4 1. LOCAL COHOMOLOGY FUNCTORS
(i ≥ 0). Applying T
and the long exact cohomology
sequence we obtain exact sequences
0 −→ T
) for p ≥ 1,
and commutative triangles
from which one deduces that
H0(T 0(G •))
Hi+1(T 0(G •))
for i 0
) = T
) = ··· = T
which is what we had to show.
The lemma shows in particular: If an abelian sheaf F has a flasque resolution
0 −→ F −→ G
−→ ··· ,
(X, F ) =
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-
(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
(X, F ) are modules over ΓY (X, OX) and
all homomorphisms HY
(X, F ) → HY
(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
(X, F ) →
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
F ) →
F|U ) (i ≥ 0).