1. LOCAL COHOMOLOGY FUNCTORS 5
Proposition 1.11. There exists a functorial long exact sequence
0 −→ ΓY (X, F ) −→ Γ(X, F ) −→ Γ(U, F|U ) −→
HY
1
(X, F ) −→
H1(X,
F ) −→
H1(U,
F|U ) −→
HY
2
(X, F ) −→ ···
Proof. For each flasque sheaf G on X the canonical sequence
0 −→ ΓY (X, G ) −→ Γ(X, G ) −→ Γ(U, G
|U
) −→ 0
is exact, as is easily seen. Assume now that a flasque resolution (1.1) of F is given.
Then there is an exact sequence of complexes
0 −→ ΓY (X, G
•)
−→ Γ(X, G
•)
−→ Γ(U, (G
|U
)•)
−→ 0
and by 1.9 the long exact cohomology sequence associated with it gives the sequence
we are looking for.
Now let V X be an open set such that Y V .
Lemma 1.12. There is a canonical isomorphism of δ-functors
HY
i
(X, F )

=
HY
i
(V, F|V ).
In other words: The cohomology with support in Y depends only on the neighbor-
hoods of Y .
Proof. The canonical map
ΓY (X, F ) ΓY (V, F|V ) (s s|V )
defines a functorial isomorphism. It is clear that the map is injective. In order to
show surjectivity let a section s ΓY (V, F|V ) be given, and set U := X \ Y . Then
s and the zero section on U define a section t Γ(X, F ) with Supp(t) Y and
t|V = s.
Passing to derived functors leads to the desired isomorphism since F F|V
is an exact functor.
Lemma 1.13. Let X X be a closed subset with Y X , and let j : X X
denote the inclusion map. Then for abelian sheaves F on X there is an isomor-
phism of δ-functors
HY
i
(X , F )

= HY
i
(X, j∗F ) (i 0).
Proof. The canonical isomorphism Γ(X , F )

= Γ(X, j∗F ) induces an iso-
morphism
ΓY (X , F )

= ΓY (X, j∗F ).
From a flasque resolution (1.1) of F we obtain a flasque resolution
0 j∗F j∗G
0
j∗G
1
···
of j∗F since the functor j∗ is exact in our situation. Thus we have
HY
i
(X , F )

=
Hi(ΓY
(X , G
•))

=
Hi(ΓY
(X, j∗G
•))

=
HY
i
(X, j∗F ).
Previous Page Next Page