Category of schemes
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| Category of schemes | |
|---|---|
| Name | Category of schemes |
| Type | Category |
| Objects | Schemes |
| Morphisms | Morphisms of schemes |
| Introduced | 1960s |
| Founders | Alexander Grothendieck |
Category of schemes
The category of schemes is the mathematical category whose objects are schemes and whose morphisms are morphisms of schemes. It provides the foundational setting for modern algebraic geometry, linking constructions arising from Spec of rings to global objects such as projective schemes, and enabling comparisons with categories appearing in étale and derived contexts. The category is central in the work of Alexander Grothendieck, used throughout the theories developed by Jean-Pierre Serre, Pierre Deligne, John Tate, and Alexander Beilinson.
Definition and basic properties
Objects are locally ringed spaces that are covered by open affines isomorphic to Spec of a commutative ring. Basic properties include existence of fibered products, limits indexed by directed systems of rings, and the fact that morphisms reflect local algebraic structure as in morphisms induced by ring homomorphisms such as those between Noetherian rings, PIDs, and DVRs. The category admits subcategories such as affine schemes and projective schemes and interacts with cohomological theories like étale topology and Zariski topology. Foundational results involve representability criteria like those in Yoneda lemma and conditions from EGA and SGA.
Morphisms and functoriality
Morphisms correspond locally to ring homomorphisms between coordinate rings; famous classes include open immersions, closed immersions, finite morphisms, proper morphisms as formulated using valuative criteria applied by Grothendieck and separatedness akin to the diagonal embedding used in definitions of separated schemes. Functoriality manifests in pushforward and pullback operations on sheaves, exemplified in the adjunction between direct image and inverse image functors as used in étale and Hodge settings. Base change theorems involve comparisons across morphisms such as those appearing in studies by Nicholas Katz and Pierre Deligne and are crucial in the use of schemes in proving results like the Weil conjectures.
Constructions (limits, colimits, products, fibered products)
The category has all fibered products: given morphisms to a base from schemes modeled on Spec A, the fibered product is constructed from tensor products of rings as in classical constructions involving affine pullbacks. Finite limits and colimits exist under specified conditions; direct limits correspond to inductive limits of rings used by Nagata and others in compactification problems such as those addressed in Nagata compactification. Products produce objects like projective products, and fibered products underlie constructions in descent theory as in algebraic stacks and moduli problems studied by David Mumford and Aise Johan de Jong.
Subcategories and variants (affine, projective, separated, Noetherian)
Prominent full subcategories include affine schemes, those equivalent to the opposite category of commutative rings, and projective schemes embedded via Proj constructions from graded rings used by Oscar Zariski and André Weil. The Noetherian subcategory, built from Noetherian rings, is central in the work of Jean-Pierre Serre and in coherence theorems appearing in EGA. Separatedness conditions parallel notions familiar from Hausdorff in topology and are formalized by diagonal morphisms studied by Grothendieck and Michael Artin. Other variants include smooth schemes, étale schemes, and proper schemes used throughout Deligne–Mumford theory.
Topos-theoretic and categorical perspectives
Schemes sit inside a larger topos-theoretic framework: the small Zariski topos, the étale topos, and other Grothendieck topoi facilitate cohomological tools used by Pierre Deligne, Alexander Grothendieck, and Jean-Louis Verdier. The functor of points viewpoint identifies a scheme with its functor from schemes to Sets satisfying sheaf conditions, making representability and Yoneda-style arguments central in representability theorems such as those by Michael Artin and in the development of algebraic spaces and algebraic stacks by Deligne and Mumford. Higher-categorical enhancements lead to derived and spectral schemes as in the work of Jacob Lurie and Bertrand Toën.
Examples and classification results
Basic examples include affine schemes Spec Z, Spec k for a field k, projective lines P1 and projective spaces studied since André Weil, and elliptic curves as in the theory of modular curves and Taniyama–Shimura. Classification results exist in restricted settings: classification of curves over algebraically closed fields by Riemann–Roch and moduli of curves by Deligne–Mumford, classification of surfaces in work of Enriques and Bombieri–Mumford, and structure theorems for Noetherian schemes influenced by Zariski and Mumford. Counterexamples demonstrating pathologies in the category were constructed by Nagata, Serre, and Grothendieck illustrating limits of representability and the need for refined notions like algebraic spaces and stacks.