38:
818:, 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
2007:
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
1806:
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
1163:
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
1732:
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
1725:-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
1729:, 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.
1811:
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
1623:, 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
1969:
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
1819:
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
1851:-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
1733:
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
1171:
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
1143:
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
2012:-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
1807:
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
1644:) 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).
1946:
operators, and they may act nontrivially on the Witt vectors. Dieudonne and
Cartier constructed an antiequivalence of categories between finite commutative group schemes over
483:
458:
421:
1799:. 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
1045:. 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
2160:
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
2139:
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
2118:
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
1757:
and throughout algebraic geometry. A complete group scheme over a field need not be commutative, however; for example, any finite group scheme is complete.
1076:. There are several other equivalent conditions, such as conjugation inducing a trivial action, or inversion map Îč being a group scheme automorphism.
1988:, and then completing. Many properties of commutative group schemes can be seen by examining the corresponding Dieudonné modules, e.g., connected
1753:
is proper and smooth with geometrically connected fibers. They are automatically projective, and they have many applications, e.g., in geometric
1041:. Right actions are defined similarly. Any group scheme admits natural left and right actions on its underlying scheme by multiplication and
785:
1164:
homomorphisms are group schemes, by taking a fiber product over the unit map from the base. Base change sends group schemes to group schemes.
960:
A homomorphism of group schemes is a map of schemes that respects multiplication. This can be precisely phrased either by saying that a map
1824:-torsion of an elliptic curve in characteristic zero is locally isomorphic to the constant elementary abelian group scheme of order
1300:
is said to be normal, and the quotient scheme admits a natural group law. Representability holds in many other cases, such as when
1333:
to the multiplicative group of invertible global sections of the structure sheaf. It can be described as the diagonalizable group
343:
2310:
2256:
2227:
293:
1895:
can be studied by transferring their geometric structure to a (semi-)linear-algebraic setting. The basic object is the
1237:) can be formed as the spectrum of a group ring. More generally, one can form groups of multiplicative type by letting
778:
288:
1440:+ 1 variables by an ideal encoding the invertibility of the determinant. Alternatively, it can be constructed using 2
2184:
2167:
2146:
2125:
1280:) is in general not a sheaf, and even its sheafification is in general not representable as a scheme. However, if
2017:
704:
1532:
to the underlying additive group of global sections of the structure sheaf. Over an affine base such as Spec
1502:
is not invertible in the base, then this scheme is not smooth. In particular, over a field of characteristic
771:
1148:, and these are affine of infinite type. More generally, by taking a locally constant sheaf of groups on
925:, satisfying the usual compatibilities of groups (namely associativity of Ό, identity, and inverse axioms)
1385:
form an important class of commutative group schemes, defined either by the property of being locally on
388:
202:
2034:
1014:
120:
2351:
2336:
2272:
2029:
1160:
1145:
827:
586:
320:
197:
85:
466:
441:
404:
2346:
2341:
2155:
2134:
2113:
1800:
1665:
be the connected component of the identity, i.e., the maximal connected subgroup scheme. Then
1168:
1107:
981:
851:
819:
736:
526:
2286:
843:
610:
1962:)-length. The Dieudonné module functor in one direction is given by homomorphisms into the
1843:
connected components (if the curve is ordinary) or one connected component (if the curve is
1670:
2320:
2266:
2237:
2100:
1852:
1396:, or as groups of multiplicative type associated to finitely generated free abelian groups.
1184:
811:
550:
538:
156:
90:
2251:, Publications de l'Institut Mathématique de l'Université de Nancago , 7, Paris: Hermann,
8:
2211:
2049:
988:
875:
835:
125:
20:
1882:
1926:), which is a quotient of the ring of noncommutative polynomials, with coefficients in
1870:
1754:
1707:
1042:
933:
839:
831:
807:
110:
82:
1325:
has the punctured affine line as its underlying scheme, and as a functor, it sends an
2306:
2252:
2244:
2223:
2180:
1609:
The automorphism group of the affine line is isomorphic to the semidirect product of
1408:
is an affine algebraic variety that can be viewed as the multiplicative group of the
941:
937:
515:
358:
252:
1896:
681:
2298:
2059:
1963:
1436:. Over an affine base, one can construct it as a quotient of a polynomial ring in
1173:
666:
658:
650:
642:
634:
622:
562:
502:
492:
334:
276:
151:
1444:
variables, with relations describing an ordered pair of mutually inverse matrices.
