1233:
1388:
1017:
242:
600:
1069:
105:
522:
412:
334:
691:
841:
consists of the origin with multiplicity three. That is, a scheme-theoretic multiplicity of an intersection may differ from an intersection-theoretic multiplicity, the latter given by
457:
920:
164:
1304:
749:
936:
67:
793:
1285:
1046:
839:
276:
131:
1259:
850:
1055:
For example, two divisors (codimension-one cycles) on a smooth variety intersect properly if and only if they share no common irreducible component.
1464:
1059:(on a smooth variety) says that an intersection can be made proper after replacing a divisor by a suitable linearly equivalent divisor (cf.
181:
1228:{\displaystyle X=\operatorname {Spec} k/(xz-yw),\,V=V({\overline {x}},{\overline {y}}),\,W=V({\overline {z}},{\overline {w}})}
1511:
1476:
538:
1048:
is proper if every irreducible component of it is proper (in particular, the empty intersection is considered proper.) Two
72:
474:
342:
284:
927:
842:
631:
427:
415:
1503:
815:
intersect at the origin with multiplicity two. On the other hand, one sees the scheme-theoretic intersection
880:
143:
1537:
1383:{\displaystyle \operatorname {codim} (P,X)\geq \operatorname {codim} (V,X)+\operatorname {codim} (W,X).}
1012:{\displaystyle \operatorname {codim} (P,X)\leq \operatorname {codim} (V,X)+\operatorname {codim} (W,X)}
846:
1498:
804:
704:
39:
1066:
Serre's inequality above may fail in general for a non-regular ambient scheme. For example, let
1056:
754:
1399:
1264:
1025:
818:
255:
110:
1521:
1486:
1060:
614:
8:
1238:
625:
1404:
610:
460:
17:
1507:
1472:
1493:
1052:
are said to intersect properly if the varieties in the cycles intersect properly.
1517:
1482:
1468:
1049:
1531:
1290:
Some authors such as Bloch define a proper intersection without assuming
237:{\displaystyle \operatorname {Spec} (R/I),\operatorname {Spec} (R/J)}
1467:. 3. Folge., vol. 2 (2nd ed.), Berlin, New York:
845:. Solving this disparity is one of the starting points for
873:
closed integral subschemes. Then an irreducible component
803:
at the origin with multiplicity one, by the linearity of
524:
is a hypersurface defined by some homogeneous polynomial
595:{\displaystyle X\cap H=\operatorname {Proj} (S/(I,f)).}
1307:
1267:
1241:
1072:
1028:
939:
883:
821:
757:
707:
634:
541:
477:
430:
345:
287:
258:
184:
146:
113:
100:{\displaystyle X\hookrightarrow W,Y\hookrightarrow W}
75:
42:
1294:is regular: in the notations as above, a component
799:is the union of two planes, each intersecting with
1382:
1279:
1253:
1227:
1040:
1011:
914:
833:
787:
743:
685:
620:Now, a scheme-theoretic intersection may not be a
594:
516:
451:
406:
328:
270:
236:
158:
125:
99:
61:
517:{\displaystyle H=\{f=0\}\subset \mathbb {P} ^{n}}
1529:
1465:Ergebnisse der Mathematik und ihrer Grenzgebiete
407:{\displaystyle R/I\otimes _{R}R/J\simeq R/(I+J)}
329:{\displaystyle \operatorname {Spec} (R/(I+J)).}
624:intersection, say, from the point of view of
69:, the fiber product of the closed immersions
496:
484:
1506:, vol. 52, New York: Springer-Verlag,
1492:
1423:
686:{\displaystyle W=\operatorname {Spec} (k)}
1186:
1144:
504:
452:{\displaystyle X\subset \mathbb {P} ^{n}}
439:
849:, which aims to introduce the notion of
701:closed subschemes defined by the ideals
1530:
1458:
1447:
1435:
915:{\displaystyle V\cap W:=V\times _{X}W}
856:
159:{\displaystyle \operatorname {Spec} R}
463:with the homogeneous coordinate ring
609:is linear (deg = 1), it is called a
13:
416:tensor product of modules#Examples
252:. Thus, locally, the intersection
14:
1549:
1022:is an equality. The intersection
744:{\displaystyle (x,y)\cap (z,w)}
1441:
1429:
1417:
1374:
1362:
1350:
1338:
1326:
1314:
1222:
1196:
1180:
1154:
1138:
1120:
1112:
1088:
1006:
994:
982:
970:
958:
946:
782:
758:
738:
726:
720:
708:
680:
677:
653:
647:
586:
583:
571:
560:
401:
389:
320:
317:
305:
294:
231:
217:
205:
191:
91:
79:
1:
1504:Graduate Texts in Mathematics
1410:
62:{\displaystyle X\times _{W}Y}
22:scheme-theoretic intersection
1426:, Appendix A: Example 1.1.1.
