314:
202:
402:
867:, Interscience Tracts in Pure and Applied Mathematics, vol. 13, New York-London: Interscience Publishers, a division of John Wiley & Sons
874:
814:
99:
470:, page 203 example 2), who found a 2-dimensional Noetherian local domain that is catenary but not universally catenary.
309:{\displaystyle {\text{height}}(P)\leq {\text{height}}(p)+{\text{tr.deg.}}_{A}(B)-{\text{tr.deg.}}_{\kappa (p)}(\kappa (P)).}
458:
It is delicate to construct examples of
Noetherian rings that are not universally catenary. The first example was found by
926:
416:
that appear in algebraic geometry are universally catenary. In particular the following rings are universally catenary:
921:
146:
128:
134:
764:
443:
906:
102:
attached to a prime ideal will decrease as the prime ideal becomes bigger, the length of such a chain
740:
by the dimension formula for universally catenary rings, but the latter ideal has height equal to dim(
748:
901:
478:
432:
794:
Lectures on integral closure, the Briançon–Skoda theorem and related topics in commutative algebra
852:
436:
152:
870:
810:
896:
834:
459:
32:
848:
844:
426:
413:
140:
786:
346:
752:
915:
856:
332:
17:
822:
886:
708:, so is catenary because all 2-dimensional local domains are catenary. The ring
43:
28:
339:
and tr.deg. means the transcendence degree (of quotient fields). In fact, when
891:
839:
421:
629:
is a regular local ring of dimension 1 (the proof of this uses the fact that
90:
of prime ideals are contained in maximal strictly increasing chains from
669:
is a 2-dimensional
Noetherian semi-local ring with 2 maximal ideals,
98:
of the same (finite) length. In a geometric situation, in which the
449:
Any finitely generated algebra over a universally catenary ring.
751:, so gives an example of a quasi-excellent ring that is not an
724:
is not universally catenary, because if it were then the ideal
113:
if all finitely generated algebras over it are catenary rings.
349:
205:
704:
is a local domain of dimension 2 with maximal ideal
454:
A ring that is catenary but not universally catenary
767:(which is the same as a universally catenary ring).
646:is a regular Noetherian local ring of dimension 2.
116:The word 'catenary' is derived from the Latin word
637:are algebraically independent) and the local ring
396:
308:
321:dimension formula for universally catenary rings
473:Nagata's example is as follows. Choose a field
8:
869:; reprinted by R. E. Krieger Pub. Co (1975)
123:There is the following chain of inclusions.
657:with respect to all elements not in either
571:be the (non-Noetherian) ring generated by
838:
385:
366:
348:
270:
265:
246:
241:
223:
206:
204:
106:is usually the difference in dimensions.
777:
823:"On the chain problem of prime ideals"
467:
463:
53:, any two strictly increasing chains
7:
616:, with residue fields isomorphic to
612:. These are both maximal ideals of
25:
343:is not universally catenary, but
100:dimension of an algebraic variety
327:is universally catenary. Here κ(
176:that is finitely generated over
522:are algebraically independent.
732:would have the same height as
716:is Noetherian and is a finite
446:of a universally catenary ring
391:
359:
300:
297:
291:
285:
280:
274:
258:
252:
234:
228:
217:
211:
1:
785:Hochster, Mel (Winter 2014),
787:"Lecture of January 8, 2014"
404:, then equality also holds.
323:says that equality holds if
747:Nagata's example is also a
684:be the Jacobson radical of
168:is a Noetherian domain and
147:complete intersection rings
943:
863:Nagata, Masayoshi (1962),
821:Nagata, Masayoshi (1956),
599:be the ideal generated by
506:of formal power series in
129:Universally catenary rings
840:10.1017/S0027763000000076
796:, University of Michigan
603:–1 and all the elements
172:is a domain containing
653:be the localization of
120:, which means "chain".
907:Gorenstein local rings
765:Formally catenary ring
712:is Noetherian because
398:
310:
192:its intersection with
575:and all the elements
399:
311:
902:Cohen-Macaulay rings
749:quasi-excellent ring
460:Masayoshi Nagata
433:Cohen-Macaulay rings
420:Complete Noetherian
347:
203:
184:is a prime ideal of
135:Cohen–Macaulay rings
111:universally catenary
74:⊂ ... ⊂
18:Universally catenary
927:Commutative algebra
807:Commutative algebra
479:formal power series
437:regular local rings
397:{\displaystyle B=A}
153:regular local rings
42:if for any pair of
922:Algebraic geometry
673:(of height 1) and
394:
306:
897:Commutative rings
720:-module. However
620:. The local ring
268:
244:
226:
209:
160:Dimension formula
109:A ring is called
16:(Redirected from
934:
868:
859:
842:
798:
797:
791:
782:
427:Dedekind domains
414:Noetherian rings
403:
401:
400:
395:
390:
389:
371:
370:
315:
313:
312:
307:
284:
283:
269:
266:
251:
250:
245:
242:
227:
224:
210:
207:
141:Gorenstein rings
33:commutative ring
21:
942:
941:
937:
936:
935:
933:
932:
931:
912:
911:
883:
862:
827:Nagoya Math. J.
820:
802:
801:
789:
784:
783:
779:
774:
761:
677:(of height 2).
