ADDITIVE INVARIANTS OF TORIC AND TWISTED PROJECTIVE HOMOGENEOUS VARIETIES

VIA NONCOMMUTATIVE MOTIVES

GONÇALO TABUADA

Abstract. I. Panin proved in the nineties that the algebraic K-theory of twisted projective homogeneous varieties can be expressed in terms of central simple algebras. Later, Merkurjev and Panin described the algebraic K-theory of toric varieties as a direct summand of the algebraic K-theory of separable algebras. In this article, making use the recent theory of noncommutative motives, we extend Panin and Merkurjev-Panin computations from algebraic K-theory to every additive invariant. As a first application, we fully compute the cyclic homology (and all its variants) of twisted projective homogeneous varieties. As a second application, we show that the noncommutative motive of a twisted projective homogeneous variety is trivial if and only if the Brauer classes of the associated central simple algebras are trivial. Along the way we construct a fully-faithful ⊗-functor from Merkurjev-Panin’s motivic category to Kontsevich’s category of noncommutative Chow motives, which is of independent interest.

Dedicated to the memory of Daniel Kan.

1. Introduction

Algebraic K-theory of twisted projective homogeneous varieties. Let G be a split semisimple algebraic group over a field k, P ⊂ G a parabolic subgroup, and γ : 𝔤 := Gal(k_{sep}/k) → G(k_{sep}) a 1-cocycle. Out of this data one can construct the projective homogeneous variety F := G/P as well as its twisted form _{γ}F. Let G̃ and P̃ be the universal covers of G and P, R(G̃) and R(P̃) the associated representation rings, n the index [W (G̃) : W (P̃)] of the Weyl groups, Z̃ the center of G̃, and finally Ch := Hom(Z̃, 𝔾_{m}) the character group. Under these notations, Panin proved in [21, Thm. 4.2] that every Ch-homogeneous basis ρ_{1}, …, ρ_{n} of R(P̃) over R(G̃) gives rise to an isomorphism

⨁_{i=1}^{n} K_{∗}(A_{χ(i),γ}) ⟶^{∼} K_{∗}(_{γ}F) , (1.1)

where A_{χ(i),γ} stands for the Tits’ central simple algebra associated to ρ_{i}. Panin’s computation (1.1) is a landmark in algebraic K-theory. It generalizes previous results of Grothendieck [2] on flag varieties, of Quillen [17] on Severi-Brauer varieties, of Swan [19] and Kapranov [6] on quadrics hypersurfaces, and of Levine-Srinivas-Weyman [15] on twisted Grassmann varieties.

Date: October 21, 2013. 2000 Mathematics Subject Classification. 11E81, 14A22, 14L17, 14M25, 18F25, 19D55. Key words and phrases. Homogeneous varieties, toric varieties, twisted forms, torsors, non-

commutative motives, algebraic K-theory, cyclic homology, noncommutative algebraic geometry. The author was partially supported by the NEC Award-2742738.

Algebraic K-theory of toric varieties. Let S be a reductive algebraic group over k and A the associated division separable algebra ∏_{ρ} A_{ρ} := ∏_{ρ} End_{S}(W_{ρ}), where the product is taken over all irreducible representation ρ : S → GL(W_{ρ}). Given an S-torsor π : U → X over a smooth projective variety X, assume that there exists an S-equivariant open imbedding of U into an affine space on which S acts linearly. Under these assumptions, Merkurjev and Panin proved in [18, Thm. 4.2] that K_{∗}(X) is a direct summand of K_{∗}(A). Examples include toric models (see Example 3.16) and more generally toric varieties (see Remark 7.4).

Additive invariants. A dg category A, over a field k, is a category enriched over complexes of k-vector spaces; see §4. Let dgcat be the category of (small) dg categories. Every (dg) algebra A gives naturally rise to a dg category with a single object. Another source of examples is provided by schemes since the derived category of perfect complexes perf(X) of every quasi-compact quasi-separated scheme X admits a canonical dg enhancement^{1} perf_{dg}(X); see Keller [7, §4.6].

Given a dg category A, let T (A) be the dg category of pairs (i, x), where i ∈ {1, 2} and x ∈ A. The complex of morphisms in T (A) from (i, x) to (i′, x′) is given by A(x, x′) if i ≤ i′ and is zero otherwise. Composition is induced from A. Intuitively speaking, T (A) “dg categorifies” the notion of upper triangular matrix. Note that we have two inclusion dg functors i_{1} : A ↪ T (A) and i_{2} : A ↪ T (A).

Definition 1.2. Let E : dgcat → D be a functor with values in an additive category. We say that E is an additive invariant if it satisfies the following two conditions:

(i) it sends derived Morita equivalences (see §4) to isomorphisms; (ii) given any dg category A, the inclusion dg functors induce an isomorphism^{2}

[E(i_{1}) E(i_{2})] : E(A) ⊕ E(A) ⟶^{∼} E(T (A)) .

Examples of additive invariants include:

(i) The mixed complex functor C : dgcat → D(Λ) with values in the derived category of mixed complexes; see Keller [8, §1.5 Thm. b) and §1.12].

(ii) The Hochschild homology functor HH : dgcat → D(k) (with values in the derived category of k), the cyclic homology functor HC : dgcat → D(k), the periodic cyclic homology functor HP : dgcat → D_{ℤ/2}(k) (with values in the derived category of ℤ/2-graded complexes of k-vector spaces), and the negative cyclic homology functor HN : dgcat → D(k); see Keller [9, §2.2].

(iii) The connective algebraic K-theory functor K : dgcat → Ho(Spt) with values in the homotopy category of spectra; see Thomason-Trobaugh [31, Thm. 1.9.8] and Waldhausen [32, Thm. 1.4.2].

(iv) The mod-l algebraic K-theory functor K(−; ℤ/l) : dgcat → Ho(Spt), with l

an integer ≥ 2; see [30, §3].

(v) The nonconnective algebraic K-theory functor IK : dgcat → Ho(Spt); see Schlichting [20, §7 Thm. 4 and §12.3 Prop. 3].

(vi) The homotopy algebraic K-theory functor KH : dgcat → Ho(Spt); see [30,

Prop. 3.3]. (vii) The topological Hochschild homology functor T HH : dgcat → Ho(Spt) and

the topological cyclic homology functor T C : dgcat → Ho(Spt); see Blumberg-Mandell [3, Prop. 3.10 and Thm. 10.8] and [26, Prop. 8.9].

1When X is quasi-projective this dg enhancement is unique; see Lunts-Orlov [16, Thm. 2.12].

2Condition (ii) can be equivalently formulated in terms of semi-orthogonal decompositions in the sense of Bondal-Orlov; see [28, Thm. 6.3(4)].

(viii) The universal additive invariant U : dgcat → Hmo_{0} with values in the category

of noncommutative motives; see §5. When applied to A, respectively to perf_{dg}(X), the above additive invariants (i)-(vii) reduce to the classical invariants of (dg) algebras, respectively of schemes: consult [7, Thm. 5.2] for (i)-(ii), [7, Thm. 5.1] for (iii) and (v), [23, Example 2.13] for (iv), [25, Prop. 2.3] for (vi), and [3, Thm. 1.3] for (vii). For this reason, we will write E(X) instead of E(perf_{dg}(X)).

2. Statement of results

Theorem 2.1. (i) Let G, P, γ be as above and E : dgcat → D an additive invariant. Under these assumptions, every Ch-homogeneous basis ρ_{1}, …, ρ_{n} of R(P̃) over R(G̃) gives rise to an isomorphism

(2.2)  ⨁_{i=1}^{n} E(A_{χ(i),γ}) ⟶^{∼} E(_{γ}F) .

(ii) Let S, π, X be as above and E : dgcat → D an additive invariant. Under these assumptions, E(X) is a direct summand of E(A).

Remark 2.3 (Quasi-split case). When G is a quasi-split algebraic group, Panin proved in [21, Thm. 12.4] that a computation similar to (1.1) also holds. In this generality, the algebras A_{χ(i),γ} are no longer central simple but only separable. The analogue of isomorphism (2.2) (with exactly the same proof) also holds.

