J. K-Theory 3 - Laboratory of Mathematical Logic

J. K-Theory 3 (2009), 85–102doi:10.1017/is008001021jkt046©2009 ISOPPMotivic decomposition of anisotropic varieties of type F 4 intogeneralized Rost motivesbyS. NIKOLENKO, N.SEMENOV AND K. ZAINOULLINE AbstractWe prove that the Chow motive of an anisotropic projective homogeneousvariety of type F 4 is isomorphic to the direct sum of twisted copies of ageneralized Rost motive. In particular, we provide an explicit constructionof a generalized Rost motive for a generically splitting variety for a symbol inK M 3 .k/=3. We also establish a motivic isomorphism between two anisotropicnon-isomorphic projective homogeneous varieties of type F 4 . All our resultshold for Chow motives with integral coefficients.Key Words: Chow motives, algebraic groups of type F 4Mathematics Subject Classification 2000: 20G15, 14F421. IntroductionThe subject of the present paper begins with the celebrated result of M. Rost [Ro98]devoted to the motivic decomposition of a norm quadric. The existence of sucha decomposition became one of the main ingredients in the proof of the Milnorconjecture by V. Voevodsky. The generalization of this conjecture to other primesp>2, known as the Bloch-Kato conjecture, was proven recently by M. Rost andV. Voevodsky. One of the ingredients of the proof is the fact that the motive withZ=pZ-coefficients of a splitting (norm) variety X contains as a direct summand acertain geometric motive M p 1 called generalized Rost motive [Vo03, Sect. 5]. Thismotive is indecomposable and splits as a direct sum of twisted Lefschetz motivesover the separable closure of the base field.Note that Voevodsky’s construction of M p 1 relies heavily on the language oftriangulated category of motives. The main goal of the present paper is to providean explicit and shortened construction of the motive M 2 (p D 3) working onlywithin the classical category of Chow motives. More precisely, we provide such aconstruction for an exceptional projective homogeneous variety of type F 4 whichsplits the symbol (in K M 3 .k/=3) given by the Rost-Serre invariant g 3 (see 4.5). Supported partially by DAAD, INTAS 00-566, Alexander von Humboldt foundation, RTN-Network HPRN-CT-2002-00287.

86 S. NIKOLENKO, N.SEMENOV &K.ZAINOULLINENote that if X is generically cellular and splits a pure symbol, it is expected thatthe motive of X is isomorphic to the direct sum of twisted copies of the motiveM p 1 . The motivic decomposition that we obtain confirms these expectations.Namely, we prove the following1.1 Theorem Let X be an anisotropic variety over a field k such that over a cubicfield extension k 0 of k it becomes isomorphic to the projective homogeneous varietyG=P ,whereG is a split simple group of type F 4 and P its maximal parabolicsubgroup corresponding to the last or the first three vertices of the Dynkin diagram4.1.Then the Chow motive of X (with integral coefficients) is isomorphic to thedirect sum of twisted copies of an indecomposable motive RM.X/ Š7MR.i/ (1)which has the property that over k 0 it becomes isomorphic to the direct sum oftwisted Lefschetz motives Z ˚ Z.4/ ˚ Z.8/.The next result provides the first known “purely exceptional” example of twodifferent anisotropic varieties with isomorphic motives. Recall that the similar resultfor groups of type G 2 obtained in [Bo03] provides a motivic isomorphism betweenquadric and an exceptional Fano variety.1.2 Theorem Let G be an anisotropic algebraic group over k such that over acubic field extension k 0 of k it becomes isomorphic to a split simple group G oftype F 4 .LetX 1 and X 4 be two projective G-homogeneous varieties which over k 0become isomorphic to G=P 1 and G=P 4 respectively, where P 1 (resp. P 4 )isthemaximal parabolic subgroup corresponding to the last (resp. first) three vertices ofthe Dynkin diagram 4.1.Then the motives of X 1 and X 4 are isomorphic.The main motivation for our work was the result of N. Karpenko where he gavea shortened construction of a Rost motive for a norm quadric [Ka98]. The keyidea is to produce enough idempotents in the ring CH.X X/ considered over theseparable closure of k and then lift them to k using the Rost Nilpotence Theorem(see [CGM]). Contrary to the techniques used by Voevodsky, the proof of Theorem1.1 is based on well-known and elementary facts about linear algebraic groups,projective homogeneous varieties, and Chow groups.We expect that similar methods can be applied to other projective homogeneousvarieties, thus providing analogous motivic decompositions. In particular, applyingour arguments to a Pfister quadric one obtains the celebrated decomposition intoiD0

