Editing Scheme (mathematics) (section)

==Motivation for schemes==
Here are some of the ways in which schemes go beyond older notions of algebraic varieties, and their significance.

*'''Field extensions.''' Given some polynomial equations in ''n'' variables over a field ''k'', one can study the set ''X''(''k'') of solutions of the equations in the product set ''k''<sup>''n''</sup>. If the field ''k'' is algebraically closed (for example the complex numbers), then one can base algebraic geometry on sets such as ''X''(''k''): define the Zariski topology on ''X''(''k''), consider polynomial mappings between different sets of this type, and so on. But if ''k'' is not algebraically closed, then the set ''X''(''k'') is not rich enough. Indeed, one can study the solutions ''X''(''E'') of the given equations in any field extension ''E'' of ''k'', but these sets are not determined by ''X''(''k'') in any reasonable sense. For example, the plane curve ''X'' over the real numbers defined by ''x''<sup>2</sup> + ''y''<sup>2</sup> = −1 has ''X''('''R''') empty, but ''X''('''C''') not empty. (In fact, ''X''('''C''') can be identified with '''C''' − 0.) By contrast, a scheme ''X'' over a field ''k'' has enough information to determine the set ''X''(''E'') of ''E''-rational points for every extension field ''E'' of ''k''. (In particular, the closed subscheme of A{{supsub|2|'''R'''}} defined by ''x''<sup>2</sup> + ''y''<sup>2</sup> = −1 is a nonempty topological space.)
*'''Generic point.''' The points of the affine line A{{supsub|1|'''C'''}}, as a scheme, are its complex points (one for each complex number) together with one generic point (whose closure is the whole scheme). The generic point is the image of a natural morphism Spec('''C'''(''x'')) → A{{supsub|1|'''C'''}}, where '''C'''(''x'') is the field of [[rational function]]s in one variable. To see why it is useful to have an actual "generic point" in the scheme, consider the following example.
*Let ''X'' be the plane curve ''y''<sup>2</sup> = ''x''(''x''−1)(''x''−5) over the complex numbers. This is a closed subscheme of A{{supsub|2|'''C'''}}. It can be viewed as a [[ramified covering|ramified]] double cover of the affine line A{{supsub|1|'''C'''}} by projecting to the ''x''-coordinate. The fiber of the morphism ''X'' → A<sup>1</sup> over the generic point of A<sup>1</sup> is exactly the generic point of ''X'', yielding the morphism <math display="block">\operatorname{Spec} \mathbf{C}(x) \left (\sqrt{x(x-1)(x-5)} \right )\to \operatorname{Spec}\mathbf{C}(x).</math> This in turn is equivalent to the [[degree of a field extension|degree]]-2 extension of fields <math display="block">\mathbf{C}(x) \subset \mathbf{C}(x) \left (\sqrt{x(x-1)(x-5)} \right ).</math> Thus, having an actual generic point of a variety yields a geometric relation between a degree-2 morphism of algebraic varieties and the corresponding degree-2 extension of [[function field of an algebraic variety|function fields]]. This generalizes to a relation between the [[fundamental group]] (which classifies [[covering space]]s in topology) and the [[Galois group]] (which classifies certain [[field extension]]s). Indeed, Grothendieck's theory of the [[étale fundamental group]] treats the fundamental group and the Galois group on the same footing.

*'''Nilpotent elements'''. Let ''X'' be the closed subscheme of the affine line A{{supsub|1|'''C'''}} defined by ''x''<sup>2</sup> = 0, sometimes called a '''fat point'''. The ring of regular functions on ''X'' is '''C'''[''x'']/(''x''<sup>2</sup>); in particular, the regular function ''x'' on ''X'' is [[nilpotent]] but not zero. To indicate the meaning of this scheme: two regular functions on the affine line have the same restriction to ''X'' if and only if they have the same value ''and first [[derivative]]'' at the origin. Allowing such non-'''[[reduced scheme|reduced]]''' schemes brings the ideas of [[calculus]] and [[infinitesimal]]s into algebraic geometry.
*Nilpotent elements arise naturally in [[intersection theory]]. For example in the plane <math>\mathbb{A}^2_k</math> over a field <math>k</math>, with coordinate ring <math>k[x,y]</math>, consider the ''x-''axis, which is the variety <math>V(y)</math>, and the parabola <math>y=x^2</math>, which is <math>V(x^2-y)</math>. Their scheme-theoretic intersection is defined by the ideal <math>(y)+(x^2-y)=(x^2,\, y)</math>. Since the intersection is not [[Transversality (mathematics)|transverse]], this is not merely the point <math>(x,y) = (0,0)</math> defined by the ideal <math>(x,y)\subset k[x,y]</math>, but rather a fat point containing the ''x-''axis tangent direction (the common tangent of the two curves) and having coordinate ring:<math display="block">\frac{k[x,y]}{(x^2,\,y)} \cong \frac{k[x]}{(x^2)}.</math>The [[Intersection number|intersection multiplicity]] of 2 is defined as the [[Length of a module|length]] of this <math>k[x,y]</math>-module, i.e. its dimension as a <math>k</math>-vector space.
*For a more elaborate example, one can describe all the zero-dimensional closed subschemes of degree 2 in a [[smooth scheme|smooth]] complex variety ''Y''. Such a subscheme consists of either two distinct complex points of ''Y'', or else a subscheme isomorphic to ''X'' = Spec '''C'''[''x'']/(''x''<sup>2</sup>) as in the previous paragraph. Subschemes of the latter type are determined by a complex point ''y'' of ''Y'' together with a line in the [[tangent space]] T<sub>''y''</sub>''Y''.{{sfn|Eisenbud|Harris|1998|loc=Example II-10}} This again indicates that non-reduced subschemes have geometric meaning, related to derivatives and tangent vectors.