Amitsur subgroup and noncommutative motives

Sa$\mathrm{\check{s}}$a Novakovi$\mathrm{\acute{c}}$
January 2021

_ЗА ИЛИАНУ._

Abstract. This paper addresses the problem of calculating the Amitsur subgroup
of a proper $k$-scheme. Under mild hypothesis, we calculate this subgroup for
proper $k$-varieties $X$ with $\mathrm{Pic}(X)\simeq\mathbb{Z}^{\oplus m}$,
using a classification of so called absolutely split vector bundles
($AS$-bundles for short). We also show that the Brauer group of $X$ is
isomorphic to $\mathrm{Br}(k)$ modulo the Amitsur subgroup, provided $X$ is
geometrically rational. Our results also enable us to classify $AS$-bundles on
twisted flags. Moreover, we find an alternative proof for a result due to
Merkurjev and Tignol, stating that the Amitsur subgroup of twisted flags is
generated by a certain subset of the set of classes of Tits algebras of the
corresponding algebraic group. This result of Merkurjev and Tignol is actually
a corollary of a more general theorem that we prove. The obtained results have
also consequences for the noncommutative motives of the twisted flags under
consideration. In particular, we show that a certain noncommutative motive of
a twisted flag is a birational invariant, generalizing in this way a result of
Tabuada. We generalize this result for $X$ having a certain type of
semiorthogonal decomposition.

##  1\. Introduction

Let $f\colon X\rightarrow S$ be a scheme that is seperated and of finite type
over a Noetherian scheme $S$ and assume
$\mathcal{O}_{S}\xrightarrow{\simeq}f_{*}\mathcal{O}_{X}$. Then, for each
$S$-scheme $T$ there is an exact sequence

$\displaystyle
0\longrightarrow\mathrm{Pic}(T)\longrightarrow\mathrm{Pic}(X_{T})\longrightarrow\mathrm{Pic}_{(X/S)(\mathrm{fppf})}(T)\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}\mathrm{Br}^{\prime}(T)\longrightarrow\mathrm{Br}^{\prime}(X_{T}).$

Here $\mathrm{Pic}_{(X/S)}$ denotes the Picard functor and
$\mathrm{Pic}_{(X/S)(\mathrm{fppf})}$ the associated sheaf in the fppf
topology. Specializing the above sequence to the case $X$ a proper variety
over a field $k$ and $T=\mathrm{Spec}(k)$, Liedtke [18] called the group
$\delta(\mathrm{Pic}_{(X/S)(\mathrm{fppf})}(k))$ the _Amitsur subgroup_ of $X$
in $\mathrm{Br}(k)$. This subgroup is denoted by $\mathrm{Am}(X)$. If $X$ is
smooth and proper over $k$, it follows that
$\mathrm{Am}(X)=\mathrm{ker}(\mathrm{Br}(k)\rightarrow\mathrm{Br}(k(X)))$. The
group $\mathrm{ker}(\mathrm{Br}(k)\rightarrow\mathrm{Br}(k(X)))$ is also
denoted by $\mathrm{Br}(k(X)/k)$ and was studied for instance in [20] and [8].
If $X$ is Brauer–Severi corresponding to a central simple algebra $A$, it is a
classical result due to Châtelet that $\mathrm{Am}(X)=\langle[A]\rangle$. In
this case $\mathrm{Br}(X)\simeq\mathrm{Br}(k)/\mathrm{Am}(X)$. It is shown in
[18] that $\mathrm{Am}(X)$ is a birational invariant for $X$ smooth and proper
over $k$. Note that if $X$ admits a $k$-rational point, then
$\mathrm{Am}(X)=0$. On the other hand, there are proper varieties with trivial
Amitsur subgroup without rational points (see [18], Proposition 5.4). Special
varieties for which $\mathrm{Am}(X)$ is calculated can be found in [8], [18]
and [20]. In this paper, we want to calculate $\mathrm{Am}(X)$ and
$\mathrm{Br}(X)$ for a certain class of proper $k$ schemes $X$. Furthermore,
we want to explain the consequences of our results for the noncommutative
motives.

To state our results, we fix some notations and recall some facts. Let $X$ be
a proper and geometrically integral $k$-scheme and denote by $X_{s}$ the base
change $X\otimes_{k}k^{s}$ to the separable closure. If
$\mathrm{Pic}(X_{s})\simeq\mathbb{Z}^{\oplus m}$, then $\mathrm{Pic}(X)\simeq
r_{1}\mathbb{Z}\oplus\cdots\oplus r_{m}\mathbb{Z}$. Let us fix a basis
$\mathcal{L}_{1},...,\mathcal{L}_{m}$ of $\mathrm{Pic}(X_{s})$. Using a basis
of $\mathrm{Pic}(X)$ one can show that there are line bundles
$\mathcal{J}_{i}\in\mathrm{Pic}(X)$ satisfying
$\mathcal{J}_{i}\otimes_{k}k^{s}\simeq\mathcal{L}_{i}^{\otimes c_{i}}$ for
some integers $c_{i}\geq 1$. Now choose the line bundles $\mathcal{J}_{i}$
such that the $c_{i}$ are minimal. According to [23], Proposition 3.4, these
$\mathcal{J}_{i}$ are unique up to isomorphism. Assume there are pure vector
bundles $\mathcal{M}_{i}$ of type $\mathcal{L}_{i}\in\mathrm{Pic}(X_{s})$. A
vector bundle $\mathcal{E}$ on a proper $k$-scheme $X$ is called _pure of
type_ $\mathcal{W}$ if there is an indecomposable vector bundle $\mathcal{W}$
on $X\otimes_{k}\bar{k}$ such that
$\mathcal{E}\otimes_{k}{\bar{k}}\simeq\mathcal{W}^{\oplus m}$. Being pure of
type $\mathcal{L}_{i}$ is equivalent to
$\mathcal{L}_{i}\in\mathrm{Pic}_{\Gamma}(X_{s})$, where $\Gamma$ denotes the
absolute Galois group (see [23], Theorem 4.5). We know from [23], Proposition
3.5 that the bundle $\mathcal{M}_{i}$ is unique up to isomorphism. We set
$\mathcal{M}_{\mathcal{L}_{i}}:=\mathcal{M}_{i}$. It is easy to see that for
any line bundle $\mathcal{L}_{i}^{\otimes a}\in\mathrm{Pic}(X_{s})$ there is
an indecomposable pure bundle of type $\mathcal{L}_{i}^{\otimes a}$. Indeed,
let $s_{i}=\mathrm{rk}(\mathcal{M}_{\mathcal{L}_{i}})$ and consider
$(\mathcal{L}_{i}^{\oplus s_{i}})^{\otimes a}\simeq(\mathcal{L}_{i}^{\otimes
a})^{\oplus s_{i}^{a}}$. Then we get $\mathcal{M}_{\mathcal{L}_{i}}^{\otimes
a}\otimes_{k}k^{s}\simeq(\mathcal{L}_{i}^{\oplus s_{i}})^{\otimes
a}\simeq(\mathcal{L}_{i}^{\otimes a})^{\oplus s_{i}^{a}}$.

Considering the Krull–Schmidt decomposition of
$\mathcal{M}_{\mathcal{L}_{i}}^{\otimes a}$ and taking into account that all
indecomposable direct summands are isomorphic (see [23], proof of Proposition
3.6 and Remark 3.7), we get an, up to isomorphism, unique indecomposable
vector bundle $\mathcal{M}_{\mathcal{L}_{i}^{\otimes a}}$ such that
$\mathcal{M}_{\mathcal{L}_{i}^{\otimes
a}}\otimes_{k}k^{s}\simeq(\mathcal{L}_{i}^{\otimes a})^{\oplus s_{i}(a)}$,
where $s_{i}(a)=\mathrm{rank}(\mathcal{M}_{\mathcal{L}_{i}^{\otimes a}})$.
Using Krull–Schmidt decomposition again, we can use our indecomposable vector
bundles $\mathcal{M}_{\mathcal{L}_{i}^{\otimes a}}$ and take the tensor
product of these to get an indecomposable vector bundle
$\mathcal{M}_{(a_{1},...,a_{m})}$ of type $\mathcal{L}_{1}^{\otimes
a_{1}}\otimes\cdots\otimes\mathcal{L}_{m}^{\otimes a_{m}}$. Again, the bundles
$\mathcal{M}_{(a_{1},...,a_{m})}$ are unique up to isomorphism. Recall that a
vector bundle $\mathcal{E}$ on a $k$-scheme $X$ is called _absolutely split_
if it splits after base change as a direct sum of line bundles on
$X\otimes_{k}\bar{k}$. For an absolutely split vector bundle we shortly write
_AS-bundle_. Over an algebraically closed field, a classical result of
Grothendieck classifies all $AS$-bundles on $\mathbb{P}^{1}$. Note that on
$\mathbb{P}^{1}$ the result of Grothendieck shows that actually all vector
bundles are $AS$-bundles. In [22] the author classifies vector bundles on
twisted forms of $\mathbb{P}^{1}$. The twisted forms of $\mathbb{P}^{1}$ are
Brauer–Severi curves (or smooth non-degenerate quadrics without rational
point). Generalizing these results further, in [23], the author clssifies
$AS$-bundles on proper $k$-schemes. Moreover, for $X$ with cyclic Picard group
the indecomposable $AS$-bundles are determined explicitely. In the present
paper we generalize this result to the case
$\mathrm{Pic}(X_{s})\simeq\mathbb{Z}^{\oplus m}$.

######  Theorem (Theorem 4.5).

Let $X$ be a proper and geometrically integral $k$-scheme with
$\mathrm{Pic}(X_{s})\simeq\mathbb{Z}^{\oplus m}$ and let
$\mathcal{L}_{1},...,\mathcal{L}_{m}\in\mathrm{Pic}(X_{s})$ be the basis from
above. Let $\mathcal{J}_{i}\in\mathrm{Pic}(X)\simeq\mathbb{Z}^{\oplus m}$ be
the up to isomorphism unique line bundles satisfying
$\mathcal{J}_{i}\otimes_{k}k^{s}\simeq\mathcal{L}_{i}^{\otimes c_{i}}$ with
$c_{i}$ being minimal. Assume there are indecomposable pure bundles
$\mathcal{M}_{\mathcal{L}_{i}}$ of type $\mathcal{L}_{i}$. Then all
indecomposable $AS$-bundles $\mathcal{E}$ are of the form

$\displaystyle\mathcal{J}_{1}^{\otimes
b_{1}}\otimes\cdots\otimes\mathcal{J}_{m}^{\otimes
b_{m}}\otimes\mathcal{M}_{(a_{1},...,a_{m})}$

with unique $b_{i}\in\mathbb{Z}$ and $0\leq a_{j}\leq c_{j}-1$.

Notice that Theorem 4.5 is a generalization of [23], Theorem 5.1. Using
Theorem 4.5 we will prove the following result about the Amitsur subgroup and
the Brauer group $\mathrm{Br}(X)$.

######  Theorem 1.1.

Let $X$ be proper and geometrically integral $k$-scheme with
$\mathrm{Pic}(X_{s})\simeq\mathbb{Z}^{\oplus m}$. With the notation and
assumption from above, the Amitsur subgroup $\mathrm{Am}(X)$ is generated by
the classes of central simple algebras $\mathrm{End}(\mathcal{M}_{e_{i}})$,
where $i\in\\{1,...,m\\}$ for which $c_{i}\geq 2$. In particular, if
$\mathrm{Pic}(X_{s})\simeq\mathbb{Z}$ is generated by an ample line bundle
$\mathcal{L}$ and if there exists a pure vector bundle
$\mathcal{M}_{\mathcal{L}}$ of type $\mathcal{L}$, then $\mathrm{Am}(X)$ is
cyclic and generated by $[\mathrm{End}(\mathcal{M}_{\mathcal{L}})]$. Moreover,
if $X$ is geometrically rational, then
$\mathrm{Br}(X)\simeq\mathrm{Br}(k)/\mathrm{Am}(X)$.

