of a symmetric monoidal triangulated category equipped with a weight structure in terms of the Picard group of the associated heart. As an application, we compute the Picard group of several categories of motivic nature – mixed Artin motives, mixed Artin-Tate motives, bootstrap motivic spectra, noncommutative mixed Artin motives, noncommutative mixed motives of central simple algebras – as well as the Picard group of certain derived categories of symmetric ring spectra.

2010 Mathematics Subject Classification: 14A22, 14C15, 14F42, 18E30, 55P43 Keywords and Phrases: Picard group, weight structure, mixed motives, motivic spectra, noncommutative mixed motives, symmetric ring spectra, noncommutative algebraic geometry

1. Introduction and statement of results

The computation of the Picard group Pic(T ) of a symmetric monoidal (triangulated) category T is, in general, a very difficult task. The goal of this article is to explain how the theory of weight structures allows us to greatly simplify this task. Let (T , ⊗, 1) be a symmetric monoidal triangulated category equipped with a weight structure w = (T^{w≥0}, T^{w≤0}); consult §3 for details. Assume that the symmetric monoidal structure − ⊗ − (as well as the ⊗-unit 1) restricts to the heart H := T^{w≥0} ∩ T^{w≤0} of the weight structure. We say that the category T has the w-Picard property if the group homomorphism Pic(H)×ℤ → Pic(T ), (a, n) ↦ a[n], is invertible. Our first main result provides sufficient conditions for this property to hold:

1M. Bondarko was supported by RFBR (grants no. 14-01-00393-a and 15-01-03034-a), by Dmitry Zimin’s Foundation “Dynasty”, and by the Scientific schools grant no. 9721.2016.1.

G. Tabuada was partially supported by the NSF CAREER Award #1350472 and by the Portuguese Foundation for Science and Technology grant PEst-OE/MAT/UI0297/2014.

Theorem 1.1. Assume that the weight structure w on T is bounded, i.e. T = ∪_{n∈ℤ}T^{w≥0}[−n] = ∪_{n∈ℤ}T^{w≤0}[−n], and that there exists a full, additive, conservative, symmetric monoidal functor from H into a symmetric monoidal semi-simple abelian category A which is moreover local in the sense that if a ⊗ b = 0 then a = 0 or b = 0. Under these assumptions, the category T has the w-Picard property.

As explained in [7, §4.3], every bounded weight structure is uniquely determined by its heart. Concretely, given any additive subcategory H′ ⊂ T which generates T and for which we have Hom_{H′}(a, b[n]) = 0 for every n > 0 and a, b ∈ H′, there exists a unique bounded weight structure on T with heart the Karoubi-closure of H′ in T . Roughly speaking, the construction of a bounded weight structure on a triangulated category amounts simply to the choice of an additive subcategory with trivial positive Ext-groups. Our second main result formalizes the conceptual idea that the w-Picard property satisfies a “global-to-local” descent principle:

Theorem 1.2. Assume the following:

(A1) The heart H of the bounded weight structure w is essentially small and R-linear for some commutative indecomposable Noetherian ring R. Moreover, Hom_{H}(a, b) is a finitely generated flat R-module for any two objects a, b ∈ H;

(A2) For every residue field κ(𝔭), with 𝔭 ∈ Spec(R), there exists a symmetric monoidal triangulated category (T_{κ(𝔭)}, ⊗, 1) equipped with a weight structure w_{κ(𝔭)} and with a weight-exact symmetric monoidal functor ι_{κ(𝔭)} : T → T_{κ(𝔭)}. Moreover, the functor ι_{κ(𝔭)} induces an equivalence of categories between the Karoubization of H ⊗_{R} κ(𝔭) and H_{κ(𝔭)}.

Under assumptions (A1)-(A2), if the categories T_{κ(𝔭)} have the w_{κ(𝔭)}-Picard property, then the category T has the w-Picard property.

Remark 1.3. (i) At assumption (A1) we can consider more generally the

case where R is possibly decomposable; consult Remark 5.3(i). (ii) As it will become clear from the proof of Theorem 1.2, at assumption (A2) it suffices to consider the residue fields κ(𝔪) associated to the maximal and minimal prime ideals of R; consult Remark 5.3(ii).

Due to their generality and simplicity, we believe that Theorems 1.1-1.2 will soon be part of the toolkit of every mathematician interested in Picard groups of triangulated categories. In the next section, we illustrate the usefulness of these results by computing the Picard group of several important categories of motivic nature; consult also §2.6 for a topological application.

Picard Groups, Weight Structures, and (NC) Motives 47

2. Applications

Let k be a base field, which we assume perfect, and R a commutative ring of coefficients, which we assume indecomposable and Noetherian. Voevodsky’s category of geometric mixed motives DM_{gm}(k; R) (see [14, 24]), Morel-Voevodsky’s stable 𝔸^{1}-homotopy category SH(k) (see [26, 28, 40]), and Kontsevich’s category of noncommutative mixed motives KMM(k; R) (see [19, 20, 21, 34]), play nowadays a central role in the motivic realm. A major challenge, which seems completely out of reach at the present time, is the computation of the Picard group of these symmetric monoidal triangulated categories^{2}. In what follows, making use of Theorems 1.1-1.2, we achieve this goal in the case of certain important subcategories.

2.1. Mixed Artin motives. The category of mixed Artin motives DMA(k; R) is defined as the thick triangulated subcategory of DM_{gm}(k; R) generated by the motives M(X)_{R} of zero-dimensional smooth k-schemes X. The smallest additive, Karoubian, full subcategory of DMA(k; R) containing the objects M(X)_{R} identifies with the (classical) category of Artin motives AM(k; R).

Theorem 2.1. When the degrees of the finite separable field extensions of k are invertible in R, we have Pic(DMA(k; R)) ≃ Pic(AM(k; R)) × ℤ.

Example 2.2. Theorem 2.1 holds, in particular, in the following cases: (i) The field k is arbitrary and R is a ℚ-algebra; (ii) The field k is formally real (e.g. k = ℝ) and 1/2 ∈ R; (iii) Let p be a (fixed) prime number, l a perfect field, and H a Sylow

pro-p-subgroup of Gal(\overline{l}/l). Theorem 2.1 also holds with k := \overline{l}^{H} and 1/p ∈ R.

Whenever R is a field, the R-linearized Galois-Grothendieck correspondence induces a symmetric monoidal equivalence of categories between AM(k; R) and the category Rep_{R}(Γ) of continuous finite dimensional R-linear representations of the absolute Galois group Γ := Gal(\overline{k}/k). Since the ⊗-invertible objects of Rep_{R}(Γ) are the 1-dimensional Γ-representations, Pic(AM(k; R)) ≃ Pic(Rep_{R}(Γ)) identifies with the group of continuous characters from Γ^{ab} to R^{×}. In the particular case where k = ℚ, the profinite group Γ^{ab} agrees with ℤ̂^{×}. Consequently, all the elements of Rep_{R}(Γ) can be represented by Dirichlet characters. Moreover, in the cases where char(k) ≠ 2 and R = ℚ, we have the following computation

k^{×}/(k^{×})^{2} ⟶^{≃} Pic(Rep_{ℚ}(Γ))  λ ↦ (Γ ↠ Gal(k(\sqrt{λ})/k) ⟶^{σ↦−1} ℚ^{×}) ,

where σ stands for the generator of the Galois group Gal(k(\sqrt{λ})/k) ≃ ℤ/2ℤ; see Peter [30, pages 340-341]. A similar computation holds in characteristic 2 with k^{×}/(k^{×})^{2} replaced by k/{λ + λ^{2} | λ ∈ k}.

