NOTE ON MOTIVIC COHOMOLOGY OF ANISOTROPIC REAL QUADRICS NOBUAKI YAGITA Abstract. In0this paper, we compute the mod 2 motivic coho- mology H*,*(X; Z=2) for the anisotropic quadric X over R the field of real numbers. 1. introduction Each anisotropic quadratic form of dimension n over R is written uniquely by qn = x21+ ... + x2n. Let Xd be the d-dimensional projective 0 * quadric defined by qd+2 = 0. Let H*,*(Xd; Z=2) (resp. Het(Xd; Z=2)) be the mod(2) motivic cohomology (resp. etale cohomology) con- structed by Suslin and Voevodsky [Vo1,2]. It is shown in [Ya1] that 0 * -1 (1.1) H*,*(Xd; Z=2) Het(Xd; Z=2)[o, o ] where o is the nonzero elemant in H0,1(Spec(R); Z=2) ~=Z=2. The above fact is showed by using the decomposition of its motives, i.e., M(Xd) ~= jMijfor some ij > 0. Here Mijis the Rost motive (over 0 R) of dimension 2ij- 1. The cohomology H*,*(Mij; Z=2) is computed in [Ya1] by using arguments by Voevodsky [Vo1,2]. Hence we know 0 the additive structure of H*,*(Xd; Z=2) completely. Moreover, the ring structure of CH*(Xd)=2 = H2*,*(Xd; Z=2) is given in [Ya2] by using the algebraic cobordism theory *(Xd). On the other hand, there is the homeomorphism Xd(C)=Z=2 ~= G2(Rd+2). Here Xd(C)=Z=2 is the quotient space of the manifold Xd(C) of C-rational points of Xd by the free Gal(C=R) = Z=2 action, and G2(R2+d) is the Grassmannian of 2-planes in Rd+2. Hence we know as cohomology rings H*et(Xd; Z=2) ~=H*(G2(Rd+2); Z=2). ____________ 1991 Mathematics Subject Classification. Primary 11E04, 14C15; Secondary 55R35, 57T05. Key words and phrases. real quadric, motivic cohomology, real Grassmannian. 1 2 N.YAGITA Let wi 2 H*(G2(R1 ); Z=2) be the i-th Stiefel-Whitney class of the canonical 2-dimensional bundle. Define fi 2 Z=2[w1, w2] inductively f0 = w1, f1 = w21+ w2, and fn+1 = w1fn + w2fn-1 for n 0. Then by Borel, Hiller [Bo], [Hi] there is the isomorphism H*(G2(Rd+2); Z=2) ~=Z=2[w1, w2]=(fd, w2fd-1). 0 Hence we can write down H*,*(Xd; Z=2) as a subring of Z=2[w1, w2]=(fd, w2fd-1) Z=2[o, o -1] as follows. Let us write the alternating 2-adic expansion of d + 2 by d + 2 = 2n0+1 - 2n1+1 + ... + (-1)r2nr+1 for n0 > n1 > ....nr-1 > nr + 1 0. For 0 j r, let sj = sj(d) be the number defined by ( 2n0 - 2n1 + ... + 2nj-2- 2nj-1 j : even sj = 2n0 - 2n1 + ... - 2nj-2+ 2nj-1- 2nj+1 + ... + (-1)r2nr+1 j : odd. We identity w1 2 H1,1(Xd; Z=2) and w2 2 H2,2(Xd; Z=2). Moreover let h = w2o -12 H2,1(Xd; Z=2). Let ff(n) be the number of 1 in the 2-adic expansion of n, e.g., ff(d) = n0 - n1 + ...(-1)rnr. Let us write X QT = { o -nwf1|f 2n + ff(n)} Z=2[o -1, w1]. n 0 Theorem 1.1. For d 1, the motivic cohomology H*,*(Xd; Z=2) is isomorphic to the Z=2[h]-subalgebra of Z=2[w1, h, o, o -1]=(fd, hfd-1) generated by Z=2[o ] and QT o -1wnj+11hsj for 0 j r. By using the above embedding, we will give the ring structure of CH*(Xd)=2 without using the algebraic cobordism theory. 2.Chow rings of excellent quadrics 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 equiva- lence. For a non zero symbol a = {a0, ..., an} in the mod 2 Milnor K-theory KMn+1(k)=2, let OEa = <> be the (n + 1)-fold Pfister form. Let XOEabe the projective quadric of dimension 2n+1 - 2 defined by OEa. The Rost motive Ma(= MOEa) is a direct summand of the motive M(XOEa) representing XOEaso that n-1 (2.1) M(XOEa) ~=Ma M(P2 ). MOTIVIC COHOMOLOGY OF REAL QUADRICS 3 The Chow ring of the Rost motive is well known. 