# Grothendieck classes of
quadrics and involution varieties

Gonçalo Tabuada  Gonçalo Tabuada, Mathematics Institute, Zeeman Building,
University of Warwick, Coventry CV4 7AL UK<EMAIL_ADDRESS>https://homepages.warwick.ac.uk/ u1972846/ To Yuri I. Manin on the occasion of
his $85^{\mathrm{th}}$ birthday, with my deepest admiration and gratitude.

###### Abstract.

In this article, by combining the recent theory of noncommutative motives with
the classical theory of motives, we prove that if two quadrics (or, more
generally, two involution varieties) have the same Grothendieck class, then
they have the same even Clifford algebra and the same signature. As an
application, we show in numerous cases (e.g., when the base field is a local
or global field) that two quadrics (or, more generally, two involution
varieties) have the same Grothendieck class if and only if they are
isomorphic.

Adoramos a perfeição, porque não a podemos ter; repugna-la-íamos se a
tivéssemos.
O perfeito é o desumano porque o humano é imperfeito. Bernardo Soares, Livro
do Desassossego.

##  1\. Introduction

Let $k$ be a field and $\mathrm{Var}(k)$ the category of varieties, i.e.,
reduced separated $k$-schemes of finite type. The Grothendieck ring of
varieties $K_{0}\mathrm{Var}(k)$, introduced in a letter from Grothendieck to
Serre (consult [6, 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
“scissor” relations $[X]=[Z]+[X\backslash Z]$, where $Z$ 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 articles [3, 4, 18, 21, 22, 24, 25] and the references therein),
the structure of the Grothendieck ring of varieties still remains nowadays
poorly understood. In this article, in order to better understand the
structure of the Grothendieck ring of varieties, we address the following
question: Question: For which pairs of varieties $X$ and $Y$ does the
implication $[X]=[Y]\Rightarrow X\simeq Y$ holds? Such implication does not
holds in general because there are numerous identifications that occur in the
Grothendieck ring of varieties. For example, when $X$ and $Y$ are piecewise
isomorphic111A concrete example is given by the ordinary cusp
$X:=\mathrm{Spec}(k[x,y]/\langle y^{2}-x^{3}\rangle)$ and the affine line
$Y:=\mathbb{A}^{1}$. Although $X$ and $Y$ are not isomorphic, we have
$[X]=[Y]$ because they are piecewise isomorphic., we have $[X]=[Y]$. Moreover,
as explained in [5, Chap. 2 §6.3], there also exist varieties which are not
piecewise isomorphic but which still have the same Grothendieck class222A
concrete example, in characteristic zero, is the following: let $V$ be a
$k$-vector space of dimension $7$ and $W\subset\bigwedge^{2}V^{\vee}$ a
generic subspace of dimension $7$. Note that, by definition, an element $w$ of
$W$ corresponds to a skew-symmetric bilinear form $\varphi_{w}$ on $V$. Under
these identifications, we can consider the varieties $X:=\\{P\subset
V\,|\,{\varphi_{w}}_{|P}=0\,\,\,\forall w\in W\\}\subset\mathrm{Gr}(2,V)$ and
$Y:=\\{w\in W\,|\,\mathrm{rank}(\varphi_{w})<6\\}\subset\mathbb{P}(W)$, where
$\mathrm{Gr}(2,V)$ stands for the Grassmannian of $2$-dimensional subspaces in
$V$. As proved by Borisov in [4], we have
$[X\times\mathbb{A}^{6}]=[Y\times\mathbb{A}^{6}]$ although
$X\times\mathbb{A}^{6}$ and $Y\times\mathbb{A}^{6}$ are not piecewise
isomorphic.. In this article we show that, surprisingly, the aforementioned
implication holds nevertheless for numerous pairs of quadrics and involution
varieties!

### Statement of results - quadrics

Let $k$ be a field of characteristic zero. Given a (finite-dimensional) non-
degenerate quadratic form $q\colon V\to k$, let us write $\mathrm{dim}(q)$ for
its dimension, $\delta(q)\in k^{\times}/(k^{\times})^{2}$ for its discriminant
(when $\mathrm{dim}(q)$ is even), $C_{0}(q)$ for its even Clifford algebra,
and $Q_{q}\subset\mathbb{P}(V)$ for the associated quadric; consult [17, §V].
Recall from [17, §V Thm. 2.4] that when $\mathrm{dim}(q)$ is odd, $C_{0}(q)$
is a central simple $k$-algebra; that when $\mathrm{dim}(q)$ is even and
$\delta(q)\notin(k^{\times})^{2}$, $C_{0}(q)$ is a central simple algebra over
its center $k(\sqrt{\delta(q)})$; and that when $\mathrm{dim}(q)$ is even and
$\delta(q)\in(k^{\times})^{2}$, $C_{0}(q)\simeq C_{0}(q)^{+}\times
C_{0}(q)^{-}$ is a product of two isomorphic central simple $k$-algebras.
Recall also that $\mathrm{dim}_{k}(C_{0}(q))=2^{\mathrm{dim}(q)-1}$ and
$\mathrm{dim}(Q_{q})=\mathrm{dim}(q)-2$. Finally, when $k$ is formally-real,
we will write $\mathrm{sgn}_{P}(q)\in\mathbb{Z}$ for the signature of $q$ with
respect to an ordering $P$ of $k$; consult [17, §VIII].

######  Theorem 1.1.

Let $q$ and $q^{\prime}$ be two non-degenerate quadratic forms. If
$[Q_{q}]=[Q_{q^{\prime}}]$, then the following holds:

  * (i)

We have $\mathrm{dim}(q)=\mathrm{dim}(q^{\prime})$.

  * (ii)

We have
$(\delta(q)\in(k^{\times})^{2})\Leftrightarrow(\delta(q^{\prime})\in(k^{\times})^{2})$.

  * (iii)

We have $C_{0}(q)\simeq C_{0}(q^{\prime})$.

  * (iv)

When $k$ is formally-real, we have
$|\mathrm{sgn}_{P}(q)|=|\mathrm{sgn}_{P}(q^{\prime})|$ for every ordering $P$
of $k$.

######  Remark 1.2.

Note that if $C_{0}(q)\simeq C_{0}(q^{\prime})$, then
$\delta(q)=\delta(q^{\prime})$.

Intuitively speaking, Theorem 1.1 shows that the dimension, the discriminant,
the even Clifford algebra, and the absolute value of the signature, of a
quadratic form are preserved by the “scissor” relations. Among other
ingredients, the proof of items (ii)-(iii), resp. item (iv), makes use of the
recent theory of noncommutative motives, resp. of the classical theory of
motives333The first article on the theory of motives was written by Yuri I.
Manin [23] in the sixties.; consult §2.2-§2.6 below. Theorem 1.1 enables the
following applications:

######  Theorem 1.3.

Let $q$ and $q^{\prime}$ be two non-degenerate quadratic forms. If
$[Q_{q}]=[Q_{q^{\prime}}]$, then the following holds:

  * (i)

When $\mathrm{dim}(q)\leq 4$ (or, equivalently, $\mathrm{dim}(q^{\prime})\leq
4$), we have $Q_{q}\simeq Q_{q^{\prime}}$.

  * (ii)

When $I(k)^{3}$ is torsion-free, where $I(k)$ stands for the fundamental ideal
of the Witt ring of quadratic forms $W(k)$, we have $Q_{q}\simeq
Q_{q^{\prime}}$. In the case where $k$ is formally-real and $\mathrm{dim}(q)$
is even (or, equivalently, $\mathrm{dim}(q^{\prime})$ is even), we assume
moreover that the Hasse-number $\tilde{u}(k)$ of $k$ is finite.

######  Remark 1.4.

  * (a)

When $k$ is not formally-real, the Witt ring $W(k)$ is torsion; consult [7,
§5]. Consequently, $I(k)^{3}$ is torsion-free if and only if $I(k)^{3}=0$.

  * (b)

All the assumptions of Theorem 1.3 hold when $k$ is a local or global field,
or a field of transcendence degree $\leq 1$ over a real-closed field or over
an algebraically closed field; consult [7, §V-§VI].

Item (i) of Theorem 1.3 shows that, in dimensions $\leq 2$, two quadrics have
the same Grothendieck class if and only if they are isomorphic! In other
words, we have the following inclusion:

(1.5) $\frac{\\{\text{Quadrics}\,\,Q_{q}\,|\,\mathrm{dim}(Q_{q})\leq
2\\}}{\text{isomorphism}}\subset K_{0}\mathrm{Var}(k)\,.$

######  Remark 1.6 (Quaternion algebras).

The assignment $(a,b)\mapsto
C(a,b):=(-ax^{2}-by^{2}+abu^{2}=0)\subset\mathbb{P}^{2}$ induces a bijection
between the set of quaternion $k$-algebras up to isomorphism and the set of
quadrics in $\mathbb{P}^{2}$ up to isomorphism (a.k.a. conics). In the same
vein, the assignment
$(a,b)\mapsto(x^{2}-ay^{2}-bu^{2}+(ab)w^{2}=0)\subset\mathbb{P}^{3}$ induces a
bijection between the set of quaternion $k$-algebras up to isomorphism and the
set of quadrics in $\mathbb{P}^{3}$ with trivial discriminant up to
isomorphism.

######  Example 1.7 (Quadrics of dimension $\leq 2$ over $\mathbb{R}$).

When $k=\mathbb{R}$, there are two quadrics in $\mathbb{P}^{2}$ up to
isomorphism, namely $C(1,1)\simeq\mathbb{P}^{1}$ and $C({\bf H})$, where ${\bf
H}:=(-1,-1)$ stands for Hamilton’s $\mathbb{R}$-algebra of quaternions. In
$\mathbb{P}^{3}$ there are three quadrics up to isomorphism, namely
$(x^{2}-y^{2}-u^{2}+w^{2}=0)\subset\mathbb{P}^{3}$,
$(x^{2}+y^{2}+u^{2}+w^{2}=0)\subset\mathbb{P}^{3}$ and
$(x^{2}+y^{2}+u^{2}-w^{2}=0)\subset\mathbb{P}^{3}$. Making use of the above
inclusion (1.5), we hence conclude that the Grothendieck classes of these
quadrics remain distinct in $K_{0}\mathrm{Var}(\mathbb{R})$. Note that since
$\mathbb{R}^{\times}/(\mathbb{R}^{\times})^{2}\simeq\\{\pm 1\\}$, the
discriminant of the quadric $(x^{2}+y^{2}+u^{2}-w^{2}=0)\subset\mathbb{P}^{3}$
is non-trivial.

######  Example 1.8 (Quadrics of dimension $\leq 2$ over $\mathbb{Q}_{p}$).

When $k=\mathbb{Q}_{p}$, with $p\neq 2$, there are two quadrics in
$\mathbb{P}^{2}$ up to isomorphism, namely $C(1,1)\simeq\mathbb{P}^{1}$ and
$C(\epsilon,p)$, where $\epsilon$ is a(ny) unit of $\mathbb{Q}_{p}^{\times}$
such that $\overline{\epsilon}$ is not a square in
$(\mathbb{Z}/p\mathbb{Z})^{\times}$. In $\mathbb{P}^{3}$ there are six
quadrics up to isomorphism:

(1.9) $\displaystyle(x^{2}-y^{2}-u^{2}+w^{2}=0)\subset\mathbb{P}^{3}$
$\displaystyle(x^{2}-\epsilon y^{2}-pu^{2}+(\epsilon
p)w^{2}=0)\subset\mathbb{P}^{3}$ (1.10)
$\displaystyle(x^{2}-y^{2}-u^{2}+\epsilon w^{2}=0)\subset\mathbb{P}^{3}$
$\displaystyle(x^{2}-y^{2}-u^{2}+pw^{2}=0)\subset\mathbb{P}^{3}$ (1.11)
$\displaystyle(x^{2}-y^{2}-u^{2}+(\epsilon p)w^{2}=0)\subset\mathbb{P}^{3}$
$\displaystyle(x^{2}+y^{2}-\epsilon u^{2}-pw^{2}=0)\subset\mathbb{P}^{3}\,.$

Making use of the above inclusion (1.5), we hence conclude that the
Grothendieck classes of these quadrics remain distinct in
$K_{0}\mathrm{Var}(\mathbb{Q}_{p})$. Note that since
$\mathbb{Q}_{p}^{\times}/(\mathbb{Q}_{p}^{\times})^{2}=\\{1,\epsilon,p,\epsilon
p\\}$, the discriminant of the quadrics (1.10)-(1.11) is non-trivial.

######  Example 1.12 (Quadrics of dimension $\leq 2$ over $\mathbb{Q}$).

When $k=\mathbb{Q}$, there are infinitely many quadrics in $\mathbb{P}^{2}$ up
to isomorphism. More specifically, following Remark 1.6, there is a bijection
between the set of quaternion $\mathbb{Q}$-algebras up to isomorphism and the
set of those positive integers which are not squares; consult [28, §III-IV].
Under such bijection, the prime numbers $p$ which are congruent to $3$ modulo
$4$ correspond to the quaternions algebras $(-1,-p)$. Consequently, we have,
for example, the infinite family of non-isomorphic quadrics
$\\{C(-1,-p)\\}_{p\equiv 3(\mathrm{mod}\,4)}$. In the same vein, we have, for
example, the infinite family of non-isomorphic quadrics with trivial
discriminant
$\\{(x^{2}+y^{2}+pu^{2}+pw^{2}=0)\subset\mathbb{P}^{3}\\}_{p\equiv
3(\mathrm{mod}\,4)}$. Note that we have also, for example, the infinite family
of non-isomorphic quadrics with non-trivial discriminant
$\\{(x^{2}+y^{2}+u^{2}+pu^{2}=0)\subset\mathbb{P}^{3}\\}_{p}$ parametrized by
all prime numbers $p$. Making use of the above inclusion (1.5), we hence
conclude that the Grothendieck classes of all the aforementioned quadrics
remain distinct in $K_{0}\mathrm{Var}(\mathbb{Q})$.

Item (ii) of Theorem 1.3 shows that when $I(k)^{3}$ is torsion-free (and
$\tilde{u}(k)<\infty$), two quadrics have the same Grothendieck class if and
only if they are isomorphic! Consequently, the above inclusion (1.5) admits
the following extension to all dimensions:

(1.13)
$\displaystyle\frac{\\{\text{Quadrics}\,\,Q_{q}\\}}{\text{isomorphism}}\subset
K_{0}\mathrm{Var}(k)$
$\displaystyle\mathrm{with}\,\,I(k)^{3}\,\,\mathrm{torsion}\text{-}\mathrm{free}\,\,\,(\mathrm{and}\,\,\tilde{u}(k)<\infty)\,.$

######  Example 1.14 (Quadrics over $\mathbb{R}$).

When $k=\mathbb{R}$, there are $\lfloor\frac{n+3}{2}\rfloor$ quadrics in
$\mathbb{P}^{n}$ up to isomorphism, namely the following family
$\\{(x^{2}_{1}+\cdots+x^{2}_{i}-x^{2}_{i+1}-\cdots-x^{2}_{n+1}=0)\subset\mathbb{P}^{n}\\}_{i}$,
with $\frac{n+1}{2}\leq i\leq n+1$ when $n$ is odd, and with
$\lfloor\frac{n+3}{2}\rfloor\leq i\leq n+1$ when $n$ is even. Making use of
the above inclusion (1.13), we hence conclude that the Grothendieck classes of
these quadrics remain distinct in $K_{0}\mathrm{Var}(\mathbb{R})$.

######  Example 1.15 (Quadrics over $\mathbb{Q}_{p}$).

When $k=\mathbb{Q}_{p}$, with $p\neq 2$, there are finitely many quadrics in
$\mathbb{P}^{n}$ up to isomorphism. More specifically, since the $u$-invariant
$u(\mathbb{Q}_{p})$ of $\mathbb{Q}_{p}$ is equal to $4$ (consult [7, §VI]),
there are two, resp. six, quadrics in $\mathbb{P}^{n}$ up to isomorphism when
$n$ is even, resp. odd. The corresponding quadratic forms (well-defined up to
similarity) are obtained by taking the orthogonal sum of the corresponding
quadratic forms of Example 1.20 with a finite number of copies of the
hyperbolic plane. Making use of the inclusion (1.13), we hence conclude that
the Grothendieck classes of these quadrics remain distinct in
$K_{0}\mathrm{Var}(\mathbb{Q}_{p})$.

######  Example 1.16 (Quadrics over $\mathbb{Q}$).

When $k=\mathbb{Q}$, there are infinitely many quadrics in $\mathbb{P}^{n}$ up
to isomorphism. Note that the assignment
$m\mapsto(\mathrm{sign}(m),\\{r_{p}\\}_{p})$, where
$m=\pm\prod_{p\,\mathrm{prime}}p^{r_{p}}$ is the prime factorization, gives
rise to a group isomorphism
$\mathbb{Q}^{\times}/(\mathbb{Q}^{\times})^{2}\simeq(\\{\pm
1\\}\times\oplus_{p}\mathbb{Z})/(\\{1\\}\times\oplus_{p}2\mathbb{Z})$.
Consequently, when $n$ is odd, we have, for example, the following infinite
family of non-isomorphic quadrics $\\{(x^{2}_{1}+\cdots+x^{2}_{n}+\lambda
x^{2}_{n+1}=0)\subset\mathbb{P}^{n}\\}_{\lambda}$ parametrized by the square
classes $\lambda\in\mathbb{Q}^{\times}/(\mathbb{Q}^{\times})^{2}$. Making use
of the above inclusion (1.13), we hence conclude that the Grothendieck classes
of all these quadrics remain distinct in $K_{0}\mathrm{Var}(\mathbb{Q})$.

