Motivic cohomology of twisted flag varieties

TL;DR

This paper investigates the mod(p) motivic cohomology of twisted flag varieties over specific fields where higher Milnor K-theory vanishes, providing new insights into their algebraic structure.

Contribution

It offers a novel computation of motivic cohomology for twisted flag varieties over fields with restricted Milnor K-theory, expanding understanding in algebraic geometry.

Findings

Computed mod(p) motivic cohomology for twisted flag varieties

Identified conditions on fields affecting cohomology calculations

Enhanced understanding of algebraic structures in motivic cohomology

Abstract

In this paper, we study the mod(p) motivic cohomology of twisted complete flag varieties over some restricted fields k. Here we take k such that the mod(p) Milnor K-theory KM_i(k)/p=0 for i>3.

Equations

$$H^{\ast }(G_k/B_k;\mathbb{Z}/p)\cong H^{2\ast }(G/T;\mathbb{Z}/p)\otimes \mathbb{Z}/p[\tau ]$$

$$M(X)\cong R({\mathbb{G}}_k)\otimes ({\oplus }_s{\mathbb{T}}^{\otimes s})$$

$$C{H}^{\ast }(R_a)\cong {\mathbb{Z}}_{(p)}\{1\}{\oplus }_{i=1}^{p-1}{M}_n(y^i).$$

$$gr{H}^{\ast }(G;\mathbb{Z}/p)\cong \mathbb{Z}/p[y]/(y^p)\otimes \Lambda (x_1,...,x_{\ell })$$

$$(G,p)=\{\begin{array}{l}{G}_2,F_4,E_6,Spin(7),Spin(8),Spin(9)forp=2\\ F_4,E_6,E_{7}forp=3\\ E_{8}p=5.\end{array}$$

$$E_2^{\ast ,{\ast }^{\prime }}\cong H^{\ast }(G;\mathbb{Z}/p)\otimes H^{\ast }(BT)$$

$$\cong \mathbb{Z}/p[y]/(y^p)\otimes \Lambda (x_1,...,x_{\ell })\otimes S(t)⟹H^{\ast }(G/T;\mathbb{Z}/p)$$

$$gr{H}^{\ast }(G/T;\mathbb{Z}/p)\cong \mathbb{Z}/p[y]/(y^p)\otimes S(t)/(b_1,...,b_{\ell }).$$

$$H^{\ast }(G/T;\mathbb{Z}/p)\cong (\mathbb{Z}/p[y]\otimes S(t)/(b,f_p),f_p=y^{p}mod(S(t{)}^{+}).$$

$$M(X)\cong R_2\otimes ({\oplus }_s{\mathbb{T}}^{\otimes s}).$$

$$H^{\ast ,{\ast }^{\prime }}({\oplus }_s{\mathbb{T}}^{\otimes s};\mathbb{Z}/p)\cong H^{\ast ,{\ast }^{\prime }}(pt.;\mathbb{Z}/p)\otimes S(t)/(b).$$

$$C{H}^{\ast }(X)/p\cong \mathbb{Z}/p\{1,c_0(y),c_1(y),...,c_0(y^{p-1}),c_1(y^{p-1})\}\otimes S(t)/(b).$$

$$S(t)/(b_i{b}_j,b_k)\cong \mathbb{Z}/p\{1,b_1,...,b_{2p-2}\}\otimes S(t)/(b)$$

$$C{H}^{\ast }(X)/p\cong S(t)/(p,b_i{b}_j,b_k).$$

$$(3.1)0\to H^{\ast +1,\ast }({\chi }_a;\mathbb{Z}/p)\stackrel{\times \tau }{\to }{K}_{\ast +1}^M(k)/p\to K_{\ast +1}^M(k(V_a))/p$$

$$(3.2)M({\chi }_a(b_n))[2b_n]\to M_a\to M({\chi }_a)\stackrel{{\delta }_a=Q_0...Q_{n-1}(a^{\prime })}{\to }M({\chi }_a)(b_n)[2b_n+1]$$

$$M_a^i=S^i(M_a)=q_i(M_a^{\otimes i})\subset M_a^{\otimes i}$$

$$(3.3)M_a^{i-1}(b_n)[2b_n]\to M_a^i\to M({\chi }_a)\stackrel{s_i}{\to }{M}_a^{i-1}(b_n)[2b_n+1]$$

$$(3.4)M({\chi }_a)(b_{n}i)[2b_{n}i]\to M_a^i\to M_a^{i-1}\stackrel{r_i}{\to }M({\chi }_a)(b_{n}i)[2b_{n}i+1].$$

$$Q_i:H^{\ast ,{\ast }^{\prime }}(X;\mathbb{Z}/p)\to H^{\ast +2p^i-1,{\ast }^{\prime }+p^i}(X;\mathbb{Z}/p).$$

$$H^{\ast ,{\ast }^{\prime }}({\tilde{\chi }}_a;\mathbb{Z}/p)\cong K_{\ast }^M(k)/(Ker(a))\otimes Q(n)\otimes \mathbb{Z}/p[{\xi }_a]\{a^{\prime }\}$$

$$C{H}^{\ast }(M_a^{p-1})/p\cong \mathbb{Z}/p\{1\}\oplus \mathbb{Z}/p[t]/(t^{p-1})\{c_0,...,c_{n-1}\}$$

$${\tilde{H}}^{\ast ,{\ast }^{\prime }}=H^{\ast ,{\ast }^{\prime }}-\mathbb{Z}/p\{1\}={\oplus }_{(\ast ,{\ast }^{\prime })\ne (0,0)}{H}^{\ast ,{\ast }^{\prime }}(Spec(k),\mathbb{Z}/p).$$

$$Q_0(\tau )={\zeta }_p=0in{H}^{\ast ,{\ast }^{\prime }}(pt.;\mathbb{Z}/p).$$

$$H^{\ast ,{\ast }^{\prime }}(R_n;\mathbb{Z}/p)\cong H^{\ast ,{\ast }^{\prime }}\oplus (\tilde{Q}(n-1)\{a^{\prime }\}\oplus {\tilde{H}}^{\ast ,{\ast }^{\prime }}\{t\})\{1,t,...,t^{p-2}\}$$

$$0\to \tilde{Q}(n-1)\{a^{\prime }\}\otimes \mathbb{Z}/p[t]/(t^{p-1})\to H^{\ast ,{\ast }^{\prime }}(R_n;\mathbb{Z}/p)$$

$$\to {\tilde{H}}_{et}^{\ast }(R_n;\mathbb{Z}/p)\otimes \mathbb{Z}/p[\tau ]\cong H^{\ast ,{\ast }^{\prime }}[t]/(t^p)\to \mathbb{Z}/p[t{]}^{+}/(t^p)\to 0.$$

$$w(x)=2s.deg(x)-f.deg(x),d(x)=f.deg(x)-s.deg(x).$$

$$D_i^{\prime }=\{deg(x)|0\le w(x),2b_n(i-1)\le \ast <2b_{n}i,b_n(i-1)<d(x)\}$$

$$H^{\ast ,{\ast }^{\prime }}(R_n;\mathbb{Z}/p)\cong K\{1\}\oplus (K_a\otimes Q(n-1)\{a^{\prime }\}\oplus {}_{a}K\{t\})\otimes K[t]/(t^{p-1}).$$

Full text

1. Introduction

Let $G$ be a simply connected compact Lie group (and $T$ its maximal torus). For a field $k\subset{\mathbb{C}}$, let $G_{k}$ be the corresponding split reductive group over $k$ and $T_{k}$ its split torus. Let us write ${\mathbb{G}}_{k}$ a nontrivial $G_{k}$-torsor. Then ${\mathbb{G}}_{k}/B_{k}$ is a (twisted form of) the complete flag variety for a Borel subgoup $B_{k}\supset T_{k}$. Let us simply write by $X$ the flag variety ${\mathbb{G}}_{k}/B_{k}$. We will study the motivic cohomology $H^{*,*^{\prime}}(X;{\mathbb{Z}}/p)$ for twisted $X$.

For non-twisted cases, note that it is well known

[TABLE]

where $0\not=\tau\in H^{0,1}(Spec(\bar{k});{\mathbb{Z}}/p).$ The cohomology $H^{*}(G/T;{\mathbb{Z}}/p)$ (for the topological flag manifold) are completely known by Toda, Watanabe, Nakagawa [Tod], [Tod-Wa], [Na].

In this paper, we give examples (which are the first for odd prime $p$) such that the ring structures of $H^{*,*^{\prime}}(X;{\mathbb{Z}}/p)$ are computed for twisted complete flag varieties $X$.

Petrov, Semenov and Zainoulline [Pe-Se-Za] show that the motive $M(X)$ of $X$ is decomposed as a sum of (degree changing) generalized Rost motive $R({\mathbb{G}}_{k})$, namely,

[TABLE]

where ${\mathbb{T}}^{\otimes s}$ is the (degree changing) Tate motive.

Hence we can compute $H^{*,*^{\prime}}(X;{\mathbb{Z}}/p)$ additively if $H^{*,*^{\prime}}(R({\mathbb{G}}_{k});{\mathbb{Z}}/p)$ can be computed, while it is a difficult problem in general cases. We say that a Lie group $G$ is of type $(I)$ if it is simply connected and $H^{*}(G;{\mathbb{Z}}/p)$ contains only one even dimensional truncated polynomial generator. In this case, $R({\mathbb{G}}_{k})$ is the original Rost motive $R_{2}$ ([Ro1],[Vo1,4],[Su-Jo]) defined from a pure symbol in the Milnor $K$-theory $K_{3}^{M}(k)/p$), and their Chow groups are known ([Me-Su], [Ya1,4]). Moreover we can compute also ring structure of $CH^{*}(X)/p$ (see [Ya7]).

