ALGEBRAIC BP-THEORY AND NORM VARIETIES
NOBUAKI YAGITA
Abstract. Let X be a smooth variety over a field k of0ch(k) = 0.
For a fixed prime p, the algebraic BP -theory ABP *,*(X) is the
algebraic version of the topological BP -theory. Given a nonzero
symbol a 2 KMn+1(k)=p, the norm variety Va is a variety such that
a = 0 2 KMn+1(k(Va))=p and Va(C) = vn. In this paper, we mainly
study ABP *,*0(Va) for p odd prime case.
1. introduction
A.Suslin and V.Voevodsky0constructed and developed the motivic
cohomology theory H*,*(X; Z=p) for algebraic sets over the base field
k. This theory is the counter part in algebraic geometry of the usual
mod p singular cohomology in algebraic topology. Let ch(k) = 0 and
fix an embedding k C. As the counter part of the complex cobordism
theory MU*(X), Voevodsky [Vo1,2] defined the algebraic cobordism
0
theory MGL*,*(X) and used it in the first proof of the Milnor conjec-
ture.
Given a nonzero symbol a 2 KMn+1(k)=p, the norm variety Va is a
variety such that a = 0 2 KMn+1(k(Va))=M and Va(C) = vn. Here vn is
the 2(pn - 1) complex manifold generating
Z(p)[v1, v2, ...] ~=BP *(pt.) MU*(pt.)(p)~= MGL2*,*(Spec(k))(p)
the coefficient ring of the BP -theory in algebraic topology.
For p = 2, we can take the norm variety by the smallest neighbor Qa
of the Pfister quadric defined by a. Voevodsky proved [Vo1] the Mil-
0
nor conjecture by studying cohomology operations on H*,*(Qa; Z=2).
Moreover MGL2*,*(Qa) is studied by Vishik and Yagita [Vi-Ya].
Recently Rost ([Ro],[Su-Jo]) announced the constructions of the
norm variety Va also for p odd, and Voevodsky ([Vo4]) gives the proof
of the Bloch-Kato conjecture (which is the odd prime version of the
0
Milnor conjecture) by studying H*,*(Va; Z=p).
____________
2000 Mathematics Subject Classification. Primary 14C15, 57T25; Secondary
55R35, 57T05.
Key words and phrases. motivic cobordism, Chow ring, motivic cohomology,
BP -theory, Atiyah-Hirzebruch spectral sequence.
1
2 N.YAGITA
0
In this paper we write down the properties of ABP *,*(X) which is
the algebraic counter part of BP -theory. For example we give a con-
struction of the Atiyah-Hirzebruch spectral sequence for ABP *,*(X; Z=p)
; its existence (of MGL-version) was announced by Hopkins and Morel
more that several years before, however any proof (or even statement)
does not appear yet. We study the cohomology0operations, products
and Gysin maps explicitly0in ABP *,*-theory. Using these results, we
compute ABP *,*(Va) which extends some parts of the results by Vishik
and Yagita to odd p cases.
2. cohomology operations
Let p be a fixed prime number. Let k be a field with ch(k) = 0,
which contains a primitive p-th root of unity. In this paper, the mod(p)
motivic cohomology HmZar(X; Z=p(n)) is written by Hm,n(X; Z=p). We
fix an emmbedding k C and denote by tC the realization map
0 *
tC : H*,*(X; Z) ! H (X(C); Z)
where the right hand side is the usual (singular) cohomology of the
C-rational points of X.
In motivic mod(p) cohomology, we have the Bockstein and the re-
duced powers operations
0 *+1,*0
(2.1) fi : H*,*(X; Z=p) ! H (X; Z=p),
0 *+2(p-1)i,*0+(p-1)i
(2.2) P i: H*,*(X; Z=p) ! H (X; Z=p)
which are compatible with the usual Bockstein and the reduced powers
operations via the realization map tC.
Let o 2 H0,1(pt.; Z=p) ~= Z=p and ae 2 H1,1(pt.; Z=p) ~= k*=(k*)p be
elements corresponding to the primitive root i of unity. Then fio =
ae. Reduced powers operations have the following properties (Lemma
9.7,Lemma 9.8 in [Vo4]),
(2.3) P 0= Identity, P n(x) = xp if x 2 H2n,n(X; Z=p),
(2.4) P i(x) = 0 if x 2 Hm,n(X; Z=p), i > m - n and i n.
When p > 2, the Cartan formula
X
P i(xy) = P j(x)P i-j(x)
0 j i
and the Adem relations are also satisfied as the topological cases. How-
ever when p = 2 we need some modification for o and ae (P i= Sq2i
ALGEBRAIC BP-THEORY 3
and fi = Sq1). For example
X X
(2.5) Sq2i(uv) = Sq2j(u)Sq2i-2j(v) + o Sq2j+1(u)Sq2i-2j-1(v).
0 i i 0 j i-1
Moreover we have the Milnor operation
0 *+2pi-1,*+pi-1
(2.6) Qi : H*,*(X; Z=p) ! H (X; Z=p).
i
When p 3, we have Q0 = fi and Qi+1 = [Qi, P p]. But for p = 2
the above property holds only with mod(ae) (see [Vo3] for details). We
note Q2i= 0 and QiQj = -QjQi. But Qi is not a derivation when
ae 6= 0 and p = 2 (while it is a derivation whenever p 3). In fact from
Proposition 13.4 in [Vo3],
(2.7) Qi(xy) = Qi(x)y + xQi(y) + aeQi-1(x)Qi-1(y)
X
+ae aJJ0QJ(x)QJ0(y)
0 ffl0
where aIJ 2 H*,*(pt.; Z=2), |aJJ0| > 0 and QJ = Q0 ... for J =
(ffl0, ..., ffli-1), fflj = 0 or 1.
For a non zero element x in Hm,n(X; Z=p) or each cohomology op-
eration (or differential in the spectral sequence), we define the weight
and the difference by
w(x) = 2s.deg(x) - f.deg(x) = 2n - m
d(x) = f.deg(x) - s.deg(x) = m - n
(here f.deg(x) (resp. s.deg(x)) is the first degree (resp. second degree)
of x) so that if X is smooth, then
w(x) 0, d(x) dim(X).
We also note w(fi) = -1, w(P i) = 0, w(Qi) = -1.
The solution of the Bloch-Kato conjecture by Voevodsky implies
0 * 0
H*,*(X; Z=p) ~=Het(X; Z=p) for * * ,
H*,*(pt.; Z=p) ~=KM*(k)=p ~=H*et(pt.; Z=p).
0
Since d(x) 0 for non zero x 2 H*,*(pt.; Z=p), we have
0 M
Lemma 2.1. H*,*(pt.; Z=p) ~=Z=p[o ] K* (k)=p.
0
Corollary 2.2. Let p 3. In H*,*(pt.; Z=p), we see Qi(x) = 0 and
P j(x) = 0 for all i, j 1.
4 N.YAGITA
Proof. By dimensional reason, P n(x) = 0 for x 2 Hn,n(pt.; Z=p) ~=
KMn(k) or x = o. When p > 2, the Cartan formula holds, hence
0
P n(x) = 0 for all x 2 H*,*(pt; Z=p) and n > 0. Thus we see also __
Qn(x) = 0 for n > 0. |__|
Remark. However when p = 2, in general, P n(x) 6= 0 and Qn(x) 6= 0
0
for x 2 H*,*(pt, ; Z=p) which, for example, see x14 bellow.
V.Voevodsky ([Vo3,4] in particular Lemma 2.2 in [Vo4]) and G.Powell
0
[Po] showed that the mod p motivic Steenrod algebra A*,*pis generated
0 i
as an H*,*(pt, Z=p)-module by products of P and fi. Moreover they
also prove
0 *,*0
(2.8) A*,*p~=H (pt; Z=p) RP (Q0, Q1, ...)
where RP is the Z=p-module generated by products of reduced powers
0
P i1....P in(without the Bockstein). Hence each element a in A*,*pis
represented by
X X
a = aIJP IQJ = aIJP i1...P inQffl00...Qfflmm
with I = (i1, ..., in), ik > 0, and J = (ffl0, ....fflm ), ffli = 0 or 1, and
0
aIJ 2 H*,*(pt.; Z=p).
In this paper, we assume and consider (generalized) cohomology the-
ories in stable homotopy categories which hold the following Whitehead
type theorem. Let X and Y be connected. If f : X ! Y is a map
such that for each extension F of k, there is the isomorphism
0 *,*0
f*|F : H*,*(Y |F ; Z(p)) ~=H (X|F ; Z(p)) X|F = X F,
then f induces the equivalence in this (satble homotopy) category over
the field k.
3. ABP theories
Recall that MU*(-) is the complex cobordism theory defined in the
usual (topological) spaces and
MU* = MU*(pt.) = Z[x1, x2, ...] |xi| = -2i.
Here each xi is represented by sum of hypersurfaces of dim(xi) = 2i
defined by polynomials with the coeffients in Z, in some product of
complex projective0spaces.
Let MGL*,*(-) be the motivic cobordism theory defined by Vo-
evodsky. By the Thom isomorphism, it is easily proved that ([Hu-Kr],
0
[Ve]) that there is the H*,*(pt)-module isomorphism
0 *,*0 *,*0
H*,*(MGL) ~=H (BGL) ~=H (pt)[c1, c2, ...] with |ci| = 2i.
ALGEBRAIC BP-THEORY 5
0
This isomorphism induces the A*,*p-module isomorphism
0 *,*0 j
H*,*(MGL; Z=p) ~=H RP Z=p[mi|i 6= p - 1]
0
with H*,*= H*,*(pt.; Z=p) and |mi| = 2i.
Let us write by AMU the spectrum MGL(p)representing the mo-
0 *,*0
tivic cobordism theory, i.e., MGL*,*(-)(p) = AMU (-). Since
0
MGL*,*(X) is a multiplicative cohomology theory, we know it is an
0 * 2*,*
MGL*,*(pt.)-algebra. Moreover we can embeds MU into MGL (pt.)
([Vo1]). Hence MGL*,*(X) is also an MU*-algebra.
Given a regular sequence Sn = (s1, ..., sn) with si 2 MU*(p), we can
inductively construct the AMU-module spectrum by the cofibering of
spectra
xsi
(3.1) T-1=2|si|^ AMU(Si-1) -! AMU(Si-1) ! AMU(Si)
where T = A=(A-{0} ~=S1t^S1sis the Tate object. It is also immediate
that tC(AMU(Sn)) ~=MU(Sn) with
MU(Sn)* = MU*=(Ideal(Sn)).
Recall BP *~= Z(p)[v1, ...] with identifying vi = xpi-1. We can con-
struct spectra
ABP = AMU(xi|i 6= pj - 1), AP (n), Ak(n), AHZ, AHZ=p
so that tC(Ah) ~=h for h = BP, P (n), .... Here P (n)* = BP *=(p, ..., vn-1)
and k(n)* = Z=p[vn] ~=BP *=(p, ..^vn, ...).
For S = (vi1, ..., vin), let us write
ABP (S) = AMU(S [ {xi|i 6= pj})
so that tC(ABP (S)) = BP (S) with BP (S)* = BP *=(S). By using
the long exact sequence induced from (3.1)
Lemma 3.1. ([Ya4]) Let S = (vi1, ..., vin). Then
0 *,*0 *
H*,*(ABP (S); Z=p) ~=H (pt.; Z=p) H (BP (S); Z=p)
~=H*,*0(pt.; Z=p) RP (Qi1, ..., Qin).
By the above lemma for S = (p, v1, ...) and the Whitehead type
theorem, we see that in the A1-stable homotopy category,
(3.2) HZ=p ~=AHZ=p = AMU(p, x1, x2, ...),
0 *,*0
e.g., AHZ=p*,*(X) ~= H (X; Z=p). More strongly, Hopkins-Morel
0 *,*0
showed that AHZ ~= HZ, namely, AHZ*,*(X) ~= H (X, Z) ; the
motivic cohomology.
6 N.YAGITA
Theorem 3.2. ([Ya4],[Ho-Mo]) Let Ah = ABP (S) for S = (vi1, vi2, ...).
Then there is the Atiyah-Hirzebruch spectral sequence
0) m,n 2n0 m+2n0,n+n0
E(Ah)(m,n,2n2= H (X; h ) =) Ah (X)
0) (m+2r+1,n+r,2n0-2r)
with the differential d2r+1 : E(m,n,2n2r+1! E2r+1 .
Remarks.
0
1) The cohomology Hm,n(X, h2n ) here is the usual motivic coho-
0 2n0
mology with coefficients in the0abelian group h2n , e.g.,0if h
is Z=p-module, then Hm,n(X; h2n ) ~= Hm,n(X; Z=p) h2n . In
0
particular, if X is smooth, then Em,n,2nr~=0 for m > 2n.
2) The convergence in AHss means that there is the filtration
0 *,*0 *,*0 *,*0
Ah*,*(X) = F0 F1 F2 ...
0 *,*0 *+2i,*0+i,-2i
such that Fi*,*=Fi+1 ~=E1 .
3) Let S R = (vj1, ...). Then the induced map ABP (S) !
ABP (R) of spectra induces the natural BP(*p)- module map of
AHss
E(ABP (S))*,*,*r! E(ABP (R))*,*,*r.
From the above theorem and dimensional reason, we see
(3.3) ABP (S)2*,*(pt) ~=BP (S)* = BP *=(S).
From (1), we also have for smooth X,
(3.4) ABP (S)2*,*(X) BP* Z(p)~= H2*,*(X) ~=CH*(X)(p).
Let vj 62 S. Consider the long exact sequence
vj 2*,*
! ABP (S)2*,*(X) ! ABP (S) (X)
! ABP (S, vj)2*,*(X) ! ABP (S)2*+1,*(X) ! .
Here the last term is zero when X is smooth. Hence this case
ABP (S, vj)2*,*(X) ~=ABP (S)2*,*(X)=(vj).
More generally
Lemma 3.3. Let X be smooth and S R (p, v1, ...). Then
ABP (R)2*,*(X) ~=ABP (S)2*,*(X)=(R).
Of course, the case R = (p, v1, ...) of the above lemma is the isomor-
phism (3.4).
0
Here we give a proof of Theorem 2.1 for ABP (S)*,*(X) with p 2 S.
Let I = (i0, i1, i2, ...) with i0 = 0 < i1 < i2 < ... and SI =
0
(p, vi1, vi2, ...). We consider the AHss for the theory ABP (SI)*,*(X).
ALGEBRAIC BP-THEORY 7
We first study the construction for the following motivic Adams
spectral sequence. We consider an (injective) resolution of ABP (SI)
by the (motivic) Eilenberg-MacLane spectrum HZ=p, namely, the se-
quence
(1) ABP (SI) --ffi0-!H1 --d1-! H2 --d2-! H3 --d3-! H4...
where Hi is the product of T*HZ=p and induced cohomology sequence
0 *,*0
is the resolution of H*,*(ABP (SI); Z=p) over Ap .
