Zero cycles on projective varieties and the norm principle
aa r X i v : . [ m a t h . AG ] S e p Zero-cycles on pro jective varieties and thenorm principle
Philippe Gille, Nikita Semenov ∗ Abstract
Using the Gille-Merkurjev norm principle we compute in a uni-form way the image of the degree map for quadrics (Springer’s theo-rem), for twisted forms of maximal orthogonal Grassmannians (the-orem of Bayer-Fluckiger and Lenstra), for E - (Rost’s theorem), andE -varieties.Keywords: norm principle, algebraic groups, zero-cycles. Let G be a simple algebraic group over a field k and X a projective G -homogeneous variety. Consider the degree mapdeg : CH ( X ) → Z . The goal of the present paper is to provide a method to compute the imageof this map (see [PSZ08] for the computation of its kernel).This problem has a long history starting probably with the Springer the-orem which says that an anisotropic quadratic form remains anisotropic overodd degree field extensions. This statement is equivalent to the fact that theimage of the degree map when X is an anisotropic quadric equals 2 Z .To stress the difficulty of the above problem note that a computationof the degree map for the varieties of Borel subgroups of groups of type E implies Serre’s Conjecture II for fields of cohomological dimension ≤ ∗ Supported partially by DFG, project GI706/1-1. ( k, G ) → Y H ( K i , G )has trivial kernel when K i are finite field extensions of k such that gcd[ K i : k ] = 1and G is a split group.The image of the degree map is known in the following cases: X is aquadric (Springer’s theorem), X is a twisted form of a maximal orthogonalGrassmannian (theorem of Bayer-Fluckiger and Lenstra [BFL90]), X is thevariety of Borel subgroups of a group of type F , E (a theorem of Rost,where cohomological invariants of Albert algebras are used), and E (Gille’stheorem [Gi97], where the norm principle is used). Note also that there arenumerous papers of M. Florence, R. Parimala, B. Totaro, and many othersconcerning closely related problems.In the present paper we apply the Gille-Merkurjev norm principle [Gi97],[Me96], [BM02] and give a uniform proof of the above results. Apart fromthis, we compute the image of the degree map for the varieties of parabolicsubgroups of type 7 of groups G of type E and prove that anisotropic groupsof type E remain anisotropic over odd degree field extensions. Note that thisproperty is used in [PS07, Corollary 6.10] to relate the Rost invariant of G and its isotropity. Let k be a perfect field with char k = 2 ,
3, Γ = Gal(¯ k/k ) the absoluteGalois group, G a connected reductive algebraic group over k , G ′ = [ G, G ]the commutator subgroup, ∆ its Dynkin diagram, and ∆ its Tits index (see[Ti65]). (Special cocharacters) . Let G be a reductive algebraic group over k and1 → G → G f → T = G m → T ∗ can be canonically identifiedwith the group Z . A cocharacter ϕ ∈ T ∗ is called f -special , if there is a k -homomorphism g : G m → G such that f ◦ g = ϕ . (Set X ( ϕ )) . Denote Z ′ = G/G ′ , C the center of the simply connectedcover of G ′ , Z the center of G , and µ the center of G ′ .2e can represent the homomorphism f as a composition G → Z ′ → T .In particular, there is the induced homomorphism α : Z ′ Γ ∗ → T ∗ between thecocharacter groups. The exact sequence1 → µ → Z → Z ′ → β : Z ′ Γ ∗ → µ ( − Γ , and the canonical epimorphism C → µ induces a map γ : C Γ ∗ → µ ( − Γ , where µ ( −
1) is the Tate twist, i.e., µ ( −
