Jacques Tits motivic measure

In this article we construct a new motivic measure called the ${\it Jacques}$ ${\it Tits}$ ${\it motivic}$ ${\it measure}$. As a first main application of the Jacques Tits motivic measure, we prove that two Severi-Brauer varieties (or, more generally, two twisted Grassmannian varieties), associated to $2$-torsion central simple algebras, have the same class in the Grothendieck ring of varieties if and only if they are isomorphic. In addition, we prove that if two Severi-Brauer varieties, associated to central simple algebras of period $\{3, 4, 5, 6\}$, have the same class in the Grothendieck ring of varieties, then they are necessarily birational to each other. As a second main application of the Jacques Tits motivic measure, we prove that two quadric hypersurfaces (or, more generally, two involution varieties), associated to quadratic forms of dimension $6$ or to quadratic forms of arbitrary dimension defined over a base field $k$ with $I^3(k)=0$, have the same class in the Grothendieck ring of varieties if and only if they are isomorphic. In addition, we prove that the latter main application also holds for products of quadric hypersurfaces.


Introduction
Let k be a field and Var(k) the category of varieties, i.e., reduced separated k-schemes of finite type. The Grothendieck ring of varieties K 0 Var(k), introduced in a letter from Grothendieck to Serre (consult [7, letter of 16/08/1964]), is defined as the quotient of the free abelian group on the set of isomorphism classes of varieties [X] by the "cut-and-paste" relations [X] = [Y ] + [X\Y ], where Y is a closed subvariety of X. The multiplication law is induced by the product of varieties. Despite the efforts of several mathematicians (consult, for example, the works of Bittner [3] and Larsen-Lunts [19]), the structure of the Grothendieck ring of varieties still remains nowadays poorly understood. In order to capture some of its flavor, a few motivic measures, i.e., ring homomorphisms µ : K 0 Var(k) → R, have been built. For example, when k is finite the assignment X → #X(k) gives rise to the counting motivic measure µ # : K 0 Var(k) → Z, and when k = C the assignment X → χ(X) := Σ n (−1) n dim Q H n c (X an ; Q) gives rise to the Euler characteristic motivic measure µ χ : K 0 Var(k) → Z. In this article we construct a new motivic measure µ JT called the Jacques Tits motivic measure. Making use of it, we then establish several new properties of the Grothendieck ring of varieties.
Statement of results. Let k be a field of characteristic zero and Γ := Gal(k/k) its absolute Galois group.
Recall that given a split semi-simple algebraic group G over k, a parabolic subgroup P ⊂ G, and a 1cocycle γ : Γ → G(k), we can consider the projective homogeneous variety F := G/P as well as its twisted form γ F . Let us write G and P for the universal covers of G and P , respectively, R( G) and R( P ) for the associated representation rings, n(F ) for the index [W ( G) : W ( P )] of the Weyl groups, Z for the center of G, and Ch for the character group Hom( Z, G m ). As proved by Steinberg in [24] (consult also [22, §12.5- §12.8]), we have R( P ) = ⊕ i R( G)ρ i , where {ρ i } i is a canonical Ch-homogeneous basis of cardinality n(F ). Let us denote by A ρi the Jacques Tits central simple k-algebra associated to ρ i ; consult [17, §27] [29] for details. Our main result is the following: Theorem 1.2. The assignment γ F → Σ i [A ρi ] gives rise to a motivic measure µ JT : K 0 Var(k) tw → R B (k).
Intuitively speaking, Theorem 1.2 shows that the Jacques Tits central simple algebras associated to a twisted projective homogeneous variety are preserved by the "cut-and-paste" relations. Motivated by this fact, we decided to call µ JT the Jacques Tits motivic measure. The proof of Theorem 1.2 makes use, among other ingredients, of the recent theory of noncommutative motives; consult §3- §4 below.

