4288:
4097:
3258:
2722:
2296:
3487:
3723:
338:
3899:
2879:
1898:
4417:
Despite its abstraction, this general definition has useful consequences. For example, in section 1.8, Grothendieck proves that a functor is pro-representable if and only if it is left exact, under some mild conditions on the category
4076:
2424:
197:
3973:
2170:
4283:{\displaystyle :\mathbf {Z} /12\mathbf {Z} \otimes _{Z}P\cong (\mathbf {Z} \otimes _{Z}P)/(12\mathbf {Z} \otimes _{Z}P)=(\mathbf {Z} \otimes _{Z}P)/(3\mathbf {Z} \otimes _{Z}P)\cong \mathbf {Z} P/3\mathbf {Z} P}
2587:
1815:
1538:
57:. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors that
3068:
2604:
2186:
3351:
3552:
216:
2512:
3728:
2929:
1990:
1955:
2983:
3030:
4330:
3343:
3301:
4351:
into colimits; a contravariant functor is left exact iff it turns finite colimits into limits; a contravariant functor is right exact iff it turns finite limits into colimits.
2731:
4394:
3063:
1849:
1920:
1844:
1773:
1570:
1464:
1430:
1408:
1386:
1364:
1672:
1642:
3528:
1496:
1701:
3978:
1616:
2319:
1741:
1721:
1590:
4397:, tome I, section 1, the notion of left (right) exact functors are defined for general categories, and not just abelian ones. The definition is as follows:
120:
3904:
2099:
2520:
3253:{\displaystyle x_{i}\in R,\sum _{i}x_{i}(r_{i}\otimes p_{i})=\sum _{i}1\otimes (r_{i}x_{i}p_{i})=1\otimes (\sum _{i}r_{i}x_{i}p_{i})}
1778:
1501:
4504:
2717:{\displaystyle I\otimes _{R}P{\stackrel {f\otimes P}{\to }}R\otimes _{R}P{\stackrel {g\otimes P}{\to }}R/I\otimes _{R}P\to 0}
2291:{\displaystyle A\otimes _{R}P{\stackrel {f\otimes P}{\to }}B\otimes _{R}P{\stackrel {g\otimes P}{\to }}C\otimes _{R}P\to 0}
3482:{\displaystyle P=\mathbf {Z} :=\{a/2^{k}:a,k\in \mathbf {Z} \},P\otimes \mathbf {Z} /m\mathbf {Z} \cong P/k\mathbf {Z} P}
17:
4536:
3718:{\displaystyle (12z)\otimes (a/2^{k})\in (12\mathbf {Z} \otimes _{Z}P).(12z)\otimes (a/2^{k})=(3z)\otimes (a/2^{k-2})}
333:{\displaystyle 0\to F(A)\ {\stackrel {F(f)}{\longrightarrow }}\ F(B)\ {\stackrel {F(g)}{\longrightarrow }}\ F(C)\to 0}
3894:{\displaystyle (3z)\otimes (a/2^{k})\in (3\mathbf {Z} \otimes _{Z}P),(3z)\otimes (a/2^{k})=(12z)\otimes (a/2^{k+2})}
2453:
2895:
1960:
1925:
30:
This article is about exact functors in homological algebra. For exact functors between regular categories, see
4587:
2938:
518:
2988:
4293:
3306:
4648:
4643:
3267:
2874:{\displaystyle R/I\otimes _{R}P\cong (R\otimes _{R}P)/Image(f\otimes P)=(R\otimes _{R}P)/(I\otimes _{R}P)}
1618:. The given tensor products only have pure tensors. Therefore, it suffices to show that if a pure tensor
4602:
1041:
730:→0 to have some exactness preserved. The following definitions are equivalent to the ones given above:
4290:. The last congruence follows by a similar argument to one in the proof of the corollary showing that
1893:{\displaystyle 5\mathbf {Z} \hookrightarrow \mathbf {Z} \twoheadrightarrow \mathbf {Z} /5\mathbf {Z} }
1824:
is not flat, then tensor product is not left exact. For example, consider the short exact sequence of
1146:
987:
3035:
4365:
Left and right exact functors are ubiquitous mainly because of the following fact: if the functor
1903:
1827:
1746:
1543:
1447:
1413:
1391:
1369:
1347:
4344:
2066:
While tensoring may not be left exact, it can be shown that tensoring is a right exact functor:
1651:
1621:
4529:
3492:
2007:
1469:
4617:
530:
4343:
A covariant (not necessarily additive) functor is left exact if and only if it turns finite
4071:{\displaystyle P=\mathbf {Z} ,A=12\mathbf {Z} ,B=\mathbf {Z} ,C=\mathbf {Z} /12\mathbf {Z} }
4567:
4413:
is left (resp. right) exact if it commutes with finite projective (resp. inductive) limits.
2419:{\displaystyle f\otimes P(a\otimes p):=f(a)\otimes p,g\otimes P(b\otimes p):=g(b)\otimes p}
2173:
1231:
1173:
1165:
203:
54:
1677:
8:
2886:
2439:
1595:
1244:
1188:
1108:
42:
4358:; the degree to which a right exact functor fails to be exact can be measured with its
1726:
1706:
1575:
1219:
78:
4436:
are sometimes called exact functors and generalize the exact functors discussed here.
4653:
4597:
4522:
4500:
1184:
1062:
96:
4354:
The degree to which a left exact functor fails to be exact can be measured with its
4612:
4582:
4577:
4562:
4557:
4433:
4405:
be a category with finite projective (resp. injective) limits. Then a functor from
4370:
2090:
2011:
1645:
1284:. This is a covariant right exact functor; in other words, given an exact sequence
1158:
1097:
99:
31:
4090:
modules by the usual multiplication action and satisfy the conditions of the main
4607:
4572:
4492:
4359:
4355:
4622:
4426:
4347:
into limits; a covariant functor is right exact if and only if it turns finite
4094:. By the exactness implied by the theorem and by the above note we obtain that
2004:
1256:
192:{\displaystyle 0\to A\ {\stackrel {f}{\to }}\ B\ {\stackrel {g}{\to }}\ C\to 0}
4637:
1441:
1192:
1116:
3968:{\displaystyle (12\mathbf {Z} \otimes _{Z}P)=(3\mathbf {Z} \otimes _{Z}P)}
2165:{\displaystyle A\ {\stackrel {f}{\to }}\ B\ {\stackrel {g}{\to }}\ C\to 0}
718:
It is not always necessary to start with an entire short exact sequence 0→
4429:
generalize the exact functors between abelian categories discussed here.
4340:
A functor is exact if and only if it is both left exact and right exact.
1341:
994:
38:
1176:, and any additive functor turns split sequences into split sequences.)
3535:
3261:
3032:. So, the kernel of this map cannot contain any nonzero pure tensors.
