Homologie, cohomologie - IMJ-PRG

. . . −→ Cn+1
ϕn+1
−→ Cn
ϕn
−→ Cn−1−→ . . . −→ 0
∀i Ciϕi
ϕi◦ϕi+1 = 0 Im ϕi+1 ⊂ker ϕi
C i
Bi= Im ϕi+1 Zi= ker ϕi
i, CiBi ZiHi=Zi/Bi
0−→ . . . −→ Cn−1
ϕn−1
−→ Cn
ϕn
−→ Cn+1 −→ . . .
Hi=Zi/BiZi= ker ϕiBi=
Im ϕi−1

G A G
R R[G]RG
G R
R[G] = {X
g∈G
agg|ag∈R, g ag= 0R}
G
R
X
g∈G
agg+X
g∈G
bgg=X
g∈G
(ag+bg)g
(X
g∈G
agg)(X
g∈G
bgg) = X
k∈G
(X
g∈G
akbk−1gg)
R=
Z
R[G]G
R G
G G
g0∗(X
k∈G
akk) = X
k∈G
akg0k
G=Z/3Z={a0, a1, a2}
Z[G] = {n0a0+n1a1+n2a2}
A=n0a0+n1a1+n2a2B=m0a0+m1a1+m2a2
A+B= (n0+m0)a0+ (n1+m1)a1+ (n2+m2)a2
AB = (n0m0+n1m2+n2m1)a0+(n0m1+n1m0+n2m2)a1+(n0m2+n1m1+n2m0)a2

Z[G]
. . . −→ Z[Gi+1]ϕi+1 −→ Z[Gi]ϕi −→ Z[Gi−1]−→ . . . −→ 0
ϕi:Z[Gi]→Z[Gi−1]
ϕi(g1, . . . , gi) =
i
X
j=1
(−1)i(g1,..., ˆgj, . . . , gi)
(g1,..., ˆgj, . . . , gi)i(g1, . . . , gi)gj
ϕi◦ϕi+1 =
0
(g1, . . . , gi+1)∈Gi+1
ϕi◦ϕi+1(g1, . . . , gi+1) =
i+1
X
j=0
(−1)j(
j−1
X
k=1
(−1)k(g1,..., ˆgk,..., ˆgj,...gi+1) +
i
X
k=j
(−1)k(g1,..., ˆgj,..., ˆgk+1, . . . , gi+1))
(g1,..., ˆgk,..., ˆgj,...gi+1)
j ϕi+1 k ϕi
(−1)j(−1)kk ϕi+1 j ϕi
(−1)k(−1)j−1
C
0−→ Ai −→ Bπ −→ C−→ 0
i π Z[G]A B C
0−→ Z[Gj]⊗Z[G]A−→ Z[Gj]⊗Z[G]B−→ Z[Gj]⊗Z[G]C−→ 0

˜
iZ[Gj]⊗AZ[Gj]⊗B˜ i(g⊗a) = g⊗i(a)
˜πZ[Gj]⊗BZ[Gj]⊗C˜π(g⊗b) = g⊗π(b)
ker ˜π= Im ˜π
Cker π= Im i
g⊗b∈ker ˜π
˜π(g⊗b) = 0 ⇒g⊗π(b) = 0, g 6= 0
⇒π(b) = 0
b∈Im i g ⊗b∈Im˜ i
g⊗b∈Im˜ i
g⊗b∈Im˜ i⇒b∈Im i
⇒π(b) = 0
⇒˜π(g⊗b)=0
ker ˜π= Im˜ i
Ci=Z[Gi]
. . . −→ Ci+1
ϕi+1
−→ Ci
ϕi
−→ Ci−1−→ . . . −→ 0
Bi= Im ϕi+1 Zi= ker ϕiHi=Zi/Bi
Ci(G, A) = Z[Gi]⊗Z[G]A
˜ϕ=ϕ⊗
Bi(G, A) = Im ˜ϕi+1
Zi(G, A) = ker ˜ϕi
Hi(G, A) = Zi(G, A)/Bi(G, A)
C
· · · → Hj+1(G, C)→Hj(G, A)→Hj(G, B)→Hj(G, C)→ · · · → H0(G, C)→0

0−→ Cj+1(G, A)˜
i
−→ Cj+1(G, B)˜π −→ Cj+1(G, C)−→ 0
↓˜ϕj+1 ↓˜ϕj+1 ↓˜ϕj+1
0−→ Cj(G, A)˜
i
−→ Cj(G, B)˜π −→ Cj(G, C)−→ 0
˜
i◦˜ϕj+1 =ϕj+1 ⊗i
=˜ i◦˜ϕj+1
π
Cj+1(G, A)/Bj+1(G, A)−→ Cj+1(G, B)/Bj+1(G, B)−→ Cj+1(G, C)/Bj+1(G, C)−→ 0
↓ ↓ ↓
0−→ Zj(G, A)−→ Zj(G, B)−→ Zj(G, C)−→ 0
Hj+1(G, A)→Hj+1(G, B)→Hj+1(G, C)→Hj(G, A)→Hj(G, B)→Hj(G, C)
0−→ . . . −→ Cn−1
ϕn−1
−→ Cn
ϕn
−→ Cn+1 −→ . . .
Hi=Zi/BiZi= ker ϕiBi=
Im ϕi−1