NOTE ON CHOW RINGS OF NONTRIVIAL
G-TORSORS OVER A FIELD
NOBUAKI YAGITA
Abstract. Let Gk be a split reductive group over a field k cor-
responding to a compact Lie group G. Let E be a nontrivial
Gk-torsor over a field k. In this paper we study the Chow ring of
nontrivial Gk-torsors. For example when (G, p) = (F4, 3), we have
the isomorphism CH*(E)(3)~=Z(3).
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 Gk
the split reductive group over k which corresponds G. By definition, a
Gk-torsor E over k is a variety over k with a free Gk-action such that
the quotient variety is Spec(k). A Gk-torsor over k is called trivial, if
it is isomorphic to Gk or equivalently it has a k-rational point.
Let H be a subgroup of G. Given a torsor E over k, we can form
the twisted form of G=H by
(E x Gk=Hk)=Gk ~=E=Hk.
In this paper, we always assume that E is a (inner) nontrivial torsor
over k. We mainly study the cases that G are exceptional Lie groups
and the (p component) torsion index t(G)(p)= p. In particular, when
(G, p) = (G2, 2), (F4, 3) and H = T ; the maximal torus, H = P maxi-
mal parabolic subroups, we compute CH*(E=Hk)(p)explicitly. More-
over we show CH*(E)(p)= Z(p)for these cases. We also study the case
(G, p) = (SO2n+1-1, 2), n 3. This case CH*(E)(2) CH*(Gk)(2)but
it is not isomorphic to Z(2)nor CH*(Gk)(2).
For these groups, Petrov, Semenov and Zainoulline [Pe-Se-Za] showed
that the Chow motive of E=Pk and E=Tk are isomorphic to direct
sums of the generalized Rost motive ([Vo4],[Ro2],[Su-Jo],[Vi-Za]). The
algebraic cobordism MGL2*,*(-) of the Rost motives are given in
[Vi-Ya],[Ya3]. From this, we show the multiplicative structures of
____________
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
CH*(E=Pk) and CH*(E=Tk). The algebraic cobordism MGL2*,*(Gk)
is studied in [Ya1]. By using arguments in [Ya1], we can compute
CH*(E)(p).
The author thanks to Burt Totaro and Kirill Zainoulline who teach
him theories of torsors and algebraic groups.
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 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
n-1
M(XOEa) representing XOEaso that M(XOEa) ~=Ma M(P2 ).
Moreover for an odd prime p and nonzero symbol 0 6= a 2 KMn+1=p,
we can define ([Ro],[Vo],[Su],[Vi-Za]) the generalized Rost motive Ma,
which is irreducible and is split over K=k if and only if a|K = 0 as the
case p = 2.
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],[Vo4], [Vi-Ya], [Ya3]) The Chow ring CH*(Ma)
is only dependent on n. There are isomorphisms
CH*(Ma) ~=Z{1} (Z{c0} Z=p{c1, ..., cn-1})[y]=(ciyp-1)
and CH*(Ma|~k) ~=Z[y]=(yp)
where |y| = 2(pn-1 + ... + p + 1) and |ci| = |y| + 2 - 2pi. Here the
multiplications are given by ci.cj = 0 for all 0 i, j n-1. Moreover
the restriction map is given by i~k(c0) = py and i~k(ci) = 0 for i > 0.
Remark. The element y does not exist in CH*(Ma) while ciy exists.
Usually CH*(Ma) is defined only additively, however we give the ring
structure as above in this paper.
Remark. In this paper the degree |x| of an element x 2 CH*(X)
means the 2-times of the usual degree of the Chow ring so that it is
compatible with the degree of the (topological) cohomology H*(X(C)).
Let us use notation *(X) for the motivic cobordism MGL2*,*(X)(p)
defined by Voevodsky (but not the algebraic cobordism of Levine and
Morel, because Lemma 2.2 bellow is not proved for this theory). It is
known that
* = *(pt.) ~=MU2*(pt.)(p)~= Z(p)[x1, x2, ...]
ANISOTROPIC ALGEBRAIC GROUP 3
where MU2*(pt.) is the complex cobordism ring and |xi| = -2i. There
is the relation ([Ya2])
(2.1) *(X) * Z(p)~= CH*(X)(p).
We can take for xpi-1 the cobordism class of a 2(pi- 1)-dimensional
manifold whose characteristic numbers are divisible by p but the ad-
ditive characteristic number spi-1 is not by p2. Let us denote xpi-1 as
vi. Let us denote by
(2.2) In = (p = v0, v1, ..., vn-1) *
the ideal of * generated by p, ..., vn-1. Then it is well known that In
and I1 are the only prime ideals stable under the Landweber-Novikov
cohomology operations ([Ra]) in *.
The category of cobordism motives is defined and studied in [Vi-Ya].
In particular, we can define the algebraic cobordism of motives. The
following is the main result in [Vi-Ya] (in [Ya3] for odd primes).