2316:
2294:
2262:
2233:
2219:
2096:
2092:
2044:
1864:
1734:
1382:
859:
815:
810:
equipped with a composition law. Group schemes arise naturally as symmetries of
750:
743:
729:
686:
574:
497:
327:
241:
181:
61:
2084:
2064:
1641:
1122:, where the number of copies is equal to the number of connected components of
855:
847:
757:
693:
383:
363:
300:
265:
186:
176:
161:
146:
100:
77:
2302:
2330:
1873:
taking finite commutative group schemes to finite commutative group schemes.
1844:
953:
823:
676:
598:
432:
305:
171:
2039:
2013:
1943:
1927:
1711:
949:
871:
531:
230:
219:
166:
141:
136:
95:
66:
29:
834:. Group schemes that are not algebraic groups play a significant role in
2054:
1046:
799:
1855:, where supersingular points are replaced by discs of positive radius.
1703:. The quotient scheme is the spectrum of a local ring of finite rank.
698:
426:
2000:
is nilpotent, and Ă©tale group schemes correspond to modules for which
1153:
1102:, and by choosing an identification of these copies with elements of
850:. The initial development of the theory of group schemes was due to
519:
1791:-module of finite rank. The rank is a locally constant function on
1737:. This generalizes to the notion of abelian scheme; a group scheme
56:
1288:, then the quotient is representable, and admits a canonical left
2293:, Graduate Texts in Mathematics, vol. 66, Berlin, New York:
1167:
Group schemes can be formed from smaller group schemes by taking
398:
312:
1341:) associated to the integers. Over an affine base such as Spec
1156:
on the base can induce non-trivial automorphisms on the fibers.
1106:, one can define the multiplication, unit, and inverse maps by
37:
2174:
2153:
2132:
2111:
1292:-action by translation. If the restriction of this action to
2206:
Shatz, Stephen S. (1986), "Group schemes, formal groups, and
1816:-adic rings can be found in Raynaud's work on prolongations.
1049:, and the algebra of left-invariant differential operators.
984:
of functors from schemes to groups (rather than just sets).
1887:
Finite flat commutative group schemes over a perfect field
1152:, one obtains a locally constant group scheme, for which
822:
of group schemes is somewhat better behaved than that of
2089:
Passage au quotient par une relation d'Ă©quivalence plate
2177:
Introduction to algebraic geometry and algebraic groups
1699:
is a smooth group variety that is a subgroup scheme of
1640:
A smooth genus one curve with a marked point (i.e., an
1524:
as its underlying scheme. As a functor, it sends any
469:
444:
407:
2008:
version of the theory that could be used to analyze
1847:). If we consider a family of elliptic curves, the
1556: â 1, and the inverse is given by sending
1206:) to be the set of abelian group homomorphisms from
1098:. As a scheme, it is a disjoint union of copies of
1869:Cartier duality is a scheme-theoretic analogue of
477:
452:
415:
890:equipped with one of the equivalent sets of data
2328:
1544:to zero, the multiplication is given by sending
1416:matrix ring variety. As a functor, it sends an
1718:, this is given by the relative spectrum of an
944:is equivalent to the presheaf corresponding to
1657:is a group scheme of finite type over a field
1432:matrices whose entries are global sections of
878:that has fiber products and some final object
1353: − 1), which is also written
1241:be a non-constant sheaf of abelian groups on
779:
1835:, it is a finite flat group scheme of order
1760:
1536:, it is the spectrum of the polynomial ring
1745:is abelian if the structural morphism from
1361:to one, multiplication is given by sending
2285:
786:
772:
2216:Arithmetic geometry (Storrs, Conn., 1984)
2175:Gabriel, Peter; Demazure, Michel (1980).