Schemes satisfying the assuptions of Theorem 1.1 include twisted flag
varieties. In order to apply Theorem 1.1 to twisted flags and to obtain in
this way an alternative proof of a result of Merkurjev and Tignol [20],
Theorem B, we recall the definition and some facts on twisted flags and refer
to [19] for details.

Let $G$ be a semisimple algebraic group over a field $k$ and
$G_{s}=G\otimes_{k}k^{s}$. For a parabolic subgroup $P$ of $G_{s}$, one has a
homogeneous variety $G_{s}/P$. A _twisted flag_ is variety $X$ such that
$X\otimes_{k}k^{s}$ is $G_{s}$-isomorphic to $G_{s}/P$ for some $G$ and some
parabolic $P$ in $G_{s}$. Any twisted flag is smooth, absolutely irreducible
and reduced. An algebraic group $G^{\prime}$ is called twisted form of $G$ iff
$G^{\prime}_{s}\simeq G_{s}$ iff $G^{\prime}={{}_{\gamma}}G$ for some
$\gamma\in Z^{1}(k,\mathrm{Aut}(G_{s}))$. The group $G^{\prime}$ is called an
_inner form_ of $G$ if there is a $\delta\in Z^{1}(k,\bar{G}(k^{s}))$ with
$G^{\prime}={{}_{\delta}}G$. Here $\bar{G}=G/Z(G)$ where $Z(G)$ denotes the
center. For an arbitrary semisimple $G$ over $k$, there is a unique (up to
isomorphism) split semisimple group $G^{d}$ such that $G_{s}\simeq G_{s}^{d}$.
If $G$ is an inner form of $G^{d}$, then $G$ is said to be of _inner type_.
For instance, let $A$ be a central simple algebra over $k$ of degree $n$ and
$G=\mathrm{PGL}_{1}(A)$, then $G_{s}\simeq\mathrm{PGL}_{n}$ over $k^{s}$.
Hence $G$ is an inner form of $\mathrm{PGL}_{n}$. Since $\mathrm{PGL}_{n}$ is
split, $G=\mathrm{PGL}_{1}(A)$ is of inner type. For a classification of
simple groups of classical type we refer to [14], p.366-373.

Now let $G$ be a semisimple (so connected) simply connected algebraic group
over $k$ and $P$ a parabolic subgroup. Let $G/P$ be a flag variety and note
that $G/P=\bar{G}/\bar{P}$. Let $\gamma\colon\mathrm{Gal}(k^{s}|k)\rightarrow
G(k^{s})$ be a 1-cocycle. We denote by $X:={{}_{\gamma}}(G/P)$ the twisted
form of $G/P$ corresponding to $\gamma$. Notice that
${{}_{\gamma}}(G/P)\otimes_{k}k^{s}\simeq G_{s}/P_{s}$ for a suitable
parabolic subgroup $P_{s}$ of $G_{s}$. The next corollary is essentially [20],
Theorem B. Below $\mathrm{Ch}(P_{s})$ denotes the character group and
$\mathrm{Ch}(P_{s})^{\Gamma}$ the character group of Galois invariant
characters.

######  Corollary 1.2.

Let $G$ be a semisimple (so connected) simply connected algebraic group over
$k$ and $P$ a parabolic subgroup. Denote by $X={{}_{\gamma}}(G/P)$ a twisted
flag. Then $\mathrm{Am}(X)$ is generated by the Brauer classes of Tits
algebras of $G$ corresponding to the elements of a basis of
$\mathrm{Ch}(P_{s})^{\Gamma}$. Moreover,
$\mathrm{Br}(X)\simeq\mathrm{Br}(k)/\mathrm{Am}(X)$.

The proof of Theorem 1.1 actually uses Theorem 4.5 which enables us to
classify all indecomposable $AS$-bundles on $X$. It is a non-trivial fact that
these bundles are in one-to-one correspondence with the closed points of the
Picard scheme $\mathrm{Pic}_{(X/k)(\mathrm{fppf})}$ (see Theorem 4.4). Notice
that Theorem 4.5 generalizes [23], Theorem 5.1.

######  Corollary 1.3.

Let $X_{i}$ be a twisted form of $G_{i}/P_{i}$ with $G_{i}$ and $P_{i}$ as in
Theorem 1.1 and let $X=X_{1}\times\cdots\times X_{n}$. Let $D_{i}$ be the set
of generators of $\mathrm{Am}(X_{i})$ obtained from Corollary 1.2. Then
$\mathrm{Am}(X)$ is generated by $\cup D_{i}$. Moreover,
$\mathrm{Br}(X)\simeq\mathrm{Br}(k)/\mathrm{Am}(X)$.

Theorem 1.1 from above has also a motivic consequence. Noncommutative motives
are by construction closely related to semiorthogonal decompositions. In the
last decades, the bounded derived category $D^{b}(X)$ of coherent sheaves on a
smooth projective variety $X$ has been recognized as an interesting invariant,
encoding a lot of geometric information. For instance, there are links between
the semiorthogonal decomposition of $D^{b}(X)$ and the birational geometry of
$X$ (see for instance [16], [2], [3], [24], [26] and references therein). From
a motivic point of view, it is quite natural to ask how birational geometry of
a given variety $X$ is detected by its noncommutative motive. And indeed,
there are results in this direction for (generalized) Brauer–Severi varieties
[30] and [31]. In the present paper we want to consider twisted flags and shed
some light to the case of arbitrary proper $k$-schemes admitting a certain
type of semiorthogonal decomposition. Our main results are Theorems 1.4 and
1.7.

Recall from the book [29] that the category dgcat of small dg categories with
dg functors carries a Quillen model structure whose weak equivalences are
Morita equivalences. Denote by $\mathrm{Hmo}$ the homotopy category obtained
from the Quillen model structure and by $\mathrm{Hmo}_{0}$ its additivization.
To any small dg category $\mathcal{A}$ one can associate functorially its
_noncommutative motive_ $U(\mathcal{A})$ which takes values in
$\mathrm{Hmo}_{0}$. This functor
$U\colon\textbf{dgcat}\rightarrow\mathrm{Hmo}_{0}$ is an _universal additive
invariant_. Recall that an universal additive invariant is any functor
$E\colon\textbf{dgcat}\rightarrow D$ taking values in an additive category $D$
such that

  * (i)

it sends derived Morita equivalences to isomorphisms,

  * (ii)

for any pre-triangulated dg category $\mathcal{A}$ admitting full pre-
triangulated dg subcategories $\mathcal{B}$ and $\mathcal{C}$ such that
$H^{0}(\mathcal{A})=\langle H^{0}(\mathcal{B}),H^{0}(\mathcal{C})\rangle$ is a
semiorthogonal decomposition, the morphism $E(\mathcal{B})\oplus
E(\mathcal{C})\rightarrow E(\mathcal{A})$ induced by the inclusions is an
isomorphism.

A source of examples for dg categories is provided by schemes since the
derived category of perfect complexes $\mathrm{perf}(X)$ of any quasi-
projective scheme $X$ admits a canonical (unique) dg enhancement
$\mathrm{perf}_{dg}(X)$. In [30] it is proved that if two Brauer–Severi
varieties $X$ and $Y$ (see Section 2 for a definition) are birational, then
$U(\mathrm{perf}_{dg}(X))=U(\mathrm{perf}_{dg}(Y))$. In view of the Amitsur
conjecture for central simple algebras (two Brauer–Severi varieties $X$ and
$Y$ are birational if and only if the corresponding central simple algebras
$A$ and $B$ generate the same subgroup in $\mathrm{Br}(k)$), it is conjectured
in _loc.cite_ that $U$ is actually a complete birational invariant for
Brauer–Severi varieties. As a Brauer–Severi variety is a special case of a
twisted flag, Theorem 1.4 from below is a generalization.

We fix some notation: Let $X$ be a twisted flag as in Corollary 1.2 and denote
by $A_{g}$ the central simple division algebra corresponding to
$g\in\mathrm{Am}(X)$. Analogously, let $B_{h}$ denote the central simple
division algebra corresponding to $h\in\mathrm{Am}(Y)$. Set
$M_{X}:=\bigoplus_{g\in\mathrm{Am}(X)}U(A_{g})$ and
$M_{Y}:=\bigoplus_{h\in\mathrm{Am}(Y)}U(B_{h})$. Note that these sums are
finite according to Theorem 1.1. Furthermore, let $\mathrm{Sep}(k)$ be the
full subcategory of the category of noncommutative Chow motives (see [30] for
a definition) consisting of objects $U(F)$ with $F$ a separable $k$-algebra.
Now let $\mathrm{CSA}(k)$ be the full subcategory of $\mathrm{Sep}(k)$
consisting of objects $U(A)$ with $A$ a central simple $k$-algebra (see
Section 2 for a definition of central simple algebra) and denote by
$\mathrm{CSA}(k)^{\oplus}$ its closure under finite sums. It is an additive
symmetric monoidal subcategory. We write shortly $U(X)$ for
$U(\mathrm{perf}_{dg}(X))$.

Let $G$, $P$ and $\gamma$ be as above and let $\rho_{1},...,\rho_{n}$ be a
$\mathrm{Ch}$-homogeneous basis of $R(P)$ over $R(G)$ (see [27],§2), where
$R(P)$ and $R(G)$ denote the corresponding representation rings. Let
$A_{\chi(i),\gamma}$ be the Tits central simple algebras associated to
$\rho_{i}$ (see Section 3 for a definition) and let $\mathrm{Ti}(X):=\langle
A_{\chi(1),\gamma},...,A_{\chi(n),\gamma}\rangle$ be the subgroup of
$\mathrm{Br}(k)$ generated by these Tits algebras. Denote by

$\displaystyle M\mathrm{Ti}(X):=\bigoplus_{f\in\mathrm{Ti}(X)}U(A_{f}),$

where $A_{f}$ are the central simple division algebras corresponding to $f$.

######  Theorem 1.4.

Let $X$ be a twisted flags as in Corollary 1.2. Then there are direct summands
$N,N^{\prime}\in\mathrm{CSA}(k)^{\oplus}$ of $M\mathrm{Ti}(X)$ such that
$M_{X}\oplus N=U(X)\oplus N^{\prime}$.

######  Remark 1.5.

In the case of a Brauer–Severi variety $X$ corresponding to a central simple
algebra of period $m$, one has $M_{X}=U(k)\oplus U(A)\oplus\cdots\oplus
U(A^{\otimes m-1})$. Note that the period $m$ divides the degree $n$. So if
$m\cdot r=n$, we get $M_{X}^{\oplus r}=U(\mathrm{perf}(X))$ (see Example 6.1
for details). In this case we have $N=M_{X}^{\oplus(r-1)}$ and $N^{\prime}=0$.
With the help of Corollary 1.6 we get back [30], Proposition 3.15 (see p.14
for detailed explanation).

######  Corollary 1.6.

Let $X$ and $Y$ be twisted flags as in Corollary 1.2 and let $N,N^{\prime}$ be
the direct summands of $M\mathrm{Ti}(X)$ and $Q,Q^{\prime}$ of
$M\mathrm{Ti}(Y)$ obtained from Theorem 1.4. If $X$ and $Y$ are birational,
then $M_{X}\simeq M_{Y}$ and $U(X)\oplus N^{\prime}\oplus Q\simeq U(Y)\oplus
Q^{\prime}\oplus N$.

