What exactly does it mean for a scheme to be proper? I can't seem to find an actual definition of this anyway despite the term being frequently used.
2026-04-07 11:32:16.1775561536
What does it mean for a scheme to be proper?
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A scheme $X$ over a ring $A$ is said to be proper if its structural morphism $f:X\rightarrow\mathrm{Spec}(A)$ is a proper morphism of schemes. I think our subject is just that way that one just states it for the relative case (i.e. for morphisms), and omits the definitions in the "absolute" case (where it is just the relative case applied to the structural morphism).
In the most general situation one says that a scheme $X$ over a base scheme $S$ is proper if the structural morphism $X\rightarrow S$ is proper. Actually at this generality it becomes clear (if it wasn't clear before) that you should think of "$X$ over $S$" as the same thing as a morphism $X\rightarrow S$. It is just a different use of language.