INDEX 157

integral

for

Pd

, 55

for a projective variety, 101

interpolation problem, 119, 127

inverse systems, 117, 127

invertible sheaf belonging to a divisor, 108

irregularity of a surface, 103

Jacobian

determinant, 43, 126

formula, 62

ideal, 70

K¨ ahler

different, 70

differential forms, 35

Koszul complex, 23

lattice, 130

lattice polytope, 131

Laurent

expansion of a differential, 107

series, 36

linear map induced by the residue, 40

Lipman, J., 86

local cohomology

at a point x, 1

and completion, 56

local duality theorem

for power series rings, 45

for complete reduced local rings, 94

on integral varieties, 95

local criterion of flatness, 75

Macaulay’s inverse systems, 117, 127

Matlis dual, 127

maximal Cohen-Macaulay module, 31

membership

problem, 123

in the radical of a complete intersection,

124

in zero-dimensional ideals, 124

midpoint, 99

minimal number of generators of the

canonical module, 68

Minkowski sum, 144

module of differentials, 35, 52

M -quasiregular sequence, 26

M -regular sequence, 26

multiplicative group, 131

multiplicity of a variety at a point, 96

Nagata’s conjecture, 127

Newton, 99

polytope, 144

Noether different, 70

Noether normalization, 81, 123

of an analytic algebra, 82

Noether position, 123

normal

lattice polytope, 132

fan of a polytope, 138

numerator of a generalized fraction, 32

permutation of (quasi-)regular sequences,

26

polyhedral cone, 130

polynomial solutions

of differential equations, 116

presentation as a complete intersection, 71

principal part of a Laurent series, 36

projective

closure, 63

toric variety, 132

punctured spectrum, 9

quantum cohomology ring, 119

quasicoherent sheaf, 8

quasiregular sequence, 26

quotient property of residues, 40

ramification index, locus, point, 109

rational differential form, 52

reducedness of zero-dimensional complete

intersections, 61

refinement of a covering, 19

regular differential forms, 84

at regular points, 84

at Cohen-Macaulay (Gorenstein) points,

86

at complete intersection points, 87

on the aﬃne cone, 101

on toric varieties, 131, 137, 139

regular differentials in the sense of

Rosenlicht, 87

regular sequence, 26

Reiss relation, 64

relation

between the arithmetic and the

geometric genus, 103

between the

ˇ

Cech and the Koszul

complex, 29

between toric and local residues, 143

to residues on smooth curves, 96

representative of the B´ ezoutian, 120

residual complex, 114

residue

at a closed point of a variety, 95

at a point x, 56

at a singular point of a curve, 107

at the vertex of the aﬃne cone, 56, 101

for convergent power series, 46

for power series, 39

map, 39, 93

of an exact differential, 37

on smooth curves, 96

sum, 57

symbol, 39, 79, 115