Non-Existence of Full Exceptional Collections on Twisted Flags

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So the main results in this context will be Theorem 6.3, Corollaries 6.9 and 6.11 and Propositions 6.12 and 6.13 below.

Theorem (Proposition 5.1, Proposition 5.3). Let X = X1 Xn be a finite product, ×···× where all the factors are either Brauer–Severi varieties over an arbitrary field or generalized Brauer–Severi varieties over a field of characteristic zero. Then X admits a full exceptional collection if and only if all the Xi are split, i.e, if all Xi are either projective spaces or Grassmannians.

Theorem (Corollary 5.5, Theorem 5.7). Let X = X1 Xn be a ﬁnite product of ×···× twisted forms of smooth quadrics over k (char(k) = 2) where all factors Xi are either 6 associated with involution algebras (Ai,σi) of orthogonal type having trivial discriminants δ(Ai,σi) or with involution algebras (Ai,σi) of orthogonal type over R. Then X admits a full exceptional collection if and only if all the Xi are split, i.e, if all Xi are smooth projective quadrics. Using noncommutative motives again and exploiting general facts from [32] and [27] we show that non-trivial twisted forms of homogeneous spaces of type An and Cn can not have full exceptional collections.

Corollary (Corollary 5.10). Let Gi be a split simply connected simple algebraic group s s of classical type An or Cn (n = 4) over k and γi : Gal(k k) Gi(k ), 1 i n, 1- 6 | → ≤ ≤ cocycles. Given parabolic subgroups Pi Gi, let γ (Gi/Pi) be the twisted forms of Gi/Pi. ⊂ i If X = γ (G1/P1) ... γ (Gn/Pn) admits a full exceptional collection, then all 1-cocycles 1 × × n γi must be trivial. In other words, non-trivial twisted flags of the considered type do not have full exceptional collections. Closely related to the problem of the existence of full exceptional collections is the question whether the considered variety admits a k-rational point. A potential measure for rationality was introduced by Bernardara and Bolognesi [7] with the notion of categorical representability. We use the definition given in [2]. A k-linear triangulated category T is said to be representable in dimension m if there is a semiorthogonal decomposition (see Section 3 for the definition) = 1, ..., n and for each i = 1, ..., n there exists a T hA A i smooth projective connected variety Yi with dim(Yi) m, such that i is equivalent to b ≤ A an admissible subcategory of D (Yi). We use the following notations rdim( ) := min m is representable in dimension m , T { | T } whenever such a finite m exists. Let X be a smooth projective k-variety. One says X is representable in dimension m if Db(X) is representable in dimension m. We will use the following notation: rdim(X) := rdim(Db(X)), rcodim(X) := dim(X) rdim(X). − Note that when the base field k of a variety X is not algebraically closed, the existence of k-rational points on X is a major open question in arithmetic geometry. We recall the following question that was asked by H. Esnault and stated in [3]. Question (H. Esnault). Let X be a smooth projective variety over k. Can the bounded derived category Db(X) detect the existence of a k-rational point?

Theorem (Theorem 6.3). If X is a ﬁnite product of Brauer–Severi varieties, then the following are equivalent: (i) rdim(X) = 0. (ii) X admits a full exceptional collection. (iii) X is rational over k. (iv) X admits a k-rational point. 3

Note that this theorem is a generalization of Proposition 6.1 in [3] and that it is proved in a completely different way than the result in [3]. For the product of generalized Brauer– Severi varieties we will show: Theorem (Theorem 6.5, Corollary 6.6). Let X be a finite product of generalized Brauer– Severi varieties over a field k of characteristic zero. Then rdim(X) = 0 if and only if X splits as the finite product of Grassmannians over k. We want to point out that Theorem 6.5 in not equivalent to the statement that X admits a k-rational point (see Remark 6.7 for an explanation). Moreover, for the finite product of certain twisted forms of smooth quadrics we show the following:

Theorem (Corollary 5.5, Theorem 5.7). Let X = X1 Xn be a finite product of ×···× twisted forms of smooth quadrics over k where all factors Xi are either associated with involution algebras (Ai,σi) of orthogonal type over a field k (char(k) = 2) with trivial 6 discriminants δ(Ai,σi) or with involution algebras (Ai,σi) of orthogonal type over R. Then rdim(X) = 0 if and only if X splits as the product of smooth quadrics, i.e. if all (Ai,σi) split. In particular, X admits a full exceptional collection if and only if X splits as the product of smooth quadrics If all the factors in the product of twisted quadrics are isotropic, we find:

Theorem (Corollary 6.9, Corollary 6.11). Let X = X1 Xn be a finite product of ×···× twisted forms of smooth quadrics over k where all factors Xi are either associated with isotropic involution algebras (Ai,σi) of orthogonal type over a field k (char(k) = 2) with 6 trivial discriminants δ(Ai,σi) or with isotropic involution algebras (Ai,σi) of orthogonal type over R. Then the following are equivalent: (i) rdim(X) = 0. (ii) X admits a full exceptional collection. (iii) X is rational over k. (iv) X admits a k-rational point. The equivalence of (ii) and (iii) (resp. (iv)) however, is not true in general. Indeed there exists a smooth anisotropic quadric without rational point that admits a full exceptional collection (see [8], proof of Proposition 1.7). Note that a smooth anisotropic quadric is rational if and only if it admits a rational point (see [15], Theorem 1.1). We want to stress that the existence of rational points seems, in general, not to be related to categorical representability in dimension zero. For instance, an elliptic curve over a number field is categorical representable in dimension one (see [2]) although it has rational points. Indeed, the rationality of a given variety X seems to be related to categorical representability in codimension 2. We recall the following question which was formulated in [9]: Question. Let X be a smooth projective variety over k of dimension at least 2. Suppose X is k-rational. Do we have rcodim(X) 2 ? ≥ There are several results suggesting that this question has a positive answer, see [9], [2] and references therein. In this context, Theorem 6.3 and Corollaries 6.9 and 6.11 from above have the following consequence:

Proposition (Proposition 6.12). Let X = X1 Xn be a ﬁnite product over k ×···× (char(k) = 2) of dimension at least two. Assume Xi is either a Brauer–Severi variety or 6 a twisted form of a quadric associated with an isotropic involution algebra of orthogonal type with trivial discriminant. If X is k-rational, then rcodim(X) 2. ≥ Proposition (Proposition 6.13). Let X = X1 Xn be a ﬁnite product over R of ×···× dimension at least two. Assume Xi is either a Brauer–Severi variety or a twisted form of a quadric associated with an isotropic involution algebra of orthogonal type. If X is k-rational, then rcodim(X) 2. ≥ 4

