The Motivic Cohomology of BSO N

Home , Motivic cohomology

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∗,∗ H (Spec k, Z)[ c1,...,cn] where ci is the Chern class of the standard rep- ∗,∗ resentation. N. Yagita determined H (BSO4) over C [24], where cases for G = On, SO2m+1, G2, Spin7 and some other groups in different coefficients are treated. It completes his preceding results in [22], [23]. We will recall his argument in 1.3. Note that SO = GL for √ 1 k. Therefore the new 2 1 − ∈ results in this paper are cases G = SO2m for m 3. We know that D. Edidin and W. Graham [1]≥ constructed an element in m CH (SO2m) for each m 2 such that the mod 2 reduction of it, y0,m in 2m,m ≥ H (BSO2m), becomes 0 through cycle class mapping t∗,∗ (y ) = 0. BSO2m 0,m ∗ (The entire structure of the group CH (SO2m) is known by R. Pandhari- pande [11] in the case m = 2 and R. Field [4] [3] in general m.) L. Molina- Rojas and A. Vistoli [8] gave a new insight on the element using the stratified method. We will review it in 1.2. This phenomenon is first observed by B.Totaro [15], that a generator in cobordism groups of BSO4(C) studied in [7] would not map to a generator of ordinary cohomology by his refined cycle class mapping. Note that the cobordism group of BSOn(C) is determined in [6]. 2m,i+m N. Yagita found a series of elements yi,m in H (BSO2m) for any i from 0 to m 2 in the the kernel of t∗,∗ [24, 9.3], where it was denoted BSO2m − ∗,∗ by ui+1. These elements will be defined in 3.3. Since H (Spec C)= Z/2[τ] (τ H0,1(Spec C)) [19, 7.8], H∗,∗(BSO ) is a Z/2[τ]-algebra. Ker(t∗,∗ ) ∈ n BSO2m becomes the ideal of τ-torsion elements exactly [14, 6]. Our result is that Ker t∗,∗ is generated by Yagita’s§ elements y over BSO2m i,m the polynomial ring of Chern classes of even indices; 0 = τyi,m = zyi,m if z is not a polynomial of c2i’s and yi,myj,m = 0: ∗,∗ ∗,∗ Theorem 0.1. The kernel of realization mapping Ker t in H (BSO2m) is ∗,∗ Ker(t )= Z/2[c , c ,...,c ] y ,..., y − . BSO2m 2 4 2m { 0,m m 2,m} 2m,2m−1 Note that Ker(tX ) = 0 for any X (It is a direct consequence of the theorems of Voevodsky [19, Theorem 6.1, Lemma 6.9, Theorem 7.4]). 0.1. Conventions. Since we only consider coefficients in Z/2, we will omit it. We have H∗,∗(Spec C)= Z/2[τ] where τ has degree 0 and weight 2. The motivic cohomology groups H∗,∗(X) is a bi-graded algebra over Z/2[τ]. The group of quadratic roots of unity µ2 has the motivic cohomology ∗,∗ 1,1 H (Bµ2) = Z/2[τ,y] Λ(x) = Z/2[τ,y] 1,x where x H (Bµ2) and x2 = τy [20, 6.10]. Using⊗ the Beilinson-Lichtenbaum{ } conjecture,∈ we will identify Hi(X) = Hi,i(X). We use the equivariant motivic cohomology in ∗,∗ ∗,∗ 1.1, which is denoted by HG (X). We use a simplified notation HG = ∗,∗ C ∗,∗ ∗ ∗ HG (Spec )= H (BG). Similarly HG = HG(pt). The orthogonal group On is the algebraic subgroup of GLn consisting of 2 elements preserving the quadratic from q(x)= xi . The special orthogonal P THE MOTIVIC COHOMOLOGY OF BSOn 3 group SOn is the subgroup of On consisting of elements with determinant one, and the inclusion is denoted ǫ : SO O . We regard O − as the n ⊂ n n 1 subgroup fixing the last coordinate in On. Similarly for ι : SOn−1 SOn. We denote by ι any of those canonical inclusions. We use the same notation⊂ for an induced morphism between classifying spaces as ι : BSOn−1 BSOn. det : O µ G will be the determinant morphism. ⊂ n → 2 ⊂ m Let us denote by γn the standard real n-dimensional representation of On(R), or its associated bundle on BOn(R). For G = On, SOn, wi Hi(BG(R)) = Hi(BG(C)) = Hi(BG) = Hi,i(BG) will mean the i-th∈ −1 ∗ Stiefel-Whitney class of γn or ǫ γn. It is classical that H (BOn)= Z/2[w1, ∗ w2, . . . , wn] and H (BSOn) = Z/2[w2, . . . , wn]. The i-th Chern class ci of γ C or ǫ−1-image of it, is an element in CHi(BG) Z/2= H2i,i(BG) so ⊗ 2 i ⊗ that (wi) = τ ci.

