622:
345:
451:
177:
2453:
617:{\displaystyle 0\longrightarrow M{\overset {\varepsilon }{\longrightarrow }}C^{0}{\overset {d^{0}}{\longrightarrow }}C^{1}{\overset {d^{1}}{\longrightarrow }}C^{2}{\overset {d^{2}}{\longrightarrow }}\cdots {\overset {d^{n-1}}{\longrightarrow }}C^{n}{\overset {d^{n}}{\longrightarrow }}\cdots ,}
340:{\displaystyle \cdots {\overset {d_{n+1}}{\longrightarrow }}E_{n}{\overset {d_{n}}{\longrightarrow }}\cdots {\overset {d_{3}}{\longrightarrow }}E_{2}{\overset {d_{2}}{\longrightarrow }}E_{1}{\overset {d_{1}}{\longrightarrow }}E_{0}{\overset {\varepsilon }{\longrightarrow }}M\longrightarrow 0.}
2297:
1689:
71:
characterizing the structure of a specific module or object of this category. When, as usually, arrows are oriented to the right, the sequence is supposed to be infinite to the left for (left) resolutions, and to the right for right resolutions. However, a
1905:
1336:
685:
412:
2650:
2448:{\displaystyle 0\rightarrow R\subset {\mathcal {C}}^{0}(M){\stackrel {d}{\rightarrow }}{\mathcal {C}}^{1}(M){\stackrel {d}{\rightarrow }}\cdots {\stackrel {d}{\rightarrow }}{\mathcal {C}}^{\dim M}(M)\rightarrow 0.}
1774:
635:-module (it is common to use superscripts on the objects in the resolution and the maps between them to indicate the dual nature of such a resolution). For succinctness, the resolution above can be written as
2553:
1554:
1802:. But, both terms are locally free, and locally flat. Both classes of sheaves can be used in place for certain computations, replacing projective resolutions for computing some derived functors.
2183:
1502:
2496:
2285:
1435:
1526:
1997:
1839:
1396:
2229:
1800:
2102:
1366:
1546:
1462:
641:
371:
2562:
1954: ≥ 0. Dually, a left resolution is acyclic with respect to a right exact functor if its derived functors vanish on the objects of the resolution.
1260:
80:; it is usually represented by a finite exact sequence in which the leftmost object (for resolutions) or the rightmost object (for coresolutions) is the
1810:
In many cases one is not really interested in the objects appearing in a resolution, but in the behavior of the resolution with respect to a given
964:. The behavior of these dimensions reflects characteristics of the ring. For example, a ring has right global dimension 0 if and only if it is a
1697:
2509:
1684:{\displaystyle \bigoplus _{i,j=0}{\mathcal {O}}_{X}(s_{i,j})\to \bigoplus _{i=0}{\mathcal {O}}_{X}(s_{i})\to {\mathcal {M}}\to 0.}
1218:, and, accordingly, projective and injective resolutions. However, such resolutions need not exist in a general abelian category
836:-module possesses an injective resolution. Projective resolutions (and, more generally, flat resolutions) can be used to compute
881:
2901:
2872:
2851:
17:
902:
does not admit a finite projective resolution then the projective dimension is infinite. For example, for a commutative
1060:
1238:). Even if they do exist, such resolutions are often difficult to work with. For example, as pointed out above, every
2814:
2785:
2687:
2114:
2802:
1467:
2936:
2461:
2250:
2677:
2672:
1409:
1019:
1900:{\displaystyle 0\rightarrow M\rightarrow E_{0}\rightarrow E_{1}\rightarrow E_{2}\rightarrow \cdots }
1507:
898:). For example, a module has projective dimension zero if and only if it is a projective module. If
1972:
1064:
969:
60:
2241:
68:
2941:
2195:
418:
1406:
One class of examples of
Abelian categories without projective resolutions are the categories
961:
1779:
1371:
2911:
2824:
2080:
1442:
1344:
890:
56:
2882:
2832:
680:{\displaystyle 0\longrightarrow M{\overset {\varepsilon }{\longrightarrow }}C^{\bullet }.}
407:{\displaystyle E_{\bullet }{\overset {\varepsilon }{\longrightarrow }}M\longrightarrow 0.}
8:
2667:
2655:
2645:{\displaystyle \mathrm {H} ^{i}(M,\mathbf {R} )=\mathrm {H} ^{i}({\mathcal {C}}^{*}(M)).}
1143:
993:
926:
795:, every module also admits projective and flat resolutions. The proof idea is to define
32:
1044:
and their degrees are the same for all the minimal free resolutions of a graded module.
2896:. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press.
