Chow rings of excellent quadrics

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1 Journal of Pure and Applied Algebra 212 (2008) Contents lists available at ScienceDirect Journal of Pure and Applied Algebra journal homepage: Chow rings of excellent quadrics Nobuaki Yagita Department of Mathematics, Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan a r t i c l e i n f o a b s t r a c t Article history: Received 10 December 2006 Received in revised form 25 February 2008 Available online 2 May 2008 Communicated by A.S. Merkurjev In this paper we compute the multiplicative structure of the Chow ring of an excellent anisotropic quadric by using the algebraic cobordism theory Elsevier B.V. All rights reserved. MSC: Primary: 11E04 14C15 secondary: 55R35 57T05 1. Introduction Let k be a field of ch(k) = 0. A quadratic form ξ over k is called excellent if for every field extension K/k, the anisotropic part of the form ξ K is defined over k. Pfister forms, norm forms, and all forms over R are typical examples of excellent forms. The quadric X ξ defined by an excellent form ξ is called excellent. The structure of the motive M(X ξ ) is known by Rost, Hoffmann, Karpenko Merkurjev [11,12,3,5,14,15]. This gives the additive structure of the Chow ring CH (X ξ ). In this paper, we determine the multiplicative structures for the cases dim(x ξ ) 2 mod(4). For these cases, they depend only on dim(x ξ ). Moreover, we see that some torsion elements in CH (X ξ ) are divisible by the large power of the hyperplane h. Theorem 1.1. Let X ξ be an excellent anisotropic quadric with 2 n 1 dim(x ξ ) = d 2 n+1 2. Then there are elements u 1 (d),..., u n 1 (d) (and u 0 (d) when d = 2 mod(4)) in CH (X ξ ) and positive integers d 1 (d) d n 1 (d) such that there is the Z[h]-algebra isomorphism CH (X ξ ) = F n 1 i=1 Z/2[h]/(hd i(d) )u i (d)} Z[h]/(h d+1 ) Zu where F = 0 (d)} for d = 2 mod(4) Z[h]/(h d+1 ) otherwise with multiplication hu 0 (d) = h d/2+1 mod(u i (d) 1 i n 1) and u i (d)u j (d) = 0 for all i, j. Here h is the hyperplane section and deg(h) = 1. The degree u i (d) and d i (d) are somewhat complex integers, and will be given in Section 5 (see Theorem 5.1 and (3.1) (3.3)). The fact that the torsion elements in CH (X ξ ) are divisible by the large power of h implies that its subquadric of not large codimension also has torsion in the Chow ring. Corollary 1.2. Let ψ be a subform of an anisotropic excellent form ξ of codim = c. Let emb : X ψ X ξ be the induced embedding. Then the induced map emb : CH (X ξ )/(h d+1 c, h d i(d) c u i (d) 1 i n 1) CH (X ψ ) is injective for d = odd and Ker(emb ) Zh d/2 u 0 (d)} for d = even. address: yagita@mx.ibaraki.ac.jp /$ see front matter 2008 Elsevier B.V. All rights reserved. doi: /j.jpaa

2 N. Yagita / Journal of Pure and Applied Algebra 212 (2008) The above corollary is almost immediate consequence from the fact emb emb = h c. Notice that the above corollary does not follow from the motivic decomposition, as the motive of the specified subquadrics do not necessarily contain any Rost motive as a direct summand. The arguments for the proof of the above theorem are the following. Let Ω (X) be the algebraic cobordism defined by Levine and Morel [8 10] or the motivic cobordism MGL 2, (X) defined by Voevodsky [16,17]. Let k be an algebraic closure of k and X k = X k k its extension. We will prove that the restriction map i k : Ω (X ξ ) Ω (X ξ k) is injective by using the result of Vishik Yagita [15]. Each quadric X ξ k is cellular and its ring structure of CH (X ξ k) is given by Rost [12]. From this, we also know some multiplicative structures of Ω (X ξ k). The multiplicative structure of CH (X ξ ) is deduced from CH (X ξ ) = Ω (X ξ ) Ω Z = i k(ω (X ξ )) Ω Z. Alexander Vishik taught me the theory of quadratic forms. Many ideas in this paper are due to him. The referee also corrected serious errors in the first versions of this paper. I am very glad to thank them very much. 2. Rost motive Let k be a field of ch(k) = 0 and X the smooth variety. We consider the Chow ring CH (X) generated by cycles modulo rational equivalence. For a non zero symbol a = a 0,..., a n } in the mod 2 Milnor K-theory Kn+1 M (k)/2, let φ a = a 0,..., a n be the (n + 1)-fold Pfister form. Let X φa be the projective quadric of dimension 2 n+1 2 defined by φ a. The Rost motive M a (= M φa ) is a direct summand of the motive M(X φa ) representing X φa so that M(X φa ) = M a M(P 2n 1 ). The Chow ring of the Rost motive is well known (in fact, CH (M a ) has the natural ring structure which is explicitly computed in [15]). Let k be an algebraic closure of k, X k = X k k, and i k : CH (X) CH (X k) the restriction map. Lemma 2.1 (Rost [11,12,15]). The Chow ring CH (M a ) is only dependent on n. There are isomorphisms CH (M a ) = Z1, c n,0 } Z/2c n,1,..., c