The Stacks project

Lemma 32.42.2. Let $X$ be an irreducible scheme of dimension $> 0$ over a field $k$. Let $x \in X$ be a closed point. The open subscheme $X \setminus \{ x\} $ is not proper over $k$.

Proof. Namely, choose a specialization $x' \leadsto x$ with $x' \not= x$ (for example take $x'$ to be the generic point). By Schemes, Lemma 25.20.4 there exists a morphism $a : \mathop{\mathrm{Spec}}(A) \to X$ where $A$ is a valuation ring with fraction field $K$ such that the generic point of $\mathop{\mathrm{Spec}}(A)$ maps to $x'$ and the closed point of $\mathop{\mathrm{Spec}}(A)$ maps to $x$. The morphism $\mathop{\mathrm{Spec}}(K) \to X \setminus \{ x\} $ does not extend to a morphism $b : \mathop{\mathrm{Spec}}(A) \to X \setminus \{ x\} $ since by the uniqueness in Schemes, Lemma 25.22.1 we would have $a = b$ as morphisms into $X$ which is absurd. Hence the valuative criterion (Schemes, Proposition 25.20.6) shows that $X \to \mathop{\mathrm{Spec}}(k)$ is not universally closed, hence not proper. $\square$


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