1. LOCAL COHOMOLOGY FUNCTORS 3

Observe that

OU,x =

Ox for x ∈ U

0 otherwise.

Therefore the sequence 0 → OU → OU is exact for U ⊂ U, U ∈ U(X). Since I is

injective, the sequence

HomOX (OU , I ) −→ HomOX (OU , I ) −→ 0

is also exact. However HomOX (OU , I ) = Γ(U, I ) = I (U) canonically, and since

I (U) −→ I (U ) −→ 0

is exact, the sheaf I is flasque.

In particular each injective abelian sheaf on a topological space X is flasque.

Proposition 1.8. Let X be a topological space and Y ⊂ X a closed subset.

Then for each flasque sheaf F on X we have

HY

i

(X, F ) = 0 for i 0.

Proof. There is an exact sequence 0 → F → I → G → 0 with an injective

sheaf I . By 1.7 I is flasque, hence so is G by 1.6 b). Furthermore, the sequence

0 −→ ΓY (X, F ) −→ ΓY (X, I ) −→ ΓY (X, G ) −→ 0

is exact by 1.6 a) and HY i (X, I ) = 0 for i 0 by the construction of derived

functors. The long exact cohomology sequence

0

HY 1 (X, F ) HY 1 (X, I ) HY 1 (X, G ) HY 2 (X, F ) HY 2 (X, I ) ···

0 0

shows that HY 1 (X, F ) = 0 and

HY+1(X, i

F )

∼

=

HY i (X, G ) for i ≥ 1. Since G is

flasque it follows by induction that HY i (X, F ) = 0 for all i ≥ 1.

If T = {T

i}i≥0

is a δ-functor on an abelian category A with values in another

abelian category, then an object F of A is called T -acyclic if T

i(F

) = 0 for i 0.

Flasque sheaves are acyclic for the cohomology on X with support in Y . Assume

(1.1) 0 −→ F −→ G

0

−→ G

1

−→ ···

is a T -acyclic resolution of F ∈ A, that is an exact sequence for which all G

i

(i ≥ 0)

are T -acyclic. Applying T

0

we obtain a complex

T

0(G •)

: 0 −→ T

0(G 0)

−→ T

0(G 1)

−→ T

0(G 2)

−→ ··· .

Lemma 1.9. T i(F )

∼

=

Hi(T 0(G •)).

Proof. Set F =: F

0

and decompose (1.1) into short exact sequences

0 −→ F

0

−→ G

0

−→ F

1

−→ 0

0 −→ F

1

−→ G

1

−→ F

2

−→ 0

.

.

.

0 −→ F

i

−→ G

i

−→ F

i+1

−→ 0

.

.

.