introduction to motives and algebraic cycles on projective

Introduction to motives andalgebraic cycles on projective

homogeneous varieties

Stefan Gille, Victor Petrov, Nikita Semenov,

Kirill Zainoulline

Contents

1 Introduction 4

2 Algebraic groups and homogeneous varieties 52.1 Linear algebraic groups . . . . . . . . . . . . . . . . . . . . . . . 52.2 Root systems and Dynkin diagrams. . . . . . . . . . . . . . . . . 72.3 The root system of a split group . . . . . . . . . . . . . . . . . . 92.4 Classification of split groups . . . . . . . . . . . . . . . . . . . . . 102.5 Projective homogeneous varieties . . . . . . . . . . . . . . . . . . 112.6 Twisted forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Chow motives and Rost nilpotence 163.1 Chow groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Chow motives with coefficients . . . . . . . . . . . . . . . . . . . 183.3 Splitting fields and rational cycles . . . . . . . . . . . . . . . . . 203.4 Rost nilpotence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.5 Lifting of coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 243.6 Rost type motivic decompositions . . . . . . . . . . . . . . . . . . 283.7 Integral decompositions . . . . . . . . . . . . . . . . . . . . . . . 32

4 Motives of isotropic homogeneous varieties 334.1 Relative cellular spaces . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Bruhat decomposition . . . . . . . . . . . . . . . . . . . . . . . . 354.3 Hasse diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4 The case of an isotropic group . . . . . . . . . . . . . . . . . . . . 384.5 Motives of fibered spaces . . . . . . . . . . . . . . . . . . . . . . . 39

5 Motives of generically split homogeneous varieties. 415.1 The J-invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Properties of the J-invariant . . . . . . . . . . . . . . . . . . . . 435.3 Motives of complete flags . . . . . . . . . . . . . . . . . . . . . . 455.4 Motives of generically split flags . . . . . . . . . . . . . . . . . . . 52

2

CONTENTS 3

6 Splitting properties of linear algebraic groups 556.1 Tits indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.2 Rational cycles and rational bundles . . . . . . . . . . . . . . . . 556.3 The J-invariant in degree one and the Tits algebras . . . . . . . 556.4 Higher Tits indices . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7 Applications of motivic decompositions 567.1 Canonical dimensions . . . . . . . . . . . . . . . . . . . . . . . . 567.2 Degrees of splitting fields . . . . . . . . . . . . . . . . . . . . . . 577.3 Examples of decompositions . . . . . . . . . . . . . . . . . . . . . 587.4 Non-hyperbolicity of orthogonal involutions . . . . . . . . . . . . 627.5 Classification of algebras with involutions of small degrees . . . . 64

8 Cobordism cycles 658.1 Algebraic cobordism and mod-p operations . . . . . . . . . . . . 658.2 The main tool lemma . . . . . . . . . . . . . . . . . . . . . . . . 688.3 F4-varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

9 Cohomological invariants versus motivic invariants 759.1 Cohomological invariants . . . . . . . . . . . . . . . . . . . . . . 759.2 Basic correspondence of a splitting variety (after M. Rost) . . . . 759.3 Symbols and Rost motives . . . . . . . . . . . . . . . . . . . . . . 75

Chapter 1

Introduction

To be written

4

Chapter 2

Algebraic groups andhomogeneous varieties

2.1 Linear algebraic groups

2.1.1. Fix a base field F . An affine group scheme over F is an affine algebraicvariety G over F endowed with a group structure, i.e. endowed with an identityelement and two algebraic morphisms over F

mult : G×G→ G and inv : G→ G

defining the multiplication and the inverse respectively and satisfying the usualgroup axioms. Alternatively, using the language of functors of points an affinegroup scheme G can be identified with a functor

G : AlgF −→ Gr

from the category of commutative associative F -algebras to the category ofgroups represented by a commutative finitely generated Hopf algebra H over F .This identification is given by G = SpecH and G(−) = HomF (−, G).

2.1.2. Here are basic examples of affine group schemes

• The additive group Ga = SpecF [t];

• The multiplicative group Gm = SpecF [t, t−1].

• The group of n-th roots of unity µn = SpecF [t]/(tn − 1). This is a closedsubgroup of Gm.

• The general linear group GLn = SpecF [tij , D−1], where 1 ≤ i, j ≤ n andD is the determinant of the n× n matrix (tij).

5

6CHAPTER 2. ALGEBRAIC GROUPS AND HOMOGENEOUS VARIETIES

2.1.3. One can show that G is an affine group scheme over F if and only if itis a closed subgroup of a general linear group GLn over F . To stress this factwe call G by a linear algebraic group over F .

2.1.4. If the base field F has characteristic 0 any linear algebraic group G issmooth as a variety over F . In the language of functors of points it means that forany commutative F -algebra R and any nilpotent ideal I of R the induced mapG(R) → G(R/I) is surjective or, equivalently, the Hopf algebra representing G isreduced. For instance, the group G = µn is smooth if and only if n is invertiblein F .

2.1.5. A linear algebraic group G over F is called connected, if it is irreducible asan algebraic variety over F . For example, the orthogonal group On of isometryclasses of the quadratic form x2

1 + . . . + x2n is not connected. It consists of

two connected components, where the connected component of the identity isisomorphic to the special orthogonal group O+

n .

2.1.6. A connected non-trivial linear algebraic group G is called simple (resp.semisimple) if it does not have any non-trivial closed (resp. solvable) connectednormal subgroups over the algebraic closure Fa of F . For instance, the productof two copies of Gm is not semisimple, since it contains a copy of Gm as thediagonal. The group SLn or O+

n , n > 1 provides an example of a simple linearalgebraic group. A product of simple groups provides an example of a semi-simple group.

2.1.7. A linear algebraic group T over F is called a torus, if over the algebraicclosure Fa it becomes isomorphic to a product of several copies of Gm, i.e.

TFa' Gm,Fa

× . . .×Gm,Fa.

If this isomorphisms is defined already over F , then T is called a split torus.

2.1.8 Example. Let F ′/F be a finite separable field extension. Consider thefunctor

G : R→ (R⊗F F ′)×

which maps a commutative F -algebra R to the group of units of its base change.One can show that G is representable by a Hopf algebra and, therefore, definesa linear algebraic group over F denoted by RF ′/F (Gm). The usual norm mapinduces a morphism of group schemes RF ′/F (Gm) → Gm over F . Its kernelprovides an example of a non-split torus over F .

2.1.9. A semisimple linear algebraic group G over F is called split, if it containsa split maximal torus (maximal with respect to the inclusion). In particular, ifthe field F is algebraically closed, then all semisimple algebraic groups over Fare split.

In the present notes we will mainly study non-split simple groups over arbi-trary fields. Here is a typical example

2.1.10 Example. Let A be a central division algebra over F . Its group of auto-morphisms is a non-split simple linear algebraic group over F called a projectivegeneral linear group of A and denoted by PGLA.

2.2. ROOT SYSTEMS AND DYNKIN DIAGRAMS. 7

2.2 Root systems and Dynkin diagrams.

In the present section we introduce a combinatorial language which will be usedto classify all split semisimple linear algebraic groups.

2.2.1. Let V be a non-trivial R-vector space together with an Euclidean scalarproduct (−,−) : V ⊗V → R. Given a vector v ∈ V we denote by sv the reflectionorthogonal to v with respect to this product. By definition sv is a linear mapgiven by

sv : w 7→ w − 2(w, v)(v, v)

· v, w ∈ V.

A finite set Φ of non-zero vectors of V is called a root system if

• The set Φ spans the R-vector space V ;

• For all α ∈ Φ we have Rα ∩ Φ = ±α;

• If α, β ∈ Φ, then sα(β) ∈ Φ;

• For all α, β ∈ Φ we have 2 (β,α)(α,α) ∈ Z.

The elements of Φ are called roots. The group generated by reflections

W (Φ) = 〈sα | α ∈ Φ〉

is a finite group and is called the Weyl group of Φ.

2.2.2. Let Φ ⊂ V be a root system. Then there exists a subset Π ⊂ Φ of linearindependent vectors, called a basis of Φ, such that any root α ∈ Φ is a linearcombination of basis vectors with integer coefficients, where either all coefficientsare non-negative or non-positive. In the first case α is called a positive root andin the second case it is called a negative root.

The number of elements of Π is called the rank of Φ and is denoted by n.Observe that n = rkV . The elements of Π = α1, . . . , αn are called simpleroots. The corresponding generators si := sαi

of the Weyl group W (Φ) arecalled simple reflections.

2.2.3. Chosen a basis Π = α1, . . . , αn of the root system Φ we define anoriented graph called a Dynkin diagram of Φ as follows.

• The vertices are in one-to-one correspondence with elements of Π.

• Given two different vertices αi, αj ∈ Π we connect them by the followingnumber of edges:

number of edges angle between αi and αj

1 120

2 135

3 150

0 otherwise


• If (αi, αi) 6= (αj , αj), then we orient the edges between αi and αj asfollows:

toward αi if (αi, αi) < (αj , αj)toward αj otherwise

2.2.4. It can be shown that the Dynkin diagram of Φ does not depend onthe choice of the basis Π ⊂ Φ and, moreover, uniquely determines Φ. TheDynkin diagram corresponding to a root system Φ will be denoted by D(Φ).In particular, the Weyl group W (Φ) can be restored from the Dynkin diagramD(Φ) as follows:

Let nij ∈ 0, 1, 2, 3 denote the number of edges between two different ver-tices αi, αj ∈ Π. Then

W (Φ) ' 〈s1, . . . , sn | s2i = 1, (sisj)nij+2 = 1 if nij < 3 and (sisj)6 = 1 if nij = 3〉

2.2.5. A root system Φ ⊂ V is called irreducible, if V can not be decomposedas V = V1 ⊥ V2 such that Vi ∩ Φ is a root system in Vi, i = 1, 2.

2.2.6. The key result concerning root systems says that all possible Dynkindiagrams of irreducible root systems are classified and can be subdivided intofour classical types An (n ≥ 1), Bn (n ≥ 2), Cn (n ≥ 2), Dn (n ≥ 3) and fiveexceptional types E6, E7, E8, F4 and G2. Here the lower index means the rankof the respective root system. In the sequel we say that a root system Φ hastype D if and only if the respective Dynkin diagram has type D.

2.2.7. Our enumeration of vertices of Dynkin diagrams follows Bourbaki andlooks as follows:

An : •1 •2 •3 •n−1 •n G2 : •1 < •2

Bn : •1 •2 •n−2 •n−1> •n F4 : •1 •2 > •3 •4

Cn : •1 •2 •n−2 •n−1< •n

Dn : •n−1

•1 •2 •n−3 •n−2

xxxxxxx

KKKKKK

K

•n

E6 : •1 •3 •4 •5 •6

•2

E7 : •1 •3 •4 •5 •6 •7

•2

E8 : •1 •3 •4 •5 •6 •7 •8

•2

2.2.8. Let Φ be a root system in V . For each α ∈ Φ there is a linear map

α∨ : V → R given by v 7→ 2(v, α)(α, α)

called the coroot of α. The set of coroots Φ∨ is again a root system in the dualspace V ∗ := HomR(V,R). It is called the dual root system of Φ.

2.3. THE ROOT SYSTEM OF A SPLIT GROUP 9

There are two important lattices in V associated to the root system Φ. Thefirst lattice Λr is given by the Z-linear span of Φ and is called the root lattice ofΦ. The second lattice Λw is given by

Λw := v ∈ V | α∨(v) ∈ Z ∀α ∈ Φ

and is called the weight lattice of Φ. We have Λr ⊂ Λw and the quotient Λw/Λr

is a finite abelian group.

2.3 The root system of a split group

Let G be a split group over a field F . We associate a root system to G as follows

2.3.1. Choose a split maximal torus T inside G. We denote the character andthe cocharacter groups of T by T∗ = Hom(T,Gm) and T∗ = Hom(Gm,T),respectively. The abelian groups T∗ and T∗ are isomorphic to Zn for somenumber n, called the rank of G, and there is a perfect pairing

〈−,−〉 : T ∗ × T∗ −→ Z = Hom(Gm,Gm) ,

which is defined on points by the equation

χ(λ(x)) = x〈χ,λ〉 for all χ ∈ T∗, λ ∈ T∗ and x ∈ k×.

2.3.2. Consider the adjoint representation Ad: G→ GL(L) of the affine groupscheme G on the associated Lie algebra L = L(G) over F . A character α ∈ T∗is called a weight of the adjoint representation if the respective weight subspaceLα defined on F -points by

Lα(F ) := v ∈ L(F ) |Ad(t)v = α(t)v ∀t ∈ T(F )

is non-trivial. Note that all non-trivial weight subspaces are one dimensionaland there is a direct sum decomposition L =

⊕weights α Lα over F .

We denote the set of all non-zero weights by Φ(G,T).

2.3.3. We define the Weyl group W (G) of G to be the quotient

W (G) = NG(T)/T,

where NG(T) is the normalizer of T in G. It is a constant finite group schemewhich acts on T by the conjugation, i.e. w(t) := w · t · w−1, where w is arepresentative of w in NG(T). And it also acts on T∗ and T∗ by

w(χ)(t) := χ(w−1(t)) and w(λ)(ξ) := w(λ(ξ)) , where w ∈W,χ ∈ T∗, λ ∈ T∗.

By definition W (G) permutes elements of Φ(G,T) ⊂ T∗ and preserves thepairing 〈−,−〉, i.e. 〈w(χ), w(λ)〉 = 〈χ, λ〉.


2.3.4. Consider a vector space T∗R := R⊗Z T∗ with a scalar product ρ. Definea new scalar product (−,−) as

(x, y) :=∑

w∈W (G)

ρ(w(x), w(y)),

where the action of W (G) on T∗ is as above. Then the set Φ(G,T) forms a rootsystem in T∗R with respect to (−,−). It can be shown that Φ(G,T) does notdepend on the choice of the split torus T. This justifies to call Φ(G) := Φ(G,T)a root system of G.

2.3.5. One of the main observations concerning W (G) is that it is isomorphicto the Weyl group of the root system Φ(G) of G, i.e.

W (G) 'W (Φ(G)).

2.3.6. As for lattices, consider the root lattice Λr and the weight lattice Λw ofthe root system Φ(G). Then we have Λr ⊆ T∗ ⊆ Λw and the respective dualroot system Φ(G)∨ can be identified with a subset of T∗ in such a way thatα∨(β) = 〈β, α∨〉.

2.4 Classification of split groups

It turns out that a root system together with an intermediate lattice subgroupclassifies all split groups over a field F .

2.4.1. Namely, for any root system Φ and any (additive) subgroup Λ, Λr ⊆Λ ⊆ Λw, there exists a split group G over F such that (Φ(G),T∗) ' (Φ,Λ).Moreover, two split groups G1 and G2 are isomorphic over F if and only if(Φ(G1),T∗1) and (Φ(G2),T∗2) are isomorphic.

2.4.2. Observe that a split group G is simple over F if and only if the respectiveroot system Φ(G) is irreducible. If T∗ = Λr then G is called adjoint, and ifT∗ = Λw then G is called simply connected.

2.4.3. In general, a surjective morphism ϕ : H → G of group schemes over Fis called a (central) isogeny if for any F -algebra R the kernel of ϕ(R) : H(R) →G(R) is finite and central in H(R). Two group schemes G, G′ over F are calledisogeneous if there is a group scheme H over F and two isogenies ϕ : H → Gand ϕ′ : H → G′.

Note that two split algebraic groups G, G′ over F are isogeneous if andonly if their root systems are isomorphic, i.e. they have the same Dynkin dia-grams. Moreover, all isogenies G → G′ with fixed G are uniquely determinedby subgroups of the quotient T∗/Λr. In particular, for any group G there ex-ists a simply connected group Gsc and an adjoint group Gad together with twoisogenies Gsc → G→ Gad.

2.4.4 Example. We now provide a list of all split simple groups of classicaltypes:

2.5. PROJECTIVE HOMOGENEOUS VARIETIES 11

An: Here Λw/Λr ' Z/(n+ 1)Z and G = SLn+1 /µl, where l divides n+ 1.For l = n + 1 we obtain the projective linear group Gad = PGLn+1 (adjoint),and for l = 1 the special linear group Gsc = SLn+1 (simply connected).

Bn: In this case Λw/Λr ' Z/2Z has only two subgroups. The trivial sub-group corresponds to the special orthogonal group O+

2n+1 (adjoint) while thewhole group corresponds to the spinor group Spin2n+1 (simply connected).

Cn: As in the previous case Λw/Λr ' Z/2Z and, therefore, there are twoisogenies: the projective symplectic group PGSp2n (adjoint) and the symplecticgroup Sp2n (simply connected).

Dn: In this case Λw/Λr depends on n.

• If n is odd, then Λw/Λr ' Z/4Z and the respective isogenies are givenby the projective orthogonal group PGO+

2n (adjoint), the special orthogo-nal group O+

2n (corresponds to the subgroup Z/2Z) and the spinor groupSpin2n (simply connected).

• If n is even, then Λw/Λr ' Z/2Z× Z/2Z and the respective isogenies aregiven by PGO+

2n (adjoint), two semispinor groups Ss±2n (corresponding tothe subgroups Z/2Z×1 and 1×Z/2Z respectively), the special orthogonalgroup O+

2n (the diagonal subgroup Z/2Z) and the spinor group Spin2n

(simply connected).

2.4.5. A celebrated result due to Chevalley says that given an irreducible rootsystem Φ and a lattice subgroup Λr ⊂ Λ ⊂ Λw there exists a group scheme Gdefined over Spec Z called a Chevalley group such that for any split simple linearalgebraic group G over F with (Φ(G),T∗) ' (Φ,Λ) there is an isomorphismG ' G×Spec Z SpecF .

2.5 Projective homogeneous varieties

In the present section we study projective homogeneous G-varieties, where G isa split group over a field F . We start with the following general definition.

2.5.1. Let G be a group scheme over a field F . A variety X over F is called ahomogeneous G-variety if there is a morphism ρ : G ×X → X of varieties overF such that

ρ(g · f, x) = f(g(x)) for all f , g ∈ G(Fa), x ∈ X(Fa),

where Fa denotes the algebraic closure of F , and the action of G(Fa) on X(Fa)is transitive, i.e. for any x, y ∈ X(Fa) there exists g ∈ G(Fa) such that g ·x = y,where g · x denotes ρ(g, x).

From now on let G be a split group over F . We fix a split maximal torusT ⊂ G and the respective set of simple roots Π.


2.5.2. It is known that the set of isomorphism classes of projective homogeneousG-varieties is in one-to-one correspondence with the set of conjugacy classes ofstabilizer subgroups of G. Namely, an F -point x of a projective homogeneousvariety X corresponds to its stabilizer subgroup Gx and for any two F -points xand y of X we have Gy = g−1Gxg, where g ∈ G(F ) is such that g · x = y.

On the other hand this set of conjugacy classes is in one-to-one correspon-dence with the set of subsets of Π. Namely, given a subset Θ ⊂ Π we define therepresentative PΘ of a conjugacy class as follows: We set PΘ to be the subgroupof G generated by T and all unipotent subgroups corresponding to all positiveroots and to all roots in the linear span of Θ with no Θ terms.

Here by the unipotent subgroup corresponding to α we mean the image ofan embedding xα : Ga → G such that txα(ξ)t−1 = xα(α(t)ξ) for all t ∈ T(F ),ξ ∈ Gm(F ).

2.5.3. The representative PΘ is called the standard parabolic subgroup of G. Inparticular, if Θ = ∅, the respective P∅ is called the Borel subgroup of G. Wehave T ⊂ P∅ ⊂ G.

By Pi1,...,ir we denote the standard parabolic subgroup PΘ′ of the comple-mentary subset Θ′ = Θ r αi1 , . . . , αir

.

2.5.4. Observe that if we choose another T′ inside G, then the respective rep-resentatives PΘ and P ′Θ will be conjugate to each other. Therefore, the cor-respondence above doesn’t depend on the choice of a split maximal torus T.Moreover, since any two isogeneous split groups have the same root systems,it doesn’t depend on the isogeny class of G. In particular, when dealing withprojective homogeneous G-varieties we may always assume that G is adjoint.

2.5.5. Summarizing the above discussion, all projective homogeneousG-varietiesare classified (up to an isomorphism) by subsets Θ of the set of vertices of theDynkin diagram of G. An isomorphism class of projective homogeneous G-varieties corresponding to a subset Θ will be denoted by D/PΘ, where D is thetype of the root system of G.