1950:of order a power of "p" and modules over
1089:, one can form the constant group scheme
471:
446:
409:
2281:Modular Forms and Fermat's Last Theorem
2249:Groupes algébriques et corps de classes
2083:
1681:has a unique maximal reduced subscheme
1486:= 1. Over an affine base such as Spec
1466:to itself. As a functor, it sends any
2329:
2210:-divisible groups", in Cornell, Gary;
1590:, and the kernel is the group scheme α
1373:, and the inverse is given by sending
344:Classification of finite simple groups
2243:
2205:
2201:Transformation de Fourier généralisée
1583:th powers induces an endomorphism of
2291:Introduction to affine group schemes
2179:. Amsterdam: North-Holland Pub. Co.
1876:
1540:. The unit map is given by sending
1357:. The unit map is given by sending
1118:to a product of copies of the group
842:, since they come up in contexts of
2195:Théorie de Dieudonné Crystalline II
1648:
1594:. Over an affine base such as Spec
1194:), defined as a functor by setting
13:
1858:
1795:, and is called the order of
1773:is finite and flat if and only if
14:
2363:
1345:, it is the spectrum of the ring
1210:to invertible global sections of
1183:, one can form the corresponding
936:, such that composition with the
1474:to the group of global sections
1079:
36:
2166:(in French). Berlin; New York:
2145:(in French). Berlin; New York:
2124:(in French). Berlin; New York:
1706:Any affine group scheme is the
1630:with the automorphism group of
1284:is finite, flat, and closed in
826:, since all homomorphisms have
2077:
1110:. As a functor, it takes any
1068:) is an abelian group for all
830:, and there is a well-behaved
705:Infinite dimensional Lie group
1:
2070:
1992:-group schemes correspond to
989:left action of a group scheme
865:
1256:, the functor that takes an
1060:is commutative if the group
928:a functor from schemes over
478:{\displaystyle \mathbb {Z} }
453:{\displaystyle \mathbb {Z} }
416:{\displaystyle \mathbb {Z} }
7:
2023:
2018:ShimuraâTaniyama conjecture
1891:of positive characteristic
1424:to the group of invertible
1312:
203:List of group theory topics
10:
2368:
2193:Berthelot, Breen, Messing
2035:Geometric invariant theory
1880:
1862:
2303:10.1007/978-1-4612-6217-6
2277:Finite flat group schemes
1769:over a noetherian scheme
1761:Finite flat group schemes
1671:finite Ă©tale group scheme
1447:For any positive integer
1399:The general linear group
1318:The multiplicative group
1161:fiber products of schemes
894:a triple of morphisms Ό:
806:is a type of object from
2030:Fundamental group scheme
1598:, it is the spectrum of
1490:, it is the spectrum of
1146:fundamental group scheme
321:Elementary abelian group
198:Glossary of group theory
1389:a product of copies of
964:satisfies the equation
2156:Alexandre Grothendieck
2135:Alexandre Grothendieck
2114:Alexandre Grothendieck
1942:are the Frobenius and
1801:restriction of scalars
1575:for some prime number
1248:For a subgroup scheme
1179:For any abelian group
1169:restriction of scalars
1108:transport of structure
982:natural transformation
852:Alexander Grothendieck
844:Galois representations
814:, and they generalize
737:Linear algebraic group
479:
454:
417:
1669:is an extension of a
1579:, then the taking of
1455:is the kernel of the
976:), or by saying that
480:
455:
418:
2218:, Berlin, New York:
2212:Silverman, Joseph H.
2170:. pp. vii, 529.
2091:, Berlin, New York:
1853:rigid analytic space
1520:has the affine line
1308:and both are affine.
1185:diagonalizable group
1013:that induces a left
882:. That is, it is an
870:A group scheme is a
862:in the early 1960s.
467:
442:
405:
2287:Waterhouse, William
2149:. pp. ix, 654.
2128:. pp. xv, 564.
2050:Group-scheme action
2004:is an isomorphism.
