##  [Nonseparated Scheme](/nonseparated-scheme-0) 

 Definition

A scheme that fails the separation axiom: the diagonal morphism Δ: X → X×_S X is not a closed immersion (or equivalently not proper/closed), so points or morphisms that would be uniquely determined in a separated setting may fail uniqueness. Nonseparated schemes exhibit gluing pathologies such as multiple origins or failure of uniqueness of limits.

 

 

 

 

 

 





## Principle

Principle

Separation formalizes an algebraic analogue of Hausdorff-ness: the diagonal being a closed immersion ensures that two S-morphisms agreeing on a dense open coincide, and ensures uniqueness properties for limits and valuative criteria. Failure of this property reflects overly generous gluing of local pieces.

 

 

 

 

 





## Demonstration

Demonstration

Classic example: the affine line with doubled origin obtained by gluing two copies of A^1 along A^1 \\{0} gives a nonseparated scheme: the two origin points cannot be separated by disjoint open neighbourhoods and two sections may agree away from the origin but differ at it. This produces explicit failures of uniqueness for morphisms from valuation rings.

 

 

 

 

## Misapplication

Misapplication

Confusing nonseparated with nonproper or non-Hausdorff in the analytic sense; nonseparated is a precise algebraic condition about the diagonal and does not imply or preclude other pathologies like nonreducedness. Also, treating constructions that implicitly use separation (Picard functors, moduli) without checking can lead to incorrect conclusions.

 

 

 

 

 





## Consequence

Consequence

Nonseparated schemes break valuative criteria uniqueness, complicate moduli interpretations, produce nonunique limits, and can make representability of functors (e.g., Picard, Hom) fail. They force extra care in descent and in forming quotients or gluing constructions.

 

 

 

 

## Reversal

Reversal

A separated scheme has a closed diagonal: points and morphisms behave with the expected uniqueness properties, limits are unique when they exist, and many representability and descent statements simplify and hold under standard hypotheses.

 

 

 

 

 





## Boundary

Boundary

Separation is a condition on morphisms of schemes relative to a base S; nonseparatedness is usually pathological in the context of moduli and geometric constructions but can appear naturally in intermediate gluing steps. Algebraic spaces and stacks have their own separation notions; nonseparatedness in schemes should be distinguished from topological non-Hausdorff phenomena in underlying topologies.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension arises between geometric intuition of 'separation' (Hausdorff-like) and the categorical formulation (diagonal as closed immersion). Users sometimes misread topological non-Hausdorff examples as algebraic nonseparatedness or vice versa; the scheme-theoretic diagonal criterion is the authoritative check.

 

 

 

 

 





## Synthesis

Synthesis

A Nonseparated Scheme is one where the algebraic diagonal fails to be a closed immersion, reflecting an overgluing of local pieces that destroys uniqueness of morphisms and limits. The condition has concrete realizations (e.g., doubled origin) and significant consequences for moduli, descent and representability.