THE IMAGE OF THE MAP FROM GROUP COHOMOLOGY TO GALOIS COHOMOLOGY

M.TEZUKA AND N.YAGITA

Abstract. We study the image of the natural map from group cohomology to Galois cohomology by using motivic cohomology of classifying spaces.

1. Introduction

Let k be a field of ch(k) = 0, which contains a p-th root of unity. Let G be a split affine algebraic group over k and W a faithful representation of G. Then G acts also on the function field k(W). Let k(W)^{G} be the invariant filed. Then we have the natural quotient map of groups q : Gal(k̄(W̄)/k(W)^{G}) → G. This induces the map of cohomologies

q^{∗} : H^{∗}(G; ℤ/p) → H^{∗}(k(W)^{G}; ℤ/p).

The purpose of this paper is to study the image Im(q^{∗}) by using the motivic cohomology defined by Suslin and Voevodsky [Vo1,3]. The image Im(q^{∗}) is called the stable cohomology in [Bo], [Bo-Pe-Ts]. The kernel Ker(q^{∗}) = Ng is called the (geometric) negligible ideal [Pe],[Sa].

Let H^{∗,∗′}(X; ℤ/p) be the mod(p) motivic cohomology. Let 0 ≠ τ ∈ H^{0,1}(Spec(k); ℤ/p) ≅ ℤ/p. Using affirmative solution of the Bloch-Kato conjecture by Voevodsky (and hence Beilinson-Lichtenbaum conjecture), the map q^{∗} is decomposed as

q^{∗} : H^{∗,∗}(BG; ℤ/p) → H^{∗,∗}(BG; ℤ/p)/(τ) → H^{∗,∗}(Spec(k(W)^{G}); ℤ/p).

where H^{∗,∗}(BG; ℤ/p)/(τ) = H^{∗,∗}(BG; ℤ/p)/(τH^{∗,∗−1}(BG; ℤ/p)) and BG is the classifying space of G defined by Totaro and Voevodsky ([To], [Vo1,4]).

By the Belinson-Lichtenbaum conjecture and the work of Bloch-Ogus [Bl-Og], we know

H^{∗,∗}(BG; ℤ/p)/(τ) ⊂ H_{Zar}^{0}(BG; H_{ℤ/p}^{∗}) ⊂ H^{∗}(k(W)^{G}; ℤ/p).

1991 Mathematics Subject Classification. Primary 11E72, 12G05; Secondary 55R35.

Key words and phrases. cohomolgy invariant, classifying spaces, motivic cohomology.

Here H_{ℤ/p}^{∗} is the Zarisky sheaf induced from the presheaf H_{et}^{∗}(V; ℤ/p) for open subset V of BG. Therefore we see ([Or-Vi-Vo])

Im(q^{∗}) = H^{∗}(BG; ℤ/p)/(Ng) ≅ H^{∗,∗}(BG; ℤ/p)/(τ).

Note that the right hand side ring does not depend on the choice of W. We also note that the ideal (Ng) coincides the coniveau filtration N^{1}H^{∗}(BG; ℤ/p) defined by Grothendieck.

In this paper we compute Im(q^{∗}) when k = ℂ for abelian p-groups, symmetric group S_{n}, O_{n}, SO_{n}, Spin_{n}, P GL_{p} and exceptional groups. Extra special p-groups are also studied. For example, we see H^{∗}(BSpin_{n}; ℤ/2)/(Ng) ≅ ℤ/2 for all n ≥ 6.

Recall that the cohomology invariant Inv^{∗}(G; ℤ/p) is a ring of natural maps H^{1}(F; G) → H^{∗}(F; ℤ/p) for finitely generated fields F over k. This ring is very well studied for example see [Ga-Me-Se]. In particular, it is very useful to compute the essential dimension ed(G) of G ([Re], [Br-Re-Vi]). Moreover, Totaro proved that

Inv^{∗}(G; ℤ/p) ≅ H_{Zar}^{0}(BG; H_{ℤ/p}^{∗})

in a letter to Serre [Ga-Me-Se]. Hence Im(q^{∗}) ⊂ Inv^{∗}(G; ℤ/p). We use these results for some parts of this paper, however we also give new explanations of Inv^{∗}(G; ℤ/p) for the case k = ℂ. For example,the image of (topological) Stiefel-Whitney class w_{i} of the map

H^{∗}(BO_{n}; ℤ/2) → H^{∗}(BO_{n}; ℤ/2)/(Ng) ⊂ Inv^{∗}(O_{n}; ℤ/2)

is indeed the Stiefel-Whitney class w_{i} defined by Milnor and Serre as the natural function from quadratic forms to Milnor K-theories.

All examples stated above are detected by abelian p-subgroups A of G, i.e., the restriction map

Res : H^{∗}(BG; ℤ/p)/(Ng) → Π_{A}H^{∗}(BA; ℤ/p)/(Ng)

is injective. (Indeed, most of the above cases are detected by only one elemenary abelian p-subgroup.)

Of course this detected property does not hold for general G. However to give examples is not so easy. Indeed, for a p-group G of exponent p, if H^{2}(BG; ℤ/p)/(Ng) is not detected by any A ≅ ℤ/p ⊕ ℤ/p, then G is a counter example of the Noether’s problem, namely, k(W)^{G} is not purely transcendental over k. The examples of Saltman and Bogomolov are essentially of these types [Sa].

For each n > 1, we give an example G_{n} of a p-group p ≥ 3, such that H^{2n}(G_{n}; ℤ/p) is not detected by abelian p-subgroups, while it does not implies a counter example of Noether’s problem. Here the composition Q_{2n−2}...Q_{0} of Milnor operations is used to see x ∉ Ng given x ∈ H^{2n}(BG; ℤ/p).

The second author learned theories of cohomology invariants from the discussions with Burt Totaro and Patrick Brosnan. Discussions with Alexander Vishik, Zinovy Reichstein, Kirill Zainoulline, Nikita Semenov, Masaharu Kaneda, Charles Vial and Dinesh Deshpande are very helpful to write this paper. The authors thank them very much.

2. motivic cohomology

Let X be a smooth (quasi projective) variety. Let H^{∗,∗′}(X; ℤ/p) be the mod(p) motivic cohomology defined by Voevodsky and Suslin.

Recall that the (mod p) B(n, p) condition holds if

H^{m,n}(X; ℤ/p) ≅ H_{et}^{m}(X; μ_{p}^{⊗n}) for all m ≤ n.

It is known that the B(n, p) condition holds for p = 2 or n = 2 by Voevodsky([Vo1,2]), and Merkurjev-Suslin respectively. Quite recently M.Rost and V.Voevodsky ([Vo5],[Su-Jo],[Ro]) announced that B(n, p) condition holds for each p and n. Hence the Bloch-Kato conjecture also holds. Therefore in this paper, we always assume the B(n, p)- condition and so the Bloch-Kato conjecture for all n, p.

Moreover we always assume that k contains a primitive p-th root of unity. For these cases, we see the isomorphism H_{et}^{m}(X; μ_{p}^{⊗n}) ≅ H_{et}^{m}(X; ℤ/p). Let τ be a generator of H^{0,1}(Spec(k); ℤ/p) ≅ ℤ/p. Hence

colim_{i}τ^{i}H^{∗,∗′}(X; ℤ/p) ≅ H_{et}^{∗}(X; ℤ/p).

Recall that ℤ/p(n) ([Vo1,2,3]) is the complex of sheaves in Zarisky topology such that H^{m,n}(X; ℤ/p) ≅ H_{Zar}^{m}(X; ℤ/p(n)). Let α be the obvious map of sites from etale topology to Zarisky topology so that

H_{et}^{m}(X; ℤ/p) ≅ H_{et}^{m}(X; μ_{p}^{⊗n}) ≅ H_{Zar}^{m}(X, Rα_{∗}α^{∗}ℤ/p(n)).

For k ≤ n, let τ_{≤k}Rα_{∗}α^{∗}ℤ/p(n) be the canonical truncation of Rα_{∗}α^{∗}ℤ/p(n) of level k. Then we have the short exact sequence of sheaves

τ_{≤n−1}Rα_{∗}α^{∗}ℤ/p(n) → τ_{≤n}Rα_{∗}α^{∗}ℤ/p(n) → H_{ℤ/p}^{n}[−n]

where H_{ℤ/p}^{n} is the Zarisky sheaf induced from the presheaf H_{et}^{n}(V; ℤ/p) for open subset V of X. The Beilinson and Lichtenbaum conjecture ( hence B(n, p)-condition ) (see [Vo2,5]) implies

ℤ/p(k) ≅ τ_{≤k}Rα_{∗}α^{∗}ℤ/p(n).

Hence we have ;

Lemma 2.1. ([Or-Vi-Vo], [Vo5]) There is the long exact sequence

→ H^{m,n−1}(X; ℤ/p) →^{×τ} H^{m,n}(X; ℤ/p)

→ H_{Zar}^{m−n}(X; H_{ℤ/p}^{n}) → H^{m+1,n−1}(X; ℤ/p) → .

In particular, we have

Corollary 2.2. The graded ring grH_{Zar}^{m−n}(X; H_{ℤ/p}^{n}) is isomorphic to

H^{m,n}(X; ℤ/p)/(τ) ⊕ Ker(τ)|H^{m+1,n−1}(X; ℤ/p)

where H^{m,n}(X; ℤ/p)/(τ) = H^{m,n}(X; ℤ/p)/(τH^{m,n−1}(X; ℤ/p)).

Note that the above long exact sequence induces the τ-Bockstein spectral sequence

E(τ)_{1} = H_{Zar}^{m−n}(X; H_{ℤ/p}^{n}) ⟹ colim_{i}τ^{i}H^{∗,∗′}(X; ℤ/p) ≅ H_{et}^{∗}(X; ℤ/p).

On the other hand, the filtration coniveau is given by

N^{c}H_{et}^{m}(X; ℤ/p) = ∪_{Z}Ker{H_{et}^{m}(X; ℤ/p) → H_{et}^{m}(X − Z; ℤ/p)}

where Z runs in the set of closed subschemes of X of codim = c. Grothendieck wrote down the E_{1}-term of the spectral sequence induced from the above coniveau filtration.

E(c)_{1}^{c,m−c} ≅ Π_{x∈X^{(c)}}H_{et}^{m−c}(k(x); ℤ/p) ⟹ H_{et}^{m}(X; ℤ/p)

where X^{(c)} is the set of primes of codimension c and k(x) is the residue field of x. We can regard i_{x∗}H_{et}^{m−c}(k(x); ℤ/p) as a constant sheaf H_{et}^{m−c}(k(x); ℤ/p) on ¯ {x} and extend it by zero to X. Then the differentials of the spectral sequence give us a complex on sheaves on X

(2.9) 0 → H_{ℤ/p}^{q} → Π_{x∈X^{(0)}}i_{x∗}H_{et}^{q}(k(x); ℤ/p) → Π_{x∈X^{(1)}}i_{x∗}H_{et}^{q−1}(k(x); ℤ/p)

→ ... → Π_{x∈X^{(q)}}i_{x∗}H_{et}^{0}(k(x); ℤ/p) → 0.