Using the theory of semiorthogonal decompositions one can try to generalize
Corollary 1.6 for arbitrary proper and geometrically integral $k$-schemes. For
the definition of w-exceptional objects and semiorthogonal decompositions we
refer to p.15. Below, $D^{b}(X)$ denotes the bounded derived category of
coherent sheaves on $X$ and
$D^{b}(X)=\langle\mathcal{D}_{1},...,\mathcal{D}_{m}\rangle$ a semiorthogonal
decomposition. We want to call a smooth, proper and geomerically integral
$k$-scheme $X$ a scheme of _pure weak exceptional type_ if it satisfies the
following conditions:

  * (i)

$D^{b}(X)=\langle\mathcal{E}_{1},...,\mathcal{E}_{m}\rangle$ is a
semiorthogonal decomposition induced by a full w-exceptional collection,

  * (ii)

The bundles $\mathcal{E}_{i}$ be pure of type $\mathcal{K}_{i}$. Furthermore,
$\mathrm{End}(\mathcal{K}_{i})\simeq k^{s}$ and some of these
$\mathcal{K}_{i}$ form a basis of $\mathrm{Pic}(X_{s})$.

Note that if $\mathcal{E}_{i}$ is pure of type $\mathcal{K}_{i}$ with
$\mathrm{End}(\mathcal{K}_{i})\simeq k^{s}$, the base change of the
semiorthogonal decomposition (i) actually implies that
$D^{b}(X_{s})=\langle\mathcal{K}_{1},...,\mathcal{K}_{m}\rangle$ is a full
exceptional collection (see p.15 for a definition) and hence
$\mathrm{Pic}(X_{s})\simeq\mathbb{Z}^{\oplus m}$. Indeed, there are schemes
satisfying these two conditions. For instance (generalized) Brauer-Severi
varieties or certain involution varieties. Conjecturally, all twisted flags
are of pure weak exceptional type. It is interesting to investigate whether
twisted flags are characterized by (i) and (ii). For a result in this
direction see [25], Theorem 1.2. One can show that schemes $X$ of pure weak
exceptional type satisfy the assumptions of Theorem 4.5 (see proof of Theorem
1.7). Therefore, we can classify indecomposable $AS$-bundles on such $X$.

Now let $D_{X}$ denote the set of indecopmposable $AS$-bundles on a scheme of
pure weak exceptional type. Note that for any $AS$-bundle $\mathcal{E}$ on $X$
satisfying the assumptions of Theorem 4.5 one has
$[\mathrm{End}(\mathcal{E})]\in\mathrm{Br}(k)$ (see Proposition 4.7). From
Theorem 4.5 and Proposition 4.7 we conclude that there are only finitely many
Brauer-classes $[\mathrm{End}(\mathcal{E})]\in\mathrm{Br}(k)$ of
indecomposable $AS$-bundles. Now let $C_{X}\subset\mathrm{Br}(k)$ be the
subgroup generated by these finitely many Brauer-classes. Denote by $A(g)$ the
central simple division algebra in $C_{X}$ corresponding to $g$. Set

$\displaystyle M_{T(X)}:=\bigoplus_{g\in C_{X}}U(A(g))$

######  Theorem 1.7.

Let $X$ and $Y$ be schemes of pure weak exceptional type. If $X$ and $Y$ are
birational, then there are direct summands
$N,N^{\prime}\in\mathrm{CSA}^{\oplus}$ of $M_{T(X)}$ and
$Q,Q^{\prime}\in\mathrm{CSA}^{\oplus}$ of $M_{T(Y)}$ such that $U(X)\oplus
N^{\prime}\oplus Q\simeq U(Y)\oplus Q^{\prime}\oplus N$.

######  Corollary 1.8.

Let $X$ and $Y$ be as in Theorem 1.7. Assume $X$ and $Y$ are birational and
let $A_{i},1\leq i\leq n$ and $B_{j},1\leq j\leq m$ be the central simple
algebras occuring in $U(X)\oplus N^{\prime}\oplus Q$ and $U(Y)\oplus
Q^{\prime}\oplus N$ respectively, then
$\langle[A_{i}]\rangle=\langle[B_{j}]\rangle$ in $\mathrm{Br}(k)$.

Notations Throughout the paper $k$ is an arbitrary field and $k^{s}$ a
separable closure. For a variety/algebraic group over $k$, we write $X_{s}$
and $G_{s}$ for the base changes $X\otimes_{k}k^{s}$ and $G\otimes_{k}k^{s}$
respectively.

##  2\. Examples of inner twisted flags

Recall that a finite-dimensional $k$-algebra $A$ is called _central simple_ if
it is an associative $k$-algebra that has no two-sided ideals other than $0$
and $A$ and if its center equals $k$. If the algebra $A$ is a division algebra
it is called _central division algebra_. Note that $A$ is a central simple
$k$-algebra if and only if there is a finite field extension $k\subset L$,
such that $A\otimes_{k}L\simeq M_{n}(L)$. This is also equivalent to
$A\otimes_{k}\bar{k}\simeq M_{n}(\bar{k})$. An extension $k\subset L$ such
that $A\otimes_{k}L\simeq M_{n}(L)$ is called splitting field for $A$. The
_degree_ of a central simple algebra $A$ is defined to be
$\mathrm{deg}(A):=\sqrt{\mathrm{dim}_{k}A}$. According to the _Wedderburn
Theorem_ , for any central simple $k$-algebra $A$ there is an unique integer
$n>0$ and a division $k$-algebra $D$ such that $A\simeq M_{n}(D)$. The
division algebra $D$ is also central and unique up to isomorphism. The degree
of the unique central division algebra $D$ is called the _index_ of $A$ and is
denoted by $\mathrm{ind}(A)$. Two central simple algebras $A$ and $B$ are said
to be _Brauer-equivalent_ if there are positive integers $r,s$ such that
$M_{r}(A)\simeq M_{s}(B)$.

For a central simple $k$-algebra $A$, the inner twisted forms arising from
$G=\mathrm{PGL}_{1}(A)$ can be described very explicitly. This will be done in
the sequel. One of these inner twisted forms is the _generalized Brauer–Severi
variety_. So let $m\leq n$. The generalized Brauer–Severi variety
$\mathrm{BS}(m,A)$ is defined to be the subset of $\mathrm{Grass}_{k}(mn,A)$
consisting of those subspaces of $A$ which are right ideals of dimension
$m\cdot n$ (see [14] or [7]). Recall that $\mathrm{Grass}_{k}(mn,A)$ is given
the structure of a projective variety via the Plücker embedding
$\mathrm{Grass}_{k}(mn,A)\rightarrow\mathbb{P}(\wedge^{mn}(A))$. This gives an
embedding $\mathrm{BS}(m,A)\rightarrow\mathbb{P}(\wedge^{mn}(A))$ and a very
ample line bundle $\mathcal{M}$ on $\mathrm{BS}(m,A)$. Note that for any
$\mathrm{BS}(m,A)$ there exists a finite Galois field extension $E$ of $k$
such that
$\mathrm{BS}(m,A)\otimes_{k}E\simeq\mathrm{Grass}_{E}(mn,n^{2})\simeq\mathrm{Grass}_{E}(m,n)$.
The Picard group $\mathrm{Pic}(\mathrm{Grass}_{E}(m,n))$ is isomorphic to
$\mathbb{Z}$ and has ample generator
$\mathcal{O}(1)\simeq\mathrm{det}(\mathcal{Q})$ with $\mathcal{Q}$ being the
universal quotient bundle on $\mathrm{Grass}_{E}(m,n)$. Recall that
$\mathrm{Pic}(\mathrm{BS}(m,A))\simeq\mathbb{Z}$ and that it has a positive
generator $\mathcal{L}$ such that
$\mathcal{L}\otimes_{k}E\simeq\mathcal{O}(r)$ for a suitable $r>0$. Since
$\mathrm{Pic}(\mathrm{BS}(m,A))$ is cyclic, we have $\mathcal{L}^{\otimes
s}\simeq\mathcal{M}$ for a suitable $s>0$. Therefore, $\mathcal{L}$ is ample.
From the definition of $\mathrm{BS}(m,A)$ it is clear that $\mathcal{L}$ is
also very ample. If $m=1$, $\mathrm{BS}(1,A)$ is called _Brauer–Severi
variety_.

We also recall the basics of generalized Brauer–Severi schemes (see [17]). Let
$X$ be a noetherian $k$-scheme and $\mathcal{A}$ a sheaf of Azumaya algebras
of rank $n^{2}$ over $X$ (see [11], [12] for details on Azumaya algebras). For
an integer $1\leq m_{1}<n$ the generalized Brauer–Severi scheme
$p:\mathrm{BS}(m_{1},\mathcal{A})\rightarrow X$ is defined as the scheme
representing the functor $F:\mathrm{Sch}/X\rightarrow\mathrm{Sets}$, where
$(\psi:Y\rightarrow X)$ is mapped to the set of left ideals $\mathcal{J}$ of
$\psi^{*}\mathcal{A}$ such that $\psi^{*}\mathcal{A}/\mathcal{J}$ is locally
free of rank $n(n-m_{1})$. By definition, there is an étale covering
$U\rightarrow X$ and a locally free sheaf $\mathcal{E}$ of rank $n$ with the
following trivializing diagram:

$\textstyle{\mathrm{Grass}(m_{1},\mathcal{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{q}$$\textstyle{\mathrm{BS}(n_{1},\mathcal{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{X}$

In the same way one defines the twisted relative flag
$\mathrm{BS}(m_{1},...,m_{r},\mathcal{A})$ as the scheme representing the
functor $F:\mathrm{Sch}/X\rightarrow\mathrm{Sets}$, where $(\psi:Y\rightarrow
X)$ is mapped to the set of left ideals
$\mathcal{J}_{1}\subset...\subset\mathcal{J}_{r}$ of $\psi^{*}\mathcal{A}$
such that $\psi^{*}\mathcal{A}/\mathcal{J}_{i}$ is locally free of rank
$n(n-m_{i})$. As for the generalized Brauer–Severi schemes, there is an étale
covering $U\rightarrow X$ and a locally free sheaf $\mathcal{E}$ of rank $n$
with diagram

$\textstyle{\mathrm{Flag}_{U}(m_{1},...,m_{r},\mathcal{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{q}$$\textstyle{\mathrm{BS}(m_{1},...,m_{r},\mathcal{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{X}$

Note that the usual Brauer–Severi schemes are obtained from the generalized
one by setting $m_{1}=1$. In this case one has a well known one-to-one
correspondence between sheaves of Azumaya algebras of rank $n^{2}$ on $X$ and
Brauer–Severi schemes of relative dimension $n-1$ via
$\check{H}^{1}(X_{et},\mathrm{PGL}_{n})$ (see [11]). Note that if the base
scheme $X$ is a point a sheaf of Azumaya algebras on $X$ is a central simple
$k$-algebra and the generalized Brauer–Severi schemes are the generalized
Brauer–Severi varieties from above. Consider a twisted flag
$X=\mathrm{SB}(m_{1},...,m_{r},A)\rightarrow\mathrm{Spec}(k)$. Such an $X$ is
an _inner form_ of a partial flag variety. That is, there is a cartesian
square of the form

$\textstyle{\mathrm{Grass}_{L}(m_{1},...,m_{r},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{q}$$\textstyle{\mathrm{BS}(m_{1},...,m_{r},A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{\mathrm{Spec}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathrm{Spec}(k)}$

where $L/k$ is a Galois extension and the 1-cocycle

$\displaystyle\mathrm{Gal}(L/k)\longrightarrow\mathrm{Aut}(\mathrm{Grass}_{L}(m_{1},...,m_{r},V))$

factors through $\mathrm{PGL}(V)$.

