# Coniveau filtrations and Milnor operations $Q_{n}$

Nobuaki Yagita  [N. Yagita] Department of Mathematics, Faculty of Education,
Ibaraki University, Mito, Ibaraki, Japan<EMAIL_ADDRESS>
###### Abstract.

Let $BG$ be the classifying space of an algebraic group $G$ over the field
${\mathbb{C}}$ of complex numbers. We compute a new stable rational invariant
defined by the difference of two coniveau filtrations ([Be-Ot]) of a
(projective) approximation of $BG\times{\mathbb{P}}^{\infty}$.

###### Key words and phrases:

coneveau filtration, Milnor operation, classifying spaces

###### 2010 Mathematics Subject Classification:

Primary 20G10, 55R35, 14C15 ; Seconary 57T25

##  1\. Introduction

Let $p$ be a prime number and $A={\mathbb{Q}},{\mathbb{Z}},{\mathbb{Z}}_{(p)}$
or ${\mathbb{Z}}/p^{i}$ for $i\geq 1$. Let $X$ be a smooth algebraic variety
over $k={\mathbb{C}}$. Let us recall the coniveau filtration of the (Betti)
cohomology with coefficients in $A$,

$N^{c}H^{i}(X;A)=\sum_{Z\subset X}ker(j^{*}:H^{i}(X;A)\to H^{i}(X-Z,A))$

where $Z\subset X$ runs through the closed subvarieties of codimension at
least $c$ of $X$, and $j:X-Z\subset X$ is the complementary open immersion.

Similarly, we can define the $strong$ coniveau filtration by

$\tilde{N}^{c}H^{i}(X;A)=\sum_{f:Y\to X}im(f_{*}:H^{i-2r}(Y;A)\to H^{i}(X,A))$

where the sum is over all proper morphism $f:Y\to X$ from a smooth complex
variety $Y$ of $dim(Y)=dim(X)-r$ with $r\geq c$, and $f_{*}$ its transfer
(Gysin map). It is immediate $\tilde{N}^{c}H^{*}(X;A)\subset N^{c}H^{*}(X;A)$.

It was hoped that the strong coniveau filtration was just the coniveau
filtration, i.e., $\tilde{N}^{c}H^{i}(X;A)=N^{c}H^{i}(X;A)$. In fact Deligne
shows they are the same for $A={\mathbb{Q}}$. However, Benoist and Ottem ([Be-
Ot]) show that they are not equal for $A={\mathbb{Z}}$.

Let $G$ be an algebraic group such that $H^{*}(BG;{\mathbb{Z}})$ has
$p$-torsion for the classifying space $BG$. The purpose of this paper is to
compute the stable birational invariant of $X$ (Proposition 2.4 in [Be-Ot])

$DH^{*}(X;A)=N^{1}H^{*}(X;A)/(p,\tilde{N}^{1}H^{*}(X;A))$

for a projective approximation $X$ for $BG$ ([Ek]). That is the projective
variety $X=X(N)$ such that there is a map $g:X\to
BG\times{\mathbb{P}}^{\infty}$ with

$g^{*}:H^{*}(BG\times{\mathbb{P}}^{\infty};A)\cong H^{*}(X;A)\quad for\ *<N.$

(In this paper, we say $X$ is an approximation $for$ $BG$ when it is that $of$
$BG\times{\mathbb{P}}^{\infty}$ strictly.)

Here we give an example we can compute a nonzero $DH^{*}(X)$. For
$G=({\mathbb{Z}}/p)^{3}$ the elementary abelian $p$-group of $rank=3$, we know
(for $p$ odd)

$H^{*}(BG;{\mathbb{Z}}/p)\cong{\mathbb{Z}}/p[y_{1},y_{2},y_{3}]\otimes\Lambda(x_{1},x_{2},x_{3}),\quad|x_{i}|=1,\
Q_{0}(x_{i})=y_{i}$
$H^{*}(BG;{\mathbb{Z}})/p\cong{\mathbb{Z}}/p[y_{1},y_{2},y_{3}](1,Q_{0}(x_{i}x_{j}),Q_{0}(x_{1}x_{2}x_{3})|1\leq
i<j\leq 3)$

where $Q_{0}=\beta$ is the Bockstein operation, and the notation $R(a,...,b)$
(resp. $R\\{a,...,b\\}$) means the $R$-submodule (resp. free $R$-module)
generated by $a,...,b$.

######  Theorem 1.1.

Let $G=({\mathbb{Z}}/p)^{3}$. Given $N>2p+3$, there are approximations $X$ for
$BG$ such that we have for $\alpha_{ij}=Q_{0}(x_{i}x_{j}),$
$\alpha=Q_{0}(x_{1}x_{2}x_{3})$

$DH^{*}(X;{\mathbb{Z}})\cong{\mathbb{Z}}/p\\{\alpha_{ij},\alpha|1\leq i<j\leq
3\\}\quad when\ *<N.$

But we have $DH^{*}(X;{\mathbb{Z}}/p)=0$.

Benoist and Ottem give many examples, where the above invariants are nonzero
for $A={\mathbb{Z}}_{(2)}$ by using the Steenrod squares. We note that some of
their arguments can be extended for $A={\mathbb{Z}}_{(p)}$ by using Milnor
operation $Q_{1}$ for odd primes $p$. However it seems not so easy to give a
nontrivial example for $A={\mathbb{Z}}/p$ in the case $X$ is an approximation
for $BG$ as the above example.

For connected groups. we have

######  Theorem 1.2.

Let $G$ be a simply connected group such that $H^{*}(BG;{\mathbb{Z}})$ has
$p$-torsion. Then there is an approximation $X$ for $BG$ such that
$DH^{4}(X;{\mathbb{Z}})\not=0.$

######  Theorem 1.3.

Let $p=2$ and $BG=BSO_{n},\ n\geq 3$, or, $p:odd$ and $G=PGL_{p}$. Then there
is an approximation $X$ for $BG$ such that $DH^{3}(X;{\mathbb{Z}})\not=0$.

##  2\. transfer and $Q_{n}$

The Milnor operation (in $H^{*}(-;{\mathbb{Z}}/p)$) is defined by
$Q_{0}=\beta$ and for $n\geq 1$

$Q_{n}=P^{\Delta_{n}}\beta-\beta
P^{\Delta_{n}},\quad\Delta_{n}=(0,..,0,\stackrel{{\scriptstyle
n}}{{1}},0,...)$

where $\beta$ is the Bockstein operation and $P^{\alpha}$ for
$\alpha=(\alpha_{1},\alpha_{2},...)$ is the fundamental base of the the module
of sums of products of reduced powers.

######  Lemma 2.1.

Let $f_{*}$ be the transfer (Gysin) map for $f:Y\to X$. Then $Q_{n}$ and
$f_{*}$ are commutative, i.e. $Q_{n}f_{*}=f_{*}Q_{n}$.

The above lemma is known (see the proof Lemma 7.1 in [Ya2]). The transfer
$f_{*}$ is expressed as $g^{*}f_{*}^{\prime}$ such that

$f_{*}^{\prime}(x)=i^{*}(Th(1)\cdot x)$

for some maps $g,f^{\prime},i$ and the Thom class $Th(1)$. Since
$Q_{n}(Th(1))=0$, we can get the lemma. However, we give here the other
computational proof.

###### Proof of Lemma 2.1..

It is well known as the Grothendieck formula

$(1)\quad P_{t}(f_{*}(x))=f_{*}(c_{t}\cdot P_{t}(x)).$

Here the total reduced powers $P_{t}(x)$ are defined

$P_{t}(x)=\sum_{\alpha}P^{\alpha}(x)t^{\alpha}\in
H^{*}(X;{\mathbb{Z}}/p)[t_{1},t_{2},...]\quad with\
t^{\alpha}=t_{1}^{\alpha_{1}}t_{2}^{\alpha_{2}}...$

where $\alpha=(\alpha_{1},\alpha_{2},...)$ and
$degree(t^{\alpha})=\sum_{i}2\alpha_{i}(p^{i}-1)$. The total Chern class
$c_{t}$ is defined similarly, for the Chern classes of the normal bundle of
the map $f$.