Bloch-Ogus [Bl-Og] proved that the above sequence of sheaves is exact and the E_{2}-term is given by

E(c)_{2}^{c,m−c} ≅ H_{Zar}^{c}(X, H_{ℤ/p}^{m−c}).

In particular, we have ;

Corollary 2.3.

H_{Zar}^{0}(X; H_{ℤ/p}^{m}) ≅ Ker{H_{et}^{m}(k(X); ℤ/p) → Π_{x∈X^{(1)}}H_{et}^{m−1}(k(x); ℤ/p)}.

By Deligne ( foot note (1) in Remark 6.4 in [Bl-Og]) and Paranjape (Corollary 4.4 in [Pj]), it is proven that there is an isomorphism of the coniveau spectral sequence with the Leray spectral sequence for the map α. Hence we have ;

Theorem 2.4. (Deligne, Parajape) There is the isomorphism E(c)_{r}^{c,m−c} ≅ E(τ)_{r−1}^{m,m−c} of spectral sequences. Hence the filtrations are the same N^{c}H_{et}^{m}(X; ℤ/p) = F_{τ}^{m,m−c} where

F_{τ}^{m,m−c} = Im(×τ^{c} : H^{m.m−c}(X; ℤ/p) → H^{m,m}(X; ℤ/p)).

3. cohomology of groups

Let G be a reductive algebraic group over k acting on an affine variety W. A point x ∈ W is called stable if the orbit Gx is closed and the stabilizer group Stab(x) is a finite group. Let us write by X^{s} the set of stable points in X. Then X^{s} is an open subset of X and there is the commutative diagram

W ⟶^{ϕ} W//G
↑^{open}  ↑^{open}
W^{s} ⟶^{ϕ^{s}} W^{s}/G,

where W//G = Spec(k[W]^{G}), the geometric quotient of X. We also note that the invariant field k(W)^{G} is a quotient field of k[W]^{G} when X^{s} ≠ ∅.

Suppose X^{s} ≠ ∅. Then H^{∗,∗′}(W//G; ℤ/p) = H^{∗,∗′}(Spec(k[W]^{G}); ℤ/p) and k(W^{s}/G) = k(W//G) = k(W)^{G}. Hence we have the diagram

H^{∗,∗′}(W//G; ℤ/p) ⟶ H^{∗,∗′}(Spec(k(W)^{G}); ℤ/p)
↓  ^{=}↓
H^{∗,∗′}(W^{s}/G; ℤ/p) ⟶^{ψ} H^{∗,∗′}(Spec(k(W^{s}/G); ℤ/p)

Restrict W as an affine space W = ⊕k and let ρ : G → W = ⊕k a faithful representation. Let U_{n} = W −S be an open set of W such that G act freely U where codim_{W} S = n. (Of course U is an open subset of W^{s}.) Then the classifying space of G is defined as colim_{n→∞}(U_{n}/G). Then the mod(p) motivic cohomology (for degree ∗ < 2n) of BG is given by ([Vo4],[To])

H^{∗,∗′}(BG; ℤ/p) ≅ lim_{n→∞}H^{∗,∗′}(U_{n}/G; ℤ/p).

In particular, by BL(p, ∗) condition, we have

H^{∗,∗}(BG; ℤ/p) ≅ H_{et}^{∗}(BG; ℤ/p) = H^{∗}(G; ℤ/p) ⊗ H^{∗}(k; ℤ/p)

where the last group is the cohomology group of the Galois group G (when G is finite). Thus from the above diagram, we have the map

ψ : H^{∗,∗′}(BG; ℤ/p) → H^{∗,∗′}(Spec(k(W)^{G}); ℤ/p).

This map ψ^{∗,∗} = ψ_{et}^{∗} is explains also as follows. Let Γ is the absolute Galois group Γ = Gal(k̄(W)/k(W)^{G}). Then the group G = Gal(k(W)/k(W)^{G}) is a quotient group of the absolute Galois group Γ.

Then the map ψ_{et} is the induced map from the quotient q : Γ → G, i.e.,

ψ_{et}^{∗} = q^{∗} : H^{∗}(G; ℤ/p) → H^{∗}(Γ; ℤ/p) = H^{∗}(k(W)^{G}; ℤ/p).

Lemma 3.1.

Im(ψ^{∗}) ≅ H^{∗,∗}(BG; ℤ/p)/(τ).

Proof. For each field F, by the Bloch-Kato conjecture, H^{∗}(F; ℤ/p) is generated by elements in H_{et}^{1}(F; ℤ/p) ≅ H^{1,1}(Spec(F); ℤ/p). So

ψ^{∗,∗−1} : H^{∗,∗−1}(BG; ℤ/p) → H^{∗,∗−1}(Spec(k(W)^{G}); ℤ/p) = 0.

Hence the map ψ^{∗} is expressed as a composition

H^{∗,∗}(BG; ℤ/p) → H^{∗,∗}(BG; ℤ/p)/(τ) → H_{et}^{∗}(k(W)^{G}; ℤ/p).

The first map is of course surjective and we only need the injectivity of the second map. Indeed, from Corollary 2.2 and 2.3, we see

H^{∗,∗}(BG; ℤ/p)/(τ) ⊂ H_{Zar}^{0}(BG; H_{ℤ/p}^{∗}) ⊂ H^{∗}(k(W)^{G}; ℤ/p).

□

Recall the coniveau filtration given in §2

N^{c}H_{et}^{m}(X; ℤ/p) = ∪_{Z}Ker{H_{et}^{m}(X; ℤ/p) → H_{et}^{m}(X − Z; ℤ/p)}

where Z runs in the set of closed subschemes of X of codim = c. From Theorem 2.4, we see

Corollary 3.2. Im(q^{∗}) ≅ H_{et}^{∗}(BG; ℤ/p)/(N^{1}H_{et}^{∗}(BG; ℤ/p)).

According to Saltman (and Serre), we say an element x ∈ H^{∗}(G; ℤ/p) is geometrically negligible if ψ^{∗}(x) = 0. Let us write Ng = Ng(G) = Ker(ψ^{∗}). From the above lemma, it is immediate

Ng(G) = N^{1}H_{et}^{∗}(BG; ℤ/p)
= Im(×τ|H^{∗,∗−1}(BG; ℤ/p) → H^{∗,∗}(BG; ℤ/p))

and we have

Im(ψ^{∗}) = H^{∗}(BG; ℤ/p)/(Ng) ≅ H^{∗}(BG; ℤ/p)/(N^{1}) ≅ H^{∗,∗}(BG; ℤ/p)/(τ).

Lemma 3.3. For each element x in H_{et}^{∗}(BG; ℤ/p), the images of cohomology (Bockstein, reduced) operations β(x), P^{i}(x) for i > 0 are in Ng(G) (hence x^{p} ∈ Ng(G)). The image of Gysin map g_{∗}(x) are also in Ng(G).

Proof. The cohomology operations act as ([Vo2,4])

β : H^{∗,∗}(X; ℤ/p) → H^{∗+1,∗}(X; ℤ/p)
P^{i} : H^{∗,∗}(X; ℤ/p) → H^{∗+2i(p−1),∗+i(p−1)}(X; ℤ/p).

For an element x ∈ H^{m,n}(X; ℤ/p), define the difference degree d(x) = m − n. Then if d(x) = 0, then d(β(x)) > 0 and d(P^{i}(x)) > 0. Hence these elements are in Im(τ) as a subset of H^{∗,∗}(X; ℤ/p).

For the embedding X ⊂ Y of codimension c, the Gysin map is defined on

g_{∗} : H^{∗,∗}(X; ℤ/p) → H^{∗+2c,∗+c}(Y; ℤ/p).

By the same reason as the cases of cohomology operations, we get lemma. □

Here we give a sufficient condition for x ∉ Ng(G). Voevodsky define the Milnor operation Q_{i} also in the mod p motivic cohomology

Q_{i} : H^{∗,∗′}(−; ℤ/p) → H^{∗+2p^{n}−1,∗′+p^{n}−1}(−; ℤ/p).

Define the weight w(x) = 2 ∗′ −∗ for element (or operation) x ∈ H^{∗,∗′}(X; ℤ/p), e.g., w(τ) = 2, w(Q_{i}) = −1 and w(P^{i}) = 0.

Lemma 3.4. Let x ∈ H_{et}^{n}(BG; ℤ/p) and Q_{n−2}...Q_{0}(x) ≠ 0. Then 0 ≠ x ∈ H_{et}^{n}(BG; ℤ/p)/(Ng).

Proof. Identify x as an element in H^{n,n}(BG; ℤ/2). Suppose that x = τx̄. So w(x̄) = n − 2 since w(τ) = 2. Then t_{ℂ}(x̄) = t_{ℂ}(x) = x where t_{ℂ} : H^{∗,∗′}(X; ℤ/p) → H^{∗,∗′}(X(ℂ); ℤ/p) is the realization map.

The operation Q_{i} descends the weight one. Let ψ̄ = Q_{n−2}...Q_{0}(x̄). Then w(ψ̄) = −1 but t_{ℂ}(ψ̄) = Q_{n−2}...Q_{0}(x) ≠ 0. This is a contradiction since w(y) ≥ 0 for each nonzero element y ∈ H^{∗,∗′}(Y; ℤ/p) and for smooth Y. □

We have the Kunneth formula for the etale cohomology of coefficient ℤ/p. Since Ng is an ideal, we have the surjection

H_{et}^{∗}(BG_{1}; ℤ/p)/(Ng) ⊗_{H^{∗}(k;ℤ/p)} H_{et}^{∗}(BG_{2}; ℤ/p)/(Ng)
→ H_{et}^{∗}(B(G_{1} × G_{2}); ℤ/p)/(Ng).

However it does not need isomorphic, because there is the possibility that x_{1} ⊗ x_{2} ∈ Ng(G_{1} × G_{2}) but x_{1} ∉ Ng(G_{1}), x_{2} ∉ Ng(G_{2}).

4. cohomology invariant

Recall that H^{1}(k; G) is the first non abelian Galois cohomology set of G, which represents the set of G-torsors over k. It is very important to study H^{1}(k; G), for example H^{1}(k; O_{n}) is isomorphic to the set of isomorphism classes of non generate quadratic forms over k of rank n. (For details, see the excellent book [Ga-Me-Se].)

