Smooth morphism

Smooth morphism
Name	Smooth morphism
Field	Algebraic geometry
Introduced	1960s
Related	Étale morphism, Flat morphism, Unramified morphism, Regular morphism

Smooth morphism

A smooth morphism is a morphism of schemes that generalizes the notion of a submersion in differential topology and a non-singular map in algebraic geometry, used extensively in the work of Grothendieck, Serre, and Artin. It combines properties studied by Grothendieck in the context of the SGA seminars with techniques from Zariski, Hironaka, and Deligne to control local structure, deformation theory, and cohomological behavior. Smooth morphisms are central to modern treatments by Hartshorne, Mumford, and EGA authors in moduli problems, descent theory, and intersection theory.

Definition

A morphism f: X → S of schemes is smooth if it is locally of finite presentation, flat, and has geometrically regular fibers. This definition refines notions introduced by Grothendieck in the Éléments de Géométrie Algébrique and used by Serre and Tate for arithmetic questions. In practice, the condition can be checked using local criteria due to Nakayama, Zariski, and EGA techniques, and is compatible with base change in the sense studied by Chevalley and Artin.

Basic Properties

Smooth morphisms satisfy many stability and functoriality properties developed by Grothendieck, EGA authors, and later exposited by Hartshorne and Mumford. They are stable under composition and base change as in the formalism of Serre duality and Verdier duality, and they are open maps in the topological sense used by Zariski and Chevalley. Fibers of smooth morphisms are regular schemes studied by Hironaka in resolution of singularities and by Deligne in the context of ℓ-adic cohomology. The sheaf of relative differentials Ω_{X/S} for a smooth morphism is locally free, a fact used by Kodaira, Spencer, and Griffiths in deformation theory and moduli of complex structures. Smooth morphisms admit étale local sections in contexts treated by Grothendieck, Artin, and Deligne–Mumford, linking to the theory of stacks and the work of Behrend.

Examples

Basic examples include projection maps A^n_S → S studied by Eisenbud, Fulton, and Fulton–MacPherson, and open immersions that appear in classical work of Noether and Weil. Smooth projective morphisms arise in the theory of abelian varieties investigated by Grothendieck and Mumford, and in families of curves studied by Deligne, Mumford, and Igusa. Étale morphisms of schemes, as in Serre and Grothendieck’s treatments, are smooth of relative dimension zero, while certain morphisms with Jacobian nonvanishing—examples considered by Newton, Gauss, and later by Tate—are smooth. Local complete intersections that are regular maps studied by Zariski and Samuel give further examples, and smooth group schemes include reductive groups treated by Chevalley and Borel.

Relation to Other Morphisms

Smooth morphisms sit in a hierarchy with several morphism types elaborated by Grothendieck and EGA: étale morphisms (Grothendieck, Serre) are smooth of relative dimension zero; unramified morphisms (Zariski, Nagata) are characterized by vanishing of Ω_{X/S}; flat morphisms (Matsumura, Serre) are a weaker condition present in the smooth definition; and regular morphisms appear in Grothendieck’s work on alteration by de Jong. The notion interacts with proper morphisms in the context of Kodaira’s vanishing, with projective morphisms in the work of Grothendieck and Mumford, and with Cohen–Macaulay maps in contributions by Auslander and Buchsbaum.

Local Structure and Jacobian Criterion

Locally on the source and target one can often present a smooth morphism as a standard smooth map given by polynomial equations with a Jacobian matrix of maximal rank, a technique rooted in the work of Kähler, Weierstrass, and Zariski. The Jacobian criterion, formulated in algebraic terms by Serre and Kunz, gives an effective test for smoothness when source and target are schemes of finite type over a field as in the treatments by Eisenbud and Harris. Formal smoothness in the sense of Grothendieck and Schlessinger captures infinitesimal lifting properties used by Artin in deformation theory and by Illusie in cotangent complex theory. These local descriptions connect to Newton’s method historically and to modern stratifications studied by Thom and Mather.

Applications and Uses in Algebraic Geometry

Smooth morphisms underpin constructions in the theory of moduli spaces by Deligne–Mumford and Artin, in the formulation of smooth base change theorems by Grothendieck and Deligne, and in the study of étale cohomology used by Deligne in proving the Weil conjectures. They play a role in arithmetic geometry through Néron models (Néron, Raynaud), in the study of fibration methods by Faltings and Arakelov, and in Hodge theory as in the work of Griffiths and Schmid. Smoothness is essential in intersection theory developed by Fulton, in resolution techniques by Hironaka, and in stack-theoretic constructions by Laumon, Moret-Bailly, and Olsson. It also appears in applications to number theory via modular curves investigated by Shimura, Deligne, and Drinfeld, and in geometric representation theory through Beilinson–Bernstein and Lusztig.

Category:Algebraic geometry