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

Home , Pfister form

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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 .

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 . For n ≥ 2, let Iq (F ) denote the subgroup of Iq(F ) gener- n ated by the n-Pfister forms. For a quadratic form ϕ over F , we will write ϕ ∈ Iq (F ) if n ϕ is nondegenerate, of even dimension, and its Witt class lies in Iq (F ). We also have the Witt ring W (F ) of nondegenerate symmetric bilinear forms over F , in which the classes of even-dimensional forms form an ideal I(F ), called the fundamental ideal. For each n ≥ 1, let In(F ) be the n-th power of the ideal I(F ) and put I0(F ) = n n−1 W (F ). The group Iq(F ) has a W (F )-module structure, and we have Iq (F )= I (F ) · Iq(F ) for all n ≥ 1. The Galois cohomology group H1(F, Z/2) can be identified with F/℘(F ) by Artin– Schreier theory, where ℘ denotes the map x 7→ x2 − x. For any b ∈ F , we write (b ] for its canonical image in F/℘(F )= H1(F, Z/2). The map

1 1 e1 : Iq (F ) −→ H (F, Z/2) ; hha]] 7−→ (a ] is a well deﬁned homomorphism, often called the discriminant or Arf invariant. It is 2 2 well known that e1 is surjective with Ker(e1) = Iq (F ). A 6-dimensional form in Iq (F ) (i.e. a 6-dimensional nondegenerate form with trivial Arf invariant) is called an Albert form. For n ≥ 2, by using the Kato–Milne group Hn F, Z/2(n − 1) (cf. [Kat80], [Mil76]), a generalization of which will be discussed in (3.1), one can also deﬁne a functorial homomorphism (see [Sah72] for n = 2 and [Kat82b] for general n)

n n en : Iq (F ) −→ H F, Z/2(n − 1)

n+1 which is surjective with Ker(en)= Iq (F ) such that

en hha1, ··· , an−1 ; an]] =(a1) ∪···∪ (an−1) ∪ (an ] , 4 ∗ ∗ ∗2 where for any a ∈ F , (a) denotes its canonical image in F /F . (The maps e2 and e3 are more classical, called the Cliﬀord invariant and the Arason invariant respectively.) Note that for all n ≥ 1,

n ∗ (2.1.1) en(ϕ)= en(c.ϕ) for all ϕ ∈ Iq (F ) and all c ∈ F

n+1 because ϕ − cϕ = hh c iiϕ ∈ Iq (F ) = Ker(en). The group H2 F, Z/2(1) may be identiﬁed with the 2-torsion subgroup of the Brauer group Br(F ) of F . For any a ∈ F ∗ and b ∈ F , let (a, b ] be the quaternion algebra generated by two elements i, j subject to the relations

i2 = a , j2 + j = b , ij = ji + i .

Its reduced norm is the 2-Pfister form hha; b]], and its Brauer class is (a)∪(b ], the Clifford invariant of hha; b]]. The plane conic defined by the ternary form hai⊥hhb]] is called the conic associated to the quaternion algebra (a, b ]. For any a, c ∈ F ∗ and b, d ∈ F , the form [1, b+d]⊥a.[1, b]⊥c.[1, d] is an Albert form. Its Clifford invariant is the Brauer class of the biquaternion algebra (a, b] ⊗F (c, d].

(2.2) Now we recall some known facts about Chow groups of projective quadrics (which are valid in arbitrary characteristic). More details can be found in [Kar90, § 2], [EKM08, § 68] and [HLS21, § 5]. Let ϕ be a quadratic form defined on a (finite dimensional) F -vector space V of dimension ≥ 3. Let X = Xϕ be the projective quadric defined by ϕ. It is a closed subvariety in the projective space P(V ). Unless otherwise stated, we always assume ϕ is nondegenerate, which means the projective quadric X = Xϕ is smooth as an algebraic variety over F . In fact, X is a smooth geometrically integral F -variety. It is well known that if K/F is a field extension such that ϕK is isotropic, then XK is K-rational. In particular, X is rational over F . For each i ∈ N, let CHi(X) denote the Chow group of codimension i cycles of X. Let h ∈ CH1(X) be the class of a hyperplane section (i.e., the pullback of the hyperplane class in CH1(P(V ))). Using the intersection pairing as multiplication in the Chow ring ([EKM08, § 57]), we get elements hi ∈ CHi(X) for each i. d−j Set d = dim X. For every integer j ∈ [0, d/2], let ℓj ∈ CH (X) be the class of a j-dimensional linear subspace contained in X. Then, for each 0 ≤ i ≤ d, we have

i d Z.h if 0 ≤ i< 2 , i d CH (X) = CHd−i(X)= Z.ℓd−i if 2 < i ≤ d ,  i d Z.h ⊕ Z.ℓi if i = 2 . If dim X = 2m is even, there are exactly two diﬀerent classes of m-dimensional linear ′ m m subspaces ℓm, ℓm in CH (X) and the sum of these two classes is equal to h .

The next two propositions can be proved in arbitrary characteristic.

5 Proposition 2.3 ([Kah99, Lemma 8.2]). Let X = Xϕ be a smooth projective quadric of even dimension dim X =2m ≥ 2. m Then CH (X) is a trivial Galois module if and only if e1(ϕ)=0. When e1(ϕ) =06 , ′ the Galois action permutes the two classes ℓm and ℓm. Proposition 2.4 ([Kar90, (2.4)]). Let X be the smooth projective quadric deﬁned by an anisotropic form ϕ with dim ϕ ≥ 3.

1. If dim X ≥ 3, then

CH1(X) = CH1(X)Gal(k/k) = CH1(X)= Z.h .

2. If dim X =2 and e1(ϕ) =06 , then

1 1 Gal(k/k) 1 CH (X)= Z.h = CH (X) ⊆ CH (X)= Z.h ⊕ Z.ℓ1 .

3. If dim X =2 and e1(ϕ)=0 is trivial, then

1 1 Gal(k/k) 1 CH (X)= Z.2ℓ1 ⊕ Z.h ⊆ CH (X) = CH (X)= Z.ℓ1 ⊕ Z.h .

4. If dim X =1, then

1 1 1 Gal(k/k) CH (X)= Z.h = Z.2ℓ0 ⊆ CH (X) = CH (X) = Z.ℓ0 .

3 Unramiﬁed cohomology in positive characteristic

Throughout this section, we fix a positive integer r and a field k of characteristic p> 0. i (3.1) For each i ∈ N, let νr(i) = WrΩlog be the i-th logarithmic Hodge–Witt sheaf on r the big étale site of k ([Ill79], [Shi07]). Define Z/p (i) := νr(i)[−i], as an object in the derived category of étale sheaves. This object can also be viewed as an étale motivic complex ([GL00]). For every integer b, we have the cohomology functor on k-schemes

b r b r b−i H · , Z/p (i) := H´et · , Z/p (i) = H´et · , νr(i) .

b b r For short hand, we sometimes write Hpr (·, i) instead of the precise notation H · , Z/p (i) . b r b The Zariski sheaf associated to the presheaf U 7→ H U, Z/p (i) is denoted by H r (i). p For a smooth connected k-variety X, we deﬁne the unramiﬁed cohomology group b r 0 H b Hnr X, Z/p (i) := HZar X, pr (i) .

This group can also be described by using the Cousin complex of X. It is naturally a subgroup of Hb k(X), Z/pr(i) . In the theorem below, we collect some well known results that are needed in this paper.

6 Theorem 3.2. Let X be a smooth connected k-variety.

1. We have the Bloch–Ogus spectral sequence

a, b a H b a+b a+b r (3.2.1) E2 = HZar X, pr (i) =⇒ E = H X, Z/p (i) with

a, b (3.2.2) E2 =0 if b∈{ / i, i +1} , or if a > b = i and

i,i i H i ∼ i r (3.2.3) E2 = HZar X, pr (i) = CH (X)/p . 2. There are natural isomorphisms

i r ∼ 0 i i r H X, Z/p (i) −→ H X, H r (i) = H X, Z/p (i) , (3.2.4) Zar p nr 2i+j r ∼ j+i−1 H i+1 H X, Z/p (i) −→ HZar X, pr (i) for j ≥ 1 , and

2 r 0 H 2 ∼ r (3.2.5) Hnr(X, Z/p (1)) = HZar X, pr (1) = Br(X)[p ] .

b r 3. For smooth proper connected k-varieties, the group Hnr X, Z/p (i) is a k-birational invariant. 4. Let π : X → Y be a proper morphism between smooth connected k-varieties whose generic ﬁber is k(Y )-rational. ∗ b r b r Then the natural map π : Hnr Y, Z/p (i) → Hnr X, Z/p (i) is an isomorphism. Proof. (1) These are standard consequences of Gersten’s conjecture for smooth varieties ([CTHK97, Cor. 5.1.11 and § 7.4 (3)], [Shi07, Thm. 4.1]) and the Bloch–Kato–Gabber theorem ([GS17, Thm. 9.5.2]). The nontrivial part of (3.2.2) is proved in [Gro85, (3.5.3)] and [Shi07, Cor. 3.4]. (3) The isomorphisms in (3.2.4) follow easily from the vanishing results in (3.2.2). The isomorphism (3.2.5) is a consequence of purity for Brauer groups ([C19ˇ , Thm. 1.2]). (4) See [CTHK97, Thm. 8.5.1]. (5) See [CTHK97, Thm. 8.6.1]. (3.3) Let X be a smooth connected k-variety. For every j ≥ 1, we have a natural restriction map

j j r j r (3.3.1) ηr : H k, Z/p (j − 1) −→ Hnr X, Z/p (j − 1) . Let i ∈ N be such that i ≤j. We have the cup product map