### Statement of results - involution varieties

Let $k$ be a field of characteristic zero. Given a central simple $k$-algebra
$A$, let us write $\mathrm{deg}(A)$ for its degree and $\mathrm{SB}(A)$ for
the associated Severi-Brauer variety. In the same vein, given a central simple
$k$-algebra with involution of orthogonal type $(A,\ast)$, let us write
$\delta(A,\ast)\in k^{\times}/(k^{\times})^{2}$ for its discriminant (when
$\mathrm{deg}(A)$ is even), $C_{0}(A,\ast)$ for its even Clifford $k$-algebra,
and $\mathrm{Iv}(A,\ast)\subset\mathrm{SB}(A)$ for the associated involution
variety; consult [16, §II] [35]. Recall from [16, Thm. 8.10] that when
$\mathrm{deg}(A)$ is odd, $C_{0}(A,\ast)$ is a central simple $k$-algebra;
that when $\mathrm{deg}(A)$ is even and
$\delta(A,\ast)\notin(k^{\times})^{2}$, $C_{0}(A,\ast)$ is a central simple
algebra over its center $k(\sqrt{\delta(A,\ast)})$; and that when
$\mathrm{deg}(A)$ is even and $\delta(A,\ast)\in(k^{\times})^{2}$,
$C_{0}(A,\ast)\simeq C_{0}(A,\ast)^{+}\times C_{0}(A,\ast)^{-}$ is a product
of two central simple $k$-algebras. Recall also that
$\mathrm{dim}_{k}(C_{0}(A,\ast))=2^{\mathrm{deg}(A)-1}$ and
$\mathrm{dim}(\mathrm{Iv}(A,\ast))=\mathrm{deg}(A)-2$. Finally, when $k$ is
formally-real, we will write $\mathrm{sgn}_{P}(A,\ast)\in\mathbb{N}$ for the
signature of $(A,\ast)$ with respect to an ordering $P$ of $k$; consult [16,
§11].

######  Example 1.17 (Quadrics).

In the particular case where $A$ is split, the central simple $k$-algebra with
involution of orthogonal type $(A,\ast)$ becomes isomorphic to
$(\mathrm{M}_{\mathrm{deg}(A)\times\mathrm{deg}(A)}(k),\ast_{q})$, where
$\ast_{q}$ is the adjoint involution of a uniquely determined quadratic form
$q$ (up to similarity). Hence, in this particular case, the involution variety
$\mathrm{Iv}(A,\ast)$ reduces to the quadric $Q_{q}$. Moreover,
$\mathrm{deg}(A)=\mathrm{dim}(q)$, $\delta(A,\ast)=\delta(q)$ and
$C_{0}(A,\ast)=C_{0}(q)$. Furthermore, when $k$ is formally-real,
$\mathrm{sgn}_{P}(A,\ast)$ reduces to $|\mathrm{sgn}_{P}(q)|$.

######  Example 1.18 (Odd dimensional involution varieties).

Given a central simple $k$-algebra $A$ of odd degree, it is well-known that
$A$ admits an involution of orthogonal type if and only if $A$ is split.
Thanks to Example 1.17, this shows that the odd dimensional involution
varieties are the odd dimensional quadrics.

######  Example 1.19 (Forms of quadrics).

Following Example 1.17, note that every involution variety
$\mathrm{Iv}(A,\ast)$ becomes isomorphic to a quadric after extension of
scalars to a splitting field of $A$. Hence, involution varieties may be
understood as “forms of quadrics”. Moreover, as explained in [35, §2], the
involution varieties admit the following characterization: a smooth projective
$k$-variety $X$ of dimension $n-2$ is an involution variety if and only if
$X\times_{k}\overline{k}$ is isomorphic to the (unique) quadric
$(x_{1}^{2}+\cdots+x^{2}_{n}=0)\subset\mathbb{P}_{\overline{k}}^{n-1}$.
Furthermore, two involution varieties $\mathrm{Iv}(A,\ast)$ and
$\mathrm{Iv}(A^{\prime},\ast^{\prime})$ are isomorphic if and only if the
central simple $k$-algebras with involution of orthogonal type $(A,\ast)$ and
$(A^{\prime},\ast^{\prime})$ are isomorphic.

######  Example 1.20 (Products of two conics).

When $\mathrm{deg}(A)=4$ and $\delta(A,\ast)\in(k^{\times})^{2}$, we have an
isomorphism $(A,\ast)\simeq(Q_{1},\ast_{1})\otimes(Q_{2},\ast_{2})$, where
$Q_{1}=(a_{1},b_{1})$ and $Q_{2}=(a_{2},b_{2})$ are uniquely determined
quaternion $k$-algebras (up to isomorphism) and $\ast_{1}$ and $\ast_{2}$ are
the conjugation involutions of $Q_{1}$ and $Q_{2}$, respectively. Following
[35, Thm. 4.15], this leads to an isomorphism $\mathrm{Iv}(A,\ast)\simeq
C(a_{1},b_{1})\times C(a_{2},b_{2})$. This shows that the involution varieties
of dimension $2$ with trivial discriminant are the products of two conics.
Note that in the particular case where $A$ is split, we have
$Q_{1}=Q_{2}=(a,b)$. Consequently, we conclude from Remark 1.6 that
$(x^{2}-ay^{2}-bu^{2}+(ab)w^{2}=0)\simeq C(a,b)\times C(a,b)$ for every
quaternion $k$-algebra $(a,b)$.

Example 1.20 motivates the following notation:

######  Notation 1.21 (Condition $(\star)$).

Let $(A,\ast)$ be a central simple $k$-algebra with involution of orthogonal
type, with $\mathrm{deg}(A)=4$ and $\delta(A,\ast)\in(k^{\times})^{2}$. We
will say that $(A,\ast)$ satisfies condition $(\star)$ if $A$ is split.

######  Theorem 1.22.

Let $(A,\ast)$ and $(A^{\prime},\ast^{\prime})$ be two central simple
$k$-algebras with involutions of orthogonal type. If
$[\mathrm{Iv}(A,\ast)]=[\mathrm{Iv}(A^{\prime},\ast^{\prime})]$, then the
following holds:

  * (i)

We have $\mathrm{deg}(A)=\mathrm{deg}(A^{\prime})$.

  * (ii)

We have
$(\delta(A,\ast)\in(k^{\times})^{2})\Leftrightarrow(\delta(A^{\prime},\ast^{\prime})\in(k^{\times})^{2})$.

  * (iii)

We have $C_{0}(A,\ast)\simeq C_{0}(A^{\prime},\ast^{\prime})$.

  * (iv)

We have $A\simeq A^{\prime}$

  * (v)

When $k$ is formally-real, we have
$\mathrm{sgn}_{P}(A,\ast)=\mathrm{sgn}_{P}(A^{\prime},\ast^{\prime})$ for
every ordering $P$ of $k$.

In the particular case where $\mathrm{deg}(A)=4$ and
$\delta(A,\ast)\in(k^{\times})^{2}$ (or, equivalently,
$\mathrm{deg}(A^{\prime})=4$ and
$\delta(A^{\prime},\ast^{\prime})\in(k^{\times})^{2}$), we assume moreover in
items (iii)-(v) that $(A,\ast)$ and $(A^{\prime},\ast^{\prime})$ satisfy
condition $(\star)$.

######  Remark 1.23.

Note that if $C_{0}(A,\ast)\simeq C_{0}(A^{\prime},\ast)$, then
$\delta(A,\ast)=\delta(A^{\prime},\ast^{\prime})$.

######  Remark 1.24 (Severi-Brauer varieties).

It is well-known that two central simple $k$-algebras $A$ and $A^{\prime}$ are
isomorphic if and only if the associated Severi-Brauer varieties
$\mathrm{SB}(A)$ and $\mathrm{SB}(A^{\prime})$ are isomorphic. Consequently,
item (iv) of Theorem 1.22 shows that if two involution varieties
$\mathrm{Iv}(A,\ast)\subset\mathrm{SB}(A)$ and
$\mathrm{Iv}(A^{\prime},\ast^{\prime})\subset\mathrm{SB}(A^{\prime})$ have the
same Grothendieck class, then the corresponding “ambient” Severi-Brauer
varieties $\mathrm{SB}(A)$ and $\mathrm{SB}(A^{\prime})$ are necessarily
isomorphic!

Theorem 1.22 enables the following applications:

######  Theorem 1.25.

Let $(A,\ast)$ and $(A^{\prime},\ast^{\prime})$ be two central simple
$k$-algebras with involutions of orthogonal type. If
$[\mathrm{Iv}(A,\ast)]=[\mathrm{Iv}(A^{\prime},\ast^{\prime})]$, then the
following holds:

  * (i)

When $\mathrm{deg}(A)\leq 4$ (or, equivalently, $\mathrm{deg}(A^{\prime})\leq
4$), we have $\mathrm{Iv}(A,\ast)\simeq\mathrm{Iv}(A^{\prime},\ast^{\prime})$.

  * (ii)

When $I(k)^{3}$ is torsion-free, we have
$\mathrm{Iv}(A,\ast)\simeq\mathrm{Iv}(A^{\prime},\ast^{\prime})$. In the case
where $k$ is formally-real and $\mathrm{deg}(A)$ is even (or, equivalently,
$\mathrm{deg}(A^{\prime})$ is even), we assume moreover that
$\tilde{u}(k)<\infty$.

In the particular case where $\mathrm{deg}(A)=4$ and
$\delta(A,\ast)\in(k^{\times})^{2}$ (or, equivalently,
$\mathrm{deg}(A^{\prime})=4$ and
$\delta(A^{\prime},\ast^{\prime})\in(k^{\times})^{2}$), we assume moreover in
items (i)-(ii) that $(A,\ast)$ and $(A^{\prime},\ast^{\prime})$ satisfy
condition $(\star)$.

Note that Theorems 1.22 and 1.25 are far reaching generalizations of Theorems
1.1 and 1.3, respectively. Consequently, the above inclusions (1.5) and (1.13)
admit extensions to the realm of involution varieties. In particular, the
above Example 1.8 admits the following extension:

######  Example 1.26 (Involution varieties of dimension $\leq 2$ over
$\mathbb{Q}_{p}$).

As mentioned in Example 1.18, there is no difference between involution
varieties of dimension $1$ and quadrics of dimension $1$. Hence, we address
solely the $2$-dimensional case. First, recall from [27, §XII-XIII] that given
any local field $l$, there is a unique division quaternion $l$-algebra ${\bf
D}$ up to isomorphism; when $l=\mathbb{Q}_{p}$, we have ${\bf
D}\simeq(\epsilon,p)$. As explained in [16, §15.B], the assignment
$(A,\ast)\mapsto C_{0}(A,\ast)$ gives rise to a bijection between the set of
central simple $\mathbb{Q}_{p}$-algebras of degree $4$ with involution of
orthogonal type up to isomorphism and the set of quaternion algebras over some
étale quadratic extension of $\mathbb{Q}_{p}$ up to isomorphism. The inverse
bijection is induced by the assignment
$Q\mapsto(N_{l/\mathbb{Q}_{p}}(Q),N_{l/\mathbb{Q}_{p}}(\ast))$, where $l$ is
the center of $Q$, $\ast$ is the conjugation involution of $Q$, and
$N_{l/k}(-)$ is the norm construction. Consequently, since
$\mathbb{Q}^{\times}_{p}/(\mathbb{Q}_{p}^{\times})^{2}=\\{1,\epsilon,p,\epsilon
p\\}$, there are nine involution varieties of dimension $2$ over
$\mathbb{Q}_{p}$ up to isomorphism. Note that under the assignment
$\mathrm{Iv}(A,\ast)\mapsto(A,\ast)\mapsto C_{0}(A,\ast)$, the involution
varieties (1.9)-(1.11) correspond to the following quaternion algebras:

$\displaystyle(1,1)\times(1,1)\,\,\mathrm{over}\,\,\mathbb{Q}_{p}\times\mathbb{Q}_{p}$
$\displaystyle(\epsilon,p)\times(\epsilon,p)\,\,\mathrm{over}\,\,\mathbb{Q}_{p}\times\mathbb{Q}_{p}$
$\displaystyle(1,1)\,\,\mathrm{over}\,\,\mathbb{Q}_{p}(\sqrt{\epsilon})$
$\displaystyle(1,1)\,\,\mathrm{over}\,\,\mathbb{Q}_{p}(\sqrt{p})$
$\displaystyle(1,1)\,\,\mathrm{over}\,\,\mathbb{Q}_{p}(\sqrt{\epsilon p})$
$\displaystyle{\bf D}\,\,\mathrm{over}\,\,\mathbb{Q}_{p}(\sqrt{\epsilon
p})\,.$

Therefore, in addition to (1.9)-(1.11), we can also consider the following two
involution varieties:

(1.27)
$\displaystyle\mathrm{Iv}(N_{\mathbb{Q}_{p}(\sqrt{\epsilon})/\mathbb{Q}_{p}}({\bf
D}),N_{\mathbb{Q}_{p}(\sqrt{\epsilon})/\mathbb{Q}_{p}}(\ast))$
$\displaystyle\mathrm{Iv}(N_{\mathbb{Q}_{p}(\sqrt{p})/\mathbb{Q}_{p}}({\bf
D}),N_{\mathbb{Q}_{p}(\sqrt{p})/\mathbb{Q}_{p}}(\ast))\,.$

Making use of item (i) of Theorem 1.25, we hence conclude that the
Grothendieck classes of the eight involution varieties, namely (1.9)-(1.11)
and (1.27), remain distinct in $K_{0}\mathrm{Var}(\mathbb{Q}_{p})$.

The following result of Kollár444Consult also the subsequent work [8]. [12]
explains why condition $(\star)$ is needed in Theorems 1.22 and 1.25:

######  Theorem 1.28 (Products of two conics).

Let $Q_{1}$ and $Q_{2}$, resp. $Q^{\prime}_{1}$ and $Q^{\prime}_{2}$, be
quaternion $k$-algebras and $C(Q_{1})$ and $C(Q_{2})$, resp.
$C(Q^{\prime}_{1})$ and $C(Q^{\prime}_{2})$, the associated conics. The
following conditions are equivalent:

  * (a)

$[C(Q_{1})\times C(Q_{2})]=[C(Q^{\prime}_{1})\times C(Q^{\prime}_{2})]$

  * (b)

$C(Q_{1})\times
C(Q_{2})\,\,\mathrm{is}\,\,\mathrm{birational}\,\,\mathrm{to}\,\,C(Q^{\prime}_{1})\times
C(Q^{\prime}_{2})$

  * (c)

$\langle[Q_{1}],[Q_{2}]\rangle=\langle[Q^{\prime}_{1}],[Q^{\prime}_{2}]\rangle$
in the Brauer group $\mathrm{Br}(k)$.

######  Example 1.29 (Products of two conics over $\mathbb{R}$).

Let $k=\mathbb{R}$, $Q_{1}={\bf H}$, $Q_{2}=(1,1)$, and
$Q^{\prime}_{1}=Q^{\prime}_{2}={\bf H}$. Following Example 1.20, let
$(A,\ast):=(Q_{1},\ast_{1})\otimes(Q_{2},\ast_{2})$ and
$(A^{\prime},\ast^{\prime}):=(Q^{\prime}_{1},\ast^{\prime}_{1})\otimes(Q^{\prime}_{2},\ast^{\prime}_{2})$.
Note that $(A^{\prime},\ast^{\prime})\simeq(\mathrm{M}_{4\times
4}(\mathbb{R}),\ast^{\prime}_{q^{\prime}})$, where
$\ast^{\prime}_{q^{\prime}}$ is the adjoint involution of the quadratic form
$q^{\prime}=\langle 1,1,1,1\rangle$. On the one hand, since $\langle[{\bf
H}],[(1,1)]\rangle=\langle[{\bf H}],[{\bf H}]\rangle$ in the Brauer group
$\mathrm{Br}(\mathbb{R})$, Theorem 1.28 implies that
$[\mathrm{Iv}(A,\ast)]=[C({\bf H})\times\mathbb{P}^{1}]=[C({\bf H})\times
C({\bf H})]=[\mathrm{Iv}(A^{\prime},\ast^{\prime})]$. On the other hand, the
even Clifford algebra $C_{0}(A,\ast)\simeq{\bf H}\times(1,1)$ is not
isomorphic to $C_{0}(A^{\prime},\ast^{\prime})\simeq{\bf H}\times{\bf H}$, the
central simple $\mathbb{R}$-algebra $A\simeq\mathrm{M}_{2\times 2}({\bf H})$
is not isomorphic to $A^{\prime}\simeq\mathrm{M}_{4\times 4}(\mathbb{R})$, the
signature $\mathrm{sgn}(A,\ast)=0$ is different from
$\mathrm{sgn}(A^{\prime},\ast^{\prime})=|\mathrm{sgn}(\langle
1,1,1,1\rangle)|=4$, and the involution variety $C({\bf
H})\times\mathbb{P}^{1}$ is not isomorphic to $C({\bf H})\times C({\bf H})$.
This shows, in particular, that the above Theorems 1.22 and 1.25 are false
without assuming condition $(\star)$.