We consider the above equation with the assumption such that $t_{n}^{2}=0$ and
$t_{j}=0$ for $j\not=n$, i.e., $P_{t}(x)\in
H^{*}(X;{\mathbb{Z}}/p)\otimes\Lambda(t_{n})$. That means

$(2)\quad P_{t}(f_{*}(x))=(1+P^{\Delta_{n}}t_{n})(f_{*}(x))$ $(3)\quad
f_{*}(c_{t}\cdot
P_{t}(x))=f_{*}((1+c_{p^{n}-1}t_{n})(x+P^{\Delta_{n}}(x)t_{n}))=f_{*}(x+(c_{p^{n}-1}x+P^{\Delta_{n}}(x))t_{n}).$

From (1), we see $(2)=(3)$ and we have

$(4)\quad P^{\Delta_{n}}(f_{*}(x))=f_{*}(c_{p^{n}-1}x+P^{\Delta_{n}}(x)).$

By the definition, $\beta$ commutes with $f_{*}$, and we have

$(5)\quad P^{\Delta_{n}}\beta(f_{*}(x))=P^{\Delta_{n}}f_{*}(\beta
x)=f_{*}(c_{p^{n}-1}\beta x+P^{\Delta_{n}}(\beta x)).$

On the other hand

$(6)\quad\beta P^{\Delta_{n}}f_{*}(x)\stackrel{{\scriptstyle(4)}}{{=}}\beta
f_{*}(c_{p^{n}-1}x+P^{\Delta_{n}}(x))=f_{*}(c_{p^{n}-1}\beta x+\beta
P^{\Delta_{n}}(x)).$

Then $(5)-(6)$ gives that $(P^{\Delta_{n}}\beta-\beta
P^{\Delta_{n}})f_{*}(x)=f_{*}(P^{\Delta_{n}}\beta-\beta P^{\Delta_{n}})(x).$ ∎

By the definition, each cohomology operation $h$ (i.e., an element in the
Steenrod algebra) is written (with $Q^{B}=Q_{0}^{b_{0}}Q_{1}^{b_{1}}...$) by

$h=\sum_{A,B}P^{A}Q^{B}\quad with\ A=(a_{1},...),\ B=(b_{0},...)\ b_{i}=0\ or\
1.$

######  Corollary 2.2.

We have $P_{t}Q^{B}(f_{*}(x))=f_{*}(c_{t}\cdot P_{t}Q^{B}(x)).$

Hence cohomology operations $h$ (for $H^{*}(-;{\mathbb{Z}}/p))$ which commute
with all transfer $f_{*}$ are cases $c_{t}=1$, i.e. $A=0$ which are only
products $Q^{B}$ of Milnor operations $Q_{i}$.

##  3\. coniveau filtrations

Bloch-Ogus [Bl-Og] give a spectral sequence such that its $E_{2}$-term is
given by

$E(c)_{2}^{c,*-c}\cong H_{Zar}^{c}(X,\mathcal{H}_{A}^{*-c})\Longrightarrow
H_{et}^{*}(X;A)$

where $\mathcal{H}_{A}^{*}$ is the Zariski sheaf induced from the presheaf
given by $U\mapsto H_{et}^{*}(U;A)$ for an open $U\subset X$.

Remark. When $X$ is complete, it is known

$E(c)_{2}^{0,*}\cong H_{Zar}^{0}(X,\mathcal{H}_{{\mathbb{Z}}/p}^{*})\cong
H^{*}_{nr}(k(X):{\mathbb{Z}}/p)$

as the unramified cohomology which is a retract rationally invariant for $X$
[Me], [Re-Sc].

The filtration for this spectral sequence is defined as the coniveau
filtration, that is

$N^{c}H_{et}^{*}(X;A)=F(c)^{c,*-c}$

where the infinite term $E(c)^{c,*-c}_{\infty}\cong
F(c)^{c,*-c}/F(c)^{c+1,*-c-1}$ and

$N^{c}H^{*}_{et}(X;A)=\sum_{Z\subset X\ ;\ codim_{X}(Z)\leq
c}ker(j^{*}:H^{*}_{et}(X;A)\to H^{*}_{et}(X-Z,A)).$

Here we recall the motivic cohomology $H^{*,*^{\prime}}(X;{\mathbb{Z}}/p)$
defined by Voevodsky and Suslin ([Vo1,Vo3,Vo4]) so that

$H^{i,i}(X;{\mathbb{Z}}/p)\cong H^{i}_{et}(X;{\mathbb{Z}}/p)\cong
H^{i}(X;{\mathbb{Z}}/p).$

In this paper, let us write the $p$-localized cohomology
$H^{*}(X;{\mathbb{Z}}_{(p)})$ (resp. Chow ring
$CH^{*}(X)\otimes{\mathbb{Z}}_{(p)}$) simply by $H^{*}(X)$ (resp.
$CH^{*}(X)$). Note that in general, $H^{*}_{et}(X)\not\cong H^{*}(X)$ while we
have the natural map $H_{et}^{*}(X)\to H^{*}(X)$.

Let $0\not=\tau\in H^{0,1}(Spec({\mathbb{C}});{\mathbb{Z}}/p)$. Then by the
multiplying by $\tau$, we can define a map
$H^{*,*^{\prime}}(X;{\mathbb{Z}}/p)\to H^{*,*^{\prime}+1}(X;{\mathbb{Z}}/p)$.
By Deligne ( foot note (1) in Remark 6.4 in [Bl-Og]) and Paranjape (Corollary
4.4 in [Pj]), it is proven that there is an isomorphism of the coniveau
spectral sequence with the $\tau$-Bockstein spectral sequence
$E(\tau)_{r}^{*,*^{\prime}}$ (see also [Te-Ya], [Ya1]).

######  Theorem 3.1.

(Deligne) Let $A={\mathbb{Z}}/p$. Then we have the isomorphism of spectral
sequence

$E(c)_{r}^{c,*-c}\cong E(\tau)_{r-1}^{*,*-c}\quad for\ r\geq 2.$

Hence the filtrations are the same, i.e.
$N^{c}H_{et}^{*}(X;{\mathbb{Z}}/p)=F_{\tau}^{*,*-c}$ where

$F_{\tau}^{*,*-c}=Im(\times\tau^{c}:H^{*,*-c}(X;{\mathbb{Z}}/p)\to
H^{*,*}(X;{\mathbb{Z}}/p)).$

######  Lemma 3.2.

Let $x\in H^{*,*}(X)$ and $x\in N^{c}H^{*}(X;{\mathbb{Z}}/p)$. Then if the map
$H^{*+1,*-c}(X)\to H^{*+1,*}(X)$ is injective, then $x\in N^{c}H^{*}(X)$.

###### Proof.

Consider the exact sequences

$\begin{CD}@>{p}>{}>H^{*,*-c}(X)@>{r_{1}}>{}>H^{*,*-c}(X;{\mathbb{Z}}/p)@>{\delta_{1}}>{}>H^{*+1,*-c}(X)\\\
@V{}V{f_{1}}V@V{}V{f_{2}}V@V{}V{f_{3}}V\\\
@>{p}>{}>H^{*,*}(X)@>{r_{2}}>{}>H^{*,*}(X;{\mathbb{Z}}/p)@>{\delta_{2}}>{}>H^{*+1,*}(X)\end{CD}$

By the assumption of this lemma, we take $x^{\prime}\in
H^{*,*-c}(X;{\mathbb{Z}}/p)$ such that $r_{2}(x)=f_{2}(x^{\prime})$. So
$\delta_{2}f_{2}(x^{\prime})=0$. Since $f_{3}$ is injective, we see
$\delta_{1}(x^{\prime})=0$, Hence there is $x^{\prime\prime}\in
H^{*.*-c^{\prime}}(X)$ such that $r_{1}(x^{\prime\prime})=x^{\prime}$. ∎

######  Lemma 3.3.

If $y\in H^{2*,*}(X;A)$, then $y\in\tilde{N}^{*}H^{2*}(X;A)$.

