Cours sur l`homologie stable des groupes à coefficients polynomiaux

C−
Mod Ck−Mod
H∗(G∞;F∞)→H∗(C;F)H∗(G∞;F∞)→
H∗(G∞× C;F)
H∗(C;F)H∗(G∞;k)H∗(G∞× C;F)
(C,⊕,0) X

H(A)→Hdeg
(A)
(Gn)n∈N
Gn→Gn+1
Σn{1, . . . , n}
ΣnΣn+1
n+ 1
GLn(A)A GLn(A)
GLn+1(A)M7→ M0
0 1 
On,n(A)
Sp2n(A)A
n
0 1
1 0   0 1 −1 0 
H∗(G0)→H∗(G1)→ · · · → H∗(Gn)→H∗(Gn+1)→. . .

d∈NN
Hd(Gn)→Hd(Gn+1)n≥N
H∗(Gn)n
Fp
(C,⊕,0) ⊕

0
CXC
⊕
X
Gn:= AutC(X⊕n).
⊕
Σn→Gn
Gnu
X⊕nu⊕IdXX⊕n+1
Gn→Gn+1
IdX⊕u u u ⊕IdX
(1 n+ 1) X⊕n+1
H∗(Gn)→H∗(Gn+1)
A
X
AP(A)A
X=A
GLn(A)
A
A  ∈ {−1,1}H(A)
A
 D
HomA(−, A)A
A A
HomA(M, A)M A
DHomA(M, DN )'
HomA(N, DM )
M=N  M
S2
(M) HomA(M, DM )
x7→ ¯x−x A
= 1 =−1
X A A2A⊕DA
0 1
0 0 
1−1
0cC
0c0
c0c
C

GLn(A)→SLn+1(A)M7→ M0
0 (det M)−1
F:C → Ab
0C C
X⊕n=X⊕n⊕0Id⊕(0→X) −−−−−−−→ X⊕n⊕X=X⊕n+1
n
Gn→Gn+1 F(X⊕n)→F(X⊕n+1)
GnGn+1 Gn→Gn+1
H∗(Gn;F(X⊕n)) →
H∗(Gn+1;F(X⊕n+1)) F
(Gn)CF
H∗(G∞;F∞)F∞F(X⊕n)
G∞Gn
H∗(G∞)
ZG∞F
H∗(C;F)CF
C
k
C − Mod Ck
F∈Ob C − Mod
H∗(C;F)H0(C;F)F
C − Mod →ModkF∈Ob C − Mod
G∈Ob Mod−C := Cop−Mod TorC
∗(G, F )
CBC
(Cop ×C)−Mod
CB HH0(C;B)