474:
1825:
have similar looking singularities at the origin when viewing their graphs (both look like a plus sign). Notice that in the second case, any
Zariski neighborhood of the origin is still an irreducible curve. If we use completions, then we are looking at a "small enough" neighborhood where the node has
234:
2293:
Since both rings are given by the intersection of two ideals generated by a homogeneous degree 1 polynomial, we can see algebraically that the singularities "look" the same. This is because such a scheme is the union of two non-equal linear subspaces of the affine plane.
2288:
1584:
752:
219:
2779:
469:{\displaystyle {\widehat {E}}=\varprojlim (E/F^{n}E)=\left\{\left.({\overline {a_{n}}})_{n\geq 0}\in \prod _{n\geq 0}(E/F^{n}E)\;\right|\;a_{i}\equiv a_{j}{\pmod {F^{i}E}}{\text{ for all }}i\leq j\right\}.\,}
2477:
2381:
2655:
1069:
862:
1273:
1669:
969:
1823:
1311:
2523:
2000:
1413:
2116:
1916:
1116:
1734:
631:
1160:
2822:
2555:
2108:
2914:
1182:
2062:
507:, then its completion is again an object with the same structure that is complete in the topology determined by the filtration. This construction may be applied both to
543:
1856:
2703:
2020:
1453:
1433:
1352:
1464:
663:
2324:
149:
2710:
2420:
2600:
992:
791:
3064:
1216:
17:
1593:
920:
3097:
3089:
1742:
1278:
2484:
1921:
2306:
The completion of a
Noetherian local ring with respect to the unique maximal ideal is a Noetherian local ring.
1357:
3081:
28:
2283:{\displaystyle u=x{\sqrt {1+x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{(1-2n)(n!)^{2}(4^{n})}}x^{n+1}.}
3157:
3152:
638:
1861:
1085:
2661:
Together with the previous property, this implies that the functor of completion on finitely generated
1685:
880:
606:
1136:
101:
centered at the point are convergent. An algebraic completion is constructed in a manner analogous to
2803:
2528:
2067:
1473:
2673:. In particular, taking quotients of rings commutes with completion, meaning that for any quotient
141:
121:
2844:
1165:
3114:
2788:
2025:
63:
879:
is injective if and only if this intersection reduces to the zero element of the ring; by the
546:
2948:
102:
518:
3103:
2670:
1829:
1679:
903:
500:
51:
8:
2680:
1318:
575:
564:
512:
3119:
2963:
2005:
1675:
1438:
1418:
1337:
1079:
75:
59:
47:
1579:{\displaystyle {\begin{cases}R]\to {\widehat {R}}_{I}\\x_{i}\mapsto f_{i}\end{cases}}}
224:
of subgroups. One then defines the completion (with respect to the filtration) as the
3093:
3085:
3060:
2943:
118:
71:
2303:
The completion of a
Noetherian ring with respect to some ideal is a Noetherian ring.
1455:
is an image of a formal power series ring, specifically, the image of the surjection
3133:
3128:
3052:
568:
508:
496:
67:
55:
35:
3100:
2958:
2562:
1202:
1128:
888:
884:
747:{\displaystyle F^{0}R=R\supset I\supset I^{2}\supset \cdots ,\quad F^{n}R=I^{n}.}
110:
491:
has additional algebraic structure compatible with the filtration, for instance
3073:
3048:
634:
582:
3146:
2938:
2825:
2666:
778:
598:
590:
558:
225:
137:
98:
70:. Complete commutative rings have a simpler structure than general ones, and
2983:
2953:
504:
106:
633:
is especially important, for example the distinguished maximal ideal of a
3110:
2569:
1737:
782:
1826:
two components. Taking the localizations of these rings along the ideal
2797:
892:
1275:
be the maximal ideal generated by the variables. Then the completion
871:
from the ring to its completion is the intersection of the powers of
214:{\displaystyle E=F^{0}E\supset F^{1}E\supset F^{2}E\supset \cdots \,}
2774:{\displaystyle {\widehat {R/I}}\cong {\widehat {R}}/{\widehat {I}}.}
3115:"On Hausdorff completions of commutative rings in rigid geometry"
43:
2472:{\displaystyle {\widehat {f}}:{\widehat {M}}\to {\widehat {N}},}
2376:{\displaystyle {\widehat {f}}:{\widehat {R}}\to {\widehat {S}}.}
66:, and together they are among the most basic tools in analysing
2321:
of topological rings gives rise to a map of their completions,
1674:
Completions can also be used to analyze the local structure of
2650:{\displaystyle {\widehat {M}}=M\otimes _{R}{\widehat {R}}.}
2309:
The completion is a functorial operation: a continuous map
1572:
302:
3078:
Commutative algebra. With a view toward algebraic geometry
1064:{\displaystyle {\widehat {M}}_{I}=\varprojlim (M/I^{n}M).}
910:-adic topology. A basis of open neighborhoods of a module
3084:, 150. Springer-Verlag, New York, 1995. xvi+785 pp.
