---- > [!theorem] Theorem. ([[valuative criterion for separatedness]]) > Let $f:X \to Y$ be a [[morphism of locally ringed spaces|scheme morphism]] with $X$ [[locally Noetherian scheme|Noetherian]]. Then the following are equivalent: > 1. $f$ is [[separated scheme morphism|separated]]; > 2. For any [[field]] $k$ and any [[valuation ring]] $R$ of $k$, given a commutative [[diagram]][^1] > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12cYAPHYAMpoYAYwAEAXwDWICaXSZc+QijIBGKrUYs2nbn0HDxEsQCVZ8kBmx4CRNeU31mrRCAAaFhTeX3SG6mcdNwBNWU0YKABzeCJQADMAJwgAWyQAJmocCCQAZkDtVxB4r2LktMQyEGykBy0XXS5efiFRCQAKMU58HDpEMy72AAsICClErCih3sTkgHdB-X4AMUS6cVMJAF4pAEoQagY6ACMYBgAFRVsVEAmpnFKk1KQqmsR0uQTy2qycxFyJBQJEA > \begin{tikzcd} > \text{Spec }k \arrow[d, "\text{Spec}( \iota: R \hookrightarrow \text{Frac R}=k)"'] \arrow[r] & X \arrow[d, "f"] \\ > \text{Spec } R \arrow[r] & Y > \end{tikzcd} > \end{document} > ``` > > there exists *at most* one lift $\text{Spec }R \to X$ making the diagram commute: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12cYAPHYAMpoYAYwAEAXwDWICaXSZc+QijIBGKrUYs2nbn0HDxEsQCVZ8kBmx4CRNeU31mrRCAAaFhTeX3SG6mcdNwBNWU0YKABzeCJQADMAJwgAWyQAJmocCCQAZkDtVxB4r2LktMQyEGykBy0XXS5efiFRCQAKMU58HDpEMy72AAsICClErCih3sTkgHdB-X4AMUS6cVMJAF4pAEoQagY6ACMYBgAFRVsVEAmpnFKk1KQqmsR0uQTy2qycxFzPmVnog6m9MiAGFgwEUoHQ4ENIgd6sEOOxeFg4Dg4GIAISLZrAMQQRKSQaQHCDdGYhCHE5nS4+OxuKHYWDhCRAA > \begin{tikzcd} > \text{Spec }k \arrow[d, "\text{Spec}( \iota: R \hookrightarrow \text{Frac R}=k)"'] \arrow[r] & X \arrow[d, "f"] \\ > \text{Spec } R \arrow[r] \arrow[ru, "\exists ! \text{ or } \not \exists" description, dashed] & Y > \end{tikzcd} > \end{document} > ``` > ^theorem > [!proof]- Proof. ([[valuative criterion for separatedness]]) > - [ ] Omitted in our course. (See Hartshorne eventually.) ---- #### [^1]: Here, $\text{Spec }\iota$ is the morphism of [[affine scheme|affine schemes]] induced by the [[ring homomorphism]] $\iota: R \hookrightarrow \text{Frac }R=k$, cf. [[the category of affine schemes is dual to that of rings]], [[spec functor]]. ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```