THE ORDER of HIGHER BRAUER GROUPS 1. Introduction Let X Be A

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 étale 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 étale sheaves ZX (r)et given by Bloch’s cycle complex [1] on the small étale 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 étale cohomology. However, in Urabe’s case a crucial role is played by the cohomology class of the canonical 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 étale 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 étale 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 étale cohomology m m 2m G cQℓ : CH (Y ) ⊗ Qℓ → Het (Y, Qℓ(m)) 4 THOMAS JAHN is surjective. Tate has shown that for a surface Y over a finite field the con- 1 jecture TC (Y )Qℓ holds if and only if the order of the ℓ-primary part of the Brauer group Br(Y )(ℓ) is finite [13, Theorem 5.2]. Using our notion of higher Brauer groups, Rosenschon and Srinivas [11, Theorem 1.3] proved that over a m m finite field and Y of arbitrary dimension TC (Y )Qℓ ⇔ Br (Y )(ℓ) < ∞; we remark that assuming the existence of complexes Γ(r) as above, Milne has m m shown in [9, Remark 4.5(g)] that TC (Y )Qℓ ⇔ BrΓ (Y )(ℓ) < ∞. r 3. Bilinear Form on Br (X)(ℓ)nd r In this section we prove that the order of Br (X)(ℓ)nd is a square, provided that ℓ =6 2. Our strategy is to construct a non-degenerate skew-symmetric r r bilinear form Br (X)(ℓ)nd × Br (X)(ℓ)nd → Qℓ/Zℓ, and to show that the order r of Br (X)(ℓ)nd is odd. Given this, we use the general facts that each skew- symmetric bilinear form A × A → Q/Z on an abelian group A of odd order with values in Q/Z is alternating, and that the existence of a non-degenerate alternating bilinear form A × A → Q/Z on a finite group A implies that the order of A is a square (cf. [15, Introduction]), which implies our claim. To construct this bilinear form, we start with the usual cup product pairings i ⊗r 4r−i ⊗r 4r ⊗2r ∼ m Het(X,µℓm ) × Het (X,µℓm ) → Het (X,µℓm )) = Z/ℓ Z which by Poincaréduality are non-degenerate pairings of finite G-modules. Since G operates trivially on Z/ℓmZ, these cup product pairings induce bilinear i ⊗r 4r−i ⊗r G m forms Het(X,µℓm )G × Het (X,µℓm ) → Z/ℓ Z. From the Hochschild-Serre spectral sequence we obtain, using that the cohomological dimension of G is at most one, the exact sequences i ⊗2r i+1 ⊗2r i+1 ⊗2r G 0 → Het(X,µℓm )G → Het (X,µℓm ) → Het (X,µℓm ) → 0 which, together with the bilinear form from above, yields the pairing 2r ⊗r 2r+1 ⊗r 4r+1 ⊗2r (2) Het (X,µℓm ) × Het (X,µℓm ) → Het (X,µℓm ) . 4r+1 ⊗2r m Since dim X = 2r, the codomain Het (X,µℓm ) is isomorphic to Z/ℓ Z; 4r ⊗2r ∼ this follows from the Hochschild-Serre spectral sequence, since Het (X,µℓm ) = m 4r+1 ⊗2r Z/ℓ Z and Het (X,µℓm ) = 0 (cf. [8, Ch. VI, Lemma 11.3 and Theorem 1.1]). Before continuing the construction of the required bilinear form, we check that we are still dealing with a non-degenerate form. Lemma 3.1. The pairing (2) is a non-degenerate bilinear form. Proof. It follows immediately from the following commutative diagram

i ⊗r 4r−i ⊗r G m H (X,µ m ) / Hom(H (X,µ m ) , Z/ℓ Z) et _ ℓ G et _ ℓ i ⊗r / 4r−i ⊗r m Het(X,µℓm ) Hom(Het (X,µℓm ), Z/ℓ Z) i ⊗r 4r−i ⊗r G m that the bilinear forms Het(X,µℓm )G × Het (X,µℓm ) → Z/ℓ Z are non- degenerate. From the short exact sequences given by the Hochschild-Serre spectral sequence (in bidegrees (i, r) = (2r − 1,r) and (i, r) = (2r, r)) we obtain the commutative diagram with exact columns THE ORDER OF HIGHER BRAUER GROUPS 5

