Proper morphism

Proper morphism
Name	Proper morphism
Field	Algebraic geometry

Proper morphism A proper morphism is a central notion in algebraic geometry, characterizing maps of schemes or varieties with finiteness and completeness properties analogous to compact continuous maps in topology. It plays a foundational role in the work of Grothendieck, Serre, Deligne and others, underpinning theorems of Zariski, Nagata, Hironaka, Artin, and many developments in modern arithmetic geometry and algebraic number theory.

Definition

In the language of schemes due to Grothendieck, a morphism f: X → Y is proper if it is separated, of finite type, and universally closed. This definition refines earlier formulations used by Zariski and Chevalley and interacts with concepts introduced by Noether, Krull, Dedekind, Weil, and Riemann. Properness generalizes the notion of projective morphisms studied in works of Castelnuovo, Enriques, Kodaira, and Iitaka, and is distinguished from separatedness arising in the theories of Serre and EGA developed by Grothendieck and Dieudonné.

Basic properties

Properness is stable under base change and composition, properties proved in EGA and exploited by Grothendieck, Deligne, and Mumford. For schemes over a field, properness implies completeness analogous to proper maps in the contexts of Hilbert by Hilbert and Chow by Chow; when combined with flatness studied by Hilbert, it yields cohomological finiteness results in the style of Serre duality and Grothendieck duality. Proper morphisms satisfy the valuative criterion, a formulation appearing in work of Nagata, valuative tests used by Zariski and Abhyankar, and are preserved under formation of fiber products as in constructions appearing in the stacks literature of Laumon and Moret-Bailly.

Examples

Projective morphisms such as those arising from Proj constructions associated to graded rings in the works of Grothendieck, Mumford, and Hartshorne provide primary examples; classical projective varieties studied by Euler, Riemann, and Plücker fit this paradigm. Finite morphisms, including normalization maps in Dedekind domain theory and integral closures used by Krull and Noether, are proper. Complete curves of genus g studied by Riemann, Abel, Jacobi, and Weierstrass give proper schemes over fields; arithmetic models like minimal models in the sense of Kodaira, Tate, and Néron illustrate properness in arithmetic geometry. Compactifications such as those of Deligne and Mumford for moduli spaces, Zariski compactifications following Nagata, and Baily–Borel compactifications in Shimura variety theory are key proper maps. Counterexamples include affine line maps which are not proper, highlighted in Serre’s foundational examples.

Stability and functoriality

Properness is functorial under composition and stable under base change, results central to the formalism of Grothendieck’s six operations developed by Verdier, Deligne, and Kashiwara. Proper pushforward of coherent sheaves and higher direct images in theorems of Grothendieck and Hartshorne rely on properness; these interplay with duality theories by Serre, Grothendieck, and Verdier and with vanishing theorems due to Kodaira and Kawamata. Descent properties studied by faithfully flat descent in Grothendieck’s work, and stack-theoretic generalizations in Artin’s and Laumon–Moret-Bailly’s frameworks, extend properness to algebraic spaces and Deligne–Mumford stacks used by Kontsevich and Behrend.

Relation to compactness and valuative criteria

Proper morphisms mirror compact maps in topology as in Tychonoff’s theorem, Alexandrov’s work on compactness, and the compactifications constructed by Stone and Čech. The valuative criterion for properness, formulated by Zariski and refined by Nagata and Grothendieck, gives a test using discrete valuation rings as in the work of Dedekind, Krull, and Huber; it is instrumental in proofs by Deligne, Mumford, and Abramovich. Properness implies that images of closed sets remain closed under specialization, a property used in proofs by Chevalley and in the constructibility results of Grothendieck and Chevalley.

Applications in algebraic geometry

Proper morphisms underpin moduli theory as in the Deligne–Mumford compactification of moduli of curves, the construction of Hilbert and Quot schemes by Grothendieck, and the use of stable maps in Gromov–Witten theory developed by Kontsevich and Ruan. They are essential in arithmetic geometry, appearing in Néron models studied by Néron and Tate, in the study of Diophantine geometry by Faltings and Vojta, and in the formulation of L-functions and étale cohomology used by Grothendieck, Deligne, and Serre. Proper base change theorems by Grothendieck and Verdier enable calculations in étale cohomology used by Grothendieck, Deligne, and Milne; the finiteness of cohomology, duality theorems, and intersection theory by Fulton, Thomason, and Gillet rely on proper morphisms. Compactifications in Shimura variety theory by Baily–Borel and toroidal methods of Ash, Mumford, Rapoport and Tai, and the use in birational geometry by Mori, Kawamata, and Kollár further demonstrate the pervasive role of proper morphisms across schemes, stacks, and complex analytic spaces.

Category:Algebraic geometry