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 [R1]) The Chow ring CH*(Ma) is only dependent on n. There are isomorphisms CH*(Ma) ~=Z{1, cn,0} Z=2{cn,1, ..., cn,n-1} |cn,i| = 2n - 2i, and CH*(Ma|~k) ~= Z{1, ~ffn} with |~ffn| = 2n - 1. The restriction map is given by i~k(cn,0) = 2~ffnand i~k(cn,i) = 0 for i > 0. Here we consider the quadrics defined by a subform , of the Pfister form OEa. Recall that T is the Tate motive i.e., M(P1) = T0 T. By using results of Rost and Hoffmann, we see the following theorem. Theorem 2.2. (Rost, Hoffmann [Ro],[Ho],[Ka-Me],[Vi-Ya]) Let , be a subform of the Pfister form OEa = <> of dim(,) = 2n+m, 2n m > 0 (i.e., , is a neighbor of OEa). Let j be a complementary form (OEa = , j). Then M(X,) = Ma M(Pm-1 ) M(Xj) T m . The typical example of this theorem is the isomorphism (2.1) which is the case m = 2n. Now we recall the definition of excellent quadrics ([Kn],[Ka-Me]). A quadratic form , over k is called excellent if for every field extension K=k, the anisotropic part of ,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 (Knebusch [Kn],[Ka-Me]). It is known (see x3 bellow) that all quadrics over R is excellent. Suppose that , = ,0 is excellent. Then we have a decreasing sequence ss0 ss1 ... ssr of embedded Pfister forms such that the class [,k] in Witt ring is given by [,k] = [ssk]-[ssk+1]+...+(-1)r-k[ssr], namely, , = ,0 and [,i]+[,i+1] = [ssi]. Let us write dim(ssi) = 2ni+1. Then n0 > n1 > .... > nr + 1 0 and (2.2) dim(,) = 2n0+1 - 2n1+1 + ... + (-1)r2nr+1. Thus ni is the places changing zero to one (or one to zero) in the 2-adic expansion of dim(X,) + 2 = d + 2. Let us write (2.3) mj = 1=2(dim(,j)-dim(,j+1)) = 2nj-2nj+1+1+...+(-1)r-j2nr+1, (2.4) sj = m0 + ... + mj-1 = 1=2(dim(,0) - dim(,j)) 4 N.YAGITA ( 2n0 - 2n1 + ... + 2nj-2- 2nj-1 j : even = 2n0 - 2n1 + ... - 2nj-2+ 2nj-1- 2nj+1 + ... + (-1)r2nr+1 j : odd. Note that sr+1 = [1=2dim(,)]. Iterating Theorem 2.2, we can see ; Lemma 2.3. (Rost [Ro], [Ka-Me]) There is an isomorphism of mo- tives M(X,) ~= ri=0Mssi M(Pmi-1 ) T si. When dim(,) = odd, we see ssr = <1> and dim(Xssr) = -1. So the respective term should be omitted. Corollary 2.4. Let r0 = r for d = even and r0 = r - 1 otherwise. There is an additive isomorphism 0 * m s CH*(X,) ~= ri=0CH (Mssi)[t]=(t i){t i}. Here ti is a generator of CHi(Ti) (but it does not mean the real product of t in CH1(X,).) Let e = d=2 for d = even and e = (d - 1)=2 for d = odd. Recall that sr0+1 = 1=2deg(,) = e + 1 for d = even, and sr0+1 = 1=2(deg(,) - 1) = e + 1. Hence we have the additive isomorphism 0 m s e+1 ri=0Z[t]=(t i){t i} ~=Z[h]=(h ) deg(h) = 1, 0 m s e+1 e+ffl ri=0Z[t]=(t i){t icn,0} ~=Z[h]=(h ){t } where ffl = 0 or 1 for d = even or d = odd respectively. Let us write s0i= |tsicni,0| = si+ 2ni- 1. We write the picture for si, s0iand mi for small i0s. 0=s0 m0 s1 m1 s2m2s3 e+1 s02m2s01m1s00 m0 d+1 . - -! .! .$ . -- . -- .$ .! .- -! . In fact, we can compute s0i-1- s0i= mi. Let _ be a (not assumed to be excellent) quadratic form. For each quadric X_, the Chow ring of X_|~k= X_ ~kis given by Rost. Let dim(X_) = d. Let ~h (resp. ~ff) be an element of CH*(X_|~k) which is represented by a hyperplane section (resp. a maximal projec- tive space) in X_|~k. So |~h| = 1 and |~ff| = e if d = even (|~ff| = e + 1 for odd). Lemma 2.5. ([Ro2]) There is an isomorphism of rings CH*(X_|~k) ~=Z{1, ~h, ..., ~he} Z[~h]=(~he+1){~ff}. The multiplication of CH*(X_|~k) is given by (1) ~he+1= 2~h~fffor d = even (~he+1= 2~fffor d = odd), (2) ~ff2= ~hd= 2~he~ffif d = 0 mod(4) (~ff2= 0 otherwise). MOTIVIC COHOMOLOGY OF REAL QUADRICS 5 Recall that X, is an excellent anisotropic quadric with 2n - 1 dim(X,) = d 2n+1 - 2. Let h 2 CH1(X,) be