0.2. Acknowledgments. The authors would like to thank Nobuaki Yagita for several discussions, and to the referees for many useful suggestions.

1. Preliminaries 1.0. Motivic cohomology and the filtration by weight. The realization mapping t = t = ti,j : Hi,j(X) Hi,i(X)= Hi(X) is just a multipli- X X → cation of τ i−j [14, 6]. We can introduce a filtration by weight on Hi(X), so that F jHi(X) consists§ of elements which are the realization of weight less than j. It means x has weight exactly 2j i if there exists x′ as x = τ i−jx′ and j is the minimum of such numbers.− We denote such x′ as x, which is not determined uniquely if there exists non-trivial element of τ-torsion. But in all the cases appear in our paper we have a unique choice of such element for a homogeneous x by reasoning on bi-degree. It defines a filtered ring structure on Hi(X) [24, 2, 6, 7], which is equivalent to the coniveau filtration of Grothendieck, after§ § shifts§ on degrees and indices of filtration. The precise argument is in [24, 7]. A morphism f : X Y induces§ a pullback f ∗ : H∗,∗(Y ) H∗,∗(X) that is a homomorphism of→Z/2[τ]-algebras; and a filtered ring→ homomorphism t(f ∗) : H∗(Y ) H∗(X). → There are motivic Milnor operations Qk on the motivic cohomology groups [18, 13.4]: k+1 k Q : Hi,j(X) Hi+2 −1,j+2 −1(X). k → (Note that we assume √ 1 k so that ρ there is trivial.) Q are derivations − ∈ i on the motivic cohomology groups, QiQj = QjQi if i = j and QiQi = 0. They commute with pullbacks and Gysin morphisms. If6 x has weight i> 0, Qjx has weight i 1 if it is not zero. These are compatible with the classical − ∗ i−j Milnor operations Qi on H (X) in the sense that t(Qix)= Qit(τ x). ( [18, 3.9]). We denote it by the same notation Qi. 4 MASANAHARADAANDMASAYUKINAKADA

1.1. Equivariant motivic cohomology. Edidin and Graham extended the construction of the Chow ring of BG to the equivariant Chow ring of a scheme with G-action, which can be generalized to motivic cohomology [2]. Let us recall the equivariant motivic cohomology [24, 8]. Suppose that G acts on a smooth quasi-affine scheme X. For each m § 0, we can choose a representation V of G with an open subscheme E on≥ which G acts freely, and codimV (V E) > m. The quotient (E X)/G exists as a smooth scheme. We set− × Hm,∗(X)= Hm,∗((E X)/G). G × It is independent of the choice of such V for fixed m and . For a morphism of groups φ : H G, a G-schemes X, an H-scheme Y , and∗ an H-equivariant morphism f : Y→ X where H acts on X through φ, there is a pull-back f ∗ : H∗,∗(X) H→∗,∗(Y ). For a subgroup H of G, we see that [8, 2.1] G → H (2) H∗,∗((X G)/H)= H∗,∗(X). G × H Let U be a unipotent group and consider a U-torsor X Y , we have an isomorphism H∗,∗(Y )= H∗,∗(X), for U is isomorphic to→ an affine space and any U-torsor is locally trivial [5, Prop.1]. In particular for a semi-direct product G ⋉ U the projection onto the component G ⋉ U G induces an isomorphism H∗,∗(B(G ⋉ U)) = H∗,∗(BG) [8, 2.3]. → Take any two elements u0, u1 of U(C). G G U by x (x, ui) induce the same morphism on H∗,∗(B(G U))→ H×∗,∗(BG) for7→ both are × → the section of the projection. Similarly for the multiplicative group Gm: H∗,∗(B(G G )) = H∗,∗(BG)[y] H∗,∗(BG). Let G be connected. For × m → any element g G(C) the conjugation by g defines a morphism cg : BG ∈ ∗ ∗ → BG, but we know that g is contained in a Borel subgroup, and cg = ce on H∗,∗(BG) for e is the unit of G [5, Lemma.1 of Proposition 4].

