375:
602:
194:
862:
269:
506:
732:
87:
902:
649:
426:
764:
277:
547:
1055:
980:
119:
773:
1019:
1011:
207:
1047:
528:
preserves fibered products (e.g. if it is a scheme), the tangent space may be given the structure of a vector space over
431:
1081:
1001:
1042:
665:
24:
42:
867:
615:
1005:
396:
1051:
1015:
976:
737:
370:{\displaystyle \delta _{p}^{v}:u\mapsto u(p)+\epsilon v(u),\quad v\in {\mathcal {O}}_{p}^{*}.}
1037:
1025:
1065:
990:
1061:
1029:
986:
972:
597:{\displaystyle F=\operatorname {Hom} _{\operatorname {Spec} k}(\operatorname {Spec} -,X)}
997:
659:
98:
1075:
964:
971:, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Berlin, New York:
23:
generalizes the classical construction of a tangent space such as the
189:{\displaystyle ({\mathfrak {m}}_{X,p}/{\mathfrak {m}}_{X,p}^{2})^{*}}
857:{\displaystyle f^{\#}(\delta _{p}^{v})=\delta _{f(p)}^{df_{p}(v)}}
658:
The construction may be thought of as defining an analog of the
27:. The construction is based on the following observation. Let
348:
214:
264:{\displaystyle {\mathcal {O}}_{p}\to k/(\epsilon )^{2}}
204:(To see this, use the fact that any local homomorphism
870:
776:
740:
668:
618:
550:
434:
399:
280:
210:
122:
45:
896:
856:
758:
726:
643:
596:
500:
420:
369:
263:
188:
81:
501:{\displaystyle \pi :F(k/(\epsilon )^{2})\to F(k)}
389:-algebras to the category of sets. Then, for any
608:as above may be identified with a derivation at
940:
8:
1050:, vol. 52, New York: Springer-Verlag,
928:
727:{\displaystyle T_{X}=X(k/(\epsilon )^{2})}
908:induces is precisely the differential of
888:
875:
869:
837:
829:
815:
799:
794:
781:
775:
739:
715:
700:
673:
667:
623:
617:
561:
549:
474:
459:
433:
398:
358:
353:
347:
346:
290:
285:
279:
255:
240:
219:
213:
212:
209:
180:
170:
159:
153:
152:
146:
134:
128:
127:
121:
73:
58:
44:
921:
655:and we recover the usual construction.
952:
612:and this gives the identification of
7:
154:
129:
782:
385:be a functor from the category of
14:
651:with the space of derivations at
82:{\displaystyle k/(\epsilon )^{2}}
912:under the above identification.
338:
93:is the same thing as to give a
897:{\displaystyle T_{X}\to T_{Y}}
881:
849:
843:
825:
819:
805:
787:
750:
721:
712:
705:
697:
691:
685:
638:
632:
591:
576:
495:
489:
483:
480:
471:
464:
456:
450:
444:
415:
409:
332:
326:
314:
308:
302:
252:
245:
237:
231:
225:
177:
123:
116:) together with an element of
70:
63:
55:
49:
1:
1048:Graduate Texts in Mathematics
644:{\displaystyle \pi ^{-1}(p)}
196:; i.e., a tangent vector at
108:(i.e., the residue field of
19:In algebraic geometry, the
1098:
941:Eisenbud & Harris 1998
864:; this shows that the map
662:in the following way. Let
21:tangent space to a functor