Motivic decomposition of anisotropic varieties 87Rost motives (see [KM02, Example 7.3]). For exceptional groups of type G 2 oneimmediately obtains the motivic decomposition of the Fano variety together withthe motivic isomorphism constructed in [Bo03].The paper is organized as follows. In Section 2 we provide backgroundinformation on Chow motives and rational cycles. Section 3 is devoted tocomputational matters of Chow rings. Namely, we introduce Pieri and Giambelliformulae and discuss their relationships with Hasse diagrams. In Section 4 we applythe formulae introduced in Section 3 to projective homogeneous varieties X 1 and X 4of type F 4 . In Section 5 we prove Theorems 1.1 and 1.2. Finally, in the Appendixwe explain the intermediate technical steps of computations used in the proofs.2. Motives and rational cyclesIn the present section we introduce the category of Chow motives over a field kfollowing [Ma68] and [Ka01]. We remind the notion of a rational cycle and statethe Rost Nilpotence Theorem for idempotents following [CGM].2.1 Let k be a field and Var k be a category of smooth projective varietiesover k. First, we define the category of correspondences (over k) denoted byCor k . Its objects are smooth projective varieties over k. For morphisms, calledcorrespondences, we set Mor.X;Y / WD `nlD1 CH d l.X l Y/, where X 1 ;:::;X n are theirreducible components of X of dimensions d 1 ;:::;d n . For any two correspondences˛ 2 CH.X Y/and ˇ 2 CH.Y Z/ we define their composition ˇ ı˛ 2 CH.X Z/asˇ ı ˛ D pr 13 .pr 12 .˛/ pr 23 .ˇ//; (2)where pr ij denotes the projection on the i-th and j -th factors of X Y Zrespectively and pr , ij pr ijdenote the induced push-forwards and pull-backs forChow groups.The pseudo-abelian completion of Cor k is called the category of Chow motivesand is denoted by M k . The objects of M k are pairs .X;p/, where X is a smoothprojective variety and p is an idempotent, that is, p ı p D p. The morphismsbetween two objects .X;p/ and .Y;q/ are the compositions q ı Mor.X;Y / ı p. Themotive .X;id/ will be denoted by M.X/ and called the Chow motive of X.2.2 By construction, M k is a tensor additive category, where the tensor product isgiven by the usual product .X;p/ ˝ .Y;q/ D .X Y;p q/. For any cycle ˛ wedenote as ˛t the corresponding transposed cycle.Moreover, the covariant Chow functor CHWVar k ! Z-Ab (to the category of Z-graded abelian groups) factors through M k , that is, one has a commutative diagram

88 S. NIKOLENKO, N.SEMENOV &K.ZAINOULLINEof functorsCHVar kZ-Ab M k CHwhere Wf 7! f is the graph functor and CHW.X;p/ 7! Im.p ? / is the realization.2.3 Observe that the composition product ı induces the ring structure on the abeliangroup CH dimX .X X/. The unit element of this ring is the class of the diagonal map X , which is defined by X ı ˛ D ˛ ı X D ˛ for all ˛ 2 CH dimX .X X/.2.4 Consider the morphism .e;id/WfptgP 1 ! P 1 P 1 . The image by meansof the induced push-forward .e;id/ .1/ does not depend on the choice of the pointeWfptg!P 1 and defines the projector in CH 1 .P 1 P 1 / denoted by p 1 . The motiveZ.1/ D .P 1 ;p 1 / is called Lefschetz motive. For a motive M and a nonnegativeinteger i we denote its twist by M.i/ D M ˝ Z.1/˝i .2.5 Let G be a split simple linear algebraic group over k. Let X be a projectiveG-homogeneous variety, that is, X ' G=P , where P is a parabolic subgroup of G.The abelian group structure of CH.X/, as well as its ring structure, is well-known.Namely, X has a cellular filtration and the generators of Chow groups of the basesof this filtration correspond to the free additive generators of CH.X/. Note that theproduct of two projective homogeneous varieties X Y has a cellular filtration aswell, and CH .X Y/Š CH .X/ ˝ CH .Y / as graded rings. The correspondenceproduct of two cycles ˛ D f˛ g˛ 2 CH.X Y/and ˇ D fˇ gˇ 2 CH.Y X/ isgiven by (cf. [Bo03, Lemma 5]).fˇ gˇ / ı .f˛ g˛/ D deg.g˛ fˇ /.f˛ gˇ /; (3)where degWCH.Y / ! CH.fptg/ D Z is the degree map.2.6 Let X be a projective variety of dimension n over a field k. Let k 0 beafieldextension of k and X 0 D X k k 0 . WesayacycleJ 2 CH.X 0 / is rational if it liesin the image of the natural homomorphism CH.X/ ! CH.X 0 /. For instance, thereis an obvious rational cycle X 0 on CH n .X 0 X 0 / that is given by the diagonalclass. Clearly, all linear combinations, intersections and correspondence productsof rational cycles are rational.2.7 Several techniques allow to produce rational cycles (cf. [Ka04, Prop. 3.3] forthe case of quadrics). We shall use the following:(i) Consider a variety Y and a morphism X ! Y such that X 0 D Y 0 Y X, whereY 0 D Y k k 0 . Then any rational cycle on CH.Y 0 / gives rise to a rational cycleon CH.X 0 / by the induced pull-back CH.Y 0 / ! CH.X 0 /.

