数理解析研究所講究録
第 1784 巻 2012 年 113-118

113

EXAMPLES OF MOTIVIC COHOMOLOGY OF CLASSIFYINGSPACES (II)
茨城大学教育学部 柳田 伸顕 (NOBUAKI YAGITA)

FACULTY OF EDUCATION, IBARAKI UNIVERSITY

1. INTRODUCTIONLet
be a compact Lie group. Taking complexffication, we can identify thegroupas the reductive algebraic group over the complex number field .
The main result of this paper is the computation of the $mod p$ motivic cohomology$H^{**’}(BG;Z/p)$ of the classifying spaces of algebraic groups (over C) corresponding
Lie groups . We compute $H^{**}’(BG;Z/2)$ for $G=O_{n},$.
and$G$$G=G_{\mathbb{C}}$$\mathbb{C}$$G$$SO_{4},$

2. THE MOTIVIC COHOMOLOGY OF$Q_{8}$$D_{8}$$B\mathbb{Z}/p$

In this section we consider the relation between the motivic and the usual ordinary cohomologies. Let be or $Z/p$ . The motivic cohomology has the following
$R$

$Z$

properties ([6],[7],[9]).
$H^{**}’(X;R)$ is a bigraded multiplicative cohomology theory in (some(Cl)good) category $Spc$ of pointed (algebraic) spaces (the cohomology of a space meansthe reduced cohomology of a pointed space); For any map: $Xarrow Y$ in thecategory $Spc$ , we have the cofiber sequence $Xarrow Yarrow Y/X$ , which induces thelong exact sequence$f$$arrow H^{**^{l}}(X;R)arrow H^{**}’(Y;R)arrow H^{**’}(Y/X:R)arrow H^{*-1,*}’(X;R)arrow\ldots$(In particular, we get the Mayer-Vietoris, Gysin and blow up long exact sequences.)(C2) There are maps (realization maps)$t_{\mathbb{C}}^{m,n}$: $H^{m,n}(X;R)arrow H^{m}(X(\mathbb{C});R)$

which sum upthe natural ring homomorphism.
(C3) There are (the Bockstein, the reduced powers) operations$t_{\mathbb{C}’}^{**^{l}}=\oplus_{m,n}t_{\mathbb{C}}^{m,n}$$\beta:H^{**}(X;Z/p.)’arrow H^{*+1,*’}(X;Z/p)$’$P^{i_{:H^{**}(X;Z/p)}’},arrow H^{*+2(p-1)i,*+(p-1)i}(X;Z/p)$

which commutes with the realization map .
(C4) For the projective space “, there is an isomorphism$t_{\mathbb{C}}$$\mathbb{P}$$H^{**}(X\wedge(\mathbb{P}^{n}/\mathbb{P}^{n-1});R)\cong H^{**}(X;R)\{y’\}$with $deg(y’)=(2n, n)$ and.
(C5) For a smooth $X$ , if $H^{m,n}(X;R)\not\cong 0$ , then$t_{\mathbb{C}}(y’)\neq 0$$m\leq n+dim(X),$$m\leq 2n$

and $m\geq 0$ .

2000 Mathematics Subject Classification. Primary $14F42,20J06$ ; Secondary $55N22,57R77$ .
Key words and phrases. classifying space, motivic cohomology.

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N. YAGITA

For an element $x\in H^{m,n}(X;\mathbb{Z}/p)$ , we define the (weight and difference degree)$w(x)=2n-m$ , $d(x)=m-n$Hence for smooth $X,$ $w(x)\geq 0$ , and $d(x)\leq dim(X)$ for nonzero $x\in H^{**}’(X;Z/p)$ .
, we can define the weight degree $w(x’)$ which
Remark. For$w(x)$.
such thatis the least number ofLichtenbaum defined the similar cohomology $H_{L’}^{**}’(X;R)$ by using the \’etaletopology, while $H^{**’}(X;R)$ is defined by using Nisnevich topology. Since Nisnevich:
covers are some restricted \’etale covers, there is the natural (cyclic) map$H^{**}’(X;R)arrow H_{L’}^{**}’(X;R)$ . We say that the condition $BL(n,p)$ holds if$x^{l}\in H^{*}(X(\mathbb{C});\mathbb{Z}/p)$$t_{\mathbb{C}}(x)=x’$$d^{**’}$$BL(n,p)$:$H^{m,n}(X;Z_{(p)})\cong H_{L}^{m,n}(X;\mathbb{Z}_{(p)})$for $allm\leq n+1$
for $allm\leq n)$hence $H^{m,n}(X;Z/p)\cong H_{L}^{m,n}(X;Z/p)$
and all smooth $X$ . The Beilinson-Lichtenbaum conjecture is that $BL(n,p)$ holds
for all $n,$ . It is proved that the $BL(n,p)$ condition is equivalent the Bloch-Katoand prime . Recently, V.Voevodsky proved theconjecture (BK) for degree
Bloch-Kato conjecture [10]. Hence $BL(n,p)$ holds for all and .
.
Moreover Suslin-Voevodsky proves
Rom the dimensional condition (C5) and the above isomorphism, we have isomorphisms$($$p$