1142:
2431:
2582:{\displaystyle I{\stackrel {f}{\to }}R{\stackrel {g}{\to }}R/I\to 0}
1092:) is a contravariant left-exact functor; it is exact if and only if
1168:. Alternatively, one can argue that every short exact sequence of
4545:
4348:
4091:
50:
4514:
347:. (The maps are often omitted and implied, and one says: "if 0→
1810:{\displaystyle M\otimes \mathbb {Q} \to N\otimes \mathbb {Q} }
1533:{\displaystyle M\otimes \mathbb {Q} \to N\otimes \mathbb {Q} }
2313:-modules and not merely of abelian groups). Here, we define
1498:, then the corresponding map between the tensor products
993:
The most basic examples of left exact functors are the
61:
to be exact, but in ways that can still be controlled.
1199:. The covariant functor that associates to each sheaf
1145:). This yields a contravariant exact functor from the
4296:
4100:
3981:
3907:
3731:
3555:
3495:
3354:
3309:
3270:
3071:
3038:
2991:
2941:
2935:-linearly extending the map defined on pure tensors:
2898:
2734:
2607:
2523:
2456:
2322:
2189:
2102:
1963:
1928:
1906:
1852:
1830:
1781:
1749:
1729:
1709:
1680:
1654:
1624:
1598:
1578:
1546:
1504:
1472:
1450:
1416:
1394:
1372:
1350:
219:
123:
4324:
4282:
4070:
3967:
3893:
3717:
3522:
3481:
3337:
3295:
3252:
3057:
3024:
2977:
2923:
2873:
2716:
2581:
2506:
2418:
2290:
2164:
1984:
1949:
1914:
1892:
1838:
1809:
1767:
1735:
1715:
1695:
1666:
1636:
1610:
1584:
1564:
1532:
1490:
1458:
1424:
1402:
1380:
1358:
517:) is exact. This is distinct from the notion of a
332:
191:
4635:
1957:gives a sequence that is no longer exact, since
2309:is commutative, this sequence is a sequence of
1187:, we can consider the abelian category of all
1153:to itself. (Exactness follows from the above:
1036:) defines a covariant left-exact functor from
4530:
3427:
3386:
2507:{\displaystyle P\otimes _{R}(R/I)\cong P/IP}
27:Functor that preserves short exact sequences
3260:. So, this map is injective. It is clearly
2924:{\displaystyle R\otimes _{R}P\rightarrow P}
2597:is the projection, is an exact sequence of
4537:
4523:
4335:
3348:As another application, we show that for,
1985:{\displaystyle \mathbf {Z} /5\mathbf {Z} }
1950:{\displaystyle \mathbf {Z} /5\mathbf {Z} }
2978:{\displaystyle r\otimes p\mapsto rp.rp=0}
2601:-modules. By the above we get that :
1803:
1789:
1526:
1512:
1452:
1418:
1396:
1374:
1352:
4491:
3025:{\displaystyle 0=rp\otimes 1=r\otimes p}
1992:is not torsion-free and thus not flat.
1300:modules, the sequence of abelian groups
3065:is composed only of pure tensors: For
14:
4636:
4325:{\displaystyle I\otimes _{R}P\cong IP}
3338:{\displaystyle I\otimes _{R}P\cong IP}
4518:
4499:. Vol. 2 (2nd ed.). Dover.
4425:The exact functors between Quillen's
3296:{\displaystyle R\otimes _{R}P\cong P}
4469:Jacobson (2009), p. 99, Theorem 3.1.
4451:Jacobson (2009), p. 98, Theorem 3.1.
2885:is the inclusion. Now, consider the
2096:having multiplicative identity. Let
4460:Jacobson (2009), p. 149, Prop. 3.9.
1674:is an element of the kernel. Then,
1388:-module. Therefore, tensoring with
24:
4388:
2724:is also a short exact sequence of
2301:is also a short exact sequence of
25:
4665:
4544:
1141:) (this is commonly known as the
877:turns cokernels into cokernels");
4273:
4257:
4233:
4201:
4174:
4142:
4118:
4105:
4064:
4051:
4037:
4023:
3989:
3945:
3915:
3781:
3605:
3472:
3453:
3440:
3423:
3362:
2017:consisting of all functors from
1978:
1965:
1943:
1930:
1908:
1886:
1873:
1865:
1857:
1832:
1648:, then it is zero. Suppose that
1540:is injective. One can show that
990:of abelian categories is exact.
977:turns kernels into cokernels").
927:turns cokernels into kernels");
4472:
4463:
4454:
4445:
4250:
4226:
4218:
4197:
4191:
4167:
4159:
4138:
4007:
3993:
3962:
3938:
3932:
3908:
3888:
3861:
3855:
3846:
3840:
3819:
3813:
3804:
3798:
3774:
3768:
3747:
3741:
3732:
3712:
3685:
3679:
3670:
3664:
3643:
3637:
3628:
3622:
3598:
3592:
3571:
3565:
3556:
3549:Proof: Consider a pure tensor
3380:
3366:
3247:
3204:
3192:
3159:
3137:
3111:
3058:{\displaystyle R\otimes _{R}P}
2951:
2915:
2868:
2849:
2841:
2822:
2816:
2804:
2781:
2762:
2708:
2666:
2628:
2573:
2550:
2531:
2484:
2470:
2407:
2401:
2392:
2380:
2359:
2353:
2344:
2332:
2282:
2248:
2210:
2156:
2138:
2113:
1869:
1861:
1793:
1690:
1684:
1516:
1482:
519:topological half-exact functor
324:
321:
315:
302:
296:
288:
278:
272:
259:
253:
245:
235:
229:
223:
183:
165:
140:
127:
64:
13:
1:
4484:
4432:The regular functors between
3345:. This proves the corollary.
1432:-module is an exact functor.
1203:the group of global sections
827:turns kernels into kernels");
343:is a short exact sequence in
1915:{\displaystyle \mathbf {Z} }
1839:{\displaystyle \mathbf {Z} }
1768:{\displaystyle m\otimes q=0}
1565:{\displaystyle m\otimes q=0}
1459:{\displaystyle \mathbb {Z} }
1436:It suffices to show that if
1425:{\displaystyle \mathbb {Z} }
1403:{\displaystyle \mathbb {Q} }
1381:{\displaystyle \mathbb {Z} }
1359:{\displaystyle \mathbb {Q} }
7:
3542:. We prove a special case:
1999:is an abelian category and
1001:is an abelian category and
981:
10:
4670:
2050:by evaluating functors at
1667:{\displaystyle m\otimes q}
1637:{\displaystyle m\otimes q}
1234:, we can define a functor
29:
4553:
3523:{\displaystyle k=m/2^{n}}
102:(so that, in particular,
4478:Jacobson (2009), p. 156.
4439:
2728:-modules. By exactness,
2033:, then we get a functor
1592:is a torsion element or
1491:{\displaystyle i:M\to N}
1336:is exact if and only if
1057:is exact if and only if
4336:Properties and theorems
1743:is torsion. Therefore,
106:(0) = 0). We say that
4588:Essentially surjective
4356:right derived functors
4326:
4284:
4072:
3969:
3895:
3719:
3524:
3483:
3339:
3297:
3254:
3059:
3026:
2979:
2925:
2875:
2718:
2583:
2508:
2420:
2292:
2166:
2010:, we can consider the
1986:
1951:
1916:
1894:
1840:
1811:
1769:
1737:
1717:
1697:
1668:
1638:
1612:
1586:
1566:
1534:
1492:
1460:
1426:
1404:
1382:
1360:
988:equivalence or duality
541:, we similarly define
533:additive functor from
383:)→0 is also exact".)