Finally, note that by combining Kollár’s Theorem 1.28 with item (i) of Theorem
1.25, we obtain a complete description of the Grothendieck classes of all the
involution varieties of dimension $2$ (over any base field $k$). Here is one
illustrative example:

######  Example 1.30 (Grothendieck classes of the involution varieties of
dimension $2$ over $\mathbb{Q}_{p}$).

As explained in Example 1.26, there are nine involution varieties of dimension
$2$ over $\mathbb{Q}_{p}$, with $p\neq 2$, up to isomorphism. Eight of them
were described in Examples 1.8 and 1.26, namely (1.9)-(1.11) and (1.27), and
these have distinct Grothendieck classes. The remaining involution variety
(which has trivial discriminant) is the following one
$\mathrm{Iv}(((\epsilon,p),\ast)\otimes((1,1),\ast)\simeq C(\epsilon,p))\times
C(1,1)\simeq C(\epsilon,1)\times\mathbb{P}^{1}$. Following Example 1.20, note
that the right-hand side of (1.9) is isomorphic to the product
$C(\epsilon,p)\times C(\epsilon,p)$. Since
$\langle[(\epsilon,p)],[(\epsilon,p)]\rangle=\langle[(\epsilon,p)],[(1,1)]\rangle$
in the Brauer group $\mathrm{Br}(\mathbb{Q}_{p})$, Theorem 1.28 implies that
$[C(\epsilon,p)\times C(\epsilon,p)]=[C(\epsilon,p)\times\mathbb{P}^{1}]$.
Consequently, we conclude that the nine involution varieties of dimension $2$
give rise to eight distinct Grothendieck classes.

##  2\. Preliminaries

Throughout the article $k$ denotes a base field of characteristic zero, and we
will write $G:=\mathrm{Gal}(\overline{k}/k)$ for its absolute Galois group.
Given a central simple $k$-algebra $A$, we will write $\mathrm{ind}(A)$ for
its index and $[A]$ for its class in the Brauer group $\mathrm{Br}(k)$. Recall
that $\mathrm{Br}(k)\simeq\oplus_{p\,\mathrm{prime}}\mathrm{Br}(k)\\{p\\}$,
where $\mathrm{Br}(k)\\{p\\}$ stands for the $p$-power torsion subgroup.
Finally, in order to simplify the exposition, we will write $\mathrm{pt}$
instead of $\mathrm{Spec}(k)$.

###  2.1. Dg categories

Throughout the article, we will assume some basic familiarity with the
language of dg categories; consult, for example, Keller’s survey [10]. We will
write $\mathrm{dgcat}(k)$ for the category of (small) dg categories and
$\mathrm{dgcat}_{\mathrm{sp}}(k)$ for the full subcategory of smooth proper dg
categories in the sense of Kontsevich [13, 14, 15]. Examples of smooth proper
dg categories include, for example, the finite-dimensional $k$-algebras of
finite global dimension $A$ as well as the dg categories of perfect complexes
$\mathrm{perf}_{\mathrm{dg}}(X)$ associated to smooth proper $k$-schemes $X$.
As explained in [31, §1.7], the symmetric monoidal category
$(\mathrm{dgcat}_{\mathrm{sp}}(k),\otimes)$ is rigid555Recall that a symmetric
monoidal category is called rigid if all its objects are dualizable., with the
dual of a smooth proper dg category ${\mathcal{A}}$ being its opposite dg
category ${\mathcal{A}}^{\mathrm{op}}$.

###  2.2. Chow motives

Given a commutative ring of coefficients $R$, recall from [1, §4.1][23] the
definition of the category of Chow motives $\mathrm{Chow}(k)_{R}$. This
category is $R$-linear, rigid symmetric monoidal and idempotent complete.
Moreover, it comes equipped with a symmetric monoidal functor
$\mathfrak{h}(-)_{R}\colon\mathrm{SmProj}(k)\to\mathrm{Chow}(k)_{R}$ defined
on the category of smooth projective $k$-schemes. The Chow motive
$\mathfrak{h}(\mathbb{P}^{1})_{R}$ of the projective line $\mathbb{P}^{1}$
decomposes into a direct sum $\mathfrak{h}(\mathrm{pt})_{R}\oplus R(-1)$. The
direct summand $R(-1)$ is called the Lefschetz motive and its
$\otimes$-inverse $R(1)$ the Tate motive. Given a smooth projective $k$-scheme
$X$ and an integer $i\in\mathbb{Z}$, we will write $\mathfrak{h}(X)_{R}(i)$
instead of $\mathfrak{h}(X)_{R}\otimes R(1)^{\otimes i}$. Under these
notations, given two smooth projective $k$-schemes $X$ and $Y$ and two
integers $i,j\in\mathbb{Z}$, we have an isomorphism

$\mathrm{Hom}_{\mathrm{Chow}(k)_{R}}(\mathfrak{h}(X)_{R}(i),\mathfrak{h}(Y)_{R}(j))\simeq{\mathcal{Z}}^{\mathrm{dim}(X)-i+j}_{\sim\mathrm{rat}}(X\times
Y)_{R}\,,$

where the right-hand side stands for the $R$-module of algebraic cycles on
$X\times Y$ of codimension $\mathrm{dim}(X)-i+j$ up to rational equivalence.
Finally, in the particular case where $R=\mathbb{Z}$, we will write
$\mathrm{Chow}(k)$ instead of $\mathrm{Chow}(k)_{\mathbb{Z}}$ and
$\mathfrak{h}(-)$ instead of $\mathfrak{h}(-)_{\mathbb{Z}}$.

###  2.3. Noncommutative Chow motives

Given a commutative ring of coefficients $R$, recall from [31, §4.1] the
definition of the category of noncommutative Chow motives
$\operatorname{NChow}(k)_{R}$. This category is $R$-linear, rigid symmetric
monoidal, idempotent complete, and comes equipped with a symmetric monoidal
functor
$U(-)_{R}\colon\mathrm{dgcat}(k)_{\mathrm{sp}}\to\operatorname{NChow}(k)_{R}$.
Given smooth proper dg categories ${\mathcal{A}}$ and
${\mathcal{A}}^{\prime}$, we have an isomorphism

$\mathrm{Hom}_{\operatorname{NChow}(k)_{R}}(U({\mathcal{A}})_{R},U({\mathcal{A}}^{\prime})_{R})\simeq
K_{0}({\mathcal{A}}^{\mathrm{op}}\otimes{\mathcal{A}}^{\prime})_{R}\,,$

where the right-hand side stands for the $R$-linearized Grothendieck group of
${\mathcal{A}}^{\mathrm{op}}\otimes{\mathcal{A}}^{\prime}$. Moreover, the
composition law on $\operatorname{NChow}(k)$ is induced by the (derived)
tensor product of bimodules, and the identity of $U({\mathcal{A}})_{R}$ is the
Grothendieck class of the diagonal
${\mathcal{A}}\text{-}{\mathcal{A}}$-bimodule ${\mathcal{A}}$. In the
particular case where $R=\mathbb{Z}$, we will write $\operatorname{NChow}(k)$
instead of $\operatorname{NChow}(k)_{\mathbb{Z}}$ and $U(-)$ instead of
$U(-)_{\mathbb{Z}}$.

######  Theorem 2.1.

(see [32, Thm. 9.1]) Given two central simple $k$-algebras $A$ and
$A^{\prime}$, we have the equivalence:

$\displaystyle U(A)\simeq
U(A^{\prime})\,\,\mathrm{in}\,\,\operatorname{NChow}(k)$
$\displaystyle\Leftrightarrow$
$\displaystyle[A]=[A^{\prime}]\,\,\mathrm{in}\,\,\mathrm{Br}(k)\,.$

######  Theorem 2.2.

(see [33, Thm. 2.20(iv)]) Given two families of central simple $k$-algebras
$\\{A_{i}\\}_{1\leq i\leq n}$ and $\\{A^{\prime}_{j}\\}_{1\leq j\leq m}$, we
have an isomorphism $\oplus_{i=1}^{n}U(A_{i})\simeq\oplus_{j=1}^{m}U(A_{j})$
in $\operatorname{NChow}(k)$ if and only if $n=m$ and for every prime number
$p$ there exists a permutation $\sigma_{p}$ such that
$[A^{\prime}_{i}]^{p}=[A_{\sigma_{p}(i)}]^{p}$ in $\mathrm{Br}(k)\\{p\\}$ for
every $i$.

######  Remark 2.3 (Central simple algebras over field extensions).

Let $l/k$, resp. $l^{\prime}/k$, be a finite Galois field extension and $A$,
resp. $A^{\prime}$, a central simple $l$-algebra, resp. central simple
$l^{\prime}$-algebra. Let us denote by $H\subseteq G$, resp.
$H^{\prime}\subseteq G$, the (unique) subgroup such that $\overline{k}^{H}=l$,
resp. $\overline{k}^{H^{\prime}}=l^{\prime}$. Given a commutative ring of
coefficients $R$, recall from [33, Thm. 2.13] that
$\mathrm{Hom}_{\operatorname{NChow}(k)_{R}}(U(A)_{R},U(A^{\prime})_{R})$ can
be identified with the $R$-module $\mathrm{Map}^{G}(G/H\times G/H^{\prime},R)$
of $G$-invariant maps from the finite $G$-set $G/H\times G/H^{\prime}$
(equipped with the diagonal $G$-action) to $R$. Under these identifications,
the composition map

$\mathrm{Hom}_{\operatorname{NChow}(k)_{R}}(U(A)_{R},U(A^{\prime})_{R})\times\mathrm{Hom}_{\operatorname{NChow}(k)_{R}}(U(A^{\prime})_{R},U(A)_{R})\longrightarrow\mathrm{Hom}_{\operatorname{NChow}(k)_{R}}(U(A)_{R},U(A)_{R})$

corresponds to the bilinear pairing

$\mathrm{Map}^{G}(G/H\times
G/H^{\prime},R)\times\mathrm{Map}^{G}(G/H^{\prime}\times
G/H,R)\longrightarrow\mathrm{Map}^{G}(G/H\times G/H,R)$

that sends a pair of $G$-invariant maps $(\alpha,\beta)$ to the following
$G$-invariant map:

$(\overline{g_{1}},\overline{g_{3}})\mapsto\sum_{\overline{g_{2}}\in
G/H^{\prime}}\alpha(\overline{g_{1}},\overline{g_{2}})\beta(\overline{g_{2}},\overline{g_{3}})\mathrm{ind}\big{(}(A\otimes_{\overline{k}^{H}}\overline{k}^{(H\cap
H^{\prime})})^{\mathrm{op}}\otimes_{\overline{k}^{(H\cap
H^{\prime})}}(A^{\prime}\otimes_{\overline{k}^{H^{\prime}}}\overline{k}^{(H\cap
H^{\prime})})\big{)}^{2}\,.$

Moreover, the identity of $U(A)_{R}$ corresponds to the $G$-invariant map
$G/H\times G/H\to R$ with $1$ in the diagonal and $0$ elsewhere.

###  2.4. Numerical motives

Given an additive rigid symmetric monoidal category ${\mathcal{C}}$, recall
from [2, §7] that its ${\mathcal{N}}$-ideal is defined as follows

${\mathcal{N}}(a,b):=\\{f\in\mathrm{Hom}_{\mathcal{C}}(a,b)\,|\,\forall
g\in\mathrm{Hom}_{\mathcal{C}}(b,a)\,\,\mathrm{we}\,\,\mathrm{have}\,\,\mathrm{tr}(g\circ
f)=0\\}\,,$

where $\mathrm{tr}(g\circ f)$ is the categorical trace of $g\circ f$. Via the
adjunction isomorphism
$\mathrm{Hom}_{\mathcal{C}}(a,b)\simeq\mathrm{Hom}_{\mathcal{C}}({\bf
1},a^{\vee}\otimes b)$, where ${\bf 1}$ stands for the $\otimes$-unit and
$a^{\vee}$ for the dual of $a$, the ${\mathcal{N}}$-ideal can also be
described as follows:

${\mathcal{N}}(a,b)=\\{f\in\mathrm{Hom}_{\mathcal{C}}({\bf 1},a^{\vee}\otimes
b)\,|\,\forall g\in\mathrm{Hom}_{\mathcal{C}}(a^{\vee}\otimes b,{\bf
1})\,\,\mathrm{we}\,\,\mathrm{have}\,\,g\circ f=0\\}\,.$

Given a commutative ring of coefficients $R$, recall from [1, §4.1][23] that
the category of numerical motives $\operatorname{Num}(k)_{R}$ is defined as
the idempotent completion of the quotient of $\mathrm{Chow}(k)_{R}$ by the
$\otimes$-ideal ${\mathcal{N}}$. This category is $R$-linear, rigid symmetric
monoidal and idempotent complete. Moreover, given two smooth projective
$k$-schemes $X$ and $Y$ and two integers $i,j\in\mathbb{Z}$, we have an
isomorphism

$\mathrm{Hom}_{\mathrm{Num}(k)_{R}}(\mathfrak{h}(X)_{R}(i),\mathfrak{h}(Y)_{R}(j))\simeq{\mathcal{Z}}_{\sim\mathrm{num}}^{\mathrm{dim}(X)-i+j}(X\times
Y)_{R}\,,$

where the right-hand side stands for the $R$-module of algebraic cycles up to
numerical equivalence.

###  2.5. Noncommutative numerical motives

Given a commutative ring of coefficients $R$, recall from [31, §4.6] that the
category of noncommutative numerical motives $\operatorname{NNum}(k)_{R}$ is
defined as the idempotent completion of the quotient of
$\operatorname{NChow}(k)_{R}$ by the $\otimes$-ideal ${\mathcal{N}}$. This
category is $R$-linear, rigid symmetric monoidal and idempotent complete.

###  2.6. Noncommutative radical motives

Given an additive category ${\mathcal{C}}$, its ${\mathcal{R}}$-ideal is
defined as follows:

${\mathcal{R}}(a,b):=\\{f\in\mathrm{Hom}_{\mathcal{C}}(a,b)\,|\,\forall
g\in\mathrm{Hom}_{\mathcal{C}}(b,a)\,\,\mathrm{the}\,\,\mathrm{endomorphism}\,\,\operatorname{id}_{a}-g\circ
f\,\,\mathrm{is}\,\,\mathrm{invertible}\\}\,.$

Given a commutative ring of coefficients $R$, the category of noncommutative
radical motives $\mathrm{NRad}(k)_{R}$ is defined as the idempotent completion
of the quotient of $\operatorname{NChow}(k)_{R}$ by the ideal ${\mathcal{R}}$.
By construction, this category is $R$-linear, additive, and idempotent
complete.

###  2.7. Motivic measures

Let $K_{0}(\operatorname{Num}(k))$ and $K_{0}(\operatorname{NChow}(k))$ be the
Grothendieck rings of the additive symmetric monoidal categories
$\operatorname{Num}(k)$ and $\operatorname{NChow}(k)$, respectively. The
following motivic measures will be used throughout the article:

######  Proposition 2.4.

(see [1, Cor. 13.2.2.1]) The assignment $X\mapsto\mathfrak{h}(X)$, with $X$ a
smooth projective $k$-scheme, gives rise to a motivic measure
$\mu_{\mathrm{c}}\colon K_{0}\mathrm{Var}(k)\to K_{0}(\operatorname{Num}(k))$.

######  Proposition 2.5.

(see [30, Prop. 4.1]) The assignment $X\mapsto
U(\mathrm{perf}_{\mathrm{dg}}(X))$, with $X$ a smooth projective $k$-scheme,
gives rise to a motivic measure $\mu_{\mathrm{nc}}\colon
K_{0}\mathrm{Var}(k)\to K_{0}(\operatorname{NChow}(k))$.

##  3\. Cancellation property

We start by recalling the following cancellation result:

######  Proposition 3.1.

(see [30, Prop. 4.9]) Let $\\{A_{i}\\}_{1\leq i\leq n}$ and
$\\{A^{\prime}_{j}\\}_{1\leq j\leq m}$ be two families of central simple
$k$-algebras. Given any noncommutative Chow motive
$N\\!\\!M\in\operatorname{NChow}(k)$, we have the following implication:

(3.2) $\displaystyle N\\!\\!M\oplus\oplus_{i=1}^{n}U(A_{i})\simeq
N\\!\\!M\oplus\oplus_{j=1}^{m}U(A^{\prime}_{j})$ $\displaystyle\Rightarrow$
$\displaystyle
n=m\,\,\,\mathrm{and}\,\,\oplus_{i=1}^{n}U(A_{i})\simeq\oplus_{j=1}^{m}U(A^{\prime}_{j})\,.$

The following result, which is of independent interest, will play a key role
in the sequel:

######  Theorem 3.3 (Cancellation).

Let $l/k$, resp. $l^{\prime}/k$, be a field a extension of degree $2$ and $A$,
resp. $A^{\prime}$, a central simple $l$-algebra, resp. central simple
$l^{\prime}$-algebra, such that $\mathrm{ind}(A)$, resp.
$\mathrm{ind}(A^{\prime})$, is a power of $2$. Moreover, let $B$ and
$B^{\prime}$ be two central simple $k$-algebras. Under these assumptions,
given any noncommutative Chow motive $N\\!\\!M\in\operatorname{NChow}(k)$ and
integer $n\geq 0$, we have the following implication:

$\displaystyle N\\!\\!M\oplus U(B)^{\oplus n}\oplus U(A)\simeq N\\!\\!M\oplus
U(B^{\prime})^{\oplus n}\oplus U(A^{\prime})$ $\displaystyle\Rightarrow$
$\displaystyle U(A)\simeq U(A^{\prime})\,.$

In order to prove Theorem 3.3, we need several ingredients.

