575:
805:
818:
556:
258:
81:
2084:
124:
1485:
1802:. Then the connecting homomorphism can still be defined, and one can write down a sequence as in the statement of the snake lemma. This will always be a chain complex, but it may fail to be exact. Exactness can be asserted, however, when the vertical sequences in the diagram are exact, that is, when the images of
582:
The maps between the kernels and the maps between the cokernels are induced in a natural manner by the given (horizontal) maps because of the diagram's commutativity. The exactness of the two induced sequences follows in a straightforward way from the exactness of the rows of the original diagram.
1949:
253:{\displaystyle \ker a~{\color {Gray}\longrightarrow }~\ker b~{\color {Gray}\longrightarrow }~\ker c~{\overset {d}{\longrightarrow }}~\operatorname {coker} a~{\color {Gray}\longrightarrow }~\operatorname {coker} b~{\color {Gray}\longrightarrow }~\operatorname {coker} c}
1667:
2277:
779:
Once that is done, the theorem is proven for abelian groups or modules over a ring. For the general case, the argument may be rephrased in terms of properties of arrows and cancellation instead of elements. Alternatively, one may invoke
384:
1197:
1478:
1368:
540:
489:
438:
2079:{\displaystyle {\begin{matrix}&1&\to &C_{3}&\to &C_{3}&\to 1\\&\downarrow &&\downarrow &&\downarrow \\1\to &1&\to &S_{3}&\to &A_{5}\end{matrix}}}
811:
is a commutative diagram with exact rows, then the snake lemma can be applied twice, to the "front" and to the "back", yielding two long exact sequences; these are related by a commutative diagram of the form
328:
1941:
1685:, hold for abelian categories as well as in the category of groups, the snake lemma does not. Indeed, arbitrary cokernels do not exist. However, one can replace cokernels by (left) cosets
1800:
1761:
1722:
1493:
1264:
2225:
2190:
968:
341:
2217:
2144:
2114:
1884:
1854:
1220:
1057:
1028:
1008:
988:
936:
907:
887:
867:
847:
1065:
1373:
562:
and then the exact sequence that is the conclusion of the lemma can be drawn on this expanded diagram in the reversed "S" shape of a slithering
2425:
1269:
2543:
494:
443:
392:
2474:
2451:
285:
781:
804:
772:), that it is a homomorphism, and that the resulting long sequence is indeed exact. One may routinely verify the exactness by
2392:
817:
2466:
2538:
1893:
1662:{\displaystyle \ker(t_{M})\to \ker(t_{N})\to \ker(t_{P})\to M\otimes _{k}k\to N\otimes _{k}k\to P\otimes _{k}k\to 0}
99:
1766:
1727:
1688:
1225:
2435:
2382:
2297:
2272:{\displaystyle 1\longrightarrow 1\longrightarrow 1\longrightarrow 1\longrightarrow C_{2}\longrightarrow 1}
793:
2342:
1669:
by applying the snake lemma. Thus, the snake lemma reflects the tensor product's failure to be exact.
792:
In the applications, one often needs to show that long exact sequences are "natural" (in the sense of
594:
2387:
773:
72:
32:
28:
2153:
598:
40:
574:
379:{\displaystyle \operatorname {coker} b~{\color {Gray}\longrightarrow }~\operatorname {coker} c}
2493:
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2447:
2421:
2413:
1827:
941:
2439:
56:
36:
2195:
2122:
2092:
1862:
1832:
1480:
is not exact in general. Hence, a natural question arises. Why is this sequence not exact?
1857:
1815:
68:
2292:
1205:
1033:
1013:
993:
973:
912:
892:
872:
852:
832:
88:
2510:
2532:
2310:
60:
555:
2367:
2147:
1887:
1192:{\displaystyle V\otimes _{k}k=V\otimes _{k}(k/(t))=V/tV=\operatorname {coker} (t).}
279:
64:
39:
and is a crucial tool in homological algebra and its applications, for instance in
80:
796:). This follows from the naturality of the sequence produced by the snake lemma.