###### Proof.

Recall that

$y\in H^{2*,*}(X;A)\cong CH^{*}(X)\otimes A.$

Hence $y$ is represented by closed algebraic set supported $Y$, while $Y$ may
be singular. However by Totaro [To], we have the modified cycle map $\bar{c}l$

$cl:CH^{*}(X)\otimes
A\stackrel{{\scriptstyle\bar{c}l}}{{\to}}MU^{2*}(X)/p\otimes_{MU^{*}}A\to
H^{2*}(X;A)$

for the complex cobordism theory $MU^{*}(X)$. It is known [Qu1] that elements
in $MU^{2*}(X)$ is written by a proper map to X from a smooth complex
manifold. Hence we can take $Y$ smooth. ∎

######  Lemma 3.4.

Let $cl$ be the cycle map. If $H^{*}(X;A)$ is generated by elements of
$deg\leq m$, as an $Im(cl)\otimes A$-module, then we have

$DH^{*}(X;A)=N^{1}H^{*}(X;A)/(p,\tilde{N}^{1}H^{*}(X;A))=0\quad when\ m<*.$

###### Proof.

Each element $x\in H^{*}(X;A)$ of $m<*$ is written as

$x=\sum ag\qquad where\ \ a\in Im(cl)^{+},\ g:Im(cl)-module\ generator.$

We have $f:Y\to X$ with $f_{*}(1)=a$ from the preceding lemma. Hence

$f_{*}f^{*}(g)=f_{*}(1\cdot f^{*}(g))=f_{*}(1)g=ag$

by Frobenius reciprocity law. ∎

The following lemma is proved by Colliot-Thelene and Voisin [Co-Vo] by using
the Bloch and Ogus spectral sequence and the affirmative answer of the Bloch-
Kato conjecture by Voevodsky. ([Vo3]. [Vo4])

######  Lemma 3.5.

( [Co-Vo]) Let $X$ be a smooth complex variety. Then any torsion element in
$H^{*}(X)$ is in $N^{1}H^{*}(X)$.

##  4\. the main lemma

The following lemma is the $Q_{i}$-version of one of results by Benoist and
Ottem.

######  Lemma 4.1.

Let $\alpha\in N^{1}H^{s}(X)$ for $s=3$ or $4$. If $Q_{i}(\alpha)\not=0\in
H^{*}(X;{\mathbb{Z}}/p)$ for some $i\geq 1$, then

$DH^{s}(X)\supset{\mathbb{Z}}/p\\{\alpha\\},\quad
DH^{s}(X;{\mathbb{Z}}/p^{t})\supset{\mathbb{Z}}/p\\{\alpha\\}\ \ for\ t\geq
2.$

###### Proof.

Suppose $\alpha\in\tilde{N}^{1}H^{s}(X)$ for $s=3$ or $4$, i.e. there is a
smooth $Y$ with $f:Y\to X$ such that the transfer
$f_{*}(\alpha^{\prime})=\alpha$ for $\alpha^{\prime}\in H^{*}(Y)$. Then for
$s=4$

$Q_{i}(\alpha^{\prime})=(P^{\Delta_{i}}\beta-\beta
P^{\Delta_{i}})(\alpha^{\prime})=(-\beta
P^{\Delta_{i}})(\alpha^{\prime})=-\beta(\alpha^{\prime})^{p^{i}}$
$=-p^{i}(\beta\alpha^{\prime})(\alpha^{\prime})^{p^{i}-1}=0\quad(by\
Cartanformula)$

since $\beta(\alpha^{\prime})=0$ and $P^{\Delta_{i}}(y)=y^{p^{i}}$ for
$dim(y)=2$. (For $s=3$, we get also $Q_{i}(\alpha^{\prime})=0$ since
$P^{\Delta_{i}}(x)=0$ for $dim(x)=1$.) This is a contradiction

$0\not=Q_{i}(\alpha)=Q_{i}(f_{*}\alpha^{\prime})=f_{*}(Q_{i}\alpha^{\prime})=0.$

The case $A={\mathbb{Z}}/p^{t}$, $t\geq 2$ is proved similarly, since for
$\alpha^{\prime}\in H^{*}(X;A)$ we see $\beta\alpha^{\prime}=0\in
H^{*}(X;{\mathbb{Z}}/p)$. ∎

######  Corollary 4.2.

As an $\Lambda(Q_{0},Q_{i})$-module, let (for $i\geq 1$)

$H^{*}(X;{\mathbb{Z}}/p)\supset\Lambda(Q_{0},Q_{i})\\{a\\}\quad|a|=2\ or\ 3.$

Then $DH^{*}(X)\supset{\mathbb{Z}}/p\\{Q_{0}a\\}$.

##  5\. classifying spaces for finite groups

Let $G$ be a finite group or an algebraic group, and $BG$ its classifying
space. For example, when $G=G_{m}$ is the multiple group, we see

$BG_{m}=BS^{1}\cong{\mathbb{P}}^{\infty},\quad
H^{*}({\mathbb{P}}^{\infty})\cong{\mathbb{Z}}[y]\ \ with\ |y|=2,$

for the infinite (complex) projective space ${\mathbb{P}}^{\infty}$. Note that
$BG_{m}$ is a colimit of complex projective spaces. Though $BG$ itself is not
a colimit of complex projective varieties, we can take a complex projective
variety $X(N)$ ([Ek]) for $N\geq 3$ such that there is a map $j:X(N)\to
BG\times{\mathbb{P}}^{\infty}$ with

$H^{*}(BG\times{\mathbb{P}}^{\infty};A)\stackrel{{\scriptstyle
j^{*}}}{{\cong}}H^{*}(X(N);A)\quad for\ *<N.$

Note that $N^{n}H^{*}(X;A)/\tilde{N}^{n}H^{*}(X;A)$ is invariant under
replacing $X$ with $X\times{\mathbb{P}}^{m}$ for all $n$ and all abelian
groups $A$. In fact, from Kunneth formula,

$H^{*}(X\times{\mathbb{P}}^{m};A)\cong
H^{*}(X;A)\otimes{\mathbb{Z}}[y]/(y^{m+1}),$

here $y\in CH^{1}({\mathbb{P}}^{m})$, and we see
$Ideal(y)\subset\tilde{N}^{*}H^{*}(X\times{\mathbb{P}}^{m};A)$ by the
Frobenius reciprocity law (Lemma 3.4).

Moreover Benoist and Ottem show the above quotient is the stable birational
invariant (Proposition 2.4 in [Be-Ot]).

In this paper, we say the above $X(N)$ as an (complex projective)
approximation $for$ $BG$ (which is an approximation $of$
$BG\times{\mathbb{P}}^{\infty}$ strictly), and let us write the stable
rational invariant by

$DH^{*}(X;A)=N^{1}H^{*}(X;A)/(p,\tilde{N}^{1}H^{*}(X;A)).$

Hereafter, we consider $DH^{*}(X;A)$ when $A={\mathbb{Z}}_{(p)}$.

Let $p$ be an odd prime. (When $p=2$, the different but the similar argument
works.) Let $G=({\mathbb{Z}}/p)^{3}$ the $rank=3$ elementary abelian
$p$-group. Then the $mod(p)$ cohomology is

$H^{*}(BG;{\mathbb{Z}}/p)\cong{\mathbb{Z}}/p[y_{1},y_{2},y_{3}]\otimes\Lambda(x_{1},x_{2},x_{3}).$

Here $|y_{i}|=2,|x_{i}|=1,\beta(x_{i})=y_{i}$, and $\Lambda(a,...,b)$ is the
${\mathbb{Z}}/p$-exterior algebra generated by $a,...,b$. The integral
cohomology (modulo $p$) is isomorphic to

$H^{*}(BG)/p\cong Ker(Q_{0})\cong H(H^{*}(BG;{\mathbb{Z}}/p);Q_{0})\oplus
Im(Q_{0})$

where $H(-;Q_{0})$ is the homology with the differential $Q_{0}$. It is
immediate
$H(H(B{\mathbb{Z}}/p;{\mathbb{Z}}/p);Q_{0})\cong{\mathbb{Z}}/p$. By the
Kunneth formula, we have
$H(H(BG;{\mathbb{Z}}/p);Q_{0})\cong{\mathbb{Z}}/p$. Thus we have

$H^{*}(BG)/p\cong{\mathbb{Z}}/p\\{1\\}\oplus Im(Q_{0})$
$\cong{\mathbb{Z}}/p[y_{1},y_{2},y_{3}](1,Q_{0}(x_{i}x_{j}),Q_{0}(x_{1}x_{2}x_{3})|i<j)$

where the notation $R(a,...,b)$ (resp. $R\\{a,...,b\\}$) means the
$R$-submodule (resp. the free $R$-module) generated by $a,...,b$. Here we note
$H^{+}(BG)$ is just $p$-torsion.

