How is a scheme different from a variety?

A variety is essentially an integral, reduced scheme of finite type over a field; a general scheme may have nilpotent functions, infinitely many or generic points, and may be defined over any commutative ring, including the integers.

Why do points of a scheme include prime ideals, not just maximal ones?

Prime ideals that are not maximal give generic points that lie in the closure of subvarieties, capturing the inclusion structure of irreducible subschemes and making the geometry functorial under ring maps.

Schemes

Schemes are Grothendieck's vast generalization of varieties, built by gluing spectra of arbitrary commutative rings, which lets algebraic geometry work over any ring and keep track of infinitesimal and arithmetic information.

Definition

A scheme is a locally ringed space that is locally isomorphic to the spectrum of a commutative ring (an affine scheme), where points are prime ideals and the structure sheaf records the ring of functions on each open set.

Scope

This topic constructs the spectrum of a commutative ring as a locally ringed space, defines affine schemes and general schemes by gluing, and develops morphisms of schemes and the relative point of view. It treats key properties — reduced, integral, separated, proper, and smooth schemes — fiber products and base change, and the functor-of-points perspective. The role of nilpotents in capturing nonreduced structure and the use of schemes over the integers for arithmetic geometry are emphasized.

Core questions

How does the prime spectrum of a ring turn arbitrary commutative algebra into geometry?
What do nilpotent elements and generic points let schemes express that varieties cannot?
How do relative schemes and base change support a uniform theory over any base?
How does the functor-of-points viewpoint characterize a scheme by the maps into it?

Key concepts

Spectrum of a ring and the Zariski topology on primes
Structure sheaf and locally ringed spaces
Affine schemes and gluing to general schemes
Morphisms, fiber products, and base change
Functor of points and nonreduced (nilpotent) structure

Clinical relevance

Scheme theory is the foundational language of modern algebraic geometry and arithmetic geometry; it made possible the cohomological proofs of the Weil conjectures and the modularity results behind Fermat's Last Theorem, and it frames moduli problems and deformation theory.

History

Building on Serre's sheaf-theoretic algebraic geometry, Grothendieck introduced schemes in the Éléments de géométrie algébrique (1960s), generalizing varieties to spectra of arbitrary rings and rebuilding the entire field on cohomological and categorical foundations.

Key figures

Alexander Grothendieck
Jean-Pierre Serre
David Mumford

Seminal works

hartshorne1977
eisenbud1995

Frequently asked questions

How is a scheme different from a variety?: A variety is essentially an integral, reduced scheme of finite type over a field; a general scheme may have nilpotent functions, infinitely many or generic points, and may be defined over any commutative ring, including the integers.
Why do points of a scheme include prime ideals, not just maximal ones?: Prime ideals that are not maximal give generic points that lie in the closure of subvarieties, capturing the inclusion structure of irreducible subschemes and making the geometry functorial under ring maps.