2248:
Every
Zariski covering is Nisnevich but the converse doesn't hold in general. This can be easily proven using any of the definitions since the residue fields will always be an isomorphism regardless of the Zariski cover, and by definition a Zariski cover will give a surjection on points. In addition,
1541:
2100:
1231:
2478:
850:
1328:
is the representable functor over the category of presheaves with transfers. For the
Nisnevich topology, the local rings are Henselian, and a finite cover of a Henselian ring is given by a product of Henselian rings, showing exactness.
1006:
1429:
1710:
449:
Assume the category consists of smooth schemes over a qcqs (quasi-compact and quasi-separated) scheme, then the original definition due to
Nisnevich, which is equivalent to the definition above, for a family of morphisms
1986:
1803:
2257:
Nisnevich introduced his topology to provide a cohomological interpretation of the class set of an affine group scheme, which was originally defined in adelic terms. He used it to partially prove a conjecture of
1880:
1323:
757:
513:
661:
2350:
1991:
1434:
2552:
346:
being a
Nisnevich morphism. The Nisnevich covers are the covering families of a pretopology on the category of schemes and morphisms of schemes. This generates a topology called the
2766:
Nisnevich, Yevsey A. (1989). "The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory". In J. F. Jardine and V. P. Snaith (ed.).
1116:
1921:
1401:
340:
310:
2754:
1958:
2358:
885:
1108:
765:
688:
544:
2165:
93:
2198:
1564:
1421:
1065:
2513:
694:
The following yet another equivalent condition for
Nisnevich covers is due to Lurie: The Nisnevich topology is generated by all finite families of étale morphisms
2238:
2218:
2123:
1978:
1032:
587:
567:
2768:
Algebraic K-theory: connections with geometry and topology. Proceedings of the NATO Advanced Study
Institute held in Lake Louise, Alberta, December 7--11, 1987
1536:{\displaystyle {\begin{aligned}(R,{\mathfrak {p}})^{h}&\rightsquigarrow \kappa \\(R,{\mathfrak {p}})^{sh}&\rightsquigarrow \kappa ^{sep}\end{aligned}}}
2770:. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences. Vol. 279. Dordrecht: Kluwer Academic Publishers Group. pp. 241–342.
893:
1580:
2095:{\displaystyle {\begin{aligned}i:\mathbb {A} ^{1}-\{a\}\hookrightarrow \mathbb {A} ^{1}\\f:\mathbb {A} ^{1}-\{0\}\to \mathbb {A} ^{1}\end{aligned}}}
1721:
1811:
1242:
697:
453:
1715:
If we look at the associated morphism of residue fields for the generic point of the base, we see that this is a degree 2 extension
592:
405:
The
Nisnevich topology has several variants which are adapted to studying singular varieties. Covers in these topologies include
2737:
1078:
One of the key motivations for introducing the
Nisnevich topology in motivic cohomology is the fact that a Zariski open cover
2302:
2352:
is the sheafification of these groups with respect to the
Nisnevich topology, there is a convergent spectral sequence
2803:
2566:
406:
48:
2522:
2691:
1226:{\displaystyle \cdots \to \mathbf {Z} _{tr}(U\times _{X}U)\to \mathbf {Z} _{tr}(U)\to \mathbf {Z} _{tr}(X)\to 0}
2578:
1885:
1546:
so the residue field of the strict
Henselization gives the separable closure of the original residue field
1367:
henselizations. One of the important points between the two cases can be seen when looking at a local ring
2259:
2270:
under a reductive group scheme over an integral regular Noetherian base scheme is locally trivial in the
1370:
315:
285:
2808:
32:
2588:
2562:
2473:{\displaystyle E_{2}^{p,q}=H^{p}(X_{\text{cd}},{\tilde {G}}_{q}^{\,{\text{cd}}})\Rightarrow G_{q-p}(X)}
44:
1934:
2583:
36:
845:{\displaystyle \varnothing =Z_{n+1}\subseteq Z_{n}\subseteq \cdots \subseteq Z_{1}\subseteq Z_{0}=X}
858:
158:
2618:
2617:
Antieau, Benjamin; Elmanto, Elden (2016-11-07). "A primer for unstable motivic homotopy theory".
2558:
1081:
666:
522:
366:
has as underlying category the same as the small étale site, that is to say, objects are schemes
40:
20:
2128:
2716:
2748:
2275:
2263:
438:
100:
66:
2170:
1549:
1406:
1037:
2519:, then there is an analogous convergent spectral sequence for K-groups with coefficients in
2271:
2498:
1882:
to get a Nisnevich cover since there is an isomorphism of points for the generic point of
1363:
in the Zariski topology. This differs from the Etale topology where the local rings are
51:. It was originally introduced by Yevsey Nisnevich, who was motivated by the theory of
2593:
2223:
2203:
2108:
1963:
1423:. In this case, the residue fields of the Henselization and strict Henselization differ
1338:
1034:-points, this implies the map is a surjection. Conversely, taking the trivial sequence
1017:
1001:{\displaystyle \coprod _{\alpha \in A}p_{\alpha }^{-1}(Z_{m}-Z_{m+1})\to Z_{m}-Z_{m+1}}
572:
552:
2797:
2274:. One of the key properties of the Nisnevich topology is the existence of a descent
1356:
264:
154:
131:
1705:{\displaystyle {\text{Spec}}(\mathbb {C} /(x^{2}-t))\to {\text{Spec}}(\mathbb {C} )}
1808:
This implies that this étale cover is not Nisnevich. We can add the étale morphism
415:
2668:"Section 10.154 (0BSK): Henselization and strict henselization—The Stacks project"
225:
if each morphism in the family is étale and for every (possibly non-closed) point
282:) is an isomorphism. If the family is finite, this is equivalent to the morphism
433:
allows morphisms as in the conclusion of Gabber's local uniformization theorem.
52:
378:
and the morphisms are morphisms of schemes compatible with the fixed maps to
2782:
2667:
759:
such that there is a finite sequence of finitely presented closed subschemes
1798:{\displaystyle \mathbb {C} (t)\to {\frac {\mathbb {C} (t)}{(x^{2}-t)}}}
2773:
2267:
2623:
1875:{\displaystyle \mathbb {A} ^{1}-\{0,1\}\to \mathbb {A} ^{1}-\{0\}}
350:. The category of schemes with the Nisnevich topology is notated
2295:) be the Quillen K-groups of the category of coherent sheaves on
2557:
The Nisnevich topology has also found important applications in
1318:{\displaystyle \mathbf {Z} _{tr}(Y)(Z):={\text{Hom}}_{cor}(Z,Y)}
402:-schemes. The topology is the one given by Nisnevich morphisms.
2240:-points. In this case, the covering is only an Etale covering.
752:{\displaystyle \{p_{\alpha }:U_{\alpha }\to X\}_{\alpha \in A}}
508:{\displaystyle \{p_{\alpha }:U_{\alpha }\to X\}_{\alpha \in A}}
656:{\displaystyle p_{k}:\coprod _{\alpha }U_{\alpha }(k)\to X(k)}
2692:"counterexamples - A Nisnevich cover which is not Zariski"
2282:
be a Noetherian scheme of finite Krull dimension, and let
437:
The cdh and l′ topologies are incomparable with the
394:
has as underlying category schemes with a fixed map to
441:, and the h topology is finer than the étale topology.
161:, locally of finite presentation, and for every point
2525:
2515:
is a prime number not equal to the characteristic of
2501:
2361:
2305:
2226:
2206:
2173:
2131:
2111:
1989:
1966:
1937:
1888:
1814:
1724:
1583:
1552:
1432:
1409:
1373:
1245:
1119:
1084:
1040:
1020:
896:
861:
768:
700:
669:
595:
575:
555:
525:
456:
318:
288:
69:
2220:. Otherwise, the covering cannot be a surjection on
2717:"Triangulated categories of motives over a field k"
2345:{\displaystyle {\tilde {G}}_{n}^{\,{\text{cd}}}(X)}
2546:
2507:
2472:
2344:
2232:
2212:
2192:
2159:
2117:
2094:
1972:
1952:
1915:
1874:
1797:
1704:
1558:
1535:
1415:
1395:
1317:
1225:
1102:
1059:
1026:
1000:
879:
844:
751:
682:
655:
581:
561:
538:
507:
334:
304:
87:
2167:, then this covering is Nisnevich if and only if
2753:: CS1 maint: bot: original URL status unknown (
419:allows proper birational morphisms as coverings.
382:. Admissible coverings are Nisnevich morphisms.
103:such that for every (possibly non-closed) point
2249:Zariski inclusions are always Etale morphisms.
1014:Notice that when evaluating these morphisms on
2547:{\displaystyle \mathbf {Z} /\ell \mathbf {Z} }
1110:does not yield a resolution of Zariski sheaves
8:
2070:
2064:
2021:
2015:
1910:
1904:
1869:
1863:
1842:
1830:
1067:gives the result in the opposite direction.
734:
701:
515:of schemes to be a Nisnevich covering is if
490:
457:
2743:. Archived from the original on 2017-09-23.
445:Equivalent conditions for a Nisnevich cover
426:allows De Jong's alterations as coverings.
2622:
2539:
2531:
2526:
2524:
2500:
2449:
2432:
2431:
2430:
2425:
2414:
2413:
2403:
2390:
2371:
2366:
2360:
2326:
2325:
2324:
2319:
2308:
2307:
2304:
2225:
2205:
2178:
2172:
2151:
2130:
2110:
2082:
2078:
2077:
2055:
2051:
2050:
2033:
2029:
2028:
2006:
2002:
2001:
1990:
1988:
1965:
1944:
1940:
1939:
1936:
1895:
1891:
1890:
1887:
1854:
1850:
1849:
1821:
1817:
1816:
1813:
1777:
1746:
1745:
1742:
1726:
1725:
1723:
1687:
1670:
1669:
1661:
1640:
1628:
1616:
1593:
1592:
1584:
1582:
1551:
1517:
1497:
1487:
1486:
1457:
1447:
1446:
1433:
1431:
1408:
1384:
1383:
1372:
1288:
1283:
1252:
1247:
1244:
1199:
1194:
1172:
1167:
1151:
1132:
1127:
1118:
1083:
1045:
1039:
1019:
986:
973:
951:
938:
922:
917:
901:
895:
860:
830:
817:
798:
779:
767:
737:
721:
708:
699:
674:
668:
623:
613:
600:
594:
574:
554:
530:
524:
493:
477:
464:
455:
326:
317:
296:
287:
68:
2604:
2266:which states that a rationally trivial
769:
589:-points, the (set-theoretic) coproduct
2746:
1916:{\displaystyle \mathbb {A} ^{1}-\{0\}}
1333:Local rings in the Nisnevich topology
7:
2612:
2610:
2608:
149:) is an isomorphism. Equivalently,
2654:Lecture Notes on Motivic Cohomology
1488:
1448:
1396:{\displaystyle (R,{\mathfrak {p}})}
1385:
335:{\displaystyle \coprod X_{\alpha }}
305:{\displaystyle \coprod u_{\alpha }}
1574:Consider the étale cover given by
14:
1355:in the Nisnevich topology is the
2540:
2527:
1953:{\displaystyle \mathbb {A} ^{1}}
1248:
1195:
1168:
1128:
409:or weaker forms of resolution.
398:and morphisms the morphisms of
16:Structure in algebraic geometry
2467:
2461:
2442:
2439:
2419:
2396:
2339:
2333:
2313:
2141:
2135:
2073:
2024:
1845:
1789:
1770:
1765:
1759:
1756:
1750:
1739:
1736:
1730:
1699:
1696:
1674:
1666:
1658:
1655:
1652:
1633:
1625:
1597:
1589:
1570:Examples of Nisnevich Covering
1510:
1494:
1477:
1467:
1454:
1437:
1390:
1374:
1312:
1300:
1276:
1270:
1267:
1261:
1217:
1214:
1208:
1190:
1187:
1181:
1163:
1160:
1141:
1123:
1094:
966:
963:
931:
727:
650:
644:
638:
635:
629:
483:
79:
29:completely decomposed topology
1:
880:{\displaystyle 0\leq m\leq n}
130:such that the induced map of
2656:. example 6.13, pages 39-40.
2640:Lectures on Algebraic Cycles
407:resolutions of singularities
370:with a fixed étale morphism
1103:{\displaystyle \pi :U\to X}
683:{\displaystyle p_{\alpha }}
539:{\displaystyle p_{\alpha }}
169:, there must exist a point
2825:
2160:{\displaystyle f(x)=x^{k}}
1980:, then a covering given by
1336:
663:of all covering morphisms
2642:. Cambridge. pp. ix.
1960:as a scheme over a field
1351:, then the local ring of
2672:stacks.math.columbia.edu
359:small Nisnevich site of
88:{\displaystyle f:Y\to X}
2784:Motivic Homotopy Theory
2579:Presheaf with transfers
2193:{\displaystyle x^{k}=a}
1559:{\displaystyle \kappa }
1416:{\displaystyle \kappa }
1347:is a point of a scheme
1060:{\displaystyle Z_{0}=X}
263:and the induced map of
203:A family of morphisms {
111:, there exists a point
39:which has been used in
27:, sometimes called the
2548:
2509:
2474:
2346:
2260:Alexander Grothendieck
2234:
2214:
2194:
2161:
2119:
2103:
2096:
1974:
1954:
1917:
1876:
1799:
1706:
1560:
1544:
1537:
1417:
1397:
1326:
1319:
1234:
1227:
1104:
1061:
1028:
1009:
1002:
881:
853:
846:
753:
684:
657:
583:
563:
540:
509:
387:big Nisnevich site of
336:
306:
89:
63:A morphism of schemes
2781:Levine, Marc (2008),
2715:Voevodsky, Vladimir.
2549:
2510:
2508:{\displaystyle \ell }
2475:
2347:
2235:
2215:
2195:
2162:
2125:is the inclusion and
2120:
2097:
1982:
1975:
1955:
1918:
1877:
1800:
1707:
1561:
1538:
1425:
1418:
1398:
1359:of the local ring of
1320:
1238:
1228:
1112:
1105:
1062:
1029:
1003:
889:
882:
847:
761:
754:
685:
658:
584:
564:
541:
510:
337:
307:
200:) is an isomorphism.
90:
33:Grothendieck topology
2738:"Nisnevich Topology"
2726:. Proposition 3.1.3.
2584:Mixed motives (math)
2523:
2499:
2359:
2303:
2224:
2204:
2200:has a solution over
2171:
2129:
2109:
1987:
1964:
1935:
1927:Conditional covering
1886:
1812:
1722:
1581:
1550:
1430:
1407:
1371:
1243:
1117:
1082:
1038:
1018:
894:
859:
766:
698:
667:
593:
573:
553:
523:
454:
316:
286:
67:
47:, and the theory of
2774:Nisnevich's website
2724:Journal of K-Theory
2438:
2382:
2332:
1403:with residue field
930:
35:on the category of
2804:Algebraic geometry
2589:A¹ homotopy theory
2565:and the theory of
2563:A¹ homotopy theory
2559:algebraic K-theory
2544:
2505:
2470:
2412:
2362:
2342:
2306:
2230:
2210:
2190:
2157:
2115:
2092:
2090:
1970:
1950:
1913:
1872:
1795:
1702:
1556:
1533:
1531:
1413:
1393:
1315:
1223:
1100:
1057:
1024:
1011:admits a section.
998:
913:
912:
877:
842:
749:
680:
653:
618:
579:
569:, on the level of
559:
536:
505:
348:Nisnevich topology
332:
302:
97:Nisnevich morphism
85:
45:A¹ homotopy theory
41:algebraic K-theory
25:Nisnevich topology
21:algebraic geometry
2435:
2422:
2406:
2329:
2316:
2276:spectral sequence
2264:Jean-Pierre Serre
2244:Zariski coverings
2233:{\displaystyle k}
2213:{\displaystyle k}
2118:{\displaystyle i}
1973:{\displaystyle k}
1793:
1664:
1587:
1286:
1027:{\displaystyle S}
897:
609:
582:{\displaystyle k}
562:{\displaystyle k}
431:l′ topology
2816:
2790:
2789:
2771:
2759:
2758:
2752:
2744:
2742:
2734:
2728:
2727:
2721:
2712:
2706:
2705:
2703:
2702:
2688:
2682:
2681:
2679:
2678:
2664:
2658:
2657:
2650:
2644:
2643:
2638:Bloch, Spencer.
2635:
2629:
2628:
2626:
2614:
2553:
2551:
2550:
2545:
2543:
2535:
2530:
2514:
2512:
2511:
2506:
2494:
2490:
2486:
2479:
2477:
2476:
2471:
2460:
2459:
2437:
2436:
2433:
2429:
2424:
2423:
2415:
2408:
2407:
2404:
2395:
2394:
2381:
2370:
2351:
2349:
2348:
2343:
2331:
2330:
2327:
2323:
2318:
2317:
2309:
2272:Zariski topology
2239:
2237:
2236:
2231:
2219:
2217:
2216:
2211:
2199:
2197:
2196:
2191:
2183:
2182:
2166:
2164:
2163:
2158:
2156:
2155:
2124:
2122:
2121:
2116:
2101:
2099:
2098:
2093:
2091:
2087:
2086:
2081:
2060:
2059:
2054:
2038:
2037:
2032:
2011:
2010:
2005:
1979:
1977:
1976:
1971:
1959:
1957:
1956:
1951:
1949:
1948:
1943:
1922:
1920:
1919:
1914:
1900:
1899:
1894:
1881:
1879:
1878:
1873:
1859:
1858:
1853:
1826:
1825:
1820:
1804:
1802:
1801:
1796:
1794:
1792:
1782:
1781:
1768:
1749:
1743:
1729:
1711:
1709:
1708:
1703:
1695:
1694:
1673:
1665:
1662:
1645:
1644:
1632:
1624:
1623:
1596:
1588:
1585:
1565:
1563:
1562:
1557:
1542:
1540:
1539:
1534:
1532:
1528:
1527:
1505:
1504:
1492:
1491:
1462:
1461:
1452:
1451:
1422:
1420:
1419:
1414:
1402:
1400:
1399:
1394:
1389:
1388:
1324:
1322:
1321:
1316:
1299:
1298:
1287:
1284:
1260:
1259:
1251:
1232:
1230:
1229:
1224:
1207:
1206:
1198:
1180:
1179:
1171:
1156:
1155:
1140:
1139:
1131:
1109:
1107:
1106:
1101:
1066:
1064:
1063:
1058:
1050:
1049:
1033:
1031:
1030:
1025:
1007:
1005:
1004:
999:
997:
996:
978:
977:
962:
961:
943:
942:
929:
921:
911:
886:
884:
883:
878:
851:
849:
848:
843:
835:
834:
822:
821:
803:
802:
790:
789:
758:
756:
755:
750:
748:
747:
726:
725:
713:
712:
689:
687:
686:
681:
679:
678:
662:
660:
659:
654:
628:
627:
617:
605:
604:
588:
586:
585:
580:
568:
566:
565:
560:
545:
543:
542:
537:
535:
534:
514:
512:
511:
506:
504:
503:
482:
481:
469:
468:
341:
339:
338:
333:
331:
330:
311:
309:
308:
303:
301:
300:
183:
129:
94:
92:
91:
86:
2824:
2823:
2819:
2818:
2817:
2815:
2814:
2813:
2794:
2793:
2787:
2780:
2772:, available at
2765:
2762:
2745:
2740:
2736:
2735:
2731:
2719:
2714:
2713:
2709:
2700:
2698:
2690:
2689:
2685:
2676:
2674:
2666:
2665:
2661:
2652:
2651:
2647:
2637:
2636:
2632:
2616:
2615:
2606:
2602:
2575:
2521:
2520:
2497:
2496:
2493:p - q ≥ 0
2492:
2488:
2484:
2445:
2399:
2386:
2357:
2356:
2301:
2300:
2290:
2255:
2246:
2222:
2221:
2202:
2201:
2174:
2169:
2168:
2147:
2127:
2126:
2107:
2106:
2089:
2088:
2076:
2049:
2040:
2039:
2027:
2000:
1985:
1984:
1962:
1961:
1938:
1933:
1932:
1929:
1889:
1884:
1883:
1848:
1815:
1810:
1809:
1773:
1769:
1744:
1720:
1719:
1683:
1636:
1612:
1579:
1578:
1572:
1548:
1547:
1530:
1529:
1513:
1506:
1493:
1474:
1473:
1463:
1453:
1428:
1427:
1405:
1404:
1369:
1368:
1341:
1335:
1282:
1246:
1241:
1240:
1193:
1166:
1147:
1126:
1115:
1114:
1080:
1079:
1076:
1070:
1041:
1036:
1035:
1016:
1015:
982:
969:
947:
934:
892:
891:
857:
856:
826:
813:
794:
775:
764:
763:
733:
717:
704:
696:
695:
670:
665:
664:
619:
596:
591:
590:
571:
570:
551:
550:
526:
521:
520:
489:
473:
460:
452:
451:
447:
322:
314:
313:
292:
284:
283:
254:
247:
233:, there exists
223:Nisnevich cover
216:
209:
174:
120:
65:
64:
61:
17:
12:
11:
5:
2822:
2820:
2812:
2811:
2806:
2796:
2795:
2792:
2791:
2777:
2776:
2761:
2760:
2729:
2707:
2683:
2659:
2645:
2630:
2603:
2601:
2598:
2597:
2596:
2594:Henselian ring
2591:
2586:
2581:
2574:
2571:
2542:
2538:
2534:
2529:
2504:
2481:
2480:
2469:
2466:
2463:
2458:
2455:
2452:
2448:
2444:
2441:
2428:
2421:
2418:
2411:
2402:
2398:
2393:
2389:
2385:
2380:
2377:
2374:
2369:
2365:
2341:
2338:
2335:
2322:
2315:
2312:
2286:
2254:
2251:
2245:
2242:
2229:
2209:
2189:
2186:
2181:
2177:
2154:
2150:
2146:
2143:
2140:
2137:
2134:
2114:
2085:
2080:
2075:
2072:
2069:
2066:
2063:
2058:
2053:
2048:
2045:
2042:
2041:
2036:
2031:
2026:
2023:
2020:
2017:
2014:
2009:
2004:
1999:
1996:
1993:
1992:
1969:
1947:
1942:
1928:
1925:
1912:
1909:
1906:
1903:
1898:
1893:
1871:
1868:
1865:
1862:
1857:
1852:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1824:
1819:
1806:
1805:
1791:
1788:
1785:
1780:
1776:
1772:
1767:
1764:
1761:
1758:
1755:
1752:
1748:
1741:
1738:
1735:
1732:
1728:
1713:
1712:
1701:
1698:
1693:
1690:
1686:
1682:
1679:
1676:
1672:
1668:
1660:
1657:
1654:
1651:
1648:
1643:
1639:
1635:
1631:
1627:
1622:
1619:
1615:
1611:
1608:
1605:
1602:
1599:
1595:
1591:
1571:
1568:
1555:
1526:
1523:
1520:
1516:
1512:
1509:
1507:
1503:
1500:
1496:
1490:
1485:
1482:
1479:
1476:
1475:
1472:
1469:
1466:
1464:
1460:
1456:
1450:
1445:
1442:
1439:
1436:
1435:
1412:
1392:
1387:
1382:
1379:
1376:
1339:Henselian ring
1334:
1331:
1314:
1311:
1308:
1305:
1302:
1297:
1294:
1291:
1281:
1278:
1275:
1272:
1269:
1266:
1263:
1258:
1255:
1250:
1222:
1219:
1216:
1213:
1210:
1205:
1202:
1197:
1192:
1189:
1186:
1183:
1178:
1175:
1170:
1165:
1162:
1159:
1154:
1150:
1146:
1143:
1138:
1135:
1130:
1125:
1122:
1099:
1096:
1093:
1090:
1087:
1075:
1072:
1056:
1053:
1048:
1044:
1023:
995:
992:
989:
985:
981:
976:
972:
968:
965:
960:
957:
954:
950:
946:
941:
937:
933:
928:
925:
920:
916:
910:
907:
904:
900:
876:
873:
870:
867:
864:
855:such that for
841:
838:
833:
829:
825:
820:
816:
812:
809:
806:
801:
797:
793:
788:
785:
782:
778:
774:
771:
746:
743:
740:
736:
732:
729:
724:
720:
716:
711:
707:
703:
692:
691:
690:is surjective.
677:
673:
652:
649:
646:
643:
640:
637:
634:
631:
626:
622:
616:
612:
608:
603:
599:
578:
558:
549:For all field
547:
533:
529:
502:
499:
496:
492:
488:
485:
480:
476:
472:
467:
463:
459:
446:
443:
439:étale topology
435:
434:
427:
420:
329:
325:
321:
299:
295:
291:
265:residue fields
252:
245:
214:
207:
132:residue fields
101:étale morphism
84:
81:
78:
75:
72:
60:
57:
15:
13:
10:
9:
6:
4:
3:
2:
2821:
2810:
2807:
2805:
2802:
2801:
2799:
2786:
2785:
2779:
2778:
2775:
2769:
2764:
2763:
2756:
2750:
2739:
2733:
2730:
2725:
2718:
2711:
2708:
2697:
2693:
2687:
2684:
2673:
2669:
2663:
2660:
2655:
2649:
2646:
2641:
2634:
2631:
2625:
2620:
2613:
2611:
2609:
2605:
2599:
2595:
2592:
2590:
2587:
2585:
2582:
2580:
2577:
2576:
2572:
2570:
2568:
2564:
2560:
2555:
2536:
2532:
2518:
2502:
2464:
2456:
2453:
2450:
2446:
2426:
2416:
2409:
2400:
2391:
2387:
2383:
2378:
2375:
2372:
2367:
2363:
2355:
2354:
2353:
2336:
2320:
2310:
2298:
2294:
2289:
2285:
2281:
2277:
2273:
2269:
2265:
2261:
2252:
2250:
2243:
2241:
2227:
2207:
2187:
2184:
2179:
2175:
2152:
2148:
2144:
2138:
2132:
2112:
2102:
2083:
2067:
2061:
2056:
2046:
2043:
2034:
2018:
2012:
2007:
1997:
1994:
1981:
1967:
1945:
1926:
1924:
1907:
1901:
1896:
1866:
1860:
1855:
1839:
1836:
1833:
1827:
1822:
1786:
1783:
1778:
1774:
1762:
1753:
1733:
1718:
1717:
1716:
1691:
1688:
1684:
1680:
1677:
1649:
1646:
1641:
1637:
1629:
1620:
1617:
1613:
1609:
1606:
1603:
1600:
1577:
1576:
1575:
1569:
1567:
1553:
1543:
1524:
1521:
1518:
1514:
1508:
1501:
1498:
1483:
1480:
1470:
1465:
1458:
1443:
1440:
1424:
1410:
1380:
1377:
1366:
1362:
1358:
1357:Henselization
1354:
1350:
1346:
1340:
1332:
1330:
1325:
1309:
1306:
1303:
1295:
1292:
1289:
1279:
1273:
1264:
1256:
1253:
1237:
1233:
1220:
1211:
1203:
1200:
1184:
1176:
1173:
1157:
1152:
1148:
1144:
1136:
1133:
1120:
1111:
1097:
1091:
1088:
1085:
1073:
1071:
1068:
1054:
1051:
1046:
1042:
1021:
1012:
1008:
993:
990:
987:
983:
979:
974:
970:
958:
955:
952:
948:
944:
939:
935:
926:
923:
918:
914:
908:
905:
902:
898:
888:
874:
871:
868:
865:
862:
852:
839:
836:
831:
827:
823:
818:
814:
810:
807:
804:
799:
795:
791:
786:
783:
780:
776:
772:
760:
744:
741:
738:
730:
722:
718:
714:
709:
705:
675:
671:
647:
641:
632:
624:
620:
614:
610:
606:
601:
597:
576:
556:
548:
546:is étale; and
531:
527:
518:
517:
516:
500:
497:
494:
486:
478:
474:
470:
465:
461:
444:
442:
440:
432:
428:
425:
421:
418:
417:
412:
411:
410:
408:
403:
401:
397:
393:
391:
388:
383:
381:
377:
373:
369:
365:
363:
360:
355:
353:
349:
345:
327:
323:
319:
297:
293:
289:
281:
277:
273:
269:
266:
262:
258:
251:
244:
240:
236:
232:
228:
224:
220:
213:
206:
201:
199:
195:
191:
187:
181:
177:
173:in the fiber
172:
168:
164:
160:
156:
152:
148:
144:
140:
136:
133:
127:
123:
119:in the fiber
118:
114:
110:
106:
102:
98:
82:
76:
73:
70:
58:
56:
54:
50:
46:
42:
38:
34:
30:
26:
22:
2809:Topos theory
2783:
2767:
2732:
2723:
2710:
2699:. Retrieved
2696:MathOverflow
2695:
2686:
2675:. Retrieved
2671:
2662:
2653:
2648:
2639:
2633:
2556:
2516:
2482:
2296:
2292:
2287:
2283:
2279:
2256:
2253:Applications
2247:
2104:
1983:
1930:
1807:
1714:
1573:
1545:
1426:
1364:
1360:
1352:
1348:
1344:
1342:
1327:
1239:
1235:
1113:
1077:
1069:
1013:
1010:
890:
854:
762:
693:
448:
436:
430:
423:
416:cdh topology
414:
404:
399:
395:
392:
389:
386:
384:
379:
375:
371:
367:
364:
361:
358:
356:
351:
347:
343:
279:
275:
271:
267:
260:
256:
249:
242:
238:
237:and a point
234:
230:
226:
222:
218:
211:
204:
202:
197:
193:
189:
185:
179:
175:
170:
166:
162:
150:
146:
142:
138:
134:
125:
121:
116:
112:
108:
104:
99:if it is an
96:
95:is called a
62:
28:
24:
18:
2489:q ≥ 0
2485:p ≥ 0
1931:If we take
2798:Categories
2701:2021-01-25
2677:2021-01-25
2624:1605.00929
2600:References
1337:See also:
1074:Motivation
424:h topology
184:such that
159:unramified
59:Definition
2537:ℓ
2503:ℓ
2454:−
2443:⇒
2420:~
2314:~
2074:→
2062:−
2025:↪
2013:−
1902:−
1861:−
1846:→
1828:−
1784:−
1740:→
1689:−
1659:→
1647:−
1618:−
1554:κ
1515:κ
1511:⇝
1471:κ
1468:⇝
1411:κ
1218:→
1191:→
1164:→
1149:×
1124:→
1121:⋯
1095:→
1086:π
980:−
967:→
945:−
924:−
919:α
906:∈
903:α
899:∐
872:≤
866:≤
824:⊆
811:⊆
808:⋯
805:⊆
792:⊆
770:∅
742:∈
739:α
728:→
723:α
710:α
676:α
639:→
625:α
615:α
611:∐
532:α
498:∈
495:α
484:→
479:α
466:α
328:α
320:∐
298:α
290:∐
80:→
2749:cite web
2573:See also
210: :
153:must be
2567:motives
221:} is a
49:motives
37:schemes
31:, is a
2495:. If
2491:, and
2278:. Let
2268:torsor
2105:where
1365:strict
519:Every
53:adeles
23:, the
2788:(PDF)
2741:(PDF)
2720:(PDF)
2619:arXiv
2299:. If
1236:where
312:from
248:s.t.
2755:link
2483:for
2262:and
1663:Spec
1586:Spec
429:The
422:The
413:The
385:The
357:The
274:) →
259:) =
192:) →
155:flat
141:) →
1343:If
1285:Hom
352:Nis
342:to
19:In
2800::
2751:}}
2747:{{
2722:.
2694:.
2670:.
2607:^
2569:.
2561:,
2554:.
2487:,
2434:cd
2405:cd
2328:cd
1923:.
1566:.
1280::=
887:,
374:→
354:.
241:∈
229:∈
217:→
165:∈
157:,
115:∈
107:∈
55:.
43:,
2757:)
2704:.
2680:.
2627:.
2621::
2541:Z
2533:/
2528:Z
2517:X
2468:)
2465:X
2462:(
2457:p
2451:q
2447:G
2440:)
2427:q
2417:G
2410:,
2401:X
2397:(
2392:p
2388:H
2384:=
2379:q
2376:,
2373:p
2368:2
2364:E
2340:)
2337:X
2334:(
2321:n
2311:G
2297:X
2293:X
2291:(
2288:n
2284:G
2280:X
2228:k
2208:k
2188:a
2185:=
2180:k
2176:x
2153:k
2149:x
2145:=
2142:)
2139:x
2136:(
2133:f
2113:i
2084:1
2079:A
2071:}
2068:0
2065:{
2057:1
2052:A
2047::
2044:f
2035:1
2030:A
2022:}
2019:a
2016:{
2008:1
2003:A
1998::
1995:i
1968:k
1946:1
1941:A
1911:}
1908:0
1905:{
1897:1
1892:A
1870:}
1867:0
1864:{
1856:1
1851:A
1843:}
1840:1
1837:,
1834:0
1831:{
1823:1
1818:A
1790:)
1787:t
1779:2
1775:x
1771:(
1766:]
1763:x
1760:[
1757:)
1754:t
1751:(
1747:C
1737:)
1734:t
1731:(
1727:C
1700:)
1697:]
1692:1
1685:t
1681:,
1678:t
1675:[
1671:C
1667:(
1656:)
1653:)
1650:t
1642:2
1638:x
1634:(
1630:/
1626:]
1621:1
1614:t
1610:,
1607:t
1604:,
1601:x
1598:[
1594:C
1590:(
1525:p
1522:e
1519:s
1502:h
1499:s
1495:)
1489:p
1484:,
1481:R
1478:(
1459:h
1455:)
1449:p
1444:,
1441:R
1438:(
1391:)
1386:p
1381:,
1378:R
1375:(
1361:x
1353:x
1349:X
1345:x
1313:)
1310:Y
1307:,
1304:Z
1301:(
1296:r
1293:o
1290:c
1277:)
1274:Z
1271:(
1268:)
1265:Y
1262:(
1257:r
1254:t
1249:Z
1221:0
1215:)
1212:X
1209:(
1204:r
1201:t
1196:Z
1188:)
1185:U
1182:(
1177:r
1174:t
1169:Z
1161:)
1158:U
1153:X
1145:U
1142:(
1137:r
1134:t
1129:Z
1098:X
1092:U
1089::
1055:X
1052:=
1047:0
1043:Z
1022:S
994:1
991:+
988:m
984:Z
975:m
971:Z
964:)
959:1
956:+
953:m
949:Z
940:m
936:Z
932:(
927:1
915:p
909:A
875:n
869:m
863:0
840:X
837:=
832:0
828:Z
819:1
815:Z
800:n
796:Z
787:1
784:+
781:n
777:Z
773:=
745:A
735:}
731:X
719:U
715::
706:p
702:{
672:p
651:)
648:k
645:(
642:X
636:)
633:k
630:(
621:U
607::
602:k
598:p
577:k
557:k
528:p
501:A
491:}
487:X
475:U
471::
462:p
458:{
400:X
396:X
390:X
380:X
376:X
372:U
368:U
362:X
344:X
324:X
294:u
280:y
278:(
276:k
272:x
270:(
268:k
261:x
257:y
255:(
253:α
250:u
246:α
243:X
239:y
235:α
231:X
227:x
219:X
215:α
212:X
208:α
205:u
198:y
196:(
194:k
190:x
188:(
186:k
182:)
180:x
178:(
176:f
171:y
167:X
163:x
151:f
147:y
145:(
143:k
139:x
137:(
135:k
128:)
126:x
124:(
122:f
117:Y
113:y
109:X
105:x
83:X
77:Y
74::
71:f
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.