NOTE ON THE COUNTEREXAMPLES FOR THE INTEGRAL TATE CONJECTURE OVER FINITE FIELDS 4 1 0 ALENA PIRUTKA AND NOBUAKI YAGITA 2 n Abstract. In this note we discuss some examples of non torsion a andnonalgebraiccohomologyclassesforvarietiesoverfinitefields. J The approach follows the construction of Atiyah-Hirzebruch and 8 Totaro. ] G A 1. Introduction . h at Let k be a finite field and let X be a smooth and projective variety m over k. Denote k¯ an algebraic closure of k and g = Gal(k¯/k). Let ℓ be [ a prime, ℓ 6= char(k). The Tate conjecture [19] predicts that the cycle 1 class map v CHi(X )⊗Q → H2i(X ,Q (i))U, 0 k¯ ℓ e´t k¯ ℓ 2 U [ 6 where the union is over all open subgroups U of g, is surjective. 1 In the integral version one is interested in the cokernel of the cycle . 1 class map 0 4 (1.1) CHi(X )⊗Z → H2i(X ,Z (i))U. 1 k¯ ℓ e´t k¯ ℓ v: [U i This map is not surjective in general: the counterexamples of Atiyah- X Hirzebruch [1], revisited by Totaro [20], to the integral version of the r a Hodge conjecture, provide also counterexamples to the integral Tate conjecture [3]. More precisely, one constructs an ℓ-torsion class in H4(X ,Z (2)), which is not algebraic, for some smooth and projective e´t k¯ ℓ variety X. However, one then wonders if there exists an example of a variety X over a finite field, such that the map (1.2) CHi(X )⊗Z → H2i(X ,Z (i))U/torsion k¯ ℓ e´t k¯ ℓ U [ is not surjective ([12, 3]). In the context of an integral version of the Hodge conjecture, Koll´ar [11] constructed such examples of curve 2000 Mathematics Subject Classification. Primary 14C15; Secondary 14L30, 55R35. 1 classes. Over a finite field, Schoen [17] has proved that the map (1.2) is always surjective for curve classes, if the Tate conjecture holds for divisors on surfaces. In this notewe follow the approach of Atiyah-Hirzebruch andTotaro and we produce examples where the map (1.2) is not surjective for ℓ = 2,3 or 5. Theorem 1.1. Let ℓ be a prime from the following list: ℓ = 2,3 or 5. There exists a smooth and projective variety X over a finite field k, chark 6= ℓ, such that the cycle class map CH2(X )⊗Z → H4(X ,Z (2))U/torsion k¯ ℓ e´t k¯ ℓ U [ is not surjective. As in the examples of Atiyah-Hirzebruch and Totaro, our counterex- amples are obtained as a projective approximation of the cohomology of classifying spaces of some simple simply connected groups, having ℓ- torsion in its cohomology. The non algebraicity of a cohomology class is obtained by means of motivic cohomology operations: one estab- lishes that the operation Q does not vanish on some class of degree 4, 1 but it always vanishes on the algebraic classes. This is done in section 2. Next, in section 3 we discuss some properties of classifying spaces in our context and finally we construct a projective variety approxi- mating the cohomology of these spaces in small degrees in section 4. Acknowledgements. This work has started during the Spring School and Workshop on Torsors, Motives and Cohomological Invariants in the Fields Institute, Toronto, as a part of a Thematic Program on Tor- sors, Nonassociative Algebras and Cohomological Invariants (January- June 2013), organized by V. Chernousov, E. Neher, A. Merkurjev, A. Pianzola and K. Zainoulline. We would like to thank the organizers and the Institute for their invitation, hospitality and support. We are very grateful to B. Totaro for his interest and for generously commu- nicating his construction of a projective algebraic approximation in theorem 1.1. The first author would like to thank B. Kahn and J. Lannes for useful discussions. 