Motivic decomposition of anisotropic varieties 89(ii) Consider a variety Y and a projective morphism Y ! X such that Y 0 D X 0 XY . Then any rational cycle on CH.Y 0 / gives rise to a rational cycle on CH.X 0 /by the induced push-forward CH.Y 0 / ! CH.X 0 /.(iii) Let X and Y be projective homogeneous varieties over k such that X splitscompletely (i.e., the respective group splits) over the function field k.Y /.Consider the following pull-back diagramCH i .X Y/fCH i .X k.Y / /g CH i .X 0 Y 0 /f 0D CH i .X 0 k 0 .Y 0 / /where the vertical arrows are surjective by [IK00, §5]. Now take any cycle˛ 2 CH i .X 0 Y 0 /, i dimX. Letˇ D g.f1 .f 0 .˛///. Thenf 0 .ˇ/ D f 0 .˛/and ˇ is rational. Hence, ˇ D ˛ C J , where J 2 Kerf 0 , and we conclude that˛ C J 2 CH i .X 0 Y 0 / is rational.2.8 (Rost Nilpotence) Finally, we shall also use the following fact (see [CGM,Cor. 8.3]) that follows from the Rost Nilpotence Theorem. Let X beatwistedflagvariety and p 0 be a non-trivial rational idempotent on CH n .X 0 X 0 /, i.e., p 0 ıp 0 D p 0 .Then there exists a non-trivial idempotent p on CH n .X X/ such that p k k 0 D p 0 .Hence, the existence of a non-trivial rational idempotent p 0 on CH n .X 0 X 0 / givesrise to the decomposition of the Chow motive of XM.X/ Š .X;p/ ˚ .X;id Xp/:3. Hasse diagrams and Chow ringsTo each projective homogeneous variety X we may associate an oriented labeledgraph H called Hasse diagram. It is known that the ring structure of CH.X/is determined by H. In the present section we remind several facts concerningrelations between Hasse diagrams and Chow rings. For detailed explanations ofthese relations see [De74], [Hi82a] and [Ko91].3.1 Let G be a split simple algebraic group defined over a field k. We fix a maximalsplit torus T in G and a Borel subgroup B of G containing T and defined over k. Wedenote by ˆ the root system of G,by… the set of simple roots of ˆ corresponding toB,byW the Weyl group, and by S the corresponding set of fundamental reflections.Let P D P ‚ be a (standard) parabolic subgroup corresponding to a subset ‚ …, i.e., P D BW ‚ B, where W ‚ Dhs ; 2 ‚i. DenoteW ‚ Dfw 2 W j8s 2 ‚ l.ws/D l.w/C 1g;