ConsiderPsequences RP = (r1, r2, ...) with ri = 0 for i 2 I. Let
l(R) = riand |R| = 2ri(pi-1) (be finite). Let Vn be a Z=p-vector
space generated by sequences with l(R) = n - 1. We can construct an
resolution by
Y
Hn = HZ=p Vn ~= HZ=p{vR }
l(R)=n-1, and ri=0 for i2I
where {vR } is just the base with |vR | = |R|. Hence
0 *,*0 *,*0
H*,*(Hn; Z=p) ~=Ap Vn ~= l(R)=n-1Ap {vR }.
0
(Note that for fixed degree (*, *0), each H*,*(Hn; Z=p) is finite, while
Vn itself is infinite dimensional vector space.) The differential map
0 *+1,*0
d*n: H*,*(Hn+1; Z=p) ! H (Hn; Z=p) is give by
X i
(2) d*n(vR ) = Qi vR- i i = (0, ..., 0, 1, 0, ...).
If there are maps dn with (2), then by standard arguments, (2) induces
the projective resolution of
0 *,*0
H*,*(ABP (SI); Z=p) ~=H (pt.; Z=p) RP (Qi|i 2 I)
0
for A*,*pwhich is isomorphic to
0 *,*0
H*,*(HZ=p; Z=p) ~=H (pt.; Z=p) RP (Q0, ...).
(For topological case, this is well known by Milnor and Novikov. See
page 513 in [Mi].)
Note that the homotopy group
0 *,*0
Hom(HZ=p, HZ=p) ~=H*,*(HZ=p; Z=p) ~=Ap .
0
Hence we have Hom(Hn, Hn+1) ~= A*,*p Hom(Vn+1, Vn). (Note also
that for fixed (i, j) the A1-homotopy [Si,jHn, Hn+1] is finite.) Hence
there exists unique map dn : Hn ! Hn+1 satisfying (2). Thus we have
the resolution (1).
8 N.YAGITA
dn-1 pn
Let Bn be the cofiber of dn-1, namely, Hn-1 ! Hn ! Bn is the
cofiber sequence. Since dndn-1 = 0, there is a map ffin : Bn ! Hn+1
such that ffinpn = dn+1
dn-1 dn dn+1
Hn-1 --- ! Hn --- ! Hn+1 --- ! Hn+2
?
pn?y %ffin
Bn
0 * *
Here H*,*(Bn; Z=p) ~= Kerdn-1 ~=Cokerdn+1. Hence the sequence
dn+1
Bn ffin!Hn+1 ! Hn+2 is the cofiber sequence, by the Whitehead theo-
rem for motivic theories, in fact, the induced cohomologies make exact
sequence. Similarly, the exact sequence
0 *
0 Kerd*n-1 H*,*(Hn+1; Z=p) Kerdn 0
pn+1 in+1
induces the cofiber sequence S-1 Bn ffin!Hn ! Bn+1 ! Bn.
i
Thus we get the following diagram such that the triangles ffi &" p
p"&ffi
is cofiber sequence and !d is commutative
ABP (SI) = B0 - -i1- B1 - -i2- B2 - -i3- B3
x x x
ffi0& p1??ffi1&p2?? ffi2& p3??
H1 --d1-! H2 --d2-! H3
Taking Hom(X, -), we have the diagram of cohomology theories such
i*
p*"&ffi*
that the triangles ffi* &" p*is exact and !d* is commutative
0 i1* *,*0 i2* *,*0 i3* *,*0
ABP (SI)*,*(X) - - - B1 (X) - - - B2 (X) - - - B3 (X)
x x x
ffi0*& p1*?? ffi1*& p2*?? ffi2*& p3*??
0 d1* *,*0 d2* *,*0
H*,*1(X) -- - ! H2 (X) --- ! H3 (X)
0 *,*0 R R
where H*,*n(X) ~= l(R)=n-1H (X; Z=p){v } with degree |v | =
-|vR | = |vr11vr22...|.
0
For an element x 2 H*,*n(X) with pn*(x) 2 Image(in+1*...in+r*), we
can define the differential
d(A)r(x) = ffin+r*(in+1*...in+r*)-1pn*(x).
If we give th filtration by
0 *,*0
F iltAdamss( nH*,*n(X)) = s nHn (X; Z=p).
ALGEBRAIC BP-THEORY 9
0
Then we have the Adams spectral sequence converging ABP (SI)*,*(X),
namely
0 *,*0
Ext (Qj|j62I)(Z=p, H*,*(X; Z=p)) =) ABP (SI)(X).
Here we give the another filtration
0 *,*0 R
F iltAHsss( nH*,*n(X)) = s |vR|H (X; Z=p){v }.
Writing vR = vr11vr22..., we have the isomorphism
0 s
F ils=F ils-1 ~=H*,*(X; Z=p) (BP=SI) .
0
Thus we get the desired AHss converging ABP (SI)*,*(X).
Remark. V.Voevodsky define a slice filtration fq(E) for a spectrum
E in the stable homotopy theory SH(k) (for details, see Theorem 2.2
[Vo5])
E = f0(E) f1(E) f2(E) ....
The slice sq(E) is defined by the cofibering
fq+1(E) ! fq(E) ! sq(E)
and sq(E) belong to qTSH(k)eff. Here SH(k)eff is the smallest trian-
gulated subcategory in SH(k) which is closed under direct sums and
contains suspension spectra of spaces but not their T-desuspensions.
The spectrum E is called slice wise cellular, if any q 2 Z the slice
sq(E) belongs to the smallest triangulated category of SH(k) closed
under direct sums which contain the spectrum qTHZ. Note that when
ch(k) = 0, the category of slice wise cellular spectra contains the sphere
spectrum and therefore T-cellular spectra (Corollary 4.5 in [Vo5]). In-
deed sq(HZ) = HZ for q = 0 and sq(HZ) = 0 otherwise (Conjecture
1 in [Vo5]), and it is also proved s0(1) = HZ where 1 is the sphere
spectrum
If E is slice wise cellular, then we can construct the motivic Adams
spectral sequence
0
ExtA*,*p(H*,*(E; Z=p), H*,*(X; Z=p)) =) [X, E](p)
similar to the case above ; exchanging Bi 7! fi(E), Hi 7! si-1(E). We
note that we can take sq(E) = 0,qHZ q(E) (see x5 in [Vo5]) where
HZ q(E) = HZ q(E) in the Voevodsky's notation.
The above arguments for ABP (SI) showed that ABP (SI) is mod p
slice wise cellular, namely,
sq(ABP (SI)) = qTHZ=pBP=(SI)-2q= qTHZ=p BP=(SI)-2q
( Conjecture 5 in [Vo5]).
10 N.YAGITA
As a most simple example, we consider the case SI = (p, v1, ...., ^vn, ...),
namely, BP (SI) = k(n) the connected Morava K-theory
k(n)* ~=BP *=(p, v1, ...., ^vn, ...) ~=Z=p[vn].
This case Hi = HZ=p{vi n}. The map f : X ! HZ=p{vi n} repre-
0 * *,*0
sents the element x vin2 H*,*i(X) if f (vi n) = x 2 H (X; Z=p).
The differential d2pn-1(x vin) is represented by the composition map
f di
X ! HZ=p{vi n} ! HZ=p{v(i+1) n}.
Here we have
f*d*ivi+1 n = f*(Qn vi n) = Qn(x).
This means that d2pn-1(x vin) = Qn(x) vi+1nin AHss.
0
Lemma 3.4. The first nonzero differential in AHss for Ak(n)*,*(X)
is given by
d2pn+1(x) = vn Qn(x).
From the above lemma and the exact sequence
0 vn *,*0 *,*0
! Ak(n)*,*(X) ! Ak(n) (X) ! H (X; Z=p) ! .
0 *,*0
Lemma 3.5. If vny = 0 2 Ak(n)*,*(X), then there is x 2 H (X; Z=p)
such that Qn(x) = ae(yn) where ae : ABP ! AHZ=p is the natural
(Thom) map.
We also note the following lemma ([Ya2]).
P 0 0
Lemma 3.6. If vnyn = 0 2 ABP *,*(X), then there is x 2 H*,*(X; Z=p)
such that Qn(x) = ae(yn) where ae : ABP ! AHZ=p is the natural
(Thom) map.
Proof. (It is just the motivic version of the arguments of Tamanoi [Ta].)
Let AL be the spectrum defined by the cofiber sequence
qi pi-1 ~ `
S-1sAL --- ! T ABP --- ! ABP -- - ! AL
where the map ~ is defined by
i-1 _vi W folding
Tp ABP --- ! ABP -- - - !ABP
P 0
so that ~*(b0, b1, ...) = vibi for bi 2 ABP *,*(X).
0
Since ~* = 0 on H*,*(-; Z=p), we have
i-1 q*i*,*0 *,*0
0 ! H*-1,*( Tp ABP ; Z=p) ! H (AL; Z=p) ! H (ABP ; Z=p) ! 0.
0 *,*0
Recall H*,*(ABP ; Z=p) ~= H (pt.; Z=p) RP (see Lemma 3.1).
Hence the mod p cohomology is easily seen
0 *,*0 * *
H*,*(AL; Z=p) ~=H (pt.; Z=p) RP {1, q0(10), q1(11), ...}
ALGEBRAIC BP-THEORY 11
i-1,pi-1 pi-1 *
where 1i 2 H2p (T ABP ; Z=p). Here we can prove that qi(1i) =
Qi(1) for 1 2 H0,0(AL; Z=p). Because this holds for topological case
0
(see [Ta] for details), and the w(x) = -1 parts in H*,*(AL; Z=p)) are
written as RP {1, q*0(10), q*1(11), ...} which maps injectivity to the
topological case by the realization map tC.
Let j : AL ! AHZ=p be the map of spectra representing 1 2
H0,0(AL; Z=p). The above equality means
i-1,pi-1
aeqi = Qij : AL ! S2p AHZ=p
as homotopyPmaps. 0
Suppose viyi = 0 2 ABP *,*(X). Then ~( (yi)) = 0. So there is
z 2 AL*-1,*(X) with (qi(z)) = (yi). Take x = j(z) and we get
ae(yi) = aeqi(z) = Qij(z) = Qi(x).
__
|__|
0,0 *,*0
Corollary 3.7. Let z 2 E*,*1 H (X; Z=p) in AHss converg-
0 *,*0,*00
ing to ABP *,*(X) such that viz = 0 2 E1 . Then there is x 2
0 P *,*
H*,*(X; Z=p) such that vnyn = 0 in ABP (X) with ae(yn) = Qn(x)
for all n and z = ae(yi) = Qi(x).
0
4. cohomology operations in ABP *,*(-)-theory
Recall that
0 *,*0 *
(4.1) H*,*(MGL) ~=H (pt.; Z) H (MU)
where additively H*(MU) ~=H*(BU) ~=Z[c1, ...] and where ci is the i-
th Chern class with deg(ci) = (2i, i) in H*,*(MGL; Z). Since w(ci) = 0,
the elements ci are infinite cycles in AHss for X = MGL
0,*00 *,*0 *00 *,*0
E(X)*,*2 = H (X, Z) MU =) MGL (X).
Hence we have the isomorphism of spectral sequences E(MGL)*,*,*r~=
E(pt)*,*,*r H*(MU). This means
0 *.*0 *
MGL*,*(MGL) ~=MGL (pt) H (MU).
0
Hence the Steenrod algebra of MGL-theory is generated as an MGL*,*(pt)-
module by the Landweber-Novikov operation sffcorresponding cff=
cff11cff22... for ff = (ff1, ff2, ...).
0
Lemma 4.1. (Theorem 7.2 in [Ya4]) The theory ABP *,*(-) is a mul-
tiplicative theory and there exists the map ABP ! AMGL(p)such that
0 *,*0 *
ABP *,*(X) ~=MGL (X)(p) MU*(p)BP ,
0 *,*0 *
MGL*,*(X)(p)~= ABP (X) BP* MU(p) .
12 N.YAGITA
0
The Steenrod algebra of ABP -theory is generated as an ABP *,*(pt)-
module by the Quillen operation rfffor ff = (ff1, ...).
Proposition 4.2. Let us write R"P ~= Z(p){rff|ff = (ff1, ff2, ...), ffi
0, }. Then there is the isomorphism
0 *,*0 * *,*0
ABP *,*(ABP ) ~=ABP H (BP ) ~=ABP (pt) R"P.
Remark. The Landweber-Novikov operation0sffis also defined as
the cohomology operations in ABP *.*(-) theory by
ABP ! MGL(p)sff!MGL(p)! ABP.
0
We also use the same letter sfffor the operation in ABP *,*(-). Then
for each sequence ff = (ff1, ff2, ...) such that ffi = 0 if i 6= pk - 1 for
0
some k, the Landweber-Novikov operation generates ABP *,*(ABP )
as an ABP *.*(pt)-module.
Each multiplicative operation o(-) in BP *(X) theory is determined
by an element
o(y) 2 (BP *[[y]])2 ~=BP 2(CP 1) |y| = 2.
The total Quillen operation rt (resp. st) in BP *(-)[t1, t2, ...] ( |ti| =
2(pi- 1)) is defined by
FBPX
n -1 X pn
r-1t(y) = tnyp (resp. st (y) = tny )
P F
where BP means sum of the formal group law of BP *(-) theory.
Then rffis defined
X
rt(x) = rff(x)tff with tff= tff11...
and sffis defined similarly.
We note that the Quillen operation rff(and the Landweber-Novikov
operation sff) satisfies the Cartan formula
Lemma 4.3.
X
rff(x) = rff0(x)rff00(y).
ff=ff0+ff00
Proof. The Cartan formula holds if
X
(*) ~*(rff) = rff0 rff00
ff=ff0+ff00
0 *,*0
for the coproduct map ~* : ABP *,*(ABP ) ! ABP (ABP ^ ABP ).
0 *,*0
Here note for X = ABP, ABP ^ABP , we have ABP *,*(X) ~=ABP (pt)
H*(X(C))(p). In particular
ABP 2*,*(X) ~=BP *(X(C)).
ALGEBRAIC BP-THEORY 13
The Cartan formular holds in BP *(-) theory and the formula (*)
holds in BP *-theory and so does in ABP 2*,*(ABP ^ ABP ), indeed __
rff2 ABP 2*,*(ABP ). |__|
By the similar arguments, we have
0 *
Lemma 4.4. ABP *,*(ABP ) is a BP (BP )-module.
i-1
Recall that H*(BP ) ~= Z(p)[m1, m2, ...] where mi = 1=(pi)CP p .
The Quillen operation rffon mn is explicitly written.