1819:
1531:
1447:
1231:
914:
145:
2730:
1331:{\displaystyle 0\rightarrow M\rightarrow I_{*},\ \ 0\rightarrow M'\rightarrow I'_{*},}
2915:
2897:
2868:
2847:
2810:
2781:
2682:
2288:
1235:
1211:
1139:
1052:
762:
112:
2878:
2828:
2559:, which is the derived functor of the global section functor Γ is computed as
2556:
1215:
1178:
1124:
1105:
953:
781:
124:
64:
2049:
The importance of acyclic resolutions lies in the fact that the derived functors
2907:
2839:
2820:
2806:
1923:
1128:
1056:
965:
918:
2889:
2794:
2503:
2234:
1966:
1438:
1401:
1113:
1101:
989:
934:
848:
164:
52:
2930:
2031:) are the projective resolutions and those that are acyclic for the functors
985:
2919:
87:
Generally, the objects in the sequence are restricted to have some property
2741:
1135:
944:
The injective and projective dimensions are used on the category of right
837:
766:
116:
108:
81:
77:
28:
2860:
2499:
1769:{\displaystyle H^{n}(\mathbb {P} _{S}^{n},{\mathcal {O}}_{X}(s))\neq 0}
1159:
1109:
903:
792:
2015:). Every flat resolution is acyclic with respect to this functor. A
705:
1200:
884:. The minimal length of a finite projective resolution of a module
821:-module generated by the elements of the kernel of the natural map
76:
is one where only finitely many of the objects in the sequence are
2073:
of a right exact functor) can be obtained from as the homology of
2046:-acyclic for any left exact (right exact, respectively) functor.
1811:
1341:
there is in general no functorial way of obtaining a map between
1243:
2233:
This situation applies in many situations. For example, for the
1242:-module has an injective resolution, but this resolution is not
968:, and a ring has weak global dimension 0 if and only if it is a
694:
if only finitely many of the modules involved are non-zero. The
2548:{\displaystyle \Gamma :{\mathcal {F}}\mapsto {\mathcal {F}}(M)}
2023:. Similarly, resolutions that are acyclic for all the functors
441:, a right resolution is a possibly infinite exact sequence of
1210:
The analogous notion of projective and injective modules are
1402:
Abelian categories without projective resolutions in general
710:
In many circumstances conditions are imposed on the modules
2745:
2734:
365:. For succinctness, the resolution above can be written as
2799:
Commutative algebra. With a view toward algebraic geometry
2658:
are acyclic with respect to the global sections functor.
1026:
are those for which the number of basis elements of each
107:, which are left resolutions consisting, respectively of
1694:
The first two terms are not in general projective since
1100:
A classic example of a free resolution is given by the
2565:
2512:
2464:
2300:
2253:
2198:
2117:
2083:
1975:
1842:
1782:
1700:
1557:
1534:
1510:
1470:
1450:
1412:
1374:
1347:
1263:
644:
454:
374:
180:
2502:, which are known to be acyclic with respect to the
1226:
has a projective (resp. injective) resolution, then
1168:
909:, the projective dimension is finite if and only if
702:
labeling a nonzero module in the finite resolution.
1075:such that the degrees of the basis elements of the
752:resolutions are left resolutions such that all the
2867:(Third ed.), Reading, Mass.: Addison-Wesley,
2780:, University Mathematical Texts, Oliver and Boyd,
2644:
2547:
2490:
2447:
2279:
2223:
2177:
2096:
2077:-acyclic resolutions: given an acyclic resolution
1991:
1899:
1794:
1768:
1683:
1540:
1520:
1496:
1456:
1429:
1390:
1360:
1330:
772:-modules, respectively. Injective resolutions are
679:
616:
406:
339:
1035:is minimal. The number of basis elements of each
706:Free, projective, injective, and flat resolutions
2928:
2039:, â‹… ) are the injective resolutions.
1000:has a free resolution in which the free modules
2775:
1814:. Therefore, in many situations, the notion of
948:-modules to define a homological dimension for
91:(for example to be free). Thus one speaks of a
1548:has a presentation given by an exact sequence
975:
960:. Similarly, flat dimension is used to define
731:is a left resolution in which all the modules
1833:between two abelian categories, a resolution
877:there exists a chain homotopy between them.
698:of a finite resolution is the maximum index
123:, which are right resolutions consisting of
2019:is acyclic for the tensor product by every
1173:The definition of resolutions of an object
1022:. Among these graded free resolutions, the
791:-module possesses a free left resolution.
1112:or of a homogeneous regular sequence in a
2042:Any injective (projective) resolution is
1716:
1479:
996:by its elements of positive degree. Then
851:, i.e., given two projective resolutions
130:
2838:
2793:
2759:
2706:
2178:{\displaystyle R_{i}F(M)=H_{i}F(E_{*}),}
1504:is projective space, any coherent sheaf
2846:(Second ed.), Dover Publications,
2061:(of a left exact functor, and likewise
917:and in this case it coincides with the
14:
2929:
2894:An introduction to homological algebra
2888:
2718:
1497:{\displaystyle X=\mathbb {P} _{S}^{n}}
1254:, together with injective resolutions
2491:{\displaystyle {\mathcal {C}}^{*}(M)}
2280:{\displaystyle {\mathcal {C}}^{*}(M)}
1805:
1009:may be graded in such a way that the
806:-module generated by the elements of
2859:
2805:, vol. 150, Berlin, New York:
2192:-th homology object of the complex
1150:is a free resolution of the module
357:are called boundary maps. The map
24:
2616:
2600:
2568:
2531:
2521:
2513:
2468:
2413:
2355:
2316:
2257:
1737:
1670:
1638:
1583:
1513:
843:Projective resolution of a module
95:. In particular, every module has
25:
2953:
1169:Resolutions in abelian categories
1116:finitely generated over a field.
