NOTE ON THE MOD p MOTIVIC COHOMOLOGY OF ALGEBRAIC GROUPS NOBUAKI YAGITA Abstract. Let Gk be a split reductive group over a field k0of ch(k) = 0 corresponding to a compact Lie group G. Let H*,*(Gk; Z=p) (resp. H*(G; Z=p)) be the mod p motivic (resp. singular (topo- logical)) cohomology. Then we show the isomorphism 0 * *,*0 grH*,*(Gk; Z=p) ~=grH (G; Z=p) H (pt.; Z=p) for G = SOn, G2, F4, E6. Here the bidegree of the right hand side cohomology is given by deg(y) = (2*, *) (resp. deg(x) = (2*-1, *)) for even (resp. odd) dimensional ring generators y (resp. x). 1. Introduction Let k be a subfield of C which contains primitive p-th root of the unity. Let G be a compact connected Lie group. Let us denote by Gk0the split reductive group over k which corresponds G. Let H*,*(Gk; Z=p) (resp. H*(G; Z=p)) be the mod p motivic (resp. singu- lar (topological)) cohomology. Let T1 ... T` = T be a sequence of tori of G where Ti ~= (S1)xi. Note that the flag variety Gk=Tk is cellular. If the sequence of tori satisfies some0good condition, then we can compute0the motivic cohomology H*,*(Gk=(Ti)k; Z=p) inductively from H*,*(G=T ; Z=p). In fact, we will show if the ring H*(G=T ; Z=p) satisfies some condition (stated in Assumption(*) in x2), then there is the isomorphism 0 * *,*0 grH*,*(Gk; Z=p) ~=grH (G; Z=p) H (pt.; Z=p) Here the bidegree of the right hand side cohomology is given by deg(y) = (2*, *) (resp. deg(x) = (2 * -1, *)) for even (resp. odd) dimensional ring generators y (resp. x). The ring structure of H*(G=T ; Z) is completely determined by Toda, Watanabe and Nakagawa for G = SOn, G2, F4, E6, E7. We easily check ____________ 1991 Mathematics Subject Classification. Primary 55P35, 57T25; Secondary 55R35, 57T05. Key words and phrases. motivic cobordism, Rost motive, compact Lie groups. 1 2 NOBUAKI YAGITA that the condition is satisfied for these cases except for E7 and p = 3. Thus we have the isomorphisms above for these cases. If the group Gk is nonsplit, then the situation is completely differ- ent. In the last section we consider the mod(2) motivic cohomology of nonsplit (twisted) form of G2 for k = R. 2. compact Lie group G Let G be a compact connected Lie group. By the Borel therorem, we have the ring isomorphism for p odd (3.1) H*(G; Z=p) ~=P (y)=(p) (x1, ..., xl) r1 prk with P (y) = Z[y1, ..., yk]=(yp1 , ...yk ) where |yi| = even and |xj| = odd. When p=2, for each yi, there is xj with x2j= yi. Hence we have grH*(G; Z=2) ~=P (y)=(2) (x1, ..., xl). Let T be the maximal torus of G and BT the classifying space of T . We consider the fibering (3.2) G !ssG=T !i BT and the induced spectral sequence E*,*2= H*(BT ; H*(G; Z=p)) =) H*(G=T ; Z=p). The cohomology of the classifying space of the torus is given by H*(BT ) ~=S(t) = Z[t1, ..., t`] with |ti| = 2. where ` is also the number of the odd degree generators xiin H*(G; Z=p). It is known that yi are permanent cycles and that there is a regular se- quence ([Tod],[Mi-Ni]) (b1, ..., b`) in H*(BT )=(p) such that d|xi|+1(xi) = bi. Thus we get 0 * E*,*1~=grH (G=T ; Z=p) ~=P (y) Z=p[t1, ..., t`]=(b1, ..., b`). Moreover we know that G=T is a manifold of torsion free, and we get (3.3) H*(G=T )(p)~= Z(p)[y1, .., yk, t1, ...t`]=(f1, ..., fk, "b1, ..., "* *b`) i where "bi= bi mod(p) and fi = ypi mod(p, t1, ..., t`). Here