We know $Q_{i}(y_{j})=y_{j}^{p^{i}}$ and $Q_{j}$ is a derivation. Let us write

$\alpha=Q_{0}(x_{1}x_{2}x_{3})=y_{1}x_{2}x_{3}-y_{2}x_{1}x_{3}+y_{3}x_{1}x_{2}.$

Note $\alpha\in H^{4}(X)$, $p\alpha=0$, and $\alpha\in N^{1}H^{*}(X)$ from
Lemma 3.5. Moreover

$Q_{1}(\alpha)=y_{1}y_{2}^{p}x_{3}-y_{1}y_{3}^{p}x_{2}-...\not=0\in
H^{*}(X;{\mathbb{Z}}/p).$

Similarly, for $\alpha_{ij}=Q_{0}(x_{i}x_{j})$, we see
$Q_{1}(\alpha_{ij})\not=0$. Hence from Lemma 3.5 and
$y_{i}\in\tilde{N}^{1}H^{*}(X)$, we see $y_{i}\alpha\in\tilde{N}^{1}H^{*}(X)$
by Lemma 3.4. Thus we have

######  Theorem 5.1.

Let $X=X(N)$ with $N>2p+3$ be an approximation for $(B{\mathbb{Z}}/p)^{3}$.
Then we have

$DH^{*}(X)\cong{\mathbb{Z}}/p\\{\alpha_{ij},\alpha|1\leq i<j\leq
3\\}\quad*<N.$

###### Proof.

We see
$H^{*}(BG)/(p,y_{1},y_{2},y_{3})\cong{\mathbb{Z}}/p\\{1,\alpha_{ij},\alpha\\}$.
Of course $1\not\in N^{1}H^{*}(X)$, we have the theorem from Lemma 4.1. ∎

######  Corollary 5.2.

Let $X=X(N)_{n}$ with $N>2p+3$ be an approximation for
$(B{\mathbb{Z}}/p)^{n}$. Then we have

$DH^{*}(X)\supset{\mathbb{Z}}/p\\{\alpha_{ij},\alpha_{i^{\prime}j^{\prime}k^{\prime}}|1\leq
i<j\leq n,1\leq i^{\prime}<j^{\prime}<k^{\prime}\leq n\\}$

for $*<N$ where $\alpha_{ij}=Q_{0}(x_{i}x_{j})$ and
$\alpha_{i^{\prime}j^{\prime}k^{\prime}}=Q_{0}(x_{i^{\prime}}x_{j^{\prime}}x_{k^{\prime}})$.

######  Corollary 5.3.

If $n\not=m\geq 3$, then $X(N)_{n}$ and $X(N)_{m}$ are not stable birational
equivalent.

Let $rank_{p}(G)=r$, i.e., there is an injection
$j:({\mathbb{Z}}/p)^{r}\subset G$. Let $j^{*}:H^{*}(BG)\to
H^{*}(B({\mathbb{Z}}/p)^{r})$ be the induced map.

######  Corollary 5.4.

Let $rank_{p}(G)\geq s-1$ for $s=4$ (or $s=3$). If the image $Im(j^{*})$
contains $\alpha$ (or $\alpha_{12}$ ), then for the approximation $X=X(N)$ for
$BG$, we have $DH^{s}(X)\not=0$ for $N>2p+3$.

Let $G$ be a non abeian group of order $p^{3}$ (see $\S 8$, for details). Then
$H^{even}(BG)$ is generated by Chern classes, and $H^{odd}(BG)$ is a
$p$-torsion, and the operation $Q_{1}$ induces the injection

$0\to H^{odd}(BG)\stackrel{{\scriptstyle Q_{1}}}{{\to}}H^{even}(BG).$

Such groups are four types (see $\S 8$ below), and they are called
extraspecial $p$ groups $G=p_{\pm}^{1+3}$ order $p^{3}$. When
$G=Q_{8}=2_{-}^{1+2}$ the quaternion group of order $8$, we know
$H^{odd}(X)=0$. However when $G=D_{8}=2_{+}^{1+2}$ the dihedral group of order
$8$, the cohomology $H^{odd}(BG)$ is generated as an $H^{even}(BG)$ by an
element $e$ of $deg(e)=3$. When $G=E=p_{+}^{1+2}$ for $p\geq 3$, $H^{odd}(BG)$
is generated by $e_{1},e_{2}$ with $deg(e_{i})=3$. When $G=M=p_{-}^{1+2}$ for
$p\geq 3$, $H^{odd}(BG)$ is generated by $e$ but $deg(e)=2p+1$.

From Lemma 3.4 (Frobenius reciprocity) and the main lemma (Lemma 4.1), we have
the following theorem.

######  Theorem 5.5.

Let $X=X(N)$ with $N>2p+3$ be an approximation for an extraspecial $p$-group
$G$ of order $p^{3}$. Then we have for $*<N$

$DH^{*}(X)\cong\begin{cases}0\quad for\ G=Q_{8}\\\ {\mathbb{Z}}/2\\{e\\}\quad
for\ G=D_{8}\\\ 0\ or\ {\mathbb{Z}}/p\\{e\\}\quad for\ G=M\\\
{\mathbb{Z}}/p\\{e_{1},e_{2}\\}\quad for\ G=E.\end{cases}$

##  6\. connected groups

At first, we consider when $G=U_{n}$, $SU_{n}$ or $Sp_{2n}$ for all $p$, the
cohomology $H^{*}(BG)$ has no torsion. Then $H^{*}(BG)$ is generated by Chern
classes, e.g.,

$H^{*}(BU_{n})\cong CH^{*}(BU_{n})\cong{\mathbb{Z}}_{(p)}[c_{1},...,c_{n}],$
$H^{*}(BSp_{2n})\cong
CH^{*}(BSp_{2n})\cong{\mathbb{Z}}_{(p)}[c_{2},c_{4},...,c_{2n}].$

Hence $DH^{*}(X)=0$ for the approximations $X$ for these groups.

Next we consider the case $G=SO_{3}$ and $p=2$. Then

$H^{*}(BG;{\mathbb{Z}}/2)\cong{\mathbb{Z}}/2[w_{1},w_{2},w_{3}]/(w_{1})\cong{\mathbb{Z}}/2[w_{2},w_{3}],$

where $w_{i}$ is the $i$-th Stiefel-Whitney class for $SO_{3}\subset O_{3}$
and $w_{i}^{2}=c_{i}$ is the $i$-th Chern class for $SO_{3}\subset U_{3}$.
Here we know $Q_{0}(w_{2})=w_{3},$ and $Q_{1}(w_{3})=w_{3}^{2}$. Therefore we
have [Ya1]

$H^{*}(BG;{\mathbb{Z}}/2)\cong{\mathbb{Z}}/2[c_{2},c_{3}]\\{1,w_{2},w_{3}=Q_{0}(w_{2}),w_{2}w_{3}=Q_{1}w_{2}\\}$
$\cong{\mathbb{Z}}/2[c_{2},c_{3}]\\{w_{2},Q_{0}(w_{2}),Q_{1}(w_{2}),Q_{0}Q_{1}(w_{2})=c_{3}\\}\oplus{\mathbb{Z}}/2[c_{2}]$
$\cong{\mathbb{Z}}/2[c_{2},c_{3}]\otimes\Lambda(Q_{0},Q_{1})\\{w_{2}\\}\oplus{\mathbb{Z}}/2[c_{2}].$

In particular $H^{*}(BG)/2\cong
Ker(Q_{0})\cong{\mathbb{Z}}/2[c_{2},c_{3}]\\{1,w_{3}\\}.$

From Corollary 4.2, we have the following theorem.

######  Theorem 6.1.

Let $X_{n}=X_{n}(N)$ be approximations for $BSO_{n}$ for $n\geq 3$. Then

$DH^{*}(X_{3})\cong{\mathbb{Z}}/2\\{w_{3}\\}\subset DH^{*}(X_{n}).$

###### Proof.

We have seen $H^{*}(BG)/(2,c_{2},c_{3})\cong{\mathbb{Z}}/2\\{1,w_{3}\\}$. Here
$1\not\in N^{1}H^{*}(X)$ but $w_{3}=Q_{0}w_{2}\in N^{1}H^{*}(X)$. Therefore
from Corollary 4.2, the approximation $X$ for $BSU_{3}$ satisfies

$DH^{*}(X)\cong{\mathbb{Z}}/2\\{Q_{0}w_{2}\\}\cong{\mathbb{Z}}/2\\{w_{3}\\}.$

∎

We consider simply connected groups. Let us write by $X$ an approximation of
$BG_{2}$ for the exceptional simple group $G_{2}$ of $rank=2$. The $mod(2)$
cohomology is generated by the Stiefel-Whitney classes $w_{i}$ of the real
representation $G_{2}\to SO_{7}$

$H^{*}(BG_{2};{\mathbb{Z}}/2)\cong{\mathbb{Z}}/2[w_{4},w_{6},w_{7}],\quad
P^{1}(w_{4})=w_{6},\ Q_{0}(w_{6})=w_{7},$ $H^{*}(BG_{2})\cong(D^{\prime}\oplus
D^{\prime}/2[w_{7}]^{+})\quad where\ \
D^{\prime}={\mathbb{Z}}_{(2)}[w_{4},c_{6}].$

Then we have $Q_{1}w_{4}=w_{7},Q_{2}(w_{7})=w_{7}^{2}=c_{7}$ (the Chern
class).