In the special case where X is a Brauer–Severi variety or a twisted form of a smooth quadric associated to an isotropic central simple algebra with involution of orthogonal type, k-rationality is equivalent to the existence of a k-rational point. So it is clear that Theorem 6.3 and Corollaries 6.9 and 6.11 reflect a very special behavior for the varieties under examination. The number rdim is closely related to the motivic categorical dimension which is introduced in [10] for arbitrary fields. In char(k) = 0, motivic categorical dimension of a smooth projective variety X can be defined as the smallest d such that µ([X]) lies in PTd(k). Here PT (k) denotes the Grothendieck ring of dg categories (see [14]), PTd(k) PT (k) the ad- ⊂ ditive subgroup generated by the smallest saturated monoid containing classes of pretriangulated dg categories of categorical dimension at least d and µ: K0(V ar(k)) PT (k) the → motivic measure defined by Bondal, Larsen and Lunts in [14] (in char(k) = 0). If one denotes the motivic categorical dimension of Db(X) by mcd(X), one has mcd(X) rdim(X) ≤ (see [10]). And a natural problem is then to calculate the numbers mcd(X) and rdim(X) and find conditions for which X equality mcd(X) = rdim(X) holds. And indeed, results from [10] suggest that motivic categorical dimension can be used to define a birational invariant. It turns out that determining rdim(X) for X a non-split (generalized) Brauer– Severi or an involution variety is indeed a challenging problem. We determine rdim(X) in the case of generalized Brauer–Severi varieties of index 3 and generalize in this way an ≤ earlier result [30], Theorem 1.4. Theorem (Theorem 6.15). Let X = BS(r, A) be a generalized Brauer–Severi variety over a field k of characteristic zero with ind(A) 3. Then rdim(X) = ind(A) 1. In particular ≤ − mcd(X) ind(A) 1. ≤ − For involution varieties we obtain: Theorem (Theorem 6.16). Let X be one of the following varieties: (i) a twisted quadric associated to a central simple algebra (A,σ) with involution of orthogonal type having trivial discriminant δ(A,σ). (ii) a twisted quadric associated to a central simple R-algebra (A,σ) with involution of orthogonal type. Then rdim(X) ind(A) 1. Moreover, if ind(A) 3, then rdim(X) = ind(A) 1. In ≤ − ≤ − particular, mcd(X) ind(A) 1. ≤ − Acknowledgement. Nearly all the proofs make use of noncommutative motives and I thank Theo Raedschelders for his explanations in this respect and notes on literature. I also like to thank Marcello Bernardara for useful comments. I would like to thank the Heinrich Heine University for financial support via the SFF-grant.

Conventions. Throughout this work k denotes an arbitrary ground ﬁeld and ks and k¯ a separable respectively algebraic closure.

2. Examples of twisted forms of homogeneous varieties As references we use [27], 1 and [20], Chapter VI. By an algebraic group over the field § k we will always mean an affine algebraic group. Let G be an algebraic group over k and X an algebraic variety such that G acts on X over k. The variety X is then called G- variety. For a closed (and reduced) subgroup H of G one has the associated homogeneous G-variety Y = G/H, which is also called a homogeneous space. Now let G be a semi- simple (so connected) algebraic group. A projective G-variety X is called twisted flag if s s s s X k k (G k k )/P where P is a closed subgroup of G k k . As(G k k )/P is ⊗ ≃ ⊗ ⊗ ⊗ projective, P must be a parabolic subgroup. Any twisted flag is a smooth, absolutely irreducible and reduced variety. 5

It is well-known that the twisted forms of a homogeneous space G/P are in one-to-one 1 s correspondence with elements in H (k, Aut(G k k )/P )). Let G¯ denote the adjoint group s ⊗ s s G/Z(G). The Gal(k k)-group homomorphism G¯(k ) Aut((G k k )/P ) induces a map | → ⊗ of pointed sets 1 s 1 s α: H (k, G¯(k )) H (k, Aut((G k k )/P )). −→ ⊗ For a twisted form we write γ X for α(γ)X where X = G/P . So in the present work we will always take some 1-cocycle γ : Gal(ks k) G(ks), the projective homogeneous space | → s s X = G/P and its twisted form γ X. Note that γ X k k (G/P ) k k . Furthermore, ⊗ ≃ ⊗ let G and P be the universal covers of G and P respectively. Denote by R(G) and R(P ) the associated representation rings and by Z G the center of G. Finally, let Ch := ⊂ Hom(e Z, Gme) be the character group. Under these notations we can give somee examplese of twisted forms of homogeneous spaces whiche wille occur quite frequentlye throughout this work. e

Example 2.1. Let G = PGLn. In this case we have G = SLn and Z µn. Then × ≃ Ch Z/nZ. Let a k and c GLn−1 and consider the following parabolic subgroup ≃ ∈ ∈ a b e e P = a det(c) = 1 SLn. { 0 c · }⊂ Pn−1 s The associated projectivee homogeneous variety is G/P G/P k . Let γ : Gal(k k) s n−≃1 ≃ | → PGLn(k ) be a 1-cocycle, then the twisted form γ P is called Brauer–Severi variety. e e Example 2.2. Let G = PGLn as in Example 2.1. We ﬁx a number 1 d n 1 and ≤ ≤ − let a GLd and c GLn−d. Consider the parabolic subgroup ∈ ∈ a b P = a det(c) = 1 SLn. { 0 c · }⊂

The associated projectivee homogeneous variety is G/P G/P Grassk(d, n). Given a s s ≃ ≃ 1-cocycle γ : Gal(k k) PGLn(k ), the twisted form γ Grassk(d, n) is called generalized | → Brauer–Severi variety. e e

Example 2.3. Let G = PSOn with n even. In this case G = Spinn. Consider the action of G on Pn−1 given by projective linear transformations. We write P G for the stabilizer ⊂ of the point [1 : 0 : : 0]. The projective homogeneous varietye G/P is a smooth quadric · ·n ·−1 s s hypersurface Q P . Given a 1-cocycle γ : Gal(k k) PSOn(k ), it is well known ⊂ | → that γ determines a central simple k-algebra A with an involution σ of orthogonal type. The associated twisted homogeneous space γ (G/P ) is a twisted form of the quadric G/P . In all examples from above the twisted forms can be described in terms of central simple k-algebras. 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 L, ⊂ such that A k L Mn(L). This is also equivalent to A k k¯ Mn(k¯). An extension ⊗ ≃ ⊗ ≃ k L such that A k L Mn(L) is called splitting field for A. ⊂ ⊗ ≃ The degree of a central simple algebra A is defined to be deg(A) := √dimkA. 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 Mn(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 ind(A). Two central simple algebras A and B are said to be Brauer-equivalent if there are positive integers r, s such that Mr(A) Ms(B). ≃ Note that a Brauer–Severi variety of dimension n can also be defined as a scheme X of n finite type over k such that X k L P for a finite field extension k L. A field extension n⊗ ≃ ⊂s k L for which X k L P is called splitting field of X. Clearly, k and k¯ are splitting ⊂ ⊗ ≃ 6

fields for any Brauer–Severi variety. In fact, every Brauer–Severi variety always splits over a finite Galois extension. It follows from descent theory that X is projective, integral and smooth over k. Via non-commutative Galois cohomology, Brauer–Severi varieties of dimension n are in one-to-one correspondence with central simple algebras A of degree n + 1. For details and proofs on all mentioned facts we refer to [4] and [16]. To a central simple k-algebra A one can also associate twisted forms of Grassmannians. Let A be of degree n and 1 d n. Consider the subset of Grassk(d n, A) consisting of ≤ ≤ · those subspaces of A that are left ideals I of dimension d n. This subset can be given the · structure of a projective variety which turns out to be a generalized Brauer–Severi variety. It is denoted by BS(d,A). After base change to some splitting field L of A the variety BS(d,A) becomes isomorphic to GrassL(d, n). If d = 1 the generalized Brauer–Severi variety is the Brauer–Severi variety associated to A. Note that BS(d,A) is a Fano variety. For details see [11]. To a central simple algebra A of degree n with involution σ of the first kind over a field k of char(k) = 2 one can associate the involution variety IV(A,σ). This variety can be 6 described as the variety of n-dimensional right ideals I of A such that σ(I) I = 0. If A ∗ ∗ · is split so (A,σ) (Mn(k), q ), where q is the adjoint involution defined by a quadratic ≃ Pn−1 form q one has IV(A,σ) V (q) k . Here V (q) is the quadric determined by q. By ≃ ⊂ Pn−1 construction such an involution variety IV(A,σ) becomes a quadric in L after base change to some splitting field L of A. In this way the involution variety is a twisted form of a smooth quadric in the sense of Example 2.3. Recall from [40] that a splitting field ∗ L splits A isotropically if (A,σ) k L (Mn(L), q ) with q an isotropic quadratic form ⊗ ≃ over L. Although the degree of A is arbitrary, (when char(k) = 2), the case where degree 6 of A is odd does not give anything new, since central simple algebras of odd degree with involution of the first kind are split (see [20], Corollary 2.8). For details on the construction and further properties on involution varieties and the corresponding algebras we refer to [40]. In fact, all the twisted flags from Examples 2.1, 2.2 and 2.3 appear as Brauer–Severi varieties, generalized Brauer–Severi varieties or as twisted smooth quadrics and are associated to some central simple algebra A in the sense described above.

