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 ≠ char(k). The Tate conjecture [20] predicts that the cycle class map

CH^{i}(Xk̄) ⊗ ℚ_{ℓ} → ⋃_{H} H_{ét}^{2i}(Xk̄, ℚ_{ℓ}(i))^{H},

where the union is over all open subgroups H of Gal(k̄/k), is surjective. In the integral version one is interested in the cokernel of the cycle class map

(1.1)  CH^{i}(Xk̄) ⊗ ℤ_{ℓ} → ⋃_{H} H_{ét}^{2i}(Xk̄, ℤ_{ℓ}(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]. More precisely, one constructs an ℓ-torsion class in H_{ét}^{4}(Xk̄, ℤ_{ℓ}(2)), which is not algebraic, for some smooth and projective 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)  CH^{i}(Xk̄) ⊗ ℤ_{ℓ} → ⋃_{H} H_{ét}^{2i}(Xk̄, ℤ_{ℓ}(i))^{H}/torsion

is not surjective ([13, 3]). In the context of an integral version of the Hodge conjecture, Kollár [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 ≠ ℓ, such that the cycle class map

CH^{2}(Xk̄) ⊗ ℤ_{ℓ} → ⋃_{H} H_{ét}^{4}(Xk̄, ℤ_{ℓ}(2))^{H}/torsion,

where the union is over all open subgroups H of Gal(k̄/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 Q_{1} 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 Institute, 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, hospitality and support. We are very grateful to B. Totaro for his interest and for generously communicating 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. 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) ≠ ℓ and let H_{·}(k) be the motivic homotopy category of pointed k-spaces (see [15]). For X ∈ H_{·}(k), 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 →^{∼} H^{2∗,∗}(X, ℤ/ℓ). Voevodsky ([23], see also [17]) defined the reduced power operations P^{i} and the Milnor’s operations Q_{i} on H^{∗,∗′}(X, ℤ/ℓ):

P^{i} : H^{∗,∗′}(X, ℤ/ℓ) → H^{∗+2i(ℓ−1),∗′+i(ℓ−1)}(X, ℤ/ℓ), i ≥ 0
Q_{i} : H^{∗,∗′}(X, ℤ/ℓ) → H^{∗+2ℓ^{i}−1,∗′+(ℓ^{i}−1)}(X, ℤ/ℓ), i ≥ 0,

where Q_{0} = β is the Bockstein operation of degree (1, 0) induced from the short exact sequence 0 → ℤ/ℓ → ℤ/ℓ^{2} → ℤ/ℓ → 0. One of the key ingredients for this construction is the following computation of the motivic cohomology of the classifying space B_{ét}μ_{ℓ} ∈ H_{·}(k):

Lemma 2.1. ([23, §6]) For each object X ∈ H_{·}(k), the graded algebra H^{∗,∗′}(X × B_{ét}μ_{ℓ}, 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

x^{2} = \begin{cases}0 ℓ is odd \\ τy + ρx ℓ = 2 \end{cases}

where τ is a generator of H^{0,1}(Spec(k), ℤ/2) ≅ μ_{2} and ρ is the class of (−1) in H^{1,1}(Spec(k), ℤ/2) ≃ k^{∗}/(k^{∗})^{2}.

For what follows, we assume that k contains a primitive ℓ^{2}-th root of unity ξ, so that B_{ét}ℤ/ℓ →^{∼} B_{ét}μ_{ℓ} and β(τ) = ξ^{ℓ} (= ρ for p = 2) is zero in k^{∗}/(k^{∗})^{ℓ} = H_{ét}^{1,1}(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 ∈ H^{m,n}(X, ℤ/ℓ); (ii) P^{i}(x) = x^{ℓ} for x ∈ H^{2i,i}(X, ℤ/ℓ); (iii) if X is a smooth variety over k, the operation

Q_{i} : CH^{m}(X)/ℓ = H^{2m,m}(X, ℤ/ℓ) → H^{2m+2ℓ^{i}−1,m+(ℓ^{i}−1)}(X, ℤ/ℓ)

is zero ; (iv) Op.(τx) = τOp.(x) for Op. = β, Q_{i} or P^{i};

(v) Q_{i} = [P^{ℓ^{i−1}}, Q_{i−1}].

