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Note on the Counterexamples for the Integral Tate Conjecture Over Finite Fields

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Note on the Counterexamples for the Integral Tate Conjecture Over Finite Fields Documenta Math. 501 Note on the Counterexamples for the Integral Tate Conjecture over Finite Fields Alena Pirutka and Nobuaki Yagita Received: March 17, 2014 Abstract. In this note we discuss some examples of non-torsion and non-algebraic cohomology classes for varieties over finite fields. The approach follows the construction of Atiyah-Hirzebruch and Totaro. A.S. Merkurjev die natali oblatum. 2010 Mathematics Subject Classification: Primary 14C15; Secondary 14L30, 55R35 Keywords and Phrases: Chow groups, classifying spaces, cohomology operations, integral Tate conjecture 1. Introduction Let k be a finite field and let X be a smooth and projective variety over k. Let ℓ be a prime, ℓ 6= char(k). The Tate conjecture [20] predicts that the cycle class map i 2i H CH (Xk¯) ⊗ Qℓ → Het´ (Xk¯, Qℓ(i)) , H [ where the union is over all open subgroups H of Gal(k/k¯ ), is surjective. In the integral version one is interested in the cokernel of the cycle class map i 2i H (1.1) CH (Xk¯) ⊗ Zℓ → Het´ (Xk¯, Zℓ(i)) . H [ This map is not surjective in general: the counterexamples of Atiyah- Hirzebruch [1], revisited by Totaro [21], to the integral version of the Hodge conjecture, provide also counterexamples to the integral Tate conjecture [3]. 4 More precisely, one constructs an ℓ-torsion class in Het´ (Xk¯, Zℓ(2)), which is not algebraic, for some smooth and projective variety X. However, one then Documenta Mathematica · Extra Volume Merkurjev (2015) 501–511 502 Alena Pirutka and Nobuaki Yagita wonders if there exists an example of a variety X over a finite field, such that the map i 2i H (1.2) CH (Xk¯) ⊗ Zℓ → Het´ (Xk¯, Zℓ(i)) /torsion H [ is not surjective ([13, 3]). In the context of an integral version of the Hodge conjecture, Koll´ar [12] constructed such examples of curve classes. Over a finite field, Schoen [18] has proved that the map (1.2) is always surjective for curve classes, if the Tate conjecture holds for divisors on surfaces. In this note we follow the approach of Atiyah-Hirzebruch and Totaro 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 2 4 H CH (Xk¯) ⊗ Zℓ → Het´ (Xk¯, Zℓ(2)) /torsion, H [ where the union is over all open subgroups H of Gal(k/k¯ ), is not surjective. As in the examples of Atiyah-Hirzebruch and Totaro, our counterexamples 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: the operation Q1 always vanishes on the algebraic classes and one establishes that it does not vanish on some class of degree 4. This is discussed in section 2. Next, in section 3 we investigate some properties of classifying spaces in our context and finally, following a suggestion of B. Totaro, we construct a projective variety approximating 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 In- stitute, Toronto, as a part of a Thematic Program on Torsors, 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, hospi- tality and support. We are very grateful to B. Totaro for his interest and for generously communicating his construction of a projective algebraic approxi- mation in theorem 1.1. The first author would like to thank B. Kahn and J. Lannes for useful discussions. We are also very grateful to the referee for the comments and corrections. 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 category of pointed k-spaces (see [15]). For X ∈ H·(k), Documenta Mathematica · Extra Volume Merkurjev (2015) 501–511 Note on the Counterexamples for the ... 