Algebraic space

Algebraic space
Name	Algebraic space
Field	Mathematics
Introduced	1960s

Algebraic space is a generalization of scheme introduced to address limitations of Grothendieck's theory of Grothendieck-style étale-local behavior and to provide a flexible framework for moduli problems encountered in Mumford's work on moduli of curves and in attempts to construct quotients by group actions such as those studied by Mumford and Geometric Invariant Theory. It refines notions present in treatments by Grothendieck in EGA and SGA and serves as an intermediate between schemes and algebraic stacks used in modern Deligne-style moduli theories and in constructions influenced by Artin and Faltings. Algebraic spaces play a central role in the work of Artin, Raynaud, and Mumford on deformation theory and representability.

Definition and basic properties

An algebraic space is defined using sheaves on the étale site of a base scheme such as Spec Z or Spec C, often appearing as a sheaf F with the property that there exists a scheme U and a representable, surjective, étale morphism U → F; this perspective is elaborated in sources by Artin, Grothendieck in SGA 1, and in treatments by Hartshorne and Knutson. Basic properties parallel those of schemes: notions of separatedness, properness, and finiteness conditions are defined via diagonal morphisms and valuative criteria as in work of Deligne and Matsumura. Algebraic spaces admit fiber products, and representability criteria by Artin's criteria relate deformation functors appearing in Grothendieck's FGA programs to algebraic spaces.

Examples and constructions

Standard examples include quotients of schemes by group actions studied in GIT by Mumford, such as coarse quotients under actions of GL_n or of finite groups discussed in treatments by Mumford and Newstead. The construction of the Hilbert scheme and the Picard scheme in the literature of Grothendieck and Mumford often proceeds through algebraic spaces; moduli spaces like the Deligne–Mumford moduli space and variants built by Deligne and Mumford are typical. Examples also arise from gluing affine charts as in Zariski-local constructions used by Nagata and in the treatment of Néron models in the work of Bosch, Lütkebohmert, and Raynaud. Nonseparated algebraic spaces appear in examples due to Artin and in pathological constructions influenced by counterexamples from Mumford.

Morphisms and local structure

Morphisms between algebraic spaces are defined via morphisms of sheaves on the étale site; properties such as smoothness, étaleness, and flatness are tested after pullback along étale presentations U → X where U is a scheme, in a manner analogous to criteria developed by Grothendieck in EGA and by Matsumura. Local structure theorems, including étale-local descriptions and results on the existence of sections or local charts, are established in expositions by Knutson and Laumon & Moret-Bailly; these build on deformation-theoretic techniques of Schlessinger and representability results of Artin. The diagonal morphism and its properties link to representability by Schemes and control separatedness as in work by Deligne.

Relationship to schemes and stacks

Algebraic spaces strictly generalize schemes—every scheme gives an algebraic space via its sheaf of sections—while algebraic stacks, especially Deligne–Mumford stacks introduced by Deligne and Mumford and more general Artin stacks studied by Artin, further generalize algebraic spaces by allowing nontrivial stabilizer group sheaves such as G_m and finite group schemes like μ_n. Comparison theorems linking coarse moduli spaces to algebraic spaces appear in work by Keel and Mori and in foundational treatments by Olsson and Laumon. The passage from algebraic spaces to stacks uses groupoid presentations and 2-categorical techniques developed by Deligne and Brylinski in various contexts.

Cohomology and sheaf theory

Sheaf cohomology on algebraic spaces uses the étale site and techniques from SGA 4 and SGA 1 pioneered by Grothendieck and worked out for algebraic spaces by Artin and Mumford. Cohomological tools such as Čech cohomology, derived functor cohomology, and spectral sequences appear in treatments by Hartshorne and Gelfand & Manin; duality theories parallel those for schemes and interact with the construction of Picard schemes and Brauer group computations as in work by Grothendieck and Milne. Étale cohomology on algebraic spaces plays a role in arithmetic applications linked to results of Weil, Deligne's proof, and later developments by Faltings and Tamagawa.

Applications and further developments

Algebraic spaces underpin modern constructions in moduli theory such as the moduli of vector bundles on curves analyzed by Narasimhan and Seshadri, compactification strategies by Deligne and Mumford, and the construction of coarse moduli spaces in works by Keel and Mori. They appear in arithmetic geometry contexts related to Néron models (see Néron), in intersection-theoretic developments stemming from Fulton and MacPherson, and in derived and homotopical refinements considered by Lurie and Toën & Vezzosi. Ongoing research connects algebraic spaces to advances by Kontsevich in enumerative geometry, to categorical approaches of Bondal and Orlov, and to arithmetic applications in work by Scholze and Bhatt.

Category:Algebraic geometry