90 S. NIKOLENKO, N.SEMENOV &K.ZAINOULLINEwhere l is the length function. The pairingW ‚ W ‚ ! W.w;v/7! wvis a bijection and l.wv/ D l.w/ C l.v/. It is easy to see that W ‚ consists of allrepresentatives in the left cosets W=W ‚ which have minimal length. Sometimes itis also convenient to consider the set of all representatives of maximal length. Weshall denote this set as ‚ W . Observe that there is a bijection W ‚ ! ‚ W givenby v 7! vw , where w is the longest element of W ‚ . The longest element of W ‚corresponds to the longest element w 0 of the Weyl group.3.2 To a subset ‚ of the finite set … we associate an oriented labeled graph, whichwe call a Hasse diagram and denote by H W .‚/. This graph is constructed asfollows. The vertices of this graph are the elements of W ‚ . There is an edgefrom a vertex w to a vertex w 0 labelled with i if and only if l.w/ < l.w 0 / andw 0 D s i w. A sample Hasse diagram is provided in 4.7. Observe that the diagramH W .;/ coincides with the Cayley graph associated to the pair .W;S/.3.3 Lemma The assignment H W W‚ 7! H W .‚/ is a contravariant functor fromthe category of subsets of the finite set … (with embeddings as morphisms) to thecategory of oriented graphs.Proof: It is enough to embed the diagram H W .‚/ to the diagram H W .;/. Wedothis as follows. We identify the vertices of H W .‚/ with the subset of vertices ofH W .;/ by means of the bijection W ‚ ! ‚ W . Then the edge from w to w 0 of‚ W W has a label i if and only if l.w/ < l.w 0 / and w 0 D s i w (as elements ofW ). Clearly, the obtained graph will coincide with H W .‚/.3.4 Now consider the Chow ring of a projective homogeneous variety G=P ‚ .Itiswell known that CH.G=P ‚ / is a free abelian group with a basis given by varietiesŒX w that correspond to the vertices w of the Hasse diagram H W .‚/. The degreeof the basis element ŒX w corresponds to the minimal number of edges neededto connect the respective vertex w with w (which is the longest word). Themultiplicative structure of CH.G=P ‚ / depends only on the root system of G andthe diagram H W .‚/.By definition one immediately obtains3.5 Lemma The contravariant functor CHW‚ 7! CH.G=P ‚ / factors through thecategory of Hasse diagrams H W , i.e., the pull-back (ring inclusion)CH.G=P ‚ 0/,! CH.G=P ‚ /arising from the embedding ‚ ‚ 0 is induced by the embedding of the respectiveHasse diagrams H W .‚ 0 / H W .‚/.

Motivic decomposition of anisotropic varieties 913.6 Corollary Let B be a Borel subgroup of G and P its (standard) parabolicsubgroup. Then CH.G=P / is a subring of CH.G=B/. The generators of CH.G=P /are ŒX w ,wherew 2 ‚ W W . The cycle ŒX w in CH.G=P / has the codimensionl.w 0 / l.w/.Proof: Apply the lemma to the case B D P ; and P D P ‚ 0.Hence, in order to compute CH.G=P / it is enough to compute CH.X/, whereX D G=B is the variety of complete flags. The following results provide tools toperform such computations.3.7 In order to multiply two basis elements h and g of CH.G=P / such that degh Cdegg D dimG=P we use the following formula (see [Ko91, 1.4]):ŒX w ŒX w 0 D ı w;w0 w 0 w Œpt:3.8 (Pieri formula) In order to multiply two basis elements of CH.X/ one of whichis of codimension 1 we use the following formula (see [De74, Cor. 2 of 4.4]):XŒX w0 s˛ŒX w Dhˇ_;!˛iŒX wsˇ;ˇ2ˆC;l.wsˇ/Dl.w/ 1where the sum runs through the set of positive roots ˇ 2 ˆC,s˛ denotes the simplereflection corresponding to ˛ and N!˛ is the fundamental weight corresponding to ˛.Here ŒX w0 s˛ is the element of codimension 1.3.9 (Giambelli formula) Let P D P.ˆ/ be the weight space. We denote as N! 1 ;::: N! lthe basis of P consisting of fundamental weights. The symmetric algebra S .P /is isomorphic to ZŒ N! 1 ;::: N! l . The Weyl group W acts on P , hence, on S .P /.Namely, for a simple root ˛i,(N! i ˛i; i D j;. N! w˛i j / DN! j ; otherwise:We define a linear map cWS .P / ! CH .G=B/ as follows. For a homogeneousu 2 ZŒ N! 1 ;:::; N! l Xc.u/ D w .u/ŒX w0 w;w2W;l.w/Ddeg.u/where for w D w˛1we denote by :::w˛k w the composition of derivations ˛1ı:::ıand the derivation WS .P / ! S 1 .P / is defined by .u/ D u w˛i .u/.˛k˛i˛i ˛iThen (see [Hi82a, ch. IV, 2.4])ŒX w D c. w1. djW j //;