$n$

$p$

$n$

$p$

$H_{L}^{m,n}(X;Z/p)\cong H_{et}^{m}(X;\mu_{p}^{\otimes n})$$H^{m,n}(Spec(k);\mathbb{Z}/p)=$$H^{m,n}(pt;Z/p)\cong H_{et}^{m}(pt;\mu_{p}^{\otimes n})\cong H_{et}^{m}(pt;\mathbb{Z}/p)$$ifm\leq n$

and $H^{m,n}(pt;\mathbb{Z}/p)\cong 0$ otherwise. Let $\tau\in H^{0,1}(pt;Z/p)$ be the element corresponding a generator of $H_{et}^{0}(pt;\mu_{p})\cong Z/p$. Then we get the isomorphism$H^{**}(Spec(k);Z’/p)\cong H_{et}^{*}(Spec(k);Z/p)\otimes \mathbb{Z}/p[\tau]$

since$\tau$:$H_{et}^{m}(pt;\mu_{p}^{\otimes n})\cong H_{et}^{m}(pt;\mu_{p}^{\otimes(n+1)})$$H^{**}(Spec(\mathbb{R});Z/2)\cong Z/2[\rho, \tau]’$,. For examples, with $deg(\rho)=(1,1)$ ,$H^{**’}(Spec(\mathbb{C});\mathbb{Z}/p)\cong \mathbb{Z}/p[\tau]$.

and $BZ/p$ . By the cofiber map
Next we compute the motivic cohomology ofand (C4), we can inductively prove that$\mathbb{P}^{\infty}$$\mathbb{P}^{n-1}arrow \mathbb{P}^{n}arrow \mathbb{P}^{n}/\mathbb{P}^{n-1}$$H^{**’}(\mathbb{P}^{\infty};\mathbb{Z}/p)\cong H^{**^{l}}(pt.;Z/p)\otimes Z/p[y]$with $deg(y)=(2,1)$ . The Lens space is identified with the sphere bundle associatedwith the line bundle. This induces the ring isomorphism for $p=odd$’$H^{**’}(BZ/p;\mathbb{Z}/p)\cong Z/p[y]\otimes\Lambda(x)\otimes H^{**}(pt;Z/p)$with $deg(x)=(1,1)$ . However note that when $p=2$ , we see ([8])$x^{2}=y\tau+x\rho$

where $\rho\in H^{1,1}(pt;Z/p)\cong k^{*}/k^{2*}$ represents $-1$ .
satisfy theandBy the above cofiber sequence, we can easily see thatgeneralholdfornot(whileformuladoes
KunnethKunneth formula for all spaces$X,$
in the $mod p$ motivic cohomology). In particular, we have the ring isomorphisms$\mathbb{P}^{\infty}$$B\mathbb{Z}/p$$Y$$H^{**’}((\mathbb{P}^{\infty})^{n};\mathbb{Z}/p)\cong \mathbb{Z}/p[y_{1}, \ldots, y_{n}]\otimes H^{**’}(pt;\mathbb{Z}/p)$,$H^{**}((BZ/p)^{n};Z/p)\cong Z/p[y_{1}, \ldots, y_{n}]\otimes\Lambda(x_{1}, \ldots, x_{n})\otimes H^{**’}(pt;\mathbb{Z}/p)’$$($when $p=2,$$x_{i}^{2}=y_{i}\tau+x_{i}\rho)$.

