Editing Scheme (mathematics) (section)

==The category of schemes==
Schemes form a [[category theory|category]], with morphisms defined as morphisms of locally ringed spaces. (See also: [[morphism of schemes]].) For a scheme ''Y'', a scheme ''X'' '''over''' ''Y'' (or a ''Y''-'''scheme''') means a morphism ''X'' → ''Y'' of schemes. A scheme ''X'' '''over''' a commutative ring ''R'' means a morphism ''X'' → Spec(''R'').

An algebraic variety over a field ''k'' can be defined as a scheme over ''k'' with certain properties. There are different conventions about exactly which schemes should be called varieties. One standard choice is that a '''variety''' over ''k'' means an [[glossary of algebraic geometry|integral separated]] scheme of [[finite morphism#Morphisms of finite type|finite type]] over ''k''.<ref name=St020D>{{Citation | title=Stacks Project, Tag 020D | url=http://stacks.math.columbia.edu/tag/020D}}.</ref>

A morphism ''f'': ''X'' → ''Y'' of schemes determines a '''pullback homomorphism''' on the rings of regular functions, ''f''*: ''O''(''Y'') → ''O''(''X''). In the case of affine schemes, this construction gives a one-to-one correspondence between morphisms Spec(''A'') → Spec(''B'') of schemes and ring homomorphisms ''B'' → ''A''.{{sfn|Hartshorne|1997|loc=Proposition II.2.3}} In this sense, scheme theory completely subsumes the theory of commutative rings.

Since '''Z''' is an [[initial object]] in the [[category of commutative rings]], the category of schemes has Spec('''Z''') as a [[terminal object]].

For a scheme ''X'' over a commutative ring ''R'', an ''R''-'''point''' of ''X'' means a [[section (category theory)|section]] of the morphism ''X'' → Spec(''R''). One writes ''X''(''R'') for the set of ''R''-points of ''X''. In examples, this definition reconstructs the old notion of the set of solutions of the defining equations of ''X'' with values in ''R''. When ''R'' is a field ''k'', ''X''(''k'') is also called the set of ''k''-[[rational point]]s of ''X''.

More generally, for a scheme ''X'' over a commutative ring ''R'' and any commutative ''R''-[[algebra over a ring|algebra]] ''S'', an ''S''-'''point''' of ''X'' means a morphism Spec(''S'') → ''X'' over ''R''. One writes ''X''(''S'') for the set of ''S''-points of ''X''. (This generalizes the old observation that given some equations over a field ''k'', one can consider the set of solutions of the equations in any [[field extension]] ''E'' of ''k''.) For a scheme ''X'' over ''R'', the assignment ''S'' ↦ ''X''(''S'') is a [[functor]] from commutative ''R''-algebras to sets. It is an important observation that a scheme ''X'' over ''R'' is determined by this [[functor of points]].{{sfn|Eisenbud|Harris|1998|loc=Proposition VI-2}}

The [[fiber product of schemes]] always exists. That is, for any schemes ''X'' and ''Z'' with morphisms to a scheme ''Y'', the [[pullback (category theory)|categorical fiber product]] <math>X\times_Y Z</math> exists in the category of schemes. If ''X'' and ''Z'' are schemes over a field ''k'', their fiber product over Spec(''k'') may be called the '''product''' ''X'' × ''Z'' in the category of ''k''-schemes. For example, the product of affine spaces <math>\mathbb{A}^m</math> and <math>\mathbb{A}^n</math> over ''k'' is affine space <math>\mathbb{A}^{m+n}</math> over ''k''.

Since the category of schemes has fiber products and also a terminal object Spec('''Z'''), it has all finite [[Limit (category theory)|limit]]s.