The cohomology invariant is defined by

Inv^{i}(G, ℤ/p) = F unc(H^{1}(F; G) → H^{i}(F; ℤ/p))

where F unc means natural functions for each fields F over k. The cohomology invariant is studied by many authors. The cohomology invariants Inv^{∗}(G; ℤ/p) are computed (for example in [Ga-Me-Se]) for groups elementary abelian 2-groups, O_{n},SO_{n}, G_{2},... It is also stated in [Ga-Me-Se] that for many G (but not all) Inv^{∗}(G; ℤ/p) are detected by Inv^{∗}(H; ℤ/p) for elementary abelian p-subgroups H.

Let x ∈ H^{0}(BG; H_{ℤ/p}^{i}). Given a G-torsor E over F, we can construct x(E) ∈ H_{et}^{i}(F; ℤ/p). Roughly speaking, we can identify E as the pullback of some map f : Spec(F) → BG. So we can take x(E) = f^{∗}(x) ∈ H^{0}(Spec(F); H_{ℤ/p}^{i}) = H_{et}^{i}(F; ℤ/p). Indeed, Totaro proved [Ga-Me-Se] the following theorem in a letter to Serre.

Theorem 4.1. Inv^{∗}(G; ℤ/p) ≅ H^{0}(BG; H_{ℤ/p}^{∗}).

Therefore we see

Corollary 4.2. Im(ψ^{∗}) ≅ H^{∗}(BG; ℤ/p)/(Ng) ⊂ Inv^{∗}(G; ℤ/p).

5. abelian p-groups

Let us write H^{∗,∗′} = H^{∗,∗′}(Spec(k); ℤ/p) and H^{∗} = H^{∗,∗} = K_{M}^{∗}(k)/p so that H^{∗,∗′} ≅ H^{∗}[τ]. First consider the case G = ℤ/p^{r}. The mod(p) motivic cohomology is computed (as the case ℤ/p in [Vo])

H^{∗,∗′}(Bℤ/p^{r}; ℤ/p) ≅ H^{∗,∗′}[y(r)] ⊗ Λ(x(r)) |y(r)| = 2, |x(r)| = 1.

(When p = 2 and r = 1, we see by Voevodsky ([Vo2,4])

x(1)^{2} = τy(1) + ρx(1) with ρ = (−1) ∈ H^{1} = k^{∗}/(k^{∗})^{2}.)

For the inclusion i : ℤ/p^{r} ⊂ ℤ/p^{s} and quotient map q : ℤ/p^{s} → ℤ/p^{r}, for s ≥ r, we have

i^{∗}(y(s)) = y(r), i^{∗}(x(s)) = 0, q^{∗}(y(r)) = 0, q^{∗}(x(r)) = x(s).

Moreover we still know x(r) ∈ H^{1,1}(BG; ℤ/p) and y(r) ∈ H^{2,1}(BG; ℤ/p). Thus we see ((Ker(τ)|H^{∗,∗′}(Bℤ/p^{r}; ℤ/p) = 0).

Inv^{∗}(ℤ/p^{r}; ℤ/p) ≅ H^{∗,∗}(ℤ/p^{r}; ℤ/p)/(τ) = H^{∗}{1, x(r)}.

Next consider their product G = ℤ/p^{r_{1}} × ... × ℤ/p^{r_{s}}. The cohomology H^{∗,∗′}(Bℤ/p^{r}; ℤ/p) has the Kunneth formula. Hence the motivic cohomology is given

H^{∗,∗′}(BG; ℤ/p) ≅ H^{∗,∗′}[y(r_{1}), ..., y(r_{s})] ⊗ Λ(x(r_{1}), .., x(r_{s}))

where x(r_{i}) ∈ H^{1,1}(BG; ℤ/p) and y(r_{i}) ∈ H^{2,1}(BG; ℤ/p).

Recall that H^{∗}(BG; ℤ/p)/(Ng) ≅ H^{∗,∗}(BG; ℤ/p)/(τ) (Lemma 3.1). Then we get

Lemma 5.1. Let G be an abelian p-group, i.e., G = ⊕_{i}ℤ/(p^{r_{i}}). Then

Inv^{∗}(G; ℤ/p) ≅ H^{∗}(G; ℤ/p)/(Ng) ≅ H^{∗} ⊗ Λ(x(r_{1}), ..., x(r_{s}))

when p = 2 r_{i} = 1, x(r_{i})^{2} = ρx(r_{i}).

The elementary 2-groups cases are stated in Theorem 16.4 in [Ga-Me-Se].

The Q_{i}-operation acts on H^{∗,∗′}(Bℤ/p; ℤ/p) by Q_{i}(x) = y^{p^{i}} (while Q_{i}(x(j)) = 0 for all j > 1). We consider Q_{i} action on

H^{∗,∗′}(B(ℤ/p)^{s}; ℤ/p) ≅ H^{∗,∗′}[y_{1}, ..., y_{s}] ⊗ Λ(x_{1}, ..., x_{s}).

Each Q_{i} is a derivation, and hence

Q_{0}...Q_{s−1}(x_{1}...x_{s}) = ∑ sgn(j_{1}, ..., j_{s})y_{1}^{p^{j_{1}}}y_{2}^{p^{j_{2}}}...y_{s}^{p^{js}} ≠ 0

where (j_{1}, ..., j_{s}) are permutations of (0, ..., s − 1). Thus we see that this case satisfies the sufficient condition of Lemma 3.3 while the other cases does not), in fact x_{1}...x_{s} ∉ Ng(G).

Let us say that an element x ∈ H^{∗}(BG; ℤ/p)/(Ng) is detected by an elemntary abelian p-subgroup A if Res(x) ≠ 0 for

Res : H^{∗}(BG; ℤ/p)/(Ng) → H^{∗}(BA; ℤ/p)/(Ng).

The following lemma is immediate from the above arguments.

Lemma 5.2. If x ∈ H^{n}(BG; ℤ/p)/(Ng) is detected by elementary abelian p-subgroups, then Q_{n−1}...Q_{0}(x) ≠ 0 in H^{∗}(BG; ℤ/p).

6. Cases G = O_{n} and SO_{n}

Hereafter, in this paper (except for §11), we assume that k = ℂ otherwise stated.

lt is well known that

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

where S_{n} is the n-th symmetric group, and w_{i} is the Stiefel-Whitney class representing the i-the elementary symmetric function. we easily see

Q_{i−1}...Q_{0}(w_{i}) = y_{1}^{p^{i−1}}y_{2}^{p^{p−2}}...y_{i}+... ≠ 0 ∈ ℤ/2[y_{1}, ..., y_{n}] ⊂ H^{∗}(B(ℤ/2)^{n}; ℤ/2)

and hence w_{i} ∉ Ng(G). Recall the Wu formula

Sq^{i}w_{k} = ∑_{j}^{i} ( \begin{matrix}k − j − 1 \\ i − j \end{matrix} ) w_{k+i−j}w_{j} (0 ≤ i ≤ k).

Many cases of product of w_{i}w_{j} are in Ng(G), e.g.,w_{i}^{2} ∈ Ng(G). More precisely, the motivic cohomology of BO_{n} is computed for k = ℂ (Theorem 8.1 in [Ya3])

H^{∗,∗′}(BO_{n}; ℤ/2) ≅ H^{∗,∗′}((Bℤ/2)^{×n}; ℤ/p)^{S_{n}}.

Given x ∈ H^{∗}(BG; ℤ/p), let us define the weight w(x) as the smallest weight w(x′) such that t_{ℂ}(x′) = x with x′ ∈ H^{∗,∗′}(BG; ℤ/p). Indeed, the weight of the symmetric polynomial

t = ∑ x_{1}^{2i_{1}+1}...x_{k}^{2k+1}x_{k+1}^{2j_{1}}...x_{k+q}^{2j_{q}} in H^{∗}((Bℤ/2)^{×n}; ℤ/2)

(with 0 ≤ i_{1} ≤ ... ≤ i_{k}, 0 ≤ j_{1} ≤ ... ≤ j_{q}) is given by w(t) = k. Hence if t ∉ Ng, then w(t) = deg(t) and this implies t = x_{1}...x_{i} = w_{i}.

Theorem 6.1.

Inv^{∗}(O_{n}; ℤ/2) ≅ H^{∗}(BO_{n}; ℤ/2)/(Ng) ≅ ℤ/2{1, w_{1}, ..., w_{n}}.

In fact, Inv(O_{n}; ℤ/2) is well known (Theorem 17.3 in [Ga-Me-Se]) for general k ;

Inv^{∗}(O_{n}; ℤ/2) ≅ H^{∗}{1, w_{1}, ..., w_{n}}.

We consider the multiplicative structure of Inv^{∗}(O_{n}; ℤ/2). From the Wu formula, we see

Sq^{1}(w_{2i}) = w_{2i+1} + w_{2i}w_{1} ∈ Ng(O_{n}).

Hence w_{2i+1} = w_{2i}w_{1} in Inv^{∗}(O_{n}; ℤ/2). By Rost and Kahn [Ka], the divided power operation can be defined in K_{∗}^{M}(F)/p compatible with fields F over k (and hence Inv^{∗}(G; ℤ/p)) if √−1 ∈ k. Vial showed [Via] that the divided power operations are only compatible maps (natural maps) with field extensions over k. Moreover Becher [Be] showed that γ_{n}(w_{2}) = w_{2n}. (See also Milnor p133 in [Mi].)

Theorem 6.2. (Becher [Be], [Via]) Let √−1 ∈ k. Then Inv^{∗}(O_{n}; ℤ/2) is generated by w_{1} and w_{2} as an H^{∗}-ring with divided powers by

γ_{i}(w_{2}) = w_{2i}, w_{2i+1} = w_{2i}w_{1}.

Next consider the case G = SO_{n}. It is well known that

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

Let n = 2m + 1 = odd. For this case, there is the isomorphism

O_{2m+1} ≅ SO_{2m+1} × ℤ/2.

Let p : O_{n} → SO_{n} is the projection and i : SO_{n} → O_{n} the inclusion. We consider the induced map p^{∗} and i^{∗} on the mod 2 motivic cohomology of their classifying spaces. Since p^{∗}(w_{1}) = 0, we see w_{2i+1} ∈ Ng(G) from the above theorem (in fact, Sq^{1}(w_{2i}) = w_{2i+1} in H^{∗}(BSO_{n}; ℤ/2)). Moreover i^{∗}w_{2i} = w_{2i} mod(Ng(G)). Thus we have

Theorem 6.3. For G = SO_{2m+1}, we have

Inv^{∗}(G; ℤ/2) ≅ H^{∗}(BG; ℤ/2)/(Ng) ≅ ℤ/2{1, w_{2}, ..., w_{2m}}.

Moreover Inv^{∗}(SO_{2m+1}; ℤ/2) is still computed (Theorem 19.1 in [Ga-Me-Se]) for general k ;

Inv^{∗}(SO_{2m+1}; ℤ/2) ≅ H^{∗}{1, w_{2}, ..., w_{2m}}.

