Proper morphism

Remarks

Let $k$ be a field. The structure morphism $\mathbb{P}^n_k \to \operatorname{Spec} k$ of projective $n$-space is proper. It is separated and of finite type, and it is universally closed by the main theorem of elimination theory, which can also be derived from the valuative criterion. The structure morphism $\mathbb{A}^1_k \to \operatorname{Spec} k$ is separated and of finite type but not proper. After base change along $\mathbb{A}^1_k \to \operatorname{Spec} k$ it becomes the projection $\mathbb{A}^2_k \to \mathbb{A}^1_k$, $(x, y) \mapsto x$. This projection maps the closed curve $xy = 1$ onto $\mathbb{A}^1_k \setminus \set{0}$, which is not closed, so clause 3 fails. Closed immersions and finite morphisms are proper, and proper morphisms are stable under composition and base change.