997:
and the depth of a module over a commutative
Noetherian local ring are complementary to each other. This is the content of the Auslander–Buchsbaum formula, which is not only of fundamental theoretical importance, but also provides an effective way to compute the depth of a module. Suppose that
554:
1216:
116:
848:
1345:
1389:
1268:
1047:
963:
919:
759:
625:
601:
1461:
145:
895:
1507:
370:
307:
1481:
1365:
1312:
1292:
1244:
1107:
1087:
1067:
1016:
983:
939:
868:
799:
779:
728:
689:
669:
645:
577:
438:
414:
390:
347:
327:
284:
261:
241:
221:
201:
169:
446:
1119:
1543:
1509:) with an embedded double point at the origin, has depth zero at the origin, but dimension one: this gives an example of a ring which is not
1568:
1110:
48:
1392:
1531:
36:. Although depth can be defined more generally, the most common case considered is the case of modules over a commutative
1588:
57:
802:
699:
695:
264:
1510:
648:
1583:
1563:. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp.
807:
1317:
1370:
1249:
1028:
994:
944:
900:
740:
606:
582:
44:
33:
1023:
735:
172:
21:
17:
1401:
29:
1564:
1539:
1271:
124:
1553:
873:
1549:
1535:
171:. Depth is used to define classes of rings and modules with good properties, for example,
148:
37:
1486:
549:{\displaystyle \mathrm {depth} _{I}(M)=\min\{i:\operatorname {Ext} ^{i}(R/I,M)\neq 0\}.}
352:
289:
1523:
1466:
1350:
1297:
1277:
1229:
1092:
1072:
1052:
1001:
968:
924:
853:
784:
764:
713:
674:
654:
630:
562:
423:
399:
375:
332:
312:
269:
246:
226:
206:
186:
154:
1577:
1019:
731:
40:
1211:{\displaystyle \mathrm {pd} _{R}(M)+\mathrm {depth} (M)=\mathrm {depth} (R).}
698:, the depth can also be characterized using the notion of a
43:. In this case, the depth of a module is related with its
1528:
Commutative algebra with a view toward algebraic geometry
1391:). This means, essentially, that the closed point is an
51:. A more elementary property of depth is the inequality
1489:
1469:
1404:
1373:
1353:
1320:
1300:
1280:
1252:
1232:
1122:
1095:
1075:
1055:
1031:
1004:
971:
947:
927:
903:
876:
856:
810:
787:
767:
743:
716:
677:
657:
633:
609:
585:
565:
449:
426:
402:
378:
355:
335:
315:
292:
272:
249:
229:
209:
189:
157:
127:
60:
1274:, or, equivalently, when there is a nonzero element
1501:
1475:
1455:
1383:
1359:
1339:
1306:
1286:
1262:
1238:
1210:
1101:
1081:
1061:
1041:
1010:
977:
957:
933:
913:
889:
862:
842:
793:
773:
753:
722:
683:
663:
639:
619:
595:
571:
548:
432:
408:
384:
364:
341:
321:
301:
278:
255:
235:
215:
195:
163:
139:
110:
1246:has depth zero if and only if its maximal ideal
486:
111:{\displaystyle \mathrm {depth} (M)\leq \dim(M),}
8:
540:
489:
1488:
1468:
1435:
1423:
1403:
1375:
1374:
1372:
1352:
1325:
1324:
1319:
1299:
1279:
1254:
1253:
1251:
1231:
1179:
1150:
1132:
1124:
1121:
1094:
1074:
1054:
1033:
1032:
1030:
1003:
970:
949:
948:
946:
926:
905:
904:
902:
881:
875:
855:
834:
815:
809:
786:
766:
745:
744:
742:
715:
676:
656:
632:
611:
610:
608:
587:
586:
584:
564:
559:By definition, the depth of a local ring
517:
502:
468:
451:
448:
425:
401:
377:
354:
334:
314:
291:
271:
248:
228:
208:
188:
156:
126:
61:
59:
1089:-module. If the projective dimension of
175:and modules, for which equality holds.