2.5.6 Example. We now provide basic examples of projective homogeneous G-varieties, where G is a split simple linear algebraic group over F (enumerationof roots follows 2.2.7):

An: We have An/P1 ' An/Pn ' Pn and, more generally, An/Pi ' An/Pn−i 'Gr(i, n+1), where Gr(i, n+1) is the Grassmannian of i-dimensional linear sub-spaces in An+1.

The variety An/P1,n is called the incidence variety. A point on this varietyis given by the pair (l,H), where l is a line and H is a hyperplane in An+1 suchthat l ⊂ H. In geometric terms it is given by the equation

∑ni=0 xiyi = 0 in

Pn × Pn, where xi (resp. yi) are the projective coordinates of the first (resp.second) factor.

Finally, An/P∅ is isomorphic to the variety of complete flags: points aregiven by n-tuples of linear subspaces (l1, l2, . . . , ln) in An+1 such that dim li = iand l1 ⊂ l2 ⊂ . . . ⊂ ln.

2.6. TWISTED FORMS 13

Bn, Dn: The variety Bn/P1 (resp. Dn/P1) is isomorphic to a smoothprojective quadric of dimension 2n− 1 (resp. 2n− 2).

The variety Bn/Pn (resp. Dn/Pn or Dn/Pn−1) is isomorphic to a (resp. aconnected component of) maximal orthogonal Grassmannian that is a variety ofmaximal totally-isotropic linear subspaces in the quadratic space of rank 2n+1(resp. 2n).

G2, F4 and E6: The variety G2/P2 is isomorphic to a 5-dimensionalsmooth projective quadric and the variety G2/P1 is a 5-dimensional Fano vari-ety. The variety E6/P6 is isomorphic to the so called Cayley plane OP2 that isthe octonionic projective plane of dimension 16 (see [IM05]). The variety F4/P4

can be identified with a hyperplane section of E6/P6.

2.6 Twisted forms

In the present section we recall the notions of twisted forms of Chevalley groupsand study the respective projective homogeneous varieties.

2.6.1. A group scheme G over F is called a twisted form of a linear algebraicgroup G′ if there is an isomorphism of group schemes G×F Fa ' G′×F Fa overthe algebraic closure Fa. The general descent theory provides an isomorphismof pointed sets

H1et(F,AutF (G′)) ' TwistF (G′)

ξ ∈ Z1(F,AutF (G′)) 7→ G = ξG′

between the set of cohomology classes of the first etale cohomology group of theautomorphism group scheme AutF (G′) and the set of twisted forms of G′.

Since any simple linear algebraic group G becomes split over Fa, it can beviewed as a twisted form of the respective Chevalley group G, i.e. as an elementof H1

et(F,AutF (G)).

2.6.2. Consider an action of the adjoint group Gad on G by means of innerautomorphisms Gad → AutF (G). It induces the map

Inn : H1et(F, G

ad) → H1et(F,AutF (G)).

A linear algebraic group G is called an inner form of G if it is a twisted form ofG such that the corresponding cohomology class lies in the image of this map.

The isogeny Gsc → Gad, where Gsc is the simply connected group, inducesthe map

s : H1et(F, G

sc) → H1et(F, G

ad).

Twisted forms that correspond to cohomology classes in the image of the com-position Inn s are called strongly inner forms of G.


2.6.3. A twisted form of G is called an outer form, if the respective cohomologyclass is not in the image of the map Inn. Such cohomology classes correspondto non-trivial graph-automorphisms of the Dynkin diagram of G. In particular,if D(G) has no non-trivial automorphisms, then the map Inn is surjective. Thisimplies that outer forms exist only for groups of types An, Dn and E6.

We will write the order of the respective graph-automorphism of D(G) as anupper-left index. For example, the notation 1An means an inner form of a groupof type An, and 6D4 means an outer form of a group of type D4 correspondingto an automorphism of order 6.

In general a twisted form G of a Chevalley group G is not necessary a splitgroup. To measure how far is G from being split we use the Tits index of G.

2.6.4. Let G be an inner form of G. The Tits index of G is a set of Dynkindigrams of G where certain vertices are circled. Each circled vertex correspondsto the copy of a split one dimensional torus Gm sitting inside G. For instance,if all vertices are circled then the group G is split. If non of them are circled,then the group G is called anisotropic. If at least one of the vertices is circled,then the group G is called isotropic. For each type of G the list of all possibleTits (diagrams) indices can be found in the table (see [Ti66]).

2.6.5 Example. A group G of type F4 can have three possible Tits indices:

(i) •1 •2 > •3 •4

(ii) •1 •2 > •3⊙•4

(iii)⊙•1

⊙•2 >

⊙•3

⊙•4

where the index (i) corresponds to the anisotropic case, (ii) to the non-splitisotropic case and (iii) to the split case.

2.6.6. Consider a projective homogeneous G-variety X over F . By the very def-inition the variety X becomes isomorphic over Fa to a projective homogeneousG-variety corresponding to a standard parabolic subgroup of G and, therefore,to a subset Θ of the Dynkin diagram of G. To stress this fact we will denote Xas ξD/PΘ, where G = ξG and D is the type of G.

2.6.7. The projective homogeneous G-variety X = ξD/PΘ has a rational pointif and only if all non-circled vertices of the Tits index of G belong to the subsetΘ. Moreover, a projective homogeneous variety is rational if and only if it hasa rational point.

2.6.8 Example. We now provide a list of examples of projective homogeneousG-varieties over F , where G is a twisted form of an adjoint split group G.

An: Let G = PGLn+1 be the projective linear group. The pointed setH1

et(F,PGLn+1) is isomorphic to the set of isomorphism classes of central simplealgebras A of degree n+ 1 over F . Moreover, cohomology classes in the imageof

H1et(F,SLn+1 /µr) → H1

et(F,PGLn+1), r | n+ 1,

2.6. TWISTED FORMS 15

correspond to central simple algebras A of index r. In particular, PGLn+1 hasno non-trivial strongly inner forms.

For an algebra A the respective inner twisted form G is given by the auto-morphism group PGLA = AutF A of A (see Example 2.1.10). The respectiveprojective homogeneous G-variety X can be identified with the variety of flagsof (right) ideals in A. For instance, if X = A(An/Pi), then X ' SBi(A) is thegeneralized Severi-Brauer variety of ideals of reduced dimension i in A.

The set of outer forms of PGLn+1 is in one-to-one correspondence with theset of isomorphism classes of central simple algebras with unitary involutions.In this case any projective homogeneous varieties becomes isomorphic over Fa

to the variety An/Pi1,...,im, where ij = n+ 1− im−j+1 for all j = 1 . . .m.

Bn,Dn: Let G be the projective orthogonal group. Depending on the typeG is either O+

2n+1 or PGO+2n. We assume char(F ) 6= 2.

In the first case (Bn) the pointed set H1et(F,O

+2n+1) is isomorphic to the set

of isometry classes of quadratic forms q of rank 2n + 1. The respective innerform G is given by the special orthogonal group O+(q) of q and the respectiveprojective homogeneous G-variety X = q(Bn/P1) is the projective quadric givenby the equation q = 0. Observe that for O+

2n+1 all twisted forms are inner.In the second case (Dn) the pointed set H1

et(F,PGO+2n) is parametrised by

the set of isomorphism classes of certain central simple algebras A with or-thogonal involutions σ. For an inner form G = (A,σ) PGO+

2n the projectivehomogeneous G-variety X = (A,σ)Dn/P1 is a projective quadric if the algebra Ais split and is an involution variety if A is non-split. More precisely, if A is split,then the variety X is given by the equation q = 0, where q is the quadratic formof rank 2n associated to the involution σ. All outer forms of PGO+

2n correspondto algebras with involutions which have non-trivial discriminant.

Pfister case: An important example of a twisted inner form of the pro-jective orthogonal group PGO+

2n (resp. O+2n+1) is given by the so called Pfister

quadratic form φ (resp. its maximal neighbor). By an n-fold Pfister form φ wecall the tensor product of n binary quadratic forms φ =

⊗ni=1〈1,−ai〉, ai ∈ F ,

where the notation 〈1,−ai〉 stands for the form x2 − aiy2.

Chapter 3

Chow motives and Rostnilpotence

3.1 Chow groups

In the present section we introduce the notion of a Chow ring of an algebraicvariety X and provide some of its properties. We follow the notation from thebook [Fu98].

3.1.1. By an algebraic variety over a field F we mean a reduced scheme of finitetype over a field F . By an algebraic cycle of codimension m on a variety X wemean a finite formal sum

∑V nV [V ] of irreducible subvarieties of codimension m

of X taken with integral coefficients. The group (with respect to the addition)of algebraic cycles of codimension m on X modulo the rational equivalencerelation is called the Chow group of codimension m cycles on X and is denotedby CHm(X). We will also use the dimension (low index) notation meaningthat CHm(Xi) = CHdim Xi−m(Xi) for each irreducible component Xi of X.Replacing the integral coefficient ring by a commutative ring Λ we obtain theChow group with coefficients in Λ denoted by CHm(X; Λ).

An equivalent definition of CH(X) can be given using the Gersten complexfor the K-theory. Namely, we define CHm(X) to be the cokernel of the residuemap dX

CHm(X) = coker( ∐

x∈X(m−1)

K1(F (x)) dX−−→∐

x∈X(m)

K0(F (x))),

where x ∈ X(m) denotes a point of codimension m in X, F (x) denotes itsresidue field, K1(F (x)) is equal to the group of invertible elements F (x)× andK0(F (x)) = Z.

3.1.2. The assignment X 7→ CHm(X) satisfies the following two importantproperties:

16

3.1. CHOW GROUPS 17

• Given a projective morphism between two varieties f : Y → X there isan induced group homomorphism which preserves the dimension of cyclesf∗ : CHm(Y ) → CHm(X) called the push-forward.

• Given a flat morphism f : Y → X there is an induced group homomor-phism in the opposite direction which preserves the codimension of cyclesf∗ : CHm(X) → CHm(Y ) called the pull-back. Moreover, according to[Fu98, §6] if X and Y are smooth then the pull-back exists for any mor-phism f .

The latter means that the assignment X 7→ CHm(X) can be viewed as a con-travariant functor with respect to pull-backs from the category of smooth vari-eties over a field F to the category of graded abelian groups.

3.1.3. By [Fu98] the Chow group CH(X) has a product structure, which turnsCH(X) into a graded commutative ring. This (intersection) product has thefollowing properties:

• It is preserved by pull-backs, i.e. f∗ is a ring homomorhism;

• There is the projection formula

f∗(f∗(α) · β) = α · f∗(β), where α ∈ CH(X), β ∈ CH(Y ).

3.1.4. Let i : Z → X be a closed subvariety and j : U = X \ Z ⊂ X, be itsopen complement. Then there is an exact sequence of Chow groups preservingthe low indices

CHm(Z) i∗−→ CHm(X)j∗−→ CHm(U) → 0

called the localization sequence (see [Fu98, Prop. 1.8]). In particular, the in-duced pull-back f∗ : CH(X) → CH(SpecF (X)) is surjective, where SpecF (X)is the generic point of X.

Let Y be a vector bundle over X with the projection f : Y → X. Thenthe induced pull-back f∗ : CH(X)

∼=−→ CH(Y ) is an isomorphism (see [Fu98,Thm. 3.3.(a)]). This property is usually called the homotopy invariance forChow groups.

3.1.5. Assume that the Chow group CH(X) is a free Z-module. Then there isthe Kunneth decomposition CH(X ×X) = CH(X)⊗Z CH(X) and the Poincareduality (see [KM06, Rem. 5.6]). The latter means that for a given Z-basis ofCH(X) there is a dual one with respect to the pairing α ⊗ β 7→ deg(α · β),where deg : CH(X) → Z denotes the push-forward p∗ induced by the structuremorphism p : X → SpecF and is called the degree map.

3.1.6. Consider the Chern character ch : Ko(X) → CH(X)⊗Q, where Ko(X)is the Grothendieck group of vector bundles on X (see [Fu98, (15.1.2)]). Bydefinition it respects pull-backs and by the Riemann-Roch theorem (see [Fu98,Example 15.2.16.(b)]) it induces an isomorphism Ko(X)⊗Q → CH(X)⊗Q.

18 CHAPTER 3. CHOW MOTIVES AND ROST NILPOTENCE

3.2 Chow motives with coefficients

Let X be a variety over a field F . We say X is essentially smooth over Fif it is an inverse limit of smooth varieties Xi over F taken with respect toopen embeddings. Let CHm(X; Λ) = CHm(X) ⊗Z Λ denote the Chow groupof codimension m cycles on X with coefficients in a commutative ring Λ. If Xis essentially smooth, then CHm(X; Λ) = lim−→CHm(Xi; Λ), where the limit istaken with respect to the pull-backs induced by open embeddings.

In the present section we introduce the category of Chow motives over anessentially smooth variety X with Λ-coefficients. Our arguments follow thepapers [Ma68] and [VZ08].

3.2.1. First, we define the category of correspondences C(X; Λ). The objects ofC(X; Λ) are smooth projective maps Y → X. The morphisms are given by

Hom([Y → X], [Z → X]) = ⊕i CHdim(Zi/X)(Y ×X Zi; Λ),

where the sum is taken over all irreducible components Zi of Z of relativedimensions dim(Zi/X). The composition of two morphisms is given by theusual correspondence product

ψ φ = (pY,T )∗((pY,Z)∗(φ) · (pZ,T )∗(ψ)

),

where φ ∈ Hom([Y → X], [Z → X]), ψ ∈ Hom([Z → X], [T → X]) andpY,T , pY,Z , pZ,T are projections Y ×X Z ×X T → Y ×X T , Y ×X Z, Z ×X T .The category C(X; Λ) is a tensor additive category, where the direct sum isgiven by [Y → X] ⊕ [Z → X] := [Y

∐Z → X] and the tensor product by

[Y → X]⊗ [Z → X] := [Y ×X Z → X] (cf. [Ma68, §2-4]). As usual we denoteby φt ∈ CH(Z ×X Y ; Λ) the transposition of a cycle φ ∈ CH(Y ×X Z; Λ).

3.2.2 Example. Assume we have the Kunneth decomposition, i.e., the map

CH(Y ; Λ)⊗CH(X;Λ) CH(Z; Λ)p∗Y (−)·p∗Z(−)−−−−−−−−→ CH(Y ×X Z; Λ)

is an isomorphism. Then the respective correspondence product in C(X; Λ) isgiven by the formula

(α1 ⊗ β1) (α2 ⊗ β2) = (pZ)∗(α1β2) · (α2 ⊗ β1).

3.2.3. The category of effective Chow motives Choweff (X; Λ) can be definedas the pseudo-abelian completion of C(X; Λ). Namely, the objects are pairs(U, ρ), where U is an object of C(X; Λ) and ρ ∈ EndC(X;Λ)(U) is a projector, i.e.ρ ρ = ρ. The morphisms between (U1, ρ1) and (U2, ρ2) are given by the groupρ2 HomC(X;Λ)(U1, U2) ρ1. The composition of morphisms is induced by thecorrespondence product. In the case X = Spec(F ) and Λ = Z we obtain theusual category of effective Chow motives over F with integral coefficients (cf.[Ma68, §5]).

3.2.4 Example. Consider the projective line P1 over F . The projector ρ =[Spec(F )×P1] ∈ CH1(P1×P1) defines an object (P1, ρ) in Choweff (F,Z) calledthe Tate motive over F and denoted by Z1 (cf. [Ma68, §6]).

3.2.5. We have two types of restriction functors.

3.2. CHOW MOTIVES WITH COEFFICIENTS 19

1) For any morphism f : X1 → X2 of essentially smooth varieties we havea tensor additive functor

resX2/X1 : C(X2; Λ) → C(X1; Λ)

given on the objects by [Y2 → X2] 7→ [Y2 ×X2 X1 → X1] and on the morphismsby φ 7→ (id × f)∗(φ), where id × f : (Y2 ×X2 Z2) ×X2 X1 → Y2 ×X2 Z2 is thenatural map. It induces a functor on pseudo-abelian completions

resX2/X1 : Choweff (X2; Λ) → Choweff (X1; Λ).

2) For any homomorphism of commutative rings h : Λ → Λ′ we have atensor additive functor

resΛ′/Λ : C(X,Λ) → C(X; Λ′)

which is identical on objects and is given by id⊗h : CH(Y ×XZ; Λ) → CH(Y ×X

Z; Λ′) on morphisms. Again, it induces a functor on pseudo-abelian completions

resΛ′/Λ : Choweff (X; Λ) → Choweff (X; Λ′).

Observe that the functor resΛ′/Λ commutes with resX2/X1 . We denote byresX2/X1,Λ′/Λ the composite resX2/X1 resΛ′/Λ. To simplify the notation weommit X1 (resp. Λ), if X1 = SpecF (resp. Λ = Z).

3.2.6. Let f : X → SpecF and h : Z → Λ be the structure maps. ThenresX,Λ : Choweff (F ; Z) → Choweff (X; Λ). Given a motive N over F we de-note by NX,Λ its image resX,Λ(N) in Choweff (X; Λ). The image Z1X,Λ ofthe Tate motive is denoted by T and is called the Tate motive over X. Let Mbe a motive from Choweff (X; Λ) and l ≥ 0 be an integer. The tensor productM ⊗ T⊗l is denoted by Ml and is called the twist of M .

The same arguments as in the proof of [Ma68, Lemma of §8] show that forany motives U and V from Choweff (X; Λ) and l ≥ 0 the natural map

HomChoweff (X;Λ)(U, V ) → HomChoweff (X;Λ)(Ul, V l) (3.1)

given by φ 7→ φ⊗ idT is an isomorphism.

3.2.7. We define the category Chow(X; Λ) of Chow motives over X with Λ-coefficients as follows. The objects are pairs (U, l), where U is an object ofChoweff (X; Λ) and l is an integer. The morphisms are given by

Hom((U, l), (V,m)) := limN→+∞

HomChoweff (X;A)(UN + l, V N +m).

This is again a tensor additive category, where the sum and the product aregiven by

(U, l)⊕ (V,m) := (Ul − n ⊕ V m− n, n), where n = min(l,m),


(U, l)⊗ (V,m) := (U ⊗ V, l +m).

Observe that the Tate motive T is isomorphic to ([id : X → X], 1) and, hence,it is invertible in (Chow(X; Λ),⊗). Moreover, we can say that Chow(X; Λ) isobtained from Choweff (X; Λ) by inverting T (cf. [Ma68, §8]).

According to (3.1) the natural functor Choweff (X; Λ) → Chow(X; Λ) givenby U 7→ (U, 0) is fully faithful and the restriction resX,Λ descend to the respec-tive functor resX,Λ : Chow(F ) → Chow(X; Λ).

3.2.8. For a smooth projective morphism Y → X we denote by M(Y → X)its effective motive ([Y → X], id) considered as an object of Chow(X; Λ). IfX = SpecF , then we denote the motive M(Y → X) simply by M(Y ; Λ). IfΛ = Z, then we denoteM(Y ; Z) byM(Y ) having in mind the integer coefficients.By definition there is a natural identification

HomChow(X;Λ)(M(Y → X)i,M(Z → X)j) = CHdim(Z/X)+j−i(Y ×X Z; Λ).

3.2.9. Let M be an object of Chow(X; Λ). We define the Chow group with lowindex CHm(M) of M as

CHm(M) := HomChow(X;Λ)(Tm,M)

and the Chow group with upper index CHm(M) as

CHm(M) := HomChow(X;Λ)(M,Tm).

Observe that if M = M(Y → X), then we obtain the usual Chow groupsCHdim(Y/X)−m(Y ; Λ) and CHm(Y ; Λ) of a variety Y . A composite with amorphism f : M → N induces a homomorphism between the Chow groupsRm(f) : CHm(M) → CHm(N) and Rm(f) : CHm(N) → CHm(M) called therealization map.

3.2.10. Assume that a motive M is a direct sum of twisted Tate motives. Inthis case its Chow group CH(M) is a free abelian group. We define its Poincarepolynomial as

P (M, t) =∑i≥0

aiti,

where ai is the rank of CHi(M).

3.3 Splitting fields and rational cycles

3.3.1 Definition. Let M be an object of Chow(F,Λ). We say that M is Λ-splitover a field extension L/F if ML = resL(M) is isomorphic to a direct sum oftwisted Tate motives over L. Such a field L will be called a Λ-splitting field ofM . In particular, we say that a smooth projective variety X is Λ-split over afield extension L/F if and only if the motive M(X; Λ) is split over L.

Clearly, if M is Z-split over L, then it is Λ-split over F for any commutativering Λ. We will say that M is split over L (resp. L is a splitting field) if it isZ-split over L (resp. L is a Z-splitting field). We will denote a motive M (resp.a variety X) considered over its Λ-splitting field by M (resp. by X).

