Outline (mathematics)

In mathematics, a schema is a mathematical structure that relaxes the definition of an algebraic variety to include, among other things, multiplicities (e.g. the equations x = 0 y x² = 0 define the same algebraic variety but different schemes) and "varieties" defined on rings (eg Fermat curves are defined on the ring of integers).

Schemes consider geometric, algebraic, and number theory ideas. The notion of schema dates back to the 1960s, when Alexander Grothendieck formulated the concept in his treatise Éléments de géométrie algébrique. One of the goals was to develop the formalism necessary to solve deep problems in algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Likewise, schema theory allows the systematic use of methods of topology and homological algebra. By including considerations about rational points, schema theory introduces a strong connection between algebraic geometry and number theory, which eventually allowed Wiles to prove Fermat's Last Theorem.

Schemas are considered by many mathematicians to be the basic objects of study in modern algebraic geometry. Technically, a schema is a topological space provided with commutative rings for each of its open sets, which arises from sticking spectra (spaces of prime ideals) along its open sets. In other words, in a locally ringed space that is locally the spectrum of a commutative ring.

Any schema S presents a unique map to Spec(Z), the schema associated with integers. Thus, a schema can be identified with its map to Spec(Z), similar to how rings can be identified with associative algebras over integers. This is the starting point of the relative point of view, consisting of studying only the morphisms between schemas. This does not restrict generality, and allows certain schema properties to be easily specified. For example, an algebraic variety over a field K defines a schema map to Spec(K), with which the variety can be identified.

Definition

A schema is a locally ringed space (X, O_X ) locally isomorphic to an affine schema, i.e. for which there is a covering by open U_i such that (U_i, O_X|_Ui ) is isomorphic –as a locally ringed space– to (Spec (A), Â), where A is a commutative ring and  is its bundle of locations.