2607:) is well understood; this determines the algebraic fundamental group. More generally, the fundamental group of a proper scheme over any algebraically closed field of characteristic zero is known, because an extension of algebraically closed fields induces isomorphic fundamental groups.
2343:
1645:
1422:
1093:
1189:
2982:
2200:
648:
1261:
894:
1849:
1950:
2864:
2184:
520:
2050:
1986:
2672:
3009:
2552:
2451:
1344:
431:
1799:
1733:
1500:
106:
3454:
2599:
2396:
551:
2516:
1304:
2899:
1455:
677:
967:
399:
138:
3080:. For geometrically unibranch schemes (e.g., normal schemes), the two approaches agree, but in general the pro-étale fundamental group is a finer invariant: its
940:
700:
578:
458:
347:
2919:
2815:
2793:
2771:
2749:
2727:
2705:
2634:
2574:
2476:
2418:
2372:
2118:
2097:
2077:
2010:
1893:
1872:
1757:
1691:
1671:
1540:
1520:
1011:
991:
914:
827:
807:
787:
767:
747:
727:
367:
320:
296:
259:
237:
216:
162:
1548:
2637:
of positive characteristic, the results are different, since Artin–Schreier coverings exist in this situation. For example, the fundamental group of the
1349:
1016:
3149:
1101:
3369:
3167:
3142:
Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Documents Mathématiques
3077:
198:
of algebraic varieties are the appropriate analogue of covering spaces of topological spaces. Unfortunately, an algebraic variety
2053:
2338:{\displaystyle 1\to \pi _{1}(X^{sep},{\overline {x}})\to \pi _{1}(X,{\overline {x}})\to \operatorname {Gal} (k^{sep}/k)\to 1.}
2924:
590:
1198:
835:
1812:
1902:
3361:
2829:
2124:
3103:
2675:
2519:
2015:
188:
3076:. It is constructed by considering, instead of finite étale covers, maps which are both étale and satisfy the
1959:
3449:
3182:
3137:
3040:
2643:
2455:
1989:
3209:
1897:
706:
3432:
465:
3337:
3081:
2987:
2526:
2479:
2425:
554:
43:
1309:
404:
3191:
The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme
2012:, and the étale fundamental group with respect to that base point identifies with the Galois group
1852:
1762:
1696:
1463:
184:
141:
69:
2582:
2379:
3327:
3288:
3262:
3236:
3218:
3153:
3052:
3044:
2485:
1269:
63:
39:
2869:
1427:
3365:
3280:
3163:
3098:
3093:
830:
434:
299:
195:
169:
51:
47:
525:
3406:
3272:
3228:
656:
173:
3420:
3390:
3349:
3055:, seeks to identify classes of varieties which are determined by their fundamental groups.
945:
372:
111:
3416:
3386:
3345:
3194:
3020:
2604:
2191:
1953:
323:
192:
3043:
asks what groups can arise as fundamental groups (or Galois groups of field extensions).
3341:
2399:, the complex numbers, there is a close relation between the étale fundamental group of
922:
682:
560:
440:
329:
2904:
2800:
2778:
2756:
2734:
2712:
2690:
2619:
2559:
2461:
2403:
2357:
2187:
2103:
2082:
2062:
1995:
1878:
1857:
1742:
1676:
1656:
1525:
1505:
996:
976:
899:
812:
792:
772:
752:
732:
712:
703:
352:
305:
281:
244:
222:
201:
180:
147:
3061:
studies higher étale homotopy groups by means of the étale homotopy type of a scheme.
3443:
3292:
1640:{\displaystyle \pi _{1}(X,x)=\varprojlim _{i\in I}{\operatorname {Aut} }_{X}(X_{i}),}
1424:
which produces a projective system of automorphism groups from the projective system
263:
3240:
3048:
917:
267:
3123:
2478:. The algebraic fundamental group, as it is typically called in this case, is the
1417:{\displaystyle \operatorname {Aut} _{X}(X_{j})\to \operatorname {Aut} _{X}(X_{i})}
2638:
1088:{\displaystyle \#\operatorname {Aut} _{X}(X_{i})=\operatorname {deg} (X_{i}/X)}
3428:
3411:
3378:
3276:
3186:
2611:
Schemes over a field of positive characteristic and the tame fundamental group
3383:
Lectures on an introduction to
Grothendieck's theory of the fundamental group
3322:
Bhatt, Bhargav; Scholze, Peter (2015), "The pro-étale topology for schemes",
3284:
17:
3253:
Achinger, Piotr (November 2017). "Wild ramification and K(pi, 1) spaces".
3232:
3072:, §7) have introduced a variant of the étale fundamental group called the
1184:{\displaystyle F(Y)=\varinjlim _{i\in I}\operatorname {Hom} _{C}(X_{i},Y)}
460:
165:
3397:
Tamagawa, Akio (1997), "The
Grothendieck conjecture for affine curves",
3158:
240:, so one must consider the entire category of finite étale coverings of
581:
2818:. For example, the tame fundamental group of the affine line is zero.
3223:
2708:
which takes into account only covers that are tamely ramified along
3427:
This article incorporates material from étale fundamental group on
3267:
3207:
Schmidt, Alexander (2002), "Tame coverings of arithmetic schemes",
3332:
164:. This definition works well for spaces such as real and complex
183:, it is shown that the fundamental group is exactly the group of
2577:. In particular, as the fundamental group of smooth curves over
1095:. It also means that we have given an isomorphism of functors:
3193:, Lecture Notes in Mathematics, Vol. 208, Berlin, New York:
218:
often fails to have a "universal cover" that is finite over
2921:
is entirely determined by its etale homotopy group. Note
262:. One can then define the étale fundamental group as an
2059:
More generally, for any geometrically connected variety
2052:. This interpretation of the Galois group is known as
1874:. Essentially by definition, the fundamental group of
2990:
2977:{\displaystyle \pi =\pi _{1}^{et}(X,{\overline {x}})}
2927:
2907:
2901:-space, in the sense that the etale homotopy type of
2872:
2832:
2803:
2781:
2759:
2737:
2715:
2693:
2646:
2622:
2585:
2562:
2529:
2488:
2464:
2428:
2406:
2382:
2360:
2203:
2127:
2106:
2085:
2065:
2018:
1998:
1962:
1905:
1881:
1860:
1815:
1765:
1745:
1699:
1679:
1659:
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1508:
1466:
1430:
1352:
1312:
1272:
1201:
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979:
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902:
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468:
443:
407:
375:
355:
332:
308:
284:
247:
225:
204:
150:
114:
72:
3023:point of view, the fundamental group is a functor:
643:{\displaystyle F(Y)=\operatorname {Hom} _{X}(x,Y);}
3385:, Bombay: Tata Institute of Fundamental Research,
3003:
2976:
2913:
2893:
2858:
2809:
2787:
2765:
2743:
2721:
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2666:
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2366:
2337:
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2091:
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2004:
1980:
1944:
1887:
1866:
1843:
1793:
1751:
1727:
1685:
1665:
1639:
1534:
1514:
1494:
1449:
1416:
1338:
1298:
1256:{\displaystyle P\in \varprojlim _{i\in I}F(X_{i})}
1255:
1183:
1087:
1005:
985:
961:
934:
908:
889:{\displaystyle \{X_{j}\to X_{i}\mid i<j\in I\}}
888:
821:
801:
781:
761:
741:
721:
694:
671:
642:
572:
545:
514:
452:
425:
393:
361:
341:
314:
290:
253:
231:
210:
156:
132:
100:
2421:and the usual, topological, fundamental group of
3433:Creative Commons Attribution/Share-Alike License
3360:, Annals of Mathematics Studies, vol. 104,
2686:is a quotient of the usual fundamental group of
2522:, which says that all finite étale coverings of
1844:{\displaystyle \pi _{1}(\operatorname {Spec} k)}
2822:Affine schemes over a field of characteristic p
1945:{\displaystyle \operatorname {Gal} (k^{sep}/k)}
1457:. We then make the following definition: the
2859:{\displaystyle X\subset \mathbf {A} _{k}^{n}}
584:to the category of sets, namely the functor:
8:
3069:
2179:{\displaystyle X^{sep}:=X\times _{k}k^{sep}}
1444:
1431:
883:
839:
3058:
2349:Schemes over a field of characteristic zero
1759:and the category of finite and continuous
3455:Topological methods of algebraic geometry
3410:
3331:
3266:
3222:
3157:
3152:, pp. xviii+327, see Exp. V, IX, X,
2991:
2989:
2961:
2943:
2938:
2926:
2906:
2871:
2850:
2845:
2840:
2831:
2802:
2780:
2758:
2736:
2714:
2692:
2658:
2653:
2648:
2645:
2621:
2587:
2586:
2584:
2561:
2537:
2536:
2528:
2493:
2487:
2463:
2436:
2435:
2427:
2405:
2384:
2383:
2381:
2359:
2318:
2306:
2277:
2262:
2242:
2227:
2214:
2202:
2164:
2154:
2132:
2126:
2105:
2084:
2064:
2045:{\displaystyle \operatorname {Gal} (K/k)}
2031:
2017:
1997:
1961:
1931:
1919:
1904:
1880:
1859:
1820:
1814:
1770:
1764:
1744:
1704:
1698:
1693:to the category of finite and continuous
1678:
1658:
1625:
1612:
1607:
1591:
1581:
1556:
1550:
1527:
1507:
1471:
1465:
1438:
1429:
1405:
1389:
1373:
1357:
1351:
1330:
1317:
1311:
1290:
1277:
1271:
1244:
1219:
1209:
1200:
1166:
1150:
1131:
1121:
1103:
1074:
1068:
1043:
1027:
1018:
998:
978:
953:
947:
924:
901:
859:
846:
837:
814:
794:
774:
754:
734:
714:
684:
658:
613:
592:
562:
527:
467:
442:
406:
374:
354:
331:
307:
283:
246:
224:
203:
149:
113:
77:
71:
27:Topological concept in algebraic geometry
3306:
1981:{\displaystyle \operatorname {Spec} (k)}
58:Topological analogue/informal discussion
3114:
168:, but gives undesirable results for an
2826:It turns out that every affine scheme
1195:In particular, we have a marked point
789:; however, it is pro-representable in
144:of homotopy classes of loops based at
1896:can be shown to be isomorphic to the
7:
3358:Étale homotopy of simplicial schemes
2667:{\displaystyle \mathbf {A} _{k}^{1}}
1851:, the étale fundamental group of a
653:geometrically this is the fiber of
2615:For an algebraically closed field
1952:. More precisely, the choice of a
1020:
993:, i.e., finite étale schemes over
769:is typically not representable in
25:
3078:valuative criterion of properness
1650:with the inverse limit topology.
3084:is the étale fundamental group.
2841:
2649:
729:in the category of schemes over
515:{\displaystyle (Y,f)\to (Y',f')}
3065:The pro-étale fundamental group
3004:{\displaystyle {\overline {x}}}
2547:{\displaystyle X(\mathbb {C} )}
2518:. This is a consequence of the
2446:{\displaystyle X(\mathbb {C} )}
522:in this category are morphisms
108:of a pointed topological space
3431:, which is licensed under the
3150:Société Mathématique de France
2971:
2952:
2888:
2876:
2541:
2533:
2505:
2499:
2440:
2432:
2329:
2326:
2299:
2290:
2287:
2268:
2255:
2252:
2220:
2207:
2039:
2025:
1975:
1969:
1939:
1912:
1838:
1826:
1788:
1776:
1722:
1710:
1631:
1618:
1574:
1562:
1489:
1477:
1411:
1398:
1382:
1379:
1366:
1339:{\displaystyle X_{j}\to X_{i}}
1323:
1250:
1237:
1178:
1159:
1114:
1108:
1082:
1061:
1049:
1036:
852:
809:, in fact by Galois covers of
663:
634:
622:
603:
597:
532:
509:
487:
484:
481:
469:
426:{\displaystyle f\colon Y\to X}
417:
388:
376:
127:
115:
95:
83:
1:
3356:Friedlander, Eric M. (1982),
2752:is some compactification and
1809:The most basic example of is
1794:{\displaystyle \pi _{1}(X,x)}
1728:{\displaystyle \pi _{1}(X,x)}
1495:{\displaystyle \pi _{1}(X,x)}
1346:induces a group homomorphism
101:{\displaystyle \pi _{1}(X,x)}
3140:; Raynaud, Michèle (2003) ,
3124:Lectures on Étale Cohomology
2996:
2966:
2594:{\displaystyle \mathbb {C} }
2391:{\displaystyle \mathbb {C} }
2375:that is of finite type over
2282:
2247:
2054:Grothendieck's Galois theory
829:. This means that we have a
3074:pro-étale fundamental group
3029:Pointed algebraic varieties
2511:{\displaystyle \pi _{1}(X)}
1299:{\displaystyle X_{i},X_{j}}
298:be a connected and locally
36:algebraic fundamental group
3471:
3362:Princeton University Press
2186:is connected) there is an
1988:is equivalent to giving a
1263:of the projective system.
191:. This is more promising:
3277:10.1007/s00222-017-0733-5
3070:Bhatt & Scholze (2015
2894:{\displaystyle K(\pi ,1)}
2520:Riemann existence theorem
1737:equivalence of categories
1735:-sets and establishes an
1450:{\displaystyle \{X_{i}\}}
702:and abstractly it is the
369:be the category of pairs
179:In the classification of
3255:Inventiones Mathematicae
3104:Fundamental group scheme
189:universal covering space
66:, the fundamental group
3412:10.1023/A:1000114400142
3183:Grothendieck, Alexander
3138:Grothendieck, Alexandre
1459:étale fundamental group
546:{\displaystyle Y\to Y'}
3399:Compositio Mathematica
3041:inverse Galois problem
3011:is a geometric point.
3005:
2978:
2915:
2895:
2860:
2811:
2789:
2767:
2745:
2723:
2701:
2680:tame fundamental group
2668:
2630:
2595:
2570:
2548:
2512:
2472:
2456:complex analytic space
2447:
2414:
2392:
2368:
2339:
2180:
2114:
2093:
2073:
2046:
2006:
1982:
1946:
1889:
1868:
1845:
1795:
1753:
1729:
1687:
1673:is now a functor from
1667:
1641:
1542:is the inverse limit:
1536:
1516:
1496:
1451:
1418:
1340:
1300:
1257:
1185:
1089:
1007:
987:
963:
936:
910:
890:
823:
803:
783:
763:
743:
723:
696:
673:
672:{\displaystyle Y\to X}
644:
574:
547:
516:
454:
427:
395:
363:
343:
316:
292:
255:
233:
212:
158:
134:
102:
3233:10.1007/s002080100262
3210:Mathematische Annalen
3127:, version 2.21: 26-27
3006:
2979:
2916:
2896:
2861:
2812:
2790:
2774:is the complement of
2768:
2746:
2724:
2702:
2674:is not topologically
2669:
2631:
2596:
2571:
2549:
2513:
2473:
2448:
2415:
2393:
2369:
2340:
2181:
2115:
2094:
2074:
2047:
2007:
1983:
1947:
1898:absolute Galois group
1890:
1869:
1846:
1805:Examples and theorems
1796:
1754:
1730:
1688:
1668:
1642:
1537:
1517:
1497:
1452:
1419:
1341:
1301:
1258:
1186:
1090:
1008:
988:
964:
962:{\displaystyle X_{i}}
937:
911:
891:
824:
804:
784:
764:
744:
724:
697:
674:
645:
575:
548:
517:
455:
435:finite étale morphism
428:
396:
394:{\displaystyle (Y,f)}
364:
344:
317:
293:
256:
234:
213:
159:
135:
133:{\displaystyle (X,x)}
103:
3082:profinite completion
2988:
2925:
2905:
2870:
2830:
2801:
2779:
2757:
2735:
2713:
2691:
2644:
2620:
2583:
2560:
2527:
2486:
2480:profinite completion
2462:
2426:
2404:
2380:
2358:
2201:
2125:
2104:
2083:
2063:
2016:
1996:
1960:
1903:
1879:
1858:
1813:
1763:
1743:
1697:
1677:
1657:
1549:
1526:
1506:
1464:
1428:
1350:
1310:
1270:
1199:
1102:
1017:
997:
977:
946:
923:
900:
836:
813:
793:
773:
753:
733:
713:
683:
657:
591:
580:This category has a
561:
526:
466:
441:
405:
373:
353:
330:
306:
282:
245:
223:
202:
185:deck transformations
148:
112:
70:
3342:2013arXiv1309.1198B
2951:
2855:
2663:
3059:Friedlander (1982)
3053:section conjecture
3045:Anabelian geometry
3021:category-theoretic
3001:
2974:
2934:
2911:
2891:
2856:
2839:
2807:
2785:
2763:
2741:
2719:
2697:
2676:finitely generated
2664:
2647:
2626:
2591:
2566:
2555:stem from ones of
2544:
2508:
2468:
2443:
2410:
2388:
2364:
2335:
2176:
2110:
2089:
2069:
2042:
2002:
1978:
1942:
1885:
1864:
1841:
1791:
1749:
1725:
1683:
1663:
1637:
1602:
1589:
1532:
1512:
1492:
1447:
1414:
1336:
1296:
1253:
1230:
1217:
1181:
1142:
1129:
1085:
1003:
983:
959:
935:{\displaystyle I,}
932:
906:
886:
819:
799:
779:
759:
739:
719:
695:{\displaystyle x,}
692:
669:
640:
573:{\displaystyle X.}
570:
543:
512:
453:{\displaystyle Y.}
450:
423:
391:
359:
342:{\displaystyle X,}
339:
312:
288:
251:
229:
208:
154:
140:is defined as the
130:
98:
64:algebraic topology
52:topological spaces
40:algebraic geometry
38:is an analogue in
3371:978-0-691-08288-2
3169:978-2-85629-141-2
3099:Fundamental group
2999:
2969:
2914:{\displaystyle X}
2810:{\displaystyle X}
2788:{\displaystyle U}
2766:{\displaystyle D}
2744:{\displaystyle X}
2722:{\displaystyle D}
2700:{\displaystyle U}
2629:{\displaystyle k}
2569:{\displaystyle X}
2471:{\displaystyle X}
2413:{\displaystyle X}
2367:{\displaystyle X}
2285:
2250:
2113:{\displaystyle X}
2092:{\displaystyle k}
2072:{\displaystyle X}
2005:{\displaystyle K}
1888:{\displaystyle k}
1867:{\displaystyle k}
1752:{\displaystyle C}
1686:{\displaystyle C}
1666:{\displaystyle F}
1582:
1580:
1535:{\displaystyle x}
1515:{\displaystyle X}
1210:
1208:
1122:
1120:
1006:{\displaystyle X}
986:{\displaystyle X}
909:{\displaystyle C}
831:projective system
822:{\displaystyle X}
802:{\displaystyle C}
782:{\displaystyle C}
762:{\displaystyle F}
742:{\displaystyle X}
722:{\displaystyle x}
362:{\displaystyle C}
315:{\displaystyle x}
300:noetherian scheme
291:{\displaystyle X}
274:Formal definition
254:{\displaystyle X}
232:{\displaystyle X}
211:{\displaystyle X}
170:algebraic variety
157:{\displaystyle x}
48:fundamental group
16:(Redirected from
3462:
3423:
3414:
3393:
3374:
3352:
3335:
3310:
3303:
3297:
3296:
3270:
3250:
3244:
3243:
3226:
3204:
3198:
3197:
3179:
3173:
3172:
3161:
3134:
3128:
3119:
3033:Profinite groups
3010:
3008:
3007:
3002:
3000:
2992:
2983:
2981:
2980:
2975:
2970:
2962:
2950:
2942:
2920:
2918:
2917:
2912:
2900:
2898:
2897:
2892:
2865:
2863:
2862:
2857:
2854:
2849:
2844:
2816:
2814:
2813:
2808:
2794:
2792:
2791:
2786:
2772:
2770:
2769:
2764:
2750:
2748:
2747:
2742:
2728:
2726:
2725:
2720:
2706:
2704:
2703:
2698:
2673:
2671:
2670:
2665:
2662:
2657:
2652:
2635:
2633:
2632:
2627:
2605:Riemann surfaces
2600:
2598:
2597:
2592:
2590:
2575:
2573:
2572:
2567:
2553:
2551:
2550:
2545:
2540:
2517:
2515:
2514:
2509:
2498:
2497:
2477:
2475:
2474:
2469:
2452:
2450:
2449:
2444:
2439:
2419:
2417:
2416:
2411:
2397:
2395:
2394:
2389:
2387:
2373:
2371:
2370:
2365:
2344:
2342:
2341:
2336:
2322:
2317:
2316:
2286:
2278:
2267:
2266:
2251:
2243:
2238:
2237:
2219:
2218:
2192:profinite groups
2185:
2183:
2182:
2177:
2175:
2174:
2159:
2158:
2143:
2142:
2119:
2117:
2116:
2111:
2098:
2096:
2095:
2090:
2078:
2076:
2075:
2070:
2051:
2049:
2048:
2043:
2035:
2011:
2009:
2008:
2003:
1992:extension field
1990:separably closed
1987:
1985:
1984:
1979:
1951:
1949:
1948:
1943:
1935:
1930:
1929:
1894:
1892:
1891:
1886:
1873:
1871:
1870:
1865:
1850:
1848:
1847:
1842:
1825:
1824:
1800:
1798:
1797:
1792:
1775:
1774:
1758:
1756:
1755:
1750:
1734:
1732:
1731:
1726:
1709:
1708:
1692:
1690:
1689:
1684:
1672:
1670:
1669:
1664:
1646:
1644:
1643:
1638:
1630:
1629:
1617:
1616:
1611:
1601:
1590:
1561:
1560:
1541:
1539:
1538:
1533:
1521:
1519:
1518:
1513:
1501:
1499:
1498:
1493:
1476:
1475:
1456:
1454:
1453:
1448:
1443:
1442:
1423:
1421:
1420:
1415:
1410:
1409:
1394:
1393:
1378:
1377:
1362:
1361:
1345:
1343:
1342:
1337:
1335:
1334:
1322:
1321:
1305:
1303:
1302:
1297:
1295:
1294:
1282:
1281:
1262:
1260:
1259:
1254:
1249:
1248:
1229:
1218:
1190:
1188:
1187:
1182:
1171:
1170:
1155:
1154:
1141:
1130:
1094:
1092:
1091:
1086:
1078:
1073:
1072:
1048:
1047:
1032:
1031:
1012:
1010:
1009:
1004:
992:
990:
989:
984:
968:
966:
965:
960:
958:
957:
941:
939:
938:
933:
915:
913:
912:
907:
895:
893:
892:
887:
864:
863:
851:
850:
828:
826:
825:
820:
808:
806:
805:
800:
788:
786:
785:
780:
768:
766:
765:
760:
748:
746:
745:
740:
728:
726:
725:
720:
701:
699:
698:
693:
678:
676:
675:
670:
649:
647:
646:
641:
618:
617:
579:
577:
576:
571:
552:
550:
549:
544:
542:
521:
519:
518:
513:
508:
497:
459:
457:
456:
451:
432:
430:
429:
424:
400:
398:
397:
392:
368:
366:
365:
360:
348:
346:
345:
340:
321:
319:
318:
313:
297:
295:
294:
289:
260:
258:
257:
252:
238:
236:
235:
230:
217:
215:
214:
209:
174:Zariski topology
163:
161:
160:
155:
139:
137:
136:
131:
107:
105:
104:
99:
82:
81:
21:
3470:
3469:
3465:
3464:
3463:
3461:
3460:
3459:
3440:
3439:
3396:
3377:
3372:
3355:
3321:
3318:
3313:
3305:(Tamagawa
3304:
3300:
3252:
3251:
3247:
3206:
3205:
3201:
3195:Springer-Verlag
3187:Murre, Jacob P.
3181:
3180:
3176:
3170:
3159:math.AG/0206203
3136:
3135:
3131:
3120:
3116:
3112:
3090:
3067:
3017:
2986:
2985:
2923:
2922:
2903:
2902:
2868:
2867:
2828:
2827:
2824:
2799:
2798:
2777:
2776:
2755:
2754:
2733:
2732:
2711:
2710:
2689:
2688:
2682:of some scheme
2642:
2641:
2618:
2617:
2613:
2581:
2580:
2558:
2557:
2525:
2524:
2489:
2484:
2483:
2460:
2459:
2424:
2423:
2402:
2401:
2378:
2377:
2356:
2355:
2351:
2302:
2258:
2223:
2210:
2199:
2198:
2160:
2150:
2128:
2123:
2122:
2102:
2101:
2081:
2080:
2061:
2060:
2014:
2013:
1994:
1993:
1958:
1957:
1954:geometric point
1915:
1901:
1900:
1877:
1876:
1856:
1855:
1816:
1811:
1810:
1807:
1766:
1761:
1760:
1741:
1740:
1700:
1695:
1694:
1675:
1674:
1655:
1654:
1621:
1606:
1552:
1547:
1546:
1524:
1523:
1504:
1503:
1467:
1462:
1461:
1434:
1426:
1425:
1401:
1385:
1369:
1353:
1348:
1347:
1326:
1313:
1308:
1307:
1286:
1273:
1268:
1267:
1240:
1197:
1196:
1162:
1146:
1100:
1099:
1064:
1039:
1023:
1015:
1014:
995:
994:
975:
974:
949:
944:
943:
921:
920:
916:, indexed by a
898:
897:
855:
842:
834:
833:
811:
810:
791:
790:
771:
770:
751:
750:
731:
730:
711:
710:
681:
680:
655:
654:
609:
589:
588:
582:natural functor
559:
558:
535:
524:
523:
501:
490:
464:
463:
439:
438:
403:
402:
371:
370:
351:
350:
328:
327:
324:geometric point
304:
303:
280:
279:
276:
243:
242:
221:
220:
200:
199:
196:étale morphisms
181:covering spaces
146:
145:
110:
109:
73:
68:
67:
60:
46:, of the usual
28:
23:
22:
15:
12:
11:
5:
3468:
3466:
3458:
3457:
3452:
3442:
3441:
3438:
3437:
3424:
3405:(2): 135–194,
3394:
3375:
3370:
3353:
3317:
3314:
3312:
3311:
3298:
3261:(2): 453–499.
3245:
3199:
3174:
3168:
3129:
3113:
3111:
3108:
3107:
3106:
3101:
3096:
3094:étale morphism
3089:
3086:
3066:
3063:
3047:, for example
3037:
3036:
3016:
3015:Further topics
3013:
2998:
2995:
2973:
2968:
2965:
2960:
2957:
2954:
2949:
2946:
2941:
2937:
2933:
2930:
2910:
2890:
2887:
2884:
2881:
2878:
2875:
2853:
2848:
2843:
2838:
2835:
2823:
2820:
2806:
2784:
2762:
2740:
2718:
2696:
2661:
2656:
2651:
2625:
2612:
2609:
2589:
2565:
2543:
2539:
2535:
2532:
2507:
2504:
2501:
2496:
2492:
2467:
2442:
2438:
2434:
2431:
2409:
2386:
2363:
2350:
2347:
2346:
2345:
2334:
2331:
2328:
2325:
2321:
2315:
2312:
2309:
2305:
2301:
2298:
2295:
2292:
2289:
2284:
2281:
2276:
2273:
2270:
2265:
2261:
2257:
2254:
2249:
2246:
2241:
2236:
2233:
2230:
2226:
2222:
2217:
2213:
2209:
2206:
2188:exact sequence
2173:
2170:
2167:
2163:
2157:
2153:
2149:
2146:
2141:
2138:
2135:
2131:
2109:
2088:
2068:
2041:
2038:
2034:
2030:
2027:
2024:
2021:
2001:
1977:
1974:
1971:
1968:
1965:
1941:
1938:
1934:
1928:
1925:
1922:
1918:
1914:
1911:
1908:
1884:
1863:
1840:
1837:
1834:
1831:
1828:
1823:
1819:
1806:
1803:
1790:
1787:
1784:
1781:
1778:
1773:
1769:
1748:
1724:
1721:
1718:
1715:
1712:
1707:
1703:
1682:
1662:
1648:
1647:
1636:
1633:
1628:
1624:
1620:
1615:
1610:
1605:
1600:
1597:
1594:
1588:
1585:
1579:
1576:
1573:
1570:
1567:
1564:
1559:
1555:
1531:
1511:
1491:
1488:
1485:
1482:
1479:
1474:
1470:
1446:
1441:
1437:
1433:
1413:
1408:
1404:
1400:
1397:
1392:
1388:
1384:
1381:
1376:
1372:
1368:
1365:
1360:
1356:
1333:
1329:
1325:
1320:
1316:
1293:
1289:
1285:
1280:
1276:
1252:
1247:
1243:
1239:
1236:
1233:
1228:
1225:
1222:
1216:
1213:
1207:
1204:
1193:
1192:
1180:
1177:
1174:
1169:
1165:
1161:
1158:
1153:
1149:
1145:
1140:
1137:
1134:
1128:
1125:
1119:
1116:
1113:
1110:
1107:
1084:
1081:
1077:
1071:
1067:
1063:
1060:
1057:
1054:
1051:
1046:
1042:
1038:
1035:
1030:
1026:
1022:
1002:
982:
956:
952:
931:
928:
905:
885:
882:
879:
876:
873:
870:
867:
862:
858:
854:
849:
845:
841:
818:
798:
778:
758:
749:. The functor
738:
718:
704:Yoneda functor
691:
688:
668:
665:
662:
651:
650:
639:
636:
633:
630:
627:
624:
621:
616:
612:
608:
605:
602:
599:
596:
569:
566:
541:
538:
534:
531:
511:
507:
504:
500:
496:
493:
489:
486:
483:
480:
477:
474:
471:
449:
446:
437:from a scheme
422:
419:
416:
413:
410:
390:
387:
384:
381:
378:
358:
338:
335:
311:
287:
275:
272:
250:
228:
207:
153:
129:
126:
123:
120:
117:
97:
94:
91:
88:
85:
80:
76:
59:
56:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3467:
3456:
3453:
3451:
3450:Scheme theory
3448:
3447:
3445:
3436:
3434:
3430:
3425:
3422:
3418:
3413:
3408:
3404:
3400:
3395:
3392:
3388:
3384:
3380:
3376:
3373:
3367:
3363:
3359:
3354:
3351:
3347:
3343:
3339:
3334:
3329:
3325:
3320:
3319:
3315:
3308:
3302:
3299:
3294:
3290:
3286:
3282:
3278:
3274:
3269:
3264:
3260:
3256:
3249:
3246:
3242:
3238:
3234:
3230:
3225:
3220:
3216:
3212:
3211:
3203:
3200:
3196:
3192:
3188:
3184:
3178:
3175:
3171:
3165:
3160:
3155:
3151:
3147:
3145:
3139:
3133:
3130:
3126:
3125:
3121:J. S. Milne,
3118:
3115:
3109:
3105:
3102:
3100:
3097:
3095:
3092:
3091:
3087:
3085:
3083:
3079:
3075:
3071:
3064:
3062:
3060:
3056:
3054:
3050:
3046:
3042:
3034:
3030:
3026:
3025:
3024:
3022:
3014:
3012:
2993:
2963:
2958:
2955:
2947:
2944:
2939:
2935:
2931:
2928:
2908:
2885:
2882:
2879:
2873:
2851:
2846:
2836:
2833:
2821:
2819:
2817:
2804:
2795:
2782:
2773:
2760:
2751:
2738:
2729:
2716:
2707:
2694:
2685:
2681:
2677:
2659:
2654:
2640:
2636:
2623:
2610:
2608:
2606:
2602:
2601:
2576:
2563:
2554:
2530:
2521:
2502:
2494:
2490:
2481:
2465:
2457:
2453:
2429:
2420:
2407:
2398:
2374:
2361:
2353:For a scheme
2348:
2332:
2323:
2319:
2313:
2310:
2307:
2303:
2296:
2293:
2279:
2274:
2271:
2263:
2259:
2244:
2239:
2234:
2231:
2228:
2224:
2215:
2211:
2204:
2197:
2196:
2195:
2193:
2189:
2171:
2168:
2165:
2161:
2155:
2151:
2147:
2144:
2139:
2136:
2133:
2129:
2121:is such that
2120:
2107:
2086:
2079:over a field
2066:
2057:
2055:
2036:
2032:
2028:
2022:
2019:
1999:
1991:
1972:
1966:
1963:
1955:
1936:
1932:
1926:
1923:
1920:
1916:
1909:
1906:
1899:
1895:
1882:
1861:
1854:
1835:
1832:
1829:
1821:
1817:
1804:
1802:
1785:
1782:
1779:
1771:
1767:
1746:
1738:
1719:
1716:
1713:
1705:
1701:
1680:
1660:
1651:
1634:
1626:
1622:
1613:
1608:
1603:
1598:
1595:
1592:
1586:
1583:
1577:
1571:
1568:
1565:
1557:
1553:
1545:
1544:
1543:
1529:
1509:
1486:
1483:
1480:
1472:
1468:
1460:
1439:
1435:
1406:
1402:
1395:
1390:
1386:
1374:
1370:
1363:
1358:
1354:
1331:
1327:
1318:
1314:
1291:
1287:
1283:
1278:
1274:
1266:For two such
1264:
1245:
1241:
1234:
1231:
1226:
1223:
1220:
1214:
1211:
1205:
1202:
1175:
1172:
1167:
1163:
1156:
1151:
1147:
1143:
1138:
1135:
1132:
1126:
1123:
1117:
1111:
1105:
1098:
1097:
1096:
1079:
1075:
1069:
1065:
1058:
1055:
1052:
1044:
1040:
1033:
1028:
1024:
1000:
980:
972:
971:Galois covers
954:
950:
929:
926:
919:
903:
880:
877:
874:
871:
868:
865:
860:
856:
847:
843:
832:
816:
796:
776:
756:
736:
716:
708:
705:
689:
686:
666:
660:
637:
631:
628:
625:
619:
614:
610:
606:
600:
594:
587:
586:
585:
583:
567:
564:
556:
539:
536:
529:
505:
502:
498:
494:
491:
478:
475:
472:
462:
447:
444:
436:
420:
414:
411:
408:
385:
382:
379:
356:
336:
333:
325:
309:
301:
285:
273:
271:
269:
265:
264:inverse limit
261:
248:
239:
226:
205:
197:
194:
190:
186:
182:
177:
175:
171:
167:
151:
143:
124:
121:
118:
92:
89:
86:
78:
74:
65:
57:
55:
53:
49:
45:
41:
37:
33:
19:
3426:
3402:
3398:
3382:
3379:Murre, J. P.
3357:
3323:
3301:
3258:
3254:
3248:
3224:math/0005310
3214:
3208:
3202:
3190:
3177:
3143:
3141:
3132:
3122:
3117:
3073:
3068:
3057:
3049:Grothendieck
3038:
3032:
3028:
3018:
2825:
2797:
2775:
2753:
2731:
2709:
2687:
2683:
2679:
2616:
2614:
2603:(i.e., open
2579:
2578:
2556:
2523:
2458:attached to
2422:
2400:
2376:
2354:
2352:
2100:
2058:
1875:
1808:
1736:
1653:The functor
1652:
1649:
1458:
1265:
1194:
970:
918:directed set
652:
277:
268:automorphism
241:
219:
178:
61:
35:
31:
29:
18:Galois cover
3217:(1): 1–18,
3031:} → {
2639:affine line
707:represented
3444:Categories
3429:PlanetMath
3326:: 99–201,
3324:Astérisque
3316:References
3268:1701.03197
1013:such that
942:where the
401:such that
266:of finite
3333:1309.1198
3293:119146164
3285:0020-9910
3148:, Paris:
2997:¯
2967:¯
2936:π
2929:π
2880:π
2837:⊂
2491:π
2330:→
2297:
2291:→
2283:¯
2260:π
2256:→
2248:¯
2212:π
2208:→
2152:×
2023:
1967:
1910:
1833:
1818:π
1768:π
1702:π
1604:
1596:∈
1587:←
1554:π
1469:π
1396:
1383:→
1364:
1324:→
1232:
1224:∈
1215:←
1206:∈
1157:
1144:
1136:∈
1127:→
1059:
1034:
1021:#
878:∈
866:∣
853:→
664:→
620:
533:→
485:→
461:Morphisms
418:→
412::
172:with the
166:manifolds
75:π
3381:(1967),
3241:29899627
3189:(1971),
3088:See also
2730:, where
1739:between
1306:the map
540:′
506:′
495:′
349:and let
270:groups.
3421:1478817
3391:0302650
3350:3379634
3338:Bibcode
3019:From a
2099:(i.e.,
1801:-sets.
555:schemes
187:of the
44:schemes
3419:
3389:
3368:
3348:
3291:
3283:
3239:
3166:
2984:where
2678:. The
2454:, the
302:, let
193:finite
42:, for
3328:arXiv
3289:S2CID
3263:arXiv
3237:S2CID
3219:arXiv
3154:arXiv
3110:Notes
2866:is a
1853:field
679:over
557:over
433:is a
322:be a
142:group
32:étale
3366:ISBN
3307:1997
3281:ISSN
3164:ISBN
3039:The
1964:Spec
1830:Spec
969:are
872:<
278:Let
30:The
3407:doi
3403:109
3273:doi
3259:210
3229:doi
3215:322
3051:'s
2796:in
2482:of
2294:Gal
2190:of
2020:Gal
1956:of
1907:Gal
1609:Aut
1584:lim
1522:at
1502:of
1387:Aut
1355:Aut
1212:lim
1148:Hom
1124:lim
1056:deg
1025:Aut
973:of
896:in
709:by
611:Hom
553:as
326:of
62:In
50:of
34:or
3446::
3417:MR
3415:,
3401:,
3387:MR
3364:,
3346:MR
3344:,
3336:,
3287:.
3279:.
3271:.
3257:.
3235:,
3227:,
3213:,
3185:;
3162:,
3035:}.
2333:1.
2194::
2145::=
2056:.
176:.
54:.
3435:.
3409::
3340::
3330::
3309:)
3295:.
3275::
3265::
3231::
3221::
3156::
3146:)
3144:3
3027:{
2994:x
2972:)
2964:x
2959:,
2956:X
2953:(
2948:t
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2932:=
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2883:,
2877:(
2874:K
2852:n
2847:k
2842:A
2834:X
2805:X
2783:U
2761:D
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2717:D
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2534:(
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2500:(
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2466:X
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2172:p
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2040:)
2037:k
2033:/
2029:K
2026:(
2000:K
1976:)
1973:k
1970:(
1940:)
1937:k
1933:/
1927:p
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1913:(
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1827:(
1822:1
1789:)
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1777:(
1772:1
1747:C
1723:)
1720:x
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1714:X
1711:(
1706:1
1681:C
1661:F
1635:,
1632:)
1627:i
1623:X
1619:(
1614:X
1599:I
1593:i
1578:=
1575:)
1572:x
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1566:X
1563:(
1558:1
1530:x
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1490:)
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1478:(
1473:1
1445:}
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1432:{
1412:)
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1080:X
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1050:)
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884:}
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840:{
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607:=
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488:(
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152:x
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122:,
119:X
116:(
96:)
93:x
90:,
87:X
84:(
79:1
20:)
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