Chow Groups of Quadrics in Characteristic Two
aa r X i v : . [ m a t h . N T ] J a n Chow Groups of Quadrics in CharacteristicTwo
Yong Hu, Ahmed Laghribi, and Peng SunJanuary 11, 2021
Abstract
Let X be a smooth projective quadric defined over a field of characteristic 2.We prove that in the Chow group of codimension 2 or 3 of X the torsion subgrouphas at most two elements. In codimension 2, we determine precisely when thistorsion subgroup is nontrivial. In codimension 3, we show that there is no torsionif dim X ≥
11. This extends the analogous results in characteristic different from2, obtained by Karpenko in the nineteen-nineties.
Key words:
Quadratic forms, Chow groups, K theory of quadrics, Clifford algebras
MSC classification 2020:
Let X be a smooth projective variety over a field k . For each natural number p , denoteby CH p ( X ) the Chow group of codimension p cycles on X modulo rational equivalence([Ful98]). When p ≥
2, determining the structure of the group CH p ( X ), especiallythat of the torsion subgroup, is an interesting but often difficult problem in algebraicgeometry. A closely related notion is the Grothendieck ring K ( X ) of vector bundles on X . A consequence of the Grothendieck–Riemann–Roch theorem (cf. [Ful98, § K ( X ) ⊗ Q ∼ = L p ≥ CH p ( X ) ⊗ Q .Consider the special case where X is a smooth projective quadric. Chow groupsand K -theory of X were first studied by Swan in [Swa85] and [Swa89]). In the 1990’s,Karpenko made a systematic study on the structure of CH p ( X ) for p ≤ K -theory of X is established in arbitrarycharacteristic, Karpenko’s theorems are stated only in characteristic different from 2.Among others he proves the following results in codimensions 2 and 3 (in characteristic = 2):1. ([Kar90, Thm. 6.1]) The torsion subgroup CH ( X ) tors of CH ( X ) is finite of orderat most 2, and it is nontrivial if and only if the quadratic form defining X is1n anisotropic 3-fold Pfister neighbor (i.e., isomorphic to a scalar multiple of asubform of dimension ≥ ( X ) tors = 0 if dim X > ( X ) tors of CH ( X ) is finite of order at most 2 ([Kar91b]).It is trivial if dim X >
10 ([Kar95, Thm. 6.1]). The proof of the latter resultdepends on a theorem of Rost about 14-dimensional forms with trivial discriminantand trivial Clifford invariant ([Ros99], [Ros06]). Without using Rost’s theorem,one can show CH ( X ) tors = 0 when dim X >
14 ([Kar95, Thm. 7.1]).Still in characteristic different from 2, Izhboldin has further developed Karpenko’smethods and obtained more precise information about CH ( X ) tors when 7 ≤ dim X ≤ § u -invariant 9 incharacteristic = 2 ([Izh01, Thm. 0.1]).It is natural to expect the same results as above in characteristic 2. Basically, one canfollow the same methods as in Karpenko’s papers. But on the one hand, at some pointsthe original proofs need appropriate modifications, where quite a few details are worthclarifying with special care. On the other hand, it does seem (at least to us) that someother arguments in Karpenko’s work (for example, those in [Kar90, §
6] and [Kar95, § ( X ) tors in characteristic2 ([BCL20, Thm. A.1]). Their proof of this vanishing result provides an example ofadapting Karpenko’s arguments in characteristic 2.In this paper, we make a further study of the Chow groups CH ( X ) and CH ( X )in characteristic 2 and extend the other results of Karpenko mentioned above. In par-ticular, we show that the group CH ( X ) tors has at most two elements, as in the case ofcharacteristic = 2. We also prove CH ( X ) tors = 0 as soon as dim X >
10 (Theorem 7.12).Here we need to extend Rost’s theorem to characteristic 2 (Theorem 7.11), which we dousing a specialization argument. The same result when dim
X >
14 (Theorem 7.8) isindependent of Rost’s theorem.Karpenko’s results for CH ( X ) tors and some of Izhboldin’s results for CH ( X ) tors relyon the computation of degree 4 unramified cohomology groups established in [KRS98].As we would like to leave out discussions on unramified cohomology in this paper, wewill not investigate full generalizations of these results in characteristic 2. We onlyprovide a few examples where CH ( X ) tors = 0 for some quadrics in lower dimensions(see Prop. 7.13 and Remark 7.17). A study of CH ( X ) tors is likely to be the theme of afurther work. 2t is interesting and a bit surprising that in proving the result for CH ( X ) tors ofa 5 or 6 dimensional quadric, the method in the appendix of [BCL20] is not enough(cf. Remark 5.4), and that our approach of using Kato–Milne cohomology (and also K -cohomology implicitly) gives a new uniform proof in all dimensions ≥ ≤ Notation and terminology.
For an algebraic variety Y over a field F , we write Y L = Y × F L for any field extension L/F , and Y = Y × F F , where F is a separableclosure of F .Unless otherwise stated explicitly, k denotes a field of characteristic 2, with a fixedseparable closure k . Let ℘ ( k ) be the image of the Artin–Schreier map ℘ : k → k ; x x − x . Frequently used notations about bilinear and quadratic forms over k will beexplained in (2.2).For an abelian group M , we denote by M tors the subgroup of torsion elements in M . (2.1) Let X be an algebraic variety over a field F . For each p ∈ N , the codimension p Chow group CH p ( X ) can be viewed as a special case of K -cohomology groups. The directsum CH ∗ ( X ) := L p ∈ N CH p ( X ) has a structure of commutative ring with multiplicationgiven by the intersection pairing (cf. [Ful98, Chap. 6] or [EKM08, § f (which induce pullback maps f ∗ )and covariant with respect to proper morphisms g (which yield pushforward maps g ∗ ).In particular, if E/F is a field extension, the natural projection f : X E = X × F E → X induces a natural restriction map Res E/F = f ∗ : CH p ( X ) → CH p ( X E ). If E/F is a finiteextension, we have a norm (or corestriction) map N E/F = f ∗ : CH p ( X E ) → CH p ( X ) andthe composition N E/F ◦ Res
E/F is the multiplication by [ E : F ].We will frequently use two exact sequences in the study of Chow groups (cf. [Kar90,(1.3)]).First, let i : Y ֒ → X be a closed immersion of pure codimension 1 and let j : U = X \ Y ֒ → X be its open complement. Then there is an exact sequence(2.1.1) CH p − ( Y ) i ∗ −→ CH p ( X ) j ∗ −→ CH p ( U ) −→ . This sequence will be referred to as the excision sequence associated to the closed im-mersion
Y ֒ → X .Second, let π : X → C be a flat morphism where C is an irreducible curve. For eachclosed point Q ∈ C , the natural map i Q : X Q → X from the closed fiber X Q of π over Q is a closed immersion; and if X η denotes the generic fiber of π , the natural morphism j : X η → X is flat. The sequence(2.1.2) M Q ∈ C CH p − ( X Q ) P ( i Q ) ∗ −−−−→ CH p ( X ) j ∗ −→ CH p ( X η ) −→
03s exact, and will be called the fibration sequence associated to π . (2.2) Since we are mostly interested in the case of quadrics, for the reader’s conveniencewe now recall some basic definitions and facts about quadratic forms. We will work overa field k , which has characteristic 2 according to our convention stated in the end of theIntroduction. We mainly follow the terminology of [EKM08].For a quadratic form ϕ defined on a (finite dimensional) k -vector space V ϕ , we denoteby b ϕ : ( x, y ) ϕ ( x + y ) − ϕ ( x ) − ϕ ( y )its polar bilinear form. The radical of b ϕ is the subspacerad( b ϕ ) := { v ∈ V ϕ | b ϕ ( v, x ) = 0 for all x ∈ V ϕ } . We say ϕ is nonsingular (resp. nondegenerate ) if rad( b ϕ ) = 0 (resp. dim rad( b ϕ ) ≤ ϕ will be denoted by X ϕ . It is a closed subvarietyin the projective space P ( V ). When dim ϕ ≥ ϕ is nondegenerate if and only if theprojective quadric X ϕ is smooth as an algebraic k -variety.For elements a , · · · , a n ∈ k ∗ , let h a , · · · , a n i bil denote the diagonal bilinear form(of dimension n ) represented by the diagonal matrix with a , · · · , a n as diagonal entries(i.e., the bilinear form (( x , · · · , x n ) , ( y , · · · , y n )) P ni =1 a i x i y i ).For a, b ∈ k , let [ a, b ] denote the binary quadratic form ( x, y ) ax + xy + by , and h a i the 1-dimensional quadratic form x ax . A nondegenerate quadratic form ϕ hasthe following normal form ϕ ∼ = [ a , b ] ⊥ · · · ⊥ [ a m , b m ] , if dim ϕ = 2 m ,ϕ ∼ = [ a , b ] ⊥ · · · ⊥ [ a m , b m ] ⊥h c i , if dim ϕ = 2 m + 1 , where a i , b i ∈ k and c ∈ k ∗ . In the even dimensional case, the Arf invariant (or discriminant ) of ϕ is defined as the image of the element P mi =1 a i b i in the quotientgroup k/℘ ( k ). It is uniquely determined by ϕ and denoted by Arf( ϕ ). The k -algebra k [ T ] / ( T − T − α ), where α ∈ k is a representative of the Arf invariant Arf( ϕ ) ∈ k/℘ ( k ),is uniquely determined. It will be called the discriminant algebra of ϕ .A (quadratic) 1 -fold Pfister form is a binary quadratic form of the shape hh a ]] :=[1 , a ]. If n ≥
2, a quadratic form is called an n -fold Pfister form if it is isomorphic to hh a , · · · , a n − ; a n ]] := h , a i bil ⊗ · · · ⊗ h , a n − i bil ⊗ hh a n ]]for some a , · · · , a n − ∈ k ∗ and a n ∈ k . If λ ∈ k ∗ and ϕ is a Pfister form, the scalarmultiple λϕ is called a general Pfister form .Let W ( k ) be the Witt ring of nondegenerate bilinear forms over k . This ring has afiltration as follows: I F = W ( F ), and for n ≥ I n F is the n -th power ofthe ideal IF of even dimensional bilinear forms over k . For any integer n ≥
1, let I nq k bethe subgroup I n − F ⊗ W q ( F ) of the Witt group W q ( k ) of nonsingular quadratic formsover k . It is clear that I nq k is additively generated by n -fold quadratic Pfister forms.4e will need the Arason-Pfister Hauptsatz, simply called the Hauptsatz, that assertsthe following: If an anisotropic quadratic form ϕ belongs to I nq k , then it has dimension ≥ n ([EKM08, (23.7)], [HL04, Thm. 4.2 (iv)]).For two quadratic forms ϕ and ψ over k , we say ψ is a subform of ϕ if ψ ∼ = ϕ | W forsome subspace W in the vector space V ϕ of ϕ . When this happens we write ψ ⊆ ϕ . For n ≥
2, an n -fold Pfister neighbor is a subform of dimension > n − of a general n -foldPfister form. (2.3) Now we recall some known facts about Chow groups of projective quadrics (whichare valid in arbitrary characteristic). More details can be found in [Kar90, §
2] and[EKM08, § ϕ be a nondegenerate quadratic form of dimension ≥ k , defined on a k -vector space V = V ϕ . Let X = X ϕ be the projective quadric defined by ϕ , which isa closed subvariety in the projective space P ( V ). Let h ∈ CH ( X ) be the pullback ofthe class of a hyperplane in P ( V ). For each p ∈ N , the power h p generates a torsion-freesubgroup Z .h p in CH p ( X ), called the elementary part of CH p ( X ). We say CH p ( X ) is elementary if it is equal to its elementary part.Let d = dim X and m = (cid:2) d (cid:3) = (cid:2) dim ϕ (cid:3) −
1. If
F/k is a field extension such thatthe Witt index of ϕ F is equal to m + 1 (which is the largest possible), we say that X is completely split over F . For example, X is completely split over the separable closure k of k .Suppose X is completely split over an extension F/k . For every integer j ∈ [0 , m ], X F contains some j -dimensional linear subspaces in P ( V ) F . These j -dimensional linearsubspaces all correspond to the same class ℓ j in CH d − j ( X F ) unless d = 2 j = 2 m . Inthe latter case, the m -dimensional linear subspaces give precisely two different classes ℓ m , ℓ ′ m ∈ CH m ( X F ) and ℓ m + ℓ ′ m = h m in CH m ( X F ). In fact, we have the followingstructure results for the Chow groups of X F :(2.3.1) CH p ( X F ) = Z .h p if 0 ≤ p < d , Z .ℓ d − p if d < p ≤ d , Z .h p ⊕ Z .ℓ p = Z .ℓ p ⊕ Z .ℓ ′ p if d = 2 m = 2 p . Moreover,(2.3.2) if d/ < p ≤ d , h p = 2 ℓ d − p in CH p ( X F ) . Notice that (2.3.1) and (2.3.2) hold in particular for X = X × k k .In general, by the standard restriction-corestriction argument, the kernel of the nat-ural map CH p ( X ) → CH p ( X ) is the torsion subgroup CH p ( X ) tors of CH p ( X ). So, if wedenote by CH p ( X ) the image of CH p ( X ) in CH p ( X ), there is a natural exact sequence(2.3.3) 0 −→ CH p ( X ) tors −→ CH p ( X ) −→ CH p ( X ) −→ . If X is isotropic, i.e. the quadratic form ϕ has a nontrivial zero over k , then we mayassume ϕ = ψ ⊥ H , where ψ is a subform of ϕ and H denotes the hyperbolic plane (bywhich we mean the binary form ( x, y ) xy ). In this case we have(2.3.4) CH p ( X ) ∼ = CH p − ( Y ) when 1 ≤ p ≤ d − , Y denotes the projective quadric defined by ψ . This formula often allows us toreduce our problems to the anisotropic case.Now let us assume X is anisotropic. Then(2.3.5) CH p ( X ) = Z .h p = ( CH p ( X ) if 0 ≤ p < d/ , Z . ℓ d − p ⊆ Z .ℓ d − p = CH p ( X ) if d/ < p ≤ d , and when d = 2 p ,(2.3.6) CH p ( X ) = Z .h p ⊆ CH p ( X ) = Z .h p ⊕ Z .ℓ p if Arf( ϕ ) = 0 , Z .h p ⊕ Z . r ℓ p ⊆ CH p ( X ) = Z .h p ⊕ Z .ℓ p for some 1 ≤ r ≤ d/ ϕ ) = 0 . The integer r here is not easy to determine in general. But when dim X = 4, it is knownthat r = 2 (cf. Theorem 5.2 below).The absolute Galois group of k/k acts naturally on CH p ( X ). It is not hard to showthat this Galois action is nontrivial if and only if d = 2 p and Arf( ϕ ) = 0 (cf. [Kah99,Lemma 8.2]). In that case the Galois action permutes the two classes ℓ p and ℓ ′ p .The following result is immediate from (2.3.3), (2.3.5) and (2.3.6). Proposition 2.4.