n,n 1 } c n,i = 2 n 2 i, and CH (M a k) = Z1, ᾱ n } with ᾱ n = 2 n 1. Here the multiplications are given by ᾱ 2 = 0 and c n,i c n,j = 0 for all 0 i, j n 1. Moreover the restriction map is given by i k(c n,0 ) = 2ᾱ n and i k(c n,i ) = 0 for i > 0. Let us use notation Ω (X) for the algebraic cobordism defined by Morel and Levine or the motivic cobordism MGL 2, (X) defined by Voevodsky. It is known ([9], Corollary 3.7 in [18]) that Ω = Ω (pt.) = MU 2 (pt.) = Z[x 1, x 2,...] where MU 2 (pt.) is the complex cobordism ring and x i = i. There is the relation ([9], Corollary 3.8 in [18]) Ω (X) Ω Z = CH (X). Let s i be the additive characteristic class (for details, see [1,13,16]). The ring generator x j of Ω is characterized (e.g. page 128 in [13]) by ±p mod(p2 ) if j = p s j (x j ) = n 1 for n, prime p ±1 otherwise. Moreover we can take x 2 n 1 such that all characteristic numbers are divided by 2. Let us write by v n such a generator x 2 n 1. We note that v n is represented by a quadric of dimension 2 n 1 and defined only mod(i n ) where I n = (2 = v 0, v 1,..., v n 1 ) Ω is the ideal of Ω generated by 2, v 1,..., v n 1. It is well known that I n and I are the only prime ideals (which do not contain odd numbers) stable under the Landweber Novikov cohomology operations [7] in Ω (see the proof of Lemma 4.4). The category of cobordism motives is defined and studied in [15]. In particular, we can define the algebraic cobordism of motives. The following is the main result in [15]. Lemma 2.2 ([15]). There is the Ω -module isomorphism Ω (M a ) = Ω 1} I n α n } Ω 1, α n } α n = 2 n 1 such that v i α n = c n,i in Ω (M a ) Ω Z = CH (M a ) in (2.2). Moreover the restriction map i k : Ω (M a ) Ω (M a k) = Ω 1, ᾱ n } is injective and is given by i k(v i α n ) = v i ᾱ n. Note that v i α n is defined over k but α n itself is not. We also note that Ω (M a ) is torsion free but CH (M a ) has torsion. (2.1) (2.2)

3 2442 N. Yagita / Journal of Pure and Applied Algebra 212 (2008) Excellent forms Now we consider the quadrics defined by a subform ξ of the Pfister form φ a. Recall that T is the Tate motive i.e., M(P 1 ) = T 0 T. By using results of Rost and Hoffmann, we see the following theorem. Theorem 3.1 (Rost, Hoffmann [11,12,3,5,15]). Let ξ be a subform of the Pfister form φ a = a 0,..., a n of dim(ξ) = 2 n +m, 2 n m > 0 (i.e., ξ is a neighbor of φ a ). Let η be a complementary form (φ a = ξ η). Then M(X ξ ) = M a M(P m 1 ) M(X η ) T m. The direct corollary of this theorem is the isomorphism given in (2.1). Let us write CH (M a ) simply by CM n. Let ξ = ξ 0 and m = m 0. Suppose that η = ξ 1 is itself a neighbor of a 0,..., a n1, n 1 < n = n 0 and ξ 2 be its complementary form. Applying the above theorem, we have the additive isomorphisms CH (X ξ0 ) = CM n0 [t]/(t m 0 ) CH (X ξ1 )t m 0 } = CM n0 [t]/(t m 0 ) CM n1 [t]/(t m 1 )t m 0 } CH (X ξ2 )t m 0+m 1 }. Here t i is a generator of CH i (T i ) = Z but the above isomorphisms do not preserve the product structure. Now we recall the definition of excellent quadrics [5,6]. A quadratic form ξ over k is called excellent if for every field extension K/k, the anisotropic part of ξ K is defined over k. An anisotropic form is excellent if and only if it is a Pfister neighbor whose complementary form is excellent also [5,6]. Suppose that ξ = ξ 0 is excellent. Then we have a decreasing sequence π 0 π 1 π r of embedded Pfister forms such that the class [ξ k ] in the Witt ring is given by [ξ k ] = [π k ] [π k+1 ]+ +( 1) r k [π r ], namely, [ξ i ] + [ξ i+1 ] = [π i ]. Let us write dim(π i ) = 2 n i+1. Then n 0 > n 1 > > n r and dim(ξ) = 2 n n ( 1) r 2 nr+1. (3.1) Thus n i are the places changing zero to one (or one to zero) in the 2-adic expansion of dim(x ξ ) + 2 = d + 2. Let us write m j = 1/2(dim(ξ j ) dim(ξ j+1 )) = 2 n j 2 n j ( 1) r j 2 nr+1, (3.2) s j = m m j 1 = 1/2(dim(ξ 0 ) dim(ξ j )) 2 n 0 2 n n j 2 2 n j 1 j : even = 2 n 0 2 n n j n j 1 2 nj ( 1) r 2 nr+1 j : odd. Note that s r+1 = [1/2 dim(ξ)]. Then we can see inductively Lemma 3.2 (Rost [11,5]). There is an isomorphism of motives M(X ξ ) = r i=0 M π i M(P m i 1 ) T s i. (3.3) When dim(ξ) = odd, we see π r = 1 and dim(x πr ) = 1. So the respective term should be omitted. Corollary 3.3. Let r = r for d = even and r = r 1 otherwise. There is an additive isomorphism CH (X ξ ) = r i=0 CM n i [t]/(t m i )ts i }. Let CM n = Z1} J n, namely J n = Zc n,0 } Z/2c n,1,..., c n,n 1 }. Let e = d/2 for d = even and e = (d 1)/2 for d = odd. Recall that s r +1 = 1/2 dim(ξ) = e + 1 for d = even, and s r +1 = 1/2(dim(ξ) 1) = e + 1 for d = odd. Hence we have the additive isomorphisms r i=0 Z[t]/(tm i )ts i } = Z[t]/(t e+1 ), J ni [t]/(t m i )ts i } = J ni t s s i s < s i+1 }. Corollary 3.4. There is an additive isomorphism CH (X ξ ) = Z[t]/(t e+1 ) r i=0 J n i t s s i s < s i+1 }.