3. Exceptional collections and semiorthogonal decompositions Let be a triangulated category and a triangulated subcategory. The subcategory D C is called thick if it is closed under isomorphisms and direct summands. For a subset A C of objects of we denote by A the smallest full thick subcategory of containing the D h i D elements of A. For a smooth projective variety X over k, we denote by Db(X) the bounded derived category of coherent sheaves on X. Moreover, if B is an associated k-algebra, we write Db(B) for the bounded derived category of ﬁnitely generated left B-modules.

Deﬁnition 3.1. Let A be a division algebra over k, not necessarily central. An object • Db(X) is called A-exceptional if End( •)= A and Hom( •, •[r]) = 0 for r = 0. By E ∈ E E E 6 generalized exceptional object, we mean A-exceptional for some division algebra A over k. If A = k, the object • is called exceptional. E Deﬁnition 3.2. A totally ordered set •, ..., • of generalized exceptional objects on {E1 En} X is called an generalized exceptional collection if Hom( •, •[r]) = 0 for all integers r Ei Ej whenever i > j. A generalized exceptional collection is full if •, ..., • = Db(X) and h{E1 En}i strong if Hom( •, •[r]) = 0 whenever r = 0. If the set •, ..., • consists of exceptional Ei Ej 6 {E1 En} objects it is called exceptional collection.

Remark 3.3. If the ring A in Definition 3.1 is required to be a semisimple algebra, the object is also called semi-exceptional object in the literature (see [31]). Consequently, one can also define (full) semi-exceptional collections. We also want to mention that a 7 general exceptional object is also called weak exceptional object in the literature (see [31], Definition 1.16) Example 3.4. Let Pn be the projective space and consider the ordered collection of invertible sheaves Pn , Pn (1), ..., Pn (n) . In [5] Beilinson showed that this is a full {O O O } strong exceptional collection.

1 1 Example 3.5. Let X = P P . Then X , X (1, 0), X (0, 1), X (1, 1) is a full strong × {O O O O } exceptional collection on X. We write X (i, j) for (i) ⊠ (j). O O O The notion of a full exceptional collection is a special case of what is called a semiorthogonal decomposition of Db(X). Recall that a full thick triangulated subcategory of Db(X) D .is called admissible if the inclusion ֒ Db(X) has a left and right adjoint functor D → Deﬁnition 3.6. Let X be a smooth projective variety over k. A sequence 1, ..., n b D D of full admissible triangulated subcategories of D (X) is called semiorthogonal if j ⊥ • b • • • D ⊂ = D (X) Hom( , ) = 0, i for i > j. Such a sequence deﬁnes a Di {F ∈ | G F ∀ G ∈ D } semiorthogonal decomposition of Db(X) if the smallest thick full subcategory containing b all i equals D (X). D b For a semiorthogonal decomposition we write D (X)= 1, ..., n . hD D i • • Remark 3.7. Let 1 , ..., n be a full generalized exceptional collection on X. It is easy E E • b to verify that by setting i = one gets a semiorthogonal decomposition D (X) = D hEi i 1, ..., n . hD D i For a wonderful and comprehensive overview of the theory on semiorthogonal decompositions and its relevance in algebraic geometry we refer to [23].

4. Recollections on noncommutative motives We refer to the book [34] (alternatively see [35] and [26] for a survey on noncommutative motives). Let be a small dg category (for details see [19]). The homotopy category 0 A 0 H ( ) has the same objects as and as morphisms H (HomA(x,y)). A source of ex- A A amples is provided by schemes since the derived category of perfect complexes perf(X) of any quasi-projective scheme X admits a canonical (unique) dg enhancement perfdg(X) (see [25], Theorem 7.9). Denote by dgcat the category of small dg categories. The op- op posite dg category has the same objects as and HomAop (x,y) := HomA(y,x). A A op A right -module is a dg functor Cdg(k) with values in the dg category Cdg(k) of A A → complexes of k-vector spaces. We write C( ) for the category of right -modules. Recall A A form [19] that the derived category D( ) of is the localization of C( ) with respect to A A A quasi-isomorphisms. A dg functor F : is called derived Morita equivalence if the A → B restriction of scalars functor D( ) D( ) is an equivalence. The tensor product B → A A ⊗ B of two dg categories is deﬁned as follows: the set of objects is the cartesian product of the sets of objects in and and HomA⊗B((x,w), (y,z)) := HomA(x,y) HomB(w,z) (see A B ⊗ [19]). Given two dg categories and , let rep( , ) be the full triangulated subcategory op A B A B of D( ) consisting of those -bimodules M such that M(x, ) is a compact A ⊗ B A − B − object of D( ) for every object x . The category dgcat of all (small) dg categories B ∈ A and dg functors carries a Quillen model structure whose weak equivalences are Morita equivalences. Let us denote by Hmo the homotopy category hence obtained and by Hmo0 its additivization. Now to any small dg category one can associate functorially its non- A commutative motive U( ) which takes values in Hmo0. This functor U : dgcat Hmo0 A → is proved to be the universal additive invariant (see [38]). An additive invariant is any functor E : dgcat taking values in an additive category such that → D D (i) it sends derived Morita equivalences to isomorphisms, 8

(ii) for any pre-triangulated dg category admitting full pre-triangulated dg sub- A categories and such that H0( ) = H0( ),H0( ) is a semiorthogonal de- B C A h B C i composition, the morphism E( ) E( ) E( ) induced by the inclusions is an B ⊕ C → A isomorphism. The category NChow(k) of noncommutative motives is the pseudo-Abelian envelope of the full subcategory of Hmo0(k) consisting of smooth and proper dg categories (see [36] for details). Recall that every additive invariant E : dgcat factors through U : dgcat → D → Hmo0. The noncommutative Chow motive of a smooth projective k-scheme X is defined to be U(perf(X)). In the present paper we will apply the three theorems stated below. We use the following notation: Let G split simply connected semi-simple algebraic group over the field k and P a parabolic subgroup. Recall from [32], Theorem 2.10 that there exits a finite free Ch-homogeneous basis of R(P ) over R(G). Moreover, associated to a 1-cocycle γ : Gal(ks k) G(ks) and each character χ Ch one has the Tits’ algebras | → ∈ Aχ,γ (see [32], 3.1 or [20], p.377). If ρ1, ..., ρn ise the Ch-homogeneouse R(G) basis of R(P ) χ(i) we write χ(i) for the character such that ρi R (P ). Under this notation one has the ∈ following theorem: e e e Theorem 4.1. [38, Theorem 2.1 (i)] Let G, P and γ be as above and E : dgcat D an → additive invariant. Then every Ch-homogeneous basis ρ1, ..., ρn of R(P ) over R(G) give rise to an isomorphism n e e ∼ E(A ) E(γ X), χ(i),γ −→ i=1 M where Aχ(i),γ stands for the Tits’ central simple algebras associated to ρi. Theorem 4.2. [38, Theorem 3.3] Let G, P and γ as in Theorem 4.1. Then n U(k) i=1 ≃ U(γ X) if and only if the Brauer classes [Aχ i ,γ ] are trivial. ( ) L For central simple k-algebras one has the following comparison theorem:

Theorem 4.3. [37, Theorem 2.19] Let A1, ..., An and B1, ..., Bm be central simple k- algebras, then the following are equivalent: (i) There is an isomorphism n m

