4277:
4086:
3247:
2711:
2285:
3476:
3712:
327:
3888:
2868:
1887:
4406:
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
4065:
2413:
186:
3962:
2159:
4272:{\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}
2576:
1804:
1527:
46:. 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
3057:
4383:
2593:
2175:
3340:
3541:
205:
2501:
3717:
2918:
1979:
1944:
2972:
3019:
4319:
3332:
3290:
4340:
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.
2720:
3052:
1838:
1909:
1833:
1762:
1559:
1453:
1419:
1397:
1375:
1353:
1661:
1631:
3517:
1485:
1690:
3967:
1605:
2308:
1730:
1710:
1579:
4386:, 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:
109:
3893:
2088:
4493:
2509:
4525:
3242:{\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})}
1767:
1490:
4576:
2706:{\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}
507:
2280:{\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}
3471:{\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}
4637:
4632:
3707:{\displaystyle (12z)\otimes (a/2^{k})\in (12\mathbf {Z} \otimes _{Z}P).(12z)\otimes (a/2^{k})=(3z)\otimes (a/2^{k-2})}
322:{\displaystyle 0\to F(A)\ {\stackrel {F(f)}{\longrightarrow }}\ F(B)\ {\stackrel {F(g)}{\longrightarrow }}\ F(C)\to 0}
3883:{\displaystyle (3z)\otimes (a/2^{k})\in (3\mathbf {Z} \otimes _{Z}P),(3z)\otimes (a/2^{k})=(12z)\otimes (a/2^{k+2})}
4591:
1030:
2442:
2884:
1135:
976:
1949:
1914:
19:
This article is about exact functors in homological algebra. For exact functors between regular categories, see
2927:
2977:
4333:
4282:
3295:
3256:
4518:
2863:{\displaystyle R/I\otimes _{R}P\cong (R\otimes _{R}P)/Image(f\otimes P)=(R\otimes _{R}P)/(I\otimes _{R}P)}
1996:
1607:. The given tensor products only have pure tensors. Therefore, it suffices to show that if a pure tensor
4606:
519:
719:→0 to have some exactness preserved. The following definitions are equivalent to the ones given above:
4279:. The last congruence follows by a similar argument to one in the proof of the corollary showing that
1882:{\displaystyle 5\mathbf {Z} \hookrightarrow \mathbf {Z} \twoheadrightarrow \mathbf {Z} /5\mathbf {Z} }
1813:
is not flat, then tensor product is not left exact. For example, consider the short exact sequence of
4556:
2162:
1220:
1162:
1154:
192:
43:
3024:
2875:
2428:
1233:
1177:
1097:
31:
4354:
Left and right exact functors are ubiquitous mainly because of the following fact: if the functor
1892:
1816:
1735:
1532:
1436:
1402:
1380:
1358:
1336:
2055:
While tensoring may not be left exact, it can be shown that tensoring is a right exact functor:
1208:
67:
1640:
1610:
3481:
4642:
4586:
4511:
4489:
1458:
1173:
1051:
85:
4601:
4571:
4566:
4551:
4546:
4422:
4359:
4332:
A covariant (not necessarily additive) functor is left exact if and only if it turns finite
4060:{\displaystyle P=\mathbf {Z} ,A=12\mathbf {Z} ,B=\mathbf {Z} ,C=\mathbf {Z} /12\mathbf {Z} }
2079:
2000:
1634:
1147:
1086:
88:
20:
4402:
is left (resp. right) exact if it commutes with finite projective (resp. inductive) limits.
2408:{\displaystyle f\otimes P(a\otimes p):=f(a)\otimes p,g\otimes P(b\otimes p):=g(b)\otimes p}
4596:
4561:
4481:
4348:
4344:
1666:
1584:
4611:
4415:
4347:; the degree to which a right exact functor fails to be exact can be measured with its
1993:
1715:
1695:
1564:
1245:
4425:
are sometimes called exact functors and generalize the exact functors discussed here.
4626:
1430:
1181:
4343:
The degree to which a left exact functor fails to be exact can be measured with its
4394:
be a category with finite projective (resp. injective) limits. Then a functor from
1273:. This is a covariant right exact functor; in other words, given an exact sequence
1105:
4079:
modules by the usual multiplication action and satisfy the conditions of the main
1330:
983:
27:
4336:
into limits; a covariant functor is right exact if and only if it turns finite
4083:. By the exactness implied by the theorem and by the above note we obtain that
181:{\displaystyle 0\to A\ {\stackrel {f}{\to }}\ B\ {\stackrel {g}{\to }}\ C\to 0}
3524:
3250:
1131:
2420:
3957:{\displaystyle (12\mathbf {Z} \otimes _{Z}P)=(3\mathbf {Z} \otimes _{Z}P)}
2154:{\displaystyle A\ {\stackrel {f}{\to }}\ B\ {\stackrel {g}{\to }}\ C\to 0}
707:
It is not always necessary to start with an entire short exact sequence 0→
4418:
generalize the exact functors between abelian categories discussed here.
4329:
A functor is exact if and only if it is both left exact and right exact.
1165:, and any additive functor turns split sequences into split sequences.)
4534:
4337:
4080:
3021:. So, the kernel of this map cannot contain any nonzero pure tensors.
39:
2571:{\displaystyle I{\stackrel {f}{\to }}R{\stackrel {g}{\to }}R/I\to 0}
1081:) is a contravariant left-exact functor; it is exact if and only if
1157:. Alternatively, one can argue that every short exact sequence of
4503:
336:. (The maps are often omitted and implied, and one says: "if 0→
4507:
1799:{\displaystyle M\otimes \mathbb {Q} \to N\otimes \mathbb {Q} }
1522:{\displaystyle M\otimes \mathbb {Q} \to N\otimes \mathbb {Q} }
2302:-modules and not merely of abelian groups). Here, we define
1487:, then the corresponding map between the tensor products
982:
The most basic examples of left exact functors are the
50:
to be exact, but in ways that can still be controlled.
