NOTE ON THE FILTRATIONS OF THE K-THEORY

NOBUAKI YAGITA

Abstract. Let X be a (colimit of) smooth algebraic variety over a subfield of ℂ. Let K_{alg}^{0}(X) (resp. K_{top}^{0}(X(ℂ))) be the algebraic (resp. topological) K-theory of k (resp. complex) vector bundles over X ( resp. X(ℂ))). When K_{alg}^{0}(X) ≅ K_{top}^{0}(X(ℂ)), we study the differences of its three (gamma, geometrical and topological) filtrations. In particular, we consider the cases X = BG for an algebraically closed field k, and X = 𝔾_{k}/T_{k} the twisted form of flag varieties G/T for non-algebraically closed field k.

1. Introduction

Let X be a (colimit of) smooth algebraic variety over a subfield k of ℂ. We consider the cases that

(1.1) K_{alg}^{0}(X) ≅ K_{top}^{0}(X(ℂ))

where K_{alg}^{0}(X) (resp. K_{top}^{0}(X(ℂ))) is the algebraic (resp. topological) K-theory generated by algebraic k-bundles (complex bundles) over X (resp. X(ℂ)). In this assumption, we study the typical three filtrations

F_{γ}^{i}(X) ⊂ F_{geo}^{i}(X) ⊂ F_{top}^{i}(X(ℂ))

namely, the gamma and the geometric filtrations defined by Grothendieck [Gr], and the topological filtration defined by Atiyah [At]. Namely, we study induced maps of associated rings

gr_{γ}^{∗}(X) → gr_{geo}^{∗}(X) → gr_{top}^{∗}(X(ℂ)).

Atiyah showed that gr_{top}^{∗}(X(ℂ)) is isomorphic to the infitite term E_{∞}^{∗,0} of the AHss (Atiyah-Hirzebruch spectral sequence) converging to K-theory K^{∗}(X(ℂ)). Moreover he showed that gr_{top}^{∗}(X(ℂ)) ≅ gr_{γ}^{∗}(X) if and only if E_{∞}^{∗,0} is generated by Chern classes in H^{∗}(X(ℂ)). We will see that similar facts hold for gr_{geo}^{∗}(X). Namely, gr_{geo}^{2∗}(X) ≅ AE_{∞}^{2∗,∗,0} of the motivic AHss converging to motivic K-theory AK^{∗,∗′}(X). Moreover we show that gr_{geo}^{∗}(X) ≅ gr_{γ}^{∗}(X) if and only if AE_{∞}^{2∗,∗,0} is generated by Chern classes in the Chow ring CH^{∗}(X) ≅ H^{2∗,∗}(X).

Let G be a compact Lie group (e.g., a finite group) and G_{k} be the corresponding algebraic group over an algebraically closed field k. Then by Merkurjev and Totaro ([To]), we have the isomorphisms

K_{alg}^{0}(BG_{k}) ≅ R(G_{k})^{∧} ≅ R(G)^{∧} ≅ K_{top}^{0}(BG),

where R(G_{k})^{∧} (resp. R(G)^{∧}) is the k-representation (resp. complex representation) ring completed by the augmentation ideal, and BG_{k} and BG are their classifying spaces.

2000 Mathematics Subject Classification. Primary 11E72, Secondary 12G05.

Key words and phrases. γ-filtration, K-theory, representation ring.

Atiyah had conjectured in [At] that F_{γ}^{i}(BG) = F_{top}^{i}(BG) for all finite groups. Weiss [Th] showed this does not hold for G = A_{4}. For counter examples of p-groups were given by Leary-Yagita [Le-Ya] when G is rank_{p}(G) = 2 of class 3 with p ≥ 5. We will see for the same group G, F_{γ}^{2p+2}(BG_{k}) ≠ F_{geo}^{2p+2}(BG_{k}) = F_{top}^{2p+2}(BG_{k}).

We study these filtrations detailedly for connected groups (O_{n}, SO_{n},...). In particular we show

Theorem 1.1. (Let k be an algebraically closed field.) For G = Spin_{7}, there is an element x in K_{alg}^{0}(BG_{k}) such that

0 ≠ x ∈ gr_{γ}^{4}(BG_{k}), 0 ≠ x ∈ gr_{geo}^{6}(BG_{k}), 0 ≠ x ∈ gr_{top}^{8}(BG).

These facts also hold for the extraspecial 2-group 2_{+}^{1+6}.

We consider the different type of examples, which satisfy (1.1). (See also [Ga-Za], [Za].) Here we do not assume that k is algebraically closed. Let us write by M(X) the (pure) motive of X, and by M_{a} = (M_{n}) the Rost motive for a nonzero pure symbol a ∈ K_{n+1}^{M}(k)/p ([Ro1,2], [Su-Jo]). We consider the cases X such that

(1.2) M(X) ≅ M_{n} ⊗ A(X)

where A(X) is a sum of k-Tate motives. Then we can see that (1.1) is satisfied by the result from([Vi-Ya],[Ya]).

Some cases of flag manifolds G/P satisfy (1.2) ([Ca-Pe-Se-Za], [Ni-Se-Za], [Pe-Se-Za]). In particular (for p = 2) X = 𝔾_{2,k}/T_{k} is a such example, where 𝔾_{2,k} is the nontrivial G_{2,k}-torsor (induced from a Rost cohomological invariant 0 ≠ a ∈ K_{3}^{M}(k)/2, [Ga-Me-Se]) for the exceptional Lie group G_{2}, and T a maximal torus in G ([Bo], [Pe-Se-Za]). (Namely, 𝔾_{2,k}/T_{k} is a twisted form of G_{2}/T.) Note that H^{∗}(G_{2}/T) is torsion free, and we have

gr_{geo}^{∗}(G_{2,k}/T_{k}) ≅ gr_{top}^{∗}(G_{2}/T) ≅ H^{∗}(G_{2}/T).

By using the fact that CH^{∗}(𝔾_{2,k}/T_{k}) is generated by Chern classes, we can show

Theorem 1.2. Let 𝔾_{2,k} be the nontrivial G_{2,k}-torsor for the Rost cohomological invariant in K_{3}^{M}(k)/2. Then we have

gr_{γ}^{2∗}(G_{2}/T) ≅ gr_{geo}^{2∗}(𝔾_{2,k}/T_{k}) ≅ CH^{∗}(𝔾_{2,k}/T_{k}).

From (1.1), the gamma filtration is defined purely topologically. Thus we see that this topological invariant is isomorphic to a purely algebraic geometric object such as the Chow ring of twisted form.

2. filtrations

We first recall the topological filtration defined by Atiyah. Let Y be a topological space (e.g., finitely generated CW -complex). Let K^{∗}(Y) be the complex K-theory ; the Grothendieck group generated by complex bundles over Y. Let Y^{i} be an i-dimensional skeleton of Y. Define the topological filtration of K^{∗}(Y) by

F_{top}^{i}(Y) = Ker(K^{∗}(Y) → K^{∗}(Y^{i}))

and the associated graded algebra gr_{top}^{i}(Y) = F_{top}^{i}(Y)/F_{top}^{i+1}(Y).

We consider the long exact sequence (exact couple)

... → K^{∗}(Y^{i}/Y^{i−1}) → K^{∗}(Y^{i}) → K^{∗}(Y^{i−1}) →^{δ} K^{∗+1}(Y^{i}/Y^{i−1}) → ...

Here we have K^{∗}(Y^{i}/Y^{i−1}) ≅ K^{∗} ⊗H^{∗}(Y^{i}/Y^{i−1}), which induces the (well known) AHss

E_{2}^{∗,∗′}(Y) ≅ H^{∗}(Y) ⊗ K^{∗} ⟹ K^{∗}(Y).

By the construction of the spectral sequence, we have

Lemma 2.1. (Atiyah [At]) gr_{top}^{∗}(Y) ≅ E_{∞}^{∗,0}(Y).

Next we consider the geometric filtration. Let X be a smooth algebraic variety over a subfield k of ℂ. Let K_{alg}^{0}(X) be the algebraic K-theory which is the Grothendiek group generated by k-vector bundles over X. It is also isomorphic to the Grothendieck group genrated by coherent sheaves over X (we assumed X smooth). This K -theory can be written by the motivic K-theory AK^{∗,∗′}(Y) ([Vo1,2], i.e.,

K_{alg}^{i}(X) = ⊕_{∗}AK^{2∗−i,∗}(X).

In particular K_{alg}^{0}(X) = ⊕_{∗}AK^{2∗,∗}(X).

The geometric filtration ([Gr]) is defined as

F_{geo}^{i}(X) = {[O_{V}]|codim_{X}V ≥ i}

(and F_{geo}^{2i−1}(X) = F_{geo}^{2i}(X)) where O_{V} is the structural sheaf of closed subvariety V of X.

We recall the algebraic cobordism MGL^{∗,∗′}(−) [Vo1] and let us write MGL^{2∗,∗}(X) = Ω^{∗}(X), in fact, this is isomorphic to the algebraic cobordism defined by Levine and Morel ([Le-Mo1,2], [Vo1,2]). Recall

Ω^{∗}(Spec(k)) = Ω^{∗}(pt.) ≅ MU^{2∗}(pt.) = MU^{∗}.

Then we have the isomorphism

Ω^{∗}(X) ⊗_{MU^{∗}} ℤ ≅ CH^{∗}(X), Ω^{∗}(X) ⊗_{MU^{∗}} K^{∗} ≅ K_{alg}^{0}(X)

where the MU^{∗} module structure of K^{∗} is given by Todd genus (see § below). Each element x ∈ Ω^{∗}(X) is represented by a projective map x = [f : M → X] with codim_{X}M = i and M smooth ([Le-Mo1,2]), namely, x = f_{∗}(1_{M}) for 1_{M} ∈ Ω^{0}(M) and f_{∗} is the Gysin map. Then the geometric filtration is also defined as

F_{geo}^{2i}(X) = {f_{∗}(1_{M})|f : M → X and codim_{X}M ≥ i}.

Here we recall the motivic AHss ([Ya2, 4])

AE_{2}^{∗,∗′,∗′′}(X) ≅ H^{∗,∗′}(X; K^{∗′′}) ⟹ AK^{∗,∗′}(X).

(Of course this spectral sequence is not defined using skeleton as the topological case. But we assume the existence of the AHss converging to the motivic K-theory AK^{∗,∗′}(X).) Note that

AE_{2}^{2∗,∗,∗′′}(X) ≅ H^{2∗,∗}(X; K^{∗′′}) ≅ CH^{∗}(X) ⊗ K^{∗′′}.