4 N. Yagita / Journal of Pure and Applied Algebra 212 (2008) The decomposition of the (Chow) motive in Lemma 3.2 also gives that of the cobordism motive from [15]. Hence from Lemma 2.2, we get the Ω -module isomorphisms Ω (X ξ ) = r i=0 (Ω 1} I ni α ni )[t]/(t m i )ts i }, Ω (X ξ k) = r i=0 (Ω 1} Ω ᾱ ni )[t]/(t m i )ts i }. Since I ni [t]/(t m i )t s iαni } = I ni t s α ni s i s < s i+1 }, we also have; Corollary 3.5. The map i k : Ω (X ξ ) Ω (X ξ k) is injective and there is an Ω -module isomorphism Ω (X ξ ) = Ω [t]/(t e+1 ) r i=0 I n i t s α ni s i s < s i+1 }. Let us write s i = t s i α n i = t s i c ni,0 = s i + 2 n i 1. We write the picture for s i, s i and m i for small i s; 0=s 0 m 0 s m 1 s m 2 s e+1 s m 2 s 1 m s 1 0 m 0 d+1 In fact, we see that s i 1 s i = (s i n i 1 1) (s i + 2 n i 1) = m i 1 + (2 n i 1 2 n i ) = ( 2 n i n i+1 2 n i ) + (2 n i 1 2 n i ) = m i. 4. Algebraic cobordism of quadrics Let ψ be a (not assumed to be excellent) quadratic form. For each quadric X ψ, the Chow ring of X ψ k = X ψ k is given by Rost. Let dim(x ψ ) = d. Let h (resp. ᾱ) be an element of CH (X ψ k) which is represented by a hyperplane section (resp. a maximal projective space) in X ψ k. So h = 1 and ᾱ = e if d = even ( ᾱ = e + 1 for d = odd). Theorem 4.1 ([12]). There is an isomorphism of rings CH (X ψ k) = Z1, h,..., he } Z[ h]/( he+1 )ᾱ}. The multiplication of CH (X ψ k) is given by (1) he+1 = 2 hᾱ for d = even ( he+1 = 2ᾱ for d = odd), (2) ᾱ 2 = heᾱ if d = 0 mod(4)(ᾱ 2 = 0 otherwise). Here notice that h Im(i k) but ᾱ Im(i k), in general. Next consider Ω ( ) version. We denote the respective cobordism classes by the same symbols h and ᾱ. Since Xψ k is cellular, we know from [9,10] that Ω (X ψ k) is Ω -free. From (2.2) and the above theorem, we see Corollary 4.2. There is the Ω -algebra isomorphism Ω (X ψ k) = Ω 1, h,..., he } Ω [ h]/( he+1 )ᾱ}. The multiplication of Ω (X ψ k) is given by (2) in the theorem and h e+1 = 2 hᾱ mod(ω <0 ) if d = even, (= 2ᾱ mod(ω <0 ) d = odd) where Ω <0 = Ideal(x 1, x 2,..) the negative degree part of Ω. Let F i be the Ω -submodule of Ω (X ψ k) generated by all elements of degree i, i.e., Ω hi,..., he, ᾱ,..., heᾱ} for i e F i = Ω hi ᾱ ᾱ,..., heᾱ} for i > e. Here note that F i coincides with the Ω -subalgebra generated by elements of degree i, and moreover is an ideal of Ω (X ψ k). Then F i gives the filtration of Ω (X ψ k) and let (see also in [10]) gr Ω (X ψ k) = F i /F i+1. Of course, as Ω -modules, gr Ω (X ψ k) = Ω (X ψ k), because these are Ω -free modules.