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MOTIVIC COHOMOLOGYThis fact is used to define the reduced power operationin (C3). Since a
Sylow subgroup of the symmetric groupof p-letters is isomorphic to $Z/p$ , weknow the isomorphism$P^{i}$$p$$S_{p}$$H^{**}’(BS_{p};Z/p)\cong H^{**’}(BZ/p;\mathbb{Z}/p)^{F_{\rho}^{*}}\cong Z/p[Y]\otimes\Lambda(W)\otimes H^{**’}(pt;Z/p)$with identifying $Y=y^{p-1}$ and $W=xy^{p-2}$ . If $X$ is smooth (and suppose isodd to simplify arguments), we can define the reduced powers (of Chow rings) as
follows. Consider maps$p$$H^{2*,*}(X;Z/p)arrow^{i_{!}}H^{2p*,p*}(X^{p}\cross s_{p}ES_{p})arrow^{\Delta^{*}}$$H^{2*,*}(X\cross BS_{p};\mathbb{Z}/p)\cong H^{2*,*}(X;Z/p)\otimes_{H^{2*,*}(pt;Z/p)}H^{2*,*}(BS_{p};Z/p)$

is the Gysin map for p-th external power, and$deg(x)=(2n, n)$ , the reduced powers are defined aswhere$i_{!}$$\triangle$

is the diagonal map. For$\triangle^{*}i_{!}(x)=\sum P^{i}(x)\otimes Y^{n-i}+\beta P^{i}(x)\otimes WY^{n-i-1}$Hence note $deg(P^{i})=deg(Y^{i})=deg(y^{i(p-1)})=(2i(p-1), i(p-1))$ .
Voevodsky defined for non smooth $X$ also. By using suspensions maps, hedefined reduced powers for all degree elements in $H^{**}(X;\mathbb{Z}/p)$ for all $X[8]$ . Thus
we get the operations in (C3).
Moreover Voevodsky defined the motivic Milnor operation such that $Q_{i}=$$[Q_{i-1}, P^{p^{-1}}]mod(\rho)$ (fordetails see [8])$i_{!}$$Q_{i}$: $H^{**’}(X;Z/p)arrow H^{*+,*+}2p^{i}-1’p^{t}-1(X;Z/p)$

which is derivative, $Q_{i}(xy)=Q_{i}(x)y+xQ_{i}(y)$ if $\rho=0$ . For the caseor [7].

3. MOTIVIC COHOMOLOGY OF$BO_{n}$$\rho\neq 0$

see [14]

AND $BSO_{4}$ .

The motivic cohomology of the classifying space is defined as follows. Let be alinear algebraic group over . Let be a representation of such that acts freely
on $V-S$ for some closed subset . Then $(V-S)/G$ exists as a quasi-projective
variety over . According to Totaro [?] and V.Voevodsky ([6]), we define
$G$

$k$

$V$

$G$

$G$

$S$

$k$$($$)$$H^{**^{l\prime}}(BG; \mathbb{Z}/p)=\lim_{dim(V),codim(S)arrow\infty}H^{**}((V-S)/G;Z/p)$.

We still know the motivic cohomologies of $BG_{m}$ and $BZ/p$ . Since $BGL_{n}$ iscellular, we have$H^{**’}(BGL_{n};Z/p)\cong Z/p[c_{1}, \ldots, c_{n}]\otimes H^{**}’(pt;Z/p)$

where the Chern classwith $deg(c_{i})=(2i, i)$ are identified with the elementary
symmetric polynomial in. So we can define the Chern class$\rho^{*}(c_{i})\in H^{2*,*}(BG;\mathbb{Z}/p)$ foreach representation : $Garrow GL_{n}$ .
Hereafter we assume
throughout this paper.
The $mod 2$ cohomology of the classifying space $BO_{n}$ of the n-th orthogonalgroup is$c_{i}$$H^{2*,*}((\mathbb{P}^{\infty})^{n};\mathbb{Z}/p)$$\rho$$k=\mathbb{C}$$H^{*}(BO_{n};Z/2)\cong H^{*}((BZ/2)^{n};Z/2)^{s_{n}}\cong Z/2[w_{1}, \ldots, w_{n}]$whereis the n-th symmetry group,is the Stiefel-Whiteney class which restricts the elementary symmetric polynomial in. Each element
isrepresented by Chern class$O(n)\subsetU(n)$of the induced representation. Let uswriteby .
$S_{n}$$w_{i}$$Z/2[x_{1}, \ldots, x_{n}]$$c_{i}$$w_{i}^{2}$$c_{i}$$w_{i}^{2}$

116
N YAGITA, we see each $w(w_{i})=i$ . However even the module
seems complicated. W.S.Wilson ([11],[1]) found astructure ofgood $Q(i)=\Lambda(Q_{0}, \ldots, Q_{i})$ -module decomposition for $BO_{n}$ , namely,Since$Q_{i-1}\ldots Q_{0}(w_{i})\neq 0$$gr^{*}H^{*}(BO_{n};Z/2)$$H^{*}(BO_{n};\mathbb{Z}/2)=\oplus_{i=-1}Q(i)G_{i}$Here$G_{k-1}$with.

