Quelques généralités sur la cohomologie des faisceaux.

FM
UMF(U)

F U
V ⊂ U
rVU :F(U)−→ F(V),
rUU =IdF(U)W ⊂ V ⊂ U
rWU ◦rVU =rWU.
U UαsαF(Uα)
rUα∩Uβ,Uα(sα) = rUα∩Uβ,Uβ(sβ),
sF(U)α
rUαU(s) = sα.
rUV(s)s|VU=∅
F(∅) = 0 Γ(U,F)F(U)
Γ(U,F)F U
Γ(M, F)F
C∞(U)C∞U
GU
M G(U) = G G(∅)=0 U ⊂ V rUV =
U−→ F(U) = C
MOMO∗
M
OM(U) = {f:U −→ C:f},
O∗
M(U) = {f:U −→ C:f , f(z)6= 0,∀z∈ U},
UMU ⊂ V rUV(f) = f|U
OM(U)

OM(E)
E M
Ωp
Mp
M
Ap,q
M(p, q)
M
K p ∈K
Kp
Kp(U) = ½K p ∈ U
0
UM p ∈ U ⊂ V rUV(p) = pU ⊂ V p /∈ U
rUV(p)=0 K0
ϕ:M−→ N
FM ϕ∗FN
Fϕ
ϕ∗F(U) = F(ϕ−1(U)),
UN
F G
ϕ:F −→ G ϕU:F(U)−→ G(U)
UMU ⊂ V
F(V)ϕV −→ G(V)
rUV↓ ↓rUV
F(U)−→
rU
G(U)
ϕ:F −→ G
F U
U
U 7−→ ker(ϕ)(U) = ker (ϕ|U:F(U)−→ G(U)) .
ϕ
ϕ
G F

gG(U) ( ϕ)(U) (Uα)U
sαF(Uα) (sα) = g|Uαα ϕ
U 7−→ (ϕ)(U) = (ϕ|U:F(U)−→ G(U)) .
(ϕ)
(Uα)UsαF(Uα)
(Uα, sα)α β sα|Uα∩Uβ−
sβ|Uα∩Uβϕ(G(Uα∩ Uβ)
( (ϕ))(U) (ϕ)
U 7−→ ( )(U) = (ϕ|U:F(U)−→ G(U)) .
G F ⊂ G
sU
(Uα)Usα∈ G(Uα)
sα|Uα∩Uβ=sβ|Uα∩Uβ.
F G H M
Fϕ −→ G ψ −→ H,
Gϕ= ker ψ
0−→ F ϕ −→ G ψ −→ H −→ 0
F G H
0−→ F −→ G −→ G/F −→ 0
F G
0−→ IN−→ OM−→ ON−→ 0
N⊂M M IN
OMM
N

0−→ Zi −→ OM
exp
−→ O∗
M−→ 0,
OMMO∗
M
M i
exp exp f=e2π√−1f
0−→ F −→ G −→ H −→ 0
0−→ F(U)−→ G(U)−→ H(U)−→ 0
U
M=C∗
C∗
0−→ F ϕ −→ G ψ −→ H −→ 0
δ∗:Hk(M, H)−→ Hk+1(M, H),
0−→ H0(M, F)ϕ∗ −→ H0(M, G)ψ∗ −→ H0(M, H)
δ∗
−→ H1(M, F)ϕ∗ −→ H1(M, G)ψ∗ −→ H1(M, H)
δ∗
−→ Hk(M, F)ϕ∗ −→ Hk(M, G)ψ∗ −→ Hk(M, H)δ∗ −→ · · ·
λ1, λ2, µ1, µ2, ν1, ν2
0−→ F1
ϕ1
−→ G1
ψ1
−→ H1−→ 0
↓λ1↓µ1↓ν1
0−→ F2
ϕ2
−→ G2
ψ2
−→ H2−→ 0
↓λ2↓µ2↓ν2
0−→ F3
ϕ3
−→ G3
ψ3
−→ H3−→ 0