Applications
In this section we describe several applications of the Jacques Tits motivic measure to Severi-Brauer varieties, twisted Grassmannian varieties, quadric hypersurfaces, and involution varieties. 2.1. Severi-Brauer varieties. Let G be the projective general linear group P GL n , with n ≥ 2. In this case, we have G = SL n . Consider the following parabolic subgroup: The associated projective homogeneous variety F := G/P ≃ G/ P is the projective space P n−1 and we have R( P ) = ⊕ n−1 i=0 R( G)ρ i . Given a 1-cocycle γ : Γ → P GL n (k), let A be the corresponding central simple k-algebra of degree n. Under these notations, the twisted form γ P n−1 is the Severi-Brauer variety SB(A) and the Jacques Tits central simple k-algebra A ρi is the tensor product A ⊗i . Note that item (i), resp. item (ii), shows that the dimension of a Severi-Brauer variety, resp. the subgroup generated by the Brauer class, is preserved by the "cut-and-paste" relations. Item (iii) shows that when the Brauer class is 2-torsion (i.e., when per(A) ∈ {1, 2}), two Severi-Brauer varieties have the same Grothendieck class in K 0 Var(k) if and only if they are isomorphic! In other words, item (iii) yields the following inclusion: (2.3) {Severi-Brauer varieties SB(A) with [A] ∈ 2 Br(k)} isomorphism ⊂ K 0 Var(k) .
Note that thanks to the Artin-Wedderburn theorem, the left-hand side of (2.3) is in bijection with 2 Br(k)×N via the assignment SB(A) → ([A], deg(A)). Note also that by restricting the inclusion (2.3) to central simple k-algebras of degree 2, i.e., to quaternion algebras Q = (a, b), we obtain the following inclusion where C(a, b) stands for the smooth conic associated to the quaternion algebra Q.
Example 2.5 (Conics over Q). When k = Q, there are infinitely many smooth conics in P 2 up to isomorphism. For example, given any two primes numbers p = q which are congruent to 3 modulo 4, the conics C(−1, p) and C(−1, q) are not isomorphic. Consequently, since there are infinitely many prime numbers p which are congruent to 3 modulo 4, the inclusion (2.4) yields the following infinite family of distinct Grothendieck classes {[(−x 2 + py 2 − z 2 = 0)]} p≡3 (mod 4) ⊂ K 0 Var(Q).
Finally, item (iv) shows that, for small values of the period, if two Severi-Brauer varieties have the same Grothendieck class in K 0 Var(k) then they are necessarily birational to each other. , we obtain from item (ii) the following variant of item (iv): (iv") The Severi-Brauer varieties SB(A) and SB(A ′ ) are stably birational to each other.

Products of conics.
Recall that two quaternion k-algebras Q and Q ′ are called unlinked in the sense of Albert [1] if their tensor product Q ⊗ Q ′ is a division k-algebra.
, then the following holds: Note that item (i) shows that if two products of conics have the same Grothendieck class in K 0 Var(k), then they necessarily share a common conic! Moreover, item (ii) provides a sufficient condition for these two products of conics to be isomorphic.
Example 2.10 (Unlinked quaternion algebras). When k = R(x, y) is the field of rational functions on two variables over R, the quaternion algebras (−1, −1) and (x, y), as well as the quaternion algebras (x, −1) and (−x, y), are unlinked; consult [18, §VI Examples 1.11 and 1.13]. In the same vein, when k = Q(x, y) is the field of rational functions on two variables over Q, the quaternion algebras (a, x) and (b, y), where a, b ∈ k × represent two independent square classes in Q × /(Q × ) 2 , are unlinked; consult [18,§VI Example 1.15]. Further examples exist for every field k with u-invariant equal to 6 or > 8; consult [18,§XIII].
2.3. Twisted Grassmannian varieties. Let G = P GL n , with n ≥ 2. Recall that in this case we have G = SL n . Choose an integer 1 ≤ d < n and consider the following parabolic subgroup: The associated projective homogeneous variety F := G/P ≃ G/ P is the Grassmannian variety Gr(d) and Note that, similarly to Theorem 2.2, item (iii) shows that when the Brauer class is 2-torsion, two twisted Grassmannian varieties have the same Grothendieck class in K 0 Var(k) if and only if they are isomorphic! In other words, item (iii) yields the following inclusion: (2.14) {Twisted Grassmannian varieties Gr(d; Following Remark 2.12, note that (2.14) extends the above inclusion (2.3).
