Scheme (algebraic geometry)
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| Scheme (algebraic geometry) | |
|---|---|
| Name | Scheme (algebraic geometry) |
| Field | Algebraic geometry |
| Introduced | 1960s |
| Founders | Alexander Grothendieck |
| Examples | Affine scheme, Projective scheme, Smooth scheme |
Scheme (algebraic geometry) A scheme is a foundational object in modern algebraic geometry that generalizes varieties, allowing a unified treatment of arithmetic and geometric phenomena. Introduced by Alexander Grothendieck and developed with collaborators at the Institut des Hautes Études Scientifiques, schemes synthesize concepts from Évariste Galois-inspired arithmetic, Bernhard Riemann-inspired complex geometry, and structural ideas formalized in the work of David Hilbert and André Weil. Schemes underpin modern approaches used by mathematicians associated with institutions like University of Paris, Harvard University, University of Cambridge, Princeton University, and Massachusetts Institute of Technology.
Definition and Basic Properties
A scheme is a locally ringed space obtained by gluing affine schemes, each of which is the prime spectrum of a commutative ring. The construction relies on the functor Spec from the category of commutative rings as studied by Emmy Noether and Oscar Zariski to the category of locally ringed spaces as formalized in the seminars of Jean-Pierre Serre and Alexander Grothendieck. Basic properties include being locally affine, admitting a structure sheaf, and possessing topological notions influenced by the work of Henri Cartan, Jean Leray, and Jean-Louis Koszul. The notion of reduced, irreducible, integral, and nilpotent behavior in schemes traces historical connections to David Hilbert and Richard Dedekind.
Examples and Constructions
Affine schemes Spec A arise from rings A studied in classical algebra by Emmy Noether, Richard Dedekind, and Ernst Kummer. Projective schemes are constructed via Proj of graded rings as in the traditions of Jean-Pierre Serre and Oscar Zariski; examples include projective space influenced by Bernard Riemann-type ideas. Smooth schemes and regular schemes reflect concepts examined by André Weil and John Tate in arithmetic geometry. Other constructions include fibered products, closed and open immersions, blowups used by Enrico Bombieri and Heisuke Hironaka, and formal schemes appearing in the work of Alexander Grothendieck and Masayoshi Nagata.
Morphisms and Functoriality
Morphisms of schemes generalize polynomial maps and rational maps from classical geometry, formalized in the Grothendieck school and deployed by mathematicians at École Normale Supérieure and Collège de France. Important classes include finite morphisms, étale morphisms inspired by Évariste Galois-flavored monodromy, flat morphisms developed alongside work of Jean-Pierre Serre and Alexander Grothendieck, and proper morphisms connected to analogues of compactness studied by André Weil. Functorial viewpoints tie schemes to representable functors, Yoneda-style embeddings used in seminars led by Alexander Grothendieck and Jean-Louis Verdier.
Sheaf-theoretic and Local Properties
The structure sheaf on a scheme encodes local algebraic data, drawing on sheaf theory advanced by Henri Cartan and cohomological methods popularized by Jean-Pierre Serre and Alexander Grothendieck. Local properties such as regularity, Cohen–Macaulay conditions, and Gorenstein conditions were systematized by researchers including Jean-Pierre Serre, Oscar Zariski, and Masayoshi Nagata. Étale cohomology and flat cohomology link to arithmetic themes pursued by Pierre Deligne, Jean-Pierre Serre, and Alexander Grothendieck, while local study of singularities connects to work by Heisuke Hironaka and Shigefumi Mori.
Schemes Over a Base and Fibered Products
Schemes are routinely considered over a base scheme, reflecting arithmetic contexts such as schemes over Spec Z pioneered by Alexander Grothendieck and Jean-Pierre Serre. The fiber product of schemes generalizes base change and pullback operations central to comparative studies performed at institutions like Princeton University and University of California, Berkeley. Notions of families, moduli problems, and descent theory interweave with contributions from Mumford, Michael Artin, Pierre Deligne, and Grothendieck himself; stacks provide further generalization as developed by researchers at Harvard University and Institut des Hautes Études Scientifiques.
Cohomology, Divisors, and Line Bundles
Cohomological methods for schemes extend sheaf cohomology used by Jean-Pierre Serre and Grothendieck’s development of derived functors; powerful tools include Čech cohomology, derived categories studied by Alexandre Grothendieck-era mathematicians, and spectral sequences explored by Jean Leray. Cartier divisors, Weil divisors, and line bundles on schemes generalize classical divisor theory from the work of Bernhard Riemann and André Weil; the Picard group and class group relate to investigations by Philip Griffiths, Joe Harris, and David Mumford. Intersection theory on schemes owes to foundational contributions by William Fulton and later developments connected to Robert MacPherson and Mark Goresky.
Relation to Other Geometric Objects and Generalizations
Schemes generalize varieties, formal schemes, and analytic spaces; links to complex analytic geometry echo the influence of Bernard Riemann, while arithmetic applications connect to modular curves and Shimura varieties studied by Goro Shimura and Yutaka Taniyama. Further generalizations include algebraic stacks formulated by Jean-Louis Verdier-school collaborators such as Deligne and Mumford, as well as derived schemes and spectral algebraic geometry explored by researchers at Princeton University, University of California, Berkeley, and Institute for Advanced Study. Interactions with number theory and representation theory reflect the work of Andrew Wiles, Pierre Deligne, Michael Harris, and Richard Taylor.