0 0 2r−1 ⊗r / 2r+1 ⊗r G m Het (X,µℓm )G Hom(Het (X,µℓm ) , Z/ℓ Z) 2r ⊗r / 2r+1 ⊗r m Het (X,µℓm ) Hom(Het (X,µℓm ), Z/ℓ Z) 2r ⊗r G / 2r ⊗r m Het (X,µℓm ) Hom(Het (X,µℓm )G, Z/ℓ Z) 0 which implies that the middle horizontal morphism is injective; since this map is induced by the pairing (2), this establishes our claim. We continue the construction of a non-degenerate skew-symmetric bilinear r r form Br (X)(ℓ)nd × Br (X)(ℓ)nd → Qℓ/Zℓ. As described in [15, p. 560f], one obtains from the cup product a bilinear form 2r 2r+1 (3) Het (X, Qℓ/Zℓ(r)) × Het (X, Zℓ(r)) → Qℓ/Zℓ with the property that the following diagram commutes.

2r ⊗r 2r+1 ⊗r m (4) H (X,µ m ) × H (X,µ m ) / Z/ℓ Z et ℓ et O ℓ 2r 2r+1 / Het (X, Qℓ/Zℓ(r)) × Het (X, Zℓ(r)) Qℓ/Zℓ It follows from lemma 3.1 that the bilinear form (3) is non-degenerate. Fur- thermore, it is easy to see that this form induces a well-defined bilinear form 2r 2r+1 (5) Het (X, Qℓ/Zℓ(r))nd × Het (X, Zℓ(r))tor → Qℓ/Zℓ, ([x],y) 7→ x ∪ y , and we use the isomorphisms from lemma 2.1 to rewrite this bilinear form as r r (6) Br (X)(ℓ)nd × Br (X)(ℓ)nd → Qℓ/Zℓ . Proposition 3.2. The form (6) is non-degenerate and skew-symmetric. Proof. It suffices to prove these properties for the form (5). Non-degeneracy follows easily from a direct computation and the fact that the form (3) is non- degenerate. For skew-symmetry we use that the boundary maps δ′ and δ of ⊗r ⊗r ⊗r the long exact sequences associated with 0 → µℓm → µℓ2m → µℓm → 0 and 0 → Zℓ(r) → Qℓ(r) → Qℓ/Zℓ(r) → 0 fit into the commutative diagram

′ 2r ⊗r δ 2r+1 ⊗r H (X,µ m ) / H (X,µ m ) et ℓ et O ℓ 2r δ / 2r+1 Het (X, Qℓ/Zℓ(r)) Het (X, Zℓ(r)) Since the diagram (4) is commutative, this also holds for the diagram

(x,y)7→x∪δ′(y) 2r ⊗r 2r ⊗r / m Het (X,µℓm ) × Het (X,µℓm ) Z/ℓ Z (x,y)7→x∪δ(y) 2r 2r / Het (X, Qℓ/Zℓ(r)) × Het (X, Qℓ/Zℓ(r)) Qℓ/Zℓ . 6 THOMAS JAHN

2r If x, y ∈ Het (X, Qℓ/Zℓ(r)), we need to verify that x ∪ δ(y)+ y ∪ δ(x) = 0. ′ ′ 2r ⊗r Choose m large enough so that there exist preimages x ,y ∈ Het (X,µℓm ); then it suffices to show that x′ ∪ δ′(y′)+ y′ ∪ δ′(x′)= δ′(x′ ∪ y′)=0. But δ′ in 4r+1 ⊗2r ∼ m 2m ∼ 4r+1 ⊗2r degree 4r is trivial since Het (X,µℓm ) = Z/ℓ Z → Z/ℓ Z = Het (X,µℓ2m ) is injective, which proves our claim. From the bilinear form (6) we obtain Theorem 1.1 in case ℓ =6 2. r Proposition 3.3. For ℓ =26 the order of Br (X)(ℓ)nd is a square. r 2r+1 Proof. The group Br (X)(ℓ)nd is isomorphic to Het (X, Zℓ(r))tor which is finite as a quotient of a finite group (cf. [8, Ch. VI, Corollary 2.8]). Since the r r order of every element x ∈ Br (X)(ℓ)nd is a power of ℓ, Br (X)(ℓ)nd cannot r contain a subgroup of even order and the order of Br (X)(ℓ)nd must be odd. Hence (6) is a skew-symmetric non-degenerate bilinear form on a finite group r of odd order. This implies that the order of Br (X)(ℓ)nd is a square number, as already mentioned at the beginning of this section.