j−i r i H i ∪ i H j (3.3.2) Hnr X, Z/p (j − i) ⊗ HZar X, pr (i) −→ HZar X, pr (j) 7 and a natural map j−i r i r j−i r i H i (3.3.3) H k , Z/p (j−i) ⊗CH (X)/p −→ Hnr X, Z/p (j−i) ⊗HZar X, pr (i) i r i H i induced from the identification CH (X)/p = HZar X, pr (i) in (3.2.3 ) and the natural map j−i r 0 H j−i j−i r H k, Z/p (j − i) −→ HZar X, pr (j − i) = Hnr X, Z/p (j − i) . Composing (3.3.2) with (3.3.3 ) yields a natural map i,j j−i r i r i H j (3.3.4) µr : H k , Z/p (j − i) ⊗ CH (X)/p −→ HZar X, pr (j) . For i = 0 this coincides with the natural map j j r j r j r η˜r : H k , Z/p (j) → Hnr X, Z/p (j) = H X, Z/p (j) . Proposition 3.4. With notation as in (3.3 ), suppose that X is proper and geometrically rational. 1 1 r 1 r Then the natural map ηr : H (k, Z/p ) → Hnr X, Z/p is an isomorphism. 1 r 1 r Proof. By (3.2.4), we have Hnr(X, Z/p ) = H (X, Z/p ). The geometric rationality of X implies that X = X ×k k (where k denotes a separable closure of k) has trivial étale fundamental group. By the homotopy exact sequence for étale fundamental groups ∼ ([Gro03, (IX.6.1)]) we get π1(X) = Gal(k/k). Hence the natural map 1 r r 1 r r H (k, Z/p ) = Hom Gal(k/k) , Z/p −→ H (X, Z/p ) = Hom(π1(X) , Z/p ) is an isomorphism. Proposition 3.5. Let X be a smooth, proper, k-rational variety. Then for every j ≥ 1, the map j j r j r ηr : H k, Z/p (j − 1) −→ Hnr X, Z/p (j − 1) is an isomorphism. Proof. This is a special case of Theorem 3.2 (4). (3.6) Let X be a smooth connected k-variety. The Bloch–Ogus spectral sequence (3.2.1) is concentrated in two horizontal lines (cf. (3.2.2)). So we can obtain natural homomorphisms a a, i a H i i+a i+a r e (i): E2 = HZar X, pr (i) −→ E = H X, Z/p (i) , 1 ≤ a ≤ i which fit into a long exact sequence 1 1,i e (i) i+1 0,i+1 0 H i+1 0 −→ E2 −−→ E −→ E2 = HZar X, pr (i) 2 d 2,i e (i) i+2 1,i+1 1 H i+1 −→ E2 −−→ E −→ E2 = HZarX, pr (i) →··· ·················· (3.6.1) a d a, i e (i) i+a a−1,i+1 a−1 H i+1 −→ E2 −−→ E −→ E2 = HZar X, pr (i) →··· ·················· i d i,i e (i) 2i i−1,i+1 i−1 H i+1 −→ E2 −−→ E −→ E2 = HZar X, pr (i) −→ 0 .

8 In particular, we have a natural map

i i i i,i i r 2i 2i r (3.6.2) clX = cl := e (i): E2 = CH (X)/p −→ E = H (X, Z/p (i)) called the cycle class map. i i,j Let j ∈ N be another integer with j ≥ i. By composing e (j) with the map µr in (3.3.4) we get a natural map

i,j j−i r i r i+j r (3.6.3) νr : H k, Z/p (j − i) ⊗ CH (X)/p −→ H X, Z/p (j) .

Compatibility of the Bloch–Ogus spectral sequence with cup products (cf. [KRS98 , p.868]) gives the commutative diagram (3.6.4) Hj−i k, Z/pr(j − i) ⊗ CHi(X)/pr ❃ ◗ ♠♠♠ ✁ ❃ ◗◗◗ ♠♠♠ ✁✁ ❃❃ ◗◗◗ j−i ♠♠ ✁ ❃ ◗◗ j−i i η˜ ⊗Id ♠♠ ✁ ❃ ◗◗η˜r ⊗cl r ♠♠♠ ✁✁ ❃❃ ◗◗◗ X ♠♠♠ ✁ ❃ ◗◗◗ ♠♠♠ ✁✁ ❃❃ ◗◗◗ ♠♠♠ ✁✁ ❃❃ ◗◗◗ ♠♠♠ ✁ ❃ ◗◗◗ ♠♠ ✁ ⊗ i ❃❃ ◗◗ ♠v ♠♠ ✁✁ Id e (i) ❃ ◗◗/( S ✁✁ ❃❃ T ✁✁ ❃❃ ✁ ❃❃ ✁✁ i, j i, j ❃ ✁ µr νr ❃ ✁ ❃❃ ✁✁ ❃ ✁ ❃❃ ✁✁ ❃ ∪ ✁ ❃❃ ∪ ✁✁ ❃❃ ✁✁ ❃❃ ✁✁ ❃❃ ✁✁ ❃❃ ✁✁ ❃❃ ✁Ð i i H j e (j) / i+j r HZar X, pr (j) H X, Z/p (j)

where j−i r i H i S = Hnr X, Z/p (j − i) ⊗ HZar X, pr (i) , j−i r 2i r T = Hnr X, Z/p (j − i) ⊗ H X, Z/p (i) . Proposition 3.7. Let X be a smooth connected k-variety. Then there is an exact sequence 1 1 r clX 2 r 0 −→ CH (X)/p −→ H´et(X, 1) −→ Br(X)[p ] −→ 0 . 1 In particular, the cycle class map clX is injective, and if X is proper and k-rational, we 1 ∼ r have Coker(clX ) = Br(k)[p ]. 2 H 1 Proof. Note that HZar X, (1) =0 by (3.2.2). Thus, taking i = 1 in (3.6.1) and noticing the isomorphism (3.2.5) we get the desired exact sequence. Alternatively, we can use the long cohomoloy sequence for the distinguished triangle

r ×p r +1 Z(1) = Gm[−1] −→ Z(1) −→ Z/p (1) −→ of motivic complexes, which is an analogue of the Kummer sequence in characteristic =6 p.

9 Proposition 3.8. Let X be a smooth proper k-rational variety. Then there is an exact sequence 1 H 2 3 r 3 r 0 −→ HZar X, pr (2) −→ H X, Z/p (2) −→ H k, Z/p (2) (3.8.1) 2 2 r clX 4 r 1 H 3 −→ CH (X)/p −→ H X, Z/p (2) −→ HZar X, pr (2) −→ 0 . In particular, if H3 k, Z/pr(2) = 0 (e.g. if k is separably closed), then the cycle class map 2 2 r 4 r clX : CH (X)/p −→ H X, Z/p (2) is injective. Proof. Taking i = 2 in (3.6.1) we obtain an exact sequence 1 H 2 3 r 3 r 0 −→ HZar X, pr (2) −→ H X, Z/p (2) −→ Hnr X, Z/p (2) (3.8.2) 2 2 r cl 4 r 1 H 3 −→ CH (X)/p −→ H X, Z/p (2) −→ H Zar X, pr (2) −→ 0 . The desired exact sequence follows from (3.8.2), because for the (smooth proper) k- 3 r ∼ 3 r rational variety X we have Hnr X, Z/p (2) = H k, Z/p (2) (Prop. 3.5). (3.9) Let X be a smooth connected k-variety. For each i ∈ Nwe have the Hochschild– Serre spectral sequence (3.9.1) Ha k,Hb(X, Z/pr(i)) =⇒ Ha+b X, Z/pr(i) .

a b r Since cdp(k) ≤ 1, we have H k,H (X, Z/p (i)) = 0 for all a>1. Thus, the spectral sequence (3.9.1) yields an isomorphism (3.9.2) H0 k, Hi(X, Z/pr(i)) =∼ Hi X, Z/pr(i) and an exact sequence 0 −→ H1 k,Hj(X, Z/pr(i)) −→ Hj+1 X, Z/pr(i) (3.9.3) 0 j+1 r −→ H k, H (X, Z/p (i)) −→ 0 for each j ≥ i. Now assume X is proper and geometrically rational (e.g. X is a smooth projective quadric of dimension ≥ 1). Then by (3.2.4) and Thm. 3.2 (4), we have i r ∼ i r i r M r (3.9.4) H X, Z/p (i) = Hnr X, Z/p (i) = H k , Z/p (i) = Ki (k)/p . It is proved in [Izh91, Cor. 6.5] that

i r M r 0 M r H k, Z/p (i) = Ki (k)/p = H k,Ki (k)/p , (3.9.5) i+1 r 1 M r H k, Z/p (i) = H k,Ki (k)/p . Hence, combining (3.9.4 ), (3.9.5) and the case j = i of (3.9.3), we obtain an exact sequence

i+1 i+1 r 0 i+1 r (3.9.6) 0 −→ Hpr (k, i) −→ H X, Z/p (i) −→ H k,H (X, Z/p (i)) −→ 0 . 10 On the other hand, since 0 H i+1 i+1 r i+1 r HZar X, pr (i) = Hnr X, Z/p (i) = H k, Z/p (i) =0 by Thm. 3.2 (4), it follows from (3.6.1) that i+1 r 1 H i (3.9.7) H X, Z/p (i) = HZar X, pr (i) . For i = 0, we can deduce from (3.9.7) and (3.2.2) that H1(X,Z/pr) = 0. Hence (3.9.6) yields H1(k, Z/pr) ∼= H1(X, Z/pr) in this case, recovering the result of Prop. 3.4. For i = 1, from (3.9.7) and (3.2.3) we get

(3.9.8) H2 X, Z/pr(1) ∼= CH1(X)/pr . (This can also be deduced from Prop. 3.7.) So the case i =1of (3.9.6) gives an exact sequence

(3.9.9) 0 −→ H2 k, Z/pr(1) −→ H2 X, Z/pr(1) −→ H0 k , CH1(X)/pr −→ 0 .