Let ϕ be a nondegenerate quadratic form of dimension ≥ over k ,and let X ϕ be the projective quadric defined by ϕ . Assume that ϕ is anisotropic.1. If dim ϕ = 2 p + 2 , then CH p ( X ϕ ) is elementary if and and only if CH p ( X ϕ ) istorsion free.2. If dim ϕ = 2 p + 2 , then CH p ( X ϕ ) is elementary if and and only if CH p ( X ϕ ) istorsion free and Arf( ϕ ) = 0 . We have some known examples of torsion-free Chow groups.
Proposition 2.5.
Let X be a smooth projective quadric of dimension d ≥ over k .1. The groups CH ( X ) , CH ( X ) and CH d ( X ) are torsion-free.2. If X is isotropic, then CH ( X ) is torsion-free.Proof. (1) For p = 0 or 1, it is classical that CH p ( X ) is torsion-free. The case p = d isdue to Totaro [Tot08, Lemma 4.1]. (In characteristic = 2 this was proved in [Swa89].)(2) Combine (2.3.4) with the assertion for CH .6 Clifford algebra and splitting index
Throughout this section, let ϕ be a nondenegerate quadratic form of dimension ≥ k , and let C ( ϕ ) and C ( ϕ ) be its Clifford algebra and even Clifford algebra respectively. (3.1) We will need some standard facts about Clifford algebras, which can be found in[Knu88, Chapters 4 and 5] and [EKM08, § C ( ϕ ) is a central simple k -algebra if and only if dim ϕ is odd, andthat C ( ϕ ) is a central simple k -algebra if and only if dim ϕ is even. When dim ϕ iseven, the center of C ( ϕ ) is isomorphic to the discriminant algebra K of ϕ and we have C ( ϕ ) ⊗ k K ∼ = M (cid:0) C ( ϕ ) (cid:1) . When ϕ has even dimension and trivial Arf invariant, wedefine its Clifford invariant e ( ϕ ) to be the Brauer class of C ( ϕ ) (in the Brauer groupBr( k ) of k ).The following facts are very useful: • For any c ∈ k ∗ we have C ( cϕ ) ∼ = C ( ϕ ) and C ( ϕ ⊥h c i ) ∼ = C ( − cϕ ) (cf. [EKM08,(11.4)]). • Suppose dim ϕ is even and let K be the discriminant algebra of ϕ . If c ∈ k ∗ liesin the image of the norm map N K/k : K → k , then C ( cϕ ) ∼ = C ( ϕ ) (cf. [EKM08,(11.8)]).In particular, if ϕ is nonsingular with trivial Arf invariant, then C ( cϕ ) ∼ = C ( ϕ ) forall c ∈ k ∗ . • If dim ϕ is even, then C ( ϕ ⊥ ψ ) ∼ = C ( ϕ ) ⊗ C ( ψ ) for any nondegenerate form ψ (cf.[Knu88, Chap. 5, Lemma 8]). • For any a ∈ k , the Clifford algebra of the binary form [1 , a ] is isomorphic to M ( k )(cf. [EKM08, (11.2) (4)]). • Suppose ϕ = c [1 , a ] ⊥ ρ , where a ∈ k , c ∈ k ∗ and ρ is an even-dimensional nonde-generate form. Set ψ = h c i⊥ ρ .Then by the above statements we have C ( cϕ ) = C ([1 , a ] ⊥ cρ ) = C ([1 , a ]) ⊗ C ( cρ ) = M ( k ) ⊗ C ( ψ ) = M ( C ( ψ ))and if K/k denotes the discriminant algebra of ϕ , C ( cϕ ) ⊗ k K ∼ = M ( C ( cϕ )) ∼ = M ( C ( ϕ )) . Comparing the above isomorphisms we find M ( C ( ϕ )) ∼ = M ( C ( ψ ) K ). Since C ( ϕ ) and C ( ψ ) K are central simple algebras over K , it follows that C ( ϕ ) ∼ = C ( ψ ) K = C ( ψ ) ⊗ k K .
7n the sequel we will frequently use a simple k -algebra C ′ ( ϕ ) defined as follows: If ϕ has even dimension and trivial Arf invariant, then C ( ϕ ) ∼ = A × A for a unique (up toisomorphism) central simple k -algebra A and C ( ϕ ) ∼ = M ( A ) (cf. [EKM08, (13.9)]). Inthis case we set C ′ ( ϕ ) = A . Otherwise (dim ϕ is odd, or dim ϕ is even but Arf( ϕ ) = 0),we put C ′ ( ϕ ) = C ( ϕ ).In any case, we can write C ′ ( ϕ ) ∼ = M s ( D ) for some s ∈ N and some division algebra D with the same center as C ′ ( ϕ ). We write s ( ϕ ) for the integer s here and defineind( ϕ ) = ind( C ′ ( ϕ )), the Schur index of C ′ ( ϕ ) over its center. Following [Kar90] and[Izh01], we call ind( ϕ ) and s ( ϕ ) the index and the splitting index of ϕ respectively.From the definitions we find easily the relation(3.1.1) s ( ϕ ) + log (cid:0) ind( ϕ ) (cid:1) = (cid:20) dim ϕ − (cid:21) . Also, it is easy to see(3.1.2) ( i W ( ϕ ) ≤ s ( ϕ ) ≤ (cid:2) dim ϕ − (cid:3) if ϕ is not hyperbolic ,s ( ϕ ) = i W ( ϕ ) − (cid:2) dim ϕ − (cid:3) if ϕ is hyperbolic , where i W ( ϕ ) denotes the Witt index of ϕ .We have some auxiliary results where the splitting index is used to detect the struc-ture of quadratic forms in low dimensions. Lemma 3.2.
Suppose dim ϕ = 5 (so that ≤ s ( ϕ ) ≤ by (3.1.2) ).1. s ( ϕ ) = 2 ⇐⇒ i W ( ϕ ) = 2 .2. Assume that ϕ is anisotropic. Then the following conditions are equivalent:(a) s ( ϕ ) = 1 .(b) For some quadratic separable extension K/k , the form ϕ K splits completely,i.e., i W ( ϕ K ) = 2 .(c) ϕ is similar to ψ ⊥h c i for some c ∈ k ∗ and some -fold Pfister form ψ .(d) ϕ is a Pfister neighbor.Proof. Let us write ϕ = ψ ⊥h c i with c ∈ k ∗ . Then we have C ( − cψ ) = C ( cψ ) ∼ = C ( ϕ )(noticing that char( k ) = 2) and hence ind( ϕ ) = ind( C ( cψ )). From (3.1.1) we see s ( ϕ ) = 0 ⇐⇒ C ( cψ ) is a central division k -algebra of degree 4 ,s ( ϕ ) = 1 ⇐⇒ C ( cψ ) ∼ = M ( Q ) for some quaternion division k -algebra Q ,s ( ϕ ) = 2 ⇐⇒ C ( cψ ) ∼ = M ( k ) = C (2 H ) . (3.2.1)If i W ( ϕ ) = 2, then clearly (3.1.2) yields s ( ϕ ) = 2. Conversely, if s ( ϕ ) = 2, then by(3.2.1) we have C ( cψ ) ∼ = C (2 H ). Hence, by [Knu88, §
9, Thm. 7], cψ is similar to 2 H ,i.e., ψ is hyperbolic, giving i W ( ϕ ) = 2. This proves (1).8o prove (2), let δ ∈ k represent Arf( ψ ) ∈ k/℘ ( k ) and put ϕ ′ := c. [1 , δ ] ⊥ ψ . ThenArf( ϕ ′ ) = 0. By basic properties of Clifford algebras (which we have reviewed in (3.1)),we have[ C ( ϕ ′ )] = [ C ( cϕ ′ )] = [ C ([1 , δ ] ⊥ cψ )] = [ C ([1 , δ ])] + [ C ( cψ )] = 0 + [ C ( cψ )] = [ C ( ϕ )]in the Brauer group Br( k ). Thus, if s ( ϕ ) = 1, which means C ( ϕ ) ∼ = C ( − cψ ) splits overa separable quadratic extension K/k , then C ( ϕ ′ K ) ∼ = C ( ϕ ′ ) K splits. This means that ϕ ′ K ∈ I q ( K ). But dim ϕ ′ K = 6 < . So the Hauptsatz implies that ϕ ′ K is hyperbolic.Hence, ϕ ′ K ∼ = 3 . H . So we get h c i ⊥ ϕ ′ K ∼ = h c i ⊥ . H . Since h c i ⊥ c. [1 , δ ] ∼ = h c i ⊥ H ,it follows that ϕ K ⊥ H ∼ = h c i ⊥ . H , which implies by Witt cancellation that ϕ K ∼ = h c i ⊥ . H , proving i W ( ϕ K ) = 2.This shows (a) ⇒ (b).The implication (b) ⇒ (c) follows from [EKM08, (34.8)].If (c) holds, then C ( cψ ) ∼ = C ( ψ ) is Brauer equivalent to the quaternion divisionalgebra whose reduced norm is the form ψ . So by (3.2.1), s ( ϕ ) = 1. We have thusshown (c) ⇒ (a).We have (c) ⇒ (d) since ψ ⊥h c i is a subform of the 3-fold Pfister form ψ ⊥ c.ψ when ψ is a 2-fold Pfister form.It remains to show (d) ⇒ (a).Assume ϕ is an anisotropic Pfister neighbor. Then up to similarity, we may assume ϕ is a subform of an anisotropic 3-fold Pfister form π . By [HL04, Lemma 3.1], we canwrite ϕ = ψ ⊥h c i and π = ψ ⊥ [ c, d ] ⊥ τ for some nondegenerate forms ψ, τ and some c ∈ k ∗ , d ∈ k . After scaling ϕ and π by c , we may assume c = 1. (Note that π ∼ = cπ since for the Pfister form π , every nonzero value is a similarity factor.)The 6-dimensional form ϕ ′ := ψ ⊥ [ c, d ] = ψ ⊥ [1 , d ] has the same Arf invariant as thebinary form τ . Let K/k be their discriminant algebra, which is a separable quadraticextension such that τ K ∼ = H . Now π K is hyperbolic and by Witt cancellation, ϕ ′ K = ψ K ⊥ [1 , d ] K is hyperbolic. By [Knu88, §
11, Prop. 8], this means that C ( cψ K ) = C ( cψ ) K splits. This last condition implies s ( ϕ ) ≥
1, in view of (3.2.1). As ϕ is anisotropic, using(1) we conclude that s ( ϕ ) = 1.In characteristic different from 2, the implication (d) ⇒ (c) in Lemma 3.2 (2) is a wellknown fact about 5-dimensional forms (cf. [Lam05, Prop. X.4.19]). Lemma 3.3.
Suppose dim ϕ = 6 . (Thus ≤ s ( ϕ ) ≤ by (3.1.2) .)1. Assume that ϕ is an Albert form, i.e., Arf( ϕ ) = 0 . Then s ( ϕ ) = 0 ⇐⇒ ind( C ( ϕ )) = 4 ⇐⇒ i W ( ϕ ) = 0 ,s ( ϕ ) = 1 ⇐⇒ ind( C ( ϕ )) = 2 ⇐⇒ i W ( ϕ ) = 1 ,s ( ϕ ) = 2 ⇐⇒ ind( C ( ϕ )) = 1 ⇐⇒ i W ( ϕ ) = 3 .
2. Assume that
Arf( ϕ ) = 0 , so that the discriminant algebra of ϕ is a separablequadratic field extension K of k . hen s ( ϕ ) = 0 ⇐⇒ ind( C ( ϕ )) = 4 ⇐⇒ i W ( ϕ K ) = 0 ,s ( ϕ ) = 1 ⇐⇒ ind( C ( ϕ )) = 2 ⇐⇒ i W ( ϕ K ) = 1 ,s ( ϕ ) = 2 ⇐⇒ ind( C ( ϕ )) = 1 ⇐⇒ i W ( ϕ K ) = 3 . Proof.
Combine (3.1.2) with [Knu88, §
11, Cor. 5 and Remark 13] (see also [KMRT98,(16.5)]).The following lemma includes a characteristic 2 version of [Kar90, (5.4)].
