Scheme (mathematics)

Scheme (mathematics)
Name	Scheme (mathematics)
Field	Algebraic geometry
Introduced by	Alexander Grothendieck
Year	1960s

Scheme (mathematics) A scheme is a unifying geometric object in Algebraic geometry introduced in the 1960s by Alexander Grothendieck and developed in the context of the Séminaire de Géométrie Algébrique and the work of Jean-Pierre Serre and Jean-Louis Verdier. Schemes generalize varieties and affine varieties by gluing spectra of rings with structure sheaves, allowing powerful connections between Number theory and geometric methods used in the proofs of the Weil conjectures and the development of Arakelov theory. The language of schemes underpins modern results such as the Modularity theorem and the proof of the Fermat's Last Theorem.

Introduction

Schemes arise from the functor Spec that assigns to a commutative ring R the prime spectrum endowed with the Zariski topology and a canonical structure sheaf; this construction was systematized by Alexander Grothendieck in the framework of the Éléments de géométrie algébrique and expanded in the SGA seminars. Early motivations included unifying Dedekind rings, elliptic curves, and projective varieties while enabling techniques from Galois theory, Étale cohomology, and flatness to be used in arithmetic contexts such as the Langlands program.

Definitions and Basic Examples

A scheme is a locally ringed space locally isomorphic to Spec R for rings R; basic examples include Spec of a principal ideal domain like Z, affine n-space Spec of a polynomial ring over a field such as A^n over C or F_p, and projective schemes constructed via Proj from graded rings, as in the description of projective curves and Hilbert schemes. Classical examples that illustrate pathology vs. regularity include schemes over Z exhibiting nonreduced structure, nilpotent elements seen in thickenings like the Dual numbers Spec of k[ε]/(ε^2), and glued schemes modeling degenerations pertinent to the Néron model and Semistable reduction theorem.

Morphisms, Fibre Products, and Functorial Properties

Morphisms of schemes generalize morphisms of varieties and correspond to ring homomorphisms on affines; they include closed immersions, open immersions, finite morphisms, and projective morphisms as studied in EGA and SGA. Fibre products of schemes provide base change, used in descent theory and representability questions such as those addressed by Grothendieck's existence theorem and in the construction of the Picard scheme and Jacobian variety. Functorial viewpoints relate schemes to representable functors and Yoneda's lemma, and are central in the formulation of the Yoneda lemma, descent, and the construction of moduli via the Hilbert scheme and Artin's criteria.

Local Properties and Sheaf-Theoretic Aspects

Local properties of schemes are encoded in the stalks of the structure sheaf and their local rings; notions like regular local rings, Cohen–Macaulay rings, and Gorenstein rings connect to singularity theory and resolutions studied by Heisuke Hironaka and later techniques by Mark Haiman and others. The étale, flat, and fppf topologies refine the Zariski topology and enable cohomological tools such as Étale cohomology and comparison theorems used by Pierre Deligne in the proof of the Weil conjectures. Sheaf-theoretic methods and derived categories introduced by Jean-Louis Verdier and formalized in the work of Alexander Grothendieck provide the language for constructible sheaves and perverse sheaves applied to intersection cohomology in the sense of Goresky–MacPherson.

Dimension, Regularity, and Singularities

Krull dimension of the underlying rings yields the dimension theory for schemes, with special attention to properties like regularity, normality, and integrality studied in Zariski's and Noetherian contexts. Resolution of singularities for schemes over fields of characteristic zero due to Heisuke Hironaka and advances in positive characteristic by researchers such as Temkin and Abhyankar address desingularization and alterations in the spirit of Deligne–Mumford stacks and de Jong's alterations. Intersection theory on schemes, developed by William Fulton and others, leverages Chow groups and Chern classes to quantify singular behavior and relate to the Riemann–Roch theorem and Grothendieck–Riemann–Roch.

Cohomology and Derived Functors on Schemes

Cohomology theories for schemes include sheaf cohomology, derived functors R^i f_*, and specialized theories like étale cohomology, crystalline cohomology, and de Rham cohomology developed by Jean-Pierre Serre, Pierre Deligne, and Alexander Grothendieck. Derived categories and the formalism of derived functors enabled breakthroughs in duality theory such as Grothendieck duality, and in categorical approaches epitomized by the work of Maxim Kontsevich on homological mirror symmetry and derived algebraic geometry advanced by Jacob Lurie. Spectral sequences like the Leray and Hodge-to-de Rham sequences are standard tools in analyzing cohomology on schemes and in proving comparison theorems with topological or arithmetic invariants.

Applications and Advanced Topics (Moduli, Arithmetic, and Algebraic Geometry)

Schemes serve as the foundational language for moduli problems (e.g., Moduli of curves, vector bundle moduli, and Moduli of abelian varieties), arithmetic geometry (e.g., Spec Z in Diophantine problems, Néron model constructions, and the proof of the Taniyama–Shimura conjecture), and for modern intersections with categorical and homotopical methods in Derived algebraic geometry and the Langlands program. Key successes including Grothendieck's six operations, the proof of the Weil conjectures by Pierre Deligne, and the development of Tate conjecture techniques illustrate how schemes mediate interactions between Galois representations, automorphic forms, and arithmetic of elliptic curves.

Category:Algebraic geometry