2.4. Quadric hypersurfaces. Assume that char(k) = 2. Let G be the special orthogonal group SO n , with n ≥ 3, with respect to the hyperbolic form n 2 H when n is even or to the form ⌊ n 2 ⌋H ⊥ 1 when n is odd. In this case, we have G = Spin n . Consider the action of G on P n−1 given by projective linear transformations, the stabilizer P ⊂ G of the isotropic point [1 : 0 : · · · : 0], and the pre-image P ⊂ G of P . The associated projective homogeneous variety F := G/P ≃ G/ P is the following smooth quadric hypersurface Q := (x 1 y 1 + · · · + x n 2 y n 2 = 0) ⊂ P n−1 n even (x 1 y 1 + · · · + x ⌊ n 2 ⌋ y ⌊ n 2 ⌋ + z 2 = 0) ⊂ P n−1 n odd and we have R( P ) = ⊕ n−1 i=0 R( G)ρ i when n is even or R( P ) = ⊕ n−2 i=0 R( G)ρ i when n is odd. Given a 1cocycle γ : Γ → SO n (k), let q be the corresponding non-degenerate quadratic form with trivial discriminant of dimension n. Under these notations, the twisted form γ Q is the smooth quadric hypersurface Q q ⊂ P n−1 and the Jacques Tits central simple k-algebra A ρi is given as follows where C 0 (q) stands for the even Clifford algebra of q and C + 0 (q) and C − 0 (q) for the (isomorphic) simple components of C 0 (q).
Theorem 2.15. Let q and q ′ be two non-degenerate quadratic forms with trivial discriminant of dimensions n and n ′ , respectively. If [Q q ] = [Q q ′ ] in the Grothendieck ring of varieties K 0 Var(k), then the following holds: (i) We have dim(Q q ) = dim(Q q ′ ). Equivalently, we have n = n ′ .
(ii) We have C + 0 (q) ≃ C + 0 (q ′ ) when n is even or C 0 (q) ≃ C 0 (q ′ ) when n is odd. (iii) When n = 6, we have Q q ≃ Q q ′ . (iv) When I 3 (k) = 0, where I(k) ⊂ W (k) is the fundamental ideal of the Witt ring, we have Q q ≃ Q q ′ .
Note that item (i), resp. item (ii), shows that the dimension of the quadric hypersurface, resp. the Brauer class of the (simple components of the) even Clifford algebra, is preserved by the "cut-and-paste" relations. Item (iii) shows that when the dimension is equal to 6, two quadric hypersurfaces have the same Grothendieck class in K 0 Var(k) if and only if they are isomorphic! Recall from [17, §16.4] that, up to similarity, a nondegenerate quadratic form q with trivial discriminant of dimension 6 is given by a, b, −ab, −a ′ , −b ′ , a ′ b ′ with a, b, a ′ , b ′ ∈ k × . Therefore, item (iii) yields the following inclusion: Example 2.17 (Quadric hypersurfaces over Q). When k = Q, there are infinitely many quadric hypersurfaces in P 5 up to isomorphism. For example, we have the following infinite family of non-isomorphic quadric hypersurfaces {(u 2 + v 2 − w 2 + x 2 − py 2 − pz 2 = 0)} p≡3 (mod 4) parametrized by the prime numbers p which are congruent to 3 modulo 4. Making use of (2.16), we hence obtain the following infinite family of distinct Grothendieck classes {[(u 2 + v 2 − w 2 + x 2 − py 2 − pz 2 = 0)]} p≡3 (mod 4) ⊂ K 0 Var(Q). Note that since all these quadric hypersurfaces have a rational k-point, it follows from [13, Thm. 1.11] that they are all birational to P 4 . This shows that, in the case of quadric hypersurfaces, the Grothendieck class in K 0 Var(k) contains much more information than the birational equivalence class.
Finally, item (iv) shows that when I 3 (k) = 0, two quadric hypersurfaces have the same Grothendieck class in K 0 Var(k) if and only if they are isomorphic! Consequently, in this case, the above inclusion (2.16) admits the following far-reaching extension: {Quadric hypersurfaces Q q with trivial discriminant} isomorphism Recall that I 3 (k) = 0 when k is a C 2 -field or, more generally, when k is a field of cohomological dimension ≤ 2. Examples include fields of transcendence degree ≤ 2 over algebraically closed fields, p-adic fields, non formally real global fields, etc.