4. Alternating form We consider the case ℓ = 2. For simplicity, we write H for the group 2r r Het (X, Zℓ(r))free. We will show that the order of Br (X)(2)nd can be written 2 as a product |(HG)tor|·|D| of the order of (HG)tor and the square of the order of a group D. To prove that |(HG)tor| is a square, we construct a non-degenerate alternating bilinear form h , i4 : (HG)tor × (HG)tor → Qℓ/Zℓ which again is induced by the cup product using auxiliary bilinear forms h , i1, h , i2 and h , i3. We construct these bilinear forms in the ﬁrst part of this section. To show that the bilinear form h , i4 is alternating, we need to exhibit a cohomology class with certain properties, which is done in the second part of this section. Again we start with the cup product 2r 2r 4r ∼ ∪ : Het (X, Zℓ(r)) × Het (X, Zℓ(r)) → Het (X, Zℓ(2r)) = Zℓ which by Poincar´eduality is a skew-symmetric unimodular bilinear form. Pass- ing to the quotient H, we obtain the skew-symmetric unimodular bilinear form

h , i1 : H × H → Zℓ, hx, yi1 = x ∪ y which in turn induces the bilinear form

h , i2 : H ⊗ (Qℓ/Zℓ) × H → Qℓ/Zℓ, (x ⊗ q,y) 7→ hx, yi1 ⊗ q . From the well-deﬁned bilinear form G ( , ):(H ⊗ Qℓ/Zℓ) × HG → Qℓ/Zℓ, (x, [y]) = hx, yi2 we obtain G h , i3 : ((H ⊗ Qℓ/Zℓ) )nd × (HG)tor → Qℓ/Zℓ, h[x],yi3 = hx, yi2 which we rewrite as −1 h , i4 :(HG)tor × (HG)tor → Qℓ/Zℓ, (x, y) 7→ hδ (x),yi3

G using the isomorphism δ : ((H ⊗Qℓ/Zℓ) )nd → (HG)tor of the following lemma. THE ORDER OF HIGHER BRAUER GROUPS 7

G Lemma 4.1. There is an isomorphism δ : ((H ⊗ Qℓ/Zℓ) )nd → (HG)tor.

Proof. Since H is torsion free, tensoring 0 → Zℓ → Qℓ → Qℓ/Zℓ → 0 with H yields the following commutative diagram with exact rows

/ ϕ / ψ / / 0 H H ⊗ Qℓ H ⊗ Qℓ/Zℓ 0 σ−1 σ−1 σ−1 / ϕ / ψ / / 0 H H ⊗ Qℓ H ⊗ Qℓ/Zℓ 0

G G δ and the exact sequence (H ⊗ Qℓ) → (H ⊗ Qℓ/Zℓ) → HG → (H ⊗ Qℓ)G. G Since H ⊗ Qℓ/Zℓ is torsion, the image of δ equals (HG)tor, and since (H ⊗ Qℓ) G G G is divisible, the image of ψ :(H ⊗Qℓ) → (H ⊗Qℓ/Zℓ) equals (H ⊗Qℓ/Zℓ)div. Thus the boundary map δ induces an isomorphism δ.

We still have to prove that h , i4 is non-degenerate and alternating.

