80:
While divisors on higher-dimensional varieties continue to play an important role in determining the structure of the variety, on varieties of dimension two or more there are also higher codimension cycles to consider. The behavior of these cycles is strikingly different from that of divisors. For
1416:
There are several variants of the above definition. We may substitute another ring for integers as our coefficient ring. The case of rational coefficients is widely used. Working with families of cycles over a base, or using cycles in arithmetic situations, requires a relative setup. Let
2027:
611:
under the map that, in one direction, takes each subscheme to its generic point, and in the other direction, takes each point to the unique reduced subscheme supported on the closure of the point. Consequently
502:
1107:
1008:
61:
The most trivial case is codimension zero cycles, which are linear combinations of the irreducible components of the variety. The first non-trivial case is of codimension one subvarieties, called
1220:
934:
579:
832:
1149:
221:
138:
1565:
1603:
1888:
1447:
1366:
1305:
1854:
900:
697:
1757:
1489:
69:
are formal linear combinations of points on the curve. Classical work on algebraic curves related these to intrinsic data, such as the regular differentials on a compact
389:
257:
182:
667:
1399:
640:
756:
729:
1518:
58:
that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the variety.
776:
2115:
438:
1628:, as well as all of the above modulo torsion. These equivalence relations have (partially conjectural) applications to the theory of
1613:
287:
65:. The earliest work on algebraic cycles focused on the case of divisors, particularly divisors on algebraic curves. Divisors on
2146:
The arithmetic and geometry of algebraic cycles: proceedings of the CRM summer school, June 7–19, 1998, Banff, Alberta, Canada
1020:
2153:
2127:
948:
298:
proved that the existence of a category of motives implies the standard conjectures. Additionally, cycles are connected to
85:
such that every divisor of degree zero is linearly equivalent to a difference of two effective divisors of degree at most
145:
294:
and would imply that algebraic cycles play a vital role in any cohomology theory of algebraic varieties. Conversely,
2095:
1157:
274:, predicts that the topology of a complex algebraic variety forces the existence of certain algebraic cycles. The
17:
905:
525:
781:
1663:
of some constant relative dimension (i.e. all fibers have the same dimension), we can define for any subvariety
1115:
187:
104:
2107:
1864:
1523:
2022:{\displaystyle f^{*}\colon Z^{k}(X')\to Z^{k}(X)\quad {\text{and}}\quad f_{*}\colon Z_{k}(X)\to Z_{k}(X')\,\!}
1570:
2045:
262:
The behavior of algebraic cycles ranks among the most important open questions in modern mathematics. The
62:
2171:
1629:
291:
1420:
267:
1316:
306:
by Bloch's formula, which expresses groups of cycles modulo rational equivalence as the cohomology of
1258:
271:
1791:
837:
676:
1677:
1468:
2144:
Gordon, B. Brent; Lewis, James D.; MĂĽller-Stach, Stefan; Saito, Shuji; Yui, Noriko, eds. (2000),
348:
283:
226:
151:
1640:
There is a covariant and a contravariant functoriality of the group of algebraic cycles. Let
652:
1374:
615:
2137:
1410:
734:
702:
323:
184:
contains transcendental information, and in effect
Mumford's theorem implies that, despite
2118:. Third series. A Series of Modern Surveys in Mathematics, vol. 2, Berlin, New York:
1494:
8:
295:
279:
144:
is false. The hypothesis that the geometric genus is positive essentially means (by the
761:
299:
51:
2149:
2123:
2091:
36:
263:
74:
2032:(the latter by virtue of the convention) are homomorphisms of abelian groups. See
2133:
2119:
1876:
1774:
1624:
for a fixed cohomology theory (such as singular cohomology or Ă©tale cohomology),
275:
98:
70:
66:
223:
having a purely algebraic definition, it shares transcendental information with
2050:
2165:
1660:
90:
43:
1882:
By linearity, these definitions extend to homomorphisms of abelian groups
28:
2088:
Chow Rings, Decomposition of the
Diagonal, and the Topology of Families
2036:
for a discussion of the functoriality related to the ring structure.