2.5. Products of quadrics. Surprisingly, Theorem 2.15 admits the following generalization: , then the following holds: , and the following extra condition holds (consult Notation 2.21 below) (2.20) Σ 1 even (m, n, l) > Σ 2 even (m, n, l) for every 2 ≤ l ≤ m − 3 n even Σ 1 odd (m, n, l) > Σ 2 odd (m, n, l) for every 2 ≤ l ≤ m − 3 n odd , Given integers m, n, l ≥ 0, consider the following sums of (n − 2)-powers: . Note that item (iii) shows that when the dimension is equal to 6, two products of quadrics (with m ≤ 5) have the same Grothendieck class in K 0 Var(k) if and only if they are isomorphic! This implies that the above inclusion (2.16) holds more generally for products of quadrics (with m ≤ 5). In the same vein, items (iv)-(iv') show that when I 3 (k) = 0, two products of quadrics have the same Grothendieck class in K 0 Var(k) if and only if they are isomorphic! This implies that the above inclusion (2.18) holds more generally for products of quadrics. Finally, note that the extra condition (2.20) holds whenever the dimension n is ≫ than the number m of quadrics because the highest power of (n − 2) in the sums Σ 1 even (m, n, l) and Σ 1 odd (m, n, l) is (n − 2) m−1 while the highest power of (n − 2) in the sums Σ 2 even (m, n, l) and Σ 2 odd (m, n, l) is (n − 2) m−2 . 2.6. Involution varieties. Assume that char(k) = 2. Let G be the projective special orthogonal group P SO n , with n ≥ 6 even, with respect to the hyperbolic form n 2 H. In this case, we have G = Spin n . Similarly to §2.4, consider the projective homogeneous variety F given by Q := (x 1 y 1 + · · · + x n 2 y n 2 = 0) ⊂ P n−1 and recall from loc. cit. that R( P ) = ⊕ n−1 i=0 R( G)ρ i . Given a 1-cocycle γ : Γ → P SO n (k), let (A, * ) be the corresponding central simple k-algebra of degree n with involution of orthogonal type and trivial discriminant. Under these notations, the twisted form γ Q is the involution variety Iv(A, * ) ⊂ P n−1 and the Jacques Tits central simple k-algebra A ρi is given as follows where C + 0 (A, * ) and C − 0 (A, * ) stand for the simple components of the even Clifford algebra C 0 (A, * ) of (A, * ). Remark 2.22 (Generalization). In the particular case where (A, * ) is split, i.e., isomorphic to (M n (k), * q ) with * q the adjoint involution associated to a quadratic form q, the involution variety Iv(A, * ) ⊂ P n−1 reduces to the quadric hypersurface Q q ⊂ P n−1 . Hence, involution varieties may be understood as "forms of quadrics".
Theorem 2.23. Let (A, * ) and (A ′ , * ′ ) be two central simple k-algebras with involutions of orthogonal type and trivial discriminant.
, then the following holds: Note that, similarly to Theorem 2.15, item (iii), resp. (iv), shows that when the degree is equal to 6, resp. I 3 (k) = 0, two involution varieties have the same Grothendieck class in K 0 Var(k) if and only if they are isomorphic! In other words, items (iii)-(iv) yield the following inclusions: Following Remark 2.22, note that (2.24), resp. (2.25), extends the above inclusion (2.16), resp. (2.18).

Preliminaries
Throughout the article k denotes a base field of characteristic zero. Dg categories. A differential graded (=dg) category A is a category enriched over complexes of k-vector spaces; consult Keller's survey [11] (and Bondal-Kapranov's original article [4]). Every (dg) k-algebra A gives naturally rise to a dg category with a single object. Another source of examples is provided by schemes since the category of perfect complexes perf(X) of every k-scheme X admits a canonical dg enhancement perf dg (X); consult [11, §4.6]. Let us denote by dgcat(k) the category of (small) dg categories.
Let A be a dg category. The opposite dg category A op has the same objects as A and A op (x, y) := A(y, x). A right dg A-module is a dg functor M : A op → C dg (k) with values in the dg category of complexes of k-vector spaces. Let us denote by C(A) the category of right dg A-modules. Following [11, §3.2], the derived category D(A) of A is defined as the localization of C(A) with respect to the objectwise quasi-isomorphisms. In what follows, we will write D c (A) for the subcategory of compact objects.
A dg functor F : A → B is called a Morita equivalence if it induces an equivalence on derived categories D(A) ≃ D(B); consult [11, §4.6]. As explained in [25, §1.6.1], dgcat(k) admits a Quillen model structure whose weak equivalences are the Morita equivalences. Let Hmo(k) be the associated homotopy category.