1.2. The stratified method. The equivariant motivic cohomology theory has the following localization sequence; if we write an equivariant closed ,immersion i: Y ֒ X of codimension s and open immersion j : X Y ֒ X there is a long exact→ sequence − → ∗ i∗ j H∗−1,∗(X Y ) δ H∗−2s,∗−s(Y ) H∗,∗(X) H∗,∗(X Y ) δ → G − −→ G −→ G −→ G − −→ where i∗ is the Gysin mapping. If we know a stratification of a G-scheme X by G-subschemes, we may ∗,∗ calculate HG (X) by repeated use of the localization sequence above and ∗ (1.1). This method is introduced by Vezzosi to compute CH (BPGL3) [17]. Molina-Rojas and Vistoli [8] calculated the Chow ring of BG using it in cases G = GLn, SLn and Spn over arbitrary fields, and G = On and SOn. over fields of characteristic different from 2. The following stratification associated with On or SOn are used to calculate the Chow groups and motivic cohomology in [8] and [24, 9]. Consider the standard representation of SO . It makes An an SO -scheme.§ We set B = x An r 0 q(x) = 0 n n { ∈ { } | 6 } THE MOTIVIC COHOMOLOGY OF BSOn 5 and C = x An r 0 q(x) = 0 where q(x) = n x2. There is a { ∈ { } | } k=1 k stratification 0 0 C An, and An r ( 0 C)=P B. Then we get{ two} ⊂ {localization}∪ ⊂ sequences: { }∪ ∗ s∗ t (3) H∗−2n,∗−n( 0 ) H∗,∗ (An) H∗,∗ (An r 0 ) → SOn { } −→ SOn −→ SOn { } −→ and ∗ i∗ j δ (4) H∗−2,∗−1(C) H∗,∗ (An r 0 ) H∗,∗ (B) . → SOn −→ SOn { } −→ SOn −→ ∗,∗ ∗,∗ n We have H ( 0 ) = H (A ) and s∗ is the multiplication of the n- SOn { } SOn th Chern class c H2n,n( 0 ) = H2n,n(BSO ) by the self-intersection n ∈ SOn { } n formula [13]. Since in the end of proof in 3.3 we will see that cn is injective, we will denote ∪ (5) H∗,∗ (An r 0 )= H∗,∗ /c . SOn { } SOn n We can define an action of G SO on B such that (t, g) G SO m × n ∈ m × n sends v B to tg(v), that is, the matrix multiplication of scalar t Gm and g(v)∈ B. This action is transitive and the stabilizer of e is a subgroup∈ ∈ 1 isomorphic to O − . The inclusion κ : O − G SO is given by n 1 1 n 1 → m × n g (det g, κ) where κ : O − SO is 7→ n 1 → n det(g) 0 (6) g . 7→ 0 g G We have B ∼= m SOn/On−1. Choose E as in× 1.1. Then we obtain the following isomorphisms G ∗,∗ ∗,∗ E ( m SOn/On−1) HSOn (B)= H × × SOn G ∗,∗ E m ∗,∗ G =H × = HOn−1 ( m). On−1

Gm is an On−1-scheme via det, and c1(γn C)= c1(det(γn C)), we get the identiﬁcation similar to (5) ⊗ ⊗ ∗,∗ G ∗,∗ (7) HOn−1 ( m)= HOn−1 /c1 for we will see in the next subsection that c1 is a monomorphism. Note ∗,∗ ∪ ∗ ∗ that these are HSOn -module isomorphism through κ . j in (4) is identiﬁed with the induced morphism of κ∗. On the other hand, we know that SOn transitively acts on C and the stabilizer of e +√ 1e is isomorphic to a semi-direct product of the stabilizer of 1 − 2 a pair e1 +√ 1e2, e1 √ 1e2 and a unipotent subgroup, so that it is isomor- − ⋉An−−1 − ⋉An−1 phic to SOn−2 [8, 5.2]. Thus we get that C ∼= SOn/(SOn−2 ) and this deduces the following isomorphisms [8, 2.3]: ∗,∗ ∗,∗ ∗,∗ H (C)= H n−1 = H . SOn SOn−2⋉A SOn−2 From the above calculations we can rewrite (4) as follows: ∗ ∗−2,∗−1 i∗ ∗,∗ κ ∗,∗ δ (8) H H /cn H /c1 . → SOn−2 −→ SOn −→ On−1 −→ 6 MASANAHARADAANDMASAYUKINAKADA

