611:
334:
440:
166:
2442:
606:{\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 ,}
329:{\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.}
2286:
1678:
60:
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
1894:
1325:
674:
401:
2639:
2437:{\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.}
1763:
624:-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
2542:
1543:
1791:. 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.
2172:
1491:
2485:
2274:
1424:
1515:
1986:
1828:
1385:
2218:
1789:
2091:
1355:
1535:
1451:
630:
360:
2551:
1943: ≥ 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.
1249:
69:; it is usually represented by a finite exact sequence in which the leftmost object (for resolutions) or the rightmost object (for coresolutions) is the
1799:
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
953:. 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
1686:
2498:
1673:{\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.}
1207:, and, accordingly, projective and injective resolutions. However, such resolutions need not exist in a general abelian category
825:-module possesses an injective resolution. Projective resolutions (and, more generally, flat resolutions) can be used to compute
870:
2890:
2861:
2840:
891:
does not admit a finite projective resolution then the projective dimension is infinite. For example, for a commutative
1049:
1227:). Even if they do exist, such resolutions are often difficult to work with. For example, as pointed out above, every
2803:
2774:
2676:
2103:
2791:
1456:
2925:
2450:
2239:
2666:
2661:
1398:
1008:
1889:{\displaystyle 0\rightarrow M\rightarrow E_{0}\rightarrow E_{1}\rightarrow E_{2}\rightarrow \cdots }
1496:
887:). For example, a module has projective dimension zero if and only if it is a projective module. If
1961:
1053:
958:
49:
2230:
57:
2930:
2184:
407:
1395:
One class of examples of
Abelian categories without projective resolutions are the categories
950:
1768:
1360:
2900:
2813:
2069:
1431:
1333:
879:
45:
2871:
2821:
669:{\displaystyle 0\longrightarrow M{\overset {\varepsilon }{\longrightarrow }}C^{\bullet }.}
396:{\displaystyle E_{\bullet }{\overset {\varepsilon }{\longrightarrow }}M\longrightarrow 0.}
8:
2656:
2644:
2634:{\displaystyle \mathrm {H} ^{i}(M,\mathbf {R} )=\mathrm {H} ^{i}({\mathcal {C}}^{*}(M)).}
1132:
982:
915:
784:, every module also admits projective and flat resolutions. The proof idea is to define
21:
1033:
and their degrees are the same for all the minimal free resolutions of a graded module.
2885:. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press.
1808:
1520:
1436:
1220:
903:
134:
2719:
1320:{\displaystyle 0\rightarrow M\rightarrow I_{*},\ \ 0\rightarrow M'\rightarrow I'_{*},}
2904:
2886:
2857:
2836:
2799:
2770:
2671:
2277:
1224:
1200:
1128:
1041:
751:
101:
2867:
2817:
2548:, which is the derived functor of the global section functor Γ is computed as
2545:
1204:
1167:
1113:
1094:
942:
770:
113:
53:
2038:
The importance of acyclic resolutions lies in the fact that the derived functors
2896:
2828:
2809:
2795:
1912:
1117:
1045:
954:
907:
2878:
2783:
2492:
2223:
1955:
1427:
1390:
1102:
1090:
978:
923:
837:
153:
41:
2919:
2020:) are the projective resolutions and those that are acyclic for the functors
974:
2908:
76:
Generally, the objects in the sequence are restricted to have some property
2730:
1124:
933:
The injective and projective dimensions are used on the category of right
826:
755:
105:
97:
70:
66:
17:
2849:
2488:
1758:{\displaystyle H^{n}(\mathbb {P} _{S}^{n},{\mathcal {O}}_{X}(s))\neq 0}
1148:
1098:
892:
781:
2004:). Every flat resolution is acyclic with respect to this functor. A
694:
1189:
873:. The minimal length of a finite projective resolution of a module
810:-module generated by the elements of the kernel of the natural map
65:
is one where only finitely many of the objects in the sequence are
2062:
of a right exact functor) can be obtained from as the homology of
2035:-acyclic for any left exact (right exact, respectively) functor.
1800:
1330:
there is in general no functorial way of obtaining a map between
1232:
2222:
This situation applies in many situations. For example, for the
1231:-module has an injective resolution, but this resolution is not
957:, and a ring has weak global dimension 0 if and only if it is a
683:
if only finitely many of the modules involved are non-zero. The
2537:{\displaystyle \Gamma :{\mathcal {F}}\mapsto {\mathcal {F}}(M)}
2012:. Similarly, resolutions that are acyclic for all the functors
430:, a right resolution is a possibly infinite exact sequence of
1199:
The analogous notion of projective and injective modules are
1391:
Abelian categories without projective resolutions in general
699:
In many circumstances conditions are imposed on the modules
2734:
2723:
354:. For succinctness, the resolution above can be written as
2788:
Commutative algebra. With a view toward algebraic geometry
2647:
are acyclic with respect to the global sections functor.