Lemma 4.2. The bilinear forms h , i2, ( , ), h , i3 and h , i4 are non-degenerate. Proof. Consider the commutative diagram with exact rows

/ / / / 0 / H / H ⊗ Qℓ / H ⊗ Qℓ/Zℓ / 0 / / / 0 Hom(H, Zℓ) Hom(H, Qℓ) Hom(H, Qℓ/Zℓ) where the outer vertical homomorphisms are induced by the bilinear forms h , i1 and h , i2 respectively, and the middle vertical homomorphism is induced by 4r the bilinear form (H ⊗ Qℓ) × H → Het (X, Zℓ(2r)) ⊗ Qℓ,(x ⊗ q,y) 7→ hx, yi1 ⊗ q that is unimodular. Since h , i1 is unimodular, the ﬁrst vertical map is an isomorphism. Thus it follows from the snake lemma that the last vertical homomorphism is injective, i.e. h , i2 is non-degenerate. G For ( , ) assume x ∈ (H ⊗ Qℓ/Zℓ) such that (x, [y]) = 0 for all [y] ∈ HG. Then hx, yi2 = 0 for all y ∈ H, and since h , i2 is non-degenerate, we have x = 0. Thus ( , ) is non-degenerate as well. G Finally, let y ∈ (HG)tor with h[x],yi3 = 0 for every [x] ∈ ((H ⊗ Qℓ/Zℓ) )nd. This means (x, y) = 0 for all x ∈ H⊗Qℓ/Zℓ. Hence, y = 0 and thus h , i3 is non- degenerate. The non-degeneracy of h , i4 follows immediately from this.

Proposition 4.3. The form h , i4 is alternating.

Proof. We have to verify that hz, zi4 = 0 for each z ∈ (HG)tor. We show first that to prove hz, zi4 = 0 it suffices to show that the cup product of two particular classes lies in 2Zℓ; for this we reverse the construction of h , i4. G First, we find a y ∈ (H ⊗Qℓ/Zℓ) such that δ(y)= z and therefore hz, zi4 = hy, zi3. Second, we find a representative w of z in H to obtain hy, zi3 = hy,wi2. Third, we use h , i1 to define the bilinear form h , i5 :(H ⊗Qℓ)×(H ⊗Qℓ) → Qℓ, hu ⊗ p, v ⊗ qi5 = hu, vi1 ⊗ pq and show by a short computation that hy,wi2 = hx, (σ − 1)(x)i5 + Zℓ in Qℓ/Zℓ for a x ∈ H ⊗ Qℓ such that ψ(x)= y. Hence in order to prove hz, zi4 = 0 it is sufficient to show that hx, (σ − 1)(x)i5 ∈ Zℓ. 8 THOMAS JAHN

Because of hw,wi1 = h(σ − 1)x, (σ − 1)xi5 = −2 ·hx, (σ − 1)xi5 it is even enough to prove hw,wi1 ∈ 2Zℓ. For ℓ =6 2 this follows from Zℓ = 2Zℓ and we are left with the case ℓ = 2. 2r G By lemma 4.4 below there exists an element ωr ∈ Het (X, Z2(r)) such that ωr ∪ w + w ∪ w ∈ 2Z2. Since hw,wi1 = w ∪ w, it suﬃces to prove ωr ∪ w = 0. Let π : H → HG = H/(σ−1)H denote the canonical projection and consider −1 −1 the preimage π ((HG)tor)=((σ − 1)H ⊗ Qℓ) ∩ H. We have w ∈ π ((HG)tor) since π(w) = z ∈ (HG)tor. Because of the orthogonality of (σ − 1)H and 2r G 2r Het (X, Z2(r)) , considered as subgroups of Het (X, Z2(r)), we also have the −1 2r G orthogonality of π ((HG)tor)=((σ − 1)H ⊗ Qℓ) ∩ H and Het (X, Z2(r)) . In particular ωr ∪ w = 0.

We still have to show the existence of ωr which is done in the following lemma; a similar result has been stated and used in a diﬀerent context in [2].

2r G Lemma 4.4. For every integer r ≥ 0 there exists a ωr ∈ Het (X, Z2(r)) such 2r that ωr ∪ x + x ∪ x ∈ 2Z2 for each x ∈ Het (X, Z2(r)).