1261:have codimension one, while
1217:
1204:
1175:
1162:
7:
1393:
10:
1554:
847:derived algebraic geometry
928:inequality (due to Serre)
805:intersection multiplicity
788:{\displaystyle (x-z,y-w)}
693:= the affine 4-space and
471:is a polynomial ring. If
1459:Fulton, William (1998),
865:be a regular scheme and
414:(for this identity, see
1287:has codimension three.
1280:{\displaystyle V\cap W}
1041:{\displaystyle V\cap W}
834:{\displaystyle X\cap Y}
271:{\displaystyle X\cap Y}
126:{\displaystyle X\cap Y}
1384:
1281:
1255:
1229:
1042:
1013:
916:
835:
789:
745:
687:
596:
518:
453:
408:
330:
272:
238:
160:
127:
101:
63:
1400:complete intersection
1385:
1282:
1256:
1230:
1043:
1014:
917:
836:
790:
746:
688:
597:
519:
454:
409:
331:
273:
239:
161:
128:
102:
64:
24:of closed subschemes
1305:
1265:
1239:
1070:
1026:
937:
881:
851:derived intersection
819:
755:
705:
632:
539:
475:
428:
343:
285:
256:
182:
144:
111:
73:
40:
1461:Intersection theory
1254:{\displaystyle V,W}
1057:Chow's moving lemma
857:Proper intersection
843:Serre's Tor formula
628:. For example, let
626:intersection theory
107:. It is denoted by
1538:Algebraic geometry
1499:Algebraic Geometry
1405:Gysin homomorphism
1380:
1277:
1251:
1225:
1038:
1009:
912:
831:
785:
741:
683:
611:hyperplane section
592:
514:
461:projective variety
449:
404:
326:
268:
234:
156:
123:
97:
59:
18:algebraic geometry
1513:978-0-387-90244-9
1494:Hartshorne, Robin
1478:978-3-540-62046-4
1220:
1207:
1178:
1165:
1061:Kleiman's theorem
615:Bertini's theorem
1545:
1524:
1489:
1451:
1450:, Example 7.1.6.
1445:
1439:
1433:
1427:
1421:
1389:
1387:
1386:
1381:
1286:
1284:
1283:
1278:
1260:
1258:
1257:
1252:
1234:
1232:
1231:
1226:
1221:
1213:
1208:
1200:
1179:
1171:
1166:
1158:
1119:
1050:algebraic cycles
1047:
1045:
1044:
1039:
1018:
1016:
1015:
1010:
921:
919:
918:
913:
908:
907:
840:
838:
837:
832:
794:
792:
791:
786:
750:
748:
747:
742:
692:
690:
689:
684:
601:
599:
598:
593:
570:
523:
521:
520:
515:
513:
512:
507:
458:
456:
455:
450:
448:
447:
442:
413:
411:
410:
405:
388:
374:
366:
365:
353:
335:
333:
332:
327:
304:
277:
275:
274:
269:
244:for some ideals
243:
241:
240:
235:
227:
201:
165:
163:
162:
157:
132:
130:
129:
124:
106:
104:
103:
98:
68:
66:
65:
60:
55:
54:
1553:
1552:
1548:
1547:
1546:
1544:
1543:
1542:
1528:
1527:
1514:
1479:
1469:Springer-Verlag
1455:
1454:
1446:
1442:
1434:
1430:
1424:Hartshorne 1977
1422:
1418:
1413:
1396:
1306:
1303:
1302:
1266:
1263:
1262:
1240:
1237:
1236:
1212:
1199:
1170:
1157:
1115:
1071:
1068:
1067:
1027:
1024:
1023:
938:
935:
934:
903:
899:
882:
879:
878:
859:
820:
817:
816:
756:
753:
752:
706:
703:
702:
633:
630:
629:
566:
540:
537:
536:
508:
503:
502:
476:
473:
472:
443:
438:
437:
429:
426:
425:
384:
370:
361:
357:
349:
344:
341:
340:
300:
286:
283:
282:
257:
254:
253:
223:
197:
183:
180:
179:
145:
142:
141:
112:
109:
108:
74:
71:
70:
50:
46:
41:
38:
37:
12:
11:
5:
1551:
1541:
1540:
1526:
1525:
1512:
1490:
1477:
1453:
1452:
1440:
1428:
1415:
1414:
1412:
1409:
1408:
1407:
1402:
1395:
1392:
1391:
1390:
1379:
1376:
1373:
1370:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1276:
1273:
1270:
1250:
1247:
1244:
1224:
1219:
1216:
1211:
1206:
1203:
1198:
1195:
1192:
1189:
1185:
1182:
1177:
1174:
1169:
1164:
1161:
1156:
1153:
1150:
1147:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1118:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1037:
1034:
1031:
1020:
1019:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
963:
960:
957:
954:
951:
948:
945:
942:
911:
906:
902:
898:
895:
892:
889:
886:
858:
855:
830:
827:
824:
784:
781:
778:
775:
772:
769:
766:
763:
760:
740:
737:
734:
731:
728:
725:
722:
719:
716:
713:
710:
682:
679:
676:
673:
670:
667:
664:
661:
658:
655:
652:
649:
646:
643:
640:
637:
603:
602:
591:
588:
585:
582:
579:
576:
573:
569:
565:
562:
559:
556:
553:
550:
547:
544:
511:
506:
501:
498:
495:
492:
489:
486:
483:
480:
446:
441:
436:
433:
403:
400:
397:
394:
391:
387:
383:
380:
377:
373:
369:
364:
360:
356:
352:
348:
339:Here, we used
337:
336:
325:
322:
319:
316:
313:
310:
307:
303:
299:
296:
293:
290:
267:
264:
261:
233:
230:
226:
222:
219:
216:
213:
210:
207:
204:
200:
196:
193:
190:
187:
166:for some ring
155:
152:
149:
122:
119:
116:
96:
93:
90:
87:
84:
81:
78:
58:
53:
49:
45:
9:
6:
4:
3:
2:
1550:
1539:
1536:
1535:
1533:
1523:
1519:
1515:
1509:
1505:
1501:
1500:
1495:
1491:
1488:
1484:
1480:
1474:
1470:
1466:
1462:
1457:
1456:
1449:
1444:
1437:
1432:
1425:
1420:
1416:
1406:
1403:
1401:
1398:
1397:
1377:
1371:
1368:
1365:
1359:
1356:
1353:
1347:
1344:
1341:
1335:
1332:
1329:
1323:
1320:
1317:
1311:
1308:
1301:
1300:
1299:
1298:is proper if
1297:
1293:
1288:
1274:
1271:
1268:
1248:
1245:
1242:
1214:
1209:
1201:
1193:
1190:
1187:
1183:
1172:
1167:
1159:
1151:
1148:
1145:
1141:
1135:
1132:
1129:
1126:
1123:
1116:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1085:
1082:
1079:
1076:
1073:
1064:
1062:
1058:
1053:
1051:
1035:
1032:
1029:
1003:
1000:
997:
991:
988:
985:
979:
976:
973:
967:
964:
961:
955:
952:
949:
943:
940:
933:
932:
931:
929:
925:
909:
904:
900:
896:
893:
890:
887:
884:
876:
872:
868:
864:
854:
852:
848:
844:
828:
825:
822:
814:
810:
806:
802:
798:
779:
776:
773:
770:
767:
764:
761:
735:
732:
729:
723:
717:
714:
711:
700:
696:
674:
671:
668:
665:
662:
659:
656:
650:
644:
641:
638:
635:
627:
623:
618:
616:
612:
608:
589:
580:
577:
574:
567:
563:
557:
554:
551:
548:
545:
542:
535:
534:
533:
531:
527:
509:
499:
493:
490:
487:
481:
478:
470:
466:
462:
444:
434:
431:
423:
419:
417:
398:
395:
392:
385:
381:
378:
375:
371:
367:
362:
358:
354:
350:
346:
323:
314:
311:
308:
301:
297:
291:
288:
281:
280:
279:
265:
262:
259:
251:
247:
228:
224:
220:
214:
211:
208:
202:
198:
194:
188:
185:
177:
173:
169:
153:
150:
147:
139:
134:
120:
117:
114:
94:
88:
85:
82:
76:
56:
51:
47:
43:
35:
31:
27:
23:
19:
1497:
1460:
1443:
1431:
1419:
1295:
1291:
1289:
1065:
1054:
1021:
923:
874:
870:
866:
862:
860:
812:
808:
807:, we expect
800:
796:
698:
694:
621:
619:
613:. See also:
606:
604:
529:
525:
468:
464:
421:
420:
338:
278:is given as
249:
245:
175:
171:
167:
140:is given as
137:
135:
33:
32:of a scheme
29:
25:
21:
15:
1448:Fulton 1998
1436:Fulton 1998
1411:References
922:is called
1438:, § 20.4.
