Valuative criterion

Valuative criterion
Name	Valuative criterion
Field	Algebraic geometry
Introduced	1960s
Related	Proper morphism, Separated morphism, Discrete valuation ring

Valuative criterion is a family of results in algebraic geometry that characterize geometric properties of morphisms—such as properness, separatedness, and completeness—by testing maps from spectra of discrete valuation rings. The statements reduce global geometric conditions to local lifting and uniqueness problems for maps from Spec of a Discrete valuation ring relative to targets like Schemes, Algebraic spaces, or Stacks. These criteria play a central role in the work of Grothendieck, Zariski, Nagata, and others in the development of the EGA and the theory surrounding the Deligne–Mumford stack of moduli problems.

Definition and statement

The basic formulation involves a commutative diagram with the inclusion of the generic point and the closed point of the spectrum of a Discrete valuation ring (DVR). Given a morphism f: X → Y of Schemes, one asks whether every morphism from the generic point extends to the whole DVR and whether such an extension is unique. The existence part asserts that for every diagram with Spec of the fraction field mapping into X and the whole DVR mapping into Y there exists a lift to X after possibly replacing the DVR by a finite extension; the uniqueness part requires that two lifts that agree on the generic point must agree on the whole DVR. These statements are used to translate statements about morphisms into verifiable conditions using Discrete valuation rings, finite extensions, and properties of points like specializations.

Variants (existence and uniqueness)

There are several closely related variants: - Existence-only criteria: used to characterize properness or completeness for morphisms to projective targets, often requiring a finite extension of the DVR as in the Nagata-style arguments. - Uniqueness-only criteria: characterize separatedness by imposing that two lifts coinciding on the generic point coincide on the DVR without base change. - Existence and uniqueness together: give a criterion for proper morphisms, relating to notions used in the works of Grothendieck, Serre, and Deligne. - Stack-theoretic versions: for stacks one considers 2-commutative diagrams and allows inertia; the relevant uniqueness becomes uniqueness up to unique 2-isomorphism, appearing in the study of Deligne–Mumford and Artin stacks.

Applications in algebraic geometry

The criteria are foundational in many constructions: proving that a morphism is proper in the context of moduli problems, verifying separatedness for Hilbert and Picard functors, and establishing completeness of maps in compactification theorems related to Nagata, Hironaka resolution contexts, and Mumford's geometric invariant theory. They are instrumental in verifying that the projection and base-change theorems apply for proper maps in the cohomological frameworks developed by Grothendieck, Serre, Hartshorne, and Deligne. The DVR-testing approach also appears in the arithmetic geometry of Shimura and modular compactifications and in verifying extension properties in the work of Faltings and Fontaine.

Examples and counterexamples

Standard examples include: - Proper morphisms: the structure morphism of a projective scheme over a field satisfies existence and uniqueness; classical examples include Projective space and abelian varieties. - Separated but not proper morphisms: the inclusion of an affine open in a noncompact variety (e.g., Affine line into Projective line minus a point) satisfies uniqueness but not existence. - Non-separated morphisms: examples constructed by gluing two copies of Affine line along the punctured line fail the uniqueness condition. Counterexamples in stack-theory involve non-representable morphisms where lifts exist only up to nontrivial automorphism, illustrating the need to replace uniqueness by uniqueness up to 2-isomorphism in treatments of stacks.

Proofs and criteria for schemes and algebraic spaces

Proofs for schemes proceed by reduction to affine situations, using properties of finitely generated algebras over DVRs, valuative criteria for integrality and closure, and specialization arguments drawn from Zariski-local behavior. For algebraic spaces one uses étale covers by Schemes and descent to transfer existence and uniqueness through the cover; proofs exploit the compatibility of DVR maps with étale morphisms and the sheaf condition for the étale topology. For stacks the arguments involve 2-descent, stabilizer group actions, and rigidification techniques as in the work of Laumon and Moret-Bailly.

Relationship with properness, separatedness, and completeness

The valuative criteria give equivalences: for morphisms of Noetherian schemes, properness is equivalent to being separated, of finite type, and universally closed, and this can be tested by the valuative criterion combining existence and uniqueness. Separatedness alone corresponds to the uniqueness condition, while universal closedness and finite type aspects connect to existence after finite extension. Completeness notions—for example, completeness of moduli spaces in the sense of extension of families—are often checked via the existence clause and are central in compactification theorems used by Deligne, Mumford, and Kempf.

Category:Algebraic geometry