Proof. The group 2Z2 is the intersection of the kernel of the canonical reduction 2i 2i 4r homomorphism ε : i Het(X, Z2(i)) → i Het(X, Z/2Z) and Het (X, Z2(2r)). 2r Thus we look for a PG-invariant element Pωr ∈ Het (X, Z2(r)) such that we have 2r the equation ε(ωr) ∪ ε(x)+ ε(x) ∪ ε(x) = 0 for every x ∈ Het (X, Z2(r)). Since this is an equation in Z/2Z, it suffices to show ε(ωr) ∪ ε(x)= ε(x) ∪ ε(x). Given a scheme Z and a closed subscheme Y , we consider the cohomology m n groups with support and the Steenrod operations Sq : Het,Y (Z, Z/2Z) → m+n Het,Y (Z, Z/2Z); for the construction and properties of we refer to [3]. In 2r 2r particular, if y ∈ Het (X, Z/2Z) we have the formula Sq (y) = y ∪ y. More- 2r over, by Poincaréduality there is a class v2r ∈ Het (X, Z/2Z) such that the 2r 2r maps Sq (−) and v2r ∪ − coincide as homomorphisms from Het (X, Z/2Z) to 4r Het (X, Z/2Z). Thus we only need to find a G-invariant ε-preimage ωr of v2r. To obtain this preimage we need some more preparation. If E is a vector bundle on X with total space T , we have in each degree i a canonical isomorphism ϕ : Hi (X, Z/2Z) → Hi+2r(T, Z/2Z). Write 1 for the generator of et et,X 0 ∼ −1 i r Het(X, Z/2Z) = Z/2Z and set wi(E) = ϕ (Sq (ϕ(1))) ∈ Het(X, Z/2Z). Let 2k ck(E) be the k-th Chern class of E and let ε(ck(E)) be its class in Het (X, Z/2Z). It is known that wi(E) = 0 in odd degrees i and that w2k(E) = ε(ck(E)) oth- erwise, see [15, Proposition 2.8]. It follows that polynomials in the wk(N ), where N = NX/X×FX is the normal bundle, have a G-invariant ε-preimage. Thus our goal is to show that v2r can in fact be written as such a polynomial. The class w2r is related to the classes vi by the Wu formula w2r(N ) = r 2s 0 s=0 Sq (v2r−2s) [15, Theorem 0.5]. Since Sq is the identity map, v2r can be r 2s writtenP as w2r(N )+ s=1 Sq (v2r−2s) and one can compute v2r recursively. We need two moreP equations. First, the Cartan formula Sqj(x ∪ y) = j i j−i i=0 Sq (x) ∪ Sq (y) [3] which allows us to only consider Steenrod homo- Pmorphisms on a class wi(E) rather than products of those. Second, we have THE ORDER OF HIGHER BRAUER GROUPS 9 for each vector bundle E and every pair of natural numbers i, j the equation j j j − i (7) Sq w (E)= w − (E) ∪ w (E) i k j k i+k Xk=0 n n(n−1)···(n−k+1) (cf. [10, Problem 8-A]), where k denotes the class of k! in Z/2Z. This last equation can be shown by induction on the rank of the vector bundle as follows: Given that E is a line bundle the equation is easily checked using elementary properties of Steenrod homomorphisms (see [3]) and the classes ′ ′′ wi(E) (see [15]). For an exact sequence 0 → E →E→E → 0 of vector i ′ ′′ bundles it is known that wi(E) = k=0 wk(E ) ∪ wi−k(E ) (see again [15]). A straightforward computation involvingP this shows that if the equation (7) is satisfied for E ′ and E ′′, it also is satisfied for E. Now the assertion follows from the splitting principle (cf. [6], [4, p. 51]).

Remark 4.5. In the case considered by Urabe (i.e. r = 1) our element v2 2 equals w2(N ) + Sq (v0) = ε(c1(N )). Thus, the ﬁrst Chern class c1(N ) of the normal bundle is a canonical choice for our element ω1. Urabe proved in [15, Proposition 2.1] that the image of the canonical line bundle KX ∈ Pic(X) under 1 2 the cycle map Pic(X) = Het(X, Gm) → Het(X, Z2(1)) deﬁnes an element ω1 with the desired property. In particular, the element ω1 considered by him and our ω1 constructed here coincide up to a sign.