The proof of Theorem 2.1 is based on the recent theory of noncommutative motives (see §5) and on a fully-faithful ⊗-functor from Merkurjev-Panin’s motivic category to Kontsevich’s category of noncommutative Chow motives (see §6).

3. Applications

Twisted projective homogeneous varieties. By applying (2.2) to the above examples (i)-(viii) of additive invariants we obtain the (concrete) isomorphisms:

⨁_{i=1}^{n} C(A_{χ(i),γ}) ≃ C(_{γ}F)

⨁_{i=1}^{n} HH(A_{χ(i),γ}) ≃ HH(_{γ}F)  ⨁_{i=1}^{n} HH_{∗}(A_{χ(i),γ}) ≃ HH_{∗}(_{γ}F)
⨁_{i=1}^{n} HC(A_{χ(i),γ}) ≃ HC(_{γ}F)  ⨁_{i=1}^{n} HC_{∗}(A_{χ(i),γ}) ≃ HC_{∗}(_{γ}F)
⨁_{i=1}^{n} HP (A_{χ(i),γ}) ≃ HP (_{γ}F)  ⨁_{i=1}^{n} HP_{∗}(A_{χ(i),γ}) ≃ HP_{∗}(_{γ}F)
⨁_{i=1}^{n} HN(A_{χ(i),γ}) ≃ HN(_{γ}F)  ⨁_{i=1}^{n} HN_{∗}(A_{χ(i),γ}) ≃ HN_{∗}(_{γ}F)
⨁_{i=1}^{n} K(A_{χ(i),γ}) ≃ K(_{γ}F)  ⨁_{i=1}^{n} K_{∗}(A_{χ(i),γ}) ≃ K_{∗}(_{γ}F)
⨁_{i=1}^{n} K(A_{χ(i),γ}; ℤ/l) ≃ K(_{γ}F; ℤ/l)  ⨁_{i=1}^{n} K_{∗}(A_{χ(i),γ}; ℤ/l) ≃ K_{∗}(_{γ}F; ℤ/l)
⨁_{i=1}^{n} IK(A_{χ(i),γ}) ≃ IK(_{γ}F)  ⨁_{i=1}^{n} IK_{∗}(A_{χ(i),γ}) ≃ IK_{∗}(_{γ}F)
⨁_{i=1}^{n} KH(A_{χ(i),γ}) ≃ KH(_{γ}F)  ⨁_{i=1}^{n} KH_{∗}(A_{χ(i),γ}) ≃ KH_{∗}(_{γ}F)
⨁_{i=1}^{n} T HH(A_{χ(i),γ}) ≃ T HH(_{γ}F)  ⨁_{i=1}^{n} T HH_{∗}(A_{χ(i),γ}) ≃ T HH_{∗}(_{γ}F)
⨁_{i=1}^{n} T C(A_{χ(i),γ}) ≃ T C(_{γ}F)  ⨁_{i=1}^{n} T C_{∗}(A_{χ(i),γ}) ≃ T C_{∗}(_{γ}F)

⨁_{i=1}^{n} U(A_{χ(i),γ}) ≃ U(_{γ}F)

Note that the left-hand-side isomorphisms enhance the right-hand-side ones. In particular, we obtain a spectral enhancement ⨁_{i=1}^{n} K(A_{χ(i),γ}) ≃ K(_{γ}F) of Panin’s original computation (1.1). In what concerns cyclic homology (and all its variants), we have the following complete computation:

Theorem 3.1. Under the assumptions of Theorem 2.1(i), we have:

⨁_{i=1}^{n} C(k) ≃ C(_{γ}F)
⨁_{i=1}^{n} HH(k) ≃ HH(_{γ}F)  ⨁_{i=1}^{n} HC(k) ≃ HC(_{γ}F)
⨁_{i=1}^{n} HP (k) ≃ HP (_{γ}F)  ⨁_{i=1}^{n} HN(k) ≃ HN(_{γ}F) .

As a consequence of Theorem 3.1 one obtains the isomorphisms:

HH_{j}(_{γ}F) ≃ \{ \begin{matrix}⊕_{i=1}^{n}k & j = 0 \\ 0 & otherwise \end{matrix}  HC_{j}(_{γ}F) ≃ \{ \begin{matrix}⊕_{i=1}^{n}k & j ≥ 0 even \\ 0 & otherwise \end{matrix}

HP_{j}(_{γ}F) ≃ \{ \begin{matrix}⊕_{i=1}^{n}k & j even \\ 0 & n odd \end{matrix}  HN_{j}(_{γ}F) ≃ \{ \begin{matrix}⊕_{i=1}^{n}k & j ≤ 0 even \\ 0 & otherwise \end{matrix} .

Intuitively speaking, Theorem 3.1 shows that cyclic homology (and all its variants) only measures the index n := [W (G̃) : W (P̃)] of the Weyl groups. Under the following restrictions, the same holds for every additive invariant.

Theorem 3.2. The above isomorphism (2.2) reduces to ⨁_{i=1}^{n} E(k) ≃ E(_{γ}F) whenever D is ℤ[\frac{1}{r}]-linear, with r := ∏_{i=1}^{n} degree(A_{χ(i),γ}).

The richest additive invariant is the universal one. In this case we have the following optimal result:

Theorem 3.3. Under the assumptions of Theorem 2.1(i), we have an isomorphism ⨁_{i=1}^{n} U(k) ≃ U(_{γ}F) if and only if the Brauer classes [A_{χ(i),γ}] are trivial.

Example 3.4 (Severi-Brauer varieties). Let G = P GL_{n}. In this case, G̃ = SL_{n}, Z̃ ≃ μ_{n} and Ch ≃ ℤ/nℤ. Consider the following parabolic subgroup

P̃ := \{( \begin{matrix}a & b \\ 0 & c \end{matrix} ) | a · det(c) = 1\} ⊂ SL_{n} ,

where a ∈ k^{×} and c ∈ GL_{n−1}. The associated projective homogeneous variety is F := G/P ≃ G̃/P̃ = ℙ^{n−1}. Given a 1-cocycle 𝔤 → P GL_{n}(k_{sep}) and an additive invariant E : dgcat → D, we conclude from Panin [21, §10.2] and (2.2) that

(3.5)  E(k) ⊕ E(A_{γ}) ⊕ ⋯ ⊕ E(A_{γ}^{⊗n−1}) ⟶^{∼} E(_{γ}ℙ^{n−1}) ,

where A_{γ} is the central simple algebra of degree n associated to γ. Thanks to Theorem 3.2, (3.5) reduces to ⨁_{i=1}^{n} E(k) ≃ E(_{γ}ℙ^{n−1}) whenever D is ℤ[\frac{1}{n}]-linear.

Example 3.6 (Twisted Grassmann varieties). Let G := P GL_{n} as above. Fix a number 1 ≤ m ≤ n − 1 and consider the parabolic subgroup

P̃ := \{( \begin{matrix}a & b \\ 0 & c \end{matrix} ) | det(a) · det(c) = 1\} ⊂ SL_{n} ,

where a ∈ GL_{m} and c ∈ GL_{n−m}. The associated projective homogeneous variety is F := G/P ≃ G̃/P̃ = Gr(m, n). Given a 1-cocycle γ : 𝔤 → P GL_{n}(k_{sep}) and an additive invariant E : dgcat → D, one concludes from [21, §10.2] and (2.2) that

(3.7)  ⨁_{α} E(A_{γ}^{⊗d(α)}) ⟶^{∼} E(_{γ}Gr(m, n)) ,

where α runs over all sequences α_{1}, …, α_{m} such that n − m > α_{1} − 1 > ⋯ > α_{m} − m > −m, and d(α) := α_{1} + ⋯ + α_{m}. Thanks to Theorem 3.2, (3.7) reduces to ⨁_{α} E(k) ≃ E(_{γ}Gr(m, n)) whenever D is ℤ[\frac{1}{n}]-linear.

