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[Math.NT] 28 Feb 2021 Unramified Cohomology of Quadrics in Characteristic

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[Math.NT] 28 Feb 2021 Unramified Cohomology of Quadrics in Characteristic Unramified Cohomology of Quadrics in Characteristic Two Yong HU and Peng SUN Abstract Let F be a field of characteristic 2 and let X be a smooth projective quadric of dimension ≥ 1 over F . We study the unramified cohomology groups with 2- primary torsion coefficients of X in degrees 2 and 3. We determine completely the kernel and the cokernel of the natural map from the cohomology of F to the unramified cohomology of X. This extends the results in characteristic different from 2 obtained by Kahn, Rost and Sujatha in the nineteen-nineties. Key words: Quadratic forms, quadrics, unramified cohomology, cycle class map MSC classification 2020: 11E04, 14F20, 19E15 1 Introduction Let F be a field. Let m be a positive integer not divisible by the characteristic of ⊗(j−1) F . For each j ≥ 1, the tensor product Z/m(j − 1) := µm of m-th roots of unity can be viewed as ´etale sheaves on F -schemes. Let X be a proper smooth connected j variety over F . The unramified cohomology group Hnr X, Z/m(j − 1) is the group H0 X, H j(j − 1) , where H j(j − 1) denotes the Zariski sheaf associated to the Zar m m presheaf U 7→ Hj (U, Z/m(j − 1)). By taking the direct limit, we can also define ´et Hj X, (Q/Z)′(j − 1) := lim Hj X, Z/m(j − 1) . nr −→ nr char(F )∤m These groups can also be described in terms of residue maps in Galois cohomology, arXiv:2103.00426v1 [math.NT] 28 Feb 2021 thanks to the Bloch–Ogus theorem on Gersten’s conjecture ([BO74]). As important birational invariants, they have found many important applications, for instance to the rationality problem (see e.g. [Sal84], [CTO89], [CTP16]), and have been extensively studied in the literature. With the development of the motivic cohomology theory (by Beilinson, Lichtenbaum, Suslin, Voevodsky, et al.), even more machinery can be applied to compute unramified cohomology nowadays. When F has characteristic different from 2 and X is a smooth projective quadric, the above unramified cohomology groups are computed up to degree j ≤ 4 by Kahn, Rost and Sujatha in a series of papers ([Kah95], [KRS98], [KS00], [KS01]). Some of their results are further developed and used by Izhboldin [Izh01] to solve a number of problems 1 on quadratic forms, including a construction of fields of u-invariant 9 in characteristic =6 2. It has been noticed for decades that unramified cohomology theory can be formulated in a more general setting (see [CT95], [CTHK97], [Kah04]). In particular, when F has positive characteristic p, the aforementioned groups have p-primary torsion variants. j r Indeed, the unramified cohomology functors Hnr · , Z/p (j − 1) for all r ≥ 1 and their limit Hj · , Q /Z (j − 1) can be defined by using the Hodge–Witt cohomology (see nr p p (3.1) and (3.10)). In contrast to the prime-to-p case, there has been much less work on these p-primary unramified cohomology groups. In this paper, we are interested in the case of a smooth projective quadric X over a field F of characteristic 2. We investigate the unramified cohomology groups via the natural maps j j r j r ηr : H F, Z/2 (j − 1) −→ Hnr X, Z/2 (j − 1) , r ≥ 1 and j j j η∞ : H F, Q2/Z2(j − 1) −→ Hnr X, Q2/Z2(j − 1) . j For each j ≥ 1, it is not difficult to see that the maps ηr for different r ≥ 1 have j j essentially the same behavior (Lemma 4.3). So we may focus on the two maps η := η1 j and η∞. They are both isomorphisms if j = 1 (Prop. 3.4) or X is isotropic (Prop. 4.1 (1)). In our main results, we determine completely the kernel and the cokernel of ηj and j η∞ for j =2, 3. The following theorem extends Kahn’s results in [Kah95] to characteristic 2. Theorem 1.1. Let F be a field of characteristic 2 and let X be the smooth projective quadric defined by a nondegenerate quadratic form ϕ with dim ϕ ≥ 3. Assume that ϕ is anisotropic. 