The tensor product A⊗B of dg categories is defined as follows: the set of objects is the cartesian product of the sets of objects and (A⊗B)((x, w), (y, z)) := A(x, y)⊗B(w, z). As explained in [11, §2.3], this construction gives rise to a symmetric monoidal structure on dgcat(k) which descends to the homotopy category Hmo(k).
A dg A-B-bimodule is a dg functor B : A ⊗ B op → C dg (k). For example, given a dg functor F : Finally, following Kontsevich [14,15,16], a dg category A is called smooth if the dg A-A bimodule id A belongs to D c (A op ⊗ A) and proper if Σ n dim H n A(x, y) < ∞ for any pair of objects (x, y). Examples include finite-dimensional k-algebras of finite global dimension A as well as the dg categories of perfect complexes perf dg (X) associated to smooth proper k-schemes X. 2 The short notation C ± 0 (A, * ) ≃ C ± 0 (A ′ , * ′ ) stands for Noncommutative motives. For a book on noncommutative motives, we invite the reader to consult [25]. As explained in [25, §1.6.3], given any two dg categories A and B, there is a natural bijection between Hom Hmo(k) (A, B) and the set of isomorphism classes of the category rep(A, B). Under this bijection, the composition in Hmo(k) corresponds to the (derived) tensor product of bimodules. The additivization of Hmo(k) is the additive category Hmo 0 (k) with the same objects and with abelian groups of morphisms Hom Hmo0(k) (A, B) given by the Grothendieck group K 0 rep(A, B) of the triangulated category rep(A, B).
The composition law is induced by the (derived) tensor product of bimodules. Given a commutative ring of coefficients R, the R-linearization of Hmo 0 (k) is the R-linear category Hmo 0 (k) R obtained by tensoring the morphisms of Hmo 0 (k) with R. Note that we have the following (composed) symmetric monoidal functor The category of noncommutative Chow motives NChow(k) R is defined as the idempotent completion of the full subcategory of Hmo 0 (k) R consisting of the objects U (A) R with A a smooth proper dg category. This category is R-linear, additive, rigid symmetric monoidal, and idempotent complete.
Given an additive rigid symmetric monoidal category C, recall that its N -ideal is defined as follows where tr(g • f ) stands for the categorical trace of g • f . The category of noncommutative numerical motives NNum(k) R is defined as the idempotent completion of the quotient of NChow(k) R by the ⊗-ideal N . By construction, this category is R-linear, additive, rigid symmetric monoidal, and idempotent complete.
Using the fact that the functor U (−) R is symmetric monoidal, we hence conclude that the noncommutative 4. Proof of Theorem 1.2 Let K 0 (NChow(k)) be the Grothendieck ring of the additive symmetric monoidal category of noncommutative Chow motives NChow(k). We start by constructing a motivic measure with values in this ring.
Proof. Thanks to Bittner's presentation [3, Thm. 3.1] of the Grothendieck ring of varieties K 0 Var(k), it suffices to prove the following two conditions: (i) Let X be a smooth projective k-scheme, Y ֒→ X a smooth closed subscheme of codimension c, Bl Y (X) the blow-up of X along Y , and E the exceptional divisor of the blow-up. Under these notations, we have the following equality in the Grothendieck ring K 0 (NChow(k)): (ii) Given smooth projective k-schemes X and Y , we have the following equality in K 0 (NChow(k)): Let us write f : Bl Y (X) → X for the blow-up map, i : E ֒→ Bl Y (X) for the embedding map, and p : E → Y for the projection map (= restriction of f to E). Under these notations, recall from Orlov [21,Thm. 4.3] that we have the following semi-orthogonal decompositions These equalities imply condition (i). Condition (ii) follows now from the above Remark 3.2.