Example 3.8 (Quadrics). Let G = SO_{n}. In this case, G̃ = Spin_{n}, Z̃ ≃ μ_{2}, and Ch ≃ {±}. Consider the action of G on ℙ^{n−1} by projective linear transformations and let P ⊂ G be the stabilizer of the isotropic point [1 : 0 : ⋯: 0]. The projective homogeneous variety F := G/P is the quadric hypersurface Q in ℙ^{n−1} given by x_{1}y_{1} + ⋯ + x_{[n/2]}y_{[n/2]} + z^{2} = 0 when n is odd and by x_{1}y_{1} + ⋯ + x_{[n/2]}y_{[n/2]} = 0 when n is even. Given a 1-cocycle γ : 𝔤 → SO_{n}(k_{sep}) and an additive invariant E : dgcat → D, we conclude from Panin [21, §10.3] and (2.2) that

(3.9)  E(k)^{⊕(n−2)} ⊕ E(C_{0}(q)) ⟶^{∼} E(_{γ}Q)  n odd
(3.10)  E(k)^{⊕(n−2)} ⊕ E(C_{0}^{+}(q)) ⊕ E(C_{0}^{−}(q)) ⟶^{∼} E(_{γ}Q)  n even ,

where q is the quadric form associated to γ and C_{0}^{+}(q), C_{0}^{−}(q) the simple components of the even Clifford algebra C_{0}(q). Since the degree of the Clifford algebras is a power of 2, we conclude from Theorem 3.2 that the left-hand-side of (3.9) (resp. of (3.10)) reduces to ⨁_{i=1}^{n−1} E(k) (resp. to ⨁_{i=1}^{n} E(k)) whenever D is ℤ[\frac{1}{2}]-linear.

Example 3.11 (Forms of quadrics). Let G = P SO_{n} with n even. Given a 1-cocycle γ : � → P SO_{n}(k_{sep}) and an additive invariant E : dgcat → D, we conclude from [21, §10.3] and (2.2) that

(3.12) (⨁_{i>0_{even}}^{n−3} E(k)) ⊕ (⨁_{i>0_{odd}}^{n−3} E(A)) ⊕ E(C_{0}^{+}(A, σ)) ⊕ E(C_{0}^{−}(A, σ)) ⟶^{∼} E(_{γ}Q) ,

where (A, σ) is the algebra with involution associated to γ. Since A is of degree n, we conclude from Theorem 3.2 that the left-hand-side of (3.12) reduces to ⨁_{i=1}^{n} E(k) whenever D is ℤ[\frac{1}{2}]-linear.

Quasi-split case. As explained in Remark 2.3, when G is a quasi-split algebraic group the algebras A_{χ(i),γ} are only separable. In this generality, we also have the following complete computation (which generalizes Theorem 3.1).

Theorem 3.13. Under the assumptions of Theorem 2.1(i) (with G quasi-split), we have the following isomorphisms: ⨁_{i} C(k) ⊗ HH_{0}(A_{χ(i),γ}) ≃ C(_{γ}F)

⨁_{i} HH(k) ⊗ HH_{0}(A_{χ(i),γ}) ≃ HH(_{γ}F) ⨁_{i} HC(k) ⊗ HH_{0}(A_{χ(i),γ}) ≃ HC(_{γ}F)
⨁_{i} HP (k) ⊗ HH_{0}(A_{χ(i),γ}) ≃ HP (_{γ}F) ⨁_{i} HN(k) ⊗ HH_{0}(A_{χ(i),γ}) ≃ HN(_{γ}F)

where HH_{0}(A_{χ(i),γ}) is the k-vector space A_{χ(i),γ}/[A_{χ(i),γ}, A_{χ(i),γ}].

Intuitively speaking, Theorem 3.13 shows us that in this generality cyclic homology (and all its variants) not only measures the index of the Weyl groups but also the “noncommutativity” of the Tits’ algebras.

Toric varieties. By applying Theorem 2.1(ii) to the above examples (i)-(viii) of additive invariants we obtain several (concrete) direct summands. In the case of cyclic homology (and all its variants) we have the following computation.

Theorem 3.14. Under the assumptions of Theorem 2.1(ii), we have:

C(X) d.s. C(k) ⊗ HH_{0}(A)
HH(X) d.s. HH(k) ⊗ HH_{0}(A)  HC(X) d.s. HC(k) ⊗ HH_{0}(A)
HP (X) d.s. HP (k) ⊗ HH_{0}(A)  HN(X) d.s. HN(k) ⊗ HH_{0}(A) ,

where HH_{0}(A) = ⨁_{ρ}(A_{ρ}/[A_{ρ}, A_{ρ}]) and d.s. stands for “direct summand”.

Remark 3.15. When S is a quasi-split reductive group (e.g. an algebraic torus), the algebras A_{ρ} are commutative; see [18, Remark 4.3].

Example 3.16 (Toric models). Let T be an algebraic torus and X a smooth projective toric T -model. As proved by Merkurjev-Panin in [18, Prop. 5.6], one can construct out of this data an algebraic torus S, an S-torsor π : U → X, and an S-equivariant open embedding of U into an affine space on which S acts linearly. As a consequence, Theorem 2.1(ii) and the above Theorem 3.14 hold in these cases. In what concerns toric varieties, please consult Remark 7.4.

To the best of the author’s knowledge, all the computations obtained in §2-3 are new in the literature.

Notations: We will reserve the letter k for the base field, the letters X, Y, Z for smooth projective k-schemes, and the letters A, B, C for separable k-algebras. Given a small category C, we will write Iso C for the set of isomorphism classes of objects. Finally, (unless stated differently) all tensor products will be taken over k.

Acknowledgments: The author is very grateful to Asher Auel for discussions about twisted projective homogeneous varieties, to Christian Haesemeyer for discussions about toric varieties, to Bjorn Poonen for discussions about the Brauer group, and to Alexander Merkurjev for pointing out his beautiful work with Panin [18]. The author would like also to thank Mikhail Kapranov and Yale’s Department of Mathematics for their hospitality.

4. Differential graded categories

Let C(k) be the category of complexes of k-vector spaces. A differential graded (=dg) category A is a category enriched over C(k). A dg functor F : A → B is a functor enriched over C(k); consult Keller’s ICM survey [7] for details. In what follows, we will write dgcat for the category of (small) dg categories and dg functors.

Let A be a dg category. The category H^{0}(A) has the same objects as A and H^{0}(A)(x, y) := H^{0}A(x, y). The opposite dg category A^{op} has the same objects as A and A^{op}(x, y) := A(y, x). A right A-module is a dg functor A^{op} → C_{dg}(k) with values in the dg category C_{dg}(k) of complexes of k-vector spaces. Let us write C(A) for the category of right A-modules. As explained in [7, §3.1], the dg structure of C_{dg}(k) makes C(A) into a dg category C_{dg}(A). Recall from [7, §3.2] that the derived category D(A) of A is the localization of C(A) with respect to quasi-isomorphisms. Its subcategory of compact objects will be denoted by D_{c}(A).

A dg functor F : A → B is called a derived Morita equivalence if the restriction of scalars functor D(B) →^{∼} D(A) is an equivalence. As proved in [28, Thm. 5.3], dgcat admits a Quillen model structure whose weak equivalences are the derived Morita equivalences. Let Hmo be the associated homotopy category.

The tensor product A ⊗ B of two dg categories A and B is defined as follows: the set of objects is the cartesian product of the sets of objects of A and B and (A ⊗ B)((x, w), (y, z)) := A(x, y) ⊗ B(w, z). As explained in [7, §2.3], this construction gives rise to a symmetric monoidal category (dgcat, − ⊗ −, k).

Given dg categories A and B, an A-B-bimodule B is a dg functor B : A ⊗ B^{op} → C_{dg}(k), i.e. a right (A^{op} ⊗ B)-module. Standard examples are the A-A-bimodule

(4.1)  A ⊗ A^{op} ⟶ C_{dg}(k)  (x, y) ↦ A(y, x)

and more generally the A-B-bimodule

(4.2)  _{F} B : A ⊗ B^{op} ⟶ C_{dg}(k)  (x, w) ↦ B(w, F (x))

associated to a dg functor F : A → B.

Notation 4.3. Given dg categories A and B, let rep(A, B) be the full triangulated subcategory of D(A^{op}⊗B) consisting of those A-B-bimodules B such that B(x, −) ∈ D_{c}(B) for every object x ∈ A. In the same vein, let rep_{dg}(A, B) be the full dg subcategory of C_{dg}(A^{op} ⊗ B) consisting of those A-B-bimodules which belong to rep(A, B). By construction, we have H^{0}(rep_{dg}(A, B)) ≃ rep(A, B).

Kontsevich’s smooth and proper dg categories. Recall from Kontsevich [10, 11, 12] that a dg category A is called smooth if the above A-A-bimodule (4.1) belongs to D_{c}(A^{op} ⊗ A) and proper if for each ordered pair of objects (x, y) we have ∑_{i} dim H^{i}A(x, y) < ∞. The standard examples are the finite dimensional k-algebras of finite global dimension (when k is perfect) and the dg categories perf_{dg}(X) associated to smooth projective k-schemes X. As proved in [4, Thm. 5.8], the smooth and proper dg categories can be characterized as being precisely the rigid (or dualizable) objects of the symmetric monoidal category (Hmo, − ⊗ −, k). Moreover, the dual of A is its opposite dg category A^{op}. This gives rise ([4, Lemma 5.9]) to the following equivalence and derived Morita equivalence

(4.4)  rep(A, B) ≃ D_{c}(A^{op} ⊗ B)  rep_{dg}(A, B) ≃ A^{op} ⊗ B .

5. Noncommutative (Chow) motives

Let A and B be two dg categories. As proved in [28, Cor. 5.10], we have a natural bijection Hom_{Hmo}(A, B) ≃ Iso rep(A, B) under which the composition law of Hmo corresponds to the derived tensor product of bimodules

(5.1)  rep(A, B) × rep(B, C) ⟶ rep(A, C)  (B, B′) ↦ B ⊗_{B}^{L} B′ .

Moreover, the identity of an object A corresponds to the isomorphism class of the A-A-bimodule (4.1). Since the above A-B-bimodules (4.2) clearly belong to rep(A, B), we have a well-defined ⊗-functor

(5.2)  dgcat ⟶ Hmo  F ↦ _{F} B .

The additivization of Hmo is the additive symmetric monoidal category Hmo_{0} with the same objects as Hmo and with abelian groups of morphisms given by Hom_{Hmo_{0}}(A, B) := K_{0}rep(A, B). The composition law is induced from the above bi-triangulated functor (5.1) and the symmetric monoidal structure by bilinearity from Hmo. We have also a well-defined ⊗-functor

(5.3)  Hmo ⟶ Hmo_{0}  B ↦ [B] .

As proved in [28, Thms. 5.3 and 6.3], the composition

U : dgcat ⟶^{(5.2)} Hmo ⟶^{(5.3)} Hmo_{0}

is the universal additive invariant, i.e. given any additive category D there is an induced equivalence of categories

(5.4)  U^{∗} : Fun_{add}(Hmo_{0}, D) ⟶^{∼} Fun_{A}(dgcat, D) ,

where the left-hand-side denotes the category of additive functors and the righthand-side the category of additive invariants in the sense of Definition 1.2. Because of this universal property, which is reminiscent from motives, Hmo_{0} is called the (additive^{3}) category of noncommutative motives; consult the survey article [24]. Since the functors (5.2)-(5.3) are the identity on objects, we will often make no notational distinction between a dg category and its image in Hmo_{0}.

Kontsevich’s noncommutative Chow motives. Kontsevich introduced in [10, 11, 12] the category NChow of noncommutative Chow motives. This category identifies with smallest full additive subcategory of NChow generated by the smooth and proper dg categories. Note that NChow is a rigid symmetric monoidal category. Moreover, thanks to the left-hand-side of (4.4), we have the isomorphisms

(5.5)  Hom_{NChow}(A, B) ≃ K_{0}D_{c}(A^{op} ⊗ B) .

6. From Merkurjev-Panin to Kontsevich

In this section we construct a fully-faithful ⊗-functor Θ from Merkurjev-Panin’s motivic category \underline{C} to Kontsevich’s category of noncommutative Chow motives NChow; see Theorem 6.10. This functor will play a key role in the sequel.

Notation 6.1. In what follows we will write SmProj for the category of smooth projective k-schemes and Sep for the category of separable k-algebras. Given X, Y, Z ∈ SmProj, the projection map from X × Y × Z to X × Y, X × Z, Y × Z will be denoted by p_{XY}^{XY Z}, p_{XZ}^{XY Z}, p_{Y Z}^{XY Z}, respectively.

Merkurjev-Panin’s motivic category \underline{C}. Recall from [18, §1] the construction of the category \underline{C}. The objects are the pairs (X, A) with X ∈ SmProj and A ∈ Sep. The morphisms are given by the Grothendieck groups

Hom\underline{C}((X, A), (Y, B)) := K_{0}vect(X × Y, A^{op} ⊗ B) ,

where vect(X ×Y, A^{op}⊗B) is the exact category of those right (O_{X×Y} ⊗(A^{op}⊗B))- modules which are locally free and of finite rank as O_{X×Y} -modules. Given [F] ∈ K_{0}vect(X × Y, A^{op} ⊗ B) and [F′] ∈ K_{0}vect(Y × Z, B^{op} ⊗ C), their composition [F′] ∘ [F] is defined as

(p_{XZ}^{XY Z})_{∗} ((p_{XY}^{XY Z})_{∗}(F) ⊗_{B} (p_{Y Z}^{XY Z})_{∗}(F′)) ∈ K_{0}vect(X × Z, A^{op} ⊗ C) ,

where the direct image (p_{XZ}^{XY Z})_{∗} is defined only at the K_{0}-theoretical level; consult [18, §1.3] for details. The identity of an object (X, A) is the class [O_{Δ} ⊗ A] ∈ K_{0}vect(X × X, A^{op} ⊗ A), where Δ is the diagonal of X × X. The category \underline{C} comes equipped with a symmetric monoidal structure

\underline{C} × \underline{C} ⟶ \underline{C} ((X, A), (Y, B)) ↦ (X × Y, A ⊗ B)

and with two ⊗-functors

Φ : SmProj^{op} ⟶ \underline{C}  Ψ : Sep ⟶ \underline{C} .

The (contravariant) functor Φ sends X to the pair (X, k) and a map f : X → Y to [O_{Γ_{f}^{t}}] ∈ K_{0}vect(Y × X), where Γ_{f}^{t} stands for the transpose of the graph Γ_{f} := {(x, f(x)) | x ∈ X} ⊂ X × Y of f. On the other hand, the functor Ψ sends A to (Spec(k), A) and a k-algebra homomorphism h : A → B to [_{h}B] ∈ K_{0}(A^{op} ⊗ B).

3A triangulated version also exists in the literature; see [27].

Perfect complexes. Given X ∈ SmProj and A ∈ Sep, let Mod(O_{X} ⊗ A) be the Grothendieck category of right (O_{X} ⊗ A)-modules, D(O ⊗ A) := D(Mod(O_{X} ⊗ A)) its derived category, and perf(X, A) the full subcategory of D(O_{X} ⊗ A) consisting of those complexes of right (O_{X} ⊗ A)-modules which are perfect as complexes of O_{X}-modules. Note that perf(X, A) = rep(A^{op}, perf_{dg}(X)).

Lemma 6.2. The canonical inclusion of categories vect(X, A) ⊂ perf(X, A) gives rise to an isomorphism K_{0}vect(X, A) ≃ K_{0}perf(X, A) of abelian groups.

