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has the intuitive property that any closed proper subset has smaller dimension. Since dimension can only 'jump down' a finite number of times, and
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is a
Noetherian topological space. The converse does not hold, since there are non-Noetherian rings with only one prime ideal, so that Spec(
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condition, namely that every open subset of such a space is compact, and in fact it is equivalent to the seemingly stronger statement that
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Every subset of X is compact in a
Hausdorff space, hence closed. So X has the discrete topology, and being compact, it must be finite.
47:, since they are the complements of the closed subsets. The Noetherian property of a topological space can also be seen as a strong
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are made up of finite unions of irreducible sets, descending chains of
Zariski closed sets must eventually be constant.
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1347:"general topology - $ V$ is Noetherian space if only if every open subset of $ V$ is compact"
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572:. That follows because the rings of algebraic geometry, in the classical sense, are
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877:{\displaystyle I(Y_{1})\subseteq I(Y_{2})\subseteq I(Y_{3})\subseteq \cdots }
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A finite union of
Noetherian subspaces of a topological space is Noetherian.
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This article incorporates material from
Noetherian topological space on
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is an example of a
Noetherian topological space. By properties of the
603:) consists of exactly one point and therefore is a Noetherian space.
785:{\displaystyle Y_{1}\supseteq Y_{2}\supseteq Y_{3}\supseteq \cdots }
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is hereditarily compact), and if and only if every open subset of
1401:"general topology - Question about Noetherian topological spaces"
43:. Equivalently, we could say that the open subsets satisfy the
576:. This class of examples therefore also explains the name.
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Many examples of
Noetherian topological spaces come from
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The continuous image of a
Noetherian space is Noetherian.
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A more algebraic way to see this is that the associated
1117:{\displaystyle I(Y_{m})=I(Y_{m+1})=I(Y_{m+2})=\cdots .}
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is a descending chain of
Zariski-closed subsets, then
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140:{\displaystyle Y_{1}\supseteq Y_{2}\supseteq \cdots }
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Every subspace of a
Noetherian space is Noetherian.
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1464:Creative Commons Attribution/Share-Alike License
1000:is a Noetherian ring, there exists an integer
1321:{\displaystyle Y_{m}=Y_{m+1}=Y_{m+2}=\cdots }
583:is a commutative Noetherian ring, then Spec(
502:{\displaystyle X=X_{1}\cup \cdots \cup X_{n}}
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568:defining algebraic sets must satisfy the
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1383:"Lemma 5.9.4 (0053)—The Stacks project"
1365:"Lemma 5.9.3 (04Z8)—The Stacks project"
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265:{\displaystyle Y_{m}=Y_{m+1}=\cdots .}
405:. If the irreducible components are
7:
887:is an ascending chain of ideals of
723:{\displaystyle \mathbb {A} _{k}^{n}}
638:{\displaystyle \mathbb {A} _{k}^{n}}
380:Noetherian space is finite with the
39:in which closed subsets satisfy the
300:is Noetherian if and only if every
25:
1224:{\displaystyle V(I(Y_{i}))=Y_{i}}
1485:Properties of topological spaces
450:{\displaystyle X_{1},...,X_{n}}
1462:, which is licensed under the
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1433:Graduate Texts in Mathematics
509:, and none of the components
29:Noetherian topological space
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1405:Mathematics Stack Exchange
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89:descending chain condition
41:descending chain condition
570:ascending chain condition
45:ascending chain condition
1387:stacks.math.columbia.edu
1369:stacks.math.columbia.edu
1158:{\displaystyle V(I(Y))}
541:From algebraic geometry
401:has a finite number of
397:Every Noetherian space
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529:{\displaystyle X_{i}}
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170:{\displaystyle Y_{i}}
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280:A topological space
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87:if it satisfies the
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63:A topological space
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55:subset is compact.
1480:Algebraic geometry
1428:Algebraic Geometry
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1247:{\displaystyle i.}
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1165:is the closure of
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730:, we know that if
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547:algebraic geometry
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150:of closed subsets
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27:In mathematics, a
1442:978-0-387-90244-9
1423:Hartshorne, Robin
1013:{\displaystyle m}
993:{\displaystyle k}
681:{\displaystyle k}
658:{\displaystyle n}
597:Noetherian scheme
382:discrete topology
357:{\displaystyle X}
337:{\displaystyle X}
317:{\displaystyle X}
293:{\displaystyle X}
210:{\displaystyle m}
190:{\displaystyle X}
76:{\displaystyle X}
37:topological space
16:(Redirected from
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33:Emmy Noether
31:, named for
28:
26:
611:The space
364:is compact.
217:such that
49:compactness
1474:Categories
1460:PlanetMath
1416:References
275:Properties
95:: for any
85:Noetherian
83:is called
59:Definition
1316:⋯
1109:⋯
972:…
914:…
872:⋯
869:⊆
847:⊆
825:⊆
780:⋯
777:⊇
764:⊇
751:⊇
487:∪
484:⋯
481:∪
378:Hausdorff
257:⋯
135:⋯
132:⊇
119:⊇
1425:(1977),
1231:for all
1169:for all
645:(affine
302:subspace
97:sequence
1451:0463157
607:Example
587:), the
457:, then
389:Proof:
35:, is a
1449:
1439:
1254:Hence
1127:Since
945:Since
566:ideals
376:Every
1333:Notes
694:ideal
667:field
53:every
1437:ISBN
91:for
591:of
579:If
553:an
304:of
177:of
1476::
1447:MR
1445:,
1431:,
1403:.
1385:.
1367:.
1349:.
1173:,
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1407:.
1389:.
1371:.
1353:.
1313:=
1308:2
1305:+
1302:m
1298:Y
1294:=
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1283:m
1279:Y
1275:=
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1266:Y
1242:.
1239:i
1217:i
1213:Y
1209:=
1206:)
1203:)
1198:i
1194:Y
1190:(
1187:I
1184:(
1181:V
1171:Y
1167:Y
1153:)
1150:)
1147:Y
1144:(
1141:I
1138:(
1135:V
1112:.
1106:=
1103:)
1098:2
1095:+
1092:m
1088:Y
1084:(
1081:I
1078:=
1075:)
1070:1
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1064:m
1060:Y
1056:(
1053:I
1050:=
1047:)
1042:m
1038:Y
1034:(
1031:I
1008:m
988:]
983:n
979:x
975:,
969:,
964:1
960:x
956:[
953:k
933:.
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925:n
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911:,
906:1
902:x
898:[
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866:)
861:3
857:Y
853:(
850:I
844:)
839:2
835:Y
831:(
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822:)
817:1
813:Y
809:(
806:I
772:3
768:Y
759:2
755:Y
746:1
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716:n
711:k
706:A
676:k
653:n
631:n
626:k
621:A
601:R
593:R
585:R
581:R
522:i
518:X
495:n
491:X
476:1
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468:=
465:X
443:n
439:X
435:,
432:.
429:.
426:.
423:,
418:1
414:X
399:X
384:.
352:X
332:X
312:X
288:X
260:.
254:=
249:1
246:+
243:m
239:Y
235:=
230:m
226:Y
205:m
185:X
163:i
159:Y
127:2
123:Y
114:1
110:Y
71:X
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.