Hence AE_{∞}^{2∗,∗,0}(X) is a quotient of CH^{∗}(X) by dimensional reason of degree of differential d_{r} (i.e., d_{r}AE_{r}^{2∗,∗,∗′′}(X) = 0). Thus we have

Lemma 2.2. gr_{geo}^{2∗}(X) ≅ AE_{∞}^{2∗,∗,0}(X).

Proof. Let q : Ω^{∗}(X) ⊗ K^{∗} → K^{∗}(X). Then

F_{geo}^{2i}(X) = q{f_{∗}(1_{M}) ∈ Ω^{∗}(X)|f : M → X and codim_{X}M ≥ i}.

Let q′ : Ω^{∗}(X) → CH^{∗}(X) and q′′ : CH^{∗}(X) → E_{∞}^{2∗,∗,0}. Then q|(Ω^{∗}(X) ⊗ 1) = q′′q′. Thus we have

F_{geo}^{2i}(X)/F_{geo}^{2i+2}(X) = q′′CH^{i}(X)

since q′ is an epimorphism. □

Lemma 2.3. Let t_{ℂ} : K_{alg}^{0}(X) → K_{top}^{0}(X(ℂ)) be the realization map. Then F_{geo}^{i}(X) ⊂ (t_{ℂ}^{∗})^{−1}F_{top}^{i}(X(ℂ)).

Proof. Let us write K_{top}^{0}(X(ℂ)) simply by K(X). The Gysin map f_{∗} : K(M) → K(X) is defined by using Thom isomorphism

K(M) ≅ K(T h_{X}(M)) → K(X).

Let codim_{X}M ≥ i. For an 2i-skeleton X^{2i} of X(ℂ), we can show that the map

K(T h_{X}(M)) → K(X) → K(X^{2i})

is zero. Because the above composition map is rewritten

K(T h_{X}(M)) → K(T h_{X}(M)^{2i}) → K(X^{2i}).

Its first map is zero, because H^{∗}(T h_{X}(M)) = 0 for ∗ < 2i and the exact sequence (exact couple) for K-theory for skeletons of X (see the definition of the AHss). □

At last, we consider the gamma filtration. Let λ^{i}(x) be the exterior power of the vector bundle x ∈ K_{alg}^{0}(X) and λ_{t}(x) = ∑ λ^{i}(x)t^{i}. Let us denote

λ_{t/(1−t)}(x) = γ_{i}(x) = ∑ γ^{i}(x)t^{i}.

The Gamma filtration is defined as

F_{γ}^{2i}(X) = {γ^{i_{1}}(x_{1}) · ... · γ^{i_{m}}(x_{m})|i_{1} + ... + i_{m} ≥ i, x_{j} ∈ K_{alg}^{0}(X)}.

Then we can see F_{γ}^{i}(X) ⊂ F_{geo}^{i}(X) (Proposition 12.5 in [At], Atiyah proved F_{γ}^{i}(X) ⊂ F_{top}^{i}(X) in K_{top}(X). However the arguments work also in K_{alg}^{0}(X) and this fact is well known.) Let ϵ : K_{alg}^{0}(X) → ℤ be the augmentation map and c_{i}(x) ∈ H^{2i,i}(X) the Chern class. Recall q′′ : CH^{∗}(X) → E_{∞}^{2∗,∗,0} be the quotient map. Then (p. 63 in [At]) we have

q′′(c_{n}(x)) = [γ^{n}(x − ϵ(x))].

Lemma 2.4. (Atiyah) The condition F_{γ}^{2∗}(Y) = F_{top}^{2∗}(Y) (resp. F_{γ}^{2∗}(X) = F_{geo}^{2∗}(X)) is equivalent to that E_{∞}^{2∗,0}(Y) (resp. AE_{∞}^{2∗,∗,0}(X)) is (multiplicatively) generated by Chern classes in H^{2∗}(BG) (resp. CH^{∗}(BG)).

3. Morava K-theory (K-theory localized at p)

In this paper, we assume that p is a fixed prime number and consider only cohomology theories (Chern rings) localized at this prime p. Namely, for the notation A^{∗}(X) means A^{∗}(X)_{(p)} in this paper. In particular, ℤ always means ℤ_{(p)} and MU^{∗}(−) means MU^{∗}(X)_{(p)} throughout this paper.

Let AMU^{∗,∗′}(X) = MGL^{∗,∗′}(X) and recall MU^{∗} = ℤ[x_{1}, ..., x_{n}, ..], deg(x_{i}) = (−2i, −i). Given a sequence S = (x_{i_{1}}, x_{i_{2}}, ...) of generators, we can construct generalized cohomology theory (in the 𝔸^{1}-homotopy category) such that

t_{ℂ}AMU(S)^{∗,∗′}(X) = MU(S)^{∗}(X(ℂ)) with MU(S)^{∗} = MU^{∗}/(S).

In particular letting x_{p^{n}−1} = v_{n} and S = (x_{i}|i ≠ p^{n} − 1), we have the motivic BP -theory ([Ya2,4])

ABP^{∗,∗′}(X) with MU^{∗}/(S) ≅ BP^{∗} = ℤ[v_{1}, v_{2}, ...].

Then we have the isomorphisms ([Ya])

ABP^{∗,∗′}(X) ≅ MGL^{∗,∗′}(X) ⊗_{MU^{∗}} BP^{∗},
MGL^{∗,∗′}(X) ≅ ABP^{∗,∗′}(X) ⊗_{BP^{∗}} MU^{∗}.

Similarly, we can construct the motivic connective Morava K-theory such that

Ak(1)^{∗,∗′}(X) with k(1)^{∗} = ℤ/p[v_{n}],

and the integral connected K-theory Ak̃(1)^{∗,∗′}(X) with k̃(n) = ℤ[v_{n}]. Moreover let the (usual) motivic Morava K-theory

AK(n)^{∗,∗′}(X) = Ak(n)^{∗,∗′}(X)[v_{n}^{−1}], AK̃(n)^{∗,∗′}(X) = Ak̃(n)^{∗,∗′}(X)[v_{n}^{−1}].

By the Landweber exact functor theorem ([Ra], [Ha]), it is well known that

AK^{∗,∗′}(X) ≅ (AMU^{∗,∗′}(X) ⊗_{MU^{∗}} ℤ) ⊗ ℤ[B, B^{−1}]

where the MU^{∗}-module structure of ℤ is given by the Todd genus, and B is the Bott periodicity with deg(B) = (−2, −1). Since the Todd genus of v_{1} (resp. v_{i}, i > 1) is 1 (resp. 0), we can write

AK^{∗,∗′}(X) ≅ ABP^{∗,∗′}(X) ⊗_{BP^{∗}} ℤ[B, B^{−1}] identifying B^{p−1} = v_{1}.

Then we have

Lemma 3.1. There is a natural isomorphism

AK̃^{∗,∗′}(X) ≅ AK̃(1)^{∗,∗′}(X) ⊗_{K̃(1)^{∗}} ℤ[B, B^{−1}] identifying v_{1} = B^{p−1}.

Proof. Recall that there is the natural map (by the construction of AMU(S))

ρ : ABP^{∗,∗′}(X) ⊗_{BP^{∗}} ℤ[B, B^{−1}] → AK̃(1)^{∗,∗′}(X) ⊗_{K̃(1)^{∗}} ℤ[B, B^{−1}].

Of course, the functor

A ↦ A ⊗_{K̃(1)^{∗}} ℤ[B, B^{−1}] ≅ A ⊗ ℤ{1, B, ..., B^{p−2}}

is exact, and we have the spectral sequence

E_{2}^{∗,∗′,∗′′}(AK̃(1)) ⊗_{K̃(1)^{∗}} ℤ[B, B^{−1}] ⟹ AK̃(1)^{∗,∗′}(X) ⊗_{K̃(1)^{∗}} ℤ[B, B^{−1}].

Since for a BP^{∗}(BP) module A, the functor

A ↦ A ⊗_{BP^{∗}} ℤ[B, B^{−1}]

is exact from the Landweber exact functor theorem, we have the spectral sequence from the AHss for ABP^{∗,∗′}(X)

E_{2}^{∗,∗′,∗′′}(ABP) ⊗_{BP^{∗}} ℤ[B, B^{−1}] ⟹ ABP^{∗,∗′}(X) ⊗_{BP^{∗}} ℤ[B, B^{−1}],

which is compatible with the map ρ. The E_{2}-term of the both spectral sequences are isomorphic to

H^{∗,∗′}(X; ℤ) ⊗ ℤ[B, B^{−1}].

Therefore the two spectral sequences are isomorphic. □

We also note from the arguments in the above proof.

Lemma 3.2. Let E(ABP)_{r}^{∗,∗′,∗′′} (resp. E(AK̃(1))_{r}^{∗,∗′,∗′′}) be the AHss coverging to ABP^{∗,∗′}(X) (resp. AK̃(1)^{∗,∗′}(X)). Then we have

E(ABP)_{r}^{∗,∗′,∗′′} ⊗_{BP^{∗}} K̃(1)^{∗} ≅ E(AK̃(1))_{r}^{∗,∗′,∗′′}).

From above lemma, it is sufficient to consider the Morava K-theory AK̃(1)^{∗,∗′}(X) when we want to study AK^{∗,∗′}(X). Hereafter of this paper, we only consider the theories AK̃(1)^{∗,∗′}(X) and Ak̃(1)^{∗,∗′}(X) instead of AK^{∗,∗′}(X) or K_{alg}^{∗}(X). (We only consider the cohomology theories and Chow rings localied at p.)

We assume the following assumption

(∗) K_{alg}^{0}(X) ≅ K_{top}^{0}(X(ℂ)) (and K_{top}^{1}(X(ℂ)) = 0).

That is equivalent to

(∗) AK̃(1)^{2∗,∗}(X) ≅ K̃(1)^{2∗}(X(ℂ)) (and K̃(1)^{2∗+1}(X(ℂ)) = 0).

From Lemma 2.3, we have

F_{γ}(X) ⊂ F_{geo}^{i}(X) ⊂ F_{top}^{i}(X(ℂ)).

Here we note that the gamma filtrations of topogical and algebraic geometrical are same, i.e., F_{γ}^{∗}(X) ≅ F_{γ}^{∗}(X(ℂ)). So we have the maps of associated graded rings

gr_{γ}^{∗}(X) → gr_{geo}^{∗}(X) → gr_{top}^{∗}(X(ℂ)).

Lemma 3.3. gr_{γ}^{2}(X) = gr_{geo}^{2}(X).

Proof. If 0 ≠ x ∈ gr_{γ}^{2}(X), then x_{1} = c_{1}(ξ) ∈ AK̃(1)^{2∗,∗}(X) for some bundle ξ. In CH^{∗}(X), we know c_{1}(ξ) = c_{1}(det(ξ)) which is determined by the line bundle det(ξ). Line bundles are determined by P ic(X) = CH^{1}(X). So 0 ≠ x ∈ CH^{1}(X). □

Lemma 3.4. If an element y ∈ AK̃(1)^{2∗,∗}(X) is represented by 0 ≠ y′ (resp.y′′, y′′′) ∈ gr_{γ}^{i}(X) (resp. gr_{geo}^{j}(X), gr_{top}^{k}(X(ℂ))), then

i ≤ j ≤ k, and i = k = j mod(2(p − 1)).