5 2444 N. Yagita / Journal of Pure and Applied Algebra 212 (2008) By the definition, the ring structure of i F i /F i+1 is deduced from that of CH (X ψ k). Hence we have the Ω -algebra isomorphism gr Ω (X ψ k) = Ω CH (X ψ k). (See Corollary 4.5.8, Remark in [10] for more general treatments.) For the map i k : Ω (X ψ ) Ω (X ψ k), the Ω -subalgebras F i Im(i k) give the filtration of Im(i k), and denote by gr Im(i k) its graded algebra, i.e., gr (Im(i k)) = i (Im(i k) F i )/(Im(i k) F i+1 ) = i (((Im(i k) F i ) + F i+1 )/F i+1 ) gr Ω (X ψ k). Let us write p i : F i F i /F i+1 gr Ω (X ψ k) and p ᾱ (ᾱ) = β (let us write p 1 ( h) by the same letter h). For a = (a 1, a 2,...), a i 0, let S a be the Landweber Novikov operation [1,2,10,18] S a : Ω (X) Ω + a (X) a = a i i. Then the following Cartan formula holds S a (xy) = S a (x)s a (y). a=a +a Here if deg(y) i, then deg(s a (y)) i. This means S a (F i ) F i. Hence Landweber Novikov operations act on gr Ω (X ψ k). Moreover by the naturality of such operations, S a also acts on gr (Im(i k)). Lemma 4.3. There is an Ω -algebra isomorphism gr (Im(i k)) = Ω 1,..., he } e s=0 L s hs β} Ω CH (X ψ k) where L 0 L 1 L e is a sequence of invariant ideals in Ω with 2 L e. Moreover if X ψ is anisotropic, then L e Ω. Proof. First note that an Ω -subalgebra of Ω is an ideal L in Ω. For each 0 i d, we see gr i Ω (X ψ k) = Ω (or = Ω he } Ω β} for i = e and d = even). Since gr (Im(i k)) is an Ω -subalgebra of gr Ω (X ψ k), there exists the above decomposition for some ideals L s in Ω. We will show that each L s is invariant. Suppose that x gr Im(i k) and x = b hsᾱ with b Ω, namely b L s. By the Cartan formula S a (x) = S a (b)s a ( hsᾱ) = S a (b) hsᾱ mod(f s+ ᾱ +1 ), a=a +a which is also in gr Im(i k). Hence S a (b) L s and this means that the ideal L s is invariant. Let b L s. Then b hsᾱ gr Im(i k). If w s, then by multiplying hw s, we see b hwᾱ gr Im(i k). This means b L w. Hence L s L w. Since dim(x ψ ) = d, we still know Ω d (X ψ k) = CH d (X ψ k) = Z heᾱ}. It follows from hd = 2h e ᾱ that 2 L e. If X ψ is anisotropic, then heᾱ Im(i k) and hence L e Ω. Let us write the ideal I Ω = (2, x 1, x 2,...) = (2, Ω <0 ) Ω. Of course I I Ω. By Landweber [7], it is known that all prime invariant ideals (which do not contain odd number) are I n and I. Lemma 4.4. Let L be an invariant ideal of Ω. Let L = (L + I 2 Ω )/(I2 Ω ) be the induced ideal in Ω /(I 2 Ω ). Then L is isomorphic to one of (0), Ω /(I 2 Ω ) or Ĩ n = Z/22, v1,..., v n 1 } for n 1 or n =. Proof. Let us write by L = (L + I 2 )/(I2 ) the induced ideal in Ω /(I 2 ). Since I is invariant, L is also invariant in Ω /(I 2 ) from the Cartan formula. It is immediate Ω /(I 2 ) = Ω (2) /(I2 ) from 4 I2. Here we recall the BP ( ) theory with the coefficient ring BP = Z (2) [v 1,...]. The complex cobordism theory MU ( ) (2) is covered by BP-theory by the famous (Quillen Novikov) natural isomorphism ([1,2]) MU (X) (2) = BP (X) N where N = Z (2) [x j j 2 i 1]. Hence cohomology operations in BP-theory are N-linearly extended to those in MU (2) -theory.