Lemma 2.2. ([Vi-Ya], [Ya3]) The restriction map
i~k: *(Ma) ! *(Ma|~k) ~= *[y]=(yp)
is injective and there is the *-algebra isomorphism
*(Ma) ~= *{1} In{y, ..., yp-1} *[y]=(yp)
such that viy = ci in *(Ma) * Z(p)~= CH*(Ma)(p)in (2.2).
Remark. When n = 1, the restriction map i~k : CH*(M1) !
CH*(M1|~k) is injective and
Im(i~k) ~=Z(p){1} Z(p)[y]=(yp-1){py}
Z(p)[y]=(yp) ~=CH*(X|~k)(p).
Remark. Let BP * = Z(p)[v1, ..., vn]. Recall ([Ya2]) that
ABP 2*,*(X) = *(X) * BP *
for smooth X. Then we also see that
ik : ABP 2*,*(Ma) ! ABP 2*,*(Ma|~k)
is injective.
4 NOBUAKI YAGITA
3. 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 ) ~=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`)
ri
where bi = ~bimod(p) and fi = ypi mod(t1, ..., t`). Since G=T is cellular,
we also know
(3.4) *(G=T ) ~= *[y1, ..., yk, t1, ...t`]=(f"1, ..., "fk, "b1, ..., "*
*b`)
where "bi= bi mod( <0) and f"i= fi mod( <0).
Let Gk be the split reductive algebraic group corresponding G and
Tk the split maximal Torus. Hence
CH*(Gk) ~=CH*(GC) ~=H*(G).
Similarly CH*(Gk=Tk) ~= H*(G=T ) and CH*(BTk) ~= H*(BT ). Next
we consider the relation between CH*(Gk) and CH*(Gk=Tk).
ANISOTROPIC ALGEBRAIC GROUP 5
Theorem 3.1. (Grothendieck [Gr], [Ya1]) Let E be a Gk-torsor over
k. (Here we do not assume the nontiviality of E). Let h*(-) =
CH*(-) or *(X). Then
h*(E) ~=h*(E=Tk)=(i*h*(BTk)) ~=h*(E=Tk)=(t1, ..., t`).
Proof. Let Li ! E=Tk be the line bundle corresponding the element
ti 2 h2(E=Tk). Then we can embed the Tk-bundle E ! E=Tk into the
associated bundle iLi ! E=Tk. such that E is an open subscheme of
iLi. Consider the localization exact sequence
si* * *
ih*( j6=iLj) ! h ( iLi) ! h (E) ! 0
where si : E=Tk ! Li is the zero section. Since Li are vector bundles
h*(E) ~=h*( i6=jLj) ~=h*( iLi).
By the definition the first Chern class, we know ti = c1(Li) = s*isi*(1).
Thus we get the desired result h*(E) ~=h*(E=Tk)=(t1, .., t`).
Since CH*(Gk) ~=CH*(GC), from the result for H*(G=T ) (3.3), we
know
Corollary 3.2. ([Kac [Ka]) Ch*(Gk)(p)~= P (y)=(pyi).
The result *(GC) is one of the main result in [Ya1]. Let Qi be
the Milnor primitive operation in H*(X; Z=p) inductively defined by
i-1 pp-1
Qi = [Qi-1, P p ] and Q0 = fi; the Bockstein operation where P
is the pp-1-th reduced power operation. It is known that we can take
generators such that Qi(xodd) 2 P (yeven)=(p) for all i 0 ([Mi-Ni]).
Theorem 3.3. ([Ya1]) Take generators so that Qi(xodd) 2 P (yeven)=(p)
for all i 0. Then there is the *-module isomorphism
X
*(Gk)=I21 ~= * P (yeven)=(I21, viQi(xodd)).
i
Let P be a parabolic subgroup. Then the inclusinon T P induces
the fibering
p
(3.5) P=T ! G=T ! G=P
and the spectral sequence (see [Tod])
0 * * *
E(E=T )*,*2~=H (G=P ) H (P=T ) =) H (G=T ).
Since these spaces have no torsion and even dimensionally generated,
this spectral sequence is collapses, that is
(3.6) grH*(G=T ) ~=H*(G=P ) H*(P=T ).
Hence H*(G=P ) can be computed from H*(G=T ) (while many cases
H*(G=P ) is more easily computed than H*(G=T )).
6 NOBUAKI YAGITA
The cohomology H*(P=T ) can be compute by the fibering
P=T ! BT !i BP.
Indeed, if i* is injective, then
(3.7) H*(P=T ) ~=H*(BT )=(i*H"*(BP )).
Note that for the Borel subgroup B, we have the isomorphisms CH*(X=T ) ~=
CH*(X=B) and CH*(BB) ~=CH*(BT ).