334:
193:
4360:left derived functors
4327:
4285:
4073:
3970:
3896:
3720:
3525:
3484:
3340:
3298:
3255:
3060:
3027:
2980:
2926:
2876:
2719:
2593:is the inclusion and
2584:
2509:
2421:
2293:
2167:
2029:is a given object of
1987:
1952:
1917:
1895:
1841:
1812:
1770:
1738:
1718:
1698:
1669:
1639:
1613:
1587:
1567:
1535:
1493:
1461:
1427:
1405:
1383:
1361:
1245:category of all left
386:Further, we say that
335:
194:
55:short exact sequences
4409:to another category
4294:
4098:
3979:
3905:
3729:
3553:
3493:
3352:
3307:
3268:
3069:
3036:
2989:
2939:
2896:
2890:-module homomorphism
2732:
2605:
2521:
2454:
2320:
2187:
2174:short exact sequence
2100:
2025:; it is abelian. If
1961:
1926:
1904:
1850:
1828:
1779:
1747:
1727:
1707:
1696:{\displaystyle i(m)}
1678:
1652:
1622:
1596:
1576:
1544:
1502:
1470:
1448:
1414:
1392:
1370:
1348:
973:)→0 exact (i.e. if "
873:)→0 exact (i.e. if "
359:→0 is exact, then 0→
217:
204:short exact sequence
121:
4649:Additive categories
4644:Homological algebra
4381:is right exact and
1817:is also injective.
1611:{\displaystyle q=0}
899:→0 exact implies 0→
606:→0 is exact then 0→
564:→0 is exact then 0→
409:→0 is exact then 0→
43:homological algebra
18:Right-exact functor
4434:regular categories
4322:
4280:
4068:
3965:
3901:. This shows that
3891:
3715:
3520:
3479:
3335:
3293:
3250:
3216:
3152:
3100:
3055:
3022:
2975:
2921:
2871:
2714:
2579:
2504:
2450:is as above, then
2430:This has a useful
2416:
2288:
2162:
1982:
1947:
1912:
1890:
1836:
1807:
1765:
1733:
1713:
1703:is torsion. Since
1693:
1664:
1634:
1608:
1582:
1562:
1530:
1488:
1456:
1422:
1400:
1378:
1356:
923:) exact (i.e. if "
823:) exact (i.e. if "
330:
189:
79:abelian categories
4631:
4630:
4603:Full and faithful
4506:978-0-486-47187-7
3207:
3143:
3091:
2681:
2643:
2559:
2540:
2305:-modules. (Since
2263:
2225:
2152:
2147:
2133:
2127:
2122:
2108:
1900:. Tensoring over
1736:{\displaystyle m}
1716:{\displaystyle i}
1585:{\displaystyle m}
1243:from the abelian
1211:) is left-exact.
1185:topological space
1046:of abelian groups
937:if and only if 0→
849:→0 exact implies
787:if and only if 0→
690:→0 is exact then
648:→0 is exact then
493:→0 is exact then
451:→0 is exact then
311:
306:
283:
268:
263:
240:
179:
174:
160:
154:
149:
135:
16:(Redirected from
4661:
4539:
4532:
4525:
4516:
4515:
4510:
4493:Jacobson, Nathan
4479:
4476:
4470:
4467:
4461:
4458:
4452:
4449:
4427:exact categories
4331:
4329:
4328:
4323:
4309:
4308:
4289:
4287:
4286:
4281:
4276:
4268:
4260:
4246:
4245:
4236:
4225:
4214:
4213:
4204:
4187:
4186:
4177:
4166:
4155:
4154:
4145:
4131:
4130:
4121:
4113:
4108:
4077:
4075:
4074:
4069:
4067:
4059:
4054:
4040:
4026:
4003:
3992:
3974:
3972:
3971:
3966:
3958:
3957:
3948:
3928:
3927:
3918:
3900:
3898:
3897:
3892:
3887:
3886:
3871:
3839:
3838:
3829:
3794:
3793:
3784:
3767:
3766:
3757:
3724:
3722:
3721:
3716:
3711:
3710:
3695:
3663:
3662:
3653:
3618:
3617:
3608:
3591:
3590:
3581:
3529:
3527:
3526:
3521:
3519:
3518:
3509:
3488:
3486:
3485:
3480:
3475:
3467:
3456:
3448:
3443:
3426:
3406:
3405:
3396:
3376:
3365:
3344:
3342:
3341:
3336:
3322:
3321:
3302:
3300:
3299:
3294:
3283:
3282:
3259:
3257:
3256:
3251:
3246:
3245:
3236:
3235:
3226:
3225:
3215:
3191:
3190:
3181:
3180:
3171:
3170:
3151:
3136:
3135:
3123:
3122:
3110:
3109:
3099:
3081:
3080:
3064:
3062:
3061:
3056:
3051:
3050:
3031:
3029:
3028:
3023:
2984:
2982:
2981:
2976:
2930:
2928:
2927:
2922:
2911:
2910:
2880:
2878:
2877:
2872:
2864:
2863:
2848:
2837:
2836:
2788:
2777:
2776:
2755:
2754:
2742:
2723:
2721:
2720:
2715:
2704:
2703:
2691:
2683:
2682:
2680:
2669:
2664:
2658:
2657:
2645:
2644:
2642:
2631:
2626:
2620:
2619:
2588:
2586:
2585:
2580:
2569:
2561:
2560:
2558:
2553:
2548:
2542:
2541:
2539:
2534:
2529:
2513:
2511:
2510:
2505:
2497:
2480:
2469:
2468:
2425:
2423:
2422:
2417:
2297:
2295:
2294:
2289:
2278:
2277:
2265:
2264:
2262:
2251:
2246:
2240:
2239:
2227:
2226:
2224:
2213:
2208:
2202:
2201:
2180:-modules. Then
2171:
2169:
2168:
2163:
2150:
2149:
2148:
2146:
2141:
2136:
2131:
2125:
2124:
2123:
2121:
2116:
2111:
2106:
2091:commutative ring
2012:functor category
2003:is an arbitrary
1991:
1989:
1988:
1983:
1981:
1973:
1968:
1956:
1954:
1953:
1948:
1946:
1938:
1933:
1921:
1919:
1918:
1913:
1911:
1899:
1897:
1896:
1891:
1889:
1881:
1876:
1868:
1860:
1845:
1843:
1842:
1837:
1835:
1816:
1814:
1813:
1808:
1806:
1792:
1774:
1772:
1771:
1766:
1742:
1740:
1739:
1734:
1722:
1720:
1719:
1714:
1702:
1700:
1699:
1694:
1673:
1671:
1670:
1665:
1643:
1641:
1640:
1635:
1617:
1615:
1614:
1609:
1591:
1589:
1588:
1583:
1571:
1569:
1568:
1563:
1539:
1537:
1536:
1531:
1529:
1515:
1497:
1495:
1494:
1489:
1465:
1463:
1462:
1457:
1455:
1431:
1429:
1428:
1423:
1421:
1409:
1407:
1406:
1401:
1399:
1387:
1385:
1384:
1379:
1377:
1365:
1363:
1362:
1357:
1355:
1005:is an object of
799:exact implies 0→
339:
337:
336:
331:
309:
308:
307:
305:
291:
286:
281:
266:
265:
264:
262:
248:
243:
238:
198:
196:
195:
190:
177:
176:
175:
173:
168:
163:
158:
152:
151:
150:
148:
143:
138:
133:
100:additive functor
94:
32:regular category
21:
4669:
4668:
4664:
4663:
4662:
4660:
4659:
4658:
4634:
4633:
4632:
4627:
4549:
4543:
4513:
4507:
4487:
4482:
4477:
4473:
4468:
4464:
4459:
4455:
4450:
4446:
4442:
4391:
4389:Generalizations
4385:is left exact.