represented by a hyperplane section. Of course i~k(h) = ~h. It is known that i~k(cni,0tsi) 0 s can be written by ~hsi= 2~h i~ff. Moeover we have ; Theorem 2.6. ([Ya2]) There are elements c01, ..., c0n-1(and c00when d = 2 mod(4)) in CH*(X,) and positive integers d1 ... dn-1 such that there is the Z[h]-algebra isomorphism CH*(X,) ~=F n-1i=1Z=2[h]=(hdi){c0i} ( Z[h]=(hd+1) Z{c00} for d = 2 mod(4) where F = Z[h]=(hd+1) otherwise with multiplication c0ic0j= 0 for all i, j and hc00= hd=2+1 mod(c0j|1 j < n). The degree is given as follows ; if ni+1 j < ni then dj = si+1 and |c0j| = 2ni- 2j + si = s0i- 2j + 1. 3. cohomology of real Rost motives Hereafter0we always assume k = R the field of real numbers. Recall that H*,*(X; Z=2) (resp. H*et(X; Z=2)) is the mod 2 motivic (resp. etale) cohomology defined by A.Suslin and V.Voevodsky [Vo1,2]. It is well known that H*et(pt.; Z=2) ~=KM*(R)=2 ~=Z=2[ae] where ae = {-1} 2 KM1(R)=2 ~= R*=(R*)2. The motivic ohomology of a point is 0 * H*,*(pt.; Z=2) ~=Het(X; Z=2)[o ] ~=Z=2[ae, o ] with 0 6= o 2 H0,1(pt.; Z.2) ~=Z=2. It is also well known that KM*(R)=2 ~=grW *(R) ; graded ring from the Witt ring W (R) of constructed from anisotropic forms. Hence all anisotroic form is written as q = (d + 2)<1>. Therefore they are excellent. Let Xd = X, = X(d+2)<1>of dim(Xd) = d. We will study 0 H*,*(Xd; Z=2). At first we study the cohomology of the Rost motive for a = aen+1. Recall that 0 *+2i+1-1,*0+2i-1 Qi : H*,*(X; Z=2) ! H (X; Z=2) is the Milnor operation in the motivic cohomology defined by Voevod- sky [Vo3]. Let us write Q(n) = (Q0, ..., Qn). 6 N.YAGITA 0 Theorem 3.1. ([Ya1]) The cohomology H*,*(Ma; Z=2) is isomorphic n+1-1 to the subalgebra of H*et(Ma; Z=2)[o, o -1] ~= Z=2[ae, o, o -1]=(ae2 ), that is isomorphic to n+1-1 (Z=2[ae, o ] Z=2[ae] Q(n - 1){ao -1})=(ae2 ) identifying with Qffl(ao -1) = Qffl00...Qffln-1n-1(aoP-1) =Po -1-d(ffl)aef(ffl)* *a where ffl 6= (1, .., 1) and f(ffl) = iffli(2i+1 - 1), d(ffl) = iffli2i. The sequence ffl is the 2-adic expantion of d. So write ffl = ffl(d) and f = f(ffl(d)) = f(d). Then X f(ffl(d)) = ffli(2i+1 - 1) = 2d - ff(d) where ff(d) is the number of 1 in the expantion of d. We can identify the algebra of operations Q(n) by the Z[ae]-submodule X Z=2[ae] Q(n) = { o -daef|f 2d - ff(d)} Z=2[ae, o -1] 0 d 2n-2 In the above theorem Q(n - 1){ao -1} = Q(n - 1) . (ao -1) in the last ring. (The . in the last term means the multiplication in the ring Z=2[ae, o -1], while the first one means the actions Qfflon ao -1) Corollary 3.2. Let f(d) = f(ffl(d)) = 2d-ff(d) for d 0 and f(-1) = -n - 1. Then there is the isomorphism 0 -d m+n+1 H*,*(Ma; Z=2)=(o ) ~= 0 d 2n-1Z=2{o ae |f(d-1) m < f(d)}. The element cn,i2 CH*(Ma; Z=2) ~= H2*,*(Ma; Z=2) in Lemma 2.1 is represented by n-2i (3.1) cn,i= Q0....Q^i....Qn-1ao -1= (o -1ae2)2 . In particular, for the cycle map 0 * cl : H*,*(Ma; Z=2) ! Het(Ma; Z=2) n+1-2i+1 we have cl(cn,i) = ae2 . Corollary 3.3. There is an additive isomorphism 0 r0 *,*0 m H*,*(Xd; Z=2) ~= i=0H (Mssi; Z=2)[h]=(h i){ti} where ti = hsi mod(c0jhk) in H2*,*(Xd; Z=2). For ease of notation, let 2(i) = 2ni+1 - 1. Since we have the similar etale motives decomposition, from Corollary 3.3, we get ; Corollary 3.4. There is an H*et(pt.; Z=2) = Z=2[ae]-module isomor- phism 0 H*et(Xd; Z=2) ~= ri=0Z=2[ae, h]=(ae2(i), hmi){ti}. MOTIVIC COHOMOLOGY OF REAL QUADRICS 7 Next we recall the coniveau spectral sequence. Let X be a smooth variety over k = R. The filtration coniveau is given by NcHmet(X; Z=p) = [Z Ker{Hmet(X; Z=p) ! Hmet(X - Z; Z=p)} where Z runs in the set of closed subschemes of X of codim = c. We consider its associated (coniveau) spectral sequence. Bloch-Ogus [Bl-Og] showed that the E2-term is given by 0 * *0 *+*0 E*,*2~=HZar(X, HZ=p) =) Het (X; Z=p) 0 where H*Z=pis the (Zarisky) sheaf assopciated to the presheaf U 7! 