1483:is a field), which represents a line (
7:
1226:A commutative Noetherian local ring
1534:, vol. 150, Berlin, New York:
1376:
1326:
1255:
1034:
950:
906:
843:{\displaystyle x_{1},\ldots ,x_{n}}
746:
627:-depth as a module over itself. If
612:
588:
1340:{\displaystyle x{\mathfrak {m}}=0}
1192:
1189:
1186:
1183:
1180:
1163:
1160:
1157:
1154:
1151:
1128:
1125:
464:
461:
458:
455:
452:
74:
71:
68:
65:
62:
14:
1559:Winfried Bruns; JĂĽrgen Herzog,
1384:{\displaystyle {\mathfrak {m}}}
1263:{\displaystyle {\mathfrak {m}}}
1042:{\displaystyle {\mathfrak {m}}}
958:{\displaystyle {\mathfrak {m}}}
914:{\displaystyle {\mathfrak {m}}}
754:{\displaystyle {\mathfrak {m}}}
620:{\displaystyle {\mathfrak {m}}}
596:{\displaystyle {\mathfrak {m}}}
286:-module with the property that
1450:
1428:
1420:
1408:
1202:
1196:
1173:
1167:
1144:
1138:
989:Depth and projective dimension
531:
511:
480:
474:
102:
96:
84:
78:
1:
1532:Graduate Texts in Mathematics
671:is equal to the dimension of
329:. (That is, some elements of
28:is an important invariant of
1456:{\displaystyle k/(x^{2},xy)}
1018:is a commutative Noetherian
730:is a commutative Noetherian
1111:Auslander–Buchsbaum formula
416:, also commonly called the
49:Auslander–Buchsbaum formula
1605:
801:-module. Then all maximal
651:local ring, then depth of
309:is properly contained in
1069:is a finitely generated
781:is a finitely generated
921:, have the same length
203:be a commutative ring,
1503:
1477:
1457:
1398:For example, the ring
1385:
1361:
1341:
1308:
1288:
1264:
1240:
1212:
1103:
1083:
1063:
1043:
1012:
979:
959:
935:
915:
891:
864:
844:
795:
775:
755:
724:
685:
665:
641:
621:
597:
573:
550:
434:
410:
386:
366:
343:
323:
303:
280:
257:
237:
217:
197:
165:
141:
140:{\displaystyle \dim M}
112:
1504:
1478:
1458:
1386:
1362:
1342:
1309:
1289:
1265:
1241:
1213:
1104:
1084:
1064:
1044:
1013:
980:
960:
936:
916:
892:
890:{\displaystyle x_{i}}
865:
845:
796:
776:
756:
725:
686:
666:
642:
622:
598:
579:with a maximal ideal
574:
551:
435:
411:
387:
367:
344:
324:
304:
281:
258:
238:
218:
198:
166:
142:
113:
1561:Cohen–Macaulay rings
1487:
1467:
1402:
1371:
1351:
1318:
1298:
1278:
1250:
1230:
1120:
1109:is finite, then the
1093:
1073:
1053:
1029:
1002:
995:projective dimension
969:
945:
925:
901:
874:
854:
808:
785:
765:
741:
714:
675:
655:
631:
607:
583:
563:
447:
424:
400:
376:
353:
333:
313:
290:
270:
247:
227:
207:
187:
173:Cohen-Macaulay rings
155:
125:
58:
45:projective dimension
1589:Commutative algebra
1502:{\displaystyle x=0}
1499:
1473:
1453:
1393:embedded component
1381:
1357:
1337:
1304:
1284:
1260:
1236:
1208:
1099:
1079:
1059:
1039:
1008:
975:
955:
931:
911:
887:
860:
840:
791:
771:
751:
720:
681:
661:
637:
617:
593:
569:
546:
430:
406:
382:
365:{\displaystyle IM}
362:
339:
319:
302:{\displaystyle IM}
299:
276:
265:finitely generated
253:
233:
213:
193:
161:
137:
108:
1545:978-0-387-94269-8
1476:{\displaystyle k}
1360:{\displaystyle x}
1307:{\displaystyle R}
1287:{\displaystyle x}
1239:{\displaystyle R}
1102:{\displaystyle M}
1082:{\displaystyle R}
1062:{\displaystyle M}
1022:with the maximal
1011:{\displaystyle R}
978:{\displaystyle M}
