## Equivalent conditions for a closed immersion of schemes

In Hartshorne, a closed immersion of schemes is defined to be a scheme morphism Φ:Y→X such that Φ is a homeomorphism onto Φ(Y), Φ(Y) is closed in |X| and (∗)Φ#:OX→Φ∗OY is surjective. Can I replace (∗) by the condition that Φ#U:OX(U):→OY(Φ−1(U)) is surjective for all affine open subsets U⊆X, or does this lead to another definition? … Read more