3.3. SPLITTING FIELDS AND RATIONAL CYCLES 21

3.3.2 Definition. Given a motive M over a field F and a field extension L/Fwe say a cycle in CH(ML) is rational if it is in the image of the restriction mapresL : CH(M) → CH(ML).

Observe that the rationality of cycles is preserved by push-forward and pull-back maps. It also respects addition, intersection and correspondence productof cycles.

3.3.3. Assume that a motive M ∈ Chow(F,Λ) has a Λ-splitting field L. Thenthe subgroup of rational cycles in CH(ML) will be denoted by CH(M) (cf.[KM06, 1.2]). If L′ is another splitting field of X, then there is a chain ofcanonical isomorphisms CH(ML) ' CH(MLL′) ' CH(ML′), where LL′ is thecomposite of L and L′. Hence, the groups CH(M) and CH(M) don’t dependon the choice of L.

3.3.4 Lemma. Let X and Y be smooth projective varieties over a field F suchthat Y is irreducible, F (Y ) is a Λ-splitting field of X and Y has a Λ-splittingfield. For any r consider the projection in the Kunneth decomposition

pr0 : CHr(X × Y ; Λ) =r⊕

i=0

CHr−i(X; Λ)⊗ CHi(Y ; Λ) → CHr(X; Λ).

Then for any ρ ∈ CHr(X; Λ) we have pr−10 (ρ) ∩ CHr(X × Y ; Λ) 6= ∅.

Proof. Let L be a common Λ-splitting field of X and Y . The lemma followsfrom the commutative diagram

CHr(X ×F Y ; Λ)resL/F//

CHr(XL ×L YL; Λ)

pr0

((RRRRRRRRRRRRR

CHr(XF (Y ); Λ) ' // CHr((XL)L(YL); Λ) CHr(XL; Λ)'oo

where the left square is obtained by taking the generic fiber of the base changemorphism XL → X; the vertical arrows are taken from the localization sequencefor Chow groups and, hence, are surjective; and the bottom horizontal maps areisomorphisms since L is a Λ-splitting field.

3.3.5 Definition. We say a motive M is generically Λ-split if there exists asmooth projective variety X over F and an integer l such that M is a directsummand of the twisted motive M(X; Λ)l of X and M is split over F (X).In particular, a smooth projective variety X is called generically Λ-split if itsChow motive M(X; Λ) is split over F (X).

3.3.6 Example. The classical examples of generically (Z-)split varieties areSeveri-Brauer varieties, Pfister quadrics and (connected components of) maxi-mal orthogonal Grassmannians.


3.3.7 Example. More generally, let PΘ be the standard parabolic subgroup ofa split simple group G corresponding to a subset Θ of the respective Dynkindiagram D. Consider a twisted inner form G = ξG, where ξ ∈ Z1(F, G), and letX = ξ(G/PΘ) be the respective projective homogeneous G-variety. Let q denotethe degree of a splitting field of G and let d be the index of the associated Titsalgebra (see [Ti66, Table II]). For groups of type Dn, we set d to be the indexof the Tits algebra associated with the vector representation.

Analyzing Tits indices of G we see that G becomes split over F (X) and,hence, X is generically split, if the subset D r Θ contains one of the followingvertices k (cf. [KR94, §7]):

G 1An Bn Cn1Dn G2

k gcd(k, d) = 1 k = n; k is odd; k = n− 1, n if d = 1; anyany k in the any k in thePfister case Pfister case

G F41E6 E7 E8

k k = 1, 2, 3; k = 3, 5; k = 2, 5; k = 2, 3, 4, 5;any k if k = 2, 4 if d = 1; k = 3, 4 if d = 1; any k ifq = 3 k = 1, 6 if q is odd k 6= 7 if q = 3 q = 5

(here by the Pfister case we mean the case when the cocycle ξ corresponds to aPfister form or its maximal neighbor)

3.4 Rost nilpotence

We will extensively use the following version of the Rost nilpotence (cf. [Ro98,Prop. 9] and [VZ08, Prop. 3.1])

3.4.1 Proposition. Let N be a generically Λ-split motive over a field F . Thenfor any field extension E/F the kernel of the restriction

resE/F : EndF (N) → EndE(NE)

consists of nilpotents.

To simplify the notation we denote by EndX(M) the endomorphism groupHomChow(X;Λ)(M,M), where M is a motive over a variety X.

Proof. Recall that a motive N over F is generically split if there exists a smoothprojective variety X and l ∈ Z such that N is a direct summand of M(X; Λ)land NK = resK/F (N) is split, where K = F (X) denotes the function field ofX.

We may assume that N is a direct summand of M(X; Λ) (that is, l = 0).Since for a split motive M and a field extension E/L, the map EndL(ML) →EndE(ME) is an isomorphism, we may assume that E = K.

Consider the composite of ring homomorphisms

resK/F : EndF (N)resX/F−−−−→ EndX(NX)

resK/X−−−−−→ EndK(NK),

3.4. ROST NILPOTENCE 23

where the last map is induced by passing to the generic point SpecK → X.Observe that EndK(NK) = lim−→EndU (NU ), where the limit is taken over all

open subvarieties U ⊂ X. Then ker(resK/X) = ∪U ker(resU/X) and by Lemma3.4.2 the kernel of resK/X consists of nilpotents.

On the other hand, the map resX/F is injective. Indeed, since N is a directsummand ofM(X; Λ), EndF (N) is a subring of EndF (M(X; Λ)) and EndX(NX)is a subring of EndX(M(X; Λ)X). So, it is sufficient to prove the injectiv-ity for the case N = M(X; Λ). The restriction resX/F : EndF (M(X; Λ)) →EndX(M(X; Λ)X) coincides with the pull-back π∗1,2 : CH(X×X; Λ) → CH(X×X×X; Λ) induced by the projection on the first two coordinates. And π∗1,2 splitsby (idX×∆X)∗ : CH(X×X×X; Λ) → CH(X×X; Λ), where ∆X : X → X×Xis the diagonal. The proposition is proven.

3.4.2 Lemma. Let X be a smooth projective variety over F and Λ be a com-mutative ring. Let U ⊂ X be an open embedding. Then for any motive M fromChow(X; Λ) the kernel of the restriction map

resU/X : EndX(M) → EndU (MU )

consists of nilpotents.

Proof. If M is a direct summand of [Y → X]i, then EndX(M) is a subringof EndX(M(Y → X)) and it is sufficient to study the case M = M(Y → X).Recall that EndX(M(Y → X)) = CHdim(Y )−dim(X)(Y ×X Y ; Λ).

Let φ be an element from the kernel of resU/X . Let j : Z → X be thereduced closed complement to U in X. Then by the localization sequence forChow groups the cycle φ belongs to the image of the induced push-forward

(id(Y×XY ) × j)∗ : CH((Y ×X Y )×X Z; Λ) → CH(Y ×X Y ; Λ).

By additivity of Chow(X; Λ) we may assume Z is irreducible.Set codim(Z) := dim(X)−dim(Z) and d := [ dim(X)

codim(Z) ]+1. We claim that thed-th power φd of φ taken with respect to the correspondence product is trivial.Indeed, φd = (π1,d+1)∗(φ1 · φ2 · . . . · φd), where φi = π∗i,i+1(φ) and the mapπi,i′ : Y ×(d+1) → Y ×X Y is the projection on the i-th and i′-th components.Since π∗i,i′ (id(Y×XY ) × j)∗ coincides with (idY ×(d+1) × j)∗ (πi,i′ × idZ)∗, allcycles φi belong to the image of the push-forward

(idY ×(d+1) × j)∗ : CH(Y ×(d+1) ×X Z) → CH(Y ×(d+1)).

By [VZ08, Prop. 6.1] applied to the projection Y ×(d+1) → X and the closedembedding j : Z → X we obtain that the product

φ1 · . . . · φd ∈((idY ×(d+1) × j)∗ CH(Y ×(d+1) ×X Z)

)d

is trivial. Therefore, φd is trivial as well.


3.5 Lifting of coefficients

This section is devoted to lifting of idempotents and isomorphisms. In the ex-position we follow [PSZ, §2]. First, we treat the case of general graded algebras.The main results here are Lemma 3.5.5 and Proposition 3.5.6. Then, assumingRost Nilpotence 3.5.8 we provide conditions to lift motivic decompositions andisomorphisms (Theorem 3.5.19).

3.5.1. Let A∗ be a Z-graded ring. Assume we are given two orthogonal idempo-tents φi and φj in A0 that is φiφj = φjφi = 0. We say an element θij providesan isomorphism of degree d between idempotents φi and φj if θij ∈ φjA

−dφi

and there exists θji ∈ φiAdφj such that θijθji = φj and θjiθij = φi.

3.5.2 Example. Let X be a smooth projective irreducible variety over a fieldF and let CH∗(X ×X; Λ) be the Chow ring with coefficients in a commutativering Λ. Set A∗ = End∗(M(X; Λ)), where

End−i(M(X; Λ)) = CHdim X−i(X ×X; Λ) = CHdim X+i(X ×X; Λ), i ∈ Z

and the multiplication is given by the correspondence product. By definitionEnd0(M(X; Λ)) is the ring of endomorphisms of the motive M(X; Λ) (see 3.2.8).Note that a direct summand of M(X; Λ) can be identified with a pair (X,φi),where φi is an idempotent. Then an isomorphism θij of degree d between φi

and φj can be identified with an isomorphism between the motives (X,φi) and(X,φj)(d).

3.5.3. Let f : A∗ → B∗ be a homomorphism of Z-graded rings. We say thatf lifts decompositions if given a family φi ∈ B0 of pair-wise orthogonal idem-potents such that

∑i φi = 1B , there exists a family of pair-wise orthogonal

idempotents ϕi ∈ A0 such that∑

i ϕi = 1A and each f(ϕi) is isomorphic to φi

by means of an isomorphism of degree 0. We say f lifts decompositions strictlyif, moreover, one can choose ϕi such that f(ϕi) = φi.

We say f lifts isomorphisms if for any idempotents ϕ1 and ϕ2 in A0 andany isomorphism θ12 of degree d between idempotents f(ϕ1) and f(ϕ2) in B0

there exists an isomorphism ϑ12 of degree d between ϕ1 and ϕ2. We say f liftsisomorphisms strictly if, moreover, one can choose ϑ12 such that f(ϑ12) = θ12.

3.5.4. By definition we have the following properties of morphisms which liftdecompositions and isomorphisms (strictly):

(i) Let f : A∗ → B∗ and g : B∗ → C∗ be two morphisms. If both f and glift decompositions or isomorphisms (strictly), then so does the compositeg f .

(ii) If gf lifts decompositions (resp. isomorphisms) and g lifts isomorphisms,then f lifts decompositions (resp. isomorphisms).

3.5. LIFTING OF COEFFICIENTS 25

(iii) Assume we are given a commutative diagram with ker f ′ ⊂ imh

A∗f // //

_

h

B∗ _

h′

A′∗

f ′ // // B′∗.

If f ′ lifts decompositions strictly (resp. isomorphisms strictly), then sodoes f .

We will use the following technical fact (for the proof see [PSZ, Lemma 2.5])

3.5.5 Lemma. Let A, B be two rings, A0, B0 their subrings, f0 : A0 → B0

a ring homomorphism and f : A → B a map of sets satisfying the followingconditions:

• f(α)f(β) equals either f(αβ) or 0 for all α, β ∈ A;

• f0(α) equals f(α) if f(α) ∈ B0 or 0 otherwise;

• ker f0 consists of nilpotent elements.

Let ϕ1 and ϕ2 be two idempotents in A0 and let ψ12, ψ21 be elements in A suchthat ψ12A

0ψ21 ⊂ A0, ψ21A0ψ12 ⊂ A0, f(ψ21)f(ψ12) = f(ϕ1), f(ψ12)f(ψ21) =

f(ϕ2).Then there exist elements ϑ12 ∈ ϕ2A

0ψ12A0ϕ1 and ϑ21 ∈ ϕ1A

0ψ21A0ϕ2

such that ϑ21ϑ12 = ϕ1, ϑ12ϑ21 = ϕ2, f(ϑ12) = f(ϕ2)f(ψ12) = f(ψ12)f(ϕ1),f(ϑ21) = f(ϕ1)f(ψ21) = f(ψ21)f(ϕ2).

3.5.6 Proposition. Let f : A∗ → B∗ be a surjective homomorphism such thatthe kernel of the restriction of f to A0 consists of nilpotent elements. Then flifts decompositions and isomorphisms strictly.

Proof. The fact that f lifts decompositions strictly follows from [AF92, Propo-sition 27.4].

Let ϕ1 and ϕ2 be two idempotents in A0 and let θ12 be an isomorphismbetween f(ϕ1) and f(ϕ2). Let ψ12 ∈ A (resp. ψ21) be a homogeneous lifting ofθ12 (resp. θ21). The proposition follows now from Lemma 3.5.5.

3.5.7 Corollary. Let m be an integer and let m = pn11 . . . pnl

l be its primefactorization. Then the product of reduction maps

End∗(M(X; Z/m)) l∏

i=1

End∗(M(X; Z/pi))

lifts decompositions and isomorphisms strictly.


Proof. We apply Proposition 3.5.6 to the case A∗ = End∗(M(X; Z/pnii )), B∗ =

End∗(M(X; Z/pi)) and the reduction map fi : A∗ B∗. We obtain thatfi lifts decompositions and isomorphisms strictly for each i. To finish theproof observe that by the Chinese remainder theorem End∗(M(X; Z/m)) '∏l

i=1 End∗(M(X; Z/pnii )).

3.5.8. Let X be a smooth projective variety over a field F . Assume that Xhas a splitting field (see 3.3.1). We say that Rost Nilpotence holds for X if thekernel of the restriction map

resE : End∗(M(XE ; Λ)) → End∗(M(X; Λ))

consists of nilpotent elements for all field extensions E/F and all rings of coef-ficients Λ.

3.5.9 Example. Let X be a smooth projective variety which splits over anyfield over which it has a rational point. Then Rost Nilpotence holds for X.

Indeed, by [EKM, Theorem 67.1] if α is in the kernel of the restriction mapresE then α(dim X+1) = 0.

3.5.10. We say that a field extension E/F is rank preserving with respect toX if the restriction map resE/F : CH(X) → CH(XE) becomes an isomorphismafter tensoring with Q.

3.5.11 Lemma. Assume X has a (Z-)splitting field. Then for any rank pre-serving finite field extension E/F we have [E : F ] · CH(XE) ⊂ CH(X).

Proof. Let L be a splitting field containing E. Let γ be any element in CH(XE).By definition there exists α ∈ CH(XE) such that γ = resL/E(α). Since resE/F⊗Q is an isomorphism, there exists an element β ∈ CH(X) and a non-zero integern such that resE/F (β) = nα. By the projection formula

n · coresE/F (α) = coresE/F (resE/F (β)) = [E : F ] · β.

Applying resL/E to the both sides of the identity we obtain

n(resL/E(coresE/F (α))) = n[E : F ] · γ.

Therefore, resL/E(coresE/F (α)) = [E : F ] · γ.

3.5.12 Example. Let G be an inner form of a split group over a field F andlet X be a projective homogeneous G-variety. Then any field extension E/F isrank preserving with respect to X and X ×X.

Indeed, by [Pa94, Theorem 2.2 and 4.2] the restriction map K0(X) →K0(XE) becomes an isomorphism after tensoring with Q. Since the Cherncharacter ch : K0(X) ⊗ Q → CH(X) ⊗ Q is an isomorphism and respects pull-backs, E is rank preserving with respect to X. It remains to note that X ×Xis a homogeneous G×G-variety.

Observe that for outer forms this fails in general. Indeed, for even di-mensional quadrics with non-trivial discriminant the restriction map K0(X) →K0(X) is not surjective.

3.5. LIFTING OF COEFFICIENTS 27

3.5.13 Lemma. Assume that Rost Nilpotence holds for X. Then for any fieldextension E/F the restriction resE : End∗(M(XE ; Λ)) im(resE) onto theimage lifts decompositions and isomorphisms strictly.

Proof. Apply Proposition 3.5.6 to the homomorphism resE : A∗ B∗ betweenthe graded rings A∗ = End∗(M(XE ; Λ)) and B∗ = im(resE).

3.5.14 Corollary. Assume that Rost Nilpotence holds for X. Let m be aninteger and let E/F be a field extension of degree coprime to m which is rankpreserving with respect to X ×X (see 3.5.10). Then the restriction map

resE/F : End∗(M(X; Z/m)) → End∗(M(XE ; Z/m))

lifts decompositions and isomorphisms.

Proof. By Lemma 3.5.11 we have im(resE) = im(resF ). We apply now Lemma 3.5.13and 3.5.4(ii) with A∗ = End∗(M(X; Z/m)), B∗ = End∗(M(XE ; Z/m)) andC∗ = im(resE).

3.5.15. Let V ∗ be a free graded Λ-module of finite rank and let A∗ = End∗(V ∗)be its ring of endomorphisms, where End−d(V ∗), d ∈ Z, is the group of endo-morphisms of V ∗ decreasing the degree by d. Assume we are given a directsum decomposition of V ∗ =

⊕i imφi by means of idempotents φi in A∗. We

say that this decomposition is Λ-free if all graded components of imφi are freeΛ-modules. Observe that if Λ = Z or Z/p, where p is prime, then any decom-position is Λ-free.

3.5.16 Example. Assume X has a splitting field. Define V ∗ = CH∗(X). Thenby Poincare duality and by the Kunneth decomposition we have End∗(V ∗) =End∗(M(X)) (see Example 3.5.2).

3.5.17 Lemma. The map SLl(Z) → SLl(Z/m) induced by the reduction modulom is surjective.

Proof. Since Z/m is a semi-local ring, the group SLl(Z/m) is generated byelementary matrices (see [HOM, Theorem 4.3.9]).

3.5.18 Proposition. Consider a free graded Z-module V ∗ of finite rank andthe reduction map f : End∗(V ∗) End∗(V ∗ ⊗Z Z/m). Then f lifts Z/m-freedecompositions strictly. Moreover, if (Z/m)× = ±1, then f lifts isomorphismsof Z/m-free decompositions strictly.

Proof. We are given a decomposition V k ⊗Z Z/m = ⊕iWki , where W k

i is thek-graded component of imφi. Present V k as a direct sum V k =

⊕i V

ki of free

Z-modules such that rkZ Vki = rkZ/mW k

i . Fix a Z-basis vkijj of V k

i . For eachW k

i choose a basis wkijj such that the linear transformation Dk of V k⊗Z Z/m

sending each vkij ⊗ 1 to wk

ij has determinant 1. By Lemma 3.5.17 there is alifting Dk of Dk to a linear transformation of V k. So we obtain V k =

⊕i W

ki ,


where W ki = Dk(V k

i ) satisfies W ki ⊗Z Z/m = W k

i . It remains to define ϕi oneach V k to be the projection onto W k

i .Now let ϕ1, ϕ2 be two idempotents in End∗(V ∗). Let V k

i denote the k-gradedcomponent of imϕi. An isomorphism θ12 between ϕ1⊗ 1 and ϕ2⊗ 1 of degree dcan be identified with a family of isomorphisms θk

12 : V k1 ⊗Z/m→ V k−d

2 ⊗Z/m.In the case (Z/m)× = ±1 all these isomorphisms are given by matrices withdeterminants ±1 and, hence, can be lifted to isomorphisms ϑk

12 : V k1 → V k−d

2

by Lemma 3.5.17.

Now we are ready to state and to prove the main result of this section.

3.5.19 Theorem. Let X be a smooth projective irreducible variety over a fieldF . Assume that X has a splitting field of degree m which is rank preservingwith respect to X×X. Assume that Rost Nilpotence holds for X. Consider onlydecompositions of M(X; Z/m) which become Z/m-free over the splitting field.Then the reduction map

f : End∗(M(X; Z)) End∗(M(X; Z/m))

lifts such decompositions. If additionally (Z/m)× = ±1, then this map liftsisomorphisms of such decompositions.

Proof. Consider the diagram

End∗(M(X; Z)) // //

f

im(resF ) h //

f

End∗(M(X; Z))

f ′

End∗(M(X; Z/m)) // // im(resF ) // End∗(M(X; Z/m))

Recall that we can identify End−d(M(X)) with the group of endomorphismsof CH∗(X) which decrease the grading by d (see 3.5.16). Applying Proposi-tion 3.5.18 to the case V ∗ = CH∗(X) we obtain that the map f ′ lifts decompo-sitions strictly. Moreover, if (Z/m)× = ±1 then f ′ lifts isomorphisms strictly.