Let Ki denote the Zariski sheaf defined by Quillen’s K-theory. Using the Bloch– Kato–Gabber theorem, we can deduce an exact sequence ([GS88, Thm. 4.13]) i+1 K r i−1 H i i r (3.9.10) 0 −→ HZar X, i /p −→ HZar X, pr (i) −→ CH (X)[p ] −→ 0 for X and similarly for X . Now assume further that X is a projective homogeneous variety. Then CHi(X) is torsion-free and i−1 K i−1 HZar X, i = K1(k) ⊗ CH (X) by [Mer95, § 1, Prop. 1]. So, in this case from (3.9.10) we get i−1 H i ∼ i−1 K r ∼ r i−1 r (3.9.11) HZar X, pr (i) = HZar X, i /p = K1(k)/p ⊗ CH (X)/p . For i = 2 we can combine (3.9.11 ) and (3.9.7) to get an isomorphism 3 r ∼ r 1 r (3.9.12) H X, Z/p (2) = K1(k)/p ⊗ CH (X)/p . In particular, this holds when X is a smooth projective quadric of dimension ≥ 1. We end this section with a few remarks on cohomology with divisible coefficients. (3.10) Given integers b, i ∈ N, by taking direct limits we can define the functors Hb · , Q /Z (i) : = lim Hb · , Z/pr(i) , p p −→ r Hb ·, Q /Z (i) : = lim Hb ·, Z/pr(i) . nr p p −→ nr r It is easy to extend all the previous discussions in this section to these cohomology groups with divisible coefficients. In particular, for a smooth connected k-variety X, by taking the limits of (3.3.1) and (3.6.2) we obtain a natural map

j j j η∞ : H k, Qp/Zp(j − 1) −→ Hnr X, Qp/Zp(j − 1) . 11 for each j ≥ 1, and a cycle class map with divisible coeﬃcients

i i 2i cl∞ : CH (X) ⊗ (Qp/Zp) −→ H X, Qp/Zp(i) for each i ≥ 1. A useful standard fact (which follows from [Izh91, Lemma 6.6]) is that the sequence

r j j ×p j (3.10.1) 0 → Hpr (K, j − 1) → H K, Qp/Zp(j − 1) −−→ H K, Qp/Zp(j − 1) → 0 is exact for any ﬁeld extension K/k. As a consequence, we have an identiﬁcation

j j r (3.10.2) Ker(ηr) = Ker(η∞)[p ] .

4 Some general observations for quadrics

From now on, we work over a ﬁeld F of characteristic 2, and let X denote a smooth projective quadric of dimension ≥ 1 over F . Notation and results in § 3 will be applied with p = 2. For each j ≥ 1, we have the natural maps (cf. (3.3.1) and (3.10))

j j j r j r ηr = ηr, X : H F, Z/2 (j − 1) −→ Hnr X, Z/2 (j − 1) , r ≥ 1 and j j j j η∞ = η∞,X : H F, Q2/Z2(j − 1) −→ Hnr X, Q2/Z2(j − 1) .

Observe that the proof of the following proposition is characteristic free.

Proposition 4.1 ([KRS98, Prop. 2.5]). With notation as above, the following statements hold:

j j 1. If X is isotropic over F , then the maps ηr, X and η∞,X are all isomorphisms.

j j 2. In general, the maps ηr, X and η∞,X all have 2-torsion kernel and cokernel. 3. Let Y be another smooth projective quadric of dimension ≥ 1 over F . If X is isotropic over the function ﬁeld F (Y ), then there is a natural commutative diagram

Hj F, Z/2r(j − 1) j ❥ ❚❚ j ηr, X ❥❥❥ ❚❚❚ ηr, Y ❥❥❥❥ ❚❚❚❚ ❥❥❥❥ ❚❚❚❚ ❥t ❥❥❥ ❚❚❚* j r / j r Hnr X, Z/2 (j − 1) ρ Hnr Y, Z/2 (j − 1) If moreover Y is isotropic over F (X), the map ρ in the above diagram is an isomorphism. Similar results hold in the case of divisible coeﬃcients.

12 Corollary 4.2 ([CTS95, Lemma 1.3]). Suppose ϕ is a Pfister form of dimension ≥ 4 and let ψ be a neighbor of ϕ. Let X and Y be the projective quadrics defined by ϕ and ψ respectively. Then for every j ≥ 1 we have isomorphisms j j j ∼ j Ker(ηr, X ) = Ker(ηr, Y ) and Coker(ηr, X ) = Coker(ηr, Y ) , r ≥ 1 , and similarly in the case of divisible coefficients. Proof. Apply Prop. 4.1 (3).

j j Lemma 4.3. For every r ≥ 1 and j ≥ 1, Ker(ηr) = Ker(η∞) and there is an exact sequence j j j 0 −→ Ker(ηr) −→ Coker(ηr ) −→ Coker(η∞) −→ 0 . Proof. The ﬁrst assertion is immediate from (3.10.2) and Prop. 4.1 (2). By functoriality and (3.10.1) we have the commutative diagram with exact rows

r / j r / j ×2 / j / 0 H F, Z/2 (j − 1) H F, Q2/Z2(j − 1) H F, Q2/Z2(j − 1) 0

j j j η η∞ η∞ r r / j r / j ×2 / j 0 Hnr X, Z/2 (j − 1) Hnr X, Q2/Z2(j − 1) Hnr X, Q2/Z2(j − 1) Applying the snake lemma to this diagram yields the desired exact sequence, noticing j that Coker(η∞) is 2-torsion.

j Thanks to the above lemma, when studying the kernel and the cokernel of ηr we may restrict to the case r = 1. The following result is a special case of [Kah99, Prop. 5.2]. 3 ∼ 2 Proposition 4.4. There is a natural isomorphism Coker(η∞) = Ker(cl∞) for the smooth projective quadric X.

5 Results for conic curves

To simplify the notation, we henceforth write b b H b H b H (· , i)= H´et · , Z/2(i) , (i)= 2 (i) , i+1 i+1 i+1 i+1 H (·)= H (· , i) , Hnr (·)= Hnr (· , i) . In this section and the next, we study the maps

j j j j η := η1 : H (F ) −→ Hnr(X) , j j j η∞ : H F, Q2/Z2(j − 1)) −→ Hnr X, Q2/Z2(j − 1) for the quadric X. They are both isomorphisms if Xis isotropic (Prop.4.1 (1)) or j =1 (Prop. 3.4). So we assume X is anisotropic and j ≥ 2 in the sequel.

For a conic curve an explicit description is known for the kernel of the map ηj.

13 2 Proposition 5.1. Let X ⊆ PF be the conic associated to a quaternion F -algebra D. Then for all j ≥ 2,

j j j M Ker η : H (F ) −→ Hnr(X) = {(D) ∪ ξ | ξ ∈ Kj−2(F )} where (D) ∈ H2(F ) = Br(F )[2] denotes the Brauer class of D. j 0 H j j j Proof. Since Hnr(X)= HZar X, (j − 1) is a subgroup of H (F (X)), Ker(η ) coincides with the kernel of the natural map Hj(F ) → Hj(F (X)). The result thus follows from [AJ09, Thm. 3.6]. Remark 5.2. In Prop. 5.1, the case j = 2 amounts to an exact sequence

0 −→ Z/2 −→δ H2(F ) −→ H2(F (X)) where the map δ sends 1 to the Brauer class (D). This is in fact a special case of Amitsur’s theorem (cf. [GS17, Thm. 5.4.1]). For j = 3, the proposition gives a characteristic 2 analogue of a theorem of Arason [Ara75, Satz 5.4], i.e., we have an exact sequence

∪(D) F ∗ −−−→ H3(F ) −→ H3(F (X)) . The kernel of the ﬁrst map is the group Nrd(D∗) of reduced norms of D by [Gil00, p.94, Thm. 6]. So in this case we have an isomorphism F ∗/Nrd(D∗) ∼= Ker(η3). The following result is slightly diﬀerent from its counterpart in characteristic =6 2 (cf. [Kah95, p.246, Remarks (4)]). 2 2 ∼ Proposition 5.3. Let X ⊆ PF be a smooth anisotropic conic. Then Coker(η ) = Z/2. Proof. By (3.2.5), we can identify η2 with the natural map Br(F )[2] → Br(X)[2]. Con- sider the exact sequence

0 → Pic(X) → Pic(X)Gal(F /F ) → Br(F ) −→ Ker(Br(X) → Br(X)) −→ H1(F, Pic(X)) obtained from the Hochschild–Serre spectral sequence

p q p+q H (F, H (X, Gm)) =⇒ H (X, Gm) .

1 1 Note that Br(X) = Br(PF ) = 0, and H (F, Pic(X)) = 0 since Pic(X) = Z as a Galois module. The conic X being anisotropic, the natural map Pic(X) → Pic(X)Gal(F /F ) can be identiﬁed with the inclusion 2Z ֒→ Z. So the above exact sequence gives an exact sequence 0 −→ Z/2 −→ Br(F ) −→ Br(X) −→ 0 . The Brauer group Br(F ) is 2-divisible since char(F )=2 (cf. (3.10.1)). Thus, applying the snake lemma to the diagram 0 / Z/2 / Br(F ) / Br(X) / 0

0 ×2 ×2 0 / Z/2 / Br(F ) / Br(X) / 0

14 we obtain an exact sequence

η2 0 −→ Z/2 −→ Br(F )[2] −→ Br(X)[2] −→ Z/2 −→ 0 proving Coker(η2) ∼= Z/2 as claimed.

2 (5.4) Let X ⊆ PF be a smooth conic. Fix i ∈ N. In the Hochschild–Serre spectral sequence Hp(F, Hq(X, i)) ⇒ Hp+q(X, i) we have

i−1 M H (F , i − 1) = Ki−1(F )/2 if q = i +1 , q q 1 i M H (X, i)= H PF , i = H (F , i)= Ki (F )/2 if q = i ,  0 otherwise by [Gro85, (2.1.15)] (and the Bloch–Kato–Gabber theorem). This combined with (3.9.2), (3.9.3), (3.9.5) and (3.9.6) yields isomorphisms

Hi(F, i)= H0(F, Hi(X, i)) ∼= Hi(X, i) , Hi(F )= H1(F, Hi+1(X)) ∼= Hi+2(X, i) and an exact sequence

ρ (5.4.1) 0 −→ Hi+1(F ) −→ Hi+1(X) −→ Hi−1(F, i − 1) −→ 0 .

Since dim X =1, (3.6.1) gives a short exact sequence 1 H i ι i+1 i+1 (5.4.2) 0 −→ HZar X, (i) −→ H (X) −→ Hnr (X) −→ 0 . Also, by the Gersten resolution of Hi(i) we have

1 H i i i−1 HZar X, (i) = Coker H (F (X), i) −→ H (κ(x) , i − 1) . x∈X(1) M In particular, there is a natural surjection

i−1 1 H i H (κ(x) , i − 1) −→ HZar X, (i) . x∈X(1) M The same arguments as in the proof of [Pey95, p.377, Lemma 2.1] show that the diagram

i−1 / / 1 H i (5.4.3) H (κ(x) , i − 1) HZar X, (i) x∈X(1) L⊕Corκ(x)/F ι ρ Hi−1(F, i − 1) o Hi+1(X) is commutative. Let D be the quaternion F -algebra corresponding to the conic X. For (1) 2 each x ∈ X we have (D)κ(x) = 0 in H (κ(x)) = Br(κ(x))[2]. Therefore,

i−1 Corκ(x)/F (α) ∪ (D) = Corκ(x)/F α ∪ (D)κ(x) = 0 for all α ∈ H (κ(x) , i − 1) . 15 This together with the commutative diagram (5.4.3) shows that the composite map

1 H i ι i+1 ρ i−1 ∪(D) i+1 HZar(X, (i)) −→ H (X) −→ H (F, i − 1) −−−→ H (F ) is 0. Now, using (5.4.2) we can deﬁne

i+1 i+1 (5.4.4) N : Hnr (X) −→ H (F ) to be the unique homomorphism making the following diagram commute:

i+1 / / i+1 H (X) Hnr (X)

ρ N ∪(D) Hi−1(F, i − 1) / Hi+1(F ).

