Scheme (mathematics): Difference between revisions
imported>Giovanni Antonio DiMatteo (New page: 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 Géomé...) |
imported>Giovanni Antonio DiMatteo m (→Definition) |
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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 Scheme|affine schemes]], i.e. of spectra of rings endowed with [[Affine Scheme|Zariski topologies]]. | 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 Scheme|affine schemes]], i.e. of spectra of rings endowed with [[Affine Scheme|Zariski topologies]]. | ||
== | ==The Category of Schemes== | ||
A '''scheme''' <math>(X,\mathcal{O}_X)</math> consists of a topological space <math>X</math> together with a sheaf <math>\mathcal{O}_X</math> of rings (called the structural sheaf on <math>X</math>) such that every point of <math>X</math> has an open neighborhood <math>U</math> such that the locally ringed space <math>(U,\mathcal{O}_X\vert_U)</math> is isomorphic to an [[Affine Scheme|affine scheme]]. | A '''scheme''' <math>(X,\mathcal{O}_X)</math> consists of a topological space <math>X</math> together with a sheaf <math>\mathcal{O}_X</math> of rings (called the structural sheaf on <math>X</math>) such that every point of <math>X</math> has an open neighborhood <math>U</math> such that the locally ringed space <math>(U,\mathcal{O}_X\vert_U)</math> is isomorphic to an [[Affine Scheme|affine scheme]]. | ||
[[Projective Scheme|Projective Schemes]] constitute an important class of schemes, especially for the study of curves. | [[Projective 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 Space|locally ringed spaces]]. Many kinds of morphisms of schemes are characterized affine-locally, in the sense that | The ''category of schemes'' is defined by taking morphisms of schemes to be morphisms of [[Locally Ringed Space|locally ringed spaces]]. Many kinds of morphisms of schemes are characterized affine-locally, in the sense that | ||
==Gluing Properties== | ==Gluing Properties== | ||
Revision as of 09:04, 2 December 2007
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.
The Category of Schemes
A scheme consists of a topological space together with a sheaf of rings (called the structural sheaf on ) such that every point of has an open neighborhood such that the locally ringed space 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.