Lemma 4.5. (Quillen [Ha],[Ra])
(
i for = (0, .., 0, n-i, 0, ..)
rff(mn) = mi if ff = p n-i n-i 1
0 otherwise.
Hazewinkel showed the expression of vn by mi
X i
vn = pmn - mivpn-i
1 i n-1
identifying ss*(BP ) = Z(p)[v1, ...] H*(BP ) = Z(p)[m1, ...].
Let us write by In the ideal in BP *generated by (v0, ..., vn-1). (Let
v0 = p.) One of the important properties of rffis ;
Lemma 4.6. ([Ha],[Ra])
(
vi mod(Ii) if ff = pi n-i
rff(vn) =
0 mod(I21) otherwise.
An Ideal J in BP *is called invariant if it is so under the Quillen (or
Landweber-Novikov) operations, i.e., rff(J) J for all ff.
Lemma 4.7. (prime invariant theorem [La]) If for a 2 BP *, the ideal
J = (In, a) is invariant, then a = ~vsnmod(In) for ~ 2 Z=p and s 1.
In particular, prime invariant ideals are written as Im for m 1 or
I1 .
One of0examples of invariant ideals is following. For AHss converging
ABP *,*(X), define a filtration of the infinite term by
0,*00
Fs(X) = * sE*,*1 .
0,0 * *,*0,*00
Corollary 4.8. If x 2 E2*,*1 and BP =J{x} E1 for some ideal
J, then this ideal J is invariant.
0
Proof. Let us write x0 2 ABP *,*(X) a corresponding element to x 2
0,0 0
E*,*1. Let a 2 J so that ax = 0 mod(Fn+1). Then
X
0 rff(ax0) = rff0(a)rff00(x0) rff(a)x0 mod(Fn+1)
ff=ff0+ff00
14 N.YAGITA
since rff00(x0) 2 Fn+1 for ff006= 0 by dimensional reason. Hence rff(a)_is
also in J. |__|
5. AP (n) = ABP (In) theories
As the topological case, let us write
0 *,*0
ABP (p, v1, ..., vn-1)*,*(X) = AP (n) (X)
0 *,*0 *,*0 *,*0
e.g., AP (0)*,*(X) = ABP (X), AP (1) (X) = ABP (X; Z=p)
0 *,*0
and AP (1)*,*(X) = H (X; Z=p).
We want show the Conner-Floyd type theorem for AP (n)*,*(-) and
AK(n)*,*(-). For ease of notations, let us write Q(n) = (Q0, ..., Qn).
0
Lemma 5.1. There is an AP (n)*,*-module isomorphism
AP (n)*,*(AP (n)) ~=AP (n)*,* RP Q(n - 1).
Proof. (Compare [Wu],[Ya1]) For the cofiber sequence
n-1 vn
(1) Tp ^ AP (n) ! AP (n) ! AP (n + 1),
we get the long exact sequence
ffik *-2pn+2,*-pn+1 vn* *,*
- AP (n) (AP (n)) - AP (n) (AP (n))
- AP (n)*,*(AP (n + 1)) .
By induction, we assume the isomorphism in the lemma for n. Let
' 2 AP (n)0,0(AP (n))Prepresents the identity map of AP (n). Then
0
v*n' = vn'. Let x = arIQJ' 2 AP (n)*,*(AP (n)) for a 2 AP (n)*,*.
Then we see
v*nrIQJ(') = rIQJ(v*n') = rIQJ(vn') = vn(rIQJ') mod(In).
The last equation is shown by Lemma 5.2 bellow. Therefore we get
v*nx = vnx. Hence we have AP (n)*,*(AP (n+1)) ~=AP (n)*,*(AP (n))=(vn).
Next we consider (the Sullivan) exact sequence induced from (1)
n-1,*+2pn-1 vn *,*
AP (n)*+2p (AP (n + 1)) ! AP (n) (AP (n + 1))
aen *,* ffin
! AP (n + 1) (AP (n + 1)) ! .
0
Since vn = 0 on AP (n)*,*(AP (n + 1), we get the isomorphism
AP (n + 1)*,*(AP (n + 1)) ~=AP (n)*,*(AP (n))=(vn) (Qn)
identifying Qn = ffinaen mod(Q(n - 1)). By induction on n, we get the_
lemma. |__|
Similarly, we get
0 s
AP *,*(AP (n)^s) ~=AP (n)*,* (R Q(n - 1)) .
ALGEBRAIC BP-THEORY 15
0
Lemma 5.2. We can take rI, QJ which commute with vn, as AP (n)*,*-
module generators of AP (n)*,*(AP (n)).
Proof. Let us write by ~ : ABP ^ AP (n) ! AP (n) the product map.
From Lemma 4.3 and the preceding lemma for n, we can write
X X
~*(rI) = rI0 rI00+ aKLJ rK rLQJ
I=I0+I00
with |K|, |J| > 0. From Lemma 4.6, we still know that
rI(vn) = 0 mod(In) |I| 6= 0.
Recall vnx is defined by
n-1 vn^x ~
Tp ^ AP (n) ! ABP ^ AP (n) ! AP (n).
Hence we have
X X
rI(vnx) = rI0(vn)rI00(x) + aKLJ rK (vn)rLQJ(x)
I=I0+I00
which is equal to vnrI(x) mod(In).
Define Qi by the map
0 ffii *,*0 aei *,*0
aeiffii : AP (n)*,*(X) ! ABP (p, ..., ^vi, ...vn-1) (X) ! AP (n) (X).
0
Here the maps ffii and aei are ABP *,*-module maps. In particular __
Qi(vnx) = vnQi(x). |__|
We recall some arguments of W"urgler([Wu]). Let us say that x 2
AP (n)*,*(AP (n)) is ABP -primitive if
0
~*(x) = 1 x in AP (n)*,*(ABP ^ AP (n)).
We say that x 2 AP (n)*,*(AP (n) ^ AP (n)) is ABP ^ ABP -primitive
if "~*(x) = (1 1) (x) for
"~: ABP ^ ABP ^ AP (n) ^ AP (n) 1^T^1!ABP ^ AP (n) ^ ABP ^ AP (n)
~^~
! AP (n) ^ AP (n)
where T is the switch map.0Similarly we can define ABP ^s-primitivity
for elements in ABP (n)*,*(AP (n)^s).
Of course ' 2 AP (n)0,0(AP (n)) is ABP -primitive, and ' ' 2
AP (n)0,0(AP (n)2) is ABP ^ABP -primitive. Hereafter we simply write
' by 1.
Lemma 5.3. If p 3, then ABP -primitive elements in AP (n)0,0(AP (n))
is additively generated by 1. Similarly if p 5 and i 3, then ABP ^i-
primitive elements in AP (n)0,0(AP (n)^i) is additively generated by 1 i.
16 N.YAGITA
0
Proof. Each element x 2 AP (n)*,*(AP (n)) is represented as a sum of
aIJrIQJ. Then
~*(x) = rI aIJQJ + ....
Hence if x is primitive, then |I|=0. Suppose that |J| > 0 (so |aJ| < 0
) and
x = aJQJ 2 AP (n)0,0(AP (n)).
First note that
w(QJ) w(Q0...Qn-1) = -n, and so w(aJ) n.
In the E1 -term of AHss for AP (n)*,*(pt.), let us write
X
[aJ] = vK a0K vK 2 P (n)*, a0K2 H*,*(pt.; Z=p).
Since |aJ| < 0, we see vK 2 P (n)- , so |vK | |vn| = -2(pn - 1).
Moreover w(a0K) n and so |a0K| 2n. Thus if p 3, then
|x| = |vK a0KQJ|| |vn| + 2n + |Q0...Qn-1|
= 2(-pn + 1 + ... + pn-1) + 2 < 0.
This is a contradiction. Hence primitive elements are case |I| = |J| =
0, and the result follows from AP (n)0,0(pt.) ~=Z=p.
0
Each element x in AP (n)*,*(AP (n) ^ AP (n)) is represented by a
sum of
a . rIQJ rI0QJ0.
The result for ABP ^ ABP -primitive elements follows from that if
p 5, then
|x| |vn| + 4n + 2|Q0...Qn-1| = -2pn + 4(1 + ... + pn-1) < 0.
The ABP ^3-primitivity follows from the inequality
|vn| + 6n + 3|Q0...Qn-1| = -2pn + 6(1 + ... + pn-1) < 0.
__
|__|
0
Theorem 5.4. If p 3, then the theory AP (n)*,*(-) is independent
of the choice of generator vi. Moreover if p 5, then there is the
unique associative, commutative product compatible with the natural
map AP (n - 1) ! AP (n).
Proof. Let I0n= {p0, v01, ..., v0n} and Ideal(In) = Ideal(I0n). Let us write
AP 0(n) = ABP (I0n). Then we can see
0 0 *,*0
AP 0(n)*,*(AP (n)) ~=AP (n) R Q(n - 1)
ALGEBRAIC BP-THEORY 17
as the proof of Lemma 4.9. The element 1 2 AP 0(n)0,0(AP (n)) rep-
resents the map i : AP (n) ! AP 0(n). Similarly we get the map
i0: AP (n) ! AP 0(n). The fact that the composition i0i is identity for
p 3 follows from the preceding lemma, namely, i*i0*(1) is primitive
which represents 1 on AP 0(n)0,0(ABP ). Such element must equal to
1.
Now we consider the product structure. The element
1 1 2 AP (n)0,0(AP (n) ^ AP (n)) represents the map
~ : AP (n) ^ AP (n) ! AP (n).
To see the commutativity, we consider the map
~
~0: AP (n) ^ AP (n) T! AP (n) ^ AP (n) ! AP (n).
We see both ~*(1 1) and ~0*(1 1) are ABP ^ ABP -primitive,
since so is 1 1. Hence from the preceding lemma, the both elements
must be equal if p 5. The associativity follows from the ABP ^3-
primitivity, and the compatibility of the product for the natural map_
AP (m) ! AP (n), m n follows from the ABP -primitivity. |__|
Corollary 5.5. For p 5, the AHss is a multipliticative spectral se-
quence. In particular there is the P (n)*-algebra isomorphism
0 *,*0 *,*0
grAP (n)*,*(pt.) ~=P (n) H (pt; Z=p).
Proof. Consider AHss
0*00 *,*0 *00 *,*0
E*,*2 = H (pt.; Z=p) P (n) =) AP (n) (pt.).
0 M
Recall that H*,*(pt; Z=p) ~=K* (k)=p[o ]. By dimensional reason,
dr(o ) = 0, and dr(x) = 0 for x 2 KM*(k)=p.
__
Since the differential is a derivation, the AHss collapses. |__|
Corollary 5.6. Let p 5. Then we can take the cohomology opera-
tion Qi and rI on AP (n)*,*(-) such that
(1) Q2i= 0, QiQj = -QjQi, ~*(Qi) = Qi 1 + 1 Qi.
X X
(2) ~*(rI) = rI0 rI00+ arK QL rK0QL0
I=I0+I00
with a 2 I1 AP (n)*,*, |K + L| > 0 and |K0+ L0| > 0.
Proof. Define Qi = aeiffii as in the proof of Lemma 4.10. Then Q2i= 0
and QiQj = -QjQi are immediately seen. We can prove the primitiv-
ity from the fact that the element Qi is generates the ABP -primitive
i-1,pi-1
elements in AP (n)2p (AP (n)). This fact is proved by the argu-_
ments similar to the proof of Lemma 5.2. |__|
18 N.YAGITA
Suppose p 5. Consider the filtration
0
0 = F-1 F0 ... F1 = AP (n)*,*(X)
P 0
with Fi = { aIxI|aI 2 AP (n)*,*, w(xI) i}. Then operations rI
act on Fi=Fi-1 satisfying the Cartan formula from (2) in the preceding
corollary. Therefore using the arguments in the proof of Corollary 4.8,
we get the filtration
Fi0 Fi1 .... Fin = Fi=Fi-1
such that Fis=Fi,s-1is the P (s)*-free module. Using this fact (for
details, see [La], [Ya1]), we can prove ;
0
Lemma 5.7. (Exact functor theorem for AP (n)*,*-theory) Let p 5.
Let G be a P (n)*-module such that the map vm : G=Im ! G=Im are
monic for all m n. Then the functor
0 *,*0
AP (n)*,*(X) 7! AP (n) (X) P(n)*G
0
is an exact functor (i.e., AP (n)*,*(-) P(n)*G is the cohomology the-
ory).
Corollary 5.8. Let p 5. Let AK(n)*,*(-) = [v-1n]Ak(n)*,*(-) be
the motivic Morava K-theory. Then we get the isomorphism
AK(n)*,*(X) ~=K(n)* BP* AP (n)*,*(X).
Proof. The operation rI also acts on the AHss E*,*,*rconverging to
AP (n)*,*(X). From the exact functor theorem, we know
E*,*,*r7! E*,*,*r P(n)*K(n)*
is the exact functor. Hence
H(E*,*,*r, dr) P(n)*K(n)* ~=H(E*,*,*r P(n)*K(n)*, dr 1).
Of course the left hand side is E*,*,*r+1 P(n)*K(n)*. By induction, we
can show that the righthand side is isomorphic to the r + 1-th term of
AHss converging to AK(n)*(X) since
E*,*,*2 P(n)*K(n)* ~=H*,*(X; Z=p) K(n)*
__
which is the E2-term of AHss converging to AK(n)*,*(X). |__|
6. homology theories and ABP*,*0(ABP )
Since eighties, most Adams Novikov spectral sequences are studied
for homology theories, but not cohomology theories. In this section,
we remark a bit about the homology theories and the motivic version
of the Adams-Novikov spectral sequence which are studied by Miller,
Ravenel and Wilson [Mi-Ra-Wi].
ALGEBRAIC BP-THEORY 19
First we recall the motivic homology for a smooth projective X. It is
known by Suslin and Voevodsky that the motivic (co)homology theory
holds the Poincare duality
0
- \ uX ; H*,*(X; Z=p) ~=H2d-*,d-*0(X; Z=p)
where d = dim(X) and uX 2 H2d,d(X; Z=p) is the fundamental class
of X. Hence if 0 6= x 2 H*,*0(X; Z=p), then
f.deg(x) 2d, w(x) 0, and d(x) 0.
Let us write
0
H*,*0= H*,*0(pt.; Z=p) ~=H-*,-* (pt.; Z=p).
It is known (Conjecture [Vo5]) that the homology
H*,*0(HZ=p; Z=p) ~=H*,*0 R~P ~
where R~P = Z=p[,1, ,2, ...] deg(,i) = (2pi- 2, pi- 1)
~ = (o0, o1, ...) deg(oi) = (2pi- 1, pi- 1).
As the cohomology cases ([Hu-Kr],[Ve]), we have
H*,*0(MGL; Z=p) ~=H*,*0 R~P Z=p[mi|i 6= pi- 1].