857:{\displaystyle {\widehat {R}}_{I}=\varprojlim (R/I^{n})}
867:
pronounced "R I hat". The kernel of the canonical map
94:: heuristically, this is a neighborhood so small that
2847:
2806:
2713:
2683:
2603:
2531:
2487:
2423:
2327:
2119:
2070:
2028:
2008:
1924:
1864:
1832:
1745:
1688:
1596:
1467:
1441:
1421:
1360:
1340:
1281:
1268:{\displaystyle {\mathfrak {m}}=(x_{1},\ldots ,x_{n})}
1219:
1168:
1139:
1088:
995:
923:
794:
666:
609:
521:
237:
152:
515:. As may be expected, when the intersection of the
3108:
2908:
2816:
2773:
2697:
2649:
2549:
2517:
2471:
2375:
2282:
2102:
2056:
2014:
1994:
1910:
1850:
1817:
1728:
1664:{\displaystyle (x_{1}-f_{1},\ldots ,x_{n}-f_{n}).}
1663:
1578:
1447:
1427:
1407:
1346:
1305:
1267:
1176:
1154:
1110:
1063:
964:{\displaystyle x+I^{n}M\quad {\text{for }}x\in M.}
963:
856:
746:
625:
537:
468:
213:
3144:
1682:. For example, the affine schemes associated to
3047:
3030:
3018:
3006:
2800:Noetherian commutative ring with maximal ideal
2415:uniquely extends to the map of the completions:
2395:are two modules over the same topological ring
1818:{\displaystyle \mathbb {C} /(y^{2}-x^{2}(1+x))}
2579:The completion of a finitely generated module
1306:{\displaystyle {\widehat {R}}_{\mathfrak {m}}}
2518:{\displaystyle {\widehat {M}},{\widehat {N}}}
1995:{\displaystyle \mathbb {C} ]/((y+u)(y-u))}
393:
387:
3132:
2072:
1926:
1866:
1747:
1690:
1170:
1142:
465:
210:
1408:{\displaystyle I=(f_{1},\ldots ,f_{n}),}
1074:This procedure converts any module over
479:This is again an abelian group. Usually
3021:, Proposition 10.16. and Theorem 10.26.
883:, this is the case for any commutative
127:
14:
3145:
986:is the inverse limit of the quotients
113:, and agrees with it in the case when
78:, a completion of a ring of functions
653:and form a descending filtration on
3057:Introduction to Commutative Algebra
2809:
2110:More explicitly, the power series:
1297:
1222:
1162:is obtained by completing the ring
618:
425:
24:
2161:
1911:{\displaystyle \mathbb {C} ]/(xy)}
1111:{\displaystyle {\widehat {R}}_{I}}
25:
3169:
1729:{\displaystyle \mathbb {C} /(xy)}
914:is given by the sets of the form
626:{\displaystyle I={\mathfrak {m}}}
552:
2411:is a continuous module map then
1155:{\displaystyle \mathbb {Z} _{p}}
2817:{\displaystyle {\mathfrak {m}}}
2792:(equicharacteristic case). Let
2550:{\displaystyle {\widehat {R}}.}
943:
898:There is a related topology on
714:
418:
3134:10.1016/j.jalgebra.2011.02.001
3024:
3012:
3000:
2976:
2895:
2892:
2860:
2857:
2451:
2355:
2255:
2242:
2233:
2223:
2220:
2205:
2197:
2188:
2179:
2169:
2103:{\displaystyle \mathbb {C} ].}
2094:
2091:
2079:
2076:
2051:
2039:
1989:
1986:
1974:
1971:
1959:
1956:
1948:
1945:
1933:
1930:
1905:
1896:
1888:
1885:
1873:
1870:
1845:
1833:
1812:
1809:
1797:
1771:
1763:
1751:
1723:
1714:
1706:
1694:
1655:
1597:
1556:
1520:
1517:
1514:
1482:
1479:
1399:
1367:
1262:
1230:
1055:
1031:
851:
830:
439:
419:
384:
360:
326:
305:
290:
266:
13:
1:
3082:Graduate Texts in Mathematics
3040:
2297:
2022:is the formal square root of
545:equals zero, this produces a
2969:
2909:{\displaystyle R\simeq K]/I}
1177:{\displaystyle \mathbb {Z} }
639:basis of open neighbourhoods
581:determines the Krull (after
320:
62:. Completion is similar to
7:
3031:Atiyah & Macdonald 1969
3019:Atiyah & Macdonald 1969
3007:Atiyah & Macdonald 1969
2984:"Stacks Project — Tag 0316"
2932:
1121:
757:(Open neighborhoods of any
10:
3174:
2705:, there is an isomorphism
2057:{\displaystyle x^{2}(1+x)}
1184:of integers at the ideal (
881:Krull intersection theorem
574:by the powers of a proper
556:
42:is any of several related
26:
3113:; Kato, Fumiharu (2011).