550:
To see where the snake lemma gets its name, expand the diagram above as follows:
2496:
335:
92:
24:
2519:
2514:
2409:
2150:, the right vertical arrow has trivial cokernel. Meanwhile the quotient group
1682:
1678:
625:
2501:
1473:{\displaystyle 0\to M\otimes _{k}k\to N\otimes _{k}k\to P\otimes _{k}k\to 0}
103:
2523:
1363:{\displaystyle M\otimes _{k}k\to N\otimes _{k}k\to P\otimes _{k}k\to 0}
1484:
563:
2219:. The sequence in the statement of the snake lemma is therefore
1490:
According to the diagram above, we can induce an exact sequence
535:{\displaystyle \operatorname {coker} c=C'/\operatorname {im} c}
484:{\displaystyle \operatorname {coker} b=B'/\operatorname {im} b}
433:{\displaystyle \operatorname {coker} a=A'/\operatorname {im} a}
43:. Homomorphisms constructed with its help are generally called
323:{\displaystyle \ker a~{\color {Gray}\longrightarrow }~\ker b}
1943:. This gives rise to the following diagram with exact rows:
1886:, which in turn can be written as a semidirect product of
1677:
While many results of homological algebra, such as the
1370:
by right exactness of tensor product. But the sequence
648:. Because of the commutativity of the diagram, we have
1954:
2228:
2198:
2156:
2125:
2095:
1952:
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1865:
1835:
1769:
1730:
1691:
1496:
1376:
1272:
1228:
1208:
1068:
1036:
1016:
996:
976:
944:
915:
895:
875:
855:
835:
497:
446:
395:
344:
288:
127:
2295:'s character at the very beginning of the 1980 film
2116:
is not a normal subgroup in the semidirect product.
700:. Since the bottom row is exact, we find an element
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2108:
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1191:
1051:
1022:
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982:
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881:
861:
841:
534:
483:
432:
378:
322:
252:
2368:"The Topological Snake Lemma and Corona Algebras"
583:The important statement of the lemma is that a
1936:{\displaystyle S_{3}\simeq C_{3}\rtimes C_{2}}
1856:: this contains a subgroup isomorphic to the
578:An animation of the construction of the map d
98:Then there is an exact sequence relating the
8:
590:exists which completes the exact sequence.
2420:(3rd ed.). Springer. pp. 157–9.
2291:The proof of the snake lemma is taught by
2089:Note that the middle column is not exact:
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1015:
995:
990:-linear transformation, so we can tensor
975:
943:
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357:
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231:
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573:
2322:
1795:{\displaystyle C'/\operatorname {im} c}
1756:{\displaystyle B'/\operatorname {im} b}
1717:{\displaystyle A'/\operatorname {im} a}
1259:{\displaystyle 0\to M\to N\to P\to 0}
358:
302:
232:
210:
163:
141:
7:
2329:
35:. The snake lemma is valid in every
2444:Introduction to Commutative Algebra
2461:Hilton, P.; Stammbach, U. (1997).
1266:, we can induce an exact sequence
776:(see the proof of Lemma 9.1 in ).
14:
593:In the case of abelian groups or
2398:from the original on 2022-10-09.
2283:which indeed fails to be exact.
1483:
1202:Given a short exact sequence of
816:
803:
554:
267:is a homomorphism, known as the
79:
2463:A course in homological algebra
2375:New York Journal of Mathematics
605:can be constructed as follows:
2263:
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2014:
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186:
164:
142:
16:Theorem in homological algebra
1:
2467:Graduate Texts in Mathematics
752:). Now one has to check that
616:and view it as an element of
274:Furthermore, if the morphism
782:Mitchell's embedding theorem
728:is unique by injectivity of
2185:{\displaystyle S_{3}/C_{3}}
2560:
282:, then so is the morphism
2544:Lemmas in category theory
1673:In the category of groups
768:and not on the choice of
59:(such as the category of
2520:Proof of the Snake Lemma
2469:. Springer. p. 99.
2414:"III §9 The Snake Lemma"
2366:Schochet, C. L. (1999).
2343:"Extensions of C2 by C3"
963:{\displaystyle t:V\to V}
732: '. We then define
570:Construction of the maps
389:The cokernels here are:
45:connecting homomorphisms
794:natural transformations
756:is well-defined (i.e.,
585:connecting homomorphism
546:Explanation of the name
269:connecting homomorphism
2273:
2213:
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2110:
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1880:
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696:) is in the kernel of
579:
536:
485:
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324:
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2274:
2214:
2212:{\displaystyle C_{2}}
2187:
2141:
2139:{\displaystyle A_{5}}
2111:
2109:{\displaystyle C_{2}}
2081:
1938:
1881:
1879:{\displaystyle S_{3}}
1851:
1849:{\displaystyle A_{5}}
1797:
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994:
974:
942:
913:
893:
873:
853:
833:
684:is in the kernel of
495:
444:
393:
342:
286:
125:
33:long exact sequences
2539:Homological algebra
87:where the rows are
73:commutative diagram
63:or the category of
29:homological algebra
2494:Weisstein, Eric W.