4338:
4304:
4300:
4295:
4292:
4291:
4272:
4264:
4256:
4241:
4237:
4232:
4221:
4209:
4205:
4200:
4182:
4178:
4173:
4162:
4150:
4146:
4141:
4126:
4122:
4117:
4109:
4104:
4099:
4096:
4095:
4063:
4055:
4050:
4036:
4022:
3999:
3988:
3980:
3977:
3976:
3953:
3949:
3944:
3923:
3919:
3914:
3906:
3903:
3902:
3876:
3872:
3867:
3834:
3830:
3825:
3789:
3785:
3780:
3762:
3758:
3753:
3730:
3727:
3726:
3700:
3696:
3691:
3658:
3654:
3649:
3613:
3609:
3604:
3586:
3582:
3577:
3554:
3551:
3550:
3534:is the highest
3514:
3510:
3505:
3494:
3491:
3490:
3471:
3463:
3452:
3444:
3439:
3422:
3401:
3397:
3392:
3372:
3361:
3353:
3350:
3349:
3317:
3313:
3308:
3305:
3304:
3278:
3274:
3269:
3266:
3265:
3241:
3237:
3231:
3227:
3221:
3217:
3211:
3186:
3182:
3176:
3172:
3166:
3162:
3147:
3131:
3127:
3118:
3114:
3105:
3101:
3095:
3076:
3072:
3070:
3067:
3066:
3046:
3042:
3037:
3034:
3033:
2990:
2987:
2986:
2940:
2937:
2936:
2906:
2902:
2897:
2894:
2893:
2859:
2855:
2844:
2832:
2828:
2784:
2772:
2768:
2750:
2746:
2738:
2733:
2730:
2729:
2699:
2695:
2687:
2670:
2665:
2663:
2662:
2653:
2649:
2632:
2627:
2625:
2624:
2615:
2611:
2606:
2603:
2602:
2565:
2554:
2549:
2547:
2546:
2535:
2530:
2528:
2527:
2522:
2519:
2518:
2493:
2476:
2464:
2460:
2455:
2452:
2451:
2321:
2318:
2317:
2273:
2269:
2252:
2247:
2245:
2244:
2235:
2231:
2214:
2209:
2207:
2206:
2197:
2193:
2188:
2185:
2184:
2142:
2137:
2135:
2134:
2117:
2112:
2110:
2109:
2101:
2098:
2097:
2089:-modules for a
2062:
2054:. This functor
2041:
1977:
1969:
1964:
1962:
1959:
1958:
1942:
1934:
1929:
1927:
1924:
1923:
1907:
1905:
1902:
1901:
1885:
1877:
1872:
1864:
1856:
1851:
1848:
1847:
1831:
1829:
1826:
1825:
1820:In general, if
1802:
1788:
1780:
1777:
1776:
1748:
1745:
1744:
1728:
1725:
1724:
1708:
1705:
1704:
1679:
1676:
1675:
1653:
1650:
1649:
1623:
1620:
1619:
1597:
1594:
1593:
1577:
1574:
1573:
1572:if and only if
1545:
1542:
1541:
1525:
1511:
1503:
1500:
1499:
1471:
1468:
1467:
1451:
1449:
1446:
1445:
1417:
1415:
1412:
1411:
1395:
1393:
1390:
1389:
1373:
1371:
1368:
1367:
1351:
1349:
1346:
1345:
1344:. For example,
1335:
1324:→ 0 is exact.
1271:
1242:
1172:-vector spaces
1132:
1083:
1073:
1056:
1027:
1017:
984:
887:if and only if
837:if and only if
741:if and only if
292:
287:
285:
284:
249:
244:
242:
241:
218:
215:
214:
169:
164:
162:
161:
144:
139:
137:
136:
122:
119:
118:
82:
67:
53:that preserves
41:, particularly
35:
28:
23:
22:
15:
12:
11:
5:
4667:
4657:
4656:
4651:
4646:
4629:
4628:
4626:
4625:
4620:
4615:
4610:
4605:
4600:
4595:
4590:
4585:
4580:
4575:
4570:
4565:
4560:
4554:
4551:
4550:
4542:
4541:
4534:
4527:
4519:
4512:
4511:
4505:
4488:
4486:
4483:
4481:
4480:
4471:
4462:
4453:
4443:
4441:
4438:
4415:
4414:
4390:
4387:
4337:
4334:
4321:
4318:
4315:
4312:
4307:
4303:
4299:
4279:
4275:
4271:
4267:
4263:
4259:
4255:
4252:
4249:
4244:
4240:
4235:
4231:
4228:
4224:
4220:
4217:
4212:
4208:
4203:
4199:
4196:
4193:
4190:
4185:
4181:
4176:
4172:
4169:
4165:
4161:
4158:
4153:
4149:
4144:
4140:
4137:
4134:
4129:
4125:
4120:
4116:
4112:
4107:
4103:
4066:
4062:
4058:
4053:
4049:
4046:
4043:
4039:
4035:
4032:
4029:
4025:
4021:
4018:
4015:
4012:
4009:
4006:
4002:
3998:
3995:
3991:
3987:
3984:
3964:
3961:
3956:
3952:
3947:
3943:
3940:
3937:
3934:
3931:
3926:
3922:
3917:
3913:
3910:
3890:
3885:
3882:
3879:
3875:
3870:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3837:
3833:
3828:
3824:
3821:
3818:
3815:
3812:
3809:
3806:
3803:
3800:
3797:
3792:
3788:
3783:
3779:
3776:
3773:
3770:
3765:
3761:
3756:
3752:
3749:
3746:
3743:
3740:
3737:
3734:
3714:
3709:
3706:
3703:
3699:
3694:
3690:
3687:
3684:
3681:
3678:
3675:
3672:
3669:
3666:
3661:
3657:
3652:
3648:
3645:
3642:
3639:
3636:
3633:
3630:
3627:
3624:
3621:
3616:
3612:
3607:
3603:
3600:
3597:
3594:
3589:
3585:
3580:
3576:
3573:
3570:
3567:
3564:
3561:
3558:
3517:
3513:
3508:
3504:
3501:
3498:
3478:
3474:
3470:
3466:
3462:
3459:
3455:
3451:
3447:
3442:
3438:
3435:
3432:
3429:
3425:
3421:
3418:
3415:
3412:
3409:
3404:
3400:
3395:
3391:
3388:
3385:
3382:
3379:
3375:
3371:
3368:
3364:
3360:
3357:
3334:
3331:
3328:
3325:
3320:
3316:
3312:
3292:
3289:
3286:
3281:
3277:
3273:
3249:
3244:
3240:
3234:
3230:
3224:
3220:
3214:
3210:
3206:
3203:
3200:
3197:
3194:
3189:
3185:
3179:
3175:
3169:
3165:
3161:
3158:
3155:
3150:
3146:
3142:
3139:
3134:
3130:
3126:
3121:
3117:
3113:
3108:
3104:
3098:
3094:
3090:
3087:
3084:
3079:
3075:
3054:
3049:
3045:
3041:
3021:
3018:
3015:
3012:
3009:
3006:
3003:
3000:
2997:
2994:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2920:
2917:
2914:
2909:
2905:
2901:
2870:
2867:
2862:
2858:
2854:
2851:
2847:
2843:
2840:
2835:
2831:
2827:
2824:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2787:
2783:
2780:
2775:
2771:
2767:
2764:
2761:
2758:
2753:
2749:
2745:
2741:
2737:
2713:
2710:
2707:
2702:
2698:
2694:
2690:
2686:
2679:
2676:
2673:
2668:
2661:
2656:
2652:
2648:
2641:
2638:
2635:
2630:
2623:
2618:
2614:
2610:
2578:
2575:
2572:
2568:
2564:
2557:
2552:
2545:
2538:
2533:
2526:
2503:
2500:
2496:
2492:
2489:
2486:
2483:
2479:
2475:
2472:
2467:
2463:
2459:
2428:
2427:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2334:
2331:
2328:
2325:
2299:
2298:
2287:
2284:
2281:
2276:
2272:
2268:
2261:
2258:
2255:
2250:
2243:
2238:
2234:
2230:
2223:
2220:
2217:
2212:
2205:
2200:
2196:
2192:
2161:
2158:
2155:
2145:
2140:
2130:
2120:
2115:
2105:
2058:
2037:
1980:
1976:
1972:
1967:
1945:
1941:
1937:
1932:
1910:
1888:
1884:
1880:
1875:
1871:
1867:
1863:
1859:
1855:
1834:
1805:
1801:
1798:
1795:
1791:
1787:
1784:
1764:
1761:
1758:
1755:
1752:
1732:
1723:is injective,
1712:
1692:
1689:
1686:
1683:
1663:
1660:
1657:
1633:
1630:
1627:
1607:
1604:
1601:
1581:
1561:
1558:
1555:
1552:
1549:
1528:
1524:
1521:
1518:
1514:
1510:
1507:
1487:
1484:
1481:
1478:
1475:
1454:
1420:
1398:
1376:
1354:
1331:
1267:
1257:tensor product
1238:
1193:abelian groups
1151:-vector spaces
1128:
1127: * = Hom
1079:
1069:
1065:. The functor
1052:
1048:. The functor
1023:
1013:
983:
980:
979:
978:
949:exact implies
928:
878:
828:
778:
753:exact implies
716:
715:
678:if whenever 0→
673:
636:if whenever 0→
631:
594:if whenever 0→
589:
552:if whenever 0→
523:
522:
481:if whenever 0→
476:
439:if whenever 0→
434:
397:if whenever 0→
341:
340:
329:
326:
323:
320:
317:
314:
304:
301:
298:
295:
290:
280:
277:
274:
271:
261:
258:
255:
252:
247:
237:
234:
231:
228:
225:
222:
200:
199:
188:
185:
182:
172:
167:
157:
147:
142:
132:
129:
126:
66:
63:
26:
9:
6:
4:
3:
2:
4666:
4655:
4652:
4650:
4647:
4645:
4642:
4641:
4639:
4624:
4621:
4619:
4618:Representable
4616:
4614:
4611:
4609:
4606:
4604:
4601:
4599:
4596:
4594:
4591:
4589:
4586:
4584:
4581:
4579:
4576:
4574:
4571:
4569:
4566:
4564:
4561:
4559:
4556:
4555:
4552:
4547:
4540:
4535:
4533:
4528:
4526:
4521:
4520:
4517:
4508:
4502:
4498:
4497:Basic algebra
4494:
4490:
4489:
4475:
4466:
4457:
4448:
4444:
4437:
4435:
4430:
4428:
4423:
4421:
4412:
4408:
4404:
4400:
4399:
4398:
4396:
4386:
4384:
4380:
4376:
4372:
4368:
4363:
4361:
4357:
4352:
4350:
4346:
4341:
4333:
4319:
4316:
4313:
4310:
4305:
4301:
4297:
4277:
4269:
4265:
4261:
4253:
4247:
4242:
4238:
4229:
4222:
4215:
4210:
4206:
4194:
4188:
4183:
4179:
4170:
4163:
4156:
4151:
4147:
4135:
4132:
4127:
4123:
4114:
4110:
4101:
4093:
4089:
4085:
4081:
4060:
4056:
4047:
4044:
4041:
4033:
4030:
4027:
4019:
4016:
4013:
4010:
4004:
4000:
3996:
3985:
3982:
3959:
3954:
3950:
3941:
3935:
3929:
3924:
3920:
3911:
3883:
3880:
3877:
3873:
3868:
3864:
3858:
3852:
3849:
3843:
3835:
3831:
3826:
3822:
3816:
3810:
3807:
3801:
3795:
3790:
3786:
3777:
3771:
3763:
3759:
3754:
3750:
3744:
3738:
3735:
3707:
3704:
3701:
3697:
3692:
3688:
3682:
3676:
3673:
3667:
3659:
3655:
3650:
3646:
3640:
3634:
3631:
3625:
3619:
3614:
3610:
3601:
3595:
3587:
3583:
3578:
3574:
3568:
3562:
3559:
3547:
3545:
3541:
3537:
3533:
3515:
3511:
3506:
3502:
3499:
3496:
3476:
3468:
3464:
3460:
3457:
3449:
3445:
3436:
3433:
3430:
3419:
3416:
3413:
3410:
3407:
3402:
3398:
3393:
3389:
3383:
3377:
3373:
3369:
3358:
3355:
3346:
3332:
3329:
3326:
3323:
3318:
3314:
3310:
3303:. Similarly,
3290:
3287:
3284:
3279:
3275:
3271:
3263:
3242:
3238:
3232:
3228:
3222:
3218:
3212:
3208:
3201:
3198:
3195:
3187:
3183:
3177:
3173:
3167:
3163:
3156:
3153:
3148:
3144:
3140:
3132:
3128:
3124:
3119:
3115:
3106:
3102:
3096:
3092:
3088:
3085:
3082:
3077:
3073:
3052:
3047:
3043:
3039:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2985:implies that
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2948:
2945:
2942:
2934:
2918:
2912:
2907:
2903:
2899:
2891:
2889:
2884:
2865:
2860:
2856:
2852:
2845:
2838:
2833:
2829:
2825:
2819:
2813:
2810:
2807:
2801:
2798:
2795:
2792:
2789:
2785:
2778:
2773:
2769:
2765:
2759:
2756:
2751:
2747:
2743:
2739:
2735:
2727:
2711:
2705:
2700:
2696:
2692:
2688:
2684:
2677:
2674:
2671:
2659:
2654:
2650:
2646:
2639:
2636:
2633:
2621:
2616:
2612:
2608:
2600:
2596:
2592:
2576:
2570:
2566:
2562:
2555:
2543:
2536:
2524:
2515:
2501:
2498:
2494:
2490:
2487:
2481:
2477:
2473:
2465:
2461:
2457:
2449:
2445:
2441:
2437:
2433:
2413:
2410:
2404:
2398:
2395:
2389:
2386:
2383:
2377:
2374:
2371:
2368:
2365:
2362:
2356:
2350:
2347:
2341:
2338:
2335:
2329:
2326:
2323:
2316:
2315:
2314:
2312:
2308:
2304:
2285:
2279:
2274:
2270:
2266:
2259:
2256:
2253:
2241:
2236:
2232:
2228:
2221:
2218:
2215:
2203:
2198:
2194:
2190:
2183:
2182:
2181:
2179:
2175:
2159:
2153:
2143:
2128:
2118:
2103:
2095:
2092:
2088:
2084:
2080:
2076:
2072:
2069:Theorem: Let
2067:
2064:
2061:
2057:
2053:
2049:
2045:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2013:
2009:
2006:
2002:
1998:
1993:
1974:
1970:
1939:
1935:
1882:
1878:
1853:
1823:
1818:
1799:
1796:
1785:
1782:
1775:. Therefore,
1762:
1759:
1756:
1753:
1750:
1730:
1710:
1687:
1681:
1661:
1658:
1655:
1647:
1631:
1628:
1625:
1605:
1602:
1599:
1579:
1559:
1556:
1553:
1550:
1547:
1522:
1519:
1508:
1505:
1485:
1479:
1476:
1473:
1443:
1442:injective map
1439:
1435:
1343:
1339:
1334:
1330:
1325:
1323:
1319:
1315:
1311:
1307:
1303:
1299:
1295:
1291:
1287:
1283:
1279:
1275:
1270:
1266:
1262:
1258:
1255:by using the
1254:
1250:
1248:
1241:
1237:
1233:
1229:
1225:
1221:
1217:
1212:
1210:
1206:
1202:
1198:
1194:
1190:
1186:
1182:
1177:
1175:
1171:
1167:
1163:
1160:
1156:
1152:
1150:
1144:
1140:
1136:
1131:
1126:
1122:
1118:
1114:
1110:
1106:
1101:
1099:
1095:
1091:
1087:
1082:
1077:
1072:
1068:
1064:
1060:
1055:
1051:
1047:
1045:
1039:
1035:
1031:
1026:
1021:
1016:
1012:
1008:
1004:
1000:
996:
991:
989:
976:
972:
968:
964:
960:
956:
952:
948:
944:
940:
936:
932:
929:
926:
922:
918:
914:
910:
906:
902:
898:
894:
890:
886:
882:
879:
876:
872:
868:
864:
860:
856:
852:
848:
844:
840:
836:
832:
829:
826:
822:
818:
814:
810:
806:
802:
798:
794:
790:
786:
782:
779:
776:
772:
768:
764:
760:
756:
752:
748:
744:
740:
736:
733:
732:
731:
729:
725:
721:
713:
709:
705:
701:
697:
693:
689:
685:
681:
677:
674:
672:)→0 is exact;
671:
667:
663:
659:
655:
651:
647:
643:
639:
635:
632:
629:
625:
621:
617:
613:
609:
605:
601:
597:
593:
590:
588:)→0 is exact;
587:
583:
579:
575:
571:
567:
563:
559:
555:
551:
548:
547:
546:
544:
540:
536:
532:
531:contravariant
528:
520:
516:
512:
508:
504:
500:
496:
492:
488:
484:
480:
477:
475:)→0 is exact;
474:
470:
466:
462:
458:
454:
450:
446:
442:
438:
435:
432:
428:
424:
420:
416:
412:
408:
404:
400:
396:
393:
392:
391:
389:
384:
382:
378:
374:
370:
366:
362:
358:
354:
350:
346:
327:
318:
312:
299:
293:
275:
269:
256:
250:
232:
226:
220:
213:
212:
211:
209:
205:
186:
180:
170:
155:
145:
130:
124:
117:
116:
115:
114:if whenever
113:
112:exact functor
109:
105:
101:
98:
93:
89:
85:
80:
76:
72:
62:
60:
56:
52:
48:
47:exact functor
44:
40:
33:
19:
4592:
4568:Conservative
4496:
4474:
4465:
4456:
4447:
4431:
4424:
4419:
4416:
4410:
4406:
4402:
4392:
4382:
4378:
4374:
4371:left adjoint
4366:
4364:
4353:
4342:
4339:
4087:
4083:
4079:
3725:. Also, for
3548:
3543:
3539:
3531:
3347:
2932:
2887:
2882:
2725:
2598:
2594:
2590:
2516:
2447:
2443:
2435:
2429:
2310:
2306:
2302:
2300:
2177:
2093:
2086:
2082:
2078:
2074:
2070:
2068:
2065:
2059:
2055:
2051:
2047:
2043:
2038:
2034:
2030:
2026:
2022:
2018:
2014:
2000:
1996:
1994:
1821:
1819:
1437:
1433:
1337:
1332:
1328:
1327:The functor
1326:
1321:
1317:
1313:
1309:
1305:
1301:
1297:
1293:
1289:
1285:
1281:
1277:
1273:
1268:
1264:
1260:
1252:
1246:
1239:
1235:
1227:
1223:
1215:
1213:
1208:
1204:
1200:
1196:
1180:
1178:
1169:
1161:
1154:
1148:
1147:category of
1138:
1134:
1129:
1124:
1120:
1117:vector space
1112:
1104:
1102:
1093:
1089:
1085:
1080:
1075:
1070:
1066:
1058:
1053:
1049:
1043:
1037:
1033:
1029:
1024:
1019:
1014:
1010:
1006:
1002:
998:
995:Hom functors
992:
985:
974:
970:
966:
962:
958:
954:
950:
946:
942:
938:
934:
930:
924:
920:
916:
912:
908:
904:
900:
896:
892:
888:
884:
880:
874:
870:
866:
862:
858:
854:
850:
846:
842:
838:
834:
830:
824:
820:
816:
812:
808:
804:
800:
796:
792:
788:
784:
780:
774:
770:
766:
762:
758:
754:
750:
746:
742:
738:
734:
727:
723:
719:
717:
711:
707:
703:
699:
695:
691:
687:
683:
679:
675:
669:
665:
661:
657:
653:
649:
645:
641:
637:
633:
627:
623:
619:
615:
611:
607:
603:
599:
595:
591:
585:
581:
577:
573:
569:
565:
561:
557:
553:
549:
542:
538:
534:
526:
524:
514:
510:
506:
502:
498:
494:
490:
486:
482:
478:
472:
468:
464:
460:
456:
452:
448:
444:
440:
436:
430:
426:
422:
418:
414:
410:
406:
402:
398:
394:
387:
385:
380:
376:
372:
368:
364:
360:
356:
352:
348:
344:
342:
207:
201:
111:
107:
103:
91:
87:
83:
74:
70:
68:
58:
46:
36:
1296:→0 of left
1226:is a right
1123:, we write
935:right-exact
835:right-exact
714:) is exact.