U(Ai) U(Bj ). ≃ i=1 j=1 M M (ii) The equality n = m holds and for all 1 i n and all p ≤ ≤ [Bp] = [Ap ] Br(k) i σp(i) ∈ for some permutations σp depending on p. For the proofs of our main results we also need Theorem 4.4 below. Let CSep(k) be the full subcategory of NChow(k) consisting of objects of the form U(A) with A a commutative separable k-algebra. Analogously, Sep(k) denotes the full subcategory of NChow(k) consisting of objects U(A) with A a separable k-algebra. And finally, we write CSA(k) for the full subcategory of Sep(k) consisting of U(A) with A being a central simple k-algebra. Moreover, CSA(k)⊕ denotes the closure of CSA(k) under finite direct sums. Theorem 4.4. [37, Corollary 2.13] There is an equivalence of categories U(k)⊕n n 0 CSA(k)⊕ CSep(k) { | ≥ }≃ ×Sep(k) i.e. U(k)⊕n n 0 is a 2-pullback of categories with respect to the inclusion morphisms. { | ≥ } Proposition 4.5. Let X and Y be smooth projective varieties over a field k. Assume that X and Y admit full generalized exceptional collections 0, ..., r respectively 0, ..., s E E F F such that End( i) Ai and End( j ) Bj are central simple algebras. Then the ordered E ≃ F ≃ 9

collection i ⊠ j is a full generalized exceptional collection, where i ⊠ j precedes {E F } E 1 F 1 i ⊠ j iff (i1, j1) (i2, j2). Here stands for the lexicographic order on 0, ..., r E 2 F 2 ≺ ≺ { }× 0, ..., s . Moreover, End( i ⊠ j ) Ai Bj and rdim(X Y ) = 0 if and only if { } E F ≃ ⊗ × Ai Bj M (k). ⊗ ≃ n(i,j) Proof. That the ordered collection i ⊠ j is a full generalized exceptional collection {E F } follows from [22] and the fact that the tensor product of central simple algebras is again central simple. To show that End( i ⊠ j ) Ai Bj one uses Künneth formula. Now E F ≃ ⊗ assume rdim(X Y ) = 0. Let us denote the central simple algebra Ai Bj by Ci,j . Then × ⊗ U(perf (X Y )) U(Ci,j ). dg × ≃ M(i,j) From [3], Lemma 1.20 we obtain that there is a semiorthogonal decomposition b (1) D (X Y )= 1, ..., n × hA A i b with i D (Ki) and Ki being étale k-algebras. Therefore A ≃ U(perf (X Y )) U(Ci,j ) U(K1) U(Kn). dg × ≃ ≃ ⊕···⊕ (Mi,j) Using Theorem 4.4 and the universal property of fibre products, the above isomorphism gives rise to an isomorphism ⊕n U(perf (X Y )) U(Ci,j ) U(k) . dg × ≃ ≃ (Mi,j) Now Theorem 4.3 implies that Ai Bj = Ci,j must be split. On the other hand if ⊗ Ai Bj = Ci,j is split, the full generalized exceptional collection i ⊠ j gives rise to a ⊗ {E F } semiorthogonal decomposition Db(X Y )= Db(M (k)) × h n(i,j) i(i,j)∈{0,...,r}×{0,...,s} and therefore rdim(X Y ) = 0, according to [3], Lemma 1.20. This completes the × proof.

Remark 4.6. If we assume in Proposition 4.5 that End( 0) End( 0) k, then E ≃ F ≃ rdim(X Y ) = 0 if and only if both Ai Mn (k) and Bj Mn (k) split. This fol- × ≃ i ≃ j lows easily from the proof.

5. Non-existence of full exceptional collections In this section we show that a finite product of non-split (generalized) Brauer–Severi varieties can not have full exceptional collections. The same is true for finite products of certain twisted forms of smooth quadrics. The idea for the proofs is essentially contained in [33]. Nevertheless, we give the proofs, adding to the literature. Proposition 5.1. Let X be a finite product of Brauer–Severi varieties. Then X admits a full exceptional collection if and only if X splits as the product of projective spaces. Proof. A projective space admits a full exceptional collection according to Example 3.4. Now it is a general fact that the finite product of varieties admitting full exceptional collections have full exceptional collections, too. In fact, this follows from the Künneth formula and from general results on generating objects in the derived category of coherent sheaves. Note that this also follows from more general results proved in [22]. To prove the other implication, we use induction on the number of factors in the finite product. We restrict ourselves to prove the statement only for the case X = Y1 Y2. × Let us denote by A1 and A2 the central simple algebras corresponding to Y1 respectively Y2. Now assume that X = Y1 Y2 admits a full exceptional collection. According to [3], × Lemma 1.20 we get rdim(X) = 0. The assertion then follows from Proposition 4.5 and Remark 4.6, since a Brauer–Severi variety has a full generalized exceptional collection (see 10

[6]) which fulﬁlls the conditions of Proposition 4.5 and Remark 4.6. In particular A1 and r s A2 are split and hence X P P . ≃ k × k Remark 5.2. The above proposition implies that a product of Brauer–Severi varieties is rational over k if and only if it admits a full exceptional collection. For further varieties satisfying such a condition see for instance [3] and [41] and references therein.

Proposition 5.3. Let X be a finite product of generalized Brauer–Severi varieties over a field of characteristic zero. Then X admits a full exceptional collection if and only if X splits as the finite product of Grassmannians.

Proof. A Grassmannian over a field k of characteristic zero admits a full exceptional collection according to [17]. One can argue as in Proposition 5.1 to conclude that the finite product of such Grassmannians has a full exceptional collection, too. In order to prove the other implication, note that from [12] it follows that a generalized Brauer–Severi variety BS(d,A) associated to a central simple algebra A of degree n over a field k of characteristic zero admits a full generalized exceptional collection. To be precise, denote by P the set of partitions λ = (λ1, ..., λd) with 0 λd ... λ1 n d. One ≤ ≤ ≤ ≤ − can choose a total order on P such that λ µ means that the Young diagram of λ is ≺ ≺ not contained in that of µ. Under this notation, in loc.cit. it is proved that there is a full semi-exceptional collection (in the sense of Remark 3.3) ... λ, ..., µ... with λ µ and ⊗|λ| { V V } ≺ End( λ) being isomorphic to A . Here λ = λ1 + ... + λd, where λ = (λ1, ..., λd) V ⊗|λ| | | ∈ P . By the Wedderburn Theorem A Mn (Dλ) with some unique central division ≃ λ algebras Dλ. In particular, this implies that the Krull–Schmidt decomposition of such a ⊕nλ λ is given by λ . By construction, the ordered set ... λ, ..., µ... is a full V V ≃ Wλ { W W } generalized exceptional collection with End( λ) being isomorphic to Dλ and therefore W Brauer-equivalent to A⊗|λ|. As in the proof of Proposition 5.1 we restrict to the case when X = BS(d1,A1) BS(d2,A2) for two central simple k-algebras A1 and A2. Now × assume that X admits a full exceptional collection. According to [3], Lemma 1.20 we get rdim(X) = 0. The assertion then follows from Proposition 4.5 and Remark 4.6. In particular A1 and A2 are split and hence X is the product of two Grassmannians.

s s Now let G = PSOn be over k with n even. Given a 1-cocycle γ : Gal(k k) PSOn(k ) | → we get a twisted form of a quadric γ Q which is associated to a central simple k-algebra (A,σ) of degree n with involution of orthogonal type. Note that γ Q is isomorphic to the involution variety IV(A,σ) from Section 2. For any splitting ﬁeld L of A, the variety n−1 γ Q k L is isomorphic to a smooth quadric in P . ⊗ L

Proposition 5.4. Let γ Q a twisted form as above and assume the associated central simple algebra (A,σ) has trivial discriminant δ(A,σ). Then rdim(γ Q) = 0 if and only Pn−1 if γ Q splits, i.e. if (A,σ) splits and γ Q is a smooth quadric in k . In particular, γ Q admits a full exceptional collection if and only if it splits.