1015:
are those for which the number of basis elements of each
96:, which are left resolutions consisting, respectively of
1683:
The first two terms are not in general projective since
1089:
A classic example of a free resolution is given by the
2554:
2501:
2453:
2289:
2242:
2187:
2106:
2072:
1964:
1831:
1771:
1689:
1546:
1523:
1499:
1459:
1439:
1401:
1363:
1336:
1252:
633:
443:
363:
169:
2491:, which are known to be acyclic with respect to the
1215:
has a projective (resp. injective) resolution, then
1157:
898:, the projective dimension is finite if and only if
691:
labeling a nonzero module in the finite resolution.
1064:such that the degrees of the basis elements of the
741:resolutions are left resolutions such that all the
2856:(Third ed.), Reading, Mass.: Addison-Wesley,
2769:, University Mathematical Texts, Oliver and Boyd,
2633:
2536:
2479:
2436:
2268:
2212:
2166:
2085:
2066:-acyclic resolutions: given an acyclic resolution
1980:
1888:
1783:
1757:
1672:
1529:
1509:
1485:
1445:
1418:
1379:
1349:
1319:
761:-modules, respectively. Injective resolutions are
668:
605:
395:
328:
1024:is minimal. The number of basis elements of each
695:Free, projective, injective, and flat resolutions
2917:
2028:, â‹… ) are the injective resolutions.
989:has a free resolution in which the free modules
2764:
1803:. Therefore, in many situations, the notion of
937:-modules to define a homological dimension for
80:(for example to be free). Thus one speaks of a
1537:has a presentation given by an exact sequence
964:
949:. Similarly, flat dimension is used to define
720:is a left resolution in which all the modules
1822:between two abelian categories, a resolution
866:there exists a chain homotopy between them.
687:of a finite resolution is the maximum index
112:, which are right resolutions consisting of
2008:is acyclic for the tensor product by every
1162:The definition of resolutions of an object
1011:. Among these graded free resolutions, the
780:-module possesses a free left resolution.
1101:or of a homogeneous regular sequence in a
2031:Any injective (projective) resolution is
1705:
1468:
985:by its elements of positive degree. Then
840:, i.e., given two projective resolutions
119:
2827:
2782:
2748:
2695:
2167:{\displaystyle R_{i}F(M)=H_{i}F(E_{*}),}
1493:is projective space, any coherent sheaf
2835:(Second ed.), Dover Publications,
2050:(of a left exact functor, and likewise
906:and in this case it coincides with the
2918:
2883:An introduction to homological algebra
2877:
2707:
1486:{\displaystyle X=\mathbb {P} _{S}^{n}}
1243:, together with injective resolutions
2480:{\displaystyle {\mathcal {C}}^{*}(M)}
2269:{\displaystyle {\mathcal {C}}^{*}(M)}
1794:
998:may be graded in such a way that the
795:-module generated by the elements of
2848:
2794:, vol. 150, Berlin, New York:
2181:-th homology object of the complex
1139:is a free resolution of the module
346:are called boundary maps. The map
13:
2605:
2589:
2557:
2520:
2510:
2502:
2457:
2402:
2344:
2305:
2246:
1726:
1659:
1627:
1572:
1502:
832:Projective resolution of a module
84:. In particular, every module has
14:
2942:
1158:Resolutions in abelian categories
1105:finitely generated over a field.
2577:
1073:in a minimal free resolution of
930:) are defined for modules also.