2Consult Bachmann [4], resp. Hu [17], for the construction of ⊗-invertible objects in the motivic category DM_{gm}(k; ℤ/2ℤ), resp. SH(k), associated to quadrics.

Now, let A(k; R) be an additive, Karoubian, symmetric monoidal, full subcategory of AM(k; R), and DA(k; R) the thick triangulated subcategory of DMA(k; R) generated by the motives associated to the objects of A(k; R). Under these notations, Theorem 2.1 admits the following generalization:

Theorem 2.3. Assume that there exists a set of finite separable field extensions l_{i}/k, i ∈ I, such that the following two conditions hold:

(B1) Every object of the category A(k; R) is isomorphic to a summand of a finite direct sum of the motives associated to the field extensions l_{i}/k, i ∈ I;

(B2) For each i ∈ I, the degree of the finite field extension l_{i}/k is invertible

in R. Under assumptions (B1)-(B2), we have Pic(DA(k; R)) ≃ Pic(A(k; R)) × ℤ.

Example 2.4 (Mixed Dirichlet motives). Let R be a field. Following Wildeshaus [41, Def. 3.4], a Dirichlet motive is an Artin motive for which the corresponding Γ-representation factors through an abelian (finite) quotient. Take A(k; R) to be the category of Dirichlet motives. In this case, the associated symmetric monoidal triangulated category DA(k; R) is called the category of mixed Dirichlet motives. Since the ⊗-invertible objects of Rep_{R}(Γ) are the 1-dimensional representations, and all these representations factor through an abelian (finite) quotient, the inclusion of categories A(k; R) ⊂ AM(k; R) yields an isomorphism Pic(A(k; R)) ≃ Pic(AM(k; R)). Consequently, in the case where R is of characteristic zero, Theorem 2.3 implies that Pic(DA(k; R)) ≃ Pic(AM(k; R)) × ℤ. Intuitively speaking, the difference between (mixed) Dirichlet motives and (mixed) Artin motives is not detected by the Picard group.

2.2. Mixed Artin-Tate motives. The category DMAT(k; R) of mixed Artin-Tate motives is defined as the thick symmetric monoidal triangulated subcategory of DM_{gm}(k; R) generated by the motives M(X)_{R} of zero-dimensional smooth k-schemes X and by the Tate motives R(m), m ∈ ℤ.

Theorem 2.5. When the degrees of the finite separable field extensions of k are invertible in R, we have DMAT(k; R) ≃ Pic(AM(k; R)) × ℤ × ℤ.

Now, let A(k; R) be an additive, Karoubian, symmetric monoidal, full subcategory of AM(k; R), and DAT(k; R) the thick symmetric monoidal triangulated subcategory of DMAT(k; R) generated by the motives associated to the objects of A(k; R) and by the Tate motives R(m), m ∈ ℤ. Theorem 2.5 admits the following generalization:

Theorem 2.6. Assume that there exists a set of field extensions l_{i}/k, i ∈ I, as in Theorem 2.3. Under these assumptions, we have Pic(DAT(k; R)) ≃ Pic(A(k; R)) × ℤ × ℤ.

Example 2.7 (Mixed Tate motives). Take A(k; R) to be the smallest additive, Karoubian, full subcategory of AM(k; R) containing the ⊗-unit. In this case, the associated symmetric monoidal triangulated category DAT(k; R) is called the category of mixed Tate motives. Since A(k; R) identifies with the category of finitely generated projective R-modules^{3}, we conclude from Theorem 2.6 that the Picard group of DAT(k; R) is isomorphic to Pic(R) × ℤ × ℤ. Note that we are not imposing the invertibility of any integer in R.

Example 2.8 (Mixed Dirichlet-Tate motives). Take A(k; R) to be the category of Dirichlet motives. In this case, the associated symmetric monoidal triangulated category DAT(k; R) is called the category of mixed Dirichlet-Tate motives. Recall from Example 2.4 that the Picard group of A(k; R) is isomorphic to the Picard group of AM(k; R). Consequently, in the case where R is of characteristic zero, Theorem 2.6 implies that Pic(DAT(k; R)) ≃ Pic(AM(k; R)) × ℤ × ℤ.

2.3. Motivic spectra. The bootstrap category Boot(k) is defined as the thick triangulated subcategory of SH(k) generated by the ⊗-unit Σ^{∞}(Spec(k)_{+}). The former category contains a lot of information. For example, as proved by Levine in [22, Thm. 1], whenever k is algebraically closed and of characteristic zero, the category Boot(k) identifies with the homotopy category of finite spectra SH_{c}. In particular, we have non-trivial negative Ext-groups

(2.9) Hom_{Boot(k)}(Σ^{∞}(Spec(k)_{+}), Σ^{∞}(Spec(k)_{+})[−n]) ≃ π_{n}(𝕊)  n > 0 ,

where 𝕊 stands for the sphere spectrum. Moreover, as proved by Morel in [25, Thm. 6.2.2], whenever k is of characteristic ≠ 2, we have a ring isomorphism

(2.10)  End_{Boot(k)}(Σ^{∞}(Spec(k)_{+})) ≃ GW (k) ,

where GW (k) stands for the Grothendieck-Witt ring of k.

Theorem 2.11. Assume that char(k) ≠ 2 and that GW (k) is Noetherian. Under these assumptions, we have Pic(Boot(k)) ≃ Pic(GW (k)) × ℤ.

Remark 2.12. The ring GW (k) is Noetherian if and only if k^{×}/(k^{×})^{2} is finite.

Example 2.13. Theorem 2.11 holds, in particular, in the following cases:

(i) The field k is quadratically closed (e.g. k is algebraically closed or the field of constructible numbers). In this case, we have GW (k) ≃ ℤ; (ii) The field k is the field of real numbers ℝ. In this case, we have GW (ℝ) ≃ ℤ[C_{2}], where C_{2} stands for the cyclic group of order 2;

(iii) The field k is the finite field 𝔽_{q} with q odd. In this case, k^{×}/(k^{×})^{2} = C_{2}.

Intuitively speaking, Theorem 2.11 shows that none of the motivic spectra which are built using the non-trivial Ext-groups (2.9) is ⊗-invertible!

3Recall that the Picard group Pic(R) of a Dedekind domain R is its ideal class group C(R).

2.4. Noncommutative mixed Artin motives. The category of noncommutative mixed Artin motives NMAM(k; R) is defined as the thick triangulated subcategory of KMM(k; R) generated by the noncommutative motives U(l)_{R} of finite separable field extensions l/k. The smallest additive, Karoubian, full subcategory of NMAM(k; R) containing the objects U(l)_{R} identifies with AM(k; R). The category of noncommutative mixed Artin motives is in general much richer than the category of mixed Artin motives. For example, whenever R is a ℚ-algebra, DMA(k; R) identifies with the category Gr_{ℤ}AM(k; R) of ℤ-graded objects in AM(k; R); see [39, page 217]. This implies that DMA(k; R) has trivial higher Ext-groups. On the other hand, given any two finite separable field extensions l_{1}/k and l_{2}/k, we have non-trivial negative Ext-groups (see [33, §4])

(2.14) Hom_{NMAM(k;R)}(U(l_{1})_{R}, U(l_{2})_{R}[−n]) ≃ K_{n}(l_{1} ⊗_{k} l_{2})_{R}  n > 0 ,

where K_{n}(l_{1} ⊗_{k} l_{2}) stands for the n^{th} algebraic K-theory group of l_{1} ⊗_{k} l_{2}. Roughly speaking, NMAM(k; R) contains not only AM(k; R) but also all the higher algebraic K-theory groups of finite separable field extensions. For example, given a number field 𝔽, we have the following computation (due to Borel [12, §12])

Hom_{NMAM(ℚ;ℚ)}(U(ℚ)_{ℚ}, U(𝔽)_{ℚ}[−n]) ≃ \{ \begin{matrix}ℚ^{r_{2}} & n ≡ 3 (mod 4) \\ ℚ^{r_{1}+r_{2}} & n ≡ 1 (mod 4) \\ 0 & otherwise \end{matrix}  n ≥ 2 ,

where r_{1} (resp. r_{2}) stands for the number of real (resp. complex) embeddings of 𝔽.