2. Motivic version of Atiyah-Hirzebruch arguments, revisited 2.1. Operations. Let k be a perfect field with char(k) 6= ℓ and let H(k) be the motivic homotopy theory of pointed k-spaces (see [14]). · 2 ForX ∈ H(k),denotebyH∗,∗′(X,Z/ℓ)themotiviccohomologygroups · with Z/ℓ-coefficients (loc.cit.). If X is a smooth variety over k, note that one has an isomorphism CH∗(X)/ℓ →∼ H2∗,∗(X,Z/ℓ). Voevodsky [22] defined the reduced power operations Pi and the Milnor’s operations Q on H∗,∗′(X,Z/ℓ): i Pi : H∗,∗′(X,Z/ℓ) → H∗+2i(ℓ−1),∗′+i(ℓ−1)(X,Z/ℓ),i ≥ 0 Q : H∗,∗′(X,Z/ℓ) → H∗+2ℓi−1,∗′+(ℓi−1)(X,Z/ℓ),i ≥ 0, i where Q = β is the Bockstein operation of degree (1,0) induced from 0 the short exact sequence 0 → Z/ℓ →×ℓ Z/ℓ2 → Z/ℓ → 0 (see also [16]). One of the key ingredients for this construction is the following com- putation of the motivic cohomology of the classifying space Bµ ([22]): ℓ Lemma 2.1. ([22, §6]) For each object X ∈ H(k), the graded algebra · H∗,∗′(X ×Bµ ,Z/ℓ) is generated over H∗,∗′(X,Z/ℓ) by ℓ x, deg(x) = (1,1) and y, deg(y) = (2,1) 0 ℓ is odd with β(x) = y and x2 = τy +ρx ℓ = 2 ( where 0 6= τ ∈ H0,1(Spec(k),Z/ℓ) ∼= µ and ρ = (−1) ∈ k∗/(k∗)2 ∼= ℓ KM(k)/2 ∼= H1,1(Spec(k),Z/2). 1 For what follows, we assume that k contains a primitive ℓ2-th root ∼ of unity ξ, so that BZ/ℓ → Bµ and β(τ) = ξℓ (= ρ for p = 2) is zero ℓ in k∗/(k∗)ℓ = H1,1(Spec(k);Z/ℓ). et We will need the following properties: Proposition 2.2. (i) Pi(x) = 0 for i > m/2and x ∈ Hm,n(X,Z/ℓ); (ii) Pi(x) = xℓ for x ∈ H2i,i(X,Z/ℓ); (iii) for X smooth the operation Q : CHm(X)/ℓ = H2m,m(X,Z/ℓ) → H2m+2ℓi−1,m+(ℓi−1)(X,Z/ℓ) i is zero ; (iv) Op.(τx) = τOp.(x) for Op. = β,Q or Pj; i (v) Q = [Pℓi−1,Q ]. i i−1 Proof. See [22, §9]; for (iii) one uses that Hm,n(X,Z/ℓ) = 0 if m−2n > 0andX isasmoothvarietyoverk, (iv)followsfromtheCartanformula for the motivic cohomology. 3 2.2. Computations for BZ/ℓ. The computations in this section are similar to [1, 20, 21]. Lemma 2.3. In H∗,∗′(BZ/ℓ,Z/ℓ), we have Q (x) = yℓi and Q (y) = 0. i i Proof. By definition Q (x) = β(x) = y. Using induction and Proposi- 0 tion 2.2, we compute Q (x) = Pℓi−1Q (x)−Q Pℓi−1(x) = Pℓi−1Q (x) i i−1 i−1 i−1 = Pℓi−1(yℓi−1) = yℓi. Then Q (y) = −Q P1(y) = −β(yℓ) = 0. For i > 1, using induction 1 0 and Proposition 2.2 again, we conclude that Q (y) = −Q Pℓi−1(y) = i i−1 0. (cid:3) Let G = (Z/ℓ)3. As above, we assume that k contains a primitive ℓ2-th root of unity. From Lemma 2.1, we have an isomorphism H∗,∗′(BG,Z/ℓ) ∼= H∗,∗′(Spec(k),Z/ℓ)[y ,y ,y ]⊗Λ(x ,x ,x ) 1 2 3 1 2 3 where Λ(x ,x ,x ) is isomorphic to the Z/ℓ-module generated by 1 1 2 3 and x ...x for i < ... < i and x x = −x x (i ≤ j), with β(x ) = y i1 is 1 s i j j i i i and x2 = τy for ℓ = 2. i i Lemma 2.4. Let x = x x x in H3,3(BG,Z/ℓ). Then 1 2 3 Q Q Q (x) 6= 0 ∈ H2∗,∗(BG,Z/ℓ) for i < j < k. i j k Proof. Using Proposition 2.2(v) and Cartan formula (2.2(iv)), we get Q (x) = yℓkx x −yℓkx x +yℓkx x . k 1 2 3 2 1 3 3 1 2 Then we deduce Q Q Q (x) = ±yℓk yℓj yℓi 6= 0 ∈ Z/ℓ[y ,y ,y ]. i j k σ(1) σ(2) σ(3) 1 2 3 σX∈S3 (cid:3) 3. exceptional Lie groups Let (G,ℓ) be a simple simply connected Lie group and a prime num- ber from the following list: G ,ℓ = 2, 2 (3.1) (G,ℓ) = F ,ℓ = 3,  4 E ,ℓ = 5. 