Open Subschemes of a Proper Scheme are not Proper

In this post I will record a cute argument showing that no open subscheme of a connected proper scheme is proper. In particular, this shows that no open subscheme of projective space is projective. Intuitively, if we think that proper just means compact then this just says that an open subset $U$ of a connected, compact set $X$ cannot be compact. We could prove this in topology by taking the inclusion $i: U \to X$ and noting that $i(U)$ has to be compact and therefore closed in $X$, which contradicts that $X$ is connected.

Let’s do the same thing for schemes $U, X$ over $Z$. We have that $X \to Z$ is proper and therefore separated and $U \to Z$ is proper by assumption. Therefore by the cancellation property, $i: U \to X$ is proper and so $i(U)$ is a clopen subscheme of $X$, which contradicts connectedness.

2 thoughts on “Open Subschemes of a Proper Scheme are not Proper”

1. Daniel Haddil says:

Since X itself is a proper open subscheme of itself your title should be: Proper Open Subschemes of a Proper Scheme are not Proper 🙂

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1. phacopsrana says:

Ha! “Proper Open Subschemes of a Proper Scheme Turn Out Not to Be”

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