4. exceptional groups of type (I)
Let G be a compact connected Lie group of dim(G) = 2d. The
torsion index is defined by
t(G) = |H2d(G=T ; Z)=i*H2d(BT ; Z)|.
By Grothendieck, it is known that any Gk-torsor E splits over any
fields L over k of index dividing t(G). By Totaro all t(G) are recently
known [To1,2]. Let us write by t(G)(p)the p-component of t(G). In
this section, we restrict the cases t(G)(p)= p (for ease of arguments)
and G are exceptional Lie groups. We call such (G, p) is of type (I),
that is
(G2, 2), (F4, 2), (E6, 2)
(F4, 3), (E6, 3), (E7, 3), and (E8, 5).
Throughout this section, we assume (G, p) are type of (I).
Remark. In [Pe-Se-Za], Petrov, Semenov and Zainoulline defined
the J-invariant JG (i1, ..., ik) of G from the smallest number is such that
is
yps 2 Im(i~k). (More accurate definition, see 4.5 in [Pe-Se-Za].) Hence
type (I) group has J-invariant JG (1).
For these cases, the ordinary mod(p) cohomology is well known
H*(G; Z=p) ~=Z=p[y]=(yp) (x1, ..., x`)
where ` = rank(G) 2, |y| = 2p + 2, |x1| = 3, |x2| = 2p + 1. Moreover
Q1(x1) = y, Q0(x2) = y.
Hence we have the isomorphisms as (3.3) and (3.4)
grH*(G; Z=p) ~=Z=p[y, t1, ..., t`]=(yp, ~b1, ..., ~b`),
H*(G) ~=Z[y, t1, ..., t`]=(f1, b1, ..., b`),
*(G) ~=Z=p[y, t1, ..., t`]=(f"1, "b1, ..., "b`),
where bi = ~bimod(p) and f1 = ypimod(t1, ..., t`), and where "bi=
bi mod( <0) and f"1= f1 mod( <0). From Corolary 3.2,we see
Corollary 4.1. CH*(Gk)(p)~= Z=p[y]=(yp).
ANISOTROPIC ALGEBRAIC GROUP 7
From Theorem 3.3 and the Qi-actions, we see
*(Gk)=I21 ~= *[y]=(py, v1y, yp, I21),
while we have more strong result (Theorem 5.1 in [Ya1]) than Theorem
3.3.
Corollary 4.2. *(Gk) ~= *[y]=(py, v1y, yp).
Remark. In the Atiyah-Hirzebruch spectral sequence
0,*00 *,*0 *00 *,*0
E*,*2 ~=H (Gk; MU ) =) MGL (Gk)
we know that
d2p-1(x1) = v1 Q1(x1) = v1y.
00 * p
Thus we get also E2*,*,*1~=MU [y]=(py, v1, y ).
In general, torsors E are parametrized by elements , 2 H1(k; Aut(G)).
A torsor is said to be inner if it is in the image of the canonical map
H1(k; G) to H1(k; Aut(G)). In this paper, we assume all torsors are
inner type.
Petrov, Semenov and Zainoulline [Pe-Se-Za] developed the theory of
generally splitting varieties. We say that L is splitting field of a variety
of X if M(X|L) is isomorphic to a direct sum of twisted Tate motives
T i and the restriction map iL : M(X) ! M(X|L) is isomorphic after
tensoring Q. A smooth scheme X is said to be generically split over k
if its function field L = k(X) is a splitting field. Petrov Semenov and
Zainoulline showed that torsors of type G of (I), (restricting E6 to the
adjoint type Ead6), are all generally split.
Theorem 4.3. (Theorem 4.9 in [Pe-Se-Za]) There is a mod(p) motive
Rp(G) such that
(1) CH*(Rp(G)|~k))=p ~=Z=p[y]=(y)
(2) M(E=Tk; Z=p) ~= sRp(G) T is ~=Rp(G) H*(G=T ; Z=p)=(y)
where we identify H*(G=T ; Z=p)=(y) as the sum of Tate motives T is.
Theorem 4.4. (Theorem 3.8 in [Pe-Se-Za]) Let Qk Pk be para-
bolic subgroups of Gk which are generically split over k. There is a
decompostion of motive M(E=Qk)(p)~= M(E=Pk)(p) H*(P=Q).
For p = 2, 3 (i.e., except for E8, p = 5), from Proposition 5.5 (for
m = p) and x7 in [Pe-Se-Za], we have the integral motivic decopostion
which deduces the mod(p) decomposition in Theorem 4.3. Moreover
from Corollary 6 in [Vi-Za] (see also [Se],[Bo]), we see the integral
motive corresponding Rp(G) is really generalized Rost motive.
8 NOBUAKI YAGITA
Theorem 4.5. Let G of type (I) except for E8 and E6. Then there is
a parabolic subgroup Pk such that E=Pk is generically split and
CH*(Gk=Pk)(p)~= Z[y]=(yp) A and M(E=Pk)(p)~= M2 A
where A is a sum of twisted Tate motives and M2 = Ma is the gener-
alized Rost motive for some 0 6= a 2 KM3(k)=p.