##  3\. Tits algebras

We refer to [27], Section 3.1 for details (see also [14], p.376-379). Now let
$G$ be a simply connected semi-simple algebraic group over the field $k$ and
$P$ a parabolic subgroup. We denote by $\widetilde{G}$ and $\widetilde{P}$
their universal covers. For the center $\widetilde{Z}\subset\widetilde{G}$ let
$\mathrm{Ch}:=\mathrm{Hom}(\widetilde{Z},\mathbb{G}_{m})$ be the character
group. Furthermore, let $R(\widetilde{G})$ and $R(\widetilde{P})$ be the
associated representation rings. Recall from [27],§2 that there exits a finite
free $\mathrm{Ch}$-homogeneous basis of $R(\widetilde{P})$ over
$R(\widetilde{G})$. Furthermore, let $\chi\in\mathrm{Ch}$ and denote by
$\mathrm{Rep}_{k}^{\chi}(\tilde{G})$ the full subcategory of
$\mathrm{Rep}_{k}(\tilde{G})$ consisting of those $V$ such that $\tilde{Z}$
acts on $V$ by $\chi$. Now for a Galois-invariant $\chi\in\mathrm{Ch}$, choose
a non-trivial representation $V_{\chi}\in\mathrm{Rep}_{k}^{\chi}(\tilde{G})$.
Put $A_{\chi}=\mathrm{End}(V_{\chi})$. Then $A_{\chi}$ is an $k$-algebra
equipped with a $G$-action by $k$-algebra automorphism. Using a 1-cocycle
$\gamma\colon\mathrm{Gal}(k^{s}|k)\rightarrow G(k^{s})$ one gets a new
$\mathrm{Gal}(k^{s}|k)$-action on $A_{\chi}\otimes_{k}k^{s}$ and hence a
twisted form $A_{\chi,\gamma}$. In this way, one obtains the _Tits map_ (see
[27], §3 or [14], p.377)
$\beta_{\gamma}\colon\mathrm{Ch}^{\Gamma}\rightarrow\mathrm{Br}(k)$ which is a
group homomorphism and assigns to each character $\chi\in\mathrm{Ch}^{\Gamma}$
a central simple algebra $A_{\chi,\gamma}\in\mathrm{Br}(k)$, called _Tits
algebra_.

######  Example 3.1 (Type $A_{n}$).

Let $G=\mathrm{SL}_{1}(A)$ where $A$ is a central simple algebra of degree
$n+1$. Then $\bar{G}=\mathrm{PGL}_{1}(A)$ and
$\mathrm{Ch}(Z)=\mathbb{Z}/(n+1)\mathbb{Z}$ with trivial
$\mathrm{Gal}(k^{s}|k)$-action. For any $i=0,1,...,n$, consider the
representation $p_{i}\colon\bar{G}\rightarrow\mathrm{GL}_{1}(\lambda^{i}A)$,
where $\lambda^{i}A$ are external powers of $A$. In the split case, the $i$-th
exterior power representation are known to be minimal representations (see
[14]). Hence $A^{\otimes i}$ are the Tits algebras for $G$.

######  Example 3.2 (Type $C_{n}$).

Let $G=\mathrm{Sp}(A,\sigma)$ where $A$ is a central simple algebra of degree
$2n$ with symplectic involution $\sigma$. Then
$\bar{G}=\mathrm{PGSp}(A,\sigma)$ and
$\mathrm{Ch}(Z)=\mathbb{Z}/2\mathbb{Z}=\\{0,\chi\\}$. The embedding
$\bar{G}\rightarrow\mathrm{GL}_{1}(A)$ is in the split case a minimal
representation. Hence $A$ is the Tits algebra.

A complete list of the (minimal) Tits algebras for the simple $k$-split
algebraic groups of classical type can be found for instance in [14], p.
378-379.

##  4\. AS-bundles on proper $k$-schemes

######  Definition 4.1.

A vector bundle $\mathcal{E}$ on a proper $k$-scheme $X$ is called _pure of
type_ $\mathcal{W}$ if there is an indecomposable vector bundle $\mathcal{W}$
on $X\otimes_{k}\bar{k}$ such that
$\mathcal{E}\otimes_{k}{\bar{k}}\simeq\mathcal{W}^{\oplus m}$.

Recall from [23] the following definition.

######  Definition 4.2.

Let $X$ be a $k$-scheme. A vector bundle $\mathcal{E}$ on $X$ is called
_absolutely split_ (_separably split_) if it splits after base change as a
direct sum of invertible sheaves on $X\otimes_{k}\bar{k}$ (resp.
$X\otimes_{k}k^{sep}$). For an absolutely split vector bundle we shortly write
_AS-bundle_.

######  Proposition 4.3 ([23], Proposition 4.2).

Let $X$ be a proper $k$-scheme and $\mathcal{E}$ a vector bundle on $X$. Then
$\mathcal{E}$ is absolutely split if and only if it is separably split.

######  Theorem 4.4 ([23], Theorem 4.6).

Let $X$ be a proper $k$-scheme with $H^{0}(X,\mathcal{O}_{X})=k$. Then the
closed points of the Picard scheme $\mathrm{Pic}_{X/k}$ are in one-to-one
correspondence with isomorphism classes of indecomposable $AS$-bundles on $X$.

In [23] the author classified all indecomposable $AS$-bundles on a proper
$k$-scheme $X$ with cyclic Picard group. We want to generalize this result for
the case $\mathrm{Pic}(X_{s})\simeq\mathbb{Z}^{\oplus m}$. Theorem 4.5 below
is interesting in its own right and can be used for instance to classify
indecomposable $AS$-bundles on twisted flags. In the proof of Theorem 1.1 it
is implicitely shown that twisted flags of classical type satisfy the
assumption of Theorem 4.5. Note that the classification of indecomposable
$AS$-bundles on the twisted flags under consideration is a vast generalization
of the main theorem of [6].

Let us repeat the notations and facts from the introduction. $X$ is still a
proper and geometrically integral $k$-scheme. From [23], Proposition 3.4 it
follows that $\mathrm{Pic}(X)$ is a subgroup of $\mathrm{Pic}(X_{s})$. In
particular, if $\mathrm{Pic}(X_{s})\simeq\mathbb{Z}^{\oplus m}$, then
$\mathrm{Pic}(X)\simeq r_{1}\mathbb{Z}\oplus\cdots\oplus r_{m}\mathbb{Z}$. Let
us fix a basis $\mathcal{L}_{1},...,\mathcal{L}_{m}$ of
$\mathrm{Pic}(X_{s})\simeq\mathbb{Z}^{\oplus m}$. Using a basis of
$\mathrm{Pic}(X)$ and an easy computation from linear algebra involving
matrices over the integers, one can show that there are line bundles
$\mathcal{J}_{i}\in\mathrm{Pic}(X)$ satisfying
$\mathcal{J}_{i}\otimes_{k}k^{s}\simeq\mathcal{L}_{i}^{\otimes c_{i}}$ for
some integers $c_{i}\geq 1$. Now we choose the $\mathcal{J}_{i}$ such that the
$c_{i}$ are minimal. According to [23], Proposition 3.4 these line bundles
$\mathcal{J}_{i}$ are unique up to isomorphism. Assume there are pure vector
bundles $\mathcal{M}_{i}$ of type $\mathcal{L}_{i}\in\mathrm{Pic}(X_{s})$. We
know from [23], Proposition 3.5 that the bundle $\mathcal{M}_{i}$ is unique up
to isomorphism. We set $\mathcal{M}_{\mathcal{L}_{i}}:=\mathcal{M}_{i}$. It is
easy to see that for any line bundle $\mathcal{L}_{i}^{\otimes
a}\in\mathrm{Pic}(X_{s})$ there is an indecomposable pure bundle of type
$\mathcal{L}_{i}^{\otimes a}$. Indeed, let
$s_{i}=\mathrm{rk}(\mathcal{M}_{\mathcal{L}_{i}})$ and consider
$(\mathcal{L}_{i}^{\oplus s_{i}})^{\otimes a}\simeq(\mathcal{L}_{i}^{\otimes
a})^{\oplus s_{i}^{a}}$. Then we get $\mathcal{M}_{\mathcal{L}_{i}}^{\otimes
a}\otimes_{k}k^{s}\simeq(\mathcal{L}_{i}^{\oplus s_{i}})^{\otimes
a}\simeq(\mathcal{L}_{i}^{\otimes a})^{\oplus s_{i}^{a}}$.

Considering the Krull–Schmidt decomposition of
$\mathcal{M}_{\mathcal{L}_{i}}^{\otimes a}$ and taking into account that all
indecomposable direct summands are isomorphic (see [23], proof of Proposition
3.6 and Remark 3.7), we get an, up to isomorphism, unique indecomposable
vector bundle $\mathcal{M}_{\mathcal{L}_{i}^{\otimes a}}$ such that
$\mathcal{M}_{\mathcal{L}_{i}^{\otimes
a}}\otimes_{k}k^{s}\simeq(\mathcal{L}_{i}^{\otimes a})^{\oplus s_{i}(a)}$,
where $s_{i}(a)=\mathrm{rank}(\mathcal{M}_{\mathcal{L}_{i}^{\otimes a}})$.
Using Krull–Schmidt decomposition again, we can use our indecomposable vector
bundles $\mathcal{M}_{\mathcal{L}_{i}^{\otimes a}}$ and take the tensor
product of these to get an indecomposable vector bundle
$\mathcal{M}_{(a_{1},...,a_{m})}$ of type $\mathcal{L}_{1}^{\otimes
a_{1}}\otimes\cdots\otimes\mathcal{L}_{m}^{\otimes a_{m}}$. Again, the bundles
$\mathcal{M}_{(a_{1},...,a_{m})}$ are unique up to isomorphism.

######  Theorem 4.5.

Let $X$ be a proper, geometrically integral $k$-scheme with
$\mathrm{Pic}(X_{s})\simeq\mathbb{Z}^{\oplus m}$ and let
$\mathcal{L}_{1},...,\mathcal{L}_{m}\in\mathrm{Pic}(X_{s})$ be the basis from
above. Let $\mathcal{J}_{i}\in\mathrm{Pic}(X)\simeq\mathbb{Z}^{\oplus m}$ be
up to isomorphism unique line bundles satisfying
$\mathcal{J}_{i}\otimes_{k}k^{s}\simeq\mathcal{L}_{i}^{\otimes c_{i}}$ with
$c_{i}$ being minimal. Assume there are indecomposable pure bundles
$\mathcal{M}_{\mathcal{L}_{i}}$ of type $\mathcal{L}_{i}$. Then all
indecomposable $AS$-bundles $\mathcal{E}$ are of the form

$\displaystyle\mathcal{J}_{1}^{\otimes
b_{1}}\otimes\cdots\otimes\mathcal{J}_{m}^{\otimes
b_{m}}\otimes\mathcal{M}_{(a_{1},...,a_{m})}$

with unique $b_{i}\in\mathbb{Z}$ and $0\leq a_{j}\leq c_{j}-1$.