######  Lemma 3.4.

Given two field extensions $l/k$ and $l^{\prime}/k$ of degree $2$, we have the
following equivalences:

$l\simeq l^{\prime}\quad\Leftrightarrow\quad U(l)_{\mathbb{Q}}\simeq
U(l^{\prime})_{\mathbb{Q}}\,\,\mathrm{in}\,\,\operatorname{NChow}(k)_{\mathbb{Q}}\quad\Leftrightarrow\quad
U(l)_{\mathbb{Q}}\simeq
U(l^{\prime})_{\mathbb{Q}}\,\,\mathrm{in}\,\,\operatorname{NNum}(k)_{\mathbb{Q}}\,.$

###### Proof.

Both implications $(\Rightarrow)$ are clear. We start by proving the right-
hand side implication $(\Leftarrow)$. Note that since the field extension
$l/k$ is of degree $2$, the composition map

$\mathrm{Hom}_{\operatorname{NChow}(k)_{\mathbb{Q}}}(U(k)_{\mathbb{Q}},U(l)_{\mathbb{Q}})\times\mathrm{Hom}_{\operatorname{NChow}(k)_{\mathbb{Q}}}(U(l)_{\mathbb{Q}},U(k)_{\mathbb{Q}})\longrightarrow\mathrm{Hom}_{\operatorname{NChow}(k)_{\mathbb{Q}}}(U(k)_{\mathbb{Q}},U(k)_{\mathbb{Q}})$

corresponds to the bilinear pairing
$\mathbb{Q}\times\mathbb{Q}\to\mathbb{Q},(\alpha,\beta)\mapsto 2\alpha\beta$.
Note also that since the field extension $l/k$ is Galois, we have the
following isomorphism of $k$-algebras
$l\otimes_{k}l\stackrel{{\scriptstyle\simeq}}{{\to}}l\times
l,\lambda_{1}\otimes\lambda_{2}\mapsto(\lambda_{1}\lambda_{2},\sigma(\lambda_{1})\lambda_{2})$,
where $\sigma$ stands for the generator of the Galois group of $l/k$. This
implies that

$U(l)^{\vee}_{\mathbb{Q}}\otimes U(l)_{\mathbb{Q}}\simeq
U(l^{\mathrm{op}})_{\mathbb{Q}}\otimes U(l)_{\mathbb{Q}}\simeq
U(l^{\mathrm{op}}\otimes_{k}l)_{\mathbb{Q}}=U(l\otimes_{k}l)_{\mathbb{Q}}\simeq
U(l)_{\mathbb{Q}}\oplus U(l)_{\mathbb{Q}}\,.$

Therefore, following the definition of the category of noncommutative
numerical motives, we observe that
$\mathrm{End}_{\operatorname{NChow}(k)_{\mathbb{Q}}}(U(l)_{\mathbb{Q}})=\mathrm{End}_{\operatorname{NNum}(k)_{\mathbb{Q}}}(U(l)_{\mathbb{Q}})$.
All the above holds mutatis mutandis with $l$ replaced by $l^{\prime}$. Hence,
we conclude that if $U(l)_{\mathbb{Q}}\simeq
U(l^{\prime})_{\mathbb{Q}}\,\,\mathrm{in}\,\,\operatorname{NNum}(k)_{\mathbb{Q}}$,
then $U(l)_{\mathbb{Q}}\simeq
U(l^{\prime})_{\mathbb{Q}}\,\,\mathrm{in}\,\,\operatorname{NChow}(k)_{\mathbb{Q}}$.
We now prove the left-hand side implication $(\Leftarrow)$. Let $H\subseteq G$
and $H^{\prime}\subseteq G$ be the subgroups of index $2$ such that
$\overline{k}^{H}=l$ and $\overline{k}^{H^{\prime}}=l^{\prime}$, respectively.
Recall from Remark 2.3 that
$\mathrm{Hom}_{\operatorname{NChow}(k)_{\mathbb{Q}}}(U(l)_{\mathbb{Q}},U(l^{\prime})_{\mathbb{Q}})$
can be identified with the $\mathbb{Q}$-vector space
$\mathrm{Map}^{G}(G/H\times G/H^{\prime},\mathbb{Q})$ of $G$-invariant maps
from the finite $G$-set $G/H\times G/H^{\prime}$ to $\mathbb{Q}$. Recall also
that the composition map

$\mathrm{Hom}_{\operatorname{NChow}(k)_{\mathbb{Q}}}(U(l)_{\mathbb{Q}},U(l^{\prime})_{\mathbb{Q}})\times\mathrm{Hom}_{\operatorname{NChow}(k)_{\mathbb{Q}}}(U(l^{\prime})_{\mathbb{Q}},U(l)_{\mathbb{Q}})\longrightarrow\mathrm{Hom}_{\operatorname{NChow}(k)_{\mathbb{Q}}}(U(l)_{\mathbb{Q}},U(l)_{\mathbb{Q}})$

corresponds to the bilinear pairing

$\mathrm{Map}^{G}(G/H\times
G/H^{\prime},\mathbb{Q})\times\mathrm{Map}^{G}(G/H^{\prime}\times
G/H,\mathbb{Q})\longrightarrow\mathrm{Map}^{G}(G/H\times G/H,\mathbb{Q})$

that sends $(\alpha,\beta)$ to the $G$-invariant map
$(\overline{g_{1}},\overline{g_{3}})\mapsto\sum_{\overline{g_{2}}\in
G/H^{\prime}}\alpha(\overline{g_{1}},\overline{g_{2}})\beta(\overline{g_{2}},\overline{g_{3}})$.
Moreover, the identity of $U(l)_{\mathbb{Q}}$ corresponds to the $G$-invariant
map $G/H\times G/H\to\mathbb{Q}$ with $1$ in the diagonal and $0$ elsewhere.
This shows, in particular, that the $\mathbb{Q}$-algebra of endomorphisms
$\mathrm{End}_{\operatorname{NChow}(k)_{\mathbb{Q}}}(U(l)_{\mathbb{Q}})$
corresponds to the group $\mathbb{Q}$-algebra $\mathbb{Q}[G/H]$. Now, let us
assume that $l\not\simeq l^{\prime}$, i.e., that the $k$-algebras $l$ and
$l^{\prime}$ are not isomorphic. Thanks to Galois theory, we have $H\neq
H^{\prime}$. Moreover, since the field extensions $l/k$ and $l^{\prime}/k$ are
of degree $2$, we have $H\not\subseteq H^{\prime}$ and
$H^{\prime}\not\subseteq H$. This implies that the (diagonal) $G$-action on
the set $G/H\times G/H^{\prime}$ is transitive. Therefore, it follows from the
above description of the category of noncommutative Chow motives that
$U(l)_{\mathbb{Q}}\simeq U(l^{\prime})_{\mathbb{Q}}$ in
$\operatorname{NChow}(k)_{\mathbb{Q}}$ if and only if there exist rational
numbers $\alpha,\beta\in\mathbb{Q}$ such that

(3.5) $\begin{cases}\alpha\beta+\alpha\beta=1\\\
\alpha\beta+\alpha\beta=0\end{cases}\,.$

Since the system of equations (3.5) is impossible, we hence conclude that
$U(l)_{\mathbb{Q}}\not\simeq U(l^{\prime})_{\mathbb{Q}}$ in
$\operatorname{NChow}(k)_{\mathbb{Q}}$. ∎

######  Lemma 3.6.

Given a field extension $l/k$ of degree $2$ and two central simple
$l$-algebras $A$ and $A^{\prime}$, we have the following equivalence:

$\displaystyle U(A)\simeq
U(A^{\prime})\,\,\mathrm{in}\,\,\operatorname{NChow}(k)$
$\displaystyle\Leftrightarrow$
$\displaystyle[A]=[A^{\prime}]\,\,\mathrm{in}\,\,\mathrm{Br}(l)\,.$

###### Proof.

Let $H\subseteq G$ be the subgroup of index $2$ such that
$\overline{k}^{H}=l$. Given a commutative ring of coefficients $R$, recall
from Remark 2.3 that
$\mathrm{Hom}_{\operatorname{NChow}(k)_{R}}(U(A)_{R},U(A^{\prime})_{R})$ can
be identified with the $R$-module $\mathrm{Map}^{G}(G/H\times G/H,R)$ of
$G$-invariant maps from the finite $G$-set $G/H\times G/H$ to $R$. Recall also
that, under these identifications, the composition map

$\displaystyle\mathrm{Hom}_{\operatorname{NChow}(k)_{R}}(U(A)_{R},U(A^{\prime})_{R})\times\mathrm{Hom}_{\operatorname{NChow}(k)_{R}}(U(A^{\prime})_{R},U(A)_{R})\longrightarrow\mathrm{Hom}_{\operatorname{NChow}(k)_{R}}(U(A)_{R},U(A)_{R})$

corresponds to the bilinear pairing

$\mathrm{Map}^{G}(G/H\times G/H,R)\times\mathrm{Map}^{G}(G/H\times
G/H,R)\longrightarrow\mathrm{Map}^{G}(G/H\times G/H,R)$

that sends $(\alpha,\beta)$ to the $G$-invariant map
$(\overline{g_{1}},\overline{g_{3}})\mapsto\sum_{\overline{g_{2}}\in
G/H}\alpha(\overline{g_{1}},\overline{g_{2}})\beta(\overline{g_{2}},\overline{g_{3}})\mathrm{ind}(A^{\mathrm{op}}\otimes_{l}A^{\prime})^{2}$.
Moreover, the identity of $U(A)_{R}$ corresponds to the $G$-invariant map
$G/H\times G/H\to R$ with $1$ in the diagonal and $0$ elsewhere. This shows,
in particular, that the $R$-algebra of endomorphisms
$\mathrm{End}_{\operatorname{NChow}(k)_{R}}(U(A)_{R})$ corresponds to the
group $R$-algebra $R[G/H]$. Note that since the field extension $l/k$ is of
degree $2$, the $G$-set $G/H\times G/H$ has two orbits (the elements in the
diagonal and the elements outside the diagonal). Therefore, thanks to the
above description of the category of noncommutative Chow motives, we observe
that $U(A)_{R}\simeq U(A^{\prime})_{R}$ in $\operatorname{NChow}(k)_{R}$ if
and only if there exist elements $\alpha^{+},\alpha^{-}\in R$ and
$\beta^{+},\beta^{-}\in R$ such that

(3.7)
$\begin{cases}\alpha^{+}\beta^{+}\mathrm{ind}(A^{\mathrm{op}}\otimes_{l}A^{\prime})^{2}+\alpha^{-}\beta^{-}\mathrm{ind}(A^{\mathrm{op}}\otimes_{l}A^{\prime})^{2}=1\\\
\alpha^{+}\beta^{-}\mathrm{ind}(A^{\mathrm{op}}\otimes_{l}A^{\prime})^{2}+\alpha^{-}\beta^{+}\mathrm{ind}(A^{\mathrm{op}}\otimes_{l}A^{\prime})^{2}=0\end{cases}\Leftrightarrow\begin{cases}(\alpha^{+}\beta^{+}+\alpha^{-}\beta^{-})\mathrm{ind}(A^{\mathrm{op}}\otimes_{l}A^{\prime})^{2}=1\\\
(\alpha^{+}\beta^{-}+\alpha^{-}\beta^{+})\mathrm{ind}(A^{\mathrm{op}}\otimes_{l}A^{\prime})^{2}=0\,.\end{cases}$

Now, let us restrict ourselves to the case where $R=\mathbb{Z}$. Note that if
the system of equations (3.7) holds, then
$\mathrm{ind}(A^{\mathrm{op}}\otimes_{l}A^{\prime})=1$. Conversely, if
$\mathrm{ind}(A^{\mathrm{op}}\otimes_{l}A^{\prime})=1$, then the system of
equations (3.7) holds; take, for example, $\alpha^{+}=\beta^{+}=1$ and
$\alpha^{-}=\beta^{-}=0$. Consequently, the proof follows now from the
classical fact that $\mathrm{ind}(A^{\mathrm{op}}\otimes_{l}A^{\prime})=1$ if
and only if $[A]=[A^{\prime}]$ in $\mathrm{Br}(l)$. ∎

######  Lemma 3.8.

Let $B$ a central simple $k$-algebra, $l/k$ a field extension of degree $2$
and $A$ a central simple $l$-algebra. Under these assumptions, we have
$\mathrm{Hom}_{\mathrm{NRad}(k)_{\mathbb{F}_{2}}}(U(B)_{\mathbb{F}_{2}},U(A)_{\mathbb{F}_{2}})=0$.

###### Proof.

Let $H\subseteq G$ be the subgroup of index $2$ such that
$\overline{k}^{H}=l$. Given a commutative ring of coefficients $R$, recall
from Remark 2.3 that
$\mathrm{Hom}_{\operatorname{NChow}(k)_{R}}(U(B)_{R},U(A)_{R})$ can be
identified with the $R$-module $\mathrm{Map}^{G}(G/G\times G/H,R)$ of
$G$-invariant maps from the finite set $G$-set $G/G\times G/H=G/H$ to $R$.
Recall also that, under these identifications, the composition map

(3.9)
$\mathrm{Hom}_{\operatorname{NChow}(k)_{R}}(U(B)_{R},U(A)_{R})\times\mathrm{Hom}_{\operatorname{NChow}(k)_{R}}(U(A)_{R},U(B)_{R})\longrightarrow\mathrm{Hom}_{\operatorname{NChow}(k)_{R}}(U(B)_{R},U(B)_{R})$

corresponds to the bilinear pairing $R\times R\to R,(\alpha,\beta)\mapsto
2\alpha\beta\mathrm{ind}((B\otimes_{k}l)^{\mathrm{op}}\otimes_{l}A)^{2}$.
Consequently, in the particular case where $R=\mathbb{F}_{2}$, the composition
map (3.9) is equal to zero. By definition of the category of noncommutative
radical motives, this hence implies that
$\mathrm{Hom}_{\mathrm{NRad}(k)_{\mathbb{F}_{2}}}(U(B)_{\mathbb{F}_{2}},U(A)_{\mathbb{F}_{2}})=0$.
∎

######  Lemma 3.10.

Let $l/k$ be a field extension of degree $2$ and $A$ a central simple
$l$-algebra. Given any noncommutative Chow motive
$N\\!\\!M\in\operatorname{NChow}(k)$, we have a direct sum decomposition
$N\\!\\!M_{\mathbb{F}_{2}}\simeq M\\!\\!N\oplus U(A)_{\mathbb{F}_{2}}^{\oplus
m}$ in the category $\mathrm{NRad}(k)_{\mathbb{F}_{2}}$ for some integer
$m\geq 0$, where $M\\!\\!N$ is a noncommutative radical motive which does not
contains $U(A)_{\mathbb{F}_{2}}$ as a direct summand.