8 4   Then G is 2-connected and H3(G,Z) ∼= Z. Hence BG, viewed as a topological space, is 3-connected and H4(BG,Z) ∼= Z (see [13] for example). We write x (G) for a generator of H4(BG,Z). 4 Given a field k with char(k) 6= ℓ, let us denote by G the (split) k reductive algebraic group over k corresponding to the Lie group G. The Chow ring CH∗(BG ) has been defined by Totaro [21]. More k precisely, one has (3.2) BG = lim(U/G ), k −→ k where U ⊂ W is an open set in a linear representation W of G , such k that G acts freely on U. One can then identify CHi(BG ) with the k k group CHi(U/G ) if codim (W \ U) > i, the group CHi(BG ) is k W k then independent of a choice of such U. Similarly, one can define the ´etale cohomology groups Hi (BG ,Z (j)) and the motivic cohomology e´t k ℓ groups H∗,∗′(BG ,Z/ℓ) (see [7]), the latter coincide with the motivic k cohomology groups of [14] (cf. [7, Proposition 2.29 and Proposition 3.10]). We also have the cycle class map (3.3) cl : CH∗(BG )⊗Z → H2∗(BG ,Z (∗))U, k¯ ℓ e´t k¯ ℓ U [ ¯ where the union is over all open subgroups U of Gal(k/k). The following proposition is known. Proposition 3.1. Let (G,ℓ) be a group and a prime number from the list (3.1). Then (i) the group G has a maximal elementary non toral subgroup of rank 3: i : A ≃ (Z/ℓ)3 ⊂ G; (ii) H4(BG,Z/ℓ) ≃ Z/ℓ, generated by the image x of the generator 4 x (G) of H4(BG,Z) ≃ Z; 4 (iii) Q (i∗x ) = Q Q (x x x ), in the notations of Lemma 2.4. In 1 4 1 0 1 2 3 particular, Q (i∗x ) is non zero. 1 4 Proof. For (i) see [5], for the computation of the cohomology groups with Z/ℓ-coefficients in (ii) see [13] VII 5.12; (iii) follows from [10] for ℓ = 2 and [8, Proposition 3.2] for ℓ = 3,5 (see [9] as well). (cid:3) 4. Algebraic approximation of BG Write (4.1) BG = lim(U/G ) k −→ k as in the previous section. Using proposition 3.1 and a specialization argument, we will first construct a quasi-projective algebraic variety X 5 over k as a quotient X = U/G (where codim (W \U) is big enough), k W such that the cycle class map (1.2) is not surjective for such X. How- ever, if one is interested only in quasi-projective counterexamples for the surjectivity of the map (1.2), one can produce more naive exam- ples, for instance as a complement of some smooth hypersurfaces in a projective space. Hence we are interested to find an approximation of Chow groups and the ´etale cohomology of BG as a smooth and k¯ projective variety. In the case when the group G is finite, this is done in [3, Th´eor`eme 2.1]. In this section we give such an approximation for the groups we consider here, this construction is suggested by B. Totaro. Proposition 4.1. Let G be a compact Lie group as in (3.1). For all but finitely many primes p there