Proof. See [23, §9]. For (iii) one uses that H^{m,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.

2.2. Computations for B_{ét}ℤ/ℓ. The computations in this section are similar to [1, 21, 22].

Lemma 2.3. In H^{∗,∗′}(B_{ét}ℤ/ℓ, ℤ/ℓ), we have Q_{i}(x) = y^{ℓ^{i}} and Q_{i}(y) = 0.

Proof. By definition Q_{0}(x) = β(x) = y. Using induction and Proposition 2.2, we compute

Q_{i}(x) = P^{ℓ^{i−1}}Q_{i−1}(x) − Q_{i−1}P^{ℓ^{i−1}}(x) = P^{ℓ^{i−1}}Q_{i−1}(x)
= P^{ℓ^{i−1}}(y^{ℓ^{i−1}}) = y^{ℓ^{i}}.

Then Q_{1}(y) = −Q_{0}P^{1}(y) = −β(y^{ℓ}) = 0. For i > 1, using induction and Proposition 2.2 again, we conclude that Q_{i}(y) = −Q_{i−1}P^{ℓ^{i−1}}(y) = 0.

□

Let G = (ℤ/ℓ)^{3}. As above, we view B_{ét}G as an object of the category H_{·}(k) and we assume that k contains a primitive ℓ^{2}-th root of unity. From Lemma 2.1, we have an isomorphism of modules over H^{∗,∗′}(Spec(k), ℤ/ℓ) :

H^{∗,∗′}(B_{ét}G, ℤ/ℓ) ≅ H^{∗,∗′}(Spec(k), ℤ/ℓ)[y_{1}, y_{2}, y_{3}] ⊗ Λ(x_{1}, x_{2}, x_{3})

where Λ(x_{1}, x_{2}, x_{3}) is isomorphic to the ℤ/ℓ-module generated by 1 and x_{i_{1}}...x_{i_{s}} for i_{1} < ... < i_{s}, with relations x_{i}x_{j} = −x_{j}x_{i} (i ≤ j), β(x_{i}) = y_{i} and x_{i}^{2} = τy_{i} for ℓ = 2.

Lemma 2.4. Let x = x_{1}x_{2}x_{3} in H^{3,3}(B_{ét}G, ℤ/ℓ). Then

Q_{i}Q_{j}Q_{k}(x) ≠ 0 ∈ H^{2∗,∗}(B_{ét}G, ℤ/ℓ) for i < j < k.

Proof. Using Proposition 2.2(v) and Cartan formula for the operations on cupproducts ([23] Proposition 9.7 and Proposition 13.4), we first get Q_{k}(x) = y_{1}^{ℓ^{k}}x_{2}x_{3} − y_{2}^{ℓ^{k}}x_{1}x_{3} + y_{3}^{ℓ^{k}}x_{1}x_{2} and one then deduces

Q_{i}Q_{j}Q_{k}(x) = ∑_{σ∈S_{3}} ±y_{σ(1)}^{ℓ^{k}}y_{σ(2)}^{ℓ^{j}}y_{σ(3)}^{ℓ^{i}} ≠ 0 ∈ ℤ/ℓ[y_{1}, y_{2}, y_{3}].

□

3. exceptional Lie groups

Let (G, ℓ) be a simple simply connected Lie group and a prime number from the following list:

(3.1)  (G, ℓ) = \begin{cases}G_{2}, ℓ = 2, \\ F_{4}, ℓ = 3, \\ E_{8}, ℓ = 5. \end{cases}

Then G is 2-connected and we have H^{3}(G, ℤ) ≅ ℤ for its (singular) cohomology group in degree 3. Hence BG, viewed as a topological space, is 3-connected and H^{4}(BG, ℤ) ≅ ℤ (see [14] for example). We write x_{4}(G) for a generator of H^{4}(BG, ℤ).