1. (See 5.1, 5.3, 5.6 and 5.7) Suppose dim ϕ =3, so that X is the conic associated to 1 a quaternion division algebra D, or dim ϕ = 4 and e1(ϕ)=0 ∈ H (F, Z/2), that is, ϕ is similar to the reduced norm of a quaternion division algebra D. Then Coker(η2) ∼= Ker(η2)= {0, (D)} ∼= Z/2 , Coker(η3) ∼= Ker(η3)= {(a) ∪ (D) | a ∈ F ∗} , where (D) denotes the Brauer class of D. 1 2. (See 5.8 and 6.7) If dim ϕ =4 and e1(ϕ) =06 ∈ H (F, Z/2), then Coker(η2) = Ker(η2)=0 , Coker(η3) ∼= Ker(η3) , Ker(η3)= {0}∪{(a) ∪ (b) ∪ (c] | ϕ is similar to a subform of hha, b ; c]]} . 3. (See 6.7, 6.11, 6.12 and 8.4) Suppose dim ϕ > 4. Then Coker(η2) = Ker(η2)=0 and 2 (a) if ϕ is an Albert form (i.e. a 6-dimensional form with trivial Arf invariant), then Coker(η3) ∼= Z/2 and Ker(η3)=0; (b) if ϕ is a neighbor of a 3-Pfister form hha, b ; c]], then Coker(η3) ∼= Ker(η3)= {0, (a) ∪ (b) ∪ (c]} =∼ Z/2 . (c) in all the other cases (e.g. dim ϕ> 8), Coker(η3) = Ker(η3)=0. The counterpart in characteristic =6 2 of the next theorem appeared in [KRS98, Thms. 4 and 5]. Theorem 1.2 (See 4.3, 6.8 and 8.5). Let F be a field of characteristic 2 and let X be the smooth projective quadric defined by a nondegenerate quadratic form ϕ with dim ϕ ≥ 3. j j 2 Then Ker(ηr ) = Ker(η∞) for all r ≥ 1, j ≥ 1, and Coker(η∞)=0. 2 ∼ 3 If ϕ is an anisotropic Albert form, then Coker(η ) = Z/2. Otherwise Coker(η∞)=0. Main tools in our proofs include the Bloch–Ogus and the Hochschild–Serre spectral sequences. A key difference between the p-primary torsion cohomology and the prime- to-p case is the lack of homotopy invariance. This results in the phenomenon that our spectral sequences look different from their analogues in characteristic different from 2. Fortunately, due to vanishing theorems for the local cohomology and the fact that the field F has 2-cohomological dimension at most 1, these spectral sequences still have many vanishing terms. The cycle class maps with finite or divisible coefficients are also studied and used in the paper. In this respect we need information about the structure of Chow groups in low codimension. This information can be found in Karpenko’s work [Kar90] in characteristic =6 2, and recently the paper [HLS21] has provided the corresponding results in characteristic 2. In the above two theorems the case of Albert quadrics is more subtle than the others. In that case we have to utilize more techniques from the algebraic theory of quadratic forms, especially residue maps on Witt groups of discrete valuation fields of characteristic 2 ([Ara18]). Notation and conventions. For any field k, denote by k a separable closure of k. For an algebraic variety Y over k, we write YL = Y ×k L for any field extension L/k, and Y = Y ×k k. We say Y is k-rational if it is integral and birational to the dim Y projective space Pk over k. We say Y is geometrically rational if YL is L-rational for the algebraic closure L of k. M Milnor K-groups of a field k are denoted by Ki (k), i ∈ N. For an abelian group M, we denote by Mtors the subgroup of torsion elements in M. For any positive integer n, we define M[n] and M/n via the exact sequence × 0 −→ M[n] −→ M −→n M −→ M/n −→ 0 . 2 For any scheme X, let Br(X)= H´et(X, Gm) denote its cohomological Brauer group. In the rest of the paper, F denotes a field of characteristic 2. 3 2 Quadrics and their Chow groups (2.1) We recall some basic definitions and facts about quadratic forms in characteristic 2. For general reference we refer to [EKM08]. We work over a field F , which has characteristic 2 according to our convention. Let a ∈ F ∗. We denote by hai the 1-dimensional quadratic form x 7→ ax2, and let [1, a] or hha]] denote the binary quadratic form (x, y) 7→ x2 + xy + ay2. A (quadratic) 1-Pfister form is a binary quadratic form isomorphic to hha]] = [1, a] for some a ∈ F ∗. Let h1, aibil denote the binary bilinear form ((x1, x2), (y1, y2)) 7→ x1y1 + ax2y2. For n ≥ 2, a quadratic form is called an n-Pfister form if it is isomorphic to hha1, ··· , an−1 ; an]] := h1, a1ibil ⊗···⊗h1, an−1ibil ⊗ hhan]] ∗ ∗ for some a1, ··· , an−1 ∈ F and an ∈ F . If λ ∈ F and ϕ is a Pfister form, the scalar multiple λϕ is called a general Pfister form. ∼ For two quadratic forms ϕ and ψ over F , we say ψ is a subform of ϕ if ψ = ϕ|W for some subspace W in the vector space V of ϕ. For n ≥ 2, an n-Pfister neighbor is a subform of dimension > 2n−1 of a general n-Pfister form. 1 We write Iq(F ) or Iq (F ) for the Witt group of even-dimensional nondegenerate n quadratic forms over F .
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