Now consider the case n = 2m = even. This case the mod 2 motivic cohomology is not computed even k = ℂ for n > 4. However we compute H^{∗,∗}(BSO_{n}; ℤ/2)/(τ) easily. Consider the inclusion

SO_{2m−1} →^{i_{1}} SO_{2m} →^{i_{2}} SO_{2m+1}.

Since the restriction map for i < m

i_{1}^{∗}(w_{2i}) ≠ 0 ∈ H^{∗,∗}(BSO_{2m−1}; ℤ/2),

we see w_{2i} ∉ Ng(SO_{2m}). Moreover we know that w(w_{n}) = n − 2 in Lemma 9.2 in [Ya3]. On the other hand, the monomial w_{I} of not w_{2i} are all in Ng(SO_{2m+1}) and so i_{2}^{∗}(w_{I}) ∈ Ng(SO_{2m}).

Thus we have

Theorem 6.4. H^{∗}(BSO_{2m}; ℤ/2)/(Ng) ≅ ℤ/2{1, w_{2}, ..., w_{2m−2}}.

Next we study Inv^{∗}(SO_{2m}; ℤ/2). There is an element (Lemma 9.3 in [Ya3]) in the motivic cohomology

u_{m−1} ∈ H^{n,n−2}(BSO_{n}; ℤ/2) with τu_{m−1} = 0.

So there is the nonzero element

u ∈ Ker(τ|H^{n,n−2}(BG; ℤ/2)) ⊂ H^{0}(BG; H_{ℤ/2}^{n−1}).

On the other hand in [Ga-Me-Su], it is proved (Theorem 20.6) for general k

Inv^{∗}(SO_{2m}; ℤ/2) ≅ H^{∗}{1, w_{2}, ..., w_{n−2}} ⊕ (Im(I_{δ})).

Here when k = ℂ, Im(I_{δ}) ≅ ℤ/2{u} with deg(u) = n − 1 from Proposition 20.1 in [Ga-Me-Se]. Thus we see

Theorem 6.5. Let G = SO_{2m}and m ≥ 2. Then for deg(u) = 2m − 1,

Inv^{∗}(G; ℤ/2) ≅ H^{∗}(BG; ℤ/2)/(Ng) ⊕ ℤ/2{u}.

From 22.10 in [Ga-Me-Se], it is known that

Res : Inv^{∗}(G; ℤ/p) → Inv^{∗}(H; ℤ/p)

is injective for p = 2, G = SO_{n} and H = (ℤ/2)^{n−1}. Hence for

Res : H^{0}(BSO_{n}; H_{ℤ/2}^{∗}) → H^{0}(B(ℤ/2)^{n−1}; H_{ℤ/2}^{∗})

we have Res(u) = x_{1}...x_{n−1}. Of course u ∉ H^{∗,∗}(BSO_{n}; ℤ/2)/(τ) but x_{1}...x_{n−1} ∈ H^{∗,∗}(B(ℤ/2)^{n−1}; ℤ/2)/(τ).

Recall that in Corollary 2.2 grH^{0}(BG; H_{ℤ/p}^{∗}) is defined by the filtration H^{∗,∗}(BG; ℤ/p)/(τ) ⊂ H^{0}(BG; H_{ℤ/p}^{∗}). So note that

Res : grH^{0}(BSO_{n}; H_{ℤ/2}^{∗}) → grH^{0}(B(ℤ/2)^{n−1}; H_{ℤ/2}^{∗})

is not injective (in Corollary 2.2).

7. Spin_{n} and exceptional groups

The mod(2) cohomology of BSpin_{n} is computed by Quillen

H^{∗}(BSpin_{n}; ℤ/2) ≅ ℤ/2[w_{2^{h}}(Δ)] ⊗ ℤ/2[w_{2}, ..., w_{n}]/(Q_{i}w_{2}|0 ≤ i ≤ h)

where w_{i}(Δ) (resp. w_{i}) is the Stiefel-Whitney class of a spin representation Δ (resp. usual representation Spin_{n} → SO_{n}), and 2^{h} is the the Radon-Hurwitz number (See [Qu] p.210). By the result of Becher (Theorem 6.2), we have

Theorem 7.1. H^{∗}(BSpin_{n}; ℤ/2)/(Ng) = ℤ/2 for n > 4.

Proof. Let us write representations j : Spin_{n} → SO_{n} and Δ : Spin_{n} → SO_{N}. We consider the induced map in Galois cohomolgy

j^{∗} : H_{et}^{∗}(k(W)^{SO_{n}}; ℤ/2) → H_{et}^{∗}(k(W)^{Spin_{n}}; ℤ/2).

By the Quillen’s result, we see j^{∗}(w_{2}) = 0. By Rost and Kahn [Ka], the divided powers naturally act on K_{∗}^{M}(F)/p for field F over k. Hence from Becher theorem (Theorem 6.2),we get

j^{∗}(w_{2i}) = j^{∗}(γ_{i}(w_{2})) = γ_{i}(j^{∗}(w_{2})) = 0.

Similarly w_{2}(Δ) = 0 implies w_{2^{h}}(Δ) = 0 if n > 4. □

Corollary 7.2. For ∗ > 0, there is the isomorphism

Inv^{∗}(Spin_{n}; ℤ/2) ≅ Ker(τ)|H^{∗+1,∗−1}(BSpin_{n}; ℤ/2).

The mod(2) motivic cohomology of BSpin_{7} is computed in (Theorem 9.6 in [Ya3]). We can easily see the above theorem also from the concrete computation. Moreover there are τ-torsion elements

y_{2} ∈ H^{4,2}(Spin_{7}; ℤ/2), y′_{2} ∈ H^{5,3}(Bspin_{7}; ℤ/2).

Therefore we can take u ∈ H^{0}(BSpin_{7}, H_{ℤ/2}^{3}), v ∈ H^{0}(BSpin_{7}, H_{ℤ/2}^{4}).

Theorem 7.3.

Inv^{∗}(BSpin_{7}; ℤ/2) ≅ ℤ/2{1, u, v} |u| = 3, |v| = 4.

We consider exceptional Lie group types G_{2}, F_{4} and split E_{6}. In [Ga-Me-Su], it is proved (Teorem 18.1, Theorem 22.5)

Inv^{∗}(G_{2}; ℤ/2) ≅ Inv^{∗}(E_{6}; ℤ/2) ≅ H^{∗}{1, u},
Inv^{∗}(F_{4}; ℤ/2) ≅ H^{∗}{1, u, f_{5}}.

(Unfortunately, we can not reexplain f_{5} by using H^{∗,∗′}(BF_{4}; ℤ/2), which is not computed yet.)

Moreover restriction image for elementary abelin 2-subgroup of rank 3 (resp. rank 5) is injective for G_{2}, E_{6} (resp. for F_{4}), see 22.10 in [Ga-Me-Se].

Theorem 7.4. Let G = G_{2}, F_{4}. Then H^{∗}(BG; ℤ/2)/(Ng) = ℤ/2.

Proof. It is known that the inclusion i : G_{2} → Spin_{7} induces the epimorphism i^{∗} : H^{∗}(BSpin_{7}; ℤ/2) → H^{∗}(BG_{2}; ℤ/2). Hence the result for G = G_{2} follows from H^{∗}(BSpin_{7}; ℤ/2)/(Ng) = ℤ/2.

It is also known that the inclusion i′ : Spin_{9} → F_{4} induces the injection i′^{∗} : H^{∗}(BF_{4}; ℤ/2) → H^{∗}(BSpin_{9}; ℤ/2). The groups Spin_{9} and F_{4} has the same maximal abelin 2-group H (of rank 5). We consider the restriction to this H. The fact Res(H^{∗}(BSpin_{9}; ℤ/2)/(Ng)) = ℤ/2 implies the results for F_{4}, because the restriction is injective from [Ga-Me-Se]. □

Here we give an example. The mod(2) cohomology is well known

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

The motivic cohomology is given in Theorem 7.5 in [Ya3]. In particular, c_{4}, c_{6}, c_{7} are the Chern classes (in H^{2∗,∗}(BG_{2}; ℤ/2)) so that τ^{i}c_{i} = w_{i}^{2}. Let us write simply by Q(n) the exterior algebra Λ(Q_{0}, ..., Q_{n}).

Theorem 7.5. The motivic cohomology H^{∗,∗′}(BG_{2}; ℤ/2) is isomorphic to

ℤ/2[c_{6}, c_{4}]⊗(ℤ/2{y_{2}}⊕ℤ/2[τ]⊗(ℤ/2{1}⊕(ℤ/2[c_{7}]Q(2)−ℤ/2{1}){a})

where deg(y) = (4, 2), and a is a virtual element with deg(a) = (3, 3) so that c_{7}a = w_{6}w_{7}w_{4}, Q_{0}a = w_{4}, Q_{1}a = w_{6} and Q_{2}a = w_{6}w_{4}.

Let us write Q_{i_{1}}...Q_{i_{s}}(a) by Q_{i_{1}...i_{s}} and w_{i_{1}}...w_{i_{s}} by w_{i_{1}...i_{s}}. Then (ℤ/2[c_{7}]Q(2) − ℤ/2{1}){a} is written as

ℤ/2[c_{7}]{c_{7}a = w_{467}, Q_{0} = w_{4}, Q_{1} = w_{6}, Q_{01} = w_{7},
Q_{2} = w_{46}, Q_{02} = w_{47}, Q_{12} = w_{67}, Q_{012} = c_{7}}.

Note that all w_{i_{1}...i_{s}} (which are generators of the ℤ/2[c_{4}, c_{6}, c_{7}]-module H^{∗}(G_{2}; ℤ/2)) appeared indeed in the above. Moreover it is immediate that all elements except for 1 and a are in Ng(G_{2}), and y ∈ Ker(τ)|H^{4,2}(BG_{2}; ℤ/2).

Next we consider odd prime case. In 22.10 in [Ga-Me-Se] it is stated that the restriction image of Inv^{∗}(G; ℤ/p) to some (maximal) elementary abelian p-group H is injective for (G, p) = (F_{4}, 3), (E_{6}, 3), (E_{7}, 3) or (E_{8}, 5).

In these cases, each exceptional Lie group has two conjugacy classes of maximal elementary abelian p-groups. One is the subgroup of a maximal torus and the other is a nontoral A. Let us write i_{A} : A → G and i_{T} : T → B be the inclusions. Tezuka and Kameko proves ([Ka], [Ka-Ya]) that the following map is injective

i_{A}^{∗} × i_{T}^{∗} : H^{∗}(BG; ℤ/p) → H^{∗}(BA; ℤ/p) × H^{∗}(BT; ℤ/p),

namely, H^{∗}(BG; ℤ/p) is detected by A and T. Since H^{∗}(BT; ℤ/p) is non-nilpotent, the above group H must be A.