2033:
1402:
607:
are in one-to-one correspondence with the scheme-theoretic points of
73:, and to extrinsic properties, such as embeddings of the curve into
1612:
Rational equivalence can also be replaced by several other coarser
497:{\displaystyle Z_{r}X=\bigoplus _{V\subseteq X}\mathbf {Z} \cdot ,}
642:
can also be described as the free abelian group on the points of
1252:, the above groups are sometimes reindexed cohomologically as
140:
of rational equivalence classes of codimension two cycles in
2143:
93:
proved that, on a smooth complete complex algebraic surface
945:. The cycles rationally equivalent to zero are a subgroup
259:. Mumford's theorem has since been greatly generalized.
1457:-cycle is a formal sum of closed integral subschemes of
1102:{\displaystyle A_{r}(X)=Z_{r}(X)/Z_{r}(X)_{\text{rat}}.}
1635:
1409:
because it has a multiplication operation given by the
1003:{\displaystyle Z_{r}(X)_{\text{rat}}\subseteq Z_{r}(X)}
1891:
1794:
1680:
1573:
1526:
1497:
1471:
1423:
1377:
1319:
1261:
1160:
1118:
1023:
951:
908:
840:
784:
764:
737:
705:
679:
655:
618:
528:
441:
351:
229:
190:
154:
107:
2090:, Annals of Mathematics Studies 187, February 2014,
1014:-cycles modulo rational equivalence is the quotient
2148:, Providence, R.I.: American Mathematical Society,
1616:. Other equivalence relations of interest include
2021:
1848:
1751:
1597:
1559:
1512:
1483:
1441:
1393:
1360:
1299:
1241:if they can be represented by an effective cycle.
1214:
1143:
1101:
1002:
928:
894:
826:
770:
750:
723:
691:
661:
634:
573:
496:
383:
251:
215:
176:
132:
2018:
1845:
1748:
507:where the sum is over closed integral subschemes
290:yield enough cycles to construct his category of
2163:
2116:Ergebnisse der Mathematik und ihrer Grenzgebiete
1762:which by assumption has the same codimension as
1215:{\displaystyle A_{*}(X)=\bigoplus _{r}A_{r}(X)}
936:denotes the divisor of a rational function on
1248:is smooth, projective, and of pure dimension
2071:Rational equivalence of 0-cycles on surfaces
929:{\displaystyle \operatorname {div} _{W_{i}}}
574:{\displaystyle Z_{*}X=\bigoplus _{r}Z_{r}X.}
827:{\displaystyle r_{i}\in k(W_{i})^{\times }}
1144:{\displaystyle \operatorname {CH} _{r}(X)}
600:if all its coefficients are non-negative.
216:{\displaystyle \operatorname {CH} ^{2}(S)}
133:{\displaystyle \operatorname {CH} ^{2}(S)}
2017:
1844:
1747:
1614:equivalence relations on algebraic cycles
1560:{\displaystyle k({\overline {\phi (Y)}})}
288:standard conjectures on algebraic cycles
101:, the analogous statement for the group
14:
2164:
2106:
1598:{\displaystyle {\overline {\phi (Y)}}}
1636:Flat pullback and proper pushforward
1453:is a regular Noetherian scheme. An
515:. The groups of cycles for varying
81:example, every curve has a constant
24:
1863:is the degree of the extension of
1442:{\displaystyle \phi \colon X\to S}
699:, if there are a finite number of
432:-cycles is the free abelian group
326:which is finite type over a field
146:Lefschetz theorem on (1,1)-classes
42:is a formal linear combination of