###### Proof.

Let $\mathcal{E}$ be an arbitrary, not necessarily indecomposable, $AS$-bundle
and let $\pi:X\otimes_{k}k^{s}\rightarrow X$ the projection. By assumption,
there are indecomposable pure vector bundles $\mathcal{M}_{\mathcal{L}_{i}}$
of type $\mathcal{L}_{i}$. Above we showed that there exist (up to
isomorphism) unique indecomposable pure vector bundles of type
$\mathcal{L}_{i}^{\otimes a}$ for all $a\in\mathbb{Z}$. Let
$d=\mathrm{lcm}(\mathrm{rk}(\mathcal{M}_{(a_{1},...,a_{m})})$, $0\leq
a_{j}\leq c_{j}-1$, be the least common multiple and consider the vector
bundle $\pi^{*}(\mathcal{E}^{\oplus d})$. Since $\mathcal{E}$ is an
$AS$-bundle, the vector bundle $\mathcal{E}^{\oplus d}$ is an $AS$-bundle,
too. Therefore $\pi^{*}(\mathcal{E}^{\oplus d})$ decomposes into a direct sum
of invertible sheaves. Below we give the proof for $m=2$ to simplify the
notation. So after reordering $(\mathrm{mod}\ r_{1},\mathrm{mod}\ r_{2})$ in
lexicographical order, we find that $\pi^{*}(\mathcal{E}^{\oplus d})$ is
isomorphic to the bundle

$\displaystyle\left(\bigoplus(\mathcal{L}_{1}^{\otimes{s_{i_{0}}^{(1)}\cdot
c_{1}+0}}\otimes\mathcal{L}_{2}^{\otimes{t_{i_{0}}^{(1)}\cdot
c_{2}}+0})^{\oplus
d}\right)\oplus\left(\bigoplus(\mathcal{L}_{1}^{\otimes{s_{i_{1}}^{(1)}\cdot
c_{1}+0}}\otimes\mathcal{L}_{2}^{\otimes{t_{i_{1}}^{(1)}\cdot
c_{2}}+1})^{\oplus d}\right)\oplus\cdots$
$\displaystyle\oplus\left(\bigoplus(\mathcal{L}_{1}^{\otimes{s_{i_{(c_{2}-1)}}^{(1)}\cdot
c_{1}+0}}\otimes\mathcal{L}_{2}^{\otimes{t_{i_{(c_{2}-1)}}^{(1)}\cdot
c_{2}}+(r_{2}-1)})^{\oplus d}\right)\oplus\cdots$
$\displaystyle\oplus\left(\bigoplus(\mathcal{L}_{1}^{\otimes{s_{i_{(c_{2}-1)}}^{(c_{1}-1)}\cdot
c_{1}+(c_{1}-1)}}\otimes\mathcal{L}_{2}^{\otimes{t_{i_{(c_{2}-1)}}^{(c_{1}-1)}\cdot
c_{2}}+(c_{2}-1)})^{\oplus d}\right)$

By definition of $d$, there are $h_{(p,q)}$ such that
$h_{(p,q)}\cdot\mathrm{rk}(\mathcal{M}_{(p,q)})=d$ for $0\leq p\leq c_{1}-1$
and $0\leq q\leq c_{2}-1$. Furthermore, the sheaves $\mathcal{M}_{(p,q)}$
satisfy

$\displaystyle\pi^{*}\mathcal{M}_{(p,q)}\simeq(\mathcal{L}_{1}^{\otimes
p}\otimes\mathcal{L}_{2}^{\otimes q})^{\oplus d_{(p,q)}},$

where $d_{(p,q)}=\mathrm{rk}(\mathcal{M}_{(p,q)})$. Now for the direct
summands

$\displaystyle\left(\mathcal{L}_{1}^{\otimes{s_{i_{m}}^{(l)}\cdot
c_{1}+p}}\otimes\mathcal{L}_{2}^{\otimes{t_{i_{m}}^{(l)}\cdot
c_{2}}+q}\right)^{\oplus d}$

where $0\leq p\leq c_{1}-1$ and $0\leq q\leq c_{2}-1$, we have

$\displaystyle\left(\left(\mathcal{L}_{1}^{\otimes{s_{i_{m}}^{(l)}\cdot
c_{1}+p}}\otimes\mathcal{L}_{2}^{\otimes{t_{i_{m}}^{(l)}\cdot
c_{2}}+q}\right)^{\oplus d_{(p,q)}}\right)^{\oplus h_{(p,q)}}.$

Considering the vector bundle
$(\mathcal{J}_{1}^{\otimes{s_{i_{m}}^{(l)}}}\otimes\mathcal{J}_{2}^{\otimes{t_{i_{m}}^{(l)}}}\otimes\mathcal{M}_{(p,q)})^{\oplus
h_{(p,q)}}$ on $X$, we find

$\displaystyle\pi^{*}\left(\mathcal{J}_{1}^{\otimes{s_{i_{m}}^{(l)}}}\otimes\mathcal{J}_{2}^{\otimes{t_{i_{m}}^{(l)}}}\otimes\mathcal{M}_{(p,q)}\right)^{\oplus
h_{(p,q)}}\simeq\left(\mathcal{L}_{1}^{\otimes{s_{i_{m}}^{(l)}\cdot
c_{1}+p}}\otimes\mathcal{L}_{2}^{\otimes{t_{i_{m}}^{(l)}\cdot
c_{2}}+q}\right)^{\oplus d}$

Now consider the vector bundle

$\displaystyle\left(\bigoplus(\mathcal{J}_{1}^{\otimes
s_{i_{0}}^{(1)}}\otimes\mathcal{J}_{2}^{t_{i_{0}}^{(1)}}\otimes\mathcal{M}_{(0,0)})^{\oplus
d}\right)\oplus\left(\bigoplus(\mathcal{J}_{1}^{\otimes
s_{i_{1}}^{(1)}}\otimes\mathcal{J}_{2}^{t_{i_{1}}^{(1)}}\otimes\mathcal{M}_{(0,1)})^{\oplus
h_{(0,1)}}\right)\oplus\cdots$
$\displaystyle\oplus\left(\bigoplus(\mathcal{J}_{1}^{\otimes
s_{i_{(c_{1}-1)}}^{(1)}}\otimes\mathcal{J}_{2}^{t_{i_{(c_{1}-1)}}^{(1)}}\otimes\mathcal{M}_{(0,c_{2}-1)})^{\oplus
h_{(0,c_{2}-1)}}\right)\oplus\cdots$
$\displaystyle\oplus\left(\bigoplus(\mathcal{J}_{1}^{\otimes
s_{i_{(c_{2}-1)}}^{(c_{1}-1)}}\otimes\mathcal{J}_{2}^{t_{i_{(c_{2}-1)}}^{(c_{1}-1)}}\otimes\mathcal{M}_{(c_{1}-1,c_{2}-1)})^{\oplus
h_{(c_{1}-1,c_{2}-1)}}\right)$

which is denoted by $\mathcal{R}$. We immediately see that
$\pi^{*}\mathcal{R}\simeq\pi^{*}(\mathcal{E}^{\oplus d})$. Applying [23],
Proposition 3.4 implies that $\mathcal{E}^{\oplus d}$ is isomorphic to
$\mathcal{R}$. Because Krull–Schmidt Theorem holds for vector bundles on $X$,
we conclude that $\mathcal{E}$ is isomorphic to the direct sum of vector
bundles of the form

$\displaystyle\mathcal{J}_{1}^{\otimes b_{1}}\otimes\mathcal{J}_{2}^{\otimes
b_{2}}\otimes\mathcal{M}_{(a_{1},a_{2})}$

with unique $b_{i}\in\mathbb{Z}$ and $0\leq a_{j}\leq c_{j}-1$. Furthermore,
since all these bundles are indecomposable by definition, we finally get that
all the indecomposable $AS$-bundles have the desired form. This completes the
proof. ∎

######  Lemma 4.6.

Let $X$ be a proper and geometrically integral $k$-scheme and
$\mathcal{L},\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ line bundles.

  * (i)

If $\mathcal{M}$ is pure of type $\mathcal{L}$, then
$\mathrm{End}(\mathcal{M})$ is a central simple $k$-algebra.

  * (ii)

There is an (up to isomorphism) unique indecomposable pure vector bundle
$\mathcal{M}_{\mathcal{L}}$ of type $\mathcal{L}$.

  * (iii)

Let $\mathcal{M}_{\mathcal{L}_{1}}$ and $\mathcal{M}_{\mathcal{L}_{2}}$ be
pure vector bundles of type $\mathcal{L}_{1}$ resp. $\mathcal{L}_{2}$. Then
$\mathrm{End}(\mathcal{M}_{\mathcal{L}_{1}})\otimes\mathrm{End}(\mathcal{M}_{\mathcal{L}_{2}})$
is Brauer-equivalent to
$\mathrm{End}(\mathcal{M}_{\mathcal{L}_{1}\otimes\mathcal{L}_{2}})$.

###### Proof.

Since $X$ is geometrically integral, we have
$H^{0}(X_{s},\mathcal{O}_{X_{s}})\simeq k^{s}$ (see [23], Proposition 4.2).
This implies $\mathcal{M}\otimes_{k}k^{s}\simeq\mathcal{L}^{\oplus r}$ and
therefore

$\displaystyle\mathrm{End}(\mathcal{M})\otimes_{k}k^{s}\simeq\mathrm{Mat}_{r}(k^{s}).$

Then [14], Theorem (1.1) shows that $\mathrm{End}(\mathcal{M})$ must be
central simple over $k$. This shows (i). Assertion (ii) follows directly from
[13], Lemma 8. It remains to show (iii). For this, let $\Delta\colon
X\rightarrow X\times X$ be the diagonal and $\pi_{i}\colon X\times
X\rightarrow X$, $i=1,2$, the two projections. For
$\mathcal{L}:=\pi_{1}^{*}\mathcal{L}_{1}\otimes\pi_{2}^{*}\mathcal{L}_{2}$ one
has $\Delta^{*}\mathcal{L}\simeq\mathcal{L}_{1}\otimes\mathcal{L}_{2}$. From
[13], Corollary 11 it follows
$\mathcal{M}_{\Delta^{*}\mathcal{L}}\simeq\Delta^{*}\mathcal{M}_{\mathcal{L}}$.
Moreover, the proof of Corollary 11 in _loc.cite_ shows
$\mathrm{End}(\mathcal{M}_{\mathcal{L}})\simeq\mathrm{End}(\Delta^{*}\mathcal{M}_{\mathcal{L}})\simeq\mathrm{End}(\mathcal{M}_{\mathcal{L}_{1}\otimes\mathcal{L}_{2}})$.
Now [13], p.14 explains that
$\mathrm{End}(\mathcal{L}_{1})\otimes\mathrm{End}(\mathcal{L}_{2})$ is Brauer-
equivalent to
$\mathrm{End}(\mathcal{M}_{\mathcal{L}_{1}\otimes\mathcal{L}_{2}})$. This
completes the proof. ∎

######  Proposition 4.7.

Let $\mathcal{E}=\mathcal{J}_{1}^{\otimes
b_{1}}\otimes\cdots\otimes\mathcal{J}_{m}^{\otimes
b_{m}}\otimes\mathcal{M}_{(a_{1},...,a_{m})}$ be an indecomposable $AS$-bundle
as in Theorem 4.5. Then $\mathrm{End}(\mathcal{E})$ is a central simple
$k$-algebra.

###### Proof.

Since
$\mathrm{End}(\mathcal{E})\simeq\mathrm{End}(\mathcal{M}_{(a_{1},...,a_{m})})$,
the assertion follows from Lemma 4.6 (i). ∎

######  Proposition 4.8.

Let $\mathcal{E}=\mathcal{J}_{1}^{\otimes
b_{1}}\otimes\cdots\otimes\mathcal{J}_{m}^{\otimes
b_{m}}\otimes\mathcal{M}_{(a_{1},...,a_{m})}$ and
$\mathcal{E}^{\prime}=\mathcal{J}_{1}^{\otimes
b^{\prime}_{1}}\otimes\cdots\otimes\mathcal{J}_{m}^{\otimes
b^{\prime}_{m}}\otimes\mathcal{M}_{(a^{\prime}_{1},...,a^{\prime}_{m})}$ be
two indecopmosable $AS$-bundles as in Theorem 4.5. Then there is a unique
positive integer $s$ such that

$\displaystyle(\mathcal{E}\otimes\mathcal{E}^{\prime})\simeq\mathcal{J}_{1}^{\otimes(b_{1}+b_{1}^{\prime})}\otimes\cdots\otimes\mathcal{J}_{m}^{\otimes(b_{m}+b_{m}^{\prime})}\otimes\mathcal{M}_{((a_{1}+a_{1}^{\prime}),...,(a_{m}+a_{m}^{\prime}))}^{\oplus
s}.$

Moreover, $\mathrm{End}(\mathcal{E}\otimes\mathcal{E}^{\prime})$ is Brauer-
equivalent to
$\mathrm{End}(\mathcal{E})\otimes\mathrm{End}(\mathcal{E}^{\prime})$ in
$\mathrm{Br}(k)$.

