Scheme (mathematics): Difference between revisions
imported>Jitse Niesen m (Scheme (Mathematics) moved to Scheme (mathematics): consistency) |
imported>Barry R. Smith No edit summary |
||
Line 1: | Line 1: | ||
{{subpages}} | {{subpages}} | ||
'''Schemes''', and [[function (mathematics)|function]]s between them, are the principal objects of study in modern [[algebraic geometry]]. Algebraic geometry began as the study of [[variety (mathematics)|varieties]], geometric figures described by polynomial equations with coefficents in a [[field (mathematics)|field]], such as the [[real number]]s. The geometric properties of an [[affine variety]] is reflected in algebraic properties in its [[ring (mathematics)|ring]] of functions, which is the [[quotient ring|quotient]] of a [[polynomial ring]]. These algebraic properties can be defined in the context of arbitrary [[commutative ring]]s, and schemes are the corresponding geometric objects. | |||
Schemes have superceded varieties as the main objects of interest in algebraic geometry for several reasons: they give a uniform way to treat all previous disparate definitions of varieties, including [[affine variety|affine]], [[projective variety|projective]], [[quasi-projective variety| quasi-projective]], and [[abstract variety|abstract]] varieties, and there is a huge variety of schemes that are not one of these earlier defined objects. Also, the theory of varieties is most successful when the points on the varieties have values in an [[algebraically closed]] field. Making this assumption makes it impossible to study the arithmetic properties of the points, which are important in [[number theory]] and [[arithmetic geometry]]. Schemes have proven to be a powerful tool for studying the arithmetic of varieties. | |||
The '''theory of schemes''' was pioneered by [[Alexander Grothendieck]]. The foundations of scheme theory were initially organized in Grothendieck's multi-volume work [[EGA|Éléments de Géométrie Algébrique]] with the assistance of [[Jean Dieudonné]]. | The '''theory of schemes''' was pioneered by [[Alexander Grothendieck]]. The foundations of scheme theory were initially organized in Grothendieck's multi-volume work [[EGA|Éléments de Géométrie Algébrique]] with the assistance of [[Jean Dieudonné]]. | ||
Revision as of 21:00, 14 November 2008
Schemes, and functions between them, are the principal objects of study in modern algebraic geometry. Algebraic geometry began as the study of varieties, geometric figures described by polynomial equations with coefficents in a field, such as the real numbers. The geometric properties of an affine variety is reflected in algebraic properties in its ring of functions, which is the quotient of a polynomial ring. These algebraic properties can be defined in the context of arbitrary commutative rings, and schemes are the corresponding geometric objects.
Schemes have superceded varieties as the main objects of interest in algebraic geometry for several reasons: they give a uniform way to treat all previous disparate definitions of varieties, including affine, projective, quasi-projective, and abstract varieties, and there is a huge variety of schemes that are not one of these earlier defined objects. Also, the theory of varieties is most successful when the points on the varieties have values in an algebraically closed field. Making this assumption makes it impossible to study the arithmetic properties of the points, which are important in number theory and arithmetic geometry. Schemes have proven to be a powerful tool for studying the arithmetic of varieties.
The theory of schemes was pioneered by Alexander Grothendieck. The foundations of scheme theory were initially organized in Grothendieck's multi-volume work Éléments de Géométrie Algébrique with the assistance of Jean Dieudonné.
Roughly speaking, a scheme is a topological space which is locally affine; that is, a scheme has the local structure of the so-called affine schemes, i.e. of spectra of rings endowed with Zariski topologies.
A number of technical definitions and procedures are outlined in the glossary of scheme theory.
The Category of Schemes
A scheme Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,\mathcal{O}_X)} consists of a topological space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} together with a sheaf Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{O}_X} of rings (called the structural sheaf on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} ) such that every point of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} has an open neighborhood Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} such that the locally ringed space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (U,\mathcal{O}_X\vert_U)} is isomorphic to an affine scheme.
Projective Schemes constitute an important class of schemes, especially for the study of curves.
The category of schemes is defined by taking morphisms of schemes to be morphisms of locally ringed spaces. Many kinds of morphisms of schemes are characterized affine-locally, in the sense that
Gluing Properties
The notion of "gluing" is one of the central ideas in the theory of schemes. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} be a scheme, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X_i)_{i\in I}} a family of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} -schemes. If we're given families Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X_{ij})_{j\in I}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} -isomorphisms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{ij}:X_{ij}\to X_{ji}} such that: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{ii}=id_{X_i}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{ij}(X_{ij}\cap X_{ik})=X_{ji}\cap X_{jk}} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{ik}=f_{jk}\circ f_{ij}} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_{ij}\cap X_{ik}} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i,j,k\in I} , then there is an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} -scheme Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} together with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} -immersions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_i:X_i\to X} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_i=g_j\circ f_ij} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_{ij}} and so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X=\bigcup_{i\in I} g_i(X_i)} . This scheme Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is called the gluing over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_i} along the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_{ij}} .
The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} -scheme Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is universal for the property above: i.e., for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} -scheme Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} and family of morphisms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_i:X_i\to Z} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_i=u_j\circ f_{ij}} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_{ij}} , then there is a unique morphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u:X\to Z} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_i=u\circ g_i} . Moreover, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} is a scheme, then giving a morphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u:Z\to X} is equivalent to giving an open covering Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (Z_i)_{i\in I}} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} and morphisms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_i:Z_i\to X_i} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_j=f_{ij}\circ u_i} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_i\cap Z_j} .