92 S. NIKOLENKO, N.SEMENOV &K.ZAINOULLINEwhere d is the product of all positive roots in S .P /. In other words, the element w 1. djW j / 2 c 1 .ŒX w /.Hence, in order to multiply two basis elements h;g 2 CH.X/ one may taketheir preimages under the map c and multiply them in S .P / ˝Z Q D QŒ N! 1 ;::: N! l ,finally applying c to the product.4. Homogeneous varieties of type F 4In the present section we remind several well-known facts concerning Albertalgebras, groups of type F 4 and respective projective homogeneous varieties (see[PR94] and [Inv]). At the end we provide partial computations of Chow rings ofthese varieties.4.1 From now on let X i be an anisotropic projective homogeneous variety over afield k that over a cubic field extension k 0 of k becomes isomorphic to the projectivehomogeneous variety Xi 0 D G=P i, where G is a split group of type F 4 and P i D P iis a maximal parabolic subgroup of G corresponding to the subset i Df1;2;3;4gnfig of the Dynkin diagramı1ı2> ı34.2 Remark The variety X i is a G-variety over k, where G is an algebraic groupover k that is the twisted form of a split group G of type F 4 by means of a 1-cocycle 2 H 1 .k;G.k 0 // (see [De77, Prop. 4]).Ifthebasefieldk has characteristic not 3 and 3 k, then all such groups Gare automorphism groups of Albert algebras coming from the first Tits construction.The next two important properties of the varieties X i will be extensively used inthe sequel.4.3 Lemma The Picard group Pic.Xi 0 / is a free abelian group of rank 1 with arational generator.Proof: Since P is maximal, Pic.Xi 0 / is a free abelian group of rank 1. Consider thefollowing exact sequence (see [Ar82] and [MT95, 2.3])0 ! PicX i ! .PicX 0 i / ˛! Br.k/;where D Gal.k 0 =k/is the Galois group and Br.k/ the Brauer group of k. The map˛ is explicitly described in [MT95] in terms of Tits classes. Since all groups of typeF 4 are simply-connected and adjoint their Tits classes are trivial and so is ˛. Since acts trivially on Pic.X 0 i / and the image of ˛ is trivial, we have Pic.X i/ ' Pic.X 0 i /.ı4

Motivic decomposition of anisotropic varieties 95for CH.X 0 4 /:g 1 1 g1 1 D g2 1 ; g1 1 g2 1 D g3 1 ; g1 1 g3 1 D g4 1 C g4 2 ; g1 1 g4 2 D g5 2 ;g 1 1 g4 1 D g5 1 C g5 2 ; g1 1 g5 2 D g6 1 C g6 2 ; g1 1 g5 1 D g6 1 ; g1 1 g6 2 D g7 1 C g7 2 ;g 1 1 g6 1 D g7 1 ; g1 1 g7 1 D 2g8 1 C g8 2 ; g1 1 g7 2 D g8 1 C 2g8 2 ; g1 1 g8 2 D g9 2 ;g1 1 g8 1 D g9 1 C g9 2 ; g1 1 g9 2 D g10 2 ; g1 1 g9 1 D g10 1 C g10 2 ; g1 1 g10 2 D g11 1 C g11 2 ;g1 1 g10 1 D g11 1 ; g1 1 g11 2 D g12 1 ; g1 1 g11 1 D g12 1 ; g1 1 g12 1 D g13 1 ;g1 1 g13 1 D g14 1 ; g1 1 g14 1 D g15 1 :4.11 Observe that the multiplication tables 4.10 can be visualized by means ofslightly modified Hasse diagrams. Namely, for the variety X1 0 consider the followinggraph which is obtained from the respective Hasse diagram by adding a few moreedges and erasing all the labels:and for X 0 4 :ıı ıı ı ı ı ı ı ı ı ı ı ı ı ı ı ıı ı ıı ı ı ıı ıı ıı ı ı ı ı ı ı ı ı ı ı ı ıı ıııııThen the multiplication rules can be restored as follows: for a vertex u (thatcorresponds to a basis element of the Chow group) we setH u D X u!vv;where H denotes either h 1 1 or g1 1and the sum runs through all the edges going fromu one step to the left (cf. [Hi82b, Cor. 3.3]).4.12 Applying Giambelli formula 3.9 we obtain the following products (for detailssee Appendix):h 4 1 h4 1 D 8h8 1 C 6h8 2 ; g4 1 g4 1 D 4g8 1 C 3g8 2 :