Given a field k with char(k) ≠ ℓ, let us denote by G_{k} the (split) reductive algebraic group over k corresponding to the Lie group G. The Chow ring CH^{∗}(BG_{k}) has been defined by Totaro [22]. More precisely, one has

(3.2)  BG_{k} = \underrightarrow{lim}(U/G_{k}),

where U ⊂ W is an open set in a linear representation W of G_{k}, such that G_{k} acts freely on U. One can then identify CH^{i}(BG_{k}) with the group CH^{i}(U/G_{k}) if codim_{W} (W \ U) > i, the group CH^{i}(BG_{k}) is then independent of a choice of such U and W. Similarly, one can define the étale cohomology groups H_{ét}^{i}(BG_{k}, ℤ_{ℓ}(j)) and the motivic cohomology groups H^{∗,∗′}(BG_{k}, ℤ/ℓ) (see [8]), the latter coincide with the motivic cohomology groups of B_{ét}G as in [15] (cf. [8, Proposition 2.29 and Proposition 3.10]). We also have the cycle class map

(3.3)  cl : CH^{∗}(BGk̄) ⊗ ℤ_{ℓ} → ⋃_{H} H_{ét}^{2∗}(BGk̄, ℤ_{ℓ}(∗))^{H},

where the union is over all open subgroups H of Gal(k̄/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) H^{4}(BG, ℤ/ℓ) ≃ ℤ/ℓ, generated by the image x_{4} of the generator x_{4}(G)

of H^{4}(BG, ℤ) ≃ ℤ; (iii) Q_{1}(i^{∗}x_{4}) = Q_{1}Q_{0}(x_{1}x_{2}x_{3}), in the notations of Lemma 2.4. In partic-

ular, Q_{1}(i^{∗}x_{4}) is nonzero.

Proof. For (i) see [5], for the computation of the cohomology groups with ℤ/ℓ-coefficients in (ii) see [14] VII 5.12; (iii) follows from [11] for ℓ = 2 and [9, Proposition 3.2] for ℓ = 3, 5 (see [10] as well). The class Q_{1}(i^{∗}x_{4}) is nonzero by Lemma 2.4 (see also [8, Théorème 4.1]). □

4. Algebraic approximation of BG

Write

(4.1)  BG_{k} = \underrightarrow{lim}(U/G_{k})

as in the previous section. Using proposition 3.1 and a specialization argument, we will first construct a quasi-projective algebraic variety X over a finite field k as a quotient X = U/G_{k} (where codim_{W} (W \ U) is big enough), such that the cycle class map (1.2) is not surjective for such X. However, if one is interested only in quasi-projective counterexamples for the surjectivity of the map (1.2), one can produce more naive examples, 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 étale cohomology of BGk̄ as a smooth and projective variety. In the case when the group G is finite, this is done in [3, Théorème 2.1]. In this section we give such an approximation for the groups we consider here, this construction is suggested by B. Totaro. We will proceed in three steps. First, we construct a quasi-projective approximation in a family parametrized by Spec ℤ. Then, for the geometric generic fibre we produce a projective approximation, by a topological argument. We finish the construction by specialization.

Let G be a compact Lie group as in (3.1). Let G be a split reductive group over Spec ℤ corresponding to G, such a group exists by [SGA3] XXV 1.3.

Lemma 4.1. For any fixed integer s ≥ 0 there exists a projective scheme Y/Spec ℤ and an open subscheme W ⊂ Y such that

(i) W → Spec ℤ is smooth and the complement of W is of codimension at least s in each fiber of Y → Spec ℤ;

(ii) for any point t ∈ Spec ℤ with residue field κ(t) there is a natural map W_{t} → B(𝔾_{m} × G)_{t} inducing an isomorphism

(4.2)  H_{ét}^{i}(Wt̄, ℤ_{ℓ}) →^{∼} H_{ét}^{i}(B(𝔾_{m} × G)t̄, ℤ_{ℓ}) for i ≤ s, ℓ ≠ char κ(t).