Proof. The element y is represented

y = v_{1}^{s}y′ ∈ AK̃(1)^{2∗,∗}(X)/F_{γ}^{2i+1} y = v_{1}^{t}y′′ ∈ AK̃(2)^{2∗,∗}(X)/F_{geo}^{2j+1}

for some s, t ∈ ℤ/p. □

Remark. The above fact does not hold for y ∈ K_{top}^{0}(X) (which is a sum of K̃(1)^{2∗,∗}(X), 0 ≤ ∗ ≤ p − 2). Let us write

y = B^{k}y_{k} + B^{k+1}y_{k+1} + ... + B^{k+p−2}y_{k+p−2},

with y_{i} ∈ K̃(1)^{2i}(Y) and, y_{i} ∈ F_{top}^{2i}(Y). Suppose j > k. Then this means that there is s such that 0 ≠ y_{s} ∈ gr_{geo}^{s′}(X) with s − s′ = 0 mod(2p − 2). Of course if s ≠ k, then k − s′ ≠ 0 mod(2p − 2).

To study the difference of F_{geo}^{∗}(X) and F_{top}^{∗}(X(ℂ)), we consider AHss E_{r}^{∗,∗′}(BP) converging to BP^{∗}(X). Suppose that

[v_{1} ⊗ x] ∈ BP^{∗′} ⊗ H^{∗}(X(ℂ)) ≅ E(BP)_{2}^{∗,0}

is an permanent cycle, but [x] ∈ H^{∗}(X(ℂ)) itself is not (i.e., d_{r}(x) ≠ 0 for some r). Let x′ ∈ BP^{∗}(X(ℂ)) be a corresponding element for [v_{1} ⊗ x] in E_{∞}^{∗,∗′}

Lemma 3.5. Let x ∈ H^{2∗}(X(ℂ)) and x′ ∈ BP^{∗′}(X(ℂ)) be elements with the assumption above. Suppose that

0 ≠ x′ ∈ BP^{∗′}(X(ℂ)) ⊗_{BP^{∗}} ℤ[v_{1}, v_{1}^{−1}] ≅ K̃(1)^{∗}(X(ℂ))

and that x′ ∈ BP^{∗′}(X(ℂ)) ⊗_{BP^{∗}} ℤ is in the image of the Totaro cycle map

CH^{∗′}(X) → BP^{2∗′}(X(ℂ)) ⊗_{BP^{∗}} ℤ.

Then 0 ≠ x′ ∈ gr_{top}^{2∗}(X(ℂ)), but 0 ≠ x′ ∈ gr_{geo}^{2(∗−p+1)}(X)).

Proof. This case ∗′ = ∗ − (p − 1) in the above arguments. Let x ∈ H^{2i}(X(ℂ)). In fact x′ ∈ Im(CH^{i−p+1}(X)) and 0 ≠ x′ ∈ gr_{geo}^{2(i−p+1)}(X(ℂ)), but 0 ≠ x′ = [v_{1} ⊗x] ∈ gr_{top}^{2i}(X(ℂ)). □

Next we consider the cases gr_{γ}^{∗}(X) ≅ gr_{top}(X(ℂ)). From the Atiyah theorem (Lemma 2.4), the following lemma is immediate.

Lemma 3.6. Suppose (∗) and suppose that the infinity term E_{∞}^{2∗,0}(K̃(1)) (of the AHss for K̃(1)^{∗}(X(ℂ))) is generated by Chern classes in H^{∗}(X) for all ∗ ≥ N. Then for all ∗ ≥ N, we have

gr_{γ}^{2∗}(X) ≅ E_{∞}^{2∗,0}(K̃(1)^{∗}(X(ℂ))) for all ∗ ≥ N.

Lemma 3.7. (Lemma 2.8 in [Ya3]) Suppose (∗) and that H^{∗}(X(ℂ)) is generated by Chern classes. Then we have

CH^{∗}(X) ≅ H^{∗}(X(ℂ)) for ∗ ≤ p − 1.

Moreover if X(ℂ) is simply connected (resp. 3-connected), then we have the isomorphisms for ∗ ≤ p (resp. ∗ ≤ p + 1)

CH^{∗}(X) ⊗ ℤ_{p} ≅ H^{2∗}(X(ℂ); ℤ_{p}).

Proof. By the assumption, we see

gr_{γ}^{2∗}(X) ≅ gr_{geo}^{2∗}(X) ≅ gr_{top}^{2∗}(X(ℂ)).

To compute the last graded ring, we consider AHss

E_{2}^{∗,∗′}(K̃(1)) ≅ H^{∗}(X; K̃(1)^{∗′}) ⟹ K̃(1)^{∗}(X(ℂ)).

Here K̃(1)^{∗} ≅ ℤ[v_{1}, v_{1}] with |v_{1}| = −2p + 2. It is well known that the first non zero differential is

d_{2p−1}(x) = v_{1} ⊗ Q_{1}(x) mod(p).

So each element in H^{2∗}(X(ℂ)) is not tergent of any differential d_{r} when ∗ ≤ p−1. (Of course d_{r}(x) = 0 for Chern classes x.) Hence we have

gr_{top}^{2∗}(X(ℂ)) ≅ H^{2∗}(X(ℂ)) for ∗ ≤ p − 1.

Similarly, considering AHss converging to AK̃(1)^{∗,∗′}(X), we have the isomorphism

gr_{geo}^{2∗}(X) ≅ CH^{2∗}(X) for ∗ ≤ p − 1.

Here we use the fact E_{2}^{2∗,∗,0}(AK̃(1)) ≅ CH^{∗}(X). Thus the isomorphism of the geometric and toplogical filtrations, gives the first statements.

From the isomorphism

H^{1,1}(X; ℤ/p) ≅ H^{1}(X(ℂ); ℤ/p) = 0.

we see that H^{1,1}(X; ℤ) is p-divisible. Since the image of the differential of p-divisible elements are also p-divisible,

H^{2p}(X(ℂ)) ≅ gr_{top}^{2p}(X)
≅ gr_{geo}^{2p}(X) ≅ CH^{2p}(X)/(p − divisible).

Hence we have the second isomorphism. The third isomorphism is seen similarly. □

Remark. The first statement in the above lemma is also proved by the Riemann-Roch formula without denominators, namely, the composition map

CH^{i}(X) → gr_{geo}^{i}(X) →^{c_{i}} CH^{i}(X)

is multiplication by (−1)^{i−1}(i − 1)!. Hence we get CH^{i}(X) ≅ gr_{geo}^{i}(X) for i ≤ p. Moreover we know that CH^{i}(X) is represented by the i-th Chern class c_{i}(ξ) for some bundle ξ.

Remark. The assumption of Lemma 2.8 in [Ya3] is not sufficient, and it should be changed as above.

4. classifying spaces BG for finite groups

Let G be a compact Lie group (e.g., a finite group) and G_{k} be the corresponding algebraic group over an algebraically closed field k in ℂ. Then by Merkurjev and Totaro ([To]), we have the isomorphisms

(1.1) K_{alg}^{0}(BG_{k}) ≅ R(G_{k})^{∧} ≅ R(G)^{∧} ≅ K_{top}^{0}(BG),

where R(G_{k})^{∧} (resp. R(G)^{∧}) is the k-representation (resp. complex representation) ring completed by the augmentation ideal and K_{alg}^{0}(BG_{k}) (resp. K_{top}^{0}(BG)) is the K-theory generated by k-bundles (resp. complex bundles) of the classifying space BG_{k} (resp. BG).

When k is algebraically closed, we write BG_{k} by BG simply. For Section 4-6, we assume k is algebraically closed.

In this section, we consider cases that G are finite groups. At first, we consider the case G = ℤ/p^{r}. Then H^{∗}(BG) ≅ ℤ[y]/(p^{r}y), |y| = 2 and y_{1} = c_{1}(e) for a nonzero linear representation e. So all three filtrations are the same. The similar fact holds for its product.

Theorem 4.1. (p=2,r=1 case by Atiyah [At]) Let q = p^{r} and G = ⊕^{n}ℤ/q. Then

gr_{top}^{∗}(BG) ≅ ℤ[y_{1}, ..., y_{n}]/(qy_{i}, y_{i}^{q}y_{j} − y_{i}y_{j}^{q}).

Hence the three filtrations are the same.

Proof. Let Q′_{0} = β_{q} be the higher Bockstein. The integral cohomology is isomorphic to a subring of the mod q cohomology

H^{∗}(BG) ⊂ H^{∗}(BG; ℤ/q), when ∗ > 0.

Here H^{∗}(BG; ℤ/q) ≅ ℤ/q[y_{1}, ..., y_{n}] ⊗ Λ(x_{1}, ..., x_{n}) with Q′_{0}(x_{i}) = y_{i}, and we know

H^{∗}(BG) ≅ ℤ/q[y_{1}, ..., y_{n}]{Q′_{0}(x_{i_{1}}...x_{i_{s}})|1 ≤ i_{1} < ..., i_{s} ≤ n}

with Q′_{0}(x_{i_{1}}...x_{i_{s}}) = ∑_{k}(−1)^{k−1}y_{i_{k}}x_{i_{1}}...x̂_{i_{k}}...x_{i_{s}}.

We consider the AHss converging to K̃(1)^{∗}(BG). We define the weight degree for elements in this AHss by

w(v_{1}) = 0, w(y_{i}) = 0, w(x_{i}) = 1

so that w(Q′_{0}(x_{i_{1}}...x_{i_{s}})) = s − 1. We will prove

(1) (weight = 0) ∩ E_{2q}^{∗,∗′} ≅ ℤ/q[y_{1}, ..., y_{n}]/(y_{i}^{q}y_{j} − y_{i}y_{j}^{q}) for ∗ > 0,
(2) (weight = 1) ∩ E_{2q}^{∗,∗′} = 0.

Then we can prove this theorem by the following arguments.

We consider the AHss converging to the motivic AK̃(1)^{∗}(BG). The weight w(x) of an element x ∈ H^{∗,∗′}(X : ℤ/q) is defined as 2 ∗′ −∗. Since x_{i} ∈ H^{1,1}(BG; ℤ/q) and y_{i} ∈ H^{2,1}(BG; ℤ/q), their weights are in fact w(x_{i}) = 1 and w(y_{i}) = 0. The degree of the motivic AHss is

deg(d_{2r−1}) = (2r − 1, r − 1, −2(r − 1)) with (r − 1) = 0 mod(p − 1),

namely, w(d_{2r−1}) = −1 which means

d_{2r−1}(weight = s) = (weight = s − 1).