Theorem 2.15. When the degrees of the finite separable field extensions of k are invertible in R, we have Pic(NMAM(k; R)) ≃ Pic(AM(k; R)) × ℤ.

Example 2.16. Theorem 2.15 holds in the cases (i)-(iii) of Example 2.2.

Theorem 2.15 shows that although the category NMAM(k; R) is much richer than DMA(k; R), this richness is not detected by the Picard group. Now, let A(k; R) be an additive, Karoubian, symmetric monoidal, full subcategory of AM(k; R), and NMA(k; R) the thick triangulated subcategory of NMAM(k; R) generated by the noncommutative motives associated to the objects of A(k; R). Theorem 2.15 admits the following generalization:

Theorem 2.17. Assume that there exists a set of field extensions l_{i}/k, ∈ I, as in Theorem 2.3. Under these assumptions, we have Pic(NMA(k; R)) ≃ Pic(A(k; R)) × ℤ.

Example 2.18 (Noncommutative mixed Dirichlet motives). Take A(k; R) to be the category of Dirichlet motives. In this case, the associated symmetric monoidal triangulated category NMA(k; R) is called the category of noncommutative mixed Dirichlet motives. Recall from Example 2.4 that the Picard group of A(k; R) is isomorphic to Pic(AM(k; R)). Consequently, in the case where R is of characteristic zero, Theorem 2.15 implies that Pic(NMA(k; R)) ≃ Pic(AM(k; R)) × ℤ. Roughly speaking, the difference between mixed Dirichlet motives and noncommutative mixed Dirichlet motives is not detected by the Picard group.

Example 2.19 (Bootstrap category). Take A(k; R) to be the smallest additive, Karoubian, full subcategory of AM(k; R) containing the ⊗-unit. In this case, the associated symmetric monoidal triangulated category NMA(k; R) is called the bootstrap category. Since A(k; R) identifies with the category of finitely generated projective R-modules, we conclude from Theorem 2.17 that Pic(NMA(k; R)) ≃ Pic(R) × ℤ. Similarly to Example 2.7, we are not imposing the invertibility of any integer in R.

2.5. Noncommutative mixed motives of central simple algebras. Let us denote by NMCSA(k; R) the thick triangulated subcategory of KMM(k; R) generated by the noncommutative motives U(A)_{R} of central simple k-algebras A. In the same vein, let CSA(k; R) be the smallest additive, Karoubian, full subcategory of NMCSA(k; R) containing the objects U(A)_{R}. As proved in [35, Thm. 9.1], given any two central simple k-algebras A and B, we have the following equivalence

(2.20)  U(A)_{ℤ} ≃ U(B)_{ℤ} ⇔ [A] = [B] ∈ Br(k) ,

where Br(k) stands for the Brauer group of k. Intuitively speaking, (2.20) shows that the noncommutative motive U(A)_{ℤ} and the Brauer class [A] contain exactly the same information. We have moreover non-trivial negative Extgroups:

(2.21) Hom_{NMCSA(k;ℤ)}(U(A)_{ℤ}, U(B)_{ℤ}[−n]) ≃ π_{n}(K(A^{op} ⊗_{k} B) ∧ Hℤ) n > 0 ,

where Hℤ stands for the Eilenberg-MacLane spectrum of ℤ. Roughly speaking, the category NMCSA(k; ℤ) contains information not only about the Brauer group but also about all the higher algebraic K-theory of central simple algebras.

Theorem 2.22. The following holds:

(i) We have an isomorphism Pic(NMCSA(k; R)) ≃ Pic(CSA(k; R)) × ℤ; (ii) We have an isomorphism Pic(CSA(k; ℤ)) ≃ Br(k).

Remark 2.23. Let R be a field. As explained in Remark 10.6, the Picard group of the category Pic(CSA(k; R)) is trivial when char(R) = 0 and isomorphic to Br(k){p} when char(R) = p > 0.

Intuitively speaking, Theorem 2.22 shows that none of the noncommutative mixed motives which are built using the non-trivial negative Ext-groups (2.21) is ⊗-invertible!

2.6. A topological application. Let E be a commutative symmetric ring spectrum and D_{c}(E) the associated derived category of compact E-modules; see [15, 31].

Theorem 2.24. Assume that the ring spectrum E is connective, i.e. π_{n}(E) = 0 for every n < 0, and that π_{0}(E) is an indecomposable Noetherian ring. Under these assumptions, we have Pic(D_{c}(E)) ≃ Pic(π_{0}(E)) × ℤ.

Example 2.25 (Finite spectra). Let E be the sphere spectrum 𝕊. In this case, the category D_{c}(𝕊) is equivalent to the homotopy category of finite spectra SH_{c} and π_{0}(𝕊) ≃ ℤ. Consequently, we obtain Pic(SH_{c}) ≃ ℤ. This computation was originally established by Hopkins-Mahowald-Sadofsky in [16] using different tools. Note that this computation may be understood as a particular case of Theorem 2.11.

Example 2.26 (Ordinary rings). Let E be the Eilenberg-MacLane spectrum HR of a commutative indecomposable Noetherian ring R. In this case, D_{c}(HR) ≃ D_{c}(R) and π_{0}(HR) ≃ R. Consequently, we obtain Pic(D_{c}(R)) ≃ Pic(R) × ℤ; consult Remark 5.3(i) for the case where R is decomposable. This computation was originally established in [13]. Although Fausk did not use weight structures, one observes that by applying our arguments (see §5) to the triangulated category D_{c}(R), equipped with the weight structure whose heart consists of the finitely generated projective R-modules, one obtains a reasoning somewhat similar to his one.

3. Weight structures

In this section we briefly review the theory of weight structures. This will give us the opportunity to fix some notations that will be used throughout the article.

Definition 3.1. (see [7, Def. 1.1.1]) A weight structure w on a triangulated category T , also known as a co-t-structure in the sense of Pauksztello [29], consists of a pair of additive subcategories (T^{w≥0}, T^{w≤0}) satisfying the following conditions^{4}:

(i) The categories T^{w≥0} and T^{w≤0} are closed under taking summands in T ;

(ii) We have inclusions of categories T^{w≥0} ⊂ T^{w≥0}[1] and T^{w≤0}[1] ⊂

T^{w≤0}; (iii) For every a ∈ T^{w≥0} and b ∈ T^{w≤0}[1], we have Hom_{T} (a, b) = 0; (iv) For every a ∈ T there exists a distinguished triangle c[−1] → a → b → c

in T with b ∈ T^{w≤0} and c ∈ T^{w≥0}.

Given an integer n ∈ ℤ, let T^{w≥n} := T^{w≥0}[−n], T^{w≤n} := T^{w≤0}[−n], and T^{w=n} := T^{w≥n} ∩ T^{w≤n}. The objects belonging to ∪_{n∈ℤ}T^{w=n} are called wpure and the additive subcategory H := T^{w=0} is called the heart of the weight structure. Finally, a weight structure w is called bounded if T = ∪_{n∈ℤ}T^{w≥n} = ∪_{n∈ℤ}T^{w≤n}.

Assumption: Let (T , ⊗, 1) be a symmetric monoidal triangulated category equipped with a weight structure w. Throughout the article, we will always assume that the symmetric monoidal structure is w-pure in the sense that the tensor product − ⊗ − (as well as the ⊗-unit 1) restricts to the heart H.