2588:
1084:in a minimal free resolution of
941:) are defined for modules also.
433:). Specifically, given a module
2688:Matrix factorizations (algebra)
2247:can be resolved by the sheaves
1430:{\displaystyle {\text{Coh}}(X)}
880:Resolutions are used to define
690:A (co)resolution is said to be
2753:
2724:
2700:
2636:
2633:
2627:
2610:
2592:
2578:
2542:
2536:
2526:
2485:
2479:
2439:
2436:
2430:
2398:
2379:
2372:
2366:
2340:
2333:
2327:
2304:
2274:
2268:
2215:
2202:
2169:
2156:
2137:
2131:
1891:
1878:
1865:
1852:
1846:
1757:
1754:
1748:
1711:
1675:
1665:
1662:
1649:
1616:
1613:
1594:
1521:{\displaystyle {\mathcal {M}}}
1424:
1418:
1309:
1298:
1273:
1267:
1184:is the same as above, but the
1061:Castelnuovo–Mumford regularity
656:
648:
593:
560:
540:
513:
486:
466:
458:
398:
387:
331:
320:
293:
266:
239:
219:
186:
135:
119:. Similarly every module has
13:
1:
2803:Graduate Texts in Mathematics
2769:
2188:where right hand side is the
1992:{\displaystyle \otimes _{R}M}
1246:, i.e., given a homomorphism
2778:Elementary rings and modules
1199:, and all maps involved are
992:, which is generated over a
7:
2661:
1095:
976:Graded modules and algebras
719:resolving the given module
31:, and more specifically in
10:
2958:
2224:{\displaystyle F(E_{*}).}
1999:is a right exact functor
67:) that is used to define
2776:Iain T. Adamson (1972),
2693:
2678:Hilbert's syzygy theorem
1065:projective algebraic set
1024:minimal free resolutions
970:von Neumann regular ring
59:(or, more generally, of
2242:differentiable manifold
1154:not only over the ring
1071:is the minimal integer
167:(possibly infinite) of
2717:is more common, as in
2646:
2549:
2492:
2449:
2281:
2225:
2179:
2098:
1993:
1950: > 0 and
1901:
1796:
1795:{\displaystyle s>0}
1770:
1685:
1542:
1522:
1498:
1458:
1431:
1392:
1391:{\displaystyle I'_{*}}
1362:
1332:
882:homological dimensions
681:
618:
408:
341:
131:Resolutions of modules
101:projective resolutions
2731:projective resolution
2673:Hilbert–Burch theorem
2647:
2550:
2493:
2450:
2282:
2226:
2180:
2099:
2097:{\displaystyle E_{*}}
1994:
1957:For example, given a
1902:
1797:
1771:
1686:
1543:
1523:
1499:
1459:
1432:
1393:
1363:
1361:{\displaystyle I_{*}}
1333:
1222:. If every object of
962:weak global dimension
682:
619:
409:
342:
121:injective resolutions
18:Projective resolution
2656:Godement resolutions
2563:
2510:
2462:
2298:
2251:
2196:
2115:
2081:
1973:
1840:
1780:
1698:
1555:
1532:
1508:
1468:
1448:
1410:
1372:
1345:
1261:
891:projective dimension
744:-modules. Likewise,
642:
452:
372:
178:
2937:Homological algebra
2668:Standard resolution
1816:acyclic resolutions
1730:
1493:
1387:
1324:
1146:) chain complex of
1088:are all lower than
927:injective dimension
925:. Analogously, the
832:etc. Dually, every
33:homological algebra
2890:Weibel, Charles A.