By Lemma 3.5.11 ker f ′ ⊂ imh and, therefore, applying 3.5.4(iii) we obtainthat f lifts decompositions strictly and, moreover, f lifts isomorphisms strictlyif (Z/m)× = ±1.

Now by Lemma 3.5.13 the horizontal arrows of the left square lift decompo-sitions and isomorphisms strictly. It remains to apply 3.5.4(i) and (ii).

3.5.20 Example. Let Q be an odd dimensional quadric or an even dimensionalquadric with trivial discriminant. Then the reduction map End(M(Q; Z)) End(M(Q; Z/2)) lifts decompositions and isomorphisms strictly.

3.6 Rost type motivic decompositions

In the present section we provide motivic decompositions which are similar tothose obtained by Rost and Voevodsky. The main object is a generically split

3.6. ROST TYPE MOTIVIC DECOMPOSITIONS 29

smooth projective variety X over a field F with the property that the greatestcommon divisor of degrees of all closed points on X is equal to a prime integerp. The main result is Theorem 3.6.1 which provides a motivic decompositionwith Z/p-coefficients of such a variety.

To simplify the notation we set A∗ = Ch∗(X), where Ch(X) = CH(X; Z/p)is the Chow ring with Z/p-coefficients. The subring of rational cycles in A∗ willbe denoted by A∗rat.

3.6.1 Theorem. Let X be a generically split variety over a field F such thatthe greatest common divisor of degrees of all closed points on X is equal to aprime integer p. Assume there exists an element ρ ∈ Ar for some r such that A∗

is generated by elements 1, ρ, . . . , ρp−1 as a graded module over A∗rat. Thenthe Chow motive of X with Z/p-coefficients is isomorphic to the direct sum

M(X; Z/p) '⊕i≥0

Rpi⊕ci ,

where the motive Rp satisfies the following properties:

• it is indecomposable

• over a splitting field it becomes isomorphic to a direct sum of twisted Lef-schetz motives

Rp 'p−1⊕j=0

(Z/p)jr

• the integers ci are the coefficients of the polynomial∑cit

i = P (A∗, t) · 1− tr

1− trp,

where P (A∗, t) is the Poincare polynomial.

Assume we are under the hypothesis of Theorem 3.6.1.

3.6.2 Lemma. There exists ε ∈ (Z/p)× such that the cycle ρ⊗ 1 + ε(1⊗ ρ) isrational.

Proof. Since X is generically split by Lemma 3.3.4 applied to the projectionpr0 : Chr(X × X) → Chr(X) there exists a rational preimage of ρ ⊗ 1 of theform

ρ⊗ 1 + δ(1⊗ ρ) +∑

k

µk ⊗ νk + 1⊗ γ,

where δ ∈ Z/p, codimensions of µk and νk are strictly less than r and the cycleγ is rational.

Since 1, ρ, . . . , ρp−1 generates A∗ as an A∗rat-module and codim ρ = r, thecycles µk and νk are rational. Therefore τ := ρ ⊗ 1 + δ(1 ⊗ ρ) is rational. Ifδ ∈ (Z/p)×, then we take ε = δ. Otherwise, the cycle τ+τ t = (1+δ)(ρ⊗1+1⊗ρ)is rational and 1 + δ is invertible. Therefore, the cycle ρ⊗ 1 + 1⊗ ρ is rationaland we take ε = 1.


3.6.3 Lemma. Let α, β ∈ A∗rat be two cycles with the property p - deg(ρlαβ).Assume that the following cycle is rational

ξ =l∑

j=0

xj(ρj ⊗ ρl−j) · (α⊗ β), where xj ∈ Z/p.

Then p divides the number of invertible coefficients xj.

Proof. Consider the rational cycle

ξ(p−2) · ξt = (l∑

j=0

xp−1j )ρlαβ ⊗ ρlαβ.

Taking the push-forward induced by the first projection we obtain a rationalzero-cycle (

∑lj=0 x

p−1j )ρlαβ. Since p divides the degree of any rational zero-

cycle on X and p - deg(ρlαβ), we have p |∑l

j=0 xp−1j . Finally, observe that

xp−1 ≡ 1 mod p if x ∈ (Z/p)× and xp−1 = 0 mod p otherwise.

3.6.4 Lemma. For all l = 0, . . . , p − 2 and homogeneous µ ∈ A∗rat we havep | deg(ρlµ).

Proof. Assume p - deg(ρlµ). Consider the rational cycle

ξ = (ρ⊗ 1 + ε(1⊗ ρ))l(µ⊗ 1) =l∑

j=0

εj

(l

j

)(ρl−j ⊗ ρj)(µ⊗ 1).

Note that all coefficients εj(

lj

)on the right hand side are invertible. Since the

number of coefficients is l+ 1 < p, we obtain the contradiction by Lemma 3.6.3applied to α = µ and β = 1.

3.6.5 Lemma. We have a perfect Z/p-linear pairing

Akrat ×A

dim X−r(p−1)−krat → Z/p

µ× ν 7→ deg(ρp−1µν) mod p.

Proof. It suffices to show that for an element µ ∈ Akrat there exists a cycle

ν ∈ Adim X−r(p−1)−krat such that deg(ρp−1µν) ≡ 1 mod p. By the Poincare

duality there exists a cycle µ∨ ∈ Adim X−k such that deg(µµ∨) = 1 mod p.Expand µ∨ as µ∨ =

∑p−1l=0 ρ

lαl for some αl ∈ A∗rat. By Lemma 3.6.4 we havep | deg(ρlµαl) for all l = 0 . . . p−2. Therefore, deg(ρp−1µαp−1) = deg(µµ∨) = 1mod p. To finish the proof we set ν = αp−1.

3.6.6 Lemma. Let α and β ∈ A∗rat be two homogeneous cycles such that p -deg(ρp−1αβ). Then the cycle σ · (α ⊗ β), where σ =

∑p−1j=0 ρ

j ⊗ ρp−1−j, isrational.

3.6. ROST TYPE MOTIVIC DECOMPOSITIONS 31

Proof. Consider the rational cycle

λ = (ρ⊗ 1 + ε(1⊗ ρ))p−1(α⊗ β) =p−1∑j=0

εj

(p− 1j

)(ρp−1−j ⊗ ρj)(α⊗ β).

Since all coefficients εj(p−1

j

)on the right hand side are invertible, we have the

equality λp−1 = σ · (α⊗ β).

3.6.7 Lemma. For all l = p, p+ 1, . . . , 2(p− 1) and homogeneous µ ∈ Arat wehave p | deg(ρlµ).

Proof. Assume that p - deg(ρlµ). By Lemma 3.6.5 there exists a rational cycleα such that deg(ρp−1α) = 1 mod p. Consider the composite of two rationalcycles: (

(ρ⊗ 1 + ε(1⊗ ρ))l(µ⊗ 1))

(σ · (α⊗ 1)

),

where σ =∑p−1

j=0 ρj ⊗ ρp−1−j is the cycle from Lemma 3.6.6. It is equal to

( l∑j=0

εj

(l

j

)(ρl−jµ⊗ ρj)

)

( p−1∑j=0

ρjα⊗ ρp−1−j)

=p−1∑j=0

εj

(l

j

)ρp−1−jα⊗ ρj .

Note that the coefficient εj(

lj

)at the summand j = 0 is equal to 1 and all other

coefficients are divisible by p. By Lemma 3.6.3 we obtain a contradiction.

3.6.8 Lemma. Let αii∈I be a homogeneous Z/p-basis of A∗rat. Then thereexists a family βii∈I of homogeneous elements of A∗rat such that for all l =0, 1, . . . , 2(p− 1) we have

deg(ρlαiβj) =

1, if l = p− 1 and i = j

0, otherwise.(*)

Moreover, the family ρlαi | i ∈ I, l = 0, . . . , p− 1 forms a Z/p-basis of A∗.

Proof. By Lemma 3.6.5 there exists a family of homogeneous elements βii∈Iin A∗rat such that

deg(ρp−1αiβ′j) mod p =

1, if i = j

0, otherwise.

By Lemma 3.6.4 and Lemma 3.6.7 the family βii∈I satisfies the condition (∗).The family ρlαi | i ∈ I, l = 0, . . . p − 1 generates A∗, since the family

ρl, l = 0, . . . , p− 1 generates A∗ as an A∗rat-module. Assume we have a linearrelation

∑ailρ

lαi = 0. Multiplying by ρp−1−lβi and taking the degree map wesee that ail = 0. Hence, the family ρlαi is linear independent.


Proof of Theorem 3.6.1. Choose a homogeneous Z/p-basis αii∈I in A∗rat. Letβii∈I be the family of elements from Lemma 3.6.8. Consider the cycles φi =σ · (αi ⊗ βi), where i ∈ I and σ =

∑p−1j=0 ρ

j ⊗ ρp−1−j . They are rational byLemma 3.6.6 and they form a family of pair-wise orthogonal idempotents by(∗). Moreover, their sum

∑i∈I

φi =p−1∑l=0

∑i∈I

ρlαi ⊗ ρq−1−lβi

is the diagonal. The cycle θij = σ · (αj ⊗ βi) provides a rational isomorphismbetween (X, φi) and (X, φj). We apply now Proposition 3.4.1 and Lemma 3.5.13and obtain the desired motivic decomposition

M(X; Z/p) '⊕i∈I

Rp(codimαi),

where Rp = (X,ϕi0) with codimαi0 = 0.It remains to show that Rp is indecomposable. Observe that its ring of

endomorphisms CHdim X(X × X) is additively generated by ρlαi0 ⊗ ρp−1−lβi0 ,l = 0, . . . , p − 1. Consider a nonzero rational idempotent φ =

∑p−1l=0 al(ρlαi0 ⊗

ρp−1−lβi0). We have al = 0 or 1 for all l. By Lemma 3.6.3 al = 1 for all l. Soφ = ∆X is the identity.

3.6.9 Proposition. Let X1 and X2 be varieties satisfying the hypothesis ofTheorem 3.6.1 with the respective ρ1 ∈ Chr(X1) and ρ2 ∈ Chr(X2). Assumethat F (X2) is a splitting field of X1 and F (X1) is a splitting field of X2. Thenthe motives RX1 and RX2 appearing in the decomposition of M(X1; Z/p) andM(X2; Z/p) respectively are isomorphic.

Proof. Following the proof of Lemma 3.6.2 we show that there exists ε ∈ (Z/p)×such that the cycle ρ1 ⊗ 1 + ε(1 ⊗ ρ2) is rational. Let αii∈I and βii∈I bethe familes of elements in Ch

∗(X1), let αjj∈J , βjj∈J be the families in

Ch∗(X2) chosen as in Lemma 3.6.8. The cycle

θi0,j0 = (ρ1 ⊗ 1 + ε(1⊗ ρ2))p−1 · (αi0 ⊗ βj0).

is rational and produces an isomorphism between (X1, φi0) and (X2, φj0). Itremains to apply Proposition 3.4.1 and Lemma 3.5.13.

3.7 Integral decompositions

to be filled

Chapter 4

Motives of isotropichomogeneous varieties

4.1 Relative cellular spaces

4.1.1. A scheme E together with a flat morphism p : E → X is called an affinefibration of relative dimension r if, for every point x ∈ X, there is a Zariski openneighborhood U ⊂ X such that p−1(U) ∼= U × Ar over U (see [NZ06, §4]).

4.1.2. (see [EKM, §66]) Let X be a smooth projective variety over a field F . Wecall X a relative cellular space over F if there exists a finite decreasing filtrationby closed proper subvarieties

X = X0 ⊃ X1 ⊃ . . . ⊃ Xn ⊃ ∅

such that each complement Ei = Xi r Xi+1 is an affine fibration of a relativedimension ri over a smooth projective variety Yi (we assume Yn = Xn). Wedenote by pi : Ei → Yi the projection morphism.

In particular, if the base Yi consists of points, i.e., each Ei is isomorphic to adisjoint union of affine spaces Ari

F , then X is called an (absolute) cellular spaceover F .

4.1.3 Example. The basic example of an absolute cellular filtration is givenby the descreasing filtration consisting of projective linear subspaces

Pn ⊃ Pn−1 ⊃ . . . ⊃ P1 ⊃ pt,

where on each step Pi r Pi−1 ' Ai.

4.1.4 Example (Rost filtration). Let Q be an isotropic smooth projectivequadric of dimension 2d − 1. It can be identified with a hypersurface given bythe equation

Q = [x0 : x1 : . . . : x2d] ∈ P2d | x0x1 + q(x2, . . . , x2d) = 0,

33

34CHAPTER 4. MOTIVES OF ISOTROPIC HOMOGENEOUS VARIETIES

where q is a quadratic form. SetQ = X0 and consider the closed subvarietyX1 ofQ given by the equation x0 = 0. Then X0 rX1 is given by two equations x0 6= 0and x0x1 + q(x2, . . . , x2d) in P2d. Dividing by x0 we see that X0 r X1 can beidentified with the affine subvariety (y1, . . . , y2d) ∈ A2d | y1 + q(y2, . . . , y2d) =0, where yi = xi/x0, which is isomorphic to the affine space A2d−1.

Let X2 = pt be the point on X1 given by x1 = 1 and xi = 0 for all i 6=1. Then the map forgetting the first two coordinates induces a well-definedmorphism π : X1 r X2 → Q1, where Q1 is the projective quadric given by theequation

Q1 = [x2 : . . . : x2d] ∈ P2d−2 | q(x2, . . . , x2d) = 0.

It is easy to see that π is, indeed, an affine fibration with fibers isomorphic toA1.

Hence, Q is a relative cellular space with the bases Y0 = pt, Y1 = Q1 andY2 = X2 = pt. Observe that X1 is not necessary smooth at the point X2.

4.1.5 Example. (cf. [NZ06, §7.1]) Let Q be a split projective quadric of di-mension 2d. Then Q can be thought of as the hypersurface given by

Q = [x0 : y0 : . . . : xd : yd] ∈ P2d+1 |∑

i=0,d

xiyi = 0.

We claim that Q has a smooth filtration defined as follows.Consider two linear subspaces of P2d+1 of dimension d defined by the equa-

tions

Z1 = [x0 : y0 : . . . : xd : yd] ∈ P2d+1 | x0 = x1 = . . . = xd = 0 and

Z2 = [x0 : y0 : . . . : xd : yd] ∈ P2d+1 | y0 = y1 = . . . = yd = 0.

Clearly Z1∼= Z2

∼= Pd and Z1 ∩ Z2 = ∅. Consider the linear map π1 : P2d+1 rZ2 → Z1 given by the projection to the y-coordinates. Both subspaces Z1 andZ2 lie in Q. It can be easily seen that the restriction π1 : QrZ2 → Z1 naturallyhas a structure of a vector bundle of rank d. Hence, we get a smooth filtrationon Q:

X0 = Q ⊃ X1 = Z2, Y0 = Z1, Y1 = Z2.

4.1.6 Example. (cf. [NZ06, §7.3]) Let Gr(d, n) be the Grassmannian of d-dimensional linear subspaces in affine space An. Then Gr(d, n) has a smoothfiltration defined as follows.

Fix a (n−1)-dimensional linear subspace V of An. Then there is the inducedclosed embedding Gr(d, n− 1) → Gr(d, n) of smooth projective varieties, whereGr(d, n − 1) is identified with the variety of d-planes in V . Consider the com-plement E = Gr(d, n)rGr(d, n− 1). We claim that this is a vector bundle overGr(d − 1, n − 1). Indeed, an element of E is a d-plane that intersects V alonga linear subspace of dimension d − 1, i.e., it gives a point of Gr(d − 1, n − 1).Moreover, the set of all elements of E that intersect V along a fixed linear sub-space of dimension d − 1 can be identified with the affine space of dimensionn− d. This gives the structure of an affine fibration.

4.2. BRUHAT DECOMPOSITION 35

Applying this procedure recursively we obtain a smooth cellular filtration

X0 = Gr(d, n) ⊃ X1 = Gr(d, n− 1) ⊃ . . . ⊃ Xn−d = Gr(d, d) = pt

with the base Y0 = Gr(d− 1, n− 1), Y1 = Gr(d− 1, n− 2), . . . , Yn−d = pt.

The following two facts (Lemmas 4.1.7 and 4.1.8) play an important role incomputations of motives of relative cellular spaces:

4.1.7 Lemma. (cf. [NZ06, Thm. 6.5]) Let X be a relative cellular space overa field F . Then we have an isomorphism of motives in Chow(F,Z)

M(X) '⊕i≥0

M(Yi)ri,

where Yi is the base and ri is the rank of the respective fibration from 4.1.2.

To formulate the next fact we use the following notion

4.1.8. Let X be a smooth projective variety over a field F . We say a smoothprojective morphism f : Y → X is a cellular fibration if it is a locally trivialfibration whose fiber F is an absolute cellular space has a decomposition intoaffine cells.

4.1.9 Lemma. (cf. [PSZ, Lemma 3.2]) Let f : Y → X be a cellular fibration.Then M(Y ) is isomorphic to M(X)⊗M(F).

Proof. We follow the proof of [EG97, Proposition 1]. Define the morphism

ϕ :⊕i∈I

M(X)codimBi →M(Y )

to be the direct sum ϕ =⊕

i∈I ϕi, where each ϕi is given by the cycle [pr∗Y (Bi) ·Γf ] ∈ CH(X × Y ) produced from the graph cycle Γf and the chosen (non-canonical) basis Bii∈I of CH(Y ) over CH(X). The realization of ϕ coincidesexactly with the isomorphism of abelian groups CH(X) ⊗ CH(F) → CH(Y )constructed in [EG97, Proposition 1]. Then by Manin’s identity principle (see[Ma68, §3]) ϕ is an isomorphism.

4.2 Bruhat decomposition

In the present section we will see that any projective homogeneous G-variety,where G is a split linear algebraic group, admits an absolute cellular filtration.

4.2.1. Let G be a split group of type D over a field F . Choose a split maximaltorus T and a Borel subgroup B = P∅. Let W be the Weyl group of G. TheBruhat decomposition of G is the following disjoint union of double cosets

G =∐

w∈W

B · w′ ·B,

where w′ ∈ NG(T) is a representative of w.


4.2.2. Consider the variety X of Borel subgroups of G, i.e. the projective ho-mogeneous variety D/P∅. Let x be an F -rational point on X with the stabilizersubgroup B (see 2.5.2). Then by Bruhat decomposition

X =∐

w∈W

B · w′ · x.

4.2.3. Let Φ ⊂ V be a root system and W = W (Φ) be its Weyl group. LetΠ = α1, . . . , αn be a basis of Φ. Denote by si = sαi

∈ W the correspondinggenerators (simple reflections). We define the length `(w) of an element w ∈Was

`(w) := minm | w = si1 · si2 · . . . · sim.

4.2.4. Let Θ be a subset of Π and let WΘ ⊂ W be the subgroup generatedby all si with αi ∈ Θ. Define WΘ ⊂ W to be the set of the representativesof minimal length in the left cosets W/WΘ. Observe that for each coset thereexists only one such representative. Moreover, one can show that

WΘ = w ∈W | ∀ i ∈ Θ `(wsi) = `(w) + 1 .

4.2.5. Let X be a projective homogeneous G-variety of dimension d correspond-ing to a subset Θ ⊂ D. By the Bruhat decomposition we have

X =∐

w∈WΘ

B · w′ · x,

where x is an F -rational point on X. We define the cellular filtration as

Xi =∐

w∈WΘ,l(w)≤d−i

B · w′ · x.

We have

X = X0, Xi rXi−1 '∐

w∈WΘ,l(w)=d−i

Al(w)F and Xd = x,

i.e. there are exactly #w ∈WΘ | l(w) = j affine cells of dimension j.Observe that for a projective space An/P1 this cellular filtration coincides

with the one from Example 4.1.3

4.3 Hasse diagram

The cellular structure of a projective homogeneous variety for a split groupcan be naturally described using the Hasse diagram. The latter is a purelycombinatorial object depending only on a root system and a subset of verticesof the respective Dynkin diagram.

4.3. HASSE DIAGRAM 37

4.3.1. Given a root system Φ, its basis Π and a subset Θ ⊂ Π we define alabeled graph called the Hasse diagram as follows

• The vertices are given by elements of WΘ;

• We draw an edge from w1 toward w2 and mark it by a number i if andonly if `(w2) = `(w1) + 1 and w2 = siw1.

It can be shown that this diagram (graph) doesn’t depend on the choice of Πin Φ, hence, it depend only on the type D of Φ and the subset Θ of the set ofvertices of the Dynkin diagram D.

4.3.2. By a distance between two vertices v1 and v2 we call a minimal numberof edges needed to connect v1 and v2. Hence, the distance between a vertex vand the starting vertex v = 1 of the Hasse diagram is equal to the length l(v).We make the convention to draw all vertices of the same length on one verticalline.

