1244:
1399:
1028:
253:
611:
1080:
116:
533:
423:
345:
702:
852:
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
468:
931:
175:
1315:
760:
947:
78:
804:
1296:
1057:
850:
287:
142:
1270:
861:
1066:
For example, two divisors (codimension-one cycles) on a smooth variety intersect properly if and only if they share no common irreducible component.
1475:
1070:(on a smooth variety) says that an intersection can be made proper after replacing a divisor by a suitable linearly equivalent divisor (cf.
192:
1239:{\displaystyle X=\operatorname {Spec} k/(xz-yw),\,V=V({\overline {x}},{\overline {y}}),\,W=V({\overline {z}},{\overline {w}})}
1522:
1487:
549:
1059:
is proper if every irreducible component of it is proper (in particular, the empty intersection is considered proper.) Two
83:
17:
485:
353:
295:
938:
853:
642:
438:
426:
1514:
826:
intersect at the origin with multiplicity two. On the other hand, one sees the scheme-theoretic intersection
891:
154:
1548:
1394:{\displaystyle \operatorname {codim} (P,X)\geq \operatorname {codim} (V,X)+\operatorname {codim} (W,X).}
1023:{\displaystyle \operatorname {codim} (P,X)\leq \operatorname {codim} (V,X)+\operatorname {codim} (W,X)}
857:
1509:
815:
715:
50:
1077:
Serre's inequality above may fail in general for a non-regular ambient scheme. For example, let
1067:
765:
1410:
1275:
1036:
829:
266:
121:
1532:
1497:
1071:
625:
8:
1249:
636:
1415:
621:
471:
28:
1518:
1483:
1504:
1063:
are said to intersect properly if the varieties in the cycles intersect properly.
1528:
1493:
1479:
1060:
1542:
1301:
Some authors such as Bloch define a proper intersection without assuming
248:{\displaystyle \operatorname {Spec} (R/I),\operatorname {Spec} (R/J)}
1478:. 3. Folge., vol. 2 (2nd ed.), Berlin, New York:
856:. Solving this disparity is one of the starting points for
884:
closed integral subschemes. Then an irreducible component
814:
at the origin with multiplicity one, by the linearity of
535:
is a hypersurface defined by some homogeneous polynomial
606:{\displaystyle X\cap H=\operatorname {Proj} (S/(I,f)).}
1318:
1278:
1252:
1083:
1039:
950:
894:
832:
768:
718:
645:
552:
488:
441:
356:
298:
269:
195:
157:
124:
111:{\displaystyle X\hookrightarrow W,Y\hookrightarrow W}
86:
53:
1305:is regular: in the notations as above, a component
810:is the union of two planes, each intersecting with
1393:
1290:
1264:
1238:
1051:
1022:
925:
844:
798:
754:
696:
631:Now, a scheme-theoretic intersection may not be a
605:
527:
462:
417:
339:
281:
247:
169:
136:
110:
72:
528:{\displaystyle H=\{f=0\}\subset \mathbb {P} ^{n}}
1540:
1476:Ergebnisse der Mathematik und ihrer Grenzgebiete
418:{\displaystyle R/I\otimes _{R}R/J\simeq R/(I+J)}
340:{\displaystyle \operatorname {Spec} (R/(I+J)).}
635:intersection, say, from the point of view of
80:, the fiber product of the closed immersions
507:
495:
1517:, vol. 52, New York: Springer-Verlag,
1503:
1434:
697:{\displaystyle W=\operatorname {Spec} (k)}
1197:
1155:
515:
463:{\displaystyle X\subset \mathbb {P} ^{n}}
450:
860:, which aims to introduce the notion of
712:closed subschemes defined by the ideals
14:
1541:
1469:
1458:
1446:
926:{\displaystyle V\cap W:=V\times _{X}W}
867:
170:{\displaystyle \operatorname {Spec} R}
474:with the homogeneous coordinate ring
620:is linear (deg = 1), it is called a
24:
427:tensor product of modules#Examples
263:. Thus, locally, the intersection
25:
1560:
1033:is an equality. The intersection
755:{\displaystyle (x,y)\cap (z,w)}
1452:
1440:
1428:
1385:
1373:
1361:
1349:
1337:
1325:
1233:
1207:
1191:
1165:
1149:
1131:
1123:
1099:
1017:
1005:
993:
981:
969:
957:
793:
769:
749:
737:
731:
719:
691:
688:
664:
658:
597:
594:
582:
571:
412:
400:
331:
328:
316:
305:
242:
228:
216:
202:
102:
90:
13:
1:
1515:Graduate Texts in Mathematics
1421:
73:{\displaystyle X\times _{W}Y}
33:scheme-theoretic intersection
1437:, Appendix A: Example 1.1.1.