Arguments similar to x3, we get
Lemma 6.1. Let S = (vi1, ..., vis). Then
0
H*,*(ABP (S); Z=p) ~=H*,*0 R~P (oi1, ..., ois)
ABP (S)*,*0(ABP (S)) ~=ABP*,*0 R~P (oi1, ..., ois).
Moreover we have the AHss for homology theory
Theorem 6.2. Let Ah = ABP (S) for S = (vi1, vi2, ...). Then there is
the Atiyah-Hirzebruch spectral sequence
E(Ah)2(m,n,2n0)= Hm,n(X; h2n0) =) Ahm+2n0,n+n0(X)
with the differential d2r+1 : E2r+1(m,n,2n0)! E2r+1(m-2r-1,n-r,2n0+2r).
For ease of arguments, let B be ABP or AP (n) for p 5 so that
they have the good product. Hence we also have the Kunneth map
B*,*0(X) B*,*0B*,*0(Y ) ! BP*,*0(X x Y ).
Since |x| 2d for nonzero x 2 H*,*0(X; Z=p), we see each element in
H2d,d(X; Z=p) is permanent in the above AHss. In particular we can
take the fundamental class uX also in B2d,d(X). Hence we can define
20 N.YAGITA
0
the Poincare dual map - \ uX : B*,*(X) ! B2d-*,d-*0(X) by the map
of spectra
uX ^1 1^x^1 1^~B
x \ uX : T* ! X ^ B ! X ^ X ^ B ! X ^ B ^ B ! X ^ B.
Theorem 6.3. For smooth X, there holds the Poincare duality
0
- \ uX : B*,*(X) ~=B2d-*,d-*0(X).
Proof. Consider spectral sequences Er*,*,*and E*,*,*rwhich converge to
B*,*(X) and B*,*,*(X) respectively. Since we can define the Poincare
map in B*,*(-) theories, we can define the Poincare map of AHss's.
0,*0 r
- \ uX : E*,*r ! E2d-*,d-*0,-*00
The isomorphic of these maps follow from the isomorphic of E2-terms,
which follows from the Poincare duality of the motivic (co)homology __
theory. |__|
Lemma 6.4. B*,*0(B x X) ~=B*,*0(B) B*,*0B*,*0(X).
Proof. First note that it is well known ([Vo3]) that the Kunneth for-
mular
H*,*0(Y x X; Z=p) ~=H*,*0(Y ; Z=p) H*,*0H*,*0(X; Z=p)
satisfies for all X, when Y = Pn. This induces that the Kunneth
formular holds, when Y = P1 , BGL, MGL. By induction on n we
can prove the Kunneth formula for AP (n),
H*,*0(AP (n) x X; Z=p) ~=H*,*0(AP (n); Z=p) H*,*0H*,*0(X; Z=p).
__
The isomorphism in the lemma follows from AHss. |__|
Then we can define B*-Adams Novikov resolution for X ( see Definition
2.1.1 in [Ra]), that is the injective resolution
0 ! B*,*0(X) ! B*,*0(B ^ X) ! B*,*0(B ^ B ^ X) ! ...
Thus we can construct the B*,*0Adams-Novikov spectral sequence ;
Theorem 6.5. There is the spectral sequence
E(B)2s,*,*0= ExtsB*,*0(B)(B*,*0, B*,*0(X)) =) ss*,*0(X)(p)
with the differential dr : Ers,*,*0! Ers+r,*,*0and
gr(ss*,*0(X)(p)) ~= sEs,*,*0=Es+1,*,*0.
ALGEBRAIC BP-THEORY 21
Now we restrict the case that B = ABP and X is celluler. Then for
Ks = ABP ^s^ X, we have isomorphisms
ABP*,*0(Ks) ~=ABP*,*0 BP* ABP2*,*(Ks)
with ABP2*,*(Ks) ~=BP*(Ks(C)) ~=BP2* (R~P ) s H2*(X(C))(p).
The differentials in the B*,*0-Adams-Novikov spectral sequence are
defined by the alternated sum of the natural (diagonal) inclusions
ij : Ks ! Ks+1. The map ij*,*0on ABP2*,*(Ks) is the just the map on
BP2*,*(Ks(C)). Hence if ABP*,*0satisfies the condition of the exact
functor theorem (Lemma 5.7), then
E(ABP )2s,*,*0~=ABP*,*0 BP* ExtsBP*(BP)(BP*, BP*(X(C)).
However the condition does not satisfied most cases. As for the case
B = AP (n), n 1, we see ABP2*,*(Ks) 6~= BP2*(K(C)). We only
know for k = C.
Corollary 6.6. Let X be celluler. Let k = C, p 5 and n 1. Then
E(AP (n))2s,*,*0~=Z=p[o ] ExtsP(n)*(P(n))(P (n)*, P (n)*(X(C)).
7. Gysin maps
First we recall the Thom isomorphism. Let V be an m-dimensional
vector bundle over X and T hX (V ) be the induced Thom space. Then
it is well known that there is the Thom isomorphism
0 *+2m,*0+m
T h : H*,*(X; Z) ~=H" (T hX (V ); Z).
The element T h(1) 2 H2m,m (T hX (V )) is called its Thom class and the
0
above isomorphism is that of H*,*(X; Z)-modules ; the right hand
0
module is a free H*,*(X; Z)-module generated by the Thom class
T h(1).
0
Lemma 7.1. The Thom isomorphism also holds in ABP *,*(X) for
smooth X
0 *+2m,*0+m
T h : ABP *,*(X) ~=ABP (T hX (V )).
Proof. Consider E(T hX (V ))r (resp. E(X)r) the AHss converging to
0 *,*0
ABP *,*(T hX (V )) (resp. ABP (X)). Since w(T h(1)) = 0, we see
the Thom class T h(1) is a permanent cycle in E(T hX (V ))r. Then we
see inductively that E(T HX (V ))r is the free E(X)r-module generated_
by T h(1). Hence we get the lemma. |__|
22 N.YAGITA
For a projective map f : Y ! X of smooth projective varieties such
that c = codimX (Y ) is constant, we will define the Gysin map
0 *+2c,*0+c
f* : ABP *,*(Y ) ! ABP (X).
By the definition, the projective map is factored as
p
f : Y !i Pm x X ! X
where i is a closed embedding to the product Pm x X and p is the
projection.
For a close embedding i : Y ! Z of codimZ (Y ) = c, we define the
Gysin map i* by
0 *+2c,*0+c q* *+2c,*0+c
i* : ABP *,*(Y ) ~=ABP (T hY (NZ=Y )) ! ABP (Z)
where NZ=Y is the normal bundle of Y in Z and q : Z ! T hY (NZ=Y )
is the quotient map.
For p : Z x X ! X, the Gysin map p* is defined as follows. There
is an m dimensional vector bundle V on Z with dim(Z) = d (Theorem
2.11 [Vo3]) such that there is a map i : Tm+d ! T hZ (V ) having the
property that the composition of maps
* 2(m+d),m+d m+d 0,0
H2d,d(Z) ~=H2(m+d),m+d(T hZ (V )) i! H (T ) ~=H (pt.) = Z
coincides the degree map. Then we can define the Gysin map
0 *+2m,*0+m
p* : ABP *,*(Z x X) ~=ABP (T hZ (V ) x X)
i*!ABP *+2m,*0+m(Tm+d x X) ~ *-2d,*0-d
= ABP (X).
Of course for a projective map f, we define the Gysin map by f* = p*i*.
Recall that we still defined the homological map f* : ABP*,*0(Y ) !
ABP*,*0(X) in the preceding section. We can easily show that
0
f*(y) \ uX = f*(y \ uY ) for y 2 ABP *,*(Y )
by using the fact that i*uY = q*T hY (1) \ uZ for an embedding and
p*(T hZ (1) \ uXxZ ) = uX for a projection p : X x Z ! X.
From Lemma 3.3, we know that
ABP 2*,*(X) BP* Z=p ~=H2*,*(X; Z=p) ~=CH*(X)=p.
By using the resolution of singularities, we can show that each element
x 2 ABP 2*,*(X) is represented by f*(1Y ) = [f : Y ! X] such that
codimX (Y ) = c is constant and f is projective. (This fact is also
showen by using the algebraic cobordism *(X) defined by Levine and
Morel).
ALGEBRAIC BP-THEORY 23
Recall that st (resp. ct) is the total Landweber-Novikov operation
(resp. total Chern class). Let us write
f = -f*(TX ) + TY 2 K(Y )
for the tangent bundles TX and TY . Then on ABP 2*,*(X), the Landweber-
Novikov operations are written ([Q]) by
st(f*(1Y )) = f*(ct( f)).
Example. Consider the inclusion i : Pd ! Pd+1. Then the total
Chern class of the normal bundle i is
X n
c-1t( i) = ( tnyp -1) with e( i) = y
i-1
in fact c i(L) = e(L)p for line bundles L. On the other hand, the
total Landweber-Novikov operation is
X n
s-1t(i*1) = s-1t(y) = tnyp
from the definition of st (see the explanation before Lemma 4.3). In-
deed, we show
X n X n
i*(c-1t( i)) = i*( tnyp -1) = tnyp
since i*i*(1) = e( i) = y.
Lemma 7.2. (Quillen [Q],[Me]) Let x 2 ABP 2*,*(Y ) and f : Y ! X
be projective. Then st(f*(x)) = f*(ct( f)st(x)).
Proof. Let x = [g : Z ! Y ]. By the definition
fg = -g*f*TX + TZ = g*(-f*TX + TY ) - g*TY + TZ = g* f + g.
This implies ct( fg) = g*(ct( f))ct( g). Hence we have
st(f*x) = st(f*g*(1)) = f*g*(ct( fg))
= f*g*(g*(ct( f)ct( g)) = f*(ct( f)g*(ct( g)) = f*(ct( f)st(x)).
__
|__|
Let us write
I(X) = ss*ABP 2*,*(X) ABP 2*,*(pt.) = BP *.
From the Quillen's lemma, it is immediate
Lemma 7.3. The ideal I(X) is generated by elements x with -dim(X)
|x| 0 as a BP *-module. Moreover I(X) is an invariant ideal of BP *.
24 N.YAGITA
Proof. Since ABP 2*,*(X) is generated as a BP *module by elements y
with 0 |y| dim(X), we have the first statement. If a 2 I(X), then
a = ss*(x) for some x = f*(1Y ) = [f : Y ! X] 2 ABP 2*,*(X). Then __
st(a) = ss*(ct( ss)st(x)) 2 I(X)[t]. |__|
0 *,*0
Let ~kbe an algebraic closure of k. Let i~k: ABP *,*(X) ! ABP (X|~k)
be the induced map from the base change. Let p~t.= Spec(~k) and
j : ~pt.! X|~kbe an inclusion. Recall the definition ([Mo-Le]) of
deg(f) 2 BP *;
deg(f){1p~t.} = j*i~kf*(1Y ) 2 ABP 2d,d(p~t.) ~=BP 2d{1p~t.}.
(When dim(X) = dim(Y ), deg(f) 2 ABP 0,0(p~t.) ~= Z(p)is the usual
degree of the map f. When X = pt., we see deg(f) = [Y ].)
Let ss : X ! pt. is the projection. Let us write by I+ (X) the sub
BP *-module of I(X) generated by ss*-images of elements in ABP 2*,*(X)
of positive degree. The degree formula for the cobordism by Levine and
Morel is following ;
Theorem 7.4. ( Levin-Morel [Le-Mo])
sff[Y ] - deg(f)sff[X] 2 I+ (X).
Proof. Consider the element
z = f*(1Y ) - deg(f)(1X ) in ABP 2d,d(X).
Then by the definition, we see j*~ki~k(z) = 0 since j*~ki~k(1X ) = 1p~t.. On
the other hand, the kernel of the map j*~ki~kis the sub BP *-module
of ABP 2*,*(X) generated by positive degree elements. Hence ss*(z) 2
I+ (X). Of course ss*(z) = [Y ] - deg(f)[X]. From the proceeding __
lemma, we have the desired result. |__|
Let V be (a stable normal) m-dimensional bundle of X used to define
0 0
the Gysin map ; there is the map i : Tm ! T hX (V ), m = d+m which
induces the degree on the motivic cohomology. Consider the cofibering
0 i q m0 @ 1,0 m0
! Tm ! T hX (V ) ! T hX (V )=T ! S T !,
and the induced long exact sequence on the mod p motivic cohomology.
The Thom class T h(1) 2 H2m,m (T hX (V ); Z=p) restricts to zero in
0 m0 *,*0 m0
H*,*(T ; Z=p), and there exists t 2 H (T hX (V )=T ; Z=p) with
0+1,m0 *
q*(t) = T h(1). Let ~oe2 H2m (T hX (V ); Z=p) be the image of @ of
0+1,m 1,0 m0
a generator oe of H2m (S T ; Z=p).
Proposition 7.5. (Lemma 4.1 in [Vo3])
Qn(t) = ~(deg(r n (X))=p)~oe where ~ 6= 0 2 Z=p.
ALGEBRAIC BP-THEORY 25
Proof. We consider the exact sequence for Ak(n)*(-) theory
@* Ak(n)*,*0(Tm0) i* Ak(n)*,*0(T h q* *,*0 m0 @*
X (V )) Ak(n) (T hX (V )=T ) .
0
Recall that the Gysin map is defined ss* = i*T h where T h : Ak(n)*,*(X) ~=
Ak(n)*+2m,*+m (T hX (V )) is the Thom isomorphism. Hence we have the
equivalents conditions
0
vn 2 I(X) () vn 2 ss*(Ak(n)*,*(X))
() vnoe 2 Im(i*) () vn~oe= 0 in Ak(n)*(T hX (V ))
0
() Qnz = ~oe for some z in H*,*(T hX (V ); Z=p).
The last equivalence follows from Lemma 3.5. By dimensional reason,
we can take z = ~t, ~ 6= 0 2 Z=p.
On the other hand
ss*(1X ) = [X] = vn () r n ([X])=p 6= 0 mod(p).
Since k(n)* is generated non-positive elements, Ak(n)2*,*(X) is gener-
ated by positive degree elements and 1X . Hence we see
ss*(Ak(n)0,0(X)) is generated by ss*(1X ) mod(I21).
Therefore vn 2 I(X) if and only if vn = ~ss*(1X ) mod(I21) for ~ 6= 0_2_
Z=p. |__|
8. In+1-torsion spaces
Recall that In+1 = (p, v1, ..., vn). In this section, we consider In+1-
torsion spaces and their applications according to V.Voevodsky. Re-
call that BP *(X) be the cohomology theory with the coefficient
BP * = Z(p)[v1, ..., vn] so that BP <- 1>*(X) = H*(X; Z=p) and
BP <1>*(X) = BP *(X).