###### Proof.

Recall from the proof of Lemma 3.6 that
$\mathrm{End}_{\operatorname{NChow}(k)}(U(A))$ corresponds to
$\mathbb{Z}[G/H]$, where $H\subseteq G$ is the subgroup of index $2$ such that
$\overline{k}^{H}=l$. Note that since the field extension $l/k$ is of degree
$2$, we have
$\mathbb{Z}[G/H]=\\{\alpha^{+}1+\alpha^{-}\sigma\,|\,\alpha^{+},\alpha^{-}\in\mathbb{Z}\\}$,
where $\sigma$ stands for the generator of the Galois group of $l/k$. By
definition, the category $\mathrm{NRad}(k)_{\mathbb{F}_{2}}$ is idempotent
complete. Therefore, we can inductively split the direct summand
$U(A)_{\mathbb{F}_{2}}$ from $N\\!\\!M_{\mathbb{F}_{2}}$. We claim that this
inductive process stops. Let us suppose by absurd that it does not. If this is
the case, then, given any integer $n\geq 1$, $U(A)_{\mathbb{F}_{2}}^{\oplus
n}$ is a direct summand of $N\\!\\!M_{\mathbb{F}_{2}}$. Using the fact that
the quotient functor
$\operatorname{NChow}(k)_{\mathbb{F}_{2}}\to\mathrm{NRad}(k)_{\mathbb{F}_{2}}$
reflects (co)retractions (consult [2, Prop. 1.4.4]), we conclude that
$U(A)_{\mathbb{F}_{2}}^{\oplus n}$ is a direct summand of
$N\\!\\!M_{\mathbb{F}_{2}}$ in the category
$\operatorname{NChow}(k)_{\mathbb{F}_{2}}$. Hence, there exist morphisms
$\rho\colon U(A)^{\oplus n}\to N\\!\\!M$ and $\varrho\colon N\\!\\!M\to
U(A)^{\oplus n}$ in the category $\operatorname{NChow}(k)$ such that:

(3.11)
$\displaystyle\varrho\circ\rho=\begin{bmatrix}\alpha^{+}_{11}1+\alpha^{-}_{11}\sigma&\cdots&\alpha^{+}_{1n}1+\alpha^{-}_{1n}\sigma\\\
\vdots&&\vdots\\\
\alpha^{+}_{n1}1+\alpha^{-}_{n1}\sigma&\cdots&\alpha^{+}_{nn}1+\alpha^{-}_{nn}\sigma\end{bmatrix}_{n\times
n}$ $\displaystyle\mathrm{with}$
$\displaystyle\begin{cases}\alpha^{+}_{ij}\,\,\mathrm{odd}\quad i=j\\\
\alpha^{+}_{ij}\,\,\mathrm{even}\,\,\,\,i\neq
j\end{cases}\,\,\,\,\alpha^{-}_{ij}\,\,\mathrm{even}\,.$

Recall from [31, §2.2.8] that Hochschild homology gives rise to a functor
$HH_{0}(-):\mathrm{dgcat}_{\mathrm{sp}}(k)\to\mathrm{vect}(k)$, with values in
the category of finite-dimensional $k$-vector spaces, such that
$HH_{0}(U({\mathcal{A}}))\simeq HH_{0}({\mathcal{A}})$ for every smooth proper
dg category ${\mathcal{A}}$. As proved in [34, Thm. 2.1], the canonical map
$l\to A$ induces an isomorphism
$HH_{0}(l)\stackrel{{\scriptstyle\simeq}}{{\to}}HH_{0}(A)$. Moreover, it is
well-known that $HH_{0}(l)\simeq l/[l,l]\simeq l$. Note that since the field
extension $l/k$ is of degree $2$, we have $l\simeq k(\sqrt{\lambda})$ for some
$\lambda\in k^{\times}/(k^{\times})^{2}$. Therefore, we conclude that
$HH_{0}(A)\simeq k(\sqrt{\lambda})$. Under the identification
$\mathrm{End}_{\operatorname{NChow}(k)}(U(A))\simeq\mathbb{Z}[G/H]$, the
additive functor $HH_{0}(-)$ sends the element $\sigma$ to the conjugation
automorphism $\sqrt{\lambda}\mapsto-\sqrt{\lambda}$ of the $k$-vector space
$k(\sqrt{\lambda})$. Consequently, making use of the identification
$\mathrm{End}_{\mathrm{vect}(k)}(HH_{0}(A))\simeq M_{2\times 2}(k)$ induced by
the basis $\\{1,\sqrt{\lambda}\\}$ of $k(\sqrt{\lambda})$, we conclude that
the functor $HH_{0}(-)$ induces the following ring homomorphism:

$\displaystyle\mathbb{Z}[G/H]\longrightarrow M_{2\times 2}(k)$
$\displaystyle\alpha^{+}1+\alpha^{-}\sigma\mapsto\begin{bmatrix}\alpha^{+}+\alpha^{-}&0\\\
0&\alpha^{+}-\alpha^{-}\end{bmatrix}_{2\times 2}\,.$

This implies that the composition of $HH_{0}(\rho)\colon
k(\sqrt{\lambda})^{\oplus n}\to HH_{0}(N\\!\\!M)$ with $HH_{0}(\varrho)\colon
HH_{0}(N\\!\\!M)\to k(\sqrt{\lambda})^{\oplus n}$, in the category
$\mathrm{vect}(k)$, admits the following block matrix representation:

(3.12)
$HH_{0}(\varrho\circ\rho)=\left[\begin{array}[]{cc|cc|cc}\alpha^{+}_{11}+\alpha^{-}_{11}&0&\cdots&\cdots&\alpha^{+}_{1n}+\alpha^{-}_{1n}&0\\\
0&\alpha^{+}_{11}-\alpha^{-}_{11}&\cdots&\cdots&0&\alpha^{+}_{1n}-\alpha^{-}_{1n}\\\
\hline\cr\vdots&\vdots&&&\vdots&\vdots\\\ \vdots&\vdots&&&\vdots&\vdots\\\
\hline\cr\alpha^{+}_{n1}+\alpha^{-}_{n1}&0&\cdots&\cdots&\alpha^{+}_{nn}+\alpha^{-}_{nn}&0\\\
0&\alpha^{+}_{n1}-\alpha^{-}_{n1}&\cdots&\cdots&0&\alpha^{+}_{nn}-\alpha^{-}_{nn}\end{array}\right]_{2n\times
2n}\,.$

Note that (3.11) implies that all the elements in the diagonal of the matrix
(3.12) are odd and that all the elements outside the diagonal are even.
Therefore, a simple inductive argument shows that the determinant of (3.12) is
odd. In particular, the determinant of (3.12) is non-zero. This implies that
the homomorphism $HH_{0}(\rho)$ is injective and consequently that
$\mathrm{dim}_{k}HH_{0}(N\\!\\!M)\geq 2n$. Since the integer $n\geq 1$ is
arbitrary and the dimension of the $k$-vector space $HH_{0}(N\\!\\!M)$ is
finite, we hence arrive to a contradiction. Therefore, there exists an integer
$m\geq 0$ such that $N\\!\\!M_{\mathbb{F}_{2}}$ contains $U(A)^{\oplus
m}_{\mathbb{F}_{2}}$, but not $U(A)^{\oplus(m+1)}_{\mathbb{F}_{2}}$, as a
direct summand. ∎

We now have all the ingredients necessary for the proof of Theorem 3.3.

### Proof of Theorem 3.3

Note first that we have induced isomorphisms:

(3.13) $\displaystyle N\\!\\!M_{\mathbb{Q}}\oplus U(B)^{\oplus
n}_{\mathbb{Q}}\oplus U(A)_{\mathbb{Q}}\simeq N\\!\\!M_{\mathbb{Q}}\oplus
U(B^{\prime})^{\oplus n}_{\mathbb{Q}}\oplus
U(A^{\prime})_{\mathbb{Q}}\,\,\mathrm{in}\,\,\operatorname{NNum}(k)_{\mathbb{Q}}$
(3.14) $\displaystyle N\\!\\!M_{\mathbb{F}_{2}}\oplus
U(B)_{\mathbb{F}_{2}}^{\oplus n}\oplus U(A)_{\mathbb{F}_{2}}\simeq
N\\!\\!M_{\mathbb{F}_{2}}\oplus U(B^{\prime})_{\mathbb{F}_{2}}^{\oplus
n}\oplus
U(A^{\prime})_{\mathbb{F}_{2}}\,\,\mathrm{in}\,\,\mathrm{NRad}(k)_{\mathbb{F}_{2}}\,.$

Moreover, as explained in [34, Thm. 2.1], we have isomorphisms
$U(B)_{\mathbb{Q}}\simeq U(k)_{\mathbb{Q}}$ and $U(A)_{\mathbb{Q}}\simeq
U(l)_{\mathbb{Q}}$ in the category $\operatorname{NChow}(k)_{\mathbb{Q}}$;
similarly for $B^{\prime}$ and $A^{\prime}$. Consequently, the above
isomorphism (3.13) reduces to $N\\!\\!M_{\mathbb{Q}}\oplus U(k)^{\oplus
n}_{\mathbb{Q}}\oplus U(l)_{\mathbb{Q}}\simeq N\\!\\!M_{\mathbb{Q}}\oplus
U(k)^{\oplus n}_{\mathbb{Q}}\oplus U(l^{\prime})_{\mathbb{Q}}$. Since the
category $\operatorname{NNum}(k)_{\mathbb{Q}}$ is abelian semi-simple (consult
[31, Thm. 4.27]), we hence conclude that $U(l)_{\mathbb{Q}}\simeq
U(l^{\prime})_{\mathbb{Q}}$ in $\operatorname{NNum}(k)_{\mathbb{Q}}$. Thanks
to Lemma 3.4, this implies that the $k$-algebras $l$ and $l^{\prime}$ are
isomorphic. Therefore, in the remainder of the proof we can (and will) assume
without loss of generality that $l=l^{\prime}$. Let $H\subseteq G$ be the
subgroup of index $2$ such that $\overline{k}^{H}=l$. As explained in the
proof of Lemma 3.6, the $\mathbb{F}_{2}$-algebras of endomorphisms
$\mathrm{End}_{\operatorname{NChow}(k)_{\mathbb{F}_{2}}}(U(A)_{\mathbb{F}_{2}})$
and
$\mathrm{End}_{\operatorname{NChow}(k)_{\mathbb{F}_{2}}}(U(A^{\prime})_{\mathbb{F}_{2}})$
can be identified with the group $\mathbb{F}_{2}$-algebra
$\mathbb{F}_{2}[G/H]$. Consequently, by definition of the category of
noncommutative radical motives, we obtain induced identifications:

(3.15)
$\displaystyle\mathrm{End}_{\mathrm{NRad}(k)_{\mathbb{F}_{2}}}(U(A)_{\mathbb{F}_{2}})\simeq\mathbb{F}_{2}$
$\displaystyle\mathrm{End}_{\mathrm{NRad}(k)_{\mathbb{F}_{2}}}(U(A^{\prime})_{\mathbb{F}_{2}})\simeq\mathbb{F}_{2}\,.$

Let us assume by absurd that $U(A)\not\simeq U(A^{\prime})$ in
$\operatorname{NChow}(k)$. Thanks to Lemma 3.6, this implies that
$[A]\neq[A^{\prime}]$ in $\mathrm{Br}(l)$ or, equivalently, that
$\mathrm{ind}(A^{\mathrm{op}}\otimes_{l}A^{\prime})\neq 1$. Since
$\mathrm{ind}(A^{\mathrm{op}}\otimes_{l}A^{\prime})$ divides the product
$\mathrm{ind}(A^{\mathrm{op}})\mathrm{ind}(A)$ and
$\mathrm{ind}(A^{\mathrm{op}})=\mathrm{ind}(A)$ and $\mathrm{ind}(A^{\prime})$
are powers of $2$, the index
$\mathrm{ind}(A^{\mathrm{op}}\otimes_{l}A^{\prime})$ is also a power of $2$.
Therefore, following the proof of Lemma 3.6, we observe that the composition
map

$\mathrm{Hom}_{\operatorname{NChow}(k)_{\mathbb{F}_{2}}}(U(A)_{\mathbb{F}_{2}},U(A^{\prime})_{\mathbb{F}_{2}})\times\mathrm{Hom}_{\operatorname{NChow}(k)_{\mathbb{F}_{2}}}(U(A^{\prime})_{\mathbb{F}_{2}},U(A)_{\mathbb{F}_{2}})\longrightarrow\mathrm{Hom}_{\operatorname{NChow}(k)_{\mathbb{F}_{2}}}(U(A)_{\mathbb{F}_{2}},U(A)_{\mathbb{F}_{2}})$

is equal to zero; similarly with $A$ and $A^{\prime}$ interchanged. By
definition of the category of noncommutative radical motives, this hence
implies that

(3.16)
$\mathrm{Hom}_{\mathrm{NRad}(k)_{\mathbb{F}_{2}}}(U(A)_{\mathbb{F}_{2}},U(A^{\prime})_{\mathbb{F}_{2}})=\mathrm{Hom}_{\mathrm{NRad}(k)_{\mathbb{F}_{2}}}(U(A^{\prime})_{\mathbb{F}_{2}},U(A)_{\mathbb{F}_{2}})=0\,.$

Thanks to Lemma 3.10, we have isomorphisms $N\\!\\!M_{\mathbb{F}_{2}}\simeq
M\\!\\!N\oplus U(A)^{\oplus m}_{\mathbb{F}_{2}}$ and
$N\\!\\!M_{\mathbb{F}_{2}}\simeq M\\!\\!N^{\prime}\oplus U(A^{\prime})^{\oplus
m^{\prime}}_{\mathbb{F}_{2}}$ in the category
$\mathrm{NRad}(k)_{\mathbb{F}_{2}}$ for some integers $m,m^{\prime}\geq 0$,
where $M\\!\\!N$, resp. $M\\!\\!N^{\prime}$, is a noncommutative radical
motive which does not contains $U(A)_{\mathbb{F}_{2}}$, resp.
$U(A^{\prime})_{\mathbb{F}_{2}}$, as a direct summand. Note that, thanks to
(3.16), these direct sum decompositions also yield a direct sum decomposition
$N\\!\\!M_{\mathbb{F}_{2}}\simeq M\\!\\!N^{\prime\prime}\oplus U(A)^{\oplus
m}_{\mathbb{F}_{2}}\oplus U(A^{\prime})^{\oplus m^{\prime}}_{\mathbb{F}_{2}}$
in $\mathrm{NRad}(k)_{\mathbb{F}_{2}}$, where $M\\!\\!N^{\prime\prime}$ is a
noncommutative radical motive which does not contains $U(A)_{\mathbb{F}_{2}}$
neither $U(A^{\prime})_{\mathbb{F}_{2}}$ as a direct summand. Consequently,
the above isomorphism (3.14) can be re-written as follows:

(3.17) $M\\!\\!N^{\prime\prime}\oplus U(B)_{\mathbb{F}_{2}}^{\oplus n}\oplus
U(A)^{\oplus(m+1)}_{\mathbb{F}_{2}}\oplus U(A^{\prime})^{\oplus
m^{\prime}}_{\mathbb{F}_{2}}\simeq M\\!\\!N^{\prime\prime}\oplus
U(B^{\prime})_{\mathbb{F}_{2}}^{\oplus n}\oplus U(A)_{\mathbb{F}_{2}}^{\oplus
m}\oplus U(A^{\prime})^{\oplus(m^{\prime}+1)}_{\mathbb{F}_{2}}\,.$

Now, by combining the above computations (3.15)-(3.16) with Lemma 3.8 and with
the fact that $M\\!\\!N^{\prime\prime}$ does not contains
$U(A)_{\mathbb{F}_{2}}$ neither $U(A^{\prime})_{\mathbb{F}_{2}}$ as a direct
summand, we observe that the above isomorphism (3.17) restricts to an
isomorphism $U(A)^{\oplus(m+1)}_{\mathbb{F}_{2}}\simeq U(A)^{\oplus
m}_{\mathbb{F}_{2}}$. By applying the functor
$\mathrm{Hom}_{\mathrm{NRad}(k)_{\mathbb{F}_{2}}}(U(A)_{\mathbb{F}_{2}},-)$ to
this latter isomorphism, we hence obtain an induced isomorphism
$\mathbb{F}_{2}^{\oplus(r+1)}\simeq\mathbb{F}_{2}^{\oplus r}$ of
$\mathbb{F}_{2}$-vector spaces, which is a contradiction. Therefore, we
conclude that $U(A)\simeq U(A^{\prime})$ in $\operatorname{NChow}(k)$.

######  Remark 3.18 (Motivation for noncommutative radical motives).

The category of noncommutative numerical motives has several good properties.
In particular, it is abelian semi-simple. However, given a field extension
$l/k$ of degree $2$ and a central simple $l$-algebra $A$, we observe that
$U(A)_{\mathbb{F}_{2}}$ is isomorphic to the zero object in the category
$\operatorname{NNum}(k)_{\mathbb{F}_{2}}$. Consequently, the proof of Theorem
3.3 fails miserably when the category $\mathrm{NRad}(k)_{\mathbb{F}_{2}}$ is
replaced by the category $\operatorname{NNum}(k)_{\mathbb{F}_{2}}$. This was
our main motivation for the use of the category of noncommutative radical
motives (instead of the category of noncommutative numerical motives), which
“interpolates” between noncommutative Chow motives and noncommutative
numerical motives.

