Lemma 33.43.2. Let $X$ be a separated, 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. Since $X$ is irreducible, $U = X \setminus \{ x\}$ is not closed in $X$. In particular, the immersion $U \to X$ is not proper. By Morphisms, Lemma 29.41.7 (here we use $X$ is separated), $U \to \mathop{\mathrm{Spec}}(k)$ is not proper either. $\square$

Comment #7492 by WhatJiaranEatsTonight on

I think we need $X$ to be separated here. Otherwise \the projecitve line with double original point is a counterexample.

Comment #7514 by David Holmes on

Dear WhatJiaranEatsTonight,

I quite agree with your counterexample. Moreover, Lemma 26.22.1 refered to in the proof uses separatedness. Though happily this tag is only applied in once place, and in that place it is applied to a separated scheme (over a field).

Best wishes, David

Comment #7516 by Laurent Moret-Bailly on

Here is a more elementary (and, I think, more natural) proof: since $X$ is irreducible, $X\setminus\{x\}$ is not closed in $X$. In particular, the immersion $X\setminus \{ x \}\to X$ is not proper. By Lemma 29.41.7 (and because $X$ is separated), $X\setminus\{x\}\to \mathrm{Spec}(k)$ is not proper either.

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