we consider the following assumption Assumption(*) We can take the Torus T1 ... T` = T with Ti ~= (S1)xi and the corresponding basis ti in S(t) such that for all 1 i ` (1) b1, ..., bi is regular in S(t)=(ti+1, ..., t`), (2) bi = tigi in S(t)=(b1, ..., bi-1, ti+1, ..., t`) for some gi 2 S(t). MOTIVIC COHOMOLOGY OF SPLIT GROUPS 3 Remark. Note that if above assumption is satisfied, then we can take b0 2 S(t) such that bi = b0imod(b1, ..., bi-1) and b0i= tig0 in S(t)=(ti+1, ..., t`) but not only S(t)=(b1, ..., bi-1, ti+1, ..., t`). Lemma 2.1. Suppose the Assumption (*). Then grH*(G=Ti; Z=p) ~=H*(G=T )=(ti+1, ..., t`) (xi+1, ..., x`). Proof. Compare the two spectral sequences E*,*2= S(t) P (y) (x1, ..., x`) =) H*(G=T ; Z=p). E(i)*,*2= S(t) P (y)=(ti+1, ..., t`) (x1, ..., x`) =) H*(G=Ti; Z=p) 0 *,*0 and the induced map j : E*,*r! E(i)r . Since dr(xk) = bk in Er for r = |bk|, so is in E(i)r. By the Assumption (1), b1, ..., bi is regular. Hence we see E(i)|bi|+1~=H*(G=T ; Z=p)=(ti+1, ..., t`) (xi+1, ..., x`). In fact, from Assumption(2), bi+1, ..., b` are all zero in S(t)=(ti+1, .., t`). Hence we get E(i)|bi|+1~=E(i)1 . Thus we get the lemma. Corollary 2.2. grH*(G=Ti-1; Z=p) ~=grH*(G=Ti; Z=p)=(ti) (xi). The fibering S1 ! G=Ti-1 ! G=Ti induces the Gysin exact sequence !ffiH*-2(G=T j*=xti * * ffi i; Z=p) ! H (G=Ti; Z=p) ! H (G=Ti-1; Z=p) ! . Hence we see grH*(G=Ti-1; Z=p) ~=grH*(G=Ti)=(ti) Ker(xti). From the above corollary, we get Corollary 2.3. We have Ker(xti) ~=H*(G=Ti; Z=p){xi} and ffi(xi) = gi in Assuption(2). Indeed, xti|H*(G=Ti; Z=p){gi} = 0. Moreover H*(G=Ti; Z=p)=(ti){gi} H*(G=Ti; Z=p). 4 NOBUAKI YAGITA 3. motivic cohomology 0 Let X be an algebraic variety over k. Let H*,*(X; Z=p) be the mod(p) motivic cohomology constructed by Suslin and Voevodsky [Vo]. For nonzero element x 2 Hm,n(X; Z=p), we define the weight degree and the different degree by w(x) = 2n - m, d(x) = m - n. When X is smooth, it is known that w(x) 0, d(x) dim(X). Moreover from the affirmative answer of the Bloch-Kato conjecture (and hence Beilinson-Lichtenbaum conjecture) implies 0 M H*,*(pt.; Z=p) ~=Z=p[o ] K* (k) where 0 6= o 2 H0,1(pt.; Z=p) ~=Z=p and the Milnor's K-theory is KM*(k)=p ~=H*,*(pt.; Z=p). pt. Let us denote by Gk the split reductive group over k corresponding to the compact Lie group G and Ti = (Ti)k the spit torus. We give here the main theorem of this paper Theorem 3.1. Suppose the Assumption(*) in the preceding section. Then 0 * *,*0 grH*,*(Gk=Ti; Z=p) ~=grH (G=Ti; Z=p) H (pt.; Z=p) where the bidegree in H*(G=Ti; Z=p) is given for nonzero element x 2 H*(G=T ; Z=p) by w(x) = 0 and w(xi) = 1 Corollary 3.2. Suppose the Assumption(*). Then 0 *,*0 H*,*(G; Z=p) ~=H (pt.; Z=p) P (y) (x1, ..., x`) where w(P (y)) = 0 and w(xi) = 1. For the motivic theory, there is the Thom isomorphism and hence the Gysin exact sequence !ffiH*-2,*0-1(G=T j*=xti *,*0 *,*0 ffi i; Z=p) ! H (G=Ti; Z=p) ! H (G=Ti-1; Z=p) ! . By descending induction on i, we easily show H2*,*(Gk=Ti; Z=p) ~=H2*(G=T ; Z=p)=(ti+1, ..., t`) and there is xi 2 H2*-1,*(Gk=Ti-1; Z=p) with ffi(xi) = bi as Corollary 3.3. MOTIVIC COHOMOLOGY OF SPLIT GROUPS 5 We will prove the main theorem also by descending induction on i. Indeed when i = `, the space Gk=T is cellular and there is the isomorphism 