###### Proof.

Note that the $AS$-bundle
$\mathcal{M}_{(a_{1},...,a_{m})}\otimes\mathcal{M}_{(a^{\prime}_{1},...,a^{\prime}_{m})}$
is a pure vector bundle of type
$\mathcal{L}_{1}^{\otimes(a_{1}+a_{1}^{\prime})}\otimes\cdots\otimes\mathcal{L}_{m}^{\otimes(a_{m}+a_{m}^{\prime})}$.
The first assertion follows from Lemma 4.6 (ii). Now we want to prove that
$\mathrm{End}(\mathcal{E}\otimes\mathcal{E}^{\prime})$ is Brauer-equivalent to
$\mathrm{End}(\mathcal{E})\otimes\mathrm{End}(\mathcal{E}^{\prime})$. As
mentioned in the proof of Proposition 4.7, we have
$\mathrm{End}(\mathcal{E})\simeq\mathrm{End}(\mathcal{M}_{(a_{1},...,a_{m})})$
and
$\mathrm{End}(\mathcal{E}^{\prime})\simeq\mathrm{End}(\mathcal{M}_{(a^{\prime}_{1},...,a^{\prime}_{m})})$.
We conclude with Lemma 4.6 (iii). ∎

##  5\. Proof of Theorem 1.1

###### Proof.

According to Theorem 4.4, the closed points of $\mathrm{Pic}_{(X/k)(et)}$ are
in one-to-one correspondence with isomorphism classes of indecomposable
$AS$-bundles. Since $X$ is proper over $k$, we have
$\mathrm{Pi}_{(X/k)(\mathrm{et})}\simeq\mathrm{Pic}_{(X/k)(\mathrm{fppf})}$ as
abelian groups. And because $\mathrm{Pic}(X_{s})\simeq\mathbb{Z}^{\oplus m}$,
we can use Theorem 4.5 to obtain a classification of all indecomposable
$AS$-bundles on $X$. In particular, the $k$-rational points of
$\mathrm{Pic}_{(X/k)(et)}$ correspond to indecomposable $AS$-bundles. As
mentioned in the introduction, being pure vector bundle of type
$\mathcal{L}\in\mathrm{Pic}(X_{s})$ is equvalent to
$\mathcal{L}\in\mathrm{Pic}_{\Gamma}(X_{s})$. By assumption, there are pure
vector bundles of type $\mathcal{L}_{i}$. This actually implies that the basis
$\mathcal{L}_{1},...,\mathcal{L}_{m}$ of $\mathrm{Pic}(X_{s})$ consists of
Galois invariant line bundles. Therefore, the indecomposable $AS$-bundles are
in one-to-one correspondence with the $k$-rational points of
$\mathrm{Pic}_{(X/k)(et)}$. Now consider the exact sequence from the
introduction and specialize it to the case $T=S=\mathrm{Spec}(k)$. We get the
following exact sequence

$\displaystyle
0\longrightarrow\mathrm{Pic}(X)\longrightarrow\mathrm{Pic}_{(X/S)(\mathrm{fppf})}(k)\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}\mathrm{Br}(k)\longrightarrow\mathrm{Br}^{\prime}(X)$

where $\delta(\mathcal{E})=[\mathrm{End}(\mathcal{E})]\in\mathrm{Br}(k)$ for
an indecomposable $AS$-bundle $\mathcal{E}$. Finally, use Theorem 4.5, Lemma
4.6, Lemma 4.8, and [27], Lemma 3.4 to conclude that $\mathrm{Am}(X)$ is
indeed generated by the set $D$ which consists of the classes
$\mathrm{End}(\mathcal{M}_{e_{i}})$, where $i\in\\{1,...m\\}$ satisfying
$c_{i}\geq 2$. The above arguments also show
$\\#D\leq\mathrm{rank}(\mathrm{Pic}(X))$. It remains to show that if $X$ is
geometrically rational, one has
$\mathrm{Br}(X)\simeq\mathrm{Br}(k)/\mathrm{Am}(X)$. For this, we consider the
Hochschild–Serre spectral sequence
$H^{p}(k,H^{q}(X_{s},\mathbb{G}_{m}))\Rightarrow H^{p+q}(X,\mathbb{G}_{m})$.
Since $X$ is geometrically rational, we have $\mathrm{Br}(X_{s})=0$. The
spectral sequence then yields

$\displaystyle\mathrm{Pic}_{\Gamma}(X_{s})\longrightarrow\mathrm{Br}(k)\longrightarrow\mathrm{Br}(X)\longrightarrow
H^{1}(k,\mathrm{Pic}(X_{s}))$

where $\Gamma$ denotes the absolute Galois group. By assumption, there are
pure vector bundles of type $\mathcal{L}_{i}$. This actually implies that the
basis $\mathcal{L}_{1},...,\mathcal{L}_{m}$ of $\mathrm{Pic}(X_{s})$ consists
of Galois invariant line bundles. Therefore $\mathrm{Pic}(X_{s})$ is a
permutation $\Gamma$-module and hence $H^{1}(k,\mathrm{Pic}(X_{s}))=0$. The
above exact sequence yields
$\mathrm{Br}(X)\simeq\mathrm{Br}(k)/\mathrm{Am}(X)$. This completes the proof.
∎

(proof of Corollary 1.2):
Recall that for any flag $G/P$ of classical type associated to some semisimple
simply connected and $k$-split $G$, one has
$\mathrm{Pic}(G/P)\simeq\mathbb{Z}^{\oplus m}$. Now, let $G$ be an semisimple
simply connected algebraic group over $k$. Consider a twisted flag
$X:={{}_{\gamma}}(G/P)$. Note that after base change to a separable closure
$k^{s}$ we have $X_{s}\simeq G_{s}/P_{s}$. Denote by
$\mathcal{A}_{1},...,\mathcal{A}_{m}$ the basis of $\mathrm{Pic}(X_{s})$ given
in [20], p.55f. From [20], p.37 we conclude that
$\mathrm{End}(\mathcal{B}_{\mathcal{A}_{i}})$ is a Tits algebra. Using these
$\mathcal{A}_{i}$, we can construct a basis
$\mathcal{L}_{1},...,\mathcal{L}_{m}$ of
$\mathrm{Pic}(X_{s})\simeq\mathbb{Z}^{\oplus m}$ consisting of Galois-
invariant line bundles. Let $\mathcal{J}_{1},...\mathcal{J}_{m}$ be the line
bundles in $\mathrm{Pic}(X)$ satisfying
$\mathcal{J}_{j}\otimes_{k}k^{s}\simeq\mathcal{L}_{j}^{\otimes c_{j}}$ with
minimal $c_{i}$. Notice that $H^{0}(X_{s},\mathcal{O}_{X_{s}})\simeq k^{s}$.
According to Theorem 4.4, the closed points of $\mathrm{Pic}_{(X/k)(et)}$ are
in one-to-one correspondence with isomorphism classes of indecomposable
$AS$-bundles. Since $X$ is proper over $k$, we have
$\mathrm{Pic}_{(X/k)(\mathrm{et})}\simeq\mathrm{Pic}_{(X/k)(\mathrm{fppf})}$
as abelian groups. And because
$\mathrm{Pic}(G_{s}/P_{s})\simeq\mathbb{Z}^{\oplus m}$ and since we have a
basis $\mathcal{L}_{1},...,\mathcal{L}_{m}$ of $\mathrm{Pic}(X_{s})$
consisting of Galois-invariant line bundles, it follows from [23], Proposition
4.3 that there are (unique) pure indecomposable vector bundles
$\mathcal{W}_{\mathcal{L}_{i}}$ of type $\mathcal{L}_{j}$. Moreover, we have
that the indecomposable $AS$-bundles are in one-to-one correspondence with the
$k$-rational points of $\mathrm{Pic}_{(X/k)(et)}$. Now we can use Theorem 4.5
from above to obtain a classification of all indecomposable $AS$-bundles on
the twisted flags of classical type. By construction, and by applying Lemma
4.6 (iii) we conclude that there exist integers $a_{1},...,a_{m}$ such that
$\mathrm{End}(\mathcal{W}_{\mathcal{L}_{i}})$ is Brauer equivalent to
$\mathrm{End}(\mathcal{B}_{\mathcal{A}_{1}})^{\otimes
a_{1}}\otimes\cdots\otimes\mathrm{End}(\mathcal{B}_{\mathcal{A}_{m}})^{\otimes
a_{m}}$.

Now consider the exact sequence from the introduction and specialize it to the
case $T=S=\mathrm{Spec}(k)$. We get the following exact sequence

$\displaystyle
0\longrightarrow\mathrm{Pic}(X)\longrightarrow\mathrm{Pic}_{(X/S)(\mathrm{fppf})}(k)\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}\mathrm{Br}(k)\longrightarrow\mathrm{Br}^{\prime}(X)$

where $\delta(\mathcal{E})=[\mathrm{End}(\mathcal{E})]\in\mathrm{Br}(k)$ for
an indecomposable $AS$-bundle $\mathcal{E}$. Finally, use Theorem 4.5, Lemma
4.6, Lemma 4.8 and [27], Lemma 3.4 to conclude that $\mathrm{Am}(X)$ is indeed
generated by the subset $D$ which consists of classes of Tits algebras coming
from a basis of $\mathrm{Ch}(P_{s})^{\Gamma}$ which is given by some of the
$\mathcal{L}_{i}$. For these $\mathcal{L}_{i}$ one has $c_{i}\geq 2$. The
above arguments also show $\\#D\leq\mathrm{rank}(\mathrm{Pic}(X_{s}))$. This
completes the proof.

(proof of Corollary 1.3)

We sketch the proof for $X=X_{1}\times X_{2}$. Let
$X_{1}={{}_{\gamma_{1}}}(G_{1}/P_{1})$ and
$X_{2}={{}_{\gamma_{2}}}(G_{2}/P_{2})$ and notice that
$\mathrm{Pic}(X_{s})\simeq\mathrm{Pic}((X_{1})_{s})\times\mathrm{Pic}((X_{2})_{s})$
since $(X_{i})_{s}$ are rational over $k^{s}$. Therefore
$\mathrm{Pic}(X_{s})\simeq\mathbb{Z}^{\oplus m_{1}}\oplus\mathbb{Z}^{\oplus
m_{2}}$. This implies $\mathrm{Pic}(X)\simeq\mathbb{Z}^{\oplus m_{1}+m_{2}}$.
Now let $\mathcal{L}_{1},...,\mathcal{L}_{m_{1}}$ be the generators of
$\mathrm{Pic}((X_{1})_{s})$ and $\mathcal{K}_{1},...,\mathcal{K}_{m_{2}}$ the
generators of $\mathrm{Pic}((X_{2})_{s})$. Now proceed as in the proof of
Theorem 1.1 and use [19], Corollary 2.3 to obtain the desired assertion.

######  Remark 5.1.

We wonder whether the Amitsur subgroup is a complete birational invariant for
twisted flags of classical type. In the special case of Brauer–Severi
varieties, this problem is known as the _Amitsur conjecture for central simple
algebras_ [1]. This conjecture is still open in general. For details and
results in this direction we refer to [10] and [15] and references therein.