Pn−1 Proof. If (A,σ) is split, i.e. if γ Q is a smooth quadric in k , the existence of a full exceptional collection follows from [21] (also, see the proof of Proposition 7.2 in [12]). Hence rdim(γ Q) = 0 according to [3], Lemma 1.20. Now assume rdim(γ Q) = 0. Again from [3], Lemma 1.20 we obtain that there is a semiorthogonal decomposition

b (2) D (γ Q)= 1, ..., r hA A i b with i D (Ki) and Ki being ´etale k-algebras. Therefore, the noncommutative motive A ≃ of perfdg (γ Q)) decomposes as

U(perf (γ Q)) = U(K1) ... U(Kr). dg ⊕ ⊕ 11

On the other hand (see [38], Example 3.11), one has

n−3 n−3 + − U(perf (γ Q)) U(k) U(A) U(C (A,σ)) U(C (A,σ)). dg ≃   ⊕   ⊕ 0 ⊕ 0 i≥0 i>0 evenM  Modd      Therefore, we have the following isomorphism

n−3 n−3 + − U(k) U(A) U(C (A,σ)) U(C (A,σ)) U(K1) ... U(Kr).   ⊕   ⊕ 0 ⊕ 0 ≃ ⊕ ⊕ i≥0 i>0 evenM  Modd      Since the discriminant δ(A,σ) is trivial, the even Cliﬀord algebra C0(A,σ) decomposes + − + − as C0(A,σ) C (A,σ) C (A,σ), where C (A,σ) and C (A,σ) are central simple ≃ 0 × 0 0 k-algebras (see [20], Theorem 8.10). Using Theorem 4.4 and the universal property of ﬁbre products, the above isomorphism gives rise to an isomorphism

n−3 n−3 U(k) U(A) U(C+(A,σ)) U(C−(A,σ)) U(k)⊕r.   ⊕   ⊕ 0 ⊕ 0 ≃ i≥0 i>0 evenM  Modd      Now Theorem 4.3 implies that A must be split. The same argument shows γ Q admits a full exceptional collection if and only if it splits. This completes the proof.

+ − Note that in Proposition 5.4 the central simple algebras k,A,C0 (A,σ) and C0 (A,σ) are the minimal Tits’ algebras of γ Q. Recall that the Tits’ algebras of the product of adjoint semi-simple algebraic groups are Brauer equivalent to the tensor product of the Tits’ algebras of its factors ([27], Corollary 2.3). Using Theorems 4.1, 4.2 and 4.4 and the fact that ﬁnite products of the considered varieties have full exceptional collections too, one can imitate the proofs of Proposition 5.1 and 5.3 to show easily:

Corollary 5.5. Let X = γ Q1 γ Qm be the ﬁnite product of twisted forms of 1 ×···× m quadrics as in Proposition 5.4. Let (Ai,σi) be the central simple algebra with involution of orthogonal type associated to the factor γi Qi. Then rdim(X) = 0 if and only if X splits as the product of smooth quadrics, i.e. if all (Ai,σi) split. In particular, X admits a full exceptional collection if and only if X splits as the product of smooth quadrics.

Theorem 5.6. Let G = PSOn be over R with n even and let γ Q be a twisted form of a quadric associated to a central simple R-algebra (A,σ) of degree n with involution of orthogonal type associated to a non-trivial 1-cocycle γ.Then rdim(γ Q) = 0 if and only if Pn−1 γ Q splits, i.e. if (A,σ) splits and γ Q is a smooth quadric in k . In particular, γ Q admits a full exceptional collection if and only if it splits.

n−1 Proof. If (A,σ) splits, i.e. if γ Q is a smooth quadric in PR , the existence of a full exceptional collection follows from [21] (see also [12]). Hence rdim(γ Q) = 0 according to b [3], Lemma 1.20. Now assume rdim(γ Q) = 0. Then the derived category D (γ Q) must have a semiorthogonal decomposition of the form b (3) D (γ Q)= 1, ..., e hA A i b with i D (R, Ki) and Ki being ´etale R-algebras (see [2], Proposition 6.1.6). We notice A ≃ ×ni ×mi that Ki R C . Now [3], Lemma 1.17 implies ≃ × b b ×ni b ×mi (4) D (Ki) D (R) D (C) . ≃ × According to [12], there is a semiorthogonal decomposition with exactly n 1 components − b b b b b b D (γ Q)= D (k), D (A), ..., D (k), D (A), D (C(A,σ)) . h i 12

Note that this semiorthogonal decomposition is induced by vector bundles 1, 2, ..., n−1 V V V on γ Q satisfying End( 1)= k, End( 2)= A, ..., End( n−1)= C(A,σ). Now the noncom- V V V mutative motive of perfdg(γ Q) decomposes as

U(perf (γ Q)) = U(k) U(A) ... U(k) U(A) U(C(A,σ)). dg ⊕ ⊕ ⊕ ⊕ ⊕ From [20], Theorem 8.10 we know that C(A,σ) is either a central simple algebra over C or that C(A,σ) splits as the direct product of two central simple R-algebras. In the ﬁrst case C(A,σ) Ms(C) whereas in the latter case C(A,σ) is isomorphic to A1 A2, where Ai is ≃ R H b × C b C isomorphic to Mn1 ( ) or Mn2 ( ). By Morita equivalence we have D (Ms( )) D ( ), b b b b ≃ D (Mn (R)) D (R) and D (Mn (H)) D (H). So there are two cases to consider. 1 ≃ 2 ≃ First case: If C(A,σ) Ms(C), one has rkK0(γ Q) = n 1 and from (3) and (4) we ≃ − conclude e e

(5) n 1= ni + mi. − i=1 i=1 X X After base change to C, the vector bundles 1 R C, 2 R C, ..., n−1 R C on the smooth V ⊗ V ⊗ V b ⊗ quadric γ Q R C give rise to a semiorthogonal decomposition of D (γ Q R C). Moreover, ⊗ ⊗ we have

End( n−1 R C)) End( n−1) R C Ms(C) R C Ms(C) Ms(C) V ⊗ ≃ V ⊗ ≃ ⊗ ≃ × and therefore rkK0(γ Q R C) = n. Since End( n−1) R C Ms(C) Ms(C), we obtain ⊗ V ⊗ ≃ × with the semiorthogonal decomposition (3) in view of (4) that e e

n = ni + 2 mi. i=1 i=1 X X Taking the diﬀerence of the latter equation with (5) we ﬁnd e

(6) 1= mi i=1 X Without loss of generality, we can assume m1 = ... = me−1 = 0,me = 1. This gives us the following isomorphism of noncommutative motives U(R)⊕(n−2) U(C) U(R) U(A) U(R) U(A) U(C). ⊕ ≃ ⊕ ⊕···⊕ ⊕ ⊕ Now [39], Proposition 4.5 implies U(R)⊕(n−2) U(R) U(A) U(R) U(A) ≃ ⊕ ⊕···⊕ ⊕ and Theorem 4.3 shows that A splits. This gives the assertion for the case C(A,σ) ≃ Ms(C). Second case: Now let C(A,σ)= A1 A2 be the product of two central simple R-algebras. × In this case the rank of the Grothendieck group K0(γ Q) must be n. Therefore e e

(7) n = ni + mi. i=1 i=1 X X After base change to C, we obtain e e

(8) n = ni + 2 mi. i=1 i=1 e X X From (7) and (8) we ﬁnd 0 = i=1 mi. This gives ⊕n U(R) U(R) UP(A) U(R) U(A) U(A1) U(A2). ≃ ⊕ ⊕···⊕ ⊕ ⊕ ⊕ Again Theorem 4.3 implies that A splits. This completes the proof. The next result generalizes Theorem 5.6. We give the proof in detail, although it is similar to the proof of Theorem 5.6. 13

Theorem 5.7. Let X = γ Q1 γ Qm be a ﬁnite product of twisted quadrics as in 1 ×···× m Theorem 5.6. Then rdim(X) = 0 if and only if X splits as the product of smooth quadrics, i.e. if all (Ai,σi) split. In particular, X admits a full exceptional collection if and only if X splits as the product of smooth quadrics.