Note that (5.4.2) also yields a homomorphism i+1 i+1 i+1 1 H i τ : Ker(η ) = Ker H (F ) → H (F (X)) −→ HZar X, (i) such that the diagram

i+1 / i+1 / i+1 η / i+1 0 Ker(η ) H (F ) Hnr (X)

τ / 1 H i ι / i+1 / i+1 / 0 HZar X, (i) H (X) Hnr (X) 0 is commutative with exact rows. Now we have the following complex

i+1 τ 1 H i ρ◦ι i−1 0 −→ Ker(η ) −→ HZar X, (i) −→ H (F, i − 1) (5.4.5) i+1 ′ ∪(D) i+1 η i+1 N i−1 −−−→ H (F ) −−→ Hnr(X) −→ H (F, i − 1) ∪ (D) −→ 0 where ∪(D) Hi−1(F, i − 1) ∪ (D) := Im Hi−1(F, i − 1) −−−→ Hi+1(F ) and N ′ is induced by the map N in (5.4.4).

We can now prove a characteristic 2 counterpart of [Pey95, p.379, Prop. 2.2]. Proposition 5.5. The complex (5.4.5) is exact except possibly at the third term Hi−1(F, i− i+1 1) and the ﬁfth term Hnr (X). Moreover,

′ Ker(N ) ∼ Ker(∪(D)) Ker(∪(D)) i+1 = = Im(η ) Im(ρ ◦ ι) i−1 i−1 Im ⊕Corκ(x)/F : H (κ(x), i − 1) → H (F, i − 1) x∈X(1) L If i ≤ 2, then (5.4.5) is exact everywhere.

16 Proof. The injectivity of τ is a consequence of the injectivity of Hi+1(F ) → Hi+1(X), and the surjectivity of N ′ follows from that of the map ρ (cf. (5.4.1)). The equality Im(τ) = Ker(ρ◦ι) can be easily shown by a diagram chase, and the equality Im(∪(D)) = Ker(ηi+1) was Prop. 5.1. Ker(N ′) ∼ Ker(∪(D)) To get the isomorphism Im(ηi+1) = Im(ρ◦ι) , it suﬃces to apply the snake lemma to the following commutative diagram

/ 1 H i ι / i+1 / i+1 / 0 HZar X, (i) H (X) Hnr (X) 0

′ (ρ◦ι) ρ N ′ ∪(D) 0 / Ker(∪(D)) / Hi−1(F, i − 1) / Hi−1(F, i − 1) ∪ (D) / 0.

From the diagram (5.4.3) we ﬁnd

i−1 i−1 Im(ρ ◦ ι) = Im ⊕Corκ(x)/F : H (κ(x), i − 1) → H (F, i − 1) . (1) x∈MX If i = 0, then Hi−1(F, i − 1)=0. If i = 1, then

Im(ρ ◦ ι) = Im(deg : CH1(X)/2 −→ Z/2)

∪(D) and it is equal to Ker Z/2 −−−→ H2(F ) . If i = 2, the equality Im(ρ ◦ ι) = Ker(∪(D)) follows from [Gil00, p.94, Thm. 6]. 2 Corollary 5.6. Let X ⊆ PF be a smooth conic with associated quaternion algebra D. Then we have isomorphisms

Coker(η3) ∼= H1(F, 1) ∪ (D) ∼= F ∗/Nrd(D∗) ∼= Ker(η3) .

Proof. The ﬁrst isomorphism follows from the i = 2 case of Prop. 5.5. The other isomorphisms have been discussed in Remark 5.2.

Corollary 5.7. Suppose ϕ is the reduced norm form of a quaternion division algebra D. Then for the quadric X = Xϕ we have isomorphisms

Ker(ηj)= Hj−2(F, j − 2) ∪ (D) , for all j ≥ 2 , Ker(η2) ∼= Coker(η2) ∼= Z/2 , Ker(η3) ∼= Coker(η3) ∼= H1(F, 1) ∪ (D) ∼= F ∗/Nrd(D∗) .

Proof. This follows from Cor. 4.2 and the corresponding results for conics (Props. 5.2, 5.3 and Cor. 5.6). Remark 5.8. Suppose 3 ≤ dim ϕ ≤ 4. Then the cycle class map with divisible coeﬃ- cients 2 2 4 cl∞ : CH (X) ⊗ (Q2/Z2) −→ H X, Q2/Z2(2) 3 3 3 is injective and the map η∞ : H F, Q2/Z2(2) → Hnr X, Q2/Z2(2) is surjective. 17 2 3 By Prop. 4.4, if one of the two groups Ker(cl∞) and Coker(η∞) is trivial, then so is 2 2 the other. If dim ϕ = 3, then dim X = 1 and CH (X) = 0, so trivially Ker(cl∞)=0. If dim ϕ = 4, as in the proof of Cor. 5.7 we may reduce to the case where ϕ is a 2-Pﬁster 3 form. Then we can apply Cor. 4.2 to obtain Coker(η∞) = 0 by passing to the case of conics. As a corollary, we have an isomorphism Ker(η3) ∼= Coker(η3) in view of Lemma 4.3. In fact, it is shown in [HLS21, Thm. 5.6] that

Ker(η3)= {0}∪{(a) ∪ (b) ∪ (c] | a, b, c ∈ F ∗, ϕ is similar to a subform of hha, b ; c]]} .

6 Quadrics of dimension ≥ 2

In this section, we prove our main results about low degree unramified cohomology for quadrics of dimension at least 2. These extend the main theorems of [Kah95] to characteristic 2. Throughout this section, let ϕ be a nondegenerate quadratic form of dimension ≥ 3 over the field F and let X = Xϕ be the projective quadric defined by ϕ. Lemma 6.1. For each i ∈ N, let ξi : CHi(X)/2 → H0(F, CHi(X)/2) be the canonical map. Assume ϕ is anisotropic.

1 ∼ 1 ∼ 1. If dim ϕ =4 and e1(ϕ)=0, then Ker(ξ ) = Coker(ξ ) = Z/2. Otherwise ξ1 is an isomorphism.

2. Suppose 4 ≤ dim ϕ ≤ 5. Then ξ2 =0, Coker(ξ2) ∼= Z/2 and

(Z/2)2 if dim ϕ =5 and ϕ is a Pﬁster neighbor , Ker(ξ2) ∼= (Z/2 otherwise .

3. Suppose dim ϕ ≥ 6.

(a) If ϕ is an Albert form, then Ker(ξ2) ∼= Coker(ξ2) ∼= Z/2. (b) If ϕ is a neighbor of a 3-Pﬁster form, then Ker(ξ2) ∼= Z/2 and Coker(ξ2)=0. (c) In all the other cases, ξ2 is an isomorphism.

1 Proof. (1) We use Prop. 2.4. If dim ϕ = 4 and e1(ϕ) = 0, then ξ can be identiﬁed with the map (0, Id) (2Z/4Z).ℓ1 ⊕ (Z/2Z)h −−−→ (Z/2Z).ℓ1 ⊕ (Z/2Z).h . 1 ′ If dim ϕ = 4 and e1(ϕ) =6 0, then the Galois action on CH (X)= Z.h⊕Z.ℓ = Z.ℓ1 ⊕Z.ℓ1 ′ 1 permutes the two line classes ℓ1 and ℓ1 (Prop. 2.3). Thus ξ can be identiﬁed with the natural isomorphism

∼ ′ (Z/2Z).h −→ (Z/2Z).h =(Z/2Z).(ℓ1 + ℓ1) .

18 If dim ϕ > 4, then CH1(X) = CH1(X) = Z.h and ξ1 is clearly the identity map on (Z/2Z).h. 2 2 2 Gal(F /F ) 2 (2) If dim ϕ = 4, then CH (X)= Z.2ℓ0 ⊆ CH (X) = CH (X) = Z.ℓ0, and ξ is the zero map from 2Z/4Z to Z/2Z. Now suppose dim ϕ = 5. We have

2 2 2 2 Gal(F /F ) CH (X) = CH (X)tors ⊕ Z.2ℓ1 and CH (X) = CH (X) = Z.ℓ1 .

The map ξ2 is thus therefore the zero map. By [Kar90, (5.3)] (see [HLS21, Thm. 5.1] in 2 ∼ 2 characteristic 2), CH (X)tors = Z/2 if ϕ is a Pﬁster neighbor, otherwise CH (X)tors = 0. In the former case, CH2(X)/2 ∼= (Z/2)2 and H0(F, CH2(X)/2) = Z/2. In the other case, CH2(X)/2 ∼= Z/2 ∼= H0(F, CH2(X)/2). (3) We will use the results in [Kar90, (5.5)] (see also [HLS21, Thm. 5.2]) if dim ϕ =6 and [HLS21, Thm. 5.3] if dim ϕ ≥ 7. If ϕ is an Albert form, we have

2 2 2 2 Gal(F /F ) 2 CH (X)= Z.h ⊕ Z.4ℓ2 ⊆ CH (X) = CH (X) = Z.h ⊕ Z.ℓ2 .

In this case, ξ2 can be identiﬁed with the natural map

2 (Id, 0) 2 (Z/2Z)h ⊕ (4Z/8Z)ℓ2 −−−→ (Z/2Z)h ⊕ (Z/2Z)ℓ2 .

If ϕ is a 6-dimensional Pﬁster neighbor, we have

2 2 ′ 2 ′2 CH (X) = CH (X)tors ⊕ Z.(ℓ2 + ℓ2) and CH (X)= Z.ℓ ⊕ Z.ℓ2 .