From (2), (weight = 0)-parts are not a target of any diffrential d_{2r−1} for r > q. By the naturality of realization map from the motivic AHss to the usual AHss, we get the same fact for the AHss for K̃(1)^{∗}(BG). Since K̃(1)^{∗}(BG) is generated by only weght = 0 elements, we have the theorem.

The first nonzero differential is known d_{2q−1}(x_{i}) = v_{1}^{1+p+...+p^{r−1}}y_{i}^{q}[Ya2]. We see (1) from

d_{2q−1}(Q′_{0}(x_{1}x_{2})) = d_{2q−1}(y_{1}x_{2} − y_{2}x_{1}) = y_{1}y_{2}^{q} − y_{1}^{q}y_{2}.

Now we prove (2). Let x ∈ Ker(d_{2s−1}) and x = ∑ a_{ij}Q′_{0}(x_{i}x_{j}). Then (since d_{r} is a derivation)

d_{2q−1}(x) = ∑ a_{ij}(y_{i}y_{j}^{q} − y_{i}^{q}y_{j}) = 0 in ℤ/q[y_{1}, ..., y_{n}].

Here we consider them in mod(x_{i}, y_{i}|i ≥ 4). Then we see a_{12} = a′_{12}y_{3} and we see (by dividing y_{1}y_{2}y_{3})

a′_{12}(y_{1}^{q−1} − y_{2}^{q−1}) + a′_{23}(y_{2}^{q−1} − y_{2}^{q−1}) + a′_{31}(y_{3}^{q−1} − y_{1}^{q−1}) = 0.

This implies that a′_{12} ∈ ideal(y_{1}^{q−1}, y_{2}^{q−1}, y_{3}^{q−1}). Moreover we see that a_{12} contains y_{3}^{q}. Similarly a_{23}, a_{13} contains y_{1}^{q} and y_{2}^{q} respectively.

On the other hand , we see

d_{2q−1}(Q′_{0}(x_{1}x_{2}x_{3})) = d_{2q−1}(∑ y_{1}x_{2}x_{3})
= ∑ y_{1}y_{2}^{q}x_{3} − ∑ y_{1}x_{2}y_{3}^{q} = ∑ y_{1}y_{2}^{q}x_{3} − ∑ y_{3}x_{1}y_{2}^{q}
= ∑ y_{2}^{q}(y_{1}x_{3} − y_{3}x_{1}) = − ∑ y_{1}^{q}Q′_{0}(x_{2}x_{3})

Taking off a′′d_{2r−1}Q′_{0}(x_{1}x_{2}x_{3}) for some adequate a′′ ∈ ℤ/q[y_{1}, ..., y_{n}], we can prove (2). □

Recall that a group G is called an extraspecial p-group if its center Z(G) ≅ ℤ/p and there is a central extension

0 → ℤ/p → G → ⊕^{2n}ℤ/p → 0.

For each prime p, such groups have only two types, namely, p_{+}^{1+2n}, p_{−}^{1+2n}. (e.g., 2_{+}^{1+2} ≅ D_{8} the dihedral group (of order 8), p_{−}^{1+2} ≅ Q_{8} the quaternion group). We here only write down the case p_{+}^{1+2} for p ≥ 3. The cohomology is known ([Ya1,3])

H^{even}(BG) ≅ (Y ⊕ B) ⊗ ℤ[c_{p}]/(p^{2}c_{p})

where Y = ℤ[y_{1}, y_{2}]/(py_{i}, y_{1}y_{2}^{p} −y_{1}^{p}y_{2}), B = ℤ/p{c_{2}, ..., c_{p−1}} and |y_{i}| = c_{1}(e_{i}) and c_{i} = c_{i}(ξ) for some linear representations e_{i} and p-dimensional representation ξ. Hence the even dimensional part of this cohomology is generated by Chern classes and all three filtrations are the same. The odd degree part is

H^{odd}(BG) ≅ Y ⊗ ℤ/p[c_{p}]{a_{1}, a_{2}}/(y_{2}a_{1} − y_{1}a_{2}, y_{2}^{p}a_{1} − y_{1}^{p}a_{2}) |a_{i}| = 3.

Theorem 4.2. Let G = p_{+}^{1+2} and p ≥ 3. Then

gr_{top}^{∗}(BG) ≅ Y ⊕ (ℤ{c_{p}} ⊕ B) ⊗ ℤ[c_{p}]/(p^{2}c_{p}).

Proof. We know the Milnor cohomology operation

v_{1}^{−1}d_{2p−1} = Q_{1} : H^{odd}(BG) → H^{even}(BG)

is injective and Q_{1}(a_{i}) = y_{i}c_{p}. Hence we see

grK̃(1)^{∗}(BG) ≅ E_{∞}^{∗,∗′} ≅ K̃(1)^{∗} ⊗ H^{even}(BG)/(Q_{1}H^{odd}(BG))
≅ K̃(1)^{∗} ⊗ H^{even}(BG)/(y_{i}c_{p}).
□

When p ≥ 5, the groups of rank_{p}G = 2 are classified by Blackburn. When groups are of class 2 (i.e., [G, [G, G]] = 1), cohomology rings are generated by Chern classes ([Le-Ya],[Ya1]), and hence all three filtrations are the same. Define the class 3 p-group (i.e., [G, [G, G]] ≠ 1) by

G(4, 1) = ⟨a, b, c|a^{p} = b^{p} = c^{p+1} = [b, c] = 1,
[a, b^{−1}] = c^{p}, [a, c] = b⟩.

Let G = G(4, 1). Then there is an element x_{p+1} ∈ H^{2p+2}(BG) [Le-Ya],[Ya] such that it is permanent in AHss for K̃(1)^{∗}(BG) and x_{p+1} is not represented by Chern class. But all elemnts in H^{even}(BG) is represented by transfers of Chern classes [Ya]. Of course Chow rings have the transfer map. Hence we have

Theorem 4.3. Let p ≥ 5 and G = G(4, 1). Then gr_{top}^{∗}(BG) ≅ gr_{geo}^{∗}(BG) but gr_{γ}^{i}(BG) ≅ g̸r_{geo}^{i}(BG) for i = 4, 2p + 2.

Proof. The first isomorphism follows from that all elements in H^{even}(BG) is represented by transfer of Chern classes. The second statement follows from x_{p+1} is not represented by Chern classes and the element x_{p+1} ∈ E_{∞}^{2p+2,0} represents a nonzero element in gr_{γ}^{4}(BG) from Lemma 3.4. □

5. connected groups with p = 2

Throughout this section, let p = 2. At first we consider the case G = O_{n}. The mod 2 cohomology of the classifying space BO_{n} of the n-th orthogonal group is

H^{∗}(BO_{n}; ℤ/2) ≅ H^{∗}((Bℤ/2)^{n}; ℤ/2)^{S_{n}} ≅ ℤ/2[w_{1}, ..., w_{n}]

where S_{n} is the n-th symmetry group, w_{i} is the Stiefel-Whiteney class which restricts the elementary symmetric polynomial in ℤ/2[x_{1}, ..., x_{n}]. Each element w_{i}^{2} is represented by Chern class c_{i} of the induced representation O_{n} ⊂ U_{n}. Let us write w_{i}^{2} by c_{i}.

Recall the Milnor operation Q_{i} which is defined Q_{0} = β and Q_{i} = [Q_{i−1}, P^{p^{i−1}}]. Let us write by Q(i) the exteria algebra Λ(Q_{0}, ..., Q_{i}). W.S.Wilson ([Wi],[Ko-Ya]) found a good Q(i)-module decomposition for BO_{n}, namely,

H^{∗}(BO_{n}; ℤ/2) = ⊕_{i=−1}Q(i)G_{i} with Q_{0}...Q_{i}G_{i} ∈ ℤ/2[c_{1}, ..., c_{n}].

Let us write by P (n)^{∗} = BP^{∗}/(p, ..., v_{i−1}). The BP^{∗}-theory is then computed

grBP^{∗}(BG)/p ≅ ⊕P (i + 1)^{∗}Q_{0}...Q_{i}G_{i}.

Hence we have K(1)^{∗}(BG) ≅ P (1)^{∗}(G_{−1} ⊕ Q_{0}G_{0}).

Moreover, by Wilson, it is known that

BP^{∗}(BO_{n}) ≅ BP^{∗}[[c_{1}, ..., c_{n}]]/(c_{1} − c_{1}^{∗}, ..., c_{n} − c_{n}^{∗})

where c_{i}^{∗} is the conjugation of c_{i}. Hence K̃(1)^{∗}(BG) is generated by Chern classes from H^{∗}(BG). Thus from Lemma 2.4, all filtrations are same.

Here G_{k−1} is quite complicated (see for details [Wi]), namely, it is generated by symmetric functions

∑ x_{1}^{2i_{1}+1}...x_{k}^{2i_{k}+1}x_{k+1}^{2j_{1}}...x_{k+q}^{2j_{q}}, k + q ≤ n,

with 0 ≤ i_{1} ≤ ... ≤ i_{k} and 0 ≤ j_{1} ≤ ... ≤ j_{q} ; and if the number of j equal to j_{u} is odd, then there is some s ≤ k such that 2i_{s} + 2^{s} < 2j_{u} < 2i_{s} + 2^{s+1}.

Thus when k ≤ 1, there is not above j_{u}, that means numbers of j = j_{u} are always even.

Theorem 5.1. Let G = O_{n}. Then all three fitrations are the same, and gr_{top}^{∗}(BG) ≅ A ⊕ B/2 with (y_{i} = x_{i}^{2} so that ∑ y_{1} = c_{1})

A = ℤ{∑(y_{1}y_{2})^{j_{1}}...(y_{2s−1}y_{2s})^{j_{s}}} B = ℤ{∑ y_{1}^{i}(y_{2}y_{3})^{j_{1}}...(y_{2s}y_{2s+1})^{j_{s}}}.

(Note A/2 = G_{−1} and B/2 = Q_{0}G_{0}.)

Example. When G = O_{2}, we have the isomorphism gr_{top}^{∗}(BG) ≅ ℤ[c_{2}] ⊕ ℤ/2[c_{1}].

When G = SO_{odd}, (since SO_{odd} × ℤ/2 ≅ O_{odd}), the situations are same. Let G = SO_{2n}. Then from Field, we have ([Fi], [Ma-Vi], [In-Ya])

CH^{∗}(BG) ≅ ℤ[c_{2}, ...c_{2n}]{y_{2n}} ⊕ CH^{∗}(BO_{2n})/(c_{1}),
BP^{∗}(BG) ≅ BP^{∗}[c_{2}, ..., c_{2n}]{y_{2n}} ⊕ BP^{∗}(BO_{2n})/(F_{1})

where F_{1} = ker(Bdet^{∗}) and y_{2n}^{2} = (−1)^{n}2^{2n−2}c_{2n}. Hence

y_{2n} = (−1)^{∗}2^{n−1}w_{2n} ∈ H^{∗}(BG)_{(2)}.

