CHOW RINGS OF NONABELIN p-GROUPS OF
ORDER p3
NOBUAKI YAGITA
Abstract. Let G be a nonabelian p group of order p3 (i.e., ex-
traspecial p-group), and BG its classifying space. Then CH*(BG) ~=
H2*(BG) where CH*(-) is the Chow ring over the field k = C.
We also compute mod(2) motivic cohomology and motivic cobor-
dism of BQ8 and BD8.
1. Introduction
For a smooth algebraic variety over k = C, let CH*(X) be the Chow
ring (over C) and BP *(X) the Brown-Peterson theory. Then Totaro
[To1] defined the modified cycle map
"cl: CH*(X)(p)! BP 2*(X) BP* Z(p)
such that the composition with the Thom map ae : BP *(X) ! H*(X),
is the usual cycle map.
Let G be an algebraic group over C and BG the classifying space. To-
taro conjectured that the map "clis an isomorphism for X = BG. This
conjecture is correct for connected groups O(n), SO(n), G2, Spin7, Sinn8,
P GLp ([To2],[Mo-Vi],[In-Ya], [Gu], [Ma], [Ka-Ya],[Vi]), and finite abelian
groups and (finite) symmetric groups (localized at 2) [To2].
We will show it holds for each non abelian p-group of order p3.
Theorem 1.1. If G is an extraspecial p-group of order p3 (i.e., p1+2+
or p1+2-for an odd prime, and Q8 or D8 for p = 2), Then
CH*(BG)(p)~= BP 2*(BG) BP* Z(p)~= H2*(BG)(p).
This is the first example for nonabelian p-group (p > 2) which sat-
isfies Totaro's conjecture. Note that the cycle map cl : CH*(BG) !
H2*(BG) is not surjective for G = (Z=p)3, and not injective for the
central product D8 . D8 x Z=2 (see [To1]).
____________
1991 Mathematics Subject Classification. Primary 55P35, 57T25; Secondary
55R35, 57T05.
Key words and phrases. Chow ring, motivic cohomology, extraspecial p-groups.
1
2 N.YAGITA
It is known [Te-Ya], that for each of the above groups, the Brown-
Peterson cohomology is given
BP *(BG) ~=BP *[[y1, y2, c1, ..., cp]]=(relations)
where y1, y2 are the first Chern classes of liner representations of G,
and ci is the i-th Chern class of some p-dimensional representation of
G. Moreover we know
BP 2*(BG) BP* Z(p)~= H2*(BG)(p).
It is shown in [Ya1] that if CH*(BG) is multiplicatively generated
by y1, y2, c1, ..., cp, then Totaro's conjecture holds. In this paper, we
will prove this fact and hence Totaro's conjecuture for the above ex-
traspecial p-groups.
Let MU*(X) be the complex cobordism theory so that MU*(X)(p)~=
0 *,*0
MU*(p) BP* BP *(X). Let MGL*,*(X) and MGL (X; Z=p) be the
motivic cobordism defined by Voevodsky and its mod(p) theory.
From the above theorem and Proposition 9.4 in [Ya3], we have
Corollary 1.2. For an extraspecial p-group G, we have the isomor-
phism MGL2*,*(BG)(p)~= MU2*(BG)(p).
0
When p = 2, we get the rather strong results. Let H*,*(X; Z=2)
be the mod(2) motivic cohomology and 0 6= o 2 H0,1(Spec(C); Z=2).
Then we prove in x6 ;
Theorem 1.3. Let G = Q8 or D8. Then there is the filtration of
H*(BG; Z=2) such that
0 *0 *
H*,*(BG; Z=2) ~=Z=2[o ] gr H (BG; Z=2).
Using this theorem, we prove in the last section ;
Theorem 1.4. Let G = Q8or D8. Then there is the isomorphism
0 2*
MGL*,*(BG; Z=2) ~=Z=2[o ] MU (BG).
2. extraspecial p-groups
Throughout this paper, let G be a non abelian p-group of order p3.
Then the group is called an extraspecial p-group so that there is the
central extension
0 ! C ! G ! V ! 0
where C ~= Z=p is the center and V ~= Z=p Z=p. We can take
a, b, c 2 G such that [a, b] = c here c generates C and a, b generate V .
(See [Lw],[Ly],[Gr-Ly],[Te-Ya] for details.)
3
These groups have two types for each prime p. For an odd prime
p, they are written as p1+2-, p1+2+where ap = c for the first type but
ap = bp = 1 for the other type. When p = 2, the groups are the
quaternion group Q8 and the dihedral group D8, where a2 = b2 = c for
Q8 but a2 = c, b2 = 1 for D8.
Let a*, b* : G ! V ! C* be the linear representation which is the
dual of a, b respectively. Let "c= IndG(c*) for G = p1+2+, for other
groups cases, let "c= IndG(a0) where a0 : ! C* is the dual of a.