0 H*et(U; Z=p). From Corollary 2.4 in [Ya1], we see the following lemma. 0 * Lemma 3.5. If the cycle map cl : H*,*(X; Z=p) ! Het(X; Z=p) is injective for each *, *0, then 0 * *0 *+*0,*0 E*,*2~=HZar(X; HZ=p) ~=H (X; Z=p)=(o ). Moreover the spectral sequence collapses from the E2-term. 0 Note deg(Qffl(d)) = (2d - ff(d), d - ff(d)) in H*,*(Ma; Z=2) but it is 0 *0 represented (d, d - ff(d)) in H*-*Zar(Ma; HZ=2). From Corollary 3.2, we can write down the E2-term for X = Ma explicitly. Lemma 3.6. There is the Z=2[ae]-module isomorphism 0 n+1 e H*Zar(Ma; H*Z=2) ~=Z=2[ae]=(ae ) 0 d 2n-2Z=2[ae]=(ae d){qd} where ed = 1 + ff(d) - ff(d + 1) and deg(qd) = (1 + d, n + 1 + d - ff(d)). 0 (Here qd = Qffl(d)(ao -1) in H*,*(Ma; Z=2)=(o ).) Corollary 3.7. The coniveau spectral sequence for X = Xd and p = 2 collapses from the E2-term and 0 * *0 m H*Zar(Xd; H*Z=2) ~= HZar(Mni; HZ=2)[h]=(h i){ti}. 4. Borel spectral sequence Let X be a smooth variety over R. The manifold X(C) of C-rational points X(C) is a Z=2-equivariant space by the Galois group Gal(C=R) action. By Cox [Co], it is known that there is a natural weak homotopy equivalence {X}^et~=(X(C) xZ=2 EZ=2)^ where {-}et means the etale homotopy type, {-}^ means the profi- nite completion and EZ=2 is a contractible space with free Z=2-action. Then we have H*et(X; Z=2) ~=H*Z=2(X(C); Z=2) = H*(X(C) xZ=2 EZ=2; Z=2). 8 N.YAGITA Here the right hand side is called the Borel cohomology or Z=2-equivariant cohomology. Thus we have the following Borel spectral sequence 0 * * * E*,*2= H (BZ=2; H (X(C); Z=2)) =) Het(X; Z=2), 0 *+r+1,*0-r dr : E*,*r! Er . Since the Borel spectral sequence is the topological one, it is multiplica- tive, in particular, the differential is a derivation. For the case X = pt., we see H*(X(C); Z=2) ~=Z=2 and the Borel spectral sequence is trivial H*et(pt.; Z=2) ~=E*,01~=H*(BZ=2; Z=2) ~=Z=2[x]. Hence we can identify ae = x 2 H1et(pt.; Z=2) ~=Z=2. Lemma 4.1. The E1 -term of the Borel spectral sequence for H*et(Xd; Z=2) is given by E*,*1~=Z=2[x, h]=(x2(i)hsi, he+1|0 i r0). Proof. From Lemma 2.5, we have H*(Xd(C); Z=2) ~=Z=2[h]=(he+1) (ff). Let 0 6= oe 2 Gal(C=R) ~=Z=2. Then it is known oe(ff) = ff + he (resp. oe(ff) = ff) if d = 0 mod(4) (resp. otherwise). Let d = 0 mod(4). (The other cases are similar but more easy.) The E2-term of this spectral sequence is 0 * *0 E*,*2= H (Z=2; H (X(C); Z=2)) ( *0(X(C); Z=2) * = 0 ~= H 0 (Ker(1 - oe)=Im(1 + oe)|H* (X(C); Z=2) * > 0. The nontrivial case is *0= d = 2e. Since oe(ff) = ff + he, we see Ker(1 - oe) = Im(1 + oe) = Z=2{he}. Hence E*>0,d2= 0 and E*,d2~=Z=2{he}. Thus we have the isomorphism 0 e+1 e e E*,*2~=Z=2[x, h]=(h , xh ) Z=2[x, h]=(h ){ffh} which is additively isomorphic to Z=2[x, h]=(he) (ffh) Z=2{he}. Note that nr = 0 and 2(r) = 2nr+1 - 1 = 1. We can decompse the above term as 0 r-1 2(r) E*,*2~= i=0A(i) (ffh) A(r)=(x ) 0<2si+1 with A(i) = E*,2si2* ~=Z=2[x]{1, ..., hmi-1 }{hsi}. MOTIVIC COHOMOLOGY OF REAL QUADRICS 9 The element h is a permanent cycle and the first nonzero differen- tial must be dt(ffh). From Corollary 3.3 (or Corollary 3.2), we see 0 x2(r-1)hsr-1= 0 in E*,*1. Hence t = 2(r - 1) and d2(r-1)(ffh) = x2(r-1)hsr-1. Note here that d2(r-1)(hmr-1ff) = x2(r-1)hsr-1+mr-1 = x2(r-1)he+1 = 0. Thus we have the isomorphism