777:-adic) completion is the
547:complete topological ring
2988:stacks.math.columbia.edu
2928:(Eisenbud, Theorem 7.7).
1590:The kernel is the ideal
1334:Given a noetherian ring
117:has a metric given by a
54:that result in complete
3049:Atiyah, Michael Francis
2789:Cohen structure theorem
2583:over a Noetherian ring
1209:variables over a field
978:-adic completion of an
906:, also called Krull or
645:is given by the powers
2910:
2835:contains a field, then
2818:
2775:
2699:
2651:
2551:
2519:
2473:
2377:
2284:
2165:
2104:
2058:
2016:
1996:
1912:
1852:
1819:
1730:
1665:
1580:
1449:
1429:
1409:
1348:
1307:
1269:
1178:
1156:
1112:
1065:
965:
858:
748:
627:
567:, the filtration on a
539:
538:{\displaystyle F^{i}E}
470:
215:
2949:Locally compact field
2911:
2819:
2776:
2700:
2671:short exact sequences
2652:
2552:
2520:
2474:
2378:
2285:
2145:
2105:
2059:
2017:
1997:
1913:
1858:and completing gives
1853:
1851:{\displaystyle (x,y)}
1820:
1731:
1666:
1581:
1450:
1430:
1410:
1349:
1308:
1270:
1179:
1157:
1113:
1066:
966:
859:
749:
628:
540:
471:
216:
3109:Fujiwara, Kazuhiro;
3033:, Proposition 10.14.
2845:
2804:
2711:
2681:
2601:
2589:extension of scalars
2561:The completion of a
2529:
2485:
2421:
2325:
2117:
2068:
2026:
2006:
2002:respectively, where
1922:
1862:
1830:
1743:
1736:and the nodal cubic
1686:
1594:
1465:
1439:
1435:-adic completion of
1419:
1358:
1338:
1279:
1217:
1166:
1137:
1086:
993:
921:
792:
765:are given by cosets
664:
607:
519:
513:noncommutative rings
235:
150:
128:General construction
74:applies to them. In
27:For other uses, see
18:Completion (algebra)
3158:Topological algebra
3153:Commutative algebra
2698:{\displaystyle R/I}
2587:can be obtained by
1319:formal power series
565:commutative algebra
445: for all
88:formal neighborhood
3120:Journal of Algebra
3059:. Westview Press.
2964:Quasi-unmixed ring
2906:
2814:
2771:
2695:
2647:
2547:
2515:
2469:
2373:
2280:
2100:
2054:
2012:
1992:
1908:
1848:
1815:
1726:
1661:
1576:
1571:
1445:
1425:
1405:
1344:
1303:
1265:
1174:
1152:
1108:
1080:topological module
1061:
1026:
961:
854:
825:
744:
623:
535:
487:abelian group. If
466:
359:
261:
211:
140:with a descending
86:concentrates on a
76:algebraic geometry
3066:978-0-201-40751-8
2944:Profinite integer
2765:
2748:
2733:
2641:
2613:
2541:
2525:are modules over
2512:
2497:
2463:
2448:
2433:
2367:
2352:
2337:
2259:
2140:
2015:{\displaystyle u}
1533:
1448:{\displaystyle R}
1428:{\displaystyle I}
1347:{\displaystyle R}
1292:
1099:
1019:
1006:
947:
818:
805:
446:
344:
323:
254:
247:
68:commutative rings
56:topological rings
16:(Redirected from
3165:
3138:
3136:
3070:
3034:
3028:
3022:
3016:
3010:
3009:, Theorem 10.26.
