A nice classical example of a projective scheme is by constructing projective morphisms which factor through rational scrolls. For example, take S = \mathbb{P}^1 and the vector bundle \mathcal{E} = \mathcal{O}_S \oplus \mathcal{O}_S \oplus \mathcal{O}_S(3). This can be used to construct a \mathbb{P}^2-bundle \mathbb{P}_S(\mathcal{E}) over S. If we want to construct a projective morphism using this sheaf we can take an exact sequence, such as