0 * *,*0 H*,*(Gk=T ; Z=p) ~=H (G=T ; Z=p) H (pt.; Z=p) identifying w(H*(G=T ; Z=p)) = 0. By induction, we assume the theorem for i. We will prove the theo- rem for i - 1, at first the case k = C. Recall that 0 H*,*(pt.; Z=p) ~=Z=p[o ], in fact, KM*(C)=p ~=Z=p. Let us write 0 m,n Hm,n(i, o ) = (H*,*(G=Ti; Z=p) Z=p[o ]) M ~= H2*,*(G=T ; Z=p)=(ti+1, ..., t`){xi1...xiso t}. (i + 1 i1 < ... < is `), m = 2 * +|xi1| + ... + |xis|, n = 1=2(m + s) + t Note here (*) Hm,n(i-1, o ) ~=Hm,n(i, o )=(ti) Hm-|xi|,n-1=2(|xi|+1)(i, o )=(ti){xi}. Next consider Ker(ti|Hm,n(i, o )). From Corollary 3,3, we still know Ker(ti|H2*(G=T ; Z=p)=(ti, ..., t`)) ~=H2*-|xi|+1(G=T ; Z=p)=(ti, ..., t`){gi}. This implies that Ker(ti|Hm,n(i, o )) is isomorphic to H2*,*(G=T ; Z=p)=(ti, ..., t`){xi1...xiso t}{gi}, where direct sum runs (i + 1 i1 < ... < is `), m - |xi| = 2 * +|xi1| + ... + |xis|, n - 1=2(|xi| - 1) = 1=2(m + s) + t. Hence this is isomorphic to Hm-|xi|,n-1=2(|xi|-1)(i, o )=(ti){xi} by the map xi 7! gi from (*). Therefore we have the exact sequence for fixed n j*=xti *,n *,n ffi (**) !ffiH*-2,n-1(i, o ) ! H (i, o ) ! H (i - 1, o ) ! . Here, by inductive assumption on i, 0 *,*0 H*,*(i, o ) ~=H (Gk=Ti; Z=p). Of course there is the natural map 0 *,*0 H*,*(j, o ) ! H (Gk=Tj; Z=p) for all 1 j `. 6 NOBUAKI YAGITA Thus we get the main result for k = C by the five lemma and the induction on i. For general field k case, we consider 0 M m,n Hm,n(i, k) = (H*,*(i, o ) k* (k)=p) ~= aH*-|a|,*0-|a|(i, o ){a} where {a} is a Z=p-base of KM*(k)=p. Then from the result for k = C, (**) !ffiH*-2-|a|,n-1-|a|(i, o ){a} j*=xti *-|a|,n-|a| *-|a|,n-|a| ffi ! H (i, o ){a} ! H (i - 1, o ){a} ! is exact sequence for each a 2 KM*(k)=p). Hence so is j*=xti *,n *,n ffi (**) !ffiH*-2,n-1|(i, k) ! H (i, k) ! H (i - 1, k) ! . By five lemma,0we also have the result of the main theorem. When H*,*(X; Z=p) ~= A Z=p[o ], we show the following theo- rem, by using the preceding arguments exchanging KM*(k)=p by A KM*(k)=p. 0 Theorem 3.3. Suppose that G satisfies Assumption(*) and H*,*(X; Z=p) is Z=p[o ]-free. Then 0 *,*0 *,*0 H*,*(X x Gk=Ti; Z=p) ~=H (X; Z=p) H*.*0(pt.;Z=p)H (Gk=Ti; Z=p). Corollary 3.4. The Kunneth formula holds for H*(Gk; Z=p). Hence it is a Hopf algebra. Since w(xi) = 1, the coproduct is given as X _(xi) = y(1)jxj y(2)j + y(3)j y(4)jxj y(s)j 2 P (y). Here the above coproduct is0determined from that of the topological case H*(G; Z=p) becase t*,*Cis injective for k = C and for each (*, *0). 4.Examples The cohomology H*(G=T ) is computed by Toda-Watanabe ([To- Wa]) for the case G = SO(n), G2, F4, E6. The case G = E7 is computed by Nakagawa [Wa], [Na]. First, we consider the orthogonal groups G = SO(m) and p = 2. The mod 2-cohomology is written as ( see for example [Ni]) grH*(SO(m); Z=2) ~= (x1, x2, ..., xm-1 ) where the multiplications are given by x2s= x2s. We write y2(odd)= x2odd. Hence we can write H*(SO(m); Z=2) ~=Z=2[y4i+2|2 4i+2 m-1]=(ys(i)4i+2) (x1, x3, ...xm~) MOTIVIC COHOMOLOGY OF SPLIT GROUPS 7 where s(i) is the smallest number such that 2s(i)(4i + 2) m and ~m = m - 1 (resp.m~ = m - 2) if m is even (resp. odd). By Toda-Watanabe [To-Wa], it is know that Theorem 4.1. (1) H*(SO(2n + 1)=Tn) ~=Z[ti, tn, y2i|1 i n]=(ci- 2y2i, Ji) X where Ji = y4i+ (-1)jy2jy4i-2j, (y2k = 0 for k n) 0