3004:
2998:
2997:
2995:
2994:
2980:
2915:
2913:
2912:
2907:
2902:
2891:
2890:
2872:
2871:
2823:
2821:
2820:
2815:
2813:
2812:
2780:
2778:
2777:
2772:
2767:
2766:
2758:
2755:
2750:
2749:
2741:
2735:
2734:
2729:
2725:
2716:
2704:
2702:
2701:
2696:
2691:
2656:
2654:
2653:
2648:
2643:
2642:
2634:
2631:
2630:
2615:
2614:
2606:
2556:
2554:
2553:
2548:
2543:
2542:
2534:
2524:
2522:
2521:
2516:
2514:
2513:
2505:
2499:
2498:
2490:
2478:
2476:
2475:
2470:
2465:
2464:
2456:
2450:
2449:
2441:
2435:
2434:
2426:
2382:
2380:
2379:
2374:
2369:
2368:
2360:
2354:
2353:
2345:
2339:
2338:
2330:
2289:
2287:
2286:
2281:
2276:
2275:
2260:
2258:
2254:
2253:
2241:
2240:
2203:
2187:
2186:
2167:
2164:
2159:
2141:
2130:
2109:
2107:
2106:
2101:
2075:
2063:
2061:
2060:
2055:
2038:
2037:
2021:
2019:
2018:
2013:
2001:
1999:
1998:
1993:
1955:
1929:
1917:
1915:
1914:
1909:
1895:
1869:
1857:
1855:
1854:
1849:
1824:
1822:
1821:
1816:
1796:
1795:
1783:
1782:
1770:
1750:
1735:
1733:
1732:
1727:
1713:
1693:
1670:
1668:
1667:
1662:
1654:
1653:
1641:
1640:
1622:
1621:
1609:
1608:
1585:
1583:
1582:
1577:
1575:
1574:
1568:
1567:
1555:
1554:
1541:
1540:
1535:
1534:
1526:
1513:
1512:
1494:
1493:
1454:
1452:
1451:
1446:
1434:
1432:
1431:
1426:
1414:
1412:
1411:
1406:
1398:
1397:
1379:
1378:
1353:
1351:
1350:
1345:
1312:
1310:
1309:
1304:
1302:
1301:
1300:
1294:
1293:
1285:
1274:
1272:
1271:
1266:
1261:
1260:
1242:
1241:
1226:
1225:
1183:
1181:
1180:
1175:
1173:
1161:
1159:
1158:
1153:
1151:
1150:
1145:
1117:
1115:
1114:
1109:
1107:
1106:
1101:
1100:
1092:
1078:into a complete
1070:
1068:
1067:
1062:
1051:
1050:
1041:
1027:
1014:
1013:
1008:
1007:
999:
970:
968:
967:
962:
948:
945:
939:
938:
878:
870:
863:
861:
860:
855:
850:
849:
840:
826:
813:
812:
807:
806:
798:
753:
751:
750:
745:
740:
739:
724:
723:
704:
703:
676:
675:
632:
630:
629:
624:
622:
621:
597:. The case of a
569:commutative ring
544:
542:
541:
536:
531:
530:
503:, or a filtered
475:
473:
472:
467:
461:
457:
447:
444:
442:
435:
434:
416:
415:
403:
402:
392:
388:
380:
379:
370:
358:
340:
339:
324:
319:
318:
309:
286:
285:
276:
262:
249:
248:
240:
220:
218:
217:
212:
200:
199:
184:
183:
168:
167:
111:Cauchy sequences
36:abstract algebra
21:
3173:
3172:
3168:
3167:
3166:
3164:
3163:
3162:
3143:
3142:
3141:
3067:
3053:Macdonald, I.G.
3043:
3038:
3037:
3029:
3025:
3017:
3013:
3005:
3001:
2992:
2990:
2982:
2981:
2977:
2972:
2959:Linear topology
2935:
2924:and some ideal
2898:
2886:
2882:
2867:
2863:
2846:
2843:
2842:
2808:
2807:
2805:
2802:
2801:
2757:
2756:
2751:
2740:
2739:
2721:
2717:
2715:
2714:
2712:
2709:
2708:
2687:
2682:
2679:
2678:
2669:: it preserves
2633:
2632:
2626:
2622:
2605:
2604:
2602:
2599:
2598:
2563:Noetherian ring
2533:
2532:
2530:
2527:
2526:
2504:
2503:
2489:
2488:
2486:
2483:
2482:
2455:
2454:
2440:
2439:
2425:
2424:
2422:
2419:
2418:
2359:
2358:
2344:
2343:
2329:
2328:
2326:
2323:
2322:
2300:
2265:
2261:
2249:
2245:
2236:
2232:
2204:
2182:
2178:
2168:
2166:
2160:
2149:
2129:
2118:
2115:
2114:
2071:
2069:
2066:
2065:
2033:
2029:
2027:
2024:
2023:
2007:
2004:
2003:
1951:
1925:
1923:
1920:
1919:
1891:
1865:
1863:
1860:
1859:
1831:
1828:
1827:
1791:
1787:
1778:
1774:
1766:
1746:
1744:
1741:
1740:
1709:
1689:
1687:
1684:
1683:
1649:
1645:
1636:
1632:
1617:
1613:
1604:
1600:
1595:
1592:
1591:
1570:
1569:
1563:
1559:
1550:
1546:
1543:
1542:
1536:
1525:
1524:
1523:
1508:
1504:
1489:
1485:
1469:
1468:
1466:
1463:
1462:
1440:
1437:
1436:
1420:
1417:
1416:
1393:
1389:
1374:
1370:
1359:
1356:
1355:
1339:
1336:
1335:
1325:variables over
1296:
1295:
1284:
1283:
1282:
1280:
1277:
1276:
1256:
1252:
1237:
1233:
1221:
1220:
1218:
1215:
1214:
1203:polynomial ring
1169:
1167:
1164:
1163:
1146:
1141:
1140:
1138:
1135:
1134:
1124:
1102:
1091:
1090:
1089:
1087:
1084:
1083:
1046:
1042:
1037:
1018:
1009:
998:
997:
996:
994:
991:
990:
944:
934:
930:
922:
919:
918:
889:integral domain
885:Noetherian ring
876:
868:
845:
841:
836:
817:
808:
797:
796:
795:
793:
790:
789:
735:
731:
719:
715:
699:
695:
671:
667:
665:
662:
661:
617:
616:
608:
605:
604:
561:
555:
526:
522:
520:
517:
516:
443:
430:
426:
417:
411:
407:
398:
394:
375:
371:
366:
348:
329:
325:
314:
310:
308:
304:
301:
300:
296:
281:
277:
272:
253:
239:
238:
236:
233:
232:
195:
191:
179:
175:
163:
159:
151:
148:
147:
130:
119:non-Archimedean
32:
23:
22:
15:
12:
11:
5:
3171:
3161:
3160:
3155:
3140:
3139:
3127:(1): 293–321.