Now we state the main theorem of this paper.
Theorem 4.6. Let G of type (I) except for E8 and E6. The chow
ring CH*(E=Tk)(p) is multiplicatively generated by t1, ..., t`. Hence
CH*(E)(p)~= Z(p).
Proof. From the above theorems, we have the decompostion of motive
M(E=Tk)(p)~= M3 H*(G=T )=(y).
We consider the restriction map
i~k: *(E=Tk) ! *(E=Tk|~k) ~= *(G=T ).
Since i~k| (M2) is injective, so is i~kabove. Let us write
Im(i~k)= i~k( *(E=Tk)) *(Gk=Tk) = *(G=T ).
Of course pyi, v1yi 2 Im(i~k) for i > 0 since so in *(M2|~k). Note
that t1, .., t` 2 Im(i~k) because they exist in CH*(E=Tk) since so in
CH*(BTk).
Recall that each element x 2 *(E=Tk|~k) ~= *(G=T ) is represented
as
p-1XX
(*) x = v(s, i)t(s, i)yi, v(s, i) 2 *, t(s, i) 2 Z(p)[t1, ..., t*
*`]
i=0 s
while if x 2 Im(i~k), then v(s, i) 2 Ideal(p, v1) for i > 0.
From Corollary 4.2, we see py = v1y = 0 in *(Gk). From Theorem
3.1, this means
py, v1y 2 (t1, ..., t`) *(Gk=Tk).
(But notePthat this does not mean 2 (t1, ..., t`)Im(i~k).) Let us write
v1y = v(s, i)t(s, i)yi as (*). The above fact implies |t(s, i)| > 0 and
hence |v(s, i)| < 0.
Now let us write
<1>*(X) = *(X) * Z(p)[v1] = ABP <1>2*,*(X).
In <1>*(Gk=Tk), the fact |v(s, i)| < 0 means
v(s, i) 2 (v1) = Z(p)[v1]<0 = <1><0.
ANISOTROPIC ALGEBRAIC GROUP 9
Hence v1y 2 (t1, ..., t`)Im(i~k) in <1>*(-) theory. Similarly we can
prove that pyi, v1yihave the same property. (We note yp 2 (p, v) *(Gk=Tk)
so in Im(i~k).)
Thus we can write
X X *
v1y = v(s, i)0t(s, i)v1yi in <1> (E=Tk).
i s
If v(s, i)0 6= 0 for i > 1, then apply the same equation to the right
hand side v1y in the above equation. Since t(s, i) = 0 when |t(s, i)| >
dim(G=T ), we can write
X
v1y = v(s, 0)t(s, 0).
s
The same property holds for pyiand v1yiwhen i > 1. Hence i~k( <1>*(E=Tk))
is generated as an <1>*-algebra by t1, ..., t`.
Since we know the isomorphisms
CH*(E=Tk)(p)~= *(E=Tk) * Z(p)~= <1>*(E=Tk) <1>*Z(p),
the elements t1, ..., t` are multiplicatively generate CH*(E=Tk)(p).
5. exceptional Lie group G2
In this section we study the case (G, p) = (G2, 2). We recall the
cohomologies from Toda-Watanabe [To-Wa]
H*(G=T ; Z) ~=Z[t1, t2, y]=(t21+ t1t2 + t22, t32- 2y, y2)
with |ti| = 2 and |y| = 6. Let P be one of the maximal parabolic
subgroups. Then from (3.6) and H*(P=T ) ~=Z{1, t1}
H*(G=P ; Z) ~=Z[t2, y]=(t32- 2y, y2) ~=Z{1, y} {1, t2, t22}.
By Bonnet, we have the decomposition
Theorem 5.1. ([Bo],Corollary 5.6 in [Pe-Se-Za])
M(E=Pk) ~=M2 M2(1) M2(2).
Theorem 5.2. There is the ring isomorphism
CH*(E=Pk)(2)~= Z(2)[t2, u]=(t62, 2u, t32u, u2)
~=Z(2)[t2]=(t62) Z=2[t2]=(t32){u}
with |t2| = 2, |u| = 4.
10 NOBUAKI YAGITA
Proof. From Lemma 2.2, we know
*(M2) ~= *{1, 2y, vy} *{1, y}.
From the preceding theorem, we have the *-algebra isomorphism
*(E=Pk) ~= *{1, v1y, 2y} {1, t2, t22} *(G=P ).
Since CH*(X)(p)~= *(X) * Z(p), we have the isomorphism
CH*(E=Pk)(2)~= Z(2){1, 2y}{1, t2, t22} Z=2{v1y}{1, t2, t22}.
Here the multiplications are given as follows. Since 2y = t32mod( <0)
in *(Gk=Tk), we can take 2y = t32so that
Z(2){1, 2y}{1, t2, t22} = Z(2)[t2]=(t62).