2 ′ 0 2 The Galois action on CH (X) permutes ℓ2 and ℓ2. In this case, H (F, CH (X)/2) = ′ 2 (Z/2Z).(ℓ2 + ℓ2) and ξ can be identiﬁed with the map

′ 0⊕Id ′ Z/2Z ⊕ (Z/2Z).(ℓ2 + ℓ2) −−→ (Z/2Z).(ℓ2 + ℓ2) . If ϕ is a 7 or 8 dimensional Pﬁster neighbor, then

2 2 2 ∼ 2 2 Gal(F /F ) 2 CH (X) = CH (X)tors ⊕ Z.h = Z/2Z ⊕ Z and CH (X) = CH (X) = Z.h .

2 2 2 The map ξ is the identity on (Z/2Z).h and 0 on CH (X)tors. If dim ϕ = 6 and ϕ is neither an Albert form nor a Pﬁster neighbor, we have

2 2 ′ 2 Gal(F /F ) 2 ′ CH (X)= Z.h = Z.(ℓ2 + ℓ2) = CH (X) ⊆ CH (X)= Z.ℓ2 ⊕ Z.ℓ2 .

2 ′ 0 2 Now the map ξ is the identity map on (Z/2Z).(ℓ2 + ℓ2)= H (F, CH (X)/2). In all the remaining case, we have CH2(X)= Z.h2 = CH2(X) and ξ2 is the identity on (Z/2).h. Recall that we have deﬁned the cycle class maps in (3.6.2). Here we only need the mod 2 case.

2 2 4 Corollary 6.2. The cycle class map clX : CH (X)/2 → H (X, 2) is injective in the following cases:

19 1. dim ϕ =6 and ϕ is neither an Albert form nor a Pﬁster neighbor.

2. 7 ≤ dim ϕ ≤ 8 and ϕ is not a Pﬁster neighbor.

3. dim ϕ> 8.

Proof. We have the following commutative diagram

cl2 CH2(X)/2 −−−→X H4(X, 2)

 cl2  2  X 4  CH (yX)/2 −−−→ H X,y 2 The left vertical map is injective, by Lemma 6.1 (3). We have seen in Prop. 3.8 that the 2 cycle class map clX for X is injective. So the above diagram yields the desired result. The next result is a characteristic two version of [Kah95, Lemma 2].

Lemma 6.3. Suppose X = Xϕ is deﬁned by the reduced norm form ϕ of a quaternion division algebra D over F . Let P be a closed point in X and let C ⊆ X be a smooth hyperplane section. 2 2 4 Then the cycle class map clX : CH (X)/2 → H (X, 2) sends the class [P ] ∈ 2 1 4 CH (X)/2 to the cup product (D) ∪ clX (C) ∈ H (X, 2).

1 1 Proof. We may assume P ∈ C and consider its image under the map clC : CH (C)/2 → H2(C, 1). For every ﬁeld extension K/F such that C has a K-rational point, the natural 1 1 1 2 map CH (C)/2 → CH (CK)/2 is the zero map. So we have clC (P ) ∈ Ker(H (C) → 2 H (CK)). Taking K = F (C) and K = F respectively we ﬁnd

1 2 2 2 2 (6.3.1) clC (P ) ∈ Ker H (C) → H (F (C)) ∩ Ker H (C) → H (C) .

On the other hand, from the Hochschild–Serre spectral sequenc e we see that H2(F )= Ker H2(C) → H2(C) (cf. (3.9.6)). This together with (6.3.1) shows

1 2 2 clC (P ) ∈ Ker H (F ) −→ H (F (C)) = {0 , (D)} .

1 1 2 2 Since the map clC is injective (Prop. 3.7), we have clC (P )=(D) ∈ H (F ) ⊆ H (C). In particular, cl1(P ) is the pullback of the class (D) ∈ H2(X). 2 4 2 Now let i∗ : H (C, 1) → H (X, 2) be the composition of the Gysin map H (C, 1) → 4 4 4 HC X, 2 and the canonical map HC X, 2 → H (X, 2). By [Gro85, Chap. II, Prop. 4.2.3], we have the commutative diagram CH1(C)/2 −−−→ CH2(X)/2

1 2 clC clX

 i∗  H2(C, 1) −−−→ H4(X, 2) y y

20 Therefore, 2 1 ∗ clX (P )= i∗clC (P )= i∗(D)= i∗(1 · i (D)) , where 1 ∈ Z/2 = H0(C, 0) and i∗ : H2(X, 1) → H2(C, 1) is the pullback map. By the projection formula for the Gysin maps (cf. [Gro85, Chap. II, Cor. 2.2.5]) we get

2 ∗ ′ 4 (6.3.2) clX (P )= i∗(1 · i (D)) = i∗(1) ∪ (D) ∈ H (X, 2) .

′ 0 2 ′ Here i∗ : H (C, 0) → H (X, 1) is deﬁned in the same way as i∗. That is, i∗ is the composite map 0 Gysin 2 2 H (C, 0) −−−→ HC X, 1 −→ H (X, 1) . From the commutative diagram

17→[C] Z/2 = CH0(C)/2 −−−→ CH1(X)/2

∼ 1 = clX

′  i∗  Z/2= H 0(C, 0) −−−→ H2(X, 1) y y ′ 1 2 it follows that i∗(1) = clX (C) ∈ H (X, 1). In view of (6.3.2) we have thus proved the lemma.

Lemma 6.4. Let X = Xϕ be as in Lemma 6.3. Then we have natural isomorphisms

H2(F,H2(X, 1) ⊗ Z/2(1)) ∼= H2(F ) ⊕ H2(F ) =∼ H1 F,H3(X, 2) .

Proof. First, we have seen in (3.9.8) that H2(X, 1) ∼= CH1(X)/2. The Galois action on CH1(X)/2 being trivial (by Prop. 2.3), the ﬁrst isomorphism follows. To obtain the second isomorphism, note that

3 ∼ 1 ∼ 2 H (X, 2) = K1(F )/2 ⊗ CH (X)/2 = (K1(F )/2) by (3.9.12). Applying the second formula in (3.9.5) with i = 1, we get

1 3 ∼ 1 2 ∼ 2 2 H (F,H (X, 2)) = H (F, K1(F )/2) = H (F ) ⊕ H (F ) .

This completes the proof.

2 2 4 Proposition 6.5. The cycle class map clX : CH (X)/2 → H (X, 2) is injective in the following two cases:

1. dim ϕ =4.

2. dim ϕ =5 and ϕ is not a Pﬁster neighbor.

Proof. First we show that the second case follows from the ﬁrst one. Indeed, CH2(X) =∼ Z.h2 by [Kar90, (5.3)] (or [HLS21, Thm. 5.1]), and if Y ⊆ X is a smooth hyperplane 2 2 2 2 section, the generator h of CH (X) restricts to the generators hY of CH (Y ). So the

21 natural pullback map i∗ : CH2(X)/2 → CH2(Y )/2 is an isomorphism. The commutative diagram ∗ CH2(X)/2 −−−→i CH2(Y )/2

2 2 clX clY  i∗  H4(X, 2) −−−→ H4(Y, 2) y y 2 2 shows that the injectivity of clX follows from that of clY . Now we can assume dim ϕ =4. If ϕ is isotropic, then the natural map CH2(X) → 2 2 2 CH (X) is an isomorphism. Since clX is injective by Prop. 3.8, it follows that clX is also injective. We may thus assume ϕ is anisotropic. 1 If e1(ϕ) is nonzero in H (F ), it correspond to a separable quadratic extension K/F . We claim that ϕK remains anisotropic. Indeed, since e1(ϕK) = 0, if ϕK is isotropic, then it is hyperbolic. By [EKM08, (34.8)], we would have ϕ = h a, b i ⊗ NK/F for some ∗ a, b ∈ F . But this contradicts the assumption e1(ϕ) =6 0. Our claim is thus proved. It 2 2 follows that CH (X) → CH (XK) is an isomorphism. Passing to the extension K/F if necessary, we may assume e1(ϕ) = 0. Then we may further assume that ϕ is the reduced norm form of a quaternion division algebra D over F . By Lemma 6.3, the generator of CH2(X)/2 ∼= Z/2 is mapped to the cup product 1 4 2 1 (D)∪clX (h) ∈ H (X, 2) by the cycle class map clX . So it suﬃces to show (D)∪clX (h) =6 0 in H4(X, 2). Taking j = 3 and i = 2 in (3.9.3) we obtain an exact sequence

0 −→ H1(F, H3(X, 2)) −→ H4(X, 2) −→ H0(F, H4(X, 2)) −→ 0 .

Thus, 1 4 4 1 3 (D) ∪ clX (h) ∈ Ker H (X, 2) −→ H (X, 2) = H (F, H (X, 2)) . By Lemma 6.4, we have H1(F, H3(X, 2)) = H2(F, H2(X) ⊗ Z/2(1)). Therefore, we 1 2 2 need only to show (D) ∪ clX (h) is nonzero in H (F, H (X) ⊗ Z/2(1)). Consider the commutative diagram

CH1(X)/2 −−−→ CH1(X)/2

1 cl1 clX X 2  2  H (X, 1) −−−→ H (X, 1) y y The top horizontal map can be identiﬁed with the map

(0, Id) : (2Z/4Z).ℓ1 ⊕ (Z/2).h −→ (Z/2).ℓ1 ⊕ (Z/2).h .

1 1 2 The map clX being an isomorphism, it follows that clX (h) =6 0 even in H (X). Now the map 2 1 Z/2 −→ H (X); 1 7−→ clX (h) can be identiﬁed with the map

ι : Z/2 −→ Z/2 ⊕ Z/2 = CH1(X)/2; 1 7−→ (0, 1) = h .

22 We have the commutative diagram 7→ Z/2= H0(F, 0) −−−→1 h H0(F, CH1(X)/2)

∪· ∪ 1 · (6.5.1) (D) (D) clX ( ) 2  ι∗ 2 2  H (F, 1) −−−→ H (F, H (X) ⊗ Z/2(1)) y y The left vertical map in (6.5.1) is injective since (D) =6 0 in H2(F ). The bottom horizontal map ι∗ in (6.5.1) can be identiﬁed with the map H2(F ) −→ H2(F ) ⊕ H2(F ); α 7−→ (0, α) via the ﬁrst isomorphism given in Lemma 6.3. Hence ι∗ is injective. Thus, the diagram 1 2 2 (6.5.1) implies that (D)∪clX (h) =6 0 in H (F, H (X)⊗Z/2(1)). As we have said before, this completes the proof. Lemma 6.6 (Compare [Kah95, Lemma 3]).