1360:
1336:
1330:≥
1312:
1272:∩
1218:¯
1205:¯
1176:¯
1163:¯
1130:−
1083:
1033:∩
992:
968:
962:≤
944:
901:×
888:∩
826:∩
777:−
765:−
724:∩
645:
558:
546:∩
500:⊂
435:⊂
379:≃
359:⊗
292:
263:∩
215:
189:
151:
136:Locally,
118:∩
92:↪
80:↪
48:×
1532:Category
1496:(1977),
1394:See also
795:. Since
467:, where
1522:0463157
1487:1644323
1235:. Then
926:if the
622:correct
532:, then
422:Example
1520:
1510:
1485:
1475:
924:proper
424:: Let
20:, the
1357:codim
1333:codim
1309:codim
989:codim
965:codim
941:codim
459:be a
1508:ISBN
1473:ISBN
1080:Spec
861:Let
811:and
751:and
642:Spec
555:Proj
289:Spec
212:Spec
186:Spec
170:and
148:Spec
1063:.)
877:of
605:If
528:in
465:S/I
418:.)
178:as
36:is
16:In
1534::
1518:MR
1516:,
1502:,
1483:MR
1481:,
1471:,
1463:,
930::
894::=
869:,
853:.
697:,
617:.
248:,
174:,
133:.
28:,
1378:.
1375:)
1372:X
1369:,
1366:W
1363:(
1354:+
1351:)
1348:X
1345:,
1342:V
1339:(
1327:)
1324:X
1321:,
1318:P
1315:(
1296:P
1292:X
1275:W
1269:V
1249:W
1246:,
1243:V
1223:)
1215:w
1210:,
1202:z
1197:(
1194:V
1191:=
1188:W
1184:,
1181:)
1173:y
1168:,
1160:x
1155:(
1152:V
1149:=
1146:V
1142:,
1139:)
1136:w
1133:y
1127:z
1124:x
1121:(
1117:/
1113:]
1110:w
1107:,
1104:z
1101:,
1098:y
1095:,
1092:x
1089:[
1086:k
1077:=
1074:X
1036:W
1030:V
1007:)
1004:X
1001:,
998:W
995:(
986:+
983:)
980:X
977:,
974:V
971:(
959:)
956:X
953:,
950:P
947:(
910:W
905:X
897:V
891:W
885:V
875:P
871:W
867:V
863:X
829:Y
823:X
813:Y
809:X
801:Y
797:X
783:)
780:w
774:y
771:,
768:z
762:x
759:(
739:)
736:w
733:,
730:z
727:(
721:)
718:y
715:,
712:x
709:(
699:Y
695:X
681:)
678:]
675:w
672:,
669:z
666:,
663:y
660:,
657:x
654:[
651:k
648:(
639:=
636:W
607:f
590:.
587:)
584:)
581:f
578:,
575:I
572:(
568:/
564:S
561:(
552:=
549:H
543:X
530:S
526:f
510:n
505:P
497:}
494:0
491:=
488:f
485:{
482:=
479:H
469:S
445:n
440:P
432:X
402:)
399:J
396:+
393:I
390:(
386:/
382:R
376:J
372:/
368:R
363:R
355:I
351:/
347:R
324:.
321:)
318:)
315:J
312:+
309:I
306:(
302:/
298:R
295:(
266:Y
260:X
250:J
246:I
232:)
229:J
225:/
221:R
218:(
209:,
206:)
203:I
199:/
195:R
192:(
176:Y
172:X
168:R
154:R
138:W
121:Y
115:X
95:W
89:Y
86:,
83:W
77:X
57:Y
52:W
44:X
34:W
30:Y
26:X
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