25:
2183:
1785:the pushforward is defined to be
1465:; here the relative dimension of
1361:{\displaystyle A^{N-r}X=A_{r}X.}
1233:. Cycle classes are said to be
778:and non-zero rational functions
475:
1957:
1951:
1491:is the transcendence degree of
1300:{\displaystyle Z^{N-r}X=Z_{r}X}
588:, and any element is called an
342:is a formal linear combination
278:makes a similar prediction for
2080:
2063:
2014:
2003:
1990:
1987:
1981:
1948:
1942:
1929:
1926:
1915:
1841:
1838:
1832:
1826:
1817:
1814:
1808:
1805:
1744:
1741:
1730:
1714:
1708:
1705:
1694:
1691:
1586:
1580:
1554:
1545:
1539:
1530:
1507:
1501:
1433:
1209:
1203:
1177:
1171:
1138:
1132:
1087:
1080:
1062:
1056:
1040:
1034:
997:
991:
969:
962:
889:
886:
873:
850:
815:
801:
718:
706:
603:Closed integral subschemes of
488:
482:
378:
365:
246:
240:
210:
204:
171:
165:
127:
121:
13:
1:
2056:
1849:{\displaystyle f_{*}()=n\,\!}
895:{\displaystyle \alpha =\sum }
692:{\displaystyle \alpha \sim 0}
671:rationally equivalent to zero
398:-dimensional closed integral
313:
50:. These are the part of the
2046:divisor (algebraic geometry)
1752:{\displaystyle f^{*}()=\,\!}
1590:
1549:
1484:{\displaystyle Y\subseteq X}
1461:whose relative dimension is
148:) that the cohomology group
7:
2039:
1112:This group is also denoted
10:
2188:
731:-dimensional subvarieties
384:{\displaystyle \sum n_{i}}
268:Clay Mathematics Institute
1567:minus the codimension of
1151:. Elements of the group
586:group of algebraic cycles
272:Millennium Prize Problems
2073:, J. Math. Kyoto Univ.
252:{\displaystyle H^{2}(S)}
177:{\displaystyle H^{2}(S)}
1652:be a map of varieties.
1622:homological equivalence
662:{\displaystyle \alpha }
2023:
1867:if the restriction of
1850:
1753:
1599:
1561:
1514:
1485:
1443:
1395:
1394:{\displaystyle A^{*}X}
1362:
1301:
1216:
1145:
1103:
1004:
930:
896:
828:
772:
752:
725:
693:
663:
636:
635:{\displaystyle Z_{*}X}
575:
519:together form a group
498:
385:
284:Alexander Grothendieck
253:
217:
178:
134:
2024:
1851:
1754:
1626:numerical equivalence
1618:algebraic equivalence
1600:
1562:
1515:
1486:
1444:
1396:
1363:
1302:
1217:
1146:
1104:
1005:
931:
897:
829:
773:
753:
751:{\displaystyle W_{i}}
726:
724:{\displaystyle (r+1)}
694:
664:
637:
576:
499:
386:
254:
218:
179:
135:
1889:
1792:
1678:
1571:
1524:
1513:{\displaystyle k(Y)}
1495:
1469:
1421:
1411:intersection product
1375:
1317:
1259:
1158:
1116:
1021:
949:
906:
838:
782:
762:
735:
703:
677:
653:
616:
526:
439:
349:
227:
188:
152:
105:
2112:Intersection theory
1010:, and the group of
584:This is called the
406:. The coefficient
296:Alexander Beilinson
2172:Algebraic geometry
2077:-2 (1969) 195–204.
2019:
1846:
1749:
1595:
1557:
1510:
1481:
1439:
1391:
1358:
1297:
1212:
1192:
1141:
1099:
1000:
926:
892:
824:
768:
748:
721:
689:
659:
632:
571:
554:
494:
473:
428:. The set of all
381:
249:
213:
174:
130:
52:algebraic topology
2155:978-0-8218-1954-8
2129:978-0-387-98549-7
1955:
1879:and 0 otherwise.
1593:
1552:
1183:
1093:
975:
771:{\displaystyle X}
545:
458:
310:-theory sheaves.
