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119:. Although these concepts may be considered for every topological space, this is rarely done outside algebraic geometry, since most common topological spaces are
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It is a fundamental theorem of classical algebraic geometry that every algebraic set may be written in a unique way as a finite union of irreducible components.
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17:
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and the closed sets are the sets of all prime ideals that contain a fixed ideal. For this topology, a closed set is
879:, for which a set of prime ideals is closed if and only if it is the set of all prime ideals that contain a fixed
628:
if it is the set of all prime ideals that contain some prime ideal, and the irreducible components correspond to
605:
implies that every algebraic set is the union of a finite number of uniquely defined algebraic sets, called its
972:
609:. These notions of irreducibility and irreducible components are exactly the above defined ones, when the
569:, explicitly exclude the empty set from being irreducible. This article will not follow that convention.
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565:(since it has no proper subsets). However some authors, especially those interested in applications to
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149:
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300:
664:
651:. So the definition of irreducibility and irreducible components extends immediately to schemes.
418:
124:
842:, its closed subsets are itself, the empty set, the singletons, and the two lines defined by
601:, is an algebraic set that cannot be decomposed as the union of two smaller algebraic sets.
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is a set of real numbers that is not a singleton, there are three real numbers such that
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is considered, since the algebraic sets are exactly the closed sets of this topology.
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is not irreducible, and its irreducible components are the two lines of equations
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of an algebraic set is an algebraic subset that is irreducible and maximal (for
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Subset (often algebraic set) that is not the union of subsets of the same nature
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The empty topological space vacuously satisfies the definition above for
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is obtained by gluing together spectra of rings in the same way that a
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is a maximal subspace (necessarily closed) that is irreducible for the
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62:) for this property. For example, the set of solutions of the equation
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and rarely considered outside this area of mathematics: consider the
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is contained in a (not necessarily unique) irreducible component of
795:{\displaystyle X=(X\cap \,]{-\infty },a])\cup (X\cap [a,\infty [).}
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288:
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is thus reducible with these two lines as irreducible components.
632:. The number of irreducible components is finite in the case of a
941:
This article incorporates material from
Irreducible component on
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is the set of all prime ideals that contain a fixed prime ideal.
663:, the irreducible subsets and the irreducible components are the
123:, and, in a Hausdorff space, the irreducible components are the
919:"Section 5.8 (004U): Irreducible componentsâThe Stacks project"
905:"Section 5.8 (004U): Irreducible componentsâThe Stacks project"
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has the corresponding property in the above sense. That is,
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if it is not the union of two proper closed subsets, and an
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is also irreducible, so irreducible components are closed.
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This article incorporates material from irreducible on
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The notion of irreducible component is fundamental in
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irreducible subset. If a subset is irreducible, its
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408:{\displaystyle F=(G_{1}\cap F)\cup (G_{2}\cap F),}
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299:, or if any two nonempty open sets have nonempty
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947:Creative Commons Attribution/Share-Alike License
936:Creative Commons Attribution/Share-Alike License
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549:is contained in some irreducible component of
342:is reducible if it can be written as a union
89:These concepts can be reformulated in purely
620:is a topological space whose points are the
667:. This is the case, in particular, for the
318:considered as a topological space via the
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585:is defined as the set of the zeros of an
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283:) if it is not reducible. Equivalently,
314:is called irreducible or reducible, if
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515:Every irreducible subset of a space
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25:
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185:{\displaystyle X=X_{1}\cup X_{2}}
287:is irreducible if all non empty
146:if it can be written as a union
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875:of the ring, endowed with the
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46:that cannot be written as the
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101:are the algebraic subsets: A
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504:of a topological space is a
475:, neither of which contains
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557:The empty topological space
448:{\displaystyle G_{1},G_{2}}
10:
989:
595:irreducible algebraic set
36:irreducible algebraic set
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583:projective algebraic set
310:of a topological space
273:A topological space is
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607:irreducible components
603:LaskerâNoether theorem
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542:{\displaystyle x\in X}
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469:
455:are closed subsets of
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54:algebraic subsets. An
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573:In algebraic geometry
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243:{\displaystyle X_{2}}
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216:{\displaystyle X_{1}}
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113:irreducible component
56:irreducible component
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630:minimal prime ideals
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973:Algebraic varieties
40:irreducible variety
18:Irreducible variety
963:Algebraic geometry
885:irreducible subset
883:. In this case an
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618:spectrum of a ring
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491:{\displaystyle F.}
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266:{\displaystyle X.}
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32:algebraic geometry
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468:{\displaystyle X}
335:{\displaystyle F}
320:subspace topology
137:topological space
103:topological space
93:terms, using the
16:(Redirected from
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968:General topology
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877:Zariski topology
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661:Hausdorff space
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634:Noetherian ring
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523:. Every point
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60:set inclusion
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873:prime ideals
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301:intersection
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626:irreducible
563:irreducible
291:subsets of
275:irreducible
131:In topology
108:irreducible
99:closed sets
91:topological
957:Categories
943:PlanetMath
932:PlanetMath
856:. The set
709:. The set
665:singletons
125:singletons
781:∞
769:∩
760:∪
744:∞
741:−
733:∩
534:∈
394:∩
378:∪
369:∩
306:A subset
170:∪
144:reducible
865:spectrum
838:For the
655:Examples
645:manifold
510:closure
506:maximal
192:of two
50:of two
695:, and
649:charts
641:scheme
579:affine
577:Every
415:where
194:closed
52:proper
42:is an
891:Notes
881:ideal
867:of a
704:<
700:<
659:In a
593:. An
589:in a
587:ideal
297:dense
48:union
34:, an
863:The
849:and
832:= 0}
828:) |
820:= {(
616:The
295:are
289:open
277:(or
76:and
854:= 0
847:= 0
636:.
581:or
500:An
250:of
142:is
105:is
81:= 0
74:= 0
67:= 0
38:or
30:In
959::
830:xy
824:,
690:â
685:,
680:â
639:A
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65:xy
949:.
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858:X
852:y
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834:.
826:y
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787:)
784:[
778:,
775:a
772:[
766:X
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730:X
727:(
724:=
721:X
711:X
706:y
702:a
698:x
692:X
688:y
682:X
678:x
673:X
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531:x
521:X
517:X
486:.
483:F
463:X
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433:,
428:1
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381:(
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360:G
356:(
353:=
350:F
330:F
316:F
312:X
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293:X
285:X
261:.
258:X
236:2
232:X
209:1
205:X
178:2
174:X
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161:X
157:=
154:X
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79:y
72:x
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.