By Field, it is shown that just (n − 1)!y_{2n} (for n > 2) is represented by Chern classes (Theorem 8, Corollary 2 in [Fi]). Thus we have

Theorem 5.2. Let G = SO_{2n} and n ≥ 3. Then

gr_{top}^{∗}(BG) = gr_{geo}^{∗}(BG) ≅ ℤ[c_{2}, c_{4}, ..., c_{2n}]{y_{2n}} ⊕ gr_{top}^{∗}(BO_{2n})/(c_{1}).

However we have gr_{γ}^{2n}(BG) ≅ g̸r_{geo}^{2n}(BG).

We note when G = SO_{4}, all the three filtrations are same, since y_{4} is represented by Chern classes.

Proposition 5.3. Let G = SO_{2(p+1)} and p ≠ 2. Then

gr_{γ}^{∗}(BG) ≅ ℤ_{(p)}[c_{2}, ..., c_{2p+2}] ⊗ (ℤ_{(p)}{1, y′} ⊕ ℤ/p{y})

with |y′| = 2(p + 1) and |y| = 4.

We consider the exceptional Lie group G_{2}. Let G = G_{2}. Its mod(2) cohomology is well known

H^{∗}(BG; ℤ/2) ≅ ℤ/2[w_{4}, w_{6}, w_{7}]

and integral cohomology is

H^{∗}(BG) ≅ ℤ[w_{4}, c_{6}] ⊗ (ℤ{1} ⊕ ℤ/2[w_{7}]{w_{7}}).

We can compute the AHss for BP^{∗}(BG) ([Ko-Ya], [Sc-Ya])

grBP^{∗}(BG) ≅ ℤ[c_{4}, c_{6}] ⊗ (BP^{∗}{1, 2w_{4}} ⊕ P (3)^{∗}[c_{7}]{c_{7}}).

Here we can show the element {2w_{4}} is represented by a Chern class c′_{2}. We see K̃(1)^{∗}(BG) ≅ K̃(1)^{∗}[c_{4}, c_{6}] ⊗ {1, 2w_{4}}), and ([Ya2], [Gu])

CH^{∗}(BG) ≅ BP^{∗}(BG) ⊗_{BP^{∗}} ℤ ≅ ℤ[c′_{2}, c_{4}, c_{6}, c_{7}]/((c′_{2})^{2} − 4c_{4}, 2c_{7}).

Theorem 5.4. Let G = G_{2}. Then all three filtrations are the same

gr_{top}^{∗}(BG) ≅ CH^{∗}(BG)/(c_{7}) ≅ ℤ[c′_{2}, c_{4}, c_{6}]/((c′_{2})^{2} − 4c_{4}).

Next we study the case G = Spin_{7}. Its mod(2) cohomology is

H^{∗}(BG; ℤ/2) ≅ ℤ/2[w_{4}, w_{6}, w_{7}, w_{8}].

The infinity term of the AHss for BP^{∗}(BG) is still computed

grBP^{∗}(BG) ≅ ℤ[c_{4}, c_{6}] ⊗ (BP^{∗}[c_{8}]{1, 2w_{4}, 2w_{8}, 2w_{4}w_{8}, v_{1}w_{8}}
⊕P (3)^{∗}[c_{7}]{c_{7}} ⊕ P (4)^{∗}[c_{7}, c_{8}]{c_{7}c_{8}}).

Hence we see

grK̃(1)^{∗}(BG) ≅ K̃(1)^{∗}[c_{4}, c_{6}, c_{8}]{1, 2w_{4}, 2w_{8}, 2w_{4}w_{8}, v_{1}w_{8}}.

Here it is known that 2w_{4}, 2w_{8}, 2w_{4}w_{8} are represented by Chern classes, and write them c′_{2}, c′_{4}, c′_{6}. But it is proved (Theorem 6.2 in [Sc-Ya]) that v_{1}w_{8} is not represented by (transfer) of Chern classes while it is in the image of cycle map. Let cl(ξ) = [v_{1}w_{8}] ([Gu], Lemma 9.6 in [Ya], §9 in [Ka-Te-Ya]). Totraro’s conjecture also holds this case

CH^{∗}(BG) ≅ BP^{∗}(BG) ⊗_{BP^{∗}} ℤ
≅ ℤ[c_{4}, c_{6}, c_{8}] ⊗ (ℤ{1, c′_{2}, c′_{4}, c′_{6}} ⊕ ℤ/2{ξ} ⊕ ℤ/2[c_{7}]{c_{7}})

with |ξ| = 6. Moreover, we can prove Lemma 5.5. Let G = Spin_{7}. Any element x ∈ BP^{∗}(BG) such that

0 ≠ x = [v_{1}w_{8}]a ∈ BP^{∗}(BP) with a ∈ ℤ[c_{4}, c_{6}, c_{8}],

can not be generated by Chern classes of BP^{∗}-theory.

Proof. Let N = Z(G) ≅ ℤ/2 be the center of G and N ⊕A is a maximal elementary abelian 2-subgroup of G, so A ≅ (ℤ/2)^{3}. A representation ξ of G is said to be a spin representation, if ξ|N ≠ 0. For a nonspin representation η, we know the total Chern class

η|_{N⊕A} = η|_{A} ∈ BP^{∗}[c_{4}, c_{6}, c_{7}].

For a spin representation χ, we have

¸(χ)|_{N} = (1 + u)^{s} ∈ BP^{∗}(BN) ≅ BP^{∗}[u]/([2](u)) |u| = 2

where [2](u) = 2u + v_{1}u^{2} + ... is the 2-th product of the BP^{∗}-formal group laws. Here we note s = 8s′ since c_{8}|_{N} = u^{8}. It is known that v_{1}w_{8}|_{N} = v_{1}u^{4} [Sc-Ya]. Then

c(χ)|_{N} = (1 + 8u + 28u^{2} + ... + u^{8})^{s}.

Here we can compute

8u = 4v_{1}u^{2} = 2v_{1}^{2}u^{3} = v_{1}^{3}u^{4}, 28u^{2} = 14v_{1}u^{3} = 7v_{1}^{2}u^{4}, ....

Thus we see that v_{1}u^{4} is not represented by the restriction of Chern classes. (However v_{1}^{2}u^{4} has its possibily, infact |v_{1}w_{8}| = 4 and it is represented by the Chern class c_{2}.)

Of course c(χ ⊕ η) = c(χ)c(η), we get the lemma. □

Theorem 5.6. Let G = Spin_{7}. Then

gr_{top}^{∗}(BG) ≅ ℤ[c_{4}, c_{6}, w_{8}]{1, c′_{2}},
gr_{α}^{∗}(BG) ≅ ℤ[c_{4}, c_{6}, c_{8}](ℤ{1, c′_{2}, c′_{4}, c′_{6}} ⊕ ℤ/2{ξ})

where deg(ξ) = 6 (resp. = 4) if α = geo ( if α = γ).

Recall that 2_{+}^{1+2n} is the extraspecial 2-group, which is isomorphic to the central product of n-copies of the dihedral group D_{8} of order 8. Let G = 2_{+}^{1+6}. There is an inclusion i : G ⊂ Spin_{7} and its induced map i^{∗} : H^{∗}(BSpin_{7}; ℤ/2) → H^{∗}(BG; ℤ/2) is also injective by Quillen [Qu]. Let j : ℤ/2 ≅ Z(G) ⊂ G. Then it is know [Qu], [Sc-Ya] j^{∗}i^{∗}(w_{8}) = u^{4} ∈ ℤ[u]/(2u) ⊂ H^{∗}(BZ(G)). Hence we have in K̃(1)^{∗}-theory

j^{∗}i^{∗}(v_{1}w_{8}) = v_{1}u^{4} ≠ 0 ∈ K̃(1)^{∗}(BZ(G)) ≅ K̃(1)^{∗}[u]/(2u − v_{1}u^{2}).

This element v_{1} ⊗ w_{8} is not generated by Chern classes also in H^{∗}(BG). Hence we have

Corollary 5.7. Let G = 2_{+}^{1+6}. Then there is an element x ∈ AK̃(1)^{∗}(BG) such that

0 ≠ x ∈ gr_{γ}^{4}(BG), x = ξ ∈ gr_{geo}^{6}(BG), and x = w_{8} ∈ gr_{top}^{8}(BG).

6. connected groups for p odd

In this section, we assume p ≥ 3. At first we consider the case G = P GL_{p}. Its mod p cohomolgy is given by Vistoli and Kameko-Yagita ([Vi], [Ka-Ya]), namely, there is a short exact sequence

0 → M/p → H^{∗}(BG) → N → 0

where M ≅ Z[x_{4}, x_{6}, ..., x_{2p}] additively (but not as rings), and N ≅ N′⊗Λ(Q_{0}, Q_{1}){u_{2}}, |u_{2}| = 2 for some ℤ/p-module N′. The BP -theory BP^{∗}(BG) is also studied There is a short exact sequence

0 → BP^{∗} ⊗ M → grBP^{∗}(BG) → N′′ → 0

where grN′′ ≅ P (3)^{∗} ⊗ N′ ⊗ Q_{0}Q_{1}(u). Therefore we see gr^{∗}K̃(1)^{∗}(BG) ≅ M additively. Totaro’s conjecture also holds this case. Thus we have

Theorem 6.1. Let G = P GL_{p}. Then

gr_{top}^{∗}(BG) ≅ gr_{geo}^{∗}(BG) (≅ M additively).

When p = 3, it is known (page 2274 in [Ka-Ya]) that M/3 ≅ ℤ/3[c_{2}, c_{3}, c_{6}]/(c_{2}^{3} − c_{6}^{2}). Hence the gamma filtration is the same when p = 3. However, for p ≥ 5, it seems unknown that M above is generated by Chern classes or not.

For exceptional Lie groups, we can compute BP^{∗}(BG) except for (G, p) = (E_{8}, p = 3). So we know gr_{top}^{∗}(BG), but it seems not so easy to compute CH^{∗}(BG) now, and gr_{geo}^{∗}(BG) seems unknown. For example, when G = F_{4} we can compute BP^{∗}(BG). The mod(3) cohomology is generated by x_{4}, x_{8}, x_{9}, x_{20,}x_{21}, ... (by Toda). The BP -theory is also computed

grBP^{∗}(BG) ≅ BP^{∗}[c_{18}, c_{24}]{1, 3x_{4}} ⊕ BP^{∗} ⊗ E ⊕ P (3)^{∗}[x_{26}]{x_{26}}

where E = ℤ[x_{4}, x_{8}]{ab|a, b ∈ {x_{4}, x_{8}, x_{20}}}. Hence we have

grK̃(1)^{∗}(BG) ≅ K̃(1)^{∗} ⊗ (ℤ[c_{18}, c_{24}]{1, 3x_{4}} ⊕ E).

