84:
74:
53:
22:
232:. The latter likely reflects local usage (somewhere in the US?); however, also in all English-language discussions I've had where étale morphisms have been discussed in the UK and the US, people have quite invariably used essentially the French pronunciation. Thus, I'd stick to the French version.
169:
Are the five defintions really equivalent without any hypothesis? J. Milne seems to imply in his lecture notes on etale cohomology (available from his website) on page 13 that the definitions involving the morphism to be locally of finite presentation are weaker than the definition flat+unramified.
185:
I changed the condition "flat+unramified" to "flat+G-unramified". The difference is that "unramified" requires locally of finite type whereas "G-unramified" requires locally of finite presentation. Milne works under the general hypothesis of all schemes being locally
Noetherian where the notions
302:
An anonymous user recently tried to correct the definition of standard étale. However, the definition formerly in the article and the definition given by the anonymous user are equivalent. I wanted to record the equivalence of the two for anyone else who came along:
655:, which are many. Is there a unique derivation generated by setting its value on elements of the ground field to 0, and on generators of the polynomial ring to 1? Even then, it would only apply to algebras over a field, and that isn't required by the article. ᛭
850:
174:
Y they are equivalent: flat + unramified (which is by definition locally of finite type) is equivalent to locally of finite presentation and formally etale (i.e. the functor Hom_X(-, Y) does only see the topological structure of -).
140:
974:
675:
896:
933:
651:
doesn't provide a definition for polynomials over arbitrary rings. If there's a standard definition, I'm unaware of it, and the best abstract definition I know of is for
562:
1025:
130:
106:
1020:
584:
is what you get by leaving out a hypersurface; you do this to remove all the bad points (where the morphism isn't étale). Localizing at
187:
97:
58:
158:
since (a) I don't know what a monic polynomial in several variables is and (b) I don't believe such a condition is necessary.
652:
33:
209:
157:
I removed the reference to monic polynomials in the section concening the localization of k/<f_1,...,f_n: -->
517:, so all the localization does is make 1 a unit, which was already true. So we make the replacement, and then
938:
254:
Then please link it as French rather than
English, and make it obvious to the reader that it isn't English.
191:
845:{\displaystyle f(x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n}\Rightarrow f'(x)=a_{1}+2a_{2}x+\cdots +na_{n}x^{n-1}.}
229:
225:
259:
39:
83:
176:
161:
21:
999:
660:
857:
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
89:
909:
73:
52:
524:
288:
255:
237:
995:
981:
656:
633:
279:
Y étale implies dim X = dim Y. Juste take Y = Spec (DVR) and X = generic point. I will
1014:
994:
Thanks. Using that information I was able to locate this derivative on
Knowledge. ᛭
284:
233:
102:
1003:
985:
664:
648:
637:
292:
263:
241:
212:
195:
179:
164:
79:
977:
629:
173:-- According to Milne's book on etale cohomology, , p.30, for f : X --: -->
854:
This is a derivation. It's canonical: The relative Kahler differentials
617:
produces the same effect; in other words, the hypersurface defined by
670:
It's what you get by formally applying the power rule and linearity:
15:
610:. What we showed above is that leaving out bad points using
575:(I'm leaving out the application of Spec here). The space
206:
Could somebody add an IPA pronunciation of the term étale?
186:
coincide, but in general one should make the distinction.
941:
912:
860:
678:
527:
101:, a collaborative effort to improve the coverage of
220:Added the correct French pronunciation of the word
968:
927:
890:
844:
556:
317:a homomorphism of commutative rings, and we know
8:
47:
940:
911:
878:
865:
859:
827:
817:
792:
776:
743:
733:
711:
698:
677:
546:
541:
526:
410:. For the other direction, assume that
969:{\displaystyle R\cdot dx\cong R\cdot 1}
571:is a one-dimensional affine space over
49:
19:
906:, and the derivative is also the map
606:and leaving out the bad points using
418:. This means that there exists some
360:. Suppose first that the derivative
7:
95:This article is within the scope of
591:simply leaves out more. The space
494:; but that doesn't matter, because
487:isn't the same as localizing it at
228:), and reverted a recent change to
38:It is of interest to the following
862:
14:
1026:Mid-priority mathematics articles
115:Knowledge:WikiProject Mathematics
458:. The trick is that we replace
118:Template:WikiProject Mathematics
82:
72:
51:
20:
599:by looking at the solutions of
135:This article has been rated as
916:
875:
869:
766:
760:
749:
688:
682:
538:
531:
213:08:09, 29 September 2007 (UTC)
1:
891:{\displaystyle \Omega _{R/R}}
165:18:57, 27 November 2006 (UTC)
109:and see a list of open tasks.