4Following [7], we will use the so-called cohomological convention for weight structures.

This differs from the homological convention used in [8, 10, 11, 41].

Remark 3.2 (Self-duality). The notion of weight structure is (categorically) selfdual. Given a triangulated category T equipped with a weight structure w, the opposite triangulated category T^{op} inherits the opposite weight structure w^{op} with (T^{op})^{w^{op}≤0} := T^{w≥0} and (T^{op})^{w^{op}≥0} := T^{w≤0}.

Definition 3.3. An exact functor F: T → T′ between triangulated categories equipped with weight structures w and w′, respectively, is called weight-exact if F (T^{w≤0}) ⊆ T′^{w′≤0} and F (T^{w≥0}) ⊆ T′^{w′≥0}.

Remark 3.4. Whenever the weight structure w is bounded, an exact functor F: T → T′ is weight-exact if and only if F (T^{w=0}) ⊆ T^{w′=0}; see [10, Prop. 1.2.3(5)].

3.1. Weight complexes. Let T be a triangulated category equipped with a weight structure w. Following [7, Def. 2.2.1] (see also [8, §2.2]), we can assign to every object a ∈ T a certain (cochain) weight H-complex t(a) : ⋯ → a^{m−1} → a^{m} → a^{m+1} → · · ·. For example, if a ∈ T^{w=0}, then we can take for t(a) the complex ⋯ → 0 → a → 0 → ⋯ supported in degree 0. As explained in loc. cit., the assignment a ↦ t(a) is well-defined only up to homotopy equivalence. Nevertheless, we will use the notation a^{p} for the p^{th} term of some choice of a weight H-complex t(a). This is justified by the next result:

Proposition 3.5. (see [10, Prop. 1.4.2(6)-(7)])

(i) Let F: T → T′ be a weight-exact functor as in Definition 3.3. If t(a) is a weight H-complex for a, then F (t(a)) is a weight H′-complex for F (a);

(ii) Given an additive functor G: H → A, with values in an abelian category, the assignment a ↦ H^{0}(G(t(a))) yields a well-defined (i.e. independent of the choice of t(a)) homological functor^{5} H_{0} : T → A. Moreover, the assignment G ↦ H_{0} is natural in the functor G.

We denote by H_{n} the precomposition of H_{0} with the n^{th} suspension functor of T .

Remark 3.6. Note that if a ∈ T^{w=m}, then H_{n}(a) = 0 for every n ≠ m.

Remark 3.7. Following the referee’s suggestion, we recall here in an informal way the construction of weight complexes. Let T be a triangulated category equipped with a weight structure w. Given a ∈ T and m ∈ ℤ, the axiom (iv) of Definition 3.1 implies the existence of a distinguished triangle b^{m} → a → c^{m} → b^{m}[1] with b^{m} ∈ T^{w≥m} and c^{m} ∈ T^{w≤m−1}. These triangles are not determined (up to isomorphism) by the couple (a, m). Nevertheless, given a morphism g: a → a′ and an integer m′ ≤ m, we can extend g to a morphism between the corresponding triangles; this extension is unique whenever m′ < m. This fact, applied to a fixed object a and to all integers m, yields connecting morphisms ∂^{m}: b^{m+1} → b^{m}. If one shifts the cone of ∂^{m} by [m], we then obtain a sequence of objects a^{m} in T^{w=0}. Moreover, the corresponding triangles give rise to connecting morphisms which yield a weight complex for a. The above considerations show that weight complexes are naturally “respected” by weight-exact functors. This naturality easily carries over to the pure functors considered in the above Proposition 3.5(ii). However, these pure functors do not depend on any choices up to canonical isomorphisms.

5The homological functors obtained this way are called pure due to their relation with Deligne’s theory of weights on cohomology; see [8, Rk. 2.4.5(5)].

3.2. Karoubization. Given a category C, let us write Kar(C) for its Karoubization. Recall that the objects of Kar(C) are the pairs (a, e), with a ∈ C and e an idempotent of the ring of endomorphisms End_{C}(a, a). The morphisms are given by Hom_{Kar(C)}((a, e), (b, e′)) := e ∘ Hom_{C}(a, b) ∘ e′. By construction, Kar(C) comes equipped with the canonical functor C → Kar(C), a ↦ (a, id). Whenever C is symmetric monoidal, resp. triangulated, the category Kar(C) is also symmetric monoidal, resp. triangulated; see [6, Thm. 1.5]. Moreover, the canonical functor C → Kar(C) becomes symmetric monoidal, resp. exact. The following result relates Karoubian categories to bounded weight structures.

Proposition 3.8. Let T be a Karoubian triangulated category. Assume that there exists a full additive subcategory H′ ⊂ T that generates^{6} T and which is negative in T in the sense that there are no T -extensions of positive degrees between objects of H′. Under these assumptions, there exists a unique bounded weight structure w on T such that its heart H contains H′. Moreover, H is equivalent to Kar(H′).

Proof. The proof is an immediate consequence of [7, Thm. 4.3.2 II and Prop. 5.2.2]; consult also [11, Cor. 2.1.2] for the generalization of this statement to the case where T is not necessarily Karoubian. □

4. Proof of Theorem 1.1

We start with the following auxiliary result:

Proposition 4.1. A symmetric monoidal triangulated category (T , ⊗, 1), equipped with a weight structure w, has the w-Picard property (see §1) if and only if all its ⊗-invertible objects are w-pure.

Proof. Let (a, n), (b, m) ∈ Pic(H) × ℤ. On the one hand, when n = m, we have a[n] ≃ b[m] in T if and only if a ≃ b in H. This follows from the fact that the suspension functor is an auto-equivalence of T . On the other hand, when n ≠ m, we have a[n] ≄ b[m] in T . This follows from the fact that Hom_{T} (a[n], b[m]), resp. Hom_{T} (b[m], a[n]), is zero whenever m < n, resp. n < m; see Definition 3.1(iii). This implies that the canonical group homomorphism

(4.2)  Pic(H) × ℤ ⟶ Pic(T )  (a, n) ↦ a[n]

6i.e. the smallest thick triangulated subcategory of T containing H′ is T itself.

is injective. Consequently, we conclude that the category T has the w-Picard property if and only if (4.2) is surjective. In other words, T has the w-Picard property if and only if all its ⊗-invertible objects are w-pure. □

Remark 4.3. Let (T , ⊗, 1) be a symmetric monoidal triangulated category equipped with a weight structure w. The arguments used in the proof of Proposition 4.1 allow us to conclude that if by hypothesis a[n] ⊗ b[m] ≃ 1 for certain objects a, b ∈ H and integers n, m ∈ ℤ, then n = −m and a is the ⊗-inverse of b.

Let us now prove Theorem 1.1. Let b ∈ T be a (fixed) ⊗-invertible object. Thanks to Proposition 4.1, it suffices to prove that b is w-pure. By assumption, there exists a full, additive, conservative, symmetric monoidal functor G: H → A into a symmetric monoidal semi-simple abelian category which is moreover local. Proposition 3.5(ii) applied to this functor G yields well-defined homological functors H_{n}: T → A, n ∈ ℤ. Consider the homological functor T → A, a ↦ H_{0}(a ⊗ b). Since by assumption the weight structure w is bounded, [7, Thm. 2.3.2] applied to the preceding homological functor yields a convergent Künneth spectral sequence

(4.4)  E_{1}^{pq} = H_{q}(a^{p} ⊗ b) ⇒ H_{p+q}(a ⊗ b) .