Let us write u = v1y in CH*(E=Tk)(2). Then t32u = 2yv1y = 0 and
u2 = v21y2 = 0 in *(E=Tk) * Z(2). Hence we have the isomorphism
in the theorem.
Next consider CH*(E=Tk)(2).
Theorem 5.3. There is the ring isomorphism
CH*(E=Tk)(2)~= Z(2)[t1, t2]=(t62, 2u, t32u, u2)
where u = t21+ t1t2 + t22.
Proof. The Chow ring is isomorphic to
(*) CH*(E=Tk)(2)~= CH*(E=Pk){1, t1}
~= (Z(2){1, 2y} Z=2{v1y}){1, t2, t22}{1, t1}.
Here 2y = t32. Since v1y 2 (t1, t2) and v1y = 0 2 CH*(G=T ), we see
v1y = ~(t21+ t1t2 + t22) mod((t1, t2) <0 *(G=T ))
for ~ 2 Z(2). We can take ~ = 1 mod(2). Otherwise v1y = 0 2
*(G=T )=2, which is a *=2-free, and this is a contradiction. Hence
we can take t21+ t1t2 + t22as v1y. (This is also proved by Lemma 4.3
in [Ya1], since Q1(x1) = y and d3(x1) = t21+ t1t2 + t22.) Hence in
CH*(E=Tk) we have the relation
(t32)2 = 0, (t32)u = 0, u2 = 0, 2u = 0.
We consider the mod 2 Poincare polynomial
X
rankZ=2(CHi(E=Tk)=2)ti = (1 + t2 + t4)(1 + t + t2)(1 + t)
i
(1 - t6)(1 - t4) 5 2
= 1 + 2t + 3t2 + 4t3 + 4t5 + 3t5 + t6 = _______________ - t (1 + t)
(1 - t)(1 - t)
ANISOTROPIC ALGEBRAIC GROUP 11
which is the (mod(2)) Poincare series of the right hand side ring of
the theorem. (Note (t62, u2) is a regular sequence in Z=2[t1, t2] but
(t62, u2, (t32)u) is not.)
The author learned the following remarks by Zainoulline.
Remark. It is well known that there is the bijection between
H1(k; G2) and the class of Cayley algebras C from the fact G2 =
Aut(C|~k). Hence each torsor E over k corresponds a Cayley algebra.
Moreover E=Tk and E=Pk correspond the following varieties [Ca-Pe-
Se-Za]. By an i-space (i = 1, 2), we mean i-dimensional subspace Vi of
C such that u.v = 0 for every u, v 2 Vi. The flag variety corresponding
E=Tk is the full flag variety
X(1, 2) = {V1 V2|Vi; i - subspaces C}
and that corresponding E=Pk is X(2) = {V2|V2; 2 - subspace C}.
Let
g : H1(k; G2) ! H3(k; Z=2) ~=KM3(k)=2
be the Arason invariant (which is know to be isomorphic). The symbol
of the Rost motive in Theorem 5.1 is g(E) i.e., M2 = Mg(E).
Remark. Similar facts hold for (G, p) = (F4, 3). This case, the
corresponding algebras are exceptional Jordan algebras of dimension
27 over k, and the symbol of the generalizedf motive is the Rost-Serre
invariant.
6. exceptional group F4 for p = 3
Let (G, p) = (F4, 3) throughout this section. Let E be a nontrivial
Gk-torsor as previous sections. Let Pk be a maximal parabolic sub-
group of Gk given by the the last vertex of the Dynkin diagram.
Theorem 6.1. ([Pa-Se-Za]) Let M2 be the generalized Rost motive.
Then there is an isomorphism M(E=Pk) ~= 7i=0M2(i).
We first recall the ordinary cohomology of G=P ([Is-To], [Du-Za]).
H*(G=P )(3)~= Z[t, y]=(r8, r12), |t| = 2, |y| = 8
where r8 = 3y2 - t8 and r12 = 26y3 - 5t12. Hence we can rewrite
H*(G=P )(3)~= Z(3){1, t, ..., t7} {1, y, y2}.
Recall the Rost motive
CH*(M2|~k) ~=Z[y]=(y3),
CH*(M2) ~=Z{1} Z{3y, 3y2} Z=3{v1y, v1y2}.
12 NOBUAKI YAGITA
Of course, the above y is that in H*(G=P )(3)by the dimensional reason.
From the above theorem, we have the decomposition
(*) CH*(E=Pk) ~=Z(3){1, t, ..., t7} (Z(3){1, 3y, 3y2} Z=3{v1y, v1y2}).
The ring structure is given as follows.
Theorem 6.2.