The Chow ring of $BG_{2}$ is also known

$CH^{*}(BG_{2})\cong(D\\{1,2w_{4}\\}\oplus D/2[c_{7}]^{+})\quad where\ \
D={\mathbb{Z}}_{(2)}[c_{4},c_{6}]\ \ c_{i}=w_{i}^{2}.$

In particular the cycle map $cl:CH^{*}(BG)\to H^{*}(BG)$ is injective.

It is known $w_{4}\in N^{1}H^{*}(X;{\mathbb{Z}}/2)$ ([Ya1]) and from Lemma
3.2, we see $w_{4}\in N^{1}H^{*}(X)$. Since $Q_{1}(w_{4})=w_{7}\not=0$, from
Lemma 4.1, we have $DH^{4}(X)\not=0$ ([Be-Ot]). Moreover we have

######  Proposition 6.2.

For $X$ an approximation for $BG_{2}$, we have the surjection

$\Lambda(w_{4},w_{7})^{+}\twoheadrightarrow DH^{*}(X)\quad for\ *<N.$

###### Proof.

We have seen $H^{*}(BG)/(c_{4},c_{6},c_{7})\cong\Lambda(w_{4},w_{7})$. ∎

######  Theorem 6.3.

Let $G$ be a simply connected group such that $H^{*}(BG)$ has $p$-torsion. Let
$X=X(N)$ be an approximation for $BG$ for $N\geq 2p+3$. Then
$DH^{4}(X)\not=0$.

###### Proof.

It is only need to prove the theorem when $G$ is a simple group having $p$
torsion in $H^{*}(BG)$. Let $p=2$. It is well known that there is an
emmbedding $j:G_{2}\subset G$ such that

$H^{4}(BG)\stackrel{{\scriptstyle
j^{*}}}{{\cong}}H^{4}(BG_{2})\cong{\mathbb{Z}}_{(2)}\\{w_{4}\\}.$

Let $x=(j^{*})^{-1}w_{4}\in H^{4}(BG)$. From Lemma 3.1 in [Ya4], we see that
$2x$ is represented by Chern classes. Hence $2x$ is the image from
$CH^{*}(X)$, and so $2x\in N^{1}H^{4}(X)$. This means there is an open set
$U\subset X$ such that $2x=0\in H^{*}(U)$ that is $x$ is $2$-torsion in
$H^{*}(U)$. Hence from Lemma 3.5, we have $x\in N^{1}H^{4}(U)$, and so there
is $U^{\prime}\subset U$ such that $x=0\in H^{4}(U^{\prime})$. This implies
$x\in N^{1}H^{4}(X)$.

Since $j^{*}(Q_{1}x)=Q_{1}w_{4}=w_{7}$, we see $Q_{1}x\not=0$. From the main
lemma (Lemma 4.1), we see $DH^{4}(X)\not=0$ for $G$.

For the cases $p=3,5$, we consider the exceptional groups $F_{4},E_{8}$
respectively. Each simply connected simple group $G$ contains $F_{4}$ for
$p=3$, $E_{8}$ for $p=5$. There is $x\in H^{4}(BG)$ such that
$Q_{1}(x)\not=0\in H^{*}(BG;{\mathbb{Z}}/p).$ In fact, there is embedding
$j:({\mathbb{Z}}/p)^{3}\subset G$ with $j^{*}(x)=Q_{0}(x_{1}x_{2}x_{3})$.
Hence we have the theorem. ∎

######  Corollary 6.4.

Let $X$ be an approximation for $BSpin_{n}$ with $n\geq 7$ or $BG$ for an
exceptional group $G$. Then $X$ is not stable rational.

Remark. Kordonskii [Ko], Merkurjev (Corollary 5.8 in [Me]), and Reinstein-
Scavia show [Re-Sc] that $BSpin_{n}$ itself is stably rational when $n\leq
14$. These facts imply that the (Ekedahl) approximation $X$ is not stable
rationally equivalent to $GL_{M}/G=BG$ where $M$ is a large number such that
$G$ acts freely on $GL_{M}$. In fact, $X$ is constructed from a quasi
projective variety $BG$ as taking intersections of subspaces of
${\mathbb{P}}^{M^{\prime}}$ for a large $M^{\prime}$. (The author thanks
Federico Scavia who pointed out this remark.)

Here we note the relations to retract rationally. (Recall the first remark in
$\S 3$ about the unramified cohomology.) By Orlov-Vishik-Voevodsky [Or-Vi-Vo],
([Te-Ya] for $p:odd$,) we know

######  Lemma 6.5.

The comology $H^{0}_{Zar}(X;\mathcal{H}^{*}_{{\mathbb{Z}}/p})$ is isomorphic
to

$H^{*,*}(X;{\mathbb{Z}}/p)/(\tau)\oplus
Ker(\tau:H^{*+1,*-1}(X;{\mathbb{Z}}/p)\to H^{*+1,*}(X;{\mathbb{Z}}/p)).$

Recall (the remark in $\S 3$) that when $X$ is complete,
$H_{nr}^{*}(X;{\mathbb{Z}}/p)\cong
H^{0}_{Zar}(X;\mathcal{H}^{*}_{{\mathbb{Z}}/p})$, and it is an invariant of
the retract rationality of $X$ [Me].

######  Theorem 6.6.

([Ya4]) Let $G$ be the groups in the preceding theorem. Then there is an
element (Rost invariant) $0\not=a\in
H^{0}_{Zar}(X;\mathcal{H}^{3}_{{\mathbb{Z}}/p}).$ Hence $X$ is not retract
rational.

###### Proof.

Recall $x\in H^{4}(X)$ such that there is a Chern class $\alpha\in
CH^{4}(X)\cong H^{4,2}(X;{\mathbb{Z}}_{(2)})$ with $\alpha=px\in H^{4}(X)$. Of
course $\alpha=0\in H^{*}(X;{\mathbb{Z}}/p)$, i.e., $\tau^{2}\alpha=0$ in
$H^{*,*}(X;{\mathbb{Z}}/p)$. But it is known [Or-Vi-Vo] that

$\tau:H^{*,*-1}(X;{\mathbb{Z}}/p)\to H^{*,*}(X;{\mathbb{Z}}/p)\ \ is\
injective.$

Hence $\alpha\in Ker(\tau:H^{4,2}(X;{\mathbb{Z}}/p)\to
H^{3,3}(X;{\mathbb{Z}}/p)).$ Take $a\in
H^{0}_{Zar}(X;\mathcal{H}^{3}_{{\mathbb{Z}}/p})$ as the corresponding element
to $\alpha$ from the preceding lemma. ∎

Remark. We do $not$ assume
$H^{0}_{Zar}(X;\mathcal{H}_{{\mathbb{Z}}/p}^{*})\cong
H^{0}_{Zar}(X^{\prime};\mathcal{H}_{{\mathbb{Z}}/p}^{*})$ for an other
approximation $X^{\prime}$.

Remark. Since stable rationality implies retract rationality, the preceding
corollary is the just corollary of the above result. However note in general,

$H_{nr}^{*}(k(GL_{N}/G);{\mathbb{Z}}/p)\not\cong
H^{0}(BG;\mathcal{H}_{Z/p}^{*}),$ $\quad k(X)\not\cong k(W)^{G}\quad
W:{\mathbb{C}}-vector\ space.$

By Voevodsky [Vo1], [Vo2], we have the $Q_{i}$ operation also in the motivic
cohomology $H^{*,*^{\prime}}(X;{\mathbb{Z}}/p)$ with
$deg(Q_{i})=(2p^{i}-1,p-1)$. Then we can take

$deg(w_{4})=(4,3),\ deg(w_{6})=(6,4),\ deg(w_{7})=(7,4),\ deg(c_{7})=(14,7).$

By Theorem 3.1, the above means

$w_{7}\in N^{7-4}H^{*}(X;{\mathbb{Z}}/2)=N^{3}H^{*}(X;{\mathbb{Z}}/2).$

We can not see here that $0\not=w_{7}\in DH^{*}(X)$, but see the following
proposition.