The object a^{p} belongs to the heart H and the functor a^{p} ⊗ −: T → T is weight-exact in the sense of Definition 3.3. Using the fact that t(b) is a weight H-complex for b, we conclude from Proposition 3.5(i) that a^{p} ⊗ t(b) is a weight H-complex for a^{p} ⊗ b. Therefore, the complex computing H_{∗}(a^{p} ⊗ b) can be obtained from the complex computing H_{∗}(b) by tensoring with G(a^{p}) (recall that G is symmetric monoidal). Since the category A is semi-simple, it follows then that H_{q}(a^{p} ⊗b) ≃ G(a^{p})⊗H_{q}(b). Furthermore, the functoriality of the assignment G ↦ H_{0} mentioned in Proposition 3.5(ii) implies that the differential E_{1}^{pq} → E_{1}^{(p+1)q} equals the corresponding morphism induced by the boundary a^{p} → a^{p+1} (tensored with b). Making use once again of the semi-simplicity of A, we conclude that E_{2}^{pq} ≃ H_{p}(a) ⊗ H_{q}(b). Recall from [7, Thm. 2.3.2] that, in contrast with the E_{1}-terms, the E_{2}-terms are essentially independent of the choice of (the terms of) the weight complex t(a). Let us denote by m_{a}, resp. m′_{a}, the smallest, resp. largest, integer such that H_{n}(a) = 0 for every n < m_{a}, resp. n > m′_{a}; the existence of such integers follows from the fact that the weight structure w is bounded. Similarly, let m_{b}, resp. m′_{b}, be the smallest, resp. largest, integer such that H_{n}(b) = 0 for every n < m_{b}, resp. n > m′_{b}. Since by assumption the category A is local, we have H_{m_{a}}(a)⊗H_{m_{b}}(b) ≠ 0 and H_{m′_{a}}(a) ⊗ H_{m′_{b}}(b) ≠ 0. Using the second page of the spectral sequence (4.4), we conclude that

(4.5)  H_{m_{a}+m_{b}}(a ⊗ b) ≠ 0 and H_{m′_{a}+m′_{b}}(a ⊗ b) ≠ 0 .

Now, recall that b is a ⊗-invertible object. Therefore, by definition, we have a⊗b ≃ 1 for some (⊗-invertible) object a ∈ T . Since H_{n}(a⊗b) ≃ H_{n}(1) = 0 for every n ≠ 0, we conclude from (4.5) that m_{b} = m′_{b}, m_{a} = m′_{a}, and m_{a} = −m_{b}.

Thanks to Proposition 4.6 below, this implies that b ∈ T^{w=m_{b}}. In particular, the object b is w-pure, and so the proof is finished.

Proposition 4.6. (Conservativity I) Let T be a triangulated category equipped with a bounded weight structure w. Assume that there exists a full, additive, conservative functor G: H → A from the heart of w into a semi-simple abelian category. Under this assumption, an object b ∈ T belongs to T^{w=m} if and only if H_{n}(b) = 0 for every n ≠ m.

Proof. Consult [8, Cor. 2.3.5]. □

Remark 4.7 (Künneth spectral sequence). (i) Let (T , ⊗, 1) be a symmetric monoidal triangulated equipped with a bounded weight structure w, and G: H → A a symmetric monoidal additive functor. Consider the associated homological functors H_{n}: T → A, n ∈ ℤ. The arguments used in the proof of Theorem 1.1 allow us to conclude that there exists a convergent Künneth spectral sequence

E_{1}^{pq} = H_{q}(a^{p} ⊗ b) ⇒ H_{p+q}(a ⊗ b) .

Assume that the (abelian) category A is moreover semi-simple and local. Then, given any ⊗-invertible object b ∈ T , there exists an integer m_{b} such that H_{n}(b) = 0 for every n ≠ m_{b} and H_{m_{b}}(b) ∈ A is ⊗-invertible. (ii) Given non-zero objects a and b as in item (i), Proposition 4.6 yields the existence of integers m_{a} and m_{b} satisfying the conditions described in the proof of Theorem 1.1. This implies that H_{m_{a}+m_{b}}(a ⊗ b) ≠ 0, and consequently that a ⊗ b ≠ 0. In particular, T is local in the sense of [5, §4]; consult Proposition 4.2 from loc. cit.

5. Proof of Theorem 1.2

Let b ∈ T be a ⊗-invertible object. Thanks to Proposition 4.1, it suffices to prove that b is w-pure. Since the functors ι_{κ(𝔭)} : T → T_{κ(𝔭)} are symmetric monoidal, and by assumption the categories T_{κ(𝔭)} have the w_{κ(𝔭)}-Picard property, the objects ι_{κ(𝔭)}(b) are w_{κ(𝔭)}-pure. Concretely, ι_{κ(𝔭)}(b) belongs to T_{κ(𝔭)}^{w=m_{κ(𝔭)}} for some integer m_{κ(𝔭)} ∈ ℤ. Our goal is to prove that all the integers m_{κ(𝔭)}, with 𝔭 ∈ Spec(R), are equal and that the object b belongs to T^{w=m_{k(𝔭)}}. We start by addressing the first goal. Since by assumption the commutative ring R is indecomposable, its spectrum Spec(R) is connected. Hence, it suffices to verify that m_{κ(𝔭)} = m_{κ(𝔓)} for every 𝔭 ∈ Spec(R) belonging to the closure of a prime ideal 𝔓 ∈ Spec(R); in the particular case where R is moreover an integral domain we can simply take 𝔓 = {0}. Note that the assumptions of Theorem 1.2, as well as the definition of the integers m_{κ(𝔭)} and m_{κ(𝔓)}, are (categorically) self-dual; see Remark 3.2. Therefore, it is enough to verify the inequalities m_{κ(𝔭)} ≥ m_{κ(𝔓)}. Given an R-algebra S, consider the abelian category PShv^{S}(H) of R-linear functors from H^{op} to the category of S-modules. Note that the Yoneda functor

(5.1)  H ⟶ PShv^{S}(H)  a ↦ (c ↦ Hom_{H}(c, a) ⊗_{R} S)

induces a fully faithful embedding of H ⊗_{R} S into the full subcategory of PShv^{S}(H) consisting of projective objects; see [24, Lem. 8.1]. Note also that every R-algebra homomorphism S → S′ gives rise to a functor − ⊗_{S} S′: PShv^{S}(H) → PShv^{S′}(H). Since PShv^{S}(H) is abelian, Proposition 3.5(ii) yields a homological functor

H_{0}^{S}: T ⟶ PShv^{S}(H)  a ↦ (c ↦ H^{0}(Hom_{H}(c, t(a)) ⊗_{R} S)) .

Recall from assumption (A2) that the functor ι_{κ(𝔭)} induces a ⊗-equivalence of categories Kar(H ⊗_{R} κ(𝔭)) ≃ H_{κ(𝔭)}. This implies that H_{0}^{κ(𝔭)} factors through ι_{κ(𝔭)}. Consequently, thanks to Remark 3.6, we have H_{n}^{κ(𝔭)}(b) = 0 for every n ≠ m_{κ(𝔭)}. Let us denote by Q the localization of R/𝔓 at the prime ideal 𝔭. Note that Q is a local Noetherian integral domain with fraction field κ(𝔓). Recall from assumption (A1) that the commutative ring R is Noetherian and that the Rmodules of morphisms of the heart H are finitely generated and flat. Thanks to the universal coefficients theorem, this implies that H_{l}^{Q}(b) ⊗_{Q} κ(𝔭) = H_{l}^{κ(𝔭)}(b), with l being the largest integer such that H_{l}^{Q}(b) ≠ 0. Consequently, by applying the Nakayama lemma to the local ring Q and to the (objectwise) finitely generated Q-module H_{l}^{Q}(b), we conclude that H_{l}^{κ(𝔭)}(b) ≠ 0. Hence, the equality m_{κ(𝔭)} = l holds. Now, since κ(𝔓) is a flat Q-module, the universal coefficients theorem yields that H_{n}^{κ(𝔓)}(b) = 0 for every n > l. This allows us to conclude that l = m_{k(𝔭)} ≥ m_{k(𝔓)}. Let us now address the second goal, i.e. prove that b ∈ T^{w=m} with m := m_{k(𝔭)}. Making use of Remark 3.2 once again, we observe that it suffices to prove that b ∈ T^{w≤m}. Thanks to Proposition 5.2 below, it is enough to verify that H_{n}^{R}(b) = 0 for every n > m. Let us denote by l the largest integer such that H_{l}^{R}(b) ≠ 0. An argument similar to the one used in the preceding paragraph, implies that H_{l}^{R}(b)⊗_{R}κ(𝔭) = H_{l}^{κ(𝔭)}(b) for every 𝔭 ∈ Spec(R). Since H_{n}^{κ(𝔭)}(b) = 0 for all n > m and 𝔭 ∈ Spec(R), we then conclude that H_{n}^{R}(b) = 0 for every n > m. This finishes the proof.