The motivic cohomology $H^{*,*^{\prime}}(R_{n};{\mathbb{Z}}/p)$, of course, depends on the field $k$. It is known for $k={\mathbb{R}}$ and $p=2$. Hence we can compute $H^{*,*^{\prime}}(X;{\mathbb{Z}}/2)$ in this case. We also compute its ring structure (Theorem 6.7) by using the result from [Ya7]. As an application, we study Barlmer’s Witt group $W^{*}(X)$ for $k={\mathbb{R}}$.

It seems no literature to compute $H^{*,*^{\prime}}(R_{n};{\mathbb{Z}}/p)$ for an odd prime $p$. Hence we write down it (Theorem 3.3) when $K_{n+2}^{M}(k)/p=0$. (Examples of such fields are high dimensional local fields defined by K. Kato [Ka].) Using these, we give the ring structure of $H^{*,*^{\prime}}(X;{\mathbb{Z}}/p)$ (Theorem 3.12) when $K_{4}^{M}(k)/p=0$ (and of course $K_{3}^{M}(k)/p\not=0$). We also compute the $mod(p)$ algebraic cobordism $MGL^{*,*^{\prime}}(X;{\mathbb{Z}}/p)$ for these fields $k$.

The plan of this paper is the following. In $\S 2$, we recall the motives of flag varieties. In $\S 3$, we compute $H^{*,*^{\prime}}(R_{n};{\mathbb{Z}}/p)$ when $K_{n+2}^{M}(k)/p=0$, and compute $H^{*,*^{\prime}}(X;{\mathbb{Z}}/p)$ for $G$ is type $(I)$. In $\S 4$, we study algebraic cobordism version. In $\S 5$ we study the $BP{\langle}n-1{\rangle}^{*}$-theory version, which gives a short another proof for the result in $\S 4$. In $\S 6$, we compute $H^{*,*^{\prime}}(X)/2$ when $k={\mathbb{R}}$, and in $\S 7$, we study the Balmer Witt group $W^{*}(X)$.

2. Lie groups and algebraic groups

Let $p$ be a prime number and $k$ field with $k\subset{\mathbb{C}}$. Let $X$ be an algebraic variety over $k$ such that $\bar{X}=X\otimes_{k}\bar{k}$ is cellular for the algebraic closure $\bar{k}$ of $k$. Let $H^{*,*^{\prime}}(X;{\mathbb{Z}}/p)$ (resp. $CH^{*}(X)=CH^{*}(X)_{(p)}$) be the mod($p$) motivic cohomology (resp. Chow ring localized at $p$) over $k$. Note that $H^{*,*^{\prime}}(\bar{X};{\mathbb{Z}}_{(p)})$ is torsion free, since $\bar{X}$ is cellular.

Let $a$ be a pure symbol in the mod $p$ Milnor $K$-theory $K_{n+1}^{M}(k)/p$. By Rost, we can construct the norm variety $V_{a}$ of $dim(V_{a})=p^{n}-1$ such that $\bar{V}_{a}\cong v_{n}$ is cellular (where $v_{n}$ is a generator of $BP^{*}$), and $a=0\in k_{*}^{M}(k(V_{a}))/p$ for the function filed $k(V_{a})$ of $V_{a}$.

Rost and Voevodsky showed that there is $y\in CH^{b_{n}}(\bar{V}_{a})$ for $b_{n}=(p^{n}-1)/(p-1)$ such that $y^{p-1}$ is the fundamental class of $\bar{V}_{a}$. We know ([Ro1], [Vo1,4], [Me-Su], [Ya4])

Theorem 2.1**.**

For a nonzero pure symbol $a\in K_{n+1}^{M}(k)/p$, there is an irreducible (Rost) motive $R_{a}$ ( write by $R_{n}$ simply) in the motive $M(V_{a})$ of the norm variety $V_{a}$ such that \ CH^{*}(\bar{R}_{a})\cong{\mathbb{Z}}_{(p)}[y]/(y^{p}),\ and

[TABLE]

Here $M_{n}(y^{i})={\mathbb{Z}}_{(p)}\{c_{0}(y^{i})\}\oplus{\mathbb{Z}}/p\{c_{1}(y^{i}),....,c_{n-1}(y^{i})\}$ with $|c_{j}(y^{i})|=|y^{i}|-2(p^{i}-1)$, where ${\mathbb{Z}}/p\{a,b,...\}$ is a ${\mathbb{Z}}$-free module generate by $a,b,...$

Let $G$ be a simply connected compact Lie group (and $T$ its maximal torus) and $G_{k}$ be the corresponding split reductive group over $k$ and $T_{k}$ its split torus. Let ${\mathbb{G}}_{k}$ be a nontrivial $G_{k}$-torsor, and ${\mathbb{G}}_{k}/B_{k}$ the (twisted form of) complete flag variety. Let us simply write by $X$ the twisted flag variety ${\mathbb{G}}_{k}/B_{k}$, We are interesting in the motivic cohomology $H^{*,*^{\prime}}(X;{\mathbb{Z}}/p)$. (For a Borel subgroup $B\supset T$, note $G_{k}/B_{k}$ is cellular and $H^{*,*^{\prime}}(G_{k}/B_{k};{\mathbb{Z}}/p)\cong H^{2*}(G/T;{\mathbb{Z}}/p)$.)

We only consider here the cases

[TABLE]

where the degree are $|y|=even$, and $|x_{i}|=odd$. The following simple Lie groups satisfy the above isomorphism ;

[TABLE]

We call that these (simply connected) groups $G$ are of type $(I)$. It is known $\ell\geq 2(p-1)$.

There is a fibering $G\to G/T\to BT$ and the induced spectral sequence (see for details, [Tod]. [To-Wa], [Mi-Ni], [Na])

[TABLE]

[TABLE]

where $S(t)=H^{*}(BT)\cong{\mathbb{Z}}_{(p)}[t_{1},...,t_{\ell}]$ with $|t_{i}|=2$. Then it is known ([Tod]) that there is a regular sequence $(b_{1},...,b_{\ell})$ in $S(t)$ such that $d_{|x_{i}|+1}(x_{i})=b_{i}$ for the differential in the spectral sequence. Hence we can write

[TABLE]

Let us write $S(t)/(p,b_{1},...,b_{\ell})$ simply by $S(t)/(b)$.

Theorem 2.2**.**

([Tod]) We have the ring isomorphism

[TABLE]

Petrov, Semenov and Zainoulline [Pe-Se-Za] prove that when $G$ is a group of type $(I)$, the motive $M(X)$ (which is localized at $p$) is decomposed as a sum of (degree changing) Rost motive $R_{2}$, namely,

[TABLE]

where ${\mathbb{T}}^{\otimes s}$ is the (degree changing) Tate motive. Hence we easily see that $CH^{*}(\oplus_{s}{\mathbb{T}}^{\otimes s})/p\cong S(t)/(b)$, and

[TABLE]

Hence additively $H^{*,*^{\prime}}(X;{\mathbb{Z}}/p)\cong H^{*,*^{\prime}}(R_{2};{\mathbb{Z}}/2)\otimes S(t)/(b).$ In particular, we have additively

[TABLE]

On the other hand, the following additive isomorphism is immediate

[TABLE]

where $i,j,k$ range $1\leq i,j\leq 2p-2<k\leq\ell$. We can take $c_{1}(y^{i})=b_{2i-1}$ and $c_{0}(y^{i})=b_{2i}$, and prove the following theorem.

Theorem 2.3**.**

([Ya7]) Let $G$ be a simple Lie group of type $(I)$. Let $X={\mathbb{G}}_{k}/T_{k}$ for a nontrivial torsor ${\mathbb{G}}_{k}$. Then we have the ring isomorphism

[TABLE]

where $i,j,k$ range $1\leq i,j\leq 2(p-1)<k\leq\ell.$

3. motivic cohomology of the Rost motive

Recall $V_{a}$ is a norm variety for a nonzero symbol $a\in K_{n+1}^{*}(k)$. Let $\chi_{a}=\check{C}(V_{a})$ be the Cech simplex sheaf for $V_{a}$ and $\tilde{\chi}_{a}$ be the object defined by the cofibering $\tilde{\chi}\to\chi\to pt$ in the stable ${\mathbb{A}}^{1}$-homotopy category.

By the solution of Bloch-Kato conjecture by Voevodsky, we see the exact sequence

[TABLE]

identifying $H^{*+1,*+1}(\chi_{a};{\mathbb{Z}}/p)\cong K_{*+1}^{M}(k)$. Since $a=0\in K_{n+1}^{M}(k(V_{a}))/p$, there is unique element $a^{\prime}\in H^{n+1,n}(\chi_{a};{\mathbb{Z}}/p)$ such that $\tau a^{\prime}=a$.

Let $M_{a}$ be the object in the derived category ${\mathbb{D}}{\mathbb{M}}$ of mixed motives [Vo2,4] defined by the following distinguished triangle

[TABLE]

where $b_{n}=(p^{n}-1)/(p-1)=p^{n-1}+...+p+1$ so that $deg(\delta_{a})=(2b_{n}+1,b_{n})$. For $i<p$, define the symmetric powers

[TABLE]

where $q_{i}=(1/i!)\sum_{\sigma\in S_{i}}\sigma$ is a projector in ${\mathbb{D}}{\mathbb{M}}$, and $\sigma$ is the symmetric group of $i$ letters. One of the important results in [Vo2,4] Voevodsky proved is that $M_{a}^{p-1}$ is a direct summand of a motive of $V_{a}$ (for details see [Vo2,4]). This $M_{a}^{p-1}$ is written by $R_{a}$ (and is called the Rost motive) in preceding sections.