$Q_{0}\ldots Q_{i}G_{i}\in \mathbb{Z}/2[c_{1}, \ldots, c_{n}]$is quite complicated, namely, it is generated by symmetric functions$\Sigma x_{1}^{2i_{1}+1}\ldots x_{k}^{2i_{k}+1}x_{k+1}^{2j_{1}}\ldots x_{k+q}^{2j_{q}}$,$k+q\leq n$ ,is; and if the number of equal toandwith$2i_{s}+2^{s}<2j_{u}<2i_{s}+2^{s+1}$$s\leqk$prove. We cansuch that
odd, then there is some$w(G_{i})=i+1$ and hence;$0\leq i_{1}\leq\ldots\leq i_{k}$$j$$0\leq j_{1}\leq\ldots\leq j_{q}$$j_{u}$

Theorem 3.1. An element $x\in H^{*}(BO_{n};Z/2)$ is $w(x)=s$ if and only if is the. Moreover we havemaximal number such thatfor somethe isomorphisms$s$$Q_{i_{1}}\ldots Q_{i_{s}}(x)\neq 0$$(i_{1}, .., i_{s})$$H^{**’}(BO_{n};Z/2)\cong H^{**’}((BZ/2);Z/2)^{s_{n}}\cong Z/2[\tau]\otimes(\oplus Q(i)G_{i})$When $n=odd$ , it is well known that there is the isomorphism
Hence we have the isomorphism$H^{**’}(BSO_{2m+1};Z/2)\cong Z/2[\tau]\otimes(\oplus Q(j)G_{j}’)$with.

$O_{n}\cong SO_{n}\cross \mathbb{Z}/2$.

$G_{j}’=i^{*}G_{j}$where : $SO_{n}arrow O_{n}$ is the inclusion. (Note $p^{*}i^{*}(w_{n})\neq w_{n}\in H^{*}(BO_{n};\mathbb{Z}/2)$ for
the projection $p:O_{n}arrow SO_{n}.$ )is complicated to write, we only write
Since the direct decomposition ofnote $O_{3}\cong SO_{3}\cross Z/2)$ .
here that of$i$$BO_{3}$$SO_{3}$$($$H^{*}(BSO_{3};Z/2)\cong Z/2[w_{1}, w_{2},w_{3}]/(w_{1})\cong Z/2[w_{2}, w_{3}]$$\cong Z/2[c_{2}, c_{3}]\{1, w_{2}, w_{3}=Q_{0}w_{2}, w_{2}w_{3}=Q_{1}w_{2}\}$$\cong \mathbb{Z}/2[c_{2}, c_{3}]\{w_{2}, Q_{0}w_{2}, Q_{1}w_{2}, c_{3}=Q_{0}Q_{1}w_{2}\}\oplus Z/2[c_{2}]$$\cong \mathbb{Z}/2[c_{2}, c_{3}]\otimes Q(1)\{w_{2}\}\oplus Z/2[c_{2}]$.

Of course, this case $w(w_{2})=2$ and we have$H^{**’}(BSO_{3};Z/2)\cong \mathbb{Z}/2[\tau]\otimes(\mathbb{Z}/2[c_{2}, c_{3}]\otimes Q(1)\{w_{2}\}\oplus Z/2[c_{2}])$.

