Index abstract aﬃne variety, 18 additive group, 55 adjunction of an integral, 143 of the exponential of an integral, 144 aﬃne n-space, 3 variety, 3 algebraic group, 55 adjoint representation of an -, 95 character of an –, 66 connected –, 57 direct product of –s, 56 identity component of an –, 57 semi-invariant of an –, 67 algebraic prevariety, 27 algebraic variety, 31 apparent singularity, 211 birational equivalence, 13 map, 13 birationally equivalent varieties, 13 categorical quotient, 69 Cauchy theorem, 167 center of a group, 91 central descending series of a group, 95 character of an algebraic group, 66 Chevalley quotient, 69 Chevalley theorem, 37 codimension of a subvariety, 35 complete variety, 46 connected algebraic group, 57 constant ring of –s, 123 constructible set, 36 coordinate ring, 10 cyclic vector, 139 derivation, 121 trivial –, 122 derived series of a group, 93 derived subgroup, 91 diagonal group, 56 diagonalizable set of matrices, 90 differential field, 122 Galois group, 141 ideal, 123 indeterminate, 123 mapping, 43 module, 138 morphism, 124 operator, 125 polynomial, 123 rational function, 123 ring, 122 differentially finitely generated, 124 dimension Krull – of a ring, 14 of a topological space, 14 direct product of algebraic groups, 56 domain of definition of a rational map, 12 dominant rational map, 13 223

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