4.3.3. Let G be a split group of type D. The relation between the Hassediagram for the subset Θ of D and the cellular decomposition of the varietyD/PΘ is given as follows:

There is an obvious one-to-one correspondence between the vertices and thecells due to the fact that they are parametrized by the same finite set WΘ.Moreover, two vertices v1, v2 ∈ WΘ with l(v1) + 1 = l(v2) are connected byan edge if and only if the cell corresponding to v1 lies in the closure of the cellcorresponding to v2.

4.3.4 Example. For Θ = α2, . . . , αn and D = An, Bn or Dn we have

An : •1•

2•

3• •

n• projective space

Bn : •1•

2• •

n−1•

n•

n−1• •

2•

1• quadric

Dn : •n

????

??

•1•

2• •

n−2•

n−1

n ????

?? •n−2

• •2•

1•

• n−1

quadric

where on the right hand side we identify the respective projective homogeneousvariety D/PΘ.

4.3.5 Example. For Θ = α1, α3, α4 and D = A4, i.e. for the variety Gr(2, 5),we have

•2 ??

???? •

????

??

•??

????

•3

•3 ??

???? •

????

??

•2

•4 ??

???? •

1

•


where all other labels are obtained by parallel translations along the edges.

4.4 The case of an isotropic group

We show that if G is an isotropic group, then a projective homogeneous G-variety X admits a relative cellular filtration. In view of Lemma 4.1.7 this factreduces the problem of computing the motivic decomposition of X to the caseof anisotropic groups.

4.4.1. Let G be an isotropic group of type D over a field F . Let X be a projec-tive homogeneous G-variety which (over the algebraic closure Fa) correspondsto a subset Θ ⊂ D. Let Ξ ⊂ D be a subset of circled vertices of the Tits indexof G. Since G is isotropic, this subset is non-emty. The main result of [CGM]and [Br05] says that the variety X is a relative cellular space where the basevarieties Yi are projective homogeneous varieties for anisotropic groups.

4.4.2. To identify the base spaces Yi we may use the following procedure. Con-sider the Hasse diagram for Θ ⊂ D, i.e. for the variety X. Erase all edgeswhich are labeled by the roots from Ξ. Then the Hasse diagram for X will splitinto several connected components. Each connected component turns out to bethe Hasse diagram for a base space Yi such that the rank ri of the fibrationXi r Xi+1 → Yi coinsides with the distance between the very left vertex ofthe connected component and the left hand side of the Hasse diagram and thecodimension of Xi in X is equal to the distance between the very right vertexof the connected component and the right hand side of the diagram.

4.4.3 Example. Let G be an isotropic group of type A3 with the set of circledvertices Ξ = α2.

Let X be a projective homogeneous G-variety with Θ = α2, α3, i.e. X isa Severi-Brauer variety of a central simple algebra A of degree 4 and index 2.Then the respective Hasse diagram (see Example 4.3.4) splits into two connectedcomponents

•1• •

3•

each corresponding to the diagram of the conic SB(D), where D is division andA = M2(D). Here Y0 is the right component and Y1 is the left component,the ranks are r0 = 2 and r1 = 0 and the codimensions are codimX0 = 0 andcodimX1 = 2. By Lemma 4.1.7 we obtain the following motivic decomposition

M(X) 'M(SB(D))⊕M(SB(D))2.

Let X be a projective homogeneous G-variety with Θ = α1, α3, i.e. X isa 4-dimensional quadric. In this case the diagram splits as

•3

????

??

• •3 ??

????

1 • •

• 1

4.5. MOTIVES OF FIBERED SPACES 39

where the middle component corresponds to a 2-dimensional anisotropic quadricQ2 and isolated vertices to rational points. Here Y0 = pt, Y1 = Q2, Y2 = pt, theranks r0 = 4, r1 = 1, r2 = 0 and the codimensions codimX0 = 0, codimX1 = 1,codimX2 = 4. Observe that the obtained relative cellular filtration coincideswith the Rost filtration from Example 4.1.4 and the motivic decompositioncoincides with the Rost decomposition:

M(X) ' Z⊕M(Q2)1 ⊕ Z4.

4.5 Motives of fibered spaces

In the present section we discuss motives of cellular fibrations. We essentiallyfollow [PSZ, §3]. The main result (Theorem 4.5.4) generalizes and uniformizesthe proofs of paper [CPSZ].

4.5.1 Lemma. Let G be a linear algebraic group over a field F and let X bea projective homogeneous G-variety. Let Y be a G-variety and let f : Y → Xbe a G-equivariant projective morphism. Assume that the fiber of f over F (X)is isomorphic to FF (X) for some variety F over F . Then f is a locally trivialfibration with fiber F .

Proof. By the assumptions, we have Y ×X SpecF (X) ' (F ×X)×X SpecF (X)as schemes over F (X). Since F (X) is a direct limit of O(U) taken over allnon-empty affine open subsets U of X, by [EGA IV, Corollaire 8.8.2.5] thereexists U such that f−1(U) = Y ×X U is isomorphic to (F ×X)×X U ' F ×Uas a scheme over U . Since G acts transitively on X and f is G-equivariant, themap f is a locally trivial fibration.

4.5.2 Corollary. Let X be a projective homogeneous G-variety and let Y bea projective variety such that YF (X) ' FF (X) for some variety F . Then theprojection map X × Y → X is a locally trivial fibration with fiber F . Moreover,if F is cellular, then M(X × Y ) 'M(X)⊗M(F).

Proof. We apply Lemma 4.5.1 to the projection map X × Y → X and useLemma 4.1.9.

4.5.3 Lemma. Let G be a semisimple linear algebraic group over F . Let X andY be projective homogeneous G-varieties corresponding to parabolic subgroups Pand Q of the split form G, Q ⊆ P . Let f : Y → X be the map induced by thequotient map. If G splits over F (X), then f is a cellular fibration with fiberF = P/Q.

Proof. Since G splits over F (X), the fiber of f over F (X) is isomorphic to(P/Q)F (X) = FF (X). Now apply Lemma 4.5.1 and note that F is cellular.

Case-by-case arguments from the paper [CPSZ] show that under certainconditions the Chow motive of a twisted flag variety X can be expressed interms of the motive of a minimal flag. These conditions cover almost all twisted


flag varieties corresponding to groups of types An and Bn together with someexamples of types Cn, G2 and F4. The following theorem together with thetable of Example 3.3.7 provides a uniform proof of these results and extendsthem to some other types.

4.5.4 Theorem. Let Y and X be taken as in Lemma 4.5.3. Then the Chowmotive M(Y ) of Y is isomorphic to a direct sum of twisted copies of the motiveM(X), i.e.

M(Y ) '⊕i≥0

M(X)i⊕ci ,

where∑cit

i = P (M(Y ), t)/P (M(X), t) is the quotient of the respective Poincarepolynomials.

Proof. Apply Lemmas 4.5.3 and 4.1.9.

4.5.5 Remark. An explicit formula for P (M(X), t) involves degrees of thebasic polynomial invariants of G and is provided in [Hi82, Ch. IV, Cor. 4.5].

Chapter 5

Motives of generically splithomogeneous varieties.

5.1 The J-invariant

Fix a prime integer p. We denote the Chow ring of a variety X with Z/p-coefficients by Ch(X) and the image of the restriction map CH(X; Z/p) →CH(X; Z/p) by Ch(X).

5.1.1. Let G be a split group over a field F with a split maximal torus T anda Borel subgroup B containing T . Let G = ξG be a twisted form of G givenby a cocycle ξ ∈ Z1(F, G). Let X = ξ(G/B) be the corresponding variety ofcomplete flags. Observe that the group G splits over any field K over which Xhas a rational point, in particular, over the function field F (X ). According to[De74] the Chow ring Ch(X ) can be expressed in purely combinatorial termsand, therefore, depends only on the type of G but not on the base field F .

5.1.2. Let T be the group of characters of T and let S(T ) be the symmetricalgebra. We denote by R the image of the characteristic map c : S(T ) → Ch(X )(see [Gr58, (4.1)]). According to [KM06, Thm.6.4] there is an embedding

R ⊆ Ch(X ), (5.1)

where the equality holds if the cocycle ξ corresponds to a generic torsor.

5.1.3. Let Ch(G) denote the Chow ring with Z/p-coefficients of the group Gover a splitting field of X . Consider the pull-back induced by the quotient map

π : Ch(X ) → Ch(G)

According to [Gr58, p. 21, Rem. 2] π is surjective with its kernel generated byR+, where R+ stands for the subgroup of the non-constant elements of R.

41

42CHAPTER 5. MOTIVES OF GENERICALLY SPLIT HOMOGENEOUS VARIETIES.

5.1.4. An explicit presentation of Ch(G) is known for all types of G and alltorsion primes p of G (in the sense of [De73, Proposition 6.(a)]) provided inTable 5.2.5. Namely, by [Kc85, Theorem 3] it is a quotient of the polynomialring in r variables x1, . . . , xr of codimensions d1 ≤ d2 ≤ . . . ≤ dr coprimeto p, modulo an ideal generated by certain p-powers xpk1

1 , . . . , xpkr

r (ki ≥ 0,i = 1, . . . , r)

Ch∗(G) = (Z/p)[x1, . . . , xr]/(xpk1

1 , . . . , xpkr

r ). (5.2)

In the case where p is not a torsion prime of G we have Ch∗(G) = Z/p, i.e.r = 0.

Note that a complete list of numbers dipkii=1...r called p-exceptional de-

grees of G is provided in [Kc85, Table II]. Taking the p-primary and p-coprimaryparts of each p-exceptional degree from this table one restores the respective ki

and di.

5.1.5. We introduce two orders on the set of additive generators of Ch(G), i.e.on the monomials xm1

1 . . . xmrr . To simplify the notation, we will denote the

monomial xm11 . . . xmr

r by xM , where M is an r-tuple of integers (m1, . . . ,mr).The codimension of xM will be denoted by |M |. Observe that |M | =

∑ri=1 dimi.

• Given two r-tuples M = (m1, . . . ,mr) and N = (n1, . . . , nr) we say xM 4xN (or equivalently M 4 N) if mi ≤ ni for all i. This gives a partialordering on the set of all monomials (r-tuples).

• Given two r-tuples M = (m1, . . . ,mr) and N = (n1, . . . , nr) we say xM ≤xN (or equivalently M ≤ N) if either |M | < |N |, or |M | = |N | andmi ≤ ni for the greatest i such that mi 6= ni. This gives a well-orderingon the set of all monomials (r-tuples) known as the DegLex order.

5.1.6. Let G = ξG be the twisted form of a split group G over a field F bymeans of a cocycle ξ ∈ Z1(F, G) and let X = ξ(G/B) be the respective varietyof complete flags. Let Ch(G) denote the image of the composite

Ch(X ) res−−→ Ch(X ) π−→ Ch(G)

Since both maps are ring homomorphisms, Ch(G) is a subring of Ch(G).For each 1 ≤ i ≤ r set ji to be the smallest non-negative integer such that

the subring Ch(G) contains an element a with the greatest monomial xpji

i withrespect to the DegLex order on Ch(G), i.e. of the form

a = xpji

i +∑

xM xpjii

cMxM , cM ∈ Z/p.

The r-tuple of integers (j1, . . . , jr) will be called the J-invariant of G modulo pand will be denoted by Jp(G).

5.2. PROPERTIES OF THE J-INVARIANT 43

Observe that if the Chow ring Ch(G) has only one generator, i.e. r = 1,then the J-invariant is equal to the smallest non-negative integer j1 such thatxpj1

1 ∈ Ch(G). Moreover, if j1 = 1, then the preimage of x in Ch(X ) satisfiesthe hypotheses for an element ρ ∈ Chd1(X ) from Theorem 3.6.1.

5.1.7 Example. By definition it follows that Jp(GE) 4 Jp(G) for any fieldextension E/F . Moreover, Jp(G) 4 (k1, . . . , kr) by (5.2).

According to (5.1) the J-invariant takes its maximal possible value Jp(G) =(k1, . . . , kr) if the cocycle ξ corresponds to a generic torsor. Later on (seeCorollary 7.2.6) it will be shown that the J-invariant takes its minimal valueJp(G) = (0, . . . , 0) if and only if the group G splits by a finite field extension ofdegree coprime to p.

The next example explains the terminology ‘J-invariant’.

5.1.8 Example. Let φ be a quadratic form with trivial discriminant. In [Vi05,Definition 5.11] A. Vishik introduced the notion of the J-invariant of φ, a set ofintegers which describes the subgroup of rational cycles on the respective max-imal orthogonal Grassmannian. This invariant provides an important tool forstudy of algebraic cycles on quadrics. In particular, it was one of the main ingre-dients used by A. Vishik in his significant progress on the solution of Kaplansky’sProblem. More precisely, in the notation of paper [Vi06] the J-invariant of aquadric corresponds to the upper row of its elementary discrete invariant (see[Vi06, Definition 2.2]).

An equivalent but slightly different definition of the J-invariant of φ can befound in [EKM, § 88]. Namely, if J(φ) denotes the J-invariant of φ from [EKM],then Vishik’s J-invariant is equal to 0, 1, . . . , n \ J(φ) in the case dimφ = 2nand 1, . . . , n \ J(φ) in the case dimφ = 2n− 1.

Using Theorem 4.5.4 one can show that J(φ) introduced in [EKM] can beexpressed in terms of J2(O(φ)) = (j1, . . . , jr) as follows:

J(φ) = 2ldi | i = 1, . . . , r, 0 ≤ l ≤ ji − 1.

Since all di are odd, J2(O(φ)) is uniquely determined by J(φ).

5.2 Properties of the J-invariant

We have the following reduction formula (cf. [EKM, Cor. 88.6] in the case ofquadrics).

5.2.1 Proposition. Let G be a semisimple group of inner type over a field Fand let X be the variety of complete G-flags. Let Y be a projective variety suchthat the map Chl(Y ) → Chl(YF (x)) is surjective for all x ∈ X and l ≤ n. Thenji(G) = ji(GF (Y )) for all i such that dip

ji(GF (Y )) ≤ n.

Proof. By [EKM, Lemma 88.5] the map Chl(X) → Chl(XF (Y )) is surjectivefor all l ≤ n and, therefore ji(G) ≤ ji(GF (Y )). The converse inequality isobvious.


5.2.2 Corollary. Jp(G) = Jp(GF (t)).

Proof. Take Y = P1 and apply Proposition 5.2.1.

5.2.3. To find restrictions on the possible values of Jp(G) we use Steenrod p-th power operations introduced by P. Brosnan. Recall (see [Br03]) that if thecharacteristic of the base field F is different from p then one can constructSteenrod p-th power operations

Sl : Ch∗(X) → Ch∗+l(p−1)(X), l ≥ 0

such that S0 = id, the restriction Sl|Chl(X) coincides with taking to the p-th power, Sl|Chi(X) = 0 for l > i, and the total operation S• =

∑l≥0 S

l isa homomorphism of Z/p-algebras compatible with pull-backs. In particular,Steenrod operations preserve rationality of cycles.

In the case of projective homogeneous varieties over the field of complexnumbers Sl is compatible with its topological counterparts: the reduced poweroperation P l if p 6= 2 and the Steenrod square Sq2l if p = 2 (over com-plex numbers Ch∗(X ) can be viewed as a subring of the singular cohomologyH2∗

sing(X ,Z/p)). Moreover, Ch∗(G) may be identified with the image of thepull-back map H2∗

sing(X ,Z/p) → H2∗sing(G,Z/p). An explicit description of this

image and formulae describing the action of P l and Sq2l on H∗sing(G,Z/p) can

be found in [MT91] for exceptional groups and in [EKM] for classical groups.The action of the Steenrod operations on Ch(X ) and on Ch(G) can be de-

scribed in purely combinatorial terms (see [DZ07]) and, hence, doesn’t dependon the choice of a base field F .

The following lemma provides an important technical tool for computingpossible values of the J-invariant of G.

5.2.4 Lemma. Assume that in Ch∗(G) we have Sl(xi) = xps

m , and for anyi′ < i Sl(xi′) < xps

m with respect to the DegLex order. Then jm ≤ ji + s.

Proof. By definition there exists a cycle α ∈ Ch(X ) such that the leading termof π(α) is xpji

i . For the total operation we have

S(xpji

i ) = S(xi)pji = S0(xi)pji + S1(xi)pji + . . .+ Sdi(xi)pji.

In particular, Slpji (xpji

i ) = Sl(xi)pji . Applying Slpji to α we obtain a rationalcycle whose image under π has the leading term xpji+s

m .

5.2.5. We summarize information about restrictions on the J-invariant intothe following table (numbers r, di and ki are taken from [Kc85, Table II]). Forgroups we use the notation from [INV].

Recall that r is the number of generators of Ch∗(G), di are their codimen-sions, ki define the p-power relations, Jp(G) = (j1, . . . , jr), 0 ≤ ji ≤ ki for all i,and G = ξ(G) for some ξ ∈ H1(F, G). If p is not in the table, then Jp(G) = ()is empty.

5.3. MOTIVES OF COMPLETE FLAGS 45

G p r ki, i = 1..r di, i = 1..r restrictions on jiSLn /µm, m|n p|m 1 pk1 ||n 1PGSpn, 2|n 2 1 2k1 ||n 1O+

n , n > 2 2 [n+14 ] [log2

n−12i−1 ] 2i− 1 ji ≥ ji+l if 2 -

(i−1

l

),

ji ≤ j2i−1 + 1Spinn, n > 2 2 [n−3

4 ] [log2n−12i+1 ] 2i+ 1 ji ≥ ji+l if 2 -

(il

),

ji ≤ j2i + 1PGO+

2n, n > 1 2 [n+22 ] 2k1 ||n 1, i = 1 ji ≥ ji+l if 2 -

(i−2

l

),

[log22n−12i−3 ] 2i− 3, i ≥ 2 ji ≤ j2i−2 + 1

Spin±2n, 2|n 2 n2 2k1 ||n 1, i = 1 ji ≥ ji+l if 2 -

(i−1

l

)[log2

2n−12i−1 ] 2i− 1, i ≥ 2 ji ≤ j2i−1 + 1

G2, F4, E6 2 1 1 3F4, Esc

6 , E7 3 1 1 4Ead

6 3 2 2, 1 1, 4Esc

7 2 3 1, 1, 1 3, 5, 9 j1 ≥ j2 ≥ j3Ead

7 2 4 1, 1, 1, 1 1, 3, 5, 9 j2 ≥ j3 ≥ j4E8 2 4 3, 2, 1, 1 3, 5, 9, 15 j1 ≥ j2 ≥ j3,

j1 ≤ j2 + 1,j2 ≤ j3 + 1.

E8 3 2 1, 1 4, 10 j1 ≥ j2E8 5 1 1 6

The upper indices sc and ad denote simply-connected and adjoint groupsrespectively. The last column of the table follows from Lemma 5.2.4 and, hence,requires char (F ) 6= p restriction. All other columns are taken directly from[Kc85, Table II] and are independent on the characteristic of the base field.

5.3 Motives of complete flags

In the present section we describe a basis of the subring of rational cycles ofCh(X × X ), where X is the variety of complete flags. The key results here arePropositions 5.3.3 and 5.3.10. As a consequence, we obtain a motivic decompo-sition of X (Theorem 5.3.13) in terms of certain motive Rp(G).

5.3.1. We use the notation of the previous section. Let G be a semisimplegroup of inner type over F and let X be the respective variety of complete flags.Let R ⊆ Ch(X ) be the image of the characteristic map. Consider the quotientmap π : Ch(X ) → Ch(G). Fix preimages ei of xi in Ch(X ). For an r-tupleM = (m1, . . . ,mr) set eM =

∏ri=1 e

mii . Set N = (pk1 − 1, . . . , pkr − 1) and

d = dimX − |N |.

5.3.2 Lemma. The Chow ring Ch(X ) is a free R-module with a basis eM,M 4 N .

Proof. Note that the subgroup R+ of the non-constant elements of R is a nilpo-tent ideal in R. Applying the Nakayama Lemma we obtain that eM generates


Ch(X ). By [Kc85, (2)] Ch(X ) is a free R-module, hence, for the Poincare poly-nomials we have

P (Ch∗(X ), t) = P (Ch∗(G), t) · P (R∗, t).

Substituting t = 1 we obtain that

rkCh(X ) = rkCh(G) · rkR.

To finish the proof observe that rk Ch(G) coincides with the number of genera-tors eM.

5.3.3 Proposition. The pairing R×R→ Z/p given by (α, β) 7→ deg(eNαβ) isnon-degenerated, i.e. for any non-zero element α ∈ R there exists β such thatdeg(eNαβ) 6= 0.