2446:. Addison–Wesley.
2287:In popular culture
2269:
2209:
2182:
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2106:
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764:) only depends on
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532:
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306:
250:
236:
214:
167:
145:
41:algebraic topology
23:is a tool used in
2427:978-0-387-95385-4
2192:is isomorphic to
1828:alternating group
1215:{\displaystyle k}
1052:{\displaystyle k}
1023:{\displaystyle k}
1003:{\displaystyle V}
983:{\displaystyle k}
931:{\displaystyle k}
902:{\displaystyle V}
882:{\displaystyle k}
862:{\displaystyle V}
842:{\displaystyle k}
688:), and therefore
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2440:Macdonald, I. G.
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1816:normal subgroups
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886:
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868:
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845:
840:
820:
807:
608:Pick an element
558:
541:
539:
538:
533:
522:
517:
490:
488:
487:
482:
471:
466:
439:
437:
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83:
57:abelian category
37:abelian category
31:, to construct
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2528:
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2370:
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2349:
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2324:
2319:
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2253:
2224:
2223:
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2017:
2011:
2004:
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1995:
1985:
1983:
1978:
1968:
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1961:
1948:
1947:
1923:
1910:
1897:
1892:
1891:
1866:
1861:
1860:
1858:symmetric group
1836:
1831:
1830:
1824:
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1726:
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1371:
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1268:
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1224:
1223:
1222:-vector spaces
1204:
1203:
1100:
1072:
1064:
1063:
1032:
1031:
1012:
1011:
992:
991:
972:
971:
940:
939:
911:
910:
891:
890:
889:-vector space.
871:
870:
851:
850:
831:
830:
827:
790:
774:diagram chasing
628:, there exists
572:
548:
510:
493:
492:
459:
442:
441:
408:
391:
390:
340:
339:
284:
283:
123:
122:
89:exact sequences
53:
27:, particularly
17:
12:
11:
5:
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2527:
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2487:
2486:External links
2484:
2482:
2481:
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2458:
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2426:
2405:
2402:
2401:
2388:10.1.1.73.1568
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2293:Jill Clayburgh
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1822:Counterexample
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979:
959:
956:
953:
950:
947:
927:
924:
921:
918:
898:
878:
858:
838:
826:
823:
822:
821:
809:
808:
789:
786:
571:
568:
560:
559:
547:
544:
531:
528:
525:
521:
516:
513:
509:
506:
503:
500:
480:
477:
474:
470:
465:
462:
458:
455:
452:
449:
429:
426:
423:
419:
414:
411:
407:
404:
401:
398:
375:
372:
369:
361:
353:
350:
347:
319:
316:
313:
305:
297:
294:
291:
261:
260:
249:
246:
243:
235:
227:
224:
221:
213:
205:
202:
199:
191:
188:
180:
177:
174:
166:
158:
155:
152:
144:
136:
133:
130:
85:
84:
71:), consider a
61:abelian groups
52:
49:
15:
13:
10:
9:
6:
4:
3:
2:
2556:
2545:
2542:
2540:
2537:
2536:
2534:
2525:
2521:
2518:
2516:
2512:
2509:
2504:
2503:
2498:
2497:"Snake Lemma"
2495:
2490:
2489:
2485:
2478:
2476:0-387-94823-6
2472:
2468:
2464:
2459:
2455:
2453:0-201-00361-9
2449:
2445:
2441:
2437:
2433:
2429:
2423:
2419:
2415:
2411:
2407:
2406:
2394:
2389:
2384:
2380:
2376:
2369:
2362:
2359:
2348:
2344:
2338:
2335:
2332:, p. 159
2331:
2326:
2323:
2316:
2312:
2311:Zig-zag lemma
2309:
2308:
2304:
2302:
2300:
2299:
2294:
2286:
2284:
2266:
2258:
2254:
2247:
2241:
2235:
2229:
2222:
2221:
2220:
2204:
2200:
2177:
2173:
2168:
2162:
2158:
2149:
2131:
2127:
2117:
2101:
2097:
2067:
2063:
2050:
2046:
2035:
2027:
2000:
1990:
1986:
1973:
1969:
1958:
1946:
1945:
1944:
1928:
1924:
1920:
1915:
1911:
1907:
1902:
1898:
1889:
1888:cyclic groups
1871:
1867:
1859:
1841:
1837:
1829:
1826:Consider the
1821:
1819:
1817:
1813:
1809:
1805:
1789:
1786:
1783:
1779:
1774:
1771:
1750:
1747:
1744:
1740:
1735:
1732:
1711:
1708:
1705:
1701:
1696:
1693:
1684:
1680:
1672:
1670:
1656:
1650:
1642:
1636:
1632:
1628:
1622:
1614:
1608:
1604:
1600:
1594:
1586:
1580:
1576:
1572:
1561:
1557:
1550:
1547:
1536:
1532:
1525:
1522:
1511:
1507:
1500:
1497:
1488:
1486:
1481:
1467:
1461:
1453:
1447:
1443:
1439:
1433:
1425:
1419:
1415:
1411:
1405:
1397:
1391:
1387:
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1377:
1357:
1351:
1343:
1337:
1333:
1329:
1323:
1315:
1309:
1305:
1301:
1295:
1287:
1281:
1277:
1273:
1253:
1247:
1241:
1235:
1229:
1209:
1186:
1180:
1174:
1171:
1168:
1165:
1162:
1158:
1154:
1151:
1142:
1135:
1128:
1122:
1111:
1105:
1101:
1097:
1094:
1091:
1083:
1077:
1073:
1069:
1062:
1061:
1060:
1043:
1037:
1017:
997:
977:
957:
951:
948:
945:
922:
916:
896:
876:
856:
836:
824:
819:
815:
814:
813:
806:
802:
801:
800:
797:
795:
787:
785:
783:
777:
775:
771:
767:
763:
759:
755:
751:
747:
743:
739:
735:
731:
727:
723:
719:
715:
711:
707:
703:
699:
695:
691:
687:
683:
680:) = 0 (since
679:
675:
671:
667:
663:
659:
655:
651:
647:
643:
639:
635:
631:
627:
623:
619:
615:
611:
606:
604:
600:
596:
591:
589:
586:
576:
569:
567:
565:
557:
553:
552:
551:
545:
543:
529:
526:
523:
519:
514:
511:
507:
504:
501:
498:
478:
475:
472:
468:
463:
460:
456:
453:
450:
447:
427:
424:
421:
417:
412:
409:
405:
402:
399:
396:
387:
373:
370:
367:
351:
348:
345:
338:, then so is
337:
333:
317:
314:
311:
295:
292:
289:
281:
277:
272:
270:
266:
247:
244:
241:
225:
222:
219:
203:
200:
197:
189:
178:
175:
172:
156:
153:
150:
134:
131:
128:
121:
120:
119:
117:
113:
109:
105:
101:
96:
94:
91:and 0 is the
90:
82:
78:
77:
76:
74:
70:
67:over a given
66:
65:vector spaces
62:
58:
50:
48:
46:
42:
38:
34:
30:
26:
22:
2524:It's My Turn
2522:in the film
2500:
2462:
2443:
2436:Atiyah, M.F.
2417:
2378:
2374:
2361:
2350:. Retrieved
2346:
2337:
2325:
2298:It's My Turn
2296:
2290:
2282:
2118:
2088:
1825:
1811:
1807:
1803:
1676:
1489:
1482:
1201:
828:
810:
798:
791:
778:
769:
765:
761:
757:
753:
749:
745:
741:
737:
733:
729:
725:
721:
717:
713:
709:
705:
701:
697:
693:
689:
685:
681:
677:
673:
669:
665:
661:
657:
653:
649:
645:
641:
637:
633:
629:
621:
617:
613:
612:in ker
609:
607:
602:
592:
587:
584:
581:
561:
549:
388:
331:
280:monomorphism
275:
273:
268:
264:
262:
115:
111:
107:
97:
86:
54:
44:
20:
18:
2511:Snake Lemma
2410:Lang, Serge
938:-module by
336:epimorphism
93:zero object
25:mathematics
21:snake lemma
2533:Categories
2515:PlanetMath
2352:2021-11-06
2347:GroupNames
2317:References
1683:nine lemma
1679:five lemma
849:be field,
788:Naturality
626:surjective
601:, the map
597:over some
2502:MathWorld
2383:CiteSeerX
2381:: 131–7.