For example when G = p1+2+, we can take
"c(c) = diag(i, ..., i), "c(a) = diag(1, i, ..., ip-1),
and "c(b) as the permutation matrix in GLp(C) where i is a primitive
p-th root of unity.
For an integer N 1, representations Nc", Na*, Nb* give the G-
action on
UN = CpN* x CN* x CN*
where CpN* = CpN - {0} and CN* = CN - {0}. Here G acts freely on
UN = CN(p+2) - HN with codim(HN ) N. Hence given G-variety X,
the Borel cohomology can be defined by
CH*G(X) = CH*(UN xG X) when * < N.
Of course CH*G(pt.) = CH*G~= CH*(BG) the Chow ring of the classi-
fying space BG.
Let us write by y1, y2 2 CH*(BG) the first Chern classes of a* and b*
respectively. Let ci be the i-th Chern class of "c. We consider CH*G(UN )
when N = 1. We use the stratified methods by Molina-Vistoli [Mo-Vi]
which was used to compute the Chow rings of BG for classical groups
G.
Lemma 2.1.
CH*G(Cp* x C* x C*) ~=CH*(BG)=(y1, y2, cp).
Proof. we first consider the localized exact sequence
CH*G({0}xCxC) i*!CH*+pG(CpxCxC) ! CH*+pG(Cp*xCxC) ! 0.
Here i* is the multiplying cp. So we have
CH*G(Cp* x C x C) ~=CH*G=(cp).
Similarly applying c1(a*) = y1, c1(b*) = y2, we have the lemma.
Corollary 2.2. The Chow ring CH*(BG) is multiplicatively generated
by elements dim p + 2.
4 N.YAGITA
Proof. First note that the G-action on Cp* x C* x C* is free. Hence
CH*G(Cp* x C* x C*) ~=CH*((Cp* x C* x C*)=G).
Of course dim(Cp* x C* x C*) = p + 2, and we see CH*G=(y1, y2, cp) is
generated by elements dim p + 2.
Recall that the Brown-Peterson theory also have Chern classes. It is
known [Te-Ya], that for each of the above groups, the Brown-Peterson
cohomology is given
BP *(BG) ~=BP *[[y1, y2, c1, ..., cp]]=(relations).
Moreover we know BP 2*(BG) BP* Z(p)~= H2*(BG). Hence H2*(BG)
is multiplicatively generated by Chern classes of dim 2p.
Lemma 2.3. ([To2]) If H2*(X)(p)is multiplicatively generated by Chern
classes for * p, then for all * p,
CH*(X)(p)~= BP 2*(X) BP* Z(p)~= H2*(X)(p).
Proof. Recall that the usual K-theory K*(X)(p)localized at p can be
decomposed to the connected Morava K-theory K"(1)*(X) with the
coefficient ring K"(1) = Z(p)[v1], |v1| = -2p + 2. We consider the
Atiyah-Hirzebruch spectral sequence ([Te-Ya])
0 * *0 *
E(K)*,*2~=H (X) "K(1) =) K"(1) (X).
The first nonzero differential is known
d2p-1(x) = v1 fiP 1(x) (= v1 Q1(x) mod(p)).
Since H2*(X)(p)is generated by Chern classes, each element is a per-
manent cycle. In fact
0 2* *0
E(K)2*,*1~=H (X) "K(1) for * p.
This implies from the definition of grigeoK0(X) ([Th], [To2])
grigeoK0(X)(p)~= H2i(X)(p) for i p.
Next consider the Atiyah-Hirzebruch spectral sequence for BP *(X)
0 * *0 *
E(BP )*,*2~=H (X) BP =) BP (X).
0 *0 2*
Similarly we have E(BP )2*,*1~=BP H (X) for * p. (The differ-
ential d2p-1 is the same as the case K"(1)*(-).) Hence we have
(BP *(X) BP* Z(p))2i~= H2i(X)(p).
On the other hand, there is the natural map
ci i
CHi(X) ! grigeoK0(X) ! CH (X),
5
which is the multiplication by (-1)i-1(i - 1)! by Riemann Roch with
denominators. Moreover the first map is epic. (See the proof of Corol-
lary 3.2 in [To2].) Hence CHi(X)(p)~= grigeoK0(X)(p).
Corollary 2.4. If the cycle map cl : CH*(BG) ! H2*(BG) is injec-
tive for * 2p - 2, then CH*(BG) ~=H2*(BG) for all * 0.
Proof. Since H2*(BG) is generated by y1, y2, ci, we see from Corollary
2.2 that CH*(BG) is generated by the same elements y1, y2, ci. All
relations between the above multiplicative generators are in dim
2p-2 (for the explicit relations, see the following results of the ordinary
cohomology). Hence we get the corollary.
Of course the usual cohomology of BG is explicitly known as follows.
Theorem 2.5. (Lewis [Le],see also [Ly],[Te-Ya])
Heven(Bp1+2+) ~=(Z[y1, y2]=(y1yp2-yp1y2, pyi) Z=p{c2, ..., cp-1}) Z[cp]=(p2cp),
Hodd(BG) ~=Heven(BG)=(p){e} |e| = 3.