Proof. Let Coh(X, A) be the abelian category of those right (O_{X} ⊗ A)-modules which are coherent as O_{X}-modules. As explained in [18, §1.1], Quillen’s resolution theorem implies that the canonical inclusion vect(X, A) ⊂ Coh(X, A) give rise to an isomorphism K_{0}vect(X, A) ≃ K_{0}Coh(X, A). Making use of it one defines

(6.3)  K_{0}perf(X, A) ⟶ K_{0}vect(X, A)  G ↦ ∑_{i}(−1)^{i}H^{i}(G) .

A simple verification shows that (6.3) is the inverse of the induced homomorphism K_{0}vect(X, A) → K_{0}perf(X, A). This achieves the proof. □

Let E be an abelian category. As explained in [7, §4.4], the derived category D_{dg}(E) of E is the dg quotient C_{dg}(E)/Ac_{dg}(E) of the dg category of complexes over E by its full dg subcategory of acyclic complexes. In what follows, we will write D_{dg}(O_{X} ⊗A) for the dg category D_{dg}(E), with E := Mod(O_{X} ⊗A), and perf_{dg}(X, A) for the full dg subcategory of those complexes of right (O_{X} ⊗ A)-modules which belong to perf(X, A). By construction, we have H^{0}(D_{dg}(O_{X} ⊗ A)) ≃ D(O_{X} ⊗ A) and H^{0}(perf_{dg}(X, A)) ≃ perf(X, A).

Lemma 6.4. The dg category perf_{dg}(X, A) is smooth and proper.

Proof. Let us start by showing that A, considered as a dg category A, is smooth and proper. Since k is a field, A is a finite dimensional k-algebra; see Knus-Ojanguren [13, §III Prop. 3.2]. This implies properness. As explained in [13, §III Thm. 1.4], A is separable if and only if A is projective as a A-A-bimodule. This implies smoothness. Making use of the right-hand-side of (4.4), one then obtains the following derived Morita equivalence

(6.5)  perf_{dg}(X, A) = rep_{dg}(A^{op}, perf_{dg}(X)) ≃ A ⊗ perf_{dg}(X) .

The proof follows now from the fact that perf_{dg}(X) is smooth and proper. □

Proposition 6.6 (Projection formula). Let f : X → Y be a flat proper morphism in SmProj and A, B, C ∈ Sep. Under these assumptions, we have canonical quasiisomorphisms of dg functors

(G, G′) ↦ (Rf)_{∗}(G) ⊗_{B}^{L} G′ ⟶ (Rf)_{∗}(G ⊗_{B}^{L} f^{∗}(G′))
(H, H′) ↦ H′ ⊗_{B}^{L} (Rf)_{∗}(H) ⟶ (Rf)_{∗}(f^{∗}(H) ⊗_{B}^{L} H) ,

where G ∈ perf_{dg}(X, A^{op} ⊗B), G′ ∈ perf_{dg}(Y, B^{op} ⊗C), H ∈ perf_{dg}(X, B^{op} ⊗C) and H′ ∈ perf_{dg}(Y, A^{op} ⊗ B).

Proof. The proof is similar to the one of Thomason-Trobaugh [31, Prop. 3.17]. □

Proposition 6.7 (Base-change formula). Let A be a separable k-algebra and

[figure: X′; f′; g′; ⌜; X; f; Y′; g; Y]

a cartesian square in SmProj with f flat proper and g flat. Under these assumptions, we have a canonical quasi-isomorphism of dg functors

perf_{dg}(X, A) ∋ G ↦ g^{∗}(Rf)_{∗}(G) → (Rf′)_{∗}(g′)^{∗}(G) .

Proof. The proof is similar to the one of Thomason-Trobaugh [31, Prop. 3.18]. □

Given X, Y ∈ SmProj and A, B ∈ Sep, every G ∈ perf(X × Y, A^{op} ⊗ B) gives rise to the following Fourier-Mukai dg functor

Φ_{G} : perf_{dg}(X, A) ⟶ perf_{dg}(Y, B)  E ↦ (Rp_{Y}^{XY})_{∗} ((p_{X}^{XY})^{∗}(E) ⊗_{A}^{L} G) .

Lemma 6.8. We have a well-defined triangulated functor

perf(X × Y, A^{op} ⊗ B) ⟶ rep(perf_{dg}(X, A), perf_{dg}(Y, B))  G ↦ _{Φ_{G}}B .

Proof. Every morphism η : G → G′ of complexes of right (O_{X×Y} ⊗ (A^{op} ⊗ B))- modules gives rise to a morphism of dg functors Φ_{η} : Φ_{G} ⇒ Φ_{G′} and consequently to a morphism of bimodules _{Φ_{η}}B : _{Φ_{G}}B ⇒ _{Φ_{G′}} B. Whenever α is a quasiisomorphism, H^{0}(Φ_{η}) is a natural isomorphism between triangulated functors. Using [1, Lemma 9.8], one then concludes that _{Φ_{η}}B is a quasi-isomorphism of bimodules. This implies that the functor is well-defined. The fact that it is triangulated is clear by now. □

The functor Θ from \underline{C} to NChow. Let X, Y ∈ SmProj and A, B ∈ Sep. By combining Lemmas 6.2 and 6.8, one obtains the following homomorphism

(6.9) K_{0}vect(X × Y, A^{op} ⊗ B) ⟶^{[F]↦[_{Φ_{F}} B]} K_{0}rep(perf_{dg}(X, A), perf_{dg}(Y, B)) .

Theorem 6.10. The assignments (X, A) ↦ perf_{dg}(X, A) and [F] ↦ [_{Φ_{F}} B] give rise to a fully-faithful ⊗-functor Θ : \underline{C} → NChow making the diagrams commute

\xymatrix{SmProj^{op} \ar[d]_{Φ} \ar[r]^{X↦perf_{dg}(X)} & dgcat \ar[d]^{U} & Sep \ar[d]_{Ψ} \ar[r]^{A↦A} & dgcat \ar[d]^{U} \\ \underline{C} \ar[r]_{Θ} & NChow ⊂ Hmo_{0} & \underline{C} \ar[r]_{Θ} & NChow ⊂ Hmo_{0} .}

Proof of Theorem 6.10. We start by showing that Θ preserves the identities.

Lemma 6.11. For every X ∈ SmProj and A ∈ Sep, the class [_{Φ_{(O_{Δ}⊗A)}}B] agrees with the identity of perf_{dg}(X, A) in NChow.

Proof. Let Id : perf_{dg}(X, A) → perf_{dg}(X, A) be the identity dg functor. Since the identity of perf_{dg}(X, A) in NChow is the class [_{Id}B], one needs to show that

(6.12)  [_{Φ_{(O_{Δ}⊗A)}}B] = [_{Id}B] in K_{0}rep(perf_{dg}(X, A), perf_{dg}(X, A)) .

Let ι be the composition X →^{∼} Δ ⊂ X ×X and p (resp. q) the projection map from X × X to the first (resp. second) component. Under these notations, we have the following natural quasi-isomorphisms

Φ_{(O_{Δ}⊗A)}(E) := (Rq)_{∗} (p^{∗}(E) ⊗_{A}^{L} (O_{Δ} ⊗ A))
≃ (Rq)_{∗} (p^{∗}(E) ⊗_{A}^{L} (Rι)_{∗}(O_{X} ⊗ A))
(6.13)  ≃ (Rq)_{∗} ((Rι)_{∗}(ι^{∗}(p^{∗}(E)) ⊗_{A}^{L} (O_{X} ⊗ A)))
(6.14)  ≃ (Rq)_{∗} ((Rι)_{∗}(ι^{∗}(p^{∗}(E))))
(6.15)  ≃ (RqRι)_{∗}((pι)^{∗}(E)) ≃ E .