Proposition 5.2 (Conservativity II). Let T be a triangulated category equipped with a bounded weight structure w whose heart H is R-linear and small. Consider the associated homological functors H_{n}^{R}: T → PShv^{R}(H), n ∈ ℤ. Under these assumptions, an object b ∈ T belongs to T^{w≤m} if and only if H_{n}^{R}(b) = 0 for every n > m.

Proof. Combine [8, Prop. 2.3.4] with [8, Rk. 2.3.6(2)]. □

Remark 5.3. (i) Suppose that in Theorem 1.2 the commutative ring R is of the form Π_{j=1}^{r}R_{j}, with R_{j} an indecomposable Noetherian ring. In this case, the corresponding idempotents e_{j} ∈ R give naturally rise to categorical decompositions T ≃ Π_{j=1}^{r}T_{j} and H ≃ Π_{j=1}^{r}H_{j}. By applying Theorem 1.2 to each one of the categories T_{j}, we conclude that

Pic(T ) ≃ Π_{j=1}^{r}Pic(T_{j}) ≃ Π_{j=1}^{r}(Pic(H_{j}) × ℤ) ≃ Pic(H) × ℤ^{r}

whenever all the triangulated categories T_{j} are non-zero; (ii) At assumption (A2) of Theorem 1.2, instead of working with all prime ideals 𝔭 ∈ Spec(R), note that it suffices to consider any connected subset of Spec(R) that contains all maximal ideals of R. For example, in the particular case where R is local, it suffices to consider the (unique) closed point 𝔭_{0} of Spec(R).

6. Proof of Theorem 2.3

Recall from [24, Part 4 and Lecture 20][39] the construction of the symmetric monoidal triangulated category DM_{gm}(k; R). Given any two zero-dimensional smooth k-schemes X and Y, we have trivial positive Ext-groups:

Hom_{DMA(k;R)}(M(X)_{R}, M(Y)_{R}[n]) = 0  n > 0 .

This implies that the subcategory AM(k; R) ⊂ DMA(k; R) is negative in the sense of Proposition 3.8. Consequently, the subcategory A(k; R) ⊂ DA(k; R) is also negative. Making use of Proposition 3.8, we then conclude that the DA(k; R) carries a bounded weight structure w_{R} with heart A(k; R). Let us now show that the category DA(k; R) has the w_{R}-Picard property; note that this automatically concludes the proof. By construction, A(k; R) is essentially small. Moreover, we have natural isomorphisms

Hom_{DA(k;R)}(M(X)_{R}, M(Y)_{R}) ≃ CH^{0}(X × Y)_{R} .

Since the R-modules CH^{0}(X × Y)_{R} are free, assumptions (A1) of Theorem 1.2 are verified. In what concerns assumptions (A2), take for T_{κ(𝔭)} the category DA(k; κ(𝔭)) and for ι_{κ(𝔭)} the functor − ⊗_{R} κ(𝔭): DA(k; R) → DA(k; κ(𝔭)). By construction, the latter functor is weight-exact (see Remark 3.4), symmetric monoidal, and induces an equivalence of symmetric monoidal categories between Kar(A(k; R) ⊗_{R} κ(𝔭)) and A(k; κ(𝔭)). This shows that assumptions (A2) are also verified. Let us now prove that the categories DA(k; κ(𝔭)) have the w_{κ(𝔭)}-Picard property; thanks to Theorem 1.2 this implies that DA(k; R) has the w_{R}-Picard property. In order to do so, we will make use of Theorem 1.1. Concretely, we will prove that the categories A(k; κ(𝔭)) are abelian semi-simple and local. Let us write L for the composite of the finite separable field extensions l_{i}/k, i ∈ I, inside \overline{k}, G for the profinite Galois group Gal(L/k), and G_{i} for the finite Galois group Gal(l_{i}/k). Thanks to assumption (B1), there is a ⊗-equivalence between A(k; κ(𝔭)) and the category of finite dimensional κ(𝔭)-linear continuous Grepresentations Rep_{κ(𝔭)}(G). Consequently, since G ≃ lim_{i∈I}G_{i}, we conclude that A(k; κ(𝔭)) ≃ colim_{i∈I}Rep_{κ(𝔭)}(G_{i}). Now, since the group G_{i} is finite, the category Rep_{κ(𝔭)}(G_{i}) may be identified with the category of finitely generated (right) κ(𝔭)[G_{i}]-modules. Thanks to assumption (B2), the degree of the field extension l_{i}/k is invertible in R and hence in κ(𝔭). The (classical) Maschke theorem then implies that the category Rep_{κ(𝔭)}(G_{i}) is abelian semi-simple. Note that this category is moreover local since the tensor product is defined on the underlying κ(𝔭)-vector spaces. The proof follows now automatically from the above description of the categories A(k; κ(𝔭)).

7. Proof of Theorem 2.6

Let us denote by AT(k; R) the smallest additive, Karoubian, full subcategory of DAT(k; R) containing the objects M(X)_{R}(m)[2m], with M(X)_{R} ∈ A and m ∈ ℤ. Under these notations, we have trivial positive Ext-groups:

Hom_{DAT(k;R)}(M(X)_{R}(m)[2m], M(Y)_{R}(m′)[2m′][n]) = 0  n > 0 .

This implies that the subcategory AT(k; R) ⊂ DAT(k; R) is negative in the sense of Proposition 3.8. The motives of the zero-dimensional smooth kschemes, as well as the Tate motives, are stable under tensor product. Therefore, AT(k; R) generates^{7} the triangulated category DAT(k; R). Making use of Proposition 3.8 once again, we then conclude that DAT(k; R) carries a bounded weight structure w_{R} with heart AT(k; R). Thanks to the equivalence of categories

Gr_{ℤ}A(k; R) ⟶^{≃} AT(k; R)  {M(X_{m})}_{m∈ℤ} ↦ ⨁_{m∈ℤ} M(X_{m})(m)[2m] ,

an argument similar to the one of the proof of Theorem 2.3 implies that the category DAT(k; R) has the w_{R}-Picard property. Consequently, we have Pic(DAT(k; R)) ≃ Pic(AT(k; R)) × ℤ. The proof follows now from the natural isomorphisms

Pic(AT(k; R)) ≃ Pic(Gr_{ℤ}A(k; R)) ≃ Pic(A(k; R)) × ℤ .