Here ciyj = cick = 0 for i < p - 1, but yjcp-1 = ypj, c2p-1= yp-11yp-12.
Theorem 2.6. (Lewis [Le], [Ly])
Heven(Bp1+2-) ~=(Z[y2]=(py2) Z=p{y1 = c1, c2, ..., cp-1}) Z[cp]=(p2cp),
Hodd(Bp1+2-) ~=Z=p[y2, cp]{e} with |e| = 2p + 1
Here ciyj = cick = 0 for i p - 1.
Theorem 2.7. (Evens [Ev])
Heven(BD8) ~=(Z[y1, y2, c2]=(y1y2, 2yi, 4c2),
Hodd(BD8) ~=Heven(BD8)=(2){e} with |e| = 3.
Theorem 2.8. (Atiyah [At])
Heven(BQ8) ~=(Z[y1, y2, c2]=(y2i, 2yi, 4c2 = y1y2),
Hodd(BQ8) ~=0.
3. the group E = p1+2+
Throughout this section, we assume p 3 and G = E = p1+2+. Recall
that E is generated by a, b, c such that [a, b] = c, ap = bp = cp = 1.
Recall also the p-dimensional representation "c= IndG(c*) so that
"c(c) = diag(i, ..., i), "c(a) = diag(1, i, ..., ip-1),
and "c(b) as the permutation matrix in GLp(C).
6 N.YAGITA
The group E does not act freely on Cp*. We consider fixed points
for small subgroups. Let W = Cp*. Since "c(a) = diag(1, i, ..., ip-1),
the fixed points of the subgroup is given by
W = {(x, 0, ..., 0)|x 2 C*} = C*{e} e = (1, 0, ..., 0).
Since b-iabi = aci in E, we see
acib-ie = b-iabib-ie = b-iae = b-ie.
i> * -i
This means W
abjg-1je = g-1je as above arguments, and so C*{g-1je} = W ,
or . But c is not a stabilizer of any element in C*. All points which
have non trivial stabilizer groups are contained in H. Thus we have
the lemma.
Let i : H Cp*. Let us write i*(yi) 2 H*E(H) by the same letter yi.
Lemma 3.2. There is the isomorphism H*E(Hi) ~=H*E(H0) and
H*E(H0; Z=p) ~=Z=p[y1] (x1, z), with |x| = |z| = 1,
H*E(H0) ~=Z[y1]=(py1){1, z}.
Proof. We consider the Hochschild-Serre spectral sequence
E*,*2~=H*(B; H***(H0; Z=p)) =) H*E(H0; Z=p).
Here we have
H*****(H0; Z=p) ~=H*****(**** x C*; Z=p) ~=H*(C*; Z=p) ~= (z).
Hence the E*,*2is isomorphic to
H*(B****; (z)) ~=Z=p[y1] (x1) (z).
7
We will see d2(z) = 0 and get the result. Consider the localized
exact sequence for the cohomology
H*+2p-1E(Cp* - H) ! H*E(H) ! H*+2pE(Cp*) ! H*+2pE(Cp* - H) ! .
Since E acts on Cp* - H freely, we see
H*+2pE(Cp* - H) ~=H*+2p((Cp* - H)=E),
which is zero if * > 0. Thus for * > 0, we have the isomorphism
H*E(H) ~=H*+2pE(Cp*) ~=H*+2p(BE; Z=p)=(cp).
Since yp1y2 - y1yp2= 0 2 H*(BE) from Theorem 2.5, we see
H2*+2pE(Cp*) ~=Z=p{y*+p1, y*+p-11y2, ..., y*+11yp-12, y*+p2}
and H2*+2p+3E(Cp*) ~=H2*+2pE(Cp*){e}. Hence
0+2p 2*0+1+2p
rankpH2*E (H) = rankpHE (H) = p + 1.
Since all elements in H*+2p(BE)=(cp) are p-torsion for * > 0, we see
0+2p 2*0+2p
rankpH2*E (H; Z=p) = 2rankpHE (H) = 2(p + 1).
For each 0 j p, we still know H*E(HjZ=p) ~= H*E(H0; Z=p).
Hence rankpH*E(H0; Z=p) = 2. Thus the spectral sequence collapses.
Lemma 3.3. The cycle map cl : CH*(BE) ! H2*(BE) is isomorphic
for * 2p.
Proof. Let * 1. Consider the diagram
iCH* *+p p* *+p p*
CH*E(H) --- ! CHE (C ) -- - ! CHE (C - H) = 0
? ? ?
cl1?y cl2?y cl?y
iH* 2*+2p p* 2*+2p p*
! H2*E(H) --- ! HE (C ) -- - ! HE (C - H) = 0.
Since H2*+2p-1E(Cp* - H) = 0, we see iH* is an isomorphism. From
the preceding lemma, H2*E(Hj) generated by Chern classes (e.g., y1 for
H0). Hence the cycle map cl1 is isomorphic for * p from Lemma 2.3.