6 N. Yagita / Journal of Pure and Applied Algebra 212 (2008) In BP-theory, there exists the cohomology operation R i (= r 2 i 1 1 in the notation in (1.8) in [4] or ( ) in [2]) such that R i (v i ) = v i 1 mod(i i 1 ) in BP. Moreover since BP <0 I, we see R i (v i ) = v i 1 mod(i 2 i 1 ). Suppose v i L. It follows v i 1 L from (4.1). Thus we see (2, v 1,..., v i 1, v i ) L. If x j N L, then s j (x j ) = ±1 mod(4) and hence L = Ω /(I 2 ). The only nonzero homogeneous elements in Ω<0 /(I 2 ) are x i, that is, Ω /(I 2 Ω ) = Z/41} Z/2x 1, x 2,...}. Hence if L (0) and Ω /(I 2 Ω ), then we have L = Z/22, v1,..., v n 1 }. Let GrΩ (X ψ ) be the Ω /I 2 Ω -subalgebra of gr Ω (X ψ k)/(i 2 Ω ) generated by elements in gr (Im(i k)). That is, GrΩ (X ψ ) = i Gr i where Gr i = (Im(i k) F i )/(Im(i k) (I 2 Ω F i + F i+1 )) gr i (Ω (X ψ k))/(i 2 Ω gr i (Ω (X ψ k))). From Lemma 4.4, we have Lemma 4.5. There is a sequence 0 l 0 l e such that there is an Ω /I 2 Ω-algebra isomorphism GrΩ (X ψ ) = Ω /I 2 Ω 1, h,..., he } e s=0 Z/22, v 1..., v ls 1} hs β}. Note that v i hs β (0 i l s 1 ) in the above lemma is not in the ideal (I Ω ) of GrΩ (X ψ ) because hs β does not exist in GrΩ (X ψ ). Indeed, we see I Ω GrΩ (X ψ ) = I Ω 1, h,..., he } GrΩ (X ψ ). (4.3) Hence we can write explicitly (1) Z/2 hi } i < e or i = e and d = odd Gr i /(I Ω ) (2) = Z/2 he } Z/22,..., v l0 1}β} i = e (d = even) (3) Z/22,..., v ls 1} hs β} s = i e > 0 (d = even) or s = i e 1 0 (d = odd). Of course, one of the most important facts is the relation between CH (X ψ ) and GrΩ (X ψ ). Theorem 4.6. There is a filtration F i of CH (X ψ )/2 such that there is an epimorphism of rings j ψ : gr CH (X ψ )/2 = i F i /F i+1 GrΩ (X ψ )/(I Ω ). Proof. Since CH (X)/2 = Ω (X) Ω Z/2, we have the surjection j ψ : CH (X ψ )/2 Im(i k) Ω Z/2 = Im(i k)/(i Ω Im(i k)). Let us write gr (Im(i k)/(i Ω Im(i k))) = F i /F i+1 where F i = (F i Im(i k))/(f i I Ω Im(i k)). We define the filtration F i = (j ψ ) 1 (F i ) of CH (X ψ ) so that there is the surjective map jψ : gr CH (X ψ )/2 gr (Im(i k)/i Ω Im(i k)). Hence for the existence of the surjective map j ψ in this theorem, we need the surjective map j ψ : gr (Im(i k)/i Ω Im(i k)) GrΩ (X ψ )/(I Ω ) so that j ψ = j ψ jψ. Here note F i /F i+1 = (F i Im(i k))/(f i I Ω Im(i k) + F i+1 Im(i k)). By the definition (4.2) of Gr i and (4.3), for the existence of the above map j ψ, it is sufficient to show that (4.1) (4.2) (4.4) F i I Ω Im(i k) I 2 Ω F i + I Ω 1, h,..., he }. (Of course F i I Ω Im(i k) Im(i k).) (4.5)

7 2446 N. Yagita / Journal of Pure and Applied Algebra 212 (2008) Let x Im(i k) and x 0 in F j /F j+1. If j > e, then from (3) in (4.4), we have x I ls F j + I 2 Ω F j + Ω x j 1 F j+1. Here x j 1 < 0, since x j. Hence I Ω x I 2 Ω F j. Similarly, when j e, from (1),(2) in (4.4), we see I Ω x I Ω hj + I 2 Ω F j. Thus we have I Ω Im(i k) I Ω 1, h,..., he } + I 2 Ω F 0. Since there is the Ω -module isomorphism gr Ω (X ψ k) = Ω (X ψ k) (in fact, they are Ω -free), we see F i I 2 Ω F 0 = I 2 Ω F i. Then we see (F i (I Ω 1, h,..., he } + I 2 Ω F 0))/(F i I 2 Ω F 0) F i /(I 2 Ω F i) (I Ω 1, h,..., he } + I 2 Ω F 0)/(I 2 Ω F 0) (4.7) = F i /(I 2 Ω F i) I Ω /(I 2 Ω )1, h,..., he } in F 0 /(I 2 Ω F 0). (4.8) Let i e. Then by the definition of F i, we see F i /(I 2 Ω F i) = Ω /(I 2 Ω ) hi,..., he, ᾱ,..., heᾱ}. Therefore the Ω /(I 2 Ω )-module (4.8) is isomorphic to I Ω /(I 2 Ω ) hi,..., he }. Thus, from (4.6) and (4.7), we have F i I Ω Im(i k) I Ω hi,..., he } + I 2 Ω F i, (4.6) which implies (4.5) for i e. When i > e, by the definition of F i, the Ω /(I 2 Ω )-module (4.8) is zero. So (4.5) follows from (4.6) also. Here we give the another expression of GrΩ (X ψ )/(I Ω ) quite explicitly. Theorem 4.7. Let l i be the numbers in the preceding Lemma 4.5. Let f j be the minimal numbers such that v j hf jβ GrΩ (X ψ ) and d j = e + 1 f j. Then we have the ring isomorphism GrΩ (X ψ )/(I Ω ) = F/2 le 1 j=1 Z/2[ h]/( hd j )u j } where F/2 = Z/2[ h]/( hd+1 ) if d = odd or l 0 = 0 Z/2[ h]/( hd+1 ) Z/2u 0 } otherwise with u j = v j hf jβ = 2j f j + e and u j Gr f j+e Ω (X ψ k), where e = e (resp. e = e + 1) for d = even (resp. odd). Here the multiplications are given by u i u j = 0 for all i, j, and hu0 = hd/2+1 mod(u i 1 i < l e ) when d = even and l 0 0. Proof. Recall that GrΩ (X ψ )/(I Ω ) is isomorphic to Z/21,..., he } e s=0 Z/22,...., v l s 1} hs β} ( ) as Ω /(I Ω )-modules. Given 0 j l e 1, let f j be the smallest f such that v j hf β GrΩ (X ψ ), that is, v j hf j β Z/22,..., v lfj 1} hf j β} but v j hf j 1 β Z/22,..., v lfj 1 1} hf j 1 β}. This means l fj 1 j < l fj. Then ( ) is rewritten by exchanging the orders of the summation F/2 le 1 j=1 Z/2[ h]/( hd j )v j hf j β} with d j = e + 1 f j. (Note he+1 = 2β (or 2 hβ) in CH (X ψ k) for d = odd (otherwise).) Writing u j = v j hf jβ Gr f j + β Ω (X ψ k) and u 0 = 2β for d = even and l 0 0, we have the isomorphism in this lemma. The relation u i u j = 0 follows from v i hf i β v j hf j β = v j v i hf i +f j β 2 = 0 mod(i 2 Ω gr Ω (X ψ k)). The relation hu0 = hd/2+1 mod(u i 1 i < l s ) follows from 2 hβ = hd/2+1 mod(ω <0 ) in gr Ω (X ψ k).

8 N. Yagita / Journal of Pure and Applied Algebra 212 (2008) Corollary 4.8. Let ψ be a subform of a form ψ of codim = c. Let emb : X ψ X ψ be the induced embedding. Then the induced map emb : GrΩ (X ψ )/(Gr d+1 c, I Ω ) GrΩ (X ψ )/(I Ω ) is injective for d = odd and Ker(emb ) Z/2 hd/2 u 0 } for d = even and l 0 0 (where Gr d+1 c = i d+1 c Gr i Ω (X ψ )). Proof. By the definition of emb, we see emb (1) = hc. Hence emb emb = hc, in fact, emb (x) = x emb (1) = x hc for x emb GrΩ (X ψ ). Since hc Gr Ω (X ψ )/(Gr d+1 c, I Ω ) GrΩ (X ψ )/(I Ω ) is injective for d = odd from the preceding theorem. The case d = even, we know Ker( hc ) Z/2 hd/2 u 0 } + Gr d c+1. Hence we get this corollary. 5. The proof of main theorem In this section, we prove the main theorem. Recall that X ξ is an excellent anisotropic quadric with 2 n 1 dim(x ξ ) = d 2 n+1 2. Theorem 5.1. There are elements u 1,..., u n 1 (and u 0 when d = 2 mod(4)) in CH (X ξ ) and positive integers d 1 d n 1 such that there is the Z[h]-algebra isomorphism CH (X ξ ) = F n 1 i=1 Z/2[h]/(hd i )u } i Z[h]/(h d+1 ) Zu 0 where F = } for d = 2 mod(4) Z[h]/(h d+1 ) otherwise with multiplication hu 0 = h d/2+1 mod(u i 1 i n) and u i u j = 0 for all i, j. The degree is given as follows ; if n i+1 j < n i then d j = s i+1 and u j = 2n i 2 j + s i (see (3.1) (3.3)). Proof. Recall that the Ω -module structure of Ω (X ξ ) is given in Corollary 3.5. Namely, there is an Ω -module (but not rings) isomorphism Ω (X ξ ) = Ω [t]/(t e+1 ) r i=0 I n i t s α ni s i s < s i+1 }. When s i s < s i+1, let us write s = s s (s) e = s + 2 n i 1 e for d = even = s e 1 = s + 2 n i 2 e for d = odd (5.1) so that t s α ni = hs β. Let us write s i = s (s i ). Since F j /F j+1 is a free Ω -module of rank = 1 (or = 2 when j = e and even), we know GrΩ (X ξ ) = Ω /(I 2 Ω )1,..., he } r i=0 Z/22, v 1,..., v ni 1} hs β s i s < s i 1 }. (Recall the picture of s i and s i in the end of Section 3.) Let n i+1 j n i 1 such that v j I ni but v j I ni+1. Then we see v j hs i β GrΩ (X ξ ). Moreover this s i is the smallest f j. In fact for s < s i, we see v jh s fβ GrΩ (X ξ ) since v j I ni+1. Hence d j = d + 1 s i = s i+1 (recall the picture after Corollary 3.5). Then the surjective map j ψ in Theorem 4.6 is really an isomorphism j ξ : gr CH (X ξ )/2 = GrΩ (X ξ )/(I Ω ) because we already know the Z/2-module structure of CH (X ξ )/2 from Corollary 3.4. The multiplicative structure of GrΩ (X ξ )/(I Ω ) is given by h d j u j = 0, u j u k = 0, hu0 = he+1 mod(u i 1 i n). For j > 0, let u j CH (X ξ )/2 be a torsion element such that it maps to u j = v j hs i β by the isomorphism j ξ. Since u j Gr d+1 d j Ω (X ψ k), we see u j F d+1 d j. So h d j u j F d+1. Hence it is zero in CH (X ξ )/2 since F d+1 = 0. Moreover u j u k F d+1 = 0 for dimensional reasons. Thus for d 2 mod(4), we see that the map j ξ induces the isomorphism jξ : CH (X ξ )/2 = GrΩ (X ξ )/(I Ω ). Note that this shows the isomorphism in the theorem with mod(2).