Lemma 3.4.
Suppose dim ϕ = 6 . Let Z be the discriminant algebra of ϕ and let N Z/k : Z → k denote the norm of Z/k regarded as a binary quadratic form.Assume that ϕ is anisotropic.1. The following conditions are equivalent:(a) ϕ ∼ = h a, b, c i bil ⊗ N K/k for some a, b, c ∈ k ∗ and some quadratic separableextension K/k .Here N K/k : K → k denotes the norm considered as a binary quadratic form.(b) ϕ is similar to h , a, b i bil ⊗ N K/k for some a, b ∈ k ∗ and some quadraticseparable extension K/k .(c) ϕ is a Pfister neighbor.(d) ϕ K is hyperbolic for some quadratic separable extension K/k .(e) ϕ is not an Albert form and s ( ϕ ) = 2 .Note that when the above conditions hold, K/k must be the discriminant algebraof ϕ and ϕ has a decomposition ϕ = ψ ⊥ θ , where ψ = h a , b i bil ⊗ N K/k is a general -fold Pfister form.2. Suppose Arf( ϕ ) = 0 . Then the following are equivalent:(a) s ( ϕ ) = 1 (i.e., C ( ϕ ) ∼ = M ( Q ) for some quaternion division Z -algebra Q ).(b) i W ( ϕ Z ) = 1 .(c) ϕ ∼ = c.N Z/k ⊥ ψ , where c ∈ k ∗ and ψ is a general -fold Pfister form such that ψ Z is anisotropic.3. Suppose Arf( ϕ ) = 0 . Then the following are equivalent:(a) s ( ϕ ) = 0 (i.e., C ( ϕ ) is a central division algebra of degree over Z ).(b) ϕ Z is anisotropic, i.e., i W ( ϕ Z ) = 0 .(c) ϕ cannot be written as ψ ⊥ θ , where ψ is a general -fold Pfister form. roof. (1) Clearly, (a) ⇒ (b) ⇒ (c). By [EKM08, (34.8)], we have (d) ⇒ (a) and hence thefield K in (d) must be the discriminant algebra of ϕ . The equivalence (d) ⇔ (e) followsfrom Lemma 3.3 (2).It remains to show (c) ⇒ (d). If ϕ is a Pfister neighbor, we may find a binary form τ such that π := ϕ ⊥ τ is a general 3-fold Pfister form. Then ϕ and τ have the sameArf invariant. Let K/k be their discriminant algebra. Since ϕ is anisotropic, π andhence also τ must be anisotropic. Hence K is a quadratic field extension of k . Now τ K is hyperbolic. It follows that π K is hyperbolic and by Witt cancellation, ϕ K is alsohyperbolic.(2) The equivalence (a) ⇔ (b) is part of Lemma 3.3 (2). Clearly, (c) ⇒ (b). If (b) holds,then ϕ ∼ = c.N Z/k ⊥ ψ with c ∈ k ∗ and ψ Z anisotropic, by [EKM08, (34.8)]. Note that theform cN Z/k has the same Arf invariant as ϕ , so ψ must have trivial Arf invariant andhence be a general Pfister form.(3) Again, the equivalence (a) ⇔ (b) is part of Lemma 3.3 (2). From (1) and (2) weget (c) ⇒ (a). Finally, if ϕ = ψ ⊥ θ , then C ( ϕ ) contains C ( ψ ) as a subalgebra. The latteris not a division algebra if ψ is a general 2-fold Pfister form. This yields immediately(a) ⇒ (c).The first assertion in the lemma below is a characteristic 2 analogue of [Kar91b,(3.3)]. A similar but slightly different proof can be found in [Lag15, Lemma 3.6]. Lemma 3.5.
Suppose that ϕ is anisotropic of dimension and Arf( ϕ ) = 0 . (Note that ≤ s ( ϕ ) ≤ by (3.1.2) .)1. If s ( ϕ ) ≥ , then there exist a, b, c ∈ k ∗ and a separable quadratic extension L/k such that ϕ is similar to ( h , a i bil ⊥ c. h , b i bil ) ⊗ N L/k . In particular, ϕ has adecomposition ϕ = ϕ ⊥ ϕ where both ϕ and ϕ are general -fold Pfister forms.2. s ( ϕ ) = 3 if and only if ϕ is a general -fold Pfister form.Proof. By (3.1.2) we have(3.5.1) s ( ϕ ) = 3 ⇐⇒ e ( ϕ ) = [ C ( ϕ )] = 0 ∈ Br( k )and(3.5.2) s ( ϕ ) ≥ ⇐⇒ ind( C ( ϕ )) ≤ . In particular, if ϕ is a generalized 3-fold Pfister form, then s ( ϕ ) = 3 by (3.5.1).Now suppose s ( ϕ ) ≥
2. Write C ( ϕ ) = M s +1 ( D ), where s = s ( ϕ ) and D is a centraldivision k -algebra. Let K/k be a quadratic separable extension such that ϕ K is isotropic.Then ϕ K = H ⊥ ϕ ′ for some 6-dimensional form ϕ ′ over K with trivial Arf invariant. If t = s ( ϕ ′ ) and D ′ is the central division K -algebra Brauer equivalent to C ( ϕ ′ ), thenM s +1 ( D K ) ∼ = C ( ϕ ) K = C ( ϕ K ) ∼ = C ( H ) ⊗ C ( ϕ ′ ) ∼ = M ( k ) ⊗ M t +1 ( D ′ ) ∼ = M t +2 ( D ′ ) . This shows that s ( ϕ ′ ) = s ( ϕ K ) − ≥ s ( ϕ ) − ≥
1. By Lemma 3.3 (1), the form ϕ ′ mustbe isotropic. Hence i W ( ϕ K ) ≥
2. Using [EKM08, (34.8)], we find that ϕ = h a , a , · · · , a n i bil ⊗ N K/k ⊥ τ , n ≥ a i ∈ k ∗ , dim τ ≤ τ K is anisotropic. Put ϕ = h a , a i bil ⊗ N K/k and ϕ = h a , · · · , a n i bil ⊗ N K/k ⊥ τ . Then ϕ is a general 2-fold Pfister form, and since Arf( ϕ ) = Arf( ϕ ) = 0, we haveArf( ϕ ) = 0. It follows that the 4-dimensional form ϕ is also a general 2-fold Pfisterform. The Clifford algebras Q i := C ( ϕ i ) , i = 1 , k -algebras, and each ϕ i is similar to the reduced norm N Q i of Q i .Note that [ C ( ϕ )] = [ Q ⊗ Q ] ∈ Br( k ). So from (3.5.2) we see that the quaternionalgebra Q ⊗ Q is not a division algebra. By a theorem of Albert, this means that Q and Q have a common separable quadratic splitting field L/k (see e.g. [Alb72] and[Dra75]). Up to similarity, we may write ϕ = h , a i bil ⊗ N L/k and ϕ = c. h , b i bil ⊗ N L/k .If s ( ϕ ) = 3, which happens precisely when e ( ϕ ) = [ C ( ϕ )] = 0 ∈ Br( k ) by (3.5.1),then we have e ( ϕ ) = e ( ϕ ) ∈ Br( k ). Therefore, Q ∼ = Q and the forms ϕ and ϕ are similar. This shows that ϕ = ϕ ⊥ ϕ is a general 3-fold Pfister from. This completesthe proof.Now we extend [Kar91b, (3.5)] to characteristic 2. Lemma 3.6.
Assume dim ϕ = 8 . Suppose that for some separable quadratic extension L/k the form ϕ L is a general -fold Pfister form.Then there exists a separable quadratic extension K/k such that i W ( ϕ K ) ≥ .Proof. If ϕ L is isotropic, then ϕ L is hyperbolic and we can just take K = L . Now assume ϕ L is anisotropic.We write ϕ = [1 , a ] ⊥ ρ (up to similarity) and let d ∈ k represent Arf( ρ ) ∈ k/℘ ( k ).Put ψ = [1 , d ] ⊥ ρ .Since ϕ L has trivial Arf invariant, we have a = Arf( ρ L ) = d ∈ L/℘ ( L ) and hence ϕ L = [1 , a ] L ⊥ ρ L ∼ = [1 , d ] L ⊥ ρ L = ψ L . It follows that [ C ( ψ )] L = [ C ( ϕ L )] = 0 ∈ Br( L ). By(3.5.2) we have s ( ψ ) ≥
2. Thus, from Lemma 3.5 (1) we find that ψ splits completely overa separable quadratic extension K/k . Since a 4-dimensional totally isotropic subspaceof ψ K must have a nonzero intersection with ρ K , the form ρ K is isotropic. Hence ρ K ∼ = H ⊥ ρ for some ρ over K . By Witt cancellation, 3 H ∼ = ρ ⊥ [1 , d ] K . By the same trick,we can decompose ρ = H ⊥ ρ for some ρ over K and then we get 2 H ∼ = ρ ⊥ [1 , d ] K .Now using [EKM08, (8.7)] we find2 H ⊥h i K ∼ = ρ ⊥ [1 , d ] K ⊥h i K ∼ = ρ ⊥ H ⊥h i K whence ρ ⊥h i K ∼ = H ⊥h i K . Now ϕ K = ρ K ⊥ [1 , a ] K = 2 H ⊥ ρ ⊥ [1 , a ] K . Since ρ ⊥h i K is isotropic, we get i W ( ϕ K ) ≥ K -theory of quadrics We will need quite a lot of facts about the K -theory of smooth projective quadrics, whichare mainly established in [Swa85], [Kar90] and [Kar95]. In this section, we briefly reviewsome most useful results and sketch a few proofs to make it clearer that the results holdin arbitrary characteristic. 12hroughout this section, let ϕ be a nondegenerate quadratic form of dimension ≥ k and let X = X ϕ be the smooth projective quadric defined by ϕ . (4.1) We will use Quillen’s K -groups as defined in [Qui73]. The quadric X underconsideration here is smooth, so K i ( X ) = K ′ i ( X ) for all i ∈ N . The group K ( X ) = K ′ ( X ) can be identified with the Grothendieck group of isomorphism classes of coherentsheaves on X modulo an equivalence relation defined short exact sequences. The naturaltopological filtration on K ( X ) will be denoted by K ( X ) ( p ) , p ∈ N . For each p ∈ N , weput K ( X ) ( p/p +1) := K ( X ) ( p ) K ( X ) ( p +1) . From the Brown–Gersten–Quillen (BGQ) spectral sequence we can deduce a spectralsequence of the form E p, q = H p ( X , K − q ) = ⇒ K − p − q ( X )where K i denotes the Zariski sheaf associated to the presheaf U K i ( U ), and E p, − p = H p ( X, K p ) ∼ = CH p ( X ) , p ∈ N . For each i ∈ N , the natural surjection E i, − i ։ E i, − i ∞ can be described explicitly as ρ i : CH i ( X ) −→ K ( X ) ( i/i +1) ; [ Z ] [ O Z ] . It is known that ρ i is an isomorphism when i ∈ { , , , , dim X } .Indeed, the discussions in [Kar90, (3.1)] are valid in arbitrary characteristic, so ρ i isan isomorphism when i ∈ { , , , dim X } (cf. Prop. 2.5 for the case i = dim X ). For i = 3, we can follow the ideas in the proof of [Kar90, (4.5)]. It is sufficient to notice thatfor any field extension E/k , the natural map H ( X, K ) → H ( X E , K ) is injective by[Mer95a, Prop. 1.5]. (4.2) By abuse of notation, let h also denote the class of the structural sheaf of ahyperplane section in X . Set H := Z . ⊕ Z .h ⊕ · · · ⊕ Z .h d ⊆ K ( X ) , where d = dim X . (It is also the subring of K ( X ) generated by h .) For each p ∈ N , we say that K ( X ) ( p/p +1) is elementary if it is generated by the image of h p . By [Kar95, (4.4)],the following conditions are equivalent:1. For every i ≤ p the group K ( X ) ( i/i +1) is elementary.2. The homomorphism K ( X ) ( p +1) −→ K ( X ) /H is surjective.As already observed in [BCL20, Appendix], we have the following variant of [Kar95,(4.5)]: 13 roposition 4.3. Suppose that ϕ can be written as ϕ := a. [1 , d ] ⊥ ρ , where a ∈ k ∗ and d ∈ k represents Arf( ρ ) . Let ψ = h a i⊥ ρ .For any p ∈ N , if the groups K ( X ϕ ) ( i/i +1) are elementary for all i ≤ p , then thegroups K ( X ψ ) ( i/i +1) are also elementary for all i ≤ p .Proof. Note that by construction we have Arf( ϕ ) = 0, that is, the discriminant algebraof ϕ is K = k × k . So, by standard facts about even Clifford algebras (cf. (3.1)), C ( ϕ ) = C ( ψ ) ⊗ k K ∼ = C ( ψ ) × C ( ψ ). This gives two projections π , π : C ( ϕ ) = C ( ψ ) × C ( ψ ) −→ C ( ψ ) . Now consider the following diagram(4.3.1) K ( X ϕ ) ( p +1) / / / / i ∗ (cid:15) (cid:15) K ( X ϕ ) /H i ∗ (cid:15) (cid:15) K (cid:0) C ( ϕ ) (cid:1) o o π ∗ + π ∗ (cid:15) (cid:15) K ( X ψ ) ( p +1) / / K (cid:0) X ψ (cid:1) /H K (cid:0) C ( ψ ) (cid:1) o o o o where the left and the middle vertical arrows are the pull-back maps by the naturalinclusion i : X ψ ֒ → X ϕ . In the left square the top horizontal map is surjective by (4.2)and the assumption. By the commutativity of the left square, we need only to show thatthe middle vertical map of (4.3.1) is surjective.In the right square of (4.3.1), the two horizontal maps are surjective by [Swa85, § π ∗ + π ∗ . Proposition 4.4 ([Kar95, (4.9)]) . Let
E/k be a finite field extension such that the normmap N E/k : K (cid:0) C ( ϕ E ) (cid:1) −→ K (cid:0) C ( ϕ ) (cid:1) is surjective (e.g. E can be a subfield in the division algebra D associated to C ( ϕ ) ).Let p ∈ N be such that K ( X ϕ E ) ( i/i +1) are elementary for all i ≤ p .Then the groups K ( ϕ ) ( i/i +1) are elementary for all i ≤ p . Now we state the characteristic 2 version of [Kar95, (4.7)].