8. Proof of Theorem 2.11

Recall from Ayoub [2, §4][3, §2.1.1] the construction of the symmetric monoidal triangulated category DA(k; ℤ) (with respect to the Nisnevich topology); in what follows, we write Boot(k; ℤ) for the thick triangulated subcategory generated by the ⊗-unit Σ^{∞}(Spec(k)_{+})_{ℤ}. By construction, we have an exact symmetric monoidal functor (−)_{ℤ}: SH(k) → DA(k; ℤ) which restricts to the bootstrap categories. Let P(k), resp. P(k; ℤ), be the smallest additive, Karoubian, full subcategory of Boot(k), resp. Boot(k; ℤ), containing the ⊗- unit Σ^{∞}(Spec(k)_{+}), resp. Σ^{∞}(Spec(k)_{+})_{ℤ}. We have trivial positive Ext-groups (see [40, Thm. 4.14]):

Hom_{Boot(k)}(Σ^{∞}(Spec(k)_{+}), Σ^{∞}(Spec(k)_{+})[n]) = 0 n > 0 ;

similarly for Boot(k; ℤ). This implies that the subcategory P(k) ⊂ Boot(k), resp. P(k; ℤ) ⊂ Boot(k; ℤ), is negative in the sense of Proposition 3.8. Making use of this latter proposition, we then conclude that the category Boot(k), resp. Boot(k; ℤ), carries a bounded weight structure w, resp. w_{ℤ}, with heart P(k), resp. P(k; ℤ).

7i.e. the smallest thick triangulated subcategory containing AT(k; R) is DAT(k; R).

Let us now show that the category Boot(k) has the w-Picard property. Thanks to the ring isomorphism (2.10), P(k) identifies with the category Proj(GW (k)) of finitely generated projective GW (k)-modules. Moreover, the functor (−)_{ℤ} restricts to an equivalence of categories P(k) →^{≃} P(k; ℤ); this is an immediate consequence of [9, Prop. 2.3.7] (this equivalence also follows easily from [27, Thm. 6.37]). Consequently, since the Grothendieck-Witt ring GW (k) is indecomposable (see [18, Prop. 2.2]), all the assumptions (A1) of Theorem 1.2 (with R = GW (k)) are verified. In what concerns assumptions (A2), take for T_{κ(𝔭)} the bounded derived category D^{b}(κ(𝔭)) of finite dimensional κ(𝔭)-vector spaces Vect(κ(𝔭)) and for ι_{κ(𝔭)} the composed functor:

(8.1) Boot(k) ⟶^{(−)_{ℤ}} Boot(k; ℤ) ⟶^{t(−)} K^{b}(Proj(GW (k))) ⟶^{−⊗_{GW (k)}κ(𝔭)} D^{b}(κ(𝔭)) .

Some explanations are in order: since the category DA(k; ℤ) is defined as the localization of a certain category of complexes, it admits a tensor differential graded (=dg) enhancement. Making use of [4, Lem. 18], we then conclude that the weight complex construction gives rise to an exact symmetric monoidal functor t(−) with values in the bounded homotopy category of Proj(GW (k)). By construction, the composed functor (8.1) is weightexact, symmetric monoidal, and induces a ⊗-equivalence of categories between Kar(P(k) ⊗_{GW (k)} κ(𝔭)) and Vect(κ(𝔭)). This shows that the assumptions (A2) are also verified. Finally, since the categories D^{b}(κ(𝔭)) clearly have the w_{κ(𝔭)}- Picard property, we conclude from Theorem 1.2 that Boot(k) has the w-Picard property. This finishes the proof.

9. Proof of Theorem 2.17

Recall from [34, §9][33, §4] the construction of the symmetric monoidal triangulated category KMM(k; R). Given any two finite separable field extensions l_{1}/k and l_{2}/k, we have trivial positive Ext-groups (see [33, Prop. 4.4]):

Hom_{NMAM(k;R)}(U(l_{1})_{R}, U(l_{2})_{R}[n]) ≃ π_{−n}(K(l_{1} ⊗_{k} l_{2}) ∧ HR) = 0  n > 0 .

This implies that the subcategory AM(k; R) ⊂ NMAM(k; R) is negative in the sense of Proposition 3.8. Consequently, the subcategory A(k; R) ⊂ NMA(k; R) is also negative. Making use of Proposition 3.8, we then conclude that the category NMA(k; R) carries a bounded weight structure^{8} w_{R} with heart A(k; R). Now, a proof similar to the one of Theorem 2.3, with DA(k; R) and DA(k; κ(𝔭)) replaced by NMA(k; R) and NMA(k; κ(𝔭)), respectively, allows us to conclude that the category NMA(k; R) has the w_{R}-Picard property. Consequently, we have Pic(NMA(k; R)) ≃ Pic(A(k; R)) × ℤ.

8A bounded weight structure on the category of noncommutative mixed motives was originally constructed in [36, Thm. 1.1].

10. Proof of Theorem 2.22

Item (i). Similarly to the proof of Theorem 2.17, given any two central simple k-algebras A and B, we have trivial positive Ext-groups (see [33, Prop. 4.4]):

Hom_{NMCSA(k;R)}(U(A)_{R}, U(B)_{R}[n]) ≃ π_{−n}(K(A^{op} ⊗_{k} B) ∧ HR) = 0 n > 0 .

This implies that the subcategory CSA(k; R) ⊂ NMCSA(k; R) is negative in the sense of Proposition 3.8. Making use of this latter proposition, we then conclude that NMCSA(k; R) carries a bounded weight structure w_{R} with heart CSA(k; R). Let us now show that the category NMCSA(k; R) has the w_{R}-Picard property. By construction, the category CSA(k; R) is essentially small. Moreover, since the K-theory spectrum K(A^{op} ⊗_{k} B) is connective, we have natural isomorphisms

Hom_{CSA(k;R)}(U(A)_{R}, U(B)_{R}) ≃ π_{0}(K(A^{op} ⊗_{k} B) ∧ HR)
(10.1)  ≃ π_{0}(K(A^{op} ⊗_{k} B)) ⊗_{ℤ} R
≃ K_{0}(A^{op} ⊗_{k} B) ⊗_{ℤ} R ≃ R ,

where (10.1) follows from the stable Hurewicz theorem. This implies, in particular, that the assumptions (A1) of Theorem 1.2 are verified. In what concerns assumptions (A2), take for T_{κ(𝔭)} the category NMCSA(k; κ(𝔭)) and for ι_{κ(𝔭)} the functor − ⊗_{R} κ(𝔭): NMCSA(k; R) → NMCSA(k; κ(𝔭)). By construction, the latter functor is weight-exact (see Remark 3.4), symmetric monoidal, and induces an equivalence of symmetric monoidal categories between Kar(CSA(k; R) ⊗_{R} κ(𝔭)) and CSA(k; κ(𝔭)). This shows that the assumptions (A2) are also verified. We now claim that the categories NMCSA(k; κ(𝔭)) have the w_{κ(𝔭)}-Picard property; thanks to Theorem 1.2 this implies that the category NMCSA(k; R) has the w_{R}-Picard property. Since the categories of finite dimensional (graded) vector spaces are local, our claim follows then from the combination of Theorem 1.1 with the following general result (with R = κ(𝔭)):

Proposition 10.2. Let R be a field.

(a) When char(R) = 0, the category CSA(k; R) is ⊗-equivalent to the category of finite dimensional R-vector spaces vect(R);

(b) When char(R) = p > 0, there exists a full, additive, conservative, symmetric monoidal functor from CSA(k; R) into the category of finite dimensional Br(k){p}-graded R-vector spaces Gr_{Br(k){p}}vect(R).