1. If either dim ϕ> 4, or dim ϕ =4 and e1(ϕ) =06 , then the map cl1 = ν1, 1 : CH1(X)/2= H0(F, 0) ⊗ CH1(X)/2 −→ H2(X) induces an isomorphism ∼ H2(F ) ⊕ CH1(X)/2 −→ H2(X) .

2. If dim ϕ> 4, then the map ν1, 2 : H1(F, 1) ⊗ CH1(X)/2 −→ H3(X) (deﬁned in (3.6.3)) induces an isomorphism ∼ H3(F ) ⊕ H1(F, 1) ⊗ CH1(X)/2 −→ H3(X) .

Proof. (1) We have the following commutative diagram

(6.6.1) CH1(X) H0(F, 0) ⊗ CH1(X)/2

1 clX 0 / H2(F ) / H2(X) / H0 F, CH1(X)/2 / 0 where the square commutes and the bottom row is the exact sequence given by (3.9.9). The right vertical map in (6.6.1) is an isomorphism by Lemma 6.1 (1). The result follows easily from these observations. (2) The method of proof is similar. This time we use the following commutative diagram (6.6.2) 1 1 0 1 H (F, 1) ⊗ CH (X)/2 H F, K1(F )/2 ⊗ CH (X)/2

ν1, 2 δ / 3 / 3 / 0 1 / 0 H (F ) H (X) H F,K1(F )/2 ⊗ CH (X)/2 0 23 Here the bottom row is obtained by combining (3.9.6) with (3.9.12). The right vertical map δ is an isomorphism since CH1(X)/2 ∼= H0(F, CH1(X)/2) = CH1(X)/2 ∼= Z/2. The following theorem is the characteristic 2 version of [Kah95, Thm. 1]. Notice however that unlike the case of characteristic diﬀerent from 2, the maps µ0,j and ηj are diﬀerent in our situation.

2 2 Theorem 6.7. If dim ϕ> 4, or dim ϕ =4 and e1(ϕ) =06 , then the map η : H (F ) → 2 Hnr(X) is an isomorphism. Proof. By Prop. 3.7, we have a natural exact sequence

1 1 cl 2 2 0 −→ CH (X)/2 −→ H (X) −→ Hnr(X) −→ 0 .

This sequence together with Lemma 6.6 (1) proves the theorem immediately.

2 Corollary 6.8. We have Coker(η∞)=0 for any smooth projective quadric of dimension ≥ 1.

Proof. Combine Lemma 4.3, Prop. 5.3, Cor. 5.7 and Thm. 6.7.

Theorem 6.9. If dim ϕ> 4, then the map

1, 2 1 1 1 H 2 µ : H (F, 1) ⊗ CH (X)/2 → HZar X, (2) is injective and there are isomorphisms

1, 2 ∼ 3 3 ∼ 2 Coker(µ ) = Ker(η ) and Coker(η ) = Ker(clX ) .

Proof. Note that the map ν1, 2 factors through µ1, 2 by construction (cf. (3.6.4)). So we have the following commutative diagram

(6.9.1) 0 / H1(F, 1) ⊗ CH1(X)/2 H1(F, 1) ⊗ CH1(X)/2 / 0 / 0

µ1, 2 ν1, 2 / 1 H 2 / 3 / / 0 HZar X, (2) H (X, 2) M 0

3 where the second row is obtained from the exact sequence (3.8.2) and M := Ker(Hnr(X, 2) → CH2(X)/2). Applying the snake lemma to the above diagram and using Lemma 6.6 (2) (and (3.8.2)), we get an exact sequence (6.9.2) 1, 2 3 1 1 µ 1 H 2 3 η 2 0 → H (F, 1) ⊗ CH (X)/2 −−→ HZar X, (2) → H (F ) −→ Ker(clX ) → 0 .

The theorem follows immediately from the above sequence. Remark 6.10. About the map µ1, 2, the cases not covered by Theorem 6.9 are: (a) dim ϕ = 3; (b) dim ϕ = 4 and e1(ϕ) = 0; (c) dim ϕ = 4 and e1(ϕ) =6 0.

24 In all these cases, we can still use the diagram (6.9.1) to get an exact sequence

1, 2 1, 2 3 2 0 −→ Coker(µ ) −→ Coker(ν ) −→ Ker Hnr(X) → CH (X)/2 −→ 0 and an equality Ker(µ1, 2) = Ker(ν1, 2). To get further information about Ker(µ1, 2), we need to analyze the right vertical map δ in (6.6.2). If ϕ is isotropic, then CH1(X)/2 ∼= CH1(X)/2 and δ is an isomorphism. So in this case the exact sequence (6.9.2) remains valid, and we get the same conclusions as in Theorem 6.9. Now we assume ϕ is anisotropic. Then we claim

Ker(µ1, 2) = Ker(ν1, 2) ∼= Nrd(D∗)/F ∗2 , in Cases (a) and (b), 1, 2 1, 2 (Ker(µ ) = Ker(ν )=0 , in Case (c).

In Case (a), the map δ in (6.6.2) can be identiﬁed with the zero map from H1(F, 1) to itself, since the map CH1(X)/2 → CH1(X)/2 is the zero map from 2Z/4Z to Z/2Z. Thus, (6.6.2) yields an exact sequence

∪(D) 0 −→ Ker(ν1, 2) −→ H1(F, 1) −−−→ H3(F ) −→ Coker(ν1, 2) −→ H1(F, 1) −→ 0 .

In Case (b), we can get an exact sequence of the same form, because the map δ can be viewed as the map

(0 , Id): H1(F, 1) ⊕ H1(F, 1) −→ H1(F, 1) ⊕ H1(F, 1) .

This proves our claim in Cases (a) and (b). 1 ∼ 2 In Case (c), the Galois action on K1(F )/2 ⊗ CH (X)/2 = (K1(F )/2) is given by

σ.(a, b)=(σ(b), σ(a)) , a, b ∈ K1(F )/2 .

The map δ in this case can be identiﬁed with the map

0 0 H (F, K1(F )/2) −→ H F, K1(F/2) ⊕ K1(F )/2 induced from the diagonal map

0 2 H (F, K1(F )/2) −→ (K1(F )/2) ; x 7−→ (x, x) .

Therefore, δ is injective in this case. We conclude that Ker(µ1, 2) = Ker(ν1, 2) = 0 in Case (c). Theorem 6.11. Assume dim ϕ > 4. The maps µ1, 2 and η3 are both isomorphisms in each of the following cases:

1. ϕ is isotropic.

2. ϕ is neither an Albert form nor a Pﬁster neighbor.

3. dim ϕ> 8.

25 Proof. We shall use Theorem 6.9. In Case (1) the result follows from Prop. 3.5. In the 3 ∼ 2 other cases we have Coker(η ) = Ker(clX ) = 0 by Cor. 6.2 and Prop. 6.5. The injectivity of η3 is immediate from [HLS21, Thm. 5.6].

Theorem 6.12. Suppose that ϕ is a neighbor of an anisotropic 3-fold Pﬁster form hh a, b ; c ]]. Then Ker(η3) is generated by (a) ∪ (b) ∪ (c] ∈ H3(F ) (so Ker(η3) ∼= Z/2), and Coker(η3) ∼= Z/2.

Proof. The assertion about Ker(η3) is immediate from [HLS21, Thm. 5.6]. To prove the other assertion, we may reduce to the 5-dimensional case thanks to Cor. 4.2. We may thus assume ϕ = h a i⊥hh b ; c ]]. Since there is an injection from Ker(η3) to Coker(η3) by Lemma 4.3, it suﬃces to show |Coker(η3)| ≤ 2. Let Y be the projective quadric deﬁned by hh b ; c ]]. It is a smooth hyperplane section in X. We have the commutative diagram

∗ CH2(X)/2 −−−→i CH2(Y )/2

2 2 clX clY 4  4  H (X, 2) −−−→ H (Y, 2) y y 2 2 The map clY is injective by Prop. 6.5. So the above diagram shows that Ker(clX ) ⊆ Ker(i∗). Here

2 2 2 2 CH (X)/2 = CH (X)tors ⊕ (Z/2).h =(Z/2).(h − 2ℓ1) ⊕ (Z/2).2ℓ1 , 2 2 CH (Y )/2=(Z/2).hY =(Z/2).2ℓ0 ,

∗ 2 2 ∗ 2 and the map i sends h − 2ℓ1 to 0 and 2ℓ1 to 2ℓ0 = hY . Hence Ker(i )=(Z/2).(h − 2 2ℓ1) = CH (X)tors, and by Thm. 6.9 we have

3 2 ∗ |Coker(η )| = |Ker(clX )|≤|Ker(i )| =2 as desired.

3 3 Now we have computed Ker(η ) and Coker(η ) for the quadric X = Xϕ except in the case where ϕ is an Albert form. This last case will be treated in § 8.

7 Unramiﬁed Witt groups in characteristic 2

We need to use residue maps on the Witt group of a discrete valuation field of characteristic 2. We recall some key definitions and facts that will be used in the next section. Throughout this section, let K be a field extension of F (so char(K) = 2) and let R be the valuation ring of a nontrivial discrete valuation v on K. Let π ∈ R be a uniformizer and let k be the residue field of R.

26 (7.1) Let Iq(R) = Wq(R) be the Witt group of nonsingular quadratic spaces over R as deﬁned in [Bae78, p.18, (I.4.8)]. It is naturally a subgroup of Iq(K). For n ≥ 2, let n Iq (R) be the subgroup of Iq(R) generated by Pﬁster forms of the following type:

∗ hha1, ··· , an−1 ; b]] where ai ∈ R , b ∈ R.