37:algebraic variety
16:(Redirected from
2179:
2158:
2140:
2099:
2086:Voisin, Claire,
2084:
2078:
2069:Mumford, David,
2067:
2028:
2026:
2025:
2020:
2013:
2002:
2001:
1980:
1979:
1967:
1966:
1956:
1953:
1941:
1940:
1925:
1914:
1913:
1901:
1900:
1855:
1853:
1852:
1847:
1804:
1803:
1781:a subvariety of
1758:
1756:
1755:
1750:
1740:
1729:
1728:
1704:
1690:
1689:
1604:
1602:
1601:
1596:
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1589:
1575:
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1564:
1563:
1558:
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1534:
1519:
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1516:
1511:
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1482:
1448:
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1400:
1398:
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1387:
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1367:
1365:
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1351:
1350:
1335:
1334:
1306:
1304:
1303:
1298:
1293:
1292:
1277:
1276:
1221:
1219:
1218:
1213:
1202:
1201:
1191:
1170:
1169:
1150:
1148:
1147:
1142:
1128:
1127:
1108:
1106:
1105:
1100:
1095:
1094:
1091:
1079:
1078:
1069:
1055:
1054:
1033:
1032:
1009:
1007:
1006:
1001:
990:
989:
977:
976:
973:
961:
960:
935:
933:
932:
927:
925:
924:
923:
922:
901:
899:
898:
893:
885:
884:
869:
868:
867:
866:
833:
831:
830:
825:
823:
822:
813:
812:
794:
793:
777:
775:
774:
769:
757:
755:
754:
749:
747:
746:
730:
728:
727:
722:
698:
696:
695:
690:
668:
666:
665:
660:
641:
639:
638:
633:
628:
627:
580:
578:
577:
572:
564:
563:
553:
538:
537:
503:
501:
500:
495:
478:
472:
451:
450:
390:
388:
387:
382:
377:
376:
364:
363:
280:Ă©tale cohomology
264:Hodge conjecture
258:
256:
255:
250:
239:
238:
222:
220:
219:
214:
200:
199:
183:
181:
180:
175:
164:
163:
139:
137:
136:
131:
117:
116:
75:projective space
67:algebraic curves
21:
18:Algebraic cycles
2187:
2186:
2182:
2181:
2180:
2178:
2177:
2176:
2162:
2161:
2156:
2130:
2120:Springer-Verlag
2108:Fulton, William
2103:
2102:
2085:
2081:
2068:
2064:
2059:
2042:
2006:
1997:
1993:
1975:
1971:
1962:
1958:
1952:
1936:
1932:
1918:
1909:
1905:
1896:
1892:
1890:
1887:
1886:
1865:function fields
1799:
1795:
1793:
1790:
1789:
1769:Conversely, if
1733:
1721:
1717:
1697:
1685:
1681:
1679:
1676:
1675:
1638:
1576:
1574:
1572:
1569:
1568:
1535:
1533:
1525:
1522:
1521:
1496:
1493:
1492:
1470:
1467:
1466:
1422:
1419:
1418:
1382:
1378:
1376:
1373:
1372:
1346:
1342:
1324:
1320:
1318:
1315:
1314:
1288:
1284:
1266:
1262:
1260:
1257:
1256:
1197:
1193:
1187:
1165:
1161:
1159:
1156:
1155:
1123:
1119:
1117:
1114:
1113:
1090:
1086:
1074:
1070:
1065:
1050:
1046:
1028:
1024:
1022:
1019:
1018:
985:
981:
972:
968:
956:
952:
950:
947:
946:
944:
918:
914:
913:
909:
907:
904:
903:
880:
876:
862:
858:
857:
853:
839:
836:
835:
818:
814:
808:
804:
789:
785:
783:
780:
779:
763:
760:
759:
742:
738:
736:
733:
732:
704:
701:
700:
678:
675:
674:
654:
651:
650:
623:
619:
617:
614:
613:
590:algebraic cycle
559:
555:
549:
533:
529:
527:
524:
523:
474:
462:
446:
442:
440:
437:
436:
427:
414:
402:-subschemes of
372:
368:
359:
355:
350:
347:
346:
316:
276:Tate conjecture
234:
230:
228:
225:
224:
195:
191:
189:
186:
185:
159:
155:
153:
150:
149:
112:
108:
106:
103:
102:
99:geometric genus
71:Riemann surface
33:algebraic cycle
23:
22:
15:
12:
11:
5:
2185:
2175:
2174:
2160:
2159:
2154:
2141:
2128:
2101:
2100:
2079:
2061:
2060:
2058:
2055:
2054:
2053:
2051:Relative cycle
2048:
2041:
2038:
2030:
2029:
2016:
2012:
2009:
2005:
2000:
1996:
1992:
1989:
1986:
1983:
1978:
1974:
1970:
1965:
1961:
1950:
1947:
1944:
1939:
1935:
1931:
1928:
1924:
1921:
1917:
1912:
1908:
1904:
1899:
1895:
1857:
1856:
1843:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1802:
1798:
1760:
1759:
1746:
1743:
1739:
1736:
1732:
1727:
1724:
1720:
1716:
1713:
1710:
1707:
1703:
1700:
1696:
1693:
1688:
1684:
1637:
1634:
1592:
1588:
1585:
1582:
1579:
1556:
1551:
1547:
1544:
1541:
1538:
1532:
1529:
1509:
1506:
1503:
1500:
1480:
1477:
1474:
1438:
1435:
1432:
1429:
1426:
1401:is called the
1390:
1385:
1381:
1371:In this case,
1369:
1368:
1357:
1354:
1349:
1345:
1341:
1338:
1333:
1330:
1327:
1323:
1308:
1307:
1296:
1291:
1287:
1283:
1280:
1275:
1272:
1269:
1265:
1223:
1222:
1211:
1208:
1205:
1200:
1196:
1190:
1186:
1182:
1179:
1176:
1173:
1168:
1164:
1140:
1137:
1134:
1131:
1126:
1122:
1110:
1109:
1098:
1089:
1085:
1082:
1077:
1073:
1068:
1064:
1061:
1058:
1053:
1049:
1045:
1042:
1039:
1036:
1031:
1027:
999:
996:
993:
988:
984:
980:
971:
967:
964:
959:
955:
940:
921:
917:
912:
891:
888:
883:
879:
875:
872:
865:
861:
856:
852:
849:
846:
843:
821:
817:
811:
807:
803:
800:
797:
792:
788:
767:
745:
741:
720:
717:
714:
711:
708:
688:
685:
682:
658:
631:
626:
622:
592:. A cycle is
582:
581:
570:
567:
562:
558:
552:
548:
544:
541:
536:
532:
505:
504:
493:
490:
487:
484:
481:
477:
471:
468:
465:
461:
457:
454:
449:
445:
423:
410:
392:
391:
380:
375:
371:
367:
362:
358:
354:
315:
312:
248:
245:
242:
237:
233:
212:
209:
206:
203:
198:
194:
173:
170:
167:
162:
158:
129:
126:
123:
120:
115:
111:
97:with positive
9:
6:
4:
3:
2:
2184:
2173:
2170:
2169:
2167:
2157:
2151:
2147:
2142:
2139:
2135:
2131:
2125:
2121:
2117:
2113:
2109:
2105:
2104:
2097:
2096:9780691160504
2093:
2089:
2083:
2076:
2072:
2066:
2062:
2052:
2049:
2047:
2044:
2043:
2037:
2035:
2010:
2007:
1998:
1994:
1984:
1976:
1972:
1968:
1963:
1959:
1945:
1937:
1933:
1922:
1919:
1910:
1906:
1902:
1897:
1893:
1885:
1884:
1883:
1880:
1878:
1874:
1870:
1866:
1862:
1835:
1829:
1823:
1820:
1811:
1800:
1796:
1788:
1787:
1786:
1784:
1780:
1776:
1772:
1767:
1765:
1737:
1734:
1725:
1722:
1718:
1711:
1701:
1698:
1686:
1682:
1674:
1673:
1672:
1670:
1667: ⊂
1666:
1662:
1658:
1653:
1651:
1647:
1643:
1633:
1631:
1627:
1623:
1619:
1615:
1610:
1608:
1583:
1577:
1542:
1536:
1527:
1504:
1498:
1478:
1475:
1472:
1464:
1460:
1456:
1452:
1436:
1430:
1427:
1424:
1414:
1412:
1408:
1404:
1388:
1383:
1379:
1355:
1352:
1347:
1343:
1339:
1336:
1331:
1328:
1325:
1321:
1313:
1312:
1311:
1294:
1289:
1285:
1281:
1278:
1273:
1270:
1267:
1263:
1255:
1254:
1253:
1251:
1247:
1242:
1240:
1236:
1232:
1228:
1227:cycle classes
1206:
1198:
1194:
1188:
1184:
1180:
1174:
1166:
1162:
1154:
1153:
1152:
1135:
1129:
1124:
1120:
1096:
1083:
1075:
1071:
1066:
1059:
1051:
1047:
1043:
1037:
1029:
1025:
1017:
1016:
1015:
1013:
994:
986:
982:
978:
965:
957:
953:
943:
939:
919:
915:
910:
881:
877:
870:
863:
859:
854:
847:
844:
841:
819:
809:
805:
798:
795:
790:
786:
765:
743:
739:
715:
712:
709:
686:
683:
680:
672:
656:
647:
645:
629:
624:
620:
610:
606:
601:
599:
595:
591:
587:
568:
565:
560:
556:
550:
546:
542:
539:
534:
530:
522:
521:
520:
518:
514:
510:
491:
485:
479:
469:
466:
463:
459:
455:
452:
447:
443:
435:
434:
433:
431:
426:
422:
418:
413:
409:
405:
401:
397:
373:
369:
360:
356:
352:
345:
344:
343:
341:
337:
335:
329:
325:
321:
311:
309:
305:
303:
297:
293:
289:
285:
281:
277:
273:
269:
266:, one of the
265:
260:
243:
235:
231:
207:
201:
196:
192:
168:
160:
156:
147:
143:
124:
118:
113:
109:
100:
96:
92:
91:David Mumford
88:
84:
78:
76:
72:
68:
64:
59:
57:
53:
49:
45:
41:
38:
34:
30:
19:
2145:
2111:
2087:
2082:
2074:
2070:
2065:
2031:
1881:
1872:
1868:
1860:
1858:
1782:
1778:
1770:
1768:
1763:
1761:
1668:
1664:
1656:
1654:
1649:
1645:
1641:
1639:
1625:
1621:
1617:
1611:
1606:
1462:
1458:
1454:
1450:
1415:
1406:
1370:
1309:
1249:
1245:
1243:
1238:
1234:
1230:
1226:
1224:
1111:
1011:
941:
937:
670:
648:
643:
608:
604:
602:
597:
593:
589:
585:
583:
516:
512:
508:
506:
429:
424:
420:
417:multiplicity
416:
411:
407:
403:
399:
395:
393:
339:
333:
331:
327:
319:
317:
307:
301:
261:
141:
94:
86:
82:
79:
60:
55:
47:
44:subvarieties
39:
32:
26:
1225:are called
29:mathematics
2057:References
834:such that
673:, written
332:algebraic
314:Definition
300:algebraic
2034:Chow ring
1991:→
1969::
1964:∗
1930:→
1903::
1898:∗
1801:∗
1723:−
1687:∗
1591:¯
1578:ϕ
1550:¯
1537:ϕ
1476:⊆
1434:→
1428::
1425:ϕ
1403:Chow ring
1384:∗
1329:−
1271:−
1235:effective
1185:⨁
1167:∗
1130:
979:⊆
871:
848:∑
842:α
820:×
796:∈
684:∼
681:α
657:α
625:∗
594:effective
547:⨁
535:∗
480:⋅
467:⊆
460:⨁
353:∑
202:
119:
2166:Category
2110:(1998),
2040:See also
2011:′
1923:′
1764:Y′
1738:′
1702:′
1644: :
1449:, where
1239:positive
902:, where
649:A cycle
598:positive
63:divisors
2138:1644323
1630:motives
415:is the
304:-theory
292:motives
2152:
2136:
2126:
2094:
1877:finite
1859:where
1777:, for
1775:proper
336:-cycle
330:. An
324:scheme
35:on an
1520:over
322:be a
31:, an
2150:ISBN
2124:ISBN
2092:ISBN
1661:flat
1310:and
318:Let
1954:and
1875:is
1871:to
1773:is
1659:is
1655:If
1605:in
1405:of
1244:If
1237:or
1229:on
1092:rat
974:rat
911:div
855:div
758:of
669:is
596:or
511:of
419:of
394:of
338:on
286:'s
282:.