We can regard the same stratiﬁcation as that of On-spaces. The induced long exact sequence is exactly the one used by Yagita. We can connect two long exact sequences by the change of groups homomorphism on ǫ : SO n → On:

∗ (i0)∗ j (9) / H∗−2,∗−1 / H∗,∗/c 0 / H∗,∗ (G ) / H∗−1,∗−1 / On−2 On n µ2×On−1 m On−2

ǫ∗ ǫ∗ ǫ∗ ǫ∗ i∗ ∗ / H∗−2,∗−1 / H∗,∗ /c κ / H∗,∗ (G ) δ / H∗−2,∗−1 // . SOn−2 SOn n On−1 m SOn−2

The standard inclusion An An+1 into the ﬁrst n components induces the following commutative long→ exact sequences (Gysin mappings and pullbacks commute in this case [20, 2.3])

∗ / ∗−2,∗−1 i∗ / ∗,∗ κ / ∗,∗ δ / ∗−1,∗−1 / (10) HSOn−1 HSOn+1 /cn+1 HOn /c1 HSOn−1

ι∗ ι∗ ι∗ ι∗ ∗ / ∗−2,∗−1 i∗ / ∗,∗ κ / ∗,∗ δ / ∗−1,∗−1 / HSOn−2 HSOn /cn HOn−1 /c1 HSOn−2 .

Later we use another inclusion An An+1 into the second to n + 1-th components. It is conjugate to the composition→ of the standard inclusions by an element of SOn+1(C), so that the induced morphisms in motivic cohomology are the same (1.1).

∗,∗ 1.3. H (BOn). It is well-known that the map induced from the restriction to the diagonal subgroup H∗(BO ) H∗((Bµ )n)Σn = Z/2[x ,...,x ]Σn n → 2 1 n is an isomorphism, where Σn is the n-th symmetric group acting by permu- tations on xi’s. We need three basis of the symmetric polynomials. Recall n−k that for any partition λ = [i1, i2,...,ik, 0,..., 0] = [i1, i2,...,ik, 0 ] of n where ij = n and ij = 0, k is called the length of λ. For each partition λ 6 i1 i2 ik we canP deﬁne a symmetric polynomial m[λ]= x1 x2 ...xk where the sum is taken on the orbit of Σn. These make theP monomial basis of symmetric polynomials. The l-th Stiefel-Whitney class wl corresponds to the elementary sym- l n−l metric polynomial of degree l : wl = m[1, 1,..., 1] = m[1 , 0 ], whose ∗ monomials gives a basis of H (BOn)= Z/2[w1, . . . , wn]. Finally we need a basis suitable to see action of Q’s. It is due to W. S. Wil- Z son. We denote by Q(i)M the /2-module generated by Qk1 ...Qkj m for all 0 k <...