Some explanations are in order: (6.13) follows from Proposition 6.6 (with f = ι and B, C equal to A), (6.15) follows from the fact that O_{X} ⊗ A is the ⊗-unit of the symmetric monoidal dg category perf_{dg}(X, A^{op} ⊗ A); and finally (6.15) follows from the equalities qι ≃ id and pι ≃ id. We obtain in this way a quasi-isomorphism Φ_{O_{Δ}⊗A} ⇒ Id of dg functors. As in the proof of Lemma 6.8, we conclude that the bimodules _{Φ_{(O_{Δ}⊗A)}}B and _{Id}B are quasi-isomorphic. This implies the above equality (6.12) and so the proof is finished. □

Given G ∈ perf(X × Y, A^{op} ⊗ B) and G′ ∈ perf(Y × Z, B^{op} ⊗ C), consider the following perfect complex

G ⋆ G′ := (Rp_{XY Z}^{XZ})_{∗} ((p_{XY Z}^{XY})^{∗}(G) ⊗_{B}^{L} (p_{XY Z}^{Y Z})^{∗}(G′)) ∈ perf(X × Z, A^{op} ⊗ C) .

Lemma 6.16. Under the above notations, we have the following equality

[_{Φ_{G⋆G′}} B] = [_{Φ_{G}}B ⊗_{perf_{dg}(Y,B)}^{L} _{Φ_{G′}} B] in K_{0}rep(perf_{dg}(X, A), perf_{dg}(Z, C)) .

Proof. Recall from §4-5 that [_{Φ_{G}}B ⊗_{perf_{dg}(Y,B)}^{L} _{Φ_{G′}} B] = [_{Φ_{G′} ∘Φ_{G}}B]. The proof will consist on showing that_{Φ_{G⋆G′}} B and_{Φ_{G′} ∘Φ_{G}}B are quasi-isomorphic, which implies automatically the above equality. We have the following natural quasi-isomorphisms

Φ_{G⋆G′}(E) := (Rp_{Z}^{XZ})_{∗} ((p_{X}^{XZ})^{∗}(E) ⊗_{A}^{L} (Rp_{XZ}^{XY Z})_{∗}((p_{XY}^{XY Z})^{∗}(G) ⊗_{B}^{L} (p_{Y Z}^{XY Z})^{∗}(G′)))

(Rp_{Z}^{Y Z})_{∗}(Rp_{Y Z}^{XY Z})_{∗} ((p_{XY}^{XY Z})^{∗}((p_{XY}^{XY Z})^{∗}(E) ⊗_{B}^{L} G) ⊗_{B}^{L} (p_{Y Z}^{XY Z})^{∗}(G′)) (6.17)

(Rp_{Z}^{Y Z}) ((Rp_{Y Z}^{XY Z})_{∗}(p_{XY}^{XY Z})^{∗}((p_{XY}^{XY Z})^{∗}(E) ⊗_{A}^{L} G) ⊗_{B}^{L} G′) (6.18)

(6.19)  ≃ (Rp_{Z}^{Y Z})_{∗} ((p_{Y}^{Y Z})^{∗}(Rp_{Y}^{XY})_{∗}((p_{XY}^{XY Z})^{∗}(E) ⊗_{A}^{L} G) ⊗_{B}^{L} G′)
= (Rp_{Z}^{Y Z})_{∗} ((p_{Y}^{Y Z})^{∗}Φ_{G}(E) ⊗_{B}^{L} G′) = Φ_{G′}(Φ_{G}(E)) = Φ_{G′∘G}(E) .

Some explanations are in order: (6.17) follows from Proposition 6.6 (with f = p_{XZ}^{XY Z}) and from the equalities p_{Z}^{XZ}p_{XZ}^{XY Z} = p_{Z}^{XY Z}, p_{X}^{XZ}p_{XZ}^{XY Z} = p_{X}^{XY} p_{XY}^{XY Z} and p_{Z}^{XY Z} = p_{Z}^{Y Z}p_{Y Z}^{XY Z}; (6.18) follows from Proposition 6.6 (with f = p_{XY}^{XY Z}); and finally (6.19) follows from Proposition 6.7 applied to the cartesian square

[figure: X × Y × Z; p_{Y Z}^{XY Z}; p_{XY}^{XY Z}; ⌜; Y × Z; p_{Y}^{Y Z}; X × Y; p_{Y}^{XY}; Y .]

We obtain in this way a zig-zag of quasi-isomorphisms of dg functors between Φ_{G⋆G′} and Φ_{G′∘G}. Similarly to the proof of Lemma 6.11, we conclude that the bimodules Φ_{G⋆G′} B and _{Φ_{G′} ∘Φ_{G}}B are quasi-isomorphic. This achieves the proof. □

Let us now show that Θ preserves the composition law.

Lemma 6.20. Given F ∈ vect(X × Y, A^{op} ⊗ B) and F′ ∈ vect(Y × Z, B^{op} ⊗ C), the image of [F′] ∘ [F] ∈ K_{0}vect(X × Z, A^{op} ⊗ C) under the above homomorphism (6.9) agrees with the class [_{Φ_{G}}B ⊗_{perf_{dg}(Y,B)}^{L} _{Φ_{G′}} B].

Proof. Note that under the isomorphism of Lemma 6.2

K_{0}vect(X × Z, A^{op} ⊗ C) ⟶^{∼} K_{0}perf(X × Z, A^{op} ⊗ C) ,

the composition [F′]∘[F] agrees with [F ⋆F′]. This implies that the image of [F′]∘ [F] under the homomorphism (6.9) agrees with [_{Φ_{F⋆F′}} B]. Thanks to Lemma 6.16, this latter class is equal to [_{Φ_{G}}B ⊗_{perf_{dg}(Y,B)}^{L} _{Φ_{G′}} B] and so the proof is finished. □

The above Lemmas 6.11 and 6.20 imply that the functor Θ is well-defined. Let us now show that it is fully-faithful.

Lemma 6.21. The above homomorphisms (6.9) are isomorphisms.

Proof. Recall first from Lemma 6.2 that the inclusion vect(X × Y, A^{op} ⊗ B) ⊂ perf(X ×Y, A^{op} ⊗B) gives rise to an isomorphism between K_{0}vect(X ×Y, A^{op} ⊗B) and K_{0}perf(X × Y, A^{op} ⊗ B). The proof follows from the sequence of isomorphisms

K_{0}perf(X × Y, A^{op} ⊗ B) = K_{0}rep(A ⊗ B^{op}, perf_{dg}(X × Y))
(6.22)  ≃ K_{0}D_{c}((A ⊗ perf_{dg}(X))^{op} ⊗ (B ⊗ perf_{dg}(Y)))
≃^{(4.4)} K_{0}rep(A ⊗ perf_{dg}(X), B ⊗ perf_{dg}(Y))
≃^{(6.5)} K_{0}rep(perf_{dg}(X, A), perf_{dg}(Y, B)) ,

where (6.22) is a consequence of the derived Morita equivalence

(6.23)  ⊠ : perf_{dg}(X) ⊗ perf_{dg}(Y) ⟶^{∼} perf_{dg}(X × Y)

(established in [29, Prop. 6.2]) and from the fact that perf_{dg}(X)^{op} ≃ perf_{dg}(X). □

By combining the derived Morita equivalences (6.5) and (6.23), we conclude that

perf_{dg}(X, A) ⊗ perf_{dg}(Y, B) ≃ perf_{dg}(X × Y, A ⊗ B)

for every X, Y ∈ SmProj and A, B ∈ Sep, i.e. that Θ is symmetric monoidal. It remains then only to show that the diagrams of Theorem 6.10 are commutative.

Lemma 6.24. For every morphism f : X → Y in SmProj, we have the equality

(6.25)  [_{Φ_{O_{Γ^{t}_{f}}}} B] = [_{f^{∗}}B] in K_{0}rep(perf_{dg}(Y), perf_{dg}(X)) .