Therefore
cl2 . iCH* = iH* . cl1
is isomorphic.
From Corollary 2.4, we have the isomorphism CH*(BE) ~=H2*(BE)
for all * 0. Thus we prove Theorem 1.1 in the introduction when
G = p1+2+.
8 N.YAGITA
4. Other groups M = p1+2-, D8 and Q8
We consider the other groups in this section. Let M = p1+2-for an
odd prime. This case ap = c and the representation "cis given as
"c(a) = diag(,, ,1+p, ,1+2p, ..., ,1+(p-1)p)
and "c(b) is the permutation matrix as the case E, where , is a p2-th
primitive root of the unity, i.e., ,p = i.
The fixed points set of the subgroup **** is given by
W ****= {(x, ..., x)|x 2 C*} = C*{e0} e0= (1, ..., 1).
i> -i 0
Since a-ibai = bci, we see W ; H****(H; Z=p)) =) HM (H; Z=p).
Since acts freely on H, we see
H= ~=C*{e0, ..., ap-1e0}= ~=C*=.
Therefore we have H(H; Z=p) ~= H*(C*=; Z=p) ~= (z) as the
case G = E. From Theorem 2.6, we know
H2*+2pM(Cp*) ~=Z=p{y*+p2}.
This implies rankpH2*+2pM(H) = 1. Therefore the spectral sequence
collapses. Lemma 3.3 holds for G = M and we see CH*(BM) ~=
H2*(BM).
Next, we consider the case G = D8 and p = 2. Then the represnta-
tion can be took as the case G = M. Take
H0 = C*{e0, ae0}, H1 = C*{g-1 e0, g-1 ae0}
whereWg 2 GL2(C) with g-1 bg = ab (note (ab)2 = 1). Let H =
H0 H1. Then D8 acts freely on C2* - H. In fact from Theorem 2.7,
we know
H2*+4D8(C2*) ~=Z=2{y*+21, y*+22}.
Hence all arguments work as the case E or M.
At last we consider the case G = Q8. The representation "cis given
` ' ` '
i 0 0 -1
"c(a) = 0 -i , "c(b) = 1 0 .
9
We can easily see that Q8 acts freely on C2*. Therefore
CHQ8(C2*) ~=CH*(C2*=Q8)
which is generated by dim 2. In fact
H*(BD8)=(c2) ~=Z=2[y1, y2]=(y2i, 2yi, y1y2).
5. motivic cohomology
We recall the motivic cohomology. Let X be a smooth (quasi pro-
0
jective) variety over a field k C. Let H*,*(X; Z=p) be the mod(p)
motivic cohomology defined by Voevodsky and Suslin ([Vo1-4]). Recall
that the Belinson-Lichtenbaum conjecture holds if
Hm,n(X; Z=p) ~=Hmet(X; ~pn ) for all m n.
Recently M.Rost and V.Voevodsky ([Vo5],[Su-Jo]) proved the Bloch-
Kato conjecture. The Bloch-Kato conjecture implies the Beilinson-
Lichtenbaum conjecture.
We assume that k contains a p-th root i of unity. Then there is
the isomorphism Hmet(X; ~pn ) ~=Hmet(X; Z=p). Let o be a generator of
H0,1(Spec(k); Z=p) ~=Z=p, so that
0 *
colimio iH*,*(X; Z=p) ~=Het(X; Z=p).
We define the weight degree w(x) = 2m - n if 0 6= x 2 Hm,n(X; Z=p).
Then it is known w(x) 0 for smooth X.
0
Let H*(X; H*Z=p) be the cohomology of the Zarisky sheaf induced
from the presheaf H*et(V ; Z=p) for open subsets V of X. This sheaf
cohomology is isomorphic to the E2-term
0 * *0 *
E*,*2~=H (X; HZ=p) =) Het(X; Z=p)
of the coniveau spectral sequence by Bloch-Ogus [Bl-Og]. We also note
0 *0
H0(X; H*Z=p) H (k(X); Z=p).
The relation between this cohomology and the motivic cohomoloy is
given as follows.
Theorem 5.1. ([Or-Vi-Vo], [Vo5]) There is the long exact sequence
! Hm,n-1 (X; Z=p) xo!Hm,n(X; Z=p)
! Hm-n (X; HnZ=p) ! Hm+1,n-1 (X; Z=p) xo!.
In particular, we have
10 N.YAGITA
Corollary 5.2. The cohomology Hm-n (X; HnZ=p) is (additively) iso-
morphic to
Hm,n(X; Z=p)=(o ) Ker(o )|Hm+1,n-1 (X; Z=p)
where Hm,n(X; Z=p)=(o ) = Hm,n(X; Z=p)=(o Hm,n-1 (X; Z=p)).
Corollary 5.3. The map xo : Hm,m-1 (X; Z=p) ! Hm,m (X; Z=p) is
injective.
By using above theorems, we can do computations for concrete cases.
