Take the <projective line with a doubled point> over $\mathbb C$, obtained by gluing two copies of $\mathbb P_{\mathbb C}^1$ along the complement of one point. After every <base change of a morphism of schemes>, a closed subset has closed image from each of the two projective-line charts, so its total image, the union of those two images, is closed. The structure morphism is therefore <universally closed morphism>[universally closed]. The two doubled points have no disjoint neighborhoods, so the scheme is not <separated scheme>[separated] and hence is not <proper morphism>[proper].

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 113 / 1 / b / Solution

Glue two copies of $\mathbb P_k^1$ by the identity away from one chosen point. The resulting finite-type scheme is universally closed over $k$: after any base change, a closed subset meets each projective-line chart in a closed subset whose image is closed, and the total image is their finite union. The two copies of the chosen point cannot be separated, so the structure morphism is not separated and therefore is not proper.


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