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valuative criterion of properness
A criteria to check if a morphism of schemes is a proper morphism.
Idea: for \(R\in \mathsf{DVR}\) with \(K = \operatorname{ff}(R)\), require 1-dimensional limits to exist.
- \(\operatorname{Spec}R\to Y\) is like a disc \({\mathbb{D}}\subseteq Y\).
- \(\operatorname{Spec}K \to Y\) is like a punctured disc \({\mathbb{D}}^\circ \subseteq Y\)
- There should be one way to lift a disc \({\mathbb{D}}\to X\) to \({\mathbb{D}}^\circ \to Y\) and extend functions over the puncture.
Concretely,
- \(R = \operatorname{Spec}{\mathbf{C}}{\left[\left[ t \right]\right] }\) is a formal open disc (expansions of analytic functions at \(z=0\) in \({\mathbf{C}}\)) and a DVR.
- Inverting \(t\) yields \(K = {\mathbf{C}}{\left(\left( t \right)\right) } = \operatorname{ff}(R)\) (formal Laurent series, expansions of meromorphic functions with poles only \(z=0\).)