Hence there are distinguished triangles ( (5.5),(5.6) in [Vo4])

[TABLE]

[TABLE]

For each prime $p$, we have the Milnor operation

[TABLE]

Let us write by $Q(n)$ the exterior algebra $\Lambda(Q_{0},...,Q_{0})$. (We see $Q_{i}Q_{j}=-Q_{j}Q_{i}$ for all $p$ and $Q_{i}^{2}=0$ also for $p=2$.)

Theorem 3.1**.**

(Theorem 8.5 in [Ya4]) Let $0\not=a=(a_{0},...,a_{n})\in K_{n+1}^{M}(k)/p$. Then there is a $K_{*}^{M}(k)\otimes Q(n)$-modules isomorphism

[TABLE]

where $\xi_{a}=Q_{n}Q_{n-1}....Q_{0}(a^{\prime})$ and $deg(a^{\prime})=(n+1,n)$.

By using the above theorem we have Theorem 2.1 in $\S 2$.

Corollary 3.2**.**

Let $c_{i}=Q_{0}...\hat{Q}_{i}...Q_{n-1}(a^{\prime})$ (hence $|c_{i}|=2(b_{n}-p^{i}+1)$). Then there is the additive isomorphism

[TABLE]

where $deg(t)=(2b_{n},b_{n})$.

Remark. Note that $t^{i}$ is a vertical element, i.e., $t^{i}$ itself does not exist in $CH^{*}(M_{a}^{p-1})/p$ but $c_{j}t^{i}$ exists.

Let $K_{1}^{M}(k)/p$ be finite, e.g., generated by $a_{1},...,a_{m}$. When $p$ is odd, $K_{*}^{M}/p$ is isomorphic to a quotient of the exterior algebra $\Lambda(a_{1},...,a_{m})$. So $K_{i}^{M}(k)/p=0$ for all $i>m$. Hereafter we consider some ease cases such that $K_{*}^{M}(k)/p=0$ for $*>n+1$ and hence $K_{*}^{M}(k)/(Ker(a))={\mathbb{Z}}/p$.

Let us write $H^{*,*^{\prime}}=H^{*,*^{\prime}}(Spec(k);{\mathbb{Z}}/p)\cong K_{*}^{M}(k)/p[\tau]$ and

[TABLE]

We assume that a primitive $p^{2}$-th root of unity exists in $k$, that is each primitive $p$-th root $\zeta_{p}$ is zero in $K_{1}^{M}(k)/p$. Hence we see

[TABLE]

We have the additive structure of $H^{*,*^{\prime}}(M_{a}^{p-1};{\mathbb{Z}}/p)\cong H^{*,*^{\prime}}(R_{n};{\mathbb{Z}}/p)$ by the following theorem.

Theorem 3.3**.**

Let $k$ contain a primitive $p^{2}$-th root of the unity. Let $a\in K_{n+1}(k)/p$ be a nonzero pure symbol, and $K_{*}^{M}(k)/p=0$ for $*>n+1$. Then there is an additive isomorphism

[TABLE]

where $\tilde{Q}(n-1)=Q(n-1)-{\mathbb{Z}}/p\{Q_{0}...Q_{n-1}\}.$

Recall that $H_{et}^{*}(X;{\mathbb{Z}}/p)\cong H^{*,*}(X;{\mathbb{Z}}/p)$. Since $\tau t\in\tilde{H}^{*,*^{\prime}}\{t\}$ and $\tau Q_{i}(a^{\prime})=Q_{i}(a)=0$, we can see

Corollary 3.4**.**

We have $H_{et}^{*}(R_{n};{\mathbb{Z}}/p)\cong K_{*}^{M}(k)[t]/(p,t^{p}).$

This fact itself is easily seen from $H^{*}_{et}(\chi_{a};{\mathbb{Z}}/p)\cong H^{*}_{et}(pt.;{\mathbb{Z}}/p)$. Let us write by $\tilde{H}_{et}^{*}(R_{n};{\mathbb{Z}}/p)$ the same ring $H_{et}^{*}(R_{n};{\mathbb{Z}}/p)\cong K_{*}^{M}(k)[t]/(t^{p})$ with $deg(t)=(2b_{n},b_{n})$ (but not $(2b_{n},2b_{n}))$.

Corollary 3.5**.**

We have the exact sequence

[TABLE]

[TABLE]

For ease of notations, for $x\in H^{*,*^{\prime}}(Y;{\mathbb{Z}}/p)$, let us write by $f.deg(x)=*,$ and $s.deg(x)=*^{\prime}$, and define

[TABLE]

Then it is well known ([Vo1,2]) that if $Y$ is smooth and $0\not=x\in H^{*,*^{\prime}}(Y;{\mathbb{Z}}/p)$, then $0\leq w(x)$ and $0\leq d(x)\leq dim(Y).$

Remark. Let us consider the following triangular domain in ${\mathbb{Z}}\times{\mathbb{Z}}$ generated by bidegree $(*,*^{\prime})$

[TABLE]

and $D^{\prime}=\cup_{j=1}^{p-1}D_{j}^{\prime}$. Then we know in Lemma 8.7 in [Ya4] the following fact. Let us write $K=K_{*}^{M}(k)$, ${}_{a}K=Ker(a))$ in $K$ and $K_{a}=K/K_{a}$. Then for bidegree $(*,*^{\prime})\in D^{\prime}$ defined above, we have the $K$-module isomorphism,

[TABLE]

So the above isomorphism shows the preceding theorem when $(*,*^{\prime})\in D^{\prime}$, since $K\cong H^{*,*^{\prime}}$ in $D^{\prime}$.

To prove the above theorem, we need some lemmas and arguments. Let us assume that $p$ is an odd prime ($p=2$ case is proved similarly). Let us write $\chi_{a}$ (resp. $M_{a}^{i},\delta_{a},\xi_{a}$) by simply $\chi$ (resp. $M^{i},\delta,\xi$). Define

[TABLE]

so that $H^{*,*^{\prime}}(Y;{\mathbb{Z}}/p)=\sum_{i=-1}^{\infty}D_{i}(Y)$ and $D_{i}\cap D_{j}=0$ for $i\not=j$. Note $|\xi|=2b_{n+1}=2pb_{n}$. To see arguments easily, here we use notations that \begin{CD}@>{D_{i}}>{a_{i}}>or@>{a_{i}}>{D_{i}}>\end{CD} simply means $a_{i}\in D_{i}$, e.g.,

[TABLE]

From Theorem 3.1 and $K_{*}^{M}(k)/Ker(a))\cong{\mathbb{Z}}/p$, we have $D_{i}(\chi)\cong D_{i+pm}(\chi)$ for $\ i\geq 1,$ and $D_{i}(\chi)\not=0$ only if $i=-1,0\ mod(p)$.

On the other hand, when $Y=M^{p-1}$, we see $D_{i}=0$ for $i\geq p-1$

[TABLE]

We first compute $H^{*}(M;{\mathbb{Z}}/p)$. Consider the exact sequence induced from (3.3)

[TABLE]

Hence we have the isomorphism

[TABLE]

The map $\delta:\chi\to\chi(b_{n})[2b_{n}+1]$ in ${\mathbb{D}}{\mathbb{M}}$ is given by the cup product $x\mapsto x\cup\delta$ for an element $\delta\in H^{2b_{n}*1,b_{n}}(\chi;{\mathbb{Z}}_{(p)})$. In fact, we note

[TABLE]

[TABLE]

Note that from $Q_{i}^{2}=0$, the map multiplying $\delta=Q_{n-1}...Q_{0}(a^{\prime})$ is a $Q(n-1)$-module map, while is not a $Q_{n}$-map.

Lemma 3.6**.**

In $H^{*,*^{\prime}}(\chi;{\mathbb{Z}}/p)$, we have $\delta\cdot Q_{n}(a^{\prime})=\xi\cdot a^{\prime}$.

Proof.

We have

[TABLE]

So $\delta\cdot Q_{n}(a^{\prime})=\xi\cdot a^{\prime}+Q_{n}(b)$ for some $b$ since $H^{*,*^{\prime}}(\tilde{\chi};{\mathbb{Z}}/p)$ is $Q(n)$-free. But $b=\sum\lambda_{J}Q_{J}(a^{\prime})$ for some $\lambda_{J}\in{\mathbb{Z}}/p$ and $Q_{J}\in Q(n-1)$. Since $|b|=|\delta a^{\prime}|$, we see $\lambda=0\in{\mathbb{Z}}/p$. ∎

The map $\times\delta:H^{*,*^{\prime}}(\chi;{\mathbb{Z}}/p)\to H^{*,*^{\prime}}(\chi;{\mathbb{Z}}/p)$ is written as the following

[TABLE]

which is not a commutative diagram, and it means only $\delta\cdot 1=\delta$, $\delta\cdot a^{\prime}=0$, $\delta\cdot\delta=0,$ .., $\delta\cdot Q_{n}(a^{\prime})=a^{\prime}\xi$,…

Recall that $Q(n-1)\{a^{\prime}\}/\delta\cong\tilde{Q}(n-1)\{a^{\prime}\}$. We also recall $\delta\cdot Q(n-1)Q_{n}\{a^{\prime}\}\cong Q(n-1)\{a^{\prime}\xi\}$ from the above lemma. Using these, we can compute

[TABLE]

[TABLE]

where $t$ is a vertical element with $deg(t)=(2b_{n},b_{n})$. We can write $H^{*,*^{\prime}}(M;{\mathbb{Z}}/p)$ as

[TABLE]

Here note $Q_{n-1}...Q_{0}(Q_{n}(a^{\prime}))=\xi$.

