Index

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affine algebra, 36, 132

affine subvariety, 44

algebra

finite, 5

finite-type, 5

algebraic group, 53

algebraic prevariety, 44

algebraic set, 14

algebraic space, 152

basic open subset, 22

Bezout’s Theorem, 98, 142

birationally equivalent, 72

category, 56

characteristic exponent, 132

Chinese Remainder Theorem, 102

Chow group, 141

complete intersection

ideal-theoretic, 114

local, 114

set-theoretic, 114

complex topology, 151

constructible set, 117

cusp, 61

cycle

algebraic, 141

degree

of a map, 122, 140

of a point, 135

total, 5

derivation, 75

Dickson’s Lemma, 10

differential, 62

dimension, 24, 25, 55

division algorithm, 7

divisor, 137

effective, 137

local equation for, 138

locally principal, 138

positive, 137

prime, 137

principal, 137

restriction of, 138

support of, 137

dominating map, 58

elliptic curve, 14, 81, 134

equivalence of categories, 38

etale, 65, 79

etale neighbourhood, 75

fibred product, 106

field

algebraically closed, 3

field of rational functions, 24, 55

finite map, 101

Frobenius map, 41

functor, 57

Groebner basis, see also standard basis

Hilbert Basis Theorem, 10, 15

Hilbert Nullstellensatz, 17

strong, 19

Hilbert polynomial, 99

homogeneous coordinate ring, 86

homomorphism

of algebras, 5

of sheaves, 131

hypersurface, 24, 90

hypersurface section, 90

ideal, 4

homogeneous, 81

maximal, 4

monomial, 9

prime, 4

immersion, 47

closed, 47

open, 47

integral, 26

integral closure, 27

integrally closed, 28

intersect properly, 138, 139, 141

irreducible components, 23

irreducible topological space, 22

Krull dimension, 26

leading coefficient, 7

leading monomial, 7

leading term, 7

linearly equivalent, 137

local system of parameters, 74

manifold

complex, 44

differentiable, 44

topological, 44

monomial, 5

morphism

of affine algebraic varieties, 36

of ringed spaces, 35

multidegree, 7

multiplicative subset, 31

multiplicity

of a point, 61

156

Algebraic Geometry: 14. Further Reading 157

Nakayama’s Lemma, 69

node, 61

Noether Normalization Theorem, 104, 136

Noetherian, 21

nondegenerate quadric, 128

nonsingular, 59, 63, 71, 136

normal, 71

ordering

grevlex, 7

lex, 6

pencil of lines, 128

Picard group, 137, 145

Picard variety, 147

point

with coordinates in a ring, 56

point with coordinates in a ring, 76

polynomial

elementary symmetric, 26

homogeneous, 80

irreducible, 6

symmetric, 26

presheaf, 131

prevariety, 134

Prime Avoidance Lemma, 113

principal open subset, 22

product

of affine varieties, 52

of algebraic varieties, 52

projection with centre, 92

projective algebraic set, 80

projectively normal, 137

pure dimension, 25

quasi-compact, 21

quasi-inverse, 38

rational function, 35

rational variety, 73

rationally equivalent, 141

regular function, 33, 37, 44

regular map, 36, 45

resultant, 95

Riemann-Roch Theorem, 150

ring

coordinate, 21

Noetherian, 5

of regular functions, 21

ring of dual numbers, 75

ringed space, 30, 131

locally, 131

section of a sheaf, 30

Segre mapping, 91

semisimple

group, 77

Lie algebra, 77

separable map, 124

separated, 46, 134

sheaf, 131

coherent, 143

invertible, 145

locally free, 143

of algebras, 30

support of, 143

singular locus, 72, 136

smooth, 59 , 63

stalk, 131

standard basis, 10

minimal, 11

Stein factorization, 130

tangent cone, 60, 78

geometric, 60, 78, 79

tangent space, 59, 62, 68

tensor product, 49

unirational variety, 73

variety, 134

abelian, 53, 97

affine algebraic, 36

algebraic, 46

complete, 93, 136

degree of,, 98

Grassmanian, 98

normal, 123, 137

projective, 80

quasi-projective, 80

Veronese mapping, 89

Yoneda Lemma, 58

Zariski topology, 16

Zariski’s Lemma, 17

Zariski’s Main Theorem, 105__