1188:. The covariant functor that associates to each sheaf
1134:). This yields a contravariant exact functor from the
4285:
4089:
3970:
3896:
3720:
3544:
3484:
3343:
3298:
3259:
3060:
3027:
2980:
2930:
2924:-linearly extending the map defined on pure tensors:
2887:
2723:
2596:
2512:
2445:
2311:
2178:
2091:
1952:
1917:
1895:
1841:
1819:
1770:
1738:
1718:
1698:
1669:
1643:
1613:
1587:
1567:
1535:
1493:
1461:
1439:
1405:
1383:
1361:
1339:
208:
112:
4313:
4271:
4059:
3956:
3882:
3706:
3511:
3470:
3326:
3284:
3241:
3046:
3013:
2966:
2912:
2862:
2705:
2570:
2495:
2407:
2279:
2153:
1973:
1938:
1903:
1881:
1827:
1798:
1756:
1724:
1704:
1684:
1655:
1625:
1599:
1573:
1553:
1521:
1479:
1447:
1413:
1391:
1369:
1347:
506:) is exact. This is distinct from the notion of a
321:
180:
1946:gives a sequence that is no longer exact, since
2298:is commutative, this sequence is a sequence of
1176:, we can consider the abelian category of all
1142:to itself. (Exactness follows from the above:
1025:) defines a covariant left-exact functor from
4519:
8:
3416:
3375:
2496:{\displaystyle P\otimes _{R}(R/I)\cong P/IP}
16:Functor that preserves short exact sequences
3249:. So, this map is injective. It is clearly
2913:{\displaystyle R\otimes _{R}P\rightarrow P}
2586:is the projection, is an exact sequence of
4526:
4512:
4504:
3337:As another application, we show that for,
1974:{\displaystyle \mathbf {Z} /5\mathbf {Z} }
1939:{\displaystyle \mathbf {Z} /5\mathbf {Z} }
4293:
4284:
4261:
4253:
4245:
4230:
4221:
4210:
4198:
4189:
4171:
4162:
4151:
4139:
4130:
4115:
4106:
4098:
4093:
4088:
4052:
4044:
4039:
4025:
4011:
3988:
3977:
3969:
3942:
3933:
3912:
3903:
3895:
3865:
3856:
3823:
3814:
3778:
3769:
3751:
3742:
3719:
3689:
3680:
3647:
3638:
3602:
3593:
3575:
3566:
3543:
3503:
3494:
3483:
3460:
3452:
3441:
3433:
3428:
3411:
3390:
3381:
3361:
3350:
3342:
3306:
3297:
3267:
3258:
3230:
3220:
3210:
3200:
3175:
3165:
3155:
3136:
3120:
3107:
3094:
3084:
3065:
3059:
3035:
3026:
2979:
2967:{\displaystyle r\otimes p\mapsto rp.rp=0}
2929:
2895:
2886:
2848:
2833:
2821:
2773:
2761:
2739:
2727:
2722:
2688:
2676:
2659:
2654:
2652:
2651:
2642:
2621:
2616:
2614:
2613:
2604:
2595:
2590:-modules. By the above we get that :
2554:
2543:
2538:
2536:
2535:
2524:
2519:
2517:
2516:
2511:
2482:
2465:
2453:
2444:
2310:
2262:
2241:
2236:
2234:
2233:
2224:
2203:
2198:
2196:
2195:
2186:
2177:
2131:
2126:
2124:
2123:
2106:
2101:
2099:
2098:
2090:
1966:
1958:
1953:
1951:
1931:
1923:
1918:
1916:
1896:
1894:
1874:
1866:
1861:
1853:
1845:
1840:
1820:
1818:
1792:
1791:
1778:
1777:
1769:
1737:
1717:
1697:
1668:
1642:
1612:
1586:
1566:
1534:
1515:
1514:
1501:
1500:
1492:
1460:
1441:
1440:
1438:
1407:
1406:
1404:
1385:
1384:
1382:
1363:
1362:
1360:
1341:
1340:
1338:
281:
276:
274:
273:
238:
233:
231:
230:
207:
158:
153:
151:
150:
133:
128:
126:
125:
111:
3014:{\displaystyle 0=rp\otimes 1=r\otimes p}
1981:is not torsion-free and thus not flat.
1289:modules, the sequence of abelian groups
4433:
3054:is composed only of pure tensors: For
4314:{\displaystyle I\otimes _{R}P\cong IP}
3327:{\displaystyle I\otimes _{R}P\cong IP}
4488:. Vol. 2 (2nd ed.). Dover.
4414:The exact functors between Quillen's
3285:{\displaystyle R\otimes _{R}P\cong P}
7:
4458:Jacobson (2009), p. 99, Theorem 3.1.
4440:Jacobson (2009), p. 98, Theorem 3.1.
2874:is the inclusion. Now, consider the
2085:having multiplicative identity. Let
4449:Jacobson (2009), p. 149, Prop. 3.9.
1663:is an element of the kernel. Then,
1377:-module. Therefore, tensoring with
2713:is also a short exact sequence of
2290:is also a short exact sequence of
14:
1130:) (this is commonly known as the
866:turns cokernels into cokernels");
4262:
4246:
4222:
4190:
4163:
4131:
4107:
4094:
4053:
4040:
4026:
4012:
3978:
3934:
3904:
3770:
3594:
3461:
3442:
3429:
3412:
3351:
2006:consisting of all functors from
1967:
1954:
1932:
1919:
1897:
1875:
1862:
1854:
1846:
1821:
1637:, then it is zero. Suppose that
1529:is injective. One can show that
979:of abelian categories is exact.
966:turns kernels into cokernels").
916:turns cokernels into kernels");
4239:
4215:
4207:
4186:
4180:
4156:
4148:
4127:
3996:
3982:
3951:
3927:
3921:
3897:
3877:
3850:
3844:
3835:
3829:
3808:
3802:
3793:
3787:
3763:
3757:
3736:
3730:
3721:
3701:
3674:
3668:
3659:
3653:
3632:
3626:
3617:
3611:
3587:
3581:
3560:
3554:
3545:
3538:Proof: Consider a pure tensor
3369:
3355:
3236:
3193:
3181:
3148:
3126:
3100:
3047:{\displaystyle R\otimes _{R}P}
2940:
2904:
2857:
2838:
2830:
2811:
2805:
2793:
2770:
2751:
2697:
2655:
2617:
2562:
2539:
2520:
2473:
2459:
2396:
2390:
2381:
2369:
2348:
2342:
2333:
2321:
2271:
2237:
2199:
2145:
2127:
2102:
1858:
1850:
1782:
1679:
1673:
1505:
1471:
508:topological half-exact functor
313:
310:
304:
291:
285:
277:
267:
261:
248:
242:
234:
224:
218:
212:
172:
154:
129:
116:
1:
4421:The regular functors between
3334:. This proves the corollary.
1421:-module is an exact functor.
1192:the group of global sections
816:turns kernels into kernels");
332:is a short exact sequence in
1904:{\displaystyle \mathbf {Z} }
1828:{\displaystyle \mathbf {Z} }
1757:{\displaystyle m\otimes q=0}
1554:{\displaystyle m\otimes q=0}
1448:{\displaystyle \mathbb {Z} }
1425:It suffices to show that if
1414:{\displaystyle \mathbb {Z} }
1392:{\displaystyle \mathbb {Q} }
1370:{\displaystyle \mathbb {Z} }
1348:{\displaystyle \mathbb {Q} }
3531:. We prove a special case:
1988:is an abelian category and
990:is an abelian category and
4659:
2039:by evaluating functors at
1656:{\displaystyle m\otimes q}
1626:{\displaystyle m\otimes q}
1223:, we can define a functor
18:
4542:
3512:{\displaystyle k=m/2^{n}}
91:(so that, in particular,
4467:Jacobson (2009), p. 156.