Lemma 3.7**.**

We have the isomorphism

[TABLE]

[TABLE]

Next, we compute $H^{*,*}(M^{i};{\mathbb{Z}}/p)$ for $1<i<p-1$. We consider the long exact sequence induced from (3.4)

[TABLE]

Recall that $D_{j}(\chi)=0$ for $j\not=-1,0\ mod(p)$. Hence

[TABLE]

The element $r_{i}^{*}$ is represented by an element in $H^{2b_{n}i+1,b_{n}i}(M^{i-1};{\mathbb{Z}}/p)$ from

[TABLE]

[TABLE]

Hence $H^{*,*}(M^{i};{\mathbb{Z}}/p)\cong Coker(r_{i}^{*})\oplus Ker(r_{i}^{*})\{t^{i}\}$. We first study in the case $i=2<p-1$. Note that the map $\times\delta t=\times Q_{n-1}...Q_{0}(a^{\prime}t)$ is a $Q(n-1)$-module map.

Lemma 3.8**.**

we have $r_{2}^{*}(1)=\delta t$ and $r_{2}^{*}(Q_{n}(a^{\prime}))=a^{\prime}t\xi$. (I.e. $\delta tQ_{n}(a^{\prime})=a^{\prime}t\xi$.)

Proof.

By the exact sequence, we know

[TABLE]

Therefore if $0\not=\delta t\in H^{*,*^{\prime}}(M^{2};{\mathbb{Z}}/p)$, then so is in $H^{*,*^{\prime}}(M^{p-1};{\mathbb{Z}}/p)$ which is a sub-motive of $V_{a}$. This is a contradiction, since $w(\delta)=(2s.deg-f.deg)(\delta)=-1<0.$

The case $r_{2}^{*}(Q_{n}(a^{\prime})$ is proved similarly, by using $w(a^{\prime}\xi)<0$. ∎

Remark. This lemma is immediate consequence from Lemma 3.5, if $t$ exists, however, which is not correct.

The map $r_{2}^{*}:H^{*,*^{\prime}}(\chi;{\mathbb{Z}}/p)\to H^{*,*^{\prime}}(M;{\mathbb{Z}}/p)$ is given as follows

[TABLE]

Using this lemma, we can compute

[TABLE]

[TABLE]

[TABLE]

For $Y=M^{2}$, we see

[TABLE]

Thus we get $H^{*,*^{\prime}}(M^{2};{\mathbb{Z}}/p)$. By using induction for $0\leq i\leq p-3$, we have $H^{*,*^{\prime}}(M^{p-2};{\mathbb{Z}}/p)$ also. We have $H^{*,*^{\prime}}(M^{p-2};{\mathbb{Z}}/p)$

[TABLE]

Lemma 3.9**.**

We have the isomorphism

[TABLE]

[TABLE]

At last we compute $H^{*,*^{\prime}}(M^{p-1};{\mathbb{Z}}/p)$ by using the exact sequence induced from (3.3) for $i=p-1$ and the following lemma. In particular $Ker(r_{p-1}^{*})\{t^{p-1}\}\cong\tilde{H}^{*,*^{\prime}}\{t^{p-1}\}$.

Lemma 3.10**.**

The map $r_{p-1}^{*}$ is given by $1\mapsto\delta t^{p-2},$

[TABLE]

Proof.

All elements in the right hand side in the above maps must disappear in $H^{*,*^{\prime}}(M^{p-1})$. For example,

[TABLE]

[TABLE]

∎

Remark. There seems a relation something like $\delta\cdot(\delta t^{p-2})=\xi$.

The map $r_{p-1}^{*}:H^{*,*^{\prime}}(\chi;{\mathbb{Z}}/p)\to H^{*,*^{\prime}}(M^{p-2};{\mathbb{Z}}/p)$ is given as follows

[TABLE]

where $1\leq i\leq p-3$. The map $r_{p-1}^{*}$ is injective only except for $H^{*,*^{\prime}}\{1\}$ (namely for $*>n+1$). We see that $Ker(r_{p-1}^{*})\cong\tilde{H}^{*,*^{\prime}}t^{p-1}$. Then we have $H^{*,*^{\prime}}(M^{p-1})\cong Coker(r_{p-1}^{*})\oplus H^{*,*^{\prime}}t^{p-1}$. So we have proved Theorem 3.3.

Lemma 3.11**.**

Let $b=Q_{n-1}...Q_{1}(a^{\prime}t^{i-1})$ in $H^{*,*^{\prime}}(M^{p-1};{\mathbb{Z}}/p)$. Then $b$ can be lift to the element $H^{*,*^{\prime}}(M^{p-1};{\mathbb{Z}}/p^{2})$ with $b=pt^{i}$.

Proof.

Recall the exact sequence

[TABLE]

The element $ht^{i}\in H^{*,*^{\prime}}(M^{i};{\mathbb{Z}}/p)$ is defined by $\delta_{i}^{*}(ht^{i})=h$ when $r_{i}^{*}(ht^{i-1})=\delta t^{i-1}h=0$.

Since $\delta t^{i-1}$ is the $Q_{0}$-image, we know $p\delta t^{i-1}=0$ in $H^{*,*^{\prime}}(M^{i};{\mathbb{Z}}/p^{2})$. Hence there is an element $pt^{i}\in H^{*}(M^{i};{\mathbb{Z}}/p^{2})$, which is ${\mathbb{Z}}/p^{2}$-module generator (since $t^{i}$ itself does not exist). So $pt^{i}$ represents nonzero element in $H^{*,*^{\prime}}(M^{i};{\mathbb{Z}}/p)$.

On the other hand, from Theorem 3.3, we see $H^{2b_{n}i,b_{n}i}(M^{i};{\mathbb{Z}}/p)\cong{\mathbb{Z}}/p\{b\}$. Hence we can take $pt^{i}=b$. ∎

Let $G$ be a Lie group of type $(I)$ and $X={\mathbb{G}}_{k}/B_{k}$. Using the case $n=2$, we see that $H^{*.*^{\prime}}(X;{\mathbb{Z}}/p)$ is additively isomorphic to

[TABLE]

Here we have $Q_{0}(a^{\prime}y^{i-1})=b_{2i}$, $Q_{1}(a^{\prime}y^{i-1})=b_{2i-1}$. By the preceding lemma, we can take the $i$-th product $y^{i}$ in $H^{*,*^{\prime}}(\bar{X};{\mathbb{Z}}/p)$ as the additive generator $t^{i}$ in $H^{*,*^{\prime}}(\bar{M}^{p-1};{\mathbb{Z}}/p)\cong{\mathbb{Z}}/p[y]/(y^{p})$.

Theorem 3.12**.**

Let $k$ be a field in Theorem 3.3. Let $G$ be a group of type of $(I)$, and $X={\mathbb{G}}_{k}/B_{k}$ be a twisted flag variety. Then $H^{*,*^{\prime}}(X;{\mathbb{Z}}/p)$ is isomorphic to the $H^{*,*^{\prime}}$-subalgebra generated by

[TABLE]

in $H^{*,*^{\prime}}[1,a^{\prime},y]\otimes S(t)/(R_{1}^{\prime},R_{2}^{\prime},R_{3}^{\prime})$ where

[TABLE]

[TABLE]

[TABLE]

Proof.

The relation $R_{1}$ holds, since so does in $CH^{*}(X)$. Since $K_{+}^{M}(k)a=0$, we have $K_{+}^{M}(k)a^{\prime}=0$ by using (3.3). Then we have

[TABLE]

We also have $\tau b_{i}=\tau Q_{i}(a^{\prime})=Q_{i}(\tau a^{\prime})=Q_{i}(a)=0.$ Since $f.deg(a^{\prime})=3$, we have $(a^{\prime})^{2}=0$ for $p$ odd, and so $b_{i}a^{\prime}=Q_{i}(a^{\prime})a^{\prime}=1/2Q_{i}((a^{\prime})^{2})=0$. When $p=2$, we have $a^{2}=a\rho=0$, and $b_{i}a^{\prime}=0$ from $|b_{i}a^{\prime}|>|y|$. Thus we have $R_{2}$.

For $R_{3}$, we only see the last element. We have in $CH^{*}(X)/p$

[TABLE]

Therefore from $R_{2}$, we see $\tilde{H}^{*,*^{\prime}}y^{p}=0$. ∎

Corollary 3.13**.**

We have an isomorphism

[TABLE]

Remark. As examples of the fields satisfying the assumption of Theorem 3.3, 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

[TABLE]

Let $k_{0}$ be a finite field, and let $k_{1},...,k_{n}$ be the sequence of complete discrete valuation fields such that the residue field of $k_{i}$ is $k_{i-1}$ for each $1\leq i\leq n$. Then the field $k=k_{n}$ obtained in this way is called an $n$-dimensional local field (see [Ka]). Then

[TABLE]

4. $Ah^{*,*^{\prime}}(Y)=ABP^{*,*^{\prime}}(Y;{\mathbb{Z}}/p)$

In this section, we study the algebraic cobordism for the field $k$ with $K_{n+2}^{M}(k)/p=0$. Moreover we assume that $k$ contains a $p^{2}$-th root of the unity. Throughout this section, we assume the above $k,p$, that is,

[TABLE]

.