96 S. NIKOLENKO, N.SEMENOV &K.ZAINOULLINE5. Construction of rational idempotentsThe goal of the present section is to prove Theorems 1.1 and 1.2. The proof consistsof several steps. First, using properties 4.3 and 4.4 we provide several importantcycles i and prove their rationality. Multiplying and composing them, we obtain aset of pairwise orthogonal idempotents pi 0 and q0 i. Then, using the Rost NilpotenceTheorem (see 2.8) we obtain the desired motivic decomposition and, hence, finishthe proof of 1.1. At the end we construct an explicit cycle which provides a motivicisomorphism of Theorem 1.2.5.0.0.1 Proof of Theorem 1.1: Let X 1 and X 4 be the varieties corresponding tothe last (resp. first) three roots of the Dynkin diagram (see 4.6). By the hypothesisof Theorem 1.1 the variety X is isomorphic either to X 1 or X 4 over k. As in 4.6 letk 0 denote the cubic field extension and let X1 0 and X 4 0 be the respective base change.We start with the following obvious observation.5.1 Since the variety X of Theorem 1.1 splits by a cubic field extension, transferarguments show that any cycle of the kind 3z 2 CH.X 0 / is rational. Hence, to provethat a cycle in CH.X 0 / is rational it is enough to prove this modulo 3. We shall writex y if x y D 3z for some cycle z.5.2 The rational cycles to start with one obtains by Lemma 4.3. Namely, those arethe classes of rational generators of the Picard groups h 1 1 and g1 1(see 4.8). Clearly,their powers .h 1 1 /i and .g1 1/i , i D 2;:::;7 are rational as well.5.3 Apply the arguments of 2.7(iii) to CH 4 .X1 0 X 4 0 / (this can be done because ofLemma 4.4). There exists a rational cycle ˛1 2 CH 4 .X1 0 X 4 0 / such that f 0 .˛1/ Dh 4 1 1. This cycle must have the following form:˛1 D h 4 1 1 C a 1h 3 1 g1 1 C a 2h 2 1 g2 1 C a 3h 1 1 g3 1 C a 41 g 4 2 C a0 1 g 4 1 ;where a i ;a 0 2f 1;0;1g. We may reduce ˛1 by adding cycles that are known to berational by 5.2 to˛1 D .h 4 1 1/ C a.1 g4 1 /;where a 2f 1;0;1g. Repeating the same procedure for a rational cycle ˛2 2CH 4 .X1 0 X 4 0 / such that f 0 .˛2/ D 1 g1 4 and reducing it we obtain the rationalcycle˛2 D b.h 4 1 1/ C .1 g4 1 /;where b 2f 1;0;1g. Hence, there is a rational cycle of the formr D h 4 1 1 C " .1 g4 1 /;where the (indefinite) coefficient " is either 1 or 1.

Motivic decomposition of anisotropic varieties 97Now combining 5.2 and 5.3 together we obtain5.4 Lemma For all i D 0;:::;7 the cyclesare rational.5.5 By the previous lemma all cycles i D r 2 ..h 1 1 /i .g 1 1 /7 i / 2 CH 15 .X 0 1 X 0 4 / t 7 i ı i 2 CH 15 .X 0 1 X 0 1 / and i ı t 7 i 2 CH15 .X 0 4 X 0 4 /where i D 0;:::;3 are rational. Direct computations (see Appendix) show that thesecycles are congruent modulo 3 to the following cycles in CH 15 .X 0 1 X 0 1 /:7 t ı 0 1 h 151 C h4 1 .h11 1 C h11 2 / C h8 1 .h7 1 C h7 2 /;6 t ı 1 h 1 1 h14 1 C .2h5 2 C h5 1 / .h10 2 h 101 / C .h9 2 h 9 1 / h6 2 ;5 t ı 2 h 2 1 h13 1 C .h6 1 C h6 2 / h9 1 C .2h10 1 h 102 / .h5 1 C h5 2 /;4 t ı 3 h 3 1 h12 1 C h7 2 .h8 2 h 8 1 / C h11 2 .h4 2 h 4 1 /and in CH 15 .X 0 4 X 0 4 /: 0 ı 7 t 1 g15 1 C g4 1 .g11 1 g2 11 / C g8 1 .g7 1 C g7 2 /; 1 ı 6 t g1 1 g14 1 C .2g5 1 g2 5 / .g10 1 C g10 2 / C .g9 1 C g9 2 / g6 2 ; 2 ı 5 t g2 1 g13 1 C .g6 1 g2 6 / g9 1 C .g10 1 C 2g10 2 / .g5 2 g1 5 /; 3 ı 4 t g3 1 g12 1 C g7 2 .g8 2 g1 8 / C g11 2 .g4 1 C g4 2 /respectively, which turn to be idempotents. Denote them by p0 0 , p0 1 , p0 2 , p0 3 and q0 0 ,q1 0 , q0 2 , q0 4respectively. Since they are congruent to rational cycles, they are rational.To complete the picture observe that the transposed cycles .pi 0/t and .qi 0/t , i D0;:::;3, are rational idempotents as well. Hence, we have produced eight rationalidempotents in each Chow ring.5.6 Now easy computations show that these idempotents are orthogonal to eachother, and their sum is equal to the respective diagonal cycle X 01D X i;j;s X 04D X i;j;sı ij h s i h15jı ij g s i g15jss3XD p 0 l C .p0 l /t 2 CH 15 .X1 0 X 1 0 /;DlD03Xq 0 l C .q0 l /t 2 CH 15 .X4 0 X 4 0 /:lD0