Proof. Write T = Spec ℤ, as it is an affine scheme of dimension 1, we can embed G as a closed subgroup of H = GL_{d,T} for some d (see [SGA3] VI_{B} 13.2). Moreover, it induces an embedding G ↪ P GL_{d,T}, as the center of G is trivial for groups we consider here. By a construction of [22, Remark 1.4] and [2, Lemme 9.2], there exists n > 0, a linear H-representation O_{T}^{⊕n} and an H-invariant open subset U ⊂ O_{T}^{⊕n}, which one can assume flat over T, such that the action of H is free on U. Let V_{N} = O_{T}^{⊕Nn}. Then the group P GL_{n,T} acts on ℙ(V_{N}) and, taking N sufficiently large, one can assume that the action is free outside a subset S of high codimension (with respect to s). By restriction, the group G acts on ℙ(V_{N}) as well, let Y = ℙ(V_{N})//G be the GIT quotient for this action [16, 19]. The scheme Y is projective over T and we fix an embedding Y ⊂ ℙ_{T}^{M}. Let

(4.3)  f : W → T

be the open set of Y corresponding to the quotient of the open set U as above where G_{T} acts freely. From the construction, one can assume that W has codimension at least s in Y in each fibre over T. For any point t ∈ T the fibre W_{t} is a smooth quasi-projective variety and if N is big enough, we have isomorphisms (cf. p. 263 in [22])

W_{t} ≅ (ℙ(V_{N}) − S)_{t}/G_{t} ≅ ((V_{N} − {0})/𝔾_{m} − S)_{t})/G_{t} ≅ (V_{N} − S′)_{t}/(𝔾_{m} × G)_{t}

where S′ = pr^{−1}S ∪ {0} for the projection pr : (V_{N} − {0}) → ℙ(V_{N}). Hence we have isomorphisms

H_{ét}^{i}(Wt̄, ℤ_{ℓ}) →^{∼} H_{ét}^{i}(B(𝔾_{m} × G)t̄, ℤ_{ℓ}) for i ≤ s, ℓ ≠ char κ(b),

induced by a natural map W_{t} → B(𝔾_{m} × G)_{t} from the presentation (4.1). □

Remark 4.2. More generally, in the statement above the map W_{t} →

B(𝔾_{m} × G)_{t} induces an isomorphism H_{ét}^{i}(W_{F} , ℤ_{ℓ}) →^{∼} H_{ét}^{i}(B(𝔾_{m} × G)_{F} , ℤ_{ℓ}),

i ≤ s, ℓ ≠ char κ(t) for any F -point of T over t.

Lemma 4.3. Let Y ⊂ ℙ_{ℂ}^{M} be a projective variety over ℂ and let W ⊂ Y be a dense open in Y. Assume that W is smooth. Then for a general linear subspace L in ℙ^{M} of codimension equal to 1 + dim(Y − W), the scheme X = L ∩ W is smooth and projective and the natural maps H^{i}(W, ℤ) → H^{i}(X, ℤ) are isomorphisms for i < dim X.

Proof. We apply a version of the Lefschetz hyperplane theorem for quasiprojective varieties, established by Hamm (as a special case of Theorem II.1.2 in [4]): for V ⊂ ℙ^{M} a closed complex subvariety of dimension d, not necessarily smooth, Z ⊂ V a closed subset, and H a hyperplane in ℙ^{M}, if V − (Z ∪ H) is local complete intersection (e.g. V − Z is smooth) then

π_{i}((V − Z) ∩ H) → π_{i}(V − Z)

is an isomorphism for i < d − 1 and surjective for i = d − 1. In particular, H^{i}(V − Z, ℤ) → H^{i}((V − Z) ∩ H, ℤ) is an isomorphism for i < d − 1 and surjective for i = d − 1 by the Whitehead theorem. Applying this statement to W and to successive intersections of W with linear forms defining L, we then deduce that H^{i}(W, ℤ) → H^{i}(X, ℤ) is an isomorphism for i < dim X.