##  4\. Proof of Theorem 1.22

Proof of item (i). Let $X$ and $Y$ be two varieties. As proved in [5, §2 Cor.
3.5.12 b)], if $[X]=[Y]$ in the Grothendieck ring of varieties
$K_{0}\mathrm{Var}(k)$, then $\mathrm{dim}(X)=\mathrm{dim}(Y)$. Consequently,
if $[\mathrm{Iv}(A,\ast)]=[\mathrm{Iv}(A^{\prime},\ast^{\prime})]$, we
conclude that
$\mathrm{dim}(\mathrm{Iv}(A,\ast))=\mathrm{dim}(\mathrm{Iv}(A^{\prime},\ast^{\prime}))$.
Using the fact that $\mathrm{dim}(\mathrm{Iv}(A,\ast))=\mathrm{deg}(A)-2$ and
$\mathrm{dim}(\mathrm{Iv}(A^{\prime},\ast^{\prime}))=\mathrm{deg}(A^{\prime})-2$,
this implies that $\mathrm{deg}(A)=\mathrm{deg}(A^{\prime})$. Proof of item
(ii). Following [32, Thm 2.1 and Rk. 2.3], we have the following computation

$\displaystyle
U(\mathrm{Iv}(A,\ast))\simeq\begin{cases}U(k)^{\oplus(\mathrm{deg}(A)-2)}\oplus
U(C_{0}(A,\ast))&\mathrm{deg}(A)\,\,\mathrm{odd}\\\
U(k)^{\oplus(\mathrm{deg}(A)/2-1)}\oplus
U(A)^{\oplus(\mathrm{deg}(A)/2-1)}\oplus
U(C_{0}(A,\ast))&\mathrm{deg}(A)\,\,\mathrm{even}\end{cases}$

in the category $\operatorname{NChow}(k)$; similarly with $(A,\ast)$ replaced
by $(A^{\prime},\ast^{\prime})$. If
$[\mathrm{Iv}(A,\ast)]=[\mathrm{Iv}(A^{\prime},\ast^{\prime})]$ in
$K_{0}\mathrm{Var}(k)$, then
$\mu_{\mathrm{nc}}(\mathrm{Iv}(A,\ast))=\mu_{\mathrm{nc}}(\mathrm{Iv}(A^{\prime},\ast^{\prime}))$
in $K_{0}(\mathrm{NChow}(k))$, where $\mu_{\mathrm{nc}}$ is the motivic
measure of Proposition 2.5. Thanks to the definition of the Grothendieck ring
$K_{0}(\mathrm{NChow}(k))$, this implies that there exists a noncommutative
Chow motive $N\\!\\!M\in\operatorname{NChow}(k)$ such that

(4.1) $\displaystyle\quad\begin{cases}N\\!\\!M\oplus U(k)^{\oplus d}\oplus
U(C_{0}(A,\ast))\simeq N\\!\\!M\oplus U(k)^{\oplus d}\oplus
U(C_{0}(A^{\prime},\ast^{\prime}))&\mathrm{deg}(A)\,\,\mathrm{odd}\\\
N\\!\\!M\oplus U(k)^{\oplus d}\oplus U(A)^{\oplus d}\oplus
U(C_{0}(A,\ast))\simeq N\\!\\!M\oplus U(k)^{\oplus d}\oplus
U(A^{\prime})^{\oplus d}\oplus
U(C_{0}(A^{\prime},\ast^{\prime}))&\mathrm{deg}(A)\,\,\mathrm{even}\,,\end{cases}$

where $d:=\mathrm{deg}(A)-2$ when $\mathrm{deg}(A)$ is odd and
$d:=\mathrm{deg}(A)/2-1$ when $\mathrm{deg}(A)$ is even. Therefore, the proof
follows now from Lemma 4.2 below.

######  Lemma 4.2.

Let $(A,\ast)$ and $(A^{\prime},\ast^{\prime})$ be two central simple
$k$-algebras with involutions of orthogonal type such that $\mathrm{deg}(A)$
and $\mathrm{deg}(A^{\prime})$ are even. Given any noncommutative Chow motive
$N\\!\\!M\in\operatorname{NChow}(k)$ and integer $n\geq 0$, we have the
following implication:

$\displaystyle N\\!\\!M\oplus U(A)^{\oplus n}\oplus U(C_{0}(A,\ast))\simeq
N\\!\\!M\oplus U(A^{\prime})^{\oplus n}\oplus
U(C_{0}(A^{\prime},\ast^{\prime}))$ $\displaystyle\Rightarrow$
$\displaystyle\big{(}\delta(A,\ast)\in(k^{\times})^{2}\Leftrightarrow\delta(A^{\prime},\ast^{\prime})\in(k^{\times})^{2}\big{)}\,.$

###### Proof.

Note first that we have an induced isomorphism

(4.3) $N\\!\\!M_{\mathbb{Q}}\oplus U(A)^{\oplus n}_{\mathbb{Q}}\oplus
U(C_{0}(A,\ast))_{\mathbb{Q}}\simeq N\\!\\!M_{\mathbb{Q}}\oplus
U(A^{\prime})^{\oplus n}_{\mathbb{Q}}\oplus
U(C_{0}(A^{\prime},\ast^{\prime}))_{\mathbb{Q}}\,\,\mathrm{in}\,\,\operatorname{NNum}(k)_{\mathbb{Q}}\,.$

Moreover, as proved in [34, Thm. 2.1], we have isomorphisms
$U(A)_{\mathbb{Q}}\simeq U(k)_{\mathbb{Q}}$ and
$U(A^{\prime})_{\mathbb{Q}}\simeq U(k)_{\mathbb{Q}}$ in
$\operatorname{NChow}(k)_{\mathbb{Q}}$. Consequently, since the category
$\operatorname{NNum}(k)_{\mathbb{Q}}$ is abelian semi-simple (consult [31,
Thm. 4.27]), we conclude from (4.3) that $U(C_{0}(A,\ast))_{\mathbb{Q}}\simeq
U(C_{0}(A^{\prime},\ast^{\prime}))_{\mathbb{Q}}$ in
$\operatorname{NNum}(k)_{\mathbb{Q}}$. Let us now prove the implication
$\delta(A,\ast)\in(k^{\times})^{2}\Rightarrow\delta(A^{\prime},\ast^{\prime})\in(k^{\times})^{2}$;
we will assume that $\delta(A,\ast)\in(k^{\times})^{2}$ and, by absurd, that
$\delta(A^{\prime},\ast^{\prime})\not\in(k^{\times})^{2}$. These assumptions
imply that $C_{0}(A,\ast)\simeq C^{+}_{0}(A,\ast)\times C^{-}_{0}(A,\ast)$ is
a product of two central simple $k$-algebras and that
$C_{0}(A^{\prime},\ast^{\prime})$ is a central simple algebra over its center
$l^{\prime}:=k(\sqrt{\delta(A^{\prime},\ast^{\prime})})$. Hence, we obtain an
isomorphism $U(C_{0}(A,\ast))\simeq U(C^{+}_{0}(A,\ast))\oplus
U(C^{-}_{0}(A,\ast))$ in the category $\operatorname{NChow}(k)$. As proved in
[34, Thm. 2.1], we have moreover isomorphisms
$U(C^{+}_{0}(A,\ast))_{\mathbb{Q}}\simeq U(k)_{\mathbb{Q}}$,
$U(C^{-}_{0}(A,\ast))_{\mathbb{Q}}\simeq U(k)_{\mathbb{Q}}$ and
$U(C_{0}(A^{\prime},\ast^{\prime}))_{\mathbb{Q}}\simeq
U(l^{\prime})_{\mathbb{Q}}$ in the category
$\operatorname{NChow}(k)_{\mathbb{Q}}$. Therefore, we obtain from the above
considerations an isomorphism $U(k)_{\mathbb{Q}}\oplus U(k)_{\mathbb{Q}}\simeq
U(l^{\prime})_{\mathbb{Q}}$ in $\operatorname{NNum}(k)_{\mathbb{Q}}$ and,
consequently, an induced isomorphism of $\mathbb{Q}$-vector spaces:

(4.4)
$\mathrm{Hom}_{\operatorname{NNum}(k)_{\mathbb{Q}}}(U(k)_{\mathbb{Q}},U(k)_{\mathbb{Q}}\oplus
U(k)_{\mathbb{Q}})\simeq\mathrm{Hom}_{\operatorname{NNum}(k)_{\mathbb{Q}}}(U(k)_{\mathbb{Q}},U(l^{\prime})_{\mathbb{Q}})\,.$

On the one hand, the left-hand side of (4.4) is isomorphic to
$\mathbb{Q}\oplus\mathbb{Q}$. On the other hand, since the field extension
$l^{\prime}/k$ is of degree $2$, the following composition map

$\mathrm{Hom}_{\operatorname{NChow}(k)_{\mathbb{Q}}}(U(k)_{\mathbb{Q}},U(l^{\prime})_{\mathbb{Q}})\times\mathrm{Hom}_{\operatorname{NChow}(k)_{\mathbb{Q}}}(U(l^{\prime})_{\mathbb{Q}},U(k)_{\mathbb{Q}})\longrightarrow\mathrm{Hom}_{\operatorname{NChow}(k)_{\mathbb{Q}}}(U(k)_{\mathbb{Q}},U(k)_{\mathbb{Q}})$

corresponds to the bilinear pairing
$\mathbb{Q}\times\mathbb{Q}\to\mathbb{Q},(\alpha,\beta)\mapsto 2\alpha\beta$.
By definition of the category $\operatorname{NNum}(k)_{\mathbb{Q}}$, this
implies that the right-hand side of (4.4) is isomorphic to $\mathbb{Q}$, which
is a contradiction! The proof of the converse implication is similar; simply
interchange $(A,\ast)$ and $(A^{\prime},\ast^{\prime})$. ∎

Proof of item (iii). We start with the following cancellation result:

######  Proposition 4.5.

Let $(A,\ast)$ and $(A^{\prime},\ast^{\prime})$ be two central simple
$k$-algebras with involutions of orthogonal type such that $\mathrm{deg}(A)$
and $\mathrm{deg}(A^{\prime})$ are even. Given any noncommutative Chow motive
$N\\!\\!M\in\operatorname{NChow}(k)$ and integer $n\geq 0$, we have the
following implication:

$\displaystyle N\\!\\!M\oplus U(A)^{\oplus n}\oplus U(C_{0}(A,\ast))\simeq
N\\!\\!M\oplus U(A^{\prime})^{\oplus n}\oplus
U(C_{0}(A^{\prime},\ast^{\prime}))$ $\displaystyle\Rightarrow$ $\displaystyle
U(C_{0}(A,\ast))\simeq U(C_{0}(A^{\prime},\ast^{\prime}))\,.$

In the particular case where $n=1$, $\mathrm{deg}(A)\equiv 0$ (mod $4$),
$\mathrm{deg}(A^{\prime})\equiv 0$ (mod $4$), and
$\delta(A,\ast)\in(k^{\times})^{2}$ (or, equivalently,
$\delta(A^{\prime},\ast^{\prime})\in(k^{\times})^{2}$), we assume moreover
that $A$ and $A^{\prime}$ are split.

###### Proof.

We consider first the case where $\delta(A,\ast)\in(k^{\times})^{2}$. Thanks
to Lemma 4.2, we also have
$\delta(A^{\prime},\ast^{\prime})\in(k^{\times})^{2}$. This implies that
$C_{0}(A,\ast)\simeq C^{+}_{0}(A,\ast)\times C^{-}_{0}(A,\ast)$ and
$C_{0}(A^{\prime},\ast^{\prime})\simeq
C^{+}_{0}(A^{\prime},\ast^{\prime})\times C^{-}_{0}(A^{\prime},\ast^{\prime})$
are products of central simple $k$-algebras. Hence, we obtain induced
isomorphisms

(4.6) $\displaystyle U(C_{0}(A,\ast))\simeq U(C_{0}^{+}(A,\ast))\oplus
U(C_{0}^{-}(A,\ast))$ $\displaystyle U(C_{0}(A^{\prime},\ast^{\prime}))\simeq
U(C_{0}^{+}(A^{\prime},\ast^{\prime}))\oplus
U(C_{0}^{-}(A^{\prime},\ast^{\prime}))\,,$

which lead to the following isomorphism of noncommutative Chow motives:

$N\\!\\!M\oplus U(A)^{\oplus n}\oplus U(C^{+}_{0}(A,\ast))\oplus
U(C^{-}_{0}(A,\ast))\simeq N\\!\\!M\oplus U(A^{\prime})^{\oplus n}\oplus
U(C^{+}_{0}(A^{\prime},\ast^{\prime}))\oplus
U(C^{-}_{0}(A^{\prime},\ast^{\prime}))\,.$

Note that thanks to Proposition 3.1, the latter isomorphism further restricts
to an isomorphism

(4.7) $U(A)^{\oplus n}\oplus U(C^{+}_{0}(A,\ast))\oplus
U(C^{-}_{0}(A,\ast))\simeq U(A^{\prime})^{\oplus n}\oplus
U(C^{+}_{0}(A^{\prime},\ast^{\prime}))\oplus
U(C^{-}_{0}(A^{\prime},\ast^{\prime}))\,\,\mathrm{in}\,\,\operatorname{NChow}(k)\,.$

When $\mathrm{deg}(A)\equiv 2$ (mod $4$), we have the following relations in
the Brauer group:

(4.8) $\displaystyle 2[C^{+}_{0}(A,\ast)]=[A]$ $\displaystyle
3[C^{+}_{0}(A,\ast)]=[C^{-}_{0}(A,\ast)]$ $\displaystyle
4[C^{+}_{0}(A,\ast)]=[k]\,.$

In the same vein, when $\mathrm{deg}(A)\equiv 0$ (mod $4$), we have the
following relations in the Brauer group:

(4.9) $\displaystyle 2[C^{+}_{0}(A,\ast)]=[k]$ $\displaystyle
2[C^{+}_{0}(A,\ast)]=[k]$
$\displaystyle[C^{+}_{0}(A,\ast)]+[C^{-}_{0}(A,\ast)]=[A]\,.$

This implies, in particular, that the Brauer classes $[C^{+}_{0}(A,\ast)]$,
$[C^{-}_{0}(A,\ast)]$, and $[A]$, belong to $\mathrm{Br}(k)\\{2\\}$; similarly
with $A$ and $\ast$ replaced by $A^{\prime}$ and $\ast^{\prime}$.
Consequently, by applying Theorem 2.2 to the above isomorphism (4.7), we
conclude that the following two sets of Brauer classes are the same up to
permutation:

$\displaystyle\\{\underbrace{[A],\ldots,[A]}_{n\text{-}\mathrm{copies}},[C_{0}^{+}(A,\ast)],[C_{0}^{-}(A,\ast)]\\}$
$\displaystyle\\{\underbrace{[A^{\prime}],\ldots,[A^{\prime}]}_{n\text{-}\mathrm{copies}},[C_{0}^{+}(A^{\prime},\ast^{\prime})],[C_{0}^{-}(A^{\prime},\ast^{\prime})]\\}\,.$

When $n\neq 1$, the above relations (4.8)-(4.9) imply that

(4.10)
$\displaystyle\begin{cases}[C_{0}^{+}(A,\ast)]=[C_{0}^{+}(A^{\prime},\ast^{\prime})]\\\
[C_{0}^{-}(A,\ast)]=[C_{0}^{-}(A^{\prime},\ast^{\prime})]\end{cases}$
$\displaystyle\mathrm{or}$
$\displaystyle\begin{cases}[C_{0}^{+}(A,\ast)]=[C_{0}^{-}(A^{\prime},\ast^{\prime})]\\\
[C_{0}^{-}(A,\ast)]=[C_{0}^{+}(A^{\prime},\ast^{\prime})]\end{cases}\,.$

Similarly, when $n=1$ and $\mathrm{deg}(A)\equiv 2$ (mod $4$) or
$\mathrm{deg}(A^{\prime})\equiv 2$ (mod $4$), the above relations (4.8)-(4.9)
yield the equalities (4.10). The same happens in the particular case where
$n=1$, $\mathrm{deg}(A)\equiv 0$ (mod $4$), $\mathrm{deg}(A^{\prime})\equiv 0$
(mod $4$), and $A$ and $A^{\prime}$ are split. Making use of Theorem 2.1, we
hence conclude from the above isomorphisms (4.6) that in all these different
cases we have $U(C_{0}(A,\ast))\simeq U(C_{0}(A^{\prime},\ast^{\prime}))$ in
$\operatorname{NChow}(k)$. We now consider the case where
$\delta(A,\ast)\notin(k^{\times})^{2}$. Thanks to Lemma 4.2, we also have
$\delta(A^{\prime},\ast^{\prime})\notin(k^{\times})^{2}$. This implies that
$C_{0}(A,\ast)$ and $C_{0}(A^{\prime},\ast^{\prime})$ are central simple
algebras over their centers $l:=k(\sqrt{\delta(A,\ast)})$ and
$l^{\prime}:=k(\sqrt{\delta(A^{\prime},\ast^{\prime})})$, respectively. Note
that since the field extension $l/k$ is of degree $2$ and
$\mathrm{dim}_{k}(C_{0}(A,\ast))=2^{\mathrm{deg}(A)-1}$, the index of the
central simple $l$-algebra $C_{0}(A,\ast)$ is a power of $2$; similarly for
the central simple $l^{\prime}$-algebra $C_{0}(A^{\prime},\ast^{\prime})$.
Consequently, the proof follows now from Theorem 3.3. ∎

######  Corollary 4.11.

Under the assumptions of Proposition 4.5, we have the following implication:

(4.12) $\displaystyle N\\!\\!M\oplus U(A)^{\oplus n}\oplus
U(C_{0}(A,\ast))\simeq N\\!\\!M\oplus U(A^{\prime})^{\oplus n}\oplus
U(C_{0}(A^{\prime},\ast^{\prime}))$ $\displaystyle\Rightarrow$ $\displaystyle
U(A)\simeq U(A^{\prime})\,.$

###### Proof.

Proposition 4.5 yields an isomorphism $U(C_{0}(A,\ast))\simeq
U(C_{0}(A^{\prime},\ast^{\prime}))$. Therefore, by applying Proposition 3.1 to
the left-hand side of (4.12), we conclude that $U(A)^{\oplus n}\simeq
U(A^{\prime})^{\oplus n}$. Thanks to Theorems 2.1 and 2.2, this hence implies
that $U(A)\simeq U(A^{\prime})$ in $\operatorname{NChow}(k)$. ∎

Recall from the proof of item (ii) of Theorem 1.22 that if
$[\mathrm{Iv}(A,\ast)]=[\mathrm{Iv}(A^{\prime},\ast^{\prime})]$, then we
obtain the above computation (4.1) in the category of noncommutative Chow
motives. Consequently, by applying Proposition 3.1 to the above isomorphism
(4.1) when $\mathrm{deg}(A)$ is odd, we conclude that $U(C_{0}(A,\ast))\simeq
U(C_{0}(A^{\prime},\ast^{\prime}))$ in $\operatorname{NChow}(k)$. In the same
vein, by applying Proposition 4.5 to the above isomorphism (4.1) when
$\mathrm{deg}(A)$ is even, we conclude that $U(C_{0}(A,\ast))\simeq
U(C_{0}(A^{\prime},\ast^{\prime}))$ in $\operatorname{NChow}(k)$; note that
Proposition 4.5 can be applied because in the particular case where
$\mathrm{deg}(A)=4$ and $\delta(A,\ast)\in(k^{\times})^{2}$ (or, equivalently,
$\mathrm{deg}(A^{\prime})=4$ and
$\delta(A^{\prime},\ast^{\prime})\in(k^{\times})^{2}$), we assume moreover
that $(A,\ast)$ and $(A^{\prime},\ast^{\prime})$ satisfy condition $(\star)$.
Therefore, since $\mathrm{deg}(A)=\mathrm{deg}(A^{\prime})$, the proof follows
now from Proposition 4.13 below.