1) = Hom( µ n , µ ) for any n with µ n = 1. For a cocharacter ϕ ∈ T ∗ wedefine a subset X ( ϕ ) ⊂ C Γ ∗ as the set γ − ( β ( α − ( { ϕ } ))). (Set Ω( ϕ )) . From now on we assume that the Dynkin diagram ∆ hasno multiple edges. Following [Me96, (5.8)] we identify C ∗ and the charactergroup C ∗ and consider X ( ϕ ) as a subset of C ∗ . Let ¯ ω i denote the i -thfundamental weight of the simply connected cover of G ′ (Enumeration ofsimple roots follows Bourbaki). Define now Ω( ϕ ) as the set of all subsetsΘ ⊂ ∆ such that the elements { P i ∈ I ¯ ω i | C , I ⊂ ∆ \ Θ a Γ-orbit } generate asubgroup of C ∗ Γ that intersects X ( ϕ ). (Type of a parabolic subgroup) . It is well-known that there is a bijectivecorrespondence between the conjugancy classes of parabolic subgroups of G ′ ¯ k and the subsets of the set ∆ of simple roots.The type of a parabolic subgroup is the corresponding subset of ∆. Underthis identification the Borel subgroup has type ∅ . If P is a maximal parabolicsubgroup of type ∆ \{ α i } , where α i is the i -th simple root, then for simplicityof notation we say that P is of type i . (Tits homomorphism) . Let β : C ∗ Γ → Br( k )be the Tits homomorphism for the simply connected cover of G ′ defined in[Ti71]. In order to compute the sets Ω( ϕ ) we need to know the restrictionsof the fundamental weights ¯ ω i to C and their images under the Tits homo-morphism.Below we describe them for groups of type D n , E , and E . We usegraphical notation, where the algebra over a vertex i of the Dynkin diagramstands for the image β (¯ ω i | C ). Apart from this, the restriction ¯ ω i | C is trivialiff the respective algebra is k . 3 ype D n : A simply connected group of inner type D n has the form Spin( A, σ ),where A is a central simple algebra of degree 2 n with an orthogonal involution σ of the first kind with trivial discriminant.For the character group C ∗ of the center of Spin( A, σ ) we have C ∗ = { , χ, χ + , χ − } , where χ (resp. χ + , χ − ) is the restriction of the fundamental weight ¯ ω (resp.¯ ω n − , ¯ ω n ) to the center.Let C ± ( A, σ ) be the direct summands of the Clifford algebra C ( A, σ ) = C + ( A, σ ) ⊕ C − ( A, σ ). We have • C + ( A,σ ) • A • k • A or k • k or A vvvvvvvvvvv HHHHHHHHHHH • C − ( A,σ ) We associate the Tits algebras to the last two vertices n − n insuch a way that for ε = + (resp. ε = − ) the algebra C ε ( A, σ ) splits over thefield of rational functions of the projective homogeneous variety of maximalparabolic subgroups of type P n − (resp. P n ). The latter are two irreduciblecomponents of the variety of 2 n -dimensional isotropic right ideals I of A with respect to σ . Type E : The Tits algebra is a certain central simple algebra A of index 1,3, 9, or 27 and of exponent 1 or 3. • A • A ⊗ • k • A • A ⊗ • k Type E : The Tits algebra is a certain central simple algebra A of index 1,2, 4, or 8 and of exponent 1 or 2. • k • k • k • A • k • A • A Under the above assumptions the following lemmas hold: ([Me96, Lemma 3.4]) . Let
K/k be a finite field extension lyingin the algebraic closure ¯ k and let ϕ ∈ T ∗ . If the cocharacter ϕ is f K -special,then the cocharacter [ K : k ] ϕ is f -special. .8 Lemma ([Me96, Theorem 5.6]) . For a cocharacter ϕ ∈ T ∗ the followingconditions are equivalent:1. ϕ is f -special;2. there exists a parabolic subgroup of G defined over k whose type iscontained in Ω( ϕ ) . ([Me96, Proposition 5.8]) . Let β : C ∗ Γ → Br( k ) be the Titshomomorphism for the simply connected cover of G ′ . Assume that the Dynkindiagram ∆ has no multiple edges. If a cocharacter ϕ ∈ T ∗ is f -special, then ∈ β ( X ( ϕ )) . Let X be an anisotropic smooth projective variety over k and p a prime number. In the above notation assume that the followingconditions hold:1. For any field extension K/k and for any coprime to p cocharacter ϕ , if ∈ β K ( X ( ϕ )) ⊂ Br( K ) and G ′ has a parabolic subgroup defined over K whose type is contained in Ω( ϕ ) , then X ( K ) = ∅ ;2. For any field extension K/k and for any coprime to p cocharacter ϕ if X ( K ) = ∅ , then there exists a parabolic subgroup of G ′ of type containedin Ω( ϕ ) defined over K .Then deg(CH ( X )) ⊂ p Z . Proof.