CH*(E=Pk)(3)~= Z(3)[t, b, a1, a2]=(t16, t8b, b2 = 3t8, bai, 3ai, t8ai, a1a2)
~=Z(3){1, t, ..., t7} (Z(3){1, p 3t4, t8} Z=3{a1, a2})
where |b| = 8 and |a1| = 4, |a2| = 12.
Proof. From the relation r8 in CH*(G=P ), we have
3y2 = t8 + vx 2 *(G=P ) for v 2 <0.
Hence we can take t8 instead of 3y2 in (*). Of course
(3y)2 = 3t8 + 3vx 2 *(Gk=Pk).
p 4
Hence we write by b = 3t the element 3y. Write by a1, a2 the ele-
ments v1y, v1y2 respectively. Elements in I1 <0 (Gk=Pk) reduces
to zero in CH*(E=Tk). Therefore we have the desired multiplicative
results.
The cohomology H*(G=T ) is given by Toda-Watanabe [To-Wa]
H*(G=T )(3)~= Z3[t1, t2, t3, t4, y]=(ae2, ae4, ae6, ae8, ae12).
Here relations aei is written by the elementary symmetric function ci =
oei(t1, t2, t3, t4), that is,
ae2 = c2 - (1=2)c21, ae4 = c4 - c3c1 + (1=2)3c41- 3y, ae6 = -c4c21+ c23,
ae8 = 3c4c41- (1=2)4c81+ 3y(24y + 23c3c1), ae12 = y3
By the arguments similar to the proof of Theorem 5.3 (or Lemma 4.3
in [Ya1]).
Theorem 6.3. Let ss : E=Tk ! E=Pk. Then
ss*(t) = c1, ss*(a1) = ae2, ss*(a2) = ae6, ss*(b) = c4 - c3c1 - (2)-3c41.
Hence there is the epimorphism
Z(3)[t1, t2, t3, t4]=(c161, c81ss*(b), ss*(b)2 - 3c81, ss*(b)aej, 3aej, c81a*
*ej, ae2ae6)
! CH*(E=Tk)(3)
where j = 2, 6.
ANISOTROPIC ALGEBRAIC GROUP 13
7. The orthogonal group SO(m) and p = 2
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~)
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).
The Qi-operations are given by Nishimoto [Ni]
Qnxodd= xodd+|Qn|, Qnxeven= Qnyeven= 0
Relations in *(SO(m)) are given by
X X
vnQn(xodd) = vnxodd+|Qn|= 0 mod(I21).
n n
For example, the relations in *(SO(m))=I21 starting with 2y6 are
given by
2y6 + v1y42+ v2y26+ v3y210+ ... = 0,
Theorem 7.1. ([Ya1]) There are *(2)-algebra isomorphisms
(1) *(SO(m))=I21 ~= *[y4i+2|2 4i + 2 m - 1]=(R, I21)
s(i) 0
where R = {relations starting with y24i+2, 2y4i+2, v1y4i0+2i 6= 0.}
For ease of arguments, we only consider the case G = SO(odd). Let
G = SO(2m0+ 1) and P = SO(2m0- 1) x SO(2). Then it is well
known [To-Wa]
0 2 0
Lemma 7.2. H*(G=P ) ~=Z[t, y]=(tm - 2y, y ) |y| = 2m .
By Toda-Watanabe [To-Wa], we also know
Theorem 7.3. ([To-Wa])
0 2
H*(G=T ) ~=Z[ti, y2i, tm0, y]=(ci- 2y2i, J2i, tmm0- 2y, y )
where 1 i m0- 1, ci = oe(t1, ..., tm0) and
X2i X
J2i= 1=4( (-1)jcjc2i-j) = y4i- (-1)jy2jy4i-2j.
j=0 0> = OEaen+1
the (n + 1)-th Pfister form associated to aen+1. (That is, q0 is the
maximal neighbor of the (n + 1)-th Pfister form.) Of course q|~k= q0|~k
and we can identify E=Pk ~=Xq0. From Lemma 7.4 (or Rost's result) ,
we know
n-1 2
CH*(Xq0|~k) ~=Z[t, y]=(t2 - 2y, y ).
The multiplicative structure of Chow ring of the anisotropic quadric is
easily compute from Lemma 2.2 [Ya4]
ANISOTROPIC ALGEBRAIC GROUP 15
Lemma 7.4. ([Ya4])
n+1-2 2n-1
CH*(E=Pk) ~=Z[t]=(t2 ) Z=2[t]=(t ){u1, ..., un-1}
where ui = viy 2 *(E=p) * Z(2)so uiuj = 0.
By the projection E=Tk ! E=Bk, Petrov, Semenov and Zainoulline
also show the following (the example of the last page in [Pe-Se-Za],
indeed, they show the J-invariant J2(G) = (0, ..., 0, 1)).
Theorem 7.5. When m = 2n+1-1. The restriction map i~k: *(E=Tk) !