Proof. We give a proof only for the case X = γ Q1 γ Q2. Let(A1,σ1)and (A2,σ2) be the 1 × 2 involution algebras associated with γ1 Q1 and γ2 Q2. Denote by n the degree of (A1,σ1) and by m the degree of (A2,σ2). We restrict ourselves to the case where C(A1,σ1) Ms(C) ≃ and C(A2,σ2) Mr(C), since the other cases are proved analogously. Recall from [12] ≃ that γ1 Q1 and γ2 Q2 have semiorthogonal decompositions b b b b b b (9) D (γ Q1)= D (k), D (A1), ..., D (k), D (A1), D (C(A1,σ1)) 1 h i and b b b b b b (10) D (γ Q2)= D (k), D (A2), ..., D (k), D (A2), D (C(A2,σ2)) . 2 h i From [12] it follows that the number of components in (9) respectively (10) is n 1, respec- − tively m 1. According to [22], the product X = γ Q1 γ Q2 has a semiorthogonal decom- − 1 × 2 position which is constructed by taking successive tensor products of the components of the semiorthogonal decompositions (9) and (10). Considering the case C(A1,σ1) Ms(C) ≃ and C(A2,σ2) Mr(C), it is an exercise to verify that ≃ (11) rkK0(X)=(n 1)(m 1) + 1. − − This is also the number of components in the semiorthogonal decomposition of Db(X), obtained by taking successive tensor products of the components of (9) and (10). The number of components in the semiorthogonal decomposition of Db(X) that are equivalent to Db(C) is (n 1)(m 1) + 1 (n 2)(m 2) = m + n 2. − − − − − − So after base change to C, we obtain

(12) rkK0(X R C)=(n 2)(m 2) + 2(m + n 2). ⊗ − − − Assuming rdim(X) = 0, there must be a semiorthogonal decomposition b (13) D (γ Q)= 1, ..., e hA A i b with i D (Ki) and Ki being ´etale R-algebras. As in the proof of Proposition 5.6, we A ≃ have

b b ×ni b ×mi (14) D (Ki) D (R) D (C) . ≃ × Comparing (11), (13) and (14), we ﬁnd e e

(15) (n 1)(m 1)+1= ni + mi. − − i=1 i=1 X X From (12) and base change of the semiorthogonal decomposition (13) to C, we get e e

(16) (n 2)(m 2) + 2(m + n 2) = nm = ni + 2 mi. − − − i=1 i=1 X X e Taking the diﬀerence of the equation (16) with equation (15) implies n+m 2= i=1 mi e − and therefore nm 2n 2m +4= ni. Now (13) gives the following isomorphism − − i=1 for the noncommutative motive of perf (X) P Pdg U(perf (X)) U(R)⊕(nm−2n−2m+4) U(C)⊕(n+m−2). dg ≃ ⊕ The semiorthogonal decomposition of the product X = γ Q1 γ Q2 gives 1 × 2 (n−2)(m−2) ⊕ 4 ⊕(n+m−2) U(perf (X)) (U(R) U(A1) U(A2) U(A1 A2) U(C) . dg ≃ ⊕ ⊕ ⊕ ⊗ ⊕ 14

Comparing both isomorphisms for U(perfdg(X)), Proposition 4.5 of [39] implies (n−2)(m−2) ⊕ 4 ⊕(nm−2n−2m+4) (U(R) U(A1) (A2) U(A1 A2) U(R) ⊕ ⊕ ⊕ ⊗ ≃ Theorem 4.3 implies that A1 and A2 split. This completes the proof. More generally, Theorem 4.2 has the following consequence. Proposition 5.8. Let G be a split simply connected simple algebraic group of classical type An or Cn over k and P G a parabolic subgroup. Moreover, let γ be a 1-cocycle, X = ⊂ G/P the homogeneous variety and γ X its twisted form. If γ X admits a full exceptional collection, then γ must be the trivial 1-cocycle. In other words, non-trivial twisted ﬂags of the considered type do not have full exceptional collections. Proof. Let G and P be the universal covers of G and P respectively and R(G) and R(P ) the associated representation rings. One has the character group Ch = Hom(Z, Gm) which is a ﬁnite group.e Nowe let ρ1, ..., ρn be a Ch-homogeneous basis of R(P ) overe R(G) ande A the Tits’ central simple algebras associated to ρi. It is clear that ne ord(Ch) χ(i),γ ≥ (see [32] for details on the basis ρ1, ..., ρn). Assuming the existence ofe a full exceptionale b collection in D (γ X), Theorem 4.1 gives an isomorphism n

(17) U(A ) U(γ X) U(k) ... U(k) χ(i),γ ≃ ≃ ⊕ ⊕ i=1 M Now Theorem 4.2 implies that the Brauer classes [Aχ(i),γ ] must be trivial for all characters χ(i). From the classiﬁcation of the minimal Tits’ algebras for simply connected classical groups (see [20], p.378) we conclude that the 1-cocycle γ must be trivial. This completes the proof. Remark 5.9. We believe that in Proposition 5.8 we actually have if and only if.

Corollary 5.10. Let G1, ..., Gn be split simply connected simple algebraic groups as in Proposition 5.8 and γi 1-cocycles. Given parabolic subgroups Pi Gi, let γ (Gi/Pi) be ⊂ i the twisted forms of Gi/Pi. If X = γ (G1/P1) ... γ (Gn/Pn) admits a full exceptional 1 × × n collection, then all 1-cocycles γi must be trivial. Proof. We just sketch the proof as it is completely analogous to that of Proposition 5.8. Consider the classiﬁcation of the minimal Tits’ algebras for simply connected classical groups given in [20] (see also [27]). Then the successive tensor products of the Tits’ algebras of each factor gives simple algebras that are Brauer equivalent to the Tits’ algebras of X. From Theorem 4.2 we conclude that all these algebras must be trivial in the respective Brauer group. Hence all 1-cocycles γi must be trivial.

6. Categorical representability in dimension zero and rational points Propositions 5.1, 5.3, 5.4 and Theorems 5.6 and 5.7 are closely related to the existence of k-rational points on the considered varieties. For details and further results in this direction see [2], [3], [7] and [9]. Recall the definition of categorical representability from [2]. Definition 6.1. A k-linear triangulated category T is representable in dimension m if it admits a semiorthogonal decomposition T = 1, ..., r and for each i = 1, ..., r there ex- hA A i ists a smooth projective connected variety Yi with dim(Yi) m, such that i is equivalent b ≤ A to an admissible subcategory of D (Yi). We will use the following notation: rdim(T ) := min m T is representable in dimension m , { | } whenever such a finite m exists. 15

Definition 6.2. Let X be a smooth projective k-variety. One says X is representable in dimension m if Db(X) is representable in dimension m. We will write rdim(X) := rdim(Db(X)), rcodim(X) := dim(X) rdim(X). − Based on the idea of the proof of Proposition 5.1, we obtain: Theorem 6.3. Let X be the finite product of Brauer–Severi varieties over a field k. Then the following are equivalent: (i) rdim(X) = 0. (ii) X admits a full exceptional collection. (iii) X is rational over k. (iv) X admits a k-rational point.