2717:-modules. By exactness,
2022:, then we get a functor
1581:is a torsion element or
1480:{\displaystyle i:M\to N}
1325:is exact if and only if
1046:is exact if and only if
4325:Properties and theorems
1732:is torsion. Therefore,
95:(0) = 0). We say that
4577:Essentially surjective
4345:right derived functors
4315:
4273:
4061:
3958:
3884:
3708:
3513:
3472:
3328:
3286:
3243:
3048:
3015:
2968:
2914:
2864:
2707:
2572:
2497:
2409:
2281:
2155:
1999:, we can consider the
1975:
1940:
1905:
1883:
1829:
1800:
1758:
1726:
1706:
1686:
1657:
1627:
1601:
1575:
1555:
1523:
1481:
1449:
1415:
1393:
1371:
1349:
977:equivalence or duality
530:, we similarly define
522:additive functor from
372:)→0 is also exact".)
323:
182:
4349:left derived functors
4316:
4274:
4062:
3959:
3885:
3709:
3514:
3473:
3329:
3287:
3244:
3049:
3016:
2969:
2915:
2865:
2708:
2582:is the inclusion and
2573:
2498:
2410:
2282:
2156:
2018:is a given object of
1976:
1941:
1906:
1884:
1830:
1801:
1759:
1727:
1707:
1687:
1658:
1628:
1602:
1576:
1556:
1524:
1482:
1450:
1416:
1394:
1372:
1350:
1234:category of all left
375:Further, we say that
324:
183:
44:short exact sequences
4398:to another category
4283:
4087:
3968:
3894:
3718:
3542:
3482:
3341:
3296:
3257:
3058:
3025:
2978:
2928:
2885:
2879:-module homomorphism
2721:
2594:
2510:
2443:
2309:
2176:
2163:short exact sequence
2089:
2014:; it is abelian. If
1950:
1915:
1893:
1839:
1817:
1768:
1736:
1716:
1696:
1685:{\displaystyle i(m)}
1667:
1641:
1611:
1585:
1565:
1533:
1491:
1459:
1437:
1403:
1381:
1359:
1337:
962:)→0 exact (i.e. if "
862:)→0 exact (i.e. if "
348:→0 is exact, then 0→
206:
193:short exact sequence
110:
4638:Additive categories
4633:Homological algebra
4370:is right exact and
1806:is also injective.
1600:{\displaystyle q=0}
888:→0 exact implies 0→
595:→0 is exact then 0→
553:→0 is exact then 0→
398:→0 is exact then 0→
32:homological algebra
4423:regular categories
4311:
4269:
4057:
3954:
3890:. This shows that
3880:
3704:
3509:
3468:
3324:
3282:
3239:
3205:
3141:
3089:
3044:
3011:
2964:
2910:
2860:
2703:
2568:
2493:
2439:is as above, then
2419:This has a useful
2405:
2277:
2151:
1971:
1936:
1901:
1879:
1825:
1796:
1754:
1722:
1702:
1692:is torsion. Since
1682:
1653:
1623:
1597:
1571:
1551:
1519:
1477:
1445:
1411:
1389:
1367:
1345:
912:) exact (i.e. if "
812:) exact (i.e. if "
319:
178:
68:abelian categories
4620:
4619:
4592:Full and faithful
4495:978-0-486-47187-7
3196:
3132:
3080:
2670:
2632:
2548:
2529:
2294:-modules. (Since
2252:
2214:
2141:
2136:
2122:
2116:
2111:
2097:
1889:. Tensoring over
1725:{\displaystyle m}
1705:{\displaystyle i}
1574:{\displaystyle m}
1232:from the abelian
1200:) is left-exact.
1174:topological space
1035:of abelian groups
926:if and only if 0→
838:→0 exact implies
776:if and only if 0→
679:→0 is exact then
637:→0 is exact then
482:→0 is exact then
440:→0 is exact then
300:
295:
272:
257:
252:
229:
168:
163:
149:
143:
138:
124:
4650:
4528:
4521:
4514:
4505:
4499:
4482:Jacobson, Nathan
4468:
4465:
4459:
4456:
4450:
4447:
4441:
4438:
4416:exact categories
4320:
4318:
4317:
4312:
4298:
4297:
4278:
4276:
4275:
4270:
4265:
4257:
4249:
4235:
4234:
4225:
4214:
4203:
4202:
4193:
4176:
4175:
4166:
4155:
4144:
4143:
4134:
4120:
4119:
4110:
4102:
4097:
4066:
4064:
4063:
4058:
4056:
4048:
4043:
4029:
4015:
3992:
3981:
3963:
3961:
3960:
3955:
3947:
3946:
3937:
3917:
3916:
3907:
3889:
3887:
3886:
3881:
3876:
3875:
3860:
3828:
3827:
3818:
3783:
3782:
3773:
3756:
3755:
3746:
3713:
3711:
3710:
3705:
3700:
3699:
3684:
3652:
3651:
3642:
3607:
3606:
3597:
3580:
3579:
3570:
3518:
3516:
3515:
3510:
3508:
3507:
3498:
3477:
3475:
3474:
3469:
3464:
3456:
3445:
3437:
3432:
3415:
3395:
3394:
3385:
3365:
3354:
3333:
3331:
3330:
3325:
3311:
3310:
3291:
3289:
3288:
3283:
3272:
3271:
3248:
3246:
3245:
3240:
3235:
3234:
3225:
3224:
3215:
3214:
3204:
3180:
3179:
3170:
3169:
3160:
3159:
3140:
3125:
3124:
3112:
3111:
3099:
3098:
3088:
3070:
3069:
3053:
3051:
3050:
3045:
3040:
3039:
3020:
3018:
3017:
3012:
2973:
2971:
2970:
2965:
2919:
2917:
2916:
2911:
2900:
2899:
2869:
2867:
2866:
2861:
2853:
2852:
2837:
2826:
2825:
2777:
2766:
2765:
2744:
2743:
2731:
2712:
2710:
2709:
2704:
2693:
2692:
2680:
2672:
2671:
2669:
2658:
2653:
2647:
2646:
2634:
2633:
2631:
2620:
2615:
2609:
2608:
2577:
2575:
2574:
2569:
2558:
2550:
2549:
2547:
2542:
2537:
2531:
2530:
2528:
2523:
2518:
2502:
2500:
2499:
2494:
2486:
2469:
2458:
2457:
2414:
2412:
2411:
2406:
2286:
2284:
2283:
2278:
2267:
2266:
2254:
2253:
2251:
2240:
2235:
2229:
2228:
2216:
2215:
2213:
2202:
2197:
2191:
2190:
2169:-modules. Then
2160:
2158:
2157:
2152:
2139:
2138:
2137:
2135:
2130:
2125:
2120:
2114:
2113:
2112:
2110:
2105:
2100:
2095:
2080:commutative ring
2001:functor category
1992:is an arbitrary
1980:
1978:
1977:
1972:
1970:
1962:
1957:
1945:
1943:
1942:
1937:
1935:
1927:
1922:
1910:
1908:
1907:
1902:
1900:
1888:
1886:
1885:
1880:
1878:
1870:
1865:
1857:
1849:
1834:
1832:
1831:
1826:
1824:
1805:
1803:
1802:
1797:
1795:
1781:
1763:
1761:
1760:
1755:
1731:
1729:
1728:
1723:
1711:
1709:
1708:
1703:
1691:
1689:
1688:
1683:
1662:
1660:
1659:
1654:
1632:
1630:
1629:
1624:
1606:
1604:
1603:
1598:
1580:
1578:
1577:
1572:
1560:
1558:
1557:
1552:
1528:
1526:
1525:
1520:
1518:
1504:
1486:
1484:
1483:
1478:
1454:
1452:
1451:
1446:
1444:
1420:
1418:
1417:
1412:
1410:
1398:
1396:
1395:
1390:
1388:
1376:
1374:
1373:
1368:
1366:
1354:
1352:
1351:
1346:
1344:
994:is an object of
788:exact implies 0→
328:
326:
325:
320:
298:
297:
296:
294:
280:
275:
270:
255:
254:
253:
251:
237:
232:
227:
187:
185:
184:
179:
166:
165:
164:
162:
157:
152:
147:
141:
140:
139:
137:
132:
127:
122:
89:additive functor
83:
21:regular category
4658:
4657:
4653:
4652:
4651:
4649:
4648:
4647:
4623:
4622:
4621:
4616:
4538:
4532:
4502:
4496:
4480:
4476:
4471:
4466:
4462:
4457:
4453:
4448:
4444:
4439:
4435:
4431:
4380:
4378:Generalizations
4374:is left exact.