16:Concept in category theory
734:. Then, for any morphism
421:{\displaystyle p\in F(k)}
31:be a scheme over a field
759:{\displaystyle f:X\to Y}
1007:The Geometry of Schemes
969:Linear algebraic groups
898:
858:
760:
728:
645:
598:
502:
422:
371:
265:
190:
83:
899:
859:
761:
729:
646:
599:
503:
423:
372:
266:
191:
84:
25:Zariski tangent space
868:
774:
738:
666:
616:
548:
432:
397:
278:
271:must be of the form
208:
120:
43:
853:
804:
363:
295:
175:
1082:Algebraic geometry
1043:Algebraic Geometry
894:
854:
811:
790:
756:
724:
641:
594:
498:
418:
367:
345:
281:
261:
186:
151:
79:
1057:978-0-387-90244-9
1038:Hartshorne, Robin
982:978-0-387-97370-8
931:, Exercise II 2.8
524:. If the functor
1089:
1068:
1033:
993:
956:
950:
944:
938:
932:
926:
903:
901:
900:
895:
893:
892:
880:
879:
863:
861:
860:
855:
852:
842:
841:
828:
803:
798:
786:
785:
766:of schemes over
765:
763:
762:
757:
733:
731:
730:
725:
720:
719:
704:
678:
677:
650:
648:
647:
642:
631:
630:
603:
601:
600:
595:
572:
571:
507:
505:
504:
499:
479:
478:
463:
427:
425:
424:
419:
376:
374:
373:
368:
362:
357:
352:
351:
294:
289:
270:
268:
267:
262:
260:
259:
244:
224:
223:
218:
217:
195:
193:
192:
187:
185:
184:
174:
169:
158:
157:
150:
145:
144:
133:
132:
88:
86:
85:
80:
78:
77:
62:
1097:
1096:
1092:
1091:
1090:
1088:
1087:
1086:
1072:
1071:
1058:
1036:
1022:
1012:Springer-Verlag
998:Eisenbud, David
996:
983:
973:Springer-Verlag
963:
960:
959:
951:
947:
939:
935:
929:Hartshorne 1977
927:
923:
918:
884:
871:
866:
865:
833:
777:
772:
771:
736:
735:
711:
669:
664:
663:
619:
614:
613:
557:
546:
545:
470:
430:
429:
428:, the fiber of
395:
394:
276:
275:
251:
211:
206:
205:
176:
126:
118:
117:
69:
41:
40:
17:
12:
11:
5:
1095:
1093:
1085:
1084:
1074:
1073:
1070:
1069:
1056:
1034:
1020:
994:
981:
958:
957:
945:
933:
920:
919:
917:
914:
891:
887:
883:
878:
874:
851:
848:
845:
840:
836:
832:
827:
824:
821:
818:
814:
810:
807:
802:
797:
793:
789:
784:
780:
755:
752:
749:
746:
743:
723:
718:
714:
710:
707:
703:
699:
696:
693:
690:
687:
684:
681:
676:
672:
660:tangent bundle
640:
637:
634:
629:
626:
622:
593:
590:
587:
584:
581:
578:
575:
570:
567:
564:
560:
556:
553:
512:is called the
497:
494:
491:
488:
485:
482:
477:
473:
469:
466:
462:
458:
455:
452:
449:
446:
443:
440:
437:
417:
414:
411:
408:
405:
402:
379:
378:
366:
361:
356:
350:
344:
341:
337:
334:
331:
328:
325:
322:
319:
316:
313:
310:
307:
304:
301:
298:
293:
288:
284:
258:
254:
250:
247:
243:
239:
236:
233:
230:
227:
222:
216:
202:
201:
183:
179:
173:
168:
165:
162:
156:
149:
143:
140:
137:
131:
125:
99:rational point
76:
72:
68:
65:
61:
57:
54:
51:
48:
15:
13:
10:
9:
6:
4:
3:
2:
1094:
1083:
1080:
1079:
1077:
1067:
1063:
1059:
1053:
1049:
1045:
1044:
1039:
1035:
1031:
1027:
1023:
1021:0-387-98637-5
1017:
1013:
1009:
1008:
1003:
999:
995:
992:
988:
984:
978:
974:
970:
966:
965:Borel, Armand
962:
961:
954:
949:
946:
942:
937:
934:
930:
925:
922:
915:
913:
911:
907:
889:
885:
876:
872:
846:
838:
834:
830:
822:
816:
812:
808:
800:
795:
791:
778:
769:
753:
747:
744:
741:
716:
708:
701:
694:
688:
682:
679:
674:
670:
661:
656:
654:
635:
627:
624:
620:
611:
607:
604:), then each
588:
585:
582:
579:
573:
568:
565:
562:
558:
554:
551:
543:
539:
535:
531:
527:
523:
519:
515:
514:tangent space
511:
492:
486:
475:
467:
460:
453:
447:
441:
438:
435:
412:
406:
403:
400:
392:
388:
384:
364:
359:
354:
342:
339:
335:
329:
323:
320:
317:
311:
305:
299:
296:
291:
286:
282:
274:
273:
272:
256:
248:
241:
234:
228:
220:
199:
181:
171:
166:
163:
160:
147:
141:
138:
135:
115:
111:
107:
103:
100:
96:
92:
74:
66:
59:
52:
46:
38:
37:
36:
34:
30:
26:
22:
1041:
1006:
968:
948:
936:
924:
909:
905:
767:
657:
652:
609:
605:
541:
537:
536:is a scheme
533:
529:
525:
521:
517:
513:
509:
390:
386:
382:
380:
203:
197:
113:
109:
105:
101:
94:
90:
32:
28:
20:
18:
1002:Harris, Joe
770:, one sees
1030:0960.14002
953:Borel 1991
916:References
89:-point of
39:To give a
967:(1991) ,
955:, AG 16.2
882:→
813:δ
792:δ
783:#
751:→
709:ϵ
695:ϵ
625:−
621:π
583:−
574:
566:
484:→
468:ϵ
454:ϵ
436:π
404:∈
360:∗
343:∈
321:ϵ
303:↦
283:δ
249:ϵ
235:ϵ
226:→
182:∗
67:ϵ
53:ϵ
1076:Category
1040:(1977),
1004:(1998).
943:, VI.1.3
1066:0463157
991:1102012
544:(i.e.,
393:-point
1064:
1054:
1028:
1018:
989:
979:
904:that
540:over
532:. If
508:over
1052:ISBN
1016:ISBN
977:ISBN
580:Spec
563:Spec
381:Let
1026:Zbl
559:Hom
520:at
516:to
112:is
104:of
1078::
1062:MR
1060:,
1046:,
1024:.
1014:.
1010:.
1000:;
987:MR
985:,
975:,
35:.
1032:.
910:f
906:f
890:Y
886:T
877:X
873:T
850:)
847:v
844:(
839:p
835:f
831:d
826:)
823:p
820:(
817:f
809:=
806:)
801:v
796:p
788:(
779:f
768:k
754:Y
748:X
745::
742:f
722:)
717:2
713:)
706:(
702:/
698:]
692:[
689:k
686:(
683:X
680:=
675:X
671:T
653:p
639:)
636:p
633:(
628:1
610:p
606:v
592:)
589:X
586:,
577:(
569:k
555:=
552:F
542:k
538:X
534:F
530:k
526:F
522:p
518:F
510:p
496:)
493:k
490:(
487:F
481:)
476:2
472:)
465:(
461:/
457:]
451:[
448:k
445:(
442:F
439::
416:)
413:k
410:(
407:F
401:p
391:k
387:k
383:F
377:)
365:.
355:p
349:O
340:v
336:,
333:)
330:u
327:(
324:v
318:+
315:)
312:p
309:(
306:u
300:u
297::
292:v
287:p
257:2
253:)
246:(
242:/
238:]
232:[
229:k
221:p
215:O
200:.
198:p
178:)
172:2
167:p
164:,
161:X
155:m
148:/
142:p
139:,
136:X
130:m
124:(
114:k
110:p
106:X
102:p
97:-
95:k
91:X
75:2
71:)
64:(
60:/
56:]
50:[
47:k
33:k
29:X
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.