Proof. Let ι be the composition X ⟶^{x↦(f(x),x)} Γ_{f}^{t} ⊂ Y × X. Under this notation, we have the following natural quasi-isomorphisms

Φ_{O_{Γ^{t}_{f}}} (E) := (Rp_{X}^{XY})_{∗}((p_{Y}^{XY})^{∗}(E) ⊗ O_{Γ_{f}^{t}})
≃ (Rp_{X}^{XY})_{∗} ((p_{Y}^{XY})^{∗}(E) ⊗ (Rι)_{∗}(O_{X}))
(6.26)  ≃ (Rp_{X}^{XY})_{∗} ((Rι)_{∗}(ι^{∗}((p_{Y}^{XY})^{∗}(E)) ⊗ O_{X}))
(6.27)  ≃ (Rp_{X}^{XY})_{∗} ((Rι)_{∗}(ι^{∗}(p_{Y}^{XY})^{∗}(E)))
(6.28)  ≃ (Rp_{X}^{XY} ι) ∗ ((p_{Y}^{XY} ι)^{∗}(E)) ≃ f^{∗}(E) .

Some explanations are in order: (6.26) follows from Proposition 6.6 (with f = ι and A, B, C equal to k), (6.27) follows from the fact that O_{X} is the ⊗-unit of the symmetric monoidal dg category perf_{dg}(X); and finally (6.28) follows from the equalities p_{X}^{XY} ι = id and p_{Y}^{XY} ι = f. We obtain in this way a quasi-isomorphism Φ_{O_{Γ^{t}_{f}}} ⇒ f^{∗} of dg functors. As in the proof of Lemma 6.11, we conclude that the bimodules _{O_{Γ^{t}_{f}}} B and _{f^{∗}}B are quasi-isomorphisms. This implies the above equality

(6.25) and so the proof is finished. □

Lemma 6.29. The right-hand-side diagram of Theorem 6.10 is commutative.

Proof. Given a morphism h : A → B in Sep, consider the Fourier-Mukai dg functor

(6.30)  Φ_{_{h}B} : perf_{dg}(Spec(k), A) → perf_{dg}(Spec(k), B) .

Making use of the derived Morita equivalence (6.5) (with X = Spec(k)), one observes that the bimodule associated to (6.30) is isomorphic in the homotopy category Hmo to the A-B-bimodule _{h}B ∈ rep(A, B). This achieves the proof. □

7. Proof of Theorem 2.1

Item (i). Since by hypothesis E : dgcat → D is an additive invariant, the equivalence of categories (5.4) furnish us a (unique) additive functor \overline{E} making the following diagram commute

(7.1)  \xymatrix{dgcat \ar[d]_{U} \ar[r]^{E} & D \\ _{Hmo_{0}} \ar@{-->}[ur]_{\overline{E}} & .}

Now, recall from [21, §6] the construction of the category 𝔸^{G} and of the functors

Φ : SmProj^{op} ⟶ 𝔸^{G}  Ψ : Sep ⟶ 𝔸^{G} .

In the particular case where G is the trivial group \{1\}, 𝔸^{\{1\}} identifies with \underline{C} and Φ, Ψ with the functors described above. As proved by Panin in [21, page 557] (after applying the functor F_{γ} : 𝔸^{G} → 𝔸^{{1}}), every Ch-homogeneous basis ρ_{1}, …, ρ_{n} of R(P̃) over R(G̃) gives rise to an isomorphism in 𝔸^{{1}}

(7.2)  ⨁_{i=1}^{n} Ψ(A_{χ(i),γ}) ≃ Ψ(∏_{i=1}^{n} A_{χ(i),γ}) ⟶^{∼} Φ(_{γ}F) .

Note that by construction of the category of noncommutative motives we have ⨁_{i=1}^{n} U(A_{χ(i),γ}) = U(∏_{i=1}^{n} A_{χ(i),γ}). Hence, by combining Theorem 6.10 with the above commutative diagram (7.1), we conclude that the image of (7.2) under

(7.3)  𝔸^{\{1\}} = \underline{C} ⟶^{Θ} NChow ⊂ Hmo_{0} ⟶^{\overline{E}} D

agrees with the desired isomorphism (2.2). This achieves the proof.

Item (ii). As proved in Merkurjev-Panin in [18, Thm. 4.2], Φ(X) is a direct summand of Ψ(A) in the motivic category \underline{C}. Using Theorem 6.10, the above commutative diagram (7.1), and the above composed functor (7.3), we then conclude that E(X) is a direct summand of E(A) in the additive category D.

Remark 7.4 (Toric varieties). Let T be an algebraic torus and X a smooth projective toric T -variety. As proved by Merkurjev-Panin in [18, Thm. 7.6] (making use of the splitting principle), there exists a separable k-algebra B and a retraction of Ψ(B) into Φ(X) in the motivic category \underline{C}. The above proof of item (ii) shows that Theorem 2.1(ii) and Theorem 3.14 also hold in these cases.

8. Cyclic homology of separable algebras

Theorem 8.1. Given a separable k-algebra A, we have:

C(A) ≃ C(k) ⊗ HH_{0}(A)
HH(A) ≃ HH(k) ⊗ HH_{0}(A)  HC(A) ≃ HC(k) ⊗ HH_{0}(A)
HP (A) ≃ HP (k) ⊗ HH_{0}(A)  HN(A) ≃ HN(k) ⊗ HH_{0}(A) .

As a consequence of Theorem 8.1 one obtains the isomorphisms:

HH_{j}(A) ≃ \{ \begin{matrix}A/[A, A] & j = 0 \\ 0 & otherwise \end{matrix}  HC_{j}(A) ≃ \{ \begin{matrix}A/[A, A] & j ≥ 0 even \\ 0 & otherwise \end{matrix}

HP_{j}(A) ≃ \{ \begin{matrix}A/[A, A] & j even \\ 0 & j odd \end{matrix}  HN_{j}(A) ≃ \{ \begin{matrix}A/[A, A] & j ≤ 0 even \\ 0 & otherwise \end{matrix} .

Proof. Recall from Loday [14, §1.1] the construction of the Hochschild homology complexes HH(A) and HH(k). In particular, the k-vector spaces in degree j are given by A^{⊗(j+1)} and k^{⊗(j+1)}, respectively. Note that the assignments

A^{⊗(j+1)} ⟶ k^{⊗(j+1)} ⊗ A/[A, A]  a_{0} ⊗ ⋯ ⊗ a_{j} ↦ 1 ⊗ ⋯ ⊗ 1 ⊗ (a_{1} ⋯ a_{j})

gives rise to a well-defined homomorphism of complexes

(8.2)  HH(A) ⟶ HH(k) ⊗ A/[A, A] .

By combining [14, Thm. 1.2.13] with [14, Cor. 1.2.14] (and their proofs), one observes that (8.2) is a quasi-isomorphism. Since A/[A, A] = HH_{0}(A), we conclude then that HH(A) is isomorphic to HH(k)⊗A/[A, A] in the derived category D(k). Now, recall from [14, §2.5] that HH(A) and HH(k) ⊗ A/[A, A] are endowed with the following cyclic action

t(a_{0} ⊗ ⋯ ⊗ a_{j}) := (−1)^{j}(a_{n} ⊗ a_{0} ⊗ ⋯ ⊗ a_{j−1})
(8.3)  t(λ_{0} ⊗ ⋯ ⊗ λ_{j} ⊗ a) := (−1)^{j}(λ_{j} ⊗ λ_{0} ⊗ ⋯ ⊗ λ_{j−1} ⊗ a) ,

where t stands for the generator of ℤ/(j + 1)ℤ. The above homomorphism (8.2) preserves this cyclic action. As a consequence, following [14, §2.5], one obtains well-defined quasi-isomorphisms:

C(A) ⟶ Mix(HH(k) ⊗ HH_{0}(A))
(8.4)  HC(A) ⟶ Tot(B(HH(k) ⊗ HH_{0}(A)))
(8.5)  HP (A) ⟶ Tot(B^{per}(HH(k) ⊗ HH_{0}(A)))
(8.6)  HN(A) ⟶ Tot(B^{−}(HH(k) ⊗ HH_{0}(A))) .