*(E=Tk|~k) = *(Gk=Tk) is injective and
grCH*(E=Tk) = grCH*(E=Pk) A,
gr *(E=Tk) = gr *(E=Pk) A
where A = Z[ti, y2i]=(c0i- 2yi, J2i).
As a corollary, we see that ti, y2i are all in CH*(E=Tk). Hence
CH*(E=Tk) is multiplicatively generated by ti, yi, t and u1, ..., un-1.
Theorem 7.6. Assume (**) and m = 2n+1 - 1. Then
CH*(E)(2)~= P (y)0=(2) P (y)0 Z=2[y]=(y2) ~=CH*(Gk)(2).
Proof. The proof is quite similar to that of Theorem 3.4. Let us write
*(X) = *(X) * Z(2)[v1, ..., vn-1] ~=ABP 2*,*(X).
By Theorem 3.1, We want to prove
(1) u1, ..., un-1 2 (t1, ..., tm0)CH*(E=Tk).
This means
u1, ..., un-1 2 ((t1, ..., tm0) + <0) *(E=Tk).
Let us write
Im(i~k)= i*~k( *(E=Tk)) *(Gk=Tk),
I(t, <0) = ((t1, ..., tm0) + <0))Im(i~k).
(Note I21 <0Im(i~k).) Thus it is sufficient for (1) to prove
(2) 2y, ..., vn-1y 2 I(t, <0).
At first we will show vn-1y 2 I(t, <0). Recall y = y2n+1-2 =
x2n+1-2. From Theorem 3.3 and Nishimoto's result, we see
(3) x = 2x2n + v1x2n+2 + ... + vn-2x2n+2n-1-2 + vn-1x2n+1-2
= 0 in *(Gk)=(I21).
So x 2 ((t1, .., tm0) + I21) *(Gk=Tk).
16 NOBUAKI YAGITA
Each element z 2 *(Gk=Tk) is written (not uniquely) by
X X
(4) z = vItJyK + vI0tJ0yK0y
with vI, vI0 2 *, tJ, tJ0 2 Z(2)[t1, .., tm0] and yK , yK0 2 P (y)0.
Note that if z 2 (t1, .., tm0) *(Gk=Tk), then we can take |tJ| > 0
and |tJ0| > 0.
Consider the case z = x in (3). Since yK 2 Im(i~k), we know
vItJyK 2 (t1, ..., tm0)Im(i~k).
Since |y| < |tJ0yK0y|, we know |vI0| < 0, i.e., vI0y 2 Im(i~k) be-
cause vI0 2 * = Z(2)[v1, ..., vn-1]. Thus we know vI0tJ0yK0y 2
(t1, ..., tm0)Im(i~k). Thereore we see
(5) x 2 I(t, <0).
In (3), x2n+2 = y2n+2, ..., x2n+2n-1-2 are in Im(i~k). So we see
v1x2n+2 + ... + vn-2x2n+2n-1-2 2 <0Imk(ik).
Hence we see
(6) 2x2n + vn-1y 2 I(t, <0).
Next we will see
(7) 2y2, ..., 2y2n-2 2 I(t, <0).
n-1 <0
Then in particular, (y2)2 = x2n implies vn-1y 2 I(t, ) from (6).
Similarly we can prove vn-2y, ..., 2y 2 I(t, <0) by using the arguments
(3) - (7). Thus we see (2) and so (1).
We prove (7) for 2y2 and the other cases are similar. By also using
Nishimoto's result and Theorem 3.3, we have the relation
x0= 2x2 + v1x4 + ...vn-1x2n = 0 2 *(Gk)=I21.
By using arguments similar to (3)-(5), we have x0 2 I(t, <0). Of
course v1x4+...vn-1x2n 2 <0Im(i~k). Thus we see 2y2 2 I(t, <0).
Remark. Note that i~k(CH*(E))(2)= CH*(Gk)(2)- {y}. The simi-
lar fact happens for -theory
Im(i~k)= *(Gk) - {y}.
Indeed 2y, ..., vn-1y 2 *(Gk) are in the *-subalgebra
generated by yi.
ANISOTROPIC ALGEBRAIC GROUP 17
References
[Bo] J.Bonnet. Un isomorphisme motivique entre deux var'et'ees homog`enes
projective sous l'action d'un groupe de type G2. Doc. Math. 8 (2003)
247-277.
[Ca-Pe-Se-Za]B.Calmes, V.Petrov, N.Semenov and K.Zainoulline. Chow motives
of twisted flag varieties. Composito Math. 142 (2006) 1063-1080
[Du-Za] H.Duan and X.Zhao The Cow ring of generalized Grassmanianns.
preprint (2005)
[Gr] A. Grothendieck. Torsion homologique et sections rationnelles. Sem.
C.Chevalley, ENS 1958,expose 5, Secreatariat Math., IHP, Paris
(1958).
[Is-To] K. Ishitoya, and H. Toda. On the cohomology of irreducible symmetric
spaces of exceptional type. J.Math. Kyoto Univ. 17 (1977), 225-243.