Let
K/k be a field extension. We show first the following
Claim. X ( K ) = ∅ if and only if any coprime to p cocharacter ϕ is f K -special. Indeed, if X ( K ) = ∅ , then by item 2 there is a parabolic subgroup of G ′ defined over K whose type is contained in Ω( ϕ ). By Lemma 2.8 ϕ is f K -special.Conversely, if ϕ is f K -special, then by Lemma 2.9 we have 0 ∈ β K ( X ( ϕ )),and by Lemma 2.8 there is a parabolic subgroup of G ′ defined over K of typecontained in Ω( ϕ ). Therefore by item 1 we have X ( K ) = ∅ .Let now K/k be a finite field extension such that X ( K ) = ∅ . To finishthe proof of the theorem it sufficies to show that [ K : k ] is divisible by p .Assume the converse.Since X ( K ) = ∅ , by Claim any coprime to p cocharacter ϕ is f K -special.By Lemma 2.7 the cocharacter [ K : k ] ϕ is f -special. Therefore by Claim X ( k ) = ∅ . Contradiction. 5 Applications (Springer’s theorem) . Let A be a central simple k -algebra ofdegree n ≥ with an orthogonal involution σ of the first kind. Let X bethe variety of isotropic with respect to σ right ideals of A of dimension n .Assume X is anisotropic.Then deg(CH ( X )) ⊂ Z . In particular, if X is an anisotropic smootheven-dimensional projective quadric, then deg(CH ( X )) = 2 Z .Proof. ([Me96, Lemma 6.2]). There is the following exact sequence of groups:1 → G = Spin( A, σ ) → G = Γ( A, σ ) f → G m → , where Γ( A, σ ) is the Clifford group and f is the spinor norm homomorphism.Let p = 2. It is easy to check that for any odd cocharacter ϕ the set X ( ϕ ) = { χ } , where χ is the restriction of ¯ ω to the center C .Let K/k be a field extension. If 0 ∈ β K ( X ( ϕ )), then the algebra A K is split (see 2.6). Thus, σ K corresponds to a quadratic form, and X K is aprojective quadric. If additionally G ′ K has a parabolic subgroup defined over K of type contained in Ω( ϕ ), then it easy to see that this quadratic form isisotropic, and thus X ( K ) = ∅ .Finally, if X ( K ) = ∅ , then G ′ has a parabolic subgroup of type ∆ \ { α } ,where α is the first simple root. But ∆ \ { α } ∈ Ω( ϕ ).Thus, we checked all conditions of Theorem 2.10. (Bayer-Fluckiger and Lenstra) . Let A be a central simplealgebra of degree n ≥ with an orthogonal involution σ of the first kind.Let Y be the variety of n -dimensional isotropic right ideals of A and X = Y × Spec( k [ t ] / ( t − disc( σ )) . Assume X is anisotropic. Then deg(CH ( X )) ⊂ Z . Proof. ([Me96, 6.3]). Consider the following exact sequence of groups:1 → G = O + ( A, σ ) → G = GO + ( A, σ ) f → G m → , where f is the multiplier map . 6et p = 2. Denote as χ + (resp. χ − ) the restriction of the fundamentalweight ¯ ω n − (resp. ¯ ω n ) to the center. It is easy to check that for any oddcocharacter ϕ we have X ( ϕ ) = ( ∅ , disc( σ ) = 1; { χ + , χ − } , disc( σ ) = 1 . Finally, if 0 ∈ β K ( X ( ϕ )), then disc( σ K ) = 1. Then the variety Y K is thedisjoint union of varieties of parabolic subgroups of G ′ K of types ∆ \ { α n − } and ∆ \ { α n } . If additinally G ′ has a parabolic subgroup defined over K oftype contained in Ω( ϕ ), then the Tits index ∆ of G ′ K contains at most oneof the roots α n − , α n (see 2.6). Therefore in this case X ( K ) = ∅ .To finish the proof of the corollary it remains to notice that condition 2of Theorem 2.10 is obvious.Using similar arguments one can show the following well-known state-ment. Opposite to the traditional approach our proof does not use cohomo-logical invariants of Albert algebras. (M. Rost) . Let G be a simply connected algebraic group oftype E over k and X the variety of its parabolic subgroups of type (resp. ). Assume X is anisotropic. Then CH ( X ) ⊂ Z .Proof. If G has a non-trivial Tits algebra, then the statement is obvious,since for a field extension K/k condition X ( K ) = ∅ implies that the Titsalgebras of ( G ) K are split.Assume that G has trivial Tits algebras. Let J be an Albert algebraassociated with G (see [Ga01a]). A k -linear map h : J → J is called a similarity if there exists α h ∈ k × (the multiplier of h ) such that N ( h ( j )) = α h N ( j )for all j ∈ J , where N stands for the cubic norm on J . Then G coincideswith the similarities of this Jordan algebra with multiplier 1. Let G bethe group of all similarities. Then G is a reductive group and there is thefollowing exact sequence of algebraic groups:1 → G → G f → T = G m → , f is defined on k -points as h α h .Let p = 3 and let ϕ ∈ T ∗ = Z be a cocharacter coprime to 3. We checknow the conditions of Theorem 2.10.First we compute X ( ϕ ). In our situation G ′ = [ G, G ] = G , Z ′ = T = G m , µ = µ , C = µ , Z = G m , and the group C ∗ ≃ C ∗ = Z / { , ¯ ω | C , ¯ ω | C = − ¯ ω | C } , where ¯ ω i | C denotes the restriction of the i -th fundamental weight of G tothe center, i = 1 ,