Proof. The equivalence of (iii) and (iv) is an application of a Theorem of Chˆatelet (see [16], Theorem 5.1.3) and the Lang–Nishimura Theorem. To be precise, if X = Y1 Yr Pn Pn 99K ×···× is birational over k to some k , we can consider the rational map k X to obtain a k-rational point on X from the Lang–Nishimura Theorem. Now assume X admits a k rational point. Note that we have the projections pi : X Yi which, again by the Lang– → Nishimura Theorem, provide us with k-rational points on every Yi. The before mentioned mi m1 mr Theorem of Chˆatelet implies Yi P for all i 1, ..., r . Thus X P ... P . ≃ k ∈ { } ≃ k × × k The equivalence of (ii) and (iii) is the content of Remark 5.2. It remains to show that (i) is equivalent to (iii). So we assume rdim(X) = 0. From Proposition 5.1 it follows that X splits as the direct product of projective spaces. There- fore, X is k-rational. On the other hand, if X is rational over k there is a birational map Ps 99K k X = Y1 Y2 and by the Lang–Nishimura Theorem X admits a k-rational point. × n m We conclude X P P . Therefore X has a full (strong) exceptional collection. Again, ≃ k × k Lemma 1.20 of [3] implies rdim(X) = 0. This completes the proof.

Remark 6.4. In [3], the statement of Theorem 6.3 is proved for the special case where X is a Brauer–Severi surface. The proof in loc.cit. however relies on the transitivity of the s 2 Braid group action on the set of full exceptional collections on X k k = P s and makes ⊗ k use of Galois descent. Theorem 6.5. Let X be a finite product of generalized Brauer–Severi varieties over a field k of characteristic zero. Then rdim(X) = 0 if and only if X splits as the finite product of Grassmannians over k.

Proof. We mentioned in Proposition 5.3 that a ﬁnite product X of Grassmannians over k admits a full exceptional collection. Lemma 1.20 of [3] immediately implies rdim(X) = 0. Now assume rdim(X) = 0. Now see the proof of Proposition 5.3 to conclude with Proposition 4.5 that X splits as the direct product of Grassmannians. Therefore, X is k-rational.

Corollary 6.6. Let X be the finite product of generalized Brauer–Severi varieties over a field of characteristic zero. Then rdim(X) = 0 if and only if it admits a full exceptional collection. Remark 6.7. Theorem 6.5 shows that if rdim(X) = 0 for a finite product X of generalized Brauer–Severi varieties over a field k of characteristic zero, then X must be rational over k and has therefore a k-rational point. Note that the other implication does not hold. Indeed, let (a,b) be a non-split quaternion algebra over a field k of characteristic zero. Consider the central simple algebra A = Mn((a,b)) for n 2 and the associated ≥ generalized Brauer–Severi variety X = BS(d,A) with for instance d = 4. As ind((a,b))=2 divides d = 4, results in [11] show that X admits a k-rational point. But Theorem 6.15 from below implies rdim(X) = 1 = 0. 6 16

Recall that a central simple algebra (A,σ) with involution is called isotropic if σ(a) a = · 0 for some nonzero element a A. So after base change to some splitting ﬁeld L of A the ∈ involution variety associated to an isotropic involution algebra (A,σ) becomes a quadric V (q) where q is an isotropic quadratic form over L (see [20], p.74 Example 6.6).

Proposition 6.8. Let γ Q be as in Proposition 5.4 and assume the associated involution algebra (A,σ) is isotropic. Then the following are equivalent:

(i) rdim(γ Q) = 0. (ii) γ Q admits a full exceptional collection. (iii) γ Q is rational over k. (iv) γ Q admits a k-rational point.

Proof. If γ Q admits a full exceptional collection, rdim(γ Q) = 0. Now if rdim(γ Q) = 0, Proposition 5.4 implies that A is split and therefore γ Q must be a smooth quadric induced by an isotropic quadratic form. Hence γ Q admits a full exceptional collection. This proves the equivalence of (i) and (ii). Assume (ii). Then Proposition 5.4 shows that A is split and hence γ Q is isomorphic to a smooth isotropic quadric which has a k-rational point. On the other hand, if γ Q admits a k-rational point, Proposition 4.3 of [40] implies that A is split and so γ Q must be a smooth quadric (which admits a full exceptional collection). This proves the equivalence of (ii) and (iii). Now it is well-known that (iii) and (iv) are equivalent (see for instance [15]). Corollary 6.9. Proposition 6.8 also holds for ﬁnite products.

Proof. Let X = X1 Xn be the ﬁnite product of twisted quadrics as in Proposition ×···× 5.4. We show the equivalence of (i) and (ii). For this, assume X admits a full exceptional collection. Then rdim(X) = 0. On the other hand, if rdim(X) = 0, Corollary 5.5 implies that X is isomorphic to the product of smooth isotropic quadrics and hence admits a full exceptional collection. Now assume X admits a full exceptional collection. Then Corollary 5.5 shows that X is isomorphic to the product of smooth isotropic quadrics and so has a k-rational point. On the other hand, the existence of a k-rational point on X gives us a k-rational point on any Xi by the Lang–Nishimura Theorem. Then Proposition 4.3 of [40] implies that any Xi must be a smooth isotropic quadric. Since a smooth quadric admits a full exceptional collection, the product X admits a full exceptional collection, too. This shows the equivalence of (ii) and (iii). Finally, if X has a k-rational point, each Xi has a k-rational point and must be a smooth isotropic quadric, which is k-rational. Therefore X is rational. Clearly, if X is k-rational it admits a k-rational point.

Proposition 6.10. Let γ Q be as in Proposition 5.6 and assume the associated involution algebra (A,σ) is isotropic. Then the following are equivalent:

(i) rdim(γ Q) = 0. (ii) γ Q admits a full exceptional collection. (iii) γ Q is rational over R. (iv) γ Q admits a R-rational point. Proof. The proof is the same as for Theorem 6.3 with the diﬀerence that one uses Theorem 5.6 instead of Proposition 5.1. Corollary 6.11. Proposition 6.10 also holds for ﬁnite products.

Proposition 6.12. Let X = X1 Xn be a ﬁnite product over k (char(k) = 2) of ×···× 6 dimension at least two. Assume Xi is either a Brauer–Severi variety or a twisted form of a quadric as in Proposition 5.4. If X is k-rational, then rcodim(X) 2. ≥ Proof. If X is k-rational, it admits a k-rational point. By Lang–Nishimura Theorem, any of the factors Xi admits a k-rational point. If Xi is a Brauer–Severi variety, Theorem 6.3 implies that Xi has a full exceptional collection. If Xi is a twisted form of a smooth quadric 17

(as in Proposition 5.4), Proposition 6.8 shows that Xi has a full exceptional collection. Therefore, the product X admits a full exceptional collection and hence rdim(X) = 0. As dim(X) 2, we conclude rcodim(X) 2. ≥ ≥

Proposition 6.13. Let X = X1 Xn be a finite product over R of dimension at ×···× least two. Assume Xi is either a Brauer–Severi variety or a twisted form of a quadric as in Proposition 5.6. If X is k-rational, then rcodim(X) 2. ≥ Finally, we want to calculate rdim(X) for generalized Brauer–Severi and certain involution varieties in the case A is non-split. We first generalize [30], Proposition 4.1. Proposition 6.14. Let X = BS(r, A) be a generalized Brauer–Severi variety over a field k of characteristic zero. Then rdim(X) ind(A) 1. In particular mcd(X) ind(A) 1. ≤ − ≤ − r Proof. Let a = (a1, ..., an) N , subject to the condition n a1 a1 0 (that ∈ ≥ ≥···≥ ≥ is Young diagrams with at most n r rows and at most r colums). By [12] we have a − semiorthogonal decomposition b (18) D (X)= 1, 2, ..., s , hA A A i b ⊗d(a) with components i being equivalent to D (A ), where d(a)= a1 + +an. According A · · · to the Wedderburn Theorem, the central simple algebra A is isomorphic to Mq(B) for a unique central division algebra B and some q > 0. Now let YB be the Brauer–Severi variety corresponding to B. As B is a division algebra, we have dim(YB) = ind(A) 1 = ind(B) 1. − − Note that Db(A⊗d(a)) Db(B⊗d(a)). Since Db(B⊗i) is an admissible subcategory of b ≃ D (YB ) for any i 0 (see [6]), we immediately conclude ≥ rdim(X) dim(YB) = ind(X) 1. ≤ −

