2 1. LOCAL COHOMOLOGY FUNCTORS

b) If F is a presheaf, we define for each U ∈ U(X) the abelian group

F (U) =

x∈U

Fx,

with restriction morphisms given by the canonical projections. In this way

we obtain a flasque sheaf on X, the flasque sheaf associated to F . If F

is already a sheaf, then the canonical morphism F → F is a monomor-

phism. One constructs an exact sequence of sheaves

0 −→ F −→ F

0

−→ F

1

−→ ···

with F

0

:= F and F

i

:= C i, where C

i

:= coker(F

i−1

→ F

i).

Here

F

−1

= F . This exact sequence is called the canonical flasque resolution

of F .

Lemma 1.6. Let 0 → F

α

→ G

β

→ H → 0 be an exact sequence in Ab(X) and

let F be flasque. Then:

a) 0 → ΓY (X, F ) → ΓY (X, G ) → ΓY (X, H ) → 0 is exact.

b) G is flasque if and only if H is.

Proof. a) We have only to show that ΓY (X, G ) → ΓY (X, H ) is surjective.

Let a section s ∈ ΓY (X, H ) be given. Consider the set M of all pairs (V, t) with

V ∈ U(X), t ∈ G (U) such that β(t) = s|V . We order M by

(V, t) ≤ (V , t ) ⇐⇒ V ⊂ V , t

|V

= t

Then M is inductively ordered and by Zorn’s lemma has a maximal element (V, t).

If V = X, then we can choose x ∈ X \ V and a neighborhood U of x with a section

t ∈ G (U) such that β(t ) = s|U . We have t|V

∩U

− t

|V

∩U

∈ F (V ∩ U), and since F

is flasque there exists t0 ∈ F (U) with t0|V

∩U

= t|V

∩U

−t

|V

∩U

. Then t +t0 ∈ G (U)

and t ∈ G (V ) define a section t∗ ∈ G (V ∪ U). It follows that β(t∗) = s|V

∪U

, which

contradicts the maximality of (V, t). Therefore V = X and there exists t ∈ G (X)

with β(t) = s.

Now set U := X \ Y . Since Supp(s) ⊂ Y we have t|U ∈ F (U), which is

regarded here as a subgroup of G (U). Choose t0 ∈ F (X) with t0|U = t|U and set

t∗

:= t − t0. Then

t∗

∈ ΓY (X, G ) and

β(t∗)

= s, which is what we had to show.

b) Applying the above in the case Y = X, it follows that for open sets V ⊂ U

the rows of the commutative diagram

0 → F (U) → G (U) → H (U) → 0

↓ ρ ↓ ρ ↓ ρ

0 → F (V ) → G (V ) → H (V ) → 0

are exact, where ρ is surjective since F is flasque. It follows that ρ is surjective if

and only ρ is.

Lemma 1.7. Let (X, OX) be a ringed space and I an injective OX-module.

Then I is a flasque sheaf.

Proof. For U ∈ U(X) let j : U → X be the inclusion and OU := j!(OX

|U

)

the extension of OX|U by zero, that is the sheaf associated to the presheaf

V −→

OX(V ) for V ⊂ U

0 otherwise.