Suppose k = C. Then the realization (cycle map)
0 * *
tC = cl : H*,*(X; Z=p) ! Het(X; Z=p) ~=H (X; Z=p)
can be identified with
0 *,*0 *,* *
xo *-* : H (X; Z=p) ! H (X; Z=p) ~=Het(X; Z=p),
from the Beilinson-Lichtembaum conjecture.
We define the motivic filtration of H*(X; Z=p) by
Fi = Im(t2*-i,*C) = ntC(H2n-i,n(X; Z=p)),
0 0
namely, x 2 Fiif x = tC(x0) for some x02 H*,*(X; Z=p) with w(x ) = i.
Let us write the associated grade ring
0 * i *
iFi=Fi-1 = gr* H (X; Z=p), Fi=Fi-1 = gr H (X; Z=p).
In [Ya2], we define
0 m,n m,n
h*,*(X; Z=p) = m,nH (X; Z=p)=(Ker(tC )),
0
and compute them for some cases of X = BG. (Note if Ker(t*,*C) = 0,
0 *,*0
then H*,*(X; Z=p) ~=h (X; Z=p).) It is immediate that
hm,n(X; Z=p) ~= i=0gr2(n+i)-mHm (X; Z=p){o i}.
We will simply write (for ease of notations) the above isomorphism
0 *0 *
h*,*(X; Z=p) ~=gr H (X; Z=p) Z=p[o ].
Lemma 5.4. Suppose dim(X) 2 and
H*(C(X); Z=p) = H*(Spec(C(X); Z=p) ~=0 for * 3.
0
Then H*,*(X; Z=p) ~= 0 for all * > 4. Moreover we have the bidegree
0 *,*0
isomorphism H*,*(X; Z=p) ~=h (X; Z=p).
Proof. It is know that
0 0
H*,*(X; Z=p) ~=0 if * -* > dim(X).
0
So we only need to show H*,*(X; Z=p) ~=0 for * > 4 when
* = *0, * = *0- 1, or * = *0- 2.
11
Let * > 4. The first case follows from H*(X; Z=p) ~= 0 and the
Beilinson-Lichtenbaum conjecture. The second case follows from that
the map H*,*-1(X; Z=p) ! H*,*(X; Z=p) is injective.
We will show the last case. Since o 2H*,*-2(X; Z=p) = 0 and the
injectivity above, we see o H*,*-2(X; Z=p) = 0. Suppose 0 6= x 2
H*,*-2(X; Z=p). Then there is y 2 H0(X; H*-1Z=p) so that d2y = x in the
coniveau spectral sequence. However we still know
H0(X; H*-1Z=p) H*-1(C(X); Z=p) = 0, for * 4.
This is a contradiction and H*,*-2(X; Z=p) = 0.
The above argument also show that Ker(o )|H4,2(X; Z=2) = 0,i.e,
0 *
* = 4 case. This means H*,*(X; Z=p) ! H (X; Z=p) is injective for
all * 0. Hence we get the lemma.
Corollary 5.5. Let X = C2*=Q8 or X = (C2*- H)=D8 for the action
0 *,*0
given in x4. Then H*,*(X; Z=2) ~=h (X; Z=2).
Proof. We only need to prove H*(C(X); Z=2) = 0 for * 3. We prove
it for G = D8, and the case Q8 is similar.
Let C2==G = Spec(C[t, s]G ) be the geometric quotient by G. Then
X = (C2 - H)=G is an open set in C2==G. So C(X) ~= C(t, s)G ; the
quotient field of the invariant ring C[t, s]G . The group G = D8 satisfies
the Nother's probem so that C(X) is purely transcendental over C, i.e.
C(X) ~=C(t0, s0).
(This fact is easily seen since
C[t, s]D8 = C[ts, t4 + s4] C[t, s],
( (
t 7! it t 7! s
where the action is given by a : , b : .)
s 7! -is s 7! t
Since it is well known H*(C(t0, s0); Z=2) ~= (x1, x2) with |xi| = 1,
we get the result.
6. motivic cohomology of BD8 and BQ8
In this section, we compute the mod(2) motivic cohomology of BD8
and BQ8.
At first, we consider the case Q8. The mod 2 (usual) cohomology is
well known (see Theorem 2.7)
H*(BQ8; Z=2) ~=Z=2{1, x1, y1, x2, y2, w} Z=2[c2]
0 *
where x2i= fixi = yiand |w| = 3. The graded algebra gr* H (BQ8; Z=2)
is given by letting the weight degree by
w(yi) = w(c2) = 0, w(xi) = w(w) = 1.
12 N.YAGITA
Theorem 6.1. There is the bidegree isomorphism
0 *0 *
H*,*(BQ8; Z=2) ~=Z=2[o ] gr H (BQ8; Z=2).