0,*00 *
Lemma 8.1. ([Ya4]) Let E*,*r be the AHss for ABP (X). If
0 *,*0,0
x = Qn...Q1Q0x0 in H*,*(X; Z=p), then x 2 E1 and x is In+1-
0,*00
torsion in E*,*1 .
Proof. There is the cofiber map of spectra
k-1 vk aek ffik
Tp ^ ABP -- - ! ABP --- ! ABP -- - !
Consider the Baas-Sullivan exact sequence, namely, the long exact se-
quence induced from the above cofiber map
k-2,*0+pk-1 vk *,*0 aek
! ABP *+2p (X) -! ABP (X) -!
0 ffik *+2pk-1,*0+pk-1
ABP *,*(X) -! ABP (X) ! .
26 N.YAGITA
The induced map
0 *,*0 *,*0
Im(ABP *,*(X) ! H (X; Z=p)) ! H (X : Z=p)
definedPby ae0...aen-1(x) 7! ae0...aenffin(x) represents the operation Qn +
aIJP IQJ with aIJ 2 Hplus,*(X; Z=p) and J 2 from the topological
case [Ya1] and (2.8).
By the Baas-Sullivan exact sequence, we can see that x00= ffin...ffi0(x0) 2
0
ABP *,*(X) is In+1-torsion since the map ffii is a map of ABP -
module spectra. In particular, x = Qn...Q0(x0) = ae0...aen(x00) is a
permanent cycle in the spectral sequence
0 * *,*0
E(ABP )2 = H*,*(X; BP ) =) ABP (X),
__
and d2pn-1(y) = vn x for y = Qn-1...Q0(x0). |__|
Compare with the spectral sequence
0 * *,*0
E(ABP )*,*,*2~=H*,*(X; BP ) =) ABP (X).
Since BP *~= BP * for * > -2pn+1 + 2, we can see that x is In+1-
0,*00
torsion also in E(ABP )*,*2pn.
0
Recall that Q(n) = (Q0, ..., Qn). If H*,*(X; Z=p) is Q(n)-free and
0
H*,*(X) is just p-torsion, we have more strong results by using the
Baas-Sullivan exact sequence in the proof of the preceding lemma.
0
Lemma 8.2. ([Ya4]) If ABP *,*(X) is Ik+1-torsion for all k n,
0 *,*0
then H*,*(X; Z=p) is a free Q(n)-module. If H (X; Z) has no (infi-
nite) p-divisible elements, then the converse is also holds.
Let the C~ech complex C~(X) be the simplicial scheme such that
~C(X)n = Xn+1 and the faces and degeneracy maps are given by partial
projections and diagonals respectively ([Vo1,2]). One of the important
properties of C~(X) is the following.
Lemma 8.3. ([Vo1,2]) Let X, Y be smooth schemes such that Hom(Y, X) 6=
;. Then the projection ~C(X)xY ! Y is a equivalence in A1-homotopy
category.
In the stable A1 homotopy category, define C"(X) by the following
cofiber sequence
(5.1) "C(X) ! ~C(X) ! Spec(k).
Lemma 8.4. (([Vo1]) Let ss : Y ! pt. be the projection and ss*([Y ]) =
y in BP *. Let Ah = ABP (Sn) for some regular sequence Sn in BP *.
0
If Hom(Y, X) 6= ;, then Ah*,*(C"(X)) is y-torsion.
ALGEBRAIC BP-THEORY 27
Proof. Let p : X x Y ! X be the projection, and consider the compo-
sition map
0 *,*0 *+|y|,*+1=2|y|
p*p* : Ah*,*(X) ! Ah (X x Y ) ! Ah (X).
Here p*p*(x) = yx, indeed,
p*p*(x) = p*(1XxY p*(x)) = p*(1XxY )x = (y1X ) . x.
0 *,*0 __
But Ah*(C"(X) x Y ) ~=0 since Ah*,*(C~(X) x Y ) ~=Ah (Y ). |__|
Recall that I(X) = ss*(ABP 2*,*(X)) for ss : X ! pt.
0
Corollary 8.5. If vn 2 I(X), then H*,*(C"(X); Z=p) is a free Q(n)-
module.
Proof. If there are maps V0 ! V1 ! ... ! Vn ! X such that
tC(p*[Vi]) = vi for all i n, then we have the result. From Lemma
3.5, we know rpi n-i(vn) = vi. . Since I(X) is invariant ideal, we see
that vi 2 I(X) for all i n. This means the existence of Vi and above_
maps. |__|
9. Chow motive
For smooth X1 and X2, an element ` 2 CHdim(X2)(X1 x X2) can be
viewed as a correspondence from X1 to X2. For more generally element
` 2 CH*(X1 x X2) gives a homomorphism
0 *,*0 *
f` : H*,*(X1; Z=p) ! H (X2; Z=p) by f`(x) = pr2*(pr1(x) [ `)
where pri are projections of X1 x X2 onto Xi.
For ` 2 CHdimX (X x X), the morphism p` = f` is called a projector
if p` O p` = p`. The object of the Chow motive (Choweff(k)) are
pairs (X, p) of smooth X and a projector p = p`, and the morphisms
are defined by morphisms f` (namely, the Chow motive is the pseudo
abelian envelop of the category of correspondences). Objects (X, p)
are simply called motives M which are direct summand of M(X) =
0
(X, idX ), and H*,*(M; Z=p) are defined as Im(p).
Lemma 9.1. Let M be a direct summand of M(X) and p` be its pro-
0 *,*0
jector, i.e., p`H*,*(X; Z=p) = H (M; Z=p). Then p` commutes with
0
Qi. Hence H*,*(M; Z=p) has the natural Q(1)-module structure.
Proof. Let ` 2 CHd(X x X) with dimX = d. Then
p`(Qi(x)) = pr2*(pr*1(Qi(x)) . `) = pr2*(Qi(pr*1(x) . `)).
The last equation follows from Qi(`) = 0 since w(`) = 0. Hence we
have the desired result if pr2*Qi = Qipr2*.
28 N.YAGITA
By definition of the Gysin map (recall x7), we know
pr2*(x) = i*(T hX (1) . x)
where T hX (1) 2 H2m,m (T hX (V ); Z=p) is the Thom class for some bun-
dle V over X and i : Tm x X T hX (V ) x X. Since w(T hX (1)) = 0,
we see Qi(T hX (1) . x) = T hX (1) . Qi(x). Therefore we see that_pr2*
commutes with Qi. (Indeed, Qi commutes with the Gysin maps.) |__|
0
Remark. The reduced powers P ido not act naturally on H*,*(M; Z=p),
see Lemma 9.2 bellow.
Let A*(X) be an oriented generalized cohomology theory on the
category of smooth varieties X over k, in the sense of Panin [P]. The
theories CH*(X) and ABP 2*,*(X) are oriented generalized cohomol-
ogy theories.
We can define the category of A-motive MA (k) as a pseudo abelian
envelop of the category of A-correspondences CorA (of degree 0). Here
objects in CorA are classes [X] of smooth varieties and its morphisms
are given by
MorCorA([X], [Y ]) = Adim(X)(X x Y ).
Theorem 9.2. ([Vi-Ya]) Let aeA : A*(X) ! CH*(X) be a map of
oriented cohomology theories such that aeA are epic and Ker(aeA ) are
nilpotent for all X. Then aeA induces the natural 1 to 1 correspon-
dence between the set of isomorphism classes of objects in MA (k) and
MCH (k).
The theory ABP 2*,*(X) satisfies the assumption of the above theo-
rem (with localized at p) from (3.4) and the fact that BP *is generated
by nonpositive degree elements.
Lemma 9.3. (Karpenko-Merkurjev) [Ka-Me] For x 2 ABP 2*,*(X)
and ` 2 ABP 2d,d(X x X), d = dim(X), we have
st(f`(x)) = fst(`)(st(x)ct( X )).
Proof. From Lemma 7.2, we have
st(f`(x)) = st(pr2*(pr*1(x) . `)) = pr2*(st(pr*1(x)`)ct( pr2)).
Here ct( pr2) = pr*1(ct( X )). Hence the above element is
pr2*(st(pr*1(x))st(`)pr*1(ct( X )) = pr2*(pr*1(st(x)ct( X ))st(`))),
__
which is fst(`)(st(x)ct( X )). |__|
Now we consider a hypersurface V in Pd+1 of degee = p. Recall that
0 d+1 *,*0 d+2
H*,*(P ; Z=p) ~=H (pt.; Z=p)[h]=(h ).
ALGEBRAIC BP-THEORY 29
We use the same letter h 2 H2,1(V ; Z=p) which is the image of h by
0 d+1 *,*0
the map H*,*(P ; Z=p) ! H (V ; Z=p).
It is well known that TPm ffl ~=(m + 1)O(1) where ffl is a trivial line
bundle. Hence there is the exact sequence of bundles
0 ! TV ! TPd+1 ! O(p) ! 0.
P i
Thus the total Chern class is ct(TV ) = ( hp -1ti)d+2.
Let us write simply
i-1 d+2
cti(TV ) = (1 + hp )
letting ti = 1 and tj = 0 for i 6= j. Similarly define the total
Landweber-Novikov operation s-1t(-) by
sti(x) = s-1t(x)|{tj=ffiij|j 1}
i
such that s(h) = s0(h) + s i(h) = h + hp . By Lemma 7.2, for a
projective map f : Y ! X, we have
sti(f*(x)) = f*(cti(f*(TX ) - TY ))sti(x)).
Recall that ~kis an algebraic closure of k, X|~k= X k ~kand i~k:
X ! X|~kthe extension map.
Lemma 9.4. Let V have no k-rational point. Then i~k(hd) = pw for a
generator w 2 CHd(V |~k)(p)~= Z(p).
Proof. Since V has no k-rational points, the degree map deg : CHd(V )(p)!
CH0(pt.)(p)is not epic. Hence from the commutative diagram
deg 0
CHd(V ) --- ! pZ(p) CH (pt)(p)~= Z(p)
? ?
i~k?y =?y
deg 0
CHd(V |~k) --- ! CH (p~t)(p)~= Z(p)
we see
(*) i~kCHd(X)(p) pCHd(X|~k)(p).
On the other hand, for the emmbedding i : V ! Pd+1, the normal
bundle is i = i*O(p), so we see i*(1V ) = c1(O(p)) = ph. Hence
i*(hd) = hdi*(1) = phd+1 in CHd+1(Pd+1).
Now consider the composition map
deg 0
CHd(V ) i*!CHd+1(Pd+1) ! CH (pt.).
Since deg = ss* for the projection ss : X ! pt., it is immediate
deg|CHd(V ) = deg O i*|CHd(V ). So we get deg(hd) = p.
30 N.YAGITA
From (*), we see that i~k(hd) = pw for a generator w in CHd(V |~k).__
|__|
Lemma 9.5. Let p` be a projector for ` 2 CHd(V x V ) for dim(V ) =
d such that p`(hd) = hd. Then for each 0 < ps - 1 < d, we see
s+1 d d-ps+1
P i(p`(hd-p )) = -h in particular, p`(h ) 6= 0.
Proof. Compare the equation given by Karpenko and Merkurjev
(*) sts(p`(hd-i)) = psts(`)(sts(hd-i)cts(-TV )).
First we consider its right hand side term ;
s d-i d-i ps-1 d-i
sts(hd-i) = (h + hp ) = h (1 + h ) ,
s-1 -d-2
cts(-TV ) = (1 + hp ) .
Here consider the case i = ps - 1 ;
s+1 d-ps+1 ps-1 -ps-1 d-ps+1 s d
sts(hd-p )c(-TV ) = h (1 + h ) = h + (-p - 1)h .
s+1 d d __
Hence P s(p`(hd-p )) = -p`(h ) = -h . |__|
s+1 d-ps+1
Corollary 9.6. If d + 1 = 0 mod(p), then p`(hd-p ) 6= h .
Proof. Note that
s+1 s d
P s(hd-p ) = (d - p + 1)h ,
__
which is zero if d + 1 = 0 mod(p). |__|
Remark. When p 3, the norm variety described in the next sections
are not hypersurfaces of Pd+1.
10. Norm Variety
Recently, Voevodsky announced the proof of the Bloch-Kato con-
jecture for all odd primes [Vo4]. For non zero a = {a0, ..., an} 2
KMn+1(k)=p, Rost ([Ro3]) constructed the norm variety Va such that
(1) ss*[1Va] = Va(C) = vn, a = 0 2 KMn+1(k(Va))=p
(2) the following sequence is exact
pr1-pr2 *
H-1,-1(Va x Va, Z) ! H-1,-1(Va; Z) ! k .
Let us write Oa = ~C(Va). By the solution of Bloch-Kato conjecture,
we see the exact sequence
(10.1) 0 ! H*+1,*(Oa; Z=p) xo!KM*+1(k)=p ! KM*+1(k(Va))=p
identifying H*+1,*+1(Oa; Z=p) ~=KM*+1(k). Since a = 0 2 KMn+1(k(Va))=p,
there is unique element a02 Hn+1,n(Oa; Z=p) such that o a0= a.
ALGEBRAIC BP-THEORY 31
Let Ma be the object in DMeff-defined by the following distin-
guished triangle
ffia=Q0...Qn-1(a0)
(10.2) M(Oa(bn))[2bn] ! Ma ! M(Oa) ! M(Oa)(bn)[2bn + 1]
where bn = (pn - 1)=(p - 1) = pn-1 + ... + p + 1 so that deg(ffia) =
(2bn + 1, bn). For i < p, define the symmetric powers
Mia= Si(Ma) = qi(Mai) Mai
P
where qi(a) = (1=i!) oe2Sioe(a) for a 2 Mai, and Si is the symmetric
group of i letters. One of the important results in [Vo4] Voevodsky
proved is that Mp-1ais a direct summand of a motive of Va (for details
see [Vo4]). Hence there are distinguished triangles ( (5.5),(5.6) in [Vo4])
si i-1
(10.3) Mi-1a(bn)[2bn] ! Mia! M(Oa) ! Ma (bn)[2bn + 1]
ri
(10.4) M(Oa)(bni)[2bni] ! Mia! Mi-1a! M(Oa)(bni)[2bni + 1].
Then we have the diagram
0
H*,*(Oa; Z=p)
?
r*p-1?y
0 s*p-1 ","0 p-2 ]-1,]0 p-1
H],](Oa; Z=p) - - - H (Ma ; Z=p)- - - H (Ma ; Z=p)
?