E*,*2(r-1)+1~= rj=r-1A(j)=(x2(j)) r-2-1i=0A(i) (ffhmr-1). Similarly, by decsending induction on j, we can prove 0 2(i) j s 0 -s E*,*2(j)~= i=ri=j+1A(i)=(x ) i=0A(i) (ffh r +1 j), d2(j)(ffhsr0+1-sj) = x2(j)hsj. In particular , we get the desired result 0 0 2(i) r0 2(i) s s -1 E*,*1~= ri=0A(i )=(x ) ~= i=0Z=2[x]=(x ){h i, ..., h i+1 } ~= Z=2[x, h]=(x2(i)hsi, he+1) Remark. In x3 in [Sa-Li], the integral Borel spectral sequence 0 * *0 * IE*,*2~=H (Z=2; H (Xd(C); Z)(2)=) H (X; Z(2)) 0 is studied, while IE*,*1seems not given there. By the filtration of multiplying 2, we have the isomorphism 0 *,*0 *0 grIE*,*2~=E2 2H (Xd(C); Z(2)) . 0 Since all elements in 2H* (Xd(C); Z(2)) are permanent cycles, we also 0 *,*0 *0 get the isomorphism grIE*,*1~=E1 2H (Xd(C); Z(2)) . Lemma 4.2. Let ti be an element in H*et(Xd; Z=2) such that ti = hsi mod(ae) and ae2(i)ti = 0. Then such ti exists uniquely. Proof. We still know the existence. From the above corollary, if sj k < sj+1, there is the isomorphism E*,2k1~=Z=2[x]=(x2(j)){hk}. So the map xm : E*,2k1! E*+m,2k1is injective if * + m < 2(j). When * + 2k = 2si and * > 0 (i.e., j < i), we see * + 2(i) = 2si+ 2(i) - 2k = 2s0i+ 1 - 2k < 2s0j+ 1 - 2k 2s0j+ 1 - 2sj = 2(j). Hence the map x2(i): E*,2k1! E*+2(i),2k1is injective when * + 2k = 2si. 10 N.YAGITA Suppose t0iis the another element which satisfies the assunption of 0 the lemma. For 0 6= t 2 H*et(Xd; Z=2), let us write by 0 6= [t] 2 E*,*1 the corresponding nonzero element. Then 0 0 0 6= [ti- t0i] 2 E*,*1 with * < 2si, and * +* = 2si which must be x2(i)-free by the above dimensional reason. However ti, t0iare ae2(i)-torsion and so is ti- t0i. This is a contradiction. 5. Cohomology of Grassmaniann G2(R2+d). Let X be a smooth variety over the real number field R. Let X(R) = ;. Then Gal(C=R) acts on X(C) freely. Hence we have the isomorphism H*Z=2(X(C); Z=2) ~=H*(X(C)=Z=2; Z=2) from the triviality of the spectral sequence induced from the cofibeing EZ=2 ! X(C) xZ=2 EZ=2 ! X(C)=Z=2. Now we consider anisotropic quadrics over R. Each anisotropic qua- dratic form q of dimension n over R is written by n<1> = qn = x21+ .... + x2n. Reacll that Xd is the projective quadric defined by qd+2 = 0 so that dim(Xd) = d. Let Gk(Rk+n ) be the Grassmannian of k-planes in Rk+n . It is well known that there is the homeomrphism Xd(C)=Z=2 ~=Gr2(R2+d) via the map from (zj = xj + iyj) 2 Xd(C) CPd+1 to the plane generated by (xj), (yj) in R2+d. (In fact, ||(xi)|| = ||(yi)|| and (xi)?(yi) in Rd+2 for (xi+ iyi) 2 Xd(C).) Thus we see that Lemma 5.1. H*et(Xd; Z=2) ~=H*(Gr2(R2+d); Z=2). The mod 2 cohomology of Grk(Rn+k ) is computed by Borel H*(Gk(Rn+k ); Z=2) ~=Z=2[w1, ..., wk, ~w1, ..., ~wn]=In,k with In,k= ((1 + w1 + w2...) . (1 + ~w1+ ~w2+ ...) in fact, wi(resp.w~i) is the i-th Stiefel-Whitney class of the universal k- plane bundle (its complementary bundle). Hiller [Hi] write down this relation explicitly. Theorem 5.2. ([Hi]) There is the isomorphism H*(Grk(Rn+k ); Z=2) ~=Z=2[w1, ..., wk]=(f1,n, ..., fk,n) MOTIVIC COHOMOLOGY OF REAL QUADRICS 11 0 1 0 1 n0 1 f1,n w1 1 0 . . 0 w1 B . C B w 0 1 . . 0 C B . C B C B 2 C B C where BB . CC = BB . CC BB . CC @ . A @ . 0 . . . 1 A @ . A fk,n wk 0 . . . 