It is now unknown whether the element x_{8}^{2} ∈ E is in the image of the cycle map. If it is so, then gr_{geo}^{∗}(BG) ≅ gr_{top}^{∗}(BG), otherwise gr_{geo}^{i}(BG) ≅ g̸r_{top}^{i}(BG) for i = 12, 16.

7. Rost motives

In this section, we do not assume that k is algebraically closed. At first, we recall the (generalized) Rost motive ([Ro1,2]). Let M(X) be the motive of (smooth) variety X. For a non zero symbol a = {a_{0}, ..., a_{n}} in the mod 2 Milnor K-theory K_{n+1}^{M}(k)/2, let ϕ_{a} = ⟨⟨a_{0}, ..., a_{n}⟩⟩ be the (n + 1)-fold Pfister form. Let X_{ϕ_{a}} be the projective quadric of dimension 2^{n+1} − 2 defined by ϕ_{a}. The Rost motive M_{a}(= M_{ϕ_{a}}) is a direct summand of the motive M(X_{ϕ_{a}}) representing X_{ϕ_{a}} so that M(X_{ϕ_{a}}) ≅ M_{a} ⊗ M(ℙ^{2^{n}−1}).

Moreover for an odd prime p and nonzero symbol 0 ≠ a ∈ K_{n+1}^{M}/p, we can define ([Ro2],[Vo4,5],[Su-Jo]) the generalized Rost motive M_{a}, which is irreducible and is split over K/k if and only if a|_{K} = 0 (as the case p = 2).

The Chow group of the Rost motive is well known. Let k̄ be an algebraic closure of k, X|k̄ = X ⊗_{k} k̄, and ik̄ : CH^{∗}(X) → CH^{∗}(X|k̄) the restriction map.

Lemma 7.1. (Rost [Ro1,2],[Vo4], [Vi-Ya], [Ya3,4]) The Chow group CH^{∗}(M_{a}) is only dependent on n. There are isomorphisms

CH^{∗}(M_{a}) ≅ ℤ{1} ⊕ (ℤ{c_{0}} ⊕ ℤ/p{c_{1}, ..., c_{n−1}})[y]/(c_{i}y^{p−1})
and CH^{∗}(M_{a}|k̄) ≅ ℤ[y]/(y^{p})

where 2deg(y) = |y| = 2(p^{n−1} + ... + p + 1) and |c_{i}| = |y| + 2 − 2p^{i}. Moreover the restriction map is given by ik̄(c_{0}) = py and ik̄(c_{i}) = 0 for i > 0.

Remark. The element y does not exist in CH^{∗}(M_{a}) while c_{i}y exists. Usually CH^{∗}(M_{a}) is defined only additively, however when CH^{∗}(M_{a}) has the natural ring structure (e.g., p = 2), the multiplications are given by c_{i} · c_{j} = 0 for all 0 ≤ i, j ≤ n − 1.

For the simplicity of notation, hereafter we always write by Ω^{∗}(X) the BP^{∗}- version of the algebraic cobordism

Ω^{∗}(X) ⊗_{MU^{∗}} BP^{∗} ≅ ABP^{2∗,∗}(X).

Hence we mean Ω^{∗} = BP^{∗}.

Let I_{n} be the ideal in Ω^{∗} generated by v_{0}, ..., v_{n−1}, i.e.,

I_{n} = (p = v_{0}, v_{1}, ..., v_{n−1}) ⊂ Ω^{∗}.

Then it is well known that I_{n} and I_{∞} are the only prime ideals stable under the Landweber-Novikov cohomology operations ([Ra]) in Ω^{∗}.

The category of cobordism motives is defined and studied in [Vi-Ya]. In particular, we can define the algebraic cobordism of motives. The following is the main result in [Vi-Ya] (in [Ya4] for odd primes).

Lemma 7.2. ([Vi-Ya], [Ya4]) The restriction map

ik̄ : Ω^{∗}(M_{a}) → Ω^{∗}(M_{a}|k̄) ≅ Ω^{∗}[y]/(y^{p})

is injective and there is an Ω^{∗}-module isomorphism

Ω^{∗}(M_{a}) ≅ Ω^{∗}{1} ⊕ I_{n}{y, ..., y^{p−1}} ⊂ Ω^{∗}[y]/(y^{p})

such that v_{i}y = c_{i} in Ω^{∗}(M_{a}) ⊗_{Ω^{∗}} ℤ ≅ CH^{∗}(M_{a}).

We consider the following assumption for X

Assumption (∗). There is an isomorphism of motives

M(X) ≅ M_{n} ⊗ A(X) with A(X) ≅ ⊕_{s}𝕋^{i_{s}}

where 𝕋 is the k-Tate module.

Lemma 7.3. Suppose Assumption (∗). Then

K_{alg}^{0}(X) ≅ K_{alg}^{0}(X|k̄) ≅ K_{top}^{0}(X(ℂ)).

Proof. Since M(X|k̄) is a sum of k̄-Tate modules, we have the isomorphism K_{alg}^{0}(X|k̄) ≅ K_{top}^{0}(X(ℂ)) from

K_{alg}^{0}(𝕋) ≅ K_{alg}^{0}(S^{2,1}|k̄) ≅ K_{top}^{0}(S^{2}).

For the first isomorphism, we only need to show K_{alg}^{0}(M_{n}) ≅ K_{alg}^{0}(M_{n}|k̄). Recall

Ω^{∗}(M_{n}) ≅ (BP^{∗} ⊕ Ideal(p, v_{1}, ..., v_{n−1}){ρ})

by c_{i} ↦ v_{i}ρ. Hence v_{i}c_{1} = v_{1}c_{i}. Therefore for i > 1, we see c_{i} = 0 in AK̃(1)^{2∗,∗}(M_{n}) where v_{i} = 0. So we have

AK̃(1)^{2∗,∗}(M_{n}) ≅ K̃(1)^{∗}{1, c_{0}, c_{1}}/(v_{1}c_{0} = pc_{1}) ≅ AK̃(1)^{2∗,∗}{1, c_{1}}

≅ AK̃(1)^{2∗,∗}{1, v_{1}ρ} ≅ AK̃(1)^{2∗,∗}{1, ρ} ≅ AK̃(1)^{2∗,∗}(M|k̄).
□

8. flag manifolds G/T

Now we consider the flag variety G/T. Let G be a simply connected Lie group and T the maximal torus. Moreover we assume its cohomology is

H^{∗}(G; ℤ/p) ≅ ℤ/p[y]/(y^{p}) ⊗ Λ(x_{1}, ..., x_{ℓ})

with |y| = 2(p + 1) and |x_{i}| = odd. Then it is well known that the cohomology of G/T is torsion free ([Tod]) and

H^{∗}(G/T) ≅ ℤ[y, t_{1}, ..., t_{ℓ}]/(f_{y}, b_{1}, ..., b_{ℓ})

where f_{y} = y^{p} mod Ideal(t_{i}) and (b_{1}, ..., b_{ℓ}) is a regular sequence in ℤ[t_{1}, ..., t_{ℓ}].

Let k be a subfield of ℂ which contains primitive p-th root of the unity. Let us denote by G_{k} the split reductive group over k which corresponds G. By definition, a G_{k}-torsor 𝔾_{k} over k is a variety over k with a free G_{k}-action such that the quotient variety is Spec(k). A G_{k}-torsor over k is called trivial, if it is isomorphic to G_{k} or equivalently it has a k-rational point. In this paper by 𝔾_{k}, we mean the nontrivial torsor at any finite extension K/k coprime to p.

Let H be a subgroup of G. Given a torsor 𝔾_{k} over k, we can form the twisted form of G/H by

(𝔾_{k} × G_{k}/H_{k})/G_{k} ≅ 𝔾_{k}/H_{k}.

Letting X = G/T, we consider cases such that Assumption hold. By [Pe-Se-Za], exceptional Lie groups (G_{2}, p = 2) and (F_{4}, p = 3) are such cases. The filtrations of such spaces are also studied by Gabrier and Zainouline ([Ga-Za], [Za], [Ju]) as the twisted gamma filtrations.

At first, we consider the case (G, p) = (G_{2}, 2). We recall the cohomology from Toda-Watanabe [To-Wa]

H^{∗}(G/T; ℤ) ≅ ℤ[t_{1}, t_{2}, y]/(t_{1}^{2} + t_{1}t_{2} + t_{2}^{2}, t_{2}^{3} − 2y, y^{2})

with |t_{i}| = 2 and |y| = 6. Let P be the maximal parabolic subgroup such that G/P is isomorphic to a quadric. Then from (3.6) and H^{∗}(P/T) ≅ ℤ{1, t_{1}} ([To-Wa])

H^{∗}(G/P; ℤ) ≅ ℤ[t_{2}, y]/(t_{2}^{3} − 2y, y^{2}) ≅ ℤ{1, y} ⊗ {1, t_{2}, t_{2}^{2}}

Of course this is isomorphic to gr_{top}^{∗}(G/P).

Since G/P is a quadric, we have the decomposition ([Bo], §7 in [Pe-Se-Za])

M(𝔾_{k}/P_{k}) ≅ M_{2} ⊕ M_{2}(1) ⊕ M_{2}(2).

Theorem 8.1. (Theorem 5.2 in[Ya5]) There is a ring isomorphism

gr_{γ}^{∗}(G/P) ≅ gr_{geo}^{∗}(𝔾_{k}/P_{k}) ≅ CH^{∗}(𝔾_{k}/P_{k})
≅ ℤ_{(2)}[t_{2}, u]/(t_{2}^{6}, 2u, t_{2}^{3}u, u^{2}) ≅ ℤ_{(2)}[t_{2}]/(t_{2}^{6}) ⊕ ℤ/2[t_{2}]/(t_{2}^{3}){u}

with |t_{2}| = 2, |u| = 4.

Proof. Recall that

Ω^{∗}(M_{2}) ≅ Ω^{∗}{1, 2y, vy} ⊂ Ω^{∗}{1, y}.

From the decomposition of the motive, we have the Ω^{∗}-module isomorphism

Ω^{∗}(𝔾_{k}/P_{k}) ≅ Ω^{∗}{1, v_{1}y, 2y} ⊗ {1, t_{2}, t_{2}^{2}} ⊂ Ω^{∗}(G_{k}/P_{k}).

