4 1. LOCAL COHOMOLOGY FUNCTORS
where F
i
:= ker(G
i
G
i+1)
(i 0). Applying T
0
and the long exact cohomology
sequence we obtain exact sequences
0 −→ T
0(F i)
−→ T
0(G i)
−→ T
0(F i+1)
−→ T
1(F i)
−→ 0,
isomorphisms
T
p
(F
p+1
)

= T
p
(F
p
) for p 1,
and commutative triangles
T
0(G i)
T
0(G i+1)
T
0(F i+1),
from which one deduces that
T
0(F i)
=
H0(T 0(G •))
T
1(F i)
=
Hi+1(T 0(G •))
for i 0
and
T
p
(F
0
) = T
p−1
(F
1
) = ··· = T
1
(F
p−1
) =
Hp(T 0
(G

)),
which is what we had to show.
The lemma shows in particular: If an abelian sheaf F has a flasque resolution
0 −→ F −→ G
0
−→ G
1
−→ ··· ,
then
HY
i
(X, F ) =
Hi(ΓY
(X, G
•)),
i.e. local cohomology can be computed by means of flasque resolutions.
Proposition 1.10. Let (X, OX) be a ringed space. The local cohomology func-
tors HY
i
(X, F ) for F Mod(X) are the derived functors of ΓY (X, F ) in the
category Mod(X) of OX-modules, that is they can be computed with injective reso-
lutions in Mod(X). In particular all HY
i
(X, F ) are modules over ΓY (X, OX) and
all homomorphisms HY
i
(X, F ) HY
i
(X, G ) associated with morphisms F G in
Mod(X) are ΓY (X, OX)-linear.
This is clear by 1.9 since injective OX-modules are flasque by 1.7.
By the universal property of derived functors, the inclusion
ΓY (X, F ) Γ(X, F )
corresponds to a homomorphism of δ-functors
HY
i
(X, F )
Hi(X,
F ) (i 0)
called the canonical homomorphism from local into global cohomology. Later this
will play a basic role in the formulation of the residue theorem.
Let U := X \ Y . The restriction Γ(X, F ) Γ(U, F|U ) likewise corresponds
to a canonical homomorphism of δ-functors
Hi(X,
F )
Hi(U,
F|U ) (i 0).
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