0 wk We also note the following fact. Lemma 5.3. The sequence (f1,n, ..., fk,n) is a regular in Z=2[w1, ..., wk]. Proof. Let Vn+k,k be the variety of orthogonal k vectors in Rn+k and O(k) be the Orthogonal group. Then we have the principal bundle O(k) ! Vn+k,k ! Gk(Rn+k ) and the induced spectral sequence E*,*2= H*(BO(k); H*(Vn+k,k; Z=2)) =) H*(Gk(Rn+k ); Z=2). Of course H*(BO(k); Z=2) ~=Z=2[w1, ..., wk] and (by Borel) grH*(Vn+k,k; Z=2) ~= (en, ..., en+k-1 ) with deg(ei) = i. Hence the differential must be d(en+i) = fn,i+1from the above the- orem. Moreover H*(Grk(Rn+k ); Z=2) is multiplicatively generated by wi. This imlies that the sequence is regular. P Let P (t) be the Poincare series iH*(Grk(Rn+k ); Z=2)ti. Since dim(wi) = i and dim(fi,n) = n + i, we get ; Corollary 5.4. (1 - tn+1)(1 - tn+2)...(1 - tn+k ) P (t) = ________________________________. (1 - t)(1 - t2)...(1 - tk) In particular, the case k = 2 of Theorem 5.2 is stated as follows by letting f1,n= fn and f2,k= w2fn-1 Corollary 5.5. There is the isomorphism H*et(Xd; Z=2) ~=Z=2[w1, w2]=(fd, w2fd-1) Here f0 = w1, f1 = w21+ w2 and fn+1 = w1fn + w2fn-1 for n 0, namely, ` ' X d + 1 - b fd = wd+1-2b1wb2. b Proof. The last equation follows from induction on d. The first iso- morphism is immediate from Theorem 5.2. 12 N.YAGITA Corollary 5.6. The elements ae, h in H*et(Xd; Z=2) correspond to w1, w2 respectively in H*(G2(R2+d); Z=2). Proof. We only need the proof for h. By dimensional reason, h cor- responds to w2 or w2 + w21 = f1. But in H*et(X1; Z=2) we know ae2 = h 6= 0. 0 Let P = P (t, E*,*1) be the Poincare series of the infinite term of the spectral sequence in lemma 3.4. Of course from above corollaries P (t) = (1 - td+1)(1 - td+2)=(1 - t)(1 - t2). This fact is also computed directly, infact (1 - t)(1 - t2)P is written X X (1 - t2(i))(1 - t2mi)t2si= (t2si- t2si+2mi+ t2si+2(i)- t2si+2mi+2(i)) i i X X 0 0 = (t2si- t2si+1) + (t2si+1- t2si-1+1) = 1 - td+1 - td+2 + t2d+3. i i Remark. In [Hi], Hiller computed some relations for w1, w2 by using above fd in the corollary. Most of them are natural consequences from the cohomology theory of quadrics. For example, Proposition 4 in [Hi] is wd26= 0, which corresponds hd 6= 0 in CH*(Xd)=2. One of the main results of [Hi] is about the Lusternik-Schnierlmann category, that is, n+1 2n+1 n+1 Cat(Gr2(R2 )) = dim(Gr2(R )) = 2 - 2 n+1-2 2n+1-2 followingnfrom w21 6= 0. This corresponds the fact ae = h2 -1 6= 0 in the norm variety (the smallest neighbour of the Pfister form) X2n-1. 6. ae2(i)-torsion elements In this section, we look for tiin Corollary 4.5. That is ti 2 Z=2[w1, w2] such that ti = wsi2mod(w1) and w2(i)1ti 2 (fd, fhd-1) where X `d + 1 - b' X `d + b' fd = wd+1-2b1wb2, w2fd-1 = wd-2b1wb+12. b b b b We first recall the famous relation about the binary coefficient P P Lemma 6.1. ([Ep-St]) If n = ni2i and m = mi2i for mi, nj = 0 or 1, then ` ' ` ' m Y mi = mod(2). n i ni MOTIVIC COHOMOLOGY OF REAL QUADRICS 13 Recall that (6.1) d + 2 = 2n0+1 - 2n1+1 + ... + (-1)r2nr0+1+ ffl. where ffl = 0, 1 or -1. For k r0, let us write (6.2) d(k) + 2 = 2n0+1 - 2n1 + ... + (-1)k2nk. Lemma 6.2. In Z=2[w1, w2], we can take ( fd(k)=w2(k)1 if d(k) d tk = d-d(k) 2(k) fd(k)w2 =w1 if d(k) < d. d-d0 Proof. We first prove k = r0and ffl = 0. Let d = 2d0. Of course d0 = 0+1 1, we see w2fd-1 = hd mod(w1). Hence we can take tr0+1= w2fd-1. Next look for tr0. Let d0- b = b0. Then ` ' ` ' ` ' ` ' d + 1 - b 2d0+ 1 - (d0- b0) d0+ 1 + b0 d0+ 1 + b0 = = = b d0- b0 d0- b0 2b0+ 1 Here d0+ 1 = 0 mod(2nr0) from (6.1). Hence the above number is zero for each b0 < 2nr0- 1 from the above lemma. (See also the proof of Lemma 9.1 bellow.) For b0 = 2nr0- 1, the nonzero value of the above binomial coefficients is one and 0)d0-2nr0+1 2(r0)s 0 2(r0)+1 fd = w2(r1w2 = w1 w2r mod(w1 ). Therefore we can take 0) tr0= fd=(w2(r1) in Z=2[w1, w2]. Next consider the case k > r0. (The case k = r0 and ffl = 1 is