422:). Specifically, given a module
2677:Matrix factorizations (algebra)
2236:can be resolved by the sheaves
1419:{\displaystyle {\text{Coh}}(X)}
869:Resolutions are used to define
679:A (co)resolution is said to be
2742:
2713:
2689:
2625:
2622:
2616:
2599:
2581:
2567:
2531:
2525:
2515:
2474:
2468:
2428:
2425:
2419:
2387:
2368:
2361:
2355:
2329:
2322:
2316:
2293:
2263:
2257:
2204:
2191:
2158:
2145:
2126:
2120:
1880:
1867:
1854:
1841:
1835:
1746:
1743:
1737:
1700:
1664:
1654:
1651:
1638:
1605:
1602:
1583:
1510:{\displaystyle {\mathcal {M}}}
1413:
1407:
1298:
1287:
1262:
1256:
1173:is the same as above, but the
1050:Castelnuovo–Mumford regularity
645:
637:
582:
549:
529:
502:
475:
455:
447:
387:
376:
320:
309:
282:
255:
228:
208:
175:
124:
108:. Similarly every module has
1:
2792:Graduate Texts in Mathematics
2758:
2177:where right hand side is the
1981:{\displaystyle \otimes _{R}M}
1235:, i.e., given a homomorphism
2767:Elementary rings and modules
1188:, and all maps involved are
981:, which is generated over a
7:
2650:
1084:
965:Graded modules and algebras
708:resolving the given module
20:, and more specifically in
10:
2947:
2213:{\displaystyle F(E_{*}).}
1988:is a right exact functor
56:) that is used to define
2765:Iain T. Adamson (1972),
2682:
2667:Hilbert's syzygy theorem
1054:projective algebraic set
1013:minimal free resolutions
959:von Neumann regular ring
48:(or, more generally, of
2231:differentiable manifold
1143:not only over the ring
1060:is the minimal integer
156:(possibly infinite) of
2706:is more common, as in
2635:
2538:
2481:
2438:
2270:
2214:
2168:
2087:
1982:
1939: > 0 and
1890:
1785:
1784:{\displaystyle s>0}
1759:
1674:
1531:
1511:
1487:
1447:
1420:
1381:
1380:{\displaystyle I'_{*}}
1351:
1321:
871:homological dimensions
670:
607:
397:
330:
120:Resolutions of modules
90:projective resolutions
2720:projective resolution
2662:Hilbert–Burch theorem
2636:
2539:
2482:
2439:
2271:
2215:
2169:
2088:
2086:{\displaystyle E_{*}}
1983:
1946:For example, given a
1891:
1786:
1760:
1675:
1532:
1512:
1488:
1448:
1421:
1382:
1352:
1350:{\displaystyle I_{*}}
1322:
1211:. If every object of
951:weak global dimension
671:
608:
398:
331:
110:injective resolutions
2645:Godement resolutions
2552:
2499:
2451:
2287:
2240:
2185:
2104:
2070:
1962:
1829:
1769:
1687:
1544:
1521:
1497:
1457:
1437:
1399:
1361:
1334:
1250:
880:projective dimension
733:-modules. Likewise,
631:
441:
361:
167:
2926:Homological algebra
2657:Standard resolution
1805:acyclic resolutions
1719:
1482:
1376:
1313:
1135:) chain complex of
1077:are all lower than
916:injective dimension
914:. Analogously, the
821:etc. Dually, every
22:homological algebra
2879:Weibel, Charles A.
2631:
2534:
2477:
2434:
2278:differential forms
2266:
2210:
2164:
2083:
1978:
1886:
1809:left exact functor
1795:Acyclic resolution
1781:
1755:
1703:
1670:
1623:
1568:
1527:
1507:
1483:
1466:
1453:. For example, if
1443:
1416:
1377:
1364:
1347:
1317:
1301:
1221:enough projectives
1147:but also over the
1048:over a field, the
1009:graded linear maps
836:is unique up to a
765:resolutions whose
666:
603:
393:
339:The homomorphisms
326:
102:projective modules
2892:978-0-521-55987-4
2863:978-0-201-55540-0
2842:978-0-486-47187-7
2672:Free presentation
2544:. Therefore, the
2396:
2377:
2338:
1935:) vanish for all
1911:-acyclic, if the
1807:is used: given a
1608:
1547:
1530:{\displaystyle X}
1446:{\displaystyle X}
1405:
1283:
1280:
1225:enough injectives
1205:injective objects
1042:homogeneous ideal
941:called the right
771:injective modules
712:. For example, a
651:
595:
568:
542:
515:
488:
461:
382:
315:
295:
268:
241:
221:
194:
114:injective modules
63:finite resolution
2938:
2912:
2874:
2845:
2833:Basic algebra II
2829:Jacobson, Nathan
2824:
2779:
2752:
2746:
2740:
2717:
2711:
2704:right resolution
2693:
2640:
2638:
2637:
2632:
2615:
2614:
2609:
2608:
2598:
2597:
2592:
2580:
2566:
2565:
2560:
2546:sheaf cohomology
2543:
2541:
2540:
2535:
2524:
2523:
2514:
2513:
2486:
2484:
2483:
2478:
2467:
2466:
2461:
2460:
2443:
2441:
2440:
2435:
2418:
2417:
2406:
2405:
2398:
2397:
2395:
2390:
2385:
2379:
2378:
2376:
2371:
2366:
2354:
2353:
2348:
2347:
2340:
2339:
2337:
2332:
2327:
2315:
2314:
2309:
2308:
2275:
2273:
2272:
2267:
2256:
2255:
2250:
2249:
2219:
2217:
2216:
2211:
2203:
2202:
2173:
2171:
2170:
2165:
2157:
2156:
2141:
2140:
2116:
2115:
2092:
2090:
2089:
2084:
2082:
2081:
2016:( â‹… ,
1987:
1985:
1984:
1979:
1974:
1973:
1913:derived functors
1895:
1893:
1892:
1887:
1879:
1878:
1866:
1865:
1853:
1852:
1790:
1788:
1787:
1782:
1764:
1762:
1761:
1756:
1736:
1735:
1730:
1729:
1718:
1713:
1708:
1699:
1698:
1679:
1677:
1676:
1671:
1663:
1662:
1650:
1649:
1637:
1636:
1631:
1630:
1622:
1601:
1600:
1582:
1581:
1576:
1575:
1567:
1536:
1534:
1533:
1528:
1516:
1514:
1513:
1508:
1506:
1505:
1492:
1490:
1489:
1484:
1481:
1476:
1471:
1452:
1450:
1449:
1444:
1428:coherent sheaves
1425:
1423:
1422:
1417:
1406:
1403:
1386:
1384:
1383:
1378:
1372:
1356:
1354:
1353:
1348:
1346:
1345:
1326:
1324:
1323:
1318:
1309:
1297:
1281:
1278:
1274:
1273:
1219:is said to have
1168:abelian category
1114:aspherical space
1095:regular sequence
943:global dimension
675:
673:
672:
667:
662:
661:
652:
644:
612:
610:
609:
604:
596:
594:
593:
581:
579:
578:
569:
567:
566:
548:
543:
541:
540:
528:
526:
525:
516:
514:
513:
501:
499:
498:
489:
487:
486:
474:
472:
471:
462:
454:
412:right resolution
402:
400:
399:
394:
383:
375:
373:
372:
352:augmentation map
335:
333:
332:
327:
316:
308:
306:
305:
296:
294:
293:
281:
279:
278:
269:
267:
266:
254:
252:
251:
242:
240:
239:
227:
222:
220:
219:
207:
205:
204:
195:
193:
192:
174:
94:flat resolutions
86:free resolutions
54:abelian category
38:right resolution
2946:
2945:
2941:
2940:
2939:
2937:
2936:
2935:
2916:
2915:
2893:
2864:
2843:
2806:
2796:Springer-Verlag
2784:Eisenbud, David
2777:
2761:
2756:
2755:
2747:
2743:
2718:
2714:
2694:
2690:
2685:
2653:
2610:
2604:
2603:
2602:
2593:
2588:
2587:
2576:
2561:
2556:
2555:
2553:
2550:
2549:
2519:
2518:
2509:
2508:
2500:
2497:
2496:
2462:
2456:
2455:
2454:
2452:
2449:
2448:
2407:
2401:
2400:
2399:
2391:
2386:
2384:
2383:
2372:
2367:
2365:
2364:
2349:
2343:
2342:
2341:
2333:
2328:
2326:
2325:
2310:
2304:
2303:
2302:
2288:
2285:
2284:
2251:
2245:
2244:
2243:
2241:
2238:
2237:
2198:
2194:
2186:
2183:
2182:
2152:
2148:
2136:
2132:
2111:
2107:
2105:
2102:
2101:
2077:
2073:
2071:
2068:
2067:
2058:
2046:
2006:flat resolution
1969:
1965:
1963:
1960:
1959:
1934:
1922:
1874:
1870:
1861:
1857:
1848:
1844:
1830:
1827:
1826:
1797:
1770:
1767:
1766:
1731:
1725:
1724:
1723:
1714:
1709:
1704:
1694:
1690:
1688:
1685:
1684:
1658:
1657:
1645:
1641:
1632:
1626:
1625:
1624:
1612:
1590:
1586:
1577:
1571:
1570:
1569:
1551:
1545:
1542:
1541:
1522:
1519:
1518:
1501:
1500:
1498:
1495:
1494:
1477:
1472:
1467:
1458:
1455:
1454:
1438:
1435:
1434:
1402:
1400:
1397:
1396:
1393:
1368:
1362:
1359:
1358:
1341:
1337:
1335:
1332:
1331:
1305:
1290:
1269:
1265:
1251:
1248:
1247:
1184:are objects in
1178:
1160:
1118:universal cover
1087:
1072:
1046:polynomial ring
1032:
1023:
1006:
997:
967:
955:semisimple ring
908:Krull dimension
883:and denoted pd(
857:
846:
816:
806:to be the free
805:
791:to be the free
790:
749:
728:
714:free resolution
707:
697:
657:
653:
643:
632:
629:
628:
589:
585:
580:
574:
570:
556:
552:
547:
536:
532:
527:
521:
517:
509:
505:
500:
494:
490:
482:
478:
473:
467:
463:
453:
442:
439:
438:
374:
368:
364:
362:
359:
358:
344:
307:
301:
297:
289:
285:
280:
274:
270:
262:
258:
253:
247:
243:
235:
231:
226:
215:
211:
206:
200:
196:
182:
178:
173:
168:
165:
164:
142:left resolution
129:Given a module
127:
122:
30:left resolution
12:
11:
5:
2944:
2934:
2933:
2928:
2914:
2913:
2891:
2875:
2862:
2846:
2841:
2825:
2804:
2780:
2775:
2760:
2757:
2754:
2753:
2741:
2712:
2687:
2686:
2684:
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2679:
2674:
2669:
2664:
2659:
2652:
2649:
2630:
2627:
2624:
2621:
2618:
2613:
2607:
2601:
2596:
2591:
2586:
2583:
2579:
2575:
2572:
2569:
2564:
2559:
2533:
2530:
2527:
2522:
2517:
2512:
2507:
2504:
2493:global section
2476:
2473:
2470:
2465:
2459:
2445:
2444:
2433:
2430:
2427:
2424:
2421:
2416:
2413:
2410:
2404:
2394:
2389:
2382:
2375:
2370:
2363:
2360:
2357:
2352:
2346:
2336:
2331:
2324:
2321:
2318:
2313:
2307:
2301:
2298:
2295:
2292:
2265:
2262:
2259:
2254:
2248:
2224:constant sheaf
2209:
2206:
2201:
2197:
2193:
2190:
2175:
2174:
2163:
2160:
2155:
2151:
2147:
2144:
2139:
2135:
2131:
2128:
2125:
2122:
2119:
2114:
2110:
2080:
2076:
2054:
2042:
1977:
1972:
1968:
1956:tensor product
1930:
1918:
1897:
1896:
1885:
1882:
1877:
1873:
1869:
1864:
1860:
1856:
1851:
1847:
1843:
1840:
1837:
1834:
1796:
1793:
1780:
1777:
1774:
1754:
1751:
1748:
1745:
1742:
1739:
1734:
1728:
1722:
1717:
1712:
1707:
1702:
1697:
1693:
1681:
1680:
1669:
1666:
1661:
1656:
1653:
1648:
1644:
1640:
1635:
1629:
1621:
1618:
1615:
1611:
1607:
1604:
1599:
1596:
1593:
1589:
1585:
1580:
1574:
1566:
1563:
1560:
1557:
1554:
1550:
1526:
1504:
1480:
1475:
1470:
1465:
1462:
1442:
1415:
1412:
1409:
1392:
1389:
1375:
1371:
1367:
1344:
1340:
1328:
1327:
1316:
1312:
1308:
1304:
1300:
1296:
1293:
1289:
1286:
1277:
1272:
1268:
1264:
1261:
1258:
1255:
1176:
1159:
1156:
1103:graded algebra
1091:Koszul complex
1086:
1083:
1068:
1028:
1019:
1002:
993:
979:graded algebra
966:
963:
924:flat dimension
877:is called its
855:
844:
838:chain homotopy
814:
803:
788:
745:
724:
703:
696:
693:
677:
676:
665:
660:
656:
650:
647:
642:
639:
636:
614:
613:
602:
599:
592:
588:
584:
577:
573:
565:
562:
559:
555:
551:
546:
539:
535:
531:
524:
520:
512:
508:
504:
497:
493:
485:
481:
477:
470:
466:
460:
457:
452:
449:
446:
404:
403:
392:
389:
386:
381:
378:
371:
367:
342:
337:
336:
325:
322:
319:
314:
311:
304:
300:
292:
288:
284:
277:
273:
265:
261:
257:
250:
246:
238:
234:
230:
225:
218:
214:
210:
203:
199:
191:
188:
185:
181:
177:
172:
154:exact sequence
126:
123:
121:
118:
42:exact sequence
9:
6:
4:
3:
2:
2943:
2932:
2931:Module theory
2929:
2927:
2924:
2923:
2921:
2910:
2906:
2902:
2898:
2894:
2888:
2884:
2880:
2876:
2873:
2869:
2865:
2859:
2855:
2851:
2847:
2844:
2838:
2834:
2830:
2826:
2823:
2819:
2815:
2811:
2807:
2805:3-540-94268-8
2801:
2797:
2793:
2789:
2785:
2781:
2778:
2776:0-05-002192-3
2772:
2768:
2763:
2762:
2750:
2749:Jacobson 2009
2745:
2739:
2737:
2732:
2728:
2726:
2721:
2716:
2709:
2705:
2701:
2697:
2696:Jacobson 2009
2692:
2688:
2678:
2675:
2673:
2670:
2668:
2665:
2663:
2660:
2658:
2655:
2654:
2648:
2646:
2641:
2628:
2619:
2611:
2594:
2584:
2573:
2570:
2562:
2547:
2528:
2505:
2494:
2490:
2471:
2463:
2431:
2422:
2414:
2411:
2408:
2392:
2380:
2373:
2358:
2350:
2334:
2319:
2311:
2299:
2296:
2290:
2283:
2282:
2281:
2279:
2260:
2252:
2235:
2232:
2228:
2225:
2220:
2207:
2199:
2195:
2188:
2180:
2161:
2153:
2149:
2142:
2137:
2133:
2129:
2123:
2117:
2112:
2108:
2100:
2099:
2098:
2096:
2093:of an object
2078:
2074:
2065:
2061:
2057:
2053:
2049:
2045:
2041:
2036:
2034:
2029:
2027:
2023:
2019:
2015:
2011:
2007:
2003:
1999:
1995:
1991:
1975:
1970:
1966:
1957:
1953:
1949:
1944:
1942:
1938:
1933:
1929:
1925:
1921:
1917:
1914:
1910:
1906:
1902:
1899:of an object
1883:
1875:
1871:
1862:
1858:
1849:
1845:
1838:
1832:
1825:
1824:
1823:
1821:
1817:
1813:
1810:
1806:
1802:
1792:
1778:
1775:
1772:
1752:
1749:
1740:
1732:
1720:
1715:
1710:
1695:
1691:
1667:
1646:
1642:
1633:
1619:
1616:
1613:
1609:
1597:
1594:
1591:
1587:
1578:
1564:
1561:
1558:
1555:
1552:
1548:
1540:
1539:
1538:
1524:
1478:
1473:
1463:
1460:
1440:
1433:
1429:
1410:
1388:
1373:
1369:
1365:
1342:
1338:
1314:
1310:
1306:
1302:
1294:
1291:
1284:
1275:
1270:
1266:
1259:
1253:
1246:
1245:
1244:
1242:
1238:
1234:
1230:
1226:
1222:
1218:
1214:
1210:
1206:
1202:
1197:
1195:
1191:
1187:
1183:
1179:
1172:
1169:
1165:
1155:
1153:
1150:
1146:
1142:
1138:
1134:
1130:
1127:. Then every
1126:
1122:
1119:
1115:
1111:
1106:
1104:
1100:
1096:
1092:
1082:
1080:
1076:
1071:
1067:
1063:
1059:
1055:
1051:
1047:
1043:
1039:
1034:
1031:
1027:
1022:
1018:
1014:
1010:
1005:
1001:
996:
992:
988:
984:
980:
976:
975:graded module
972:
962:
960:
956:
952:
948:
944:
940:
936:
931:
929:
925:
921:
917:
913:
909:
905:
901:
897:
894:
890:
886:
882:
881:
876:
872:
867:
865:
861:
854:
850:
843:
839:
835:
830:
828:
824:
820:
813:
809:
802:
798:
794:
787:
783:
779:
774:
772:
768:
764:
760:
757:
753:
748:
744:
740:
736:
732:
727:
723:
719:
715:
711:
706:
702:
692:
690:
686:
682:
663:
658:
654:
648:
640:
634:
627:
626:
625:
623:
619:
600:
597:
590:
586:
575:
571:
563:
560:
557:
553:
544:
537:
533:
522:
518:
510:
506:
495:
491:
483:
479:
468:
464:
458:
450:
444:
437:
436:
435:
433:
429:
425:
421:
417:
413:
410:is that of a
409:
390:
384:
379:
369:
365:
357:
356:
355:
353:
350:is called an
349:
345:
323:
317:
312:
302:
298:
290:
286:
275:
271:
263:
259:
248:
244:
236:
232:
223:
216:
212:
201:
197:
189:
186:
183:
179:
170:
163:
162:
161:
159:
155:
151:
147:
143:
139:
136:
132:
117:
115:
111:
107:
103:
99:
95:
91:
87:
83:
79:
74:
72:
68:
64:
59:
55:
51:
47:
43:
39:
35:
31:
27:
23:
19:
2882:
2853:
2832:
2787:
2766:
2744:
2735:
2724:
2715:
2703:
2700:coresolution
2699:
2698:, §6.5 uses
2691:
2642:
2489:fine sheaves
2447:The sheaves
2446:
2233:
2226:
2221:
2178:
2176:
2094:
2063:
2059:
2055:
2051:
2047:
2043:
2039:
2037:
2032:
2030:
2025:
2021:
2017:
2013:
2009:
2005:
2001:
1997:
1993:
1989:
1958:
1951:
1947:
1945:
1940:
1936:
1931:
1927:
1923:
1919:
1915:
1908:
1904:
1900:
1898:
1819:
1815:
1811:
1804:
1798:
1682:
1394:
1329:
1240:
1236:
1228:
1216:
1212:
1208:
1198:
1193:
1185:
1181:
1174:
1170:
1163:
1161:
1151:
1144:
1140:
1136:
1125:contractible
1120:
1116:, i.e., its
1109:
1107:
1088:
1078:
1074:
1069:
1065:
1061:
1057:
1037:
1035:
1029:
1025:
1020:
1016:
1012:
1003:
999:
994:
990:
986:
970:
968:
946:
938:
934:
932:
927:
919:
911:
899:
895:
888:
884:
878:
874:
868:
863:
859:
852:
848:
841:
833:
831:
827:Tor functors
822:
818:
811:
807:
800:
796:
792:
785:
777:
775:
766:
762:
758:
746:
742:
738:
734:
730:
725:
721:
717:
716:of a module
713:
709:
704:
700:
698:
688:
684:
680:
678:
621:
617:
615:
431:
427:
426:over a ring
423:
419:
418:, or simply
416:coresolution
415:
411:
405:
351:
347:
340:
338:
157:
149:
145:
141:
137:
130:
128:
109:
106:flat modules
98:free modules
93:
89:
85:
82:P resolution
81:
77:
75:
62:
37:
34:coresolution
33:
29:
25:
15:
2850:Lang, Serge
2708:Weibel 1994
1056:defined by
799:, and then
616:where each
408:dual notion
144:(or simply
125:Definitions
71:zero-object
32:; dually a
18:mathematics
2920:Categories
2872:0848.13001
2822:0819.13001
2759:References
2731:resolution
2643:Similarly
2276:of smooth
2097:, we have
1996:) →
1907:is called
1233:functorial
1201:projective
1149:group ring
1133:simplicial
1099:local ring
1007:and ε are
893:local ring
782:A fortiori
752:projective
735:projective
420:resolution
146:resolution
58:invariants
26:resolution
2831:(2009) ,
2710:, Chap. 2
2702:, though
2612:∗
2516:↦
2503:Γ
2464:∗
2429:→
2412:
2388:→
2381:⋯
2369:→
2330:→
2300:⊂
2294:→
2253:∗
2200:∗
2154:∗
2079:∗
1967:⊗
1884:⋯
1881:→
1868:→
1855:→
1842:→
1836:→
1750:≠
1665:→
1655:→
1610:⨁
1606:→
1549:⨁
1370:∗
1343:∗
1307:∗
1299:→
1288:→
1271:∗
1263:→
1257:→
1190:morphisms
729:are free
659:∙
649:ε
646:⟶
638:⟶
598:⋯
583:⟶
561:−
550:⟶
545:⋯
530:⟶
503:⟶
476:⟶
459:ε
456:⟶
448:⟶
434:-modules
388:⟶
380:ε
377:⟶
370:∙
321:⟶
313:ε
310:⟶
283:⟶
256:⟶
229:⟶
224:⋯
209:⟶
176:⟶
171:⋯
160:-modules
2909:36131259
2881:(1994).
2852:(1993),
2786:(1995),
2651:See also
2495:functor
1950:-module
1818:→
1374:′
1311:′
1295:′
1239:→
1129:singular
1085:Examples
817:→
769:are all
67:non-zero
40:) is an
2901:1269324
2854:Algebra
2814:1322960
2733:at the
2722:at the
1801:functor
1223:(resp.
1052:of the
977:over a
904:regular
133:over a
50:objects
46:modules
2907:
2899:
2889:
2870:
2860:
2839:
2820:
2812:
2802:
2773:
2751:, §6.5
1954:, the
1432:scheme
1282:
1279:
1166:in an
1112:be an
922:) and
776:Every
685:length
681:finite
620:is an
152:is an
52:of an
2683:Notes
2229:on a
1430:on a
1097:in a
1093:of a
1044:in a
1040:is a
983:field
973:be a
763:right
148:) of
2905:OCLC
2887:ISBN
2858:ISBN
2837:ISBN
2800:ISBN
2771:ISBN
2487:are
1776:>
1765:for
1357:and
1203:and
1180:and
1131:(or
1108:Let
969:Let
851:and
756:flat
754:and
750:are
739:flat
737:and
414:(or
406:The
140:, a
135:ring
92:and
28:(or
24:, a
2868:Zbl
2818:Zbl
2738:Lab
2727:Lab
2409:dim
2022:Hom
2014:Hom
1998:Mod
1990:Mod
1903:of
1517:on
1426:of
1404:Coh
1241:M'
1192:in
1123:is
1079:r-i
1036:If
945:of
926:fd(
918:id(
910:of
902:is
862:of
104:or
88:,
44:of
36:or
16:In
2922::
2903:.
