THE ORDER OF HIGHER BRAUER GROUPS THOMAS JAHN Abstract. Let X be a smooth projective variety of dimension 2r over a finite field F. We prove that for every prime ℓ 6= char(F) the order of the non-divisible quotient of the ℓ-primary torsion group of the higher Brauer r group Br (X)(ℓ)nd is a square number. 1. Introduction Let X be a smooth projective surface over a finite field F, and let Br(X)= 2 Het(X, Gm) be its cohomological Brauer group. If ℓ =6 char(F) is a prime, Tate has shown that the Tate conjecture for divisors on X at the prime ℓ is equiv- alent to the ℓ-primary torsion subgroup Br(X)(ℓ) being finite. Furthermore, the Artin-Tate conjecture [13, Conjecture C] relates the order of Br(X) with the zeta function of X. More precisely, the zeta function of X may be written −s 1−s P1(X, q )P1(X, q ) ζ(X,s)= −s −s 2−s , (1 − q )P2(X, q )(1 − q ) and the Artin-Tate conjecture states that Br(X) is finite, and satisfies −s |Br(X)|·|det(Di · Dj)| 1−s ρ(X) P2(X, q ) ∼ α(X) 2 · (1 − q ) , as s → 1 ; q ·|NS(X)tor| for details, see [13]. This type of formula has been generalized to smooth projective varieties of even dimension as follows. Assuming the existence of certain complexes Γ(r) of ´etale sheaves satisfying the axioms stated by Lichten- r 2r+1 baum in [7], one can define higher Brauer groups BrΓ(X) = Het (X, Γ(r)); 1 since Γ(1) ∼ Gm[−1], we have BrΓ(X) = Br(X). Under further assumptions, including the strong Tate conjecture, Milne [9, Theorem 6.6] proved in case dim X =2r the formula (we refer to [9] for precise definitions of all quantities) |Brr (X)|·| det(D · D )| P (X, q−s) ∼ ± Γ i j (1 − qr−s)ρr , as s → r . 2r αr(X) r 2 q ·|A (X)tor| If X is a surface, i.e. r = 1, the formula given by Milne coincides with the one of the Artin-Tate conjecture [9, Remark 6.7]. In particular, these formulae relating the values of zeta functions with the order of higher Brauer groups motivate the attempt to understand the order of these groups. In the classical case, i.e. for Br(X) of a surface, Tate [13] proved that there is a canonical skew-symmetric form on the non-divisible quotient Br(X)(ℓ)nd (which, assuming the Tate conjecture for divisors at ℓ coincides with Br(X)(ℓ)). Tate conjectured that this form is also alternating which would imply that the Date: June 13, 2014. 1 2 THOMAS JAHN order of Br(X)(ℓ)nd is a square. The second statement has been proven by Urabe in [15] (using a different bilinear form). We generalize this as follows. Instead of the complexes considered by Licht- enbaum we work with the (unbounded) complex of ´etale sheaves ZX (r)et given by Bloch’s cycle complex [1] on the small ´etale site Xet. Its hypercohomology m groups define the Lichtenbaum groups HL (X, Z(r)); since ZX (1)et ∼ Gm[−1] ∼ 3 we have Br(X) = HL(X, Z(1)), and the higher Brauer we consider here are r 2r+1 the groups Br (X) = HL (X, Z(r)). As in the classical case, these groups provide obstructions to the Tate conjecture. More precisely, if X is a smooth projective variety over a finite field F of arbitrary dimension, the Tate conjec- r r ture TC (X)Qℓ for X in codimension r at ℓ holds if and only if Br (X)(ℓ) < ∞ [11], see also section 2 below. Furthermore, the complexes ZX (r)et satisfy the r formal requirements needed to prove Milne’s formula with BrΓ(X) replaced by r Br (X). Thus we are interested in the (finite) order of Br(X)(ℓ)nd. With our notion of a higher Brauer group we can generalize Urabe’s result as follows. Theorem 1.1. Let X be a smooth projective variety of dimension 2r over a r finite field F. Then |Br (X)(ℓ)nd| is a square number for every ℓ =6 char(F). The proof makes use of cohomological methods analogous to the ones used by Urabe in his proof for r = 1, i.e. a pairing induced by the cup product, Steenrod operations, and the Wu Formula for ´etale cohomology. However, in Urabe’s case a crucial role is played by the cohomology class of the canoni- cal divisor, and a crucial impetus in our generalization is the construction of cohomology classes with similar