634:right-exact
630:) is exact;
437:right-exact
433:) is exact;
65:Definitions
39:mathematics
4638:Categories
4485:References
3975:. Letting
3536:power of 2
2063:is exact.
1644:is in the
1366:is a flat
1143:dual space
1063:projective
885:left-exact
785:left-exact
676:half-exact
592:left-exact
479:half-exact
395:left-exact
81:, and let
4598:Forgetful
4314:≅
4302:⊗
4254:≅
4239:⊗
4207:⊗
4180:⊗
4148:⊗
4136:≅
4124:⊗
3951:⊗
3921:⊗
3859:⊗
3817:⊗
3787:⊗
3772:∈
3745:⊗
3705:−
3683:⊗
3641:⊗
3611:⊗
3596:∈
3569:⊗
3538:dividing
3458:≅
3437:⊗
3420:∈
3327:≅
3315:⊗
3288:≅
3276:⊗
3209:∑
3202:⊗
3157:⊗
3145:∑
3125:⊗
3093:∑
3083:∈
3044:⊗
3017:⊗
3005:⊗
2952:↦
2946:⊗
2931:given by
2916:→
2904:⊗
2857:⊗
2830:⊗
2811:⊗
2770:⊗
2760:≅
2748:⊗
2709:→
2697:⊗
2675:⊗
2667:→
2651:⊗
2637:⊗
2629:→
2613:⊗
2574:→
2551:→
2532:→
2488:≅
2462:⊗
2432:corollary
2411:⊗
2387:⊗
2375:⊗
2363:⊗
2339:⊗
2327:⊗
2283:→
2271:⊗
2257:⊗
2249:→
2233:⊗
2219:⊗
2211:→
2195:⊗
2157:→
2139:→
2114:→
1870:↠
1862:↪
1846:-modules
1800:⊗
1794:→
1786:⊗
1754:⊗
1659:⊗
1629:⊗
1551:⊗
1523:⊗
1517:→
1509:⊗
1483:→
1466:-modules
1159:injective
1098:injective
1042:category
325:→
289:⟶
246:⟶
224:→
184:→
166:→
141:→
128:→
97:covariant
4654:Functors
4613:Monoidal
4583:Enriched
4578:Diagonal
4558:Additive
4495:(2009).
4411:C′
4349:colimits
2881:, since
2589:, where
2008:category
1249:-modules
982:Examples
777:) exact;
4608:Logical
4573:Derived
4563:Adjoint
4546:Functor
4377:, then
4092:theorem
4080:A,B,C,P
2517:Proof:
1189:sheaves
1078:) = Hom
1040:to the
1022:) = Hom
1009:, then
51:functor
4623:Smooth
4503:
4345:limits
3489:where
3264:. So,
2438:is an
2151:
2132:
2126:
2107:
1646:kernel
1440:is an
1434:Proof:
1232:module
1174:splits
1166:module
1157:is an
986:Every
545:to be
390:is
310:
282:
267:
239:
210:then
178:
159:
153:
134:
110:is an
4593:Exact
4548:types
4440:Notes
3546:=12.
2892:from
2440:ideal
2434:: If
2172:be a
2042:from
2005:small
1922:with
1410:as a
1259:over
1218:is a
1183:is a
1119:over
1115:is a
1109:field
1107:is a
997:: if
739:exact
550:exact
529:is a
202:is a
95:be a
49:is a
45:, an
4501:ISBN
4401:Let
4395:SGA4
4082:are
3530:and
3262:onto
2446:and
2176:of
2081:and
1342:flat
1276:) =
1222:and
1220:ring
1111:and
73:and
69:Let
59:fail
4393:In
4373:to
4369:is
4078:,
2442:of
2085:be
2046:to
2021:to
1995:If
1444:of
1340:is
1251:to
1214:If
1195:on
1191:of
1179:If
1103:If
1096:is
1061:is
933:is
883:is
833:is
783:is
737:is
537:to
525:If
206:in
77:be
37:In
4640::
4422:.
4362:.
4332:.
4171:12
4115:12
4061:12
4020:12
3912:12
3850:12
3632:12
3602:12
3560:12
3384::=
2514:.
2396::=
2348::=
1320:⊗
1316:→
1312:⊗
1308:→
1304:⊗
1280:⊗
1263::
1253:Ab
1100:.
1044:Ab
965:)→
957:)→
915:)→
907:)→
865:)→
857:)→
815:)→
807:)→
769:)→
761:)→
706:)→
698:)→
664:)→
656:)→
622:)→
614:)→
580:)→
572:)→
509:)→
501:)→
467:)→
459:)→
425:)→
417:)→
375:)→
367:)→
86::
4538:e
4531:t
4524:v
4509:.
4420:C
4407:C
4403:C
4383:G
4379:F
4375:G
4367:F
4320:P
4317:I
4311:P
4306:R
4298:I
4278:P
4274:Z
4270:3
4266:/
4262:P
4258:Z
4251:)
4248:P
4243:Z
4234:Z
4230:3
4227:(
4223:/
4219:)
4216:P
4211:Z
4202:Z
4198:(
4195:=
4192:)
4189:P
4184:Z
4175:Z
4168:(
4164:/
4160:)
4157:P
4152:Z
4143:Z
4139:(
4133:P
4128:Z
4119:Z
4111:/
4106:Z
4102::
4088:Z
4086:=
4084:R
4065:Z
4057:/
4052:Z
4048:=
4045:C
4042:,
4038:Z
4034:=
4031:B
4028:,
4024:Z
4017:=
4014:A
4011:,
4008:]
4005:2
4001:/
3997:1
3994:[
3990:Z
3986:=
3983:P
3963:)
3960:P
3955:Z
3946:Z
3942:3
3939:(
3936:=
3933:)
3930:P
3925:Z
3916:Z
3909:(
3889:)
3884:2
3881:+
3878:k
3874:2
3869:/
3865:a
3862:(
3856:)
3853:z
3847:(
3844:=
3841:)
3836:k
3832:2
3827:/
3823:a
3820:(
3814:)
3811:z
3808:3
3805:(
3802:,
3799:)
3796:P
3791:Z
3782:Z
3778:3
3775:(
3769:)
3764:k
3760:2
3755:/
3751:a
3748:(
3742:)
3739:z
3736:3
3733:(
3713:)
3708:2
3702:k
3698:2
3693:/
3689:a
3686:(
3680:)
3677:z
3674:3
3671:(
3668:=
3665:)
3660:k
3656:2
3651:/
3647:a
3644:(
3638:)
3635:z
3629:(
3626:.