Proposition 4.5.
Suppose ϕ = ρ ⊥ a. [1 , b ] , where a ∈ k ∗ and ρ is an even-dimensionalform. Let ψ = ρ ⊥h a i . Assume that the discriminant algebra of ϕ is a quadratic fieldextension K/k such that ind (cid:0) C ( ψ ) K (cid:1) = ind (cid:0) C ( ψ ) (cid:1) .If for some p ∈ N the groups K ( X ψ ) ( i/i +1) are elementary for all i ≤ p − , then thegroups K ( X ϕ ) ( i/i +1) are elementary for all i ≤ p .Proof. As in Prop. 4.3, we have C ( ϕ ) = C ( ψ ) ⊗ k K . We use the following commutativediagram K ( X ψ ) ( p ) / / / / i ∗ (cid:15) (cid:15) K ( X ψ ) /H i ∗ (cid:15) (cid:15) K (cid:0) C ( ψ ) (cid:1) o o restriction (cid:15) (cid:15) K ( X ϕ ) ( p +1) / / K (cid:0) X ϕ (cid:1) /H K (cid:0) C ( ϕ ) (cid:1) o o o o C ( ϕ )) =ind( C ( ψ ) K ) = ind( C ( ψ )) ensures that the rightmost restriction map in the abovediagram is surjective. The diagram then implies the surjectivity of the natural map K ( X ϕ ) ( p +1) → K (cid:0) X ϕ (cid:1) /H . We can thus conclude by (4.2). In this section we prove our main results about codimension two Chow groups.As in the previous section, let X = X ϕ be a smooth projective quadric of dimension d ≥
1, defined by a nondegenerate quadratic form ϕ over k . We will writeCH ∗ ( X ) := M i ≥ CH i ( X ) and Gr K ( X ) := M i ≥ K ( X ) i/i +1 . It is known that CH d ( X ) = Z . [ x ], where x ∈ X is a closed point of minimal degree([EKM08, (70.4)]). To study the Chow group CH ( X ) we need only to consider the case d = dim X ≥ X = 3 or 4 can treated in the same way asin [Kar90], using the isomorphism CH ∗ ( X ) ∼ = Gr K ( X ) (cf. (4.1)). Theorem 5.1 ([Kar90, (5.3)]) . Assume that ϕ is an anisotropic form of dimension .Then CH ( X ) tors ∼ = ( Z / s ( ϕ ) and s ( ϕ ) = 0 or .Moreover, s ( ϕ ) = 1 if and only if ϕ contains a general -fold Pfister form, if andonly if ϕ is a Pfister neighbor (cf. Lemma . ). Theorem 5.2 ([Kar90, (5.5)]) . Assume that ϕ is an anisotropic form of dimension .1. If ϕ is an Albert form, i.e., Arf( ϕ ) = 0 , then the group CH ∗ ( X ) = L i ≥ CH i ( X ) is torsion free and CH ( X ) can be identified with the subgroup Z .h ⊕ Z . ℓ of CH ( X ) = Z .h ⊕ Z .ℓ = Z .ℓ ′ ⊕ Z .ℓ .2. Assume that Arf( ϕ ) = 0 .(a) If ϕ is a Pfister neighbor, i.e., s ( ϕ ) = 2 (cf. Lemma . ), then CH ( X ) tors and CH ( X ) tors are both isomorphic to Z / .(b) If s ( ϕ ) = 1 (cf. Lemma . ), then CH ( X ) tors ∼ = Z / and CH ( X ) tors = 0 .(c) If s ( ϕ ) = 0 (cf. Lemma . ), then CH( X ) is torsion free. Our goal now is to prove the following:
Theorem 5.3 (See [Kar90, (6.1)] in characteristic = 2) . Let X = X ϕ be the projectivequadric defined by a nondegenerate quadratic form ϕ of dimension ≥ over k .Then CH ( X ) tors is either or isomorphic to Z / Z .Moreover, CH ( X ) tors ∼ = Z / if and only if ϕ is an anisotropic -fold Pfister neighbor. emark . If char( k ) = 2, Karpenko’s proof of Theorem 5.3 is based on the followingobservation (cf. [Kar90, (6.2)–(6.3)]): Assume ϕ is anisotropic of dimension ≥ . Then there exists a purely transcendentalextension L/k and a nondegenerate -dimensional quadratic form ψ over L such thatthe following properties hold: The transcendence degree trdeg(
L/k ) is equal to dim ϕ − . The form ψ is anisotropic over L . Letting X ϕ /k and X ψ /L be the projective quadrics defined by ϕ and ψ respectively,we have CH ( X ϕ ) tors ∼ = CH ( X ψ ) tors .4. ψ is a -fold Pfister neighbor if and only if ϕ is a -fold Pfister neighbor. Here properties (1) and (2) are clear from the construction. Karpenko verified prop-erty (3) by using excision and fibration arguments (cf. (2.1)), and he proved property(4) with the help of some algebraic theory of quadratic forms in characteristic = 2. In[BCL20, Appendix], Barry, Chapman and Laghribi have shown that Karpenko’s methodcan be adapted to deal with the case dim ϕ > ψ of dimension 9 (instead of 6) is used, and hence there is no need to check acondition similar to property (4) above.When dim ϕ is 7 or 8, following the method of [BCL20] we can still prove the followingin characteristic 2: Write ϕ = ρ ⊥ [ b, c ] ⊥ τ with b, c ∈ k and dim τ = 4 . Then there exists a purelytranscendental extension L/k and an element f ∈ L ∗ such that the following propertieshold: The transcendence degree trdeg(
L/k ) is equal to dim ϕ − . ′ . The form ψ := [ f, c ] ⊥ τ is anisotropic over L . Letting X ϕ /k and X ψ /L be the projective quadrics defined by ϕ and ψ respectively,we have CH ( X ϕ ) tors ∼ = CH ( X ψ ) tors .4 ′ . If ψ is a Pfister neighbor, then h c i⊥ τ is a Pfister neighbor (and hence contains ageneral -fold Pfister form by Lemma 3.2). In particular, the first assertion in Theorem 5.3 can also be proved with this method.But unfortunately, for the second assertion this seems not enough. This is becausecondition (4 ′ ) is not as strong as condition (4) above. In fact, if ϕ = h a i⊥ [ b, c ] ⊥ τ hasdimension 7, our construction chooses f = ( a + bx ) x − ∈ L = k ( x ). We don’t knowhow to show directly that condition (4) still holds for the form ψ = [ f, c ] ⊥ τ . (However,this follows incidentally by combining Thm. 5.3 with Thm. 5.1.)16ur proof of Theorem 5.3 works in a uniform way in all dimensions ≥ H ( F ) := H ( F , Z / F of characteristic 2. For basic facts about Kato–Milne cohomology, thereaders are referred to [Kat82a] or [GMS03, Appendix]. Lemma 5.5 (See [KRS98, (5.1)] in characteristic = 2) . When dim ϕ ≥ there is anatural isomorphism θ : Ker (cid:0) H ( k ) −→ H ( k ( X ) (cid:1) ∼ −→ CH ( X ) tors . Proof.
To see that such an isomorphism exists one can simply apply [Kah96, Cor. 7.1].However, it would be better to have a more explicit construction. One way to do sois to proceed in essentially the same way as in the proof of [Mer95b, §
2, Prop. 1]. Weinclude a proof for the reader’s convenience.If ϕ is isotropic over k , then the map H ( k ) → H ( k ( X )) is injective and CH ( X ) tors =0. So we may assume ϕ is anisotropic.Let L/k be a separable quadratic extension such that ϕ L is isotropic. Then we haveCH ( X L ) tors = 0 and CH ( X ) tors = Ker(CH ( X ) → CH ( X L )). The extension L/k corresponds naturally to an element ( L ] in H ( k, Z / ∪ ( L ] : K ( k ) −→ H ( k ).Now consider the following commutative diagram with exact rows (5.5.1) K ( L ) (cid:15) (cid:15) N / / K ( k ) (cid:15) (cid:15) ∪ ( L ] / / H ( k ) (cid:15) (cid:15) / / H ( L ) (cid:15) (cid:15) K L ( X ) ∂ L (cid:15) (cid:15) N / / K k ( X ) ∂ k (cid:15) (cid:15) ∪ ( L ] / / H k ( X ) / / H L ( X ) L x ∈ X (1) K k ( x ) ∂ k (cid:15) (cid:15) / / L x ∈ X (1) K L ( x ) ∂ L (cid:15) (cid:15) − σ / / L x ∈ X (1) K L ( x ) ∂ L (cid:15) (cid:15) N / / L x ∈ X (1) K k ( x ) L P ∈ X (2) K k ( P ) / / L P ∈ X (2) K L ( P ) − σ / / L P ∈ X (2) K L ( P ) Here in the first three rows N denotes the corresponding norm maps, and in the lasttwo rows σ denotes the nontrivial element in the Galois group Gal( L/k ). (The top tworows of (5.5.1) are exact by [EKM08, (101.12)].)Let α ∈ Ker( H ( k ) → H k ( X )). We define θ ( α ) ∈ CH ( X ) tors as follows:First, by the injectivity of the map H ( L ) → H L ( X ), we have α L = 0. Thus, bythe exactness of the sequence K ( k ) ∪ ( L ] −−→ H ( k ) −→ H ( L ) we can find z ∈ K ( k ) suchthat α = z ∪ ( L ]. Note that z k ( X ) ∪ ( L ] = α k ( X ) = 0. So the exactness of K L ( X ) N −→ K k ( X ) ∪ ( L ] −−→ H k ( X )17mplies that there exists y ∈ K L ( X ) such that N ( y ) = z k ( X ) . We have N ∂ L ( y ) = ∂ k ( z k ( X ) ) = 0. So by the exact sequence M x ∈ X (1) K L ( x ) − σ −−→ M x ∈ X (1) K L ( x ) N −→ M x ∈ X (1) K k ( x )we can find an element w ∈ L x ∈ X (1) K L ( x ) such that (1 − σ )( w ) = ∂ L ( y ).Now (1 − σ ) ∂ L ( w ) = ∂ L ∂ L ( y ) = 0. Therefore, the exactness of the last row of (5.5.1)shows that ∂ L ( w ) comes from a unique element ∂ L ( w ) ∈ L P ∈ X (2) K k ( P ). We define θ ( α ) to be the class of ∂ L ( w ) in the quotient groupcoker (cid:18) ∂ k : M x ∈ X (1) K k ( x ) −→ M P ∈ X (2) K k ( P ) (cid:19) = CH ( X ) . (Notice that θ ( α ) ∈ Ker(CH ( X ) → CH ( X L )).)Now we prove the injectivity of θ . Suppose θ ( α ) = 0. Then in the construction above,we have ∂ L ( w ) = ∂ F ( w ) L for some w ∈ L x ∈ X (1) K k ( x ). It follows that w − ( w ) L ∈ Ker( ∂ L ). This element represents an element in H ( X, K ) = Ker( ∂ L )Im( ∂ L ) . Now we use thenatural isomorphism (cf. [Kar90, (4.2)])(5.5.2) ρ : K ( L ) ∼ −→ H ( X L , K ) . If we fix a hyperplane section H in X and consider H as a point in X (1) , then for any f ∈ K ( L ), ρ ( f ) is represented by the family( ρ x ) x ∈ X (1) = (1 , · · · , , f, , · · · , ∈ Ker ∂ L : M x ∈ X (1) K L ( x ) −→ M P ∈ X (2) K L ( P ) with ρ x = f if x = H ∈ X (1) and ρ x = 1 otherwise. By the surjectivity of ρ , we can find y ∈ K L ( X ) such that w − ( w ) L = ρ ( f ) + ∂ L ( y ) . Then ∂ L ( y ) = (1 − σ )( w ) = (1 − σ ) ρ ( f ) + ∂ L ((1 − σ )( y )) , which implies (1 − σ )( f ) ∈ K ( L ) is mapped to 0 ∈ H ( X L , K ) by ρ . Now from theinjectivity of ρ we conclude that (1 − σ )( f ) = 0 and hence from the above computation, ∂ L ( y ) = ∂ L ((1 − σ )( y )). Since K ( L ) ∼ −→ H ( X L , K ), we have y = (1 − σ )( y ) + ( z ) L ( X ) for some z ∈ K ( L ) . It follows that z k ( X ) = N ( y ) = N ( z ) k ( X ) . Since the map K ( k ) → K k ( X ) is injective(by [Sus85] (14.3)), we get z = N ( z ). Thus, α = z ∪ ( L ] = N ( z ) ∪ ( L ] = 0.Finally, let us show that θ is surjective. Let ξ ∈ L P ∈ X (2) K k ( P ) represent anelement in CH ( X ) tors = Ker(CH ( X ) → CH ( X L )). Then ξ L = ∂ L ( w ) for some w ∈ x ∈ X (1) K L ( x ). Then ∂ L (1 − σ )( w ) = (1 − σ ) ∂ L ( w ) = (1 − σ )( ξ L ) = 0. This meansthat (1 − σ )( w ) represents an element of H ( X L , K ). Now we use the exact sequence H ( X L , K ) − σ −−→ H ( X L , K ) N −→ H ( X, K ) , which may be identified with the exact sequence K ( L ) − σ −−→ K ( L ) N −→ k ( k ) by (5.5.2).So we can find an element w ∈ L x ∈ X (1) K L ( x ) with ∂ L ( w ) = 0 such that(1 − σ )( w ) = (1 − σ )( w ) + ∂ L ( y ) for some y ∈ K L ( X ) . Now ∂ k N ( y ) = N ∂ L ( y ) = 0, and the isomorphism K ( k ) ∼ −→ H ( X, K ) (cf. [Sus85,(25.5)]) yields an element z ∈ K ( k ) such that N ( y ) = z k ( X ) . According to the diagramchase that defines the map θ , we see that the element α := z ∪ ( L ] satisfies θ ( α ) = ξ .This completes the proof.We need the following result, which is a characteristic 2 analogue of [Ara75, Satz 5.6]. Theorem 5.6.