2.1. Determination of weight. We will write down κ∗ in terms of Stiefel- Whitney classes. Recall γn is the canonical n-dimensional real vector bundle 8 MASANAHARADAANDMASAYUKINAKADA over BOn(R). We have the following isomorphism of vector bundles: −1 −1 κ ǫ γ = det(γ − ) γ − n n 1 ⊕ n 1 −1 −1 Thus the total Stiefel-Whitney class of κ ǫ γn is 1+ w (κ−1ǫ−1γ )+ w (κ−1ǫ−1γ )+ + w (κ−1ǫ−1γ ) 1 n 2 n · · · n n =1 + w (det γ − γ − )+ + w ((det γ − γ − ) 1 n 1 ⊕ n 1 · · · n n 1 ⊕ n 1 =(1 + w )(1 + w + + w − ) 1 1 · · · n 1 2 (13) =1+(w + w ) + (w + w w )+ + (w − + w w − )+ w w − . 2 1 3 1 2 · · · n 1 1 n 2 1 n 1 ∗ ∗ ∗ We will prove that the mapping t(κ ): H (BSOn) H (BOn−1) preserves weight strictly. We recall that a filtered mapping φ→: H∗(X) H∗(Y ) strictly preserves filtrations if F iH∗(X)= φ−1(F iH∗(Y )). → ∗ ∗ ∗ To see first t(κ ) is injective, we recall Ker t(κ ) Ker t(ι ) = (wn). ∗ ∗ ⊂ Suppose 0 = x HSOn is in Ker t(κ ). It is divisible by wn, and there 6 ∈ j exists xj such that x = wnxj and xj is not divisible by wn. The image ∗ j j ∗ ∗ t(κ )x = wn−1w1κ xj is zero, so t(κ )xj is zero, which is contradiction. ∗ ∗ Let x be an element of H (BSOn) such that t(κ )x is contained in the ∗ weight 0 part of H (BOn−1). We can write x = P (wi) with a polynomial ∗ ∗ of wi. t(κ )x = P (t(κ )x) is a polynomial of ci, so that each term will be a square of a monomial of wi. By (13) it is possible, by induction on lexicographic order on monomials, only if P is a square. We have proved that t(κ∗) is strict on the weight 0 part. If y = t(κ∗)x has weight k > 0, we switch to the motivic cohomology. Then we can find x H∗,∗ and y H∗,∗ such that κ∗x = τ f y for some ∈ SOn ∈ On−1 f Z≥ . We have that ∈ 0 ∗ f κ Qj0 ...Qjk−1 x = τ Qj0 ...Qjk−1 y. for some j0,...,jk−1 0,...,n 2 and Qj0 ...Qjk−1 y is a non-zero { 2h,h } ⊂ { − } element in H for it has zero image by all Qi. It means f = 0 and x ∗ has weight equal to k by the theorem of Yagita. Finally t(κ )cn = c1cn−1 implies the following. Proposition 2.1. The mapping t(κ∗): H∗ /c H∗ /c is injective SOn n → On−1 1 and it preserves weight strictly.

Corollary 2.2. (1) For odd n, the Stiefel-Whitney class wl ( 2 l n) ∗ ≤ ≤ in H (BSOn) has weight l if l is even, and l 2 if l is odd. (2) For even n, the Stiefel-Whitney class w has weight− n 2 in H∗(BSO ). n − n For other wl (2 l n), wl has weight l if l is even, and l 2 if l is odd. ≤ ≤ − (3) Q w = w for all 1 l n/2, where we regard w = 0. 0 2l 2l+1 ≤ ≤ n+1 ∗ ∗ ∗ Proof. Consider t(κ )wl in HOn−1 . If l = n, t(κ wl) = wl + w1wl−1 = l l−1 6 ∗ l−1 m[1 ] + m[1]m[1 ] from (13). If l is odd, t(κ )wl = m[2, 1 ]. If l is ∗ l l−1 even, t(κ )wl = wl + w1wl−1 = m[1 ]+ m[2, 1 ]. For l = n, we have that ∗ n−2 t(κ )wn = w1wn−1 = m[2, 1 ]. THE MOTIVIC COHOMOLOGY OF BSOn 9

2.2. The stratiﬁed method works ﬁne in ordinary cohomology, and the result is compatible with the realization:

∗ t(s∗) t(t ) (14) H∗−2 H∗ H∗ (An r 0 ) . → SOn −−−→ SOn −−−→ SOn { } −→ We can identify as in (5) H∗ (An r 0 )= H∗ /c . SOn { } SOn n The sequence (8) becomes

∗ ∗−2 t(i∗) ∗ t(κ ) t(δ) ∗−1 (15) H H /cn HO −1 /c1 H → SOn−2 −−−→ SOn −−−→ n −−→ SOn−2 → We have the following data on (15). These are direct consequences of ∗ Proposition 2.1, (13), and t(δ) is an HSOn -module mapping. We may compare the result of Yagita in [24, 8] by (9). Note we will write the image in the quotient as the same letter, as§ is w H∗ /c . i ∈ SO2m 2m