also proved similarly.) Let k = even so that d(k) d. Let si(d(k)) (resp. r0(d(k)), ti(d(k)) be the number si (resp. r0, ti) for Xd(k). Then from (2.3), we see sk(d(k)) = sk = 2n0 - 2n1 + ... + 2nk-2 - 2nk-1. Since r0(d(k)) = k, we still know tk(d(k)) = fd(k)=(w2(k)1) = wsk(d(k))2+ .... Let ik : Xd Xd(k)be the natural embedding. Then we can take tk = i*ktk(d(k)) = (fd(k))=(w2(k)1). Let k = odd so that d(k) d. Then si- si(d(k)) = 2nk+1+1 - ... + (-1)r2nr+1 = d - d(k). Let ik : Xd(k) Xd be the embedding. For the Gysin map ik*, we see ik*(1) = wd-d(k)2, and ik*(wa1wb2) = wa1wb+d-d(k)2. 14 N.YAGITA (Note w2 = h is the hyperplane section.) Thus we can take tk = ik*(fd(k))=w2(k)1= fd(k)wd-d(k)2=w2(k)1. From the Borel spectral sequence (Lemma 4.1), we see w2(k+1)1wmk2tk = w2(k+1)1wmk+12= 0 mod(w*1tj|j k). k * Moreover w21tk is nonzero and it is written as w1tk-1 by the following reason by using Theorem 2.6. Recall that each element citsk (with the notation in x2, see Corollary 2.4, Corollary 3.2,) is represented by nk-2i (o -1ae2)2 tk nk-1 2nk In particular cnk-1tsk is represent by o -2 ae tk. Using Theorem 2.6, we can prove that hmk cnk-1tk must be written as ae*cnk-1tk-1. However we show the following lemma by direct computation of fd. The proof is just a computation of binomial coefficient but not so short, hence we give it in the last section. Lemma 6.3. nk-1 mk 2(k-1)-2nk w21 w2 tk = w1 tk-1. Lemma 6.4. nk-1X nk-1+1-2j+1s -2nk-1+2j 2(k)+2nk-1+1 fd(k)= w2(k)1( w21 w2k ) mod(w1 ), j=nk+1 nk-1 s -2nk-1-1 i.e., tk = wsk2+ w21 w2k + ... 7. motivic cohomology In this section, we study the motivic cohomology. Take element w1 2 H1,1(Xd; Z=2), and w2 2 H2,2(Xd; Z=2) such that the images of the cycle map cl are the same letter elements. (So w1 = ae and w2 = o h in the notation x2 and x3.) Hence deg(fd) = (d + 1, d + 1). Take ~tk2 H2*,*(Xd; Z=2) = CH*(Xd)=2 such that cl("tk) = tk, namely, ( 2(k)o sk if d(k) d ~tk= fd(k)=ae fd(k)hd-d(k)=ae2(k)o sk if d(k) < d. Since we still know 0 * -1 H*,*(Xd; Z=2) Het(Xd; Z=2)[o, o ], MOTIVIC COHOMOLOGY OF REAL QUADRICS 15 the following theorem is the immediate consequence of Theorem 3.1, Corollary 3.2, and Lemma 5.2 0 Theorem 7.1. Given d > 0, the motivic cohomology H*,*(Xd; Z=2) is isomorphic to the Z=2[ae, h]-subalgebra of the algebra Z=2[ae, o, o -1, h]=(fd, hfd-1) generated by Z=2[o ] and Z=2[ae] Q(ni- 1){aeni+1o -1~ti} 0 i r0 P where Z=2[ae] Q(ni- 1) = { 0 m 2ni-1-2o -maef|f 2m - ff(m)}. Here we can take hsiinstead of ~ti. 0 Corollary 7.2. The motivic cohomology H*,*(Xd; Z=2) is isomorphic to the Z=2[ae, h]-subalgebra generated by Z=2[o ] and Z=2[ae] Q(ni- 1){aeni+1o -1hsii} 0 i r0. Proof. By inductin on k for k < r0, we assume Z=2[ae] Q(ni- 1){o -1aeni+1~ti} 0 i < k is generated by (1) Z=2[ae] Q(ni- 1){o -1aeni+1hsii} 0 i < k. From Lemma 6.4, we see ~tk= hsk + ae2nk-1hsk-2nk-1-1+ ... Of course 2nk-1 + nk + 1 nk-1 + 1. Hence nk-1 s -2nk-1-1 Z=2[ae] Q(nk-1 - 1){(o -1aenk+1+2 h k + ...)} is contained in (1). Now we consider the proof of Theorem 2.6 without algebraic cobor- dism theory (for the field R). Proof of Theorem 2.6. The additive structure of CH*(Xd)=2 is still known. Hence we only need the h-divisibility of element ci. Recall that each element citsk (with the notation in x2, see Corollary 2.4, Corollary 3.2,) is represented by nk-2i (o -1ae2)2 ~tk. Consider the element nk-2i -1 * hmk (citsk) = hmk (o -1ae2)2 ~tk= (o ae) ~tk-1. Here we see * = 2nk - 2i+ (2(k - 1) - 2nk)=2 - (2nk - 1)=2 16 N.YAGITA = 2nk - 2i+ 2nk-1 + 1=2 - 2nk-1 - 2nk-1 - 1=2 = 2nk-1 - 2i. Thus we get hmk citsk = citsk-1. 