Proof. Choose a homogeneous basis of Ch(X ). Let α∨ be the Poincare dual ofα with respect to this basis. By Lemma 5.3.2 Ch(X ) is a free R-module withthe basis eM, hence, expanding α∨ we obtain

α∨ =∑

M4N

eMβM , where βM ∈ R.

Note that if M 6= N then codimαβM > d, therefore, αβM = 0. So we can setβ = βN .

We fix a homogeneous Z/p-basis αi of R and the dual basis α#i with

respect to the pairing introduced in Proposition 5.3.3.

5.3.4 Corollary. For |M | ≤ |N | we have

deg(eMαiα#j ) =

1, M = N and i = j;0, otherwise.

Proof. If M = N , then it follows from the definition of the dual basis. Assume|M | < |N |. If deg(eMαiα

#j ) 6= 0, then codim(αiα

#j ) > d, in contradiction

with the fact that αiα#j ∈ R. Hence, we are reduced to the case M 6= N and

|M | = |N |. Since |M | = |N |, codim(αiα#j ) = d and, hence, R+αiα

#j = 0. On

the other hand there exists i such that mi ≥ pki and epki ∈ Ch(X ) ·R+. Hence,eMαiα

#j = 0.

5.3.5 Definition. Given two pairs (l, L) and (m,M), where L,M are r-tuplesand l,m are integers, we say (l, L) ≤ (m,M) if either l < m, or in the casel = m we have L ≤M . We introduce a filtration on the ring Ch(X ) as follows:

The (m,M)-th term Ch(X )m,M of the filtration is the Z/p-subspacespanned by the elements eIα with I ≤ M , α ∈ R homogeneous,|I|+ codimα ≤ m.


Define the associated graded ring as follows:

A∗,∗ =⊕

(m,M)

Am,M , where Am,M = Ch(X )m,M/⋃

(l,L)(m,M)

Ch(X )l,L.

By Lemma 5.3.2 if M 4 N the graded component Am,M consists of the classesof elements eMα with α ∈ R and codimα = m− |M |. In particular, rkAm,M =rkRm−|M |. Comparing the ranks we see that Am,M is trivial when M 64 N .

Consider the subring Ch(X ) of rational cycles with the induced filtration.The associated graded subring will be denoted by A∗,∗rat. From the definition ofthe J-invariant it follows that the elements epji

i , i = 1, . . . , r, belong to A∗,∗rat.Similarly, we introduce a filtration on the ring Ch(X × X ) as follows:

The (m,M)-th term of the filtration is the Z/p-subspace spannedby the elements eIα× eLβ with I + L ≤M , α, β ∈ R homogeneousand |I|+ |L|+ codimα+ codimβ ≤ m.

The associated graded ring will be denoted by B∗,∗. By definition B∗,∗ is isomor-phic to the tensor product of graded rings A∗,∗ ⊗Z/p A

∗,∗. The graded subringassociated to Ch(X × X ) will be denoted by B∗,∗

rat.

5.3.6. The key observation is that due to Corollary 5.3.4 we have

Ch(X × X )m,M Ch(X × X )l,L ⊂ Ch(X × X )m+l−dimX ,M+L−N and

(Ch(X × X )m,M )?(Ch(X )l,L) ⊂ Ch(X )m+l−dimX ,M+L−N

and, therefore, we have a correctly defined composition law

: BM,m ×BL,l → Bm+l−dimX ,M+L−N

and the realization map

? : BM,m ×AL,l → Am+l−dimX ,M+L−N

In particular, BdimX+∗,N+∗ can be viewed as a graded ring with respect to thecomposition and (α β)? = α? β?. Note also that both operations preserverationality of cycles.

The proof of the following result is based on the fact that the group G splitsover F (X ).

5.3.7 Lemma. The classes of the elements ei× 1− 1× ei in B∗,∗, i = 1, . . . , r,belong to B∗,∗

rat.

Proof. Fix an i. Since G splits over F (X ), F (X ) is a splitting field of X and byLemma 3.3.4 there exists a cycle in Chdi(X × X ) of the form

ξ = ei × 1 +∑

s

µs × νs + 1× µ,


where codimµs, codim νs < di. Then the cycle

pr∗13(ξ)− pr∗23(ξ) = (ei × 1− 1× ei)× 1 +∑

s

(µs × 1− 1× µs)× νs

belongs to Ch(X ×X ×X ), where prij denotes the projection on the product ofthe i-th and j-th factors. Applying Corollary 4.5.2 to the projection pr12 : X ×X × X → X × X we conclude that there exists a (non-canonical) Ch(X × X )-linear isomorphism Ch(X×X×X ) ' Ch(X×X )⊗Ch(X ), where Ch(X×X ) actson the left-hand side via pr∗12. This gives rise to a Ch(X ×X )-linear retractionδ to the pull-back map pr∗12. Since the construction of the retraction preservesbase change, it preserves rationality of cycles. Hence, passing to a splitting fieldwe obtain a rational cycle

δ(pr∗13(ξ)− pr∗23(ξ)) = ei × 1− 1× ei +∑

s

(µs × 1− 1× µs)δ(1× 1× νs)

whose image in B∗,∗rat is ei × 1− 1× ei.

We will write (e × 1 − 1 × e)M for the product∏r

i=1(ei × 1 − 1 × ei)mi

and(ML

)for the product of binomial coefficients

∏ri=1

(mi

li

). We assume that(

mi

li

)= 0 if li > mi. In the computations we will extensively use the following

two formulae (the first follows directly from Corollary 5.3.4 and the second oneis a well-known binomial identity).

5.3.8. Let α be an element of R∗ and α# be its dual with respect to the non-degenerate pairing from 5.3.3, i.e. deg(eNαα#) = 1. Then we have

((e× 1− 1× e)M (α# × 1))?(eLα) =(

M

M + L−N

)(−1)M+L−NeM+L−N .

Indeed, expanding the brackets in the left-hand side, we obtain( ∑I4M

(−1)I

(M

I

)eM−Iα# × eI

)?(eLα),

and it remains to apply Corollary 5.3.4.

5.3.9 (Lucas’ Theorem). The following identity holds(n

m

)≡

∏i≥0

(ni

mi

)mod p,

where m =∑

i≥0mipi and n =

∑i≥0 nip

i are the base p presentations of mand n.

Let J = Jp(G) = (j1, . . . , jr) be the J-invariant of G (see Definition 5.1.6).Set K = (k1, . . . , kr).


5.3.10 Proposition. Let αi be a homogeneous Z/p-basis of R. Then the setof elements B = epJLαi | L 4 pK−J − 1 forms a Z/p-basis of A∗,∗rat.

Proof. According to Lemma 5.3.2 the elements from B are linearly independent.Assume B does not generate A∗,∗rat. Choose an element ω ∈ Am,M

rat of the smallestindex (m,M) which is not in the linear span of B. By definition of Am,M (seeDefinition 5.3.5) ω can be written as ω = eMα, where M 4 N , α ∈ Rm−|M |

and M can not be presented as M = pJL′ for an r-tuple L′. The latter meansthat in the decomposition of M into p-primary and p-coprimary componentsM = pSL, where M = (m1, . . . ,mr), S = (s1, . . . , sr), L = (l1, . . . , lr) andp - lk for k = 1, . . . , r, we have J 64 S. Choose an i such that si < ji. DenoteMi = (0, . . . , 0,mi, 0, . . . , 0) and Si = (0, . . . , 0, si, 0, . . . , 0), where mi and si

stand at the i-th place.Set T = N −M + Mi. By Lemma 5.3.7 and 5.3.8 together with observa-

tion 5.3.6 the element

((e× 1− 1× e)T (α# × 1))?(eMα) =(pki − 1mi

)(−1)miemi

belongs to A|Mi|,Mi

rat . By 5.3.9 we have p -(pki−1

mi

)and, therefore, this element is

non-trivial. Moreover, since si < ji, this element is not in the span of B. Since(m,M) was chosen to be the smallest index and (|Mi|,Mi) ≤ (m,M) we obtainthat (m,M) = (|Mi|,Mi). Repeating the same arguments for T = N−Mi +pSi

we obtain that Mi = pSi , i.e., li = 1.Now let γ be a representative of ω = epsi

i in Ch(X ). Then its image π(γ) inCh(G) has the leading term xpsi

i with si < ji. This contradicts the definition ofthe J-invariant.

5.3.11 Corollary. The elements

(e× 1− 1× e)S(epJLαi × epJ (pK−J−1−M)α#j ) | L,M 4 pK−J − 1, S 4 pJ − 1

form a Z/p-basis of B∗,∗rat. In particular, the ones such that S = pJ − 1 and

L = M form a basis of BN,drat .

Proof. According to Lemma 5.3.2 these elements are linearly independent andtheir number is p|2K−J|(rkR)2. They are rational by Definition 5.3.5 andLemma 5.3.7. Applying Corollary 4.5.2 to the projection X × X → X weobtain that

rkB∗,∗rat = rk Ch(X × X ) = rkCh(X ) · rkCh(X ),

where the latter coincides with rkA∗,∗rat·p|K| rkR = p|2K−J|(rkR)2 by Lemma 5.3.2and Proposition 5.3.10.

5.3.12 Lemma. The elements

θL,M,i,j = (e× 1− 1× e)pJ−1(epJLαi × epJ (pK−J−1−M)α#j ), L,M 4 pK−J − 1,

belong to B∗,∗rat and satisfy the relations θL,M,i,j θL′,M ′,i′,j′ = δLM ′δij′θL′,M,i′,j

and∑

L,i θL,L,i,i = ∆X .


Proof. Expanding the brackets and using the identity(pj−1

i

)≡ (−1)i mod p,

we see thatθL,M,i,j =

∑I4pJ−1

epJL+Iαi × eN−pJM−Iα#j ,

and the composition relation follows from Corollary 5.3.4. By definition we have∑L,i

θL,L,i,i =∑

I4N,i

eIαi × eN−Iα#i .

By Corollary 5.3.4 the latter sum acts trivially on all basis elements of Ch(X )and, hence, coincides with the diagonal.

We are now ready to provide a motivic decomposition of the variety of com-plete flags.

5.3.13 Theorem. Let G be a semisimple linear algebraic group of inner typeover a field F and X be the variety of complete G-flags. Let p be a prime.Assume that Jp(G) = (j1, . . . , jr). Then the motive of X is isomorphic to thedirect sum

M(X ; Z/p) '⊕i≥0

Rp(G)(i)⊕ci ,

where the motive Rp(G) is indecomposable, its Poincare polynomial over a split-ting field is given by

P (Rp(G), t) =r∏

i=1

1− tdipji

1− tdi, (5.3)

and the integers ci are the coefficients of the quotient∑i≥0

citi = P (Ch∗(X ), t)/P (Rp(G), t).

Proof. Consider the projection map

f0 : Ch(X × X )dimX ,N → BdimX ,Nrat .

Observe that the kernel of f0 is nilpotent. Indeed, any element ξ from ker f0

belongs to Ch(X ×X )m,M for some (m,M) (dimX , N) which depends on ξ.Then by 5.3.6 its i-th composition power ξi belongs to the graded componentCh(X × X )im−(i−1) dimX ,iM−(i−1)N , and, therefore, becomes trivial for i bigenough.

By Lemma 5.3.12 the elements θL,L,i,j form a family of pairwise-orthogonalidempotents whose sum is the identity. Therefore, by Proposition 3.5.6 thereexist pair-wise orthogonal idempotents ϕL,i in Ch(X × X ) which are mappedto θL,L,i,i and whose sum is the identity.

Recall that given two correspondences φ and ψ in Ch(X × X ) of degrees cand c′ respectively its composite φ ψ has degree c + c′. Using this fact we


conclude that the homogeneous components of ϕL,i of codimension dimX arepair-wise orthogonal idempotents whose sum is the identity. Hence, we mayassume that ϕL,i belong to ChdimX (X × X ).

We now show that ϕL,i are indecomposable. By Corollary 5.3.11 and Lemma 5.3.12the ring (BdimX ,N

rat , ) can be identified with a product of matrix rings over Z/p

BdimX ,Nrat '

d∏s=0

End((Z/p)p|K−J| rk Rs

).

By means of this identification θL,L,i,i : epJMαj 7→ δL,Mδi,jepJLαi is an idem-

potent of rank 1 and, therefore, is indecomposable. Since the kernel of f0 isnilpotent, the ϕL,i are indecomposable as well.

Next we show that ϕL,i is isomorphic to ϕM,j . In the ring B∗,∗rat mutually

inverse isomorphisms between them are given by θL,M,i,j and θM,L,j,i. Let

f : Ch(X × X ) → B∗,∗rat

be the leading term map; it means that for any γ ∈ Ch(X × X ) we find thesmallest degree (s, I) such that γ belongs to Ch(X × X )s,I and set f(γ) to bethe image of γ in Bs,I

rat. Note that f is not a homomorphism but satisfies thecondition that f(ξ) f(η) equals either f(ξ η) or 0. Choose preimages ψL,M,i,j

and ψM,L,j,i of θL,M,i,j and θM,L,j,i by means of f . Applying Lemma 3.5.5 weobtain mutually inverse isomorphisms ϑL,M,i,j and ϑM,L,j,i between ϕL,i andϕM,j . By the definition of f it remains to take their homogeneous componentsof the appropriate degrees.

Applying now Lemma 3.5.9 and Corollary 3.5.13 to the restriction map

resF : End(M(X ; Z/p)) → End(M(X ; Z/p))

and the family of idempotents ϕL,i we obtain a family of pair-wise orthogonalidempotents φL,i ∈ End(M(X ; Z/p)) such that

∆X =∑L,i

φL,i.

Since resFs/F lifts isomorphisms, for the respective motives we have (X , φL,i) '(X , φ0,0)(|L| + codimαi) for all L and i (see 3.5.2). The twists |L| + codimαi

can be easily recovered from the explicit formula for θL,L,i,i (see Lemma 5.3.12).Denoting Rp(G) = (X , φ0,0) we obtain the desired motivic decomposition.

Finally, consider the motive Rp(G) over a splitting field. The idempotentθ0,0,0,0 splits into the sum of pair-wise orthogonal (non-rational) idempotentseI × eN−I1#, I 4 pJ − 1. The motive corresponding to each summand isisomorphic to (Z/p)(|I|). Therefore, we obtain the decomposition into Tatemotives

Rp(G) '⊕

I4pJ−1

(Z/p)(|I|),

which gives formula (5.3) for the Poincare polynomial.


As a direct consequence of the proof we obtain

5.3.14 Corollary. Any direct summand of M(X ; Z/p) is isomorphic to a directsum of twisted copies of Rp(G).

Proof. Indeed, in the ring BN,drat any idempotent is isomorphic to a sum of idem-

potents θL,L,i,i, and the map f0 lifts isomorphisms.

5.3.15 Remark. Corollary 5.3.14 can be viewed as a particular case of theKrull-Schmidt Theorem proven by V. Chernousov and A. Merkurjev (see [CM06,Corollary 9.7]).

5.4 Motives of generically split flags

Using motivic decompositions of cellular fibrations (see Theorem 4.5.4) we gen-eralize Theorem 5.3.13 to arbitrary generically split projective homogeneousvarieties. At the end we discuss some properties of the motive Rp(G).

5.4.1 Definition. Let G be a linear algebraic group over a field F and let X bea projective homogeneous G-variety. We say X is generically split if the groupG splits over the generic point of X.

The main result of the present section is the following

5.4.2 Theorem. Let G be a semisimple linear algebraic group of inner typeover a field F and let p be a prime integer. Let X be a generically split pro-jective homogeneous G-variety. Then the motive of X with Z/p-coefficients isisomorphic to the direct sum

M(X; Z/p) '⊕i≥0

Rp(G)(i)⊕ai ,

where Rp(G) is an indecomposable motive; Poincare polynomial P (Rp(G), t) isgiven by (5.3) and, hence, only depends on the J-invariant of G; the ai’s arethe coefficients of the quotient polynomial∑

i≥0

aiti = P (CH∗(X), t)/P (Rp(G), t).

Proof. Let X be the variety of complete G-flags. According to Theorem 4.5.4the motive of Y = X is isomorphic to a direct sum of twisted copies of the motiveof X. To finish the proof we apply Theorem 5.3.13 and Corollary 5.3.14.

We now provide several properties of Rp(G) which will be extensively usedin the applications.

5.4.3 Proposition. Let G, G′ be two semisimple algebraic groups of inner typeover F and let X , X ′ be the corresponding varieties of complete flags.

5.4. MOTIVES OF GENERICALLY SPLIT FLAGS 53

(i) (base change) For any field extension E/F we have

Rp(G)E '⊕i≥0

Rp(GE)(i)⊕ai ,

where∑ait

i = P (Rp(G), t)/P (Rp(GE), t).

(ii) (transfer argument) If E/F is a field extension of degree coprime to pthen Jp(GE) = Jp(G) and Rp(GE) = Rp(G)E. Moreover, if Rp(GE) 'Rp(G′E) then Rp(G) ' Rp(G′).

(iii) (comparison lemma) If G splits over F (X ′) and G′ splits over F (X )then Rp(G) ' Rp(G′).

Proof. The first claim follows from Theorem 5.3.13 and Corollary 5.3.14. Toprove the second claim note that E is rank preserving with respect to X andX × X by Lemma 3.5.12. Now Jp(GE) = Jp(G) by Lemma 3.5.11, and henceRp(GE) = Rp(G)E by the first claim. The remaining part of the claim followsfrom Corollary 3.5.14 applied to the variety X

∐X ′.

We now prove the last claim. The variety X ×X ′ is the variety of completeG × G′-flags. By Corollary 4.5.2 applied to the projections X × X ′ → X andX × X ′ → X ′ we can express M(X × X ′; Z/p) in terms of M(X ; Z/p) andM(X ′; Z/p). The latter motives can be expressed in terms ofRp(G) andRp(G′).The claim now follows from the Krull-Schmidt theorem (see Corollary 5.3.14).

5.4.4 Corollary. We have Rp(G) ' Rp(Gan), where Gan is the semisimpleanisotropic kernel of G.

Finally, we provide conditions which allow one to lift a motivic decompositionof a generically split homogeneous variety with Z/m-coefficients to a decompo-sition with Z-coefficients.

5.4.5. Let m be a positive integer. We say a polynomial g(t) is m-positive,if g 6= 0, P (Rp(G), t) | g(t) and the quotient polynomial g(t)/P (Rp(G), t) hasnon-negative coefficients for all primes p dividing m.

5.4.6 Proposition. Let G be a semisimple linear algebraic group of inner typeover a field F and let X be a generically split projective homogeneous G-variety.Assume that X splits by a field extension of degree m. Let f(t) be an m-positivepolynomial dividing P (M(X), t) which can not be presented as a sum of twom-positive polynomials. Then the motive of X with integer coefficients splits asa direct sum

M(X; Z) '⊕

i

Ri(ci), ci ∈ Z,

where Ri are indecomposable and P (Ri, t) = f(t) for all i. Moreover, if m =2, 3, 4 or 6, then all motives Ri are isomorphic up to twists.


Proof. First, we apply Corollary 3.5.7 to obtain a decomposition with Z/m-coefficients. By Lemma 3.5.12 our field extension is rank preserving so we canapply Theorem 3.5.19 to lift the decomposition to the category of motives withZ-coefficients.

Chapter 6

Splitting properties oflinear algebraic groups

6.1 Tits indices

To be filled

6.2 Rational cycles and rational bundles

To be filled

6.3 The J-invariant in degree one and the Titsalgebras

To be filled

6.4 Higher Tits indices

To be filled

55

Chapter 7

Applications of motivicdecompositions

7.1 Canonical dimensions

The notion of the canonical dimension cd(G) of a linear algebraic group G ap-pears naturally in the study of the splitting properties of G-torsors. It was intro-duced by Berhuy and Reichstein in [BR05] and developed further by Karpenkoand Merkurjev [KM06]. In the present section we discuss the relations betweenthe J-invariant and the values of cd(G). In particular, we provide computa-tions of canonical p-dimensions cdp(G) for all split groups G and primes p. Thisgeneralizes previously known results by Karpenko, Merkurjev and Reichstein.

7.1.1. According to [KM06, §4] the canonical dimension cd(X) of an irreducibleprojective variety X over a field F is defined to be the minimum of dimensions ofirreducible closed subvarieties Y such that YF (X) has a rational point. Assumingthat YF (X) has a closed point of degree coprime to p, where p is a fixed prime,we obtain the definition of a canonical p-dimension cdp(X) of X. Observe thatcdp(X) ≤ cd(X) ≤ dimX. The canonical (p-)dimension of a split group G isequal to the supremum of canonical (p-)dimensions of all twisted forms of thevariety of complete G-flags taken over all field extensions of F .