?
y
0 p-1
H","(Ma ; Z=p)
where
(], ]0) = (* + 2(pn + bn), *0+ pn + bn - 1) = (* + 2bn+1, *0+ bn+1 - 1),
(", "0) = (* + 2pn - 1, *0+ pn - 1).
and the vertical and horizontal arrows are exact. From the result of
Voevodsky, we know (Appendix in [S])
0
Lemma 10.1. ([Vo4]) For x 2 H*,*(Oa; Z=p), we have
s*p-1r*p-1(x) = ~Q0Q1...Qn(a0) [ x ~ 6= 0 2 Z=p.
Corollary 10.2. The following map
0 ],]0
Q0...Qn(a0) [ - : H*,*(Oa; Z=p) ! H (Oa; Z=p)
is surjective (resp. isomorphic) if the difference * - *0 0 i.e., ] - ]0
bn+1 - 1 (resp. * - *0> 0 i.e., ] - ]0> bn+1 - 1).
32 N.YAGITA
Proof. Let the difference * - *0 0. Since Mp-1 is a direct summand
of the motive of Va, we see
0 p-1 ","0 p-1
H],](Ma ; Z=p) = 0, H (Ma ; Z=p) = 0
since their difference is larger than pn - 1 = dim(Va). Hence we know
the subjectivity of s*p-1r*p-1. When the difference * - *0 > 0, we get
moreover
0 p-1 "-1,"0 p-1
H]-1,](Ma ; Z=p) = 0, H (Ma ; Z=p) = 0,
__
by the same reasons. Thus we see the injectivity. |__|
Denote by k(Qa) the function field of Qa and by (Qa)0 the set of
closed points of Qa. The main theorem of the paper ([Or-Vi-Vo]) by
Orlov,Vishik and Voevodsky is the following (for p = 2).
Theorem 10.3. ([Or-Vi-Vo]) For any a = {a0, ..., an} 2 KM*(k)=p,
the following sequence is exact
Trk(x)=kM a M M
qx2(Va)0KM*(k(x))=p - ! K* (k)=p ! K*+n+1 (k)=p ! K*+n+1 (k(Va))=p.
Outline of proof.(See for the case * = 1, A.1 in [Su-Jo]) This is just
odd primes p version of the arguments of the proof by Orlov, Vishik and
Voevodsky. From arguments by Voevodsky ([Vo4],the main theorem
in Appendix in [Su-Jo]), we see the exact sequence
Trk(x)=kM ffia*+2b +1,*+b
(10.5) qx2(Va)0KM*(k(x))=p - ! K* (k)=p ! H n n(O; Z=p).
The last map ffia is epic by the following reason. Consider the compo-
sition
Qn 2pbn+2,pbn
KM*(k)=p ffia!H*+2bn+1,*+bn(O; Z=p) ! H (Oa; Z=p).
Since Qnffia = Qn...Q0(a0), we see that Qnffia is epic from the above
0
lemma. Since H*,*(O"a; Z=p) is (Qn)-free we see that Qn above is
injective. Thus we show that the map ffia in the above sequence is epic.
We also know that the map
(10.6) KM*(k)=p ffia!H*+2bn+1,*+bn(O; Z=p)
(Qn-1...Q0)-1*+n+1,*+n xo M
-! H (Oa; Z=p) ! K*+n+1 (k)=p
is the multiplication with a because Va is a splitting variety of a. Thus_
we get the exact sequence. |__|
Corollary 10.4. (Theorem 2.10 in [Or-Vi-Vo]) For each 0 6= h 2
KMn(k)=p, there is a field E=k and a nonzero pure symbol a 2 KMn(k)=p
such that h|E = a|E in KMn(E).
ALGEBRAIC BP-THEORY 33
Proof. Let h = b1+ ... + bl and each bi a pure symbol for 1 i l. Let
Vbi be the norm varieties and Ei = k(Vb1x ... x Vbi). Then of course
h|El = 0. Take i such that h|Ei-1 6= 0 but h|Ei = 0. Then from the
above theorem,
Ker(KMn(Ei-1)=p ! KMn(Ei)=p)) = biKM0(Ei-1)=p.
__
Hence h|Ei-1 = ~bi|Ei for ~ 6= 0 2 Z=p. |__|
Theorem 10.5. (For p = 2, [Ya4]) Let 0 6= a = (a0, ..., an) 2 KMn+1(k)=p.
Then there is a KM*(k) Q(n)-modules isomorphism
0 M 0
H*,*(O"a; Z=p) ~=K* (k)=(Ker(a)) Q(n) Z=p[,a]{a }
where ,a = QnQn-1....Q0(a0) and deg(a0) = (n + 1, n).
Proof. Recall the difference d(x) = f.deg(x) - s.deg(x). Hence if 0 6=
0
x 2 H*,*(O"a; Z=2), then d(x) > 0. We prove the theorem by induction
0
on d(t) for a Q(n)-module generator t in H*,*(O"a; Z=p). From (10.6)
we already know that
(Qn-1...Q0)-1*+n+1,*+n
KM*(k)=p ffia!H*+2bn+1,*+bn+1(Oa; Z=p) - ! H (Oa; Z=p)
is an epimorphism, indeed, the map ffia is epic0from Corollary 10.2
and the map Qn-1...Q0 is isomorphic since H*,*(O"a; Z=2) is Q(n)-
free. The composition of the above map with H*+n+1,*+n(O"a; Z=p) !
KM*+n+1(k)=2 is multiplying a from (10.6). Since the last map is monic
from (10.1), we see that
H*+n+1,*+n(O"a; Z=p) ~=KM*(k)=(Ker(a)){a} KM*+n+1(k).
Hence we get the case d(t) = 1.
Suppose that the isomorphism in the theorem holds for degree d(x) <
0
d. Let t 2 H*,*(O"a; Z=p) be a smallest weight element such that it is
a Q(n)-module generator with d(t) = d. Then we see
d(Q0...Qnt) = pn + pn-1 + ... + 1 + d > bn+1 + 1.
0
From Corollary 10.2, there is an element y 2 H*,*(O"a; Z=p) such that
Q0...Qn(t) = s*p-1r*p-1(y) = ,a [ y.
Since Qi is a derivation (with some modification for p = 2).
,a [ Qi(y) = Q0...Qn(a0) [ Qi(y)
= Qi(Q0...Qn(a0) [ y) = Qi(Q0...Qn(t)) = 0
for i n. Since the map multiplying ,a is injective (indeed, isomor-
0
phic) for H*,*(O"a; Z=p) from Corollary 10.2, we see
Qi(y) = 0 for all 0 i n.
34 N.YAGITA
0
The fact that H*,*(O"a; Z=p) is Q(n)-free implies that
0
y = QnQn-1...Q0(y0) for some y0 2 H*,*(O"a; Z=p).
Since we have
Q0...Qn(t - ,ay0) = Q0...Qn(t) - ,aQ0...Qn(y0) = Q0...Qn(t) - ,ay = 0,
the element t - ,ay0 is not a Q(n)-module generator. Hence we can
take Q(n)-module generator ,ay0 instead of t.
Of course d(y0) = d(t) - d(,a) < d(t). By induction on d, we see that
0 M 0
H*,*(O"a; Z=p) is generated as a K* (k) Q(n) module by Z=p[,a]{a }.
The theorem follows from also the fact that the multiplying ,a is iso-_
morphic. |__|
Remark. For n = 1 case of the above theorem is known by A.Suslin.
0 *,*0 p-1
Lemma 10.6. If * < 4bn, then H*,*(Ma; Z=p) ~= H (Ma ; Z=p).
0 *,*0
Moreover if * < 2bn, then H*,*(Ma; Z=p) ~=H (Oa; Z=p).
0
Proof. Since H*,*(Oa; Z=p) ~= 0 for * < 0, we get this lemma from
(10.2),(10.4). For example, (10.4) induces the long exact sequence
0-bn *,*0 i *,*0 i-1
H*-2bni,* (Oa; Z=p) H (Ma; Z=p) H (Ma ; Z=p) ,
0 i *,*0 i-1
which induces the isomorphism H*,*(Ma; Z=p) ~=H (Ma ; Z=p) for __
the first degree * < 2bni. |__|
Let us consider the following triangular domain generated by bidegree
Di = {deg(x)|w(x) 0, f.deg(x) < 2bni, d(x) > bn(i - 1)}
and D = [p-1j=1Dj.
Lemma 10.7. Let us write K = KM*(k)=(Ker(a)). For bidegree (*, *0) 2
D defined above, we have the K-module (but not ring) isomorphism,
0 p-1 p-1 0
H*,*(Ma ; Z=p) ~=K[t]=(t ) Q(n - 1){a }
where deg(t) = (2bn, bn).
Proof. Consider the exact sequence induced from (10.3)
0 i-1 j1 *,*0 i j2 *,*0
H*,*(Ma (bn)[2bn]; Z=p) H (Ma; Z=p) H (Oa; Z=p) .
By induction we assume for (*, *0) 2 [i-1j=1Dj
0 i-1 i-1 0
H*,*(Ma ; Z=p) ~=K[t]=(t ) Q(n - 1){a }.
Then for (*, *0) 2 [ij=1Dj, we see
0 i-1 i-1 0
H*,*(Ma (bn)[2bn]; Z=p) ~=(K[t]=(t ) Q(n - 1){a }) {t}.
ALGEBRAIC BP-THEORY 35
In particular, both sides of the above are zero if (*, *0) 2 D1. Hence j2
is injective for this case.
Note |Qna0| = 2pn + n, and for * < 2pn + n we have the isomorphism
0 0 0 i
H*,*(Oa; Z=p) ~=K Q(n - 1){a } which is zero for (*, * ) 2 [j=2Dj.
__
Hence j2 is injective and j1 is surjective in [ij=1Dj. |__|
Question. Is the isomorphism in the above lemma is that of Q(n-1)-
modules ?
Corollary 10.8. Let ci = Q0...Q^i...Qn-1(a0). Then there is the addi-
tive isomorphism
CH*(Mp-1a)=p ~=Z=p{1} Z=p[t]=(tp-1){c0, ..., cp-1}.
Here we consider some easy cases such that KM*(k)=p = 0 for * >
n + 1. Then note KM*(k)=(Ker(a)) = Z=p.
Proposition 10.9. Let 0 6= a 2 Kn+1(k)=p and KM*(k)=p = 0 for
* > n + 1. Then there is the additive isomorphism
0 p-1 *,* p p-1 0
H*,*(Ma ; Z=p) ~=H (pt; Z=p)[t]=(t ) Z=p[t]=(t ) "Q(n - 1){a }
where Q"(n - 1) = Q(n - 1) - Z=p{Q0...Qn-1, Q1...Qn-1}.
0
Proof. For n + 1 < *, we know H*,*(pt.; Z.p) = 0. Hence for n + 1 <
* < |Qna0| = 2pn + n, there is the isomorphism
0 0
H*,*(Oa; Z=p) ~=Q(n - 1){a }.
0 n
In particular H*,*(Oa; Z=p) = 0 for n + 1 < * < 2p + n. By using
arguments similar to the proof of Lemma 10.7, we get the proof. (Here __
we identify t = Q1...Qn-1(a0).) |__|
Remark. As examples satisfying the assumption of the above propo-
sition, we can take the high dimensional local fields defined by Kato
and Parsin. Let k be a complete discrete valuation field with residue
field F . Then it is well known that
KMr(k)=p ~=KMr(F )=p KMr-1(F )=p.
Let k0 be a finite field, and let k1, ..., kn be the sequence of complete
discrete valuation fields such that the residue field of ki is ki-1 for each
1 i n. Then the field k = kn obtained in this way is called an
n-dimensional local field. (see [Ka]). Then
KMn(kn)=p ~=Z=p and KMm(kn)=p = 0 for m > n.
36 N.YAGITA
11. ABP 2*,*(X) for the norm varieties
0
Using the fact that H*,*(O"a; Z=p) is a Q(n)-free module, we compute
AHss
0 * *,*0
E(ABP )*,*,*2~=H*,*(O"a; BP ) =) ABP (O"a).
Lemma 11.1. The Er-term of the above AHss is computed for r
2pn, namely, for i n, the nonzero differentials are given by d2pi-1(x) =
vi Qi(x) and we have
E(ABP )*,*,*2pi~=BP *=Ii+1 K[,a] (Qi+1, ..., Qn){Q0...Qia0}
where K = KM*(k)=(Ker(a)).
Proof. For ease of notations, let us write Q(i, n) = (Qi, ..., Qn). By
induction, we assume the result for E(ABP )2pi-1, i 1. We note that
all elements in E(ABP )2pi-1are Q0-image and that
E(ABP )*,*,02pi-1~=K[,a] Q(i, n){Q0...Qi-1(a0)}
0 *,*0
H*,*(O"a) H (O"a; Z=p).
Moreover we can identify
00 *,*0 00 n+1
E*,*,*2pi-1 H (O"a; Z=p) Z=p[vi] for * > -2p + 2 = |vi+1|.
0
Let E(Ak(i))*,*,*rbe the AHss converging to Ak(i)*,*(O"a). The nat-
ural map ABP ! Ak(i) of spectra induces the map of AHss
00 *,*,*00 *,*,*00 *,*0 *00
j : E(ABP )*,*,*2pi-1! E(Ak(i))2pi-1~= E(Ak(i))2 ~= H (O"a; Z=p) k(i) .
Then we see that the map j is injective for *00> -2pi+1+ 2. The first
nonzero differential for E(Ak(i))r is d2pi-1(x) = vi Qi(x) in E(Ak(i))
and so in E(ABP ). In particular the image (d2pi-1E(ABP )*,*,*) is
equal to
BP *=Ii{vi} K[,a] Q(i + 1, n){QiQ0...Qi-1(a)}.
Since the sequence (p, v1, ..., vi) is a regular sequence in BP *, we_
have the result. |__|
0
The same fact holds for ABP *,*(-) theory with m n. In par-
ticular , we have for m = n
0 M +
Corollary 11.2. ABP *,*(O"a) ~=K* (k)=(Ker(a))[,a] .
0 * M +
Conjecture. ABP *,*(O"a) ~=BP =In+1 K* (k)=(Ker(a))[,a] .
ALGEBRAIC BP-THEORY 37
Lemma 11.3. Let E(Off)r (resp. E(O"ff)r, E(pt.)r) be AHss converging
0 *,*0 *,*0
to ABP *,*(Off) (resp. ABP (O"ff), ABP (pt.)). Then
E(Off)*,*,*r~=E(O"ff)*,*,*r E(pt.)*,*,*r.
Proof. We consider maps of AHss
E(O"ff)r E(Off)r E(pt.)r.
0
We have the additive decomposition H*,*(Off) ~=A B where A (resp.
0
B) is the d(x) > 0 (resp. d(x) 0) parts of H*,*(O"ff). Then we know
0 *,*0
A ~=H*,*(O"ff) and B ~=H (pt.).
By induction on r, suppose that E(Off)r ~= E(O"ff)r E(pt.)r and
E(O"ff)r (resp. E(pt.)r) is isomorphic to the d(x) > 0 (resp. d(x) 0)
parts of E(Off)r.