5. Proof of the theorem

0 2r G (Het (X, Qℓ/Zℓ(r))div) α / 2r−1 ρ˜ / 2r π˜ / 2r G / 0 Het (X, Qℓ/Zℓ(r))G Het (X, Qℓ/Zℓ(r)) Het (X, Qℓ/Zℓ(r)) 0 ′ ′′ δ δ δ / 2r ρ / 2r+1 π / 2r+1 G / 0 Het (X, Zℓ(r))G Het (X, Zℓ(r)) Het (X, Zℓ(r)) 0 β 2r (Het (X, Zℓ(r))free)G 0 Lemma 5.1. The above diagram is commutative with exact rows and columns. Proof. Both rows are induced from the Hochschild-Serre spectral sequence (with divisible and ℓ-adic coeﬃcients) and are exact. The outer columns are G exact without applying the functors •G and • and these functors are right- 10 THOMAS JAHN and left-exact, respectively. The right square is commutative since the homomorphism π andπ ˜ are induced by the covering X → X. For the commutativity of the left square, we consider the Hochschild-Serre p,q p q ⊗r p+q ⊗r spectral sequence E2 (⋆) = Het(G, Het(X,µℓ⋆ )) ⇒ Het (X,µℓ⋆ ). The con- ⊗r ⊗r ⊗r necting homomorphism associated with 0 → µℓm → µℓm+n → µℓn → 0 induces a homomorphism lim E1,2r−1(n) → lim E1,2r(m) which coincidences with δ′. −→n 2 ←−m 2 Similarly, there exists a homomorphism lim E2r(n) → lim E2r+1(m) that −→n 0 ←−m 0 agrees with δ. Since the cohomological dimension being at most one and thus p,q E2 (⋆)=0 for p =06 , 1 we gain the compositions (we omit (⋆) for a moment) ∼ q−1 ⊗r 1,q−1 1,q−1 ∼ q q = q q q ⊗r Het (X,µℓ⋆ )G = E2 = E∞ = E1 /E2 ← E1 ⊆ E0 = Het(X,µℓ⋆ ) . For q = 2r and q = 2r + 1 these compositions are the horizontal maps of the commutative diagram 1,2r−1 / 2r (8) E2 (n) E0 (n) 1,2r / 2r+1 E2 (m) E0 (m) . Application of the direct limit over all n to this composition with q = 2r 2r−1 2r induces a homomorphism Het (X, Qℓ/Zℓ(r))G → Het (X, Qℓ/Zℓ(r)) that coincidences withρ ˜. Similarly, we get a homomorphism that equals ρ when applying an inverse limit over all m with q =2r + 1. We can therefore deduce the commutativity of the left square from the diagram (8) above. We deﬁne the two groups

2r ρ 2r+1 2r+1 C := im Het (X, Z2(r))G → Het (X, Z2(r)) ∩ Het (X, Z2(r))tor and 2r−1 ρ˜ 2r 2r D := im Het (X, Q2/Z2(r))G → Het (X, Q2/Z2(r)) → Het (X, Q2/Z2(r))nd r which we consider as subgroups of B = Br (X)(2)nd in view of lemma 2.1. It follows directly from the above diagram that D ⊆ C ⊆ B. ∼ Lemma 5.2. There is an isomorphism C/D = (HG)tor. −1 ρ 2r Proof. The composition ϕ : C → im(ρ) → Het (X, Z2(r))G → HG has kernel D and image (HG)tor and therefore induces the desired isomorphism. The assertions about kernel and cokernel follow from the diagram at the beginning of this section. We are now ready to prove the following theorem which in addition with proposition 3.3 implies theorem 1.1. r Theorem 5.3. If char F =26 , the order of Br (X)(2)nd is a square. Proof. The group B is ﬁnite as we have seen in the proof of proposition 3.3. By proposition 3.2 there exists a non-degenerate bilinear form on B; this form induces a non-degenerate bilinear form C×(B/D) → Qℓ/Zℓ (recall the diagram before lemma 5.1) which implies |C| = |B/D| = |B|/|D|. Therefore, the order of B equals the product |C/D|·|D|2. By lemma 5.2 the ﬁrst factor is the order of (HG)tor and thus a square by proposition 4.3. THE ORDER OF HIGHER BRAUER GROUPS 11

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