224 Index endomorphism nilpotent –, 86 semisimple –, 86 unipotent –, 87 equivariant morphism, 60 extension generalized Liouville –, 160 Liouville –, 158 normal –, 153 of differential rings, 124 Picard-Vessiot –, 127 finite morphism, 32 Fuchsian differential equation, 166 full universal solution algebra, 128 function field of an affine variety, 10 of an irreducible prevariety, 29 fundamental set of solutions, 127 G-torsor, 66 G-variety, 60 Gauss hypergeometric equation, 172 Gauss hypergeometric function, 135 general linear group, 56 generalized Liouville extension, 160 geometric space, 17 structure sheaf of a –, 17 graded ring, 20 homogeneous element in a –, 20 homogeneous ideal of a –, 20 group closure, 58 Hilbert’s Nullstellensatz, 5 homogeneous coordinate ring, 21 homogeneous space, 65 principal –, 66 hypergeometric equation, 172 hyperplane at infinity, 21 indicial equation, 169 generalized –, 171 induced structure sheaf, 18 irreducible space, 8 irrelevant ideal, 21 isomorphic varieties, 12 isomorphism of varieties, 12 Krull dimension of a ring, 14 Lie algebra, 75 abelian –, 76 adjoint representation of a –, 77 center of a –, 77 central descending series of a –, 78 characteristic ideal of a –, 77 derivation of a –, 77 derived ideal of a –, 78 derived series of a –, 78 general linear –, 76 ideal of a –, 76 of a linear algebraic group, 79 Lie bracket, 75 Lie subalgebra, 76 Lie-Kolchin theorem, 94 linear algebraic group, 56 Liouville extension, 158 generalized –, 160 Liouvillian solution, 160 local exponent, 169 local ring of a prevariety at a point, 29 of an affine variety at a point, 11 locally closed, 28 locally finite morphism, 40 map birational –, 13 dominant rational –, 13 rational –, 12,13 regular –, 12 monodromy group, 174 morphism dominant of varieties, 33 equivariant –, 60 finite of affine varieties, 32 of affine varieties, 12 of algebraic groups, 60 of geometric spaces, 18 of Lie algebras, 77 of prevarieties, 28 multiplicative group, 55 nilpotent endomorphism, 86 nilpotent group, 95 nilpotent Lie algebra, 78 nilpotent part, 89 Noether’s normalization lemma, 5 nonsingular variety, 44 normal extension, 153 normal point of a variety, 42 normalizer of a subgroup, 59 Picard-Vessiot extension, 127 existence of –, 130
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