properties in all codimension r. Notations and Conventions. In what follows, X always denotes a smooth projective variety of dimension 2r over a finite field F of characteristic p, and ℓ a prime number different from p. If F is an algebraic closure of F, we write X = X ×F F for the base change. We let G = Gal(F/F) be the Galois group and denote by σ the Frobenius. If M is a G-Module, we define M G to be the kernel and MG to be the cokernel of (1 − σ): M → M. For an abelian group A we use the following conventions: An denotes the n- n torsion subgroup, i.e. An = ker(A → A), Ator = ∪n≥1An, and Afree = A/Ator; n if q is a prime number, A(q) = n>0 Aq is the q-primary torsion subgroup. We write Adiv for the subgroup ofS divisible elements, and set And = A/Adiv. Acknowledgement. The author is grateful to his advisor Professor Andreas Rosenschon for suggesting the problem and his helpful advice. 2. Preliminaries We first give the precise definition of the higher Brauer groups used in this paper, and explain the relation of these groups to the Tate conjecture. Let Y be a equi-dimensional smooth quasi-projective variety over a field k and let zn(Y, •) the cycle complex of abelian groups defined by Bloch in [1]. The assertion zn(−, •): U 7→ zn(U, •) defines a complex of sheaves on n the flat site of Y , thus a complex z (−, •)et of sheaves on the ´etale site of Y THE ORDER OF HIGHER BRAUER GROUPS 3 (which is unbounded on the left). Given an abelian group A, we set AY (n)et = n (z (−, •)et ⊗ A)[−2n]. The Lichtenbaum cohomology groups are defined as m m HL (Y, A(n)) = Het (Y, AY (n)et); for the definition of the hypercohomology of an unbounded complex, see [12, 3 2 p. 121]. Since ZY (1)et ∼ Gm[−1] we have HL(Y, Z(1)) = Het(Y, Gm) = r 2r+1 r Br(Y ). Set Br (Y ) = HL (Y, Z(r)). Note that the Br (Y ) are torsion groups 2r (as quotients of HL (Y, Q/Z(r)); this uses that with rational coefficients the Lichtenbaum groups coincide with the usual motivic cohomology groups, thus 2r+1 HL (Y, Q(r)) = 0). Moreover, for every prime ℓ we have the exact sequence m ℓ m 0 → ZY (n)et → ZY (n)et → (Z/ℓ Z)Y (n)et → 0 . m If ℓ =6 char(k), it follows from this and the quasi-isomorphism (Z/ℓ Z)Y (n)et ∼ ⊗n µℓm [5, Theorem 1.5] that there are short exact universal coefficient sequences ⊗ i m i n i+1 m (1) 0 → HL(Y, Z(n)) ⊗ Z/ℓ Z → Het(Y,µℓm ) → HL (Y, Z(n))ℓ → 0 . In particular, in bidegree (i, n)=(2r, r) we obtain the short exact sequence ⊗ 2r m 2r r r m 0 → HL (Y, Z(r)) ⊗ Z/ℓ Z → Het (Y,µℓm ) → Br (Y )ℓ → 0 which generalizes the short exact sequence for the classical Brauer group com- ing from the Kummer sequences. r We use this to describe the non-divisible quotient Br (Y )(ℓ)nd in terms of ´etale cohomology groups. Lemma 2.1. Let k be a field and let Y/k be a smooth quasi-projective variety. If ℓ =6 char(k) is a prime, there are isomorphisms ∼ ∼ r = 2r = 2r+1 Br (Y )(ℓ)nd ← Het (Y, Qℓ/Zℓ(r))nd → Het (Y, Zℓ(r))tor . Proof. From (1) we obtain by taking the evident direct limit the exact sequence 2r 2r r 0 → HL (Y, Z(r)) ⊗ Qℓ/Zℓ → Het (Y, Qℓ/Zℓ(n)) → Br (Y )(ℓ) → 0 which immediately implies the first isomorphism. Consider the exact sequence 2r ϕ 2r δ 2r+1 ψ 2r+1 Het (Y, Qℓ(r)) → Het (Y, Qℓ/Zℓ(r)) → Het (Y, Zℓ(r)) → Het (Y, Qℓ(r)) associated with 0 → Zℓ(r) → Qℓ(r) → Qℓ/Zℓ(r) → 0. Since im ϕ is the max- 2r imal divisible subgroup of Het (Y, Qℓ/Zℓ(r)) and ker ψ is the torsion subgroup 2r+1 of Het (Y, Zℓ(r)), the boundary map δ induces the second isomorphism. Finally, we sketch how the higher Brauer groups provide obstructions to the Tate conjecture [14]. Let k be a field which is finitely generated over its prime field, let Y/k be a smooth, projective, geometrically integral variety, and set Y = Y ×k k, where k is a separable closure of k. Denote by G = Gal(k/k) the m Galois group. The Tate conjecture TC (Y )Qℓ for Y in codimension m at the prime ℓ =6 char(k) states that the ℓ-adic cycle map from Chow groups to ´etale cohomology m m 2m G cQℓ : CH (Y ) ⊗ Qℓ → Het (Y, Qℓ(m)) 4 THOMAS JAHN is surjective.