# Difference between revisions of "Algebraic geometry"

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− | '''Algebraic geometry''' is the study of solutions of [[polynomial]] equations by means of [[abstract algebra]], and in particular [[ring theory]]. Algebraic geometry is most easily done over [[algebraically closed]] [[field]]s, but it can also be done more generally over any field or even over [[ring]]s. | + | '''Algebraic geometry''' is the study of solutions of [[polynomial]] equations by means of [[abstract algebra]], and in particular [[ring theory]]. Algebraic geometry is most easily done over [[algebraically closed]] [[field]]s, but it can also be done more generally over any field or even over [[ring]]s. It is not to be confused with [[analytic geometry]], which is use of coordinates to solve geometrical problems. |

== Affine Algebraic Varieties == | == Affine Algebraic Varieties == | ||

− | One of the first basic objects studied in algebraic geometry is a [[variety]]. Let <math>\mathbb{A}^ | + | One of the first basic objects studied in algebraic geometry is a [[variety]]. Let <math>\mathbb{A}^n</math> denote [[affine]] <math>n</math>-space, i.e. a [[vector space]] of [[dimension]] <math>n</math> over an algebraically closed field, such as the field <math>\mathbb{C}</math> of [[complex number]]s. (We can think of this as <math>n</math>-dimensional "complex Euclidean" space.) Let <math>R=\mathbb{C}[X_1,\ldots,X_n]</math> be the [[polynomial ring]] in <math>n</math> variables, and let <math>I</math> be a [[prime ideal]] of <math>R</math>. Then <math>V(I)=\{p\in\mathbb{A}^n\mid f(p)=0\mathrm{\ for\ all\ } f\in I\}</math> is called an '''affine algebraic variety'''. |

== Projective Varieties == | == Projective Varieties == | ||

− | Let k be a field. A projective variety over k is a projective scheme over k. Projective varieties are algebraic varieties. | + | Let k be a field. A projective variety over k is a projective scheme over k. Projective varieties are algebraic varieties. A projective space <math>\mathbb{P}^n</math> is a quotient set with an equivalence class satisfying |

+ | <math>(x_0, \dots, x_n) \sim \lambda (x_0, \dots, x_n)</math>. | ||

+ | |||

+ | == Quasiprojective Varieties == | ||

+ | |||

+ | The varieties most commonly used, quasiprojective varieties are algebraic varieties given as open subsets of a projective variety with respect to the Zariski topology. | ||

+ | |||

+ | == General Algebraic Varieties == | ||

+ | |||

+ | Defined in terms of sheafs and patchings. | ||

== Schemes == | == Schemes == | ||

+ | Let <math>A</math> be a ring and <math>X=\operatorname{Spec}A</math>. An affine scheme is a ringed topological space isomorphic to some <math>(\operatorname{Spec }A,\mathcal{O}_{\operatorname{Spec}A})</math>. | ||

+ | A scheme is a ringed topological space <math>(X,\mathcal{O}_X)</math> admitting an open covering <math>\{U_i\}_i</math> such that <math>(U_i,\mathcal{O}_{X|U_i})</math> is an affine scheme for every <math>i</math>. | ||

− | + | {{stub}} | |

− | + | [[Category:Algebra]] | |

+ | [[Category:Geometry]] |

## Latest revision as of 00:59, 24 January 2020

**Algebraic geometry** is the study of solutions of polynomial equations by means of abstract algebra, and in particular ring theory. Algebraic geometry is most easily done over algebraically closed fields, but it can also be done more generally over any field or even over rings. It is not to be confused with analytic geometry, which is use of coordinates to solve geometrical problems.

## Contents

## Affine Algebraic Varieties

One of the first basic objects studied in algebraic geometry is a variety. Let denote affine -space, i.e. a vector space of dimension over an algebraically closed field, such as the field of complex numbers. (We can think of this as -dimensional "complex Euclidean" space.) Let be the polynomial ring in variables, and let be a prime ideal of . Then is called an **affine algebraic variety**.

## Projective Varieties

Let k be a field. A projective variety over k is a projective scheme over k. Projective varieties are algebraic varieties. A projective space is a quotient set with an equivalence class satisfying .

## Quasiprojective Varieties

The varieties most commonly used, quasiprojective varieties are algebraic varieties given as open subsets of a projective variety with respect to the Zariski topology.

## General Algebraic Varieties

Defined in terms of sheafs and patchings.

## Schemes

Let be a ring and . An affine scheme is a ringed topological space isomorphic to some . A scheme is a ringed topological space admitting an open covering such that is an affine scheme for every .

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