######  Example 5.2.

Let $X$ be a Brauer–Severi variety corresponding to the central simple algebra
$A$. Then $X_{s}\simeq\mathbb{P}^{n}$ and $\mathrm{Pic}(X_{s})=\mathbb{Z}$ is
generated by the ample line bundle $\mathcal{O}_{\mathbb{P}^{n}}(1)$. In
Section 4 we showed that there is a (up to isomorphism) unique $AS$-bundle
$\mathcal{M}_{1}$ of type $\mathcal{O}_{\mathbb{P}^{n}}(1)$. The proof of
Theorem 1.1 actually shows that $[\mathrm{End}(\mathcal{M}_{1})]$ generates
$\mathrm{Am}(X)$. It is well known that
$[\mathrm{End}(\mathcal{M}_{1})]=[A]^{-1}$ (see for instance [27], p.571).
Therefore, $\mathrm{Am}(X)=\langle[A]\rangle$. This gives back the classical
result due to Châtelet which is mentioned in the introduction.

######  Example 5.3.

Let $X=\mathrm{BS}(d,A)$ be a generalized Brauer–Severi corresponding to a
central simple algebra $A$ of degree $n$. We have
$X_{s}\simeq\mathrm{Grass}(d,n)$ and $\mathrm{Pic}(X_{s})=\mathbb{Z}$ is
generated by the ample line bundle $\mathcal{O}(1)=\mathrm{det}(\mathcal{Q})$
where $\mathcal{Q}$ is the universal quotient bundle on $\mathrm{Grass}(d,n)$.
As explained in [23], p.16 there is a (up to isomorphism) unique $AS$-bundle
$\mathcal{N}$ of type $\mathcal{O}(1)$. Moreover, it is well known that
$\mathrm{End}(\mathcal{N})$ is Brauer-equivalent to $A^{\otimes-d}$ (see for
instance [27], p.572). Hence $\mathrm{Am}(X)=\langle[A^{\otimes d}]\rangle$.
This gives back [7], Theorem 7.

##  6\. Application to noncommutative motives

(proof of Theorem 1.4 and Corollary 1.6)

If $X$ and $Y$ are birational, we conclude from [18], Proposition 2.10 that
$\mathrm{Am}(X)=\mathrm{Am}(Y)$ in $\mathrm{Br}(k)$. According to Theorem 1.1,
both Amitsur subgroups $\mathrm{Am}(X)$ and $\mathrm{Am}(Y)$ are generated by
certain Tits algebras of the algebraic groups involved. Since the Amitsur
subgroup is a finitely generated torsion abelian subgroup of $\mathrm{Br}(k)$,
we conclude with the fundamental theorem of finitely generated abelian groups
that

$\displaystyle\mathrm{Am}(X)\simeq\mathbb{Z}/p_{1}^{r_{1}}\mathbb{Z}\times\cdots\times\mathbb{Z}/p_{s}^{r_{s}}\mathbb{Z},\
\mathrm{Am}(Y)\simeq\mathbb{Z}/q_{1}^{v_{1}}\mathbb{Z}\times\cdots\times\mathbb{Z}/q_{t}^{v_{t}}\mathbb{Z}$

with uniquely determined $p_{i}^{r_{i}}$ and $q_{j}^{v_{j}}$ where $p_{i}$ and
$q_{j}$ are prime numbers. Since $\mathrm{Am}(X)=\mathrm{Am}(Y)$, we have
$s=t$ and isomorphic factors up to permutation. Without loss of generality we
assume $p_{i}=q_{i}$ and therefore $r_{i}=v_{i}$. Let $a_{i}$ be a generator
of $\mathbb{Z}/p_{i}^{r_{i}}\mathbb{Z}$ and $b_{i}$ a generator of
$\mathbb{Z}/q_{i}^{v_{i}}\mathbb{Z}$. Denote by $e_{i}=(0,...,a_{i},0,...,0)$
$i=1,...,s$ a set of generators of
$\mathbb{Z}/p_{1}^{r_{1}}\mathbb{Z}\times\cdots\times\mathbb{Z}/p_{s}^{r_{s}}\mathbb{Z}$
and by $f_{i}=(0,...,b_{i},0,...,0)$ $i=1,...,t$ a set of generators of
$\mathbb{Z}/q_{1}^{v_{1}}\mathbb{Z}\times\cdots\times\mathbb{Z}/q_{t}^{v_{t}}\mathbb{Z}$.
The corresponding central simple algebras are denoted by $A_{e_{i}}$ and
$B_{f_{i}}$ respectively. By definition, we have
$[A_{e_{i}}]=[B_{f_{i}}^{\otimes n_{i}}]$ for a unique positive integer
$n_{i}$. Note that from [30], (2.18) it follows

$\displaystyle\bigoplus_{i=1}^{s}U(A_{e_{i}})\simeq\bigoplus_{i=1}^{t}U(B_{f_{i}}^{\otimes
n_{i}}).$

Now let $A_{g}$ be the central simple division algebra corresponding to
$g\in\mathrm{Am}(X)$. Analogously, let $B_{h}$ be the central simple division
algebra corresponding to $h\in\mathrm{Am}(Y)$. Now [30], Theorem Theorem 2.19
implies

$\displaystyle M_{X}:=\bigoplus_{g\in\mathrm{Am}(X)}U(A_{g})\
\simeq\bigoplus_{h\in\mathrm{Am}(Y)}U(B_{h})=:M_{Y}.$

Claim: Let $G$, $P$ and $\gamma$ be as in Corollary 1.2 and let
$\rho_{1},...,\rho_{n}$ be a $\mathrm{Ch}$-homogeneous basis of $R(P)$ over
$R(G)$, where $R(P)$ and $R(G)$ denote the corresponding representation rings.
Let $A_{\chi(i),\gamma}$ be the Tits central simple algebras associated to
$\rho_{i}$ and let $\mathrm{Ti}_{\rho_{1},...,\rho_{n}}(X):=\langle
A_{\chi(1),\gamma},...,A_{\chi(n),\gamma}\rangle$ be the subgroup generated by
these Tits algebras. Denote by

$\displaystyle
M\mathrm{Ti}_{\rho_{1},...,\rho_{n}}(X):=\bigoplus_{f\in\mathrm{Ti}_{\rho_{1},...,\rho_{n}}(X)}U(A_{f}),$

where $A_{f}$ are the central simple algebras corresponding to $f$. Then
$U(X)\oplus N^{\prime}=M\mathrm{Ti}_{\rho_{1},...,\rho_{n}}(X)$ with
$N^{\prime}\in\mathrm{CSA}(k)^{\oplus}$.

###### Proof.

By [28], Theorem 2.1 we conclude that
$U(X)=U(A_{\chi(1),\gamma})\oplus\cdots\oplus U(A_{\chi(n),\gamma})$.
Obviously, $U(X)$ is a direct summand of
$M\mathrm{Ti}_{\rho_{1},...,\rho_{n}}(X)$ and
$N^{\prime}\in\mathrm{CSA}(k)^{\oplus}$ by construction. ∎

Note that $\mathrm{Am}(X)$ is a subgroup of
$\mathrm{Ti}_{\rho_{1},...,\rho_{n}}(X)$. It is clear that there exists a
$N\in\mathrm{CSA}(k)^{\oplus}$ such that $M_{X}\oplus
N=M\mathrm{Ti}_{\rho_{1},...,\rho_{n}}(X)$. Now use the claim to conclude that
$U(X)$ is a direct summand of $M\mathrm{Ti}_{\rho_{1},...,\rho_{n}}(X)$. Hence
$M_{X}\oplus N=U(X)\oplus N^{\prime}$ with
$N^{\prime}\in\mathrm{CSA}(k)^{\oplus}$. By the same argument we obtain
$M_{Y}\oplus Q=U(Y)\oplus Q^{\prime}$. This completes the proof of Theorem 1.4
and Corollary 1.6.

######  Example 6.1.

Let $X$ and $Y$ be Brauer–Severi varieties corresponding to central simple
algebras $A$ and $B$. Then $\mathrm{Am}(X)=\langle A\rangle$ and
$\mathrm{Am}(Y)=\langle B\rangle$. Now if $X$ and $Y$ are birational, then
$\langle A\rangle=\langle B\rangle$ according to a theorem of Amitsur [1].
Since $\mathrm{Am}(X)$ and $\mathrm{Am}(Y)$ are cyclic of order
$\mathrm{per}(A)=\mathrm{per}(B):=m$, we conclude from [30], Theorem 3.20

$\displaystyle M_{X}=U(k)\oplus U(A)\oplus\cdots\oplus U(A^{\otimes
m-1})\simeq U(k)\oplus U(B)\oplus\cdots\oplus U(B^{\otimes m-1})=M_{Y}$

Since $m\cdot r=\mathrm{deg}(A)=\mathrm{deg}(B)$, we can use [30], Theorem
2.19 to conclude that $M_{X}^{\oplus r}\simeq U(X)$ and $M_{Y}^{\oplus
r}\simeq U(Y)$. So in the case of Brauer–Severi varieties we have
$N=M_{X}^{\oplus(r-1)}$, $N^{\prime}=0$, $Q=M_{Y}^{\oplus(r-1)}$ and
$Q^{\prime}=0$.

######  Example 6.2.

Let $X=\mathrm{BS}(d,A)$ and $Y=\mathrm{BS}(d,B)$ be generalized Brauer–Severi
varieties corresponding to central simple algebras $A$ and $B$. Then
$\mathrm{Am}(X)=\langle A^{\otimes d}\rangle$ and $\mathrm{Am}(Y)=\langle
B^{\otimes d}\rangle$ (see [7], Theorem 7). If $X$ is birational to $Y$, then
$\mathrm{Am}(X)=\mathrm{Am}(Y)$. Notice that $\mathrm{Am}(X)$ and
$\mathrm{Am}(Y)$ are cyclic of order
$m=\mathrm{per}(A)/\mathrm{gcd}(d,\mathrm{per}(A))$. According to [30],
Theorem 3.20 one has $M_{X}\simeq M_{Y}$, where

$\displaystyle M_{X}=U(k)\oplus U(A^{\otimes d})\oplus\cdots\oplus
U(A^{\otimes dm-d}),$ $\displaystyle M_{Y}=U(k)\oplus U(B^{\otimes
d})\oplus\cdots\oplus U(B^{\otimes dm-d}).$

One can use [30], Theorem 2.19 and 3.18 to conclude that $M_{X}$ is a direct
summand of $U(X)$ and $M_{Y}$ a direct summand of $U(Y)$. Again, we have
$N^{\prime}=0=Q^{\prime}$.

In particular, for birational (generalized) Brauer–Severi varieties $X$ and
$Y$ one has $U(X)\oplus Q\simeq U(Y)\oplus N$. Not that if $X$ and $Y$ are
Brauer–Severi, then $N\simeq Q$ and [31], Proposition 4.5 implies $U(X)\simeq
U(Y)$. In this way we get back [30], Proposition 3.15. Using the theory of
semiorthogonal decompositions one can try to generalize Corollary 1.6 to
arbitrary proper and geometrically integral $k$-schemes. We recall the
definition of exceptional object and semiorthogonal decomposition.

Let $\mathcal{D}$ be a triangulated category and $\mathcal{C}$ a triangulated
subcategory. The subcategory $\mathcal{C}$ is called _thick_ if it is closed
under isomorphisms and direct summands. For a subset $A$ of objects of
$\mathcal{D}$ we denote by $\langle A\rangle$ the smallest full thick
subcategory of $\mathcal{D}$ containing the elements of $A$. Furthermore, we
define $A^{\perp}$ to be the subcategory of $\mathcal{D}$ consisting of all
objects $M$ such that $\mathrm{Hom}_{\mathcal{D}}(E[i],M)=0$ for all
$i\in\mathbb{Z}$ and all elements $E$ of $A$. We say that $A$ _generates_
$\mathcal{D}$ if $A^{\perp}=0$. Now assume $\mathcal{D}$ admits arbitrary
direct sums. An object $B$ is called _compact_ if
$\mathrm{Hom}_{\mathcal{D}}(B,-)$ commutes with direct sums. Denoting by
$\mathcal{D}^{c}$ the subcategory of compact objects we say that $\mathcal{D}$
is _compactly generated_ if the objects of $\mathcal{D}^{c}$ generate
$\mathcal{D}$. One has the following important theorem (see [9], Theorem
2.1.2).