exists a smooth and projective variety X over a finite field k with chark = p, an element x ∈ k 4,k H4(B(G ×G ),Z (2)), invariant under the action of Gal(k¯/k) and e´t m k¯ ℓ a map τ : X → B(G ×G ) in the category H(k) such that k m k · (i) y = τ∗pr∗x is a non zero class in H4(X ,Z (2))/torsion, 4,k 2 4,k e´t k¯ ℓ where pr : G × G → G is the projection on the second 2 m k k factor; (ii) the operation Q (y¯ ) is non zero, where we write y¯ for the 1 4,k 4,k image of y in H4(X ,Z/ℓ). 4,k e´t k¯ Remark 4.2. For the purpose of this note, the proposition above is enough. See also [6] for a a general statement on the projective ap- proximation of the cohomology of classifying spaces. Theorem 1.1 now follows from the proposition above: Proof of theorem 1.1. For k a finite field and X as in the proposition above, we find a non- k trivial class y in its cohomology in degree 4 modulo torsion, which is 4,k not annihilated by the operation Q . This class can not be algebraic 1 (cid:3) by proposition 2.2(iii). Proof of proposition 4.1. We proceed in three steps. First, we construct a quasi-projective ap- proximation in a family parametrized by SpecZ. Then, for the geo- metric generic fibre we produce a projective approximation, by a topo- logical argument. We finish the proof by specialization. Step 1: construction of a family. Let G be a split reductive group over B = SpecZ corresponding to G, 6 such a group exists by [SGA3] XXV 1.3. As B is an affine scheme of dimension 1, we can embed G as a closed subgroup of GL for some d d,B (see [SGA3] VI 13.2 and 13.5). Moreover, one can assume that G ֒→ B PGL such thatthisembedding liftstoH = GL , uptoremplacing d,B d,B 1 0 0 B by an open subset (e.g. using the map A 7→ 0 −1 0 and   0 0 A changing d by d+2).   By a construction of [21, Remark 1.4] and [2, Lemme 9.2], there exists n > 0, a linear H-representation O⊕n and an H-invariant open B subset U ⊂ O⊕n, which one can assume flat over B, such that the B action of H is free on U. Let V = O⊕Nn. Then the group PGL N B n,B acts on P(V ) and, taking N sufficiently large, one can assume that N the action is free outside a subset S of high codimension s ≥ 4. By restriction, the group G acts on P(V ) as well, let Y = P(V )//G N N betheGITquotient forthis action[15,18]. The scheme Y isprojective over B and we fix an embedding Y ⊂ PM. Let B (4.2) f : W → B be the open set of Y corresponding to the quotient of the open set U as above where G acts freely. From the construction, Y −W has high codimension in Y. For any point b ∈ B with residue field κ(b), the fibre W is a smooth b quasi-projective variety and if N is big enough, we have isomorphisms by lifting G to GL (cf. p. 263 in [21]) n,B W ∼= (P(V )−S) /G ∼= ((V −{0})/G −S) )/G ∼= (V −S′) /(G ×G) b N b b N m b b N b m b where S′ = pr−1S ∪{0} for the projection pr : (V −{0}) → P(V ). N N Hence we have isomorphisms (4.3) Hi(W ,Z ) →∼ Hi(B(G ×G) ,Z ) for i ≤ s,ℓ 6= charκ(b), b ℓ m b ℓ induced by a natural map W → B(G × G ) from the presentation b m b (4.1). Step 2: the generic fibre. Let Y = Y and W = W be the geometric generic fibres of Y and W C C over B. Consider a general linear space L in PM