Theorem 7.6. Let (G, p) = (F_{4}, 3), (E_{6}, 3), (E_{7}, 3) or (E_{8}, 5). Then H^{∗}(BG; ℤ/p)/(Ng) = ℤ/p.

Proof. The restriciton image to A is studied in [Ka-Ya]. Images are generated as a ring by Chern classes and

Q_{I}(x_{1}..x_{s}) ∈ H^{∗}(BA; ℤ/p) ≅ ℤ/p[y_{1}, ..., y_{s}] ⊗ Λ(x_{1}, ..., x_{s})

where Q_{I} = Q_{i_{1}}...Q_{is} for some I ≠ ∅. (Note x_{1}...x_{s} ∉ Im(i_{A}^{∗}).) Hence i_{A}^{∗}(H^{+}(BG; ℤ/p)) ∈ Ng(A). □

Here we give an example. The mod 3 cohomology of BF_{4} is completely determined by Toda.

Theorem 7.7. ([Toda]) The cohomology H^{∗}(BF_{4}; ℤ/3) is isomorphic to

ℤ/3[x_{36}, x_{48}]⊗(ℤ/3[x_{4}, x_{8}]⊗{1, x_{20}, x_{20}^{2}}+ℤ/3[x_{26}]⊗Λ(x_{9})⊗{1, x_{20}, x_{21}, x_{25}})

where the above two terms have the intersection {1, x_{20}}.

Indeed, we see that x_{26}|A = Q_{0}Q_{1}Q_{2}(u_{3}), x_{36}|A = c_{3,1}, x_{48}|A = c_{3,2}, x_{4}|A = Q_{0}(u_{3}), x_{8}|A = Q_{1}(u_{3}), x_{20}|A = Q_{2}(u_{3}), x_{9}|A = Q_{0}Q_{1}(u_{3}), x_{21}|A = Q_{0}Q_{2}(u_{3}), x_{25}|A = Q_{1}Q_{2}(u_{3}). Here u_{3} = x_{1}x_{2}x_{3} and x_{36} and x_{48} are represented by Chern classes.

8. GL_{n}(𝔽_{ℓ}) and P GL_{p}

Of course, there is the isomorphism

H^{∗,∗′}(BGL_{n}; ℤ/p) ≅ H^{∗,∗′}[c_{1}, ..., c_{n}]

where c_{i} is the Chern class. Hence Inv^{∗}(GL_{n}; ℤ/p) ≅ ℤ/p (for k = ℂ). Let G be a finite group such that

(8.1) H^{∗}(BG; ℤ/p) ≅ ℤ/p[c_{i_{1}}, ..., c_{i_{n}}] ⊗ Λ(e_{i_{1}}, ..., e_{i_{n}})

where β_{j_{s}}(e_{i_{s}}) = c_{i_{s}} where β_{j_{s}} is the higher Bockstein operation, and c_{i_{s}} is a Chern class of some representation of G, e.g., G = GL_{n}(𝔽_{ℓ}) where ℓ is prime to p.

Theorem 8.1. Let G be a finite group given as (8.1) so that i_{1} ≥ 2. Then H^{∗}(BG; ℤ/p)/(Ng) ≅ ℤ/p.

Proof. First note that H^{∗}(BG; ℤ/p)/(Ng) is a quotient Λ(e_{i_{1}}, ..., e_{i_{n}}). Since the motivic cohomology has the transfer map, we know that each element x in H^{∗,∗′}(BG; ℤ) has the exponent dividing |G|. Hence there is e′_{s} ∈ H^{∗,∗−1}(BG; ℤ/p) such that

β_{s′}(e′_{s}) = τ^{i_{s}−2}c_{i_{s}}.

From Lemma 2.1, we know that τ : H^{∗,∗−1}(X; ℤ/p) → H^{∗,∗}(X; ℤ/p) is injective. Hence τe′_{s} ≠ 0. This means β_{s′′}(τe′_{s}) = τ^{i_{s}−1}c_{i_{s}} for s′′ ≤ s′. So s′′ = i_{s} and we can take τe′_{s} = e_{i_{s}} mod(Ideal(e_{i_{1}}, ..., e_{i_{s−1}})). By induction on s, we can prove all e_{i_{s}} ∈ Ng. □

Let p be an odd prime and denote by P GL_{p} the projective group which is the quotient of the general linear group GL_{p} by the center 𝔾_{m}. Its ordinary mod(p) cohomology and the Chow ring are known by Vistoli [Vi] and Kameko-Yagita [Ka-Ya].

To state the cohomology H^{∗}(BP GL_{p}; ℤ/p), we recall the Dickson algebra. Let A ≅ (ℤ/p)^{n} be an elementary abelian p-group of rank n, and H^{∗}(BA) ≅ ℤ/p[y_{1}, ..., y_{n}] ⊗ Λ(x_{1}, ..., x_{n}). The Dickson algebra is

D_{n} = ℤ/p[y_{1}, ..., y_{n}]^{GL_{n}(𝔽_{p})} ≅ ℤ/p[c_{n,0}, ..., c_{n,n−1}]

with |c_{n,i}| = 2(p^{n} − p^{i}). The invariant ring under SL_{n}(𝔽_{p}) is also given

SD_{n} = ℤ/p[y_{1}, ..., y_{n}]^{SL_{n}(𝔽_{p})} ≅ D_{n}{1, e_{n}, ..., e_{n}^{p−2}} with e_{n}^{p−1} = c_{n,0}.

We also recall the Mui’s result by using Q_{i} according to Kameko and Mimura [Ka-Mi]

grH^{∗}(BA)^{SL_{n}(𝔽_{p})} ≅ SD_{n}/(e_{n}) ⊕ SD_{n} ⊗ Q(n − 1){u_{n}}

where u_{n} = x_{1}...x_{n} and e_{n} = Q_{0}...Q_{n−1}u_{n}. (Here note

SD_{n}/(e_{n}) ≅ D_{n}/(c_{n,0}) ≅ ℤ/p[c_{n,1}, ..., c_{n,n−1}].)

Theorem 8.2. ([Vi],[Ka-Ya]) There is the isomorphism

H^{∗}(BP GL_{p}; ℤ/p) ≅ M ⊕ N

where M ≅ ℤ/p[x_{4}, x_{6}, ⋯, x_{2p}] as modules (but not rings) and

N ≅ SD_{2} ⊗ Q(1){u_{2}} ≅ ℤ/p[e_{2}, c_{2,1}] ⊗ Q(1){u_{2}}.

Theorem 8.3. ([Vi],[Ka-Ya]) There is the additive isomorphism

CH^{∗}(BP GL_{p})/p ≅ M ⊕ SD_{2}{Q_{0}Q_{1}u_{2}}.

It is also proved that Ker(τ)|H^{∗,∗′}(BP GL_{p}; ℤ/p) = 0 in Theorem 10.4 in [Ya3].

Theorem 8.4.

Inv^{∗}(P GL_{p}; ℤ/p) ≅ H^{∗}(G; ℤ/p)/(Ng) ≅ ℤ/2{1, u} |u| = 2.

This fact is also shown in [Ga-Me-Se].

9. Symmetric group S_{n}

Let S_{n} be the Symmetric group generated by permutations of nletters. The permutations induce the natural representation S_{n} → O_{n}. Let us write by w_{i} its Stiefel-Whitney class. Then it is proved for example in [Ga-Me-Se] that for general k

Theorem 9.1. Inv^{∗}(S_{n}; ℤ/2) ≅ H^{∗}{1, w_{1}, ..., w_{[n/2]}}.

Let A be a subgroup of S_{n} generated by the transpositions (2i−1, 2j) for 1 ≤ j ≤ [n/2] so that A ≅ ⊕^{[n/2]}ℤ/2. Then Inv^{∗}(S_{n}; ℤ/2) is detected by the group A.

In this section, by using the cohomology H^{∗}(BS_{n}; ℤ/2), we will reexplain above facts (for H^{∗}(BS_{n}; ℤ/2)) but when k = ℂ and n = 2^{m}. (We assume k = ℂ.)

V.Voevodsky showed (for the definition of the reduced power operation)

H^{∗,∗′}(BS_{p}; ℤ/p) ≅ H^{∗,∗′}(Bℤ/p; ℤ/p)^{S_{p}} ≅ ℤ/p[Y] ⊗ Λ(X)

where Y = y^{p−1} and X = y^{p−2}x in H^{∗,∗′}(Bℤ/p; ℤ/p) ≅ ℤ/p[τ, y]⊗Λ(x). Hence H^{∗}(BS_{p}; ℤ/p)/(Ng) = ℤ/p for p ≠ 2 but H^{∗}(BS_{2}; ℤ/2)/(Ng) ≅ Λ(x).

Now we restrict p = 2 and let n = 2^{m}. Consider the natural embedding

g_{m} : V_{m} = ⊕^{m}ℤ/2 → ℤ/2 ≀ ... ≀ ℤ/2 → S_{2^{m}}

where − ≀ − is the wreath product. Then it is known (Theorem 3.23 in [Ma-Mi]) that H^{∗}(BS_{2^{m}}; ℤ/2) is detected by S_{2^{m−1}} × S_{2^{m−1}} and V_{m}.

Let f_{m} : V_{m} →^{g_{m}} S_{n} → O_{n}. The the total Whitney class is written as

w(f_{m}) = 1 + w(m − 1) + w(m − 2) + ... + w(0)

where w(i) = w_{2^{m}−2^{i}}(f_{m}). Moreover Sq^{2^{i}}(w(i + 1)) = w(i). In fact the image of g_{m}^{∗} is just the Dickson algebra

Im(g_{m}^{∗}) = ℤ/2[x_{1}, ..., x_{m}]^{GL_{2}(𝔽_{2})} = ℤ/2[w(m − 1), ...., w(0)].

( Here w(i)^{2} corresponds to c_{m,i} for odd prime cases stated in the preceding section.)

By induction on m, we assume

H^{∗}(BS_{2^{m−1}}; ℤ/2)/(Ng) ≅ ℤ/2{1, w_{1}, ..., w_{2^{m−2}}}.

Then considering the restriction

H^{∗}(BS_{n}; ℤ/2)/(Ng) → H^{∗}(BS_{n−1}; ℤ/2)/(Ng)⊗H^{∗}(BS_{n−1}; ℤ/2)/(Ng),

we see

H^{∗}(BS_{n}; ℤ/2)/(Ng) ⊃ ℤ/2{1, w_{1}, ..., w_{2^{m−2}}, ..., w_{2^{m−1}}}.

Note by construction of maps g_{m}, f_{m}, we see g_{m}^{∗}w_{2^{m−1}} = w(m − 1). Hence

w(j)′ = Sq^{2^{j}}...Sq^{2^{m−2}}w_{2^{m−1}} ∈ Ng(S_{n})

with g_{m}^{∗}(w(j)′) = w(j). Thus we have

Proposition 9.2. H^{∗}(BS_{n}; ℤ/2)/(Ng) ≅ ℤ/2{1, w_{1}, ..., w_{2^{m−1}}}.