######  Theorem 6.3.

Let $\mathcal{D}$ be a compactly generated triangulated category. Then a set
of objects $A\subset\mathcal{D}^{c}$ generates $\mathcal{D}$ if and only if
$\langle A\rangle=\mathcal{D}^{c}$.

For a smooth projective scheme $X$ over $k$, we denote by
$D(\mathrm{Qcoh}(X))$ the derived category of quasicoherent sheaves on $X$.
The bounded derived category of coherent sheaves is denoted by $D^{b}(X)$.
Note that $D(\mathrm{Qcoh}(X))$ is compactly generated with compact objects
being all of $D^{b}(X)$. For details on generating see [9].

######  Definition 6.4.

Let $A$ be a division algebra over $k$, not necessarily central. An object
$\mathcal{E}\in D^{b}(X)$ is called _w-exceptional_ if
$\mathrm{End}(\mathcal{E})=A$ and $\mathrm{Hom}(\mathcal{E},\mathcal{E}[r])=0$
for $r\neq 0$. If $A=k$ the object is called _exceptional_. If $A$ is a
separable $k$-algebra, the object $\mathcal{E}$ is called _separable-
exceptional_.

######  Definition 6.5.

A totally ordered set $\\{\mathcal{E}_{1},...,\mathcal{E}_{n}\\}$ of
w-exceptional (resp. separable-exceptional) objects on $X$ is called an _w-
exceptional collection_ (resp. _separable-exceptional collection_) if
$\mathrm{Hom}(\mathcal{E}_{i},\mathcal{E}_{j}[r])=0$ for all integers $r$
whenever $i>j$. An w-exceptional (resp. separable-exceptional) collection is
_full_ if $\langle\\{\mathcal{E}_{1},...,\mathcal{E}_{n}\\}\rangle=D^{b}(X)$
and _strong_ if $\mathrm{Hom}(\mathcal{E}_{i},\mathcal{E}_{j}[r])=0$ whenever
$r\neq 0$. If the set $\\{\mathcal{E}_{1},...,\mathcal{E}_{n}\\}$ consists of
exceptional objects it is called _exceptional collection_.

######  Example 6.6.

Let $\mathbb{P}^{n}$ be the projective space and consider the ordered
collection of invertible sheaves
$\\{\mathcal{O}_{\mathbb{P}^{n}},\mathcal{O}_{\mathbb{P}^{n}}(1),...,\mathcal{O}_{\mathbb{P}^{n}}(n)\\}$.
In [5] Beilinson showed that this is a full strong exceptional collection.

A generalization of the notion of a full w-exceptional collection is that of a
semiorthogonal decomposition of $D^{b}(X)$. Recall that a full triangulated
subcategory $\mathcal{D}$ of $D^{b}(X)$ is called _admissible_ if the
inclusion $\mathcal{D}\hookrightarrow D^{b}(X)$ has a left and right adjoint
functor.

######  Definition 6.7.

Let $X$ be a smooth projective variety over $k$. A sequence
$\mathcal{D}_{1},...,\mathcal{D}_{n}$ of full triangulated subcategories of
$D^{b}(X)$ is called _semiorthogonal_ if all $\mathcal{D}_{i}\subset D^{b}(X)$
are admissible and
$\mathcal{D}_{j}\subset\mathcal{D}_{i}^{\perp}=\\{\mathcal{F}\in
D^{b}(X)\mid\mathrm{Hom}(\mathcal{G},\mathcal{F})=0$, $\forall$
$\mathcal{G}\in\mathcal{D}_{i}\\}$ for $i>j$. Such a sequence defines a
_semiorthogonal decomposition_ of $D^{b}(X)$ if the smallest full thick
subcategory containing all $\mathcal{D}_{i}$ equals $D^{b}(X)$.

For a semiorthogonal decomposition we write
$D^{b}(X)=\langle\mathcal{D}_{1},...,\mathcal{D}_{n}\rangle$.

######  Example 6.8.

Let $\mathcal{E}_{1},...,\mathcal{E}_{n}$ be a full w-exceptional collection
on $X$. It is easy to verify that by setting
$\mathcal{D}_{i}=\langle\mathcal{E}_{i}\rangle$ one gets a semiorthogonal
decomposition $D^{b}(X)=\langle\mathcal{D}_{1},...,\mathcal{D}_{n}\rangle$.

The noncommutative motives $M_{T(X)}$ and $M_{T(Y)}$ are defined in the
introduction.

######  Theorem (Theorem 1.7).

Let $X$ and $Y$ be schemes of pure weak exceptional type. If $X$ and $Y$ are
birational, then there are direct summands
$N,N^{\prime}\in\mathrm{CSA}^{\oplus}$ of $M_{T(X)}$ and
$Q,Q^{\prime}\in\mathrm{CSA}^{\oplus}$ of $M_{T(Y)}$ such that $U(X)\oplus
N^{\prime}\oplus Q\simeq U(Y)\oplus Q^{\prime}\oplus N$.

###### Proof.

The semiorthogonal decompositions

$\displaystyle D^{b}(X)=\langle\mathcal{E}_{1},...,\mathcal{E}_{m}\rangle$

and

$\displaystyle D^{b}(Y)=\langle\mathcal{F}_{1},...,\mathcal{F}_{n}\rangle$

imply $\mathrm{Pic}(X)\simeq\mathbb{Z}^{\oplus m}$ and
$\mathrm{Pic}(Y)\simeq\mathbb{Z}^{\oplus n}$. Since $\mathcal{E}_{i}$ is pure
of type $\mathcal{K}_{i}$, we conclude that
$\langle\mathcal{K}_{1},...,\mathcal{K}_{m}\rangle$ is a semiorthogonal
decomposition of $D^{b}(X_{s})$ which is induced from a full exceptional
collection. Therefore $\mathrm{Pic}(X_{s})\simeq\mathbb{Z}^{\oplus m}$. In the
same way it follows $\mathrm{Pic}(Y_{s})\simeq\mathbb{Z}^{\oplus n}$. Some of
the $\mathcal{K}_{i}$ form a basis of $\mathrm{Pic}(X_{s})$. Without loss of
generality, let $\mathcal{K}_{1},...,\mathcal{K}_{r}$ be a basis. Then, by
assumption, there are pure vector bundles
$\mathcal{E}_{1},...,\mathcal{E}_{r}$ of type
$\mathcal{K}_{1},...,\mathcal{K}_{r}$. Therefore, the assumptions of Theorem
4.5 are fulfilled. Let $D_{X}$ (resp. $D_{Y}$) denote the set of
indecopmposable $AS$-bundles on $X$ (resp. $Y$). Since $X$ satisfies the
assumptions of Theorem 4.5, the proof of Theorem 1.1 shows that for any
indecomposable $AS$-bundle $\mathcal{E}$ on $X$ one has
$[\mathrm{End}(\mathcal{E})]\in\mathrm{Br}(k)$. From Theorem 4.5 and
Proposition 4.7 we conclude that there are only finitely many Brauer-classes
$[\mathrm{End}(\mathcal{E})]\in\mathrm{Br}(k)$ of indecomposable $AS$-bundles.
Now let $C_{X}\subset\mathrm{Br}(k)$ be the subgroup generated by these
finitely many Brauer-classes. Analogously, we define
$C_{Y}\subset\mathrm{Br}(k)$. Denote by $A(g)$ the central simple division
algebra corresponding to $g\in C_{X}$ and by $A(h)$ the central simple
division algebra corresponding to $h\in C_{Y}$. As in the introduction, we set

$\displaystyle M_{T(X)}:=\bigoplus_{g\in
C_{X}}U(A(g))\quad\textnormal{and}\quad M_{T(Y)}:=\bigoplus_{h\in
C_{Y}}U(A(h)).$

Furthermore, we put

$\displaystyle
M_{X}:=\bigoplus_{p\in\mathrm{Am}(X)}U(A_{p})\quad\textnormal{and}\quad
M_{Y}:=\bigoplus_{q\in\mathrm{Am}(Y)}U(A_{q})$

where $A_{p}$ is the central simple division algebra corresponding to $p$ and
$A_{q}$ the central simple division algebra corresponding to $q$. Consider the
simiorthogonal decompositions

$\displaystyle D^{b}(X)=\langle\mathcal{E}_{1},...,\mathcal{E}_{m}\rangle$

and

$\displaystyle D^{b}(Y)=\langle\mathcal{F}_{1},...,\mathcal{F}_{n}\rangle.$

By assumption, the vector bundles $\mathcal{E}_{1},...,\mathcal{E}_{m}$ and
$\mathcal{F}_{1},...,\mathcal{F}_{n}$ are pure having endomorphism algebras
being isomorphic to $k^{s}$. This implies that $\mathrm{End}(\mathcal{E}_{i})$
and $\mathrm{End}(\mathcal{F}_{j})$ are central simple algebras (see proof of
[23], Proposition 3.3). From the construction of noncommutative motives we
have

$\displaystyle
U(X):=\bigoplus^{m}_{i=1}U(\mathrm{End}(\mathcal{E}_{i}))\quad\textnormal{and}\quad
U(Y):=\bigoplus^{n}_{j=1}U(\mathrm{End}(\mathcal{F}_{j})).$

Obviously, $U(X)$ is a direct summand of $M_{T(X)}$ and therefore there exists
$N^{\prime}\in\mathrm{CSA}(k)^{\oplus}$ such that $U(X)\oplus N^{\prime}\simeq
M_{T(X)}$. In the same way one shows that there is a
$Q^{\prime}\in\mathrm{CSA}(k)^{\oplus}$ such that $U(Y)\oplus Q^{\prime}\simeq
M_{T(Y)}$. Note that $\mathrm{Am}(X)$ is a subgroup of $C_{X}$. Hence there
exists a $N$ such that $M_{T(X)}\simeq M_{X}\oplus N$. The same holds for
$M_{Y}$. This gives us $U(X)\oplus N^{\prime}=M_{X}\oplus N$ and $U(Y)\oplus
Q^{\prime}=M_{Y}\oplus Q$, implying the equalities $M_{X}\oplus N\oplus
Q=U(X)\oplus N^{\prime}\oplus Q$ and $M_{Y}\oplus Q\oplus N=U(Y)\oplus
Q^{\prime}\oplus N$. Now if $X$ and $Y$ are birational, it follows
$\mathrm{Am}(X)=\mathrm{Am}(Y)$ and therefore $M_{X}\simeq M_{Y}$. Hence
$U(X)\oplus N^{\prime}\oplus Q\simeq U(Y)\oplus Q^{\prime}\oplus N$. This
completes the proof. ∎

######  Corollary (Corollary 1.8).

Let $X$ and $Y$ be as in Theorem 1.7. Assume $X$ and $Y$ are birational and
let $A_{i},1\leq i\leq n$ and $B_{j},1\leq j\leq m$ be the central simple
algebras occuring in $U(X)\oplus N^{\prime}\oplus Q$ and $U(Y)\oplus
Q^{\prime}\oplus N$ respectively, then
$\langle[A_{i}]\rangle=\langle[B_{j}]\rangle$ in $\mathrm{Br}(k)$.

###### Proof.

This follows from [31], Corollary 4.8. ∎

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