69:
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
1405:
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
2016:
600:
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
491:
1096:
997:
50:
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
1209:
923:
568:
821:
1138:
210:
127:
1554:
1592:
1877:
1436:
1355:
1294:
1843:
889:
686:
1746:
1478:
58:
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
378:
246:
171:
656:
1388:
629:
745:
718:
1507:
47:
that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the variety.
765:
2104:
427:
1617:, as well as all of the above modulo torsion. These equivalence relations have (partially conjectural) applications to the theory of
1602:
276:
54:. The earliest work on algebraic cycles focused on the case of divisors, particularly divisors on algebraic curves. Divisors on
2135:
The arithmetic and geometry of algebraic cycles: proceedings of the CRM summer school, June 7–19, 1998, Banff, Alberta, Canada
1009:
2142:
2116:
937:
287:
proved that the existence of a category of motives implies the standard conjectures. Additionally, cycles are connected to
74:
such that every divisor of degree zero is linearly equivalent to a difference of two effective divisors of degree at most
134:
283:
and would imply that algebraic cycles play a vital role in any cohomology theory of algebraic varieties. Conversely,
2084:
1146:
263:, predicts that the topology of a complex algebraic variety forces the existence of certain algebraic cycles. The
894:
514:
770:
1652:
of some constant relative dimension (i.e. all fibers have the same dimension), we can define for any subvariety
1104:
176:
93:
2096:
1853:
1512:
2011:{\displaystyle f^{*}\colon Z^{k}(X')\to Z^{k}(X)\quad {\text{and}}\quad f_{*}\colon Z_{k}(X)\to Z_{k}(X')\,\!}
1559:
2034:
251:
The behavior of algebraic cycles ranks among the most important open questions in modern mathematics. The
51:
2160:
1618:
280:
1409:
256:
1305:
295:
by Bloch's formula, which expresses groups of cycles modulo rational equivalence as the cohomology of
1247:
260:
1780:
826:
665:
1666:
1457:
2133:
Gordon, B. Brent; Lewis, James D.; MĂĽller-Stach, Stefan; Saito, Shuji; Yui, Noriko, eds. (2000),
337:
272:
215:
140:
1629:
There is a covariant and a contravariant functoriality of the group of algebraic cycles. Let
641:
1363:
604:
2126:
1399:
723:
691:
312:
173:
contains transcendental information, and in effect
Mumford's theorem implies that, despite
2107:. Third series. A Series of Modern Surveys in Mathematics, vol. 2, Berlin, New York:
1483:
8:
284:
268:
133:
is false. The hypothesis that the geometric genus is positive essentially means (by the
750:
288:
40:
2138:
2112:
2080:
25:
252:
63:
2021:(the latter by virtue of the convention) are homomorphisms of abelian groups. See
2122:
2108:
1865:
1763:
1613:
for a fixed cohomology theory (such as singular cohomology or Ă©tale cohomology),
264:
87:
59:
55:
212:
having a purely algebraic definition, it shares transcendental information with
2039:
2154:
1649:
79:
32:
1871:
By linearity, these definitions extend to homomorphisms of abelian groups
17:
2077:
Chow Rings, Decomposition of the
Diagonal, and the Topology of Families
2025:
for a discussion of the functoriality related to the ring structure.
2022:
1391:
596:
are in one-to-one correspondence with the scheme-theoretic points of
62:, and to extrinsic properties, such as embeddings of the curve into
1601:
Rational equivalence can also be replaced by several other coarser
486:{\displaystyle Z_{r}X=\bigoplus _{V\subseteq X}\mathbf {Z} \cdot ,}
631:
can also be described as the free abelian group on the points of
1241:, the above groups are sometimes reindexed cohomologically as
129:
of rational equivalence classes of codimension two cycles in
2132:
82:
proved that, on a smooth complete complex algebraic surface
934:. The cycles rationally equivalent to zero are a subgroup
248:. Mumford's theorem has since been greatly generalized.