The next result generalizes [30], Theorem 1.4. Theorem 6.15. Let X = BS(r, A) be a generalized Brauer–Severi variety over a ﬁeld k of characteristic zero with ind(A) 3. Then rdim(X) = ind(A) 1. In particular ≤ − mcd(X) ind(A) 1. ≤ − Proof. According to Theorem 6.5 a generalized Brauer–Severi variety X is split if and only if rdim(X) = 0. It is a general fact that X splits if and only if ind(A) = 1. Therefore, rdim(X) = ind(A) 1 = 0. This covers the case ind(A) = 1. So it remains to prove the − assertion for r := ind(A) 2, 3 . By Proposition 6.14 one has rdim(X) ind(X) 1= ∈ { } ≤ − r 1. Assume by contradiction that rdim(X) genus. Note that Db(C) ∈ { } cannot be present because K0(X) is torsion free. Using the described semiorthogonal decomposition (that we get assuming rdim(X) = 1), we see that the noncommutative motive U(perf dg(X)) decomposes as

n m

(19) U(perf (X)) U(Ki) U(Dj ) dg ≃ ⊕ i=1 j=1 ! M M (20) U(A⊗d(a)) ≃ a M 18

for suitable central division algebras Dj with ind(Dj ) 1, 2 and suitable separable ⊕s ∈ { } extensions Ki. Note that K0(X) Z and therefore n + m = s. After base change ¯ Z⊕s≃ b to k we also have K0(Xk¯) . Now for any D (Ki) we obtain after base change b ri b ≃ n n D (Ki)¯ D (k¯). Hence ri + m = s. But this implies ri = n and k ≃ q=1 i=1 i=1 therefore r = 1. The above isomorphisms (19) and (20) then give i Q P P n m ⊗d(a) U(k) U(Dj ) U(A ) ⊕ ≃ i=1 j=1 ! a M M M Then by Theorem 4.3 we conclude that A must be split or that A must be Brauer- equivalent to Dj for some j. This contradicts r = 3 and completes the proof. For involution varieties, we have: Theorem 6.16. Let X be one of the following varieties: (i) a twisted quadric associated to a central simple algebra (A,σ) with involution of orthogonal type having trivial discriminant δ(A,σ). (ii) a twisted quadric associated to a central simple R-algebra (A,σ) with involution of orthogonal type. Then rdim(X) ind(A) 1. Moreover, if ind(A) 3, then rdim(X) = ind(A) 1. In ≤ − ≤ − particular mcd(X) ind(A) 1. ≤ − Proof. Note that we have a semiorthogonal decomposition (see [12]) (21) Db(X)= Db(k), Db(A), ..., Db(k), Db(A), Db(C(A,σ)) . h i Let us first consider the case where X is a twisted quadric associated to a central simple R-algebra (A,σ) with involution of orthogonal type. In this case, A Mm(H) and C(A,σ) ≃ is either a central simple C-algebra or the product of two central simple R-algebras. If b C(A,σ) Mn(C) take Y = Spec(C). Then we see that D (C(A,σ)) can be embedded b ≃ b into D (Y ) as an admissible subcategory. If C(A,σ) A1 A2, we have D (C(A,σ)) = b b ≃ × D (A1) D (A2). So there is a semiorthogonal decomposition × b b b b b b b D (X)= D (k), D (A), ..., D (k), D (A), D (A1), D (A2) . h i Since A1 and A2 are central simple R-algebras, it is easy to check that there are smooth b projective connected varieties Y1 and Y2 of dimension at most one such that D (Ai) is b b embedded into D (Yi) as an admissible subcategory. And since D (Mm(H)) is admissible in Db(Z), where Z is the Brauer–Severi curve associated to H, we conclude rdim(X) ≤ 1 = ind(A) 1. Now let X be a twisted quadric associated to a central simple algebra − (A,σ) with involution of orthogonal type having trivial discriminant δ(A,σ). In this case, C(A,σ) is the product A1 A2 of two central simple k-algebras. So there is a × semiorthogonal decomposition b b b b b b b D (X)= D (k), D (A), ..., D (k), D (A), D (A1), D (A2) . h i After base change to some splitting field L of A, this semiorthogonal decomposition gives rise to the full exceptional collection from Kapranov [21] on the smooth quadric X k L. b b ⊗ This in particular implies that D (A1) k L D (L). The same for A2. According to ⊗ ≃ [3], Corollary 1.19 this implies that A1 and A2 are split by L. Now take L to be a separable splitting field of degree ind(A). Let Zi be the Brauer–Severi variety associated to Di, where

Di is the central division algebra for which Ai Mli (Di) according to the Wedderburn b ≃ theorem. Now we know that D (Ai) can be embedded as an admissible subcategory into b D (Zi). And since ind(Ai) ind(A), we obtain dim(Zi) ind(A) 1 for i = 1, 2. This ≤ ≤ − gives rdim(X) ind(A) 1. For the second statement of our theorem, we ﬁrst consider ≤ − the split case. So if A is split, i.e. if ind(A) = 1, we have rdim(X) = 0 according to Proposition 6.10 and Proposition 5.4. Therefore, rdim(X)=0=ind(A) 1. If ind(A) = 2, − the central simple algebra is non-split. So Propositions 5.4 and 6.10 imply 1 rdim(X). ≤ 19

On the other hand, the ﬁrst statement of our theorem gives rdim(X) ind(A) 1 = 1. ≤ − Hence rdim(X)=0=ind(A) 1. Finally, consider the case ind(A) = 3. Then rdim(X) − ≤ ind(A) 1 = 2. In what follows, we exclude rdim(X) 0, 1 . Since A is non-split, we − ∈ { } immediately have rdim(X) = 0. Now assume rdim(X) = 1. From [2], Proposition 6.1.6 6 and 6.1.10 we conclude that if rdim(X) = 1 there must be a semiorthogonal decomposition of Db(X) whose components are either Db(K), where K/k is a (ﬁnite) separable extension, or Db(D), where D is a central division algebra with ind(D) 1, 2 , or Db(C), where C b ∈ { } is a smooth k-curve of positive genus. Note that D (C) cannot be present because K0(X) is torsion free. Using the described semiorthogonal decomposition (that we get assuming rdim(X) = 1), we see that the noncommutative motive U(perf dg(X)) decomposes as n m

U(perf (X)) U(Ki) U(Dj ) dg ≃ ⊕ i=1 j=1 ! M M for suitable central division algebras Dj with ind(Dj ) 1, 2 and suitable separable ⊕r ∈ { } extensions Ki. Note that K0(X) Z and therefore n + m = r. After base change to k¯ Z⊕r ≃ b b we also have K0(Xk¯) . Now for any D (Ki) we obtain after base change D (Ki)k¯ ri b ¯ ≃n n ≃ q=1 D (k). Hence i=1 ri + m = r. But this implies i=1 ri = n and therefore ri = 1. Note that the semiorthogonal decomposition (21) forces the noncommutative motive to Q P P decompose as U(perf (X)) U(k) U(A) U(C(A,σ)). dg ≃ ⊕ ⊕ s t M M Since we assumed ind(A) = 3, C(A,σ) is the product of two central simple algebras A1 and A2 (note that case (ii) cannot appear since in this case index is two). But then we have n m

U(k) U(Dj ) U(k) U(A) U(A1) U(A2)) ⊕ ≃ ⊕ ⊕ ⊕ i=1 j=1 ! s t M M M M Then by Theorem 4.3 we conclude that A must be split or that A must be Brauer- equivalent to Dj for some j. This contradicts ind(A) = 3. This completes the proof.

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