1996:-modules for which
1922: −
1513:The additive group
876:category of schemes
836:arithmetic geometry
111:Group homomorphisms
21:Algebraic structure
2245:Serre, Jean-Pierre
2222:, pp. 29â78,
2154:Demazure, Michel;
2133:Demazure, Michel;
2112:Demazure, Michel;
1871:Pontryagin duality
1782:is a locally free
1755:class field theory
1459:th power map from
1252:of a group scheme
934:category of groups
840:algebraic topology
832:deformation theory
808:algebraic geometry
587:Special orthogonal
475:
450:
413:
294:Lagrange's theorem
2312:978-0-387-90421-4
2258:978-2-7056-1264-1
2229:978-0-387-96311-2
1877:Dieudonné modules
1735:abelian varieties
1710:of a commutative
1692:is perfect, then
1548:to 1 â
1296:is trivial, then
1159:The existence of
938:forgetful functor
796:
795:
371:
370:
253:Alternating group
210:
209:
2359:
2352:Duality theories
2337:Algebraic groups
2323:
2269:
2240:
2190:
2171:
2150:
2129:
2104:
2103:
2081:
2060:Invariant theory
1883:Dieudonné module
1839:that has either
1649:Basic properties
1174:Weil restriction
950:Yoneda embedding
816:algebraic groups
788:
781:
774:
730:Algebraic groups
503:Hyperbolic group
493:Arithmetic group
484:
482:
481:
476:
474:
459:
457:
456:
451:
449:
422:
420:
419:
414:
412:
335:Schur multiplier
289:Cauchy's theorem
277:Quaternion group
225:
224:
51:
50:
40:
27:
16:
15:
2367:
2366:
2362:
2361:
2360:
2358:
2357:
2356:
2327:
2326:
2313:
2295:Springer-Verlag
2259:
2230:
2220:Springer-Verlag
2187:
2168:Springer-Verlag
2158:, eds. (1970).
2147:Springer-Verlag
2137:, eds. (1970).
2126:Springer-Verlag
2116:, eds. (1970).
2108:
2107:
2093:Springer-Verlag
2085:Raynaud, Michel
2082:
2078:
2073:
2045:Groupoid scheme
2026:
2016:'s work on the
1987:
1980:
1885:
1879:
1867:
1865:Cartier duality
1861:
1859:Cartier duality
1834:
1810:
1790:
1781:
1765:A group scheme
1763:
1724:
1698:
1687:
1651:
1636:
1629:
1622:
1615:
1593:
1589:
1560:to −
1519:
1509:
1498:−1). If
1465:
1454:
1407:
1395:
1324:
1315:
1216:
1139:if and only if
1135:is affine over
1134:
1097:
1082:
1005:
901:
868:
860:Michel Demazure
848:moduli problems
824:group varieties
792:
763:
762:
751:Abelian variety
744:Reductive group
732:
722:
721:
720:
719:
670:
662:
654:
646:
638:
611:Special unitary
522:
508:
507:
489:
488:
470:
468:
465:
464:
445:
443:
440:
439:
408:
406:
403:
402:
394:
393:
384:Discrete groups
373:
372:
328:Frobenius group
273:
260:
249:
242:Symmetric group
238:
222:
212:
211:
62:Normal subgroup
48:
28:
19:
12:
11:
5:
2365:
2355:
2354:
2349:
2344:
2339:
2325:
2324:
2311:
2283:
2270:
2257:
2241:
2228:
2203:
2197:
2191:
2185:
2172:
2151:
2130:
2106:
2105:
2075:
2074:
2072:
2069:
2068:
2067:
2065:Quotient stack
2062:
2057:
2052:
2047:
2042:
2037:
2032:
2025:
2022:
1985:
1978:
1897:Dieudonné ring
1881:Main article:
1878:
1875:
1863:Main article:
1860:
1857:
1832:
1808:
1786:
1777:
1762:
1759:
1722:
1696:
1685:
1650:
1647:
1646:
1645:
1642:elliptic curve
1638:
1634:
1627:
1620:
1613:
1607:
1591:
1587:
1565:
1517:
1511:
1510:is not smooth.