Proof. Let G = Q8. In the usual mod(2) cohomology
H*G(C2*; Z=2) ~=H*(BG; Z=2)=(c2) ~=Z=2{1, x1, y1, x2, y2, w},
which is isomorphic to H*(C2*=Q8; Z=2). Hence we can use Corollary
5.5
0 2*
H*,*G(C ; Z=2) ~=Z=2[o ] Z=2{1, x1, y1, x2, y2, w}.
Here deg(w) = (3, 2) by the folowing reason. The Bockstein exact
sequence also exists in the motivic cohomology
0 *,*0 x2 *,*0
! H*-1,*(BG; Z=2) ! H (BG; Z) ! H (BG; Z) ! .
Since c2 2 H4,2(BG) and 4c2 = 0, we see w 2 H3,2(BG; Z=2).
Using above facts, we can show the lower sequence in the following
diagram is exact
0-2 c2 *,*0 *,*0 2*
! H*-4,* (BG; Z=2) -- - ! H (BG; Z=2) -- - ! HG (C ; Z=2) !
? ? ?
? ? ?
y y ~=y
0-2 c2 *,*0 *,*0 2*
! h*-4,* (BG; Z=2) -- - ! h (BG; Z=2) -- - ! hG (C ; Z=2) !
0 *0 *
where h*,*G(X; Z=2) = Z=2[o ] gr HG (X; Z=2).
By induction on * 0 and the five lemma, we easily see that the
vertical maps are isomorphic.
Now we consider the case G = D8. We recall the mod(2) cohomology.
H*(BD8; Z=2) ~=(Z=2[x1] Z=2[x2]) Z=2[u] ~=
(Z=2[y1]{y1, x1, y1u, x1u} Z=2[y2]{y2, x2, y2u, x2u} Z=2{1, u}) Z=2[c2]
Here we identify, yi = x2iand c2 = u2. The cohomology operations on
H*(BD8; Z=2) is well known, e.g.,
Q0(u) = (x1 + x2)u = e, Q1Q0(u) = (y1 + y2)c2.
Lemma 6.2. There exist u01, u022 H3,2(BD8; Z=2) with o u0i= xiu 2
H3,3(BD8; Z=2) (so u0i= o -1xiu).
Proof. First note
H3,2(BG; Z) Z=2{Q0(u)}, H4,2(BG; Z) ~=Z=2{y21, y22} Z=4{c2}.
From the Bockstein exact sequence, we see
rankpH3,2(BG; Z=2) 1 + 3 = 4.
13
From the Beilinson-Lichtenbaum0conjecture and Corollary 5.3, we
see that H*,*(X; Z=p) ! H*(X; Z=p) is injective for * 3. On the
other hand
H3(BG; Z=2) ~=Z=2{x1u, x2u, x1y1, x2y3}.
Hence each element in H3(BG; Z=2) must be in H3,2(BG; Z=2).
0 *
Therefore we get gr* H (BD8; Z=2) which is isomorphic to
(Z=2[y1]{y1, x1, x1u01, u01} Z=2[y2]{y2, x2, x2u02, u02} Z=2{1, u}) Z=2[c2]
with w(yi) = w(c2) = 0, w(xi) = w(u0i) = 1 and w(u) = 2 (note
u 62 CH*(BG)=2), and xiu0i= yiu.
Theorem 6.3. There is the the bidegree module isomorphism
0 *0 *
H*,*(BD8; Z=2) ~=Z=2[o ] gr H (BD8; Z=2).
Before the proof of this theorem, we give a lemma.
Lemma 6.4.
0
H*,*D8(H0, Z=2) ~=Z=2[o ] Z=2[y1] (x1, z) with deg(z) = (1, 1).
Proof. Let G = D8. We consider the exact sequence
0-1 y1 *,*0 *,*0 *
! H*-2,*G ({0}xH0; Z=2) ! HG (CxH0; Z=2) ! HG (C xH0; Z=2) ! .
Here G acts freely on C* x H0 and
0 * *,*0 *
H*,*G(C x H0; Z=2) ~=H (C x H0=G; Z=2)
~= H*,*0(C*=**** x C*=; Z=2)
~= H*,*0(C*=****; Z=2) Z=2[o]H*,*0(C*=; Z=2) ~=Z=2[o ] (x1, z)
since H*,*(Cn*=(Z=p); Z=p) holds the Kunneth formula.
0 *
The natural map H*,*G(H0; Z=2) ! Z=2[o ] HG (H0; Z=2) induces
the diagram for two exact sequences. We can prove the lemma by
induction on * 0 and the five lemma.
Proof of Theorem 6.3. Let G = D8. First we consider the exact se-
quence
! H*-2G(H; Z=2) i*!H*G(C2*; Z=2) ! H*G(C2* - H; Z=2) ! .
Recall that
H*G(C2*; Z=2) ~=A{1, u} with A = Z=2[y1] (x1) Z=2[y2] (x2).