503 ′ denote by H∗,∗ (X, Z/ℓ) the motivic cohomology groups with Z/ℓ-coefficients (loc.cit.). If X is a smooth variety over k (viewed as an object of H·(k)), note ∼ that one has an isomorphism CH∗(X)/ℓ → H2∗,∗(X, Z/ℓ). Voevodsky ([23], see also [17]) defined the reduced power operations P i and ∗,∗′ the Milnor’s operations Qi on H (X, Z/ℓ): ′ ′ P i : H∗,∗ (X, Z/ℓ) → H∗+2i(ℓ−1),∗ +i(ℓ−1)(X, Z/ℓ),i ≥ 0 ′ ′ ∗,∗ ∗+2ℓi−1,∗ +(ℓi−1) Qi : H (X, Z/ℓ) → H (X, Z/ℓ),i ≥ 0, where Q0 = β is the Bockstein operation of degree (1, 0) induced from the short exact sequence 0 → Z/ℓ → Z/ℓ2 → Z/ℓ → 0. One of the key ingredients for this construction is the following computation of the motivic cohomology of the classifying space Bet´ µℓ ∈ H·(k): Lemma 2.1. ([23, §6]) For each object X ∈ H·(k), the graded alge- ∗,∗′ ∗,∗′ bra H (X × Bet´ µℓ, Z/ℓ) is generated over H (X, Z/ℓ) by elements x and y, deg(x) = (1, 1) and deg(y) = (2, 1), with β(x) = y and 0 ℓ is odd x2 = (τy + ρx ℓ =2 0,1 where τ is a generator of H (Spec(k), Z/2) =∼ µ2 and ρ is the class of (−1) in H1,1(Spec(k), Z/2) ≃ k∗/(k∗)2. For what follows, we assume that k contains a primitive ℓ2-th root of unity ∼ ℓ ξ, so that Bet´ Z/ℓ → Bet´ µℓ and β(τ) = ξ (= ρ for p = 2) is zero in ∗ ∗ ℓ 1,1 k /(k ) = Het´ (Spec(k); Z/ℓ). We will need the following properties: Proposition 2.2. Let X ∈ H·(k). (i) P i(x)=0 for i>m − n and i ≥ n and x ∈ Hm,n(X, Z/ℓ); (ii) P i(x)= xℓ for x ∈ H2i,i(X, Z/ℓ); (iii) if X is a smooth variety over k, the operation m 2m,m 2m+2ℓi−1,m+(ℓi−1) Qi : CH (X)/ℓ = H (X, Z/ℓ) → H (X, Z/ℓ) is zero ; i (iv) Op.(τx)= τOp.(x) for Op. = β,Qi or P ; ℓi−1 (v) Qi = [P ,Qi−1]. Proof. See [23, §9]. For (iii) one uses that Hm,n(X, Z/ℓ) = 0 if m − 2n > 0 and X is a smooth variety over k, (iv) follows from the Cartan formula for the motivic cohomology. Documenta Mathematica · Extra Volume Merkurjev (2015) 501–511 504 Alena Pirutka and Nobuaki Yagita 2.2. Computations for Bet´ Z/ℓ. The computations in this section are similar to [1, 21, 22]. ∗,∗′ ℓi Lemma 2.3. In H (Bet´ Z/ℓ, Z/ℓ), we have Qi(x)= y and Qi(y)=0. Proof. By definition Q0(x) = β(x) = y. Using induction and Proposition 2.2, we compute ℓi−1 ℓi−1 ℓi−1 Qi(x)= P Qi−1(x) − Qi−1P (x)= P Qi−1(x) i−1 i−1 i = P ℓ (yℓ )= yℓ . 1 ℓ Then Q1(y) = −Q0P (y) = −β(y ) = 0. For i > 1, using induction and ℓi−1 Proposition 2.2 again, we conclude that Qi(y)= −Qi−1P (y)=0. 3 Let G = (Z/ℓ) . As above, we view Bet´ G as an object of the category H·(k) and we assume that k contains a primitive ℓ2-th root of unity. From Lemma ′ 2.1, we have an isomorphism of modules over H∗,∗ (Spec(k), Z/ℓ): ∗,∗′ ∗,∗′ H (Bet´ G, Z/ℓ) =∼ H (Spec(k), Z/ℓ)[y1,y2,y3] ⊗ Λ(x1, x2, x3) where Λ(x1, x2, x3) is isomorphic to the Z/ℓ-module generated by 1 and xi1 ...xis for i1 < ... < is, with relations xixj = −xj xi (i ≤ j), β(xi) = yi and 2 xi = τyi for ℓ = 2. 3,3 Lemma 2.4. Let x = x1x2x3 in H (Bet´ G, Z/ℓ). Then 2∗,∗ QiQj Qk(x) 6=0 ∈ H (Bet´ G, Z/ℓ) for i < j < k. Proof. Using Proposition 2.2(v) and Cartan formula for the operations on cup- products ([23] Proposition 9.7 and Proposition 13.4), we first get Qk(x) = ℓk ℓk ℓk y1 x2x3 − y2 x1x3 + y3 x1x2 and one then deduces ℓk ℓj ℓi QiQj Qk(x)= ±yσ(1)yσ(2)yσ(3) 6=0 ∈ Z/ℓ[y1,y2,y3]. σ∈S3 X 3. exceptional Lie groups Let (G, ℓ) be a simple simply connected Lie group and a prime number from the following list: G2,ℓ =2, (3.1) (G, ℓ)= F4,ℓ =3, E8,ℓ =5. 3 ∼ Then G is 2-connected and we have H (G, Z) = Z for its (singular) cohomol- ogy group in degree 3. Hence BG, viewed as a topological space, is 3-connected 4 and H (BG, Z) =∼ Z (see [14] for example). We write x4(G) for a generator of H4(BG, Z). Documenta Mathematica · Extra Volume Merkurjev (2015) 501–511 Note on the Counterexamples for the ... 505 Given a field k with char(k) 6= ℓ, let us denote by Gk the (split) reductive algebraic group over k corresponding to the Lie group G. ∗ The Chow ring CH (BGk) has been defined by Totaro [22].
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