Artin's criteria
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| Artin's criteria | |
|---|---|
| Name | Artin's criteria |
| Introduced | 1969 |
| Author | Michael Artin |
| Field | Algebraic geometry |
Artin's criteria are a set of conditions formulated by Michael Artin that characterize when a functor or a category fibered in groupoids is representable by an algebraic space or an algebraic stack; they form a cornerstone of modern moduli theory. The criteria give necessary and sufficient hypotheses—often phrased in terms of deformation theory, effectivity of formal objects, and openness of versality—bridging formal geometry and the existence of geometric objects. Artin's work integrates techniques from Grothendieckian algebraic geometry, deformation theory, and category theory to produce practical tests for representability used across moduli problems.
Background and statement of Artin's criteria
Artin formulated his representability criteria in the context of the development of algebraic geometry influenced by Alexander Grothendieck, Jean-Pierre Serre, Jean-Pierre Serre's students, and the seminars at Institut des Hautes Études Scientifiques and École Normale Supérieure, building on ideas from the work of Grothendieck on schemes and Séminaire de Géométrie Algébrique techniques. The formal statement asserts that a functor F (or category fibered in groupoids) on the category of schemes over a base scheme S which satisfies Schlessinger-type deformation conditions, openness of versality, and effectivity of formal objects is an algebraic space or algebraic stack. The hypotheses are motivated by examples such as the moduli of curves treated by Deligne and Mumford, the Hilbert scheme studied by Grothendieck, and the Picard functor explored by Grothendieck and Artin himself.
Artin’s original papers interacted with the work of David Mumford on geometric invariant theory, Pierre Deligne on étale cohomology and moduli of curves, and Jean-Pierre Serre on cohomological methods. Subsequent expositions connected Artin's criteria to the formal existence theorems of Alexander Grothendieck, the prorepresentability results of Michael Schlessinger, and the infinitesimal lifting conditions in the work of Grothendieck, Jean Giraud, and Jean-Louis Verdier.
Conditions and variants (effectivity, openness, infinitesimal lifting)
The core hypotheses include an effectivity condition asserting that formal objects arising from compatible systems over nilpotent thickenings are algebraizable, an openness of versality condition ensuring that versal deformations persist in families, and an infinitesimal lifting condition controlling the obstruction theory. These are closely related to Schlessinger’s conditions (H1)–(H4), the condition on tangent and obstruction spaces examined by Serre in the context of coherent cohomology, and the formal functions theorems associated with Grothendieck and Jean-Pierre Jouanolou.
Variants of Artin’s criteria adapt to different contexts: for algebraic spaces one requires conditions similar to those used by Grothendieck for the Hilbert functor and for algebraic stacks additional conditions about automorphism groups and diagonal representability are imposed, mirroring constraints seen in the works of Deligne, Mumford, and Giraud on gerbes and descent. Effectivity ties to the formal GAGA principles developed by Jean-Pierre Serre and later by Alexander Grothendieck, while openness of versality reflects geometric inputs found in the constructions of moduli stacks by David Mumford, Pierre Deligne, and Gerd Faltings.
Applications to algebraic stacks and moduli problems
Artin’s criteria have been applied to prove representability results for moduli of curves, vector bundles, principal bundles, coherent sheaves, and complex structures, connecting to landmark contributions by Deligne, Mumford, Grothendieck, Drinfeld, Laumon, and Moret-Bailly. Specific moduli problems addressed include the Deligne–Mumford stack of stable curves, the stack of principal G-bundles studied by Alexander Grothendieck and later by Drinfeld and Laumon, and the moduli of polarized varieties analyzed in works by Gerd Faltings and Faltings–Chai. Artin’s methods also underpin representability of the Picard functor, the Hilbert scheme, and Quot schemes introduced by Grothendieck and further developed by Mumford.
In arithmetic geometry and number theory, applications touch on deformation rings in Mazur’s deformation theory for Galois representations, moduli of abelian varieties in the work of André Weil and Jean-Pierre Serre, and stacks arising in the Langlands program investigated by Robert Langlands and Edward Frenkel. In complex algebraic geometry the criteria influence the study of Kuranishi spaces following Masatake Kuranishi and Kodaira–Spencer theory.
Key examples and counterexamples
Key successful applications include the Hilbert scheme of subschemes of a projective scheme (Grothendieck), the algebraicity of the Picard scheme (Grothendieck, Artin), and the algebraicity of moduli stacks of curves (Deligne–Mumford) and of polarized varieties under suitable conditions (Kollár, Viehweg). Counterexamples illuminate necessity of hypotheses: functors failing effectivity or openness yield non-algebraic functors, paralleling pathologies studied by Nagata, Zariski, and Oort in examples related to non-separatedness and failure of formal glueing. Instances where automorphism groups are too large or diagonals fail to be representable occur in gerbes analyzed by Giraud and examples of non-algebraic stacks appear in literature related to Mumford’s geometric invariant theory limits.
Proof techniques and outline
Proofs synthesize deformation-theoretic arguments, Schlessinger-style prorepresentability, coherence results for deformation and obstruction spaces rooted in cohomology theories of Grothendieck and Serre, and approximation theorems such as Artin approximation and Popescu’s theorem. Central steps include establishing formal versality, proving effectivity via algebraization theorems reminiscent of formal GAGA, constructing smooth neighborhoods using Artin’s approximation, and verifying diagonal representability through representability theorems of Grothendieck and results about automorphism functors. Techniques draw on work by Michael Artin, Alexander Grothendieck, David Mumford, Jean-Pierre Serre, Michel Demazure, and later refinements by Brian Conrad, Johan de Jong, and Johan F. de Jong.
Related results and generalizations
Generalizations and refinements include Artin’s algebraicity criteria for stacks with nontrivial stabilizers, refinements by Conrad–de Jong on approximation, Olsson’s treatments of algebraic stacks, Lieblich’s work on moduli of complexes, and Bayer–Macrí–Toda style developments in derived algebraic geometry following Jacob Lurie, Bertrand Toën, Gabriele Vezzosi, and Dominic Joyce. Connections extend to formal methods in derived deformation theory by Dennis Gaitsgory, Tony Pantev, and Max Kontsevich, and to applications in geometric representation theory by Beilinson, Bernstein, and Drinfeld. The landscape encompasses the interaction with moduli constructions by Kollár, Alexei Bondal, and Dmitri Orlov, and with recent advances in perfectoid methods by Peter Scholze.