######  Proposition 6.7.

For $X$ the approximation for $BG_{2}$, we have

${\mathbb{Z}}/2\\{w_{7}\\}\subset
D^{3}H^{*}(X)=N^{3}H^{*}(X)/(2,\tilde{N}^{3}H^{*}(X)).$

###### Proof.

Suppose $w_{7}\in\tilde{N}^{3}H^{*}(X)$. That is, there is $x\in H^{1}(Y)$
with $f_{*}(x)=w_{7}$ for $f:Y\to X$. Act $Q_{2}$ on $H^{*}(Y)$, and

$Q_{2}(x)=(P^{\Delta_{2}}\beta+\beta P^{\Delta_{2}})(x)=0$

since $\beta(x)=0$ and $P^{i}(x)=Sq^{2i}(x)=0$ for $i>0$. But
$Q_{2}w_{7}=c_{7}\not=0$. This contradicts to the commutativity of $f_{*}$ and
$Q_{2}$. ∎

At last of this section, we consider the case $G=PGL_{p}$. We have (for
example Theorem 1.5,1.7 in [Ka-Ya]) additively

$H^{*}(BG;{\mathbb{Z}}/p)\cong M\oplus N\quad with\ \ M\stackrel{{\scriptstyle
add.}}{{\cong}}{\mathbb{Z}}/p[x_{4},x_{6},...,x_{2p}],\quad$
$N=SD\otimes\Lambda(Q_{0},Q_{1})\\{u_{2}\\}\quad with\ \
SD={\mathbb{Z}}/p[x_{2p+2},x_{2p^{2}-2p}]$

where $x_{2p+2}=Q_{1}Q_{0}u_{2}$ and suffix means its degree. The Chow ring is
given as

$CH^{*}(BG)/p\cong M\oplus SD\\{Q_{0}Q_{1}(u_{2})\\}.$

From Corollary 4.2, we have ;

######  Theorem 6.8.

Let $p$ be odd. For an approximation $X$ for $BPGL_{p}$, we see
${\mathbb{Z}}/p\\{Q_{0}u_{2}\\}\subset DH^{*}(X)$, and moreover there is a
surjection

${\mathbb{Z}}/p[x_{2p^{2}-2p}]\\{Q_{0}u_{2}\\}\twoheadrightarrow
DH^{*}(X)\quad for\ *<N.$

For $G=PGL_{p}$, we easily see [Te-Ya]

$H^{2}(X;{\mathbb{Z}}/p)/N^{1}H^{2}(X;{\mathbb{Z}}/p)\cong
H^{2}(BG;{\mathbb{Z}}/p)/N^{1}H^{2}(BG;{\mathbb{Z}}/p),$
$H_{Zar}^{0}(X;\mathcal{H}^{*}_{{\mathbb{Z}}/p})\supset
H^{2}(BG;{\mathbb{Z}}/p)/N^{1}H^{2}(BG;{\mathbb{Z}}/p)\supset{\mathbb{Z}}/p\\{u_{2}\\}.$

(When $G=SO_{3}$, we see
$H^{*}(X;{\mathbb{Z}}/2)/N^{1}H^{*}(X;{\mathbb{Z}}/2)\supset{\mathbb{Z}}/2\\{1,w_{2}\\}.$)
Hence in both cases $X$ are not retract rational.

Remark. All examples $X$ such that $H^{*}(BG)$ have $p$-torsion in this
section are not retract rational.

Remark. There are primes $\ell\not=p$ such that almost all results in $\S 5,6$
hold for $X$ varieties over $\bar{k}$ with $ch(k)=\ell$ letting

$H^{*}(X)=H_{et}^{*}(X;{\mathbb{Z}}_{p})=Lim_{i}H^{*}_{et}(X;{\mathbb{Z}}/p^{i}),$
$DH^{*}(X)=N^{1}H^{*}(X)/(p,\tilde{N}^{1}H^{*}(X),Im(cl)).$

The above facts are proved by the similar arguments in [Pi-Ya] (exchanging $p$
and $\ell$). However, we do not prove that $DH^{*}(X)$ is stable birational
invariant of varieties over the field $\bar{k}$.

##  7\. ${\mathbb{Z}}/p$-coefficient cohomology for abelian groups

In the preceding sections, we have seen that cases $DH^{*}(X;A)\not=0$ are not
so rare for $A={\mathbb{Z}}_{(p)}$, ${\mathbb{Z}}/p^{i}$, $i\geq 2$. However
currently it seems no such example for $A={\mathbb{Z}}/p$.

######  Question 7.1.

We have $DH^{*}(X;{\mathbb{Z}}/p)\cong 0$ for an approximation $X$ for $BG$ ?

At first, we consider the case $G=({\mathbb{Z}}/p)^{3}$. We can take an (quasi
projective) approximation $\bar{X}(N)$ of $B{\mathbb{Z}}/p$ explicitly by the
quotient (the $N$-dimensional lens space)

$\bar{X}(N)={\mathbb{C}}^{N*}/({\mathbb{Z}}/p)\quad where\
{\mathbb{C}}^{N*}=({\mathbb{C}}^{N}-\\{0\\}).$

We consider the approximations

$X(N)\to\bar{X}(N)\times{\mathbb{P}}^{N}\to
B{\mathbb{Z}}/p\times{\mathbb{P}}^{\infty}.$

Let us write $X_{i}$ (resp. $X_{i}^{\prime}$) the above $\bar{X}(N)$ (resp.
$\bar{X}(N-1))$ for a sufficient large number $N$. Let

$i_{1}:Y_{1}=X_{1}^{\prime}\times X_{2}\times X_{3}\to X=X_{1}\times
X_{2}\times X_{3}.$

Similarly we define $Y_{2},Y_{3}$, and the disjoin union $Y=Y_{1}\sqcup
Y_{2}\sqcup Y_{3}$.

Recall that for $p:odd$

$H^{*}(X;{\mathbb{Z}}/p)\cong{\mathbb{Z}}/p[y_{1},y_{2},y_{3}]/(y_{1}^{N+1},y_{1}^{N+1},y_{3}^{N+1})\otimes\Lambda(x_{1},x_{2},x_{3}),$

and $H^{*}(Y_{i};{\mathbb{Z}}/p)\cong H^{*}(X;{\mathbb{Z}}/p)/(y_{i}^{N})$ for
$i=1,2,3$. For $p=2$, some graded ring $grH^{*}(X;{\mathbb{Z}}/2)$ is
isomorphic to the above ring (in fact $x^{2}_{i}=y_{i}$).

For the embedding $f_{i}:X_{i}^{\prime}\to X_{i}$, it is known
$f_{i*}(1)=c_{1}(N)$ where $N$ is the normal bundle for $X_{i}^{\prime}\subset
X_{i}$. Hence the transfer is given by

$f_{1*}(1)=y_{1},\quad f_{2*}(1)=y_{2}\quad f_{3*}(1)=y_{3}.$

(Since $y_{1}\in CH^{*}(X)$, there is $\tilde{i}_{*}:\tilde{Y}\to X$ such that
$\tilde{i}_{*}(1)=y_{1}$. We take $\tilde{Y}=Y_{1}$.) Therefore we have for
$x=(x_{2}x_{3}+x_{3}x_{1}+x_{1}x_{2})\in H^{*}(Y_{1}\sqcup Y_{2}\sqcup
Y_{3};{\mathbb{Z}}/p)$,

$f_{*}(x)=y_{1}x_{2}x_{3}+y_{2}x_{3}x_{1}+y_{3}x_{1}x_{2}=Q_{0}(x_{1}x_{2}x_{3})=\alpha.$

Note that the element $\alpha^{\prime}=(x_{1}x_{2}+x_{2}x_{3}+x_{3}x_{1})$ is
not in the integral cohomology $H^{*}(Y)$. Similarly we have

$i_{*}(Q_{1}\alpha^{\prime})=i_{*}(y_{1}^{p}x_{2}-y_{2}^{p}x_{1})+....=(y_{3}y_{1}^{p}x_{2}+y_{3}y_{2}^{p}x_{1})+...=Q_{1}(\alpha).$

Thus we see $\alpha\in\tilde{N}^{c}H^{*}(X;{\mathbb{Z}}/p).$ More generally,
we see

######  Theorem 7.2.

We have the approximation $X$ for $(B{\mathbb{Z}}/p)^{n}$ such that

$DH^{*}(X;{\mathbb{Z}}/p)=0\quad*<N.$

We recall here the motivic cohomology. By Voevodsky [Vo 1],
$H^{*,*^{\prime}}(B{\mathbb{Z}}/p;{\mathbb{Z}}/p)$ satisfies the Kunneth
formula so that for

$H^{*,*^{\prime}}(B({\mathbb{Z}}/p)^{n};{\mathbb{Z}}/p)\cong{\mathbb{Z}}/p[\tau,y_{1},...,,y_{n}]/(y_{1}^{N+1},...,y_{n}^{N+1})\otimes\Lambda(x_{1},...,x_{n}).$

Here $0\not=\tau\in H^{0,1}(Spec({\mathbb{C}});{\mathbb{Z}}/p)$, and
$deg(y_{i})=(2,1)$, $deg(x_{i})=(1,1)$.