1446:-cycle is a formal sum of closed integral subschemes of
1091:{\displaystyle A_{r}(X)=Z_{r}(X)/Z_{r}(X)_{\text{rat}}.}
1624:
1398:
because it has a multiplication operation given by the
992:{\displaystyle Z_{r}(X)_{\text{rat}}\subseteq Z_{r}(X)}
1880:
1783:
1669:
1562:
1515:
1486:
1460:
1412:
1366:
1308:
1250:
1149:
1107:
1012:
940:
897:
829:
773:
753:
726:
694:
668:
644:
607:
517:
430:
340:
218:
179:
143:
96:
2079:, Annals of Mathematics Studies 187, February 2014,
1003:-cycles modulo rational equivalence is the quotient
2137:, Providence, R.I.: American Mathematical Society,
1605:. Other equivalence relations of interest include
2010:
1837:
1740:
1586:
1548:
1501:
1472:
1430:
1382:
1349:
1288:
1230:if they can be represented by an effective cycle.
1203:
1132:
1090:
991:
917:
883:
815:
759:
739:
712:
680:
650:
623:
562:
485:
372:
240:
204:
165:
121:
2007:
1834:
1737:
496:where the sum is over closed integral subschemes
279:yield enough cycles to construct his category of
2152:
2105:Ergebnisse der Mathematik und ihrer Grenzgebiete
1751:which by assumption has the same codimension as
1204:{\displaystyle A_{*}(X)=\bigoplus _{r}A_{r}(X)}
925:denotes the divisor of a rational function on
1237:is smooth, projective, and of pure dimension
2060:Rational equivalence of 0-cycles on surfaces
918:{\displaystyle \operatorname {div} _{W_{i}}}
563:{\displaystyle Z_{*}X=\bigoplus _{r}Z_{r}X.}
816:{\displaystyle r_{i}\in k(W_{i})^{\times }}
1133:{\displaystyle \operatorname {CH} _{r}(X)}
589:if all its coefficients are non-negative.
205:{\displaystyle \operatorname {CH} ^{2}(S)}
122:{\displaystyle \operatorname {CH} ^{2}(S)}
2006:
1833:
1736:
1603:equivalence relations on algebraic cycles
1549:{\displaystyle k({\overline {\phi (Y)}})}
277:standard conjectures on algebraic cycles
90:, the analogous statement for the group
2153:
2095:
1587:{\displaystyle {\overline {\phi (Y)}}}
1625:Flat pullback and proper pushforward
1442:is a regular Noetherian scheme. An
504:. The groups of cycles for varying
70:example, every curve has a constant
13:
1852:is the degree of the extension of
1431:{\displaystyle \phi \colon X\to S}
688:, if there are a finite number of
421:-cycles is the free abelian group
315:which is finite type over a field
135:Lefschetz theorem on (1,1)-classes
31:is a formal linear combination of
14:
2172:
1774:the pushforward is defined to be
1454:; here the relative dimension of
1350:{\displaystyle A^{N-r}X=A_{r}X.}
1222:. Cycle classes are said to be
767:and non-zero rational functions
464:
1946:
1940:
1480:is the transcendence degree of
1289:{\displaystyle Z^{N-r}X=Z_{r}X}
577:, and any element is called an
331:is a formal linear combination
267:makes a similar prediction for
2069:
2052:
2003:
1992:
1979:
1976:
1970:
1937:
1931:
1918:
1915:
1904:
1830:
1827:
1821:
1815:
1806:
1803:
1797:
1794:
1733:
1730:
1719:
1703:
1697:
1694:
1683:
1680:
1575:
1569:
1543:
1534:
1528:
1519:
1496:
1490:
1422:
1198:
1192:
1166:
1160:
1127:
1121:
1076:
1069:
1051:
1045:
1029:
1023:
986:
980:
958:
951:
878:
875:
862:
839:
804:
790:
707:
695:
592:Closed integral subschemes of
477:
471:
367:
354:
235:
229:
199:
193:
160:
154:
116:
110:
1:
2045:
1838:{\displaystyle f_{*}()=n\,\!}
884:{\displaystyle \alpha =\sum }
681:{\displaystyle \alpha \sim 0}
660:rationally equivalent to zero
387:-dimensional closed integral
302:
39:. These are the part of the
2035:divisor (algebraic geometry)
1741:{\displaystyle f^{*}()=\,\!}
1579:
1538:
1473:{\displaystyle Y\subseteq X}
1450:whose relative dimension is
137:) that the cohomology group
7:
2028:
1101:This group is also denoted
10:
2177:
720:-dimensional subvarieties
373:{\displaystyle \sum n_{i}}
257:Clay Mathematics Institute
1556:minus the codimension of
1140:. Elements of the group
575:group of algebraic cycles
261:Millennium Prize Problems
2062:, J. Math. Kyoto Univ.