□

Proposition 4.4. 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_{k} over a finite field k with char k = p, an element x_{4,k̄} ∈ H_{ét}^{4}(B(𝔾_{m} × Gk̄), ℤ_{ℓ}(2)), invariant under the action of Gal(k̄/k) and a map ι : X_{k} → B(𝔾_{m} × G_{k}) in the category H_{·}(k) such that

(i) αk̄ = ι^{∗}x_{4,k̄} is a nonzero class in H_{ét}^{4}(Xk̄, ℤ_{ℓ}(2))/torsion; (ii) the operation Q_{1}(ᾱk̄) is nonzero, where we write ᾱk̄ for the image of

αk̄ in H_{ét}^{4}(Xk̄, μ_{ℓ}^{⊗2}).

Proof. Let W ⊂ Y ⊂ ℙ_{ℤ}^{M} be as in Lemma 4.1 for s ≥ 4. Let Y = Y_{ℂ} and W = W_{ℂ} be the geometric generic fibres of Y and W. Consider a general linear space L in ℙ^{M} of codimension equal to 1 + dim(Y − W). We deduce from Lemma 4.3 above, that the variety X := L ∩ W is smooth and projective, and

(4.4)  H^{i}(X, R) ≃ H^{i}(B(𝔾_{m} × G), R) for i ≤ s and R = ℤ or ℤ/n.

Hence H_{ét}^{i}(X, ℤ/n) ≃ H_{ét}^{i}(B(𝔾_{m} × G), ℤ/n), i ≤ s. In particular, by functoriality of the isomorphisms H_{ét}^{i}(·, ℤ/n) ≃ H_{ét}^{i}(·, μ_{n}^{⊗j}), i ≤ s, j > 0, for · = X and B(𝔾_{m} × G), we get

(4.5)  H_{ét}^{i}(X, μ_{n}^{⊗j}) ≃ H_{ét}^{i}(B(𝔾_{m} × G), μ_{n}^{⊗j}), i ≤ s.

We can assume that we have an isomorphism as above for i = 4 and i = 2ℓ + 3. Note that the cohomology of BG is a direct factor in the cohomology of B(𝔾_{m} × G) (cf. [8, Lemme 2.23]). Using Proposition 3.1, we then get an element x_{4,ℂ} generating a direct factor isomorphic to ℤ_{ℓ} in the cohomology group H_{ét}^{4}(B(𝔾_{m} × G), ℤ_{ℓ}(2)). Denote α_{ℂ} its image in H_{ét}^{4}(X, ℤ_{ℓ}(2)).

We can now specialize the construction above to obtain the statement over a finite field. Note that one can assume that L is defined over ℚ. One can then find an open T′ ⊂ Spec ℤ and a linear space L ⊂ ℙ_{T′}^{M} such that L_{ℂ} ≃ L and such that for any t ∈ T′ the fibre X_{t} of X = L ∩ T is smooth. After passing to an étale cover T′′ of T′, one can assume that the inclusion (ℤ/ℓ)^{3} ⊂ G_{ℂ} from proposition 3.1 extends to an inclusion i : A = (ℤ/ℓ)_{T′′}^{3} ↪ G_{T′′} (cf. [SGA3] XI.5.8). Let t ∈ T′′ and let k = κ(t). As the schemes X_{T′′}, W_{T′′} and U/A are smooth over T′′, we have the following commutative diagram, where the vertical maps are induced by the specialization maps (cf. [SGA4 1/2] Arcata V.3):

\xymatrix{H_{ét}^{4}(X, ℤ_{ℓ}(2)) \ar[d] & H_{ét}^{4}(W, ℤ_{ℓ}(2)) \ar[l] \ar[d] \ar[r] & H_{ét}^{4}(U_{ℂ}/(ℤ/ℓ)^{3}, ℤ/ℓ) \ar[d] & H_{ét}^{4}(B(ℤ/ℓ)_{ℂ}^{3}, ℤ/ℓ) \ar[d]^{≃} \ar[l]_{≃} \\ H_{ét}^{4}(Xk̄, ℤ_{ℓ}(2)) & H_{ét}^{4}(Wk̄, ℤ_{ℓ}(2)) \ar[l] \ar[r] & H_{ét}^{4}(Uk̄/(ℤ/ℓ)^{3}, ℤ/ℓ) & H_{ét}^{4}(B(ℤ/ℓ)3 ; k̄, ℤ/ℓ) \ar[l]_{≃}}