3106:
3074:David Eisenbud
3071:
3065:
3044:
3042:
3039:
3036:
3035:
3023:
3011:
2999:
2974:
2973:
2971:
2968:
2967:
2966:
2961:
2956:
2951:
2946:
2941:
2934:
2931:
2930:
2929:
2918:
2917:
2916:
2905:
2901:
2897:
2894:
2889:
2885:
2881:
2878:
2875:
2870:
2866:
2862:
2859:
2856:
2853:
2850:
2837:
2836:
2811:
2796:be a complete
2784:
2783:
2782:
2781:
2770:
2764:
2761:
2754:
2747:
2744:
2738:
2732:
2728:
2724:
2720:
2694:
2690:
2686:
2659:
2658:
2657:
2646:
2640:
2637:
2629:
2625:
2621:
2618:
2612:
2609:
2593:
2592:
2577:
2558:
2557:
2546:
2540:
2537:
2511:
2508:
2502:
2496:
2493:
2479:
2468:
2462:
2459:
2453:
2447:
2444:
2438:
2432:
2429:
2416:
2384:
2383:
2372:
2366:
2363:
2357:
2351:
2348:
2342:
2336:
2333:
2307:
2304:
2299:
2296:
2291:
2290:
2279:
2274:
2271:
2268:
2264:
2257:
2252:
2248:
2244:
2239:
2235:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2202:
2199:
2196:
2193:
2190:
2185:
2181:
2177:
2174:
2171:
2163:
2158:
2155:
2152:
2148:
2144:
2139:
2136:
2133:
2128:
2125:
2122:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2074:
2053:
2050:
2047:
2044:
2041:
2036:
2032:
2011:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1954:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1928:
1907:
1904:
1901:
1898:
1894:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1868:
1847:
1844:
1841:
1838:
1835:
1814:
1811:
1808:
1805:
1802:
1799:
1794:
1790:
1786:
1781:
1777:
1773:
1769:
1765:
1762:
1759:
1756:
1753:
1749:
1725:
1722:
1719:
1716:
1712:
1708:
1705:
1702:
1699:
1696:
1692:
1672:
1671:
1660:
1657:
1652:
1648:
1644:
1639:
1635:
1631:
1628:
1625:
1620:
1616:
1612:
1607:
1603:
1599:
1588:
1587:
1586:
1573:
1566:
1562:
1558:
1553:
1549:
1545:
1544:
1539:
1532:
1529:
1522:
1519:
1516:
1511:
1507:
1503:
1500:
1497:
1492:
1488:
1484:
1481:
1478:
1475:
1474:
1472:
1457:
1456:
1444:
1424:
1404:
1401:
1396:
1392:
1388:
1385:
1382:
1377:
1373:
1369:
1366:
1363:
1343:
1331:
1330:
1299:
1291:
1288:
1264:
1259:
1255:
1251:
1248:
1245:
1240:
1236:
1232:
1229:
1224:
1190:
1189:
1172:
1149:
1144:
1132:-adic integers
1123:
1120:
1105:
1098:
1095:
1072:
1071:
1060:
1057:
1054:
1049:
1045:
1040:
1036:
1033:
1030:
1025:
1022:
1017:
1012:
1005:
1002:
972:
971:
960:
957:
954:
951:
942:
937:
933:
929:
926:
865:
864:
853:
848:
844:
839:
835:
832:
829:
824:
821:
816:
811:
804:
801:
755:
754:
743:
738:
734:
730:
727:
722:
718:
713:
710:
707:
702:
698:
694:
691:
688:
685:
682:
679:
674:
670:
635:valuation ring
620:
615:
612:
583:Wolfgang Krull
557:Main article:
554:
553:Krull topology
551:
534:
529:
525:
477:
476:
464:
460:
456:
453:
450:
441:
438:
433:
429:
424:
421:
414:
410:
406:
401:
397:
391:
386:
383:
378:
374:
369:
365:
362:
357:
354:
351:
347:
343:
338:
335:
332:
328:
322:
317:
313:
307:
303:
299:
295:
292:
289:
284:
280:
275:
271:
268:
265:
260:
257:
252:
246:
243:
222:
221:
209:
206:
203:
198:
194:
190:
187:
182:
178:
174:
171:
166:
162:
158:
155:
129:
126:
122:absolute value
90:of a point of
72:Hensel's lemma
9:
6:
4:
3:
2:
3170:
3159:
3156:
3154:
3151:
3150:
3148:
3135:
3130:
3126:
3122:
3121:
3116:
3112:
3107:
3105:
3102:
3099:
3098:0-387-94269-6
3095:
3091:
3090:0-387-94268-8
3087:
3083:
3079:
3075:
3072:
3068:
3062:
3058:
3054:
3050:
3046:
3045:
3032:
3027:
3020:
3015:
3008:
3003:
2989:
2985:
2979:
2975:
2965:
2962:
2960:
2957:
2955:
2952:
2950:
2947:
2945:
2942:
2940:
2939:Formal scheme
2937:
2936:
2927:
2923:
2919:
2903:
2899:
2887:
2883:
2879:
2876:
2873:
2868:
2864:
2854:
2851:
2848:
2841:
2840:
2839:
2838:
2834:
2830:
2827:
2826:residue field
2799:
2795:
2791:
2790:
2786:
2785:
2768:
2762:
2759:
2752:
2745:
2742:
2736:
2730:
2726:
2722:
2718:
2707:
2706:
2692:
2688:
2684:
2676:
2672:
2668:
2664:
2660:
2644:
2638:
2635:
2627:
2623:
2619:
2616:
2610:
2607:
2597:
2596:
2595:
2594:
2590:
2586:
2582:
2578:
2575:
2571:
2567:
2564:
2560:
2559:
2544:
2538:
2535:
2509:
2506:
2500:
2494:
2491:
2480:
2466:
2460:
2457:
2445:
2442:
2436:
2430:
2427:
2417:
2414:
2410:
2407: →
2406:
2402:
2398:
2394:
2390:
2387:Moreover, if
2386:
2385:
2370:
2364:
2361:
2349:
2346:
2340:
2334:
2331:
2320:
2317: →
2316:
2312:
2308:
2305:
2302:
2301:
2295:
2277:
2272:
2269:
2266:
2262:
2250:
2246:
2237:
2229:
2226:
2217:
2214:
2211:
2208:
2200:
2194:
2191:
2183:
2175:
2172:
2156:
2153:
2150:
2146:
2142:
2137:
2134:
2131:
2126:
2123:
2120:
2113:
2112:
2111:
2097:
2088:
2085:
2082:
2048:
2045:
2042:
2034:
2030:
2009:
1983:
1980:
1977:
1968:
1965:
1962:
1952:
1942:
1939:
1936:
1902:
1899:
1892:
1882:
1879:
1876:
1842:
1839:
1836:
1806:
1803:
1800:
1792:
1788:
1784:
1779:
1775:
1767:
1760:
1757:
1754:
1739:
1720:
1717:
1710:
1703:
1700:
1697:
1681:
1677:
1676:singularities
1658:
1650:
1646:
1642:
1637:
1633:
1629:
1626:
1623:
1618:
1614:
1610:
1605:
1601:
1589:
1564:
1560:
1551:
1547:
1537:
1530:
1527:
1509:
1505:
1501:
1498:
1495:
1490:
1486:
1476:
1470:
1461:
1460:
1459:
1458:
1442:
1422:
1402:
1394:
1390:
1386:
1383:
1380:
1375:
1371:
1364:
1361:
1354:and an ideal
1341:
1333:
1332:
1328:
1324:
1320:
1316:
1289:
1286:
1257:
1253:
1249:
1246:
1243:
1238:
1234:
1227:
1212:
1208:
1204:
1200:
1196:
1192:
1191:
1187:
1147:
1133:
1131:
1126:
1125:
1119:
1103:
1096:
1093:
1081:
1077:
1058:
1052:
1047:
1043:
1038:
1034:
1028:
1023:
1020:
1015:
1010:
1003:
1000:
989:
988:
987:
985:
981:
977:
958:
955:
952:
949:
940:
935:
931:
927:
924:
917:
916:
915:
913:
909:
905:
901:
896:
894:
890:
886:
882:
874:
846:
842:
837:
833:
827:
822:
819:
814:
809:
802:
799:
788:
787:
786:
784:
780:
779:inverse limit
776:
772:
768:
764:
760:
741:
736:
732:
728:
725:
720:
716:
711:
708:
705:
700:
696:
692:
689:
686:
683:
680:
677:
672:
668:
660:
659:
658:
656:
652:
648:
644:
640:
636:
613:
610:
603:
601:
596:
592:
591:adic topology