8. example We consider the case d = 8. Since d + 2 = 24 - 23 + 21, we see n0 = 3, n1 = 2, n2 = 0 m0 = 2, m1 = 2, m3 = 1. 0 Hence the infinity term E*,*1of the Borel spectral sequence is Z=2{h4} Z=2[x]=(x7){h2, h3} Z=2[x]=(x15){1, h} h4 o h3 o o o o o o o h2 o o o o o o o h1 o o o o o o o o o o o o o o o 1 o o o o o o o o o o o o o o o 1 x x2 x6 x14 On the other hand f8 = w91+ w51w22+ w1w42, w2f7 = w81w2 + w61w22+ w41w32+ w52 are zero in H*et(X8; Z=2). From the first equation, we have t2 = f8=w1 = w42+ w41w22+ w81. We also know that f6w22= w71w22= w2f8 - w1w2f7, f16 = w151= w61f8 + w41f6 + w22w2f6 are zero in H*et(X8; Z=2). Thus we can take t1 = f6w22=w71= w22and t0 = 1 of course. The statement of Lemma 6.3 is described w2t2 = w61w22= w61t1, w1w22t1 = w91+ w51w22= w91t0. 0 Now we consider the motivic cohomology H*,*(X8; Z=2). It is a subalgebra of Z=2[ae, o, o -1, h]=(f8, w2f7) (identifying w1 = ae, w2 = ho ) generated as a QT Z=2[h]-module by 1, ae4o -1 and ae3o -1h2. (Here aeo -1h4 = o -1ae9 does not need as generators.) Let us write oe = o -1ae2. Then the Chow ring CH*(X8)=2 is described as Z{1, h, ..., h4} Z=2{1, h} {oe4, oe6, oe7, oe2h2, oe3h2}. MOTIVIC COHOMOLOGY OF REAL QUADRICS 17 Here the Qi actions are oe4 = Q0Q1(ae4o -1), oe6 = Q0Q2(ae4o -1), oe7 = Q1Q2(ae4o -1), oe2h2 = Q0(ae3o -1h2), oe3h2 = Q1(ae3o -1h2). Its multiplicatin is given by h5 = oe4h + oe3h2 + oe2h3, h6 = oe6 + oe3h3, h7 = oe7 + oe6h + oe5h2, h8 = oe7h. Moreover let c1 = oe2h2 and c2 = oe4. Then h2c1 = oe4h4 = oe6. Thus we have the isomorphism CH*(X8)=2 ~=Z=2[h]=(h9) Z=2[h]=(h4){c1} Z=2[h]=(h2){c2}. 9. Proofs of lemmas in x5 Proof of Lemma 6.4. Let d = even and d = d0. Recall that ` ' d0+ 1 + b0 2b0+1 d0-b0 fd = w1 w2 2b0+ 1 as described in the proof of Lemma 6.2. We will prove the case k = even and the odd case is proved similarly. Let d(k) + 2 = 2n0+1 - 2n1+1 + ... - 2nk-1+1 + 2nk+1. Here let d(k) = 2d0 and 0 b0 d0. We want to compute the 2-adic expansion ffl(-) of d0+ 1, b0, 2b0+ 1, namely, 1 nk nk-1 ffl(d0+ 1) = (0, ..., 0, 1 , 0, ..., 0, 1 , 1, ...) ffl(b0) = (*, *0, *00, .........), ffl(2b0+ 1) = (1, *, *0, *00, ....* *...). Note that the expansion of 2b0+ 1 is the shiftting of that of b0 to the right and adding010at the first entry. Assume d2+1+bb0+16= 0 mod(2). Suppose b0< 2nk. Then nk ffl(d0+ 1 + b0) = (*, *0, *00, ..., 1 , 0, ..., 0, 1, 1, ...) The fact ffl1(2b0+ 1) = 1 imlies ffl1(d0+ 1 + b0) = 1, which means ffl1(b0) = 1. That implies ffl2(2b0+ 1) = 1, and so ffl2(b0) = 1,... Thus we see b0= 2nk - 1. Next consider the case b0 2nk. By the same argument as above we see ffl1(b0) = ... = fflnk-1(b0) = 1. Suppose fflnk(b0) = 1. Since fflnk(d0+ 1) = 1, we know fflnk(d0+ 1 + b0) = 0. This implies fflnk(2b0 + 1) = 0, and so fflnk-1(b0) = 0. This is a contradiction. Hence fflnk(b0) = 0. Thus we can write nk 0 00 0 0 nk * * 0 00 ffl(b0) = (1, .., 1, 0 , *, * , * , ...), ffl(d + 1 + b ) = (1, .., 1, 1 , *,* * * , * , ..). 18 N.YAGITA Suppose that there is i such that ffli(b0) = 1, but ffli+1(b0) = 0 for i < nk-1 - 1, i.e., nk 0 00 i nk-1 000 ffl(b0) = (1, .., 1, 0 , *, * , * , 1, 0, ... 1 , * , ..). Then ffli+1(d0+ 1 + b0) = 0 but ffli+1(2b0+ 1) = ffli(b0) = 1. This is a contradiction. Thus if b0< 2nk-1, then we get nk j nk-1-1 nk-1 ffl(b0) = (1, ..., 1 0 , 0, ..., 0, 1, ..., 1 , 0 , 0, ..) i.e., b0= 2nk - 1 + 2nk-1 - 2j for nk < j nk-1. Lemma 9.1. Xk k+1-2j+1 2j 2k+1-1 fd+2k+1 = ( w21 w2 )fd + w1 w2fd-1. j=0 X `+1 j0+1 k+1 j+1 j `+1 0 fd+2k+1-2`+1= ( w21 -2 +2 -2 w22-2 +2j)fd+ 0 j0<`