9 2448 N. Yagita / Journal of Pure and Applied Algebra 212 (2008) When d = 2 mod(4), we can take u 0 = 2ᾱ in F e using the fact 2ᾱ Im(i k). (However when d = 0 mod(4), the number n r = 0 and I nr β} = 0}, so we see 2α Im(i k) and u 0 is not defined.) Then 2 hᾱ = he+1 mod(ω <0 Ω (X ψ k)) implies u 0 h = he+1 mod(u j 1 j n) in CH (X ξ )/2, since CH (X ξ )/2 is multiplicatively generated by u j and h. Thus we can prove the isomorphism in the theorem with mod(2) also for this case. We already know that in CH (X ξ ) the elements h i and u 0 are torsion free but c j, j > 0 are just (not higher) 2-torsion. Indeed ξ splits over a quadratic extension of k. Hence we get the isomorphism in the theorem. From the Corollary 4.8, we get Corollary 1.2 in the introduction. For excellent X ξ, we can compute Im(i k) (but not only grim(i k)) more directly. Recall the notation of s in proof (5.1) of the preceding theorem. (We think s = s (s) (resp. s = s(s )) as a function of s (resp. s ) here.) Proposition 5.2. As an Ω -subalgebra of Ω (X ψ k), the restriction image of Ω (X ξ ) is written as Im(i k) = Ω hs 0 s e} r i=0 I n i hs (s)ᾱ s i s < s i+1 }. Proof. From Corollary 3.5 and the arguments of the first parts of the proof of Theorem 5.1, we rather see that gr Im(i k) is isomorphic to the right hand side Ω -module in the proposition. Of course hs Im(i k). Suppose that I ni hs ᾱ} Im(i k) for s i s < s i+1. (5.2) Then the right hand side Ω -module in this proposition is contained in Im(i k). Since gr Im(i k) is isomorphic to the right hand module, we get the proposition. Hence we only need to prove (5.2). Given 0 w e, by induction, we assume (5.2) for all s > w. Let s j s(w) < s j+1. From Corollary 3.5 and Theorem 4.1, we already know that there is an Ω -module generator t w = t s(w) α nj Ω (X ξ k) which projects hwᾱ in CH (X ξ k) so that t w = hwᾱ and I nj t s Im(i k). Let x = t s hwᾱ. Then x Ω <0 Ω (X ψ k) and I nj ( hwᾱ + x) Im(i k). Here x Ω hw+1ᾱ,..., heᾱ}. For each s > w, we note I nj = I lw I ls = I ni. By the inductive assumption (5.2), I ls hs ᾱ} Im(i k). Hence we see I nj x} Im(i k). This implies that I nj hwᾱ} Im(i k). Remark. From above proposition, we can prove Theorem 5.1 more directly. In the end of this section, we note the cases of isotropic quadrics. If ψ is isotropic, then there is m > 0 such that ψ = mh ψ where ψ is anisotropic and H = 1, 1 is the hyperbolic form. By Rost, the decomposition of the motive of X ψ is given by M(X ψ ) = m 1 i=0 (Ti T dim(x ψ) i ) M(X ψ ) T m. By arguments similar to the anisotropic cases, we have the following corollary. Corollary 5.3. Let ψ = mh ξ be a form such that ξ is an anisotropic excellent form of dim(ξ) = d + 2. Then there is the Z[h]-algebra isomorphism CH (X ψ ) = IF n 1 i=1 Z/2[h]/(hd i(d) )ũ i (d)} where IF = IF Zũ 0 (d)} for d = 2 mod(4) (IF = IF otherwise), with IF = (Z[h]/(h d+2m+1 ) Z[h]/(h m )α })/(2α = h d+m+1 ) and where ũ i (d)ũ j (d) = ũ i (d)α = 0 for all i, j > 0 (and hũ 0 (d) = h d/2+m+1 mod(u i 1 i n)). The degree is given ũ i (d) = u i (d) + m, α = d + m. Proof. From the above decomposition of the motive of Ω (X ψ ), we have the Ω -module (but not ring) isomorphism Ω (X ψ ) = Ω [t]/(t m )1, t d+m+1 } Ω (X ξ )t m } with t = 1. We also know the Ω -algebra isomorphism Ω (X ψ k) = Ω 1,..., he+m } Ω [ h]/(h e+m+1 )ᾱ ψ } where ᾱ ψ is the cobordism class representing a maximal projective space in Ω (X ψ k), and so ᾱ ψ = ᾱ + m.