98 S. NIKOLENKO, N.SEMENOV &K.ZAINOULLINEHence, by 2.8 we obtain decompositions of the motives of X 1 and X 43MM.X 1 / D .X 1 ;p i / ˚ .X 1 ;pi t /;iD03MM.X 4 / D .X 4 ;q i / ˚ .X 4 ;qi t /;iD0where p i k k 0 D pi 0 and q i k k 0 D qi 0k).are pairwise orthogonal idempotents (over5.7 By a straightforward computation using the definition of the idempotents pi0and qi 0 given in 5.5 we immediately obtain that there are isomorphisms of motives.X 1 ;p i / ' .X 1 ;p 0 /.i/; .X 1 ;p t i / ' .X 1;p i /.7 i/;.X 4 ;q i / ' .X 4 ;q 0 /.i/; .X 4 ;q t i / ' .X 4;q i /.7 i/; over the cubic extension k 0 the motives .X 1 ;p 0 / and .X 4 ;q 0 / split as directsums of Lefschetz motives Z ˚ Z.4/ ˚ Z.8/, where the shifts correspond tothe codimensions of the first factors of p 0 and q 0 .5.8 To finish the proof of Theorem 1.1 we have to prove that the motives.X 1 ;p 0 / and .X 4 ;q 0 / are indecomposable. To see this observe that the group ofendomorphisms End.X1 0 ;p0 0/ is a free abelian group with the basish1 h 151 ;h4 1 .h11 1 C h11 2 /;h8 1 .h7 1 C h7 2 /i (4)Assume that .X 1 ;p 0 / is decomposable, then the motive .X1 0 ;p0 0/ is decomposableas well. The latter means that there exists a non-trivial rational idempotent inEnd.X1 0 ;p0 0/. In this case easy computations show that one of the elements of thebasis (4) must be rational. For instance, assume 1 h 151is rational. Then the cycle.1 h 151 / .h15 1 1/ D h151 h15 1is rational and so is its image h151by means ofthe push-forward CH 0 .X1 0 X 1 0 / ! CH 0.X1 0 / induced by a projection. Hence, weobtain a cycle of degree 1 on X 1 , i.e., the variety X 1 must have a rational point[PR94, Cor. p. 205]. We have arrived to a contradiction.5.9 Remark The following picture demonstrates how the realizations of motives.X1 0 ;p0 i/, i D 0;1;2;3, are supported by the generators of the Hasse diagram (the

Motivic decomposition of anisotropic varieties 99numbers i drawn inside the rectangulars correspond to the motives .X 0 1 ;p0 i /).0 1 2 3 3 ı ı ı ı 3 ı 2 ı 0 1 ı ı ı ı ı 0 1 ı ı 2 ı ı ı A similar picture for the motives .X1 0 ;pt i/ corresponding to transposed idempotentsis obtained by the reflection along the vertical dashed line of the diagram.Proof of Theorem 1.2: We use the notation of the proof of 1.1. Easy computations(see Appendix) show that. 0 C 1 C " 2 C " 3 / C . 0 C 1 C " 2 C " 3 / t X i;j;s˙ı ij h s i g15j s :Denote the right hand side by J .WehaveJ t ı J D X 01and J ı J t D X 04, i.e., Jand J t are two mutually inverse rational correspondences and the cycle J providesa rational motivic isomorphism between the motives of X1 0 and X 4 0 . By the RostNilpotence Theorem, J can be lifted to a motivic isomorphism between the motivesof twisted forms X 1 and X 4 . This completes the proof of Theorem 1.2.AppendixMost of the computations of the present section were performed and checked usingthe Maple package by J. Stembridge [St04].5.10 In the present paragraph we list the intermediate results of the computationsof 5.5. First, using 4.12 we obtainr 2 h 8 1 1 "h4 1 g4 1 C 1 g8 1 :