Let G be an inner form of a split group G. Let X be the twisted form of thevariety of complete G-flags. We obtain the following formula for the canonicalp-dimension of X

7.1.2 Proposition. In the notation of Theorem 5.3.13 we have

cdp(X ) =r∑

i=1

di(pji − 1),

where Jp(G) = (j1, . . . , jr) is the J-invariant of G.

56

7.2. DEGREES OF SPLITTING FIELDS 57

Proof. By [KM06, Theorem 5.8] we have

cdp(G) = mini | Chi(X ) 6= 0.

The proposition then follows from Proposition 5.3.10.

7.1.3 Example. (cf. [Za07, Corollary 5]) Applying the formula of the propo-sition for the maximal value of the J-invariant, i.e. for ji = ki, by Table 5.2.5we obtain the following values for the canonical p-dimensions of split groups oftypes G2, D4, F4, E6, E7 and E8

cd2 G2 = 3, cd2 Dsc4 = 3, cd2 Dad

4 = 9,cd2 F4 = 3, cd3 F4 = 8,cd2 E6 = 3, cd3 Esc

6 = 8, cd3 Ead = 16cd2 Esc

7 = 17, cd2 Ead7 = 18 cd3 E7 = 8

cd2 E8 = 60, cd3 E8 = 28 cd5 E8 = 24

7.2 Degrees of splitting fields

Let X be a smooth projective variety which has a splitting field.

7.2.1 Lemma. For any φ, ψ ∈ CH∗(X ×X) one has

deg((pr2)∗(φ · ψt)) = tr((φ ψ)?).

Proof. Choose a homogeneous basis ei of CH∗(X). Let e∨i be its Poincaredual. Since both sides of the relation under proof are bilinear, it suffices tocheck the assertion for φ = ei × e∨j and ψ = ek × e∨l . In this case both sides ofthe relation are equal to δilδjk.

We denote the greatest common divisor of degrees of all zero cycles on X byd(X) and its p-primary component by dp(X).

7.2.2 Corollary. Let m be an integer. For any φ ∈ CH(X ×X; Z/m) we have

gcd(d(X),m) | tr(φ?).

Proof. Set ψ = ∆X and apply Lemma 7.2.1.

7.2.3 Corollary. Assume that M(X; Z/p) has a direct summand M . Then

1. dp(X) | P (M, 1);

2. if dp(X) = P (M, 1) and the kernel of the restriction End(M(X)) →End(M(X)) consists of nilpotents, then M is indecomposable.

58 CHAPTER 7. APPLICATIONS OF MOTIVIC DECOMPOSITIONS

Proof. Set q = dp(X) for brevity. Let M = (X,φ). By Corollary 3.5.7 thereexists an idempotent ϕ ∈ End(M(X); Z/q) such that ϕ mod p = φ. Thenres(ϕ) ∈ End(M(X); Z/q) is a rational idempotent. Since every projectivemodule over Z/q is free, we have

tr(res(ϕ)?) = rkZ/q(res(ϕ)?) = rkZ/p(res(φ)?) = P (M, 1) mod q,

and the first claim follows from Corollary 7.2.2. The second claim follows fromthe first one, since the second assumption implies that for any non-trivial directsummand M ′ of M we have P (M ′, 1) < P (M, 1).

7.2.4. Let n(G) denote the greatest common divisor of degrees of all finitesplitting fields of G and let np(G) be its p-primary component. Note thatn(G) = d(X ) and np(G) = dp(X ).

We obtain the following estimate on np(G) in terms of the J-invariant (cf.[EKM, Prop. 88.11] in the case of quadrics).

7.2.5 Proposition. Let G be a semisimple linear algebraic group of inner typewith Jp(G) = (j1, . . . , jr). Then

np(G) ≤ p∑

i ji .

Proof. Follows from Theorem 5.3.13 and Corollary 7.2.3.

7.2.6 Corollary. The following statements are equivalent:

• Jp(G) = (0, . . . , 0);

• np(G) = 1;

• Rp(G) = Z/p.

Proof. If Jp(G) = (0, . . . , 0) then np(G) = 1 by Proposition 7.2.5. If np(G) = 1then there exists a splitting field L of degree m prime to p and, therefore,Rp(G) = Z/p by transfer argument 5.4.3(ii). The remaining implication isobvious.

7.3 Examples of decompositions

In the present section we provide examples of motivic decompositions of pro-jective homogeneous G-varieties obtained using Theorem 5.4.2, where G is asimple group of inner type over a field F . We identify certain groups of smallranks via exceptional isomorphisms, e.g. B2 = C2 (see [INV, §15]).

7.3. EXAMPLES OF DECOMPOSITIONS 59

The case r = d1 = 1. According to Table 5.2.5 Jp(G) = (j1) for some j1 ≥ 0if and only if G is a split group of type An or Cn. In this case p|(n+1) or p = 2respectively.

Let G = ξG by means of ξ ∈ Z1(F, G). Let D be a central division algebraof degree d corresponding to the class of the image of ξ in H1(F,Aut G) (see[INV, p.396 and p.404]). Observe that p | d.

LetXΘ be a projective homogeneousG-variety given by a subset Θ of verticesof the respective Dynkin diagram D. By [Ti66, p.55-56] XΘ is generically splitif D\Θ contains at least one r-th vertex with p - r (cf. Example 3.3.7). Observethat the converse is also true by the Index Reduction Formula (see [MPW,Appendix I, III]).

Then by Theorem 5.4.2 we obtain that for a generically split XΘ

M(XΘ; Z/p) '⊕i≥0

Rp(G)(i)⊕ai , (7.1)

where Rp(G) is indecomposable and

Rp(G) 'pj1−1⊕i=0

(Z/p)(i).

We now identifyRp(G). Using the comparison lemma (see Proposition 5.4.3)we conclude that Rp(G) only depends on D, so pj1 | d. On the other hand byProposition 7.2.5 we have np(G) ≤ pj1 . Since np(G) is the p-primary part of d,it coincides with pj1 (in the Cn-case it coincides with d).

We have D ' Dp ⊗F D′, where pj1 = ind(Dp) and p - ind(D′). Passingto a splitting field of D′ of degree prime to p and using Proposition 5.4.3 weconclude that the motives of XΘ and SB(Dp) are direct sums of twisted Rp(G).Comparing the Poincare polynomials we conclude that

7.3.1 Lemma. M(SB(Dp); Z/p) ' Rp(G).

7.3.2 Corollary. The motive of SB(D) with integer coefficients is indecompos-able.

Proof. We apply Proposition 5.4.6 to X = SB(D) and compare the Poincarepolynomials of M(X) and Ri.

7.3.3 Remark. Indeed, we provided a uniform proof of the results of pa-per [Ka96]. Namely, the decomposition of M(SB(Mm(D)); Z/p) (see [Ka96,Cor. 1.3.2]) and indecomposability of M(SB(D); Z) (see [Ka96, Thm. 2.2.1]).

The case r = 1 and d1 > 1. According to Table 5.2.5 this holds if and onlyif G is a split group of type (here sc denotes a simply-connected group)

p = 2: G2, F4, E6, Bsc3 , Bsc

4 , Dsc4 , Dsc

5 ;

p = 3: F4, E7, Esc6 ;


p = 5: E8.

Moreover, it follows from Table 5.2.5 that in all these cases k1 = 1.Let G = ξG, where ξ ∈ Z1(F, G). Let X be a generically split projective

homogeneous G-variety (cf. Example 3.3.7). By Theorem 5.4.2 we obtain thedecomposition

M(X; Z/p) '⊕i≥0

Rp(G)(i)⊕ai , (7.2)

where the motive Rp(G) is indecomposable and (cf. [Vo03, (5.4-5.5)])

Rp(G) 'p−1⊕i=0

(Z/p)(i · (p+ 1)).

We now identify Rp(G).Observe that for any such group G there exists a finite field extension E/F

of degree coprime to p such that GE = ξG0,E , where ξ ∈ Z1(E, Gsc). Indeed,for E6 (p = 2) and E7 (p = 3) it follows from the fact that the center of Esc

6 isµ3 and the center of Esc

7 is µ2. For all other groups it is obvious.Let r be the Rost invariant as defined in [Me03]. According to [GP07,

Lemma 2.1] if a group is given by a cocycle with values in a simply-connectedgroup, the invariant r depends only on the isomorphism class of the group butnot on the particular choice of the cocycle. Therefore, the invariant r(G) = r(ξ)is well-defined over E.

Let rp be the p-component of r. Observe that for the group G over E theinvariant rp takes values in H3(E,µ⊗2

p ). Define rp(G) = 1[E:F ]coresE/F rp(GE).

It is easy to see that rp(G) does not depend on the choice of E. Indeed, if Eand E′ are two such field extensions, we can pass to the composite field E · E′and use the functoriality of rp.

7.3.4 Lemma. Let G be a simple linear algebraic group of inner type over Fsatisfying r = 1 and d1 > 1 and let p be a prime. Then rp(G) is trivial iffRp(G) ' Z/p.

Proof. According to [Ga01, Theorem 0.5], [Ch94] and [Gi00, Theoreme 10] theinvariant rp(G) is trivial iff the group G splits over a p-primary closure of F .By Corollary 7.2.6 the latter is equivalent to the fact that Rp(G) ' Z/p.

7.3.5 Lemma. Let G and G′ be simple linear algebraic groups of inner typeover F satisfying r = 1 and d1 > 1 (observe that in this case k = 1).

If rp(G) = c · rp(G′) for some c ∈ (Z/p)×, then Rp(G) ' Rp(G′).

Proof. By transfer arguments (see Proposition 5.4.3) it is enough to prove thisover a p-primary closure of F . Let X and X ′ be the respective varieties of com-plete flags. Observe that the invariant rp(G) becomes trivial over the functionfield F (X ). Since rp(G) = c · rp(G′), the invariant becomes trivial over F (X ′)as well. By Lemma 7.3.4 X splits over F (X ′). Similarly, X ′ splits over F (X ).

7.3. EXAMPLES OF DECOMPOSITIONS 61

Therefore, by Lemma 3.3.4 there exists a rational cycle φ in ChdimX (X ×X ′

) of the form φ = 1 × pt +∑

codim αi>0 αi × βi. Observe that by definitionφ? : ptX 7→ ptX ′ . Similarly, interchanging X and X ′ we obtain a rational cycleφ′ ∈ ChdimX ′(X ′ × X ) such that φ′? : ptX ′ 7→ ptX . Restricting φ and φ′ tothe direct summands Rp(G) and Rp(G′) of M(X ) and M(X ′

) respectively weobtain the rational maps φR : Rp(G) → Rp(G′) and φ′R : Rp(G′) → Rp(G).

Since the motive Rp(G) is indecomposable and rk Chi(Rp(G)) ≤ 1 for alli, the ring of rational endomorphisms of Rp(G) is generated by the identityendomorphism ∆. The same holds for the ring of rational endomorphisms ofRp(G′). Since (φ′R)? (φR)? : ptX 7→ ptX , the composition φ′R φR = ∆.Similarly we obtain φR φ′R = ∆′. By the Rost nilpotence since φR and φ′R arerational, the motives Rp(G) and Rp(G′) are isomorphic.

Z-coefficients. Let G = ξG be a twisted form by means of a cocycle ξ ∈Z1(F, G), where G is a group of type F4 or Esc

6 . Assume that G is not split bya field extension of degree less than 4. Observe that such a group always splitsby an extension of degree 6.

Let X be a generically split projective homogeneous G-variety. Then accord-ing to Proposition 5.4.6 the Chow motive of X with integer coefficients splits asa direct sum of twisted copies of an indecomposable motive R(G) such that

R(G)⊗ Z/2 =⊕

i=0,1,2,6,7,8

R2(G)(i), P (R2(G), t) = 1 + t3,

R(G)⊗ Z/3 =⊕

i=0,1,2,3

R3(G)(i), P (R3(G), t) = 1 + t4 + t8,

P (R(G), t) = 1 + t+ t2 + . . .+ t11.

7.3.6 Remark. In particular, we provided a uniform proof of the main re-sults of papers [Bo03] and [NSZ], where the cases of G2- and F4-varieties wereconsidered.

The case r > 1. According to Table 5.2.5 this holds for split groups G ofremaining classical types Bn, Dn for p = 2; exceptional types E7, E8 for p = 2and Ead

6 , E8 for p = 3.

Projective Quadric. Consider a generically split projective quadric X. By[EKM, 25.6, 28.2] a quadric X is generically split if and only if it is a Pfis-ter quadric or its codimension one neighbor. Let φ be the k-fold Pfister form(or its codimension one neighbor) defining X. Then X = XΘ is a projectivehomogeneous G-variety, where G = O(φ) is the orthogonal group of φ, and Θis the subset of the respective Dynkin diagram obtained by removing the firstvertex.

Assume J2(G) 6= (0, . . . , 0). In view of Corollary 7.2.6 this holds if and onlyif n2(G) 6= 1. By Springer’s Theorem the latter holds if and only if φ is not


split. By Theorem 5.4.2 we obtain the decomposition

M(X; Z/2) '⊕i≥0

R2(G)(i)⊕ai

where the motive R2(G) is indecomposable. Moreover, by Theorem 3.5.19 thesame decomposition holds with Z-coefficients.

We now compute J2(G). Observe that the group G splits over the functionfield F (X) and X splits over F (x) for any x ∈ X. It is known (see [EKM, §72])that the Chow groups Chl(X) for l < 2k−1−1 consist of rational cycles, i.e. therestriction maps Chl(X) → Chl(XF (x)) are surjective for l < 2k−1 − 1. Thenby Proposition 5.2.1 and Table 5.2.5 we obtain that ji = 0 for 1 ≤ i < 2k−2.Therefore,

J2(G) = (0, . . . , 0, 1) and P (R2(G), t) = 1 + t2k−1−1. (7.3)

Finally, by Corollary 5.3.14 the motive R2(G) coincides with the motive intro-duced in [Ro98], called the Rost motive.

In this way we obtain the Rost decomposition of the motive of a Pfisterquadric and its codimension one neighbor.

Maximal orthogonal Grassmannian. Let X be a connected component of amaximal orthogonal Grassmannian. Then X = XΘ is a projective homogeneousG-variety, where G = O(q) is the orthogonal group of q, and Θ is the subset ofthe respective Dynkin diagram obtained by removing the last vertex. Accordingto [Ti66, p.55-56] X is generically split.

By Theorem 5.4.2 we obtain the decomposition

M(X; Z/2) '⊕i≥0

R2(G)(i)⊕ai ,

where the motive R2(G) is indecomposable. Comparing the Poincare polyno-mials of M(X; Z/2) and R2(G) we obtain the following two extreme cases:

• If the group G is given by a generic cocycle, i.e. q is generic, the motiveM(X; Z/2) coincides with R2(G) and, therefore, is indecomposable. Thiscorresponds to the maximal value of the J-invariant.

• If q = φ is a Pfister form or its codimension one neighbor, then by theprevious exampleR2(G) coincides with the Rost motive. This correspondsto the minimal non-trivial value of the J-invariant (7.3).

7.4 Non-hyperbolicity of orthogonal involutions

In the present section we provide an application of the J-invariant to the problemof hyperbolicity of orthogonal involutions. Namely, we prove the following

7.4. NON-HYPERBOLICITY OF ORTHOGONAL INVOLUTIONS 63

7.4.1 Theorem. (cf. [Ga07, Appendix]) Let (A, σ) be a central simple algebrawith orthogonal involution over a field F of characteristic 6= 2. If degA/ indAis odd, then the involution σ is not hyperbolic over the function field of theSeveri-Brauer variety of A.

7.4.2. The index 2 case is [PSS, Prop. 3.3]. The case of a division algebra, i.e.indA = degA, is [Ka00, Thm. 5.3]. Note that our theorem follows also from[Ka08, Thm. 3.3].

Proof. We may assume that A has index at least 4 and hence degree is divisibleby 4. Further, we may assume that (A, σ) is in I3 as in Example 1.2(3), otherwisethe conclusion is obvious.

Consider the groups G = HSpin(A, σ) and G′ = PGL(A). Let X and X ′ berespective varieties of Borel subgroups.

Assume that σ is hyperbolic over the function field F (SB(A)). (*)

Then the group G is split over F (SB(A)) and, hence, over F (X ′). Since thegroup G is split over F (X), the algebra A and the group G′ are split over F (X).

By Theorem 5.4.2 to any simple linear algebraic group of inner type over kand its torsion prime p one may associate an indecomposable Chow motive Rp

such that over the algebraic closure of F the generating function of Rp is givenby the product of r cyclotomic polynomials

r∏i=1

1− tdi2ji

1− tdi, where 0 ≤ ji ≤ ki and di > 0

the explicit values of the parameters di and bounds ki are provided in Table 5.2.5and the r-tuple of integers (j1, j2, . . . , jr) is the J-invariant.

Let R2(G) and R2(G′) be the respective motives for the groups G and G′

and for p = 2. By Proposition 5.4.3.(iii) applied to G over F (X ′) and G′ overF (X) we obtain the following motivic reformulation of the assumption (∗):

R2(G) ' R2(G′). (**)

Since the group G′ is a twisted form of the group PGLdeg A, by the firstline of Table 5.2.5 the J-invariant of G′ has only one entry (r = 1), and theparameter d1 is 1. As in the proof of Lemma 7.3.1 we obtain that the J-invariantis the list consisting of the single element s, where 2s is the index of A. Hencethe generating function of R2(G′) is (1− t2

s

)/(1− t).Similarly, since the group G is a twisted form of the group HSpindeg A, by

Table 5.2.5 the J-invariant of G has 14 degA entries with di = 2i − 1 and the

following inequality holds

j1 ≤ k1 = v2( 12 degA) = v2(2s−1 · deg A

ind A ) = s− 1 < s,

where v2 is the 2-adic valuation (here we essentially use that deg Aind A is odd).


The isomorphism (∗∗) implies the equality of the respective generating func-tions, namely

1− t2s

1− t=

r∏i=1

1− t(2i−1)2ji

1− t2i−1, where j1 < s.

We claim that it never holds. Indeed, comparing the coefficients at t2 and t3

of the polynomials at the left and the right hand side, we conclude that j1 ≥ 2and j2 = 0. Then comparing them consequently at powers t2i and t2i+1, i ≥ 2we conclude that 2j1 ≥ 2i + 1 and ji+1 = 0. Therefore, j2 = j3 = . . . = jr = 0and j1 must coincide with s, which is not the case, since j1 < s.

Hence, the assumption (∗) fails and the theorem is proven.

7.5 Classification of algebras with involutions ofsmall degrees

To be filled

Chapter 8

Cobordism cycles

In paper [Vi07] A. Vishik using the techniques of symmetric operations in alge-braic cobordism (see [Vi06]) proved that changing the base field by the functionfield of a smooth projective quadric doesn’t change the property of being ra-tional for cycles of small codimension. This fact which he calls the Main ToolLemma plays the crucial role in his construction of fields with u-invariant 2r +1.

In this chapter we prove the Main Tool Lemma for i) a class of varietiesintroduced by M. Rost in the proof of the Bloch-Kato conjecture, namely, forvarieties which possess a special correspondence (see [Ro06, Definition 5.1]), andii) for projective homogeneous F4-varieties.

As in Vishik’s proof the main technical tools are the algebraic cobordism ofM. Levine and F. Morel, the generalised degree formula and the divisibility ofChow traces of certain Landweber-Novikov operations.

In the exposition we follow paper [Za09]. We assume that the base field Fhas characteristic 0.

8.1 Algebraic cobordism and mod-p operations

In the present section we recall several facts about Landweber-Novikov opera-tions on algebraic cobordism and Rost characteristic numbers. The goal is tointroduce certain operations

φq(t)p : Ω(X) → Ch(X) parametrised by q(t) ∈ Ch(X)[[t]]

from the ring of algebraic cobordism Ω(X) of a smooth projective variety Xover F to the Chow ring Ch(X) modulo p-torsion with Z/p-coefficients of asmooth variety X, where p is a given prime.

8.1.1. The group Ωm(X) of cobordism cycles is generated by classes of propermorphisms [Z → X] of pure codimension m with Z smooth (see [LM07]). Thereare cohomological operations on Ω parametrised by partitions called Landweber-Novikov operations and denoted by SLN . These operations commute with pull-backs, satisfy projection and Cartan formulas. We will deal only with operations

65

66 CHAPTER 8. COBORDISM CYCLES

parametrized by partitions (p − 1, p − 1, . . . , p − 1). Such an operation will bedenoted by Si

LN , where i is the length of a partition.There is a commutative diagram for any integer m

Ωm(X)

SiLN

pr // CHm(X)/p-tors // Chm(X)

Si

Ωm+i(p−1)(X)

pr // CHm+i(p−1)(X)/p-tors // Chm+i(p−1)(X)

where SiLN is the Landweber-Novikov operation, pr : Ω(X) → CH(X) is the

canonical morphism of oriented theories and Si is the i-th reduced p-poweroperation.

