---- > [!definition] Definition. ([[separated scheme morphism]]) > Let $f:X\to Y$ be a [[morphism of locally ringed spaces|morphism of]] [[scheme|schemes]]. Let $\Delta: X \to X \times_{Y} X$ be the **diagonal morphism** arising via the [[categorical pullback|fiber product]] [[universal property]]: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAmAXVJADcBDAGwFcYkQANEAX1PU1z5CKcqWLU6TVuy69+2PASKiqNBizaIQATR58QGBUKJlxaqZs4ACADo28AW3gB9bVdn7DgpSgAMpXwl1aS1ZCRgoAHN4IlAAMwAnCAckfxAcCCRRSQ12OJAaRnoAIxhGAAUBRWEQBKxIgAscPXiklMQydMzEbODLfLkQROSkAGYaDKRiQeH28a7UmbakABYJ7s7SsCgxtL72YmcPVpHENYXENKLSiqrjLTrG5potncQAWlG9iwOjlqHlmd1mNzLktHYYAAPLBwHBwKwAQlsNgAImUcPQCiBGFgwJYoPQ4A0IjxKNwgA > \begin{tikzcd} > X \arrow[rrd, "1_X", bend left] \arrow[rdd, "1_X"', bend right] \arrow[rd, "\exists ! \Delta", dashed] & & \\ > & X \times_Y X \arrow[r] \arrow[d] & X \arrow[d, "f"] \\ > & X \arrow[r, "f"'] & Y > \end{tikzcd} > \end{document} > ``` > We say $f$ is **separated** if $\Delta$ is a [[subscheme|closed immersion]]. > [!equivalence] > In practice, one uses the following criterion to check separatedness: ![[valuative criterion for separatedness#^theorem]] ^equivalence > [!intuition] > Separatedness is the algebro-geometric analogue of [[Hausdorff space|Hausdorffness]]. The latter notion is not useful here because the [[Zariski topology on a ring spectrum|Zariski topology]] is generally not Hausdorff. This definition plays on the "[[Hausdorff iff diagonal is closed]]" characterization of [[Hausdorff space|Hausdorffness]]. ^intuition ---- #### ---- #### 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 > ```