Notation 4.2. Given a commutative ring of coefficients R, let us write CSA(k) R for the full subcategory of NChow(k) R consisting of the objects U (A) R with A a central simple k-algebra. The closure of CSA(k) R under finite direct sums will be denoted by CSA(k) ⊕ R . In the same vein, let us write CSA(k) R for the full subcategory of NNum(k) R consisting of the objects U (A) R with A a central simple k-algebra, and CSA(k) ⊕ R for its closure under finite direct sums. In the particular case where R = Z, we will omit the underscript (−) Z . Proposition 4.3. Given two central simple k-algebras A and A ′ and a prime number p, the following holds: Proof. Given central simple k-algebras A, A ′ , A ′′ , recall from [28,Prop. 2.25] that the composition map corresponds to the following bilinear pairing: , this implies, in particular, that the composition map corresponds to the following bilinear pairing Note that Proposition 4.3 implies, in particular, the following result: Corollary 4.5. Given a prime number p, we have an induced equivalence of categories where Vect Br(k){p} (k) stands for the category of finite-dimensional Br(k){p}-graded F p -vector spaces.
Recall from [28, Thm. 2.20(iv)] the following result: Proposition 4.7. Given two families of central simple k-algebras {A j } 1≤j≤m and {A ′ j } 1≤j≤m ′ , the following conditions are equivalent: We have m = m ′ and for every prime number p there exists a permutation σ p (which depends on p) The following result is of independent interest: Proof. Given a (fixed) prime number p, consider the induced isomorphism We claim that the following composition maps (with 1 ≤ i ≤ m and 1 ≤ j ≤ m ′ ) Fp in the category NNum(k) Fp and that the right-hand side of (4.12) is isomorphic to F p . Since the category NNum(k) Fp is F p -linear and MN does not contains the noncommutative numerical motives {U (A j ) Fp } 1≤j≤m and {U (A ′ j ) Fp } 1≤j≤m ′ as direct summands, we then conclude that the composition map (4.12) is necessarily equal to zero; otherwise MN would contain U (A i ) Fp , or equivalently U (A ′ j ) Fp , as a direct summand. Note that the triviality of the composition maps (4.12) implies that the above isomorphism (4.11) in the category NNum(k) Fp restricts to an isomorphism Recall from Corollary 4.5 that the category CSA(k) ⊕ Fp is equivalent to the category Vect Br(k){p} (k) of finite-dimensional Br(k){p}-graded F p -vector spaces. Under the equivalence (4.6), the isomorphism (4.13) corresponds to the following isomorphism: Fp ⊕ MN in the category NNum(k) Fp for some noncommutative numerical motive MN and integers r j , r ′ j ≥ 0. Note that the inductive splitting procedure stops at a finite stage. Otherwise, the following F p -vector spaces would be infinite-dimensional, which is impossible because the  Proposition 4.19. The inclusion of categories CSA(k) ⊕ ⊂ NChow(k) gives rise to an injective ring homomorphism K 0 (CSA(k) ⊕ ) → K 0 (NChow(k)).
Proof. Recall first that the group completion of an arbitrary monoid (M, +) is defined as the quotient of the product M × M by the following equivalence relation: (4.20) (m, n) ∼ (m ′ , n ′ ) := ∃ r ∈ M such that m + n ′ + r = n + m ′ + r .
Consider the (composed) ring homomorphism Therefore, making use of Proposition 4.19, we conclude that the assignment gives rise to a (well-defined) motivic measure K 0 Var(k) tw → K 0 (CSA(k) ⊕ ). Consequently, the proof of Theorem 1.2 follows now from the following result: Proof. Let us write K 0 (CSA(k) ⊕ ) + for the semi-ring of the additive symmetric monoidal category CSA(k) ⊕ . Concretely, K 0 (CSA(k) ⊕ ) + is the set of isomorphism classes of the category CSA(k) ⊕ equipped with the addition, resp. multiplication, law induced by ⊕, resp. ⊗. Consider also the semi-ring N[Br(k)] and the following (semi-ring) homomorphism: The homomorphism (4.23) is surjective. Moreover, thanks to Proposition 4.7, it yields an isomorphism where p is a prime number and σ p a permutation of the set {1, . . . , m}. Thanks to Lemma 4.26 below, the left-hand side of (4.24) may be replaced by the following semi-ring where A and A ′ are central simple k-algebras with coprime indexes. Therefore, by passing to groupcompletion, we obtain an induced (ring) isomorphism: Finally, the proof follows now from the fact that the left-hand side of (4.25) agrees with the ring R B (k).