241:{\displaystyle H^{2}(S)}
166:{\displaystyle H^{2}(S)}
1641:be a map of varieties.
1611:homological equivalence
651:{\displaystyle \alpha }
2012:
1856:if the restriction of
1839:
1742:
1588:
1550:
1503:
1474:
1432:
1384:
1383:{\displaystyle A^{*}X}
1351:
1290:
1205:
1134:
1092:
993:
919:
885:
817:
761:
741:
714:
682:
652:
625:
624:{\displaystyle Z_{*}X}
564:
508:together form a group
487:
374:
273:Alexander Grothendieck
242:
206:
167:
123:
2013:
1840:
1743:
1615:numerical equivalence
1607:algebraic equivalence
1589:
1551:
1504:
1475:
1433:
1385:
1352:
1291:
1206:
1135:
1093:
994:
920:
886:
818:
762:
742:
740:{\displaystyle W_{i}}
715:
713:{\displaystyle (r+1)}
683:
653:
626:
565:
488:
375:
243:
207:
168:
124:
1878:
1781:
1667:
1560:
1513:
1502:{\displaystyle k(Y)}
1484:
1458:
1410:
1400:intersection product
1364:
1306:
1248:
1147:
1105:
1010:
938:
895:
827:
771:
751:
724:
692:
666:
642:
605:
515:
428:
338:
216:
177:
141:
94:
2101:Intersection theory
999:, and the group of
573:This is called the
395:. The coefficient
285:Alexander Beilinson
2161:Algebraic geometry
2066:-2 (1969) 195–204.
2008:
1835:
1738:
1584:
1546:
1499:
1470:
1428:
1380:
1347:
1286:
1201:
1181:
1130:
1088:
989:
915:
881:
813:
757:
737:
710:
678:
648:
621:
560:
543:
483:
462:
417:. The set of all
370:
238:
202:
163:
119:
41:algebraic topology
2144:978-0-8218-1954-8
2118:978-0-387-98549-7
1944:
1868:and 0 otherwise.
1582:
1541:
1172:
1082:
964:
760:{\displaystyle X}
534:
447:
299:-theory sheaves.