Using fact that i* is isomorphic for * > 4, The map i* is given explicitly
i*(11) = y1, i*(12) = y2, i*(z1) = x1u, i*(z2) = x2u
14 N.YAGITA
where we use and 1i, zi are the generators in H*G(Hi-1; Z=2). (Note
i*(xizi) = yiu since yi = x2iin H*(BG; Z=2).) Therefore
H*((C2* - H)=G; Z=2) ~=Z=2{1, x1, x2, u}.
Hence from Corollary 5.5, we have
0 2*
H*,*((C - H)=G; Z=2) ~=Z=2[o ] Z=2{1, x1, x2, u}.
Here deg(u) = (2, 2) (i.e., w(u) = 2) since CH1G((C2* - H)=G) = 0.
Next we consider the following diagram
0-1 i* *,*0 2* *,*0 2*
! H*-2,*G (H; Z=2) --- ! HG (C ; Z=2) --- ! HG (C - H; Z=2) !
? ? ?
? ? ?
y y ~=y
0-1 i* *,*0 2* *,*0 2*
! h*-2,*G (H; Z=2) --- ! hG (C ; Z=2) --- ! hG (C - H; Z=2) ! .
Here the lower sequence is also exact from the above arguments for
H*G(C2* - H; Z=2), and from Lemma 6.4 and the isomorphism just
before Theorem 6.3.
By using the induction on * 0 and five lemma, we get
0 2* *0 * 2*
H*,*G(C ; Z=2) ~=Z=2[o ] gr HG (C ; Z=2)
where w(u) = 2, w(u0) = 1.
As the case G = Q8, we can see
0 *0 * 2*
H*,*(BG; Z=2) ~=Z=2[o ] gr HG (C ; Z=2) Z=2[c2].
7. motivic cobordism of BQ8 and BD8
Let MU*(X) and MU*(X; Z=p) be the usual complex cobordism
0
theory and its mod p theory. Let MGL*,*(X) be th motivic cobordism
theory defined by Voevodsky [Vo1 ]. Since tC|CH*(BG) is injective,
from Proposition 9.4 in [Ya3], we have the isomorphism
MGL2*,*(BG) ~=MU2*(BG)
for each group of order p3.
In this0section, we give rather strong results for only Q8 and D8. Let
MGL*,*(X; Z=p) be the mod p theory defined by the exact sequence
0 xp *,*0 ae *,*0 ffi
! MGL*,*(X) ! MGL (X) ! MGL (X; Z=p) ! .
Then we have the following theorem, which holds also for (Z=p)n, On, SOn.
(For accurate definition for MGL2*,*(BG) see [De].)
15
Theorem 7.1. Let G = Q8 or D8. Then there are isomorphisms
0 2*,*
MGL*,*(BG; Z=p) ~=MGL (BG; Z=p) Z=p[o ],
MGL2*,*(BG; Z=p) ~=MU2*(BG; Z=p) ~=MU2*(BG)=p.
0,*00 *,*00
Proof. Let G = Q8 or D8. Let E(MGL)*,*r (resp. E(MU)r ) be the
0
Atiyah-Hirzebruch spectral sequence converging to MGL*,*(BG; Z=2)
(resp. MU*(BG; Z=2)) (see [Ya3]), namely,
0,*00 *,*0 *00 *,*0
E(MGL)*,*2 ~=H (BG; Z=2) MU =) MGL (BG; Z=2),
0,*00 * *00 *
E(MU)*,*2 ~=H (BG; Z=2) MU =) MU (BG; Z=2).
0,*00 *,*0,*00
The realization map tC induces the map t*,*C : E(MGL)r !
00
E(MU)*,*rof spectral sequences.
From Theorem 6.1 and 6.3, we know
0 *0 *
H*,*(BG; Z=2) ~=Z=2[o ] gr H (BG; Z=2).
0 *,*00 *0 * *00
Let us write gr* E(MU)2 = gr H (BG; Z=2) MU so that we
have the the bidegree module isomorphism
0,*00 *0 *,*00
E(MGL)*,*2 ~=Z=2[o ] gr E(MU)2 .
0 *,*00 *,*0,*00
Suppose that for all x 2 gr* E(MU)2 E(MGL)2 ,
0 *,*00
(*) d2(x) 2 gr* E(MU)2 (i.e., d2(x) 6= o y for some o y 6= 0).
00
Then from the naturality of the map t*,*Cof spectral sequences, we
have
0,*00 *0 *,*00
E(MGL)*,*3 ~=Z=2[o ] gr E(MU)3
0 *,*00 *,*00
where gr* E(MU)3 is the bidegree module made from grE(MU)3
0 *,*00
giving the same second degree. Moreover, if for all x 2 gr* E(MU)r ,
r 2
0 *,*00
(**) dr(x) 2 gr* E(MU)r ,
then we have the bidegree isomorphism
0,*00 *0 *,*00
E(MGL)*,*1 ~=Z=2[o ] gr E(MU)1 ,
and we can prove this theorem. 0
To see (*), (**), we note that gr* H*(BG; Z=2) is generated by ele-
ments of degree w(x) 1 (resp. w(x) 2 e.g., w(u) = 2) for G = Q8
(resp. G = D8). Since w(o ) = 2, we have
0 *
H2*,*(BG; Z=2) H2*+1,*(BG; Z=2) gr* H (BG; bZ=2).