Let ϕ be a nondegenerate quadratic form of dimension ≥ over k and X = X ϕ its projective quadric. Then, for every α ∈ H ( k ( X ) /k ) := Ker( H ( k ) → H ( k ( X ))) , if α = 0 , there must exist elements a, b ∈ F ∗ and c ∈ F such that α =( a ) ∪ ( b ) ∪ ( c ] ∈ H ( F ) and ϕ is a neighbor of the -fold Pfister form hh a, b ; c ]] .Proof. Let ψ be a 3-dimensional nondegenerate subform of ϕ . After scaling if necessary,we may assume ψ = [1 , c ] ⊥h b i . Since ϕ is isotropic over k ( ψ ), the field extension k ( ψ )( ϕ ) /k ( ψ ) is purely transcendental. Hence H (cid:0) k ( ψ )( ϕ ) /k ( ψ ) (cid:1) := Ker (cid:0) H ( k ( ψ )) −→ H (cid:0) k ( ψ )( ϕ ) (cid:1)(cid:1) = 0 . Therefore, H ( k ( ϕ ) /k ) ⊆ H ( k ( ψ ) /k ).From [AJ09, Thm. 3.6] we know that H ( k ( ψ ) /k ) = k ∗ ∪ ( b ) ∪ ( c ]. In particular,every element α ∈ H ( k ( ϕ ) /k ) can be written as α = ( a ) ∪ ( b ) ∪ ( c ] for some a ∈ k ∗ . Let π = hh a, b ; c ]] be the Pfister form corresponding to α ∈ H ( F ). We assume α = 0, sothat π is anisotropic over k .The assumption α k ( ϕ ) = 0 implies that π k ( ϕ ) ∈ I q ( k ( ϕ )) by [Kat82b]. Using theHauptsatz, we conclude that π k ( ϕ ) is hyperbolic. Then, it follows from [HL04, Thm. 4.2(i)] that ϕ is a neighbor of the 3-fold Pfister form π . Proof of Theorem . . By Lemma 5.5, CH ( X ) tors is isomorphic to the kernel of thenatural map η : H ( k ) → H ( k ( X )). If ϕ is isotropic, then Ker( η ) = 0. If ϕ isanisotropic, then by Thm. 5.6, Ker( η ) consists of symbols whose corresponding 3-foldPfister form contains ϕ up to a scalar multiple. Since dim ϕ ≥
5, such a symbol is uniqueif it exists. Thus, if η is not injective, we have Ker( η ) ∼ = Z /
2, and this case happens ifand only if ϕ is an anisotropic neighbor of a 3-fold Pfister form. The theorem is thusproved. 19 Chow groups of affine quadrics
To prepare the proofs of our results about codimension three Chow groups, we needsome analysis on affine quadrics.We begin with a characteristic 2 variant of [Kar95, (5.3)].
Lemma 6.1.
Let ρ be an irreducible nondegenerate quadratic form of dimension n ≥ over k . Let a ∈ k and ψ = h a i⊥ ρ . Let U ⊆ A nk be the affine quadric defined by a + ρ = 0 .Then CH p ( U ) = 0 in the each of the following cases:1. The form ψ is nondegenerate (i.e. a = 0 and dim ρ is even) and CH p ( X ψ ) iselementary.2. a = 0 and CH p ( X ρ ) is elementary.Proof. The proof in [Kar95, (5.3)] works verbatim as soon as we notice that when a =0, CH p ( X ψ ) ∼ = CH p ( X ρ ) and the pushforward map CH p − ( X ρ ) → CH p ( X ψ ) may beidentified with the multiplication by h ∈ CH ( X ρ ) (cf. [EKM08, 70.2]). Corollary 6.2 (Compare [Kar95, (5.4)]) . Let ρ be an anisotropic (hence irreducible)nondegenerate quadratic form of dimension n ≥ over k . Let a ∈ k and let U ⊆ A nk bethe affine quadric defined by a + ρ = 0 .1. Suppose a = 0 . Then CH ( U ) = 0 in the following cases:(a) dim ρ > .(b) dim ρ ∈ { , , } , and ρ is not a Pfister neighbor (e.g. ρ contains an Albertform).(c) dim ρ = 6 , and ρ is neither an Albert form nor a Pfister neighbor.(d) ≤ dim ρ ≤ .2. Suppose a = 0 . Then CH ( U ) = 0 in the following cases:(a) dim ρ is even and ≥ .(b) dim ρ = 6 and ρ is not a Pfister neighbor.(c) dim ρ = 4 and ρ is not contained in a general -fold Pfister form(d) dim ρ = 2 .(e) ρ is a general -fold Pfister form.Proof. (1) By Lemma 6.1 (2), we need only to show that CH ( X ρ ) is elementary in thesecases. In cases (a)–(c), we use Theorem 5.3 and Prop. 2.4. In case (d), the result is clearfor dimensional reason.(2) The proof in the other cases being similar to the previous ones, it remains totreat case (e). 20onsider again the form ψ = h a i⊥ ρ . Note the following commutative diagram withexact rows CH ( X ρ ) (cid:15) (cid:15) / / CH ( X ψ ) (cid:15) (cid:15) / / CH ( U ) / / (cid:15) (cid:15) ( X ρ ) / / CH ( X ψ ) / / CH ( U ) / / Z . ℓ ⊕ Z .h ֒ → Z .ℓ ⊕ Z .h . The middle vertical map has cokernel ( Z / Z ) .ℓ by (2.3.5). We can rewritethe above diagram as Z . ℓ ⊕ Z .h (cid:15) (cid:15) / / CH ( X ψ ) (cid:15) (cid:15) / / CH ( U ) / / (cid:15) (cid:15) / / Z .ℓ ⊕ Z .h Z . ( h − ℓ ) ∼ / / CH ( X ψ ) = Z .ℓ / / CH ( U ) = 0 / / Z . ( h − ℓ ) −→ CH ( X ψ ) tors −→ CH ( U ) −→ ( Z / Z ) .ℓ ∼ −→ ( Z / Z ) .ℓ −→ . The first arrow in this sequence is surjective, because CH ( X ψ ) tors is generated by h − ℓ by [KM90, (1.8)]. So from the above exact sequence we obtain CH ( U ) = 0 asdesired. Lemma 6.3.
Let ρ be an irreducible nondegenerate quadratic form of dimension n ≥ over k . Let a , b ∈ k, c ∈ k ∗ and ϕ = [ ac − , b ] ⊥ ρ . Let U ⊆ A n +1 k be the affine quadricdefined by the equation a + cy + by + ρ ( x , · · · , x n ) = 0 .If CH p ( X ϕ ) is elementary, then CH p ( U ) = 0 .Proof. Let ψ = h b i⊥ ρ (which can be degenerate). Note that [ ac − , b ] is isomorphic tothe binary form ax + cxy + by . So we have the exact excision sequence (cf. (2.1.1))CH p − ( X ψ ) i ∗ −→ CH p ( X ϕ ) −→ CH p ( U ) −→ i ∗ is surjective when CH p ( X ϕ ) is elementary. Corollary 6.4.
With notation and hypotheses as in Lemma . , we have CH ( U ) = 0 in the following cases:1. dim ρ > .2. ≤ dim ρ ≤ and ρ is not a Pfister neighbor.Proof. In the two cases above CH ( X ϕ ) is elementary by Thm. 5.3 and Prop. 2.4. Thenapply Lemma 6.3. 21 Codimension three cycles on pro jective quadrics
In this section we prove our results about codimension three Chow groups.For a nondegenerate quadratic form ϕ over k , we write ϕ ∈ I q ( k ) if dim ϕ is evenand Arf( ϕ ) = 0. If ϕ ∈ I q ( k ) and ϕ has trivial Clifford invariant, we write ϕ ∈ I q ( k ). Lemma 7.1 (See [Lam05, XII.2.8] in characteristic = 2) . Let ϕ be a nondenegeneratequadratic form of dimension over k . If ϕ ∈ I q ( k ) , then ϕ is isotropic.Proof. We can write ϕ = τ ⊥ ψ with ψ nondegenerate of dimension 6. As ϕ has trivialClifford invariant, the Brauer classes [ C ( ψ )] and [ C ( τ )] coincide.If Arf( ψ ) = 0, then Arf( τ ) = 0 and hence the 4-dimensional form τ is a general2-fold Pfister form. It follows that the Brauer class [ C ( ψ )] = [ C ( τ )] has index ≤
2. Thisimplies that the Albert form ψ is isotropic, and we are done.Now we can assume Arf( ψ ) = 0 and ψ is anisotropic. Let K/k be the separablequadratic extension representing Arf( ψ ). Then the above argument shows that ψ K isisotropic. By [EKM08, (34.8)], there is a decomposition ψ = a.N K/k ⊥ τ ′ for some a ∈ k ∗ and some 4-dimensional form τ ′ . Since Arf( ψ ) = Arf( aN K/k ), Arf( τ ′ ) = 0 Setting ψ ′ = τ ⊥ a.N K/k , we are back to the situation ϕ = τ ′ ⊥ ψ ′ with ψ ′ an Albert form. Theargument in the previous paragraph shows that ψ ′ is isotropic. The lemma is thusproved. Lemma 7.2.
Let ϕ be a nondegenerate quadratic form of dimension over k . Supposethat ϕ ∈ I q ( k ) \ I q ( k ) .Then there exists an odd degree extension K/k and a separable extension
L/K with [ L : K ] = 2 − s such that ϕ L is hyperbolic, where s = s ( ϕ ) is the splitting index of ϕ .Proof. By (3.1.1), the assumption ϕ / ∈ I q ( k ) means that C ( ϕ ) does not split, whence s ( ϕ ) ≤ s ( ϕ ) = 3. Let F/k be a separable quadratic extension such that somebinary nondegenerate subform of ϕ becomes isotropic over F . Then ϕ F = H ⊥ ρ F forsome 8-dimensional form ρ F ∈ I q ( F ). Then s ( ρ F ) = s ( ϕ F ) − ≥ s ( ϕ ) − L/F such that ρ L ishyperbolic. Now [ L : k ] = 4 = 2 − s and we can take K = k .Now let us assume s = s ( ϕ ) ≤
2. By [Pie82, § K/k and a separable extension
F/K of degree 2 − s such thatind( C ( ϕ ) F ) = 2. Then s ( ϕ F ) = 3. So by the previous case we can find a separableextension L/F of degree 4 such that ϕ L is hyperbolic. Now [ L : K ] = 2 − s · − s .The lemma is thus proved. Theorem 7.3.
Let X = X ϕ be the projective quadric defined by a nondegenerate quadricform ϕ over k .Then (cid:12)(cid:12) CH ( X ) tors (cid:12)(cid:12) ≤ .Proof. If ϕ is isotropic, then CH ( X ) tors ∼ = CH ( Y ) tors for a lower dimensional smoothquadric Y (cf. (2.3.4)). In this case the theorem follows from the results for Chowgroups of codimension 2 (Theorem 5.3). 22ow we can assume ϕ is anisotropic. Note that CH ( X ) ∼ = K ( X ) (3 / (cf. (4.1)).If ϕ / ∈ I q ( k ), we can just apply [Kar90, (3.8)]. So we assume ϕ ∈ I q ( k ). In particulardim ϕ is even.If dim ϕ ≤
8, i.e., m := dim X ≤
3, then 2 m − ≤ m . With notation as in [Kar90,(3.10)], in the torsion subgroup of the second kind the dimension 2 m − T II m − is 0 and henceCH ( X ) tors ∼ = (cid:0) K ( X ) (3 / (cid:1) tors = T I m − ∼ = Z / . It remains to consider the case where ϕ ∈ I q ( k ), dim ϕ ≥
10 and ϕ is anisotropic.Now K ( X ) ( i/i +1) ∼ = CH i ( X ) is torsion free for i ≤
2. (For i = 2 we use Thm. 5.3.) Bythe last assertion in [Kar96, (3.9)], ( T I ) (3) = 0 and hence CH ( X ) tors ∼ = (cid:0) K ( X ) (3 / (cid:1) tors =( T II ) (3) is a cyclic group.It is now sufficient to show that CH ( X ) tors is killed by 2.If dim ϕ >
10, then we can write ϕ = ρ ⊥ τ with dim τ = 2 and dim ρ >
8. Choos-ing
L/k to be a quadratic separable extension with τ L ∼ = H , we get CH ( X L ) tors ∼ =CH ( Y L ) tors , where Y is the quadric defined by ρ . Here CH ( Y L ) tors = 0 by Thm. 5.3.So the standard restriction-corestriction argument shows that 2 · CH ( X ) tors = 0.So now we assume dim ϕ = 10 (and ϕ is anisotropic, belonging to I q ( k )).Since ϕ is anisotropic, Lemma 7.1 implies ϕ / ∈ I q ( k ). Let s = s ( ϕ ). By Lemma 7.2,we can find an odd degree extension K/k and a separable extension
L/K of degree 2 − s such that ϕ L is hyperbolic. Note that the splitting index does change after an odd degreebase extension. So s ( ϕ K ) = s ( ϕ ) = s . Now, by the estimate of | T II | in [Kar96, (3.9)]we have (cid:12)(cid:12) CH ( X ) tors (cid:12)(cid:12) ≤ (cid:12)(cid:12) CH ( X K ) tors (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:0) T II (cid:1) (3) (cid:12)(cid:12)(cid:12) ≤ | T II | ≤ s +(5 − s ) − = 2 . The theorem is thus proved.