In [Ga-Me-Se], it is also shown Inv^{∗}(S_{n}; ℤ/p) = ℤ/p for odd prime p. Let n = p^{m}. The symmetric group S_{n} has a subgroup isomorphic to

S(m) = ℤ/p ≀ S_{p^{m−1}} ≅ (S_{p^{m−1}})^{p} ▷ ℤ/p

of the index prime to p. Hence H^{∗}(BS_{n}; ℤ/p) ⊂ H^{∗}(BS(m); ℤ/p). We consider the Hochshild-Serre spectral sequence

E_{2}^{∗,∗′} = H^{∗}(ℤ/p; H^{∗}((BS_{p^{m−1}})^{p}; ℤ/p)) ⟹ H^{∗}(BS(m); ℤ/p).

Let us write by σ the generator of the cyclic group ℤ/p. Let T = (1−σ) and N = (1+σ+...+σ^{p−1}). For a ℤ/p-module M, the homology is written

H^{∗}(ℤ/p; M) = \begin{cases}Ker(T) = M^{ℤ/p} ∗ = 0 \\ Ker(T)/Im(N) ∗ = even > 0 \\ Ker(N)/Im(T) ∗ = odd. \end{cases}

Let {x_{i}} be a ℤ/p basis of H^{∗}(BS_{p^{m−1}}; ℤ/p). Then the basis of

S = H^{∗}((BS_{p^{m−1}})^{p}; ℤ/p) ≅ ℤ/p{x_{i_{1}} ⊗ x_{i_{2}} ⊗ ....x_{i_{p}}}.

decomposes as I ∪ F with I = {x_{i} ⊗ ... ⊗ x_{i}} and

F = {x_{i_{1}} ⊗ ... ⊗ x_{i_{s}}|i_{k} ≠ i_{ℓ} for some k ≠ ℓ}.

The generator σ acts on F freely but invariants on I. Then cohomology is computed

H^{∗}(ℤ/p; ℤ/p{I}) ≅ ℤ/p[y] ⊗ Λ(x) ⊗ ℤ/p{I} |x| = 1, |y| = 2
H^{∗}(ℤ/p; ℤ/p{F }) ≅ ℤ/p{F }^{ℤ/p}.

Since S^{ℤ/p} = ℤ/p{F }^{ℤ/p} ⊕ ℤ/p{I}, we have the isomorphism

E_{2}^{∗,∗′} ≅ S^{ℤ/p} ⊕ ℤ/p{I} ⊗ ℤ/p[y]{x, y}.

It is well known that S^{ℤ/p} ⊂ H^{∗}(BG; ℤ/p) by Nakaoka (also Totaro [To])). Hence this spectral sequence collapses from the E_{2}-term. Thus we have the well known result ;

Theorem 9.3. (Nakaoka)

H^{∗}(BS(m); ℤ/p) ≅ E_{2}^{∗,∗′} ≅ S^{ℤ/p} ⊕ H^{∗}(BS_{p^{m−1}}; ℤ/p)^{[p]} ⊗ ℤ/p[y]{x, y}

where A^{[p]} is the graded algebra whose degree is given by p-times of degree of A.

By induction on m, we may assume H^{∗}(BS_{p^{m−1}}; ℤ/p)/(Ng) = ℤ/p. Hence ℤ/p{I}/(Ng) = ℤ/p. Therefore there is the surjection

ℤ/p{1, x} → H^{∗}(BS; ℤ/p)/(Ng).

But H^{2}(BS_{n}; ℤ/p) = 0. Hence H^{+}(BS_{n}; ℤ/p)/(Ng) = 0.

Proposition 9.4. H^{∗}(BS_{p^{m}}; ℤ/p)/(Ng) = ℤ/p.

We note here about the restriction image for g_{m} : V_{m} = ℤ/p → S_{n}. The restriction image is contained in the Dickson algebra as stated in the preceding section

grH^{∗}(BA)^{GL_{n}(𝔽_{p})} ≅ D_{n}/(c_{n,0}) ⊕ D_{n} ⊗ Q(n − 1){e_{n}^{p−2}u_{n}}

where u_{n} = x_{1}...x_{n} and e_{n} = Q_{0}...Q_{n−1}u_{n}. (Here note D_{n}/(c_{n,0}) = ℤ/p[c_{n,1}, ..., c_{n,n−1}].) We know D_{n} is in the image of g_{m}^{∗}. However it is known that e_{n}^{p−2}u_{n} ∉ g_{m}^{∗} for n ≥ 3 (p.196 in [Ad-Mi]).

10. extraspecial p-groups

We assume that p is an odd prime. The extraspecial p-group E_{n} = p_{+}^{1+2n} is the group such that exponent is p, its center is C ≅ ℤ/p and there is the extension

0 ⟶ C ⟶^{i} E_{n} ⟶^{π} V ⟶ 0

with V = ⊕^{2n}ℤ/p. (For details of the cohomology of E_{n} see [Te-Ya].) We can take generators a_{1}, ..., a_{2n}, c ∈ E_{n} such that π(a_{1}), .., π(a_{2n}) (resp. c ) make a base of V (resp. C) such that

[a_{2i−1}, a_{2i}] = c and [a_{2i−1}, a_{j}] = 1 if j ≠ 2i.

We note that E_{n} is also the central product of the n-copies of E_{1}

E_{n} ≅ E_{1} ⋯ E_{1} = E_{1} ×_{⟨c⟩} E_{1}... ×_{⟨c⟩} E_{1}.

Take cohomologies

H^{∗}(BC; ℤ/p) ≅ ℤ/p[u] ⊗ Λ(z), βz = u,
H^{∗}(BV; ℤ/p) ≅ ℤ/p[y_{1}, ..., y_{2n}] ⊗ Λ(x_{1}, ...x_{2n}), βx_{i} = y_{i},

identifying the dual of a_{i} (resp.c) with x_{i} (resp. z). That means

H^{1}(E_{n}; ℤ/p) ≅ Hom(E_{n}; ℤ/p) ∋ x_{i} : a_{j} ↦ δ_{ij}.

The central extension is expressed by

f = ∑_{i=1}^{n} x_{2i−1}x_{2i} ∈ H^{2}(BV; ℤ/p).

Hence π^{∗}f = 0 in H^{2}(BE_{n}; ℤ/p). We consider the Hochshild-Serre spectral sequence

E_{2}^{∗,∗′} ≅ H^{∗}(BV; ℤ/p) ⊗ H^{∗}(BC; ℤ/p) ⟹ H^{∗}(BE_{n}; ℤ/p).

Hence the first nonzero differential is d_{2}(z) = f and the next differential is

d_{3}(u) = d_{3}(Q_{0}(z)) = Q_{0}(f) = ∑ y_{2i−1}x_{2i} − y_{2i}x_{2i−1}.

In particular

E_{4}^{0,∗} ≅ ℤ/p[y_{1}, ..., y_{2n}] ⊗ Λ(x_{1}, ...x_{2n})/(f, Q_{0}(f)).

Lemma 10.1. We have the inclusion

Λ(x_{1}, ..., x_{2n})/(f) ⊂ H^{∗}(BE_{n}; ℤ/p).

Proof. We consider similar group E′_{n} such that its center is C ≅ ℤ/p and there is the extension

0 ⟶ C ⟶^{i} E′_{n} ⟶^{π} V′ ⟶ 0

but V′ = ⊕^{2n}ℤ_{p} such that there is the quotient map q : E′_{n} → E_{n}. We also consider the spectral sequence

E_{2}^{∗,∗′} ≅ H^{∗}(BV′; ℤ/p) ⊗ H^{∗}(BC; ℤ/p) ⟹ H^{∗}(BE′_{n}; ℤ/p).

Here H^{∗}(BV′; ℤ/p) ≅ Λ(x_{1}, ...x_{2n}). The first nonzero differential is d_{2}(z) = f but the second differential is

d_{3}(u) = ∑ y_{2i−1}x_{2i} − y_{2i}x_{2i−1} = 0.

Hence we have

H^{∗}(BE′_{n}; ℤ/p) ≅ ℤ/p[u] ⊗ Λ(x_{1}, ..., x_{2n})/(f).

From the map q^{∗} : H^{∗}(BE_{n}; ℤ/p) → H^{∗}(BE′_{n}; ℤ/p), we get the result. □

However H^{∗}(BE_{n}; ℤ/p)/(Ng) ≅ Λ̸(x_{1}, ..., x_{2n})/(f), infact, when n = 1, from Theorem 3.3 in [Ya3] we see

Proposition 10.2.

H^{∗}(Bp_{+}^{1+2}; ℤ/p)/(Ng) ≅ ℤ/p{1, x_{1}, x_{2}, a′_{1}, a′_{2}} deg(a′_{i}) = 2.

When p = 2, the situation becomes well. The extraspecial 2-group D(n) = 2_{+}^{1+2n} in the n-th central extension of the dihedral group D_{8} of order 8. It has the central extension

0 → ℤ/2 → D(n) → V → 0

with V = ⊕^{2n}ℤ/2. Hence H^{∗}(BV; ℤ/2) ≅ ℤ/2[x_{1}, ..., x_{2n}]. Then using the Hochschild-Serre spectral sequence, Quillen proved [Qu]

H^{∗}(BD(n); ℤ/2) ≅ ℤ/2[x_{1}, ..., x_{2n}]/(f, Q_{0}(f), ..., Q_{n−2}(f))⊗ℤ[w_{2^{n}}(Δ)].

(In fact when n is the real case (i.e., n = −1, 0, 1 mod(8)), the cohomology H^{∗}(BSpin_{n}; ℤ/2) injects into H^{∗}(BD(n); ℤ/2).) Here w_{i} (resp. w_{2^{n}}(Δ)) is the Stiefel-Whitney class of usual representation from the above extension (resp. 2^{n}-dimensional representation which restrict nonzero on the center). Moreover Quillen proves following two theorems (Theorem 5.10-11 in [Qu])

Theorem 10.3. ([Qu]) H^{∗}(BD(n); ℤ/2) is detected by the product of cohomoloogy of maximal elementary abeian groups.

Theorem 10.4. ([Qu]) The nonzero Stiefel-Whitney w_{i}(Δ) are those of degrees 2^{n} and 2^{n} − 2^{i} for 0 ≤ i < n.

In fact w_{i}(Δ) generates the Dickson algebra in the cohomology of the maximal elementary abelian 2-group as stated in the preceding section.

Proposition 10.5. When n > 2, there is the surjection

Λ(x_{1}, ..., x_{2n})/(f) → H^{∗}(BD(n); ℤ/2)/(Ng).

Proof. By the same arguments with p = odd, we see

Λ = Λ(x_{1}, ..., x_{2n})/(f) ⊂ H^{∗}(BD(n); ℤ/2).