645:
628:
611:
583:
563:
554:
545:
531:
498:
490:
456:
410:
381:
362:
345:
344:
264:
240:
201:
200:
162:
82:
73:
66:
23:
22:
15:
12:
11:
5:
940:
938:
930:
929:
924:
914:
913:
910:
909:
904:
899:
894:
889:
882:
879:
878:
877:
860:
818:
805:H. Matsumura,
800:
799:
776:
775:
773:
770:
769:
768:
760:
757:
753:excellent ring
641:
624:
607:
591:be the ideal (
579:
559:
550:
540:
529:
494:
485:
455:
452:
451:
450:
447:
440:
430:
424:
409:
406:
393:
388:
384:
380:
377:
374:
369:
365:
361:
358:
355:
352:
317:
316:
305:
302:
299:
296:
293:
290:
287:
282:
279:
276:
273:
263:
260:
257:
254:
249:
239:
236:
233:
230:
222:
219:
216:
213:
161:
158:
157:
156:
88:
87:
78:
71:
64:
24:
14:
13:
10:
9:
6:
4:
3:
2:
939:
928:
925:
923:
920:
919:
917:
908:
905:
903:
900:
898:
895:
893:
890:
888:
885:
884:
880:
876:
875:0-88275-228-6
872:
866:
861:
858:
854:
850:
846:
841:
836:
832:
828:
824:
819:
816:
815:0-8053-7026-9
812:
808:
804:
803:
795:
788:
781:
778:
771:
766:
763:
762:
758:
756:
754:
750:
745:
743:
739:
735:
731:
727:
723:
719:
715:
711:
707:
703:
699:
695:
691:
687:
683:
678:
676:
672:
668:
664:
660:
656:
652:
647:
644:
640:
636:
632:
627:
623:
619:
615:
610:
606:
602:
598:
594:
590:
585:
582:
578:
574:
570:
565:
562:
558:
553:
549:
543:
539:
535:
528:
523:
521:
517:
513:
509:
505:
501:
497:
493:
488:
483:
480:
476:
471:
469:
465:
461:
453:
448:
445:
441:
438:
434:
431:
428:
425:
423:
419:
418:
417:
415:
407:
405:
386:
382:
378:
375:
372:
367:
363:
356:
353:
350:
342:
338:
334:
333:residue field
330:
326:
322:
303:
294:
288:
277:
271:
261:
255:
247:
237:
231:
220:
214:
199:
198:
197:
195:
191:
187:
183:
179:
175:
171:
167:
164:Suppose that
159:
155:
154:
149:
148:
143:
142:
137:
136:
131:
130:
126:
125:
124:
121:
119:
114:
112:
107:
105:
101:
97:
93:
86:
81:
77:
70:
63:
59:
56:
55:
54:
52:
48:
45:
41:
37:
34:
30:
19:
864:
830:
826:
806:
793:
780:
746:
741:
737:
733:
729:
725:
721:
717:
713:
709:
705:
701:
697:
693:
689:
685:
681:
679:
674:
670:
666:
662:
658:
654:
650:
648:
642:
638:
634:
630:
625:
621:
617:
613:
608:
604:
600:
596:
592:
588:
586:
580:
576:
572:
568:
566:
560:
556:
551:
547:
541:
537:
533:
526:
524:
519:
515:
511:
507:
503:
502:in the ring
499:
495:
491:
486:
481:
474:
472:
457:
444:localization
429:(and fields)
411:
340:
336:
328:
324:
320:
318:
193:
189:
185:
181:
177:
173:
169:
165:
163:
151:
145:
139:
133:
127:
122:
117:
115:
110:
108:
103:
95:
91:
89:
84:
79:
75:
68:
61:
57:
50:
46:
44:prime ideals
39:
35:
26:
892:Local rings
887:Ring theory
865:Local rings
700:. The ring
595:), and let
422:local rings
412:Almost all
29:mathematics
916:Categories
772:References
688:, and let
514:such that
857:122444738
833:: 51–64,
376:…
331:) is the
289:κ
272:κ
262:−
221:≤
881:See also
759:See also
408:Examples
196:, then
67:⊂
40:catenary
849:0078974
736:∩
665:. Then
525:Define
462: (
267:tr.deg.
243:tr.deg.
873:
855:
847:
813:
477:and a
225:height
208:height
118:catena
853:S2CID
809:1980
790:(PDF)
744:)=2.
510:over
489:>0
435:(and
180:. If
871:ISBN
811:ISBN
680:Let
649:Let
633:and
587:Let
567:Let
536:and
518:and
468:1962
464:1956
442:Any
319:The
188:and
31:, a
835:doi
728:of
661:or
555:/x–
335:of
132:⊃
94:to
38:is
27:In
918::
851:,
845:MR
843:,
831:10
829:,
825:,
792:,
755:.
734:mB
726:mB
692:=
675:nB
671:mB
584:.
564:.
544:+1
532:=
484:=Σ
466:,
150:⊃
144:⊃
138:⊃
83:=
60:=
49:,
837::
817:.
742:A
738:A
730:B
722:A
718:A
714:B
710:A
706:I
702:A
698:I
696:+
694:k
690:A
686:B
682:I
667:B
663:n
659:m
655:R
651:B
643:n
639:R
635:x
631:z
626:m
622:R
618:k
614:R
609:i
605:z
601:x
597:n
593:x
589:m
581:i
577:z
573:x
569:R
561:i
557:a
552:i
548:z
546:=
542:i
538:z
534:z
530:1
527:z
520:x
516:z
512:k
508:x
504:S
500:x
496:i
492:a
487:i
482:z
475:k
439:)
392:]
387:n
383:x
379:,
373:,
368:1
364:x
360:[
357:A
354:=
351:B
341:A
337:P
329:P
325:A
304:.
301:)
298:)
295:P
292:(
286:(
281:)
278:p
275:(
259:)
256:B
253:(
248:A
238:+
235:)
232:p
229:(
218:)
215:P
212:(
194:A
190:p
186:B
182:P
178:A
174:A
170:B
166:A
104:n
96:q
92:p
85:q
80:n
76:p
72:1
69:p
65:0
62:p
58:p
51:q
47:p
36:R
20:)
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