Since CH^{∗}(X) ≅ Ω^{∗}(X) ⊗_{Ω^{∗}} ℤ, we have the isomorphism

CH^{∗}(𝔾_{k}/P_{k}) ≅ ℤ{1, 2y}{1, t_{2}, t_{2}^{2}} ⊕ ℤ/2{v_{1}y}{1, t_{2}, t_{2}^{2}}.

(Note 2v_{1}y = v_{1}(2y) ∈ Ω^{<0}Ω^{∗}(𝔾_{k}/P_{k}).)

Here the multiplications are given as follows. Since 2y = t_{2}^{3} mod(Ω^{<0}) in Ω^{∗}(G_{k}/T_{k}), we can take 2y = t_{2}^{3} ∈ CH^{∗}(𝔾/P_{k}) so that

ℤ{1, 2y}{1, t_{2}, t_{2}^{2}} = ℤ[t_{2}]/(t_{2}^{6}) ⊂ CH^{∗}(𝔾/P_{k}).

Let us write u = v_{1}y in CH^{∗}(𝔾_{k}/T_{k}). Then t_{2}^{3}u = 2yv_{1}y = 0 and u^{2} = v_{1}^{2}y^{2} = 0 in Ω^{∗}(𝔾_{k}/T_{k}) ⊗_{Ω^{∗}} ℤ. Hence we have the second isomorphism for in the theorem.

Since |u| = 4, the element u is represented by Chern classes, we see the first isomorphism. □

Remark. The space 𝔾_{k}/T_{k} is isomorphic to the quadric defined by the maximal neighbor of the 3-Pfister form. Hence its Chow ring is computed in [Ya5].

It is well known that the representations (over ℂ)) are written as

R(G/T) ≅ R(T)/R(G).

Therefore each element which is represented by Chern classes is written as an element in Ω^{∗}(𝔾_{k}/T_{k})

c(ξ) = ∏(1 + λ_{1}t_{1} + λ_{2}t_{2}) ∈ Ω^{∗}[t_{1}, t_{2}] λ_{i} ∈ ℤ/2

modulo ((t_{1}, t_{2})Ω^{<0}Ω^{∗}(G_{k}/T_{k})). By the similar arguments, we have (see Theorem 5.3 in [Ya5])

Theorem 8.2. There are ring isomorphisms

gr_{γ}^{∗}(G/T) ≅ CH^{∗}(𝔾_{k}/T_{k}) ≅ ℤ[t_{1}, t_{2}]/(t_{2}^{6}, 2u, t_{2}^{3}u, u^{2})

where u = t_{1}^{2} + t_{1}t_{2} + t_{2}^{2}.

Proof. The Chow ring is isomorphic to

(∗) CH^{∗}(𝔾_{k}/T_{k}) ≅ CH^{∗}(𝔾_{k}/P_{k}){1, t_{1}}
≅ (ℤ{1, 2y} ⊕ ℤ/2{v_{1}y}){1, t_{2}, t_{2}^{2}}{1, t_{1}}.

Here 2y = t_{2}^{3}. Since v_{1}y ∈ (t_{1}, t_{2}) and v_{1}y = 0 ∈ CH^{∗}(G_{k}/T_{k}), we see

v_{1}y = λ(t_{1}^{2} + t_{1}t_{2} + t_{2}^{2}) mod((t_{1}, t_{2})Ω^{<0}Ω^{∗}(G_{k}/T_{k}))

for λ ∈ ℤ. We can take λ = 1 mod(2). Otherwise v_{1}y = 0 ∈ Ω^{∗}(G_{k}/T_{k})/2, which is an Ω^{∗}/2-free, and this is a contradiction. Hence we can take t_{1}^{2} + t_{1}t_{2} + t_{2}^{2} as v_{1}y. Hence in CH^{∗}(𝔾_{k}/T_{k}) we have the relation

(t_{2}^{3})^{2} = 0, (t_{2}^{3})u = 0, u^{2} = 0, 2u = 0.
□

Next we consider the case (G, p) = (F_{4}, 3). Let 𝔾_{k} be a nontrivial G_{k}-torsor at 3 as previous sections. Let P_{k} be a maximal parabolic subgroup of G_{k} given by the the first three vertexes

∘^{1} −− ∘^{2}⇒=∘^{3} −− ∘^{4}

of the Dynkin diagram. Then Nikolenko- Semenov-Zainoulline ([Ni-Se-Za]) showed that there is an isomorphism

M(𝔾_{k}/P_{k}) ≅ ⊕_{i=0}^{7}M_{2}(i).

We first recall the ordinary cohomology of G/P ([Is-To], [Du-Za]).

H^{∗}(G/P)_{(3)} ≅ ℤ[t, y]/(r_{8}, r_{12}), |t| = 2, |y| = 8

where r_{8} = 3y^{2} − t^{8} and r_{12} = 26y^{3} − 5t^{12}. Hence we can rewrite

H^{∗}(G/P) ≅ ℤ{1, t, ..., t^{7}} ⊗ {1, y, y^{2}}.

Recall the Rost motive CH^{∗}(M_{2}|k̄) ≅ ℤ[y]/(y^{3}),

CH^{∗}(M_{2}) ≅ ℤ{1} ⊕ ℤ{3y, 3y^{2}} ⊕ ℤ/3{v_{1}y, v_{1}y^{2}}.

Of course, the above y ∈ CH^{∗}(M_{a}) can be identified with the same named element in H^{∗}(G_{k}/P_{k}) by the restriction map CH^{∗}(M_{a}) → CH^{∗}(M_{a}|k̄) ⊂ CH^{∗}(G_{k}/P_{k}). From the above theorem, we have the decomposition

(∗) CH^{∗}(𝔾_{k}/P_{k}) ≅ ℤ{1, t, ..., t^{7}} ⊗ (ℤ{1, 3y, 3y^{2}} ⊕ ℤ/3{v_{1}y, v_{1}y^{2}}).

The ring structure is given as follows.

Proposition 8.3. ( Theorem 6.2 in [Ya5])

gr_{geo}^{∗}(𝔾_{k}/P_{k}) ≅ CH^{∗}(𝔾_{k}/P_{k})
≅ ℤ[t, b, a_{1}, a_{2}]/(t^{16}, t^{8}b, b^{2} = 3t^{8}, ba_{i}, 3a_{i}, t^{8}a_{i}, a_{1}a_{2})
≅ ℤ{1, t, ..., t^{7}} ⊗ (ℤ{1, √3t^{4}, t^{8}} ⊕ ℤ/3{a_{1}, a_{2}})

where |b| = 8 and |a_{1}| = 4, |a_{2}| = 12.

Proof. From the relation r_{8} in CH^{∗}(G/P), we have

3y^{2} = t^{8} + vx ∈ Ω^{∗}(G/P) for v ∈ Ω^{<0}.

Hence we can take t^{8} instead of 3y^{2} in (∗). Of course

(3y)^{2} = 3t^{8} + 3vx ∈ Ω^{∗}(G_{k}/P_{k}).

Hence we write by b = √3t^{4} the element 3y. Write by a_{1}, a_{2} the elements v_{1}y, v_{1}y^{2} respectively. Elements in I_{∞}Ω^{<0} ⊂ Ω(G_{k}/P_{k}) reduces to zero in CH^{∗}(𝔾_{k}/T_{k}). Therefore we have the desired multiplicative results. □

The element b = 3y is represented by a Chern class c_{4}(ξ) for some ξ by the Riemann-Roch theorem without denominators. Unfortunately, we do not know if a_{2} = v_{1}y^{2} are Chern classes in CH^{∗}(BG) or not.

Proposition 8.4. If a_{2} = v_{1}y^{2} ∈ CH^{∗}(𝔾_{k}/P_{k}) is not represented by a Chern class, then

gr_{γ}(G/P) ≅ ℤ[t, b, a_{1}]/(t^{16}, t^{8}b, b^{2} = 3t^{8}, ba_{1}, 3a_{1}, t^{8}a_{1}, a_{1}^{3})

where |b| = 8 and |a_{1}| = 4.

Proof. If v_{1}y^{2} is not represented by Chern class of CH^{∗}(𝔾_{k}/P k) ( or Ω^{∗}(𝔾_{k}/P_{k})), then the corresponding nonzero element in gr_{γ}(G/T) is v_{1}^{2}y^{2}, which is written as (v_{1}y)^{2} = (a_{1})^{2}. □

9. Filtrations of the Morava K-theory

For most groups G in the preceding sections, it is known that K(n)^{odd}(BG) = 0 (while Kriz gave some examples with K(n)^{odd}(BG) ≠ 0). Hereafter, we only consider spaces X such that

(9.1) K(n)^{odd}(X(ℂ)) = K̃(n)^{odd}(X(ℂ)) = 0,
(9.2) K(n)^{∗}(X(ℂ)) ≅ AK(n)^{2∗,∗}(X)

Then we can define the three filtrations for the Morava K(n)-theory.

F (n)_{top}^{2i} = Ker(K(n)^{∗}(X(ℂ) → K(n)^{∗}(X(ℂ)^{2i}),
F (n)_{geo}^{2i} = {f_{∗}(1_{M})|f : M → X and codim_{X}M ≥ i}
F (n)_{γ}^{2i} = {c_{i_{1}}^{K(n)}(x_{1}) · ... · c_{i_{m}}^{K(n)}(x_{m})|i_{1} + ...i_{m} ≥ i}.

Here c_{i_{s}}^{K(n)}(x_{s}) is the Chern class for AK(n)^{∗,∗′}-theory for some k-representation x_{s} : X → BGL_{N}. This Chern class is induced from the isomorphism

AK(n)^{2∗,∗}(BGL_{N}) ≅ K(n)^{∗} ⊗_{BP^{∗}} Ω^{∗}(BGL_{N}),

in fact, it is well known that in Ω^{∗}(X), we can define Chern classes canonically (see [Mo-Le] for example). However each element in K(n)^{∗}(X(ℂ)) (for n ≥ 2) need not to be represented by K(n)^{∗}-theory Chern classes. Hence we need the assumption

(9.3) F_{γ}^{0} = K(n)^{∗}(X).

(We also consider the cases where (9.3) does not hold.) Of course the assumptions are satisfied for K(1)^{∗}-theory, if they are so for K̃(1)^{∗}-theory.

Recall P (n)^{∗}(X) be the cohomology theory with the coefficient

P (n)^{∗} = BP^{∗}/(p, v_{1}, ..., v_{n−1}).

It is well known, for all X,

P (n)^{∗}(X) ⊗_{BP^{∗}} K(n)^{∗} ≅ K(n)^{∗}(X).