4327:
4289:
4281:
4280:
4226:
4194:
4167:
4135:
4111:
4085:
4084:
3966:
3965:
3938:
3908:
3892:
3891:
3861:
3819:
3774:
3747:
3716:
3715:
3685:
3643:
3598:
3571:
3540:
3539:
3523:is the highest
3499:
3480:
3479:
3386:
3339:
3338:
3302:
3294:
3293:
3263:
3255:
3254:
3226:
3216:
3206:
3171:
3161:
3151:
3116:
3103:
3090:
3061:
3056:
3055:
3031:
3023:
3022:
2976:
2975:
2926:
2925:
2891:
2883:
2882:
2844:
2817:
2757:
2735:
2719:
2718:
2684:
2638:
2600:
2592:
2591:
2508:
2507:
2449:
2441:
2440:
2307:
2306:
2258:
2220:
2182:
2174:
2173:
2087:
2086:
2078:-modules for a
2051:
2043:. This functor
2030:
1948:
1947:
1913:
1912:
1891:
1890:
1837:
1836:
1815:
1814:
1809:In general, if
1766:
1765:
1734:
1733:
1714:
1713:
1694:
1693:
1665:
1664:
1639:
1638:
1609:
1608:
1583:
1582:
1563:
1562:
1561:if and only if
1531:
1530:
1489:
1488:
1457:
1456:
1435:
1434:
1401:
1400:
1379:
1378:
1357:
1356:
1335:
1334:
1333:. For example,
1324:
1313:→ 0 is exact.
1260:
1231:
1161:-vector spaces
1121:
1072:
1062:
1045:
1016:
1006:
973:
876:if and only if
826:if and only if
730:if and only if
204:
203:
108:
107:
71:
56:
42:that preserves
30:, particularly
24:
17:
12:
11:
5:
4656:
4654:
4646:
4645:
4640:
4635:
4625:
4624:
4618:
4617:
4615:
4614:
4609:
4604:
4599:
4594:
4589:
4584:
4579:
4574:
4569:
4564:
4559:
4554:
4549:
4543:
4540:
4539:
4533:
4531:
4530:
4523:
4516:
4508:
4501:
4500:
4494:
4477:
4475:
4472:
4470:
4469:
4460:
4451:
4442:
4432:
4430:
4427:
4404:
4403:
4379:
4376:
4326:
4323:
4310:
4307:
4304:
4301:
4296:
4292:
4288:
4268:
4264:
4260:
4256:
4252:
4248:
4244:
4241:
4238:
4233:
4229:
4224:
4220:
4217:
4213:
4209:
4206:
4201:
4197:
4192:
4188:
4185:
4182:
4179:
4174:
4170:
4165:
4161:
4158:
4154:
4150:
4147:
4142:
4138:
4133:
4129:
4126:
4123:
4118:
4114:
4109:
4105:
4101:
4096:
4092:
4055:
4051:
4047:
4042:
4038:
4035:
4032:
4028:
4024:
4021:
4018:
4014:
4010:
4007:
4004:
4001:
3998:
3995:
3991:
3987:
3984:
3980:
3976:
3973:
3953:
3950:
3945:
3941:
3936:
3932:
3929:
3926:
3923:
3920:
3915:
3911:
3906:
3902:
3899:
3879:
3874:
3871:
3868:
3864:
3859:
3855:
3852:
3849:
3846:
3843:
3840:
3837:
3834:
3831:
3826:
3822:
3817:
3813:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3781:
3777:
3772:
3768:
3765:
3762:
3759:
3754:
3750:
3745:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3703:
3698:
3695:
3692:
3688:
3683:
3679:
3676:
3673:
3670:
3667:
3664:
3661:
3658:
3655:
3650:
3646:
3641:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3613:
3610:
3605:
3601:
3596:
3592:
3589:
3586:
3583:
3578:
3574:
3569:
3565:
3562:
3559:
3556:
3553:
3550:
3547:
3506:
3502:
3497:
3493:
3490:
3487:
3467:
3463:
3459:
3455:
3451:
3448:
3444:
3440:
3436:
3431:
3427:
3424:
3421:
3418:
3414:
3410:
3407:
3404:
3401:
3398:
3393:
3389:
3384:
3380:
3377:
3374:
3371:
3368:
3364:
3360:
3357:
3353:
3349:
3346:
3323:
3320:
3317:
3314:
3309:
3305:
3301:
3281:
3278:
3275:
3270:
3266:
3262:
3238:
3233:
3229:
3223:
3219:
3213:
3209:
3203:
3199:
3195:
3192:
3189:
3186:
3183:
3178:
3174:
3168:
3164:
3158:
3154:
3150:
3147:
3144:
3139:
3135:
3131:
3128:
3123:
3119:
3115:
3110:
3106:
3102:
3097:
3093:
3087:
3083:
3079:
3076:
3073:
3068:
3064:
3043:
3038:
3034:
3030:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2909:
2906:
2903:
2898:
2894:
2890:
2859:
2856:
2851:
2847:
2843:
2840:
2836:
2832:
2829:
2824:
2820:
2816:
2813:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2776:
2772:
2769:
2764:
2760:
2756:
2753:
2750:
2747:
2742:
2738:
2734:
2730:
2726:
2702:
2699:
2696:
2691:
2687:
2683:
2679:
2675:
2668:
2665:
2662:
2657:
2650:
2645:
2641:
2637:
2630:
2627:
2624:
2619:
2612:
2607:
2603:
2599:
2567:
2564:
2561:
2557:
2553:
2546:
2541:
2534:
2527:
2522:
2515:
2492:
2489:
2485:
2481:
2478:
2475:
2472:
2468:
2464:
2461:
2456:
2452:
2448:
2417:
2416:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2288:
2287:
2276:
2273:
2270:
2265:
2261:
2257:
2250:
2247:
2244:
2239:
2232:
2227:
2223:
2219:
2212:
2209:
2206:
2201:
2194:
2189:
2185:
2181:
2150:
2147:
2144:
2134:
2129:
2119:
2109:
2104:
2094:
2047:
2026:
1969:
1965:
1961:
1956:
1934:
1930:
1926:
1921:
1899:
1877:
1873:
1869:
1864:
1860:
1856:
1852:
1848:
1844:
1823:
1794:
1790:
1787:
1784:
1780:
1776:
1773:
1753:
1750:
1747:
1744:
1741:
1721:
1712:is injective,
1701:
1681:
1678:
1675:
1672:
1652:
1649:
1646:
1622:
1619:
1616:
1596:
1593:
1590:
1570:
1550:
1547:
1544:
1541:
1538:
1517:
1513:
1510:
1507:
1503:
1499:
1496:
1476:
1473:
1470:
1467:
1464:
1443:
1409:
1387:
1365:
1343:
1320:
1256:
1246:tensor product
1227:
1182:abelian groups
1140:-vector spaces
1117:
1116: * = Hom
1068:
1058:
1054:. The functor
1041:
1037:. The functor
1012:
1002:
972:
969:
968:
967:
938:exact implies
917:
867:
817:
767:
742:exact implies
705:
704:
667:if whenever 0→
662:
625:if whenever 0→
620:
583:if whenever 0→
578:
541:if whenever 0→
512:
511:
470:if whenever 0→
465:
428:if whenever 0→
423:
386:if whenever 0→
330:
329:
318:
315:
312:
309:
306:
303:
293:
290:
287:
284:
279:
269:
266:
263:
260:
250:
247:
244:
241:
236:
226:
223:
220:
217:
214:
211:
189:
188:
177:
174:
171:
161:
156:
146:
136:
131:
121:
118:
115:
55:
52:
15:
13:
10:
9:
6:
4:
3:
2:
4655:
4644:
4641:
4639:
4636:
4634:
4631:
4630:
4628:
4613:
4610:
4608:
4607:Representable
4605:
4603:
4600:
4598:
4595:
4593:
4590:
4588:
4585:
4583:
4580:
4578:
4575:
4573:
4570:
4568:
4565:
4563:
4560:
4558:
4555:
4553:
4550:
4548:
4545:
4544:
4541:
4536:
4529:
4524:
4522:
4517:
4515:
4510:
4509:
4506:
4497:
4491:
4487:
4486:Basic algebra
4483:
4479:
4478:
4473:
4464:
4461:
4455:
4452:
4446:
4443:
4437:
4434:
4428:
4426:
4424:
4419:
4417:
4412:
4410:
4401:
4397:
4393:
4389:
4388:
4387:
4385:
4377:
4375:
4373:
4369:
4365:
4361:
4357:
4352:
4350:
4346:
4341:
4339:
4335:
4330:
4324:
4322:
4308:
4305:
4302:
4299:
4294:
4290:
4286:
4266:
4258:
4254:
4250:
4242:
4236:
4231:
4227:
4218:
4211:
4204:
4199:
4195:
4183:
4177:
4172:
4168:
4159:
4152:
4145:
4140:
4136:
4124:
4121:
4116:
4112:
4103:
4099:
4090:
4082:
4078:
4074:
4070:
4049:
4045:
4036:
4033:
4030:
4022:
4019:
4016:
4008:
4005:
4002:
3999:
3993:
3989:
3985:
3974:
3971:
3948:
3943:
3939:
3930:
3924:
3918:
3913:
3909:
3900:
3872:
3869:
3866:
3862:
3857:
3853:
3847:
3841:
3838:
3832:
3824:
3820:
3815:
3811:
3805:
3799:
3796:
3790:
3784:
3779:
3775:
3766:
3760:
3752:
3748:
3743:
3739:
3733:
3727:
3724:
3696:
3693:
3690:
3686:
3681:
3677:
3671:
3665:
3662:
3656:
3648:
3644:
3639:
3635:
3629:
3623:
3620:
3614:
3608:
3603:
3599:
3590:
3584:
3576:
3572:
3567:
3563:
3557:
3551:
3548:
3536:
3534:
3530:
3526:
3522:
3504:
3500:
3495:
3491:
3488:
3485:
3465:
3457:
3453:
3449:
3446:
3438:
3434:
3425:
3422:
3419:
3408:
3405:
3402:
3399:
3396:
3391:
3387:
3382:
3378:
3372:
3366:
3362:
3358:
3347:
3344:
3335:
3321:
3318:
3315:
3312:
3307:
3303:
3299:
3292:. Similarly,
3279:
3276:
3273:
3268:
3264:
3260:
3252:
3231:
3227:
3221:
3217:
3211:
3207:
3201:
3197:
3190:
3187:
3184:
3176:
3172:
3166:
3162:
3156:
3152:
3145:
3142:
3137:
3133:
3129:
3121:
3117:
3113:
3108:
3104:
3095:
3091:
3085:
3081:
3077:
3074:
3071:
3066:
3062:
3041:
3036:
3032:
3028:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2974:implies that
2961:
2958:
2955:
2952:
2949:
2946:
2943:
2937:
2934:
2931:
2923:
2907:
2901:
2896:
2892:
2888:
2880:
2878:
2873:
2854:
2849:
2845:
2841:
2834:
2827:
2822:
2818:
2814:
2808:
2802:
2799:
2796:
2790:
2787:
2784:
2781:
2778:
2774:
2767:
2762:
2758:
2754:
2748:
2745:
2740:
2736:
2732:
2728:
2724:
2716:
2700:
2694:
2689:
2685:
2681:
2677:
2673:
2666:
2663:
2660:
2648:
2643:
2639:
2635:
2628:
2625:
2622:
2610:
2605:
2601:
2597:
2589:
2585:
2581:
2565:
2559:
2555:
2551:
2544:
2532:
2525:
2513:
2504:
2490:
2487:
2483:
2479:
2476:
2470:
2466:
2462:
2454:
2450:
2446:
2438:
2434:
2430:
2426:
2422:
2402:
2399:
2393:
2387:
2384:
2378:
2375:
2372:
2366:
2363:
2360:
2357:
2354:
2351:
2345:
2339:
2336:
2330:
2327:
2324:
2318:
2315:
2312:
2305:
2304:
2303:
2301:
2297:
2293:
2274:
2268:
2263:
2259:
2255:
2248:
2245:
2242:
2230:
2225:
2221:
2217:
2210:
2207:
2204:
2192:
2187:
2183:
2179:
2172:
2171:
2170:
2168:
2164:
2148:
2142:
2132:
2117:
2107:
2092:
2084:
2081:
2077:
2073:
2069:
2065:
2061:
2058:Theorem: Let
2056:
2053:
2050:
2046:
2042:
2038:
2034:
2029:
2025:
2021:
2017:
2013:
2009:
2005:
2002:
1998:
1995:
1991:
1987:
1982:
1963:
1959:
1928:
1924:
1871:
1867:
1842:
1812:
1807:
1788:
1785:
1774:
1771:
1764:. Therefore,
1751:
1748:
1745:
1742:
1739:
1719:
1699:
1676:
1670:
1650:
1647:
1644:
1636:
1620:
1617:
1614:
1594:
1591:
1588:
1568:
1548:
1545:
1542:
1539:
1536:
1511:
1508:
1497:
1494:
1474:
1468:
1465:
1462:
1432:
1431:injective map
1428:
1424:
1332:
1328:
1323:
1319:
1314:
1312:
1308:
1304:
1300:
1296:
1292:
1288:
1284:
1280:
1276:
1272:
1268:
1264:
1259:
1255:
1251:
1247:
1244:by using the
1243:
1239:
1237:
1230:
1226:
1222:
1218:
1214:
1210:
1206:
1201:
1199:
1195:
1191:
1187:
1183:
1179:
1175:
1171:
1166:
1164:
1160:
1156:
1152:
1149:
1145:
1141:
1139:
1133:
1129:
1125:
1120:
1115:
1111:
1107:
1103:
1099:
1095:
1090:
1088:
1084:
1080:
1076:
1071:
1066:
1061:
1057:
1053:
1049:
1044:
1040:
1036:
1034:
1028:
1024:
1020:
1015:
1010:
1005:
1001:
997:
993:
989:
985:
980:
978:
970:
965:
961:
957:
953:
949:
945:
941:
937:
933:
929:
925:
921:
918:
915:
911:
907:
903:
899:
895:
891:
887:
883:
879:
875:
871:
868:
865:
861:
857:
853:
849:
845:
841:
837:
833:
829:
825:
821:
818:
815:
811:
807:
803:
799:
795:
791:
787:
783:
779:
775:
771:
768:
765:
761:
757:
753:
749:
745:
741:
737:
733:
729:
725:
722:
721:
720:
718:
714:
710:
702:
698:
694:
690:
686:
682:
678:
674:
670:
666:
663:
661:)→0 is exact;
660:
656:
652:
648:
644:
640:
636:
632:
628:
624:
621:
618:
614:
610:
606:
602:
598:
594:
590:
586:
582:
579:
577:)→0 is exact;
576:
572:
568:
564:
560:
556:
552:
548:
544:
540:
537:
536:
535:
533:
529:
525:
521:
520:contravariant
517:
509:
505:
501:
497:
493:
489:
485:
481:
477:
473:
469:
466:
464:)→0 is exact;
463:
459:
455:
451:
447:
443:
439:
435:
431:
427:
424:
421:
417:
413:
409:
405:
401:
397:
393:
389:
385:
382:
381:
380:
378:
373:
371:
367:
363:
359:
355:
351:
347:
343:
339:
335:
316:
307:
301:
288:
282:
264:
258:
245:
239:
221:
215:
209:
202:
201:
200:
198:
194:
175:
169:
159:
144:
134:
119:
113:
106:
105:
104:
103:if whenever
102:
101:exact functor
98:
94:
90:
87:
82:
78:
74:
69:
65:
61:
53:
51:
49:
45:
41:
37:
36:exact functor
33:
29:
22:
4581:
4557:Conservative
4485:
4463:
4454:
4445:
4436:
4420:
4413:
4408:
4405:
4399:
4395:
4391:
4381:
4371:
4367:
4363:
4360:left adjoint
4355:
4353:
4342:
4331:
4328:
4076:
4072:
4068:
3714:. Also, for
3537:
3532:
3528:
3520:
3336:
2921:
2876:
2871:
2714:
2587:
2583:
2579:
2505:
2436:
2432:
2424:
2418:
2299:
2295:
2291:
2289:
2166:
2082:
2075:
2071:
2067:
2063:
2059:
2057:
2054:
2048:
2044:
2040:
2036:
2032:
2027:
2023:
2019:
2015:
2011:
2007:
2003:
1989:
1985:
1983:
1810:
1808:
1426:
1422:
1326:
1321:
1317:
1316:The functor
1315:
1310:
1306:
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1266:
1262:
1257:
1253:
1249:
1241:
1235:
1228:
1224:
1216:
1212:
1204:
1202:
1197:
1193:
1189:
1185:
1169:
1167:
1158:
1150:
1143:
1137:
1136:category of
1127:
1123:
1118:
1113:
1109:
1106:vector space
1101:
1093:
1091:
1082:
1078:
1074:
1069:
1064:
1059:
1055:
1047:
1042:
1038:
1032:
1026:
1022:
1018:
1013:
1008:
1003:
999:
995:
991:
987:
984:Hom functors
981:
974:
963:
959:
955:
951:
947:
943:
939:
935:
931:
927:
923:
919:
913:
909:
905:
901:
897:
893:
889:
885:
881:
877:
873:
869:
863:
859:
855:
851:
847:
843:
839:
835:
831:
827:
823:
819:
813:
809:
805:
801:
797:
793:
789:
785:
781:
777:
773:
769:
763:
759:
755:
751:
747:
743:
739:
735:
731:
727:
723:
716:
712:
708:
706:
700:
696:
692:
688:
684:
680:
676:
672:
668:
664:
658:
654:
650:
646:
642:
638:
634:
630:
626:
622:
616:
612:
608:
604:
600:
596:
592:
588:
584:
580:
574:
570:
566:
562:
558:
554:
550:
546:
542:
538:
531:
527:
523:
515:
513:
503:
499:
495:
491:
487:
483:
479:
475:
471:
467:
461:
457:
453:
449:
445:
441:
437:
433:
429:
425:
419:
415:
411:
407:
403:
399:
395:
391:
387:
383:
376:
374:
369:
365:
361:
357:
353:
349:
345:
341:
337:
333:
331:
196:
190:
100:
96:
92:
80:
76:
72:
63:
59:
57:
47:
35:
25:
1285:→0 of left
1215:is a right
1112:, we write
924:right-exact
824:right-exact
703:) is exact.
623:right-exact
619:) is exact;
426:right-exact
422:) is exact;
54:Definitions
28:mathematics
4627:Categories
4474:References
3964:. Letting
3525:power of 2
2052:is exact.