Proof. Note first that the set (4.28) is contained in the set (4.27). Note also that since every permutation σ p can be written as a composition of transpositions, (4.27) is equivalent to the following set of relations where q = p is a(ny) prime number. Therefore, it suffices to show that every relation in (4.29) is a particular case of a relation in (4.28). Recall from the Artin-Wedderburn theorem that A 1 , resp. A 2 , may be written as the matrix algebra of a unique central division k-algebra D 1 , resp. D 2 . Let us write D 1 = ⊗ p D p 1 , resp.
Note that (ind(A), ind(A ′ )) = 1 and that for these choices the relation in (4.28) reduces to the relation:

Making use of the equalities [D
, we hence conclude that (4.30) agrees with the relation in (4.29). This finishes the proof.

Properties of the Jacques Tits motivic measure
. Proof. Note first that an iterated application of Remark 3.2 leads to the following isomorphisms: .

Suppose that we have an isomorphism
) in the category NChow(k). Since the noncommutative Chow motives (5.2)-(5.3) belong to the subcategory CSA(k) ⊕ , it follows from the construction of the Jacques Tits motivic measure that

then it follows from the construction of the Jacques Tits motivic measure that [U (perf
). By definition of K 0 (NChow(k)), this implies that there exists a noncommutative Chow motive NM such that Since the noncommutative Chow motives (5.2)-(5.3) belong to the subcategory CSA(k) ⊕ , we hence conclude from the cancellation Proposition 4.9 (applied to (5.4)) that U (perf dg (Π m j=1 γ F j )) ≃ U (perf dg (Π m ′ j=1 γ ′ F ′ j )) in the category NChow(k). This proves item (i). Item (ii) follows now from item (i), from the fact that the noncommutative Chow motives (5.2)-(5.3) belong to the subcategory CSA(k) ⊕ , and from Corollary 4.8.

Proof of Proposition 2.9
Item (ii). When the quaternion k-algebras Q and Q ′ are unlinked, i.e., when Q⊗Q ′ is a division k-algebra,

Proof of Theorem 2.15
Item (i). Recall first that dim(Q q ) = n − 2. Following §2.4, note that µ ρ (Q q ) is equal to n when n is even or to n − 1 when n is odd. Hence, if [Q q ] = [Q q ′ ] in the Grothendieck ring of varieties K 0 Var(k), we conclude that n = n ′ when n is even or that n − 1 = n ′ − 1 when n is odd. In both cases, we have dim(Q q ) = dim(Q q ′ ). This is equivalent to the equality n = n ′ .
in the Grothendieck ring of varieties K 0 Var(k), then it follows from Proposition 5.1(ii) that [C + 0 (q)] = [C + 0 (q ′ )] when n is even or [C 0 (q)] = [C 0 (q ′ )] when n is odd. Since [C + 0 (q)], [C 0 (q)] ∈ 2 Br(k), this implies that [C + 0 (q)] = [C + 0 (q ′ )] when n is even or [C 0 (q)] = [C 0 (q ′ )] when n is odd. Using the fact that deg(C + 0 (q)) = 2 n 2 −1 , deg(C 0 (q)) = 2 ⌊ n 2 ⌋ , and n = n ′ (proved in item (i)), we hence conclude that C + 0 (q) ≃ C + 0 (q ′ ) when n is even or C 0 (q) ≃ C 0 (q ′ ) when n is odd. Item (iii). When n = 6, the assignment q → C + 0 (q) gives rise to a one-to-one correspondence between similarity classes of non-degenerate quadratic forms with trivial discriminant of dimension 6 and isomorphism classes of quaternion algebras; consult [17,Cor. 15.33]. Consequently, the proof follows from the combination of item (ii) with the general fact that two quadratic forms q and q ′ are similar if and only if the associated quadric hypersurfaces Q q and Q q ′ are isomorphic.
Item (iv). When I 3 (k) = 0, we have the following classification result: if n = n ′ and C + 0 (q) ≃ C + 0 (q ′ ) when n is even or C 0 (q) ≃ C 0 (q ′ ) when n is odd, then the quadratic forms q and q ′ are similar; consult [8,Thm. 3']. Consequently, the proof follows from the combination of items (i)-(ii) with the general fact that two quadratic forms q and q ′ are similar if and only if the quadric hypersurfaces Q q and Q q ′ are isomorphic.