100 S. NIKOLENKO, N.SEMENOV &K.ZAINOULLINEThen, using 4.10 we obtain the following congruences for the cycles i 0 1 g1 15 C "h4 1 .g11 1 g2 11 / C h8 1 .g7 1 C g7 2 / 1 h 1 1 g14 1 C ".h5 2 h 5 1 / .g10 1 C g10 2 / C .h9 2 h 9 1 / g6 2 2 h 2 1 g13 1 ".h 6 1 C h6 2 / g9 1 C .h10 1 C h10 2 / .g5 2 g1 5 / 3 h 3 1 g12 1 C "h7 2 .g8 2 g1 8 / h11 2 .g4 1 C g4 2 / 4 .h 4 1 h 4 2 / g11 2 C ".h8 2 h 8 1 / g7 2 C h12 1 g3 1 5 .h 5 1 C h5 2 / .g10 2 g1 10 / C "h9 1 .g6 2 g1 6 / h13 1 g2 1 6 h 6 2 .g9 1 C g9 2 / C ".h10 2 h 101 / .g5 1 C g5 2 / h14 1 g1 1 7 .h 7 1 C h7 2 / g8 1 C ".h11 1 C h11 2 / g4 1 h 151 15.11 The root enumeration follows Bourbaki [Bou]. To obtain the products of 4.12we apply the Giambelli formula 3.9. Let N! i , i D 1;:::;4, be the fundamentalweights. Then, the preimages of h 4 1 and g4 1 in S .P / can be expressed aspolynomials in fundamental weights as followsh 4 1 D c.11 6 N!2 1 N!2 4 C 3 44 N!2 1 N!2 23 N! 1 N! 2 N! 3 2 C 11 26 N!2 1 N!2 33 N! 1 N! 2 N! 3 N! 4 C 1112 N!4 1 C1 46 N!4 23 N! 2 N! 3 2 N! 4 C 4 3 N! 2 N! 3 N! 4 2 C 2 3 N!2 2 N! 3 N! 4 C 2 3 N! 1 N! 2 N! 42 116 N!2 1 N! 3 N! 4 C2 N! 1 N! 3 2 N! 4 2 N! 1 N! 3 N! 42 712 N!3 1 N! 1126 N!2 1 N! 2 N! 3 C 4 3 N! 1 N! 2 2 N! 3 C 2 3 N!2 2 N!2 323 N!3 2 N! 133 N! 1 N! 23 23 N!2 2 N!2 4 /;g1 4 Dc.11 76 N!4 46 N! 3 N! 4 3 C 1112 N!2 1 N!2 4 C 3 112 N!2 3 N!2 46 N! 2 N! 3 N! 4 2 C 1112 N!2 2 N!2 41112 N! 1 N! 2 N! 42 23 N!3 3 N! 142 N!2 1 N! 2 N! 4 C 1 3 N!2 1 N! 3 N! 4 C 4 3 N! 2 N! 3 2 N! 4 C 1 2 N! 1 N! 2 2 N! 423 N!2 2 N! 13 N! 43 N! 1 N! 2 N! 3 N! 4 C 1 13 N!4 33 N! 1 N! 2 2 N! 3 C 1 3 N! 1 N! 2 N! 32 13 N!2 1 N!2 3 C13 N!2 1 N! 2 N! 3 C 1 23 N!2 2 N!2 33 N! 2 N! 3 3 /:Multiplying the respective polynomials and taking the c function, we find theproducts.Acknowledgments: The authors gratefully acknowledge the hospitality and supportof Bielefeld University. We are sincerely grateful to N. Karpenko and A. Vishikfor the comments concerning rational cycles and norm varieties. The first and thesecond authors are sincerely grateful to A. Bak for his advice and hospitality, inparticular for giving all of us the opportunity of working together.

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102 S. NIKOLENKO, N.SEMENOV &K.ZAINOULLINEPR94. H.P. Petersson, M.L. Racine. Albert Algebras. In W. Kaup, K. McCrimmon,H.P. Petersson (eds). Jordan Algebras. Proc. of the Conference in Oberwolfach,Walter de Gruyter, Berlin-New York, 1994, 197–207Ro98. M. Rost. The motive of a Pfister form. Preprint, 1998.http://www.mathematik.uni-bielefeld.de/˜rost/St04. J. Stembridge. The coxeter and weyl maple packages. 2004.http://www.math.lsa.umich.edu/˜jrs/maple.htmlSu05. A. Suslin, S. Joukhovitsky. Norm Varieties. J. Pure Appl. Algebra 206 (2006), 245–276Ti66. J. Tits. Classification of algebraic semisimple groups. In Algebraic Groups and DiscontinuousSubgroups (Proc. Symp. Pure Math.), Amer. Math. Soc., Providence,R.I., 1966Vo03. V. Voevodsky. On motivic cohomology with Z=l-coefficients, Preprint, 2003.http://www.math.uiuc.edu/K-theorySERGEY NIKOLENKODepartment of the Steklov Mathematical Institute27, FontankaSt. Petersburg 191023RussiaNIKITA SEMENOVMathematisches InstitutUniversitaet MuenchenTheresienstr. 3980333 MuenchenGermanyKIRILL ZAINOULLINEMathematisches InstitutUniversitaet MuenchenTheresienstr. 3980333 MuenchenGermanysemenov@mathematik.uni-muenchen.dekirill@mathematik.uni-muenchen.deReceived: November, 2006