Let $Y$ be a smooth algebraic variety over $k$. We study the $BP$ version of the algebraic cobordism

[TABLE]

[TABLE]

[TABLE]

defined by Voevodsky and Levine-Morel ([Vo1], [Le-Mo1,2], [Ya1,4]), with the coefficient ring $BP^{*}={\mathbb{Z}}_{(p)}[v_{1},v_{2},...]$, where $|v_{n}|=-2(p^{n}-1)$.

The Chow ring $CH^{*}(Y)$ is written as a quotient of $\Omega^{*}(Y)$ ([Vo1],[Le-Mo1,2]) that is,

[TABLE]

Let us write by $ABP^{*,*^{\prime}}(Y;{\mathbb{Z}}/p)$ the mod $p$ $ABP$-theory so that

[TABLE]

and $ABP^{2*,*}(Y;{\mathbb{Z}}/p)\cong\Omega^{2*}(Y)/p$. We simply write

[TABLE]

Lemma 4.1**.**

If $n+1\leq 2p-1$, then $Ah^{*,*^{\prime}}(pt.)\cong H^{*,*^{\prime}}\otimes BP^{*}$ where $pt.=Spec(k)$ and $H^{*,*^{\prime}}=H^{*,*^{\prime}}(pt.;{\mathbb{Z}}/p)$.

Proof.

Consider the Atiyah-Hirzebruch spectral sequence

[TABLE]

It is known the first (possible) non zero differential is

[TABLE]

Here $H^{*,*^{\prime}}=0$ for $*>n+1$ from the assumption for $k$. Recall $H^{0,i}(pt.;{\mathbb{Z}}/p)$ is generated by $\tau^{i}$. When $p$ is odd, $Q_{i}$ is a derivation. and we see $Q_{1}(\tau^{i})=0$ since $Q_{1}(\tau)=0$. For $p=2$, we have $Q_{1}(\tau^{2})=\rho^{3}$ (Lemma 6.3 in [Ya1]). Since $\rho=-1=0\in K_{1}^{M}(k)/2$, we see $d_{3}=0$. Hence assumption of this lemma, this spectral sequence collapses. ∎

Hereafter this section we assume that $k$ is a field $k_{*}^{M}(k)/p\cong 0$ for $*>n+1$ and $2p-1>n+1$ and $\zeta_{p^{2}}\in k$. Therefore $h^{*,*^{\prime}}(X)$ is an $H^{*,*^{\prime}}\otimes BP^{*}$-algebra in this case.

Remark. When $p=2$ and $k={\mathbb{R}}$, the algebra $Ah^{*,*^{\prime}}(Spec({\mathbb{R}}))$ is quite complicated ([Ya1]).

For smooth $Y_{1},Y_{2}$ and $\theta\in Ah^{2*,*}(Y_{1}\times Y_{2})$ for $*=dim(Y_{2})$, we can define

[TABLE]

where $pr_{i}:Y_{1}\times Y_{2}\to Y_{i}$ is the $i$-th projection. For the projector $p$ (i.e., $p=f_{\theta}$ with $p\cdot p=p$), we can define the motive $M=(Y,p)$ with $Ah^{*,*^{\prime}}(M)=p\cdot Ah^{*,*^{\prime}}(Y)$. Thus we have the category $Ah$-motives.

Lemma 4.2**.**

For smooth $Y$ and a motive $M\subset M(Y)$, we have the Atiyah-Hirzebruch spectral sequence

[TABLE]

Proof.

(See Lemma 7.1 in [Ya4].) We only need to see that $d_{r}f_{\theta}=f_{\theta}d_{r}$ for the differential $d_{r}$ in the spectral sequence. For $\theta\in CH^{*}(Y\times Y)/p\subset Ah^{2*,*}(Y\times Y)$, we have

[TABLE]

The last equation follows $d_{r}(\theta)=0$, since $w(\theta)=0$.

We know (e.g. $\S 5$ in [Ya])

[TABLE]

where $Th_{Y}(1)\in H^{2m,m}(Th_{Y}(V);{\mathbb{Z}}/p)$ is the Thom class for some bundle $V$ over $Y$ and $i:{\mathbb{T}}^{m}\times Y\subset Th_{Y}(V)\times Y$. Since $w(Th_{Y}(1))=0$, we see $d_{r}(Th_{Y}(1)\cdot x)=Th_{Y}(1)\cdot d_{r}(x).$ Therefore $pr_{2*}$ commutes with $d_{r}$. Hence so does $f_{\theta}$. ∎

For the Rost motive $R_{n}$, we will study the Atiyah-Hirzebruch spectral sequence

[TABLE]

From Theorem 3.3, we still have

[TABLE]

where $\tilde{Q}(n-1)=Q(n-1)-{\mathbb{Z}}/p\{Q_{0}...Q_{n-1}\}$.

Lemma 4.3**.**

The nonzero differential $d_{r}$ in AHss $(*)$ has the form

[TABLE]

The proof of this lemma will give just before Corollary 4.8 below.

Lemma 4.4**.**

Let $c_{i}=Q_{0}...\hat{Q}_{i}...Q_{n-1}(a^{\prime})$ and

[TABLE]

Then the infinity term in the AHss $(*)$ is written as

[TABLE]

Proof.

Let us write $Q(i)^{\prime}=\Lambda(Q_{i},...,Q_{1})$. For $x\in\tilde{Q}(n-1)$, considering that $Q_{0}$ is contained in $x$ or not, we have the decomposition

[TABLE]

Considering $Q_{n-1}$ is contained in $x$ or not for $x\in\tilde{Q}(n-1)^{\prime}$, we have

[TABLE]

Continuing this argument, we can see that

[TABLE]

For a $BP^{*}\otimes Q(n-1)^{\prime}$-module $B$, let us write $E_{0}(B)=B$ and $E_{r+1}(B)=H(E_{r}(B);v_{r}\otimes Q_{r})$. Then we easily see by induction on $j$

[TABLE]

Hence we see that

[TABLE]

[TABLE]

[TABLE]

Therefore we get

[TABLE]

From Lemma 4.3, all differential are of the forms $d_{2p^{i}-1}=v_{i}\otimes Q_{i}$ (in particular $d_{r}(BP^{*}\otimes\tilde{H}^{*,*^{\prime}}y^{j})=0$), we have this lemma. ∎

We recall the following lemma to see the relations between $c_{i}$ in $Ah^{*,*^{\prime}}(R_{n})$.

Lemma 4.5**.**

(Corollary 3.4 in [Ya4]) Let $x\in CH^{*}(Y)\cong E_{\infty}^{2*,*,0}$ and $v_{s}x=0$ in $E_{\infty}^{2*,*,*^{\prime}}$ for the AHss converging to $ABP^{*,*^{\prime}}(Y)$. Then there is $b\in H^{*,*^{\prime}}(Y;{\mathbb{Z}}/p)$ with $Q_{s}(b)=x$ and a relation in $ABP^{2*,*}(Y)$

[TABLE]

with $x_{k}=Q_{k}(b)$ in $H^{2*,*}(Y;{\mathbb{Z}}/p)$ for all $k\geq s$.

Corollary 4.6**.**

In $ABP^{2*,*}(R_{n})$, we have $v_{i}c_{j}=v_{j}c_{i}$ $mod(I_{\infty}^{2})$. The restriction map $Res(c_{i})=v_{i}y\ mod(I_{\infty}^{2})$ in $ABP^{2*,*}(\bar{R}_{n})$.

Proof.

First note that $c_{i}$ exists in (the integral) $ABP^{2*,*}(R_{n})$, since $c_{i}$ exists in $H^{2*,*}(R_{2})$ (considering the integral AHss). Let $i>0$. Since $pc_{i}=0$ in the spectral sequence there is $x\in H^{*,*^{\prime}}(R_{2})$ such that $Q_{0}x=c_{i}$ and there is a relation

[TABLE]

By the dimensional reason, we see $x=Q_{n-1}...\hat{Q}_{i}..Q_{1}(a^{\prime})$. Hence the above equation is written as

[TABLE]

since $Q_{i}x=Q_{1}...Q_{n-1}(a^{\prime})=c_{0}$.

From lemma 3.11, we know $Res(c_{0})=py$ $mod(p^{2},BP^{<0})$. So $Res(c_{i})=v_{i}y$ since $ABP^{*,*^{\prime}}(\bar{R}_{n})$ is $BP^{*}$-free. The first relation is proved similarly by using $v_{j}c_{i}=0$ $j<i$ in the spectral sequence. ∎

Theorem 4.7**.**

We have an isomorphism

[TABLE]

Proof.

We have an isomorphism

[TABLE]

Therefore we get the result from Lemma 4.4 and Lemma 4.6. ∎

Proof of Lemma 4.3.

For an element $x=Q_{i}...Q_{1}Q_{0}(a^{\prime})$ in $E_{2p^{i}}$, the next nonzero differential is $d_{2p^{i+1}-1}$. Otherwise

[TABLE]

But by dimensional reason, there does not exist nonzero $x$ such that $|x|<|x^{\prime}|<|x|+|Q_{i+1}|$.