2330:Lang 2002
2264:⟶
2251:⟶
2245:⟶
2239:⟶
2233:⟶
2058:→
2041:→
2031:→
2021:↓
2015:↓
2009:↓
1998:→
1981:→
1964:→
1921:⋊
1908:≃
1787:
1748:
1709:
1654:→
1633:⊗
1626:→
1605:⊗
1598:→
1577:⊗
1570:→
1551:
1545:→
1526:
1520:→
1501:
1465:→
1444:⊗
1437:→
1416:⊗
1409:→
1388:⊗
1381:→
1355:→
1334:⊗
1327:→
1306:⊗
1299:→
1278:⊗
1251:→
1245:→
1239:→
1233:→
1175:
1102:⊗
1074:⊗
955:→
527:
502:
476:
451:
425:
400:
371:
360:⟶
349:
330:, and if
315:
304:⟶
293:
245:
234:⟶
223:
212:⟶
201:
187:⟶
176:
165:⟶
154:
143:⟶
132:
104:cokernels
51:Statement
2442:(1969).
2412:(2002).
2393:Archived
2305:See also
1775:′
1736:′
1697:′
970:being a
712: '(
620:; since
515:′
464:′
413:′
2418:Algebra
1681:or the
825:Example
595:modules
100:kernels
2473:
2450:
2424:
2385:
2148:simple
2119:Since
1810:, and
1763:, and
365:
355:
334:is an
309:
299:
263:where
239:
229:
217:
207:
195:
182:
170:
160:
148:
138:
114:, and
55:In an
2396:(PDF)
2371:(PDF)
1172:coker
1030:over
869:be a
708:with
672:)) =
660:)) =
636:with
564:snake
499:coker
448:coker
397:coker
368:coker
346:coker
278:is a
242:coker
220:coker
198:coker
69:field
2471:ISBN
2448:ISBN
2422:ISBN
1814:are
1010:and
829:Let
740:) =
716:) =
644:) =
599:ring
102:and
19:The
2513:at
2146:is
1548:ker
1523:ker
1498:ker
909:is
799:If
724:).
706:A'
704:in
698:g'
632:in
624:is
312:ker
290:ker
173:ker
151:ker
129:ker
106:of
2535::
2499:.
2465:.
2438:;
2416:.
2391:.
2377:.
2373:.
2345:.
2301:.
1890::
1818:.
1806:,
1784:im
1745:im
1724:,
1706:im
1059:.
784:.
746:im
744:+
650:g'
566:.
542:.
524:im
491:,
473:im
440:,
422:im
386:.
332:g'
271:.
118::
110:,
95:.
75::
47:.
2505:.
2479:.
2456:.
2430:.
2379:5
2355:.
2279:,
2267:1
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2255:C
2248:1
2242:1
2236:1
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2205:2
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2178:3
2174:C
2169:/
2163:3
2159:S
2132:5
2128:A
2102:2
2098:C
2068:5
2064:A
2051:3
2047:S
2036:1
2028:1
2001:1
1991:3
1987:C
1974:3
1970:C
1959:1
1929:2
1925:C
1916:3
1912:C
1903:3
1899:S
1872:3
1868:S
1842:5
1838:A
1812:c
1808:b
1804:a
1790:c
1780:/
1772:C
1751:b
1741:/
1733:B
1712:a
1702:/
1694:A
1657:0
1651:k
1646:]
1643:t
1640:[
1637:k
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1615:t
1612:[
1609:k
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1587:t
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1581:k
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1562:P
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1554:(
1542:)
1537:N
1533:t
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1468:0
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1181:t
1178:(
1169:=
1166:V
1163:t
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1143:t
1140:(
1136:/
1132:]
1129:t
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1120:(
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1044:t
1041:[
1038:k
1018:k
998:V
978:k
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949::
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923:t
920:[
917:k
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877:k
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770:y
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762:x
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758:d
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720:(
718:b
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512:C
508:=
505:c
479:b
469:/
461:B
457:=
454:b
428:a
418:/
410:A
406:=
403:a
374:c
352:b
318:b
296:a
276:f
265:d
248:c
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190:d
179:c
157:b
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116:c
112:b
108:a
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