26:algebraic variety
2168:
2147:
2129:
2088:
2075:Voisin, Claire,
2073:
2067:
2058:Mumford, David,
2056:
2017:
2015:
2014:
2009:
2002:
1991:
1990:
1969:
1968:
1956:
1955:
1945:
1942:
1930:
1929:
1914:
1903:
1902:
1890:
1889:
1844:
1842:
1841:
1836:
1793:
1792:
1770:a subvariety of
1747:
1745:
1744:
1739:
1729:
1718:
1717:
1693:
1679:
1678:
1593:
1591:
1590:
1585:
1583:
1578:
1564:
1555:
1553:
1552:
1547:
1542:
1537:
1523:
1508:
1506:
1505:
1500:
1479:
1477:
1476:
1471:
1437:
1435:
1434:
1429:
1389:
1387:
1386:
1381:
1376:
1375:
1356:
1354:
1353:
1348:
1340:
1339:
1324:
1323:
1295:
1293:
1292:
1287:
1282:
1281:
1266:
1265:
1210:
1208:
1207:
1202:
1191:
1190:
1180:
1159:
1158:
1139:
1137:
1136:
1131:
1117:
1116:
1097:
1095:
1094:
1089:
1084:
1083:
1080:
1068:
1067:
1058:
1044:
1043:
1022:
1021:
998:
996:
995:
990:
979:
978:
966:
965:
962:
950:
949:
924:
922:
921:
916:
914:
913:
912:
911:
890:
888:
887:
882:
874:
873:
858:
857:
856:
855:
822:
820:
819:
814:
812:
811:
802:
801:
783:
782:
766:
764:
763:
758:
746:
744:
743:
738:
736:
735:
719:
717:
716:
711:
687:
685:
684:
679:
657:
655:
654:
649:
630:
628:
627:
622:
617:
616:
569:
567:
566:
561:
553:
552:
542:
527:
526:
492:
490:
489:
484:
467:
461:
440:
439:
379:
377:
376:
371:
366:
365:
353:
352:
269:Ă©tale cohomology
253:Hodge conjecture
247:
245:
244:
239:
228:
227:
211:
209:
208:
203:
189:
188:
172:
170:
169:
164:
153:
152:
128:
126:
125:
120:
106:
105:
64:projective space
56:algebraic curves
2176:
2175:
2171:
2170:
2169:
2167:
2166:
2165:
2151:
2150:
2145:
2119:
2109:Springer-Verlag
2097:Fulton, William
2092:
2091:
2074:
2070:
2057:
2053:
2048:
2031:
1995:
1986:
1982:
1964:
1960:
1951:
1947:
1941:
1925:
1921:
1907:
1898:
1894:
1885:
1881:
1879:
1876:
1875:
1854:function fields
1788:
1784:
1782:
1779:
1778:
1758:Conversely, if
1722:
1710:
1706:
1686:
1674:
1670:
1668:
1665:
1664:
1627:
1565:
1563:
1561:
1558:
1557:
1524:
1522:
1514:
1511:
1510:
1485:
1482:
1481:
1459:
1456:
1455:
1411:
1408:
1407:
1371:
1367:
1365:
1362:
1361:
1335:
1331:
1313:
1309:
1307:
1304:
1303:
1277:
1273:
1255:
1251:
1249:
1246:
1245:
1186:
1182:
1176:
1154:
1150:
1148:
1145:
1144:
1112:
1108:
1106:
1103:
1102:
1079:
1075:
1063:
1059:
1054:
1039:
1035:
1017:
1013:
1011:
1008:
1007:
974:
970:
961:
957:
945:
941:
939:
936:
935:
933:
907:
903:
902:
898:
896:
893:
892:
869:
865:
851:
847:
846:
842:
828:
825:
824:
807:
803:
797:
793:
778:
774:
772:
769:
768:
752:
749:
748:
731:
727:
725:
722:
721:
693:
690:
689:
667:
664:
663:
643:
640:
639:
612:
608:
606:
603:
602:
579:algebraic cycle
548:
544:
538:
522:
518:
516:
513:
512:
463:
451:
435:
431:
429:
426:
425:
416:
403:
391:-subschemes of
361:
357:
348:
344:
339:
336:
335:
305:
265:Tate conjecture
223:
219:
217:
214:
213:
184:
180:
178:
175:
174:
148:
144:
142:
139:
138:
101:
97:
95:
92:
91:
88:geometric genus
60:Riemann surface
22:algebraic cycle
12:
11:
5:
2174:
2164:
2163:
2149:
2148:
2143:
2130:
2117:
2090:
2089:
2068:
2050:
2049:
2047:
2044:
2043:
2042:
2040:Relative cycle
2037:
2030:
2027:
2019:
2018:
2005:
2001:
1998:
1994:
1989:
1985:
1981:
1978:
1975:
1972:
1967:
1963:
1959:
1954:
1950:
1939:
1936:
1933:
1928:
1924:
1920:
1917:
1913:
1910:
1906:
1901:
1897:
1893:
1888:
1884:
1846:
1845:
1832:
1829:
1826:
1823:
1820:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
1791:
1787:
1749:
1748:
1735:
1732:
1728:
1725:
1721:
1716:
1713:
1709:
1705:
1702:
1699:
1696:
1692:
1689:
1685:
1682:
1677:
1673:
1626:
1623:
1581:
1577:
1574:
1571:
1568:
1545:
1540:
1536:
1533:
1530:
1527:
1521:
1518:
1498:
1495:
1492:
1489:
1469:
1466:
1463:
1427:
1424:
1421:
1418:
1415:
1390:is called the
1379:
1374:
1370:
1360:In this case,
1358:
1357:
1346:
1343:
1338:
1334:
1330:
1327:
1322:
1319:
1316:
1312:
1297:
1296:
1285:
1280:
1276:
1272:
1269:
1264:
1261:
1258:
1254:
1212:
1211:
1200:
1197:
1194:
1189:
1185:
1179:
1175:
1171:
1168:
1165:
1162:
1157:
1153:
1129:
1126:
1123:
1120:
1115:
1111:
1099:
1098:
1087:
1078:
1074:
1071:
1066:
1062:
1057:
1053:
1050:
1047:
1042:
1038:
1034:
1031:
1028:
1025:
1020:
1016:
988:
985:
982:
977:
973:
969:
960:
956:
953:
948:
944:
929:
910:
906:
901:
880:
877:
872:
868:
864:
861:
854:
850:
845:
841:
838:
835:
832:
810:
806:
800:
796:
792:
789:
786:
781:
777:
756:
734:
730:
709:
706:
703:
700:
697:
677:
674:
671:
647:
620:
615:
611:
581:. A cycle is
571:
570:
559:
556:
551:
547:
541:
537:
533:
530:
525:
521:
494:
493:
482:
479:
476:
473:
470:
466:
460:
457:
454:
450:
446:
443:
438:
434:
412:
399:
381:
380:
369:
364:
360:
356:
351:
347:
343:
304:
301:
237:
234:
231:
226:
222:
201:
198:
195:
192:
187:
183:
162:
159:
156:
151:
147:
118:
115:
112:
109:
104:
100:
86:with positive
9:
6:
4:
3:
2:
2173:
2162:
2159:
2158:
2156:
2146:
2140:
2136:
2131:
2128:
2124:
2120:
2114:
2110:
2106:
2102:
2098:
2094:
2093:
2086:
2085:9780691160504
2082:
2078:
2072:
2065:
2061:
2055:
2051:
2041:
2038:
2036:
2033:
2032:
2026:
2024:
1999:
1996:
1987:
1983:
1973:
1965:
1961:
1957:
1952:
1948:
1934:
1926:
1922:
1911:
1908:
1899:
1895:
1891:
1886:
1882:
1874:
1873:
1872:
1869:
1867:
1863:
1859:
1855:
1851:
1824:
1818:
1812:
1809:
1800:
1789:
1785:
1777:
1776:
1775:
1773:
1769:
1765:
1761:
1756:
1754:
1726:
1723:
1714:
1711:
1707:
1700:
1690:
1687:
1675:
1671:
1663:
1662:
1661:
1659:
1656: ⊂
1655:
1651:
1647:
1642:
1640:
1636:
1632:
1622:
1620:
1616:
1612:
1608:
1604:
1599:
1597:
1572:
1566:
1531:
1525:
1516:
1493:
1487:
1467:
1464:
1461:
1453:
1449:
1445:
1441:
1425:
1419:
1416:
1413:
1403:
1401:
1397:
1393:
1377:
1372:
1368:
1344:
1341:
1336:
1332:
1328:
1325:
1320:
1317:
1314:
1310:
1302:
1301:
1300:
1283:
1278:
1274:
1270:
1267:
1262:
1259:
1256:
1252:
1244:
1243:
1242:
1240:
1236:
1231:
1229:
1225:
1221:
1217:
1216:cycle classes
1195:
1187:
1183:
1177:
1173:
1169:
1163:
1155:
1151:
1143:
1142:
1141:
1124:
1118:
1113:
1109:
1085:
1072:
1064:
1060:
1055:
1048:
1040:
1036:
1032:
1026:
1018:
1014:
1006:
1005:
1004:
1002:
983:
975:
971:
967:
954:
946:
942:
932:
928:
908:
904:
899:
870:
866:
859:
852:
848:
843:
836:
833:
830:
808:
798:
794:
787:
784:
779:
775:
754:
732:
728:
704:
701:
698:
675:
672:
669:
661:
645:
636:
634:
618:
613:
609:
599:
595:
590:
588:
584:
580:
576:
557:
554:
549:
545:
539:
535:
531:
528:
523:
519:
511:
510:
509:
507:
503:
499:
480:
474:
468:
458:
455:
452:
448:
444:
441:
436:
432:
424:
423:
422:
420:
415:
411:
407:
402:
398:
394:
390:
386:
362:
358:
349:
345:
341:
334:
333:
332:
330:
326:
324:
318:
314:
310:
300:
298:
294:
292:
286:
282:
278:
274:
270:
266:
262:
258:
255:, one of the
254:
249:
232:
224:
220:
196:
190:
185:
181:
157:
149:
145:
136:
132:
113:
107:
102:
98:
89:
85:
81:
80:David Mumford
77:
73:
67:
65:
61:
57:
53:
48:
46:
42:
38:
34:
30:
27:
23:
19:
2134:
2100:
2076:
2071:
2063:
2059:
2054:
2020:
1870:
1861:
1857:
1849:
1847:
1771:
1767:
1759:
1757:
1752:
1750:
1657:
1653:
1645:
1643:
1638:
1634:
1630:
1628:
1614:
1610:
1606:
1600:
1595:
1451:
1447:
1443:
1439:
1404:
1395:
1359:
1298:
1238:
1234:
1232:
1227:
1223:
1219:
1215:
1213:
1100:
1000:
930:
926:
659:
637:
632:
597:
593:
591:
586:
582:
578:
574:
572:
505:
501:
497:
495:
418:
413:
409:
406:multiplicity
405:
400:
396:
392:
388:
384:
382:
328:
322:
320:
316:
308:
306:
296:
290:
250:
130:
83:
75:
71:
68:
49:
44:
36:
33:subvarieties
28:
21:
15:
1214:are called
18:mathematics
2046:References
823:such that
662:, written
321:algebraic
303:Definition
289:algebraic
2023:Chow ring
1980:→
1958::
1953:∗
1919:→
1892::
1887:∗
1790:∗
1712:−
1676:∗
1580:¯
1567:ϕ
1539:¯
1526:ϕ
1465:⊆
1423:→
1417::
1414:ϕ
1392:Chow ring
1373:∗
1318:−
1260:−
1224:effective
1174:⨁
1156:∗
1119:
968:⊆
860:
837:∑
831:α
809:×
785:∈
673:∼
670:α
646:α
614:∗
583:effective
536:⨁
524:∗
469:⋅
456:⊆
449:⨁
342:∑
191:
108:
2155:Category
2099:(1998),
2029:See also
2000:′
1912:′
1753:Y′
1727:′
1691:′
1633: :
1438:, where
1228:positive
891:, where
638:A cycle
587:positive
52:divisors
2127:1644323
1619:motives
404:is the
293:-theory
281:motives
2141:
2125:
2115:
2083:
1866:finite
1848:where
1766:, for
1764:proper
325:-cycle
319:. An
313:scheme
24:on an
1509:over
311:be a
20:, an
2139:ISBN
2113:ISBN
2081:ISBN
1650:flat
1299:and
307:Let
1943:and
1864:is
1860:to
1762:is
1648:is
1644:If
1594:in
1394:of
1233:If
1226:or
1218:on
1081:rat
963:rat
900:div
844:div
747:of
658:is
585:or
500:of
408:of
383:of
327:on
275:'s
271:.