The elements $\tilde{H}^{*,*^{\prime}}(t^{j})$ is generated as a $BP^{*}\otimes H^{*,*^{\prime}}$-algebra by $\tau t^{j}$ and $bt^{j}$ for $b\in K_{1}^{M}(k)/p$. Since $w(bt^{j})=1$, the differential image $w(d_{r}(bt^{i}))=0$. Hence $d_{r}(bt^{j})\in\oplus_{i,k}BP^{*}/I_{i}\{c_{i}t^{k}\}$. But if $d_{r}(bt^{j})\not=0$, then it contradicts to the existence of the restriction map $gr(Res)$ in the proof of Theorem 4.7.

At last we consider $\tau t$. Since $\tau b_{1}t^{i}=0$ in the spectral sequence, $\tau b_{1}t^{i}=vc$ for $v\in BP^{<0}$ in $Ah^{*,*^{\prime}}(R_{n})$. Restrict it to $Ah^{*,*^{\prime}}(\bar{R}_{n})$, we have $\tau v_{1}t^{i+1}=v(c|_{\bar{k}})$. Since $Ah^{*,*^{\prime}}(\bar{R}_{n})$ is $BP^{*}/p[\tau]$-free, we see $v=v_{1}$ and $c|_{\bar{k}}=\tau t^{i+1}$. This means that $\tau t^{i+1}$ exists in $Ah^{*,*^{\prime}}(R_{n})$, and it is a permanent cycle. ∎

Corollary 4.8**.**

The motivic cobordism $Ah^{*,*^{\prime}}(R_{n})$ is isomorphic to the $Ah^{*,*^{\prime}}$-submodule of $Ah^{*,*^{\prime}}[t]/(t^{p})$ (for $Ah^{*,*^{\prime}}\cong BP^{*}\otimes H^{*,*^{\prime}}$) generated by

[TABLE]

We consider here the another cobordism theory

[TABLE]

so that $Ak^{*,*^{\prime}}\cong BP{\langle}n-1{\rangle}\otimes H^{*,*^{\prime}}\cong{\mathbb{Z}}/p[v_{1},...,v_{n-1}]\otimes H^{*,*^{\prime}}.$ In this case, we note $I_{n}\cong\tilde{k}^{*}=BP{\langle}n-1{\rangle}^{*}-{\mathbb{Z}}/p\{1\}$. Hence we can see

Corollary 4.9**.**

The motivic cobordism $Ak^{*,*^{\prime}}(R_{n})$ is isomorphic to the $Ak^{*,*^{\prime}}$-submodule of $Ak^{*,*^{\prime}}[t]/(t^{p})$ (for $Ak^{*,*^{\prime}}\cong BP{\langle}n-1{\rangle}^{*}\otimes H^{*,*^{\prime}}$) generated by

[TABLE]

where $\tilde{A}k^{*,*^{\prime}}=Ak^{*,*^{\prime}}-{\mathbb{Z}}/p.$

In the next section, we give an another short proof for this fact but assuming some properties for $Ak$-motives.

Proposition 4.10**.**

Let $G$ be a group of type of $I$. Let $X={\mathbb{G}}_{k}/B_{k}$ be a twisted flag variety. Then there is a filtration such that $grAh^{*,*^{\prime}}(X)$ is isomorphic to the $BP^{*}\otimes H^{*,*^{\prime}}$-subalgebra generated by

[TABLE]

in $BP^{*}\otimes H^{*,*^{\prime}}[y]/(y^{p})\otimes S(t)/(R_{1},R_{2}^{\prime\prime})$ where

[TABLE]

[TABLE]

Note that $\tau b_{2i-1}=0$ in AHss, but $\tau b_{2i-1}=v_{1}\tau y$ in $Ak^{*,*^{\prime}}(X)$.

Define the etale cobordism theory by $Ah^{*}_{et}(Y)=lim_{N}(\tau^{N}Ah^{*,*^{\prime}}(Y)).$

Corollary 4.11**.**

We have the isomorphism

[TABLE]

Recall that $AK(n)^{*,*^{\prime}}(Y)$ be the motivic Morava K-theory such that

[TABLE]

Corollary 4.12**.**

Suppose the same assumption for $X,k$ as Theorem 4.10. Then we have

[TABLE]

where $R^{K}_{1}=(b_{2i},b_{k}|1\leq i\leq p-1,\ 2p-1\leq k\leq\ell)$ and

[TABLE]

Proof.

In $Ah^{*,*^{\prime}}(X)$, we have $\tau b_{1}=\tau v_{1}y$. (Note $\tau y$ exists in $Ah^{*,*^{\prime}}(X)$, but $y$ itself does not.) Let new $y=v_{1}^{-1}b_{1}$ in $AK(1)^{*,*^{\prime}}(X)$. Then we have $b_{2i-1}=v_{1}y^{i}=v_{1}(v_{1}^{-1}b_{1})^{i}$ in $AK(1)^{*,*^{\prime}}(X)$. Of course it holds for this $y$ that $hb_{1}=hv_{1}y$ for $h\in\tilde{H}^{*,*^{\prime}}$. ∎

Recall $grK(1)^{*}(G/T)\cong K(1)^{*}[y]/(y^{p})\otimes S(t)/(b).$ Here $b_{2i-1}=v_{1}y^{i}$, in particular, $y=v_{1}^{-1}b_{1}$ in $K(1)^{*}(G/T)$. Thefore we see

[TABLE]

Hence $AK(1)^{*,*^{\prime}}(X)\cong H^{*,*^{\prime}}\otimes K(1)^{*}(G/T)$.

The graded algebra associated to the gamma filtration of the (topological $mod$ $K$-theory) $K^{*}(G/T;{\mathbb{Z}}/p)$ is given ([Ya6])

[TABLE]

[TABLE]

The associated graded algebra in the above corollary gives more strong information for the gamma filtration than $gr(1)_{\gamma}^{*}(X)$. For example, from $b_{1}^{2}=0\in gr(1)_{\gamma}^{*}(X)$, we see $b_{1}^{2}\in F_{\gamma}^{2|b_{1}|+2}$, but from $b_{1}^{2}=v_{1}b_{3}\in gr(K(1)^{*}(X))$, we see

[TABLE]

5. $Ak^{*,*^{\prime}}(Y)=ABP{\langle}n-1{\rangle}^{*,*^{\prime}}(Y;{\mathbb{Z}}/p)$

Throughout this section, we assume

[TABLE]

as the preceding section.

In algebraic topology, $BP{\langle}m{\rangle}^{*}(Y)$ is the cohomology theory with the coefficient ring $BP{\langle}m{\rangle}^{*}={\mathbb{Z}}_{(p)}[v_{1},...,v_{m}]$. Let us write

[TABLE]

Hence $Ak^{*,*^{\prime}}(pt)\cong H^{*,*^{\prime}}[v_{1},...,v_{n-1}]$. In this section we compute $Ak^{*,*^{\prime}}(R_{n})$ assuming the existence of some category of $Ak^{*.*^{\prime}}$-$motives$. The result is get quite easily, and of course, coincides with Corollary 4.9. For ease of notations, hereafter this section, we simply write

[TABLE]

Lemma 5.1**.**

We have $Ak^{*,*^{\prime}}(\chi)\cong Ak^{*,*^{\prime}}\oplus Ak^{*,*^{\prime}}(\tilde{\chi})$, and

[TABLE]

where $c=Q_{n-1}...Q_{1}(a^{\prime})$, $\delta=Q_{0}c$ and $\xi^{\prime}=Q_{n}(c),Q_{0}(\xi^{\prime})=\xi$.

Proof.

We consider AHss

[TABLE]

Here $H^{*,*^{\prime}}(\tilde{\chi};{\mathbb{Z}}/p)\cong Q(n)\{a^{\prime}\}\otimes{\mathbb{Z}}/p[\xi]$.

We consider the decomposition

[TABLE]

We easily see all differentials are of form $d_{2p^{r}-1}(x)=v_{r}\otimes Q_{r}(x)$. Hence we have

[TABLE]

[TABLE]

Here we used $BP{\langle}n-1{\rangle}^{*}/I_{n}\cong{\mathbb{Z}}/p[v_{1},...,v_{n-1}]/(v_{1},...,v_{n-1})\cong{\mathbb{Z}}/p.$ ∎

Elements $1,\delta^{\prime},\delta,\xi^{\prime},\xi$ in $Ak^{*,*^{\prime}}(\chi)$ are contained in $D_{i}$ as follows

[TABLE]

Recall the category $Ak$-motives in the preceding section. More strongly, we assume

Assumption 5.2**.**

There is a triangular category ${\mathbb{D}}{\mathbb{M}}(Ak)$ which contains the category of $Ak$-motives such that for $X\in{\mathbb{D}}{\mathbb{M}}(Ak)$, we can define $Ak^{*,*^{\prime}}(X)$, and for a cofibering $X\to Y\to Z$ in ${\mathbb{D}}{\mathbb{M}}(Ak)$, we have the induced exact sequence

[TABLE]

Moreover, $\chi$ is an object of ${\mathbb{D}}{\mathbb{M}}(Ak)$ and

[TABLE]

From the assumption we can define the map $\delta:\chi\to\chi(*^{\prime})[*]$ in the category ${\mathbb{D}}{\mathbb{M}}(Ak)$. We still know $\delta Q_{n}(a^{\prime})=a^{\prime}\xi$, which induces $\delta\xi^{\prime}=c\xi$.