6. Therefore, X ( ϕ ) = { ¯ ω | C } or X ( ϕ ) = { ¯ ω | C } (it dependson ϕ mod 3).Let K/k be a field extension. Assume first that G ′ K is isotropic andthe type of a parabolic subgroup P of G ′ defined over K is contained inΩ( ϕ ). If the parabolic subgroup of G ′ of type 1 is not defined, then by Tits’sclassification [Ti65, p. 58] the Tits index of G ′ equals ∆ = ∆ \ { α , α } . Butthe restrictions to the center of the 2-nd and of the 4-th fundamental weightsare trivial (see [Ti90, p. 653] or 2.6). This contradicts to the assumption thatthe type of P is contained in Ω( ϕ ).Finally, condition 2 of Theorem 2.10 is obvious. If the Tits algebras of G are trivial, then the image of thedegree homomorphism CH ( X ) → Z equals 3 Z . Let G be a simply connected algebraic group of type E over k and X the variety of maximal parabolic subgroup of G of type . Assume X is anisotropic. Then CH ( X ) ⊂ Z .Proof. Let (
A, σ, π ), where A is a central simple k -algebra with a symplecticinvolution σ and π : A → A a linear map, be a gift associated with G (see [Fe72], [Ga01a] and [Ga01b]). An invertible element h ∈ A is called a similarity if there exists α h ∈ k × (the multiplier of h ) such that σ ( h ) h = α h · π ( hah − ) = α h hπ ( a ) h − for all a ∈ A . Then G coincides with the similarities of this gift withmultiplier 1. Let G be the group of all similarities. Then G is a connectedreductive group and there is the following exact sequence of algebraic groups:1 → G → G f → T = G m → , f is defined on k -points as h α h .Let p = 2 and let ϕ be an odd cocharacter.First we compute X ( ϕ ). In our situation G ′ = [ G, G ] = G , Z ′ = T = G m , µ = µ , C = µ , and Z = G m . Thus, the maps α and γ from 2.3 arethe identity maps. The map β : Z ′∗ = Z → µ ( −
1) = Z / X ( ϕ ) = { χ } , where as χ we denote a uniquenon-trivial element of C ∗ ≃ C ∗ .Let K/k be a field extension. Assume that 0 ∈ β K ( X ( ϕ )), G ′ K is isotropicand the type of a parabolic subgroup of G ′ defined over K is contained inΩ( ϕ ). The first assumption implies that the Tits algebra A of G is split.If the parabolic subgroup of G ′ of type 7 is not defined, then by Tits’sclassification [Ti65, p. 59] the Tits index of G ′ equals ∆ = ∆ \ { α } . But therestriction to the center of the 1-st fundamental weight is trivial (see [Ti90,p. 653]). Therefore we have X ( K ) = ∅ .Finally, condition 2 of Theorem 2.10 is obvious. If the Tits algebras of G are trivial, then the image of thedegree homomophism CH ( X ) → Z equals 2 Z . A group G as in the statement of the previous corollary doesnot split over an odd degree field extension. References [BM02] P. Barquero, A. Merkurjev. Norm Principle for Reductive AlgebraicGroups, Algebra, arithmetic and geometry, Part I, II (Mumbai,2000), 123–137, Tata Inst. Fund. Res. Stud. Math. (2002), Bom-bay.[BFL90] E. Bayer-Fluckiger, H. W., Jr. Lenstra. Forms in odd degree ex-tensions and self-dual normal bases. Amer. J. Math. (1990),359–373.[Fe72] J.C. Ferrar. Strictly regular elements in Freudenthal triple systems.Trans. Amer. Math. Soc., (1972), 313–331.[Ga01a] S. Garibaldi. Structurable algebras and groups of type E and E .J. Algebra (2001), no. 2, 651–691.9Ga01b] S. Garibaldi. Groups of type E over arbitrary fields. Comm. Alg. (2001), no. 6, 2689–2710.[Gi97] Ph. Gille. La R -equivalence sur les groupes alg´ebriques r´eductifsd´efinis sur un corps global. Inst. Hautes ´Etudes Sci. Publ. Math. (1997), 199–235.[Gi97] Ph. Gille. Cohomologie galoisienne des groupes quasi-d´eploy´es surdes corps de dimension cohomologique ≤
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