1633:is in the
1355:is a flat
1132:dual space
1052:projective
874:left-exact
774:left-exact
665:half-exact
581:left-exact
468:half-exact
384:left-exact
70:, and let
4587:Forgetful
4303:≅
4291:⊗
4243:≅
4228:⊗
4196:⊗
4169:⊗
4137:⊗
4125:≅
4113:⊗
3940:⊗
3910:⊗
3848:⊗
3806:⊗
3776:⊗
3761:∈
3734:⊗
3694:−
3672:⊗
3630:⊗
3600:⊗
3585:∈
3558:⊗
3527:dividing
3447:≅
3426:⊗
3409:∈
3316:≅
3304:⊗
3277:≅
3265:⊗
3198:∑
3191:⊗
3146:⊗
3134:∑
3114:⊗
3082:∑
3072:∈
3033:⊗
3006:⊗
2994:⊗
2941:↦
2935:⊗
2920:given by
2905:→
2893:⊗
2846:⊗
2819:⊗
2800:⊗
2759:⊗
2749:≅
2737:⊗
2698:→
2686:⊗
2664:⊗
2656:→
2640:⊗
2626:⊗
2618:→
2602:⊗
2563:→
2540:→
2521:→
2477:≅
2451:⊗
2421:corollary
2400:⊗
2376:⊗
2364:⊗
2352:⊗
2328:⊗
2316:⊗
2272:→
2260:⊗
2246:⊗
2238:→
2222:⊗
2208:⊗
2200:→
2184:⊗
2146:→
2128:→
2103:→
1859:↠
1851:↪
1835:-modules
1789:⊗
1783:→
1775:⊗
1743:⊗
1648:⊗
1618:⊗
1540:⊗
1512:⊗
1506:→
1498:⊗
1472:→
1455:-modules
1148:injective
1087:injective
1031:category
314:→
278:⟶
235:⟶
213:→
173:→
155:→
130:→
117:→
86:covariant
4643:Functors
4602:Monoidal
4572:Enriched
4567:Diagonal
4547:Additive
4484:(2009).
4400:C′
4338:colimits
2870:, since
2578:, where
1997:category
1238:-modules
971:Examples
766:) exact;
4597:Logical
4562:Derived
4552:Adjoint
4535:Functor
4366:, then
4081:theorem
4069:A,B,C,P
2506:Proof:
1178:sheaves
1067:) = Hom
1029:to the
1011:) = Hom
998:, then
40:functor
4612:Smooth
4492:
4334:limits
3478:where
3253:. So,
2427:is an
2140:
2121:
2115:
2096:
1635:kernel
1429:is an
1423:Proof:
1221:module
1163:splits
1155:module
1146:is an
975:Every
534:to be
379:is
299:
271:
256:
228:
199:then
167:
148:
142:
123:
99:is an
4582:Exact
4537:types
4429:Notes
3535:=12.
2881:from
2429:ideal
2423:: If
2161:be a
2031:from
1994:small
1911:with
1399:as a
1248:over
1207:is a
1172:is a
1108:over
1104:is a
1098:field
1096:is a
986:: if
728:exact
539:exact
518:is a
191:is a
84:be a
38:is a
34:, an
4490:ISBN
4390:Let
4384:SGA4
4071:are
3519:and
3251:onto
2435:and
2165:of
2070:and
1331:flat
1265:) =
1211:and
1209:ring
1100:and
62:and
58:Let
48:fail
4382:In
4362:to
4358:is
4067:,
2431:of
2074:be
2035:to
2010:to
1984:If
1433:of
1329:is
1240:to
1203:If
1184:on
1180:of
1168:If
1092:If
1085:is
1050:is
922:is
872:is
822:is
772:is
726:is
526:to
514:If
195:in
66:be
26:In
4629::
4411:.
4351:.
4321:.
4160:12
4104:12
4050:12
4009:12
3901:12
3839:12
3621:12
3591:12
3549:12
3373::=
2503:.
2385::=
2337::=
1309:⊗
1305:→
1301:⊗
1297:→
1293:⊗
1269:⊗
1252::
1242:Ab
1089:.
1033:Ab
954:)→
946:)→
904:)→
896:)→
854:)→
846:)→
804:)→
796:)→
758:)→
750:)→
695:)→
687:)→
653:)→
645:)→
611:)→
603:)→
569:)→
561:)→
498:)→
490:)→
456:)→
448:)→
414:)→
406:)→
364:)→
356:)→
75::
4527:e
4520:t
4513:v
4498:.
4409:C
4396:C
4392:C
4372:G
4368:F
4364:G
4356:F
4309:P
4306:I
4300:P
4295:R
4287:I
4267:P
4263:Z
4259:3
4255:/
4251:P
4247:Z
4240:)
4237:P
4232:Z
4223:Z
4219:3
4216:(
4212:/
4208:)
4205:P
4200:Z
4191:Z
4187:(
4184:=
4181:)
4178:P
4173:Z
4164:Z
4157:(
4153:/
4149:)
4146:P
4141:Z
4132:Z
4128:(
4122:P
4117:Z
4108:Z
4100:/
4095:Z
4091::
4077:Z
4075:=
4073:R
4054:Z
4046:/
4041:Z
4037:=
4034:C
4031:,
4027:Z
4023:=
4020:B
4017:,
4013:Z
4006:=
4003:A
4000:,
3997:]
3994:2
3990:/
3986:1
3983:[
3979:Z
3975:=
3972:P
3952:)
3949:P
3944:Z
3935:Z
3931:3
3928:(
3925:=
3922:)
3919:P
3914:Z
3905:Z
3898:(
3878:)
3873:2
3870:+
3867:k
3863:2
3858:/
3854:a
3851:(
3845:)
3842:z
3836:(
3833:=
3830:)
3825:k
3821:2
3816:/
3812:a
3809:(
3803:)
3800:z
3797:3
3794:(
3791:,
3788:)
3785:P
3780:Z
3771:Z
3767:3
3764:(
3758:)
3753:k
3749:2
3744:/
3740:a
3737:(
3731:)
3728:z
3725:3
3722:(
3702:)
3697:2
3691:k
3687:2
3682:/
3678:a
3675:(
3669:)
3666:z
3663:3
3660:(
3657:=
3654:)
3649:k
3645:2
3640:/
3636:a
3633:(
3627:)
3624:z
3618:(
3615:.