The fact w_{2}(Δ) = 0 follows from the above second Quillen’s theorem. Hence we have w_{2^{n}}(Δ) ∈ Ng from Becher’s theorem (Theorem 6.2). Thus we get the theorem. □

Let 0 ≠ x ∈ Λ. Then by the Quillen’s theorem, i_{A}^{∗}(x) ≠ 0 for

i_{A}^{∗} : H^{∗}(BD(n); ℤ/2) → H^{∗}(BA; ℤ/2)

with A ≅ ℤ/2 ⊕ ... ⊕ ℤ/2. Let H^{∗}(BA; ℤ/2)/(Ng) ≅ Λ(x′_{1}, ..., x′_{m}). Then i_{A}^{∗}(x_{i}) ∈ ℤ/2{x′_{1}, ..., x′_{m}} and hence the map is restricted

i_{A}^{∗} : Λ(x_{1}, ..., x_{2n})/(f) → Λ(x′_{1}, ..., x′_{m}).

However this map is not need injective. In fact there is a possibility of i_{A}^{∗}(x) ∈ Ng(A), e.g., i_{A}^{∗}(x_{1}x_{2}) = (x′_{1})^{2} ∈ Ng(A).

11. unramified theory

In this section, we assume that k is an algebraic closed field of ch(k) = 0.

Let K be a function field of k, that is finitely generated as a field over k. Here we recall the definition of unramifed cohomology of H^{∗}(K; ℤ/p) according to Saltman, Peyre and Colliot-Thelene. We denote by P (K/k) the set of discrete valuation rings A of rank one such that k ⊂ A ⊂ K and that the fraction field F r(A) of A is K. If A belongs to P (K/k), then for the residue field κ_{A}, we can define the residue map ∂_{A} : H^{∗}(K; ℤ/p) → H^{∗−1}(κ_{A}; ℤ/p) as follows.

Let K̂_{A} be the completion, K̂_{A}^{nr} the maximal unramified extension of K_{A} and K̄_{A} is an algebraic closure of K_{A}. Put I_{A} = Gal(K̄_{A}/K̂_{A}^{nr}) and G_{A} = Gal(K̄_{A}/K̂_{A}).

K̄_{A} −−^{I_{A}} K̂_{A} −−^{G_{A}/I_{A}} K̂_{A} −− K.

Then ∂_{A} is defined as the composition of maps

∂ : H^{∗}(K; ℤ/p) → H^{∗}(K̂_{A}; ℤ/p)

→^{proj.} H^{∗−1}(G_{A}/I_{A}) ⊗ H^{1}(I_{A}; ℤ/p) ≅ H^{∗−1}(κ_{A}; ℤ/p).

Here we used that K̂_{A} ≅ I_{A} ⊕ (G_{A}/I_{A}) ( [Sa]) and I_{A} ≅ ℤ̂. Moreover H^{∗}(ℤ̂; ℤ/p) ≅ ℤ/p if ∗ = 0, 1 and ≅ 0 otherwise.

Then we can define the unramified cohomology

H_{nr}^{∗}(K; ℤ/p) = ∩_{A∈P (K/k)}Ker(H^{∗}(K; ℤ/p) →^{∂_{A}} H^{∗−1}(κ_{A}; ℤ/p)).

Namely, when X is complete, the residue map is the same as the differential d_{1} of the coniveau spectral sequence given in §2, and hence the unramified cohomology is just the E_{2}-term H_{Zar}^{0}(X; H_{ℤ/p}^{m}) of the coniveau spectral sequence.

Corollary 11.1. When X is complete, there is the isomorphism

H_{nr}^{∗}(k(X); ℤ/p) ≅ H^{∗,∗}(X; ℤ/p)/(τ) ⊕ Ker(τ)|H^{∗+1,∗−1}(X; ℤ/p).

Now we consider the case X = BG ; non complete cases. Let W//G be the scheme which has the coordinate ring k[W]^{G}. Then it is known that W//G contains U_{n}/G as an open set. Here U_{n} is the open set given in §3, where G acts freely. Hence we have for ∗ < n

H_{Zar}^{0}(W//G; H_{ℤ/p}^{∗}) ⊂ H_{Zar}^{0}(U_{n}/G; H_{ℤ/p}^{∗}).

Since G is reductive, it is also known that the fraction field of k[W]^{G} is k(W)^{G}, that is k(W//G) = k(W)^{G}. Thus we get

Lemma 11.2. For ∗ < n, we have

H_{nr}^{∗}(K(W)^{G}; ℤ/p) = H_{Zar}^{0}(W//G; H_{ℤ/p}^{∗})
⊂ H^{0}(U_{n}/G; H_{ℤ/p}^{∗}) = H^{0}(BG; H_{ℤ/p)}^{∗}).

According to Peyre [Pe], we define a subring H_{nr}^{∗}(G; ℤ/p) of H_{et}^{∗}(BG; ℤ/p) as follows. Let P (G) be the set of elements g ∈ G such that I = ⟨g⟩ ≅ ℤ/p^{s} for some s ≥ 1 but g ≠ h^{p} for any h ∈ G. Then the centralizer is written as

Z_{G}(I) ≅ I ⊕ H with H = Z_{G}(I)/I.

Let us write by ∂_{g} the composition map

∂_{g} : H^{∗}(BG; ℤ/p) → H^{∗}(BZ_{G}(I); ℤ/p)

≅ H^{∗−∗′}(BH; ℤ/p) ⊗ H^{∗′}(BI; ℤ/p) →^{proj.} H^{∗−1}(BH; ℤ/p)

using H^{1}(BI; ℤ/p) ≅ ℤ/p. Then define the unramified cohomology by

H_{nr}^{∗}(G; ℤ/p) = ⋂_{g∈P (G)} Ker(H^{∗}(BG; ℤ/p) →^{∂_{g}} H^{∗−1}(BH; ℤ/p)).

Remark. The restriction map H^{1}(B⟨g⟩; ℤ/p) → H^{1}(B⟨g^{p}⟩; ℤ/p) is always zero. Hence we need not consider the case I = ⟨g^{p}⟩.

Theorem 11.3. (Peyre [Pe]) Let W be a faithful representaion of G. Let q is the quotient map q : Gal(k̄(W)/k(W)^{G}) → Gal(k(W)/k(W)^{G}) = G. Then

q^{∗}(H_{nr}^{∗}(G; ℤ/p)) ⊂ H_{nr}^{∗}(k(W)^{G}; ℤ/p).

Proof. Arguments of pages 203 to 206 in the proof of Proposition 3 in [Pe] work exchanging H^{3}(−; ℚ/ℤ(−)) to H^{∗}(−; ℤ/p). Indeed we have the commutative diagram ((12) in [Pe])

H^{∗}(G; ℤ/p) ⟶^{∂_{D,g}} H^{∗−1}(D; ℤ/p)
↓  ↓
H^{∗}(k(W)^{G}; ℤ/p) ⟶^{∂_{A}} H^{∗−1}(κ_{A}; ℤ/p)

where D is the decomposition subgroup of G. If x ∈ H_{nr}^{∗}(G; ℤ/p), then ∂_{D,g}(x) = 0 and hence ∂_{A}(ρ^{∗}(x)) = 0. □

It is well known (Theorem 4.1.5 in [Co-Te]) that if K is purely transcendental over k (i.e., K ≅ k(x_{1}, ..., x_{n}) for indeterminate x_{i}), then

H^{∗}(k; ℤ/p) ≅ H_{nr}^{∗}(K; ℤ/p) for ∗ > 0.

When k is algebraic closed field, it is immediate H^{+}(k; ℤ/p) = 0. Corollary 11.4. Suppose that

0 ≠ x ∈ H^{∗}(BG; ℤ/p)/(Ng) and x ∈ H_{nr}^{∗}(G; ℤ/p).

Then q^{∗}(x) ≠ 0 in H_{nr}^{∗}(k(W)^{G}; ℤ/p). Hence k(W)^{G}/k is not purely trascendental. Proof. From the preceding theorem, we have

q^{∗}(x) ∈ H_{nr}^{∗}(k(W)^{G}; ℤ/p) ⊂ H^{∗}(k(W)^{G}; ℤ/p).

It is nonzero since

H^{∗}(BG; ℤ/p)/(Ng) ⊂ H_{Zar}^{0}(BG; H_{ℤ/p}^{∗}) ⊂ H^{∗}(k(W)^{G}; ℤ/p).
□

Corollary 11.5. Let G be a p−group of exponent p. If H^{2}(BG; ℤ/p)/(Ng) is not detected by ℤ/p×ℤ/p, then k(W)^{G} is not purely transcendental. Proof. Suppose that x ∉ H_{un}^{2}(G; ℤ/p). Then we can take x = ∑ x_{1}x_{2} such that x_{1}|H^{1}(BI; ℤ/p) ≠ 0 with I ≅ ℤ/p and for H = Z_{G}(I)/I, x_{2}|H^{1}(BH; ℤ/p) ≠ 0. Here

H^{1}(BH; ℤ/p)/(Ng) = H^{1}(BH; ℤ/p) ≅ Hom(H, ℤ/p).

Hence x_{2} defines subgroup J ≅ ℤ/p of H such that x_{2}|H^{1}(BJ; ℤ/p) ≠ 0 (since H is exponent p). Thus the element x is detected by the subgroup I ⊕ J ≅ ℤ/p ⊕ ℤ/p. □

12. Saltman’s example

Let G be the group defined by

0 → ⟨c_{3}, c_{4}⟩ → G → E_{2} = p_{+}^{1+4} → 0
with [a_{1}, a_{3}] = c_{3}, and [a_{1}, a_{4}] = c_{4}.

Saltman and Bogomolov showed that H_{nr}^{2}(k(W)^{G}; ℚ/ℤ) ≠ 0 for this group. We will see the ℤ/p-coefficient case.

Lemma 12.1. The 3-dimensional cohomology H^{3}(BG; ℤ/p) contains the ℤ/p-module

A = ℤ/p{y_{i}x_{j}|1 ≤ i, j ≤ 4}/(Q_{0}(f), Q_{0}(x_{1}x_{3}), Q_{0}(x_{1}x_{4})).

Proof. Consider the central extension

0 → ⟨c, c_{3}, c_{4}⟩ → G → V = ⊕^{4}ℤ/p → 0

and induced spectral sequence

E_{2}^{∗,∗′} ≅ ℤ/p[y_{1}, ..., y_{4}, u, u_{3}, u_{4}] ⊗ Λ(x_{1}, ..., x_{4}, z, z_{3}, z_{4})

converging to H^{∗}(BG; ℤ/p). The first differential is

d_{2}(z) = f, d_{2}(z_{3}) = x_{1}x_{3}, d_{2}(z_{4}) = x_{1}x_{4}.