Remark . Our proof of Theorem 7.3 is slightly different from Karpenko’s arguments([Kar91b, §
3] or [Kar96, § For a 8-dimensional form ρ , CH ( X ρ ) tors = 0 if and only if ρ is a general 3-foldPfister form. Our proof here does not need any characterization of 8-dimensional forms with non-trivial torsion in the codimension 2 Chow group. We have only used the first assertionof Theorem 5.3 and the vanishing of CH ( X ϕ ) tors for ϕ of dimension >
10. These tworesults can be proved without using Lemma 5.5 (Remark 5.4).We now prove [Kar95, (6.2)] in characteristic 2.
Lemma 7.5.
Let p, n ∈ N with n > p + 2 . Let P ( p, n ) be the following statement: Forevery extension field F of k and every nondegenerate quadratic form ψ of dimension n over F , the group CH p ( X ψ ) is elementary.Then P ( p, n ) implies P ( p, n + 1) . roof. It is clear that P (0 , n ) holds for all n >
2. We may thus assume p ≥ F be a field extension of k and let ρ be a nondegenerate quadratic form ofdimension n − F . Then ψ = ρ ⊥ H has dimension n and CH p ( X ψ ) ∼ = CH p − ( X ρ ).So P ( p, n ) implies P ( p − , n − p we find that P ( p, n ) implies P ( p − , N ) for all N ≥ n − P ( p, n ) holds and consider a nondegenerate quadratic form ϕ of di-mension n + 1 over F . We distinguish two cases to show CH p ( X ϕ ) is elementary. Case 1. n + 1 is even.In this case we can write ϕ = [ a, b ] ⊥ ρ for some ( n − ρ over F .Put ψ = h b i⊥ ρ and let U be the affine quadric defined by a + y + by + ρ = 0. We havethe exact sequence CH p − ( X ψ ) −→ CH p ( X ϕ ) −→ CH p ( U ) −→ p − ( X ψ ) is elementary since P ( p − , n ) holds. Hence CH p ( U )coincides with the non-elementary part CH p ( X ϕ ) / Z .h p of CH p ( X ϕ ). It remains to showCH p ( U ) = 0.Let π : U → A be the projection onto the y -coordinate. We have an exact fibrationsequence (cf. (2.1.2))(7.5.1) M P ∈ A CH p − ( U P ) −→ CH p ( U ) −→ CH p ( U η ) −→ , where U η denotes the generic fiber of π and for each closed point P ∈ A , U P denotesthe closed fiber of π over P .For each closed point P ∈ A , the affine variety U P is defined over the residue field κ ( P ) by the equation α ( P ) + ρ = 0, where α ( P ) := a + y ( P ) + by ( P ) . If Y denotes theprojective quadric over κ ( P ) defined by ρ κ ( P ) , then the property P ( p − , n −
1) impliesthat CH p − ( Y ) is elementary. Similarly, if α ( P ) = 0 and Z denotes the projectivequadric defined by h α ( P ) i⊥ ρ κ ( P ) , then CH p − ( Z ) is elementary. Thus, by Lemma 6.1,CH p − ( U P ) = 0.The generic fiber U η is the affine quadric over the rational function field L = F ( y )defined by α ( y ) + ρ = 0. By the property P ( p, n ), the group CH p ( X h α ( y ) i⊥ ρ L ) is elemen-tary. Applying Lemma 6.1 once again yields CH p ( U η ) = 0. Thus, from (7.5.1) we getCH p ( U ) = 0 as desired. Case 2. n + 1 is odd.Now we can write ϕ = h a i⊥ [ b, c ] ⊥ τ for some nondegenerate form τ of dimension n − b = 0, then [ b, c ] ∼ = H and CH p ( X ϕ ) ∼ = CH p − ( X h a i⊥ τ ). The result then followsimmediately from P ( p − , n − b = 0. Let U be the affine quadric defined by a + [ b, c ] ⊥ τ = 0,that is, a + bx + xy + cy + τ = 0 . As in Case 1, it is sufficient to show CH p ( U ) = 0.24e consider the projection π : U → A onto the y -coordinate, which gives rise to anexact sequence of the form (7.5.1). We want to show CH p ( U η ) = 0 and CH p − ( U P ) = 0for all closed points P ∈ A .The generic fiber U η is defined over the rational function field L = F ( y ) by theequation ( a + cy )+ yX + bX + τ = 0. Let θ be the binary form ( a + cy ) Z + yXZ + bX over L and consider the forms ρ := h b i⊥ τ and θ := θ ⊥ τ . Then we have an exactsequence CH p − ( X ρ L ) −→ CH p ( X θ ) −→ CH p ( U η ) −→ . The groups CH p − ( X ρ L ) and CH p ( X θ ) are elementary by the properties P ( p − , n − P ( p, n ). So the above sequence shows CH p ( U η ) = 0.Now let P be a closed point of A F and write K = κ ( P ). The variety U P is definedover K by the equation ( a + cy ( P ) ) + y ( P ) X + bX + τ = 0. Let θ be the binary form( a + cy ( P ) ) Z + y ( P ) XZ + bX over K and write θ = θ ⊥ τ as above.If y ( P ) = 0, then θ is nondegenerate. As in the case for the generic fiber, we can usethe properties P ( p − , n −
1) and P ( p − , n ) to show CH p − ( U P ) = 0.It remains to treat the case y ( P ) = 0. In this case U P is defined by a + h b i⊥ τ = 0over K . Put V = U P and consider the projection onto the variable t corresponding tothe subform h b i . Then we have the following analogue of (7.5.1):(7.5.2) M Q ∈ A CH p − ( V Q ) −→ CH p − ( V ) −→ CH p − ( V η ) −→ . The generic fiber V η is the affine quadric defined by ( a + bt ) + τ = 0 over K ( t ). Since[ a + bt ] ⊥ τ is a nondegenerate form over K ( t ) and CH p − ( X h a + bt i⊥ τ ) is elementary by P ( p − , n − p − ( V η ) = 0. Similarly, if Q ∈ A K is a closed point with a + bt ( Q ) = 0, then CH p − ( V Q ) = 0 by Lemma 6.1 (1) andthe property P ( p − , n − a + bt ( Q ) = 0, then we can use Lemma 6.1 (2) andthe property P ( p − , n − p − ( U P ) =CH p − ( V ) = 0.This completes the proof. Proposition 7.6.
Let n be an odd integer > . Then the following are equivalent:1. For every field extension F/k and every nondengenerate quadratic form ψ of di-mension ≥ n over F , CH ( X ψ ) tors = 0 .2. For every field extension F/k and every nondengenerate quadratic form ψ of di-mension n over F , CH ( X ψ ) tors = 0 .3. For every field extension F/k and every nondengenerate quadratic form ψ of di-mension n + 1 over F with ψ ∈ I q ( F ) , CH ( X ψ ) tors = 0 .4. For every field extension F/k and every nondengenerate quadratic form ψ of di-mension n + 1 over F with ψ ∈ I q ( F ) , CH ( X ψ ) tors = 0 .Proof. Combine Lemma 7.5, Prop. 4.3 and Prop. 4.4.25 emma 7.7.
Let τ be a nondegenerate quadratic form of even dimension m ≥ over k , and let U ⊆ A m +3 be the affine quadric over k defined by the equation a + cY + b Y + [ a , b ] ⊥ τ = 0 , where c , a i , b i ∈ k ∗ . Assume either m ≥ or τ is an Albert form. Then CH ( U ) ∼ = CH ( U ) , where U ⊆ A m +1 is the affine quadric over the rational function field F = k ( y , x ) defined bythe equation ( a + cy + b y + a x ) + x Y + b Y + τ = 0 . Proof.
Let k be the rational function field k ( y ) and let U ⊆ A m +2 be the affine quadricover k defined by ( a + cy + b y ) + [ a , b ] ⊥ τ = 0 . By considering a fibration over A as in Case 1 of the proof of Lemma 7.5, we can useCor. 6.2 to get CH ( U ) ∼ = CH ( U ).Now letting x , y denote the variables corresponding to the binary form [ a , b ],consider the projection π : U → A k onto the x -coordinate. Then the generic fiber of π is the affine quadric U over F = k ( x ) = k ( y , x ) in the statement of the lemma.By the fibration method, to show CH ( U ) ∼ = CH ( U ), it is sufficient to prove that forevery closed point P ∈ A k , the closed fiber ( U ) P of π over P satisfies CH (( U ) P ) = 0.Let us fixe a closed point P ∈ A k and put V = ( U ) P . Writing α = a + cy + b y ∈ k , V is the affine quadric over K := k ( P ) defined by the equation( α + a x ( P ) ) + x ( P ) Y + b Y + τ = 0 . If x ( P ) = 0, we can deduce from Cor. 6.4 that CH ( V ) = 0. If x ( P ) = 0, then V isdefined by α + h b i⊥ τ = 0. In this case, the exact sequence (7.5.2) has the followingform M Q ∈ A K CH ( V Q ) −→ CH ( V ) −→ CH ( V η ) −→ . The generic fiber V η is the affine quadric defined by ( α + b y ) + τ = 0 over the rationalfunction field K ( y ). By Cor. 6.2 (3), we have CH ( V η ) = 0. For each closed point Q ∈ A K , Lemma 6.1 shows that CH ( V Q ) = 0. So we get CH ( V ) = 0 as desired. Thelemma is thus proved. Theorem 7.8.
Let ϕ be a nondengerate quadratic form of dimension ≥ over k . Then CH ( X ϕ ) is elementary.Proof. By Prop. 7.6, we may assume dim ϕ = 18.If ϕ is isotropic, then CH ( X ϕ ) ∼ = CH ( X ψ ) for some nondengerate form ψ of dimen-sion 16 and the result follows from Thm. 5.3. So we may assume ϕ is anisotropic.We can write ϕ = [ a , b ] ⊥ · · · ⊥ [ a , b ] ⊥ ρ where a i , b i ∈ k ∗ and dim ρ = 8. Thereare only two cases to discuss: Case 1 : For some c ∈ { a , · · · , a } ∪ { b , · · · , b } , the even Clifford algebra C ( h c i⊥ ρ )is not a division algebra. Without loss of generality, we may assume c = b in this case.26 ase 2 : For every c ∈ { a , · · · , a } ∪ { b , · · · , b } , the even Clifford algebra C ( h c i⊥ ρ )is a division algebra.In any case, let U be the affine quadric over k defined by the equation a + Y + b Y + [ a , b ] ⊥ · · · ⊥ [ a , b ] ⊥ ρ = 0 . Then the standard excision sequence gives CH ( X ϕ ) / Z .h ∼ = CH ( U ), and a repeatedapplication of Lemma 7.7 yields CH ( U ) ∼ = CH ( U ), where U is the affine quadric overthe rational function field F = k ( y , · · · , y , x , · · · , x ) defined by (cid:0) α + a x (cid:1) + x Y + b Y + ρ = 0 , where α = a + y + b y + X i =2 ( a i x i + x i y i + b i y i ) . It remains to prove CH ( U ) = 0.Now we distinguish the two cases mentioned above.In Case 1, we consider ψ := h b i⊥ ρ . It is anisotropic since ϕ is. As mentioned before,we may assume C ( ψ ) is not a division algebra.By Lemma 6.3, it is sufficient to show that CH ( X θ ) is elementary, where θ is thequadratic form [ αx − + a , b ] ⊥ ρ over F . The groups K ( X ψ ) ( i/i +1) ∼ = CH i ( X ψ ) , i ≤ i = 2). Therefore, using Prop. 4.5 we reduce theproblem to proving the following assertion: The discriminant algebra K of the form θ over F is a field such that ind (cid:0) C ( ψ ) K (cid:1) is equal to ind (cid:0) C ( ψ ) F (cid:1) = ind( C ( ψ )). (Notethat we have ind (cid:0) C ( ψ ) F (cid:1) = ind( C ( ψ )) since F/k is a purely transcendental extension.)Indeed, letting d ∈ k represent Arf( ρ ) ∈ k/℘ ( k ) we have K = F [ Z ] / (cid:0) Z − Z − b ( αx − + a ) − d (cid:1) = F [ T ] / (cid:0) T − x T − b ( α + a x ) − x d (cid:1) = Frac k [ Y , · · · , Y , X , · · · , X , T ] (cid:0) T − X T − b (cid:0) a + Y + b Y + P i =2 ( a i X i + X i Y i + b i Y i ) + a X (cid:1) − dX (cid:1) ! = Frac k [ Y , · · · , Y , X , · · · , X , T ] (cid:0) b (cid:0) a + Y + b Y + P i =2 ( a i X i + X i Y i + b i Y i ) (cid:1) + T − X T − ( a b + d ) X (cid:1) ! . In other words, K is the function field k ( τ ) of the projective quadric over k defined bythe quadratic form τ := b (cid:0) [ a , b ] ⊥ · · · ⊥ [ a , b ] (cid:1) ⊥ [1 , a b + d ] . Note that dim C ( ψ ) = 2 and we have assumed that C ( ψ ) is not a division algebra.Thus, the division algebra in the Brauer class of C ( ψ ) has dimension < . Sincedim τ = 10 ≥ C ( ψ ) K ) = ind( C ( ψ ) k ( τ ) ) = ind( C ( ψ )) as desired. This proves the theoremin Case 1.Now consider Case 2. By assumption, C ( h a i⊥ ρ ) is a division algebra.We will prove CH ( U ) = 0 by the fibration method. By considering the projection π : U → A F onto the y -coordinate, we obtain the exact sequence M P ∈ A F CH ( U P ) −→ CH ( U ) −→ CH ( U η ) −→ . U η is an open subset of the projective quadric X θ , where θ is thequadratic form over L := F ( y ) given by θ = h β i⊥ ρ L where β = α + a x + x y + b y = a + y + b y + X i =2 ( a i x i + x i y i + b i y i ) ∈ L = k ( y , · · · , y , x , · · · , x ) . Since β specializes to a when y = · · · = y = x = · · · = x = 0 and C ( h a i⊥ ρ ) isa division algebra by assumption, it follows that C ( θ ) = C ( h β i⊥ ρ L ) is also a divisionalgebra. By [Kar95, (4.3)], the group CH ( X θ ) ∼ = K ( X θ ) (3 / is elementary. This impliesthat CH ( U η ) = 0, by Lemma 6.1.Next consider a closed point P ∈ A F . The fiber U P is the affine quadric over K := F ( P ) defined by the equation β ( P ) + ρ = 0. If β ( P ) = 0, then we conclude fromCor. 6.2 (2.a) that CH ( U P ) = 0.Finally, assume β ( P ) = 0 ∈ K . We claim that ρ K has nontrivial Arf invariant,in particular ρ K is not a Pfister form. Thus, by Cor. 6.2 (1.b) we get CH ( U P ) = 0again. In fact, the field K = F ( P ) is nothing but the function field k ( τ ) of the form τ := [ a , b ] ⊥ · · · ⊥ [ a , b ]. As C ( h a i⊥ ρ ) is a division algebra over k , Arf( ρ ) is nonzeroin k/℘ ( k ). Since k is algebraically closed in k ( τ ) = K , Arf( ρ K ) = 0 in K/℘ ( K ). Ourclaim thus follows, and this completes the proof of the theorem. Lemma 7.9.
