COMPLEX GEOMETRY Course notes

(2) i ∗ xF = F x has finite dimension over C.
Definition 4.1.6. A sheaf of modules M on an algebraic variety (X, O X ) is said to be (quasi)-coherent
if it is locally isomorphic to the cokernel of a morphism of free sheaves (of finite rank).
Easy fact: f ∗ preserves (locally) free sheaves, rank and invertibility. In particular, f ∗ gives a homomorphism
of Picard groups.
Definition 4.1.7. A short sequence of sheaves
0 −→ F −→ G −→ H −→ 0
is said to be exact if and only if it is exact on the level of stalks.
Let (f, f ∗ ) : (X, A) −→ (Y, B) be a morphism of ringed spaces, i.e., f : X −→ Y is a continuous map, A and
B are shaves of rings. For a sheaf of A-modules F on X, the direct image sheaf f ∗ F of F on X is the sheaf
of B-modules on Y given by V ↦→ F(f −1 (V )). Recall that for a sheaf of B-modules G on Y , its pullback is
defined as follows: Set f (∗) G(U) = lim G(V ), and then set (f ∗ G) to be the sheaf associated with the
−→f(U)⊆V
presheaf f (∗) G ⊗ f (∗) B A.
Example 4.1.3. Let F be the sheaf of holomorphic sections of a vector bundle F and i : {x} −→ (X, O).
(1) i (∗) F =: F x is called the stalk of i, and it equals the set of germs of sections of F .
(2) i ∗ F = F x , the fibre of F at x, is a finite dimensional vector space.
Definition 4.1.8. Recall that a sheaf of modules on an algebraic variety (X, O X ) is said to be quasicoherent
(resp. coherent) if it is locally isomorphic to the cokernel of a morphism of free shaves (resp. of
finite rank). By a free sheaf we mean a sheaf O ⊕n , where n is a cardinal number.
X
Fact 4.1.1. f ∗ preserves local freeness invertibility, in particular f ∗ gives a homomorphism of Picard groups,
where
Pic(X) = group of invertible sheaves ∼ = holomorphic line bundles
A short sequence of shaves 0 −→ F −→ G −→ H −→ 0 is said to be exact if it is exact at the level of stalks.
Remark 4.1.3. Ker, CoKer, ⊗ and f ∗ preserve the property of being (quasi-)coherent. However, f ∗ does not.
Example 4.1.4.
(1) f : C −→ {0}. Then f ∗ O alg
C
(2) i : C ∗ −→ C. Then i ∗ O alg
C ∗
= C[z] which is not finite dimensional.
= C[z, z−1 ] is not of finite type over C[z].
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