Let us write by E(P (n))_{r}^{∗,∗′} (resp. E(K(n))_{r}^{∗,∗′}) the AHss converging to P (n)^{∗}(X) (resp. K(n)^{∗}(X)). Then we have

E(P (n))_{r}^{∗,∗′} ⊗_{BP^{∗}} K(n)^{∗} ≅ E(K(n))_{r}^{∗,∗′}.

If (9.1)-(9.3) are satisfied, then K(n)-version (exchanging BP^{∗}(X) to P (n)^{∗}(X)). of all lemmas in §2 also hold.

Lemma 9.1. Suppose that Ω^{∗}(X)/p ≅ BP^{∗}(X(ℂ))/p and it is generated by (BP^{∗}- ) Chern classes. Then (9.1)-(9,3) are satisfied.

Proof. We consider the maps

Ω^{∗}(X) ⊗_{BP^{∗}} K(n)^{∗} →^{ρ_{1}} AK^{2∗,∗}(X) →^{ρ_{2}} K(n)^{∗}(X(ℂ)).

Here the map ρ_{1} is an epimorphism because Ω^{∗}(X) (resp. AK(n)^{2∗,∗}(X)) is generated as a BP^{∗}-module (resp. K(n)^{∗}-module) by elements in CH^{∗}(X).

On the other hand by Ravenel-Wilson-Yagita [Ra-Wi-Ya], we know that (1.2) implies

K(n)^{∗}(X(ℂ)) ≅ K(n)^{∗} ⊗_{BP^{∗}} BP^{∗}(X(ℂ)).

From the supposion in the theorem, we see that ρ_{2}ρ_{1} is an isomorphism. This means that ρ_{1}, ρ_{2} are also isomorphisms. □

For X = BG, G = finite abelian, p_{±}^{1+2}, O_{n}, G_{2} and P SL_{3} (p = 3) satisfy the assumptions in the above lemma.

Of course gr_{top}^{∗}(X) and gr(n)_{top}^{∗}(X) are quite different. Let G = ℤ/p. Then

K(n)^{∗}(BG) ≅ K(n)^{∗}(y]/(y^{p^{n}}).

and this is generated by Chern classes in H^{∗}(BG; ℤ/p).

Theorem 9.2. Let G = ⊕^{s}ℤ/p. Then all three filtrations of K(n)^{∗}(BG) are same and

gr(n)_{top}^{∗}(BG) ≅ ℤ/p[y_{1}, ..., y_{s}]/(y_{1}^{p^{n}}, ..., y_{s}^{p^{n}}).

Similarly, we have

Theorem 9.3. Let G = O_{m}. Then all three filtrations of K(n)^{∗}(BG) are same and

gr(n)_{top}^{∗}(BG) ≅ {∑ y_{1}^{i_{1}}...y_{s}^{i_{s}}(y_{s+1}y_{s+2})^{j_{s+1}}...(y_{2k+1}y_{2k+2})^{j_{2k+1}}}

where 0 ≤ i_{1} ≤ ... ≤ i_{s} < 2^{n} ≤ i_{s} ≤ ... ≤ i_{k}.

For example, gr(n)_{top}^{∗} ≅ ℤ/2[c_{2}] ⊕ ℤ/2{c_{1}^{i}c_{2}^{j}|i + 2j < 2^{n}}.

Next we consider the case G = SO_{2m} Recall for m ≥ 3, y_{2m} is not represented by Chern classes

Theorem 9.4. Let G = SO_{2m} and m > 2. Then

gr(n)_{geo}^{∗}(BG) ≅ ℤ[c_{2}, c_{4}, ..., c_{2m}]{y_{2m}} ⊕ gr(n)_{geo}^{∗}(BO_{2m})/(c_{1}).

However gr(n)_{γ}^{∗}(BG) ≅ g̸r(n)_{geo}^{∗}(BG) ≅ g̸r(n)_{top}^{∗}(BG).

Proof. We only need the second nonisomorphism of the second statement. Since y_{2m} = (−1)^{∗}2^{m−1}w_{2m} ∈ H^{∗}(BG) is zero in H^{∗}(BG; ℤ/2). Hence 0 ≠ y_{2m} ∈ P (n)^{∗}(BG) is represented in the AHss converging to P (n)^{∗}(BG) as element in E_{∞}^{∗,∗′} with ∗′ < 0 and ∗ > 2m. □

Next consider the case G = G_{2} (and p = 2). By the computation of the AHss for P (1)^{∗}(BG) (= BP^{∗}(BG; ℤ/2)), we have

K(1)^{∗}(BG) ≅ K(1)^{∗}[c_{4}, c_{6}]{1, v_{1}w_{6}}.

By the direct computation of the AHss for K(2)^{∗}(BG), we see

K(2)^{∗}(BG) ≅ K(1)^{∗}[c_{4}, c_{6}]{1, w_{4}w_{6}}.

Therefore we have

Theorem 9.5. Let G = G_{2}. Then

gr(i)_{α}^{∗}(BG) ≅ ℤ/2[c_{4}, c_{6}]{1, a}

where a^{2} = \begin{cases}c_{4}c_{6} |a| = 10 if i = 2. α = top \\ c_{6} |a| = 6 if i = 1. α = top \\ 0 |a| = 4 if i = 1, 2. α ≠ top. \end{cases} .

Proof. The above a is represented as a = w_{4}w_{6} (resp. w_{6}, v_{1}w_{6}, v_{2}w_{4}w_{6}) when i = 2, α = top (resp. i = 1, α = top, i = 1 α ≠ top), and i = 2 α ≠ top)). □

When n ≥ 1, the geometric and topological filtrations are quite different.

Theorem 9.6. Let G be a simply connected simple Lie group such that H^{∗}(G) has p-torsion. Then for n ≥ 1

gr(n)_{geo}^{4}(BG) ≠ 0 but gr(n)_{top}^{4}(BG) = 0.

Proof. The space BG is 3-connected and H^{4}(BG) ≅ ℤ. Let us write by x its 4dimensional generator. By [Ka-Ya2], it is known that px ∈ H^{4}(BG) is represented as the Chern class c_{2} for some representation. Hence gr(n)_{geo}^{4}(BG) ≠ 0.

To see gr(n)_{top}^{4}(BG) = 0, we only need to show

(∗) d_{2p^{n}−1}(x) = v_{n} ⊗ Q_{n}(x) ≠ 0

in the AHss converging to K(n)^{∗}(BG).

For these groups, it is well known that there are embedding G_{2} ⊂ G for p = 2, (F_{4} ⊂ G for p = 3 and G = E_{8} for p = 5). We will prove (∗) for G = F_{4} and p = 3, then we can see (∗) the other groups when p = 3. (The other primes cases are similar).

Let G = F_{4} and p = 3. Then G has a maximal elementary p-group A ≅ (ℤ/3)^{3}. We consider the restriction map for i : A ⊂ G,

i^{∗} : H^{∗}(BG; ℤ/p) → H^{∗}(BA; ℤ/p) ≅ ℤ/p[y_{1}, y_{2}, y_{3}] ⊗ Λ(x_{1}, x_{2}, x_{3}).

The restriction image is i^{∗}(x) = Q_{0}(x_{1}x_{2}x_{3}) (see [Ka-Te-Ya]). Hence we show

i^{∗}(Q_{n}(x)) = Q_{n}Q_{0}(x_{1}x_{2}x_{3}) = ∑ y_{1}^{p^{n}}y_{2}x_{3} ≠ 0.

□

Now we recall arguments for quadrics. Let m = 2m′ + 1. and let us write the quadratic form q(x) defined by

q(x_{1}, ..., x_{m}) = x_{1}x_{2} + ... + x_{m−2}x_{m−1} + x_{m}^{2}

and the projective quadric X_{q} defined by the quadratic form q. Then it is well known that (in fact SO(m) acts on the affine quadric in 𝔸^{m} − 0)

X_{q} ≅ SO(m)/(SO(m − 2) × SO(2)).

Let G = SO(m) and P = SO(m − 2) × SO(2). Then the quadric q is always split over k and we know CH^{∗}(G_{k}/P_{k}) ≅ CH^{∗}(X_{q}).

In particular we consider the case m = 2^{n+1} − 1. Let ρ = {−1} ∈ K_{1}^{M}(k)/2 = k^{∗}/(k^{∗})^{2}. We consider fields k such that

0 ≠ ρ^{n+1} ∈ K_{n+1}^{M}(k)/2.

Define the quadratic form q′ by

q′(x_{1}, ..., x_{m}) = x_{1}^{2} + ... + x_{m}^{2}.

Then this q′ is a subform of

⟨⟨ − 1, ..., −1⟩⟩ = ϕ_{ρ^{n+1}}

the (n+1)-th Pfister form associated to ρ^{n+1}. (That is, q′ is the maximal neighbor of the (n + 1)-th Pfister form.) Of course q|k̄ = q′|k̄ and we can identify 𝔾_{k}/P_{k} ≅ X_{q′}. From Lemma 7.2 (or Rost’s result), we know

CH^{∗}(X_{q′}|k̄) ≅ ℤ[t, y]/(t^{2^{n}−1} − 2y, y^{2}).

As stated in §7, there is a decomposition of motives

M(X_{q′}) ≅ M_{n} ⊗ ℤ/2[t]/(t^{2^{n}−1}).

Hence we have the additive isomorphism

CH^{∗}(X_{ϕ′_{a}}) ≅ ℤ[t]/(t^{2^{n}−1}) ⊗ (ℤ{1, c_{n,0}} ⊕ ℤ/2{c_{n,1}, ..., c_{n,n−1}}).

With identification t^{2^{n}−1} = 2y = c_{n,0}, and u_{i} = c_{n,i} for i > 0, we also get the ring isomorphism

Theorem 9.7. ( [Ya5]) Let G_{k}/P_{k} be the above quadric X_{q′}. Then there is a ring isomorphism

CH^{∗}(𝔾_{k}/P_{k}) ≅ ℤ[t]/(t^{2^{n+1}−2}) ⊕ ℤ/2[t]/(t^{2^{n}−1}){u_{1}, ..., u_{n−1}}

where u_{i} = v_{i}y ∈ Ω^{∗}(𝔾_{k}/p) ⊗_{Ω^{∗}} Z_{(2)} so u_{i}u_{j} = 0. Hence we have for 1 ≤ i ≤ n − 1

gr(i)_{geo}(𝔾_{k}/P_{k}) ≅ ℤ[t]/(t^{2^{n+1}−2}) ⊕ ℤ/2[t]/(t^{2^{n}−1}){u_{i}}.

Proof. In K(i)^{∗}(X), we see v_{j} = 0 for i ≠ j. Since v_{j}u_{i} = v_{i}u_{j}, we see u_{j} = 0 for i ≠ j. □

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faculty of Education, Ibaraki University, Mito, Ibaraki, Japan

   E-mail address: yagita@mx.ibaraki.ac.jp,