Separated morphism

Separated morphism
Name	Separated morphism
Field	Algebraic geometry

Separated morphism.

A separated morphism is a structural property of morphisms in algebraic geometry that imposes an analogue of the Hausdorff separation axiom from topology on schemes, morphisms of schemes, and related geometric objects. It plays a central role in the foundations developed by Grothendieck and appears in the formulation of properness, representability, and moduli problems in the work of many mathematicians and institutions.

Definition

A morphism f: X → Y of schemes is defined to be separated when the diagonal morphism Δ_f: X → X ×_Y X is a closed immersion. This condition originates in the foundations set by Alexander Grothendieck and is essential in the theory described in the Éléments de géométrie algébrique and related expositions by Jean-Pierre Serre and Pierre Deligne. The diagonal criterion relates X, the fibre product X ×_Y X, and the projection maps, and it can be checked locally using affine charts associated to rings studied in commutative algebra by Emmy Noether and David Hilbert.

Basic Properties

Separatedness is stable under base change, composition, and local on the target: if f: X → Y and g: Y → Z are separated then g∘f is separated; if f is separated and Z' → Z is any morphism then the base change X×_Z Z' → Z' is separated. Affine morphisms between schemes such as those arising from Hom and Spec constructions by Grothendieck are separated, and projective morphisms introduced by Guido Castelnuovo and Oscar Zariski are separated as well. Separatedness interacts with representability criteria used in the work of Matsumura, Mumford, and Deligne–Mumford: for many moduli functors studied by David Mumford and Pierre Deligne, representability requires separatedness conditions.

Examples and Non-examples

Classic examples of separated morphisms include affine morphisms like Spec of a ring to Spec of a subring studied in algebraic number theory by Richard Dedekind, and projective morphisms such as those appearing in the theory of algebraic curves by Bernhard Riemann and algebraic surfaces by Federigo Enriques. Smooth proper curves considered by Alexander Grothendieck and André Weil provide separated and proper morphisms. Non-examples arise in constructions akin to gluing two affine lines along a punctured line similar to pathologies investigated by David Mumford, where the resulting morphism to Spec of a base ring may fail to be separated; Nagata provided counterexamples in his compactification work, and constructions related to moduli stacks by Michael Artin can exhibit failures of separatedness when automorphism groups are large.

Relations to Other Morphism Classes

Separated morphisms sit between affine/projective and proper morphisms in the hierarchy used in scheme theory: affine and projective morphisms are separated, while proper morphisms (as formalized by Grothendieck and refined by Nagata and Deligne) are separated plus universally closed and of finite type. Étale morphisms studied by Jean-Pierre Serre and Alexander Grothendieck are separated in many contexts; flat morphisms considered by Jean-Louis Koszul and Jean-Pierre Serre may or may not be separated, depending on the diagonal. The relationship with representable morphisms and algebraic stacks in the work of Deligne, Mumford, and Artin involves separatedness conditions that mirror separation axioms used in the theory of algebraic spaces by Michael Artin and David Rydh.

Criteria and Tests for Separatedness

One can test separatedness by checking that the diagonal Δ_f is a closed immersion, a condition reducible to checking ring maps on affine charts as in EGA by Grothendieck, or via valuative criteria reminiscent of the valuative criterion for properness formulated by Zariski and later used by Nagata and Temkin. For morphisms of finite type over Noetherian bases studied by Oscar Zariski and Masayoshi Nagata, separatedness can often be checked by verifying the equality of certain subrings in coordinate rings, a technique used in commutative algebra by Krull, Cohen, and Auslander. For algebraic spaces and stacks, separatedness is phrased in terms of representable diagonals, an approach central to the work of Deligne, Mumford, and Artin.

Applications and Consequences

Separatedness is a hypothesis in many foundational theorems and constructions: in the theory of moduli spaces developed by David Mumford and Pierre Deligne, separatedness enters criteria for coarse moduli schemes and for uniqueness of limits. In intersection theory pioneered by William Fulton and Serge Lang, separatedness ensures the well-behavedness of intersections and the finiteness properties needed in the Riemann–Roch theorems studied by Friedrich Hirzebruch and Alexander Grothendieck. In arithmetic geometry and Diophantine studies influenced by André Weil, Jean-Pierre Serre, and Gerd Faltings, separatedness underlies properness and compactification results crucial to proofs of finiteness theorems. Further developments by Michael Artin, Brian Conrad, and Jacob Lurie extend separatedness conditions to algebraic stacks, rigid analytic spaces examined by John Tate, and derived algebraic geometry influenced by Toën and Vezzosi, ensuring the analogues of classical separation properties persist in broader categorical settings.

Category:Algebraic geometry