2642:
2545:
2488:
2445:
2289:differential forms
2277:
2221:
2175:
2094:
1989:
1897:
1820:left exact functor
1806:Acyclic resolution
1792:
1766:
1714:
1681:
1634:
1579:
1538:
1518:
1494:
1477:
1464:. For example, if
1454:
1427:
1388:
1375:
1358:
1328:
1312:
1232:enough projectives
1158:but also over the
1059:over a field, the
1020:graded linear maps
847:is unique up to a
776:resolutions whose
677:
614:
404:
350:The homomorphisms
337:
113:projective modules
2903:978-0-521-55987-4
2874:978-0-201-55540-0
2853:978-0-486-47187-7
2683:Free presentation
2555:. Therefore, the
2407:
2388:
2349:
1946:) vanish for all
1922:-acyclic, if the
1818:is used: given a
1619:
1558:
1541:{\displaystyle X}
1457:{\displaystyle X}
1416:
1294:
1291:
1236:enough injectives
1216:injective objects
1053:homogeneous ideal
952:called the right
782:injective modules
723:. For example, a
662:
606:
579:
553:
526:
499:
472:
393:
326:
306:
279:
252:
232:
205:
125:injective modules
74:finite resolution
16:(Redirected from
2949:
2923:
2885:
2856:
2844:Basic algebra II
2840:Jacobson, Nathan
2835:
2790:
2763:
2757:
2751:
2728:
2722:
2715:right resolution
2704:
2651:
2649:
2648:
2643:
2626:
2625:
2620:
2619:
2609:
2608:
2603:
2591:
2577:
2576:
2571:
2557:sheaf cohomology
2554:
2552:
2551:
2546:
2535:
2534:
2525:
2524:
2497:
2495:
2494:
2489:
2478:
2477:
2472:
2471:
2454:
2452:
2451:
2446:
2429:
2428:
2417:
2416:
2409:
2408:
2406:
2401:
2396:
2390:
2389:
2387:
2382:
2377:
2365:
2364:
2359:
2358:
2351:
2350:
2348:
2343:
2338:
2326:
2325:
2320:
2319:
2286:
2284:
2283:
2278:
2267:
2266:
2261:
2260:
2230:
2228:
2227:
2222:
2214:
2213:
2184:
2182:
2181:
2176:
2168:
2167:
2152:
2151:
2127:
2126:
2103:
2101:
2100:
2095:
2093:
2092:
2027:( â‹… ,
1998:
1996:
1995:
1990:
1985:
1984:
1924:derived functors
1906:
1904:
1903:
1898:
1890:
1889:
1877:
1876:
1864:
1863:
1801:
1799:
1798:
1793:
1775:
1773:
1772:
1767:
1747:
1746:
1741:
1740:
1729:
1724:
1719:
1710:
1709:
1690:
1688:
1687:
1682:
1674:
1673:
1661:
1660:
1648:
1647:
1642:
1641:
1633:
1612:
1611:
1593:
1592:
1587:
1586:
1578:
1547:
1545:
1544:
1539:
1527:
1525:
1524:
1519:
1517:
1516:
1503:
1501:
1500:
1495:
1492:
1487:
1482:
1463:
1461:
1460:
1455:
1439:coherent sheaves
1436:
1434:
1433:
1428:
1417:
1414:
1397:
1395:
1394:
1389:
1383:
1367:
1365:
1364:
1359:
1357:
1356:
1337:
1335:
1334:
1329:
1320:
1308:
1292:
1289:
1285:
1284:
1230:is said to have
1179:abelian category
1125:aspherical space
1106:regular sequence
954:global dimension
686:
684:
683:
678:
673:
672:
663:
655:
623:
621:
620:
615:
607:
605:
604:
592:
590:
589:
580:
578:
577:
559:
554:
552:
551:
539:
537:
536:
527:
525:
524:
512:
510:
509:
500:
498:
497:
485:
483:
482:
473:
465:
423:right resolution
413:
411:
410:
405:
394:
386:
384:
383:
363:augmentation map
346:
344:
343:
338:
327:
319:
317:
316:
307:
305:
304:
292:
290:
289:
280:
278:
277:
265:
263:
262:
253:
251:
250:
238:
233:
231:
230:
218:
216:
215:
206:
204:
203:
185:
105:flat resolutions
97:free resolutions
65:abelian category
49:right resolution
21:
2957:
2956:
2952:
2951:
2950:
2948:
2947:
2946:
2927:
2926:
2904:
2875:
2854:
2817:
2807:Springer-Verlag
2795:Eisenbud, David
2788:
2772:
2767:
2766:
2758:
2754:
2729:
2725:
2705:
2701:
2696:
2664:
2621:
2615:
2614:
2613:
2604:
2599:
2598:
2587:
2572:
2567:
2566:
2564:
2561:
2560:
2530:
2529:
2520:
2519:
2511:
2508:
2507:
2473:
2467:
2466:
2465:
2463:
2460:
2459:
2418:
2412:
2411:
2410:
2402:
2397:
2395:
2394:
2383:
2378:
2376:
2375:
2360:
2354:
2353:
2352:
2344:
2339:
2337:
2336:
2321:
2315:
2314:
2313:
2299:
2296:
2295:
2262:
2256:
2255:
2254:
2252:
2249:
2248:
2209:
2205:
2197:
2194:
2193:
2163:
2159:
2147:
2143:
2122:
2118:
2116:
2113:
2112:
2088:
2084:
2082:
2079:
2078:
2069:
2057:
2017:flat resolution
1980:
1976:
1974:
1971:
1970:
1945:
1933:
1885:
1881:
1872:
1868:
1859:
1855:
1841:
1838:
1837:
1808:
1781:
1778:
1777:
1742:
1736:
1735:
1734:
1725:
1720:
1715:
1705:
1701:
1699:
1696:
1695:
1669:
1668:
1656:
1652:
1643:
1637:
1636:
1635:
1623:
1601:
1597:
1588:
1582:
1581:
1580:
1562:
1556:
1553:
1552:
1533:
1530:
1529:
1512:
1511:
1509:
1506:
1505:
1488:
1483:
1478:
1469:
1466:
1465:
1449:
1446:
1445:
1413:
1411:
1408:
1407:
1404:
1379:
1373:
1370:
1369:
1352:
1348:
1346:
1343:
1342:
1316:
1301:
1280:
1276:
1262:
1259:
1258:
1195:are objects in
1189:
1171:
1129:universal cover
1098:
1083:
1057:polynomial ring
1043:
1034:
1017:
1008:
978:
966:semisimple ring
919:Krull dimension
894:and denoted pd(
868:
857:
827:
817:to be the free
816:
802:to be the free
801:
760:
739:
725:free resolution
718:
708:
668:
664:
654:
643:
640:
639:
600:
596:
591:
585:
581:
567:
563:
558:
547:
543:
538:
532:
528:
520:
516:
511:
505:
501:
493:
489:
484:
478:
474:
464:
453:
450:
449:
385:
379:
375:
373:
370:
369:
355:
318:
312:
308:
300:
296:
291:
285:
281:
273:
269:
264:
258:
254:
246:
242:
237:
226:
222:
217:
211:
207:
193:
189:
184:
179:
176:
175:
153:left resolution
140:Given a module
138:
133:
41:left resolution
23:
22:
15:
12:
11:
5:
2955:
2945:
2944:
2939:
2925:
2924:
2902:
2886:
2873:
2857:
2852:
2836:
2815:
2791:
2786:
2771:
2768:
2765:
2764:
2752:
2723:
2698:
2697:
2695:
2692:
2691:
2690:
2685:
2680:
2675:
2670:
2663:
2660:
2641:
2638:
2635:
2632:
2629:
2624:
2618:
2612:
2607:
2602:
2597:
2594:
2590:
2586:
2583:
2580:
2575:
2570:
2544:
2541:
2538:
2533:
2528:
2523:
2518:
2515:
2504:global section
2487:
2484:
2481:
2476:
2470:
2456:
2455:
2444:
2441:
2438:
2435:
2432:
2427:
2424:
2421:
2415:
2405:
2400:
2393:
2386:
2381:
2374:
2371:
2368:
2363:
2357:
2347:
2342:
2335:
2332:
2329:
2324:
2318:
2312:
2309:
2306:
2303:
2276:
2273:
2270:
2265:
2259:
2235:constant sheaf
2220:
2217:
2212:
2208:
2204:
2201:
2186:
2185:
2174:
2171:
2166:
2162:
2158:
2155:
2150:
2146:
2142:
2139:
2136:
2133:
2130:
2125:
2121:
2091:
2087:
2065:
2053:
1988:
1983:
1979:
1967:tensor product
1941:
1929:
1908:
1907:
1896:
1893:
1888:
1884:
1880:
1875:
1871:
1867:
1862:
1858:
1854:
1851:
1848:
1845:
1807:
1804:
1791:
1788:
1785:
1765:
1762:
1759:
1756:
1753:
1750:
1745:
1739:
1733:
1728:
1723:
1718:
1713:
1708:
1704:
1692:
1691:
1680:
1677:
1672:
1667:
1664:
1659:
1655:
1651:
1646:
1640:
1632:
1629:
1626:
1622:
1618:
1615:
1610:
1607:
1604:
1600:
1596:
1591:
1585:
1577:
1574:
1571:
1568:
1565:
1561:
1537:
1515:
1491:
1486:
1481:
1476:
1473:
1453:
1426:
1423:
1420:
1403:
1400:
1386:
1382:
1378:
1355:
1351:
1339:
1338:
1327:
1323:
1319:
1315:
1311:
1307:
1304:
1300:
1297:
1288:
1283:
1279:
1275:
1272:
1269:
1266:
1187:
1170:
1167:
1114:graded algebra
1102:Koszul complex
1097:
1094:
1079:
1039:
1030:
1013:
1004:
990:graded algebra
977:
974:
935:flat dimension
888:is called its
866:
855:
849:chain homotopy
825:
814:
799:
756:
735:
714:
707:
704:
688:
687:
676:
671:
667:
661:
658:
653:
650:
647:
625:
624:
613:
610:
603:
599:
595:
588:
584:
576:
573:
570:
566:
562:
557:
550:
546:
542:
535:
531:
523:
519:
515:
508:
504:
496:
492:
488:
481:
477:
471:
468:
463:
460:
457:
415:
414:
403:
400:
397:
392:
389:
382:
378:
353:
348:
347:
336:
333:
330:
325:
322:
315:
311:
303:
299:
295:
288:
284:
276:
272:
268:
261:
257:
249:
245:
241:
236:
229:
225:
221:
214:
210:
202:
199:
196:
192:
188:
183:
165:exact sequence
137:
134:
132:
129:
53:exact sequence
9:
6:
4:
3:
2:
2954:
2943:
2942:Module theory
2940:
2938:
2935:
2934:
2932:
2921:
2917:
2913:
2909:
2905:
2899:
2895:
2891:
2887:
2884:
2880:
2876:
2870:
2866:
2862:
2858:
2855:
2849:
2845:
2841:
2837:
2834:
2830:
2826:
2822:
2818:
2816:3-540-94268-8
2812:
2808:
2804:
2800:
2796:
2792:
2789:
2787:0-05-002192-3
2783:
2779:
2774:
2773:
2761:
2760:Jacobson 2009
2756:
2750:
2748:
2743:
2739:
2737:
2732:
2727:
2720:
2716:
2712:
2708:
2707:Jacobson 2009
2703:
2699:
2689:
2686:
2684:
2681:
2679:
2676:
2674:
2671:
2669:
2666:
2665:
2659:
2657:
2652:
2639:
2630:
2622:
2605:
2595:
2584:
2581:
2573:
2558:
2539:
2516:
2505:
2501:
2482:
2474:
2442:
2433:
2425:
2422:
2419:
2403:
2391:
2384:
2369:
2361:
2345:
2330:
2322:
2310:
2307:
2301:
2294:
2293:
2292:
2290:
2271:
2263:
2246:
2243:
2239:
2236:
2231:
2218:
2210:
2206:
2199:
2191:
2172:
2164:
2160:
2153:
2148:
2144:
2140:
2134:
2128:
2123:
2119:
2111:
2110:
2109:
2107:
2104:of an object
2089:
2085:
2076:
2072:
2068:
2064:
2060:
2056:
2052:
2047:
2045:
2040:
2038:
2034:
2030:
2026:
2022:
2018:
2014:
2010:
2006:
2002:
1986:
1981:
1977:
1968:
1964:
1960:
1955:
1953:
1949:
1944:
1940:
1936:
1932:
1928:
1925:
1921:
1917:
1913:
1910:of an object
1894:
1886:
1882:
1873:
1869:
1860:
1856:
1849:
1843:
1836:
1835:
1834:
1832:
1828:
1824:
1821:
1817:
1813:
1803:
1789:
1786:
1783:
1763:
1760:
1751:
1743:
1731:
1726:
1721:
1706:
1702:
1678:
1657:
1653:
1644:
1630:
1627:
1624:
1620:
1608:
1605:
1602:
1598:
1589:
1575:
1572:
1569:
1566:
1563:
1559:
1551:
1550:
1549:
1535:
1489:
1484:
1474:
1471:
1451:
1444:
1440:
1421:
1399:
1384:
1380:
1376:
1353:
1349:
1325:
1321:
1317:
1313:
1305:
1302:
1295:
1286:
1281:
1277:
1270:
1264:
1257:
1256:
1255:
1253:
1249:
1245:
1241:
1237:
1233:
1229:
1225:
1221:
1217:
1213:
1208:
1206:
1202:
1198:
1194:
1190:
1183:
1180:
1176:
1166:
1164:
1161:
1157:
1153:
1149:
1145:
1141:
1138:. Then every
1137:
1133:
1130:
1126:
1122:
1117:
1115:
1111:
1107:
1103:
1093:
1091:
1087:
1082:
1078:
1074:
1070:
1066:
1062:
1058:
1054:
1050:
1045:
1042:
1038:
1033:
1029:
1025:
1021:
1016:
1012:
1007:
1003:
999:
995:
991:
987:
986:graded module
983:
973:
971:
967:
963:
959:
955:
951:
947:
942:
940:
936:
932:
928:
924:
920:
916:
912:
908:
905:
901:
897:
893:
892:
887:
883:
878:
876:
872:
865:
861:
854:
850:
846:
841:
839:
835:
831:
824:
820:
813:
809:
805:
798:
794:
790:
785:
783:
779:
775:
771:
768:
764:
759:
755:
751:
747:
743:
738:
734:
730:
726:
722:
717:
713:
703:
701:
697:
693:
674:
669:
665:
659:
651:
645:
638:
637:
636:
634:
630:
611:
608:
601:
597:
586:
582:
574:
571:
568:
564:
555:
548:
544:
533:
529:
521:
517:
506:
502:
494:
490:
479:
475:
469:
461:
455:
448:
447:
446:
444:
440:
436:
432:
428:
424:
421:is that of a
420:
401:
395:
390:
380:
376:
368:
367:
366:
364:
361:is called an
360:
356:
334:
328:
323:
313:
309:
301:
297:
286:
282:
274:
270:
259:
255:
247:
243:
234:
227:
223:
212:
208:
200:
197:
194:
190:
181:
174:
173:
172:
170:
166:
162:
158:
154:
150:
147:
143:
128:
126:
122:
118:
114:
110:
106:
102:
98:
94:
90:
85:
83:
79:
75:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
19:
2893:
2864:
2843:
2798:
2777:
2755:
2746:
2735:
2726:
2714:
2711:coresolution
2710:
2709:, §6.5 uses
2702:
2653:
2500:fine sheaves
2458:The sheaves
2457:
2244:
2237:
2232:
2189:
2187:
2105:
2074:
2070:
2066:
2062:
2058:
2054:
2050:
2048:
2043:
2041:
2036:
2032:
2028:
2024:
2020:
2016:
2012:
2008:
2004:
2000:
1969:
1962:
1958:
1956:
1951:
1947:
1942:
1938:
1934:
1930:
1926:
1919:
1915:
1911:
1909:
1830:
1826:
1822:
1815:
1809:
1693:
1405:
1340:
1251:
1247:
1239:
1227:
1223:
1219:
1209:
1204:
1196:
1192:
1185:
1181:
1174:
1172:
1162:
1155:
1151:
1147:
1136:contractible
1131:
1127:, i.e., its
1120:
1118:
1099:
1089:
1085:
1080:
1076:
1072:
1068:
1048:
1046:
1040:
1036:
1031:
1027:
1023:
1014:
1010:
1005:
1001:
997:
981:
979:
957:
949:
945:
943:
938:
930:
922:
910:
906:
899:
895:
889:
885:
879:
874:
870:
863:
859:
852:
844:
842:
838:Tor functors
833:
829:
822:
818:
811:
807:
803:
796:
788:
786:
777:
773:
769:
757:
753:
749:
745:
741:
736:
732:
728:
727:of a module
724:
720:
715:
711:
709:
699:
695:
691:
689:
632:
628:
626:
442:
438:
437:over a ring
434:
430:
429:, or simply
427:coresolution
426:
422:
416:
362:
358:
351:
349:
168:
160:
156:
152:
148:
141:
139:
120:
117:flat modules
109:free modules
104:
100:
96:
93:P resolution
92:
88:
86:
73:
48:
45:coresolution
44:
40:
36:
26:
2861:Lang, Serge
2719:Weibel 1994
1067:defined by
810:, and then
627:where each
419:dual notion
155:(or simply
136:Definitions
82:zero-object
43:; dually a
29:mathematics
2931:Categories
2883:0848.13001
2833:0819.13001
2770:References
2742:resolution
2654:Similarly
2287:of smooth
2108:, we have
2007:) →
1918:is called
1244:functorial
1212:projective
1160:group ring
1144:simplicial
1110:local ring
1018:and ε are
904:local ring
793:A fortiori
763:projective
746:projective
431:resolution
157:resolution
69:invariants
37:resolution
2842:(2009) ,
2721:, Chap. 2
2713:, though
2623:∗
2527:↦
2514:Γ
2475:∗
2440:→
2423:
2399:→
2392:⋯
2380:→
2341:→
2311:⊂
2305:→
2264:∗
2211:∗
2165:∗
2090:∗
1978:⊗
1895:⋯
1892:→
1879:→
1866:→
1853:→
1847:→
1761:≠
1676:→
1666:→
1621:⨁
1617:→
1560:⨁
1381:∗
1354:∗
1318:∗
1310:→
1299:→
1282:∗
1274:→
1268:→
1201:morphisms
740:are free
670:∙
660:ε
657:⟶
649:⟶
609:⋯
594:⟶
572:−
561:⟶
556:⋯
541:⟶
514:⟶
487:⟶
470:ε
467:⟶
459:⟶
445:-modules
399:⟶
391:ε
388:⟶
381:∙
332:⟶
324:ε
321:⟶
294:⟶
267:⟶
240:⟶
235:⋯
220:⟶
187:⟶
182:⋯
171:-modules
2920:36131259
2892:(1994).
2863:(1993),
2797:(1995),
2662:See also
2506:functor
1961:-module
1829:→
1385:′
1322:′
1306:′
1250:→
1140:singular
1096:Examples
828:→
780:are all
78:non-zero
51:) is an
2912:1269324
2865:Algebra
2825:1322960
2744:at the
2733:at the
1812:functor
1234:(resp.