1272:have codimension one, while
1228:
1215:
1186:
1173:
7:
1404:
10:
1565:
858:derived algebraic geometry
939:inequality (due to Serre)
816:intersection multiplicity
799:{\displaystyle (x-z,y-w)}
704:= the affine 4-space and
482:is a polynomial ring. If
1470:Fulton, William (1998),
876:be a regular scheme and
425:(for this identity, see
1298:has codimension three.
1291:{\displaystyle V\cap W}
1052:{\displaystyle V\cap W}
845:{\displaystyle X\cap Y}
282:{\displaystyle X\cap Y}
137:{\displaystyle X\cap Y}
1395:
1292:
1266:
1240:
1053:
1024:
927:
846:
800:
756:
698:
607:
529:
464:
419:
341:
283:
249:
171:
138:
112:
74:
1411:complete intersection
1396:
1293:
1267:
1241:
1054:
1025:
928:
847:
801:
757:
699:
608:
530:
465:
420:
342:
284:
250:
172:
139:
113:
75:
35:of closed subschemes
1316:
1276:
1250:
1081:
1037:
948:
892:
862:derived intersection
830:
766:
716:
643:
550:
486:
439:
354:
296:
267:
193:
155:
122:
84:
51:
1472:Intersection theory
1265:{\displaystyle V,W}
1068:Chow's moving lemma
868:Proper intersection
854:Serre's Tor formula
639:. For example, let
637:intersection theory
118:. It is denoted by
18:Proper intersection
1549:Algebraic geometry
1510:Algebraic Geometry
1416:Gysin homomorphism
1391:
1288:
1262:
1236:
1049:
1020:
923:
842:
796:
752:
694:
622:hyperplane section
603:
525:
472:projective variety
460:
415:
337:
279:
245:
167:
134:
108:
70:
29:algebraic geometry
1524:978-0-387-90244-9
1505:Hartshorne, Robin
1489:978-3-540-62046-4
1231:
1218:
1189:
1176:
1072:Kleiman's theorem
626:Bertini's theorem
16:(Redirected from
1556:
1535:
1500:
1462:
1461:, Example 7.1.6.
1456:
1450:
1444:
1438:
1432:
1400:
1398:
1397:
1392:
1297:
1295:
1294:
1289:
1271:
1269:
1268:
1263:
1245:
1243:
1242:
1237:
1232:
1224:
1219:
1211:
1190:
1182:
1177:
1169:
1130:
1061:algebraic cycles
1058:
1056:
1055:
1050:
1029:
1027:
1026:
1021:
932:
930:
929:
924:
919:
918:
851:
849:
848:
843:
805:
803:
802:
797:
761:
759:
758:
753:
703:
701:
700:
695:
612:
610:
609:
604:
581:
534:
532:
531:
526:
524:
523:
518:
469:
467:
466:
461:
459:
458:
453:
424:
422:
421:
416:
399:
385:
377:
376:
364:
346:
344:
343:
338:
315:
288:
286:
285:
280:
255:for some ideals
254:
252:
251:
246:
238:
212:
176:
174:
173:
168:
143:
141:
140:
135:
117:
115:
114:
109:
79:
77:
76:
71:
66:
65:
21:
1564:
1563:
1559:
1558:
1557:
1555:
1554:
1553:
1539:
1538:
1525:
1490:
1480:Springer-Verlag
1466:
1465:
1457:
1453:
1445:
1441:
1435:Hartshorne 1977
1433:
1429:
1424:
1407:
1317:
1314:
1313:
1277:
1274:
1273:
1251:
1248:
1247:
1223:
1210:
1181:
1168:
1126:
1082:
1079:
1078:
1038:
1035:
1034:
949:
946:
945:
914:
910:
893:
890:
889:
870:
831:
828:
827:
767:
764:
763:
717:
714:
713:
644:
641:
640:
577:
551:
548:
547:
519:
514:
513:
487:
484:
483:
454:
449:
448:
440:
437:
436:
395:
381:
372:
368:
360:
355:
352:
351:
311:
297:
294:
293:
268:
265:
264:
234:
208:
194:
191:
190:
156:
153:
152:
123:
120:
119:
85:
82:
81:
61:
57:
52:
49:
48:
23:
22:
15:
12:
11:
5:
1562:
1552:
1551:
1537:
1536:
1523:
1501:
1488:
1464:
1463:
1451:
1439:
1426:
1425:
1423:
1420:
1419:
1418:
1413:
1406:
1403:
1402:
1401:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1287:
1284:
1281:
1261:
1258:
1255:
1235:
1230:
1227:
1222:
1217:
1214:
1209:
1206:
1203:
1200:
1196:
1193:
1188:
1185:
1180:
1175:
1172:
1167:
1164:
1161:
1158:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1129:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1048:
1045:
1042:
1031:
1030:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
989:
986:
983:
980:
977:
974:
971:
968:
965:
962:
959:
956:
953:
922:
917:
913:
909:
906:
903:
900:
897:
869:
866:
841:
838:
835:
795:
792:
789:
786:
783:
780:
777:
774:
771:
751:
748:
745:
742:
739:
736:
733:
730:
727:
724:
721:
693:
690:
687:
684:
681:
678:
675:
672:
669:
666:
663:
660:
657:
654:
651:
648:
614:
613:
602:
599:
596:
593:
590:
587:
584:
580:
576:
573:
570:
567:
564:
561:
558:
555:
522:
517:
512:
509:
506:
503:
500:
497:
494:
491:
457:
452:
447:
444:
414:
411:
408:
405:
402:
398:
394:
391:
388:
384:
380:
375:
371:
367:
363:
359:
350:Here, we used
348:
347:
336:
333:
330:
327:
324:
321:
318:
314:
310:
307:
304:
301:
278:
275:
272:
244:
241:
237:
233:
230:
227:
224:
221:
218:
215:
211:
207:
204:
201:
198:
177:for some ring
166:
163:
160:
133:
130:
127:
107:
104:
101:
98:
95:
92:
89:
69:
64:
60:
56:
9:
6:
4:
3:
2:
1561:
1550:
1547:
1546:
1544:
1534:
1530:
1526:
1520:
1516:
1512:
1511:
1506:
1502:
1499:
1495:
1491:
1485:
1481:
1477:
1473:
1468:
1467:
1460:
1455:
1448:
1443:
1436:
1431:
1427:
1417:
1414:
1412:
1409:
1408:
1388:
1382:
1379:
1376:
1370:
1367:
1364:
1358:
1355:
1352:
1346:
1343:
1340:
1334:
1331:
1328:
1322:
1319:
1312:
1311:
1310:
1309:is proper if
1308:
1304:
1299:
1285:
1282:
1279:
1259:
1256:
1253:
1225:
1220:
1212:
1204:
1201:
1198:
1194:
1183:
1178:
1170:
1162:
1159:
1156:
1152:
1146:
1143:
1140:
1137:
1134:
1127:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1096:
1093:
1090:
1087:
1084:
1075:
1073:
1069:
1064:
1062:
1046:
1043:
1040:
1014:
1011:
1008:
1002:
999:
996:
990:
987:
984:
978:
975:
972:
966:
963:
960:
954:
951:
944:
943:
942:
940:
936:
920:
915:
911:
907:
904:
901:
898:
895:
887:
883:
879:
875:
865:
863:
859:
855:
839:
836:
833:
825:
821:
817:
813:
809:
790:
787:
784:
781:
778:
775:
772:
746:
743:
740:
734:
728:
725:
722:
711:
707:
685:
682:
679:
676:
673:
670:
667:
661:
655:
652:
649:
646:
638:
634:
629:
627:
623:
619:
600:
591:
588:
585:
578:
574:
568:
565:
562:
559:
556:
553:
546:
545:
544:
542:
538:
520:
510:
504:
501:
498:
492:
489:
481:
477:
473:
455:
445:
442:
434:
430:
428:
409:
406:
403:
396:
392:
389:
386:
382:
378:
373:
369:
365:
361:
357:
334:
325:
322:
319:
312:
308:
302:
299:
292:
291:
290:
276:
273:
270:
262:
258:
239:
235:
231:
225:
222:
219:
213:
209:
205:
199:
196:
188:
184:
180:
164:
161:
158:
150:
145:
131:
128:
125:
105:
99:
96:
93:
87:
67:
62:
58:
54:
46:
42:
38:
34:
30:
19:
1508:
1471:
1454:
1442:
1430:
1306:
1302:
1300:
1076:
1065:
1032:
934:
885:
881:
877:
873:
871:
823:
819:
818:, we expect
811:
807:
709:
705:
632:
630:
624:. See also:
617:
615:
540:
536:
479:
475:
432:
431:
349:
289:is given as
260:
256:
186:
182:
178:
151:is given as
148:
146:
44:
43:of a scheme
40:
36:
32:
26:
1459:Fulton 1998
1447:Fulton 1998
1422:References
933:is called
1449:, § 20.4.