Then (*), (**) are immediate since w(dr(x)) = w(x) - 1 1.
16 N.YAGITA
References
[At] M. Atiyah. Character and cohomology of finite groups. Publ. I.H.E.S. 9
(1964) 23-64.
[Bl-Og] S.Bloch and A.Ogus. Gersten's conjecture and the homology of schemes.
Ann.Scient.E'c.Norm.Sup. 7 (1974) 181-202.
[De] D. Deshpande Algebraic cobordism of classifying spaces. preprint (2009)
[Ev] L.Evens. On Chern classes of representation of finite groups. Trans of
A.M.S. 115 (1965), 180-193.
[Gr-Ly] D.Green and I.Leary. The spectrum of the Chern subring. Comment.
Math. Helv. 73 (1998) 406-426.
[Gu1] P.Guillot. The Chow ring of G2 and Spin(7). J. Reine Angew. Math. 604
(2007) 137-158.
[Gu2] P.Guillot. Geometric methods for cohomological invariants. Document
Math. 12 (2007), 521-545.
[In-Ya] K.Inoue and N.Yagita. The complex cobordism of BSOn.
http://hopf.math.purdue.edu/ December (2005).
[Ka-Ya] M.Kameko and N.Yagita. The Brown-Peterson cohomology of the classi-
fying spaces of the projective unitary groups P U(p) and exceptional Lie
group Trans. of A.M.S. 360 (2008), 2265-2284.
[Le] G.Lewis. The integral cohomology rings of groups order p3. Trans. A.M.S.
132 (1968), 501-529.
[Ly] I. Leary. The mod p cohomology rings of some p-groups. Math. Proc.
Camb. Phil. Soc. 112 (1992), 63-75.
[Mo] L.A. Molina. The Chow ring of classifying space of Spin8. preprint (200*
*7).
[Mo-Vi] L.Molina and A.Vistoli. On the Chow rings of classifying spaces for cla*
*s-
sical groups. Rend. Sem. Mat. Univ. Padova 116 (2006), 271-298.
[Or-Vi-Vo]D.Orlov,A.Vishik and V.Voevodsky. An exact sequence for Milnor's K-
theory with applications to quadric forms. Ann. of Math. 165 (2007)
1-13.
[Su-Jo] A.Suslin and S.Joukhovistski. Norm Variety. J.Pure and Appl. Algebra
206 (2006) 245-276.
[Te-Ya] M.Tezuka and N.Yagita. Cohomology of finite groups and Brown-
Peterson cohomology. Lect. Notes in Math. 1370, Springer, Berlin
(1989), 396-408.
[Th] C.B.Thomas. Chern classes of representations. Bull. London Math. Soc.
18 (1986), 225-240.
[To1] B. Totaro. Torsion algebraic cycles and complex cobordism. J. Amer.
Math. Soc. 10 (1997), 467-493.
[To2] B. Totaro. The Chow ring of classifying spaces. Proc.of Symposia in Pure
Math. "Algebraic K-theory" (1997:University of Washington,Seattle) 67
(1999), 248-281.
[Vi] A.Vistoli. On the cohomology and the Chow ring of the classifying space
of P GLp. J. Reine Angew. Math. 610 (2007) 181-227.
[Vo1] V. Voevodsky. The Milnor conjecture. www.math.uiuc.edu/K-
theory/0170 (1996).
17
[Vo2] V. Voevodsky (Noted by Weibel). Voevodsky's Seattle lectures : K-
theory and motivic cohomology Proc.of Symposia in Pure Math. "Alge-
braic K-theory" (1997:University of Washington,Seattle) 67 (1999), 283-
303.
[Vo3] V.Voevodsky. Reduced power operations in motivic cohomology. Publ.
Math. I.H.E.S. 98 (2003), 1-54.
[Vo4] V.Voevodsky. Motivic cohomology with Z=2-coefficients. Publ. Math.
I.H.E.S. 98 (2003), 59-104.
[Vo5] V.Voevodsky. On motivic cohomology with Z=`-coefficient.
www.math.uiuc.edu/K-theory/0639 (2003).
[Ya1] N. Yagita. Chow ring of classifying spaces of extraspecial p-groups. Co*
*n-
tem. Math. 293, (2002) 397-409.
[Ya2] N. Yagita. Examples for the mod p motivic cohomology of classifying
spaces. Trans. of A.M.S. 355 (2003),4427-4450.
[Ya3] N. Yagita. Applications of Atiyah-Hirzebruch spectral sequence for mo-
tivic cobordism. Proc. London Math. Soc. 90 (2005) 783-816.
Department of Mathematics, Faculty of Education, Ibaraki Univer-
sity, Mito, Ibaraki, Japan
E-mail address: yagita@mx.ibaraki.ac.jp
**