3612:)
3609:P
3604:Z
3595:Z
3588:(
3582:)
3577:k
3573:2
3568:/
3564:a
3561:(
3555:)
3552:z
3546:(
3533:m
3529:m
3521:n
3505:n
3501:2
3496:/
3492:m
3489:=
3486:k
3466:P
3462:Z
3458:k
3454:/
3450:P
3443:Z
3439:m
3435:/
3430:Z
3423:P
3420:,
3417:}
3413:Z
3406:k
3403:,
3400:a
3397::
3392:k
3388:2
3383:/
3379:a
3376:{
3370:]
3367:2
3363:/
3359:1
3356:[
3352:Z
3348:=
3345:P
3322:P
3319:I
3313:P
3308:R
3300:I
3280:P
3274:P
3269:R
3261:R
3237:)
3232:i
3228:p
3222:i
3218:x
3212:i
3208:r
3202:i
3194:(
3188:1
3185:=
3182:)
3177:i
3173:p
3167:i
3163:x
3157:i
3153:r
3149:(
3143:1
3138:i
3130:=
3127:)
3122:i
3118:p
3109:i
3105:r
3101:(
3096:i
3092:x
3086:i
3078:,
3075:R
3067:i
3063:x
3042:P
3037:R
3029:R
3009:p
3003:r
3000:=
2997:1
2991:p
2988:r
2985:=
2982:0
2962:0
2959:=
2956:p
2953:r
2950:.
2947:p
2944:r
2938:p
2932:r
2922:R
2908:P
2902:P
2897:R
2889:R
2877:R
2872:f
2858:)
2855:P
2850:R
2842:I
2839:(
2835:/
2831:)
2828:P
2823:R
2815:R
2812:(
2809:=
2806:)
2803:P
2797:f
2794:(
2791:e
2788:g
2785:a
2782:m
2779:I
2775:/
2771:)
2768:P
2763:R
2755:R
2752:(
2746:P
2741:R
2733:I
2729:/
2725:R
2715:R
2701:0
2695:P
2690:R
2682:I
2678:/
2674:R
2667:P
2661:g
2649:P
2644:R
2636:R
2629:P
2623:f
2611:P
2606:R
2598:I
2588:R
2584:g
2580:f
2566:0
2560:I
2556:/
2552:R
2545:g
2533:R
2526:f
2514:I
2491:P
2488:I
2484:/
2480:P
2474:)
2471:I
2467:/
2463:R
2460:(
2455:R
2447:P
2437:P
2433:R
2425:I
2415:.
2403:p
2397:)
2394:b
2391:(
2388:g
2382:)
2379:p
2373:b
2370:(
2367:P
2361:g
2358:,
2355:p
2349:)
2346:a
2343:(
2340:f
2334:)
2331:p
2325:a
2322:(
2319:P
2313:f
2300:R
2296:R
2292:R
2275:0
2269:P
2264:R
2256:C
2249:P
2243:g
2231:P
2226:R
2218:B
2211:P
2205:f
2193:P
2188:R
2180:A
2167:R
2149:0
2143:C
2133:g
2118:B
2108:f
2093:A
2083:R
2076:R
2072:P
2068:C
2066:,
2064:B
2062:,
2060:A
2049:X
2045:E
2041:X
2037:A
2033:A
2028:X
2024:E
2020:C
2016:X
2012:A
2008:C
2004:A
1990:C
1986:A
1968:Z
1964:5
1960:/
1955:Z
1933:Z
1929:5
1925:/
1920:Z
1898:Z
1876:Z
1872:5
1868:/
1863:Z
1855:Z
1847:Z
1843:5
1822:Z
1811:T
1793:Q
1786:N
1779:Q
1772:M
1752:0
1749:=
1746:q
1740:m
1720:m
1700:i
1680:)
1677:m
1674:(
1671:i
1651:q
1645:m
1621:q
1615:m
1595:0
1592:=
1589:q
1569:m
1549:0
1546:=
1543:q
1537:m
1516:Q
1509:N
1502:Q
1495:M
1475:N
1469:M
1466::
1463:i
1442:Z
1427:i
1408:Z
1386:Q
1364:Z
1342:Q
1327:T
1322:T
1318:H
1311:C
1307:T
1303:B
1299:T
1295:A
1291:T
1287:R
1283:C
1281:→
1279:B
1277:→
1275:A
1271:X
1267:T
1263:X
1261:(
1258:T
1254:H
1250:R
1236:R
1229:T
1225:H
1219:-
1217:R
1213:T
1205:R
1198:X
1196:(
1194:F
1190:F
1186:X
1170:X
1159:k
1153:-
1151:k
1144:k
1138:k
1128:k
1126:,
1124:V
1122:(
1119:k
1114:V
1110:k
1102:V
1094:k
1083:A
1079:A
1077:,
1075:X
1073:(
1070:A
1065:X
1063:(
1060:A
1056:G
1048:A
1043:A
1039:F
1027:A
1023:X
1021:,
1019:A
1017:(
1014:A
1009:X
1007:(
1004:A
1000:F
996:A
992:A
988:A
964:G
960:A
958:(
956:G
952:B
950:(
948:G
944:C
942:(
940:G
936:C
934:→
932:B
930:→
928:A
920:G
914:G
910:A
908:(
906:G
902:B
900:(
898:G
894:C
892:(
890:G
886:C
884:→
882:B
880:→
878:A
870:G
864:F
860:C
858:(
856:F
852:B
850:(
848:F
844:A
842:(
840:F
836:C
834:→
832:B
830:→
828:A
820:F
814:F
810:C
808:(
806:F
802:B
800:(
798:F
794:A
792:(
790:F
786:C
784:→
782:B
780:→
778:A
770:F
764:C
762:(
760:F
756:B
754:(
752:F
748:A
746:(
744:F
740:C
738:→
736:B
734:→
732:A
724:F
717:C
715:→
713:B
711:→
709:A
701:A
699:(
697:G
693:B
691:(
689:G
685:C
683:(
681:G
677:C
675:→
673:B
671:→
669:A
659:A
657:(
655:G
651:B
649:(
647:G
643:C
641:(
639:G
635:C
633:→
631:B
629:→
627:A
617:A
615:(
613:G
609:B
607:(
605:G
601:C
599:(
597:G
593:C
591:→
589:B
587:→
585:A
575:A
573:(
571:G
567:B
565:(
563:G
559:C
557:(
555:G
551:C
549:→
547:B
545:→
543:A
532:G
528:Q
524:P
516:G
510:.
504:C
502:(
500:F
496:B
494:(
492:F
488:A
486:(
484:F
480:C
478:→
476:B
474:→
472:A
462:C
460:(
458:F
454:B
452:(
450:F
446:A
444:(
442:F
438:C
436:→
434:B
432:→
430:A
420:C
418:(
416:F
412:B
410:(
408:F
404:A
402:(
400:F
396:C
394:→
392:B
390:→
388:A
377:F
370:C
368:(
366:F
362:B
360:(
358:F
354:A
352:(
350:F
346:C
344:→
342:B
340:→
338:A
334:Q
317:0
311:)
308:C
305:(
302:F
292:)
289:g
286:(
283:F
268:)
265:B
262:(
259:F
249:)
246:f
243:(
240:F
225:)
222:A
219:(
216:F
210:0
197:P
176:0
170:C
160:g
145:B
135:f
120:A
114:0
97:F
93:F
81:Q
79:→
77:P
73:F
64:Q
60:P
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.