Let ϕ be a nondegenerate -dimensional quadratic form over k . Supposethat ϕ contains an Albert form as a subform.Then CH ( X ϕ ) is elementary.Proof. We may assume ϕ is anisotropic and write ϕ = [ a , b ] ⊥ · · · ⊥ [ a , b ] ⊥ ρ , where a i , b i ∈ k ∗ and ρ is an Albert form. Put F = k ( y , y , x , x ) and let U be the affinequadric over F defined by the equation( α + a x ) + x Y + b Y + ρ = 0 where α = a + y + b y + X ≤ i ≤ ( a i x i + x i y i + b y i ) . As in the proof of Thm. 7.8, we have CH ( X ϕ ) / Z .h ∼ = CH ( U ).Put ψ = h b i⊥ ρ and consider the form θ := [ αx − + a , b ] ⊥ ρ over F . The form ψ is anisotropic since ϕ is, and it is not a Pfister neighbor since the Albert form ρ is nota Pfister neighbor. Therefore, CH ( X ψ ) is elementary by Thm. 5.3.As in Case 1 of the proof of Thm. 7.8, it is sufficient to show that the discriminantalgebra K of the form θ over F is a field such that ind( C ( ψ ) K ) = ind( C ( ψ )).In fact, K is the function field k ( τ ) of the quadratic form τ over k given by τ = b ([ a , b ] ⊥ [ a , b ] ⊥ [ a , b ]) ⊥ [1 + a b ] . Note that C ( ψ ) is not a division algebra since the Albert form ρ is a subform of ψ .So the division algebra Brauer equivalent to C ( ψ ) has dimension < dim C ( ψ ) = 2 .Since dim τ = 8 ≥ C ( ψ ) K ) = ind( C ( ψ ) k ( τ ) ) = ind( C ( ψ )). This completes the proof.28 We recall some facts on residue forms in the case of valued fields. Let A be aring endowed with a rank 1 discrete Henselian valuation ν . Let K and A × be its fieldof fractions and the group of units, respectively. Let π be a uniformizing parameter and k = A/πA the residue field. Let ϕ be an anisotropic quadratic form over a K -vectorspace V . Since ϕ is anisotropic and ν is Henselian, we have the following inequality:(7.10.1) ν ( B ϕ ( x, y ) ) ≥ ν ( ϕ ( x )) + ν ( ϕ ( y ))for all x, y ∈ V ([Tie74, Lemma 2.2]).For i ∈ Z , let V i = { x ∈ V | ϕ ( x ) ∈ π i A } . Using the inequality (7.10.1), we provethat V i is an A -module. The form ϕ induces two quadratic forms ϕ and ϕ , called thefirst and the second residue forms, on the k -vector space V i /V i +1 as follows: ϕ i : V i /V i +1 −→ kx + V i +1 π − i ϕ ( x )Obviously, the quadratic forms ϕ and ϕ are anisotropic. When ϕ is nonsingular,we have by [MMW91, Theorem 1]:(7.10.2) dim ϕ = dim ϕ + dim ϕ . To improve Theorem 7.8 we prove the following result which is an analogue in char-acteristic 2 of a theorem of Rost (cf. [Ros99], [Ros06]).
Theorem 7.11.
Let ϕ ∈ I q ( k ) be an anisotropic form of dimension . Then, ϕ containsan Albert form as a subform.Proof. Let A be a Henselian discrete rank 1 valuation ring of characteristic 0 whosemaximal ideal is 2 A and residue field k (see [Wad85, (1.4)]). Let K and A × be the fieldof fractions and the group of units of A , respectively.There exists a nondegenerate quadratic module θ of rank 14 defined on an A -module V that is a lifting of ϕ , i.e., ϕ is isometric to the induced quadratic form θ on the k -vectorspace V / V . The form θ is anisotropic.Let S = { ( − k a + 4 b | k ∈ Z , a ∈ A × , b ∈ A } . This is clearly a subgroup of A × .By [Wad85, Lemma 1.6], there exists a surjective group homomorphism γ : S −→ k/℘ ( k )given by: ( − k a + 4 b ba − + ℘ ( k ), and Ker γ = ± A × . Moreover, det θ ∈ S/A × and γ (det θ ) = Arf( θ ) [Wad85, Proposition 1.14].Using [BCL20, Corollary 5.4], we get a form ϕ ′ ∈ I A such that ϕ ′ is Witt-equivalentto ϕ ∼ = θ . Since A is Henselian, it follows that ϕ ′ is Witt-equivalent to θ [Kne69, Satz3.3]. Hence, θ ∈ I A . In particular, θ K ∈ I K . It follows from a theorem of Rost([Ros99], [Ros06]) that θ K contains an Albert form θ ′ as a subform.We write θ ′ ∼ = [ a , b ] ⊥ [ a , b ] ⊥ [ a , b ] for suitable a i , b i ∈ K , 1 ≤ i ≤
3. We claimthat a i , b i ∈ A × for all 1 ≤ i ≤
3, i.e., θ ′ is defined over A . In fact, let us write a i = u i ǫ i and b i = v i δ i for u i , v i ∈ A × and ǫ i , δ i ∈ Z .29i) The form ϕ is nothing but the first residue form of θ K , and thus the second residueform of θ K is the zero form by (7.10.2).(ii) By (i) we deduce that ǫ i and δ i are even for all 1 ≤ i ≤
3, otherwise the secondresidue form of θ K would be of dimension > a i , b i ] ∼ = [ u i ǫ i + δ i , v i ] for all 1 ≤ i ≤ δ i is even andthe isometry a [ b, c ] ∼ = [ ab, a − c ] for scalars a = 0 , b and c ).(iv) By the inequality (7.10.1), we have ǫ i + δ i ≤
0. Moreover, if for some i , we have ǫ i + δ i <
0, then the first residue form of [ u i ǫ i + δ i , v i ] is the degenerate form h u i , v i i , thisis excluded since ϕ is nondegenerate. Consequently, [ a i , b i ] = [ u i ǫ i , v i − ǫ i ] ∼ = 2 ǫ i [ u i , v i ] ∼ =[ u i , v i ] because ǫ i is even.Hence, θ ′ ∼ = ( θ ′′ ) K , where θ ′′ is the form [ u , v ] ⊥ [ u , v ] ⊥ [ u , v ].Now, the conditions that θ ′′ is defined over A and θ ∼ = ( θ ′′ ) K is a subform of θ K implythat θ ′′ is also a subform of θ over A . Taking the reduction modulo 2, we get that θ ′′ is asubform of ϕ . The form θ ′′ has determinant − A × because the scalar Q ≤ i ≤ (4 u i v i − ∈ A × is a representative of det( θ ′′ ) K = − K ∗ ∈ K ∗ /K ∗ . Since Ker γ = ± A × , it followsthat γ (det θ ′′ ) = Arf( θ ′′ ) = 0, which means that θ ′′ is an Albert form. Theorem 7.12.
For every nondegenerate form ϕ of dimension ≥ over k , CH ( X ϕ ) is elementary.Proof. Combine Theorem 7.11 with Lemma 7.9 and Prop. 7.6.In characteristic different from 2, Izhboldin completely determined when the groupCH ( X ϕ ) tors is trivial for all nondegenerate forms ϕ of dimension ≥ Proposition 7.13.
Let φ be a nondegenerate quadratic form over k satisfying one ofthe following conditions:1. dim φ = 12 , Arf( φ ) = 0 ∈ k/℘ ( k ) , and ind( φ ) ≤ .2. dim φ = 11 and ind( φ ) ≥ .3. dim φ = 10 , Arf( φ ) = 0 ∈ k/℘ ( k ) , and ind( φ ) = 2 .4. dim φ = 9 and ind( φ ) ≥ .Then CH ( X φ ) tors = 0 . The proof of Prop. 7.13 it may be given along the lines of the case treated in [Izh01],except possibly for the subcase with ind( φ ) = 2 in Prop. 7.13 (1), which can be provedby applying the same argument with n = 7. We shall not provide full details of the30roof, but content ourselves with the easy observation that the key ingredient we needis the characteristic 2 version of [Izh01, Lemma 1.19]. That is, it suffices to prove thefollowing: Lemma 7.14.
Let n be an integer ≥ and let φ be a nondegenerate quadratic form over k such that one of the following conditions holds:1. dim φ = 12 , Arf( φ ) = 0 ∈ k/℘ ( k ) , and ind( φ ) ≤ .2. dim φ = 11 and ind( φ ) ≥ .3. dim φ = 10 , Arf( φ ) = 0 ∈ k/℘ ( k ) , and ind( φ ) = 2 .4. dim φ = 9 and ind( φ ) ≥ .Then there exists a (2 n + 1) -dimensional nondegenerate form ˜ φ and a (2 n + 2) -dimensional nondegenerate form γ ∈ I q ( k ) such that φ ⊆ ˜ φ ⊆ γ and ind( ˜ φ ) = 1 . Below we provide a detailed the proof of Lemma 7.14, which seems to involve somemore subtleties than its counterpart in characteristic different from 2.First note that we have:
Lemma 7.15.
Let A be a central simple k -algebra of exponent ≤ , L/k a separable fieldextension of degree ≤ and m an integer. Suppose that one of the following conditionsholds:1. ind( A L ) = 1 and m = 2 .2. L = k , ind( A ) ≤ , and m = 3 .3. ind( A L ) ≤ and m = 4 .4. L = k , ind( A ) ≤ , and m = 5 .Then there exists an m -dimensional nondegenerate form µ over k such that the al-gebra C ′ ( µ ) has center L and is Brauer equivalent to A L .Proof. In Cases (1)–(3), one can use the same arguments in the proof of [Izh01, Lemma 1.17].It suffices to change the notations k ( √ d ) , hh d ii , h , − a, − b i , hh a, b ii in characteristic = 2to k [ t ] / ( t − t − d ) , hh d ]] , h a i⊥ [1 , b ] , hh a ; b ]] in characteristic 2 . In Case (4), A is Brauer equivalent to a biquaternion k -algebra, which gives rise to anAlbert form q = c [1 , a ] ⊥ ρ , where c ∈ k ∗ , a ∈ k and ρ is a 4-dimensional form with Arfinvariant a ∈ k/℘ ( k ). Then we can take µ = h c i⊥ ρ .The proof of Lemma 7.14 also relies on the lemma below.31 emma 7.16 (See [Izh98, Lemma 4.3] in characteristic = 2) . Let ϕ and ψ be nonsingular(hence even dimensional) quadratic forms over k with the same Arf invariant (so thealgebras C ′ ( ϕ ) and C ′ ( ψ ) have the same center). Suppose that C ′ ( ϕ ) and C ′ ( ψ ) areBrauer equivalent.Then there exists an element a ∈ k ∗ such that ϕ ⊥ a.ψ ∈ I q ( k ) .Proof. First assume ϕ and ψ have trivial Arf invariant, i.e. they lie in I q ( k ). Then theassumption implies that ϕ and ψ have the same Clifford invariant. So we can just take a = − ϕ and ψ have the same nontrivial Arf invariant d ∈ k/℘ ( k ). Then theirdiscriminant algebra L = k [ t ] / ( t − t − d ) is a quadratic separable field extension of k .By assumption in the Brauer group Br( L ) we have[ C ( ϕ ) L ] = [ C ( ϕ )] = [ C ( ψ )] = [ C ( ψ ) L ] . Thus the forms ˜ ϕ := ϕ ⊥ [1 , d ] and ˜ ψ = ψ ⊥ [1 , d ] lie in I q ( k ), and the forms ˜ ϕ L and ˜ ψ L have the same Clifford invariant. Thus, the Clifford invariant e ( ˜ ϕ − ˜ ψ ) of ˜ ϕ − ˜ ψ lies inBr( L/k ) = Ker(Br( k ) −→ Br( L )) . By the well known structure of the group Br(
L/k ), we have e ( ˜ ϕ ) − e ( ˜ ψ ) = ( a, d ] forsome a ∈ k ∗ . Note that e ( a ˜ ψ ) = e ( ˜ ψ ) since ˜ ψ ∈ I q ( k ). Thus e ( ˜ ϕ − a. ˜ ψ − hh a ; d ]]) = 0,and it follows that ϕ − a.ψ = ϕ − a.ψ + [1 , d ] − a [1 , d ] − hh a ; d ]] = ˜ ϕ − a. ˜ ψ − hh a ; d ]] ∈ I q ( k ) . This completes the proof.