From Theorem 3.1, we can identify
$N^{c}H_{et}^{*}(X;{\mathbb{Z}}/p)=F_{\tau}^{*,*-c}$ where
$F_{\tau}^{*,*-c}=Im(\times\tau^{c}:H^{*,*-c}(X;{\mathbb{Z}}/p)\to
H^{*,*}(X;{\mathbb{Z}}/p)).$

######  Corollary 7.3.

(Theorem 5.1 in [Te-Ya]) Let $X$ be the approximation for
$(B{\mathbb{Z}}/p)^{n}$. Then we have

$H^{*}(X:{\mathbb{Z}}/p)/N^{1}H^{*}(X;{\mathbb{Z}}/p)\cong\Lambda(x_{1},....,x_{n}).$

###### Proof.

Let $x\in Ideal(y_{1},...,y_{n})\subset H^{*,*^{\prime}}(X;{\mathbb{Z}}/p)$.
Then $deg(x)=(*,*^{\prime})$ with $*>*^{\prime}$, and $x$ is a multiplying of
$\tau$. Hence $x\in N^{1}H^{*}(X;{\mathbb{Z}}/p)$. ∎

###### Proof of Theorem 7.2..

Let $x\in N^{1}H^{*}(X;{\mathbb{Z}}/p)$. From the above corollary. $x\in
Ideal(y_{1},...,y_{n})$ which is in the image of the transfer (such as
$Y_{1}=X_{1}^{\prime}\times X_{2}\times...\times X_{n}$). That is
$x\in\tilde{N}^{1}H^{*}(X;{\mathbb{Z}}/p)$. ∎

We can extend Lemma 7.3, by using the following lemma. Let us write by $XG$ an
approximation for $BG$.

######  Lemma 7.4.

Let $G$ have a Sylow $p$-subgroup $S$. If $DH^{*}(XS;{\mathbb{Z}}/p)=0$, then
so for $BG$.

###### Proof.

Let $j:BS\to BG$ and $i;Y\to XS$. We consider maps

$H^{*}(Y;{\mathbb{Z}}/p)\stackrel{{\scriptstyle
i_{*}}}{{\to}}H^{*}(XS;{\mathbb{Z}}/p)\stackrel{{\scriptstyle
j_{*}}}{{\to}}H^{*}(XG;{\mathbb{Z}}/p).$

Here $j_{*}=cor_{S}^{G}$ is the transfer for finite groups.

Note $j^{*}N^{1}H^{*}(XG;{\mathbb{Z}}/p)\subset N^{1}H^{*}(XS;{\mathbb{Z}}/p)$
by the naturality of $j^{*}$. Hence given $x\in N^{1}H^{*}(XG;{\mathbb{Z}}/p)$
the element $y=j^{*}(x)$ is in $N^{1}H^{*}(XS;{\mathbb{Z}}/p)$.

By the assumption in this lemma, there is $y^{\prime}\in
H^{*}(Y;{\mathbb{Z}}/p)$ with $i_{*}(y^{\prime})=y$. Thus we have $\
j_{*}i_{*}(y^{\prime})=j_{*}y=j_{*}j^{*}(x)=[G;S]x.$ ∎

######  Corollary 7.5.

Let $G$ have an abelian $p$ Sylow subgroup, and $X=X(N)$ be an approximation
for $BG$. Then $DH^{*}(X;{\mathbb{Z}}/p)=0$ for $*<N$.

##  8\. the groups $Q_{8}$ and $D_{8}$

When $|G|=p^{3}$, we have the short exact sequence

$0\to C\to G\to V\to 0$

where $C\cong{\mathbb{Z}}/p$ is in the center and
$V\cong{\mathbb{Z}}/p\times{\mathbb{Z}}/p$. Let us take generators such that
$C={\langle}c{\rangle},V={\langle}a,b{\rangle}.$ Moreover $[a,b]=c$ (when $G$
is non-abelian).

There are four cases, when $p=2$, the quaternion group $Q_{8}$ and the
dihedral group $D_{8}$. When $p\geq 3$, the extraspecial groups
$p_{-}^{1+2}=M$ and $p^{1+2}_{+}=E$.

We will show

######  Theorem 8.1.

Let $X=X(N)$ be an approximation for $Q_{8}$ or $D_{8}$. Then
$DH^{*}(X;{\mathbb{Z}}/2)=0$ for $*<N$.

8.1. The case $G=Q_{8}$. Then $p=2$ and $a^{2}=b^{2}=c$. Its cohomologies are
well known (see [Qu2])

$H^{*}(BG)/2\cong{\mathbb{Z}}/2[y_{1},y_{2},c_{2}]/(y_{i}^{2},y_{1}y_{2})\ \
|y_{i}|=2,$
$H^{*}(BG;{\mathbb{Z}}/2)\cong{\mathbb{Z}}/2[x_{1},x_{2},c_{2}]/(x_{1}x_{2}+y_{1}+y_{2},x_{1}y_{2}+x_{2}y_{1})$
$\cong{\mathbb{Z}}/2\\{1,x_{1},y_{1},x_{2},y_{2},w\\}\otimes{\mathbb{Z}}/2[c_{2}]$

where $x_{i}^{2}=y_{i}$ $|x_{i}|=1$, and $w=y_{1}x_{2}=y_{2}x_{1}$, $|w|=3$.

Therefore, we see

$H^{*}(BG;{\mathbb{Z}}/2)/(y_{1},y_{2},c_{2})\cong{\mathbb{Z}}/2\\{1,x_{1},x_{2}\\}.$

Of course $deg(x_{i})=(1,1)$ in $H^{*,*^{\prime}}(BG;{\mathbb{Z}}/2)$ and they
are not in $N^{1}H^{*}(BG;{\mathbb{Z}}/2)$. Thus we have Theorem 8.1 for
$G=Q_{8}$.

8.2. The case $G=D_{8}$. That is $p=2$ and $a^{2}=c,b^{2}=1$. It is well known

$H^{*}(BG)/2\cong{\mathbb{Z}}/2[y_{1},y_{2},c_{2}]/(y_{1}y_{2})\\{1,e\\}\quad
with\ |e|=3.$

The mod $2$ cohomlogy is written [Qu2]

$H^{*}(BG;{\mathbb{Z}}/2)\cong{\mathbb{Z}}/2[x_{1},x_{2},u]/(x_{1}x_{2})\quad(with\
|u|=2)$
$\cong(\oplus_{i=1}^{2}{\mathbb{Z}}/2[y_{i}]\\{y_{i},x_{i},y_{i}u,x_{i}u\\}\oplus{\mathbb{Z}}/2\\{1,u\\})\otimes{\mathbb{Z}}/2[c_{2}].$

Here $y_{i}=x_{i}^{2},u^{2}=c_{2}$ and
$Q_{0}(u)=(x_{1}+x_{2})u=e,Q_{1}Q_{0}(u)=(y_{1}+y_{2})c_{2}$.

We note $y_{1},y_{2},c_{2}\in CH^{*}(BG)/2$ and

$H^{*}(BG;{\mathbb{Z}}/2)/(y_{1},y_{2},c_{2})\cong(\oplus_{j=1}^{2}{\mathbb{Z}}/2\\{x_{j},x_{j}u\\})\oplus{\mathbb{Z}}/2\\{1,u\\}.$

Moreover, $deg(x_{j})=(1,1)$, $deg(u)=(2,2)$ in the motivic cohomology
$H^{*,*^{\prime}}(BG;{\mathbb{Z}}/2)$ and they are not in
$N^{1}H^{*}(BG;{\mathbb{Z}}/2)$. Here we note $deg(x_{j}u)=(3,3)$, but there
is $u_{j}^{\prime}\in H^{3,2}(BG;{\mathbb{Z}}/2)$ with $x_{j}u=\tau
u_{j}^{\prime}$ from Lemma 6.2 in [Ya3] ( i.e., $x_{i}u\in
N^{1}H^{*}(X;{\mathbb{Z}}/2)$).

Hence for the proof of Lemma 8.1 (for $G=D_{8}$), it is only need to show

######  Lemma 8.2.

We have $x_{i}u\in\tilde{N}^{1}H^{*}(BG;{\mathbb{Z}}/2)$.