of codimension equal to 1+dim(Y −W). Then L∩Y = L∩W so X := L∩W is a smooth projective variety. Note that one can assume that L is defined over Q. ByaversionoftheLefschetzhyperplanetheoremforquasi-projective varieties, established by Hamm (as a special case of Theorem II.1.2 in [4]), for V ⊂ PM a closed complex subvariety of dimension d, not 7 necessarily smooth, Z ⊂ V a closed subset, and H a hyperplane in PM, if V − (Z ∪H) is local complete intersection (e.g. V − Z is smooth) then π ((V −Z)∩H) → π (V −Z) i i is an isomorphism for i < d − 1 and surjective for i = d − 1. In particular, Hi((V −Z)∩H,Z) → Hi(V −Z,Z) is an isomorphism for i < d−1 and surjective for i = d−1 by the Whitehead theorem. We then deduce that (4.4) Hi(X,R) →∼ Hi(B(G ×G),R) for i ≤ s and R = Z or Z/n. m ∼ Hence Hi (X,Z/n) → Hi (B(G ×G),Z/n),i ≤ s. Note that as the e´t e´t m cohomology of BG is a direct factor in the cohomology of B(G ×G), m wegetthatx (G)(withthenotationsoftheprevioussection)generates 4 a direct factor isomorphic to Z in the cohomology group H4(X,Z ). ℓ e´t ℓ Step 3: specialization argument. We can now specialize the construction above to obtain the statement over a finite field. More precisely, one can find a dense open set B′ ⊂ B and a linear space L ⊂ PM such that L ≃ L and such that for any b ∈ B′ the B′ C fibre X of X = L ∩Y is smooth. Up to passing to an ´etale cover of b B′, one can assume that the inclusion (Z/ℓ)3 ⊂ G from proposition C 3.1 extends an inclusion i : A = (Z/ℓ)3B′ ֒→ GB′ (cf. [SGA3] XI.5.8). Let b ∈ B′ and let k = κ(b). As the schemes X, Y and U/A are smooth over B′, we have the following commutative diagram, where the vertical maps are induced by the specialisation maps: He´4t(X,Zℓ(2))oo He´4t(Y,Zℓ(2)) //He´4t(UC/(Z/ℓ)3,Z/ℓ)oo ≃ He´4t(B(Z/ℓ)3,Z/ℓ) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) He´4t(Xk¯,Zℓ(2))oo He´4t(Yk¯,Zℓ(2)) //He´4t(Uk¯/(Z/ℓ)3,Z/ℓ)oo ≃ He´4t(B(Z/ℓ)3,Z/ℓ) The left vertical map is an isomorphism since X is proper. Hence we getaclassy ∈ H4(X ,Z (2)),correspondingtox (G) ∈ H4(X,Z (2)). 4,k e´t k¯ ℓ 4 e´t ℓ The map H4(Y,Z (2)) → H4(X,Z (2)) is an isomorphism by step 2, e´t ℓ e´t ℓ so that y comes from an element x ∈ H4(Y ,Z (2)). Let z ∈ 4,k 4,k e´t k¯ ℓ 4,k H4(B(Z/ℓ)3,Z/ℓ) be the image of x . From the diagram and propo- e´t 4,k sition 3.1 we deduce that Q (z ) = Q Q (x x x ) 6= 0, hence Q (y¯ ) 1 4,k 1 0 1 2 3 1 4,k is non zero as well. From the construction, the class y generates a 4,k subgroup of H4(X ,Z (2)), which is a direct factor isomorphic to Z , e´t k¯ ℓ ℓ and is Galois-invariant. Letting X = X this finishes the proof of the k k proposition. 8 (cid:3) Remark 4.3. Wecanalsoadapttheargumentsof[3, Th´eor`eme2.1]to produce projective examples with higher torsion non-algebraic classes, while in loc.cit. one constructs ℓ-torsion classes. 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Alena Pirutka, Centre de Math´ematiques Laurent Schwartz, UMR 7640 de CNRS, E´cole Polytechnique 91128 Palaiseau France E-mail address: [email protected] Nobuaki Yagita, Department of Mathematics, Faculty of Educa- tion, Ibaraki University, Mito, Ibaraki, Japan E-mail address: [email protected] 10