Proposition 2.3. (1) t(i∗) = 0. ∗ ∗ (2) t(κ )w = w + w − w for 2 i n 1, and t(κ )w = w − w . i i i 1 1 ≤ ≤ − n n 1 1 (3) t(δ)(w1) = 1 and t(δ)(zw1)= z for z is a polynomial of wi for 2 i n 2. ≤ ≤ − ∗,∗ . ∗ ∗ 2.3. H (BSO2m+1) Since HOn and HSOn are generated by the Stiefel- Whitney classes, the realization mapping t∗,∗ are epimorphisms in all these cases. As in 1.3, t: Hp,q Hp is injective for every pair (p,q). On → On When n = 2m + 1, we have that O − = SO − µ . This direct 2m 1 ∼ 2m 1 × 2 product structure induces the mapping π : BO − BSO − deduced 2m 1 → 2m 1 from multiplication of determinant g det(g)g if g O2m−1. Then π ǫ = 1 and (ǫ π)(ǫ π) = (ǫ π). 7→ ∈ · · · · ∗,∗ ∗,∗ ∗,∗ (16) H /c1 = HZ (Gm)= H 1,x O2m−1 /2×SO2m−1 SO2m−1 { } p,q where x = w1 has degree (1,1). It implies the realization t : H SO2m−1 → Hp is injective. We can compare (8) and (15) through monomorphisms SO2m−1 ∗ ∗ ∗ t. Because t(κ )tSO2m+1 = tO2m κ is injective, κ is injective, then we get

∗ / ∗,∗ κ / ∗,∗ δ2 / ∗−1,∗−1 / (17) 0 HSO2m+1 /c2m+1 HO2m /c1 HSO2m−1 0

∗,∗ 3. H (BSO2m) 3.1. Let us denote by Y the kernel of H∗,∗ H∗ . We will induc- m SO2m → SO2m tively determine Ym so that we assume Ym is known. We denote by Ym+1 ∗,∗ ∗ the kernel of t : H /c2m+2 H /c2m+2. Later we will see in SO2m+2 → SO2m+2 3.3 that (Ker tSO2m )/c2m+2 = Ym+1. We can combine (8) and (15) to the 10 MASANAHARADAANDMASAYUKINAKADA following commutative diagram (18) i∗ ∗ δ / H∗−2,∗−1 / H∗,∗ /c κ / H∗,∗ /c δ / H∗−1,∗−1 / SO2m SO2m+2 2m+2 O2m+1 1 SO2m

t2m t2m+2 t2m+1 t(κ∗) t(δ) / H∗−2 0 / H∗ /c / H∗ /c / H∗−1 0 / . SO2m SO2m+2 2m+2 O2m+1 1 SO2m t2m+1 on the right column is injective for each degree. w1 has degree (1, 1) and tw1 = w1. ∗,∗ We obtain that Im(i∗) = Ker(t2m+2) H /c2m+2, using diagram- ⊂ SO2m+2 tracing argument. Note t2m+1 is injective. The restriction of i∗ gives an isomorphism between Coker δ and Ym+1, and it takes Ym into Ym+1. We can see Im δ Ym = 0. For if there is u = δv such that t(u) = 0, there exists w such that ∩t(κ∗)t(w) = t(v) . Because t(κ∗) strictly preserves ﬁltrations, ′ ∗,∗ ∗ ′ there is w H /c2m+2 such that κ w = v. This means u = 0. ∈ SO2m+2 The restriction of i∗ gives an acyclic subcomplex Y Y , where the m → m second copy is in Ym+1, in (3.1), and we get (19) ∗ / ∗,∗ κ / ∗,∗ δ1 / ∗−1,∗−1 / HSO2 +2 /Ym /c2m+2 HO2 +1 /c1 HSO2 /Ym m m m where the induced t2m is now a monomorphism and Coker δ1 ∼= Ym+1/Ym. By Proposition 2.3 we know that Coker δ1 is isomorphic to the cokernel ∗ ∗,∗ ∗,∗ of the restriction ι : H H /Ym, up to permutation of co- 2m+1 SO2m+1 → SO2m ordinates, which makes a part of the following commutative exact sequences

∗ / ∗,∗ κ / ∗,∗ δ2 / ∗−1,∗−1 / (20) 0 HSO2m+1 /c2m+1 HO2m /c1 HSO2m−1 0

∗ ∗ ∗ ι2m+1 ι2m ι2m−1 ∗ / ∗,∗ κ / ∗,∗ δ1 / ∗−1,∗−1 0 (HSO2m /Ym)/c2m HO2m−1 /c1 HSO2m−2 /Ym−1 ,

∗ where ι2m is an epimorphism.