1063:of the
988:over a
915:regular
144:over a
61:objects
57:modules
2918:
2910:
2900:
2881:
2871:
2850:
2831:
2823:
2813:
2784:
2762:, §6.5
1965:, the
1443:scheme
1293:
1290:
1177:in an
1123:be an
933:) and
787:Every
696:length
692:finite
631:is an
163:is an
63:of an
2694:Notes
2240:on a
1441:on a
1108:in a
1104:of a
1055:in a
1051:is a
994:field
984:be a
774:right
159:) of
2916:OCLC
2898:ISBN
2869:ISBN
2848:ISBN
2811:ISBN
2782:ISBN
2498:are
1787:>
1776:for
1368:and
1214:and
1191:and
1142:(or
1119:Let
980:Let
862:and
767:flat
765:and
761:are
750:flat
748:and
425:(or
417:The
151:, a
146:ring
103:and
39:(or
35:, a
2879:Zbl
2829:Zbl
2749:Lab
2738:Lab
2420:dim
2033:Hom
2025:Hom
2009:Mod
2001:Mod
1914:of
1528:on
1437:of
1415:Coh
1252:M'
1203:in
1134:is
1090:r-i
1047:If
956:of
937:fd(
929:id(
921:of
913:is
873:of
115:or
99:,
55:of
47:or
27:In
2933::
2914:.
2908:MR
2906:.
2877:,
2827:,
2821:MR
2819:,
2809:,
2801:,
2740:,
2443:0.
2291::
1825::
1679:0.
1398:.
1207:.
1165:.
1092:.
972:.
869:→
858:→
840:.
784:.
402:0.
335:0.
127:.
111:,
84:.
2922:.
2747:n
2736:n
2640:.
2637:)
2634:)
2631:M
2628:(
2617:C
2611:(
2606:i
2601:H
2596:=
2593:)
2589:R
2585:,
2582:M
2579:(
2574:i
2569:H
2543:)
2540:M
2537:(
2532:F
2522:F
2517::
2486:)
2483:M
2480:(
2469:C
2437:)
2434:M
2431:(
2426:M
2414:C
2404:d
2385:d
2373:)
2370:M
2367:(
2362:1
2356:C
2346:d
2334:)
2331:M
2328:(
2323:0
2317:C
2308:R
2302:0
2275:)
2272:M
2269:(
2258:C
2245:M
2238:R
2219:.
2216:)
2207:E
2203:(
2200:F
2190:i
2173:,
2170:)
2161:E
2157:(
2154:F
2149:i
2145:H
2141:=
2138:)
2135:M
2132:(
2129:F
2124:i
2120:R
2106:M
2086:E
2075:F
2071:F
2067:i
2063:L
2059:F
2055:i
2051:R
2044:F
2037:M
2035:(
2029:M
2021:M
2013:R
2011:(
2005:R
2003:(
1987:M
1982:R
1963:M
1959:R
1952:n
1948:i
1943:n
1939:E
1937:(
1935:F
1931:i
1927:R
1920:F
1916:A
1912:M
1887:2
1883:E
1874:1
1870:E
1861:0
1857:E
1850:M
1844:0
1831:B
1827:A
1823:F
1790:0
1784:s
1764:0
1758:)
1755:)
1752:s
1749:(
1744:X
1738:O
1732:,
1727:n
1722:S
1717:P
1712:(
1707:n
1703:H
1671:M
1663:)
1658:i
1654:s
1650:(
1645:X
1639:O
1631:0
1628:=
1625:i
1614:)
1609:j
1606:,
1603:i
1599:s
1595:(
1590:X
1584:O
1576:0
1573:=
1570:j
1567:,
1564:i
1536:X
1514:M
1490:n
1485:S
1480:P
1475:=
1472:X
1452:X
1425:)
1422:X
1419:(
1377:I
1350:I
1326:,
1314:I
1303:M
1296:0
1287:,
1278:I
1271:M
1265:0
1248:M
1240:R
1228:A
1224:A
1220:A
1205:A
1197:A
1193:C
1188:i
1186:E
1182:A
1175:M
1163:Z
1156:Z
1152:Z
1148:E
1132:E
1121:X
1086:I
1081:i
1077:E
1073:r
1069:I
1049:I
1041:i
1037:E
1032:i
1028:E
1015:i
1011:d
1006:i
1002:E
998:M
982:M
958:R
950:R
946:R
939:M
931:M
923:R
911:R
907:R
900:M
896:M
886:M
875:M
871:M
867:1
864:P
860:M
856:0
853:P
845:M
834:R
830:M
826:0
823:E
819:R
815:1
812:E
808:M
804:R
800:0
797:E
789:R
778:C
770:R
758:i
754:E
742:R
737:i
733:E
729:M
721:M
716:i
712:E
700:n
675:.
666:C
652:M
646:0
633:R
629:C
612:,
602:n
598:d
587:n
583:C
575:1
569:n
565:d
549:2
545:d
534:2
530:C
522:1
518:d
507:1
503:C
495:0
491:d
480:0
476:C
462:M
456:0
443:R
439:R
435:M
396:M
377:E
359:ε
354:i
352:d
329:M
314:0
310:E
302:1
298:d
287:1
283:E
275:2
271:d
260:2
256:E
248:3
244:d
228:n
224:d
213:n
209:E
201:1
198:+
195:n
191:d
169:R
161:M
149:R
142:M
89:P
20:)
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