The map $\times\delta:Ak^{*,*^{\prime}}(\chi;{\mathbb{Z}}/p)\to Ak^{*,*^{\prime}}(\chi;{\mathbb{Z}}/p)$ is written as

[TABLE]

We ca easily see $Coker(\delta)\cong Ak^{*,*^{\prime}}\oplus{\mathbb{Z}}/p\{c\}\oplus{\mathbb{Z}}/p[\xi]\{\xi^{\prime},\xi\},$ and

[TABLE]

Lemma 5.3**.**

We have the isomorphism

[TABLE]

Next, we compute $Ak^{*,*}(M^{i};{\mathbb{Z}}/p)$ for $1<i<p-1$. As lemma 3.9, we can compute

Lemma 5.4**.**

We have the isomorphism

[TABLE]

[TABLE]

At last we compute $Ak^{*,*^{\prime}}(M^{p-1})$. Recall Lemma 3.9 and the map $r_{p-1}^{*}$ is given in $H^{*,*^{\prime}}(M^{p-2};{\mathbb{Z}}/p)$ by

[TABLE]

Hence, in $Ak^{*,*^{\prime}}(M^{p-2})$, we have $1\mapsto\delta t^{p-2}$  $c\mapsto\xi^{\prime}$,    $\delta\mapsto\xi$,  $\xi^{\prime}\mapsto c\xi,$   $\xi\mapsto\delta t^{p-2}\xi$. The map $r_{p-1}^{*}$ is given as follows.

[TABLE]

Note that $Ker(r_{p-1}^{*})\{t\}\cong\tilde{A}k^{*,*^{\prime}}t^{p-1}$. Using these, we see the following theorem with Assumption 5.2, while it also follows from Corollary 4.9 (without Assumption 5.2).

Theorem 5.5**.**

We have the isomorphism

[TABLE]

Remark. When $Ak^{*,*^{\prime}}(X)=ABP{\langle}n{\rangle}^{*,*^{\prime}}(X)$, we see that $\delta_{a}$ does not exist in $Ak^{*,*^{\prime}}(\chi)$. Hence we can not do the above arguments.

6. motivic cohomology over ${\mathbb{R}}$

When $p=2$, important cases have the property with $K_{*}^{M}(k)/2\not=0$ for all $*\geq 0$. Let ${\mathbb{R}}$ be the field of real numbers. Let $\rho=-1\in K_{1}^{M}({\mathbb{R}})/2\cong{\mathbb{R}}^{\times}/({\mathbb{R}}^{\times})^{2}.$ Then it is well known

[TABLE]

Hence the motivic cohomology is given as

[TABLE]

where $0\not=\tau\in H^{0,1}(Spec({\mathbb{R}}))/2.$

Let $a=\rho^{n+1}\in K_{n+1}^{M}({\mathbb{R}})/2$. Since $K_{*}^{M}({\mathbb{R}})/Ker(a)\cong{\mathbb{Z}}/2[\rho]$, we have from Theorem 3.1

[TABLE]

where $\xi=\delta_{a}^{2}$. Note $H_{et}^{*}(\chi_{a};{\mathbb{Z}}/2)\cong H_{et}^{*}({\mathbb{R}};{\mathbb{Z}}/2)\cong{\mathbb{Z}}/2[\rho]$.

Lemma 6.1**.**

Let $Q^{{\epsilon}}=Q_{0}^{{\epsilon}_{0}}...Q_{n-1}^{{\epsilon}_{n-1}}$ for ${\epsilon}=({\epsilon}_{0},...,{\epsilon}_{n-1})$ with ${\epsilon}_{i}=0$ or $1$. Then $H^{*}(\chi_{a};{\mathbb{Z}}/2)[\tau,\tau^{-1}]\cong{\mathbb{Z}}/2[\rho,\tau,\tau^{-1}]$, and we have

[TABLE]

where $d({\epsilon})=\sum_{i}{\epsilon}_{i}2^{i}$ and $f({\epsilon})=\sum_{i}{\epsilon}_{i}(2^{i+1}-1)$.

Proof.

By definition, we see $Q_{0}\tau=\rho$. The $Q_{0}$ action is a derivative and we have

[TABLE]

hence $Q_{0}(\tau^{-1})=\rho\tau^{-2}$. We have $Q_{1}(\tau^{-1})=\rho^{3}\tau^{-3}$ by using that the coproduct $\psi$ is given (Proposition 13.4 in [Vo])

[TABLE]

Similarly, we can prove the lemma (for details see proof of Lemma 4.6 in [Ya]). ∎

Lemma 6.2**.**

The motivic cohomology $H^{*,*^{\prime}}(\chi_{a};{\mathbb{Z}}/2)$ has no $\tau$-torsion elements and, it is the ${\mathbb{Z}}/2[\rho,\tau]$-submodule of ${\mathbb{Z}}/2[\rho,\tau,\tau^{-1}]$ generated by

[TABLE]

Proof.

We can define a $Q(n-1)$-module map $j:H^{*}(\chi_{a};{\mathbb{Z}}/2)\to{\mathbb{Z}}/2[\rho,\tau,\tau^{-1}]$ by $j(a^{\prime})=\tau^{-1}\rho^{n+1}$ and so $j(Q^{{\epsilon}}(a^{\prime}))=\tau^{-d({\epsilon})-1}\rho^{f({\epsilon})+n+1}$. This map is injective from $(*)$ and $Q(n)=\Lambda(Q_{n})\otimes Q(n-1)$ is $Q(n-1)$-free. In fact $Q^{{\epsilon}}(a^{\prime})$ generate free ${\mathbb{Z}}/2[\rho]$-modules in ${\mathbb{Z}}/2[\rho,\tau,\tau^{-1}]$. ∎

Lemma 6.3**.**

We have $\delta a^{\prime}=Q_{n}(a^{\prime})$ and $\delta Q_{n}(a^{\prime})=a^{\prime}\xi$.

Proof.

The first equation follows from the fact that $H^{*,*^{\prime}}(\tilde{\chi};{\mathbb{Z}}/2)$ is a $Q(n-1)$-free and

[TABLE]

Then $\delta Q_{n}(a^{\prime})=\delta^{2}a^{\prime}=\xi a^{\prime}$. ∎

The map $\times\delta:H^{*,*^{\prime}}(\chi;{\mathbb{Z}}/2)\to H^{*,*^{\prime}}(\chi;{\mathbb{Z}}/2)$ is written as

[TABLE]

Theorem 6.4**.**

(Theorem 1.3 in [Ya2]) The motivic cohomology $H^{*,*^{\prime}}(R_{n};{\mathbb{Z}}/2)$ is isomorphic to the ${\mathbb{Z}}/2[\rho,\tau]$-subalgebra of

[TABLE]

generated by $1,\ \ Q^{{\epsilon}}(a^{\prime})=\tau^{d({\epsilon})-1}\rho^{f({\epsilon})+n+1}$.

Corollary 6.5**.**

We have the ${\mathbb{Z}}/2[\rho,\tau]$-module isomorphism

[TABLE]

[TABLE]

identifying $a^{\prime}=\tau^{-1}\rho^{3}$, $Q_{0}a^{\prime}=\tau^{-2}\rho^{4}$, $Q_{1}a^{\prime}=\tau^{-3}\rho^{6}$.

Corollary 6.6**.**

We have a ${\mathbb{Z}}/2[\tau]$-module isomorphism

[TABLE]

The following theorem is shown for $G=G_{2}$ in Corollary 5.5 in [Ya3].

Theorem 6.7**.**

Let $X={\mathbb{G}}_{k}/B_{k}$ for $G$ of type $(I)$. Then we have the isomorphism for $*\geq*^{\prime}$

[TABLE]

where $R_{1}=(b_{i}b_{j},b_{k}|1\leq i,j\leq 2<k<\ell)$, $R_{2}=(\rho^{7},\rho^{3}b_{1},\rho b_{2})$ and

[TABLE]

Proof.

In $H^{*,*^{\prime}}(X;{\mathbb{Z}}/2)$, the relation $R_{1}$ also holds since so does in $CH^{*}(X)/2\cong H^{2*,*}(X;{\mathbb{Z}}/2)$. By dimensional reason, we can take $b_{1}=Q_{0}a^{\prime}+\lambda t$ for $0\not=t\in S(t)/(b)$ and $\lambda\in{\mathbb{Z}}$. Since $b_{1}|_{{\mathbb{C}}}=Q_{0}a^{\prime}|_{{\mathbb{C}}}=0$ but $t|_{{\mathbb{C}}}\not=0$, we have $\lambda=0\in{\mathbb{Z}}/2$. Similarly we can take $Q_{1}a^{\prime}=b_{2}$. Hence $R_{2}$ also holds in $H^{*,*^{\prime}}(X;{\mathbb{Z}}/2)$. We can take $a^{\prime}$ satifying $R_{3}$ from $H^{*,*^{\prime}}(R_{2};{\mathbb{Z}}/2)$. Hence we have the ${\mathbb{Z}}/2[\rho]$-algebra map

[TABLE]

When $*\geq*^{\prime}$, this map is additively isomorphic from Petrov-Semenov-Zainoulline theorem. So is isomorphic as ${\mathbb{Z}}/2[\rho]$-algebra map. ∎

Corollary 6.8**.**

We have the ${\mathbb{Z}}/2[\rho]$-algebra isomorphism

[TABLE]

where $R_{2}^{\prime}=(\rho^{7},\rho^{4}=b_{1},\rho^{6}=b_{2})$.