259:'s
78:.
43:of
35:of
16:In
2157::
2123:MR
2121:,
2111:,
2103:,
1755:.
1660::
1658:X'
1654:Y'
1639:X'
1637:→
1621:.
1609:,
1598:.
1402:.
1110:CH
635:.
182:CH
99:CH
66:.
2087:.
2064:9
2004:)
1997:X
1993:(
1988:k
1984:Z
1977:)
1974:X
1971:(
1966:k
1962:Z
1949:f
1938:)
1935:X
1932:(
1927:k
1923:Z
1916:)
1909:X
1905:(
1900:k
1896:Z
1883:f
1862:Y
1858:f
1850:n
1831:]
1828:)
1825:Y
1822:(
1819:f
1816:[
1813:n
1810:=
1807:)
1804:]
1801:Y
1798:[
1795:(
1786:f
1772:X
1768:Y
1760:f
1734:]
1731:)
1724:Y
1720:(
1715:1
1708:f
1704:[
1701:=
1698:)
1695:]
1688:Y
1684:[
1681:(
1672:f
1646:f
1635:X
1631:f
1596:S
1576:)
1573:Y
1570:(
1544:)
1535:)
1532:Y
1529:(
1520:(
1517:k
1497:)
1494:Y
1491:(
1488:k
1468:X
1462:Y
1452:r
1448:X
1444:r
1440:S
1426:S
1420:X
1396:X
1378:X
1369:A
1345:.
1342:X
1337:r
1333:A
1329:=
1326:X
1321:r
1315:N
1311:A
1284:X
1279:r
1275:Z
1271:=
1268:X
1263:r
1257:N
1253:Z
1239:N
1235:X
1220:X
1199:)
1196:X
1193:(
1188:r
1184:A
1178:r
1170:=
1167:)
1164:X
1161:(
1152:A
1128:)
1125:X
1122:(
1114:r
1086:.
1077:)
1073:X
1070:(
1065:r
1061:Z
1056:/
1052:)
1049:X
1046:(
1041:r
1037:Z
1033:=
1030:)
1027:X
1024:(
1019:r
1015:A
1001:r
987:)
984:X
981:(
976:r
972:Z
959:)
955:X
952:(
947:r
943:Z
931:i
927:W
909:i
905:W
879:]
876:)
871:i
867:r
863:(
853:i
849:W
840:[
834:=
805:)
799:i
795:W
791:(
788:k
780:i
776:r
755:X
733:i
729:W
708:)
705:1
702:+
699:r
696:(
676:0
633:X
619:X
610:Z
598:X
594:X
558:.
555:X
550:r
546:Z
540:r
532:=
529:X
520:Z
506:r
502:X
498:V
481:,
478:]
475:V
472:[
465:Z
459:X
453:V
445:=
442:X
437:r
433:Z
419:r
414:i
410:V
401:i
397:n
393:X
389:k
385:r
368:]
363:i
359:V
355:[
350:i
346:n
329:X
323:r
317:k
309:X
297:K
291:K
236:)
233:S
230:(
225:2
221:H
200:)
197:S
194:(
186:2
161:)
158:S
155:(
150:2
146:H
131:S
117:)
114:S
111:(
103:2
84:S
76:N
72:N
45:V
37:V
29:V
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