1507:
1463:
1452:
1445:
1403:
1397:
1393:
1383:Algebraic tori
1322:
1314:
1311:
1310:
1309:
1246:
1214:
1177:
1165:
1157:
1130:
1093:
1085:Given a group
1081:
1078:
1056:-group scheme
1003:
998:is a morphism
958:
957:
926:
899:
867:
864:
856:Michel Raynaud
794:
793:
791:
790:
783:
776:
768:
765:
764:
761:
760:
758:Elliptic curve
754:
753:
747:
746:
740:
739:
733:
728:
727:
724:
723:
718:
717:
714:
711:
707:
703:
702:
701:
696:
694:Diffeomorphism
690:
689:
684:
679:
673:
672:
668:
664:
660:
656:
652:
648:
644:
640:
636:
631:
630:
619:
618:
607:
606:
595:
594:
583:
582:
571:
570:
559:
558:
551:Special linear
547:
546:
539:General linear
535:
534:
529:
523:
514:
513:
510:
509:
506:
505:
500:
495:
487:
486:
473:
461:
448:
435:
433:Modular groups
431:
430:
429:
424:
411:
395:
392:
391:
386:
380:
379:
378:
375:
374:
369:
368:
367:
366:
361:
356:
353:
347:
346:
340:
339:
338:
337:
331:
330:
324:
323:
318:
309:
308:
306:Hall's theorem
303:
301:Sylow theorems
297:
296:
291:
283:
282:
281:
280:
274:
269:
266:Dihedral group
262:
261:
256:
250:
245:
239:
234:
223:
218:
217:
214:
213:
208:
207:
206:
205:
200:
192:
191:
190:
189:
184:
179:
174:
169:
164:
159:
157:multiplicative
154:
149:
144:
139:
131:
130:
129:
128:
123:
115:
114:
106:
105:
104:
103:
101:Wreath product
98:
93:
88:
86:direct product
80:
78:Quotient group
72:
71:
70:
69:
64:
59:
49:
46:
45:
42:
41:
33:
32:
9:
6:
4:
3:
2:
2364:
2353:
2350:
2348:
2347:Hopf algebras
2345:
2343:
2342:Scheme theory
2340:
2338:
2335:
2334:
2332:
2322:
2318:
2314:
2308:
2304:
2300:
2296:
2292:
2288:
2284:
2282:
2278:
2274:
2271:
2268:
2264:
2260:
2254:
2250:
2246:
2242:
2239:
2235:
2231:
2225:
2221:
2217:
2213:
2209:
2204:
2202:
2198:
2196:
2192:
2188:
2186:0-444-85443-6
2182:
2178:
2173:
2169:
2165:
2163:
2157:
2152:
2148:
2144:
2142:
2136:
2131:
2127:
2123:
2121:
2115:
2110:
2109:
2102:
2098:
2094:
2090:
2086:
2080:
2076:
2066:
2063:
2061:
2058:
2056:
2053:
2051:
2048:
2046:
2043:
2041:
2038:
2036:
2033:
2031:
2028:
2027:
2021:
2019:
2015:
2011:
2005:
2003:
1999:
1995:
1991:
1984:
1977:
1973:
1968:
1965:
1964:abelian sheaf
1961:
1957:
1953:
1949:
1945:
1941:
1937:
1933:
1929:
1925:
1921:
1917:
1913:
1909:
1905:
1901:
1898:
1894:
1890:
1884:
1874:
1872:
1866:
1856:
1854:
1850:
1846:
1845:supersingular
1842:
1838:
1831:
1827:
1823:
1817:
1815:
1804:
1802:
1798:
1794:
1789:
1785:
1780:
1776:
1772:
1768:
1758:
1756:
1752:
1748:
1744:
1740:
1736:
1730:
1728:
1721:
1717:
1714:(over a base
1713:
1709:
1704:
1702:
1695:
1691:
1684:
1680:
1676:
1672:
1668:
1664:
1660:
1656:
1653:Suppose that
1643:
1639:
1633:
1626:
1619:
1612:
1608:
1605:
1601:
1597:
1586:
1582:
1578:
1574:
1570:
1566:
1563:
1559:
1555:
1552: +
1551:
1547:
1543:
1539:
1535:
1531:
1527:
1523:
1516:
1512:
1505:
1501:
1497:
1493:
1489:
1485:
1481:
1477:
1473:
1469:
1462:
1458:
1451:, the group Ό
1450:
1446:
1443:
1439:
1435:
1431:
1427:
1423:
1419:
1415:
1411:
1406:
1402:
1398:
1392:
1388:
1384:
1380:
1376:
1372:
1368:
1364:
1360:
1356:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1321:
1317:
1316:
1307:
1304:is closed in
1303:
1299:
1295:
1291:
1287:
1283:
1279:
1275:
1271:
1267:
1263:
1259:
1255:
1251:
1247:
1244:
1240:
1236:
1232:
1228:
1224:
1220:
1213:
1209:
1205:
1201:
1197:
1193:
1189:
1186:
1182:
1178:
1175:
1170:
1166:
1162:
1158:
1155:
1151:
1147:
1142:
1138:
1133:
1129:
1125:
1121:
1117:
1113:
1109:
1105:
1101:
1096:
1092:
1088:
1084:
1083:
1080:Constructions
1077:
1075:
1071:
1067:
1063:
1059:
1055:
1050:
1048:
1044:
1040:
1036:
1032:
1028:
1025:) on the set
1024:
1020:
1017:of the group
1016:
1012:
1008:
1001:
997:
993:
990:
985:
983:
979:
975:
971:
967:
963:
955:
954:group functor
952:. (See also:
951:
947:
943:
939:
935:
931:
927:
924:
920:
916:
912:
908:
904:
897:
893:
892:
891:
889:
885:
881:
877:
873:
863:
861:
857:
853:
849:
845:
841:
837:
833:
829:
825:
821:
817:
813:
809:
805:
801:
789:
784:
782:
777:
775:
770:
769:
767:
766:
759:
756:
755:
752:
749:
748:
745:
742:
741:
738:
735:
734:
731:
726:
725:
715:
712:
709:
708:
706:
700:
697:
695:
692:
691:
688:
685:
683:
680:
678:
675:
674:
671:
665:
663:
657:
655:
649:
647:
641:
639:
633:
632:
628:
624:
621:
620:
616:
612:
609:
608:
604:
600:
597:
596:
592:
588:
585:
584:
580:
576:
573:
572:
568:
564:
561:
560:
556:
552:
549:
548:
544:
540:
537:
536:
533:
530:
528:
525:
524:
521:
517:
512:
511:
504:
501:
499:
496:
494:
491:
490:
462:
437:
436:
434:
428:
425:
400:
397:
396:
390:
387:
385:
382:
381:
377:
376:
365:
362:
360:
357:
354:
351:
350:
349:
348:
345:
342:
341:
336:
333:
332:
329:
326:
325:
322:
319:
317:
315:
311:
310:
307:
304:
302:
299:
298:
295:
292:
290:
287:
286:
285:
284:
278:
275:
272:
267:
264:
263:
259:
254:
251:
248:
243:
240:
237:
232:
229:
228:
227:
226:
221:
220:Finite groups
216:
215:
204:
201:
199:
196:
195:
194:
193:
188:
185:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
158:
155:
153:
150:
148:
145:
143:
140:
138:
135:
134:
133:
132:
127:
124:
122:
119:
118:
117:
116:
113:
112:
108:
107:
102:
99:
97:
94:
92:
89:
87:
84:
81:
79:
76:
75:
74:
73:
68:
65:
63:
60:
58:
55:
54:
53:
52:
47:Basic notions
44:
43:
39:
35:
34:
31:
26:
22:
18:
17:
2290:
2280:
2276:
2248:
2215:
2207:
2200:
2194:
2176:
2161:
2159:
2140:
2138:
2119:
2117:
2088:
2079:
2040:GIT quotient
2009:
2006:
2001:
1997:
1993:
1989:
1982:
1975:
1971:
1966:
1959:
1955:
1954:with finite
1951:
1947:
1944:Verschiebung
1939:
1935:
1931:
1928:Witt vectors
1923:
1919:
1915:
1911:
1907:
1903:
1899:
1892:
1888:
1886:
1868:
1848:
1840:
1836:
1829:
1825:
1821:
1818:
1813:
1805:
1796:
1792:
1787:
1783:
1778:
1774:
1770:
1766:
1764:
1750:
1746:
1742:
1741:over a base
1738:
1731:
1726:
1719:
1715:
1712:Hopf algebra
1705:
1700:
1693:
1689:
1682:
1678:
1674:
1666:
1662:
1658:
1654:
1652:
1631:
1624:
1617:
1610:
1603:
1599:
1595:
1584:
1580:
1576:
1572:
1568:
1561:
1557:
1553:
1549:
1545:
1541:
1537:
1533:
1529:
1525:
1521:
1514:
1503:
1499:
1495:
1491:
1487:
1483:
1479:
1475:
1471:
1467:
1460:
1456:
1448:
1441:
1437:
1433:
1429:
1425:
1421:
1417:
1413:
1409:
1404:
1400:
1390:
1386:
1378:
1374:
1370:
1366:
1362:
1358:
1354:
1350:
1346:
1342:
1338:
1334:
1330:
1326:
1319:
1305:
1301:
1297:
1293:
1289:
1285:
1281:
1277:
1273:
1269:
1265:
1261:
1257:
1253:
1249:
1242:
1238:
1234:
1230:
1226:
1222:
1218:
1211:
1207:
1203:
1199:
1195:
1191:
1187:
1180:
1149:
1140:
1136:
1131:
1127:
1123:
1119:
1115:
1111:
1103:
1099:
1094:
1090:
1086:
1073:
1069:
1065:
1061:
1057:
1053:
1051:
1038:
1034:
1030:
1026:
1022:
1018:
1010:
1006:
999:
995:
994:on a scheme
991:
986:
977:
973:
969:
965:
961:
959:
945:
929:
922:
918:
914:
910:
906:
902:
895:
887:
883:
879:
872:group object
869:
804:group scheme
803:
797:
626:
614:
602:
590:
578:
566:
554:
542:
313:
270:
257:
246:
235:
231:Cyclic group
109:
96:Free product
67:Group action
30:Group theory
25:Group theory
24:
2055:Group-stack
1828:, but over
1229:is affine,
1047:Lie algebra
1043:conjugation
800:mathematics
516:Topological
355:alternating
2331:Categories
2071:References
1482:such that
1033:) for any
948:under the
866:Definition
623:Symplectic
563:Orthogonal
520:Lie groups
427:Free group
152:continuous
91:Direct sum
2273:John Tate
1688:, and if
1217:for each
1154:monodromy
1072:-schemes
917:, and Îč:
687:Conformal
575:Euclidean
182:nilpotent
2289:(1979),
2247:(1984),
2214:(eds.),
2199:Laumon,
2087:(1967),
2024:See also
1708:spectrum
1528:-scheme
1470:-scheme
1420:-scheme
1329:-scheme
1313:Examples
1260:-scheme
1221:-scheme
1114:-scheme
1037:-scheme
886:-scheme
820:category
682:Poincaré
527:Solenoid
399:Integers
389:Lattices
364:sporadic
359:Lie type
187:solvable
177:dihedral
162:additive
147:infinite
57:Subgroup
2321:0547117
2279:, from
2267:0907288
2238:0861972
2101:0232781
1661:. Let
1571:= 0 in
932:to the
828:kernels
812:schemes
677:Lorentz
599:Unitary
498:Lattice
438:PSL(2,
172:abelian
83:(Semi-)
2319:
2309:
2265:
2255:
2236:
2226:
2183:
2099:
1225:. If
1015:action
968:Ό = Ό(
532:Circle
463:SL(2,
352:cyclic
316:-group
167:cyclic
142:finite
137:simple
121:kernel
2014:Wiles
980:is a
909:, e:
874:in a
716:Sp(â)
713:SU(â)
126:image
2307:ISBN
2253:ISBN
2224:ISBN
2181:ISBN
1938:and
942:sets
858:and
846:and
838:and
802:, a
710:O(â)
699:Loop
518:and
2299:doi
2162:153
2141:152
2120:151
1986:n+1
1934:.
1930:of
1918:}/(
1749:to
1697:red
1686:red
1677:.
1673:by
1616:by
1567:If
1506:, Ό
1478:of
1428:by
1412:by
1381:.