To prove above lemma, for a $G$ variety $H$, we consider the equivariant
cohomology

$H_{G}^{*}(H;{\mathbb{Z}}/p)=H^{*}(U(N)\times_{G}H;{\mathbb{Z}}/p)$

where $U(N)$ is an (approximation of) contractible free $G$-variety. Let us
write

$X_{G}H=appro.\ of\ U(N)\times_{G}H\ so\ that\
H_{G}^{*}(H;{\mathbb{Z}}/p)\cong H^{*}(X_{G}H;{\mathbb{Z}}/p).$

For the injection $i:H\subset K$ of $G$-varieties, we can define the transfer

$i_{*}:H_{G}^{*}(H;{\mathbb{Z}}/p)\to H^{*}_{G}(K;{\mathbb{Z}}/p)\quad by\
i:X_{G}H\stackrel{{\scriptstyle id\times_{G}i}}{{\to}}X_{G}K.$

Hereafter in this section, let $G=D_{8}$. We recall arguments in [Ya3]. We
define the $2$-dimensional representation $\tilde{c}:G\to U_{2}$ such that
$\tilde{c}(a)=diag(i,-i)$ and $\bar{c}(b)$ is the permutation matrix $(1,2)$.
By this representation, we identify that
$W={\mathbb{C}}^{2*}={\mathbb{C}}^{2}-\\{0\\}$ is an $G$-variety. Note $G$
acts freely on $W\times{\mathbb{C}}^{*}$ but it does not so on
$W={\mathbb{C}}^{2*}.$

The fixed points set on $W$ under $b$ is

$W^{{\langle}b{\rangle}}=\\{(x,x)|x\in{\mathbb{C}}^{*}\\}={\mathbb{C}}^{*}\\{e^{\prime}\\},\quad
e^{\prime}=diag(1,1)\in GL_{2}({\mathbb{C}}).$

Similarly $W^{{\langle}bc{\rangle}}={\mathbb{C}}^{*}\\{a^{-1}e^{\prime}\\}.$
Take

$H_{0}={\mathbb{C}}^{*}\\{e^{\prime},ae^{\prime}\\},\quad
H_{1}={\mathbb{C}}^{*}\\{g^{-1}e^{\prime},q^{-1}ae^{\prime}\\}$

where $g\in GL_{2}({\mathbb{C}})$ with $g^{-1}bg=ab$ (note $(ab)^{2}=1$). Let
us write

$H=H_{0}\sqcup H_{1}.$

Then $G$ acts on $H_{i}$ and acts freely on ${\mathbb{C}}^{2*}-H$. In fact it
does not contain fixed points of non-trivial stabilizer groups.

We consider the transfer for some $G$-set $H$ in ${\mathbb{C}}^{2*}$, and
induced equivariant cohomology

$i_{*}:H^{*}_{G}(H;{\mathbb{Z}}/2)\to
H^{*}_{G}({\mathbb{C}}^{2*};{\mathbb{Z}}/2).$

######  Lemma 8.3.

We have

$H^{*}_{G}(H_{0};{\mathbb{Z}}/2)\cong{\mathbb{Z}}/2[y]\otimes\Lambda(x,z)\quad
with\ y=x^{2},\ |x|=|z|=1.$

###### Proof.

We consider the group extension $0\to{\langle}a{\rangle}\to
G\to{\langle}b{\rangle}\to 0$ and the induced spectral sequence

$E_{2}^{*,*^{\prime}}=H^{*}(B{\langle}b{\rangle};H^{*^{\prime}}_{{\langle}a{\rangle}}(H_{0};{\mathbb{Z}}/2))\Longrightarrow
H_{G}^{*}(H_{0};{\mathbb{Z}}/2).$

Since ${\langle}a{\rangle}\cong{\mathbb{Z}}/4$ acts freely on $H_{0}$, we see

$H_{0}/{\langle}a{\rangle}\cong{\mathbb{C}}^{*}\\{e^{\prime},ae^{\prime}\\}/{\langle}a{\rangle}\cong{\mathbb{C}}^{*}/{\langle}a{\rangle}\cong{\mathbb{C}}^{*}.$

Therefore we have

$H^{*}_{{\langle}a{\rangle}}(H_{0};{\mathbb{Z}}/2)\cong
H^{*}({\mathbb{C}}^{*}/{\langle}a{\rangle};{\mathbb{Z}}/2)\cong
H^{*}({\mathbb{C}}^{*};{\mathbb{Z}}/2)\cong\Lambda(z)\quad|z|=1.$

Since ${\langle}b{\rangle}$ acts trivially on $\Lambda(z)$ we have
$H^{*}_{G}(H;{\mathbb{Z}}/2)\cong
H^{*}(B{\langle}b{\rangle};{\mathbb{Z}}/2)\otimes\Lambda(z).$ ∎

Note $H_{G}^{*}(H_{0};{\mathbb{Z}}/2)\cong H_{G}^{*}(H_{1};{\mathbb{Z}}/2)$
and hence we see

$H^{*}_{G}(H;{\mathbb{Z}}/2)\cong\oplus_{j=1}^{2}{\mathbb{Z}}/2[y_{j}]\\{1_{j},y_{j},x_{j},x_{j}z_{j},z_{j}\\}.$

We consider the long exact sequence

$(*)\quad...\to H^{*}_{G}(\\{0\\};{\mathbb{Z}}/2)\stackrel{{\scriptstyle
i_{*}=c_{2}}}{{\to}}H^{*+4}_{G}({\mathbb{C}}^{2};{\mathbb{Z}}/2)\to
H^{*+4}_{G}({\mathbb{C}}^{2*};{\mathbb{Z}}/2)\to...$

and we have $H^{*}({\mathbb{C}}^{2*};{\mathbb{Z}}/2)\cong
H^{*}(BG;{\mathbb{Z}}/2)/(c_{2})$. Hence, we get

$H^{*}_{G}({\mathbb{C}}^{2*};{\mathbb{Z}}/2)\cong(\oplus_{j=1}^{2}{\mathbb{Z}}/2[y_{j}]\\{y_{j},x_{j},x_{j}u_{j}^{\prime},u_{j}^{\prime}\\})\oplus{\mathbb{Z}}/2\\{1,u\\}$

Now we consider the transfer
$H^{*}_{G}(H;{\mathbb{Z}}/2)\stackrel{{\scriptstyle
i_{*}}}{{\to}}H^{*+2}_{G}({\mathbb{C}}^{2*};{\mathbb{Z}}/2)$. We have
explicitly (page 527 in [Ya3])

$i_{*}(1_{j})=y_{j},\ i_{*}(x_{j})=y_{j}x_{j},\
i_{*}(x_{j}z_{j})=x_{j}u_{j}^{\prime},\ i_{*}(z_{j})=u_{j}^{\prime}.$

Therefore we have Theorem 8.1 for $G=D_{8}$.

To see the above fact we recall the long exact sequence for
$i;H\subset{\mathbb{C}}^{2*}$

$(**)\quad...\to
H^{*+1}_{G}({\mathbb{C}}^{2*}-H;{\mathbb{Z}}/2)\stackrel{{\scriptstyle\delta}}{{\to}}H^{*}_{G}(H;{\mathbb{Z}}/2)\stackrel{{\scriptstyle
i_{*}}}{{\to}}H^{*+2}_{G}({\mathbb{C}}^{2*};{\mathbb{Z}}/2)$
$\stackrel{{\scriptstyle
j^{*}}}{{\to}}H^{*+2}_{G}({\mathbb{C}}^{2*}-H;{\mathbb{Z}}/2)\to...$

The transfer $i_{*}$ is determined by the following lemma.

######  Lemma 8.4.

In the above $(**)$, we see $\delta=0$, and hence $i_{*}$ is injective.

###### Proof.

Since $G$ acts freely on ${\mathbb{C}}^{2*}-H$, we have

$H^{*}_{G}({\mathbb{C}}^{2*}-H:{\mathbb{Z}}/2)\cong
H^{*}(({\mathbb{C}}^{2*}-H)/G;{\mathbb{Z}}/2),$

which is zero when $*>4$. Hence $\delta$ must be zero for $*>4$, and $i_{*}$
is injective for $*>4$. In particular,
$i_{*}(y^{2}_{j}z_{j})=y_{j}^{2}u_{j}^{\prime}$. Since
$H^{*}_{G}(H;{\mathbb{Z}}/2)$ is ${\mathbb{Z}}/2[y_{1}]$-free (or
$Z/2[y_{2}]$-free,) we see $i_{*}(z_{j})=u_{j}^{\prime}$. ∎

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