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).