Proof of Lemma . . Put m = 2 n + 2 − dim φ . We claim that we first prove the claimand then use it to deduce the lemma.In Cases (1) and (3), if φ ∈ I q ( k ), we put A = C ′ ( φ ) and L = k ; otherwise put A = C ( φ ) and let L be the center of C ′ ( φ ) = C ( φ ). Then A L is Brauer equivalentto C ′ ( φ ), and ind( φ ) = ind( A L ). By Cases (1) and (3) of Lemma 7.15, there exists an m -dimensional nondegenerate form µ over k such that C ′ ( φ ) and C ′ ( µ ) have the samecenter and are Brauer equivalent. Here φ and µ are even dimensional. So we can applyLemma 7.16 to find an element a ∈ k ∗ such that the form γ := φ ⊥ aµ lies in I q ( k ).Writing aµ = θ ⊥ c. [1 , b ] and setting ˜ φ = φ ⊥ θ ⊥h c i , we have[ C ( ˜ φ )] = [ C ( c.γ )] = [ C ( γ )] = 0 in Br( k )whence ind( ˜ φ ) = 1. Thus the forms γ and ˜ φ have the required properties, and we obtainthe desired result.Now consider Cases (2) and (4). We put A = C ′ ( φ ) and L = k . By Cases (2)and (4) of Lemma 7.15, there exists an m -dimensional nondegenerate form µ over k such that C ′ ( φ ) = C ( φ ) and C ′ ( µ ) = C ( µ ) are Brauer equivalent over k . Write φ = ρ ⊥h a i , µ = µ ⊥h ab i and choose representatives a , b ∈ k of the Arf invariantsArf( φ ) , Arf( µ ). Set φ := φ ⊥ a. [1 , a + u ] . C ( aφ )] = [ C ( φ )] and [ C ( abµ )] = [ C ( µ )] in Br( k ). Also, it is easy to see thatthe form γ := φ ⊥ bµ has trivial Arf invariant, i.e., γ ∈ I q ( k ). Now e ( aγ ) = e ( aφ ) − e ( abµ ) = [ C ( φ )] − [ C ( µ )] = 0 ∈ Br( k ) . It follows that aγ ∈ I q ( k ) and hence γ ∈ I q ( k ).Set ˜ φ := φ ⊥h a i⊥ bµ = φ ⊥ bµ . We have φ ⊆ ˜ φ = φ ⊥ bµ = φ ⊥h a i⊥ bµ ⊆ γ = φ ⊥ a. [1 , a + b ] ⊥ bµ , and 0 = [ C ( aγ )] = [ C ( φ ⊥h a i⊥ bµ )] = [ C ( ˜ φ )] ∈ Br( k ) . Hence ind( ˜ φ ) = 1. This completes the proof. Remark . One can also check that Corollary 3.10 and Lemmas 3.11 and 3.12 of[Izh01] extend to characteristic 2. Namely, for a nondegenerate quadratic form φ over k , the following statements hold:1. Suppose dim φ is even and >
8, the discriminant algebra L of φ is a field (i.e.Arf( φ ) = 0) and φ L is hyperbolic. Then CH ( X φ ) tors = 0.2. Suppose dim φ = 10, the discriminant algebra L of φ is a field (i.e. Arf( φ ) = 0)and φ = τ ⊥ c.N L/k for some c ∈ k ∗ and some subform τ . Then CH ( X φ ) tors = 0except possibly when the following conditions hold simultaneously:ind( φ ) = ind( τ L ) = 1 , ind( τ ) = 2 and φ L is not hyperbolic .
3. Suppose dim φ = 9, ind( φ ) > φ has one of the following forms:(i) φ = γ ⊥ [ a, b ], where a, b ∈ k and γ is a 7-dimensional Pfister neighbor.(ii) φ = τ ⊥h d i , where d ∈ k ∗ and τ ∈ I q ( k ).Then CH ( X φ ) tors = 0.Indeed, the above assertions follow on parallel lines along the proofs of the corre-sponding results in [Izh01], as all the necessary ingredients in characteristic 2 have beenestablished previously in this paper. Acknowledgements.
Yong Hu is supported by a grant from the National Natural ScienceFoundation of China (Project No. 11801260). Peng Sun is supported by the Fundamen-tal Research Funds for the Central Universities.33 eferences [AJ09] Roberto Aravire and Bill Jacob. H ( X, ν ) of conics and Witt kernels incharacteristic 2. In
Quadratic forms—algebra, arithmetic, and geometry ,volume 493 of
Contemp. Math. , pages 1–19. Amer. Math. Soc., Providence,RI, 2009.[Alb72] A. A. Albert. Tensor products of quaternion algebras.
Proc. Amer. Math.Soc. , 35:65–66, 1972.[Ara75] J´on Kr. Arason. Cohomologische invarianten quadratischer Formen.
J. Al-gebra , 36(3):448–491, 1975.[BCL20] Demba Barry, Adam Chapman, and Ahmed Laghribi. The descent of bi-quaternion algebras in characteristic two.
Israel J. Math. , 235(1):295–323,2020.[Dra75] Peter Draxl. ¨uber gemeinsame separabel-quadratische Zerf¨allungsk¨orper vonQuaternionenalgebren.
Nachr. Akad. Wiss. G¨ottingen Math.-Phys. Kl. II ,(16):251–259, 1975.[EKM08] Richard Elman, Nikita Karpenko, and Alexander Merkurjev.
The algebraicand geometric theory of quadratic forms , volume 56 of
American Mathe-matical Society Colloquium Publications . American Mathematical Society,Providence, RI, 2008.[Ful98] William Fulton.
Intersection theory , volume 2 of
Ergebnisse der Mathematikund ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics[Results in Mathematics and Related Areas. 3rd Series. A Series of ModernSurveys in Mathematics] . Springer-Verlag, Berlin, second edition, 1998.[GMS03] Skip Garibaldi, Alexander Merkurjev, and Jean-Pierre Serre.
Cohomologicalinvariants in Galois cohomology , volume 28 of
University Lecture Series .American Mathematical Society, Providence, RI, 2003.[HL04] Detlev W. Hoffmann and Ahmed Laghribi. Quadratic forms and Pfisterneighbors in characteristic 2.
Trans. Amer. Math. Soc. , 356(10):4019–4053,2004.[HSW21] Yong Hu, Peng Sun, and Zhengyao Wu. Unramified cohomology of quadricsin characterstic two. in preparation, 2021.[Izh98] O. T. Izhboldin. On the isotropy of quadratic forms over the function fieldof a quadric.
Algebra i Analiz , 10(1):32–57, 1998.[Izh01] Oleg T. Izhboldin. Fields of u -invariant 9. Ann. of Math. (2) , 154(3):529–587, 2001. 34Kah95] Bruno Kahn. Lower H -cohomology of higher-dimensional quadrics. Arch.Math. (Basel) , 65(3):244–250, 1995.[Kah96] Bruno Kahn. Applications of weight-two motivic cohomology.
Doc. Math. ,1:No. 17, 395–416, 1996.[Kah99] Bruno Kahn. Motivic cohomology of smooth geometrically cellular varieties.In
Algebraic K -theory (Seattle, WA, 1997) , volume 67 of Proc. Sympos. PureMath. , pages 149–174. Amer. Math. Soc., Providence, RI, 1999.[Kar90] N. A. Karpenko. Algebro-geometric invariants of quadratic forms.
Algebra iAnaliz , 2(1):141–162, 1990.[Kar91a] N. A. Karpenko. Chow groups of quadrics and the stabilization conjecture.In
Algebraic K -theory , volume 4 of Adv. Soviet Math. , pages 3–8. Amer.Math. Soc., Providence, RI, 1991.[Kar91b] N. A. Karpenko. Cycles of codimension 3 on a projective quadric.
Zap.Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) , 191(VoprosyTeor. Predstav. Algebr i Grupp. 1):114–123, 164, 1991.[Kar95] N. Karpenko. Chow groups of quadrics and index reduction formula.
NovaJ. Algebra Geom. , 3(4):357–379, 1995.[Kar96] Nikita A. Karpenko. Order of torsion in CH of quadrics. Doc. Math. , 1:No.02, 57–65, 1996.[Kat82a] Kazuya Kato. Galois cohomology of complete discrete valuation fields. In
Al-gebraic K -theory, Part II (Oberwolfach, 1980) , volume 967 of Lecture Notesin 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.[KM90] N. A. Karpenko and A. S. Merkurjev. Chow groups of projective quadrics.
Algebra i Analiz , 2(3):218–235, 1990.[KMRT98] Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tig-nol.
The book of involutions , volume 44 of
American Mathematical SocietyColloquium Publications . American Mathematical Society, Providence, RI,1998. With a preface in French by J. Tits.[Kne69] M. Knebusch. Isometrien ¨uber semilokalen Ringen.
Math. Z. , (108):255–268,1969.[Knu88] Max-Albert Knus.
Quadratic forms, Clifford algebras and spinors , volume 1of
Semin´arios de Matem´atica [Seminars in Mathematics] . Universidade Es-tadual de Campinas, Instituto de Matem´atica, Estat´ıstica e Ciˆencia da Com-puta¸c˜ao, Campinas, 1988. 35KRS98] Bruno Kahn, Markus Rost, and R. Sujatha. Unramified cohomology ofquadrics. I.
Amer. J. Math. , 120(4):841–891, 1998.[KS00] Bruno Kahn and R. Sujatha. Motivic cohomology and unramified cohomol-ogy of quadrics.
J. Eur. Math. Soc. (JEMS) , 2(2):145–177, 2000.[KS01] Bruno Kahn and R. Sujatha. Unramified cohomology of quadrics. II.
DukeMath. J. , 106(3):449–484, 2001.[Lag15] Ahmed Laghribi. Quelques invariants de corps de caract´eristique 2 li´es auˆ u -invariant. Bull. Sci. Math. , 139(7):806–828, 2015.[Lam05] T. Y. Lam.
Introduction to quadratic forms over fields , volume 67 of
Grad-uate Studies in Mathematics . American Mathematical Society, Providence,RI, 2005.[Mer95a] A. S. Merkurjev. Certain K -cohomology groups of Severi-Brauer varieties.In K -theory and algebraic geometry: connections with quadratic forms anddivision algebras (Santa Barbara, CA, 1992) , volume 58 of Proc. Sympos.Pure Math. , pages 319–331. Amer. Math. Soc., Providence, RI, 1995.[Mer95b] A. S. Merkurjev. K -theory of simple algebras. In K -theory and algebraicgeometry: connections with quadratic forms and division algebras (SantaBarbara, CA, 1992) , volume 58 of Proc. Sympos. Pure Math. , pages 65–83.Amer. Math. Soc., Providence, RI, 1995.[MMW91] P. Mammone, R. Moresi, and A. R. Wadsworth. u -invariants of fields ofcharacteristic 2. Math. Z. , (208):335–347, 1991.[Pie82] Richard S. Pierce.
Associative algebras , volume 88 of
Graduate Texts inMathematics . Springer-Verlag, New York-Berlin, 1982. Studies in the Historyof Modern Science, 9.[Qui73] Daniel Quillen. Higher algebraic K -theory. I. In Algebraic K -theory, I:Higher K -theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash.,1972) spin K -theory and the norm-residue homomorphism. J.Soviet Math. , :2556–2611, 1985.36Swa85] Richard G. Swan. K -theory of quadric hypersurfaces. Ann. of Math. (2) ,122(1):113–153, 1985.[Swa89] Richard G. Swan. Zero cycles on quadric hypersurfaces.
Proc. Amer. Math.Soc. , 107(1):43–46, 1989.[Tie74] Uwe-Peter Tietze. Zur theorie quadratischer Formen ¨uber Hensel K¨orpern.
Arch. Math. , (25):144–150, 1974.[Tot08] Burt Totaro. Birational geometry of quadrics in characteristic 2.
J. AlgebraicGeom. , 17(3):577–597, 2008.[Wad85] A. Wadsworth. Discriminants in characteristic two.