[Ka] V. G. Kac. Torsion in cohomology of compact Lie groups and Chow
rings of reductive algebraic groups. Invent. Math. 80 (1985), 69-79.
[L-M 1] M. Levine and F. Morel. Cobordisme alg'ebrique I. C. R. Acad. Sci.
Paris 332 (2001), 723-728.
[L-M 2] M. Levine and F. Morel. Cobordisme alg'ebrique II. C. R. Acad. Sci.
Paris 332 (2001), 815-820.
[Mi-Ni] M. Mimura and T. Nishimoto. Hopf algebra structure of Morava K-
theory of exceptional Lie groups. Contem. Math. 293 (2002), 195-231.
[Mo-Vi] L.Molina and A.Vistoli. On the Chow rings of classifying spaces for
classical groups. Math.AG/0505560 (2005), May.
[Ni-Se-Za]S. Nikolenko, V.Petrov, N.Semenov and K.Zainoulline. Motivic de-
composition of anisotropic varieties of type F4 into generalized Rost
motives. preprint (2005)
[Ni] T. Nishimoto. Higher torsion in Morava K-thoeory of SO(m) and
Sin(m). J.Math.Soc. Japan. 52 (2001), 383-394.
[Pe-Se-Za]V.Petrov, N.Semenov and K.Zainoulline. Chow motives of generically
split varieties. preprint (2006)
[Ra] D.Ravenel. Complex cobordism and stable homotopy groups of
spheres. Pure and Applied Mathematics, 121. Academic Press (1986).
[Ro1] M.Rost. Some new results on Chowgroups of quadrics. preprint (1990).
[Ro2] M.Rost. On the basic correspondence of a splitting variety. preprint
(2006)
[Su-Jo] A.Suslin and S.Joukhovistski. Norm Variety. J.Pure and Appl. Algebra
206 (2006) 245-276.
[Se] N.Semenov. Motivic decomposition of a compactification of a
Merkjurjev-Suslin variety. arXiv preprint (2006)
[To1] B. Totaro. The Chow ring of classifying spaces. Proc.of Symposia
in Pure Math. "Algebraic K-theory" (1997:University of Washing-
ton,Seattle) 67 (1999), 248-281.
[To2] B. Totaro. The torsion index of E8 and the other groups. Duke Math.
J. 299 (2005), 219-248.
[To3] B. Totaro. The torsion index of the spin groups. Duke Math. J. 299
(2005), 249-290.
[Tod] H.Toda. On the cohomolgy ring of some homogeneous spaces. J. Math.
Kyoto Univ. 15 (1975), 185-199.
18 NOBUAKI YAGITA
[To-Wa] H.Toda and T.Watanabe. The integral cohomology ring of F4=T and
E4=T . J. Math. Kyoto Univ. 14 (1974), 257-286.
[Vi] A.Vishik. Motives of quadrics with applications to the theory of qua-
dratic forms. Geometric methods in algebraic theory of quadratic form*
*s,
by Izhboldin, Kahn, Karpenko and Vishik LNM 1835 (2004), 25-101.
[Vi-Ya] A.Vishik and N.Yagita. Algebraic cobordisms of a Pfister quadric.
www.math.uiuc.edu/K-theory/0170 (2005).
[Vi-Za] A.Vishik and K.Zainoulline. Motivic splitting lemma.
www.mathematik..uni-bielefeld.de/LAG/237 (2007).
[Vo1] V. Voevodsky. Motivic cohomology with Z=2 coefficient. Publ. Math.
IHES 98 (2003), 59-104.
[Vo2] V. Voevodsky (Noted by Weibel). Voevodsky's Seattle lectures : K-
theory and motivic cohomology Proc.of Symposia in Pure Math. "Al-
gebraic K-theory" (1997:University of Washington,Seattle) 67 (1999),
283-303.
[Vo3] V.Voevodsky. Reduced power operations in motivic cohomology.
Publ.Math. IHES 98 (2003),1-57.
[Vo4] V.Voevodsky. On motivic cohomology with Z=l-coefficients.
www.math.uiuc.edu/K-theory/0631 (2003).
[Ya1] N. Yagita. Algebraic cobordism of simply connected Lie groups.
Math.Proc.Camb.Phil.Soc. 139 (2005), 243-260.
[Ya2] N. Yagita. Applications of Atiyah-Hirzebruch spectral sequence for
motivic cobordism. Proc. London Math. Soc. 90 (2005) 783-816.
[Ya3] N. Yagita. Chow rings of excellent quadrics. submitting (2006)
[Ya4] N. Yagita. Algebraic BP -theory and norm varieties. preprint (2007).
Department of Mathematics, Faculty of Education, Ibaraki Univer-
sity, Mito, Ibaraki, Japan
E-mail address: yagita@mx.ibaraki.ac.jp