Algèbres à involution et invariants cohomologiques

k
G k G ×G→G K/k
G(K)×G(K)→G(K)G(K)
•AnSL(V)V k n + 1
•BnSpin(V, q)SO(V, q)
q
V k
2n+ 1
•CnSp(V, b)b V k
2n
•DnSpin(V, q)q V
k2n
A k
K/k AKK
A

•
dimkV= dimKVK
•
x2+y2x2−y2
R C
X
K A k AK'X
k k
VΦ∈V⊗p⊗(V∗)⊗q
V⊗q→V⊗p
VK=
V⊗kKΦK= Φ ⊗k1K
p q
•p=q= 0
•p=q= 1
k[X]
•p= 0, q = 2
•p= 1, q = 2
k
(V, Φ) (W, Ψ)
k f :V→W f⊗p⊗(f∗)⊗qΦ Ψ
K/k G
A(V, Φ) (p, q)k G
AKσ(v⊗λ) = v⊗σ(λ)A
AK
AG
K
G K/k
V K G V
k
• ∀σ∈G, ∀λ∈K, ∀v∈V, σ(λv) = σ(λ)σ(v)
•v∈V L/k Gal(K/L)⊂G
v
G
V
G×V→V
X K k

G
X K
G XGk
XG⊗kK−→ X
X
G X
k
A k X =AK
AKG X
ϕ:G→Autk(X)G
X=AKψ:G→Autk(X)
a:G−→ AutK(X)
σ7−→ aσ=ψ(σ)◦(ϕ(σ))−1.
k
a:
G→AutK(X) AutK(X)
aστ =aσσ(aτ).
a b k
c∈AutK(X)
aσ=c−1bσσ(c).
Z1(G, AutK(X)) a∼b
H1(G, AutK(X))
Z1(G, AutK(X))/∼
A k
k AKK
A
H1(G, AutK(AK)) A
aσ=Id

AKA p =
q= 0 AKGLn(K)n
H1(G, GLn(K)) = {∗}.
Q=Px2
i
Rnn
QC/R
R
H1(G, On(C)) ' {(r, s)∈N2|r+s=n}.
A=Mn(k) (1,2)
A k
n2K
AAutK(AK)'P GLn(K)
H1(G, P GLn(K)) ←→ n2K .
H1(G, A)G
A
H1(G, •)
G
B G A G
G C =B/A
CG=BG/AG
G C
G B [b]∈CGb∈B
b∈BGσ∈G σ(b) = baσaσ∈A
(aσ)
H1(G, A) [b]b∈BG
1→AG→BG→CG→H1(G, A).
1→AG→BG→CG→H1(G, A)→H1(G, B)→H1(G, C).