10 N. Yagita / Journal of Pure and Applied Algebra 212 (2008) Then we can prove i k(ω (X ψ )) = e+m i=0 Ω h i } m i=1 Ω he+iᾱ ψ } r i=0 I n i hs ᾱ ψ s i s < s i+1 } by the same arguments as those in the proof of the preceding proposition. Let u j (d), (resp. α ) be the element in CH (X ψ ) corresponding to v j hs i ᾱ ψ (resp. he+1ᾱ ψ ) in the isomorphism CH (X ψ ) = Ω (X ψ ) Ω Z. Then we get the desired result. 6. Examples Recall φ a is the Pfister form of dim(φ a ) = 2 n+1. When ξ = φ a, we still know the additive structure CH (X φa ) = Z[t]/(t 2n ) (Z1, c n,0 } Z/2c n,1,..., c n,n 1 }). This case d + 2 = 2 n+1 and n 0 = n, r = 0. From Theorem 5.1, we get the ring isomorphism CH (X φa ) = Z[h]/(h 2n+1 1 ) Zu 0 } Z/2[h]/(h 2n )u 1,..., u n 1 }. Here we identify h = t, u i = c n,i and h 2n 1+s = t s c n,0 for s > 0. Let φ a is a maximal neighbor of φ a, i.e., dim(φ a ) = 2n+1 1. We see the additive isomorphism CH (X φ a ) = Z[t]/(t 2n 1 ) (Z1, c n,0 } Z/2c n,1,..., c n,n 1 }). With identification h 2n 1 = c n,0, we also get the ring isomorphism CH (X φ a ) = Z[h]/(h 2n+1 2 ) Z/2[h]/(h 2n 1 )u 1,..., u n 1 }. At last, we consider the norm form q a = a 0,..., a n 1 a n. This case dim(q a ) = 2 n + 1 and n 0 = n, n 1 = n 1 and n 2 = 1. Moreover m 0 = 1 and m 1 = 2 n 1 1. By Corollary 3.3, the additive structure is CH (X qa ) = CM n [t]/(t) CM n 1 [t]/(t 2n 1 1 )t} = Z1, c n,0 } Z/2c n,1,..., c n,n 1 } Z[t]/(t 2n 1 1 ) (Zt, c n 1,0 t} Z/2c n 1,1 t,..., c n 1,n 2 t}). From Theorem 5.1, we have the ring isomorphism CH (X qa ) = Z[h]/(h 2n ) Z/2[h]/(h 2n 1 )u 1,..., u n 2 } Z/2u n 1 }. (Note hu n 1 = 0.) Here we can identify t = h, c n 1,0 t = h 2n 1, c n,0 = h 2n 1, and c n 1,j t = u j for 1 j n 2, c n,n 1 = u n 1 and c n 1,j t 2n 1 = c n,j. References [1] J.F. Adams, Stable Homotopy and Generalized Homology, in: Chicago Lectures in Math., Univ. of Chicago Press, [2] M. Hazewinkel, Formal Groups and Applications, Academic Press, New York, San Francisco, London, [3] D. Hoffmann, Isotropy of quadric forms over the function field of a quadric, Math. Z. 220 (1995) [4] D. Johnson, W.S. Wilson, BP operation and Morava s extraordinary K-theory, Math. Z. 144 (1975) [5] N. Karpenko, A. Merkurjev, Rost projectors and Steenrod operations, Doc. Math. 7 (2002) [6] M. Knebush, Generic splitting of quadric forms I, Proc. London Math. Soc. 33 (1976) [7] P. Landweber, Annihilator ideals and complex bordism, Illinois J. Math. 17 (1973) [8] M. Levine, F. Morel, Cobordisme algébrique I, C. R. Acad. Sci. Paris 332 (2001) [9] M. Levine, F. Morel, Cobordisme algébrique II, C. R. Acad. Sci. Paris 332 (2001) [10] M. Levine, F. Morel, Algebraic Cobordism, in: Springer Monographs in Math., 2007, pp [11] M. Rost, Some new results on Chowgroups of quadrics, preprint, [12] M. Rost, The motive of a Pfister form, preprint, [13] R. Stong, Notes on Cobordism Theory, Princeton Univ. Press, [14] A. Vishik, Motives of quadrics with applications to the theory of quadratic forms, in: Izhboldin, Kahn, Karpenko, Vishik (Eds.), Geometric Methods in Algebraic Theory of Quadratic Forms, in: LNM, vol. 1835, 2004, pp [15] A. Vishik, N. Yagita, Algebraic cobordisms of a Pfister quadric, J. London Math. Soc. 76 (2007) [16] V. Voevodsky, The Milnor conjecture. 1996, [17] V. Voevodsky, (Noted by Weibel). Voevodsky s Seattle lectures : K-theory and motivic cohomology, in: Proc. of Symposia in Pure Math. Algebraic K-theory (1997:University of Washington, Seattle) 67, 1999, pp [18] N. Yagita, Applications of Atiyah Hirzebruch spectral sequence for motivic cobordism, Proc. London Math. Soc. 90 (2005)