1021:B-Class mathematics articles
935:followed by the isomorphism
638:14:11, 4 December 2012 (UTC)
298:Definition of standard étale
278:It is not true that X -: -->
928:{\displaystyle f\mapsto df}
293:19:50, 6 January 2011 (UTC)
242:09:05, 10 August 2008 (UTC)
196:22:46, 16 August 2017 (UTC)
1042:
1004:05:09, 23 March 2013 (UTC)
986:00:57, 23 March 2013 (UTC)
665:00:31, 22 March 2013 (UTC)
373:. Then it's obvious that
180:21:17, 17 April 2007 (UTC)
557:{\displaystyle R_{g_{1}}}
479:. Of course, localizing
449:is some other element of
264:11:22, 2 March 2010 (UTC)
134:
67:
46:
643:Definition of derivative
402:is still the inverse of
274:Comparison of dimensions
141:project's priority scale
283:change this statement.
98:WikiProject Mathematics
970:
929:
892:
846:
558:
202:Pronunciation of étale
28:This article is rated
971:
930:
893:
847:
559:
385:to be the inverse of
939:
910:
858:
676:
525:
377:′ is also a unit in
210:Baccala@freesoft.org
121:mathematics articles
902:with basis element
966:
925:
898:are isomorphic to
888:
842:
554:
90:Mathematics portal
34:content assessment
513:term vanishes in
505:is isomorphic to
155:
154:
151:
150:
147:
146:
1033:
975:
973:
972:
967:
934:
932:
931:
926:
897:
895:
894:
889:
887:
886:
882:
851:
849:
848:
843:
838:
837:
822:
821:
797:
796:
781:
780:
759:
748:
747:
738:
737:
716:
715:
703:
702:
623:
605:
563:
561:
560:
555:
553:
552:
551:
550:
504:
478:
444:
359:
316:
231:
227:
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
1041:
1040:
1036:
1035:
1034:
1032:
1031:
1030:
1011:
1010:
937:
936:
908:
907:
861:
856:
855:
823:
813:
788:
772:
752:
739:
729:
707:
694:
674:
673:
645:
618:
616:
600:
590:
583:
567:Geometrically,
542:
537:
523:
522:
521:′ is a unit in
499:
493:
469:
463:
457:
432:
430:
414:′ is a unit in
397:
381:: If we choose
372:
364:′ is a unit in
358:
346:
334:
307:
300:
276:
204:
177:Jakob.scholbach
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
1039:
1037:
1029:
1028:
1023:
1013:
1012:
1009:
1008:
1007:
1006:
989:
988:
965:
962:
959:
956:
953:
950:
947:
944:
924:
921:
918:
915:
885:
881:
877:
874:
871:
868:
864:
852:
841:
836:
833:
830:
826:
820:
816:
812:
809:
806:
803:
800:
795:
791:
787:
784:
779:
775:
771:
768:
765:
762:
758:
755:
751:
746:
742:
736:
732:
728:
725:
722:
719:
714:
710:
706:
701:
697:
693:
690:
687:
684:
681:
671:
644:
641:
614:
595:is defined in
588:
579:
549:
545:
540:
536:
533:
530:
491:
467:
453:
426:
393:
368:
354:
342:
299:
296:
275:
272:
271:
270:
269:
268:
267:
266:
247:
246:
245:
244:
203:
200:
199:
198:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
1038:
1027:
1024:
1022:
1019:
1018:
1016:
1005:
1001:
997:
993:
992:
991:
990:
987:
983:
979:
963:
960:
957:
954:
951:
948:
945:
942:
922:
919:
913:
905:
901:
883:
879:
872:
866:
853:
839:
834:
831:
828:
824:
818:
814:
810:
807:
804:
801:
798:
793:
789:
785:
782:
777:
773:
769:
763:
756:
753:
744:
740:
734:
730:
726:
723:
720:
717:
712:
708:
704:
699:
695:
691:
685:
679:
672:
669:
668:
667:
666:
662:
658:
654:
650:
642:
640:
639:
635:
631:
627:
624:doesn't meet
622:
613:
609:
603:
598:
594:
587:
582:
578:
574:
570:
565:
547:
543:
534:
528:
520:
516:
512:
508:
503:
498:localized at
497:
490:
486:
482:
477:
473:
466:
461:
456:
452:
448:
443:
439:
435:
429:
425:
421:
417:
413:
409:
405:
401:
396:
392:
388:
384:
380:
376:
371:
367:
363:
357:
353:
350:
345:
341:
337:
333:is monic and
332:
328:
324:
320:
315:
311:
304:
297:
295:
294:
290:
286:
282:
273:
265:
261:
257:
253:
252:
251:
250:
249:
248:
243:
239:
235:
223:
219:
218:
217:
216:
215:
214:
211:
207:
201:
197:
193:
189:
184:
183:
182:
181:
178:
171:
167:
166:
163:
159:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
903:
899:
647:The article
646:
625:
620:
611:
607:
601:
596:
592:
585:
580:
576:
572:
568:
566:
518:
514:
510:
506:
501:
495:
488:
484:
480:
475:
471:
464:
459:
454:
450:
446:
441:
437:
433:
427:
423:
419:
415:
411:
407:
403:
399:
394:
390:
386:
382:
378:
374:
369:
365:
361:
355:
351:
348:
343:
339:
335:
330:
326:
322:
318:
313:
309:
305:
301:
280:
277:
221:
208:
205:
188:84.59.181.40
172:
168:
160:
156:
137:Mid-priority
136:
96:
62:Mid‑priority
40:WikiProjects
653:derivations
112:Mathematics
103:mathematics
59:Mathematics
1015:Categories
649:derivative
431:such that
329:such that
996:LokiClock
657:LokiClock
162:RJChapman
445:, where
306:We have
230:/eɪtɑːl/
470:= (1 +
398:, then
139:on the
30:B-class
509:: The
440:= 1 +
281:remove
234:Stca74
226:/etal/
36:scale.
406:′ in
389:′ in
256:kwami
222:étale
1000:talk
982:talk
978:Ozob
661:talk
634:talk
630:Ozob
619:1 +
500:1 +
321:and
308:φ :
289:talk
260:talk
238:talk
192:talk
628:.
604:= 0
483:at
462:by
422:in
325:in
285:Liu
131:Mid
1017::
1002:)
984:)
976:.
961:⋅
955:≅
946:⋅
917:↦
904:dx
863:Ω
832:−
805:⋯
750:⇒
724:⋯
663:)
636:)
621:fk
564:.
511:fk
502:fk
472:fk
442:fk
347:/
338:=
312:→
291:)
262:)
240:)
194:)
998:(
980:(
964:1
958:R
952:x
949:d
943:R
923:f
920:d
914:f
900:R
884:R
880:/
876:]
873:x
870:[
867:R
840:.
835:1
829:n
825:x
819:n
815:a
811:n
808:+
802:+
799:x
794:2
790:a
786:2
783:+
778:1
774:a
770:=
767:)
764:x
761:(
757:′
754:f
745:n
741:x
735:n
731:a
727:+
721:+
718:x
713:1
709:a
705:+
700:0
696:a
692:=
689:)
686:x
683:(
680:f
659:(
632:(
626:S
615:1
612:g
608:g
602:f
597:R
593:S
589:1
586:g
581:g
577:R
573:R
569:R
548:1
544:g
539:]
535:x
532:[
529:R
519:f
515:S
507:S
496:S
492:1
489:g
485:g
481:R
476:g
474:)
468:1
465:g
460:g
455:g
451:R
447:k
438:h
436:′
434:f
428:g
424:R
420:h
416:S
412:f
408:S
404:f
400:h
395:g
391:R
387:f
383:h
379:S
375:f
370:g
366:R
362:f
356:g
352:R
349:f
344:g
340:R
336:S
331:f
327:R
323:g
319:f
314:S
310:R
287:(
258:(
236:(
224:(
190:(
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