Then there is no nonzero differential for x 2 E(Off)r such that
dr(x) = y with d(x) 0 but d(y) > 0 because if d(x) 0 then x is in
the image from E(pt.)r, and so is y. Note that d(d2s+1) = s is positive.
Hence the differential is closed in both d(x) > 0 and d(x) 0 parts._
Therefore we have the decomposition of the spectral sequence. |__|
Now we consider the AHss
0,*00 *,*0 * *,*0
E*,*2 = H (Va : BP ) =) ABP (Va).
Here recall that (Cor 10.8)
0
Z(p){1, c0} Z=p{c1, ..., cn-1} H*,*(Va; Z(p)).
Lemma 11.4. The element ci is Ii-torsion in E2*,*,01.
Proof. From Lemma 11.1, the element ci = Q0...Q^i...Qn(a0) is Ii-
0
torsion in E2*,*,02pi(O"a) converging to ABP *,*(O"a). Considering the map
of spectral sequences
E(O"a)r E(Oa)r E(Va)r
__
and the above lemma, we see that ci in E1 is also Ii-torsion. |__|
We note the following lemma.
Lemma 11.5. We have the filtration such that
grBP *{c0, ...cn-1}=(vicj + vjci|i < j) ~= 0 i n-1BP *=Ii{ci}.
Proof. Let Fi = BP *{c0, ..., ci-1}=(vjck = vkcj|0 j < k i - 1).
Then we have Fn-1 .... F0 and
Fi=Fi-1 ~=BP *{ci}=(vj~ci= 0|j < i) = BP *=Ii{ci}.
__
This induces the isomorphism of graded rings. |__|
38 N.YAGITA
Lemma 11.6. Let ~ci2 ABP 2*,*(Va) be a lift of ci 2 grABP 2*,*(Va).
There is the relation
vi~cj- vj~ci= 0 mod(I21) for all i < j.
Proof. Since ci is a Ii-torsion in grABP *(Va), it is a vk-torsion for
k < i. From Corollary 3.7, there is z such that Qk(z) = ci. Let
pr : M(Va) ! M(Va) be the projector for Mpr-1 = pM(Va) and M0 =
(1 - pr)M(Va) the orthogonal summand for pr. This z is uniquely
0
written in H*,*(Va; Z=p) as
z = Q0...Q^k...Q^i...Qn-1(ff0) + b, b 2 M0
from Theorem 10.5 (by using w(z) = 1). Then
prQi(z) = Qi(Q0...Q^k...Q^i...Qn-1(ff0)) + pQi(b) = ck + pQi(b).
Here prQi(b) = Qi(pr(b)) from Lemma 9.1 and so prQi(b) = 0. More-
over prQj(z) = 0 for j 6= k, j 6= i. It follows from Lemma 3.6 that we__
get the relation vk~ci+ vi~ck= 0 mod(I2). |__|
The following lemma is immediate by vi 7! ci.
Lemma 11.7.
Ideal(In) ~=BP *{c0, ..., cn-1}=(vicj - vjci).
Let ~k be the algebraic closure of k and X|~k= X k ~k. Let i~k:
ABP 2*,*(X) ! ABP 2*,*(X|~k) be the induced map. Of course we have
the isomorphism of BP *-modules
(11.1) ABP 2*,*(Mp-1a|~k) ~=BP * CH*(Mp-1a|~k) ~=BP *[~t]=(~tp)
for deg(~t) = (2bn, bn).
Lemma 11.8. i~k(tjc0) = p~tj+1.
Proof. First we prove i~k(tp-2c0) = p~tp-1. Here tp-2~c0(resp. ~tp-1) gen-
erates Z(p) ABP 2d,d(Vn) ~= H2d,d(Vn; Z(p)) (resp. ABP 2d,d(Va|~k) ~=
Z(p)) where d = dim(Va) = pn - 1.
Since Vk has no k-rational points,
degH2d,d(Va : Z(p)) pH0,0(Spec(k); Z(p)) = pZ(p).
On the other hand, the fact tC(Va) = vn implies that
deg(r n (-TVa)) = deg(r n (-TVa|~k)) = p mod(p2).
ALGEBRAIC BP-THEORY 39
Hence we have degH2d,d(Va; Z(p)) = pZ(p), while degH2d,d(Va|~k; Z(p)) =
Z(p). Since the following diagram is commutative
deg 0,0
H2d,d(Va; Z(p)) -- - ! H (Spec(k); Z(p))
? ?
i~k?y ~=?y
deg 0,0
H2d,d(Va|~k; Z(p))--- ! H (Spec(~k); Z(p))
we see that i~k(tp-2c0) = p~tp-1.
Consider the commutative diagram
j1|~k 2*,* i 2*,*
ABP 2*-2bn,*-bn(Mi-1a|~k)-- - ABP (Ma|~k) - - - ABP (Oa|~k)
x x x
i~k?? i~k?? i~k??
j1 2*,* i 2*,*
ABP 2*-2bn,*-bn(Mi-1a) - - - ABP (Ma) - - - ABP (Oa)
When (*, *0) = (2bni, bni), we see that j1|~kand j1 are isomorphism
0 *,*0 i i-1
since ABP *,*(Oa) = ABP (Oa|~k) = 0. Moreover j1|~k(~t) = ~t and
j1(c0ti-1) = c0ti-2. By induction starting i~k(tp-2c0) = p~tp-1, we have_
the desired result i~k(ti-2c0) = p~ti-1, from the above diagram. |__|
Lemma 11.9. In ABP 2*,*(Va|~k), we see i~k(ci) = vi~tmod(I21).
Proof. By induction on i, we assume i~k(ck) = vk~tmod(I21), for k < i.
Since (vi~c0- p~ci) 2 I21{~c0, ..., ~ci-1}, we have
i~k(vi~c0- p~ci) = p(vi~t- i~k(~ci)) mod(I3{~t}).
Hence i~k(~ci) = vi~tmod(I2), since ~tgenerates a free BP *-module, that_
is integral domain, indeed, BP *is a polynomial algebra. |__|
Theorem 11.10. The cohomology ABP 2*,*(Va) contains sub BP *-module
isomorphic to In{~t}.
Proposition 11.11. Suppose that the isomorphism in Lemma 10.7
is that of Q(n - 1)-modules. Then the map i~k: ABP 2*,*(Mp-1a) !
ABP 2*,*(Mp-1a|~k) is injective and hence
ABP 2*,*(Mp-1a) ~=BP *{1} Z(p)[t]=(tp-2) In{c}
where pc = ~c0and deg(c) = (2bn, bn).
Theorem 11.12. Suppose that the projection of the motive
CH*(Va|~k) ! CH*(Mp-1a|~k) ~=Z[~t]=(~tp-1)
induces the map of rings. Then
Im(i~k: ABP 2*,*(Mp-1a)) ~=BP *{1} In[~t]=(~tp-2).
40 N.YAGITA
Proof. From the above corollary we see Im(i~k) In~t. By the commu-
tative diagram
p 2*,* p-1
ABP 2*,*(Va) --- ! ABP (Ma )
? ?
i~k?y i~k?y
p 2*,* p-1
ABP 2*,*(Va|~k) --- ! ABP (Ma |~k),
we see Im(p . i~k) In~t. Hence
Im(i~k. p) = Im(p . i~k) I2n~t2.
In particular v21~t22 i~kABP 2*,*(Mp-1a).
Suppose that v1~t262 i~kABP 2*,*(Mp-1a). Then v21~t2is a BP *-module
generator of i~kABP 2*,*(Mp-1a). Hence there is a nonzero element
c 2 2*,*(Mp-1a) * Z ~=CH*(Mp-1a)
such that |c| = |v21~t2| = 2bn - 4(p - 1). But such element does
not exist in CH*(Mp-1a) form Corollary 10.8. Thus we have v1~t22
i~kABP 2*,*(Mp-1a).
Similarly we see that
vi~tj2 i~kABP 2*,*(Mp-1a)
for all 0 i n - 1 and 1 j p - 1. Then we can prove the __
theorem. |__|
12. Smith and Toda spectrum V (n)
The Smith and Toda spectrum V (n) is defined as a (topological)
spectrum such that
H*(V (n); Z=p) ~=Q(n), BP *(V (n)) ~=P (n + 1)* = BP *=In.
It is known that such V (n) exists for n = 1 if p 3, n = 2 if p 5 and
n = 3 if p 7. Suppose that V (n) exists. Then V (n) is constructed
from V (n - 1) by the cofiber sequence
n-2 f
S2p V (n - 1) ! V (n - 1) ! V (n)
where f is a map so that f* = vn identifying BP *(V (n-1)) ~=BP *=In.
The Greek letter element is defined as the stable map
fs q 2bn-n
G(n)s : S-s|vn| S-s|vn|V (n - 1) ! V (n - 1) ! S
where q is the projection map to the biggest cell. Usually G(n)k is
written as
G(1)s = ffs, G(2)s = fis, G(3)s = fls
ALGEBRAIC BP-THEORY 41
and called the Greek letter elements. It is known that if V (n) exists
then the stable homotopy group of sphere
sssi= limN!1 [SN+i , S]
is multiplicative generated by the Greek letter elements when i <
|G(n + 1)1|.
We note here there exists analogous spaces in the stable A1-homotopy
category. First we consider the space defined by the cone
Ma|~k! Ma ! cone = MV (n - 1).
Lemma 12.1. Let p = 2 or Suppose the assumption in Proposition
11.11. Then ABP 2*,*(MV (n - 1)) ~=P (n)*[t]+ =(tp-1).
0
Note that there are nonzero elements ABP *,*(MV (n - 1)) when
* 6= 2*0.
Another candidate for V (n) is the reduced C~eck spaces.
Lemma 12.2. Let k be a field such that KM*(k)=p = 0 for * > n + 1.
0 * +
Then ABP *,*(O"a) ~=P (n + 1) [,a] .
13. The case Mp-1aexists as a variety
In general, we can not identify the motive Mp-1aas an object in the
stable homotopy category and so we can not consider its generalized
cohomology theories. However we consider (hereafter of this section)
the cases that
(Assumption) There is a space Ua and a map Ua ! Va such that
this map induces the isomorphism
0 *,*0 p-1
H*,*(Ua; Z=p) ~=H (Ma ; Z=p).
Note when p = 2, this assumption holds (Theorem [Vi-Ya]).
Define a space Wa by the cofibering
p i @
Ua ! Oa ! cone = Wa ! Ua(1).
Remark. When p = 2, we have the isomorphism of motives M(Wa) =
M(Oa)(2n - 1)[2n+1 - 1]. However the A1-homotopy type of Wa is
n+1-1,2n-1
not that of S2 Oa. In fact, their (Qn)-module structures are
different.
We consider some more easy theory
0 *,*0 *
A~h*,*(X) = ABP (X), ~h = Z(p)[v1, ...vn-1]
and give a short another proof of the A~h*,*-version of Theorem 11.10.
By the arguments similar to the proof of Lemma 11.1 and Corollary
11.2, we get
42 N.YAGITA
0 M
Lemma 13.1. A~h*,*(O"a) ~=K* (k)=(Ker(a))[,a]{ffia, Qnffia} where ffia =
Q0...Qn-1a0.
In particular, the above BP *-module is indeed a BP *=(In) = Z=p module.
0,*00
Lemma 13.2. For * 2bn, we see grA~h*,* (Wa) = 0 and
00 * 0 * 0
grA~h2bn+1,bn,*(Wa) ~=BP {ffi } with i (ffi ) = ffia.
Proof. Consider the exact sequence
0 p* *,*0 i* *,*0
H*,*(Ua; Z=p) H (Oa; Z=p) H (Wa; Z=p) .
0 *,*0
Since H*,*(Ua; Z=p) ~= H (Oa; Z=p) for * 2bn from Lemma 6.6,
0
we have H*,*(Wa; Z=p) = 0 for these degrees. Moreover by the exact
sequence
0 H2bn+1,bn(Oa; Z(p)) = Z=p H2bn+1,bn(Wa; Z(p))
H2bn,bn(Ua; Z(p)) = Z(p) 0,
we see H2bn+1,bn(Wa; Z(p)) ~=Z(p)and write its generator by ffi0a.
Consider AHss for A~h*,*(Wa). Since w(ffia) = -1, we see p*(ffia) =
0 0 *,* * 0
0 2 A~h*,*(Ua). Hence there is ffi 2 A~h (Wa) with i (ffi ) = ffia. i.e.,
ffi0 is represented by ffi0ain AHss. By dimensional reason, there is no
nonzero differential which targets in
BP *{ffi0} BP * H2bn+1,bn(Wa; Z(p)) ~=E2bn+1,bn,*2.
__
Thus we have the lemma. |__|
We consider the map
0 i* *,* * 0 0
A~h*,*(Wa) ! A~h (Oa), i (ffi ) = ffia = Q0...Qn-1(a ).
n+1-1,2n-1 *
Here ffi0 2 A~h2 (Oa) is a free BP -module but ffia is a
In-torsion module. Thus we know
Ker(i*)|BP *{ffi0} ~=In{ffi0}
0
which must be contained in A~h*,*(Ua).
Proposition 13.3.
ABP 2*,*(Ua) In{ff} deg(ff) = (2bn, bn), @*(viff) = viffi0.
The element viff corresponds ~ciin0Lemma 11.6.
Remark. For the case ABP *,*(-), the situation is quite different.
The arguments for n - 1 does not work for this case, e.g., Qnffia 6= 0 2
ALGEBRAIC BP-THEORY 43
0 *,*0
H*,*(Oa; Z=p) and hence ffia does not exist in ABP (Oa). Consider
AHss
0 * *,*0
E2 = H*,*(Wa; BP ) =) ABP (Wa).
In E*,*,*2pn-1, the differential is given by d2pn-1(ffi0) = vnQnffi0. Here Qnf*
*fi0
generates In- torsion module but ffi0 generates a free BP *-module.
Thus we have
E2bn+1,bn,*2pn In{ffi0}.
0
Elements viffi0 represent corresponding ci in ABP *,*(Ua).
14. Real case
In the last section of this paper, we0restricted0the case k = R and p =
2. The mod 2 motivic cohomology is H*,* = H*,*(pt; Z=2) ~=Z=2[ae, o ]
0
with deg(ae) = (1, 1), deg(o ) = (0, 1). We want to study ABP *,*(Ua)
for all bidegree. Recall that
0 *,*0 0
H*,*(Oa; Z=2) ~=H (pt.; Z=2) Z=2[ae, ,a] Q(n){a }
here note ,a = ffi2a. We still know from [Ya]
0 *,*0 n+1
H*,*(Ua; Z=2) ~=H (Oa; Z=2)=(f.deg > 2 - 2).
We write down it more explicitly.
