1. LOCAL COHOMOLOGY FUNCTORS 1
1. Local Cohomology Functors
In this section some properties of local cohomology with values
in sheaves are presented. Later in § 5 and § 10 residues will
be defined for local cohomology classes of sheaves of differential
forms. The reader is assumed to be familiar with basic notions
and facts of homological algebra and sheaf theory as explained
in [39, Chap. II and III].
Let X be a topological space, Y X a closed subset, and let U(X) be the set
of open subsets of X. Let Ab(X) denote the category of sheaves of abelian groups
on X, called abelian sheaves. Let F Ab(X) be given. For a section s of F in
a neighborhood U of a point x X, the germ of s at x is denoted by sx, and the
support Supp(s) of s is the set of all y U with sy = 0.
Definition 1.1. ΓY (X, F ) := {s F (X) | Supp(s) Y } is called the group
of sections of F with support in Y .
Lemma 1.2. ΓY (X, −) is a left-exact functor.
Proof. Clearly ΓY (X, −) is a functor. Let an exact sequence of abelian
sheaves
0 −→ F −→ G −→ H −→ 0
be given. Then 0 Fx Gx Hx 0 is an exact sequence of abelian groups
for all x X. Likewise the sequence 0 F (X) G (X) H (X) is exact. Since
obviously ΓY (X, F ) = ker(ΓY (X, G ) ΓY (X, H )), the exactness of
0 −→ ΓY (X, F ) −→ ΓY (X, G ) −→ ΓY (X, H )
follows immediately.
Since Ab(X) has sufficiently many injective objects the right-derived functors
of ΓY (X, −) are defined.
Definition 1.3. Let HY
i
(X, −) be the i-th right-derived functor of ΓY (X, −).
For an abelian sheaf F on X we call HY
i
(X, F ) the i-th cohomology on X with
values in F and support in Y .
Important special cases are:
a) Y = X: Then we simply write
Hi(X,
F ) and call this the i-th cohomology
of X with values in F .
b) Y = {x} with a closed point x X: Then we write Hx(X,
i
F ) and call
this the local cohomology of X at x with values in F .
If X carries additional structure, for example is a ringed space or a scheme, and
if F is an OX-module, the cohomology is always understood in the above sense as
a derived functor of abelian sheaves.
Definition 1.4. An abelian sheaf F is called flasque if for all open sets V U
the restriction morphism F (U) F (V ) is surjective. A module M over a ring R
is called flasque if its associated sheaf M on Spec R is flasque.
Examples 1.5.
a) A constant sheaf is flasque on an irreducible space.
http://dx.doi.org/10.1090/ulect/047/01
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