1 n n Put Iq (R)= Iq(R). There is a natural homomorphism Iq (R) → Iq (k), ϕ 7→ ϕ for each n ≥ 1. Following [Ara18], we deﬁne the tame (or tamely ramiﬁed) subgroup of Iq(K) to be the subgroup Iq(K)tr := W (K) · Iq(R). For general n ≥ 1, we put

n n−1 n n−1 Iq (K)tr := I (K) · Iq(R)= Iq (R)+ h1, πibil · Iq (R) . By [Ara18, Props. 1.1 and 1.2], there is a well deﬁned residue map

∂ : Iq(K)tr −→ Iq(k); ϕ0 + π.ϕ1 7−→ ϕ1 for ϕ0, ϕ1 ∈ Iq(R) and the sequence ∂ 0 −→ Iq(R) −→ Iq(K)tr −→ Iq(k) −→ 0 n n−1 is exact. For each n ≥ 1 we have an induced residue map ∂ : Iq (K)tr → Iq (k) (with 0 1 Iq (k)= Iq (k) by convention), and putting

n n n−1 n Iq (K)nr := Ker(∂ : Iq (K)nr → Iq (k)) = Iq (K)tr ∩ Iq(R) , we get an exact sequence

n n ∂ n−1 0 −→ Iq (K)nr −→ Iq (K)tr −→ Iq (k) −→ 0 . Note that the term “tame” and the residue map depend on the discrete valuation v (or the valuation ring R).

(7.2) We use shorthand notation for cohomology functors introduced at the beginning of § 5. Given i ∈ N, localization theory in ´etale cohomology theory gives rise to a long exact sequence

q+1 q+1 δ q+2 ··· −→ H (R, i + 1) −→ H (F, i + 1) −→ Hm (R , i + 1) −→··· where m denotes the unique closed point of Spec(R). By [Shi07, Cor. 3.4] we have

q+2 Hm (R , i + 1) = 0 if q∈{ / i, i +1} .

i ∼ i+2 Moreover, [Shi07, Thm. 3.2] tells us that H (k, i) −→ Hm (R , i + 1). So the localization sequence above yields an exact sequence

0 −→ Hi+1(R, i + 1) −→ Hi+1(K, i + 1) −→δ0 Hi(k, i) −→

i+2 i+2 δ1 i+3 i+3 −→ H (R) −→ H (K) −→ Hm (R, i + 1) −→ H (R, i + 1) −→ 0 .

27 By the Bloch–Kato–Gabber theorem, the map δ0 in the above sequence can be identiﬁed M M with the residue map Ki+1(K)/2 → Ki (k)/2 in Milnor K-theory, which is surjective. Hence, we have an exact sequence

i+2 i+2 δ1 i+3 i+3 0 −→ H (R) −→ H (K) −→ Hm (R, i + 1) −→ H (R, i + 1) −→ 0 .

An element of Hi+2(K) is called unramified at v if it lies in the subgroup Hi+2(R) = Ker(δ1). For each n ≥ 2, we define the tame (or tamely ramified) part of Hn(K) to be the subgroup n n n ˆ n ˆ nr Htr(K) := Ker H (K) −→ H (K) −→ H (K ) , where Kˆ denotes the v-adic completion of K and Kˆ nr is the maximal unramified exten- ˆ H n h sion of K. Kato constructed (cf. [Kat82a], [Kat86, § 1]) a residue map ∂ : Htr(K ) → Hn−1(k) satisfying

H ∗ (7.2.1) ∂ ((uπ) ∪ (a1) ··· (an−2) ∪ (b ]) = (a1) ∪··· (an−2) ∪ (b ] , u,ai ∈ R , b ∈ R, such that the sequence

H n n ∂ n−1 0 −→ H (R) −→ H (K)tr −→ H (k) −→ 0

n n is exact.Therefore, an element α ∈ H (K) is unramiﬁed if and only if α ∈ H (K)tr and ∂H (α) = 0. For each n ≥ 2, using the formula (7.2.1) we can check through calculation that

n n n n en(Iq (R)) ⊆ H (R) , en(Iq (K)tr) ⊆ Htr(K) and that the following diagram with exact rows is commutative:

/ n / n ∂ / n−1 / (7.2.2) 0 Iq (K)nr Iq (K)tr Iq (k) 0

en en en−1 / n / n ∂H / n−1 / 0 H (R) Htr(K) H (k) 0

8 Nontrivial unramiﬁed class for Albert quadrics

In this section we investigate the case of Albert quadrics and complete our study of the map η3. Recall some standard notation. The hyperbolic plane H is the binary quadratic form (x, y) 7→ xy. For any nondegenerate form q of dimension ≥ 3 over F , let F (q) denote the function ﬁeld of the projective quadric deﬁned by q.

Lemma 8.1. Let q be an Albert form over F which represents 1, and let q1 be a form over F (q) such that qF (q) = q1⊥H. If q1 represents 1, then q must be isotropic over F .

28 Proof. We may write q = a[1, b]⊥c[1, d]⊥[1, b + d], where a, c ∈ F ∗ and b, d ∈ F . Set q′ = q⊥h1i and q′′ = a[1, b]⊥c[1, d]⊥h1i. Since

[1, b + d]⊥h1i ∼= [0, b + d]⊥h1i ∼= H⊥h1i we have q′ ∼= H⊥q′′ over F . Now ′ ∼ ′′ H⊥q1⊥h1i = qF (q)⊥h1i = qF (q) = H⊥qF (q) . ∼ ′′ By Witt cancellation, q1⊥h1i = qF (q). The assumption that q1 represents 1 implies that ′′ qF (q) is isotropic. Assume that q is anisotropic. Then q′′ is also anisotropic because it is a subform of q. By [Lag02, Thm. 1.2 (2)], q′′ must be a neighbor of a 3-Pfister form π. The isotropy of ′′ qF (q) shows that πF (q) is hyperbolic. Thus, by [Lag02, Prop. 3.4], q must be a neighbor of π. But this is absurd, because an anisotropic Albert form cannot be a Pfister neighbor (see e.g. [HLS21, Lemma 3.4]). (8.2) Now assume X is defined by an anisotropic Albert form ϕ over F . Let K = F (X) be its function field. We shall normalize ϕ so that it represents 1. Thus ϕ = [1, b + d]⊥a.[1, b]⊥c.[1, d] for some a, b, c, d ∈ F ∗ .

2 We have ϕ = hh a ; b ]] − hh c ; d ]] ∈ Iq (F ). Let ϕ1/K be the anisotropic part of ϕK . Since ϕ is anisotropic of dimension < 8, 3 the Hauptsatz [EKM08, (23.7)] implies ϕ∈ / Iq (F ). This means that ϕ has degree 2 (by [EKM08, (40.10)]). By [EKM08, (25.7)], ϕ1 is a general 2-Pfister form (and the height of ϕ is 2). Let τ/K be the quadratic 2-Pfister form similar to ϕ1, say ϕ1 = f.τ, ∗ 3 where f ∈ K . We have τ − ϕ1 = h1, fibil · τ ∈ Iq (K), so that the cohomology class 3 e3(τ − ϕ1) ∈ H (K) can be defined. 3 We claim that e3(τ − ϕ1) lies in the unramified cohomology group Hnr(X). In fact, we can prove the following: For every discrete valuation v of K that is trivial on F , e3(τ − ϕ1) is unramified at v. Proof. Let R be the valuation ring of v in K = F (X). By (7.2.2), it is sufficient to show

3 3 τ − ϕ1 ∈ Iq (K)nr = Iq(R) ∩ Iq (K)tr .

First note that the Witt class of ϕ1 is equal to that of ϕK. Hence

2 2 2 ϕ1 = hh a ; b ]] − hh c ; d ]] ∈ Im(Iq (F ) −→ Iq (K)) ⊆ Iq (R) .

−1 3 3 If f is a unit for v, then τ − ϕ1 = −h1, f ibil · ϕ1 ∈ Iq (R) ⊆ Iq (K)nr. We may thus assume π := f −1 is a uniformizer for v. Then

−1 2 3 τ − ϕ1 = −h1, f ibil · ϕ1 = −h1, πibil · ϕ1 ∈ h1, πibil · Iq (R) ⊆ Iq (K)tr .

2 It remains to prove τ − ϕ1 ∈ Iq(R). We already know ϕ1 ∈ Iq (R) ⊆ Iq(R). So it suﬃces ˆ ˆ to show τ ∈ Iq(R). This is equivalent to τKˆ ∈ Iq(R), where R denotes the completion of R and Kˆ is the fraction ﬁeld of Rˆ (see e.g. [Ara18, p.106]).

29 Indeed,

2 2 2 e2(ϕ1)= e2(hh a ; b ]] − hh c ; d ]]) ∈ Im(H (F ) −→ H (K)) ⊆ H (R) .

Therefore, the cohomology class e2(ϕ1) is unramified at v. Since e2(τ) = e2(ϕ1) by (2.1.1), e2(τ) is also unramified at v. As a 2-Pfister form, τ is the reduced norm form of a quaternion division K-algebra D. The cohomology class e2(τ) is the Brauer class of D. Thus, the fact that e2(τ) is unramified means that the quaternion algebra D is ∼ ˆ∗ ˆ ∼ unramified. Thus, DKˆ = (α, β] for some α ∈ R , β ∈ R. It follows that τKˆ = hh α ; β ]] 2 ˆ ([EKM08, (12.5)]), proving that the Witt class of τKˆ lies in Iq (R). In particular, τRˆ ∈ Iq(Rˆ) as desired.

3 3 3 Proposition 8.3. With notation as in (8.2), e3(τ − ϕ1) ∈/ Im η : H (F ) → Hnr(X) . 3 3 3 Proof. Since e3 : Iq (F ) → H (F ) is a surjection and Iq (F ) is additively generated by 3-Pfister forms, every element of H3(F ) is a sum of finitely many symbols (by a symbol 3 in H (F ) we mean the class e3(π) of a 3-fold Pfister form π). We may define the symbol length of an element β ∈ H3(F ) to be the smallest positive integer n such that β is a sum of n symbols. We use induction to show: For every n ≥ 1, there is no element β ∈ H3(F ) of symbol length n such that βF (X) = e3(τ − ϕ1). Assume the contrary. Let β ∈ H3(F ) be an element of symbol length n such that βF (X) = e3(τ − ϕ1). First assume n = 1, i.e., β = e3(π) for some 3-Pfister form π over F . Since τ⊥ − ϕ1 and πF (X) are both 3-Pfister forms, from e3(τ − ϕ1) = e3(πF (X)) we can conclude that ∼ τ⊥ − ϕ1 = πF (X), by [Kat82a, p.237, Prop. 3]. Letting E = F (π), we get that τ − ϕ1 = 3 0 in Iq (E(X)). Hence (ϕ1)E(X) = τE(X) is a 2-Pfister form. In particular, (ϕ1)E(X) represents 1. By Lemma 8.1, ϕE = ϕF (π) must be isotropic. But this contradicts an index reduction theorem of Merkurjev (cf. [EKM08, (30.9)]). Now consider the general case and write β = π + δ, where π ∈ H3(F ) is a symbol and δ ∈ H3(F ) has symbol length n − 1. Consider again the function field E = F (π). As before ϕE = ϕF (π) is anisotropic by Merkurjev’s index reduction theorem. But

3 δE(X) = βE(X) = e3(τE(X) − (ϕ1)E(X)) ∈ H (E(X)) . This contradicts the induction hypothesis. Theorem 8.4. Let X be the projective quadric deﬁned by an anisotropic Albert form over F . 3 3 3 3 ∼ Then the map η : H (F ) → Hnr(X) is injective and Coker(η ) = Z/2. Proof. The injectivity of η3 follows from [HLS21, Thm. 5.6]. We have seen in Prop. 8.3 that Coker(η3) =6 0. It remains to show |Coker(η3)| ≤ 2. 2 ∼ 2 To this end, we use the isomorphism Coker(η ) = Ker(clX ) obtained in Thm. 6.9. 2 2 By functoriality and the injectivity of clX , it follows that Ker(clX ) is contained in the kernel of the restriction map ξ2 : CH2(X)/2 → CH2(X)/2. By Lemma 6.1, we have Ker(ξ2) =∼ Z/2. This completes the proof.