Consider the Borel spectral sequence

[TABLE]

Here $H^{*}(B{\mathbb{Z}}/2;{\mathbb{Z}}/2)\cong{\mathbb{Z}}/2[\rho]$ and $H^{*}_{et}(G_{k}/T_{k};{\mathbb{Z}}/2)\cong{\mathbb{Z}}/2[y]/(y^{2})\otimes S(t)/(b)$. The element $y$ is not a permanent cycle of this spectral sequence. We can see $d_{7}(y)=\rho^{7}$ and

[TABLE]

This term becomes $E_{\infty}$-term and $grH^{*}_{et}({\mathbb{G}}_{k}/B_{k};{\mathbb{Z}}/2)\cong E_{8}^{*,*^{\prime}}$. Of course, this fact coincide the above corollary by $b_{1}=\rho^{4}$ and $b_{2}=\rho^{6}$ in $H^{*}_{et}(X;{\mathbb{Z}}/2)$.

7. Witt groups.

For a smooth variety $Y$ over a field $k$ with $1/2\in k$, let $W(Y)$ denote the Witt group of $Y$. Balmer defined the periodic Witt group $W^{i}(Y)\cong W^{i+4}(Y),\ (i\in{\mathbb{Z}})$ with $W^{0}(X)=W(Y)$ (see for details, [Ba-Wa]).

Balmer and Walter [Ba-Wa] define the Gersten-Witt complex

[TABLE]

Let $H^{*}(W(Y))$ denote the cohomology group of the above cochain complex, with $W(k(Y))$ places in degree [math]. From the above complex, we have the (Balmer-Walter) spectral sequence

[TABLE]

By the affirmative answer of the Milnor conjecture of quadratic forms by Orlov-Vishik-Voevodsky [Or-Vi-Vo], we have the isomorphism of graded rings $H^{*}(k(x);{\mathbb{Z}}/2)\cong grW^{*}(k(x))$. Using this fact, Pardon and Gille ([Pa],[Gi],[To]) defined the spectral sequence

[TABLE]

so that the differential $d_{r}$ has degree $(1,r-1)$ for $r\geq 2$. Here $H_{{\mathbb{Z}}/2}^{s}$ is the Zariski sheaf induced from the presheaf $H_{et}^{s}(V;{\mathbb{Z}}/2)$ for open subset $V$ of $Y$.

The above sheaf cohomology $H^{r}_{Zar}(Y;H_{{\mathbb{Z}}/2}^{s})$ relates the motivic cohomology $H^{*,*^{\prime}}(Y;{\mathbb{Z}}/2)$. Indeed, we get the long exact sequence from the solution of Beilinson-Lichtenbaum conjecture by Voevodsky [Or-Vi-Vo]

[TABLE]

[TABLE]

Therefore, we have

Lemma 7.1**.**

$E(GP)_{2}^{m-n,n}\cong H_{Zar}^{m-n}(Y;H_{{\mathbb{Z}}/2}^{n})\cong$**

[TABLE]

In particular, $E(GP)_{2}^{m,m}\cong H^{2m,m}(Y;{\mathbb{Z}}/2)\cong CH^{m}(Y)/2$,

and $E(GP)_{2}^{m,m+1}\cong H^{2m+1,m+1}(Y;{\mathbb{Z}}/2)$. Moreover Totaro proved

Lemma 7.2**.**

(Totaro [To]) If $x\in E(GP)_{2}^{*,*}$, then $d_{2}(x)=Sq^{2}(x)$.

Many examples ([Ya2,5] satisfy the following assumption.

Assumption 7.3**.**

If $x\in E(GP)_{2}^{*,*+1}$, then $d_{2}(x)=Sq^{2}(x)$.

It is known that the Witt group is written as the motivic Hermitian $K$-theory by Schlichting and Tripathi [Sc-Tr], namely, $W^{i}(Y)\cong KO^{i+*,*}(Y)$ for $\ i>*.$ Hence $W^{*}(X)$ has the multiplicative structure. In particular, we can see [Ya5] that the differentials in the both spectral sequences $E(GP)_{r}$ and $E(BW)_{r}$ are derivations.

Now we consider the cases $Y=X={\mathbb{G}}/B_{k}$. At first, we recall the case $\bar{X}$. Let us write $H(Y;Sq^{2})=Ker(d)/Im(d)$ the homology with the differential $d=Sq^{2}$ on $H^{*}(Y;{\mathbb{Z}}/2)$. We give dgree of $x\in H(Y;Sq^{2})$ by half of the degree $|x|\in H^{*}(Y;{\mathbb{Z}}/2)$.

Theorem 7.4**.**

([Ya5], [Zi]) There is an isomorphism

[TABLE]

[TABLE]

where $deg(y)=1/2|y|=3$ and all $deg(z_{i})$ are odd.

It is known (Theorem 5.11 in [Ya2]) that there is an open variety $U_{a}$ in some quadric $Q$ such that $R_{a}\subset M(Q)$ and $H^{*,*^{\prime}}(U_{a};{\mathbb{Z}}/2)\cong H^{*,*^{\prime}}(R_{a};{\mathbb{Z}}/2).$ So we write by $W^{*}(R_{a})$ the Witt group $W^{*}(U_{a})$.

Lemma 7.5**.**

*If Assumption 7.3 is satisfied, then $grW^{*}(R_{2})\cong{\mathbb{Z}}/2\{1,\rho,\rho^{2}\}$. *

Proposition 7.6**.**

Let $G$ be of type $(I)$ and $X={\mathbb{G}}_{k}/B_{k}$ for $k={\mathbb{R}}$. Moreover Assumption 7.3 holds. Then we have

[TABLE]

[TABLE]

To prove above lemma and proposition, we need some lemmas. First recall that

[TABLE]

where $H^{*,*^{\prime}}(R_{2};{\mathbb{Z}}/2)\cong{\mathbb{Z}}/2[\tau]\{1,\rho,\rho^{2},a^{\prime},Q_{0}(a^{\prime}),\rho Q_{0}(a^{\prime}),Q_{1}(a^{\prime})\}$. (Here note $\tau^{-1}\rho^{3}=a^{\prime},Q_{0}(a^{\prime})=b_{1},Q_{1}(a^{\prime})=b_{2}$ in $H^{*,*^{\prime}}(X;{\mathbb{Z}}/2)$. Hence

[TABLE]

Here in $H^{*}_{Zar}(X;H^{*^{\prime}}_{{\mathbb{Z}}/2})$, degree is given $deg(a^{\prime})=(1,2),deg(\rho Q_{0}(a^{\prime}))=(2,3)$, and degree of $Q_{i}(a^{\prime})$ and elements in $S(t)/(b)$ are $(*,*)$. Note that all elements in $E(GP)_{2}$ have degree $(*+1,*)$ or $(*,*)$. Hence if Assumption 7.3 holds

[TABLE]

Lemma 7.7**.**

(Lemma 6.12 in [Ya2]) In $H^{*,*^{\prime}}(X;{\mathbb{Z}}/2)$, we have

[TABLE]

Proof.

By the modified Cartan formula ([Vo])

[TABLE]

Hence $\tau Sq^{2}(a^{\prime})=\tau Sq^{1}(\tau)Sq^{1}(a^{\prime})=\tau\rho Q_{0}(a^{\prime})$. Since $H^{*,*^{\prime}}(X;{\mathbb{Z}}/2)$ is $\tau$-torsion free, we have the first equation. The second equation follows from

[TABLE]

and $Q_{0}Sq^{2}(a^{\prime})=\rho Q_{0}Q_{0}(a^{\prime})=0$. ∎

Thus we have Lemma 7.5.

Lemma 7.8**.**

If Assumption 7.3 is satisfied, then we have the isomorphism

[TABLE]

Proof.

The following submodule in $H^{*}(X;H_{{\mathbb{Z}}/2}^{*^{\prime}})$

[TABLE]

is closed under $Sq^{2}$. Its $Sq^{2}$-homology is $H(A;Sq^{2})\cong{\mathbb{Z}}/2\{1,\rho,\rho^{3}\}$. Note $S(t)/(b)$ is also closed under $Sq^{2}$.since $H^{*}(G/T;{\mathbb{Z}}/2)$ is closed under $Sq^{2}$. By the Kunneth formula, we have the lemma. ∎

Lemma 7.9**.**

If Assumption 7.3 is satisfied, then we have

[TABLE]

Proof.

We consider the map which induced from the product

[TABLE]

It is known [Ya5] that $W^{*}(G_{k})\cong W^{*}(G_{k}/T_{k})$. At first we recall

[TABLE]

since $H^{*,*^{\prime}}(G_{k}/B_{k};{\mathbb{Z}}/2)\cong H^{*,*^{\prime}}(\bar{X};{\mathbb{Z}}/2)$ and $\bar{X}$ is cellular. By using Kunneth formula, inductively we get

[TABLE]

where $E(GP)_{r}(Y)$ is the spectral sequence for a space $Y$. Similar fact holds for $E(BW)_{r}$. Thus we get $W^{*}(G_{k}\times X)\cong W^{*}(G_{k})\otimes W^{*}(X).$

Define that $x\in W^{*}(X)$ is primitive if

[TABLE]

Of course, if $x$ is primitive, then $d_{r}(x)$ is primitive. We can take generators $z_{i}$ primitive (otherwise, add some non-primitive elements). If $d_{3}(z_{i})\not=0$, then it is $\rho z$ for $z\in H^{*}(\bar{X};H_{{\mathbb{Z}}/2}^{*})$. But

[TABLE]

is not primitive. Similarly, we can prove the lemma. ∎