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Difference between revisions of "Irreducible variety"

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An [[Algebraic variety|algebraic variety]] that is an [[Irreducible topological space|irreducible topological space]] in the [[Zariski topology|Zariski topology]]. In other words, an algebraic variety is irreducible if it cannot be represented as the union of two proper closed algebraic subvarieties. Irreducibility of a scheme is defined similarly. For a smooth (and even a normal) variety the concepts of being irreducible and being connected are the same. Every irreducible variety has a unique generic point (cf. [[Point in general position|Point in general position]]).
 
  
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By analogy with the decomposition of a topological space into irreducible components, any algebraic variety is the union of finitely many irreducible closed subvarieties. The algebraic basis of this representation (which can be expressed more precisely) is the [[Primary decomposition|primary decomposition]] in commutative Noetherian rings.
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An [[algebraic variety]] that is an [[Irreducible topological space|irreducible topological space]] in the [[Zariski topology|Zariski topology]]. In other words, an algebraic variety is irreducible if it cannot be represented as the union of two proper closed algebraic subvarieties. Irreducibility of a scheme is defined similarly. For a smooth (and even a normal) variety the concepts of being irreducible and being connected are the same. Every irreducible variety has a unique [[generic point]] (cf. [[Point in general position]]).
  
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A product of irreducible varieties over an algebraically closed field is also irreducible. For an arbitrary ground field this is no longer true. The following version of the concept of an irreducible variety is also useful: A variety $X$ over a field $k$ is called geometrically irreducible if for any field extension $k'$ of $k$ the variety $X\otimes_kk'$ obtained from $X$ by [[Base change|base change]] remains irreducible.
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By analogy with the decomposition of a topological space into irreducible components, any algebraic variety is the union of finitely many irreducible closed subvarieties. The algebraic basis of this representation (which can be expressed more precisely) is the [[primary decomposition]] in commutative Noetherian rings.
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A product of irreducible varieties over an algebraically closed field is also irreducible. For an arbitrary ground field this is no longer true. The following version of the concept of an irreducible variety is also useful: A variety $X$ over a field $k$ is called geometrically irreducible if for any field extension $k'$ of $k$ the variety $X\otimes_kk'$ obtained from $X$ by [[base change]] remains irreducible.
  
  
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry", Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry", Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR>
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</table>

Latest revision as of 16:15, 3 January 2016


An algebraic variety that is an irreducible topological space in the Zariski topology. In other words, an algebraic variety is irreducible if it cannot be represented as the union of two proper closed algebraic subvarieties. Irreducibility of a scheme is defined similarly. For a smooth (and even a normal) variety the concepts of being irreducible and being connected are the same. Every irreducible variety has a unique generic point (cf. Point in general position).

By analogy with the decomposition of a topological space into irreducible components, any algebraic variety is the union of finitely many irreducible closed subvarieties. The algebraic basis of this representation (which can be expressed more precisely) is the primary decomposition in commutative Noetherian rings.

A product of irreducible varieties over an algebraically closed field is also irreducible. For an arbitrary ground field this is no longer true. The following version of the concept of an irreducible variety is also useful: A variety $X$ over a field $k$ is called geometrically irreducible if for any field extension $k'$ of $k$ the variety $X\otimes_kk'$ obtained from $X$ by base change remains irreducible.


Comments

References

[a1] R. Hartshorne, "Algebraic geometry", Springer (1977) MR0463157 Zbl 0367.14001
How to Cite This Entry:
Irreducible variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irreducible_variety&oldid=37334
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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