Some explanations are in order: Mix(−) stands for the mixed complex construction, B(−), B^{per}(−), B^{−}(−) for the cyclic, periodic and negative bicomplex constructions, and Tot(−) for the totalization functor. Thanks to the above action (8.3), one observes that Mix(HH(k) ⊗ HH_{0}(A)) identifies with the mixed complex HH_{0}(A) ⊗ A/[A, A]. As a consequence, C(A) is isomorphic to C(k) ⊗ HH_{0}(A) in the derived category of mixed complexes D(Λ). In what concerns (8.4), we have a canonical isomorphism between its right-hand-side and the complex Tot(B(HH(k))) ⊗ A/[A, A] = HC(k) ⊗ A/[A, A]. This implies that HC(A) is isomorphic to HC(k) ⊗ HH_{0}(A) in the derived category D(k). Finally, in what concerns (8.5)-(8.6), we have canonical quasi-isomorphisms

Tot(B^{per}(HP (k))) ⊗ A/[A, A] ⟶ Tot(B^{per}(HP (k) ⊗ A/[A, A]))
Tot(B^{−}(HN(k))) ⊗ A/[A, A] ⟶ Tot(B^{−}(HN(k) ⊗ A/[A, A])) .

Making use of them, we conclude that HP (A) (resp. HN(A)) is isomorphic to HP (k) ⊗ HH_{0}(k) (resp. to HN(k) ⊗ HH_{0}(k)) in the derived category D_{ℤ/2}(k) (resp. D(k)). This achieves the proof.

□

Proposition 8.7. When A is a central simple k-algebra, we have HH_{0}(A) ≃ k.

Proof. As explained in [13, §III Thm. 5.13], we have a ⊗-equivalence of categories

(8.8)  Mod(k) ⟶^{∼} Mod(A^{op} ⊗ A)  M ↦ M ⊗ A .

Since k is a field, [14, Prop. 1.1.13] implies the following isomorphisms

(8.9) HH_{∗}(k) ≃ Tor_{n}^{Mod(k)}(k, k)  HH_{∗}(A) ≃ Tor_{n}^{Mod(A^{op}⊗A)}(A, A) .

By combining (8.8)-(8.9), we then conclude that HH_{0}(A) ≃ HH_{0}(k) ≃ k. □

Proof of Theorem 3.1. The proof follows from the combination of isomorphism (2.2) (with E = C, HH, HC, HP and HN) with Theorems 8.1 and 8.7.

Proof of Theorem 3.2. Given a central simple k-algebra A of degree r, [22, Cor. 3.1] implies that E(k) ≃ E(A) whenever D is ℤ[\frac{1}{r}]-linear. The proof follows then from the combination of this fact with isomorphism (2.2).

Proof of Theorem 3.13. The proof follows from the combination of isomorphism (2.2) (with E = C, HH, HC, HP of HN) with Theorem 8.1.

Proof of Theorem 3.14. The proof follows from the combination of Theorem 2.1(ii) (with E = C, HH, HC, HP and HN) with Theorem 8.1.

9. Noncommutative motives of central simple algebras

Let us denote by Br(k) the Brauer group of k.

Theorem 9.1. Given central simple k-algebras A and B, we have:

U(A) ≃ U(B) ⇔ [A] = [B] ∈ Br(k) .

Proof. We start with some preliminaries. Given a central simple k-algebra C, let us denote by P(C) the exact category of finitely generated projective right C-modules. As explained by Gille-Szamuely in [5, Thm. 2.1.3], the Wedderburn theorem implies that C ≃ M_{n×n}(D) for some integer m ≥ 1 and division algebra D ⊃ k. As a consequence, the right C-modules M ∈ P(C) are completely classified by their length. In particular, every element of the Grothendieck group K_{0}P(C) ≃ ℤ can be written as ±[M] for a unique (up to isomorphism) right C-module M ∈ P(C).

Now, let us assume that [A] = [B] ∈ Br(k). This equality is equivalent to the existence of bimodules B ∈ P(A^{op} ⊗B) and B′ ∈ P(B^{op} ⊗A) such that B⊗_{B} B′ ≃ A in P(A^{op} ⊗ A) and B′ ⊗_{A} B ≃ B in P(B^{op} ⊗ B). In other words, the central simple k-algebras A and B are Morita equivalent. Making use of the canonical inclusions of categories P(A^{op} ⊗ B) ⊂ D_{c}(A^{op} ⊗ B) and P(B^{op} ⊗ A) ⊂ D_{c}(B^{op} ⊗ A), one then concludes that A and B are isomorphic in the homotopy category Hmo. This implies automatically that U(A) ≃ U(B) and so (⇐) is proved.

Let us now prove (⇒). Assume that U(A) ≃ U(B). By construction of Kontsevich’s category of noncommutative Chow motives NChow, there exist elements α ∈ K_{0}D_{c}(A^{op}⊗B) and β ∈ K_{0}D_{c}(B^{op}⊗A) such that β∘α = [A] in K_{0}D_{c}(A^{op}⊗A) and α ∘ β = [B] in K_{0}D_{c}(B^{op} ⊗ B). Making use of the isomorphisms

K_{0}P(A^{op} ⊗ B) ≃ K_{0}D_{c}(A^{op} ⊗ B)  K_{0}P(B^{op} ⊗ A) ≃ K_{0}D_{c}(B^{op} ⊗ A)
K_{0}P(A^{op} ⊗ A) ≃ K_{0}D_{c}(A^{op} ⊗ A)  K_{0}P(B^{op} ⊗ B) ≃ K_{0}D_{c}(B^{op} ⊗ B) ,

one then concludes from above preliminaries that α = ±[M] for a unique A-B-bimodule M in P(A^{op} ⊗ B), that β = ±[N] for a unique B-A-bimodule N in P(B^{op}⊗B), that M ⊗_{B} N ≃ A in P(A^{op}⊗A), and that N ⊗_{A}M ≃ B in P(B^{op}⊗B). This implies the equality [A] = [B] ∈ Br(k) and so the proof is finished. □

Proposition 9.2. Given central simple k-algebras A_{1}, …, A_{n}, we have

(9.3)  ⨁_{i=1}^{n} U(A_{i}) ≃ ⨁_{i=1}^{n} U(k)

if and only if [A_{1}] = ⋯ = [A_{n}] = [k] ∈ Br(k).

Proof. Assume first that [A_{1}] = ⋯ = [A_{n}] = [k]. Making use of Theorem 9.1, we conclude that U(A_{i}) ≃ U(k) for every i and consequently that (9.3) holds. This proves (⇐). Let us now prove the converse implication. Assume (9.3). By combining isomorphism (5.5) (with A = B = k) with the linearity of NChow and with the fact that k is a field, we observe that

(9.4)  End_{NChow}(⊕_{i=1}^{n}U(k)) ≃ M_{n×n}(ℤ) .

Let \{e_{i}\}_{i=1}^{n} (resp. \{\overline{e_{i}}\}_{i=1}^{n}) be the idempotent elements of End_{NChow}(⨁_{i=1}^{n} U(A_{i})) (resp. of End_{NChow}(⨁_{i=1}^{n} U(k))) associated to the n factors. Via (9.3), the idempotents {e_{i}}_{i=1}^{n} correspond to idempotent elements {ẽ_{i}}_{i=1}^{n} of End_{NChow}(⨁_{i=1}^{n} U(k)). Using (9.4) and the fact that ∑_{i} ẽ_{i} = id, we conclude that ẽ_{i} = \overline{e_{j}} for some j. This implies that U(A_{i}) = U(k) for every i. Consequently, using Theorem 9.1, we obtain the equalities [A_{1}] = ⋯ = [A_{n}] = [k]. This concludes the proof. □

Proof of Theorem 3.3. The proof follows from the combination of isomorphism (2.2) (with E = U) with Proposition 9.2.

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Gonçalo Tabuada, Department of Mathematics, MIT, Cambridge, MA 02139, USA

E-mail address: tabuada@math.mit.edu
URL: http://math.mit.edu/~tabuada/