30 Together with Prop. 4.4, the following corollary also describes the kernel of the cycle class map 2 2 4 cl∞ : CH (X) ⊗ (Q2/Z2) −→ H X, Q2/Z2(2) . Corollary 8.5. Let ϕ be a nondegenerate quadratic form of dimension ≥ 3 over F and let X = Xϕ. 3 3 3 Then the map η∞ : H F, Q2/Z2(2) → Hnr X, Q2/Z2(2) is surjective unless ϕ is an anisotropic Albert form. In the latter case Coker(η3 ) ∼= Z/2. ∞ Proof. If dim ϕ ≤ 4, this has been discussed in Remark 5.8. When dim ϕ > 4, we have already computed Ker(η3) and Coker(η3) in Theorems 6.11, 6.12 and 8.4. So it suﬃces to apply Lemma 4.3.

Acknowledgements. We thank Yang Cao, Zhengyao Wu and Yigeng Zhao for help discussions. Yong Hu is supported by a grant from the National Natural Science Foundation of China (Project No. 11801260). Peng Sun is supported by the Fundamental Research Funds for the Central Universities.

References

[AJ09] Roberto Aravire and Bill Jacob. H1(X, ν) of conics and Witt kernels in characteristic 2. In Quadratic forms—algebra, arithmetic, and geometry, volume 493 of Contemp. Math., pages 1–19. Amer. Math. Soc., Providence, RI, 2009.

[Ara75] J´on Kr. Arason. Cohomologische invarianten quadratischer Formen. J. Al- gebra, 36(3):448–491, 1975.

[Ara18] J´on Kr. Arason. The Witt group of a discretely valued ﬁeld. J. Algebra, 511:102–113, 2018.

[Bae78] Ricardo Baeza. Quadratic forms over semilocal rings. Lecture Notes in Mathematics, Vol. 655. Springer-Verlag, Berlin-New York, 1978.

[BO74] Spencer Bloch and Arthur Ogus. Gersten’s conjecture and the homology of schemes. Ann. Sci. Ecole´ Norm. Sup. (4), 7:181–201 (1975), 1974.

[CT95] J.-L. Colliot-Th´el`ene. Birational invariants, purity and the Gersten conjecture. In K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), volume 58 of Proc. Sympos. Pure Math., pages 1–64. Amer. Math. Soc., Providence, RI, 1995.

[CTHK97] Jean-Louis Colliot-Th´el`ene, Raymond T. Hoobler, and Bruno Kahn. The Bloch-Ogus-Gabber theorem. In Algebraic K-theory (Toronto, ON, 1996), volume 16 of Fields Inst. Commun., pages 31–94. Amer. Math. Soc., Provi- dence, RI, 1997.

31 [CTO89] Jean-Louis Colliot-Thélène and Manuel Ojanguren. Variétés unirationnelles non rationnelles: au-delàde l’exemple d’Artin et Mumford. Invent. Math., 97(1):141–158, 1989.

[CTP16] Jean-Louis Colliot-Thélène and Alena Pirutka. Hypersurfaces quartiques de dimension 3: non-rationalitéstable. Ann. Sci. Ec.´ Norm. Supér. (4), 49(2):371–397, 2016.

[CTS95] J.-L. Colliot-Thélène and R. Sujatha. Unramified Witt groups of real anisotropic quadrics. In K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), volume 58 of Proc. Sympos. Pure Math., pages 127–147. Amer. Math. Soc., Providence, RI, 1995.

[EKM08] Richard Elman, Nikita Karpenko, and Alexander Merkurjev. The algebraic and geometric theory of quadratic forms, volume 56 of American Mathe- matical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2008.

[Gil00] Philippe Gille. Invariants cohomologiques de Rost en caract´eristique positive. K-Theory, 21(1):57–100, 2000.

[GL00] Thomas Geisser and Marc Levine. The K-theory of ﬁelds in characteristic p. Invent. Math., 139(3):459–493, 2000.

[Gro85] Michel Gros. Classes de Chern et classes de cycles en cohomologie de Hodge- Witt logarithmique. M´em. Soc. Math. France (N.S.), (21):87, 1985.

[Gro03] A. Grothendieck. Revêtements étales et groupe fondamental (SGA 1). Société Mathématique de France, 2003.

[GS88] Michel Gros and Noriyuki Suwa. La conjecture de Gersten pour les faisceaux de Hodge-Witt logarithmique. Duke Math. J., 57(2):615–628, 1988.

[GS17] Philippe Gille and Tam´as Szamuely. Central simple algebras and Galois cohomology, volume 165 of Cambridge Studies in Advanced Mathematics. Cam- bridge University Press, Cambridge, 2017. Second edition of [ MR2266528].

[HLS21] Yong Hu, Ahmed Laghribi, and Peng Sun. Chow groups of quadrics in characteristic two. arXiv: 2101.0300, 2021.

[Ill79] Luc Illusie. Complexe de de Rham-Witt et cohomologie cristalline. Ann. Sci. Ecole´ Norm. Sup. (4), 12(4):501–661, 1979.

M [Izh91] O. Izhboldin. On p-torsion in K∗ for ﬁelds of characteristic p. In Algebraic K-theory, volume 4 of Adv. Soviet Math., pages 129–144. Amer. Math. Soc., Providence, RI, 1991.

32 [Izh01] Oleg T. Izhboldin. Fields of u-invariant 9. Ann. of Math. (2), 154(3):529–587, 2001.

[Kah95] Bruno Kahn. Lower H -cohomology of higher-dimensional quadrics. Arch. Math. (Basel), 65(3):244–250, 1995.

[Kah99] Bruno Kahn. Motivic cohomology of smooth geometrically cellular varieties. In Algebraic K-theory (Seattle, WA, 1997), volume 67 of Proc. Sympos. Pure Math., pages 149–174. Amer. Math. Soc., Providence, RI, 1999.

[Kah04] Bruno Kahn. Cohomologie non ramiﬁ´ee des quadriques. In Geometric meth- ods in the algebraic theory of quadratic forms, volume 1835 of Lecture Notes in Math., pages 1–23. Springer, Berlin, 2004.

[Kar90] N. A. Karpenko. Algebro-geometric invariants of quadratic forms. Algebra i Analiz, 2(1):141–162, 1990.

[Kat80] Kazuya Kato. A generalization of local class ﬁeld theory by using K-groups. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27(3):603–683, 1980.

[Kat82a] Kazuya Kato. Galois cohomology of complete discrete valuation ﬁelds. In Al- gebraic K-theory, Part II (Oberwolfach, 1980), volume 967 of Lecture Notes in Math., pages 215–238. Springer, Berlin, 1982.

[Kat82b] Kazuya Kato. Symmetric bilinear forms, quadratic forms and Milnor K- theory in characteristic two. Invent. Math., 66(3):493–510, 1982.

[Kat86] Kazuya Kato. A Hasse principle for two-dimensional global fields. J. Reine Angew. Math., 366:142–183, 1986. With an appendix by Jean-Louis Colliot- Thélène.

[KRS98] Bruno Kahn, Markus Rost, and R. Sujatha. Unramiﬁed cohomology of quadrics. I. Amer. J. Math., 120(4):841–891, 1998.

[KS00] Bruno Kahn and R. Sujatha. Motivic cohomology and unramiﬁed cohomology of quadrics. J. Eur. Math. Soc. (JEMS), 2(2):145–177, 2000.

[KS01] Bruno Kahn and R. Sujatha. Unramiﬁed cohomology of quadrics. II. Duke Math. J., 106(3):449–484, 2001.

[Lag02] Ahmed Laghribi. Certaines formes quadratiques de dimension au plus 6 et corps des fonctions en caract´eristique 2. Israel J. Math., 129:317–361, 2002.

1 [Mer95] A. S. Merkurjev. The group H (X, K2) for projective homogeneous varieties. Algebra i Analiz, 7(3):136–164, 1995.

[Mil76] J. S. Milne. Duality in the ﬂat cohomology of a surface. Ann. Sci. Ecole´ Norm. Sup. (4), 9(2):171–201, 1976.

33 [Pey95] Emmanuel Peyre. Products of Severi-Brauer varieties and Galois cohomology. In K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), volume 58 of Proc. Sympos. Pure Math., pages 369–401. Amer. Math. Soc., Providence, RI, 1995.

[Sah72] Chih Han Sah. Symmetric bilinear forms and quadratic forms. J. Algebra, 20:144–160, 1972.

[Sal84] David J. Saltman. Noether’s problem over an algebraically closed ﬁeld. In- vent. Math., 77(1):71–84, 1984.

[Shi07] Atsushi Shiho. On logarithmic Hodge-Witt cohomology of regular schemes. J. Math. Sci. Univ. Tokyo, 14(4):567–635, 2007.

[C19]ˇ Kestutis Cesnaviˇcius.ˇ Purity for the Brauer group. Duke Math. J., 168(8):1461–1486, 2019.

Contact information of the authors:

Yong HU Department of Mathematics Southern University of Science and Technology Shenzhen 518055, China Email: [email protected]

Peng SUN School of Mathematics Hunan University Changsha 410082, China Email: [email protected]