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 ).