659:
688:
585:
549:
108:
693:
589:
299:
291:
116:
72:
44:
573:
48:
28:
374:, i.e., a morphism which is a homeomorphism onto a closed subspace such that the associated map
64:
350:) is a scheme (both assertions can be checked locally). It is called the closed subscheme of
668:
371:
278:
168:
541:
596:
of a complex space is analytic if and only if the ideal sheaf of functions vanishing on
601:
427:
223:
682:
664:
294:, the importance of ideal sheaves lies mainly in the correspondence between closed
88:
51:. The ideal sheaves on a geometric object are closely connected to its subspaces.
32:
17:
295:
100:
552:
may be defined as the pull-back of the ideal sheaf defining the diagonal
441:
having the same underlying space, which is defined by the nilradical of O
199:
is a homomorphism between two sheaves of rings on the same space
222:, there is a natural structure of a sheaf of rings on the
426:
A particular case of this correspondence is the unique
608:
the structure of a reduced closed complex subspace.
277:is injective, but not surjective in general. (See
540:contains important information, it is called the
447:(defined stalk-wise, or on open affine charts).
403:is surjective on the stalks. Then, the kernel
123:viewed as a sheaf of abelian groups such that
576:so that the diagonal is a closed immersion.)
8:
617:
419:onto the closed subscheme defined by
411:is a quasi-coherent ideal sheaf, and
7:
25:
308:and a quasi-coherent ideal sheaf
304:ideal sheaves. Consider a scheme
660:Éléments de géométrie algébrique
528:The pull-back of an ideal sheaf
214:Conversely, for any ideal sheaf
43:) is the global analogue of an
675:. Springer-Verlag, Berlin 1984
604:. This ideal sheaf also gives
568:. (Assume for simplicity that
494:is defined by the ideal sheaf
1:
233:. Note that the canonical map
592:states that a closed subset
548:. For example, the sheaf of
415:induces an isomorphism from
710:
473:defined by an ideal sheaf
673:Coherent Analytic Sheaves
651:EGA IV, 16.1.2 and 16.3.1
332:is a closed subspace of
586:complex-analytic spaces
462:and a closed subscheme
207:is an ideal sheaf in
155:for all open subsets
550:Kähler differentials
318:. Then, the support
218:in a sheaf of rings
79:. (In other words, (
31:and other areas of
590:Oka-Cartan theorem
358:. Conversely, let
290:In the context of
286:Algebraic geometry
182:General properties
163:. In other words,
115:-modules, i.e., a
91:.) An ideal sheaf
29:algebraic geometry
584:In the theory of
580:Analytic geometry
532:to the subscheme
273:for open subsets
65:topological space
16:(Redirected from
701:
652:
649:
643:
640:
634:
631:
625:
622:
482:
467:
372:closed immersion
279:sheaf cohomology
203:, the kernel of
21:
709:
708:
704:
703:
702:
700:
699:
698:
679:
678:
656:
655:
650:
646:
641:
637:
632:
628:
624:EGA I, 4.2.2 b)
623:
619:
614:
582:
542:conormal bundle
523:
510:
489:
480:
477:, the preimage
465:
450:For a morphism
446:
436:
399:
393:
386:
345:
327:
317:
288:
184:
173:-submodules of
57:
41:sheaf of ideals
23:
22:
18:Sheaf of ideals
15:
12:
11:
5:
707:
705:
697:
696:
691:
681:
680:
677:
676:
662:
654:
653:
644:
635:
626:
616:
615:
613:
610:
581:
578:
526:
525:
519:
506:
485:
442:
434:
401:
400:
395:
391:
382:
341:
323:
313:
301:quasi-coherent
287:
284:
283:
282:
271:
270:
269:
235:
234:
224:quotient sheaf
212:
183:
180:
153:
152:
111:of sheaves of
56:
53:
24:
14:
13:
10:
9:
6:
4:
3:
2:
706:
695:
692:
690:
689:Scheme theory
687:
686:
684:
674:
670:
666:
663:
661:
658:
657:
648:
645:
639:
636:
630:
627:
621:
618:
611:
609:
607:
603:
599:
595:
591:
587:
579:
577:
575:
571:
567:
563:
560: Ă—
559:
556: →
555:
551:
547:
543:
539:
535:
531:
522:
517:
514:
509:
504:
500:
497:
496:
495:
493:
488:
483:
476:
472:
469: ⊆
468:
461:
458: →
457:
453:
448:
445:
440:
433:
429:
424:
422:
418:
414:
410:
406:
398:
390:
385:
380:
377:
376:
375:
373:
369:
366: →
365:
361:
357:
353:
349:
344:
339:
335:
331:
326:
321:
316:
311:
307:
303:
302:
297:
293:
285:
280:
276:
272:
267:
263:
259:
255:
251:
247:
243:
239:
238:
237:
236:
232:
228:
225:
221:
217:
213:
210:
206:
202:
198:
195: →
194:
190:
186:
185:
181:
179:
177:
176:
172:
166:
162:
158:
150:
146:
142:
138:
134:
130:
126:
125:
124:
122:
118:
114:
110:
106:
102:
98:
94:
90:
86:
82:
78:
74:
70:
66:
62:
54:
52:
50:
46:
42:
38:
34:
30:
19:
694:Sheaf theory
672:
647:
642:EGA I, 4.4.5
638:
629:
620:
605:
597:
593:
583:
569:
565:
561:
557:
553:
545:
537:
533:
529:
527:
520:
515:
512:
507:
502:
498:
491:
486:
478:
474:
470:
463:
459:
455:
451:
449:
443:
438:
431:
425:
420:
416:
412:
408:
404:
402:
396:
388:
383:
378:
367:
363:
359:
355:
351:
347:
342:
337:
333:
329:
324:
319:
314:
309:
305:
300:
289:
274:
265:
261:
257:
253:
249:
245:
241:
230:
226:
219:
215:
208:
204:
200:
196:
192:
188:
174:
170:
164:
160:
156:
154:
148:
144:
140:
136:
132:
128:
120:
112:
104:
96:
92:
89:ringed space
84:
80:
76:
75:of rings on
68:
60:
58:
40:
36:
26:
536:defined by
354:defined by
37:ideal sheaf
33:mathematics
683:Categories
669:R. Remmert
665:H. Grauert
633:EGA I, 5.1
612:References
430:subscheme
296:subschemes
55:Definition
574:separated
169:sheaf of
101:subobject
602:coherent
340:, O
117:subsheaf
109:category
484: Ă—
454::
428:reduced
362::
336:, and (
292:schemes
191::
107:in the
87:) is a
83:,
588:, the
490:
256:) → Γ(
143:) ⊆ Γ(
135:) · Γ(
511:= im(
370:be a
167:is a
99:is a
73:sheaf
63:be a
47:in a
45:ideal
35:, an
322:of O
312:in O
298:and
248:)/Γ(
67:and
59:Let
49:ring
39:(or
600:is
572:is
564:to
544:of
518:→ O
437:of
435:red
407:of
381:: O
187:If
159:of
119:of
103:of
95:in
27:In
685::
671::
667:,
524:).
505:)O
423:.
387:→
281:.)
260:,
252:,
244:,
240:Γ(
178:.
147:,
139:,
131:,
127:Γ(
71:a
606:A
598:A
594:A
570:X
566:X
562:X
558:X
554:X
546:Z
538:J
534:Z
530:J
521:X
516:J
513:f
508:X
503:J
501:(
499:f
492:X
487:Y
481:′
479:Y
475:J
471:Y
466:′
464:Y
460:Y
456:X
452:f
444:X
439:X
432:X
421:J
417:Z
413:i
409:i
405:J
397:Z
394:O
392:⋆
389:i
384:X
379:i
368:X
364:Z
360:i
356:J
352:X
348:J
346:/
343:X
338:Z
334:X
330:J
328:/
325:X
320:Z
315:X
310:J
306:X
275:U
268:)
266:J
264:/
262:A
258:U
254:J
250:U
246:A
242:U
231:J
229:/
227:A
220:A
216:J
211:.
209:A
205:f
201:X
197:B
193:A
189:f
175:A
171:A
165:J
161:X
157:U
151:)
149:J
145:U
141:J
137:U
133:A
129:U
121:A
113:A
105:A
97:A
93:J
85:A
81:X
77:X
69:A
61:X
20:)
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