. The motivic cohomology seems difficult to
For $n=$ even,is notcompute. Even $n=4$ it seems complicated. In fact, the realization map
in the following theorem).
injective (i.e.,$O_{n}\not\cong SO_{n}\cross \mathbb{Z}/2$$t_{\mathbb{C}}$$\tau\cross y_{2}=t_{\mathbb{C}}(y_{2})=0$

Theorem 3.2. The motivic cohomology $H^{**’}(BSO_{4};Z/2)$ is isomorphic to$\mathbb{Z}/2[c_{2}, c_{4}]\{y_{2}\}\oplus Z/2[\tau, c_{2}]\otimes(\mathbb{Z}/2[c_{4}]\{1\}$$\oplus \mathbb{Z}/2[c_{3}]\otimes Q(1)\{w_{2}\}\oplus \mathbb{Z}/2[c_{4}]\otimes(Z/2[c_{3}]Q(2)-Z/2\{1\})\{a\})$where$w_{2}w_{4}$is a virtual element so thatand $t_{C}(Q_{2}a)=c_{4}w_{2}$ .
$a$$t_{\mathbb{C}}(c_{3}a)=w_{2}w_{3}w_{4},$$t_{\mathbb{C}}(Q_{0}a)=w_{4},$$t_{C}(Q_{1}a)=$117

MOTIVIC COHOMOLOGY

4. MOTIVIC COHOMOLOGY OF$BD_{8}$

AND $BQ_{8}$In this section, we compute the mod (2) motivic cohomology of $BD_{8}$ and $BQ_{8}$ .
At first, we consider the case
. The $mod 2$ (usual) cohomology is well known(see Theorem 2.7)
$Q_{8}$$H^{*}(BQ_{8};\mathbb{Z}/2)\cong Z/2\{1, x_{1}, y_{1}, x_{2}, y_{2}, w\}\otimes \mathbb{Z}/2[c_{2}]$whereand $|w|=3$ . The graded algebra $gr^{*}H^{*}(BQ_{8};\mathbb{Z}’/2)$ is given
by letting the weight degree by$x_{i}^{2}=\beta x_{i}=y_{i}$$w(y_{i})=w(c_{2})=0$ ,$w(x_{i})=w(w)=1$ .

The facts $w(y_{i})=w(c_{2})=0$ follows from that they are Chern classes. We canprove that $w(w)=1$ (in fact, we can take $w\in H^{3,2}(BQ_{8};Z/2).$ )Theorem 4.1. We have the bidegree isomorphism.

$H^{**}’(BQ_{8};Z/2)\cong Z/2[\tau]\otimes gr^{*^{l}}H^{*}(BQ_{8};Z/2)$

Now we consider the case $G=D_{8}$ . We recall the mod (2) cohomology.
$H^{*}(BD_{8};\mathbb{Z}/2)\cong(\mathbb{Z}/2[x_{1}, x_{2}]/(x_{1}x_{2}))\otimes Z/2[u]\cong$$(\oplus_{i=1}^{2}\mathbb{Z}/2[y_{i}]\{y_{i}, x_{i}, y_{i}u, x_{i}u\}\oplus Z/2\{1, u\})\otimes Z/2[c_{2}]$Here we identify,
and $c_{2}=u^{2}$ . The cohomology operations on $H^{*}(BD_{8};Z/2)$
is well known, e.g., (see [Te-Ya])$y_{i}=x_{i}^{2}$$Q_{0}(u)=(x_{1}+x_{2})u=e$ ,Lemma 4.2. There exist$(so u_{i}’=\tau^{-1}x_{i}u)$ .

$u_{1}’,$$Q_{1}Q_{0}(u)=(y_{1}+y_{2})c_{2}$$u_{2}’\in H^{3,2}(BD_{8};Z/2)$.

with $\tau u_{i}’=x_{i}u\in H^{3,3}(BD_{8};Z/2)$

Therefore we get $gr^{*}H^{*}’(BD_{8};Z/2)$ which is isomorphic to$(\oplus_{i=1}^{2}Z/2[y_{i}]\{y_{i}, x_{i}, x_{i}u_{i}^{l}, u_{i}’\}\oplus Z/2\{1, u\})\otimes Z/2[c_{2}]$with $w(y_{i})=w(c_{2})=0,$ $w(x_{i})=w(u_{i}’)=1$ and $w(u)=w(x_{i}u_{i}’)=2$ . (Note$u,$ $x_{i}u_{i}’\not\in CH^{*}(BG)/2$ , and $x_{i}u_{i}’=y_{i}u$ ).

Theorem 4.3. We have the the bidegree module isomorphism$H^{**}’(BD_{8};Z/2)\cong Z/2[\tau]\otimes gr^{*^{l}}H^{*}(BD_{8};Z/2)$.

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$p^{3}$

DEPARTMENT OF MATHEMATICS, FACULTY OF EDUCATION, IBARAKI UNIVERSITY, MITO, IBARAKI,JAPAN. ibarak i. ac. jp
E-mail address: yag it$a\emptyset mx$