Proof of items (i)-(ii) of Theorem 2.19
Item (i). Recall first that dim(Π m j=1 Q qj ) = Σ m j=1 dim(Q qj ) = m × (n − 2). Following §2.4, note that is equal to n m when n is even or to (n − 1) m when n is odd.
in the Grothendieck ring of varieties K 0 Var(k), we conclude that n m = n m ′ when n is even or that (n − 1) m = (n − 1) m ′ when n is odd. In both cases, we have dim(Π m j=1 Q qj ) = dim(Π m ′ j=1 Q q ′ j ). This is equivalent to the equality m = m ′ .
The next result may be understood as the ⊗-analogue of Proposition 4.9.
Proposition 11.2 (⊗-cancellation). Let {A j } 1≤j≤m and {A ′ j } 1≤j≤m ′ be two families of central simple kalgebras and q a non-degenerate quadratic form with trivial discriminant of dimension n ≥ 5. If we have an Proof. We prove first the case where n ≥ 6 is even. Recall from [26, Example 3.8] that, since the central simple k-algebras C + 0 (q) and C − 0 (q) are isomorphic, we have the following computation: . Consequently, we obtain the following computation: Making use of (11.3), the given isomorphism ) in the category NChow(k) may then be re-written as the following isomorphism in the category CSA(k) ⊕ . Therefore, by applying Proposition 4.7 to the isomorphism (11.4), we conclude that (n−2)m+2m = (n−2)m ′ +2m ′ , which implies that m = m ′ . In order to prove that ⊕ m j=1 U (A j ) ≃ ⊕ m j=1 U (A ′ j ), we will also make use of Proposition 4.7. Concretely, we need to show that for every prime number p the following two sets of Brauer classes 11.5) are the same up to permutation. Recall that [C + 0 (q)] ∈ 2 Br(k). Therefore, when p = 2, the isomorphism (11.4) combined with Proposition 4.7 implies that the following two sets are the same up to permutation where the numbers below the parenthesis denote the number of copies. Clearly, this implies that the above sets (11.5) are also the same up to permutation. When p = 2, the isomorphism (11.4) combined with Proposition 4.7 implies that the following two sets are the same up to permutation: , Note that in the case where [C + 0 (q)] = [k], this also implies that the sets (11.5) are the same up to permutation. Let us then assume that [C + 0 (q)] = [k]. In this case, each one of the sets (11.6)-(11.7) is equipped with a non-trivial involution induced by tensoring with C + 0 (q) (we are implicitly ignoring the number of copies of each Brauer class). In particular, we have [A j ⊗ C + 0 (q)] 2 = [A j ] 2 for every 1 ≤ j ≤ m and if there exist integers r and s such that Consequently, there exist disjoint subsets {j 1 , j 1 , . . . , j r , j r } and {i 1 , . . . , i s } of the set {1, . . . , m} and integers n 1 , n 1 , . . . , n r , n r ≥ 1 and l 1 , . . . , l s ≥ 1 such that (11.6) agrees with the following set of distinct Brauer classes: Making use once again of the non-trivial involution on the left-hand side of (11.11) and of the precise number of copies of each Brauer class, we observe that r = r ′ and that the sets {n 1 , n 1 , . . . , n r , n r } and {n ′ 1 , n ′ 1 , . . . , n ′ r ′ , n ′ r ′ } are the same up to permutation. In the same vein, we conclude from the right-hand side of (11.11) that s = s ′ and that the sets {l 1 , . . . , l s } and {l ′ 1 , . . . , l ′ s ′ } are the same up to permutation. This implies that the following sets of Brauer classes are the same up to permutation: Consequently, by concatenating the permutations provided by (11.12), we hence obtain a permutation which identifies the left-hand side of (11.10) with the right-hand side of (11.10). In other words, the two sets in (11.10) are the same up to permutation. This proves the case where n ≥ 6 is even. The proof of the case where n ≥ 5 is odd is similar: simply replace the above isomorphism U (perf dg (Q q )) ≃ U (k) ⊕(n−2) ⊕U (C + 0 (q)) ⊕2 by the isomorphism U (perf dg (Q q )) ≃ U (k) ⊕(n−2) ⊕ U (C 0 (q)) and perform all the subsequent computations.
We now have the ingredients necessary to prove items (iii)-(iv)-(iv') of Theorem 2.19.
in the Grothendieck ring of varieties K 0 Var(k), then it follows from Theorem 2.