270:'s
89:.
54:of
46:of
27:In
2168::
2134:MR
2132:,
2122:,
2114:,
1766:.
1671::
1669:X'
1665:Y'
1650:X'
1648:→
1632:.
1620:,
1609:.
1413:.
1121:CH
646:.
193:CH
110:CH
77:.
2098:.
2075:9
2015:)
2008:X
2004:(
1999:k
1995:Z
1988:)
1985:X
1982:(
1977:k
1973:Z
1960:f
1949:)
1946:X
1943:(
1938:k
1934:Z
1927:)
1920:X
1916:(
1911:k
1907:Z
1894:f
1873:Y
1869:f
1861:n
1842:]
1839:)
1836:Y
1833:(
1830:f
1827:[
1824:n
1821:=
1818:)
1815:]
1812:Y
1809:[
1806:(
1797:f
1783:X
1779:Y
1771:f
1745:]
1742:)
1735:Y
1731:(
1726:1
1719:f
1715:[
1712:=
1709:)
1706:]
1699:Y
1695:[
1692:(
1683:f
1657:f
1646:X
1642:f
1607:S
1587:)
1584:Y
1581:(
1555:)
1546:)
1543:Y
1540:(
1531:(
1528:k
1508:)
1505:Y
1502:(
1499:k
1479:X
1473:Y
1463:r
1459:X
1455:r
1451:S
1437:S
1431:X
1407:X
1389:X
1380:A
1356:.
1353:X
1348:r
1344:A
1340:=
1337:X
1332:r
1326:N
1322:A
1295:X
1290:r
1286:Z
1282:=
1279:X
1274:r
1268:N
1264:Z
1250:N
1246:X
1231:X
1210:)
1207:X
1204:(
1199:r
1195:A
1189:r
1181:=
1178:)
1175:X
1172:(
1163:A
1139:)
1136:X
1133:(
1125:r
1097:.
1088:)
1084:X
1081:(
1076:r
1072:Z
1067:/
1063:)
1060:X
1057:(
1052:r
1048:Z
1044:=
1041:)
1038:X
1035:(
1030:r
1026:A
1012:r
998:)
995:X
992:(
987:r
983:Z
970:)
966:X
963:(
958:r
954:Z
942:i
938:W
920:i
916:W
890:]
887:)
882:i
878:r
874:(
864:i
860:W
851:[
845:=
816:)
810:i
806:W
802:(
799:k
791:i
787:r
766:X
744:i
740:W
719:)
716:1
713:+
710:r
707:(
687:0
644:X
630:X
621:Z
609:X
605:X
569:.
566:X
561:r
557:Z
551:r
543:=
540:X
531:Z
517:r
513:X
509:V
492:,
489:]
486:V
483:[
476:Z
470:X
464:V
456:=
453:X
448:r
444:Z
430:r
425:i
421:V
412:i
408:n
404:X
400:k
396:r
379:]
374:i
370:V
366:[
361:i
357:n
340:X
334:r
328:k
320:X
308:K
302:K
247:)
244:S
241:(
236:2
232:H
211:)
208:S
205:(
197:2
172:)
169:S
166:(
161:2
157:H
142:S
128:)
125:S
122:(
114:2
95:S
87:N
83:N
56:V
48:V
40:V
20:)
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