Proof. Given an R-linear, additive, Karoubian, rigid^{9} symmetric monoidal category (C, ⊗, 1), with End_{C}(1) ≃ R, recall from [1, §1.4.1 and §1.7.1] the construction of the following categorical ideals

N (a, b) := {f: a → b | ∀g: b → a tr(g ∘ f) = 0}
R(a, b) := {f: a → b | ∀g: b → a id_{a} −(g ∘ f) is invertible} ,

9Recall that a symmetric monoidal category is called rigid if all its objects are dualizable.

where tr(g ∘ f) stands for the categorical trace of the endomorphism g ∘ f. As explained in loc. cit., the categorical ideal N is moreover symmetric monoidal. Item (a). As proved in [38, Thm. 2.1], we have U(A)_{R} ≃ U(k)_{R} for every central simple k-algebra A. Using the natural ring isomorphism End(U(k)_{R}) ≃ R, we then conclude that the category CSA(k; R) is ⊗-equivalent to the category of finite dimensional R-vector spaces vect(R). Item (b). By construction, the category CSA(k; R) is R-linear, additive, and symmetric monoidal. Moreover, all its objects are dualizable and End(U(k)_{R}) ≃ R; see [34, §1.7.1]. As proved in [32, Prop. 6.11], the quotient CSA(k; R)/N is ⊗-equivalent to the category Gr_{Br(k){p}}vect(R). Consequently, we have an induced full, additive, and symmetric monoidal functor

(10.3)  CSA(k; R) ⟶ Gr_{Br(k){p}}vect(R) .

It remains then only to prove that the functor (10.3) is moreover conservative. In order to do so, we will show the inclusion N ⊆ R. Thanks to [1, Prop. 7.1.6], this implies that the quotient functor (10.3) is conservative. By definition, the categorical ideals N and R are compatible with direct sums and summands. Hence, given central simple k-algebras A and B, it suffices to show that the inclusion N (U(A)_{R}, U(B)_{R}) ⊆ R(U(A)_{R}, U(B)_{R}) holds. This inclusion follows now from the combination of the definitions of N and R with Lemma 10.4 below. □

Lemma 10.4. Given a central simple k-algebra A, the following morphism

(10.5) End_{CSA(k;R)}(U(A)_{R}) ⟶ End_{CSA(k;R)}(U(k)_{R}) ≃ R h ↦ tr(h) ,

induced by the categorical trace construction, is invertible.

Proof. By construction, the induced morphism (10.5) is R-linear. Therefore, thanks to the natural isomorphism End(U(A)_{R}) ≃ R, (10.5) is completely determined by the image of the identity of U(A)_{R}. In other words, (10.5) reduces to the morphism R → R, r ↦ r · χ(U(A)_{R}), where χ(U(A)_{R}) stands for the Euler characteristic of the noncommutative motive U(A)_{R}. As proved in [34, Prop. 2.24], the Euler characteristic χ(U(A)_{R}) agrees with the Grothendieck class [HH(A)]_{R} ∈ K_{0}(k)_{R} ≃ R of the Hochschild homology HH(A) of A. Since HH(A) ≃ A/[A, A] ≃ k (see [23, §1.2.12]), we then conclude that (10.5) is the identity. This finishes the proof. □

Remark 10.6. It follows from the proof of Proposition 10.2 that the Brauer group of the symmetric monoidal category CSA(k; R) is trivial when char(R) = 0 and isomorphic to Br(k){p} when char(k) = p > 0.

Item (ii). Thanks to equivalence (2.20), we have an injective group homomorphism

(10.7)  Br(k) ⟶ Pic(CSA(k; ℤ))  [A] ↦ U(A)_{ℤ} .

It remains then only to prove that (10.7) is moreover surjective. Recall from [34, §9][36] the construction of the symmetric monoidal triangulated category KMM(k) and of the full subcategories NMCSA(k) and CSA(k). By construction, we have an exact symmetric monoidal functor (−)_{ℤ}: KMM(k) → KMM(k; ℤ) which restricts to a ⊗-equivalence CSA(k) ≃ CSA(k; ℤ). Therefore, making use [37, Thm. 2.20(iv)], we observe that the objects U(A_{1})_{ℤ} ⊕ ⋯ ⊕ U(A_{m})_{ℤ} of CSA(k; ℤ), with m > 1 are not ⊗-invertible. Since the category CSA(k; ℤ) is Karoubian (see [37, Thm. 2.20(i)]), we then conclude that (10.7) is moreover surjective.

Remark 10.8. Given any two central simple k-algebras A and B, we have

Hom_{NMCSA(k)}(U(A)_{R}, U(B)_{R}[n]) ≃ K_{−n}(A^{op} ⊗_{k} B) = 0  n > 0 .

Therefore, a proof similar to the one of Theorem 2.22, with NMCSA(k; ℤ) replaced by NMCSA(k), allows us to conclude that Pic(NMCSA(k)) ≃ Br(k)× ℤ. In conclusion, although the categories NMCSA(k) and NMCSA(k; ℤ) are not equivalent, they have nevertheless the same Picard group!

11. Proof of Theorem 2.24

Let us denote by P(E) the smallest additive, Karoubian, full subcategory of D_{c}(E) containing the E-module E. Since by assumption the ring spectrum E is connective, we have trivial positive Ext-groups:

Hom_{D_{c}(E)}(E, E[n]) ≃ π_{−n}(E) = 0  n > 0 .

This implies that the subcategory P(E) ⊂ D_{c}(E) is negative in the sense of Proposition 3.8. Making use of this latter proposition, we then conclude that the category D_{c}(E) carries a bounded weight structure w with heart P(E). Let us now show that the category D_{c}(E) has the w-Picard property. By construction, P(E) identifies with the category of finitely generated projective π_{0}(R)-modules. Therefore, by taking R := π_{0}(E), all the assumptions (A1) of Theorem 1.2 are verified. In what concerns assumptions (A2), take for T_{k(𝔭)} the category D^{b}(k(𝔭)), equipped with the canonical bounded weight structure with heart Vect(k(𝔭)), and for ι_{k(𝔭)} the (composed) base-change functor

D_{c}(E) ⟶^{−∧_{E}Hπ_{0}(E)} D_{c}(Hπ_{0}(E)) ≃ D_{c}(R) ⟶^{−⊗_{R}k(𝔭)} D^{b}(k(𝔭)) .

By construction, the latter functor is weight-exact (see Remark 3.4), symmetric monoidal, and induces a ⊗-equivalence of categories between Kar(P(E)⊗_{R}κ(𝔭)) and Vect(k(𝔭)). Since the categories D^{b}(k(𝔭)) clearly have the w_{k(𝔭)}-property, we conclude from Theorem 1.2 that D_{c}(E) has the w-Picard property. Finally, since the category D_{c}(E) has the w-Picard property, we have an isomorphism Pic(D_{c}(E)) ≃ Pic(P(E)) × ℤ. The proof follows now from the fact that Pic(P(E)) is isomorphic to Pic(π_{0}(E)).

Acknowledgments: M. Bondarko is grateful to Vladimir Sosnilo, Qiaochu Yuan, and to the users of the Mathoverflow forum for their really helpful comments. G. Tabuada is grateful to Joseph Ayoub, Andrew Blumberg, and Niranjan Ramachandran for useful conversations. The authors are also grateful to Tom Bachmann for commenting a previous version of this article and for kindly informing us that some related results, concerning the categories DM_{gm}(k; R) and SH(k) and whose proofs use others methods, will appear in his Ph.D. thesis. Finally, the authors would like to thank the anonymous referee for his/her comments.

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Mikhail Bondarko Universitetsky pr. 28 St. Petersburg State University the Faculty of Higher Algebra and Number Theory 198504 St. Petersburg Russia m.bondarko@spbu.ru Gonçalo Tabuada Department of Mathematics MIT Cambridge MA 02139 USA; Departamento de Matemática FCT-UNL Portugal; CMA Centro de Matemática e Aplicações, Portugal. tabuada@math.mit.edu