588:
584:
580:
577:
573:
570:
566:
560:
559:Adic topology
550:
548:
532:
527:
523:
514:
510:
506:
502:
499:, a filtered
498:
497:filtered ring
494:
490:
486:
482:
462:
458:
454:
451:
448:
436:
431:
427:
422:
412:
408:
404:
399:
395:
389:
381:
376:
372:
367:
363:
355:
352:
349:
345:
341:
336:
333:
330:
315:
311:
297:
293:
287:
282:
278:
273:
269:
263:
258:
255:
250:
244:
241:
231:
230:
229:
227:
226:inverse limit
207:
204:
201:
196:
192:
188:
185:
180:
176:
172:
169:
164:
160:
156:
153:
146:
145:
144:
143:
139:
138:abelian group
135:
132:Suppose that
125:
123:
120:
116:
112:
108:
104:
100:
99:Taylor series
97:
93:
89:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
30:
19:
3124:
3118:
3111:Gabber, Ofer
3077:
3056:
3026:
3014:
3002:
2991:. Retrieved
2987:
2978:
2954:Zariski ring
2925:
2921:
2832:
2828:
2793:
2787:
2674:
2665:-modules is
2662:
2588:
2584:
2580:
2573:
2565:
2412:
2408:
2404:
2400:
2396:
2392:
2388:
2318:
2314:
2310:
2292:
1673:
1326:
1322:
1314:
1313:is the ring
1210:
1206:
1198:
1194:
1185:
1129:
1127:The ring of
1075:
1073:
983:
979:
975:
973:
911:
907:
899:
897:
887:which is an
872:
866:
783:factor rings
774:
770:
766:
762:
758:
756:
654:
650:
649:, which are
646:
642:
599:
594:
586:
578:
571:
562:
505:vector space
492:
488:
484:
480:
478:
223:
133:
131:
114:
107:metric space
95:
91:
87:
83:
79:
64:localization
39:
33:
29:Completeness
2570:flat module
1738:plane curve
509:commutative
82:on a space
3147:Categories
3041:References
2993:2017-01-14
2298:Properties
893:local ring
142:filtration
103:completion
40:completion
2970:Citations
2920:for some
2877:…
2852:≃
2763:^
2746:^
2737:≅
2731:^
2677:-algebra
2639:^
2624:⊗
2611:^
2539:^
2510:^
2495:^
2461:^
2452:→
2446:^
2431:^
2365:^
2356:→
2350:^
2335:^
2212:−
2173:−
2162:∞
2147:∑
1981:−
1785:−
1643:−
1627:…
1611:−
1557:↦
1531:^
1521:→
1499:…
1384:…
1290:^
1247:…
1097:^
1029:
1024:←
1004:^
953:∈
946:for
828:
823:←
803:^
709:⋯
706:⊃
693:⊃
687:⊃
452:≤
405:≡
353:≥
346:∏
342:∈
334:≥
321:¯
264:
259:←
245:^
208:⋯
205:⊃
189:⊃
173:⊃
3055:(1969).
2933:See also
1122:Examples
982:-module
773:.) The (
761:∈
641:of 0 in
485:additive
44:functors
3104:1322960
2403::
2313::
1201:be the
904:modules
875:. Thus
781:of the
600:maximal
60:modules
52:modules
3096:
3088:
3063:
2481:where
1680:scheme
651:nested
637:. The
501:module
483:is an
136:is an
2831:. If
2798:local
2667:exact
2572:over
2568:is a
1678:of a
1317:] of
1082:over
891:or a
602:ideal
585:) or
576:ideal
495:is a
109:with
105:of a
48:rings
3094:ISBN
3086:ISBN
3061:ISBN
2824:and
2399:and
2391:and
1918:and
1415:the
1213:and
1193:Let
974:The
511:and
58:and
50:and
38:, a
3129:doi
3125:332
2064:in
1321:in
1205:in
1118:.
1021:lim
820:lim
593:on
563:In
423:mod
256:lim
96:all
46:on
34:In
3149::
3123:.
3117:.
3101:MR
3092:;
3080:.
3076:,
3051:;
2986:.
1197:=
1188:).
895:.
785:,
769:+
657::
549:.
228::
124:.
3137:.
3131::
3069:.
2996:.
2926:I
2922:n
2904:I
2900:/
2896:]
2893:]
2888:n
2884:x
2880:,
2874:,
2869:1
2865:x
2861:[
2858:[
2855:K
2849:R
2833:R
2829:K
2810:m
2794:R
2769:.
2760:I
2753:/
2743:R
2727:I
2723:/
2719:R
2693:I
2689:/
2685:R
2675:R
2663:R
2645:.
2636:R
2628:R
2620:M
2617:=
2608:M
2591::
2585:R
2581:M
2576:.
2574:R
2566:R
2545:.
2536:R
2507:N
2501:,
2492:M
2467:,
2458:N
2443:M
2437::
2428:f
2413:f
2409:N
2405:M
2401:f
2397:R
2393:N
2389:M
2371:.
2362:S
2347:R
2341::
2332:f
2319:S
2315:R
2311:f
2278:.
2273:1
2270:+
2267:n
2263:x
2256:)
2251:n
2247:4
2243:(
2238:2
2234:)
2230:!
2227:n
2224:(
2221:)
2218:n
2215:2
2209:1
2206:(
2201:!
2198:)
2195:n
2192:2
2189:(
2184:n
2180:)
2176:1
2170:(
2157:0
2154:=
2151:n
2143:=
2138:x
2135:+
2132:1
2127:x
2124:=
2121:u
2098:.
2095:]
2092:]
2089:y
2086:,
2083:x
2080:[
2077:[
2073:C
2052:)
2049:x
2046:+
2043:1
2040:(
2035:2
2031:x
2010:u
1990:)
1987:)
1984:u
1978:y
1975:(
1972:)
1969:u
1966:+
1963:y
1960:(
1957:(
1953:/
1949:]
1946:]
1943:y
1940:,
1937:x
1934:[
1931:[
1927:C
1906:)
1903:y
1900:x
1897:(
1893:/
1889:]
1886:]
1883:y
1880:,
1877:x
1874:[
1871:[
1867:C
1846:)
1843:y
1840:,
1837:x
1834:(
1813:)
1810:)
1807:x
1804:+
1801:1
1798:(
1793:2
1789:x
1780:2
1776:y
1772:(
1768:/
1764:]
1761:y
1758:,
1755:x
1752:[
1748:C
1724:)
1721:y
1718:x
1715:(
1711:/
1707:]
1704:y
1701:,
1698:x
1695:[
1691:C
1659:.
1656:)
1651:n
1647:f
1638:n
1634:x
1630:,
1624:,
1619:1
1615:f
1606:1
1602:x
1598:(
1565:i
1561:f
1552:i
1548:x
1538:I
1528:R
1518:]
1515:]
1510:n
1506:x
1502:,
1496:,
1491:1
1487:x
1483:[
1480:[
1477:R
1471:{
1443:R
1423:I
1403:,
1400:)
1395:n
1391:f
1387:,
1381:,
1376:1
1372:f
1368:(
1365:=
1362:I
1342:R
1329:.
1327:K
1323:n
1315:K
1298:m
1287:R
1263:)
1258:n
1254:x
1250:,
1244:,
1239:1
1235:x
1231:(
1228:=
1223:m
1211:K
1207:n
1199:K
1195:R
1186:p
1171:Z
1148:p
1143:Z
1130:p
1104:I
1094:R
1076:R
1059:.
1056:)
1053:M
1048:n
1044:I
1039:/
1035:M
1032:(
1016:=
1011:I
1001:M
984:M
980:R
976:I
959:.
956:M
950:x
941:M
936:n
932:I
928:+
925:x
912:M
908:I
902:-
900:R
877:π
873:I
869:π
852:)
847:n
843:I
838:/
834:R
831:(
815:=
810:I
800:R
775:I
771:I
767:r
763:R
759:r
742:.
737:n
733:I
729:=
726:R
721:n
717:F
712:,
701:2
697:I
690:I
684:R
681:=
678:R
673:0
669:F
655:R
647:I
643:R
619:m
614:=
611:I
595:R
589:-
587:I
579:I
572:R
533:E
528:i
524:F
493:E
489:E
481:E
463:.
459:}
455:j
449:i
440:)
437:E
432:i
428:F
420:(
413:j
409:a
400:i
396:a
390:|
385:)
382:E
377:n
373:F
368:/
364:E
361:(
356:0
350:n
337:0
331:n
327:)
316:n
312:a
306:(
298:{
294:=
291:)
288:E
283:n
279:F
274:/
270:E
267:(
251:=
242:E
202:E
197:2
193:F
186:E
181:1
177:F
170:E
165:0
161:F
157:=
154:E
134:E
115:R
92:X
84:X
80:R
31:.
20:)
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