Knowlpedia

Definition

Projective morphism

A scheme morphism that factors as a closed immersion into relative projective space.

Definition

A morphism of schemes f:X→Sf:X\to S is projective if there is an integer n≥0n\geq0 and a factorization

X↪PSn⟶SX\hookrightarrow\mathbb P^n_S\longrightarrow S

in which the first map is a closed immersion and the second is the canonical projection.

Projective morphisms are proper and of finite type. They are preserved by and composition. If the base is locally Noetherian, then a projective morphism is also of finite presentation.

Relative ample line bundles

Under standard quasi-compactness hypotheses, projectivity can equivalently be characterized by the existence of a relatively ample invertible sheaf. This formulation does not change the defining factorization above.

References
  1. The Stacks Project Authors, “Projective morphisms,” Tag 01W7.