The left vertical map is an isomorphism since X is proper, by a smooth-proper base change theorem. Hence we get a class αk̄ ∈ H_{ét}^{4}(Xk̄, ℤ_{ℓ}(2)), corresponding to α_{ℂ} ∈ H_{ét}^{4}(X, ℤ_{ℓ}(2)). The map H_{ét}^{4}(W, ℤ_{ℓ}(2)) → H_{ét}^{4}(X, ℤ_{ℓ}(2)) is an isomorphism by Lemma 4.3, so that αk̄ comes from an element x_{4,k̄} ∈ H_{ét}^{4}(Wk̄, ℤ_{ℓ}(2)). Let ᾱ_{ℂ} ∈ H_{ét}^{4}(X, μ_{ℓ}^{⊗2}) be the image of α_{ℂ} and let ᾱk̄ ∈ H_{ét}^{4}(Xk̄, μ_{ℓ}^{⊗2}) be the image of αk̄. As the operation Q_{1} commutes with the isomorphisms H_{ét}^{i}(X, ℤ/ℓ) → H_{ét}^{i}(X, μ_{ℓ}^{⊗j}), we get Q_{1}(ᾱ_{ℂ}) ≠ 0 by proposition 3.1. The étale cohomology operation Q_{1} also commutes with the specialization maps (cf. [7]), since these maps are obtained as composite of the natural maps ϕ ∘ ψ^{−1} on the étale cohomology groups with torsion coefficients H_{ét}^{i}(X_{ℂ}) ←^{ψ} H_{ét}^{i}(X_{S}) →^{ϕ} H_{ét}^{i}(Xk̄), where S is the strict henselization of T′′ at t and ϕ is an isomorphism since X is smooth. Hence Q_{1}(ᾱk̄) is nonzero as well. From the construction, the class αk̄ generates a subgroup of H_{ét}^{4}(Xk̄, ℤ_{ℓ}(2)), which is a direct factor isomorphic to ℤ_{ℓ}, and is Galois-invariant. Letting X_{k} = X_{k} this finishes the proof of the proposition.

□

Remark 4.5. For the purpose of this note, the proposition above is enough. See also [6] for a a general statement on a projective approximation 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_{k} as in the proposition above, we find a nontrivial class αk̄ in its cohomology in degree 4 modulo torsion, which is not annihilated by the operation Q_{1}. This class cannot be algebraic by proposition 2.2(iii). □

Remark 4.6. We can also adapt the arguments of [3, Théorème 2.1] to produce projective examples with higher torsion non-algebraic classes, while in loc.cit. one constructs ℓ-torsion classes. Let G(n) be the finite group G(𝔽_{ℓ^{n}}), so that we have

\underleftarrow{lim} H_{ét}^{∗}(BG(n), ℤ_{ℓ}) = H_{ét}^{∗}(BGk̄, ℤ_{ℓ}).

Then, following the construction in loc.cit. one gets For any n > 0, there exists a positive integer i_{n} and a Godeaux-Serre variety X_{n,k̄} for the finite group G(i_{n}) such that

(1) there is an element x ∈ H_{ét}^{4}(X_{n,k̄}; ℤ_{ℓ}(2)) generating ℤ/ℓ^{n′} for some n′ ≥ n;

(2) x is not in the image of the cycle class map (1.1).

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Alena Pirutka Centre de Mathématiques Laurent Schwartz UMR 7640 de CNRS École Polytechnique 91128 Palaiseau France alena.pirutka @polytehcnique.edu Nobuaki Yagita Department of Mathematics Faculty of Education Ibaraki University Mito Ibaraki Japan yagita@mx.ibaraki.ac.jp 512

Documenta Mathematica · Extra Volume Merkurjev (2015)