The second nonzero differential is

d_{3}(u) = Q_{0}(f), d_{3}(u_{3}) = Q_{0}(x_{1}x_{3}), d_{3}(u_{4}) = Q_{0}(x_{1}x_{4}).

Thus we see

A ≅ E_{4}^{3,0} ≅ E_{∞}^{3,0} ⊂ H^{3}(BG; ℤ/p).
□

Theorem 12.2. ( Saltman [Sa]) We have

0 ≠ x_{1}x_{2} ∈ H_{nr}^{2}(G; ℤ/p) ∩ H^{2,2}(BG; ℤ/p)/(τ).

Hence k(W)^{G} is not purely transcendental.

Proof. From the preceding lemma, we see Q_{0}(x_{1}x_{2}) ≠ 0 in H^{3}(BG; ℤ/p). Hence we see x_{1}x_{2} ∉ Ng(G) from Lemma 3.4.

Next we will show x_{1}x_{2} ∈ H_{nr}(G; ℤ/p). Suppose this is not the case. By the definition, this means that there is an element g and h ∈ Z_{G}(⟨g⟩) such that

x_{i}|⟨g⟩ ≠ 0, x_{j}|⟨h⟩ ≠ 0 for {i, j} = {1, 2}.

Let us write

a^{Λ} = a_{1}^{λ_{1}}...a_{4}^{λ_{4}} for Λ = (λ_{1}, ..., λ_{4}),
a^{M} = a_{1}^{μ}...a_{4}^{μ_{4}} for M = (μ_{1}, ..., μ_{4}).

Note that x_{i}|⟨a^{Λ}⟩ = λ_{i}. The commutator is given by the definition

[a^{Λ}, a^{M}] = c^{d}c_{3}^{d_{3}}c_{4}^{d_{4}} where d = λ_{1}μ_{2} − λ_{2}μ_{1} + λ_{3}μ_{4} − λ_{4}μ_{3},

d_{3} = λ_{1}μ_{3} − λ_{1}μ_{3}, d_{4} = λ_{1}μ_{4} − λ_{4}μ_{1}.

Take g = a^{Λ}. Exchanging x_{1} ↦ x_{1} + λx_{2} or x_{1} ↦ x_{2} (if necessary), we can take

λ_{1} = 1, λ_{2} = 0 Λ = (1, 0, λ_{3}, λ_{4}).

(Hence x_{1}⟨g⟩ = 1 and x_{2}|⟨g⟩ = 0.) Then we can take h = a^{M} so that x_{2}|⟨h⟩ = 1 and h ∉ ⟨g⟩, that means

μ_{1} = 0, μ_{2} = 1 M = (0, 1, μ_{3}, μ_{4}).

From d_{3} = 0 and d_{4} = 0, we see

μ_{3} = λ_{3}μ_{1} = 0 and μ_{4} = λ_{4}μ_{1} = 0

(and hence M = (0, 1, 0, 0)). Therefore d = 1 × 1 − 0 + 0 − 0 = 1. This is a contradiction. Hence we have proved x_{1}x_{2} ∈ H_{nr}(G; ℤ/p). □

13. non detected examples

We consider group G′_{m} and element ξ_{m} ∈ H^{∗}(BG′_{m}; ℤ/p)/(Ng) which is not detected by elementary abelina p-groups, while it is not in H_{nr}(G′_{m}; ℤ/p).

Let G′ be the group defined by

0 → ⟨c_{3}, c_{4}⟩ → G′ → E_{2} = p_{+}^{1+4} → 0
with [a_{1}, a_{3}] = c_{3}, and [a_{2}, a_{4}] = c_{4}.

Difference of the definitions of G and G′ is just [a_{2}, a_{4}] = c_{4}. However note G′ has the rank 2 elementary abeilan p-subgroup

⟨a_{1}a_{3}^{−1}, a_{2}a_{4}^{−1}⟩ ≅ ℤ/p ⊕ ℤ/p

while G does not.

Let G′_{m} be the central product of m − 1-th copies of G′ and one G

G′_{m} = G · G′ · ... · G′ = G ×_{⟨c⟩} G′ ×_{⟨c⟩} ... ×_{⟨c⟩} G′

which is generated by a_{1}, ..., a_{4m}, c, c_{3}, c_{4}, c_{7}, ..., c_{4m−1}, c_{4m}. The comutators are given for 1 ≤ i ≤ m

[a_{4i−3}, a_{4i−2}] = [a_{4i−1}, a_{4i}] = c,
[a_{4i−3}, a_{4i−1}] = c_{4i−1}, and [a_{4i−2}, a_{4i}] = c_{4i} for i ≠ 1,
but [a_{1}, a_{4}] = c_{4}.

Let us write a^{Λ} = a_{1}^{λ_{1}}...a_{4m}^{λ_{4m}} and Λ = (λ_{1}, ..., λ_{4m}). Note also x_{i}|⟨a^{Λ}⟩ = λ_{i}. The commutator is given by the definition

[a^{Λ}, a^{M}] = c^{d}c_{3}^{d_{3}}c_{4}^{d_{4}}....c_{4m}^{d_{4m}}

where d = ∑_{i} λ_{4i−3}μ_{4i−2} − λ_{4i−2}μ_{4i−3} + λ_{4i−1}μ_{4i} − λ_{4i}μ_{4i−1},

d_{4i−1} = λ_{4i−3}μ_{4i−1} − λ_{4i−1}μ_{4i−3}, d_{4i} = λ_{4i−2}μ_{4i} − λ_{4i}μ_{4i−2} (i ≠ 1), and d_{4} = λ_{1}μ_{4} − λ_{4}μ_{1}.

Lemma 13.1. Let ξ_{m} = x_{1}x_{2}x_{5}x_{6}....x_{4m−3}x_{4m−2} ∈ H^{∗}(BG′_{m}; ℤ/p). Then ξ_{m} ∈ H^{∗}(BG′_{n}; ℤ/p)/(Ng) is not detected by elementary abelian p − groups.

Proof. Suppose that ξ_{m} is detected by

⟨g_{1}, g_{2}, ..., g_{2m}⟩ ≅ ℤ/p ⊕ ... ⊕ ℤ/p

such that ξ_{m}|⟨g_{1}, ..., g_{2m}⟩ ≠ 0, and x_{i}|⟨g_{i}⟩ ≠ 0 for some i. Take generators adequately, let g_{i} = a^{Λ_{i}} such that

Λ_{1} = (1, 0, ∗, ∗|0, 0, ∗, ∗|....|0, 0, ∗, ∗),
Λ_{2} = (0, 1, ∗, ∗|0, 0, ∗, ∗|....|0, 0, ∗, ∗),
Λ_{5} = (0, 0, ∗, ∗|1, 0, ∗, ∗|....|0, 0, ∗, ∗),
Λ_{6} = (0, 0, ∗, ∗|0, 1, ∗, ∗|....|0, 0, ∗, ∗),

.......

Λ_{4m−3} = (0, 0, ∗, ∗|0, 0, ∗, ∗|....|1, 0, ∗, ∗),
Λ_{4m−2} = (0, 0, ∗, ∗|0, 0, ∗, ∗|....|0, 1, ∗, ∗).

Let i > 1. Consider the commutativity of L = Λ_{4i−3} and M = Λ_{1}. Since d_{4i−1} = λ_{4i−3}μ_{4i−1} − λ_{4i−1}μ_{4i−3} = 0, we see

μ_{4i−1} = 0 from μ_{4i−3} = 0 and λ_{4i−3} = 1.

Similarly we have μ_{4i} = 0 from the commutativity of Λ_{4i−2} and Λ_{1}. Thus we see

Λ_{1} = (1, 0, ∗, ∗|0, 0, 0, 0|....|0, 0, 0, 0).

We also have Λ_{2} = (0, 1, ∗, ∗|0, 0, 0, 0|....|0, 0, 0, 0). Let Λ = Λ_{1} and M = Λ_{2}. By the commutativity and facts d_{3} = d_{4} = 0, we see μ_{3} = μ_{4} = 0, that is,

Λ_{2} = (0, 1, 0, 0|0, 0, 0, 0|....|0, 0, 0, 0).

However this is a contradiction, indeed,

d = ∑_{i} λ_{4i−3}μ_{4i−2} − λ_{4i−2}μ_{4i−3} + λ_{4i−1}μ_{4i} − λ_{4i}μ_{4i−1} = 1 ≠ 0.

□

Lemma 13.2. Q_{0}...Q_{2m−2}(ξ_{m}) ≠ 0 in H^{∗}(BG_{m}; ℤ/p).

Proof. We recall that G′_{m} is the central product of G and copies of G′, that means

0 → ⊕^{m}ℤ/p → G × G′ × ... × G′ →^{pr.} G′_{m} → 0.

The map induces the map of cohomologies

pr.^{∗} : H^{∗}(BG′_{m}; ℤ/p) → H^{∗}(BG; ℤ/p) ⊗ H^{∗}(BG′; ℤ/p)^{⊗n}.

The operation Q_{i} is derivative, we have

Q_{0}...Q_{2m−2}(ξ_{m}) =

Q_{0}(x_{1}x_{2})Q_{1}Q_{2}(x_{5}x_{6})...Q_{2m−3}Q_{2m−2}(x_{4m−3}x_{4m−2}) + ..... Here from Lemma 12.1,

Q_{1}(x_{1}x_{2}) = y_{1}x_{2} − y_{2}x_{1} ≠ 0 in H^{∗}(BG; ℤ/p).

For i > 1, we see

Q_{2i−3}Q_{2i−2}(x_{4i−3}x_{4i−2}) = y_{4i−3}^{p^{2i−2}}y_{4i−2}^{p^{2i−3}} − y_{4i−3}^{p^{2i−3}}y_{4i−2}^{p^{2i−2}}.

This is nonzero in H^{∗}(BG′; ℤ/p) because it is nonzero in the restriction image

H^{∗}(BG′; ℤ/p) → H^{∗}(B⟨a_{1}a_{3}^{−1}, a_{2}a_{4}^{−1}⟩; ℤ/p) ≅ ℤ/p[y′_{1}, y′_{2}] ⊗ Λ(x′_{1}, x′_{2})

where y_{4i−3}, y_{4i−1} ↦ y′_{1} and y_{4i−2}, −y_{4i} ↦ y′_{2}. □

Corollary 13.3. Elements in H^{2m}(BG′_{m}; ℤ/p)/(Ng) is not detected by abelian subgroups, namely, the restriction map

Res : H^{2m}(BG′_{m}; ℤ/p)/(Ng) → Π_{A;abelian}H^{2m}(BA; ℤ/p)/(Ng)

is not injective.

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Department of mathematics, Faculty of Science, Ryukyu University, Okinawa, Japan, Department of Mathematics, Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan

  E-mail address: tez@sci.u-ryukyu.ac.jp, yagita@mx.ibaraki.ac.jp,