1377:to
1365:to
1264:to
1126:.
1052:An
940:to
798:In
625:Sp(
613:SU(
589:SO(
553:SL(
541:GL(
2333::
2317:MR
2315:,
2305:,
2297:,
2275:,
2263:MR
2261:,
2234:MR
2232:,
2097:MR
2095:,
2020:.
1981:â
1974::
1967:CW
1920:FV
1910:){
1902:=
1803:.
1606:).
1602:/(
1494:/(
1401:GL
1369:â
1351:xy
1349:/(
1272:)/
1202:)(
1009:â
987:A
972:Ă
956:.)
921:â
913:â
905:â
854:,
601:U(
577:E(
565:O(
23:â
2301::
2208:p
2189:.
2164:)
2143:)
2122:)
2010:p
2002:F
1998:F
1994:D
1990:p
1983:W
1979:n
1976:W
1972:V
1960:k
1958:(
1956:W
1952:D
1948:k
1940:V
1936:F
1932:k
1924:p
1916:V
1914:,
1912:F
1908:k
1906:(
1904:W
1900:D
1893:p
1889:k
1849:p
1841:p
1837:p
1833:p
1830:F
1826:p
1822:p
1814:p
1809:2
1797:G
1793:S
1788:S
1784:O
1779:G
1775:O
1771:S
1767:G
1751:S
1747:G
1743:S
1739:G
1727:G
1723:S
1720:O
1716:S
1701:G
1694:G
1690:k
1683:G
1679:G
1675:G
1667:G
1663:G
1659:k
1655:G
1637:.
1635:a
1632:G
1628:m
1625:G
1621:m
1618:G
1614:a
1611:G
1604:x
1600:A
1596:A
1592:p
1588:a
1585:G
1581:p
1577:p
1573:S
1569:p
1564:.
1562:x
1558:x
1554:x
1550:x
1546:x
1542:x
1538:A
1534:A
1530:T
1526:S
1522:A
1518:a
1515:G
1508:p
1504:p
1500:n
1496:x
1492:A
1488:A
1484:f
1480:T
1476:f
1472:T
1468:S
1464:m
1461:G
1457:n
1453:n
1449:n
1442:n
1438:n
1434:T
1430:n
1426:n
1422:T
1418:S
1414:n
1410:n
1405:n
1394:m
1391:G
1387:S
1379:x
1375:x
1371:x
1367:x
1363:x
1359:x
1355:A
1347:A
1343:A
1339:Z
1337:(
1335:D
1331:T
1327:S
1323:m
1320:G
1306:G
1302:H
1298:H
1294:H
1290:G
1286:G
1282:H
1278:T
1276:(
1274:H
1270:T
1268:(
1266:G
1262:T
1258:S
1254:G
1250:H
1245:.
1243:S
1239:A
1235:A
1233:(
1231:D
1227:S
1223:T
1219:S
1215:T
1212:O
1208:A
1204:T
1200:A
1198:(
1196:D
1192:A
1190:(
1188:D
1181:A
1176:.
1150:S
1141:G
1137:S
1132:S
1128:G
1124:T
1120:G
1116:T
1112:S
1104:G
1100:S
1095:S
1091:G
1087:G
1074:T
1070:S
1066:T
1064:(
1062:G
1058:G
1054:S
1039:T
1035:S
1031:T
1029:(
1027:X
1023:T
1021:(
1019:G
1011:X
1007:X
1004:S
1002:Ă
1000:G
996:X
992:G
978:f
974:f
970:f
966:f
962:f
946:G
930:S
923:G
919:G
915:G
911:S
907:G
903:G
900:S
898:Ă
896:G
888:G
884:S
880:S
787:e
780:t
773:v
669:8
667:E
661:7
659:E
653:6
651:E
645:4
643:F
637:2
635:G
629:)
627:n
617:)
615:n
605:)
603:n
593:)
591:n
581:)
579:n
569:)
567:n
557:)
555:n
545:)
543:n
485:)
472:Z
460:)
447:Z
423:)
410:Z
401:(
314:p
279:Q
271:n
268:D
258:n
255:A
247:n
244:S
236:n
233:Z
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