3.2. Since κ∗w = w + τw w in H∗,∗ , w has weight 2m in 2m 2m 1 2m−1 O2m 2m ∗,∗ ∗ SO2m+1. On the other hand w2m in HSO2 satisﬁes κ w2m = w1w2m−1, m ∗ which implies it has weight 2m 2. That is, ι2m+1w2m = τw2m, and w2m − ∗ presents a nonzero element in Coker ι2m+1, where τw2m = 0. ∗ We want to claim that w2m is a generator of Coker ι2m+1 over the ring of even Chern classes. Since κ and ι commute, we have to consider elements having factor w2m only. Consider the long exact sequence associated with (20):

∂ (21) Ker ι∗ Ker ι∗ Ker ι∗ Coker ι∗ 0. 2m+1 → 2m → 2m−1 → 2m+1 → THE MOTIVIC COHOMOLOGY OF BSOn 11

∗ −1 ∗ −1 ∗ Connecting mapping ∂ is (κ ) ι2m (δ2) . Let us take z in Ker ι2m−1. −1 · · ∗ We can take (δ2) z = w1z. If it is in ordinary cohomology, Coker ι2m+1 = −1 ∗ 0, for we can take w2m as (δ2) w2m−1 which go to zero by ι2m, but in motivic cohomology w2m has strictly larger weight than w1w2m−1, hence ∗ w2m presents a non-zero element in Coker ι2m+1 as above. Therefore it suﬃces to prove that in H∗,∗ , zw has weight less than O2m 2m or equal to the weight of zw1w2m−1, if z is not a polynomial of even Chern classes. Let z be an element of monomial base m[i·]. First we assume not all i’s are even. We will use induction of length of z. If it has length 1, m[2i + 1]m[2, 12m−2, 0] and m[2i + 1]m[12m] have weight 2m 1 (i > 0). m[2i + 1, 2j + 1]m[2, 12m−2, 0] have weight more than 2m 2− and m[2i + 1, 2j + 1]m[12m] have weight 2m 2. If in a monomial of length− j 2 all i − ≥ are odd, zw have weight 2m j, and zw w − also have weight 2m j. 2m − 1 2m 1 − In general let 2j be the largest even member of i· which has multiplicity k. ′ ′ 2j Let z be the partition discarding 2j’s from z. Then z is equal to z (wk) ′ 2j plus elements of length less than that of z. z (wk) w2m has the same weight ′ as that of z w2m, and similarly for w1w2m−1. It ﬁnishes the inductive step. 2 If all i’s are even, it is a polynomial of (wi) and zw2m has weight 2m 2 but zw w − has weight 2m 2. If z contains w , there exists u such 1 2m 1 − 2j−1 that zw2m +uw1w2m has weight 2m 2, hence it does not represent nonzero ∗ − elements in Coker ι2m+1.

3.3. We know that Y1 = 0, and the cokernel of δ is easily seen to be the image of Z/2[c ] w , here w H2,1 in H∗,∗ /c . Let us deﬁne y = 2 { 2} 2 ∈ SO2 SO4 4 0,2 i ∗w . Then Y = Z/2[c , c ] y [24, 8]. 2 2 2 2 4 { 0,2} § Assume we know that Ym is generated by yi,m over Z/2 [ c2,...,c2m] for ∗,∗ 0 i m 2. i∗(y ) are non-zero elements in H /c (i∗ is that in ≤ ≤ − i,m SO2m+2 2m+2 (3.1).) For i y have degree 2m + 2 there exist y in H∗,∗ . We will ∗ i,m i,m+1 SO2m+2 see that (Ym+1/c2m+2)/i∗(Ym) is generated by i∗w2m over Z/2 [ c2,...,c2m]. 2m+2,2m There exists a unique element ym−1,m+1 in H (BSO2m+2) whose image by quotient of c2m+2 is i∗w2m. We get an isomorphism

Y = Z/2 [ c , c ,...,c ] y ,...,y − . m+1 2 4 2m+2 { 0,m+1 m 1,m+1} Multiplicative relations are consequence of the projection formula [13]. Finally we consider s in (3) as an H∗,∗ -module mapping. c is ∗ GL2m 2m ∗,∗ ∗ ∗,∗ An ∪ injective on Ker tSO2m and HSO2m , so that we can identify HSOn ( 0 )= ∗,∗ −{ } HSOn /cn for all n.

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Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan E-mail address: [email protected]

Kobe University Secondary School.Kyoto University E-mail address: [email protected]