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