######  Proposition 4.13.

Let $(A,\ast)$ and $(A^{\prime},\ast^{\prime})$ be two central simple
$k$-algebras with involutions of orthogonal type such that
$\mathrm{deg}(A)=\mathrm{deg}(A^{\prime})$. Under these assumptions, we have
the following implication:

$\displaystyle U(C_{0}(A,\ast))\simeq
U(C_{0}(A^{\prime},\ast^{\prime}))\,\,\mathrm{in}\,\,\operatorname{NChow}(k)$
$\displaystyle\Rightarrow$ $\displaystyle C_{0}(A,\ast)\simeq
C_{0}(A^{\prime},\ast^{\prime})\,.$

###### Proof.

Recall that $\mathrm{dim}_{k}(C_{0}(A,\ast))=2^{\mathrm{deg}(A)-1}$ and
$\mathrm{dim}_{k}(C_{0}(A^{\prime},\ast^{\prime}))=2^{\mathrm{deg}(A^{\prime})-1}$.
Therefore, since $\mathrm{deg}(A)=\mathrm{deg}(A^{\prime})$, we have
$\mathrm{dim}_{k}(C_{0}(A,\ast))=\mathrm{dim}_{k}(C_{0}(A^{\prime},\ast^{\prime}))$.
We consider first the case where $\mathrm{deg}(A)$ is odd. In this case,
$C_{0}(A,\ast)$ and $C_{0}(A^{\prime},\ast^{\prime})$ are central simple
$k$-algebras. Hence, Theorem 2.1 implies that
$[C_{0}(A,\ast)]=[C_{0}(A^{\prime},\ast^{\prime})]$. Consequently, by
combining the equality
$\mathrm{dim}_{k}(C_{0}(A,\ast))=\mathrm{dim}_{k}(C_{0}(A^{\prime},\ast^{\prime}))$
with the Wedderburn theorem, we conclude that $C_{0}(A,\ast)\simeq
C_{0}(A^{\prime},\ast^{\prime})$. We now consider the case where
$\mathrm{deg}(A)$ is even and $\delta(A,\ast)\in(k^{\times})^{2}$. Thanks to
Lemma 4.2, we also have $\delta(A^{\prime},\ast^{\prime})\in(k^{\times})^{2}$.
This implies that $C_{0}(A,\ast)\simeq C^{+}_{0}(A,\ast)\times
C^{-}_{0}(A,\ast)$ and $C_{0}(A^{\prime},\ast^{\prime})\simeq
C^{+}_{0}(A^{\prime},\ast^{\prime})\times C^{-}_{0}(A^{\prime},\ast^{\prime})$
are products of central simple $k$-algebras. Hence, we obtain induced
isomorphisms

$\displaystyle U(C_{0}(A,\ast))\simeq U(C_{0}^{+}(A,\ast))\oplus
U(C_{0}^{-}(A,\ast))$ $\displaystyle U(C_{0}(A^{\prime},\ast^{\prime}))\simeq
U(C_{0}^{+}(A^{\prime},\ast^{\prime}))\oplus
U(C_{0}^{-}(A^{\prime},\ast^{\prime}))\,,$

which lead to the following isomorphism

(4.14) $U(C^{+}_{0}(A,\ast))\oplus U(C^{-}_{0}(A,\ast))\simeq
U(C^{+}_{0}(A^{\prime},\ast^{\prime}))\oplus
U(C^{-}_{0}(A^{\prime},\ast^{\prime}))\,\,\mathrm{in}\,\,\operatorname{NChow}(k)\,.$

As explained in the proof of Proposition 4.5, the Brauer classes
$[C_{0}^{+}(A,\ast)]$ and $[C_{0}^{-}(A,\ast)]$ belong to
$\mathrm{Br}(k)\\{2\\}$; similarly with $(A,\ast)$ replaced by
$(A^{\prime},\ast^{\prime})$. Therefore, by applying Theorem 2.2 to the above
isomorphism (4.14), we conclude that the sets of Brauer classes
$\\{[C_{0}^{+}(A,\ast)],[C_{0}^{-}(A,\ast)]\\}$ and
$\\{[C_{0}^{+}(A^{\prime},\ast^{\prime})],[C_{0}^{-}(A^{\prime},\ast^{\prime})]\\}$
are the same up to permutation. Thanks to the assumption
$\mathrm{deg}(A)=\mathrm{deg}(A^{\prime})$, to the following equalities

$\displaystyle\mathrm{dim}_{k}(C^{+}_{0}(A,\ast))=\mathrm{dim}_{k}(C^{-}_{0}(A,\ast))=2^{\mathrm{deg}(A)-2}$
$\displaystyle\mathrm{dim}_{k}(C^{+}_{0}(A^{\prime},\ast^{\prime}))=\mathrm{dim}_{k}(C^{-}_{0}(A^{\prime},\ast^{\prime}))=2^{\mathrm{deg}(A^{\prime})-2}\,,$

and to the Wedderburn theorem, this implies that

$\displaystyle\begin{cases}C_{0}^{+}(A,\ast)\simeq
C_{0}^{+}(A^{\prime},\ast^{\prime})\\\ C_{0}^{-}(A,\ast)\simeq
C_{0}^{-}(A^{\prime},\ast^{\prime})\end{cases}$ $\displaystyle\mathrm{or}$
$\displaystyle\begin{cases}C_{0}^{+}(A,\ast)\simeq
C_{0}^{-}(A^{\prime},\ast^{\prime})\\\ C_{0}^{-}(A,\ast)\simeq
C_{0}^{+}(A^{\prime},\ast^{\prime})\end{cases}\,.$

In both cases, we have an isomorphism $C_{0}(A,\ast)\simeq
C_{0}(A^{\prime},\ast^{\prime})$. Finally, we consider the case where
$\mathrm{deg}(A)$ is even and $\delta(A,\ast)\notin(k^{\times})^{2}$. Thanks
to Lemma 4.2, we also have
$\delta(A^{\prime},\ast^{\prime})\notin(k^{\times})^{2}$. This implies that
$C_{0}(A,\ast)$ and $C_{0}(A^{\prime},\ast)$ are central simple algebras over
their centers $l:=k(\sqrt{\delta(A,\ast)})$ and
$l^{\prime}:=k(\sqrt{\delta(A^{\prime},\ast^{\prime})})$, respectively. Hence,
we obtain an induced isomorphism $U(C_{0}(A,\ast))_{\mathbb{Q}}\simeq
U(C_{0}(A^{\prime},\ast^{\prime}))_{\mathbb{Q}}$ in
$\operatorname{NChow}(k)_{\mathbb{Q}}$. As proved in [34, Thm. 2.1], we have
isomorphisms $U(C_{0}(A,\ast))_{\mathbb{Q}}\simeq U(l)_{\mathbb{Q}}$ and
$U(C_{0}(A^{\prime},\ast^{\prime}))_{\mathbb{Q}}\simeq
U(l^{\prime})_{\mathbb{Q}}$ in $\operatorname{NChow}(k)_{\mathbb{Q}}$. Thanks
to Lemma 3.4, this implies that $l\simeq l^{\prime}$. Therefore, we can
consider the central simple $l$-algebra
$\mathbf{C}_{0}:=C_{0}(A^{\prime},\ast^{\prime})\otimes_{l^{\prime}}l$. Note
that since the $k$-algebras $C_{0}(A^{\prime},\ast^{\prime})$ and
$\mathbf{C}_{0}$ are isomorphic, we have an isomorphism
$U(C_{0}(A^{\prime},\ast^{\prime}))\simeq U(\mathbf{C}_{0})$ in
$\operatorname{NChow}(k)$. Thanks to Lemma 3.6, this latter isomorphism,
combined with the hypothesis that $U(C_{0}(A,\ast))\simeq
U(C_{0}(A^{\prime},\ast^{\prime}))$ in $\operatorname{NChow}(k)$, implies that
$[C_{0}(A,\ast)]=[\mathbf{C}_{0}]$ in $\mathrm{Br}(l)$. Consequently, making
use of the following equalities

$\mathrm{dim}_{l}(C_{0}(A,\ast))=\mathrm{dim}_{k}(C_{0}(A,\ast))/2=\mathrm{dim}_{k}(C_{0}(A^{\prime},\ast^{\prime}))/2=\mathrm{dim}_{k}(\mathbf{C}_{0})/2=\mathrm{dim}_{l}(\mathbf{C}_{0})$

and of the Wedderburn theorem, we conclude that the $l$-algebras
$C_{0}(A,\ast)$ and $\mathbf{C}_{0}$ are isomorphic. This implies that
$C_{0}(A,\ast)\simeq C_{0}(A^{\prime},\ast^{\prime})$. ∎

Proof of item (iv). Thanks to item (i) of Theorem 1.22, we have
$\mathrm{deg}(A)=\mathrm{deg}(A^{\prime})$. Moreover, recall from Example 1.18
that when $\mathrm{deg}(A)=\mathrm{deg}(A^{\prime})$ is odd, both $A$ and
$A^{\prime}$ are isomorphic to
$\mathrm{M}_{\mathrm{deg}(A)\times\mathrm{deg}(A)}(k)$. Hence, it suffices to
consider the case where $\mathrm{deg}(A)=\mathrm{deg}(A^{\prime})$ is even.
Recall from item (ii) of Theorem 1.22 that if
$[\mathrm{Iv}(A,\ast)]=[\mathrm{Iv}(A^{\prime},\ast^{\prime})]$, then we
obtain the above computation (4.1) in the category of noncommutative Chow
motives. By applying Corollary 4.11 to the above isomorphism (4.1), we hence
conclude that $U(A)\simeq U(A^{\prime})$ in $\operatorname{NChow}(k)$; note
that Corollary 4.11 can be applied because in the particular case where
$\mathrm{deg}(A)=4$ and $\delta(A,\ast)\in(k^{\times})^{2}$ (or, equivalently,
$\mathrm{deg}(A^{\prime})=4$ and
$\delta(A^{\prime},\ast^{\prime})\in(k^{\times})^{2}$), we assume moreover
that $(A,\ast)$ and $(A^{\prime},\ast^{\prime})$ satisfy condition $(\star)$.
Therefore, since $\mathrm{deg}(A)=\mathrm{deg}(A^{\prime})$, it follows now
from Theorem 2.1 and from the Wedderburn theorem that $A\simeq A^{\prime}$.
Proof of item (v). We start with a general result of independent interest:

######  Proposition 4.15.

Let $q$ and $q^{\prime}$ be two non-degenerate quadratic forms such that
$\mathrm{dim}(q)=\mathrm{dim}(q^{\prime})$. If
$\mu_{\mathrm{c}}([Q_{q}])=\mu_{\mathrm{c}}([Q_{q^{\prime}}])$ in
$K_{0}(\operatorname{Num}(k))$, then $q$ and $q^{\prime}$ are both isotropic
or both anisotropic.

###### Proof.

Let $l/k$ be a field extension making the quadratic form $q\otimes_{k}l$
isotropic. As explained in [1, §4.2.3], extension of scalars along the
inclusion $k\subset l$ gives rise to the commutative diagram:

$\textstyle{\mathrm{Chow}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\times_{k}l}$$\textstyle{\mathrm{Chow}(l)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Num}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\times_{k}l}$$\textstyle{\operatorname{Num}(l)\,.}$

This leads to the following induced commutative diagram of abelian groups

(4.16)
$\textstyle{\mathrm{Hom}_{\mathrm{Chow}(k)}(\mathfrak{h}(\mathrm{pt})(-d),\mathfrak{h}(Q_{q}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{q}}$$\textstyle{\mathrm{Hom}_{\mathrm{Chow}(l)}(\mathfrak{h}(\mathrm{pt})(-d),\mathfrak{h}(Q_{q}\times_{k}l))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Hom}_{\operatorname{Num}(k)}(\mathfrak{h}(\mathrm{pt})(-d),\mathfrak{h}(Q_{q}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{q}}$$\textstyle{\mathrm{Hom}_{\operatorname{Num}(l)}(\mathfrak{h}(\mathrm{pt})(-d),\mathfrak{h}(Q_{q}\times_{k}l))\,,}$

where $d:=\mathrm{dim}(Q_{q})=\mathrm{dim}(q)-2$. Since the quadratic form
$q\otimes_{k}l$ is isotropic, we have an orthogonal sum decomposition
$q\otimes_{k}l=\underline{q}\perp\mathbb{H}$. Consequently, as proved by Rost
in [26, Prop. 2], the Chow motive
$\mathfrak{h}(Q_{q}\times_{k}l)=\mathfrak{h}(Q_{q\otimes_{k}l})$ admits the
direct sum decomposition
$\mathfrak{h}(\mathrm{pt})\oplus\mathfrak{h}(Q_{\underline{q}})(-1)\oplus\mathbb{Z}(-d)$
in $\mathrm{Chow}(k)$. Since $\mathrm{dim}(Q_{\underline{q}})=d-2$, this
decomposition implies that the upper and lower right corners of the
commutative diagram (4.16) are both isomorphic to $\mathbb{Z}$ and that the
(vertical) homomorphism between them is the identity. Note that by definition
(consult §2), the upper and lower left corners of the commutative diagram
(4.16) are isomorphic to ${\mathcal{Z}}^{d}_{\sim\mathrm{rat}}(Q_{q})$ and
${\mathcal{Z}}^{d}_{\sim\mathrm{num}}(Q_{q})$, respectively. Note also that
these are the abelian groups of $0$-cycles on $Q_{q}$ up to rational
equivalence and numerical equivalence, respectively. Following Karpenko [9,
§2], under the above identifications, the homomorphism $\phi_{q}$ corresponds
to the degree map
$\mathrm{deg}\colon{\mathcal{Z}}^{d}_{\sim\mathrm{rat}}(Q_{q})\to\mathbb{Z}$.
Consequently, since two $0$-cycles in $Q_{q}$ are numerically trivial if and
only if they have the same degree, we conclude that
$\mathrm{cok}(\varphi_{q})\simeq\mathrm{cok}(\phi_{q})$. All the above holds
mutatis mutandis with $q$ replaced by $q^{\prime}$. Now, let $l/k$ be a field
extension making both quadratic forms $q\otimes_{k}l$ and
$q^{\prime}\otimes_{k}l$ isotropic. By definition of the Grothendieck ring
$K_{0}(\operatorname{Num}(k))$, if
$\mu_{\mathrm{c}}([Q_{q}])=\mu_{\mathrm{c}}([Q_{q^{\prime}}])$, then there
exists a numerical motive $M\in\operatorname{Num}(k)$ such that
$M\oplus\mathfrak{h}(Q_{q})\simeq M\oplus\mathfrak{h}(Q_{q^{\prime}})$. Let us
choose an isomorphism $\theta\colon M\oplus\mathfrak{h}(Q_{q})\to
M\oplus\mathfrak{h}(Q_{q^{\prime}})$. Note that since the extension of scalars
functor $-\times_{k}l\colon\operatorname{Num}(k)\to\operatorname{Num}(l)$ is
additive, it leads to the following commutative diagram of abelian groups:

$\textstyle{\mathrm{Hom}_{\operatorname{Num}(k)}(\mathfrak{h}(\mathrm{pt})(-d),M\oplus\mathfrak{h}(Q_{q}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{\ast}}$$\scriptstyle{\simeq}$$\scriptstyle{\varphi\oplus\varphi_{q}}$$\textstyle{\mathrm{Hom}_{\operatorname{Num}(l)}(\mathfrak{h}(\mathrm{pt})(-d),(M\times_{k}l)\oplus\mathfrak{h}(Q_{q}\times_{k}l))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\theta\times_{k}l)_{\ast}}$$\scriptstyle{\simeq}$$\textstyle{\mathrm{Hom}_{\operatorname{Num}(k)}(\mathfrak{h}(\mathrm{pt})(-d),M\oplus\mathfrak{h}(Q_{q^{\prime}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi\oplus\varphi_{q^{\prime}}}$$\textstyle{\mathrm{Hom}_{\operatorname{Num}(l)}(\mathfrak{h}(\mathrm{pt})(-d),(M\times_{k}l)\oplus\mathfrak{h}(Q_{q^{\prime}}\times_{k}l))\,.}$