By properties of reduced power operations Si = 0 if i > m. By commutativ-ity of the diagram it means that the composite pr Si

LN is divisible by p in theChow ring modulo p-torsion CHm+i(p−1)(X)/p-tors. Define (cf. [Vi06, 3.3])

φt(p−1)a

p = 1p (pr Si

LN ) mod p, where a = i−m > 0. (8.1)

If r is not divisible by (p− 1), then we set φtr

p = 0. Hence, we have constructedan operation φtr

p , r > 0, which maps Ωm(X) to Chr+pm(X).Finally, given a power series q(t) ∈ Ch(X)[[t]] define

φq(t)p =

∑r≥0

qrφtr

p , where q(t) =∑r≥0

qrtr.

By the very definition operations φq(t)p are additive and respect pull-backs.

8.1.2 Definition. Let [U ] be the class of a smooth projective variety U ofdimension d in the Lazard ring Ld = Ω−d(pt). Assume that (p − 1) divides d.Then the integer φtdp

([U ]) ∈ Ch0(pt) = Z/p will be called the Rost number ofU and will be denoted by ηp(U).

Using the definition of SLN the number ηp(U) can be computed as follows.Let ξ1, ξ2, . . . , ξd be the roots of the total Chern class of the tangent bundle ofU . Define c(TU )(p) =

∏dj=1(1 + ξp−1

j ). Then

ηp(U) = 1p deg

(c(TU )(p)

)−1. (8.2)

Indeed, it coincides with the number bd(U)/p introduced in [Ro06, Sect. 9].

8.1.3 Lemma. (cf. [Vi07, Prop.2.3]) Let U be a smooth projective variety ofpositive dimension d and [U ] be its class in the Lazard ring. Let β ∈ Ωj(X).Then

φtr

p ([U ] · β) = ηp(U) · pr(SiLN (β)), where r = (p− 1)(i− j) + dp > 0.

Observe that φtr

p ([U ] · β) = 0 if d is not divisible by (p− 1).

8.1. ALGEBRAIC COBORDISM AND MOD-P OPERATIONS 67

Proof. Let q : X → pt be the structure map. Then by Cartan formula

1p (pr Si

LN )([U ] · β) =∑

α+β=i

1ppr(S

αLN (q∗[U ])) · pr(Sβ

LN (β)).

To finish the proof observe that if α 6= dp−1 , then pr(Sα

LN (q∗[U ])) = q∗(pr Sα

LN ([U ])) = 0 in CH(X)/p-tors by dimension reasons.

8.1.4 Corollary. (cf. [LM07, Lemma 4.4.20]) Let U and V be smooth projec-tive varieties of positive dimensions. Then ηp(U · V ) = 0 in Z/p.

Proof. Apply Lemma 8.1.3 to X = pt and β = [V ]. Then ηp(U · V ) = ηp(U) ·pr(Si

LN ([V ])), where the last factor is divisible by p and, hence, becomes trivialmodulo p.

According to [LM07, Remark 4.5.6] the kernel of the canonical morphismpr : Ω(X) → Ch(X) is generated by classes of positive dimensions, i.e. ker(pr) =L>0 · Ω(X). Hence, any γ ∈ ker(pr) can be written as

γ =∑

uZ∈L>0

uZ · [Z → X]. (8.3)

Let γpt ∈ L denote the class uZ corresponding to the point Z = pt.

8.1.5 Lemma. Let π ∈ Ω(X×X) be an idempotent and γ ∈ Ω(X) be such thatpr(π?(γ)− γ) = 0, where π? is the realization. Then ηp(π?(γ)pt) = ηp(γpt).

Proof. In presentation (8.3) let π?(γ)− γ =∑

uZ∈L>0uZ · [Z → X]. Since π is

an idempotent and π?, q∗ are L-module homomorphisms, we obtain

0 = q∗π?(π?(γ)− γ) =∑

uZ∈L>0

uZ · q∗π?([Z → X]).

Apply ηp to the both sides of the equality. By Corollary 8.1.4 all summandswith dimZ > 0 become trivial (modulo p). Hence, ηp(upt) = 0 mod p.

8.1.6 Corollary. Let π and γ be as above. Then ηp(q∗(π?(γ)− γ)) = 0, whereq : X → pt is the structure map.

Proof. Observe that ηp(q∗(π?(γ) − γ)) =∑

uZ∈L>0ηp(uZ · [Z]), where all sum-

mands with dimZ > 0 are trivial by Corollary 8.1.4 and the summand withZ = pt is trivial by Lemma 8.1.5.

The next important lemma is a direct consequence of the result by M. Rost[Ro06, Lemma 9.3].

8.1.7 Lemma. Let X be a variety which possesses a special correspondence.Then for any γ ∈ Ω>0(X) the deg pr(Si

LN (γ)) is divisible by p in CH(X)/p-tors.


Proof. We have S• = S• · cΩ(−TXFa), where (S•) S• are the (co-)homological

operations. Hence, deg pr(SiLN (γ)) = deg

(pr(S•(γ))·pr(cΩ(−TXFa

))). Since all

Chern classes of the tangent bundle of XFa are defined over F and X possessesa special correspondence, according to [Ro06, Lemma 9.3] we obtain that

deg(pr(S•(γ)) · pr(cΩ(−TXFa

)))

= deg pr(Si(γ)) mod p.

Since S• respect push-forwards and γ has positive dimension, deg pr(Si(γ)) isdivisible by p as well.

8.2 The main tool lemma

8.2.1. Let L/F be a field extension and X be a variety over F . We say acycle z ∈ CH(XL) is defined over F if z belongs to the image of the restrictionmap resL/F . Let Ch(X) denote the Chow group of X modulo p-torsion withZ/p-coefficients, where p is a prime. Let Fa denote the algebraic closure of F .

8.2.2. Let X be a smooth proper irreducible variety over a field F of dimensionn, p be a prime and d be an integer 0 ≤ d ≤ n. Assume that X has no zero-cycles of degree coprime to p. We say X is a d-splitting variety mod p if forany smooth quasi-projective variety Y over F , for any m < d and for any cycley ∈ Chm(YFa) the following condition holds

y is defined over k ⇐⇒ yFa(X) is defined over F (X). (8.4)

8.2.3 Example. LetQ be an anisotropic projective quadric over F of dimensionn > 2 and p = 2. Then according to A. Vishik

(a) Q is a [n+12 ]-splitting variety [Vi07, Cor. 3.5.(1)];

(b) Q is a n-splitting variety if and only if Q possesses a Rost projector (theproof is unpublished).

In the present section we prove the following generalisation of 8.2.3.(b)

8.2.4 Theorem. Let X be a smooth proper irreducible variety of dimension nover a field F of characteristic 0. Assume that X has no zero-cycles of degreecoprime to p. If X possesses a special correspondence in the sense of Rost, thenX is a n

p−1 -splitting variety and the value np−1 is optimal.

Proof. The proof consists of several steps. First, following Vishik’s argumentsfor a given y ∈ Chm(XFa

) we construct a cycle ω defined over F in the cobordismring of the product Ω(XFa

×YFa). To do this we essentially use the surjectivity

of the canonical map pr : Ω → CH. The motivic decomposition of X providesan idempotent cycle π. Applying the realization of π to ω we obtain a cycleρ defined over F which can be written in the form (8.5). To finish the proofwe apply two operations φtr

p pY ∗ and pY ∗ φtr′

p to the cycle ρ. The directcomputations which are based on the generalised degree formula and [Ro06,Lemma 9.3] show that the difference (φtr

p pY ∗ − pY ∗ φtr′

p )(ρ) is defined overF and provides the cycle y.

8.2. THE MAIN TOOL LEMMA 69

I. We start as in the proof of [Vi07, Thm. 3.1]. Let Y be a smooth quasi-projective variety over F . Let y ∈ Chm(YFa) be such that yFa(X) is defined overF (X). We want to show that y is defined over F for all m < d.

Consider the commutative diagram

ω_

Ωm(X × Y )

res

pr // // Chm(X × Y )pr∗1 // //

res

Chm(YF (X))

res

u_

ω Ωm(XFa

× YFa)

pr // // Chm(XFa× YFa

)pr∗1 // // Chm(YFa(X)) yFa(X)

where the pull-back pr∗1 is surjective by the localisation sequence and pr issurjective due to [LM07, Thm.4.5.1]. By the hypothesis there exists a preimageu of yFa(X) by means of res. By the surjectivity of pr and pr∗1 , there exists apreimage ω of u. Set ω = res(ω).

II. Let X be a variety which possesses a special correspondence and has nozero-cycles of degree coprime to p. By the results of M. Rost it follows that

(a) ηp(XFa) 6= 0 mod p (see [Ro06, Thm. 9.9]),

(b) n = ps − 1 (see [Ro06, Cor. 9.12]),

(c) the Chow motive of X contains an indecomposable summand M whichover Fa splits as a direct sum of Tate motives twisted by the multiples ofd = n

p−1 (see [Ro06, Prop. 7.14])

MFa 'p−1⊕i=0

Z/pdi.

Let π be an idempotent defining the respective Ω-motive M . Then therealization ρ = π?(ω) is defined over F and can be written as (cf. [Vi07, p.368])

ρ = xn × yn +p−2∑i=1

xdi × ydi + x0 × y0 ∈ Ωm(XFa× YFa

), (8.5)

where xj ∈ Ωj(XFa), yj ∈ Ωm−n+j(YFa), x0 = [pt → XFa ], xn = π?(1) andpr(yn) = y (cf. [Vi07, Lemma 3.2]).

III. Let p∗X and p∗Y denote the pull-backs induced by projections X × Y →X,Y . Since m < d, r = (dp −m)(p − 1) > 0. Consider the cycle φtr

p (pY ∗(ρ)).It is defined over k and has codimension m.

8.2.5 Lemma. φtr

p (pY ∗(ρ)) = ηp(XFa) · y + φtr

p (y0) in Chm(YFa).


Proof. By the projection formula

φtr

p (pY ∗(xj × yj)) = φtr

p (pY ∗(p∗X(xj) · p∗Y (yj))) = φtr

p (q∗(xj) · yj),

where xj × yj is a summand of (8.5) and q : X → pt is the structure map.Assume 0 < j < n. By Lemma 8.1.3 we obtain

φtr

p (pY ∗(xj × yj)) = ηp(q∗(xj)) · pr(SlLN (yj)), where l = n−j

p−1 .

Since codim(yj) = m − n + j ≤ m − n + (p − 2)d = m − d < 0, we havepr(Sl

LN (yi)) = Sl(pr(yj)) = 0 in Chm(XFa). Therefore, only the very right and

the left summands of (8.5) remain non-trivial after applying φtr

p pY ∗.Now by Lemma 8.1.6 the first summand is equal to

φtr

p (pY ∗(xn × yn)) = ηp(q∗(xn)) · pr(yn) = ηp(q∗(π∗(1))) · y = ηp(XFa) · y.

and the last summand φtr

p (pY ∗(x0 × y0)) = φtr

(q∗(x0) · y0) = φtr

p (y0).

Since m < d, r′ = (d−m)(p− 1) > 0. Consider the cycle pY ∗(φtr′

p (ρ)). It isdefined over k and has codimension m.

8.2.6 Lemma. pY ∗(φtr′

p (ρ)) = φtr

p (y0) in Chm(YFa).

Proof. By the very definition pY ∗(φtr′

p (xj × yj))

= 1ppY ∗pr(Sd

LN (xj × yj)). Bythe projection and Cartan formulas the latter can be written as

1p deg

(pr(Sa

LN (xj)))· pr(Sd−a

LN (yj)), where a = jp−1 .

Since m < d, the cycles yj have negative codimensions for all j < n and,therefore, pr(Sd−a

LN (yj)) is divisible by p for all j < n. On the other hand,by Lemma 8.1.7 the deg

(pr(Sa

LN (xj)))

is divisible by p for all j > 0. Hence,

pY ∗(φtr′

p (xj × yj))

= 0 for all 0 < j < n.Consider the case j = n. Recall that xn = π?(1). Then we have

1p deg

(pr(Sd

LN (xn)))

=∑

uZ∈L>0

ηp(uZ) · deg pr(Sd−aLN ([Z → X])),

where a = dim Zp−1 and xn−1 =

∑uZ [Z → X] in presentation (8.3). Observe that

it is trivial mod p, since for all dimZ > 0 the degree of the cycle pr(Sd−aLN ([Z →

X])) is trivial by Lemma 8.1.7 and for dimZ = 0 the Rost number ηp(upt) istrivial by Lemma 8.1.5.

Hence, only the very last summand, i.e. x0 × y0, remains non-trivial afterapplying pY ∗ φtr′

which gives pY ∗(φtr′

(x0 × y0)) = φtr

(y0).

8.3. F4-VARIETIES 71

IV. By Lemmas 8.2.5 and 8.2.6 the following cycle is defined over F

φtr

p (pY ∗(ρ))− pY∗(φtr′

p (ρ)) = ηp(XFa) · y.

Since ηp(XFa) 6= 0 mod p, the cycle y is defined over F , therefore, X is ad-splitting variety.

To see that d = np−1 is an optimal value take Y = X and consider the cycle

y ∈ Chd(XFa) which generates the Chow group of the Tate motive Z/pn− d

in the decomposition of MFa over Fa. Observe that y coincides with the cycle Hintroduced in [Ro06, Sect. 5]. Since M splits over F (X), yFa(X) is defined overF (X). By condition (8.4) we obtain that y is defined over F , i.e. M splits overF which contradicts to the indecomposability of M . The theorem is proven.

8.3 F4-varieties

In the present section we apply the techniques developed in the proof of 8.2.4to describe all d-splitting varieties of type F4. Namely, we prove the following

8.3.1 Corollary. Let X be a projective homogeneous variety of type F4 andp be one of its torsion primes (2 or 3). Assume that X has no zero-cycles ofdegree coprime to p. Then depending on p we have

p = 2: If X is of type F4/P4, then X is a (dimX)-splitting variety. For all othertypes X is a 3-splitting variety and this value is optimal.

p = 3: X is always a 4-splitting variety and this value is optimal.

Proof. We follow the steps of the proof of Theorem 8.2.4

I. Let X be a smooth geometrically cellular variety over F of dimension n. Asin the beginning of the previous section given a cycle y ∈ Chm(YFa), where Yis smooth quasi-projective, we construct a cobordism cycle ω ∈ Ωm(XFa

×YFa)

defined over k.

II. Assume that the motive of X contains a motive M = (X,π) such that

πFa = γ × γ∨ +p−2∑i=0

xdi × x∨di, where d > 1,

the cycle γ ∈ Ω(p−1)d(XFa) is defined over F , γ∨ denotes its Poincare dual, i.e.

pr(γ · γ∨) = pt, and xj ∈ Ωj(XFa). Then the realization ρ = π?(p∗X(γ) · ω) ∈

Ωm+n−g(XFa × YFa) is defined over F and can be written as (cf. (8.5))

ρ = xg × yg +p−2∑i=1

xdi × ydi + x0 × y0,


where g = (p− 1)d, xg = π?(γ) and pr(yg) = y.The transposed cycle πt defines an opposite direct summand M t = (X,πt)

of the motive of X (the one which contains the generic point of X over Fa).The realization ρ′ = πt

?(ω) is defined over F and can be written as

ρ′ = x(0) × y(0) +p−2∑i=1

x(di) × y(di) + x(g) × y(g) ∈ Ωm(XFa × YFa),

where x(j) ∈ Ωj(XFa) and y(g) = y0.

III. Since the Chow group of the cellular variety XFa is torsion-free, we mayuse Mod-p operations over Fa. We now apply the operations φtr pY ∗ andpY ∗ (p∗X(γ) · φtr′

) to the cycles ρ and ρ′ respectively.Applying the first operation and repeating the arguments of the proof of

Lemma 8.2.5 we obtain that for any m < d, r = (dp−m)(p− 1), the followingcycle is defined over F

φtr

p (pY ∗(ρ)) = ηp(γ) · y + φtr

p (y0) in Chm(YFa).

To apply the second operation for any m < d consider the cycle δ =pY ∗(p∗X(γ) · φtr′

p (ρ′)), where r′ = (d − m)(p − 1). It is defined over k andhas codimension m. Since we don’t have the version of [Ro06, Lemma 9.3] foran arbitrary variety, to compute δ we have to treat each torsion prime caseseparately.

For p = 2 we obtain δ = φtr

p (y0) by dimension reasons. Indeed, in this casethe cycle ρ′ consists only of two summands

ρ′ = x(0) × y(0) + γ∨ × y(g),

where the first summand vanishes, since SαLN (x(0)) = 0 if |α| > 0 and the second

summand gives the required cycle φtr

p (y0).For p = 3 the cycle ρ′ consists of three terms

ρ′ = x(0) × y(0) + x(d) × y(d) + γ∨ × y(g),

where again the first summand vanishes, the last gives φtr

(y0) and the middlegives

deg(pr(γ) · pr(Sα

LN (x(d))))· 1

ppr(SβLN (y(d))), where α = β = d/2. (8.6)

Hence, following the part IV of the previous section to prove that X is ad-splitting variety for p = 2 or 3 it is enough to assume that ηp(γ) 6= 0 mod pand that (8.6) vanishes for p = 3.

8.3. F4-VARIETIES 73

IV. We use the following notation. Let G be a simple linear algebraic groupover F . Recall that a projective homogeneous G-variety X is of type D, ifthe group G has a root system of type D. Moreover, if XFa

is the variety ofparabolic subgroups of G defined by the subset of simple roots S of D, then wesay that X is of type D/PS .

Given an F4-variety X we provide a cycle γ satisfying ηp(γ) 6= 0 mod p asfollows

p = 2: If X is generically split over the 2-primary closure of F , then wemay assume that X is of type F4/P1. In this case X has dimension 15 andby Theorem 5.4.2 the Chow motive of X with Z/p-coefficients splits as a directsum of twisted copies of a certain motive M = (X,π) with the generatingfunction P (MFa

, t) = 1 + t3. Since the Chow group Chr(XFa) has rank 1 for

r = 0 . . . 3, the idempotent πFa can be written as πFa = γ × γ∨ + pt× 1, whereγ is represented by a 3-dimensional subquadric Q3 → XFa which is an additivegenerator of Ch3(XFa

) defined over k. Since η2(Q3) = 1, X is a d-splittingvariety with d = 3.

If X is not generically split, i.e. X is of the type F4/P4, then X is a splittingvariety of the symbol given by the cohomological invariant f5. By the resultof Rost [Ro06, Rem. 2.3 and § 8] X is a variety which possesses a specialcorrespondence. Hence, by Thm. 8.2.4 X is a 15-splitting variety.

p = 3: In this case all F4-varieties are generically split over the 3-primaryclosure of F and we may assume thatX is of type F4/P4. Similar to the previouscase using the motivic decomposition of Theorem 5.4.2 we obtain an idempotent

πFa= γ × γ∨ + x4 × x∨4 + pt× 1,

where γ ∈ Ω8(XFa) is defined over F and x4 ∈ Ω4(XFa) is not. By the explicitformulae from [NSZ, 5.5] we may identify pr(γ) with the 7-th power of the gener-ator H of the Picard group of XFa

. which is the only cycle in Ch8(XFa) defined

over F . Since H is very ample, H7 can be represented by a smooth projectivesubvariety Z of XFa

. Hence, we may identify γ with the class [Z → XFa]. Then

the direct computations using the adjunction formula [Fu98, Example 3.2.12]show that η3(Z) 6= 0 mod 3.

To prove the vanishing of the cycle (8.6) it is enough to prove the vanishingof the cycle

deg(pr(γ) · S2(pr(x(4)))

)· φt12−2m

3 (y(4)).

Direct computations show that S2(Ch4(XFa)) is trivial, hence, δ = φtr

p (y0) andX is a d-splitting variety with d = 4. The Corollary 8.3.1 is proven.


8.3.2 Remark. Let X be a d-splitting geometrically cellular variety. As animmediate consequence of [KM06, Cor. 4.11] we obtain the following bound fora canonical p-dimension of X

cdp(X) ≥ d.

In the case of a variety of type F4/P4 it gives cd2(X) = dimX = 15.

Chapter 9

Cohomological invariantsversus motivic invariants

9.1 Cohomological invariants

9.2 Basic correspondence of a splitting variety(after M. Rost)

9.3 Symbols and Rost motives

75

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