3623:)
3620:P
3615:Z
3606:Z
3599:(
3593:)
3588:k
3584:2
3579:/
3575:a
3572:(
3566:)
3563:z
3557:(
3544:m
3540:m
3532:n
3516:n
3512:2
3507:/
3503:m
3500:=
3497:k
3477:P
3473:Z
3469:k
3465:/
3461:P
3454:Z
3450:m
3446:/
3441:Z
3434:P
3431:,
3428:}
3424:Z
3417:k
3414:,
3411:a
3408::
3403:k
3399:2
3394:/
3390:a
3387:{
3381:]
3378:2
3374:/
3370:1
3367:[
3363:Z
3359:=
3356:P
3333:P
3330:I
3324:P
3319:R
3311:I
3291:P
3285:P
3280:R
3272:R
3248:)
3243:i
3239:p
3233:i
3229:x
3223:i
3219:r
3213:i
3205:(
3199:1
3196:=
3193:)
3188:i
3184:p
3178:i
3174:x
3168:i
3164:r
3160:(
3154:1
3149:i
3141:=
3138:)
3133:i
3129:p
3120:i
3116:r
3112:(
3107:i
3103:x
3097:i
3089:,
3086:R
3078:i
3074:x
3053:P
3048:R
3040:R
3020:p
3014:r
3011:=
3008:1
3002:p
2999:r
2996:=
2993:0
2973:0
2970:=
2967:p
2964:r
2961:.
2958:p
2955:r
2949:p
2943:r
2933:R
2919:P
2913:P
2908:R
2900:R
2888:R
2883:f
2869:)
2866:P
2861:R
2853:I
2850:(
2846:/
2842:)
2839:P
2834:R
2826:R
2823:(
2820:=
2817:)
2814:P
2808:f
2805:(
2802:e
2799:g
2796:a
2793:m
2790:I
2786:/
2782:)
2779:P
2774:R
2766:R
2763:(
2757:P
2752:R
2744:I
2740:/
2736:R
2726:R
2712:0
2706:P
2701:R
2693:I
2689:/
2685:R
2678:P
2672:g
2660:P
2655:R
2647:R
2640:P
2634:f
2622:P
2617:R
2609:I
2599:R
2595:g
2591:f
2577:0
2571:I
2567:/
2563:R
2556:g
2544:R
2537:f
2525:I
2502:P
2499:I
2495:/
2491:P
2485:)
2482:I
2478:/
2474:R
2471:(
2466:R
2458:P
2448:P
2444:R
2436:I
2426:.
2414:p
2408:)
2405:b
2402:(
2399:g
2393:)
2390:p
2384:b
2381:(
2378:P
2372:g
2369:,
2366:p
2360:)
2357:a
2354:(
2351:f
2345:)
2342:p
2336:a
2333:(
2330:P
2324:f
2311:R
2307:R
2303:R
2286:0
2280:P
2275:R
2267:C
2260:P
2254:g
2242:P
2237:R
2229:B
2222:P
2216:f
2204:P
2199:R
2191:A
2178:R
2160:0
2154:C
2144:g
2129:B
2119:f
2104:A
2094:R
2087:R
2083:P
2079:C
2077:,
2075:B
2073:,
2071:A
2060:X
2056:E
2052:X
2048:A
2044:A
2039:X
2035:E
2031:C
2027:X
2023:A
2019:C
2015:A
2001:C
1997:A
1979:Z
1975:5
1971:/
1966:Z
1944:Z
1940:5
1936:/
1931:Z
1909:Z
1887:Z
1883:5
1879:/
1874:Z
1866:Z
1858:Z
1854:5
1833:Z
1822:T
1804:Q
1797:N
1790:Q
1783:M
1763:0
1760:=
1757:q
1751:m
1731:m
1711:i
1691:)
1688:m
1685:(
1682:i
1662:q
1656:m
1632:q
1626:m
1606:0
1603:=
1600:q
1580:m
1560:0
1557:=
1554:q
1548:m
1527:Q
1520:N
1513:Q
1506:M
1486:N
1480:M
1477::
1474:i
1453:Z
1438:i
1419:Z
1397:Q
1375:Z
1353:Q
1338:T
1333:T
1329:H
1322:C
1318:T
1314:B
1310:T
1306:A
1302:T
1298:R
1294:C
1292:→
1290:B
1288:→
1286:A
1282:X
1278:T
1274:X
1272:(
1269:T
1265:H
1261:R
1247:R
1240:T
1236:H
1230:-
1228:R
1224:T
1216:R
1209:X
1207:(
1205:F
1201:F
1197:X
1181:X
1170:k
1164:-
1162:k
1155:k
1149:k
1139:k
1137:,
1135:V
1133:(
1130:k
1125:V
1121:k
1113:V
1105:k
1094:A
1090:A
1088:,
1086:X
1084:(
1081:A
1076:X
1074:(
1071:A
1067:G
1059:A
1054:A
1050:F
1038:A
1034:X
1032:,
1030:A
1028:(
1025:A
1020:X
1018:(
1015:A
1011:F
1007:A
1003:A
999:A
975:G
971:A
969:(
967:G
963:B
961:(
959:G
955:C
953:(
951:G
947:C
945:→
943:B
941:→
939:A
931:G
925:G
921:A
919:(
917:G
913:B
911:(
909:G
905:C
903:(
901:G
897:C
895:→
893:B
891:→
889:A
881:G
875:F
871:C
869:(
867:F
863:B
861:(
859:F
855:A
853:(
851:F
847:C
845:→
843:B
841:→
839:A
831:F
825:F
821:C
819:(
817:F
813:B
811:(
809:F
805:A
803:(
801:F
797:C
795:→
793:B
791:→
789:A
781:F
775:C
773:(
771:F
767:B
765:(
763:F
759:A
757:(
755:F
751:C
749:→
747:B
745:→
743:A
735:F
728:C
726:→
724:B
722:→
720:A
712:A
710:(
708:G
704:B
702:(
700:G
696:C
694:(
692:G
688:C
686:→
684:B
682:→
680:A
670:A
668:(
666:G
662:B
660:(
658:G
654:C
652:(
650:G
646:C
644:→
642:B
640:→
638:A
628:A
626:(
624:G
620:B
618:(
616:G
612:C
610:(
608:G
604:C
602:→
600:B
598:→
596:A
586:A
584:(
582:G
578:B
576:(
574:G
570:C
568:(
566:G
562:C
560:→
558:B
556:→
554:A
543:G
539:Q
535:P
527:G
521:.
515:C
513:(
511:F
507:B
505:(
503:F
499:A
497:(
495:F
491:C
489:→
487:B
485:→
483:A
473:C
471:(
469:F
465:B
463:(
461:F
457:A
455:(
453:F
449:C
447:→
445:B
443:→
441:A
431:C
429:(
427:F
423:B
421:(
419:F
415:A
413:(
411:F
407:C
405:→
403:B
401:→
399:A
388:F
381:C
379:(
377:F
373:B
371:(
369:F
365:A
363:(
361:F
357:C
355:→
353:B
351:→
349:A
345:Q
328:0
322:)
319:C
316:(
313:F
303:)
300:g
297:(
294:F
279:)
276:B
273:(
270:F
260:)
257:f
254:(
251:F
236:)
233:A
230:(
227:F
221:0
208:P
187:0
181:C
171:g
156:B
146:f
131:A
125:0
108:F
104:F
92:Q
90:→
88:P
84:F
75:Q
71:P
34:.
20:)
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