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| In [[mathematics]], and specifically in [[algebraic geometry]], the concept of '''irreducible component''' is used to make formal the idea that a set such as defined by the equation
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| :''XY'' = 0
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| is the union of the two lines
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| :''X'' = 0
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| and | |
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| :''Y'' = 0.
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| Thus an [[algebraic set]] is '''irreducible''' if it is not the union of two proper algebraic subsets. It is a fundamental theorem of classical algebraic geometry that every algebraic set is the union of a finite number of irreducible algebraic subsets (varieties) and that this decomposition is unique if one removes those subsets that are contained in another one. The elements of this unique decomposition are called '''irreducible components'''.
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| This notion may be reformulated in [[topology|topological]] terms, using [[Zariski topology]], for which the closed sets are the subvarieties: an algebraic set is irreducible if it is not the union of two proper subsets that are closed for Zariski topology. This allows a generalization in topology, and, through it, to general [[scheme (mathematics)|scheme]]s for which the above property of finite decomposition is not necessarily true.
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| == In topology ==
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| A [[Topological spaces|topological space]] ''X'' is '''reducible''' if it can be written as a union <math>X = X_1 \cup X_2</math> of two non-empty [[Closed set|closed]] [[Subset|proper subsets]] <math>X_1</math>, <math>X_2</math> of <math>X</math>.
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| A topological space is '''irreducible''' (or '''[[hyperconnected space|hyperconnected]]''') if it is not reducible. Equivalently, all non empty [[open set|open]] subsets of ''X'' are [[dense set|dense]] or any two nonempty open sets have nonempty [[intersection (set theory)|intersection]].
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| A subset ''F'' of a topological space ''X'' is called irreducible or reducible, if ''F'' considered as a topological space via the [[subspace topology]] has the corresponding property in the above sense. That is, <math>F</math> is reducible if it can be written as a union <math>F = (G_1\cap F)\cup(G_2\cap F)</math> where <math>G_1,G_2</math> are closed subsets of <math>X</math>, neither of which contains <math>F</math>.
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| An '''irreducible component''' of a [[topological space]] is a [[maximal element|maximal]] [[Reduction (mathematics)|irreducible]] [[proper subset|subset]]. If a subset is irreducible, its [[closure (topology)|closure]] is, so irreducible components are [[topological space|closed]].
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| == In algebraic geometry ==
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| Every [[affine variety|affine]] or [[projective variety|projective algebraic set]] is defined as the set of the zeros of an [[ideal (ring theory)|ideal]] in a [[polynomial ring]]. In this case, the irreducible components are the varieties associated to the minimal primes over the ideal. This is this identification that allows to prove the uniqueness and the finiteness of the decomposition. This decomposition is strongly related with the [[primary decomposition]] of the ideal.
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| In general [[scheme theory]], every scheme is the union of its irreducible components, but the number of components is not necessary finite. However, in most cases occurring in "practice", namely for all [[noetherian scheme]]s, there are finitely many irreducible components.
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| == Examples ==
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| The irreducibility depends much on actual topology on some set. For example, possibly contradicting the intuition, the real numbers (with their usual topology) are reducible: for example the open interval (−1, 1) is not dense, its closure is the closed interval [−1, 1].
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| However, the notion is fundamental and more meaningful in [[algebraic geometry]]: consider the variety
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| :''X'' := {''x'' · ''y'' = 0}
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| (a subset of the affine plane, ''x'' and ''y'' are the variables) endowed with the ''[[Zariski topology]]''. It is reducible, its irreducible components are its closed subsets {''x = 0''} and {''y'' = 0}.
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| This can also be read off the coordinate ring ''k''[''x'', ''y'']/(''xy'') (if the variety is defined over a [[field (mathematics)|field]] ''k''), whose minimal prime ideals are (''x'') and (''y'').
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| {{PlanetMath attribution|id=1109|title=irreducible}}
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| {{PlanetMath attribution|id=3107|title=Irreducible component}}
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| {{DEFAULTSORT:Irreducible Component}}
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| [[Category:Algebraic geometry]]
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| [[Category:General topology]]
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| [[Category:Algebraic varieties]]
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| [[fr:Dimension_de_Krull#Composantes_irr.C3.A9ductibles]]
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