Theorem 14.1. ([Ya4]) Let k = R and a = aen+1. Then
0 2n+1-1 ffl0 k(ffl)f*
*fl0
H*,*(Ua; Z=2) ~=Z=2[o, ae]=(ae ) ffl6=(1,..,1)Z=[ae]{Q (a )}=(ae Q (a*
* ))
P n-1
where k(ffl) = (2n+1 - 1) - f.deg(Qffl(a0)) = 2n+1 - i=0(ffli(2i+1- 1)) -
n - 2.
Now we recall some facts for cohomology operations.
k *,*0
Lemma 14.2. ([Ya4]) Let tk = o 2 and let grH = Z=2[ae] (t0, t1, ...).
0 *,*0
Then Qn acts on H*,* as a derivation on grH with
n+1-1
Qn(ae) = 0, Qn(tn) = ae2 , Qn(tj) = 0 for n 6= j,
n+1-1 k
namely, Qn( teiiaek) = ( n6=iteii)(enae2 )ae for ei = 0 or 1.
Theorem 14.3. ([Ya4]) Let us write (^n) = (t0, ..., ^tn, ...). Then we
have
n+1-1
grAK(n)*(pt.) ~=K(n)*[ae]=(ae2 ) (^n)
44 N.YAGITA
Proof. From Lemma 3.4, the first nonzero differential is d2n+1-1(x) =
n+1-1 *,*0
vn Qn(x). The fact that Qntn = ae2 implies that E2n+1 is iso-
0,*00
morphic to the righthand module in the theorem. Since E*,*r ~= 0
0
if * 2n+1 - 1, we see that dr = 0 for all r 2n+1. Thus E*,*2n+1is_
isomorphic to the infinitive term of AHss. |__|
Theorem 14.4. For 1 i n - 1, we have the isomorphism
0 * 2i+1-1
AK(i)*,*(Ua) ~=K(i) [ae]=(ae ) (A B C)
n+1-2i+1 k(ffl)-2i+1+1 0
where A = (^i), B = (^i){ae2 ti}, and C = ffli=0Z=2{ae Qffl(a*
* )}.
Proof. Consider the motivic AHss
0 * *
E*,*,*2= H*,*(Ua; K(i) ) =) AK(i) (Ua).
First consider elements of the difference 0. The first nonzero
i-1
differential is d2i+1-1(x) = vi Qi(x). From the fact Qi(tj) = ffii,jae2
and from the above lemma, we have
i+1-1
H(E*,*,*2\ (difference 0); Qi) ~=(A B) Z=2[ae]=(ae2 ).
Since A K(i)*(Spec(R)), elements in A are permanent cycles. By
the dimensional reason such as f.deg > 2n+1 - 2i+1, elements in B are
also permanent cycles.
Next consider elements of difference > 0. We can see that
(
ffli = 1 or
QiQffl(aeka0) = 0 if P
k + 2i+1 + iffli(2i+1 - 1) + n 2n+1 - 1.
Of course, ffli = 1 case, Qffl(a0) is in the Im(Qi), and hence we have
i+1-1
H(E*,*,*2\ (difference > 0); Qi) ~=C Z=2[ae]=(ae2 ).
__
By also dimensional reason, all elements in C are permanent cycles. |__|
It seems not so easy to compute ABP *(pt), ABP *(Ua). Hence we
consider the more easy case
0 *,*0 *,*0
Ah*,*(-) = A~h=p (-) = ABP =(p) (-),
namely, throughout this section, let h* = Z=p[v1, ..., vn-1]. Let us write
for i j
ji= (ti, ..., tj), Vij= (t0, ..., ti, tj, ...)
Theorem 14.5. ([Ya4]) There is an isomorphism
0 n * n-1 + 3 2n-1
grAh*,*(pt.) ~=V0 h [ae]{1, vix|x 2 ( i+1) }=(v1ae , ..., vn-1ae ).
ALGEBRAIC BP-THEORY 45
Proof. Consider AHss
0 * *,*0
E(Ah)*,*,*2= H*,*(pt; h ) =) Ah (pt).
By induction on n, we assume
i+1-1
(*) E*,*,*2i+1= V0n (Ai (h*[ae] n-1i+1))=(v1ae3, ..., viae2 )
0,minus i+1
where Ai is an h*-module with generators in E*,*2i+1 , 0 * < 2 -
1. The next nonzero differential is
0,0
d2i+2-1(x) = vi+1Qi+1(x) mod(Ii+1) for x 2 E*,*2i+2-1.
In particular
i+2-1 n-1
d2i+2-1(ati+1) = vi+1ae2 a for a 2 Z=2[ae] i+2.
Moreover we can prove d2i+2-1(a) = 0 for a 2 Ai considering the map
of spectral sequences induced from Ah ! ABP *=(2)
where BP **=(2)* = Z=2[vi+2, ..., vn-1]. (For details, see
[Ya4].) Thus we can prove the (i + 1)-version of (*).
Moreover we see that Ai+1 is isomorphic to
n-1
Ai BP *[ae]{v1ti+1, ..., viti+1} n-1i+1=(v1ae3, ..., vn-1ae2 )
because d2i+2-1(vjti+1) = 0 for all j < i + 1. Thus we get
n-1
A1 ~=BP *[ae]{vix|x 2 ( n-1i+1)+ }=(v1ae3, ..., vn-1ae2 ).
__
|__|
Theorem 14.6. There is the isomorphism
0 n
grAh*,*(Ua) ~=(A B) V0 C with
n-1 2n+1-1
(1) A = h*[ae]{1, vix|x 2 ( n-1i+1)+ }=(v1ae3, ..., vn-1ae2 , ae ),
n+1-2i+1 2i+1-1 n-1
(2) B = n-1i=0h*=Ii[ae]{ae2 ti}=(ae ) i+1,
i+1+1 0 *
*2i+1-1
(3) C = n-1i=0(h*=Ii[ae]{ ffl0=1,...ffli-1=1,ffli=0aek(ffl)-2 Qffl(a )}=(ae*
* )).
__
Proof. The proof is similar to the proof of Theorems 6.7 and 6.8. |__|
0 *,*0
Next we also compute AK(n)*,*(Ua) and Ah (Ua) by using the
cofiber sequence
Ua -! Oa -i! Wa
given in Section 13. The cohomology of Wa is easily computed from
the long exact sequence induced from the above cofibering and the
cohomologies of Ma and Oa. Recall again that
0 *,*0 0
H*,*(Oa; Z=2) ~=H (pt.; Z=2) Z=2[ae, ,a] Q(n){a }
46 N.YAGITA
0 *,*0 n+1
H*,*(Ua; Z=2) ~=H (Oa; Z=2)=(f.deg > 2 - 2)
here note ,a = ffi2a. Hence we have
0 n+1 *,*0
H*,*(Wa; Z=2) ~={(f.deg > 2 - 2) parts of H (Oa; Z=2)}
which is isomorphic to the ideal Ideal(Qn(a0), ffia) of Q(n - 1)-algebra
0
H*,*(Oa; Z=2). So it is written explicitly
Z=2[ae] (Z=2[o ]{ffia} Q(n - 1){Qn(a0)} Z=2[ffi2a] Q(n){ffi2aa0}).
Next we consider just Q(n - 1)-module structures ignoring Qn-actions.
Note deg(ffia) = deg(Qn) and let us write Qn also by ffia, then we have
the isomorphism of Q(n - 1)-modules
0 *,*0 0
H*,*(Wa; Z=2) ~=(H (pt.; Z=2) Z=2[ae, ffia] Q(n - 1){a }){ffia}
~=H*,*0(Oa; Z=2){ffia}.
0 *,*0
Corollary 14.7. There is the isomorphism H*,*(Wa; Z=2) ~=H (Oa; Z=2)
as Q(n - 1)-modules but not as Q(n)-modules.
0
Proof. Note Qnffia = QnQn-1...Q0(a0) 6= 0 in H*,*(Wa; Z=2) but of
0 __
course Qn(1) = 0 in H*,*(Oa; Z=2). |__|
0
From Theorem 10.5, the cohomology H*,*(O"a; Z=2) is Q(n)-free.
Hence
0
H(H*,*(O"a; Z=2), Qi) ~=0 for 0 < i n.
0
By the AHss converging to AK(i)*,*(O"a), we have
0
Lemma 14.8. For 1 i n, we see AK(i)*,*(O"a) ~=0.
Corollary 14.9. For 1 i n - 1, we have
0 *,*0 *,*0 *,*0
AK(i)*,*(Oa) ~=AK(i) (pt.), AK(i) (Wa) ~=AK(i) (pt.){ffia}.
Let us write
0 *-2n+1+2,*-2n+1
AK(i)[,[(X) = AK(i) (X).
Then we can write ;
Theorem 14.10. For 0 < i n - 1, we have ;
0 *,*0 [,[0
grAK(i)*,*(Ua) ~=AK(i) (pt.) AK(i) (pt.).
Proof. We consider the exact sequence
0 *,*0 i* *.*0 ffia [-1,[0
AK(i)*,*(Ua) AK(i) (Oa) AK(i) (Wa) ~=AK(i) (Oa)
Since ffia(1) = Q0...Qn-1(a0) 2 Im(Qi) in H*(Oa; Z=2), we have i*ffia(1) =
0 __
0 in AK(i)*,*(Oa) by AHss. |__|
ALGEBRAIC BP-THEORY 47
Remark. The corresponding elements in Theorem 14.4 and Theorem
14.10 are given by the following
i+1-1 *,*0
K(i)*[ae]=(ae2 ) A ~=AK(i) (pt.)
0
K(i)* B ~=K(i)* (^i){tn, tn+1, ...} AK(i)[,[(pt.)
n+1-2i+1 2
by ae2 ti $ vitn (note tn = tn+1)
0
K(i)* C ~= K(i)* (t0, ..., ^ti, ..., tn-1) AK(i)[,[(pt.)
i+1+1 0 c(ffl) c(ffl)
by aek(ffl)-2 Qffl(a ) $ vit , with t = i6=jtj(1 - fflj).
The above corresponding is given by checking both first degree and
difference degree, by using d(Qj) = 2j = -d(tj) = -d(vj) - 1.
We also write Q1...Qn-1(a0) = ffi00a so that Q0ffi00a= ffia.
Lemma 14.11. Writing Qn also by ffia (see [Po2]), we have
0 2 00 00
Ah*,*(O"a) ~=Z=2[ae] (Q0, Qn)[ffia]{ffia} ~=Z=2[ae] (Q0)[ffia]{ffia}.
0 *,*0
Proof. Consider AHss for Ah*,*(O"a). Since H (O"a; Z=2) is Q(n -
1)-free, we can see that the right hand side module is isomorphic to
E*,*,*2i. It is In-torsion and hence E*,*,*1by dimensional reason. We note
0 __
Ah*,*(O"a) is In-torsion by also dimensional reason. |__|
0 *,*0 *,*0
Lemma 14.12. grAh*,*(Oa) ~=grAh (pt) grAh (O"a).
Proof. As Q(n - 1)-modules, we have the decomposition
0 *,*0 *,*0
H*,*(Oa; Z=2) ~=H (pt.; Z=2) H (O"a; Z=2).
We also get the decompositions0of the Er-term of AHss by induction __
on r, and hence of grA*,*(Oa). |__|
Since H*(Oa; Z=2) and H*(Wa; Z=2) are isomorphic as Q(n-1)-modules,
we can prove that
0 *,*0
Lemma 14.13. grAh*,*(Wa) ~=grAh (Oa){ffia}.
Recall Theorem 14.5. If * 2n+1 - 1 = f.deg(ffia), then the fact that
0
viae*x = 0 in Ah*,*(pt.) implies
0 n n+1
Ah*,*(pt) ~=Z=2[ae] V0 for * 2 - 1.
Let us write
0 *,*0 n
KAh*,*(pt) = Ker(Ah (pt) ! Z=2[ae] V0 )
~=h*[ae](In{1} Z=2{vix|x 2 ( n-1i+1)+ }=(viae2i+1-1|1 i n - 1) V0n
~=h*[ae]{vix|x 2 n-1i+1}=(viae2i+1-1|1 i n - 1) V0n.
48 N.YAGITA
Theorem 14.14.
0 *,*0 2n+1-1 00 [,[0
Ah*,*(Ua) ~=Ah (pt.)=(ae ) Z=2{ffia} KAh .
Proof. We consider the exact sequence
0 *,*0 i* *,*0
Ah*,*(Ma) Ah (Oa) Ah (Wa) .
n+1-1 00
Using the facts that i*(ffia) = ffia(1) = t-1nae2 and aeffia = o ffia, we *
*see
0 *,*0 2n+1-1 00
Ah*,*(Oa)=(ffia) ~=Ah (pt.)=(ae ) Z=2{ffi }.
0 *,*0
The map xffia : Ah[,[(O"a) ! Ah (O"a) is injective from Lemma 14.11.
The kernel Ker(i*) is isomorphic to the kernel
n+1-1 [,[0 *,*0 n
ae2 : Ah (pt.) ! Ah (pt.) ~=Z=2[ae] V0
__
for * 2n+1-1. Thus from Lemma 9.11, we can prove the theorem. |__|
0
Remark. The elements vi 2 In{1} KAh[,[ for i 1 correspond
ci in Section 10 and ffi00acorresponds c0. By isomorphisms given for
AK(i)*(Ua), we also know the correspondence with Theorem 14.6 ;
0 2n+1-1
A V0n~= Ah*,*(pt.)=(ae ),
n+1-2i+1
B V0n~= gr(In{1}) V0n by ae2 ti $ vitn,
i+1-1 n
C ~= h*[ae]{vix|x 2 ( n-1i+1)+ }=(viae2 ) V0
i+1+1 0 c(ffl) c(ffl)
by aek(ffl)-2 Qffl(a ) $ vit , with t = i6=jtj(1 - fflj).
j+1-1
In fact, we have the following relations. Since Qjtitj = ae2 ti and
i+1-1
Qititj = ae2 tj, we see
n+1-2i+1 2n+1-2j+1 2 *,*0
vjae2 ti = viae tj mod(I1 ) in Ah (Ua).
When ffl0 = 1, ...ffli-1 = 1, ffli = 0, we consider operations Qffl= QjQffl- j
and QiQffl- jfor j < i. Then we have the relation
i+1+1 0 k(ffl+ - )-2j+1+1 0 2
vjaek(ffl)-2 Qffl(a ) = viae i j Qffl+ i- j(a ) mod(I1 ),
since k(ffl) - 2i+1 = k(ffl + i- j) - 2j+1.
ALGEBRAIC BP-THEORY 49
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Department of Mathematics, Faculty of Education, Ibaraki Univer-
sity, Mito, Ibaraki, Japan
E-mail address: yagita@mx.ibaraki.ac.jp
*