2897:MR
2895:.
2866:,
2816:,
2810:MR
2808:,
2798:,
2790:,
2729:,
2432:0.
2280::
1814::
1668:0.
1387:.
1196:.
1154:.
1081:.
961:.
858:→
847:→
829:.
773:.
391:0.
324:0.
116:.
100:,
73:.
2911:.
2736:n
2725:n
2629:.
2626:)
2623:)
2620:M
2617:(
2606:C
2600:(
2595:i
2590:H
2585:=
2582:)
2578:R
2574:,
2571:M
2568:(
2563:i
2558:H
2532:)
2529:M
2526:(
2521:F
2511:F
2506::
2475:)
2472:M
2469:(
2458:C
2426:)
2423:M
2420:(
2415:M
2403:C
2393:d
2374:d
2362:)
2359:M
2356:(
2351:1
2345:C
2335:d
2323:)
2320:M
2317:(
2312:0
2306:C
2297:R
2291:0
2264:)
2261:M
2258:(
2247:C
2234:M
2227:R
2208:.
2205:)
2196:E
2192:(
2189:F
2179:i
2162:,
2159:)
2150:E
2146:(
2143:F
2138:i
2134:H
2130:=
2127:)
2124:M
2121:(
2118:F
2113:i
2109:R
2095:M
2075:E
2064:F
2060:F
2056:i
2052:L
2048:F
2044:i
2040:R
2033:F
2026:M
2024:(
2018:M
2010:M
2002:R
2000:(
1994:R
1992:(
1976:M
1971:R
1952:M
1948:R
1941:n
1937:i
1932:n
1928:E
1926:(
1924:F
1920:i
1916:R
1909:F
1905:A
1901:M
1876:2
1872:E
1863:1
1859:E
1850:0
1846:E
1839:M
1833:0
1820:B
1816:A
1812:F
1779:0
1773:s
1753:0
1747:)
1744:)
1741:s
1738:(
1733:X
1727:O
1721:,
1716:n
1711:S
1706:P
1701:(
1696:n
1692:H
1660:M
1652:)
1647:i
1643:s
1639:(
1634:X
1628:O
1620:0
1617:=
1614:i
1603:)
1598:j
1595:,
1592:i
1588:s
1584:(
1579:X
1573:O
1565:0
1562:=
1559:j
1556:,
1553:i
1525:X
1503:M
1479:n
1474:S
1469:P
1464:=
1461:X
1441:X
1414:)
1411:X
1408:(
1366:I
1339:I
1315:,
1303:I
1292:M
1285:0
1276:,
1267:I
1260:M
1254:0
1237:M
1229:R
1217:A
1213:A
1209:A
1194:A
1186:A
1182:C
1177:i
1175:E
1171:A
1164:M
1152:Z
1145:Z
1141:Z
1137:E
1121:E
1110:X
1075:I
1070:i
1066:E
1062:r
1058:I
1038:I
1030:i
1026:E
1021:i
1017:E
1004:i
1000:d
995:i
991:E
987:M
971:M
947:R
939:R
935:R
928:M
920:M
912:R
900:R
896:R
889:M
885:M
875:M
864:M
860:M
856:1
853:P
849:M
845:0
842:P
834:M
823:R
819:M
815:0
812:E
808:R
804:1
801:E
797:M
793:R
789:0
786:E
778:R
767:C
759:R
747:i
743:E
731:R
726:i
722:E
718:M
710:M
705:i
701:E
689:n
664:.
655:C
641:M
635:0
622:R
618:C
601:,
591:n
587:d
576:n
572:C
564:1
558:n
554:d
538:2
534:d
523:2
519:C
511:1
507:d
496:1
492:C
484:0
480:d
469:0
465:C
451:M
445:0
432:R
428:R
424:M
385:M
366:E
348:ε
343:i
341:d
318:M
303:0
299:E
291:1
287:d
276:1
272:E
264:2
260:d
249:2
245:E
237:3
233:d
217:n
213:d
202:n
198:E
190:1
187:+
184:n
180:d
158:R
150:M
138:R
131:M
78:P
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