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- Tags: - #AG/basics - Refs: - #todo/add-references - Links: - scheme - algebraic space
separated
Any morphism between affine schemes is separated. 
Idea: generalizes being Hausdorff. Clasically, \(X\in {\mathsf{Top}}\) is Hausdorff iff \(\Delta_X \subseteq X{ {}^{ \scriptscriptstyle\times^{2} } }\) is a closed subspace.
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A morphism of schemes \(f: X\to Y\) is separated if the relative diagonal \(\Delta_{X/Y}: X\to X{ {}^{ \scriptscriptstyle \underset{\scriptscriptstyle {Y} }{\times} ^{2} } }\) is a closed immersion.
- Note that the diagonal is always an immersion, so the content is that the morphism is closed.
- A relative scheme $X\in{\mathsf{Sch}}_{/ {S}} $ is separated iff \(\Delta_X \hookrightarrow X{ {}^{ \scriptscriptstyle \underset{\scriptscriptstyle {S} }{\times} ^{2} } }\) is a closed immersion.
- A general scheme \(Y\in {\mathsf{Sch}}\) is separated if the structure morphism \(f: Y\to \operatorname{Spec}{\mathbf{Z}}\) is separated, so \(\Delta_{Y/ \operatorname{Spec}{\mathbf{Z}}}: Y\to Y{ {}^{ \scriptscriptstyle \underset{\scriptscriptstyle {\operatorname{Spec}{\mathbf{Z}}} }{\times} ^{2} } }\) is a closed immersion.
quasiseparated
Idea: intersections of two affines is quasicompact, so a finite union of affines.
- A morphism of schemes \(f: X\to Y\) is quasiseparated if the diagonal morphism \(\Delta_Y: X\to X \underset{\scriptscriptstyle {Y} }{\times} X\) is quasicompact.
- A scheme \(X\in {\mathsf{Sch}}\) is quasiseparated if the structure morphism \(f: X\to \operatorname{Spec}{\mathbf{Z}}\) is quasiseparated.
- An object \(X\) in a topos is quasiseparated iff or every pair of morphisms \(U \rightarrow X \leftarrow V\), where \(U\) and \(V\) are quasi-compact, the fiber product \(U \times_{X} V\) is also quasi-compact.
Note that separated implies quasiseparated.