This implies that
$\mathrm{cok}(\varphi\oplus\varphi_{q})\simeq\mathrm{cok}(\varphi\oplus\varphi_{q^{\prime}})$
or, equivalently, that
$\mathrm{cok}(\varphi)\oplus\mathrm{cok}(\varphi_{q})\simeq\mathrm{cok}(\varphi)\oplus\mathrm{cok}(\varphi_{q^{\prime}})$.
Using the fact that these abelian groups are finitely generated (consult [1,
Prop. 3.2.7.1]), we hence conclude that
$\mathrm{cok}(\varphi_{q})\simeq\mathrm{cok}(\varphi_{q^{\prime}})$. The proof
follows now from a celebrated result of Springer (consult [7, Cor. 71.3][29]),
which asserts that $\mathrm{cok}(\varphi_{q})\simeq 0$ if $q$ is isotropic and
$\mathrm{cok}(\varphi_{q})\simeq\mathbb{Z}/2$ if $q$ is anisotropic. ∎

Let $P$ be a (fixed) ordering of $k$ and $k_{P}$ the associated real-closure
of $k$. Note first that extension of scalars along the inclusion $k\subset
k_{P}$ gives rise to a ring homomorphism $-\times_{k}k_{P}\colon
K_{0}\mathrm{Var}(k)\to K_{0}\mathrm{Var}(k_{P})$. Consequently, since by
hypothesis $[\mathrm{Iv}(A,\ast)]=[\mathrm{Iv}(A^{\prime},\ast^{\prime})]$, we
obtain the following equality in $K_{0}\mathrm{Var}(k_{P})$:

(4.17)
$[\mathrm{Iv}(A,\ast)\times_{k}k_{P}]=[\mathrm{Iv}(A^{\prime},\ast^{\prime})\times_{k}k_{P}]=[\mathrm{Iv}(A\otimes_{k}k_{P},\ast\otimes_{k}k_{P})]=[\mathrm{Iv}(A^{\prime}\otimes_{k}k_{P},\ast^{\prime}\otimes_{k}k_{P})]\,.$

Now, recall from item (iv) that if
$[\mathrm{Iv}(A,\ast)]=[\mathrm{Iv}(A^{\prime},\ast^{\prime})]$, then $A\simeq
A^{\prime}$. This yields an induced isomorphism $A\otimes_{k}k_{P}\simeq
A^{\prime}\otimes_{k}k_{P}$. On one hand, if the central simple
$k_{P}$-algebra $A\otimes_{k}k_{P}$ is not split, we have
$\mathrm{sgn}_{P}(A,\ast)=0$; consult [20, Thm. 1]. If this is the case, then
$A^{\prime}\otimes_{k}k_{P}$ is also not split and
$\mathrm{sgn}_{P}(A^{\prime},\ast^{\prime})=0$. On the other hand, if
$A\otimes_{k}k_{P}$ is split, we have an isomorphism of central simple
$k_{P}$-algebras with involutions of orthogonal type
$(A\otimes_{k}k_{P},\ast\otimes_{k}k_{P})\simeq(\mathrm{M}_{\mathrm{deg}(A)\times\mathrm{deg}(A)}(k_{P}),\ast_{q})$
(consult Example 1.17) and $\mathrm{sgn}_{P}(A,\ast)=|\mathrm{sgn}(q)|$. If
this is the case, then $A^{\prime}\otimes_{k}k_{P}$ is also split, we also
have an isomorphism of central simple $k_{P}$-algebras with involution of
orthogonal type
$(A^{\prime}\otimes_{k}k_{P},\ast^{\prime}\otimes_{k}k_{P})\simeq(\mathrm{M}_{\mathrm{deg}(A)\times\mathrm{deg}(A)}(k_{P}),\ast_{q^{\prime}})$,
and $\mathrm{sgn}_{P}(A^{\prime},\ast^{\prime})=|\mathrm{sgn}(q^{\prime})|$.
Consequently, thanks to the above equality (4.17) and to the fact that
$\mathrm{Iv}(A\otimes_{k}k_{P},\ast\otimes_{k}k_{P})\simeq Q_{q}$ and
$\mathrm{Iv}(A^{\prime}\otimes_{k}k_{P},\ast^{\prime}\otimes_{k}k_{P})\simeq
Q_{q^{\prime}}$, the proof of item (v) of Theorem 1.22 follows now from
Proposition 4.18 below.

######  Proposition 4.18.

Given two non-degenerate quadratic forms $q$ and $q^{\prime}$ over a real-
closed field $k$, we have the following implication:

$\displaystyle[Q_{q}]=[Q_{q^{\prime}}]\,\,\mathrm{in}\,\,K_{0}\mathrm{Var}(k)$
$\displaystyle\Rightarrow$
$\displaystyle|\mathrm{sgn}(q)|=|\mathrm{sgn}(q^{\prime})|\,.$

###### Proof.

Recall first from item (i) of Theorem 1.1 that if by hypothesis
$[Q_{q}]=[Q_{q^{\prime}}]$, then $\mathrm{dim}(q)=\mathrm{dim}(q^{\prime})$.
It is well-known that every real-closed field is, in particular, euclidean.
Consequently, thanks to Sylvester’s law of inertia (consult [7, Prop. 31.5]),
the quadratic forms $q$ and $q^{\prime}$ are similar to the following ones

(4.19) $\displaystyle
q=\langle\underbrace{1,\ldots,1}_{m\text{-}\text{copies}},\underbrace{-1,\ldots,-1}_{n\text{-}\text{copies}}\rangle$
$\displaystyle
q^{\prime}=\langle\underbrace{1,\ldots,1}_{m^{\prime}\text{-}\text{copies}},\underbrace{-1,\ldots,-1}_{n^{\prime}\text{-}\text{copies}}\rangle$

for uniquely determined integers $m\geq n\geq 0$ and $m^{\prime}\geq
n^{\prime}\geq 0$, respectively. Note that since
$\mathrm{dim}(q)=\mathrm{dim}(q^{\prime})$, we have
$m+n=m^{\prime}+n^{\prime}$. In what follows we will assume that
$|\mathrm{sgn}(q)|\neq|\mathrm{sgn}(q^{\prime})|$ and, by absurd, that
$[Q_{q}]=[Q_{q^{\prime}}]$. If $|\mathrm{sgn}(q)|=m-n\neq
m^{\prime}-n^{\prime}=|\mathrm{sgn}(q^{\prime})|$, then $m\neq m^{\prime}$.
Hence, we can (and will) assume without loss of generality that
$m>m^{\prime}$. Let us consider first the particular case where $n=0$. In this
case, the quadratic form $q$ is anisotropic while the quadratic form
$q^{\prime}$ is isotropic (because $n^{\prime}>0$). However, if
$[Q_{q}]=[Q_{q^{\prime}}]$, then it follows from Proposition 4.18 that $q$ and
$q^{\prime}$ are both isotropic or both anisotropic. Hence, we arrive to a
contradiction. Let us consider now the case where $n>0$. In this case, we have
orthogonal sum decompositions

$\displaystyle q=\underline{q}\perp
n\mathbb{H}\,\,\,\,\text{with}\,\,\,\,\underline{q}=\langle\underbrace{1,\ldots\ldots,1}_{(m-n)\text{-}\text{copies}}\rangle$
$\displaystyle q^{\prime}=\underline{q^{\prime}}\perp
n\mathbb{H}\,\,\,\,\text{with}\,\,\,\,\underline{q^{\prime}}=\langle\underbrace{1,\ldots\ldots,1}_{(m^{\prime}-n)\text{-}\text{copies}},\underbrace{-1,\ldots\ldots,-1}_{(n^{\prime}-n)\text{-}\text{copies}}\rangle\,,$

where $\mathbb{H}=\langle 1,-1\rangle$ stands for the hyperbolic plane. Note
that the quadratic form $\underline{q}$ is anisotropic while the quadratic
form $\underline{q^{\prime}}$ is isotropic (because $n^{\prime}-n>0$). Making
use of Lemma 4.20 below, we hence obtain the following computations in the
Grothendieck ring of varieties:

$[Q_{q}]=(1+\mathbb{L}+\cdots+\mathbb{L}^{n-1})+[Q_{\underline{q}}]\cdot\mathbb{L}^{n}+\mathbb{L}^{\mathrm{dim}(q)-2}\cdot(1+\mathbb{L}+\cdots+\mathbb{L}^{n-1})$
$[Q_{q^{\prime}}]=(1+\mathbb{L}+\cdots+\mathbb{L}^{n-1})+[Q_{\underline{q^{\prime}}}]\cdot\mathbb{L}^{n}+\mathbb{L}^{\mathrm{dim}(q)-2}\cdot(1+\mathbb{L}+\cdots+\mathbb{L}^{n-1})\,.$

This implies that
$[Q_{q}]\cdot\mathbb{L}^{n}=[Q_{q^{\prime}}]\cdot\mathbb{L}^{n}$ in
$K_{0}\mathrm{Var}(k)$. Since $\mu_{\mathrm{c}}(\mathbb{L})=[\mathbb{Z}(-1)]$
(consult [1, §13.2.2]) and the Lefschetz motive
$\mathbb{Z}(-1)\in\operatorname{Num}(k)$ is $\otimes$-invertible, this further
implies that

$\mu_{\mathrm{c}}([Q_{\underline{q}}]\cdot\mathbb{L}^{n})=\mu_{\mathrm{c}}([Q_{\underline{q^{\prime}}}]\cdot\mathbb{L}^{n})\Leftrightarrow\mu_{\mathrm{c}}([Q_{\underline{q}}])\cdot[\mathbb{Z}(-n)]=\mu_{\mathrm{c}}([Q_{\underline{q^{\prime}}}])\cdot[\mathbb{Z}(-n)]\Leftrightarrow\mu_{\mathrm{c}}([Q_{\underline{q}}])=\mu_{\mathrm{c}}([Q_{\underline{q^{\prime}}}])\,.$

Consequently, Proposition 4.18 implies that $\underline{q}$ and
$\underline{q^{\prime}}$ are both isotropic or both anisotropic, which is a
contradiction. This concludes the proof. ∎

######  Lemma 4.20.

Given a non-degenerate quadratic form $q$ and an integer $n\geq 1$, we have
the following computation in $K_{0}\mathrm{Var}(k)$ ($\mathbb{L}$ stands for
the Grothendieck class $[\mathbb{A}^{1}]$):

(4.21) $[Q_{q\perp
n\mathbb{H}}]=(1+\mathbb{L}+\cdots+\mathbb{L}^{n-1})+[Q_{q}]\cdot\mathbb{L}^{n}+\mathbb{L}^{\mathrm{dim}(q)}\cdot(\mathbb{L}^{n-1}+\cdots+\mathbb{L}^{2(n-1)})\,.$

###### Proof.

Let ${\mathcal{P}}(n)$ be the equality (4.21). The proof will be done by
induction on $n$. We start by proving the equality $\mathcal{P}(1)$ (for every
quadratic form $q$). Note that there exists a choice of coordinates
$x_{1},\ldots,x_{\mathrm{dim}(q)},u,w$ such that
$(q\perp\mathbb{H})(x_{1},\ldots,x_{\mathrm{dim}(q)},u,w)=q(x_{1},\ldots,x_{\mathrm{dim}(q)})+uw$.
Following Rost [26, Prop. 2], consider the stratification $\mathrm{pt}\subset
Z\subset Q_{q\perp\mathbb{H}}$ of $Q_{q\perp\mathbb{H}}$, where
$Z:=\\{[x_{1}:\cdots:x_{\mathrm{dim}(q)}:0:w]\,|\,q(x_{1},\ldots,x_{\mathrm{dim}(q)})+0w=0\\}$
and $\mathrm{pt}:=[0:\cdots 0:1:0]$. By definition of $K_{0}\mathrm{Var}(k)$,
we hence obtain the following computation:

(4.22) $[Q_{q\perp\mathbb{H}}]=[Z]+[Q_{q\perp\mathbb{H}}\backslash
Z]=[\mathrm{pt}]+[Z\backslash\mathrm{pt}]+[Q_{q\perp\mathbb{H}}\backslash
Z]\,.$

On the one hand, note that $Q_{q\perp\mathbb{H}}\backslash
Z\simeq\mathbb{A}^{\mathrm{dim}(q)}$. This implies that
$[Q_{q\perp\mathbb{H}}\backslash Z]=\mathbb{L}^{\mathrm{dim}(q)}$. On the
other hand, as explained in [26, Prop. 2], the following projection map

$\displaystyle Z\backslash\mathrm{pt}\longrightarrow Q_{q}$
$\displaystyle[x_{1}:\cdots:x_{\mathrm{dim}(q)}:0:w]\mapsto[x_{1}:\cdots:x_{\mathrm{dim}(q)}]$

is a $1$-dimensional vector bundle. Following [5, §2 Prop. 2.3.3], this
implies that $[Z\backslash\mathrm{pt}]=[Q_{q}]\cdot[\mathbb{A}^{1}]$.
Consequently, since $[\mathrm{pt}]=1$, we conclude that (4.22) reduces to the
following computation:

(4.23)
$[Q_{q\perp\mathbb{H}}]=1+[Q_{q}]\cdot\mathbb{L}+\mathbb{L}^{\mathrm{dim}(q)}\,.$

Let us now prove the implication $\mathcal{P}(n)\Rightarrow\mathcal{P}(n+1)$.
If the equality $\mathcal{P}(n)$ holds (for every quadratic form $q$), then we
obtain the following computation in $K_{0}\mathrm{Var}(k)$:

$[Q_{(q\perp\mathbb{H})\perp
n\mathbb{H}}]=(1+\mathbb{L}+\cdots+\mathbb{L}^{n-1})+[Q_{q\perp\mathbb{H}}]\cdot\mathbb{L}^{n}+\mathbb{L}^{\mathrm{dim}(q)+2}\cdot(\mathbb{L}^{n-1}+\cdots+\mathbb{L}^{2(n-1)})\,.$

By combining it with (4.23), we hence obtain the searched equality
$\mathcal{P}(n+1)$. ∎

##  5\. Proof of Theorem 1.25

Proof of item (i). Recall first from item (i) of Theorem 1.22 that if
$[\mathrm{Iv}(A,\ast)]=[\mathrm{Iv}(A^{\prime},\ast^{\prime})]$, then
$\mathrm{deg}(A)=\mathrm{deg}(A^{\prime})$. Also, recall from Example 1.18
that the odd dimensional involution varieties are the odd dimensional
quadrics. As proved in [16, §15.A], the assignment $q\mapsto C_{0}(q)$ gives
rise to an bijection between the similarity classes of non-degenerate
quadratic forms of dimension $3$ and the isomorphism classes of quaternion
algebras. In the same vein, as proved in [16, §15.B], the assignment
$(A,\ast)\mapsto C_{0}(A,\ast)$ gives rise to a bijection between the
isomorphism classes of central simple $k$-algebras with involution of
orthogonal type of degree $4$ and the isomorphism classes of quaternion
algebras over an étale quadratic $k$-algebra. Consequently, given two central
simple $k$-algebras with involutions of orthogonal type $(A,\ast)$ and
$(A^{\prime},\ast^{\prime})$ such that
$[\mathrm{Iv}(A,\ast)]=[\mathrm{Iv}(A^{\prime},\ast^{\prime})]$ and
$\mathrm{deg}(A)=\mathrm{deg}(A^{\prime})\leq 4$, the proof follows now from
the combination of item (iii) of Theorem 1.22 with the well-known fact that
$(A,\ast)$ and $(A^{\prime},\ast^{\prime})$ are isomorphic if and only if
$\mathrm{Iv}(A,\ast)$ and $\mathrm{Iv}(A^{\prime},\ast^{\prime})$ are
isomorphic. Proof of item (ii). Given a central simple $k$-algebra $A$ and two
involutions of orthogonal type $\ast$ and $\ast^{\prime}$ on $A$, recall from
[19, Thm. B] that if $I(k)^{3}$ is torsion-free, then we have the following
equivalence:

(5.1)
$(A,\ast)\simeq(A,\ast^{\prime})\Leftrightarrow\big{(}C_{0}(A,\ast)\simeq
C_{0}(A,\ast^{\prime})\,\,\text{and}\,\,\mathrm{sgn}_{P}(A,\ast)=\mathrm{sgn}_{P}(A,\ast^{\prime})\,\,\text{for}\,\,\text{every}\,\,\text{ordering}\,\,P\,\,\text{of}\,\,k\big{)}\,;$

in the case where $k$ is formally-real and $\mathrm{deg}(A^{\prime})$ is even,
we assume moreover that $\tilde{u}(k)<\infty$. Now, let $(A,\ast)$ and
$(A^{\prime},\ast^{\prime})$ be two central simple $k$-algebras with
involution of orthogonal type such that
$[\mathrm{Iv}(A,\ast)]=[\mathrm{Iv}(A^{\prime},\ast^{\prime})]$. Thanks to
item (iv) of Theorem 1.22, we have an isomorphism $A\simeq A^{\prime}$.
Therefore, we can (and will) assume without loss of generality that
$A=A^{\prime}$. Consequently, the proof follows now from items (iii) and (v)
of Theorem 1.22, from the above equivalence (5.1), and from the well-known
fact that $(A,\ast)$ and $(A^{\prime},\ast^{\prime})$ are isomorphic if and
only if $\mathrm{Iv}(A,\ast)$ and $\mathrm{Iv}(A^{\prime},\ast^{\prime})$ are
isomorphic.

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