 Closedness of projection from projective space

For every scheme $X$, projection $p:X\times {\mathbb{P}}^n\to X$ is a closed map, and remains so after every base change. Hence projective space is universally closed over its base. The assertion concerns the underlying scheme points, not just rational points over a non-algebraically closed field.^[ref]

The existence part of the valuative criterion explains the role of projective space. Given a valuation ring $V$ with fraction field $K$, a $K$-point $[a_0:\cdots :a_n]$ can be rescaled so that every coordinate belongs to $V$ and one coordinate is a unit. Indeed, among the finitely many nonzero coordinates choose $a_j$ whose principal fractional ideal contains all the others; then $a_i/a_j\in V$ and $a_j/a_j=1$. These coordinates define the required morphism of schemes $\mathrm{Spec}V\to {\mathbb{P}}^n$. Since projective space is also separated and of finite type, this proves that projection is a proper morphism.

On a Noetherian scheme, one can instead use finitely many homogeneous equations and the Projective Nullstellensatz to express the image as closed rank-defect conditions on graded multiplication map for a projective fiber. An arbitrary affine base need not give a finitely generated defining ideal, so that finite-equation hypothesis must not be assumed in the general statement.