934:{\displaystyle n}
863:{\displaystyle M}
803:regular sequences
794:{\displaystyle R}
774:{\displaystyle M}
734:with the maximal
723:{\displaystyle R}
684:{\displaystyle R}
664:{\displaystyle R}
640:{\displaystyle R}
572:{\displaystyle R}
440:, is defined as
433:{\displaystyle M}
409:{\displaystyle M}
385:{\displaystyle I}
342:{\displaystyle M}
322:{\displaystyle M}
279:{\displaystyle R}
256:{\displaystyle M}
236:{\displaystyle R}
216:{\displaystyle I}
196:{\displaystyle R}
164:{\displaystyle M}
1596:
1556:
1508:
1506:
1505:
1500:
1482:
1480:
1479:
1474:
1462:
1460:
1459:
1454:
1440:
1439:
1427:
1390:
1388:
1387:
1382:
1380:
1379:
1366:
1364:
1363:
1358:
1346:
1344:
1343:
1338:
1330:
1329:
1313:
1311:
1310:
1305:
1293:
1291:
1290:
1285:
1272:associated prime
1269:
1267:
1266:
1261:
1259:
1258:
1245:
1243:
1242:
1237:
1222:Depth zero rings
1217:
1215:
1214:
1209:
1195:
1166:
1137:
1136:
1131:
1108:
1106:
1105:
1100:
1088:
1086:
1085:
1080:
1068:
1066:
1065:
1060:
1048:
1046:
1045:
1040:
1038:
1037:
1017:
1015:
1014:
1009:
984:
982:
981:
976:
964:
962:
961:
956:
954:
953:
940:
938:
937:
932:
920:
918:
917:
912:
910:
909:
896:
894:
893:
888:
886:
885:
869:
867:
866:
861:
849:
847:
846:
841:
839:
838:
820:
819:
800:
798:
797:
792:
780:
778:
777:
772:
760:
758:
757:
752:
750:
749:
729:
727:
726:
721:
700:regular sequence
694:By a theorem of
690:
688:
687:
682:
670:
668:
667:
662:
646:
644:
643:
638:
626:
624:
623:
618:
616:
615:
602:
600:
599:
594:
592:
591:
578:
576:
575:
570:
555:
553:
552:
547:
521:
507:
506:
473:
472:
467:
439:
437:
436:
431:
415:
413:
412:
407:
391:
389:
388:
383:
371:
369:
368:
363:
348:
346:
345:
340:
328:
326:
325:
320:
308:
306:
305:
300:
285:
283:
282:
277:
262:
260:
259:
254:
242:
240:
239:
234:
222:
220:
219:
214:
202:
200:
199:
194:
170:
168:
167:
162:
146:
144:
143:
138:
117:
115:
114:
109:
77:
1604:
1603:
1599:
1598:
1597:
1595:
1594:
1593:
1574:
1573:
1546:
1536:Springer-Verlag
1524:Eisenbud, David
1522:
1519:
1485:
1484:
1465:
1464:
1431:
1400:
1399:
1369:
1368:
1349:
1348:
1316:
1315:
1296:
1295:
1276:
1275:
1248:
1247:
1228:
1227:
1224:
1123:
1118:
1117:
1091:
1090:
1071:
1070:
1051:
1050:
1027:
1026:
1000:
999:
991:
967:
966:
943:
942:
923:
922:
899:
898:
877:
872:
871:
852:
851:
830:
811:
806:
805:
783:
782:
763:
762:
739:
738:
712:
711:
708:
673:
672:
653:
652:
629:
628:
605:
604:
581:
580:
561:
560:
498:
450:
445:
444:
422:
421:
398:
397:
374:
373:
351:
350:
331:
330:
311:
310:
288:
287:
268:
267:
245:
244:
225:
224:
205:
204:
185:
184:
181:
153:
152:
149:Krull dimension
123:
122:
56:
55:
12:
11:
5:
1602:
1600:
1592:
1591:
1586:
1576:
1575:
1572:
1571:
1557:
1544:
1518:
1515:
1511:Cohen–Macaulay
1498:
1495:
1492:
1472:
1452:
1449:
1446:
1443:
1438:
1434:
1430:
1426:
1422:
1419:
1416:
1413:
1410:
1407:
1378:
1356:
1336:
1333:
1328:
1323:
1303:
1283:
1257:
1235:
1223:
1220:
1219:
1218:
1207:
1204:
1201:
1198:
1194:
1191:
1188:
1185:
1182:
1178:
1175:
1172:
1169:
1165:
1162:
1159:
1156:
1153:
1149:
1146:
1143:
1140:
1135:
1130:
1127:
1098:
1078:
1058:
1036:
1007:
990:
987:
974:
952:
930:
908:
884:
880:
859:
837:
833:
829:
826:
823:
818:
814:
790:
770:
748:
719:
707:
706:Theorem (Rees)
704:
680:
660:
649:Cohen-Macaulay
636:
614:
590:
568:
557:
556:
545:
542:
539:
536:
533:
530:
527:
524:
520:
516:
513:
510:
505:
501:
497:
494:
491:
488:
485:
482:
479:
476:
471:
466:
463:
460:
457:
454:
429:
405:
381:
361:
358:
338:
318:
298:
295:
275:
252:
232:
212:
192:
180:
177:
160:
151:of the module
136:
133:
130:
119:
118:
107:
104:
101:
98:
95:
92:
89:
86:
83:
80:
76:
73:
70:
67:
64:
13:
10:
9:
6:
4:
3:
2:
1601:
1590:
1587:
1585:
1584:Module theory
1582:
1581:
1579:
1570:
1569:0-521-41068-1
1566:
1562:
1558:
1555:
1551:
1547:
1541:
1537:
1533:
1529:
1525:
1521:
1520:
1516:
1514:
1512:
1496:
1493:
1490:
1470:
1447:
1444:
1441:
1436:
1432:
1424:
1417:
1414:
1411:
1405:
1396:
1394:
1354:
1334:
1331:
1321:
1301:
1281:
1273:
1233:
1221:
1205:
1199:
1176:
1170:
1147:
1141:
1133:
1116:
1115:
1114:
1112:
1096:
1076:
1056:
1025:
1021:
1005:
996:
988:
986:
972:
941:equal to the
928:
882:
878:
870:, where each
857:
835:
831:
827:
824:
821:
816:
812:
804:
788:
768:
737:
733:
717:
710:Suppose that
705:
703:
701:
697:
692:
678:
658:
650:
634:
566:
543:
537:
534:
528:
525:
522:
518:
514:
508:
503:
499:
495:
492:
483:
477:
469:
443:
442:
441:
427:
419:
403:
395:
379:
359:
356:
336:
316:
296:
293:
273:
266:
250:
230:
210:
190:
178:
176:
174:
158:
150:
134:
131:
128:
105:
99:
93:
90:
87:
81:
54:
53:
52:
50:
46:
42:
39:
35:
31:
27:
23:
19:
1560:
1527:
1397:
1367:annihilates
1225:
992:
709:
693:
558:
417:
393:
372:.) Then the
223:an ideal of
182:
147:denotes the
120:
25:
15:
897:belongs to
349:are not in
22:homological
18:commutative
1578:Categories
1517:References
1347:(that is,
1314:such that
1020:local ring
965:-depth of
732:local ring
696:David Rees
179:Definition
41:local ring
38:Noetherian
825:…
535:≠
509:
132:
94:
88:≤
24:algebra,
1526:(1995),
1554:1322960
1463:(where
1113:states
603:is its
47:by the
34:modules
1567:
1552:
1542:
1270:is an
121:where
1024:ideal
736:ideal
647:is a
418:grade
394:depth
30:rings
26:depth
1565:ISBN
1540:ISBN
1049:and
993:The
850:for
761:and
243:and
183:Let
32:and
20:and
1294:of
500:Ext
487:min
420:of
396:of
129:dim
91:dim
16:In
1580::
1550:MR
1548:,
1538:,
1530:,
1513:.
1395:.
985:.
702:.
691:.
263:a
1497:0
1494:=
1491:x
1471:k
1451:)
1448:y
1445:x
1442:,
1437:2
1433:x
1429:(
1425:/
1421:]
1418:y
1415:,
1412:x
1409:[
1406:k
1377:m
1355:x
1335:0
1332:=
1327:m
1322:x
1302:R
1282:x
1256:m
1234:R
1206:.
1203:)
1200:R
1197:(
1193:h
1190:t
1187:p
1184:e
1181:d
1177:=
1174:)
1171:M
1168:(
1164:h
1161:t
1158:p
1155:e
1152:d
1148:+
1145:)
1142:M
1139:(
1134:R
1129:d
1126:p
1097:M
1077:R
1057:M
1035:m
1006:R
973:M
951:m
929:n
907:m
883:i
879:x
858:M
836:n
832:x
828:,
822:,
817:1
813:x
789:R
769:M
747:m
718:R
679:R
659:R
635:R
613:m
589:m
567:R
544:.
541:}
538:0
532:)
529:M
526:,
523:I
519:/
515:R
512:(
504:i
496::
493:i
490:{
484:=
481:)
478:M
475:(
470:I
465:h
462:t
459:p
456:e
453:d
428:M
404:M
392:-
380:I
360:M
357:I
337:M
317:M
297:M
294:I
274:R
251:M
231:R
211:I
191:R
159:M
135:M
106:,
103:)
100:M
97:(
85:)
82:M
79:(
75:h
72:t
69:p
66:e
63:d
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.