2 1. LOCAL COHOMOLOGY FUNCTORS
b) If F is a presheaf, we define for each U U(X) the abelian group
F (U) =
x∈U
Fx,
with restriction morphisms given by the canonical projections. In this way
we obtain a flasque sheaf on X, the flasque sheaf associated to F . If F
is already a sheaf, then the canonical morphism F F is a monomor-
phism. One constructs an exact sequence of sheaves
0 −→ F −→ F
0
−→ F
1
−→ ···
with F
0
:= F and F
i
:= C i, where C
i
:= coker(F
i−1
F
i).
Here
F
−1
= F . This exact sequence is called the canonical flasque resolution
of F .
Lemma 1.6. Let 0 F
α
G
β
H 0 be an exact sequence in Ab(X) and
let F be flasque. Then:
a) 0 ΓY (X, F ) ΓY (X, G ) ΓY (X, H ) 0 is exact.
b) G is flasque if and only if H is.
Proof. a) We have only to show that ΓY (X, G ) ΓY (X, H ) is surjective.
Let a section s ΓY (X, H ) be given. Consider the set M of all pairs (V, t) with
V U(X), t G (U) such that β(t) = s|V . We order M by
(V, t) (V , t ) ⇐⇒ V V , t
|V
= t
Then M is inductively ordered and by Zorn’s lemma has a maximal element (V, t).
If V = X, then we can choose x X \ V and a neighborhood U of x with a section
t G (U) such that β(t ) = s|U . We have t|V
∩U
t
|V
∩U
F (V U), and since F
is flasque there exists t0 F (U) with t0|V
∩U
= t|V
∩U
−t
|V
∩U
. Then t +t0 G (U)
and t G (V ) define a section t∗ G (V U). It follows that β(t∗) = s|V
∪U
, which
contradicts the maximality of (V, t). Therefore V = X and there exists t G (X)
with β(t) = s.
Now set U := X \ Y . Since Supp(s) Y we have t|U F (U), which is
regarded here as a subgroup of G (U). Choose t0 F (X) with t0|U = t|U and set
t∗
:= t t0. Then
t∗
ΓY (X, G ) and
β(t∗)
= s, which is what we had to show.
b) Applying the above in the case Y = X, it follows that for open sets V U
the rows of the commutative diagram
0 F (U) G (U) H (U) 0
ρ ρ ρ
0 F (V ) G (V ) H (V ) 0
are exact, where ρ is surjective since F is flasque. It follows that ρ is surjective if
and only ρ is.
Lemma 1.7. Let (X, OX) be a ringed space and I an injective OX-module.
Then I is a flasque sheaf.
Proof. For U U(X) let j : U X be the inclusion and OU := j!(OX
|U
)
the extension of OX|U by zero, that is the sheaf associated to the presheaf
V −→
OX(V ) for V U
0 otherwise.
Previous Page Next Page