Section 2.7 - Projective Morphisms

Let A be a fixed ring, and consider the projective space PA = ProjA[x0, . . . , xn] over A. On PA we have the invertible sheaf O(1), and the homogenous coordinates x0, . . . , xn give rise to global sections ẋ0, . . . , ẋn ∈ Γ(PA,O(1)). Throughout this section we drop the dot in this notation and just write xi for the global section of O(1). One sees easily that the sheaf O(1) is generated by the global sections x0, . . . , xn, i.e., the images of these sections generate the stalk O(1)P of the sheaf O(1) as a module over the local ring OX,P for each point P ∈ PA. Definition 1. Let f : X −→ Y be a morphism of schemes and F be a OY -module. Let η : F −→ f∗f ∗F be canonical. Given s ∈ F (V ) we denote by f∗(s) the section ηV (s) ∈ f∗F (f−1V ). So f∗(s) = [V, s] ⊗̇ 1 and for x ∈ f−1V we have f∗(s)(x) = (f−1V, ̇ (V, s) ⊗ 1). If φ : F −→ G is a morphism of OY -modules then (fφ)f−1V (f∗(s)) = f(φV (s)). Lemma 1. Let f : X −→ Y be a morphism of schemes and let F be a OY -module generated by global sections x1, . . . , xn ∈ F (Y ). Then the global sections si = f(xi) generate f∗F . Proof. Our notes on the isomorphism (f∗F )x ∼= Ff(x) ⊗OY,f(x) OX,x show that there is a commutative diagram of abelian groups for x ∈ X: