49:
829:, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance. The
2018:
Dieudonné theory exists in a somewhat more general setting than finite flat groups over a field. Oda's 1967 thesis gave a connection between
Dieudonné modules and the first de Rham cohomology of abelian varieties, and at about the same time, Grothendieck suggested that there should be a crystalline
1817:
Among the finite flat group schemes, the constants (cf. example above) form a special class, and over an algebraically closed field of characteristic zero, the category of finite groups is equivalent to the category of constant finite group schemes. Over bases with positive characteristic or more
1174:
allows one to make several constructions. Finite direct products of group schemes have a canonical group scheme structure. Given an action of one group scheme on another by automorphisms, one can form semidirect products by following the usual set-theoretic construction. Kernels of group scheme
1743:
Complete connected group schemes are in some sense opposite to affine group schemes, since the completeness implies all global sections are exactly those pulled back from the base, and in particular, they have no nontrivial maps to affine schemes. Any complete group variety (variety here meaning
1736:-algebra). The multiplication, unit, and inverse maps of the group scheme are given by the comultiplication, counit, and antipode structures in the Hopf algebra. The unit and multiplication structures in the Hopf algebra are intrinsic to the underlying scheme. For an arbitrary group scheme
1740:, the ring of global sections also has a commutative Hopf algebra structure, and by taking its spectrum, one obtains the maximal affine quotient group. Affine group varieties are known as linear algebraic groups, since they can be embedded as subgroups of general linear groups.
1822:
is non-constant, because the special fiber isn't smooth. There exist sequences of highly ramified 2-adic rings over which the number of isomorphism types of group schemes of order 2 grows arbitrarily large. More detailed analysis of commutative finite flat group schemes over
1634:, where the additive group acts by translations, and the multiplicative group acts by dilations. The subgroup fixing a chosen basepoint is isomorphic to the multiplicative group, and taking the basepoint to be the identity of an additive group structure identifies
1980:
of Witt co-vectors. This sheaf is more or less dual to the sheaf of Witt vectors (which is in fact representable by a group scheme), since it is constructed by taking a direct limit of finite length Witt vectors under successive
Verschiebung maps
1830:
Commutative finite flat group schemes often occur in nature as subgroup schemes of abelian and semi-abelian varieties, and in positive or mixed characteristic, they can capture a lot of information about the ambient variety. For example, the
1862:-torsion forms a finite flat group scheme over the parametrizing space, and the supersingular locus is where the fibers are connected. This merging of connected components can be studied in fine detail by passing from a modular scheme to a
1744:
reduced and geometrically irreducible separated scheme of finite type over a field) is automatically commutative, by an argument involving the action of conjugation on jet spaces of the identity. Complete group varieties are called
1182:
with respect to some morphism of base schemes, although one needs finiteness conditions to be satisfied to ensure representability of the resulting functor. When this morphism is along a finite extension of fields, it is known as
1154:
is a finite group. However, one can take a projective limit of finite constant group schemes to get profinite group schemes, which appear in the study of fundamental groups and Galois representations or in the theory of the
2023:-divisible groups. Galois actions on the group schemes transfer through the equivalences of categories, and the associated deformation theory of Galois representations was used in
1818:
arithmetic structure, additional isomorphism types exist. For example, if 2 is invertible over the base, all group schemes of order 2 are constant, but over the 2-adic integers, Ό
1655:) has a unique group scheme structure with that point as the identity. Unlike the previous positive-dimensional examples, elliptic curves are projective (in particular proper).
1957:
operators, and they may act nontrivially on the Witt vectors. Dieudonne and
Cartier constructed an antiequivalence of categories between finite commutative group schemes over
494:
469:
432:
1810:. The order of a constant group scheme is equal to the order of the corresponding group, and in general, order behaves well with respect to base change and finite flat
1056:. Conjugation is an action by automorphisms, i.e., it commutes with the group structure, and this induces linear actions on naturally derived objects, such as its
796:
2171:
SĂ©minaire de GĂ©omĂ©trie AlgĂ©brique du Bois Marie – 1962–64 – SchĂ©mas en groupes – (SGA 3) – vol. 3 (Lecture notes in mathematics
2150:
SĂ©minaire de GĂ©omĂ©trie AlgĂ©brique du Bois Marie – 1962–64 – SchĂ©mas en groupes – (SGA 3) – vol. 2 (Lecture notes in mathematics
2129:
SĂ©minaire de GĂ©omĂ©trie AlgĂ©brique du Bois Marie – 1962–64 – SchĂ©mas en groupes – (SGA 3) – vol. 1 (Lecture notes in mathematics
1768:
and throughout algebraic geometry. A complete group scheme over a field need not be commutative, however; for example, any finite group scheme is complete.
1087:. There are several other equivalent conditions, such as conjugation inducing a trivial action, or inversion map Îč being a group scheme automorphism.
1999:, and then completing. Many properties of commutative group schemes can be seen by examining the corresponding Dieudonné modules, e.g., connected
1764:
is proper and smooth with geometrically connected fibers. They are automatically projective, and they have many applications, e.g., in geometric
1052:. Right actions are defined similarly. Any group scheme admits natural left and right actions on its underlying scheme by multiplication and
354:
1175:
homomorphisms are group schemes, by taking a fiber product over the unit map from the base. Base change sends group schemes to group schemes.
2321:
2267:
2238:
971:
A homomorphism of group schemes is a map of schemes that respects multiplication. This can be precisely phrased either by saying that a map
304:
789:
299:
2195:
1835:-torsion of an elliptic curve in characteristic zero is locally isomorphic to the constant elementary abelian group scheme of order
1311:
is said to be normal, and the quotient scheme admits a natural group law. Representability holds in many other cases, such as when
1344:
to the multiplicative group of invertible global sections of the structure sheaf. It can be described as the diagonalizable group
2178:
2157:
2136:
2028:
715:
782:
1906:
can be studied by transferring their geometric structure to a (semi-)linear-algebraic setting. The basic object is the
1248:) can be formed as the spectrum of a group ring. More generally, one can form groups of multiplicative type by letting
1451:+ 1 variables by an ideal encoding the invertibility of the determinant. Alternatively, it can be constructed using 2
399:
213:
2045:
1291:) is in general not a sheaf, and even its sheafification is in general not representable as a scheme. However, if
1025:
2362:
2347:
2283:
2040:
1171:
1156:
838:
597:
331:
208:
96:
1543:
to the underlying additive group of global sections of the structure sheaf. Over an affine base such as Spec
1513:
is not invertible in the base, then this scheme is not smooth. In particular, over a field of characteristic
2357:
2352:
2166:
2145:
2124:
1811:
1179:
1159:, and these are affine of infinite type. More generally, by taking a locally constant sheaf of groups on
1118:
992:
936:, satisfying the usual compatibilities of groups (namely associativity of Ό, identity, and inverse axioms)
862:
830:
747:
537:
1396:
form an important class of commutative group schemes, defined either by the property of being locally on
2297:
854:
621:
1681:
1863:
1195:
822:
561:
549:
167:
101:
2222:
2060:
1893:
999:
886:
846:
136:
31:
477:
452:
415:
1881:
1765:
1718:
1053:
944:
850:
842:
818:
121:
93:
1676:
be the connected component of the identity, i.e., the maximal connected subgroup scheme. Then
2317:
2263:
2255:
2234:
2191:
1907:
952:
948:
692:
526:
369:
263:
2309:
2070:
1974:
1973:)-length. The Dieudonné module functor in one direction is given by homomorphisms into the
1854:
connected components (if the curve is ordinary) or one connected component (if the curve is
1184:
677:
669:
661:
653:
645:
633:
573:
513:
503:
345:
287:
162:
131:
2331:
2277:
2248:
2111:
1407:, or as groups of multiplicative type associated to finitely generated free abelian groups.
2327:
2305:
2273:
2262:, Publications de l'Institut Mathématique de l'Université de Nancago , 7, Paris: Hermann,
2244:
2230:
2107:
2103:
2055:
1875:
1745:
1393:
870:
826:
761:
754:
740:
697:
585:
508:
338:
252:
192:
72:
2095:
2075:
1937:), which is a quotient of the ring of noncommutative polynomials, with coefficients in
1652:
866:
858:
768:
704:
394:
374:
311:
276:
197:
187:
172:
157:
111:
88:
1336:
has the punctured affine line as its underlying scheme, and as a functor, it sends an
2341:
1855:
1620:
The automorphism group of the affine line is isomorphic to the semidirect product of
1419:
is an affine algebraic variety that can be viewed as the multiplicative group of the
964:
834:
687:
609:
443:
316:
182:
2050:
2024:
1954:
1938:
1722:
1447:. Over an affine base, one can construct it as a quotient of a polynomial ring in
960:
882:
542:
241:
230:
177:
152:
147:
106:
77:
40:
17:
1455:
variables, with relations describing an ordered pair of mutually inverse matrices.
821:
equipped with a composition law. Group schemes arise naturally as symmetries of
2065:
1057:
810:
1133:, where the number of copies is equal to the number of connected components of
2313:
709:
437:
1884:
taking finite commutative group schemes to finite commutative group schemes.
1164:
530:
845:. Group schemes that are not algebraic groups play a significant role in
48:
67:
1866:, where supersingular points are replaced by discs of positive radius.
1714:. The quotient scheme is the spectrum of a local ring of finite rank.
409:
323:
2011:
is nilpotent, and Ă©tale group schemes correspond to modules for which
1113:, and by choosing an identification of these copies with elements of
861:. The initial development of the theory of group schemes was due to
1802:-module of finite rank. The rank is a locally constant function on
1748:. This generalizes to the notion of abelian scheme; a group scheme
1299:, then the quotient is representable, and admits a canonical left
2304:, Graduate Texts in Mathematics, vol. 66, Berlin, New York:
1178:
Group schemes can be formed from smaller group schemes by taking
1352:) associated to the integers. Over an affine base such as Spec
1167:
on the base can induce non-trivial automorphisms on the fibers.
1117:, one can define the multiplication, unit, and inverse maps by
1303:-action by translation. If the restriction of this action to
2217:
Shatz, Stephen S. (1986), "Group schemes, formal groups, and
1827:-adic rings can be found in Raynaud's work on prolongations.
1060:, and the algebra of left-invariant differential operators.
995:
of functors from schemes to groups (rather than just sets).
1898:
Finite flat commutative group schemes over a perfect field
1163:, one obtains a locally constant group scheme, for which
833:
of group schemes is somewhat better behaved than that of
2100:
Passage au quotient par une relation d'Ă©quivalence plate
2188:
Introduction to algebraic geometry and algebraic groups
1710:
is a smooth group variety that is a subgroup scheme of
1651:
A smooth genus one curve with a marked point (i.e., an
1535:
as its underlying scheme. As a functor, it sends any
480:
455:
418:
2019:
version of the theory that could be used to analyze
1858:). If we consider a family of elliptic curves, the
1567: â 1, and the inverse is given by sending
1217:) to be the set of abelian group homomorphisms from
1109:. As a scheme, it is a disjoint union of copies of
1880:Cartier duality is a scheme-theoretic analogue of
488:
463:
426:
901:equipped with one of the equivalent sets of data
1555:to zero, the multiplication is given by sending
1427:matrix ring variety. As a functor, it sends an
1729:, this is given by the relative spectrum of an
955:is equivalent to the presheaf corresponding to
1668:is a group scheme of finite type over a field
1443:matrices whose entries are global sections of
889:that has fiber products and some final object
1364: − 1), which is also written
1252:be a non-constant sheaf of abelian groups on
790:
8:
1846:, it is a finite flat group scheme of order
1547:, it is the spectrum of the polynomial ring
1756:is abelian if the structural morphism from
1372:to one, multiplication is given by sending
797:
783:
235:
61:
26:
2227:Arithmetic geometry (Storrs, Conn., 1984)
2186:Gabriel, Peter; Demazure, Michel (1980).
1961:of order a power of "p" and modules over
1100:, one can form the constant group scheme
482:
481:
479:
457:
456:
454:
420:
419:
417:
2292:Modular Forms and Fermat's Last Theorem
2260:Groupes algébriques et corps de classes
2087:
1692:has a unique maximal reduced subscheme
1497:= 1. Over an affine base such as Spec
1477:to itself. As a functor, it sends any
353:
119:
29:
2221:-divisible groups", in Cornell, Gary;
1601:, and the kernel is the group scheme α
1384:, and the inverse is given by sending
355:Classification of finite simple groups
2212:Transformation de Fourier généralisée
1594:th powers induces an endomorphism of
7:
2302:Introduction to affine group schemes
2190:. Amsterdam: North-Holland Pub. Co.
1551:. The unit map is given by sending
1368:. The unit map is given by sending
1129:to a product of copies of the group
853:, since they come up in contexts of
2206:Théorie de Dieudonné Crystalline II
1605:. Over an affine base such as Spec
1205:), defined as a functor by setting
1806:, and is called the order of
1784:is finite and flat if and only if
25:
1356:, it is the spectrum of the ring
1221:to invertible global sections of
1194:, one can form the corresponding
947:, such that composition with the
1485:to the group of global sections
47:
2177:(in French). Berlin; New York:
2156:(in French). Berlin; New York:
2135:(in French). Berlin; New York:
1717:Any affine group scheme is the
1641:with the automorphism group of
1295:is finite, flat, and closed in
837:, since all homomorphisms have
1121:. As a functor, it takes any
1079:) is an abelian group for all
841:, and there is a well-behaved
716:Infinite dimensional Lie group
1:
2003:-group schemes correspond to
1000:left action of a group scheme
1267:, the functor that takes an
1071:is commutative if the group
939:a functor from schemes over
489:{\displaystyle \mathbb {Z} }
464:{\displaystyle \mathbb {Z} }
427:{\displaystyle \mathbb {Z} }
2029:ShimuraâTaniyama conjecture
1902:of positive characteristic
1435:to the group of invertible
214:List of group theory topics
2379:
2204:Berthelot, Breen, Messing
2046:Geometric invariant theory
1891:
1873:
2314:10.1007/978-1-4612-6217-6
2288:Finite flat group schemes
1780:over a noetherian scheme
1772:Finite flat group schemes
1682:finite Ă©tale group scheme
1458:For any positive integer
1410:The general linear group
1329:The multiplicative group
1172:fiber products of schemes
905:a triple of morphisms Ό:
817:is a type of object from
2041:Fundamental group scheme
1609:, it is the spectrum of
1501:, it is the spectrum of
1157:fundamental group scheme
332:Elementary abelian group
209:Glossary of group theory
1400:a product of copies of
975:satisfies the equation
2167:Alexandre Grothendieck
2146:Alexandre Grothendieck
2125:Alexandre Grothendieck
1953:are the Frobenius and
1812:restriction of scalars
1586:for some prime number
1259:For a subgroup scheme
1190:For any abelian group
1180:restriction of scalars
1119:transport of structure
993:natural transformation
863:Alexander Grothendieck
855:Galois representations
825:, and they generalize
748:Linear algebraic group
490:
465:
428:
1680:is an extension of a
1590:, then the taking of
1466:is the kernel of the
987:), or by saying that
491:
466:
429:
2229:, Berlin, New York:
2223:Silverman, Joseph H.
2181:. pp. vii, 529.
2102:, Berlin, New York:
1864:rigid analytic space
1531:has the affine line
1319:and both are affine.
1196:diagonalizable group
1024:that induces a left
893:. That is, it is an
881:A group scheme is a
873:in the early 1960s.
478:
453:
416:
2298:Waterhouse, William
2160:. pp. ix, 654.
2139:. pp. xv, 564.
2061:Group-scheme action
2015:is an isomorphism.
2007:-modules for which
1933: −
1524:The additive group
887:category of schemes
847:arithmetic geometry
122:Group homomorphisms
32:Algebraic structure
18:Affine group scheme
2256:Serre, Jean-Pierre
2233:, pp. 29â78,
2165:Demazure, Michel;
2144:Demazure, Michel;
2123:Demazure, Michel;
1882:Pontryagin duality
1793:is a locally free
1766:class field theory
1470:th power map from
1263:of a group scheme
945:category of groups
851:algebraic topology
843:deformation theory
819:algebraic geometry
598:Special orthogonal
486:
461:
424:
305:Lagrange's theorem
2323:978-0-387-90421-4
2269:978-2-7056-1264-1
2240:978-0-387-96311-2
1888:Dieudonné modules
1746:abelian varieties
1721:of a commutative
1703:is perfect, then
1559:to 1 â
1307:is trivial, then
1170:The existence of
949:forgetful functor
807:
806:
382:
381:
264:Alternating group
221:
220:
16:(Redirected from
2370:
2363:Duality theories
2348:Algebraic groups
2334:
2280:
2251:
2201:
2182:
2161:
2140:
2115:
2114:
2092:
2071:Invariant theory
1894:Dieudonné module
1850:that has either
1660:Basic properties
1185:Weil restriction
961:Yoneda embedding
827:algebraic groups
799:
792:
785:
741:Algebraic groups
514:Hyperbolic group
504:Arithmetic group
495:
493:
492:
487:
485:
470:
468:
467:
462:
460:
433:
431:
430:
425:
423:
346:Schur multiplier
300:Cauchy's theorem
288:Quaternion group
236:
62:
51:
38:
27:
21:
2378:
2377:
2373:
2372:
2371:
2369:
2368:
2367:
2338:
2337:
2324:
2306:Springer-Verlag
2296:
2270:
2254:
2241:
2231:Springer-Verlag
2216:
2198:
2185:
2179:Springer-Verlag
2169:, eds. (1970).
2164:
2158:Springer-Verlag
2148:, eds. (1970).
2143:
2137:Springer-Verlag
2127:, eds. (1970).
2122:
2119:
2118:
2104:Springer-Verlag
2096:Raynaud, Michel
2094:
2093:
2089:
2084:
2056:Groupoid scheme
2037:
2027:'s work on the
1998:
1991:
1896:
1890:
1878:
1876:Cartier duality
1872:
1870:Cartier duality
1845:
1821:
1801:
1792:
1776:A group scheme
1774:
1735:
1709:
1698:
1662:
1647:
1640:
1633:
1626:
1604:
1600:
1571:to −
1530:
1520:
1509:−1). If
1476:
1465:
1418:
1406:
1335:
1326:
1227:
1150:if and only if
1146:is affine over
1145:
1108:
1093:
1016:
912:
879:
871:Michel Demazure
859:moduli problems
835:group varieties
803:
774:
773:
762:Abelian variety
755:Reductive group
743:
733:
732:
731:
730:
681:
673:
665:
657:
649:
622:Special unitary
533:
519:
518:
500:
499:
476:
475:
451:
450:
414:
413:
405:
404:
395:Discrete groups
384:
383:
339:Frobenius group
284:
271:
260:
253:Symmetric group
249:
233:
223:
222:
73:Normal subgroup
59:
39:
30:
23:
22:
15:
12:
11:
5:
2376:
2374:
2366:
2365:
2360:
2355:
2350:
2340:
2339:
2336:
2335:
2322:
2294:
2281:
2268:
2252:
2239:
2214:
2208:
2202:
2196:
2183:
2162:
2141:
2117:
2116:
2086:
2085:
2083:
2080:
2079:
2078:
2076:Quotient stack
2073:
2068:
2063:
2058:
2053:
2048:
2043:
2036:
2033:
1996:
1989:
1908:Dieudonné ring
1892:Main article:
1889:
1886:
1874:Main article:
1871:
1868:
1843:
1819:
1797:
1788:
1773:
1770:
1733:
1707:
1696:
1661:
1658:
1657:
1656:
1653:elliptic curve
1649:
1645:
1638:
1631:
1624:
1618:
1602:
1598:
1576:
1528:
1522:
1521:is not smooth.
1518:
1474:
1463:
1456:
1414:
1408:
1404:
1394:Algebraic tori
1333:
1325:
1322:
1321:
1320:
1257:
1225:
1188:
1176:
1168:
1141:
1104:
1096:Given a group
1092:
1089:
1067:-group scheme
1014:
1009:is a morphism
969:
968:
937:
910:
878:
875:
867:Michel Raynaud
805:
804:
802:
801:
794:
787:
779:
776:
775:
772:
771:
769:Elliptic curve
765:
764:
758:
757:
751:
750:
744:
739:
738:
735:
734:
729:
728:
725:
722:
718:
714:
713:
712:
707:
705:Diffeomorphism
701:
700:
695:
690:
684:
683:
679:
675:
671:
667:
663:
659:
655:
651:
647:
642:
641:
630:
629:
618:
617:
606:
605:
594:
593:
582:
581:
570:
569:
562:Special linear
558:
557:
550:General linear
546:
545:
540:
534:
525:
524:
521:
520:
517:
516:
511:
506:
498:
497:
484:
472:
459:
446:
444:Modular groups
442:
441:
440:
435:
422:
406:
403:
402:
397:
391:
390:
389:
386:
385:
380:
379:
378:
377:
372:
367:
364:
358:
357:
351:
350:
349:
348:
342:
341:
335:
334:
329:
320:
319:
317:Hall's theorem
314:
312:Sylow theorems
308:
307:
302:
294:
293:
292:
291:
285:
280:
277:Dihedral group
273:
272:
267:
261:
256:
250:
245:
234:
229:
228:
225:
224:
219:
218:
217:
216:
211:
203:
202:
201:
200:
195:
190:
185:
180:
175:
170:
168:multiplicative
165:
160:
155:
150:
142:
141:
140:
139:
134:
126:
125:
117:
116:
115:
114:
112:Wreath product
109:
104:
99:
97:direct product
91:
89:Quotient group
83:
82:
81:
80:
75:
70:
60:
57:
56:
53:
52:
44:
43:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2375:
2364:
2361:
2359:
2358:Hopf algebras
2356:
2354:
2353:Scheme theory
2351:
2349:
2346:
2345:
2343:
2333:
2329:
2325:
2319:
2315:
2311:
2307:
2303:
2299:
2295:
2293:
2289:
2285:
2282:
2279:
2275:
2271:
2265:
2261:
2257:
2253:
2250:
2246:
2242:
2236:
2232:
2228:
2224:
2220:
2215:
2213:
2209:
2207:
2203:
2199:
2197:0-444-85443-6
2193:
2189:
2184:
2180:
2176:
2174:
2168:
2163:
2159:
2155:
2153:
2147:
2142:
2138:
2134:
2132:
2126:
2121:
2120:
2113:
2109:
2105:
2101:
2097:
2091:
2088:
2081:
2077:
2074:
2072:
2069:
2067:
2064:
2062:
2059:
2057:
2054:
2052:
2049:
2047:
2044:
2042:
2039:
2038:
2034:
2032:
2030:
2026:
2022:
2016:
2014:
2010:
2006:
2002:
1995:
1988:
1984:
1979:
1976:
1975:abelian sheaf
1972:
1968:
1964:
1960:
1956:
1952:
1948:
1944:
1940:
1936:
1932:
1928:
1924:
1920:
1916:
1912:
1909:
1905:
1901:
1895:
1887:
1885:
1883:
1877:
1869:
1867:
1865:
1861:
1857:
1856:supersingular
1853:
1849:
1842:
1838:
1834:
1828:
1826:
1815:
1813:
1809:
1805:
1800:
1796:
1791:
1787:
1783:
1779:
1771:
1769:
1767:
1763:
1759:
1755:
1751:
1747:
1741:
1739:
1732:
1728:
1725:(over a base
1724:
1720:
1715:
1713:
1706:
1702:
1695:
1691:
1687:
1683:
1679:
1675:
1671:
1667:
1664:Suppose that
1659:
1654:
1650:
1644:
1637:
1630:
1623:
1619:
1616:
1612:
1608:
1597:
1593:
1589:
1585:
1581:
1577:
1574:
1570:
1566:
1563: +
1562:
1558:
1554:
1550:
1546:
1542:
1538:
1534:
1527:
1523:
1516:
1512:
1508:
1504:
1500:
1496:
1492:
1488:
1484:
1480:
1473:
1469:
1462:, the group Ό
1461:
1457:
1454:
1450:
1446:
1442:
1438:
1434:
1430:
1426:
1422:
1417:
1413:
1409:
1403:
1399:
1395:
1391:
1387:
1383:
1379:
1375:
1371:
1367:
1363:
1359:
1355:
1351:
1347:
1343:
1339:
1332:
1328:
1327:
1323:
1318:
1315:is closed in
1314:
1310:
1306:
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1266:
1262:
1258:
1255:
1251:
1247:
1243:
1239:
1235:
1231:
1224:
1220:
1216:
1212:
1208:
1204:
1200:
1197:
1193:
1189:
1186:
1181:
1177:
1173:
1169:
1166:
1162:
1158:
1153:
1149:
1144:
1140:
1136:
1132:
1128:
1124:
1120:
1116:
1112:
1107:
1103:
1099:
1095:
1094:
1091:Constructions
1090:
1088:
1086:
1082:
1078:
1074:
1070:
1066:
1061:
1059:
1055:
1051:
1047:
1043:
1039:
1036:) on the set
1035:
1031:
1028:of the group
1027:
1023:
1019:
1012:
1008:
1004:
1001:
996:
994:
990:
986:
982:
978:
974:
966:
965:group functor
963:. (See also:
962:
958:
954:
950:
946:
942:
938:
935:
931:
927:
923:
919:
915:
908:
904:
903:
902:
900:
896:
892:
888:
884:
876:
874:
872:
868:
864:
860:
856:
852:
848:
844:
840:
836:
832:
828:
824:
820:
816:
812:
800:
795:
793:
788:
786:
781:
780:
778:
777:
770:
767:
766:
763:
760:
759:
756:
753:
752:
749:
746:
745:
742:
737:
736:
726:
723:
720:
719:
717:
711:
708:
706:
703:
702:
699:
696:
694:
691:
689:
686:
685:
682:
676:
674:
668:
666:
660:
658:
652:
650:
644:
643:
639:
635:
632:
631:
627:
623:
620:
619:
615:
611:
608:
607:
603:
599:
596:
595:
591:
587:
584:
583:
579:
575:
572:
571:
567:
563:
560:
559:
555:
551:
548:
547:
544:
541:
539:
536:
535:
532:
528:
523:
522:
515:
512:
510:
507:
505:
502:
501:
473:
448:
447:
445:
439:
436:
411:
408:
407:
401:
398:
396:
393:
392:
388:
387:
376:
373:
371:
368:
365:
362:
361:
360:
359:
356:
352:
347:
344:
343:
340:
337:
336:
333:
330:
328:
326:
322:
321:
318:
315:
313:
310:
309:
306:
303:
301:
298:
297:
296:
295:
289:
286:
283:
278:
275:
274:
270:
265:
262:
259:
254:
251:
248:
243:
240:
239:
238:
237:
232:
231:Finite groups
227:
226:
215:
212:
210:
207:
206:
205:
204:
199:
196:
194:
191:
189:
186:
184:
181:
179:
176:
174:
171:
169:
166:
164:
161:
159:
156:
154:
151:
149:
146:
145:
144:
143:
138:
135:
133:
130:
129:
128:
127:
124:
123:
118:
113:
110:
108:
105:
103:
100:
98:
95:
92:
90:
87:
86:
85:
84:
79:
76:
74:
71:
69:
66:
65:
64:
63:
58:Basic notions
55:
54:
50:
46:
45:
42:
37:
33:
28:
19:
2301:
2291:
2287:
2259:
2226:
2218:
2211:
2205:
2187:
2172:
2170:
2151:
2149:
2130:
2128:
2099:
2090:
2051:GIT quotient
2020:
2017:
2012:
2008:
2004:
2000:
1993:
1986:
1982:
1977:
1970:
1966:
1965:with finite
1962:
1958:
1955:Verschiebung
1950:
1946:
1942:
1939:Witt vectors
1934:
1930:
1926:
1922:
1918:
1914:
1910:
1903:
1899:
1897:
1879:
1859:
1851:
1847:
1840:
1836:
1832:
1829:
1824:
1816:
1807:
1803:
1798:
1794:
1789:
1785:
1781:
1777:
1775:
1761:
1757:
1753:
1752:over a base
1749:
1742:
1737:
1730:
1726:
1723:Hopf algebra
1716:
1711:
1704:
1700:
1693:
1689:
1685:
1677:
1673:
1669:
1665:
1663:
1642:
1635:
1628:
1621:
1614:
1610:
1606:
1595:
1591:
1587:
1583:
1579:
1572:
1568:
1564:
1560:
1556:
1552:
1548:
1544:
1540:
1536:
1532:
1525:
1514:
1510:
1506:
1502:
1498:
1494:
1490:
1486:
1482:
1478:
1471:
1467:
1459:
1452:
1448:
1444:
1440:
1436:
1432:
1428:
1424:
1420:
1415:
1411:
1401:
1397:
1389:
1385:
1381:
1377:
1373:
1369:
1365:
1361:
1357:
1353:
1349:
1345:
1341:
1337:
1330:
1316:
1312:
1308:
1304:
1300:
1296:
1292:
1288:
1284:
1280:
1276:
1272:
1268:
1264:
1260:
1253:
1249:
1245:
1241:
1237:
1233:
1229:
1222:
1218:
1214:
1210:
1206:
1202:
1198:
1191:
1160:
1151:
1147:
1142:
1138:
1134:
1130:
1126:
1122:
1114:
1110:
1105:
1101:
1097:
1084:
1080:
1076:
1072:
1068:
1064:
1062:
1049:
1045:
1041:
1037:
1033:
1029:
1021:
1017:
1010:
1006:
1005:on a scheme
1002:
997:
988:
984:
980:
976:
972:
970:
956:
940:
933:
929:
925:
921:
917:
913:
906:
898:
894:
890:
883:group object
880:
815:group scheme
814:
808:
637:
625:
613:
601:
589:
577:
565:
553:
324:
281:
268:
257:
246:
242:Cyclic group
120:
107:Free product
78:Group action
41:Group theory
36:Group theory
35:
2066:Group-stack
1839:, but over
1240:is affine,
1058:Lie algebra
1054:conjugation
811:mathematics
527:Topological
366:alternating
2342:Categories
2082:References
1493:such that
1044:) for any
959:under the
877:Definition
634:Symplectic
574:Orthogonal
531:Lie groups
438:Free group
163:continuous
102:Direct sum
2284:John Tate
1699:, and if
1228:for each
1165:monodromy
1083:-schemes
928:, and Îč:
698:Conformal
586:Euclidean
193:nilpotent
2300:(1979),
2258:(1984),
2225:(eds.),
2210:Laumon,
2098:(1967),
2035:See also
1719:spectrum
1539:-scheme
1481:-scheme
1431:-scheme
1340:-scheme
1324:Examples
1271:-scheme
1232:-scheme
1125:-scheme
1048:-scheme
897:-scheme
831:category
693:Poincaré
538:Solenoid
410:Integers
400:Lattices
375:sporadic
370:Lie type
198:solvable
188:dihedral
173:additive
158:infinite
68:Subgroup
2332:0547117
2290:, from
2278:0907288
2249:0861972
2112:0232781
1672:. Let
1582:= 0 in
943:to the
839:kernels
823:schemes
688:Lorentz
610:Unitary
509:Lattice
449:PSL(2,
183:abelian
94:(Semi-)
2330:
2320:
2276:
2266:
2247:
2237:
2194:
2110:
1236:. If
1026:action
979:Ό = Ό(
543:Circle
474:SL(2,
363:cyclic
327:-group
178:cyclic
153:finite
148:simple
132:kernel
2025:Wiles
991:is a
920:, e:
885:in a
727:Sp(â)
724:SU(â)
137:image
2318:ISBN
2264:ISBN
2235:ISBN
2192:ISBN
1949:and
953:sets
869:and
857:and
849:and
813:, a
721:O(â)
710:Loop
529:and
2310:doi
2173:153
2152:152
2131:151
1997:n+1
1945:.
1941:of
1929:}/(
1760:to
1708:red
1697:red
1688:.
1684:by
1627:by
1578:If
1517:, Ό
1489:of
1439:by
1423:by
1392:.
1388:to
1376:to
1275:to
1137:.
1063:An
951:to
809:In
636:Sp(
624:SU(
600:SO(
564:SL(
552:GL(
2344::
2328:MR
2326:,
2316:,
2308:,
2286:,
2274:MR
2272:,
2245:MR
2243:,
2108:MR
2106:,
2031:.
1992:â
1985::
1978:CW
1931:FV
1921:){
1913:=
1814:.
1617:).
1613:/(
1505:/(
1412:GL
1380:â
1362:xy
1360:/(
1283:)/
1213:)(
1020:â
998:A
983:Ă
967:.)
932:â
924:â
916:â
865:,
612:U(
588:E(
576:O(
34:â
2312::
2219:p
2200:.
2175:)
2154:)
2133:)
2021:p
2013:F
2009:F
2005:D
2001:p
1994:W
1990:n
1987:W
1983:V
1971:k
1969:(
1967:W
1963:D
1959:k
1951:V
1947:F
1943:k
1935:p
1927:V
1925:,
1923:F
1919:k
1917:(
1915:W
1911:D
1904:p
1900:k
1860:p
1852:p
1848:p
1844:p
1841:F
1837:p
1833:p
1825:p
1820:2
1808:G
1804:S
1799:S
1795:O
1790:G
1786:O
1782:S
1778:G
1762:S
1758:G
1754:S
1750:G
1738:G
1734:S
1731:O
1727:S
1712:G
1705:G
1701:k
1694:G
1690:G
1686:G
1678:G
1674:G
1670:k
1666:G
1648:.
1646:a
1643:G
1639:m
1636:G
1632:m
1629:G
1625:a
1622:G
1615:x
1611:A
1607:A
1603:p
1599:a
1596:G
1592:p
1588:p
1584:S
1580:p
1575:.
1573:x
1569:x
1565:x
1561:x
1557:x
1553:x
1549:A
1545:A
1541:T
1537:S
1533:A
1529:a
1526:G
1519:p
1515:p
1511:n
1507:x
1503:A
1499:A
1495:f
1491:T
1487:f
1483:T
1479:S
1475:m
1472:G
1468:n
1464:n
1460:n
1453:n
1449:n
1445:T
1441:n
1437:n
1433:T
1429:S
1425:n
1421:n
1416:n
1405:m
1402:G
1398:S
1390:x
1386:x
1382:x
1378:x
1374:x
1370:x
1366:A
1358:A
1354:A
1350:Z
1348:(
1346:D
1342:T
1338:S
1334:m
1331:G
1317:G
1313:H
1309:H
1305:H
1301:G
1297:G
1293:H
1289:T
1287:(
1285:H
1281:T
1279:(
1277:G
1273:T
1269:S
1265:G
1261:H
1256:.
1254:S
1250:A
1246:A
1244:(
1242:D
1238:S
1234:T
1230:S
1226:T
1223:O
1219:A
1215:T
1211:A
1209:(
1207:D
1203:A
1201:(
1199:D
1192:A
1187:.
1161:S
1152:G
1148:S
1143:S
1139:G
1135:T
1131:G
1127:T
1123:S
1115:G
1111:S
1106:S
1102:G
1098:G
1085:T
1081:S
1077:T
1075:(
1073:G
1069:G
1065:S
1050:T
1046:S
1042:T
1040:(
1038:X
1034:T
1032:(
1030:G
1022:X
1018:X
1015:S
1013:Ă
1011:G
1007:X
1003:G
989:f
985:f
981:f
977:f
973:f
957:G
941:S
934:G
930:G
926:G
922:S
918:G
914:G
911:S
909:Ă
907:G
899:G
895:S
891:S
798:e
791:t
784:v
680:8
678:E
672:7
670:E
664:6
662:E
656:4
654:F
648:2
646:G
640:)
638:n
628:)
626:n
616:)
614:n
604:)
602:n
592:)
590:n
580:)
578:n
568:)
566:n
556:)
554:n
496:)
483:Z
471:)
458:Z
434:)
421:Z
412:(
325:p
290:Q
282:n
279:D
269:n
266:A
258:n
255:S
247:n
244:Z
20:)
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