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
.
.
.
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