1371:
1347:
1341:≥
1323:
1283:∩
1229:¯
1216:¯
1187:¯
1174:¯
1141:−
1094:
1044:∩
1003:
979:
973:≤
955:
912:×
899:∩
837:∩
788:−
776:−
735:∩
656:
569:
557:∩
511:⊂
446:⊂
390:≃
370:⊗
303:
274:∩
226:
200:
162:
147:Locally,
129:∩
103:↪
91:↪
59:×
1543:Category
1507:(1977),
1405:See also
806:. Since
478:, where
1533:0463157
1498:1644323
1246:. Then
937:if the
633:correct
543:, then
433:Example
1531:
1521:
1496:
1486:
935:proper
435:: Let
31:, the
1368:codim
1344:codim
1320:codim
1000:codim
976:codim
952:codim
470:be a
1519:ISBN
1484:ISBN
1091:Spec
872:Let
822:and
762:and
653:Spec
566:Proj
300:Spec
223:Spec
197:Spec
181:and
159:Spec
1074:.)
888:of
616:If
539:in
476:S/I
429:.)
189:as
47:is
27:In
1545::
1529:MR
1527:,
1513:,
1494:MR
1492:,
1482:,
1474:,
941::
905::=
880:,
864:.
708:,
628:.
259:,
185:,
144:.
39:,
1389:.
1386:)
1383:X
1380:,
1377:W
1374:(
1365:+
1362:)
1359:X
1356:,
1353:V
1350:(
1338:)
1335:X
1332:,
1329:P
1326:(
1307:P
1303:X
1286:W
1280:V
1260:W
1257:,
1254:V
1234:)
1226:w
1221:,
1213:z
1208:(
1205:V
1202:=
1199:W
1195:,
1192:)
1184:y
1179:,
1171:x
1166:(
1163:V
1160:=
1157:V
1153:,
1150:)
1147:w
1144:y
1138:z
1135:x
1132:(
1128:/
1124:]
1121:w
1118:,
1115:z
1112:,
1109:y
1106:,
1103:x
1100:[
1097:k
1088:=
1085:X
1047:W
1041:V
1018:)
1015:X
1012:,
1009:W
1006:(
997:+
994:)
991:X
988:,
985:V
982:(
970:)
967:X
964:,
961:P
958:(
921:W
916:X
908:V
902:W
896:V
886:P
882:W
878:V
874:X
840:Y
834:X
824:Y
820:X
812:Y
808:X
794:)
791:w
785:y
782:,
779:z
773:x
770:(
750:)
747:w
744:,
741:z
738:(
732:)
729:y
726:,
723:x
720:(
710:Y
706:X
692:)
689:]
686:w
683:,
680:z
677:,
674:y
671:,
668:x
665:[
662:k
659:(
650:=
647:W
618:f
601:.
598:)
595:)
592:f
589:,
586:I
583:(
579:/
575:S
572:(
563:=
560:H
554:X
541:S
537:f
521:n
516:P
508:}
505:0
502:=
499:f
496:{
493:=
490:H
480:S
456:n
451:P
443:X
413:)
410:J
407:+
404:I
401:(
397:/
393:R
387:J
383:/
379:R
374:R
366:I
362:/
358:R
335:.
332:)
329:)
326:J
323:+
320:I
317:(
313:/
309:R
306:(
277:Y
271:X
261:J
257:I
243:)
240:J
236:/
232:R
229:(
220:,
217:)
214:I
210:/
206:R
203:(
187:Y
183:X
179:R
165:R
149:W
132:Y
126:X
106:W
100:Y
97:,
94:W
88:X
68:Y
63:W
55:X
45:W
41:Y
37:X
20:)
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